\(\int (e x)^m (a+b x^2)^3 (A+B x^2) (c+d x^2)^2 \, dx\) [8]
Optimal result
Integrand size = 31, antiderivative size = 292 \[
\int (e x)^m \left (a+b x^2\right )^3 \left (A+B x^2\right ) \left (c+d x^2\right )^2 \, dx=\frac {a^3 A c^2 (e x)^{1+m}}{e (1+m)}+\frac {a^2 c (3 A b c+a B c+2 a A d) (e x)^{3+m}}{e^3 (3+m)}+\frac {a \left (a B c (3 b c+2 a d)+A \left (3 b^2 c^2+6 a b c d+a^2 d^2\right )\right ) (e x)^{5+m}}{e^5 (5+m)}+\frac {\left (a B \left (3 b^2 c^2+6 a b c d+a^2 d^2\right )+A b \left (b^2 c^2+6 a b c d+3 a^2 d^2\right )\right ) (e x)^{7+m}}{e^7 (7+m)}+\frac {b \left (3 a^2 B d^2+3 a b d (2 B c+A d)+b^2 c (B c+2 A d)\right ) (e x)^{9+m}}{e^9 (9+m)}+\frac {b^2 d (2 b B c+A b d+3 a B d) (e x)^{11+m}}{e^{11} (11+m)}+\frac {b^3 B d^2 (e x)^{13+m}}{e^{13} (13+m)}
\]
[Out]
a^3*A*c^2*(e*x)^(1+m)/e/(1+m)+a^2*c*(2*A*a*d+3*A*b*c+B*a*c)*(e*x)^(3+m)/e^3/(3+m)+a*(a*B*c*(2*a*d+3*b*c)+A*(a^
2*d^2+6*a*b*c*d+3*b^2*c^2))*(e*x)^(5+m)/e^5/(5+m)+(a*B*(a^2*d^2+6*a*b*c*d+3*b^2*c^2)+A*b*(3*a^2*d^2+6*a*b*c*d+
b^2*c^2))*(e*x)^(7+m)/e^7/(7+m)+b*(3*a^2*B*d^2+3*a*b*d*(A*d+2*B*c)+b^2*c*(2*A*d+B*c))*(e*x)^(9+m)/e^9/(9+m)+b^
2*d*(A*b*d+3*B*a*d+2*B*b*c)*(e*x)^(11+m)/e^11/(11+m)+b^3*B*d^2*(e*x)^(13+m)/e^13/(13+m)
Rubi [A] (verified)
Time = 0.19 (sec) , antiderivative size = 292, normalized size of antiderivative = 1.00,
number of steps used = 2, number of rules used = 1, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.032, Rules used = {584}
\[
\int (e x)^m \left (a+b x^2\right )^3 \left (A+B x^2\right ) \left (c+d x^2\right )^2 \, dx=\frac {a^3 A c^2 (e x)^{m+1}}{e (m+1)}+\frac {(e x)^{m+7} \left (A b \left (3 a^2 d^2+6 a b c d+b^2 c^2\right )+a B \left (a^2 d^2+6 a b c d+3 b^2 c^2\right )\right )}{e^7 (m+7)}+\frac {a (e x)^{m+5} \left (A \left (a^2 d^2+6 a b c d+3 b^2 c^2\right )+a B c (2 a d+3 b c)\right )}{e^5 (m+5)}+\frac {b (e x)^{m+9} \left (3 a^2 B d^2+3 a b d (A d+2 B c)+b^2 c (2 A d+B c)\right )}{e^9 (m+9)}+\frac {a^2 c (e x)^{m+3} (2 a A d+a B c+3 A b c)}{e^3 (m+3)}+\frac {b^2 d (e x)^{m+11} (3 a B d+A b d+2 b B c)}{e^{11} (m+11)}+\frac {b^3 B d^2 (e x)^{m+13}}{e^{13} (m+13)}
\]
[In]
Int[(e*x)^m*(a + b*x^2)^3*(A + B*x^2)*(c + d*x^2)^2,x]
[Out]
(a^3*A*c^2*(e*x)^(1 + m))/(e*(1 + m)) + (a^2*c*(3*A*b*c + a*B*c + 2*a*A*d)*(e*x)^(3 + m))/(e^3*(3 + m)) + (a*(
a*B*c*(3*b*c + 2*a*d) + A*(3*b^2*c^2 + 6*a*b*c*d + a^2*d^2))*(e*x)^(5 + m))/(e^5*(5 + m)) + ((a*B*(3*b^2*c^2 +
6*a*b*c*d + a^2*d^2) + A*b*(b^2*c^2 + 6*a*b*c*d + 3*a^2*d^2))*(e*x)^(7 + m))/(e^7*(7 + m)) + (b*(3*a^2*B*d^2
+ 3*a*b*d*(2*B*c + A*d) + b^2*c*(B*c + 2*A*d))*(e*x)^(9 + m))/(e^9*(9 + m)) + (b^2*d*(2*b*B*c + A*b*d + 3*a*B*
d)*(e*x)^(11 + m))/(e^11*(11 + m)) + (b^3*B*d^2*(e*x)^(13 + m))/(e^13*(13 + m))
Rule 584
Int[((g_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_.)*((c_) + (d_.)*(x_)^(n_))^(q_.)*((e_) + (f_.)*(x_)^(n_))^
(r_.), x_Symbol] :> Int[ExpandIntegrand[(g*x)^m*(a + b*x^n)^p*(c + d*x^n)^q*(e + f*x^n)^r, x], x] /; FreeQ[{a,
b, c, d, e, f, g, m, n}, x] && IGtQ[p, -2] && IGtQ[q, 0] && IGtQ[r, 0]
Rubi steps \begin{align*}
\text {integral}& = \int \left (a^3 A c^2 (e x)^m+\frac {a^2 c (3 A b c+a B c+2 a A d) (e x)^{2+m}}{e^2}+\frac {a \left (a B c (3 b c+2 a d)+A \left (3 b^2 c^2+6 a b c d+a^2 d^2\right )\right ) (e x)^{4+m}}{e^4}+\frac {\left (a B \left (3 b^2 c^2+6 a b c d+a^2 d^2\right )+A b \left (b^2 c^2+6 a b c d+3 a^2 d^2\right )\right ) (e x)^{6+m}}{e^6}+\frac {b \left (3 a^2 B d^2+3 a b d (2 B c+A d)+b^2 c (B c+2 A d)\right ) (e x)^{8+m}}{e^8}+\frac {b^2 d (2 b B c+A b d+3 a B d) (e x)^{10+m}}{e^{10}}+\frac {b^3 B d^2 (e x)^{12+m}}{e^{12}}\right ) \, dx \\ & = \frac {a^3 A c^2 (e x)^{1+m}}{e (1+m)}+\frac {a^2 c (3 A b c+a B c+2 a A d) (e x)^{3+m}}{e^3 (3+m)}+\frac {a \left (a B c (3 b c+2 a d)+A \left (3 b^2 c^2+6 a b c d+a^2 d^2\right )\right ) (e x)^{5+m}}{e^5 (5+m)}+\frac {\left (a B \left (3 b^2 c^2+6 a b c d+a^2 d^2\right )+A b \left (b^2 c^2+6 a b c d+3 a^2 d^2\right )\right ) (e x)^{7+m}}{e^7 (7+m)}+\frac {b \left (3 a^2 B d^2+3 a b d (2 B c+A d)+b^2 c (B c+2 A d)\right ) (e x)^{9+m}}{e^9 (9+m)}+\frac {b^2 d (2 b B c+A b d+3 a B d) (e x)^{11+m}}{e^{11} (11+m)}+\frac {b^3 B d^2 (e x)^{13+m}}{e^{13} (13+m)} \\
\end{align*}
Mathematica [A] (verified)
Time = 0.70 (sec) , antiderivative size = 247, normalized size of antiderivative = 0.85
\[
\int (e x)^m \left (a+b x^2\right )^3 \left (A+B x^2\right ) \left (c+d x^2\right )^2 \, dx=x (e x)^m \left (\frac {a^3 A c^2}{1+m}+\frac {a^2 c (3 A b c+a B c+2 a A d) x^2}{3+m}+\frac {a \left (a B c (3 b c+2 a d)+A \left (3 b^2 c^2+6 a b c d+a^2 d^2\right )\right ) x^4}{5+m}+\frac {\left (a B \left (3 b^2 c^2+6 a b c d+a^2 d^2\right )+A b \left (b^2 c^2+6 a b c d+3 a^2 d^2\right )\right ) x^6}{7+m}+\frac {b \left (3 a^2 B d^2+3 a b d (2 B c+A d)+b^2 c (B c+2 A d)\right ) x^8}{9+m}+\frac {b^2 d (2 b B c+A b d+3 a B d) x^{10}}{11+m}+\frac {b^3 B d^2 x^{12}}{13+m}\right )
\]
[In]
Integrate[(e*x)^m*(a + b*x^2)^3*(A + B*x^2)*(c + d*x^2)^2,x]
[Out]
x*(e*x)^m*((a^3*A*c^2)/(1 + m) + (a^2*c*(3*A*b*c + a*B*c + 2*a*A*d)*x^2)/(3 + m) + (a*(a*B*c*(3*b*c + 2*a*d) +
A*(3*b^2*c^2 + 6*a*b*c*d + a^2*d^2))*x^4)/(5 + m) + ((a*B*(3*b^2*c^2 + 6*a*b*c*d + a^2*d^2) + A*b*(b^2*c^2 +
6*a*b*c*d + 3*a^2*d^2))*x^6)/(7 + m) + (b*(3*a^2*B*d^2 + 3*a*b*d*(2*B*c + A*d) + b^2*c*(B*c + 2*A*d))*x^8)/(9
+ m) + (b^2*d*(2*b*B*c + A*b*d + 3*a*B*d)*x^10)/(11 + m) + (b^3*B*d^2*x^12)/(13 + m))
Maple [B] (verified)
Leaf count of result is larger than twice the leaf count of optimal. \(2442\) vs. \(2(292)=584\).
