\(\int (e x)^m (a+b x^2)^3 (A+B x^2) (c+d x^2) \, dx\) [1]

Optimal result

Integrand size = 29, antiderivative size = 189 \[ \int (e x)^m \left (a+b x^2\right )^3 \left (A+B x^2\right ) \left (c+d x^2\right ) \, dx=\frac {a^3 A c (e x)^{1+m}}{e (1+m)}+\frac {a^2 (3 A b c+a B c+a A d) (e x)^{3+m}}{e^3 (3+m)}+\frac {a (3 A b (b c+a d)+a B (3 b c+a d)) (e x)^{5+m}}{e^5 (5+m)}+\frac {b (3 a B (b c+a d)+A b (b c+3 a d)) (e x)^{7+m}}{e^7 (7+m)}+\frac {b^2 (b B c+A b d+3 a B d) (e x)^{9+m}}{e^9 (9+m)}+\frac {b^3 B d (e x)^{11+m}}{e^{11} (11+m)} \]

[Out]

a^3*A*c*(e*x)^(1+m)/e/(1+m)+a^2*(A*a*d+3*A*b*c+B*a*c)*(e*x)^(3+m)/e^3/(3+m)+a*(3*A*b*(a*d+b*c)+a*B*(a*d+3*b*c)
)*(e*x)^(5+m)/e^5/(5+m)+b*(3*a*B*(a*d+b*c)+A*b*(3*a*d+b*c))*(e*x)^(7+m)/e^7/(7+m)+b^2*(A*b*d+3*B*a*d+B*b*c)*(e
*x)^(9+m)/e^9/(9+m)+b^3*B*d*(e*x)^(11+m)/e^11/(11+m)

Rubi [A] (verified)

Time = 0.13 (sec) , antiderivative size = 189, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 1, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.034, Rules used = {584} \[ \int (e x)^m \left (a+b x^2\right )^3 \left (A+B x^2\right ) \left (c+d x^2\right ) \, dx=\frac {a^3 A c (e x)^{m+1}}{e (m+1)}+\frac {a^2 (e x)^{m+3} (a A d+a B c+3 A b c)}{e^3 (m+3)}+\frac {b^2 (e x)^{m+9} (3 a B d+A b d+b B c)}{e^9 (m+9)}+\frac {b (e x)^{m+7} (A b (3 a d+b c)+3 a B (a d+b c))}{e^7 (m+7)}+\frac {a (e x)^{m+5} (3 A b (a d+b c)+a B (a d+3 b c))}{e^5 (m+5)}+\frac {b^3 B d (e x)^{m+11}}{e^{11} (m+11)} \]

[In]

Int[(e*x)^m*(a + b*x^2)^3*(A + B*x^2)*(c + d*x^2),x]

[Out]

(a^3*A*c*(e*x)^(1 + m))/(e*(1 + m)) + (a^2*(3*A*b*c + a*B*c + a*A*d)*(e*x)^(3 + m))/(e^3*(3 + m)) + (a*(3*A*b*
(b*c + a*d) + a*B*(3*b*c + a*d))*(e*x)^(5 + m))/(e^5*(5 + m)) + (b*(3*a*B*(b*c + a*d) + A*b*(b*c + 3*a*d))*(e*
x)^(7 + m))/(e^7*(7 + m)) + (b^2*(b*B*c + A*b*d + 3*a*B*d)*(e*x)^(9 + m))/(e^9*(9 + m)) + (b^3*B*d*(e*x)^(11 +
m))/(e^11*(11 + 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 (e x)^m+\frac {a^2 (3 A b c+a B c+a A d) (e x)^{2+m}}{e^2}+\frac {a (3 A b (b c+a d)+a B (3 b c+a d)) (e x)^{4+m}}{e^4}+\frac {b (3 a B (b c+a d)+A b (b c+3 a d)) (e x)^{6+m}}{e^6}+\frac {b^2 (b B c+A b d+3 a B d) (e x)^{8+m}}{e^8}+\frac {b^3 B d (e x)^{10+m}}{e^{10}}\right ) \, dx \\ & = \frac {a^3 A c (e x)^{1+m}}{e (1+m)}+\frac {a^2 (3 A b c+a B c+a A d) (e x)^{3+m}}{e^3 (3+m)}+\frac {a (3 A b (b c+a d)+a B (3 b c+a d)) (e x)^{5+m}}{e^5 (5+m)}+\frac {b (3 a B (b c+a d)+A b (b c+3 a d)) (e x)^{7+m}}{e^7 (7+m)}+\frac {b^2 (b B c+A b d+3 a B d) (e x)^{9+m}}{e^9 (9+m)}+\frac {b^3 B d (e x)^{11+m}}{e^{11} (11+m)} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.45 (sec) , antiderivative size = 151, normalized size of antiderivative = 0.80 \[ \int (e x)^m \left (a+b x^2\right )^3 \left (A+B x^2\right ) \left (c+d x^2\right ) \, dx=x (e x)^m \left (\frac {a^3 A c}{1+m}+\frac {a^2 (3 A b c+a B c+a A d) x^2}{3+m}+\frac {a (3 A b (b c+a d)+a B (3 b c+a d)) x^4}{5+m}+\frac {b (3 a B (b c+a d)+A b (b c+3 a d)) x^6}{7+m}+\frac {b^2 (b B c+A b d+3 a B d) x^8}{9+m}+\frac {b^3 B d x^{10}}{11+m}\right ) \]