Time = 3.74 (sec) , antiderivative size = 2443, normalized size of antiderivative =
8.37
| | |
method | result | size |
| | |
gosper |
\(\text {Expression too large to display}\) |
\(2443\) |
risch |
\(\text {Expression too large to display}\) |
\(2443\) |
parallelrisch |
\(\text {Expression too large to display}\) |
\(3284\) |
| | |
|
|
|
[In]
int((e*x)^m*(b*x^2+a)^3*(B*x^2+A)*(d*x^2+c)^2,x,method=_RETURNVERBOSE)
[Out]
x*(B*b^3*d^2*m^6*x^12+36*B*b^3*d^2*m^5*x^12+A*b^3*d^2*m^6*x^10+3*B*a*b^2*d^2*m^6*x^10+2*B*b^3*c*d*m^6*x^10+505
*B*b^3*d^2*m^4*x^12+38*A*b^3*d^2*m^5*x^10+114*B*a*b^2*d^2*m^5*x^10+76*B*b^3*c*d*m^5*x^10+3480*B*b^3*d^2*m^3*x^
12+3*A*a*b^2*d^2*m^6*x^8+2*A*b^3*c*d*m^6*x^8+555*A*b^3*d^2*m^4*x^10+3*B*a^2*b*d^2*m^6*x^8+6*B*a*b^2*c*d*m^6*x^
8+1665*B*a*b^2*d^2*m^4*x^10+B*b^3*c^2*m^6*x^8+1110*B*b^3*c*d*m^4*x^10+12139*B*b^3*d^2*m^2*x^12+120*A*a*b^2*d^2
*m^5*x^8+80*A*b^3*c*d*m^5*x^8+3940*A*b^3*d^2*m^3*x^10+120*B*a^2*b*d^2*m^5*x^8+240*B*a*b^2*c*d*m^5*x^8+11820*B*
a*b^2*d^2*m^3*x^10+40*B*b^3*c^2*m^5*x^8+7880*B*b^3*c*d*m^3*x^10+19524*B*b^3*d^2*m*x^12+3*A*a^2*b*d^2*m^6*x^6+6
*A*a*b^2*c*d*m^6*x^6+1839*A*a*b^2*d^2*m^4*x^8+A*b^3*c^2*m^6*x^6+1226*A*b^3*c*d*m^4*x^8+14039*A*b^3*d^2*m^2*x^1
0+B*a^3*d^2*m^6*x^6+6*B*a^2*b*c*d*m^6*x^6+1839*B*a^2*b*d^2*m^4*x^8+3*B*a*b^2*c^2*m^6*x^6+3678*B*a*b^2*c*d*m^4*
x^8+42117*B*a*b^2*d^2*m^2*x^10+613*B*b^3*c^2*m^4*x^8+28078*B*b^3*c*d*m^2*x^10+10395*B*b^3*d^2*x^12+126*A*a^2*b
*d^2*m^5*x^6+252*A*a*b^2*c*d*m^5*x^6+13584*A*a*b^2*d^2*m^3*x^8+42*A*b^3*c^2*m^5*x^6+9056*A*b^3*c*d*m^3*x^8+229
02*A*b^3*d^2*m*x^10+42*B*a^3*d^2*m^5*x^6+252*B*a^2*b*c*d*m^5*x^6+13584*B*a^2*b*d^2*m^3*x^8+126*B*a*b^2*c^2*m^5
*x^6+27168*B*a*b^2*c*d*m^3*x^8+68706*B*a*b^2*d^2*m*x^10+4528*B*b^3*c^2*m^3*x^8+45804*B*b^3*c*d*m*x^10+A*a^3*d^
2*m^6*x^4+6*A*a^2*b*c*d*m^6*x^4+2037*A*a^2*b*d^2*m^4*x^6+3*A*a*b^2*c^2*m^6*x^4+4074*A*a*b^2*c*d*m^4*x^6+49881*
A*a*b^2*d^2*m^2*x^8+679*A*b^3*c^2*m^4*x^6+33254*A*b^3*c*d*m^2*x^8+12285*A*b^3*d^2*x^10+2*B*a^3*c*d*m^6*x^4+679
*B*a^3*d^2*m^4*x^6+3*B*a^2*b*c^2*m^6*x^4+4074*B*a^2*b*c*d*m^4*x^6+49881*B*a^2*b*d^2*m^2*x^8+2037*B*a*b^2*c^2*m
^4*x^6+99762*B*a*b^2*c*d*m^2*x^8+36855*B*a*b^2*d^2*x^10+16627*B*b^3*c^2*m^2*x^8+24570*B*b^3*c*d*x^10+44*A*a^3*
d^2*m^5*x^4+264*A*a^2*b*c*d*m^5*x^4+15876*A*a^2*b*d^2*m^3*x^6+132*A*a*b^2*c^2*m^5*x^4+31752*A*a*b^2*c*d*m^3*x^
6+83064*A*a*b^2*d^2*m*x^8+5292*A*b^3*c^2*m^3*x^6+55376*A*b^3*c*d*m*x^8+88*B*a^3*c*d*m^5*x^4+5292*B*a^3*d^2*m^3
*x^6+132*B*a^2*b*c^2*m^5*x^4+31752*B*a^2*b*c*d*m^3*x^6+83064*B*a^2*b*d^2*m*x^8+15876*B*a*b^2*c^2*m^3*x^6+16612
8*B*a*b^2*c*d*m*x^8+27688*B*b^3*c^2*m*x^8+2*A*a^3*c*d*m^6*x^2+753*A*a^3*d^2*m^4*x^4+3*A*a^2*b*c^2*m^6*x^2+4518
*A*a^2*b*c*d*m^4*x^4+61005*A*a^2*b*d^2*m^2*x^6+2259*A*a*b^2*c^2*m^4*x^4+122010*A*a*b^2*c*d*m^2*x^6+45045*A*a*b
^2*d^2*x^8+20335*A*b^3*c^2*m^2*x^6+30030*A*b^3*c*d*x^8+B*a^3*c^2*m^6*x^2+1506*B*a^3*c*d*m^4*x^4+20335*B*a^3*d^
2*m^2*x^6+2259*B*a^2*b*c^2*m^4*x^4+122010*B*a^2*b*c*d*m^2*x^6+45045*B*a^2*b*d^2*x^8+61005*B*a*b^2*c^2*m^2*x^6+
90090*B*a*b^2*c*d*x^8+15015*B*b^3*c^2*x^8+92*A*a^3*c*d*m^5*x^2+6280*A*a^3*d^2*m^3*x^4+138*A*a^2*b*c^2*m^5*x^2+
37680*A*a^2*b*c*d*m^3*x^4+104958*A*a^2*b*d^2*m*x^6+18840*A*a*b^2*c^2*m^3*x^4+209916*A*a*b^2*c*d*m*x^6+34986*A*
b^3*c^2*m*x^6+46*B*a^3*c^2*m^5*x^2+12560*B*a^3*c*d*m^3*x^4+34986*B*a^3*d^2*m*x^6+18840*B*a^2*b*c^2*m^3*x^4+209
916*B*a^2*b*c*d*m*x^6+104958*B*a*b^2*c^2*m*x^6+A*a^3*c^2*m^6+1670*A*a^3*c*d*m^4*x^2+25979*A*a^3*d^2*m^2*x^4+25
05*A*a^2*b*c^2*m^4*x^2+155874*A*a^2*b*c*d*m^2*x^4+57915*A*a^2*b*d^2*x^6+77937*A*a*b^2*c^2*m^2*x^4+115830*A*a*b
^2*c*d*x^6+19305*A*b^3*c^2*x^6+835*B*a^3*c^2*m^4*x^2+51958*B*a^3*c*d*m^2*x^4+19305*B*a^3*d^2*x^6+77937*B*a^2*b
*c^2*m^2*x^4+115830*B*a^2*b*c*d*x^6+57915*B*a*b^2*c^2*x^6+48*A*a^3*c^2*m^5+15080*A*a^3*c*d*m^3*x^2+47436*A*a^3
*d^2*m*x^4+22620*A*a^2*b*c^2*m^3*x^2+284616*A*a^2*b*c*d*m*x^4+142308*A*a*b^2*c^2*m*x^4+7540*B*a^3*c^2*m^3*x^2+
94872*B*a^3*c*d*m*x^4+142308*B*a^2*b*c^2*m*x^4+925*A*a^3*c^2*m^4+69518*A*a^3*c*d*m^2*x^2+27027*A*a^3*d^2*x^4+1
04277*A*a^2*b*c^2*m^2*x^2+162162*A*a^2*b*c*d*x^4+81081*A*a*b^2*c^2*x^4+34759*B*a^3*c^2*m^2*x^2+54054*B*a^3*c*d
*x^4+81081*B*a^2*b*c^2*x^4+9120*A*a^3*c^2*m^3+146108*A*a^3*c*d*m*x^2+219162*A*a^2*b*c^2*m*x^2+73054*B*a^3*c^2*
m*x^2+48259*A*a^3*c^2*m^2+90090*A*a^3*c*d*x^2+135135*A*a^2*b*c^2*x^2+45045*B*a^3*c^2*x^2+129072*A*a^3*c^2*m+13
5135*A*a^3*c^2)*(e*x)^m/(13+m)/(11+m)/(9+m)/(7+m)/(5+m)/(3+m)/(1+m)
Fricas [B] (verification not implemented)
Leaf count of result is larger than twice the leaf count of optimal. 1711 vs. \(2 (292) = 584\).
Time = 0.30 (sec) , antiderivative size = 1711, normalized size of antiderivative = 5.86
\[
\int (e x)^m \left (a+b x^2\right )^3 \left (A+B x^2\right ) \left (c+d x^2\right )^2 \, dx=\text {Too large to display}
\]
[In]
integrate((e*x)^m*(b*x^2+a)^3*(B*x^2+A)*(d*x^2+c)^2,x, algorithm="fricas")
[Out]
((B*b^3*d^2*m^6 + 36*B*b^3*d^2*m^5 + 505*B*b^3*d^2*m^4 + 3480*B*b^3*d^2*m^3 + 12139*B*b^3*d^2*m^2 + 19524*B*b^
3*d^2*m + 10395*B*b^3*d^2)*x^13 + ((2*B*b^3*c*d + (3*B*a*b^2 + A*b^3)*d^2)*m^6 + 24570*B*b^3*c*d + 38*(2*B*b^3
*c*d + (3*B*a*b^2 + A*b^3)*d^2)*m^5 + 555*(2*B*b^3*c*d + (3*B*a*b^2 + A*b^3)*d^2)*m^4 + 3940*(2*B*b^3*c*d + (3
*B*a*b^2 + A*b^3)*d^2)*m^3 + 12285*(3*B*a*b^2 + A*b^3)*d^2 + 14039*(2*B*b^3*c*d + (3*B*a*b^2 + A*b^3)*d^2)*m^2
+ 22902*(2*B*b^3*c*d + (3*B*a*b^2 + A*b^3)*d^2)*m)*x^11 + ((B*b^3*c^2 + 2*(3*B*a*b^2 + A*b^3)*c*d + 3*(B*a^2*
b + A*a*b^2)*d^2)*m^6 + 15015*B*b^3*c^2 + 40*(B*b^3*c^2 + 2*(3*B*a*b^2 + A*b^3)*c*d + 3*(B*a^2*b + A*a*b^2)*d^
2)*m^5 + 613*(B*b^3*c^2 + 2*(3*B*a*b^2 + A*b^3)*c*d + 3*(B*a^2*b + A*a*b^2)*d^2)*m^4 + 4528*(B*b^3*c^2 + 2*(3*
B*a*b^2 + A*b^3)*c*d + 3*(B*a^2*b + A*a*b^2)*d^2)*m^3 + 30030*(3*B*a*b^2 + A*b^3)*c*d + 45045*(B*a^2*b + A*a*b
^2)*d^2 + 16627*(B*b^3*c^2 + 2*(3*B*a*b^2 + A*b^3)*c*d + 3*(B*a^2*b + A*a*b^2)*d^2)*m^2 + 27688*(B*b^3*c^2 + 2
*(3*B*a*b^2 + A*b^3)*c*d + 3*(B*a^2*b + A*a*b^2)*d^2)*m)*x^9 + (((3*B*a*b^2 + A*b^3)*c^2 + 6*(B*a^2*b + A*a*b^
2)*c*d + (B*a^3 + 3*A*a^2*b)*d^2)*m^6 + 42*((3*B*a*b^2 + A*b^3)*c^2 + 6*(B*a^2*b + A*a*b^2)*c*d + (B*a^3 + 3*A
*a^2*b)*d^2)*m^5 + 679*((3*B*a*b^2 + A*b^3)*c^2 + 6*(B*a^2*b + A*a*b^2)*c*d + (B*a^3 + 3*A*a^2*b)*d^2)*m^4 + 5
292*((3*B*a*b^2 + A*b^3)*c^2 + 6*(B*a^2*b + A*a*b^2)*c*d + (B*a^3 + 3*A*a^2*b)*d^2)*m^3 + 19305*(3*B*a*b^2 + A
*b^3)*c^2 + 115830*(B*a^2*b + A*a*b^2)*c*d + 19305*(B*a^3 + 3*A*a^2*b)*d^2 + 20335*((3*B*a*b^2 + A*b^3)*c^2 +
6*(B*a^2*b + A*a*b^2)*c*d + (B*a^3 + 3*A*a^2*b)*d^2)*m^2 + 34986*((3*B*a*b^2 + A*b^3)*c^2 + 6*(B*a^2*b + A*a*b
^2)*c*d + (B*a^3 + 3*A*a^2*b)*d^2)*m)*x^7 + ((A*a^3*d^2 + 3*(B*a^2*b + A*a*b^2)*c^2 + 2*(B*a^3 + 3*A*a^2*b)*c*
d)*m^6 + 27027*A*a^3*d^2 + 44*(A*a^3*d^2 + 3*(B*a^2*b + A*a*b^2)*c^2 + 2*(B*a^3 + 3*A*a^2*b)*c*d)*m^5 + 753*(A
*a^3*d^2 + 3*(B*a^2*b + A*a*b^2)*c^2 + 2*(B*a^3 + 3*A*a^2*b)*c*d)*m^4 + 6280*(A*a^3*d^2 + 3*(B*a^2*b + A*a*b^2
)*c^2 + 2*(B*a^3 + 3*A*a^2*b)*c*d)*m^3 + 81081*(B*a^2*b + A*a*b^2)*c^2 + 54054*(B*a^3 + 3*A*a^2*b)*c*d + 25979
*(A*a^3*d^2 + 3*(B*a^2*b + A*a*b^2)*c^2 + 2*(B*a^3 + 3*A*a^2*b)*c*d)*m^2 + 47436*(A*a^3*d^2 + 3*(B*a^2*b + A*a
*b^2)*c^2 + 2*(B*a^3 + 3*A*a^2*b)*c*d)*m)*x^5 + ((2*A*a^3*c*d + (B*a^3 + 3*A*a^2*b)*c^2)*m^6 + 90090*A*a^3*c*d
+ 46*(2*A*a^3*c*d + (B*a^3 + 3*A*a^2*b)*c^2)*m^5 + 835*(2*A*a^3*c*d + (B*a^3 + 3*A*a^2*b)*c^2)*m^4 + 7540*(2*
A*a^3*c*d + (B*a^3 + 3*A*a^2*b)*c^2)*m^3 + 45045*(B*a^3 + 3*A*a^2*b)*c^2 + 34759*(2*A*a^3*c*d + (B*a^3 + 3*A*a
^2*b)*c^2)*m^2 + 73054*(2*A*a^3*c*d + (B*a^3 + 3*A*a^2*b)*c^2)*m)*x^3 + (A*a^3*c^2*m^6 + 48*A*a^3*c^2*m^5 + 92
5*A*a^3*c^2*m^4 + 9120*A*a^3*c^2*m^3 + 48259*A*a^3*c^2*m^2 + 129072*A*a^3*c^2*m + 135135*A*a^3*c^2)*x)*(e*x)^m
/(m^7 + 49*m^6 + 973*m^5 + 10045*m^4 + 57379*m^3 + 177331*m^2 + 264207*m + 135135)
Sympy [B] (verification not implemented)
Leaf count of result is larger than twice the leaf count of optimal. 11914 vs. \(2 (294) = 588\).
Time = 1.36 (sec) , antiderivative size = 11914, normalized size of antiderivative = 40.80
\[
\int (e x)^m \left (a+b x^2\right )^3 \left (A+B x^2\right ) \left (c+d x^2\right )^2 \, dx=\text {Too large to display}
\]
[In]
integrate((e*x)**m*(b*x**2+a)**3*(B*x**2+A)*(d*x**2+c)**2,x)
[Out]
Piecewise(((-A*a**3*c**2/(12*x**12) - A*a**3*c*d/(5*x**10) - A*a**3*d**2/(8*x**8) - 3*A*a**2*b*c**2/(10*x**10)
- 3*A*a**2*b*c*d/(4*x**8) - A*a**2*b*d**2/(2*x**6) - 3*A*a*b**2*c**2/(8*x**8) - A*a*b**2*c*d/x**6 - 3*A*a*b**
2*d**2/(4*x**4) - A*b**3*c**2/(6*x**6) - A*b**3*c*d/(2*x**4) - A*b**3*d**2/(2*x**2) - B*a**3*c**2/(10*x**10) -
B*a**3*c*d/(4*x**8) - B*a**3*d**2/(6*x**6) - 3*B*a**2*b*c**2/(8*x**8) - B*a**2*b*c*d/x**6 - 3*B*a**2*b*d**2/(
4*x**4) - B*a*b**2*c**2/(2*x**6) - 3*B*a*b**2*c*d/(2*x**4) - 3*B*a*b**2*d**2/(2*x**2) - B*b**3*c**2/(4*x**4) -
B*b**3*c*d/x**2 + B*b**3*d**2*log(x))/e**13, Eq(m, -13)), ((-A*a**3*c**2/(10*x**10) - A*a**3*c*d/(4*x**8) - A
*a**3*d**2/(6*x**6) - 3*A*a**2*b*c**2/(8*x**8) - A*a**2*b*c*d/x**6 - 3*A*a**2*b*d**2/(4*x**4) - A*a*b**2*c**2/
(2*x**6) - 3*A*a*b**2*c*d/(2*x**4) - 3*A*a*b**2*d**2/(2*x**2) - A*b**3*c**2/(4*x**4) - A*b**3*c*d/x**2 + A*b**
3*d**2*log(x) - B*a**3*c**2/(8*x**8) - B*a**3*c*d/(3*x**6) - B*a**3*d**2/(4*x**4) - B*a**2*b*c**2/(2*x**6) - 3
*B*a**2*b*c*d/(2*x**4) - 3*B*a**2*b*d**2/(2*x**2) - 3*B*a*b**2*c**2/(4*x**4) - 3*B*a*b**2*c*d/x**2 + 3*B*a*b**
2*d**2*log(x) - B*b**3*c**2/(2*x**2) + 2*B*b**3*c*d*log(x) + B*b**3*d**2*x**2/2)/e**11, Eq(m, -11)), ((-A*a**3
*c**2/(8*x**8) - A*a**3*c*d/(3*x**6) - A*a**3*d**2/(4*x**4) - A*a**2*b*c**2/(2*x**6) - 3*A*a**2*b*c*d/(2*x**4)
- 3*A*a**2*b*d**2/(2*x**2) - 3*A*a*b**2*c**2/(4*x**4) - 3*A*a*b**2*c*d/x**2 + 3*A*a*b**2*d**2*log(x) - A*b**3
*c**2/(2*x**2) + 2*A*b**3*c*d*log(x) + A*b**3*d**2*x**2/2 - B*a**3*c**2/(6*x**6) - B*a**3*c*d/(2*x**4) - B*a**
3*d**2/(2*x**2) - 3*B*a**2*b*c**2/(4*x**4) - 3*B*a**2*b*c*d/x**2 + 3*B*a**2*b*d**2*log(x) - 3*B*a*b**2*c**2/(2
*x**2) + 6*B*a*b**2*c*d*log(x) + 3*B*a*b**2*d**2*x**2/2 + B*b**3*c**2*log(x) + B*b**3*c*d*x**2 + B*b**3*d**2*x
**4/4)/e**9, Eq(m, -9)), ((-A*a**3*c**2/(6*x**6) - A*a**3*c*d/(2*x**4) - A*a**3*d**2/(2*x**2) - 3*A*a**2*b*c**
2/(4*x**4) - 3*A*a**2*b*c*d/x**2 + 3*A*a**2*b*d**2*log(x) - 3*A*a*b**2*c**2/(2*x**2) + 6*A*a*b**2*c*d*log(x) +
3*A*a*b**2*d**2*x**2/2 + A*b**3*c**2*log(x) + A*b**3*c*d*x**2 + A*b**3*d**2*x**4/4 - B*a**3*c**2/(4*x**4) - B
*a**3*c*d/x**2 + B*a**3*d**2*log(x) - 3*B*a**2*b*c**2/(2*x**2) + 6*B*a**2*b*c*d*log(x) + 3*B*a**2*b*d**2*x**2/
2 + 3*B*a*b**2*c**2*log(x) + 3*B*a*b**2*c*d*x**2 + 3*B*a*b**2*d**2*x**4/4 + B*b**3*c**2*x**2/2 + B*b**3*c*d*x*
*4/2 + B*b**3*d**2*x**6/6)/e**7, Eq(m, -7)), ((-A*a**3*c**2/(4*x**4) - A*a**3*c*d/x**2 + A*a**3*d**2*log(x) -
3*A*a**2*b*c**2/(2*x**2) + 6*A*a**2*b*c*d*log(x) + 3*A*a**2*b*d**2*x**2/2 + 3*A*a*b**2*c**2*log(x) + 3*A*a*b**
2*c*d*x**2 + 3*A*a*b**2*d**2*x**4/4 + A*b**3*c**2*x**2/2 + A*b**3*c*d*x**4/2 + A*b**3*d**2*x**6/6 - B*a**3*c**
2/(2*x**2) + 2*B*a**3*c*d*log(x) + B*a**3*d**2*x**2/2 + 3*B*a**2*b*c**2*log(x) + 3*B*a**2*b*c*d*x**2 + 3*B*a**
2*b*d**2*x**4/4 + 3*B*a*b**2*c**2*x**2/2 + 3*B*a*b**2*c*d*x**4/2 + B*a*b**2*d**2*x**6/2 + B*b**3*c**2*x**4/4 +
B*b**3*c*d*x**6/3 + B*b**3*d**2*x**8/8)/e**5, Eq(m, -5)), ((-A*a**3*c**2/(2*x**2) + 2*A*a**3*c*d*log(x) + A*a
**3*d**2*x**2/2 + 3*A*a**2*b*c**2*log(x) + 3*A*a**2*b*c*d*x**2 + 3*A*a**2*b*d**2*x**4/4 + 3*A*a*b**2*c**2*x**2
/2 + 3*A*a*b**2*c*d*x**4/2 + A*a*b**2*d**2*x**6/2 + A*b**3*c**2*x**4/4 + A*b**3*c*d*x**6/3 + A*b**3*d**2*x**8/
8 + B*a**3*c**2*log(x) + B*a**3*c*d*x**2 + B*a**3*d**2*x**4/4 + 3*B*a**2*b*c**2*x**2/2 + 3*B*a**2*b*c*d*x**4/2
+ B*a**2*b*d**2*x**6/2 + 3*B*a*b**2*c**2*x**4/4 + B*a*b**2*c*d*x**6 + 3*B*a*b**2*d**2*x**8/8 + B*b**3*c**2*x*
*6/6 + B*b**3*c*d*x**8/4 + B*b**3*d**2*x**10/10)/e**3, Eq(m, -3)), ((A*a**3*c**2*log(x) + A*a**3*c*d*x**2 + A*
a**3*d**2*x**4/4 + 3*A*a**2*b*c**2*x**2/2 + 3*A*a**2*b*c*d*x**4/2 + A*a**2*b*d**2*x**6/2 + 3*A*a*b**2*c**2*x**
4/4 + A*a*b**2*c*d*x**6 + 3*A*a*b**2*d**2*x**8/8 + A*b**3*c**2*x**6/6 + A*b**3*c*d*x**8/4 + A*b**3*d**2*x**10/
10 + B*a**3*c**2*x**2/2 + B*a**3*c*d*x**4/2 + B*a**3*d**2*x**6/6 + 3*B*a**2*b*c**2*x**4/4 + B*a**2*b*c*d*x**6
+ 3*B*a**2*b*d**2*x**8/8 + B*a*b**2*c**2*x**6/2 + 3*B*a*b**2*c*d*x**8/4 + 3*B*a*b**2*d**2*x**10/10 + B*b**3*c*
*2*x**8/8 + B*b**3*c*d*x**10/5 + B*b**3*d**2*x**12/12)/e, Eq(m, -1)), (A*a**3*c**2*m**6*x*(e*x)**m/(m**7 + 49*
m**6 + 973*m**5 + 10045*m**4 + 57379*m**3 + 177331*m**2 + 264207*m + 135135) + 48*A*a**3*c**2*m**5*x*(e*x)**m/
(m**7 + 49*m**6 + 973*m**5 + 10045*m**4 + 57379*m**3 + 177331*m**2 + 264207*m + 135135) + 925*A*a**3*c**2*m**4
*x*(e*x)**m/(m**7 + 49*m**6 + 973*m**5 + 10045*m**4 + 57379*m**3 + 177331*m**2 + 264207*m + 135135) + 9120*A*a
**3*c**2*m**3*x*(e*x)**m/(m**7 + 49*m**6 + 973*m**5 + 10045*m**4 + 57379*m**3 + 177331*m**2 + 264207*m + 13513
5) + 48259*A*a**3*c**2*m**2*x*(e*x)**m/(m**7 + 49*m**6 + 973*m**5 + 10045*m**4 + 57379*m**3 + 177331*m**2 + 26
4207*m + 135135) + 129072*A*a**3*c**2*m*x*(e*x)**m/(m**7 + 49*m**6 + 973*m**5 + 10045*m**4 + 57379*m**3 + 1773
31*m**2 + 264207*m + 135135) + 135135*A*a**3*c**2*x*(e*x)**m/(m**7 + 49*m**6 + 973*m**5 + 10045*m**4 + 57379*m
**3 + 177331*m**2 + 264207*m + 135135) + 2*A*a**3*c*d*m**6*x**3*(e*x)**m/(m**7 + 49*m**6 + 973*m**5 + 10045*m*
*4 + 57379*m**3 + 177331*m**2 + 264207*m + 135135) + 92*A*a**3*c*d*m**5*x**3*(e*x)**m/(m**7 + 49*m**6 + 973*m*
*5 + 10045*m**4 + 57379*m**3 + 177331*m**2 + 264207*m + 135135) + 1670*A*a**3*c*d*m**4*x**3*(e*x)**m/(m**7 + 4
9*m**6 + 973*m**5 + 10045*m**4 + 57379*m**3 + 177331*m**2 + 264207*m + 135135) + 15080*A*a**3*c*d*m**3*x**3*(e
*x)**m/(m**7 + 49*m**6 + 973*m**5 + 10045*m**4 + 57379*m**3 + 177331*m**2 + 264207*m + 135135) + 69518*A*a**3*
c*d*m**2*x**3*(e*x)**m/(m**7 + 49*m**6 + 973*m**5 + 10045*m**4 + 57379*m**3 + 177331*m**2 + 264207*m + 135135)
+ 146108*A*a**3*c*d*m*x**3*(e*x)**m/(m**7 + 49*m**6 + 973*m**5 + 10045*m**4 + 57379*m**3 + 177331*m**2 + 2642
07*m + 135135) + 90090*A*a**3*c*d*x**3*(e*x)**m/(m**7 + 49*m**6 + 973*m**5 + 10045*m**4 + 57379*m**3 + 177331*
m**2 + 264207*m + 135135) + A*a**3*d**2*m**6*x**5*(e*x)**m/(m**7 + 49*m**6 + 973*m**5 + 10045*m**4 + 57379*m**
3 + 177331*m**2 + 264207*m + 135135) + 44*A*a**3*d**2*m**5*x**5*(e*x)**m/(m**7 + 49*m**6 + 973*m**5 + 10045*m*