[In]

Integrate[(e*x)^m*(a + b*x^2)^3*(A + B*x^2)*(c + d*x^2),x]

[Out]

x*(e*x)^m*((a^3*A*c)/(1 + m) + (a^2*(3*A*b*c + a*B*c + a*A*d)*x^2)/(3 + m) + (a*(3*A*b*(b*c + a*d) + a*B*(3*b*
c + a*d))*x^4)/(5 + m) + (b*(3*a*B*(b*c + a*d) + A*b*(b*c + 3*a*d))*x^6)/(7 + m) + (b^2*(b*B*c + A*b*d + 3*a*B
*d)*x^8)/(9 + m) + (b^3*B*d*x^10)/(11 + m))

Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(1228\) vs. \(2(189)=378\).

Time = 4.37 (sec) , antiderivative size = 1229, normalized size of antiderivative = 6.50

method	result	size
gosper	\(\text {Expression too large to display}\)	\(1229\)
risch	\(\text {Expression too large to display}\)	\(1229\)
parallelrisch	\(\text {Expression too large to display}\)	\(1709\)

[In]

int((e*x)^m*(b*x^2+a)^3*(B*x^2+A)*(d*x^2+c),x,method=_RETURNVERBOSE)

[Out]

x*(B*b^3*d*m^5*x^10+25*B*b^3*d*m^4*x^10+A*b^3*d*m^5*x^8+3*B*a*b^2*d*m^5*x^8+B*b^3*c*m^5*x^8+230*B*b^3*d*m^3*x^
10+27*A*b^3*d*m^4*x^8+81*B*a*b^2*d*m^4*x^8+27*B*b^3*c*m^4*x^8+950*B*b^3*d*m^2*x^10+3*A*a*b^2*d*m^5*x^6+A*b^3*c
*m^5*x^6+262*A*b^3*d*m^3*x^8+3*B*a^2*b*d*m^5*x^6+3*B*a*b^2*c*m^5*x^6+786*B*a*b^2*d*m^3*x^8+262*B*b^3*c*m^3*x^8
+1689*B*b^3*d*m*x^10+87*A*a*b^2*d*m^4*x^6+29*A*b^3*c*m^4*x^6+1122*A*b^3*d*m^2*x^8+87*B*a^2*b*d*m^4*x^6+87*B*a*
b^2*c*m^4*x^6+3366*B*a*b^2*d*m^2*x^8+1122*B*b^3*c*m^2*x^8+945*B*b^3*d*x^10+3*A*a^2*b*d*m^5*x^4+3*A*a*b^2*c*m^5
*x^4+906*A*a*b^2*d*m^3*x^6+302*A*b^3*c*m^3*x^6+2041*A*b^3*d*m*x^8+B*a^3*d*m^5*x^4+3*B*a^2*b*c*m^5*x^4+906*B*a^
2*b*d*m^3*x^6+906*B*a*b^2*c*m^3*x^6+6123*B*a*b^2*d*m*x^8+2041*B*b^3*c*m*x^8+93*A*a^2*b*d*m^4*x^4+93*A*a*b^2*c*
m^4*x^4+4098*A*a*b^2*d*m^2*x^6+1366*A*b^3*c*m^2*x^6+1155*A*b^3*d*x^8+31*B*a^3*d*m^4*x^4+93*B*a^2*b*c*m^4*x^4+4
098*B*a^2*b*d*m^2*x^6+4098*B*a*b^2*c*m^2*x^6+3465*B*a*b^2*d*x^8+1155*B*b^3*c*x^8+A*a^3*d*m^5*x^2+3*A*a^2*b*c*m
^5*x^2+1050*A*a^2*b*d*m^3*x^4+1050*A*a*b^2*c*m^3*x^4+7731*A*a*b^2*d*m*x^6+2577*A*b^3*c*m*x^6+B*a^3*c*m^5*x^2+3
50*B*a^3*d*m^3*x^4+1050*B*a^2*b*c*m^3*x^4+7731*B*a^2*b*d*m*x^6+7731*B*a*b^2*c*m*x^6+33*A*a^3*d*m^4*x^2+99*A*a^
2*b*c*m^4*x^2+5190*A*a^2*b*d*m^2*x^4+5190*A*a*b^2*c*m^2*x^4+4455*A*a*b^2*d*x^6+1485*A*b^3*c*x^6+33*B*a^3*c*m^4
*x^2+1730*B*a^3*d*m^2*x^4+5190*B*a^2*b*c*m^2*x^4+4455*B*a^2*b*d*x^6+4455*B*a*b^2*c*x^6+A*a^3*c*m^5+406*A*a^3*d
*m^3*x^2+1218*A*a^2*b*c*m^3*x^2+10467*A*a^2*b*d*m*x^4+10467*A*a*b^2*c*m*x^4+406*B*a^3*c*m^3*x^2+3489*B*a^3*d*m
*x^4+10467*B*a^2*b*c*m*x^4+35*A*a^3*c*m^4+2262*A*a^3*d*m^2*x^2+6786*A*a^2*b*c*m^2*x^2+6237*A*a^2*b*d*x^4+6237*
A*a*b^2*c*x^4+2262*B*a^3*c*m^2*x^2+2079*B*a^3*d*x^4+6237*B*a^2*b*c*x^4+470*A*a^3*c*m^3+5353*A*a^3*d*m*x^2+1605
9*A*a^2*b*c*m*x^2+5353*B*a^3*c*m*x^2+3010*A*a^3*c*m^2+3465*A*a^3*d*x^2+10395*A*a^2*b*c*x^2+3465*B*a^3*c*x^2+91
29*A*a^3*c*m+10395*A*a^3*c)*(e*x)^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. 911 vs. \(2 (189) = 378\).