*4 + 57379*m**3 + 177331*m**2 + 264207*m + 135135) + 753*A*a**3*d**2*m**4*x**5*(e*x)**m/(m**7 + 49*m**6 + 973*
m**5 + 10045*m**4 + 57379*m**3 + 177331*m**2 + 264207*m + 135135) + 6280*A*a**3*d**2*m**3*x**5*(e*x)**m/(m**7
+ 49*m**6 + 973*m**5 + 10045*m**4 + 57379*m**3 + 177331*m**2 + 264207*m + 135135) + 25979*A*a**3*d**2*m**2*x**
5*(e*x)**m/(m**7 + 49*m**6 + 973*m**5 + 10045*m**4 + 57379*m**3 + 177331*m**2 + 264207*m + 135135) + 47436*A*a
**3*d**2*m*x**5*(e*x)**m/(m**7 + 49*m**6 + 973*m**5 + 10045*m**4 + 57379*m**3 + 177331*m**2 + 264207*m + 13513
5) + 27027*A*a**3*d**2*x**5*(e*x)**m/(m**7 + 49*m**6 + 973*m**5 + 10045*m**4 + 57379*m**3 + 177331*m**2 + 2642
07*m + 135135) + 3*A*a**2*b*c**2*m**6*x**3*(e*x)**m/(m**7 + 49*m**6 + 973*m**5 + 10045*m**4 + 57379*m**3 + 177
331*m**2 + 264207*m + 135135) + 138*A*a**2*b*c**2*m**5*x**3*(e*x)**m/(m**7 + 49*m**6 + 973*m**5 + 10045*m**4 +
57379*m**3 + 177331*m**2 + 264207*m + 135135) + 2505*A*a**2*b*c**2*m**4*x**3*(e*x)**m/(m**7 + 49*m**6 + 973*m
**5 + 10045*m**4 + 57379*m**3 + 177331*m**2 + 264207*m + 135135) + 22620*A*a**2*b*c**2*m**3*x**3*(e*x)**m/(m**
7 + 49*m**6 + 973*m**5 + 10045*m**4 + 57379*m**3 + 177331*m**2 + 264207*m + 135135) + 104277*A*a**2*b*c**2*m**
2*x**3*(e*x)**m/(m**7 + 49*m**6 + 973*m**5 + 10045*m**4 + 57379*m**3 + 177331*m**2 + 264207*m + 135135) + 2191
62*A*a**2*b*c**2*m*x**3*(e*x)**m/(m**7 + 49*m**6 + 973*m**5 + 10045*m**4 + 57379*m**3 + 177331*m**2 + 264207*m
+ 135135) + 135135*A*a**2*b*c**2*x**3*(e*x)**m/(m**7 + 49*m**6 + 973*m**5 + 10045*m**4 + 57379*m**3 + 177331*
m**2 + 264207*m + 135135) + 6*A*a**2*b*c*d*m**6*x**5*(e*x)**m/(m**7 + 49*m**6 + 973*m**5 + 10045*m**4 + 57379*
m**3 + 177331*m**2 + 264207*m + 135135) + 264*A*a**2*b*c*d*m**5*x**5*(e*x)**m/(m**7 + 49*m**6 + 973*m**5 + 100
45*m**4 + 57379*m**3 + 177331*m**2 + 264207*m + 135135) + 4518*A*a**2*b*c*d*m**4*x**5*(e*x)**m/(m**7 + 49*m**6
+ 973*m**5 + 10045*m**4 + 57379*m**3 + 177331*m**2 + 264207*m + 135135) + 37680*A*a**2*b*c*d*m**3*x**5*(e*x)*
*m/(m**7 + 49*m**6 + 973*m**5 + 10045*m**4 + 57379*m**3 + 177331*m**2 + 264207*m + 135135) + 155874*A*a**2*b*c
*d*m**2*x**5*(e*x)**m/(m**7 + 49*m**6 + 973*m**5 + 10045*m**4 + 57379*m**3 + 177331*m**2 + 264207*m + 135135)
+ 284616*A*a**2*b*c*d*m*x**5*(e*x)**m/(m**7 + 49*m**6 + 973*m**5 + 10045*m**4 + 57379*m**3 + 177331*m**2 + 264
207*m + 135135) + 162162*A*a**2*b*c*d*x**5*(e*x)**m/(m**7 + 49*m**6 + 973*m**5 + 10045*m**4 + 57379*m**3 + 177
331*m**2 + 264207*m + 135135) + 3*A*a**2*b*d**2*m**6*x**7*(e*x)**m/(m**7 + 49*m**6 + 973*m**5 + 10045*m**4 + 5
7379*m**3 + 177331*m**2 + 264207*m + 135135) + 126*A*a**2*b*d**2*m**5*x**7*(e*x)**m/(m**7 + 49*m**6 + 973*m**5
+ 10045*m**4 + 57379*m**3 + 177331*m**2 + 264207*m + 135135) + 2037*A*a**2*b*d**2*m**4*x**7*(e*x)**m/(m**7 +
49*m**6 + 973*m**5 + 10045*m**4 + 57379*m**3 + 177331*m**2 + 264207*m + 135135) + 15876*A*a**2*b*d**2*m**3*x**
7*(e*x)**m/(m**7 + 49*m**6 + 973*m**5 + 10045*m**4 + 57379*m**3 + 177331*m**2 + 264207*m + 135135) + 61005*A*a
**2*b*d**2*m**2*x**7*(e*x)**m/(m**7 + 49*m**6 + 973*m**5 + 10045*m**4 + 57379*m**3 + 177331*m**2 + 264207*m +
135135) + 104958*A*a**2*b*d**2*m*x**7*(e*x)**m/(m**7 + 49*m**6 + 973*m**5 + 10045*m**4 + 57379*m**3 + 177331*m
**2 + 264207*m + 135135) + 57915*A*a**2*b*d**2*x**7*(e*x)**m/(m**7 + 49*m**6 + 973*m**5 + 10045*m**4 + 57379*m
**3 + 177331*m**2 + 264207*m + 135135) + 3*A*a*b**2*c**2*m**6*x**5*(e*x)**m/(m**7 + 49*m**6 + 973*m**5 + 10045
*m**4 + 57379*m**3 + 177331*m**2 + 264207*m + 135135) + 132*A*a*b**2*c**2*m**5*x**5*(e*x)**m/(m**7 + 49*m**6 +
973*m**5 + 10045*m**4 + 57379*m**3 + 177331*m**2 + 264207*m + 135135) + 2259*A*a*b**2*c**2*m**4*x**5*(e*x)**m
/(m**7 + 49*m**6 + 973*m**5 + 10045*m**4 + 57379*m**3 + 177331*m**2 + 264207*m + 135135) + 18840*A*a*b**2*c**2
*m**3*x**5*(e*x)**m/(m**7 + 49*m**6 + 973*m**5 + 10045*m**4 + 57379*m**3 + 177331*m**2 + 264207*m + 135135) +
77937*A*a*b**2*c**2*m**2*x**5*(e*x)**m/(m**7 + 49*m**6 + 973*m**5 + 10045*m**4 + 57379*m**3 + 177331*m**2 + 26
4207*m + 135135) + 142308*A*a*b**2*c**2*m*x**5*(e*x)**m/(m**7 + 49*m**6 + 973*m**5 + 10045*m**4 + 57379*m**3 +
177331*m**2 + 264207*m + 135135) + 81081*A*a*b**2*c**2*x**5*(e*x)**m/(m**7 + 49*m**6 + 973*m**5 + 10045*m**4
+ 57379*m**3 + 177331*m**2 + 264207*m + 135135) + 6*A*a*b**2*c*d*m**6*x**7*(e*x)**m/(m**7 + 49*m**6 + 973*m**5
+ 10045*m**4 + 57379*m**3 + 177331*m**2 + 264207*m + 135135) + 252*A*a*b**2*c*d*m**5*x**7*(e*x)**m/(m**7 + 49
*m**6 + 973*m**5 + 10045*m**4 + 57379*m**3 + 177331*m**2 + 264207*m + 135135) + 4074*A*a*b**2*c*d*m**4*x**7*(e
*x)**m/(m**7 + 49*m**6 + 973*m**5 + 10045*m**4 + 57379*m**3 + 177331*m**2 + 264207*m + 135135) + 31752*A*a*b**
2*c*d*m**3*x**7*(e*x)**m/(m**7 + 49*m**6 + 973*m**5 + 10045*m**4 + 57379*m**3 + 177331*m**2 + 264207*m + 13513
5) + 122010*A*a*b**2*c*d*m**2*x**7*(e*x)**m/(m**7 + 49*m**6 + 973*m**5 + 10045*m**4 + 57379*m**3 + 177331*m**2
+ 264207*m + 135135) + 209916*A*a*b**2*c*d*m*x**7*(e*x)**m/(m**7 + 49*m**6 + 973*m**5 + 10045*m**4 + 57379*m*
*3 + 177331*m**2 + 264207*m + 135135) + 115830*A*a*b**2*c*d*x**7*(e*x)**m/(m**7 + 49*m**6 + 973*m**5 + 10045*m
**4 + 57379*m**3 + 177331*m**2 + 264207*m + 135135) + 3*A*a*b**2*d**2*m**6*x**9*(e*x)**m/(m**7 + 49*m**6 + 973
*m**5 + 10045*m**4 + 57379*m**3 + 177331*m**2 + 264207*m + 135135) + 120*A*a*b**2*d**2*m**5*x**9*(e*x)**m/(m**
7 + 49*m**6 + 973*m**5 + 10045*m**4 + 57379*m**3 + 177331*m**2 + 264207*m + 135135) + 1839*A*a*b**2*d**2*m**4*
x**9*(e*x)**m/(m**7 + 49*m**6 + 973*m**5 + 10045*m**4 + 57379*m**3 + 177331*m**2 + 264207*m + 135135) + 13584*
A*a*b**2*d**2*m**3*x**9*(e*x)**m/(m**7 + 49*m**6 + 973*m**5 + 10045*m**4 + 57379*m**3 + 177331*m**2 + 264207*m
+ 135135) + 49881*A*a*b**2*d**2*m**2*x**9*(e*x)**m/(m**7 + 49*m**6 + 973*m**5 + 10045*m**4 + 57379*m**3 + 177
331*m**2 + 264207*m + 135135) + 83064*A*a*b**2*d**2*m*x**9*(e*x)**m/(m**7 + 49*m**6 + 973*m**5 + 10045*m**4 +
57379*m**3 + 177331*m**2 + 264207*m + 135135) + 45045*A*a*b**2*d**2*x**9*(e*x)**m/(m**7 + 49*m**6 + 973*m**5 +
10045*m**4 + 57379*m**3 + 177331*m**2 + 264207*m + 135135) + A*b**3*c**2*m**6*x**7*(e*x)**m/(m**7 + 49*m**6 +
973*m**5 + 10045*m**4 + 57379*m**3 + 177331*m**2 + 264207*m + 135135) + 42*A*b**3*c**2*m**5*x**7*(e*x)**m/(m*
*7 + 49*m**6 + 973*m**5 + 10045*m**4 + 57379*m**3 + 177331*m**2 + 264207*m + 135135) + 679*A*b**3*c**2*m**4*x*
*7*(e*x)**m/(m**7 + 49*m**6 + 973*m**5 + 10045*m**4 + 57379*m**3 + 177331*m**2 + 264207*m + 135135) + 5292*A*b
**3*c**2*m**3*x**7*(e*x)**m/(m**7 + 49*m**6 + 973*m**5 + 10045*m**4 + 57379*m**3 + 177331*m**2 + 264207*m + 13
5135) + 20335*A*b**3*c**2*m**2*x**7*(e*x)**m/(m**7 + 49*m**6 + 973*m**5 + 10045*m**4 + 57379*m**3 + 177331*m**
2 + 264207*m + 135135) + 34986*A*b**3*c**2*m*x**7*(e*x)**m/(m**7 + 49*m**6 + 973*m**5 + 10045*m**4 + 57379*m**
3 + 177331*m**2 + 264207*m + 135135) + 19305*A*b**3*c**2*x**7*(e*x)**m/(m**7 + 49*m**6 + 973*m**5 + 10045*m**4
+ 57379*m**3 + 177331*m**2 + 264207*m + 135135) + 2*A*b**3*c*d*m**6*x**9*(e*x)**m/(m**7 + 49*m**6 + 973*m**5
+ 10045*m**4 + 57379*m**3 + 177331*m**2 + 264207*m + 135135) + 80*A*b**3*c*d*m**5*x**9*(e*x)**m/(m**7 + 49*m**
6 + 973*m**5 + 10045*m**4 + 57379*m**3 + 177331*m**2 + 264207*m + 135135) + 1226*A*b**3*c*d*m**4*x**9*(e*x)**m
/(m**7 + 49*m**6 + 973*m**5 + 10045*m**4 + 57379*m**3 + 177331*m**2 + 264207*m + 135135) + 9056*A*b**3*c*d*m**
3*x**9*(e*x)**m/(m**7 + 49*m**6 + 973*m**5 + 10045*m**4 + 57379*m**3 + 177331*m**2 + 264207*m + 135135) + 3325
4*A*b**3*c*d*m**2*x**9*(e*x)**m/(m**7 + 49*m**6 + 973*m**5 + 10045*m**4 + 57379*m**3 + 177331*m**2 + 264207*m
+ 135135) + 55376*A*b**3*c*d*m*x**9*(e*x)**m/(m**7 + 49*m**6 + 973*m**5 + 10045*m**4 + 57379*m**3 + 177331*m**
2 + 264207*m + 135135) + 30030*A*b**3*c*d*x**9*(e*x)**m/(m**7 + 49*m**6 + 973*m**5 + 10045*m**4 + 57379*m**3 +
177331*m**2 + 264207*m + 135135) + A*b**3*d**2*m**6*x**11*(e*x)**m/(m**7 + 49*m**6 + 973*m**5 + 10045*m**4 +
57379*m**3 + 177331*m**2 + 264207*m + 135135) + 38*A*b**3*d**2*m**5*x**11*(e*x)**m/(m**7 + 49*m**6 + 973*m**5
+ 10045*m**4 + 57379*m**3 + 177331*m**2 + 264207*m + 135135) + 555*A*b**3*d**2*m**4*x**11*(e*x)**m/(m**7 + 49*
m**6 + 973*m**5 + 10045*m**4 + 57379*m**3 + 177331*m**2 + 264207*m + 135135) + 3940*A*b**3*d**2*m**3*x**11*(e*
x)**m/(m**7 + 49*m**6 + 973*m**5 + 10045*m**4 + 57379*m**3 + 177331*m**2 + 264207*m + 135135) + 14039*A*b**3*d
**2*m**2*x**11*(e*x)**m/(m**7 + 49*m**6 + 973*m**5 + 10045*m**4 + 57379*m**3 + 177331*m**2 + 264207*m + 135135
) + 22902*A*b**3*d**2*m*x**11*(e*x)**m/(m**7 + 49*m**6 + 973*m**5 + 10045*m**4 + 57379*m**3 + 177331*m**2 + 26
4207*m + 135135) + 12285*A*b**3*d**2*x**11*(e*x)**m/(m**7 + 49*m**6 + 973*m**5 + 10045*m**4 + 57379*m**3 + 177
331*m**2 + 264207*m + 135135) + B*a**3*c**2*m**6*x**3*(e*x)**m/(m**7 + 49*m**6 + 973*m**5 + 10045*m**4 + 57379
*m**3 + 177331*m**2 + 264207*m + 135135) + 46*B*a**3*c**2*m**5*x**3*(e*x)**m/(m**7 + 49*m**6 + 973*m**5 + 1004
5*m**4 + 57379*m**3 + 177331*m**2 + 264207*m + 135135) + 835*B*a**3*c**2*m**4*x**3*(e*x)**m/(m**7 + 49*m**6 +
973*m**5 + 10045*m**4 + 57379*m**3 + 177331*m**2 + 264207*m + 135135) + 7540*B*a**3*c**2*m**3*x**3*(e*x)**m/(m
**7 + 49*m**6 + 973*m**5 + 10045*m**4 + 57379*m**3 + 177331*m**2 + 264207*m + 135135) + 34759*B*a**3*c**2*m**2
*x**3*(e*x)**m/(m**7 + 49*m**6 + 973*m**5 + 10045*m**4 + 57379*m**3 + 177331*m**2 + 264207*m + 135135) + 73054
*B*a**3*c**2*m*x**3*(e*x)**m/(m**7 + 49*m**6 + 973*m**5 + 10045*m**4 + 57379*m**3 + 177331*m**2 + 264207*m + 1
35135) + 45045*B*a**3*c**2*x**3*(e*x)**m/(m**7 + 49*m**6 + 973*m**5 + 10045*m**4 + 57379*m**3 + 177331*m**2 +
264207*m + 135135) + 2*B*a**3*c*d*m**6*x**5*(e*x)**m/(m**7 + 49*m**6 + 973*m**5 + 10045*m**4 + 57379*m**3 + 17
7331*m**2 + 264207*m + 135135) + 88*B*a**3*c*d*m**5*x**5*(e*x)**m/(m**7 + 49*m**6 + 973*m**5 + 10045*m**4 + 57
379*m**3 + 177331*m**2 + 264207*m + 135135) + 1506*B*a**3*c*d*m**4*x**5*(e*x)**m/(m**7 + 49*m**6 + 973*m**5 +
10045*m**4 + 57379*m**3 + 177331*m**2 + 264207*m + 135135) + 12560*B*a**3*c*d*m**3*x**5*(e*x)**m/(m**7 + 49*m*
*6 + 973*m**5 + 10045*m**4 + 57379*m**3 + 177331*m**2 + 264207*m + 135135) + 51958*B*a**3*c*d*m**2*x**5*(e*x)*
*m/(m**7 + 49*m**6 + 973*m**5 + 10045*m**4 + 57379*m**3 + 177331*m**2 + 264207*m + 135135) + 94872*B*a**3*c*d*
m*x**5*(e*x)**m/(m**7 + 49*m**6 + 973*m**5 + 10045*m**4 + 57379*m**3 + 177331*m**2 + 264207*m + 135135) + 5405
4*B*a**3*c*d*x**5*(e*x)**m/(m**7 + 49*m**6 + 973*m**5 + 10045*m**4 + 57379*m**3 + 177331*m**2 + 264207*m + 135
135) + B*a**3*d**2*m**6*x**7*(e*x)**m/(m**7 + 49*m**6 + 973*m**5 + 10045*m**4 + 57379*m**3 + 177331*m**2 + 264
207*m + 135135) + 42*B*a**3*d**2*m**5*x**7*(e*x)**m/(m**7 + 49*m**6 + 973*m**5 + 10045*m**4 + 57379*m**3 + 177
331*m**2 + 264207*m + 135135) + 679*B*a**3*d**2*m**4*x**7*(e*x)**m/(m**7 + 49*m**6 + 973*m**5 + 10045*m**4 + 5
7379*m**3 + 177331*m**2 + 264207*m + 135135) + 5292*B*a**3*d**2*m**3*x**7*(e*x)**m/(m**7 + 49*m**6 + 973*m**5
+ 10045*m**4 + 57379*m**3 + 177331*m**2 + 264207*m + 135135) + 20335*B*a**3*d**2*m**2*x**7*(e*x)**m/(m**7 + 49
*m**6 + 973*m**5 + 10045*m**4 + 57379*m**3 + 177331*m**2 + 264207*m + 135135) + 34986*B*a**3*d**2*m*x**7*(e*x)
**m/(m**7 + 49*m**6 + 973*m**5 + 10045*m**4 + 57379*m**3 + 177331*m**2 + 264207*m + 135135) + 19305*B*a**3*d**
2*x**7*(e*x)**m/(m**7 + 49*m**6 + 973*m**5 + 10045*m**4 + 57379*m**3 + 177331*m**2 + 264207*m + 135135) + 3*B*
a**2*b*c**2*m**6*x**5*(e*x)**m/(m**7 + 49*m**6 + 973*m**5 + 10045*m**4 + 57379*m**3 + 177331*m**2 + 264207*m +
135135) + 132*B*a**2*b*c**2*m**5*x**5*(e*x)**m/(m**7 + 49*m**6 + 973*m**5 + 10045*m**4 + 57379*m**3 + 177331*
m**2 + 264207*m + 135135) + 2259*B*a**2*b*c**2*m**4*x**5*(e*x)**m/(m**7 + 49*m**6 + 973*m**5 + 10045*m**4 + 57
379*m**3 + 177331*m**2 + 264207*m + 135135) + 18840*B*a**2*b*c**2*m**3*x**5*(e*x)**m/(m**7 + 49*m**6 + 973*m**
5 + 10045*m**4 + 57379*m**3 + 177331*m**2 + 264207*m + 135135) + 77937*B*a**2*b*c**2*m**2*x**5*(e*x)**m/(m**7
+ 49*m**6 + 973*m**5 + 10045*m**4 + 57379*m**3 + 177331*m**2 + 264207*m + 135135) + 142308*B*a**2*b*c**2*m*x**
5*(e*x)**m/(m**7 + 49*m**6 + 973*m**5 + 10045*m**4 + 57379*m**3 + 177331*m**2 + 264207*m + 135135) + 81081*B*a
**2*b*c**2*x**5*(e*x)**m/(m**7 + 49*m**6 + 973*m**5 + 10045*m**4 + 57379*m**3 + 177331*m**2 + 264207*m + 13513
5) + 6*B*a**2*b*c*d*m**6*x**7*(e*x)**m/(m**7 + 49*m**6 + 973*m**5 + 10045*m**4 + 57379*m**3 + 177331*m**2 + 26
4207*m + 135135) + 252*B*a**2*b*c*d*m**5*x**7*(e*x)**m/(m**7 + 49*m**6 + 973*m**5 + 10045*m**4 + 57379*m**3 +
177331*m**2 + 264207*m + 135135) + 4074*B*a**2*b*c*d*m**4*x**7*(e*x)**m/(m**7 + 49*m**6 + 973*m**5 + 10045*m**
4 + 57379*m**3 + 177331*m**2 + 264207*m + 135135) + 31752*B*a**2*b*c*d*m**3*x**7*(e*x)**m/(m**7 + 49*m**6 + 97
3*m**5 + 10045*m**4 + 57379*m**3 + 177331*m**2 + 264207*m + 135135) + 122010*B*a**2*b*c*d*m**2*x**7*(e*x)**m/(
m**7 + 49*m**6 + 973*m**5 + 10045*m**4 + 57379*m**3 + 177331*m**2 + 264207*m + 135135) + 209916*B*a**2*b*c*d*m
*x**7*(e*x)**m/(m**7 + 49*m**6 + 973*m**5 + 10045*m**4 + 57379*m**3 + 177331*m**2 + 264207*m + 135135) + 11583
0*B*a**2*b*c*d*x**7*(e*x)**m/(m**7 + 49*m**6 + 973*m**5 + 10045*m**4 + 57379*m**3 + 177331*m**2 + 264207*m + 1
35135) + 3*B*a**2*b*d**2*m**6*x**9*(e*x)**m/(m**7 + 49*m**6 + 973*m**5 + 10045*m**4 + 57379*m**3 + 177331*m**2
+ 264207*m + 135135) + 120*B*a**2*b*d**2*m**5*x**9*(e*x)**m/(m**7 + 49*m**6 + 973*m**5 + 10045*m**4 + 57379*m
**3 + 177331*m**2 + 264207*m + 135135) + 1839*B*a**2*b*d**2*m**4*x**9*(e*x)**m/(m**7 + 49*m**6 + 973*m**5 + 10
045*m**4 + 57379*m**3 + 177331*m**2 + 264207*m + 135135) + 13584*B*a**2*b*d**2*m**3*x**9*(e*x)**m/(m**7 + 49*m
**6 + 973*m**5 + 10045*m**4 + 57379*m**3 + 177331*m**2 + 264207*m + 135135) + 49881*B*a**2*b*d**2*m**2*x**9*(e
*x)**m/(m**7 + 49*m**6 + 973*m**5 + 10045*m**4 + 57379*m**3 + 177331*m**2 + 264207*m + 135135) + 83064*B*a**2*
b*d**2*m*x**9*(e*x)**m/(m**7 + 49*m**6 + 973*m**5 + 10045*m**4 + 57379*m**3 + 177331*m**2 + 264207*m + 135135)
+ 45045*B*a**2*b*d**2*x**9*(e*x)**m/(m**7 + 49*m**6 + 973*m**5 + 10045*m**4 + 57379*m**3 + 177331*m**2 + 2642
07*m + 135135) + 3*B*a*b**2*c**2*m**6*x**7*(e*x)**m/(m**7 + 49*m**6 + 973*m**5 + 10045*m**4 + 57379*m**3 + 177
331*m**2 + 264207*m + 135135) + 126*B*a*b**2*c**2*m**5*x**7*(e*x)**m/(m**7 + 49*m**6 + 973*m**5 + 10045*m**4 +
57379*m**3 + 177331*m**2 + 264207*m + 135135) + 2037*B*a*b**2*c**2*m**4*x**7*(e*x)**m/(m**7 + 49*m**6 + 973*m
**5 + 10045*m**4 + 57379*m**3 + 177331*m**2 + 264207*m + 135135) + 15876*B*a*b**2*c**2*m**3*x**7*(e*x)**m/(m**
7 + 49*m**6 + 973*m**5 + 10045*m**4 + 57379*m**3 + 177331*m**2 + 264207*m + 135135) + 61005*B*a*b**2*c**2*m**2
*x**7*(e*x)**m/(m**7 + 49*m**6 + 973*m**5 + 10045*m**4 + 57379*m**3 + 177331*m**2 + 264207*m + 135135) + 10495
8*B*a*b**2*c**2*m*x**7*(e*x)**m/(m**7 + 49*m**6 + 973*m**5 + 10045*m**4 + 57379*m**3 + 177331*m**2 + 264207*m
+ 135135) + 57915*B*a*b**2*c**2*x**7*(e*x)**m/(m**7 + 49*m**6 + 973*m**5 + 10045*m**4 + 57379*m**3 + 177331*m*
*2 + 264207*m + 135135) + 6*B*a*b**2*c*d*m**6*x**9*(e*x)**m/(m**7 + 49*m**6 + 973*m**5 + 10045*m**4 + 57379*m*
*3 + 177331*m**2 + 264207*m + 135135) + 240*B*a*b**2*c*d*m**5*x**9*(e*x)**m/(m**7 + 49*m**6 + 973*m**5 + 10045
*m**4 + 57379*m**3 + 177331*m**2 + 264207*m + 135135) + 3678*B*a*b**2*c*d*m**4*x**9*(e*x)**m/(m**7 + 49*m**6 +
973*m**5 + 10045*m**4 + 57379*m**3 + 177331*m**2 + 264207*m + 135135) + 27168*B*a*b**2*c*d*m**3*x**9*(e*x)**m
/(m**7 + 49*m**6 + 973*m**5 + 10045*m**4 + 57379*m**3 + 177331*m**2 + 264207*m + 135135) + 99762*B*a*b**2*c*d*
m**2*x**9*(e*x)**m/(m**7 + 49*m**6 + 973*m**5 + 10045*m**4 + 57379*m**3 + 177331*m**2 + 264207*m + 135135) + 1
66128*B*a*b**2*c*d*m*x**9*(e*x)**m/(m**7 + 49*m**6 + 973*m**5 + 10045*m**4 + 57379*m**3 + 177331*m**2 + 264207
*m + 135135) + 90090*B*a*b**2*c*d*x**9*(e*x)**m/(m**7 + 49*m**6 + 973*m**5 + 10045*m**4 + 57379*m**3 + 177331*
m**2 + 264207*m + 135135) + 3*B*a*b**2*d**2*m**6*x**11*(e*x)**m/(m**7 + 49*m**6 + 973*m**5 + 10045*m**4 + 5737
9*m**3 + 177331*m**2 + 264207*m + 135135) + 114*B*a*b**2*d**2*m**5*x**11*(e*x)**m/(m**7 + 49*m**6 + 973*m**5 +
10045*m**4 + 57379*m**3 + 177331*m**2 + 264207*m + 135135) + 1665*B*a*b**2*d**2*m**4*x**11*(e*x)**m/(m**7 + 4
9*m**6 + 973*m**5 + 10045*m**4 + 57379*m**3 + 177331*m**2 + 264207*m + 135135) + 11820*B*a*b**2*d**2*m**3*x**1
1*(e*x)**m/(m**7 + 49*m**6 + 973*m**5 + 10045*m**4 + 57379*m**3 + 177331*m**2 + 264207*m + 135135) + 42117*B*a
*b**2*d**2*m**2*x**11*(e*x)**m/(m**7 + 49*m**6 + 973*m**5 + 10045*m**4 + 57379*m**3 + 177331*m**2 + 264207*m +
135135) + 68706*B*a*b**2*d**2*m*x**11*(e*x)**m/(m**7 + 49*m**6 + 973*m**5 + 10045*m**4 + 57379*m**3 + 177331*
m**2 + 264207*m + 135135) + 36855*B*a*b**2*d**2*x**11*(e*x)**m/(m**7 + 49*m**6 + 973*m**5 + 10045*m**4 + 57379
*m**3 + 177331*m**2 + 264207*m + 135135) + B*b**3*c**2*m**6*x**9*(e*x)**m/(m**7 + 49*m**6 + 973*m**5 + 10045*m
**4 + 57379*m**3 + 177331*m**2 + 264207*m + 135135) + 40*B*b**3*c**2*m**5*x**9*(e*x)**m/(m**7 + 49*m**6 + 973*
m**5 + 10045*m**4 + 57379*m**3 + 177331*m**2 + 264207*m + 135135) + 613*B*b**3*c**2*m**4*x**9*(e*x)**m/(m**7 +
49*m**6 + 973*m**5 + 10045*m**4 + 57379*m**3 + 177331*m**2 + 264207*m + 135135) + 4528*B*b**3*c**2*m**3*x**9*
(e*x)**m/(m**7 + 49*m**6 + 973*m**5 + 10045*m**4 + 57379*m**3 + 177331*m**2 + 264207*m + 135135) + 16627*B*b**
3*c**2*m**2*x**9*(e*x)**m/(m**7 + 49*m**6 + 973*m**5 + 10045*m**4 + 57379*m**3 + 177331*m**2 + 264207*m + 1351
35) + 27688*B*b**3*c**2*m*x**9*(e*x)**m/(m**7 + 49*m**6 + 973*m**5 + 10045*m**4 + 57379*m**3 + 177331*m**2 + 2
64207*m + 135135) + 15015*B*b**3*c**2*x**9*(e*x)**m/(m**7 + 49*m**6 + 973*m**5 + 10045*m**4 + 57379*m**3 + 177
331*m**2 + 264207*m + 135135) + 2*B*b**3*c*d*m**6*x**11*(e*x)**m/(m**7 + 49*m**6 + 973*m**5 + 10045*m**4 + 573
79*m**3 + 177331*m**2 + 264207*m + 135135) + 76*B*b**3*c*d*m**5*x**11*(e*x)**m/(m**7 + 49*m**6 + 973*m**5 + 10
045*m**4 + 57379*m**3 + 177331*m**2 + 264207*m + 135135) + 1110*B*b**3*c*d*m**4*x**11*(e*x)**m/(m**7 + 49*m**6
+ 973*m**5 + 10045*m**4 + 57379*m**3 + 177331*m**2 + 264207*m + 135135) + 7880*B*b**3*c*d*m**3*x**11*(e*x)**m
/(m**7 + 49*m**6 + 973*m**5 + 10045*m**4 + 57379*m**3 + 177331*m**2 + 264207*m + 135135) + 28078*B*b**3*c*d*m*
*2*x**11*(e*x)**m/(m**7 + 49*m**6 + 973*m**5 + 10045*m**4 + 57379*m**3 + 177331*m**2 + 264207*m + 135135) + 45
804*B*b**3*c*d*m*x**11*(e*x)**m/(m**7 + 49*m**6 + 973*m**5 + 10045*m**4 + 57379*m**3 + 177331*m**2 + 264207*m
+ 135135) + 24570*B*b**3*c*d*x**11*(e*x)**m/(m**7 + 49*m**6 + 973*m**5 + 10045*m**4 + 57379*m**3 + 177331*m**2
+ 264207*m + 135135) + B*b**3*d**2*m**6*x**13*(e*x)**m/(m**7 + 49*m**6 + 973*m**5 + 10045*m**4 + 57379*m**3 +
177331*m**2 + 264207*m + 135135) + 36*B*b**3*d**2*m**5*x**13*(e*x)**m/(m**7 + 49*m**6 + 973*m**5 + 10045*m**4
+ 57379*m**3 + 177331*m**2 + 264207*m + 135135) + 505*B*b**3*d**2*m**4*x**13*(e*x)**m/(m**7 + 49*m**6 + 973*m
**5 + 10045*m**4 + 57379*m**3 + 177331*m**2 + 264207*m + 135135) + 3480*B*b**3*d**2*m**3*x**13*(e*x)**m/(m**7
+ 49*m**6 + 973*m**5 + 10045*m**4 + 57379*m**3 + 177331*m**2 + 264207*m + 135135) + 12139*B*b**3*d**2*m**2*x**
13*(e*x)**m/(m**7 + 49*m**6 + 973*m**5 + 10045*m**4 + 57379*m**3 + 177331*m**2 + 264207*m + 135135) + 19524*B*
b**3*d**2*m*x**13*(e*x)**m/(m**7 + 49*m**6 + 973*m**5 + 10045*m**4 + 57379*m**3 + 177331*m**2 + 264207*m + 135
135) + 10395*B*b**3*d**2*x**13*(e*x)**m/(m**7 + 49*m**6 + 973*m**5 + 10045*m**4 + 57379*m**3 + 177331*m**2 + 2
64207*m + 135135), True))
Maxima [A] (verification not implemented)
none
Time = 0.29 (sec) , antiderivative size = 550, normalized size of antiderivative = 1.88
\[
\int (e x)^m \left (a+b x^2\right )^3 \left (A+B x^2\right ) \left (c+d x^2\right )^2 \, dx=\frac {B b^{3} d^{2} e^{m} x^{13} x^{m}}{m + 13} + \frac {2 \, B b^{3} c d e^{m} x^{11} x^{m}}{m + 11} + \frac {3 \, B a b^{2} d^{2} e^{m} x^{11} x^{m}}{m + 11} + \frac {A b^{3} d^{2} e^{m} x^{11} x^{m}}{m + 11} + \frac {B b^{3} c^{2} e^{m} x^{9} x^{m}}{m + 9} + \frac {6 \, B a b^{2} c d e^{m} x^{9} x^{m}}{m + 9} + \frac {2 \, A b^{3} c d e^{m} x^{9} x^{m}}{m + 9} + \frac {3 \, B a^{2} b d^{2} e^{m} x^{9} x^{m}}{m + 9} + \frac {3 \, A a b^{2} d^{2} e^{m} x^{9} x^{m}}{m + 9} + \frac {3 \, B a b^{2} c^{2} e^{m} x^{7} x^{m}}{m + 7} + \frac {A b^{3} c^{2} e^{m} x^{7} x^{m}}{m + 7} + \frac {6 \, B a^{2} b c d e^{m} x^{7} x^{m}}{m + 7} + \frac {6 \, A a b^{2} c d e^{m} x^{7} x^{m}}{m + 7} + \frac {B a^{3} d^{2} e^{m} x^{7} x^{m}}{m + 7} + \frac {3 \, A a^{2} b d^{2} e^{m} x^{7} x^{m}}{m + 7} + \frac {3 \, B a^{2} b c^{2} e^{m} x^{5} x^{m}}{m + 5} + \frac {3 \, A a b^{2} c^{2} e^{m} x^{5} x^{m}}{m + 5} + \frac {2 \, B a^{3} c d e^{m} x^{5} x^{m}}{m + 5} + \frac {6 \, A a^{2} b c d e^{m} x^{5} x^{m}}{m + 5} + \frac {A a^{3} d^{2} e^{m} x^{5} x^{m}}{m + 5} + \frac {B a^{3} c^{2} e^{m} x^{3} x^{m}}{m + 3} + \frac {3 \, A a^{2} b c^{2} e^{m} x^{3} x^{m}}{m + 3} + \frac {2 \, A a^{3} c d e^{m} x^{3} x^{m}}{m + 3} + \frac {\left (e x\right )^{m + 1} A a^{3} c^{2}}{e {\left (m + 1\right )}}
\]
[In]
integrate((e*x)^m*(b*x^2+a)^3*(B*x^2+A)*(d*x^2+c)^2,x, algorithm="maxima")
[Out]
B*b^3*d^2*e^m*x^13*x^m/(m + 13) + 2*B*b^3*c*d*e^m*x^11*x^m/(m + 11) + 3*B*a*b^2*d^2*e^m*x^11*x^m/(m + 11) + A*
b^3*d^2*e^m*x^11*x^m/(m + 11) + B*b^3*c^2*e^m*x^9*x^m/(m + 9) + 6*B*a*b^2*c*d*e^m*x^9*x^m/(m + 9) + 2*A*b^3*c*
d*e^m*x^9*x^m/(m + 9) + 3*B*a^2*b*d^2*e^m*x^9*x^m/(m + 9) + 3*A*a*b^2*d^2*e^m*x^9*x^m/(m + 9) + 3*B*a*b^2*c^2*
e^m*x^7*x^m/(m + 7) + A*b^3*c^2*e^m*x^7*x^m/(m + 7) + 6*B*a^2*b*c*d*e^m*x^7*x^m/(m + 7) + 6*A*a*b^2*c*d*e^m*x^
7*x^m/(m + 7) + B*a^3*d^2*e^m*x^7*x^m/(m + 7) + 3*A*a^2*b*d^2*e^m*x^7*x^m/(m + 7) + 3*B*a^2*b*c^2*e^m*x^5*x^m/
(m + 5) + 3*A*a*b^2*c^2*e^m*x^5*x^m/(m + 5) + 2*B*a^3*c*d*e^m*x^5*x^m/(m + 5) + 6*A*a^2*b*c*d*e^m*x^5*x^m/(m +
5) + A*a^3*d^2*e^m*x^5*x^m/(m + 5) + B*a^3*c^2*e^m*x^3*x^m/(m + 3) + 3*A*a^2*b*c^2*e^m*x^3*x^m/(m + 3) + 2*A*
a^3*c*d*e^m*x^3*x^m/(m + 3) + (e*x)^(m + 1)*A*a^3*c^2/(e*(m + 1))
Giac [B] (verification not implemented)
Leaf count of result is larger than twice the leaf count of optimal. 3283 vs. \(2 (292) = 584\).
Time = 0.35 (sec) , antiderivative size = 3283, normalized size of antiderivative = 11.24
\[
\int (e x)^m \left (a+b x^2\right )^3 \left (A+B x^2\right ) \left (c+d x^2\right )^2 \, dx=\text {Too large to display}
\]
[In]
integrate((e*x)^m*(b*x^2+a)^3*(B*x^2+A)*(d*x^2+c)^2,x, algorithm="giac")
[Out]
((e*x)^m*B*b^3*d^2*m^6*x^13 + 36*(e*x)^m*B*b^3*d^2*m^5*x^13 + 2*(e*x)^m*B*b^3*c*d*m^6*x^11 + 3*(e*x)^m*B*a*b^2
*d^2*m^6*x^11 + (e*x)^m*A*b^3*d^2*m^6*x^11 + 505*(e*x)^m*B*b^3*d^2*m^4*x^13 + 76*(e*x)^m*B*b^3*c*d*m^5*x^11 +
114*(e*x)^m*B*a*b^2*d^2*m^5*x^11 + 38*(e*x)^m*A*b^3*d^2*m^5*x^11 + 3480*(e*x)^m*B*b^3*d^2*m^3*x^13 + (e*x)^m*B
*b^3*c^2*m^6*x^9 + 6*(e*x)^m*B*a*b^2*c*d*m^6*x^9 + 2*(e*x)^m*A*b^3*c*d*m^6*x^9 + 3*(e*x)^m*B*a^2*b*d^2*m^6*x^9
+ 3*(e*x)^m*A*a*b^2*d^2*m^6*x^9 + 1110*(e*x)^m*B*b^3*c*d*m^4*x^11 + 1665*(e*x)^m*B*a*b^2*d^2*m^4*x^11 + 555*(
e*x)^m*A*b^3*d^2*m^4*x^11 + 12139*(e*x)^m*B*b^3*d^2*m^2*x^13 + 40*(e*x)^m*B*b^3*c^2*m^5*x^9 + 240*(e*x)^m*B*a*
b^2*c*d*m^5*x^9 + 80*(e*x)^m*A*b^3*c*d*m^5*x^9 + 120*(e*x)^m*B*a^2*b*d^2*m^5*x^9 + 120*(e*x)^m*A*a*b^2*d^2*m^5
*x^9 + 7880*(e*x)^m*B*b^3*c*d*m^3*x^11 + 11820*(e*x)^m*B*a*b^2*d^2*m^3*x^11 + 3940*(e*x)^m*A*b^3*d^2*m^3*x^11
+ 19524*(e*x)^m*B*b^3*d^2*m*x^13 + 3*(e*x)^m*B*a*b^2*c^2*m^6*x^7 + (e*x)^m*A*b^3*c^2*m^6*x^7 + 6*(e*x)^m*B*a^2
*b*c*d*m^6*x^7 + 6*(e*x)^m*A*a*b^2*c*d*m^6*x^7 + (e*x)^m*B*a^3*d^2*m^6*x^7 + 3*(e*x)^m*A*a^2*b*d^2*m^6*x^7 + 6
13*(e*x)^m*B*b^3*c^2*m^4*x^9 + 3678*(e*x)^m*B*a*b^2*c*d*m^4*x^9 + 1226*(e*x)^m*A*b^3*c*d*m^4*x^9 + 1839*(e*x)^
m*B*a^2*b*d^2*m^4*x^9 + 1839*(e*x)^m*A*a*b^2*d^2*m^4*x^9 + 28078*(e*x)^m*B*b^3*c*d*m^2*x^11 + 42117*(e*x)^m*B*
a*b^2*d^2*m^2*x^11 + 14039*(e*x)^m*A*b^3*d^2*m^2*x^11 + 10395*(e*x)^m*B*b^3*d^2*x^13 + 126*(e*x)^m*B*a*b^2*c^2
*m^5*x^7 + 42*(e*x)^m*A*b^3*c^2*m^5*x^7 + 252*(e*x)^m*B*a^2*b*c*d*m^5*x^7 + 252*(e*x)^m*A*a*b^2*c*d*m^5*x^7 +
42*(e*x)^m*B*a^3*d^2*m^5*x^7 + 126*(e*x)^m*A*a^2*b*d^2*m^5*x^7 + 4528*(e*x)^m*B*b^3*c^2*m^3*x^9 + 27168*(e*x)^
m*B*a*b^2*c*d*m^3*x^9 + 9056*(e*x)^m*A*b^3*c*d*m^3*x^9 + 13584*(e*x)^m*B*a^2*b*d^2*m^3*x^9 + 13584*(e*x)^m*A*a
*b^2*d^2*m^3*x^9 + 45804*(e*x)^m*B*b^3*c*d*m*x^11 + 68706*(e*x)^m*B*a*b^2*d^2*m*x^11 + 22902*(e*x)^m*A*b^3*d^2
*m*x^11 + 3*(e*x)^m*B*a^2*b*c^2*m^6*x^5 + 3*(e*x)^m*A*a*b^2*c^2*m^6*x^5 + 2*(e*x)^m*B*a^3*c*d*m^6*x^5 + 6*(e*x
)^m*A*a^2*b*c*d*m^6*x^5 + (e*x)^m*A*a^3*d^2*m^6*x^5 + 2037*(e*x)^m*B*a*b^2*c^2*m^4*x^7 + 679*(e*x)^m*A*b^3*c^2
*m^4*x^7 + 4074*(e*x)^m*B*a^2*b*c*d*m^4*x^7 + 4074*(e*x)^m*A*a*b^2*c*d*m^4*x^7 + 679*(e*x)^m*B*a^3*d^2*m^4*x^7
+ 2037*(e*x)^m*A*a^2*b*d^2*m^4*x^7 + 16627*(e*x)^m*B*b^3*c^2*m^2*x^9 + 99762*(e*x)^m*B*a*b^2*c*d*m^2*x^9 + 33
254*(e*x)^m*A*b^3*c*d*m^2*x^9 + 49881*(e*x)^m*B*a^2*b*d^2*m^2*x^9 + 49881*(e*x)^m*A*a*b^2*d^2*m^2*x^9 + 24570*
(e*x)^m*B*b^3*c*d*x^11 + 36855*(e*x)^m*B*a*b^2*d^2*x^11 + 12285*(e*x)^m*A*b^3*d^2*x^11 + 132*(e*x)^m*B*a^2*b*c
^2*m^5*x^5 + 132*(e*x)^m*A*a*b^2*c^2*m^5*x^5 + 88*(e*x)^m*B*a^3*c*d*m^5*x^5 + 264*(e*x)^m*A*a^2*b*c*d*m^5*x^5
+ 44*(e*x)^m*A*a^3*d^2*m^5*x^5 + 15876*(e*x)^m*B*a*b^2*c^2*m^3*x^7 + 5292*(e*x)^m*A*b^3*c^2*m^3*x^7 + 31752*(e
*x)^m*B*a^2*b*c*d*m^3*x^7 + 31752*(e*x)^m*A*a*b^2*c*d*m^3*x^7 + 5292*(e*x)^m*B*a^3*d^2*m^3*x^7 + 15876*(e*x)^m
*A*a^2*b*d^2*m^3*x^7 + 27688*(e*x)^m*B*b^3*c^2*m*x^9 + 166128*(e*x)^m*B*a*b^2*c*d*m*x^9 + 55376*(e*x)^m*A*b^3*
c*d*m*x^9 + 83064*(e*x)^m*B*a^2*b*d^2*m*x^9 + 83064*(e*x)^m*A*a*b^2*d^2*m*x^9 + (e*x)^m*B*a^3*c^2*m^6*x^3 + 3*
(e*x)^m*A*a^2*b*c^2*m^6*x^3 + 2*(e*x)^m*A*a^3*c*d*m^6*x^3 + 2259*(e*x)^m*B*a^2*b*c^2*m^4*x^5 + 2259*(e*x)^m*A*
a*b^2*c^2*m^4*x^5 + 1506*(e*x)^m*B*a^3*c*d*m^4*x^5 + 4518*(e*x)^m*A*a^2*b*c*d*m^4*x^5 + 753*(e*x)^m*A*a^3*d^2*
m^4*x^5 + 61005*(e*x)^m*B*a*b^2*c^2*m^2*x^7 + 20335*(e*x)^m*A*b^3*c^2*m^2*x^7 + 122010*(e*x)^m*B*a^2*b*c*d*m^2
*x^7 + 122010*(e*x)^m*A*a*b^2*c*d*m^2*x^7 + 20335*(e*x)^m*B*a^3*d^2*m^2*x^7 + 61005*(e*x)^m*A*a^2*b*d^2*m^2*x^
7 + 15015*(e*x)^m*B*b^3*c^2*x^9 + 90090*(e*x)^m*B*a*b^2*c*d*x^9 + 30030*(e*x)^m*A*b^3*c*d*x^9 + 45045*(e*x)^m*
B*a^2*b*d^2*x^9 + 45045*(e*x)^m*A*a*b^2*d^2*x^9 + 46*(e*x)^m*B*a^3*c^2*m^5*x^3 + 138*(e*x)^m*A*a^2*b*c^2*m^5*x
^3 + 92*(e*x)^m*A*a^3*c*d*m^5*x^3 + 18840*(e*x)^m*B*a^2*b*c^2*m^3*x^5 + 18840*(e*x)^m*A*a*b^2*c^2*m^3*x^5 + 12
560*(e*x)^m*B*a^3*c*d*m^3*x^5 + 37680*(e*x)^m*A*a^2*b*c*d*m^3*x^5 + 6280*(e*x)^m*A*a^3*d^2*m^3*x^5 + 104958*(e
*x)^m*B*a*b^2*c^2*m*x^7 + 34986*(e*x)^m*A*b^3*c^2*m*x^7 + 209916*(e*x)^m*B*a^2*b*c*d*m*x^7 + 209916*(e*x)^m*A*
a*b^2*c*d*m*x^7 + 34986*(e*x)^m*B*a^3*d^2*m*x^7 + 104958*(e*x)^m*A*a^2*b*d^2*m*x^7 + (e*x)^m*A*a^3*c^2*m^6*x +
835*(e*x)^m*B*a^3*c^2*m^4*x^3 + 2505*(e*x)^m*A*a^2*b*c^2*m^4*x^3 + 1670*(e*x)^m*A*a^3*c*d*m^4*x^3 + 77937*(e*
x)^m*B*a^2*b*c^2*m^2*x^5 + 77937*(e*x)^m*A*a*b^2*c^2*m^2*x^5 + 51958*(e*x)^m*B*a^3*c*d*m^2*x^5 + 155874*(e*x)^
m*A*a^2*b*c*d*m^2*x^5 + 25979*(e*x)^m*A*a^3*d^2*m^2*x^5 + 57915*(e*x)^m*B*a*b^2*c^2*x^7 + 19305*(e*x)^m*A*b^3*
c^2*x^7 + 115830*(e*x)^m*B*a^2*b*c*d*x^7 + 115830*(e*x)^m*A*a*b^2*c*d*x^7 + 19305*(e*x)^m*B*a^3*d^2*x^7 + 5791
5*(e*x)^m*A*a^2*b*d^2*x^7 + 48*(e*x)^m*A*a^3*c^2*m^5*x + 7540*(e*x)^m*B*a^3*c^2*m^3*x^3 + 22620*(e*x)^m*A*a^2*
b*c^2*m^3*x^3 + 15080*(e*x)^m*A*a^3*c*d*m^3*x^3 + 142308*(e*x)^m*B*a^2*b*c^2*m*x^5 + 142308*(e*x)^m*A*a*b^2*c^
2*m*x^5 + 94872*(e*x)^m*B*a^3*c*d*m*x^5 + 284616*(e*x)^m*A*a^2*b*c*d*m*x^5 + 47436*(e*x)^m*A*a^3*d^2*m*x^5 + 9
25*(e*x)^m*A*a^3*c^2*m^4*x + 34759*(e*x)^m*B*a^3*c^2*m^2*x^3 + 104277*(e*x)^m*A*a^2*b*c^2*m^2*x^3 + 69518*(e*x
)^m*A*a^3*c*d*m^2*x^3 + 81081*(e*x)^m*B*a^2*b*c^2*x^5 + 81081*(e*x)^m*A*a*b^2*c^2*x^5 + 54054*(e*x)^m*B*a^3*c*
d*x^5 + 162162*(e*x)^m*A*a^2*b*c*d*x^5 + 27027*(e*x)^m*A*a^3*d^2*x^5 + 9120*(e*x)^m*A*a^3*c^2*m^3*x + 73054*(e
*x)^m*B*a^3*c^2*m*x^3 + 219162*(e*x)^m*A*a^2*b*c^2*m*x^3 + 146108*(e*x)^m*A*a^3*c*d*m*x^3 + 48259*(e*x)^m*A*a^
3*c^2*m^2*x + 45045*(e*x)^m*B*a^3*c^2*x^3 + 135135*(e*x)^m*A*a^2*b*c^2*x^3 + 90090*(e*x)^m*A*a^3*c*d*x^3 + 129
072*(e*x)^m*A*a^3*c^2*m*x + 135135*(e*x)^m*A*a^3*c^2*x)/(m^7 + 49*m^6 + 973*m^5 + 10045*m^4 + 57379*m^3 + 1773
31*m^2 + 264207*m + 135135)
Mupad [B] (verification not implemented)
Time = 6.22 (sec) , antiderivative size = 694, normalized size of antiderivative = 2.38
\[
\int (e x)^m \left (a+b x^2\right )^3 \left (A+B x^2\right ) \left (c+d x^2\right )^2 \, dx=\frac {x^7\,{\left (e\,x\right )}^m\,\left (B\,a^3\,d^2+6\,B\,a^2\,b\,c\,d+3\,A\,a^2\,b\,d^2+3\,B\,a\,b^2\,c^2+6\,A\,a\,b^2\,c\,d+A\,b^3\,c^2\right )\,\left (m^6+42\,m^5+679\,m^4+5292\,m^3+20335\,m^2+34986\,m+19305\right )}{m^7+49\,m^6+973\,m^5+10045\,m^4+57379\,m^3+177331\,m^2+264207\,m+135135}+\frac {a\,x^5\,{\left (e\,x\right )}^m\,\left (2\,B\,a^2\,c\,d+A\,a^2\,d^2+3\,B\,a\,b\,c^2+6\,A\,a\,b\,c\,d+3\,A\,b^2\,c^2\right )\,\left (m^6+44\,m^5+753\,m^4+6280\,m^3+25979\,m^2+47436\,m+27027\right )}{m^7+49\,m^6+973\,m^5+10045\,m^4+57379\,m^3+177331\,m^2+264207\,m+135135}+\frac {b\,x^9\,{\left (e\,x\right )}^m\,\left (3\,B\,a^2\,d^2+6\,B\,a\,b\,c\,d+3\,A\,a\,b\,d^2+B\,b^2\,c^2+2\,A\,b^2\,c\,d\right )\,\left (m^6+40\,m^5+613\,m^4+4528\,m^3+16627\,m^2+27688\,m+15015\right )}{m^7+49\,m^6+973\,m^5+10045\,m^4+57379\,m^3+177331\,m^2+264207\,m+135135}+\frac {A\,a^3\,c^2\,x\,{\left (e\,x\right )}^m\,\left (m^6+48\,m^5+925\,m^4+9120\,m^3+48259\,m^2+129072\,m+135135\right )}{m^7+49\,m^6+973\,m^5+10045\,m^4+57379\,m^3+177331\,m^2+264207\,m+135135}+\frac {a^2\,c\,x^3\,{\left (e\,x\right )}^m\,\left (2\,A\,a\,d+3\,A\,b\,c+B\,a\,c\right )\,\left (m^6+46\,m^5+835\,m^4+7540\,m^3+34759\,m^2+73054\,m+45045\right )}{m^7+49\,m^6+973\,m^5+10045\,m^4+57379\,m^3+177331\,m^2+264207\,m+135135}+\frac {b^2\,d\,x^{11}\,{\left (e\,x\right )}^m\,\left (A\,b\,d+3\,B\,a\,d+2\,B\,b\,c\right )\,\left (m^6+38\,m^5+555\,m^4+3940\,m^3+14039\,m^2+22902\,m+12285\right )}{m^7+49\,m^6+973\,m^5+10045\,m^4+57379\,m^3+177331\,m^2+264207\,m+135135}+\frac {B\,b^3\,d^2\,x^{13}\,{\left (e\,x\right )}^m\,\left (m^6+36\,m^5+505\,m^4+3480\,m^3+12139\,m^2+19524\,m+10395\right )}{m^7+49\,m^6+973\,m^5+10045\,m^4+57379\,m^3+177331\,m^2+264207\,m+135135}
\]
[In]
int((A + B*x^2)*(e*x)^m*(a + b*x^2)^3*(c + d*x^2)^2,x)
[Out]
(x^7*(e*x)^m*(A*b^3*c^2 + B*a^3*d^2 + 3*A*a^2*b*d^2 + 3*B*a*b^2*c^2 + 6*A*a*b^2*c*d + 6*B*a^2*b*c*d)*(34986*m
+ 20335*m^2 + 5292*m^3 + 679*m^4 + 42*m^5 + m^6 + 19305))/(264207*m + 177331*m^2 + 57379*m^3 + 10045*m^4 + 973
*m^5 + 49*m^6 + m^7 + 135135) + (a*x^5*(e*x)^m*(A*a^2*d^2 + 3*A*b^2*c^2 + 3*B*a*b*c^2 + 2*B*a^2*c*d + 6*A*a*b*
c*d)*(47436*m + 25979*m^2 + 6280*m^3 + 753*m^4 + 44*m^5 + m^6 + 27027))/(264207*m + 177331*m^2 + 57379*m^3 + 1
0045*m^4 + 973*m^5 + 49*m^6 + m^7 + 135135) + (b*x^9*(e*x)^m*(3*B*a^2*d^2 + B*b^2*c^2 + 3*A*a*b*d^2 + 2*A*b^2*
c*d + 6*B*a*b*c*d)*(27688*m + 16627*m^2 + 4528*m^3 + 613*m^4 + 40*m^5 + m^6 + 15015))/(264207*m + 177331*m^2 +
57379*m^3 + 10045*m^4 + 973*m^5 + 49*m^6 + m^7 + 135135) + (A*a^3*c^2*x*(e*x)^m*(129072*m + 48259*m^2 + 9120*
m^3 + 925*m^4 + 48*m^5 + m^6 + 135135))/(264207*m + 177331*m^2 + 57379*m^3 + 10045*m^4 + 973*m^5 + 49*m^6 + m^
7 + 135135) + (a^2*c*x^3*(e*x)^m*(2*A*a*d + 3*A*b*c + B*a*c)*(73054*m + 34759*m^2 + 7540*m^3 + 835*m^4 + 46*m^
5 + m^6 + 45045))/(264207*m + 177331*m^2 + 57379*m^3 + 10045*m^4 + 973*m^5 + 49*m^6 + m^7 + 135135) + (b^2*d*x
^11*(e*x)^m*(A*b*d + 3*B*a*d + 2*B*b*c)*(22902*m + 14039*m^2 + 3940*m^3 + 555*m^4 + 38*m^5 + m^6 + 12285))/(26
4207*m + 177331*m^2 + 57379*m^3 + 10045*m^4 + 973*m^5 + 49*m^6 + m^7 + 135135) + (B*b^3*d^2*x^13*(e*x)^m*(1952
4*m + 12139*m^2 + 3480*m^3 + 505*m^4 + 36*m^5 + m^6 + 10395))/(264207*m + 177331*m^2 + 57379*m^3 + 10045*m^4 +
973*m^5 + 49*m^6 + m^7 + 135135)