Time = 0.28 (sec) , antiderivative size = 911, normalized size of antiderivative = 4.82 \[ \int (e x)^m \left (a+b x^2\right )^3 \left (A+B x^2\right ) \left (c+d x^2\right ) \, dx=\frac {{\left ({\left (B b^{3} d m^{5} + 25 \, B b^{3} d m^{4} + 230 \, B b^{3} d m^{3} + 950 \, B b^{3} d m^{2} + 1689 \, B b^{3} d m + 945 \, B b^{3} d\right )} x^{11} + {\left ({\left (B b^{3} c + {\left (3 \, B a b^{2} + A b^{3}\right )} d\right )} m^{5} + 1155 \, B b^{3} c + 27 \, {\left (B b^{3} c + {\left (3 \, B a b^{2} + A b^{3}\right )} d\right )} m^{4} + 262 \, {\left (B b^{3} c + {\left (3 \, B a b^{2} + A b^{3}\right )} d\right )} m^{3} + 1122 \, {\left (B b^{3} c + {\left (3 \, B a b^{2} + A b^{3}\right )} d\right )} m^{2} + 1155 \, {\left (3 \, B a b^{2} + A b^{3}\right )} d + 2041 \, {\left (B b^{3} c + {\left (3 \, B a b^{2} + A b^{3}\right )} d\right )} m\right )} x^{9} + {\left ({\left ({\left (3 \, B a b^{2} + A b^{3}\right )} c + 3 \, {\left (B a^{2} b + A a b^{2}\right )} d\right )} m^{5} + 29 \, {\left ({\left (3 \, B a b^{2} + A b^{3}\right )} c + 3 \, {\left (B a^{2} b + A a b^{2}\right )} d\right )} m^{4} + 302 \, {\left ({\left (3 \, B a b^{2} + A b^{3}\right )} c + 3 \, {\left (B a^{2} b + A a b^{2}\right )} d\right )} m^{3} + 1366 \, {\left ({\left (3 \, B a b^{2} + A b^{3}\right )} c + 3 \, {\left (B a^{2} b + A a b^{2}\right )} d\right )} m^{2} + 1485 \, {\left (3 \, B a b^{2} + A b^{3}\right )} c + 4455 \, {\left (B a^{2} b + A a b^{2}\right )} d + 2577 \, {\left ({\left (3 \, B a b^{2} + A b^{3}\right )} c + 3 \, {\left (B a^{2} b + A a b^{2}\right )} d\right )} m\right )} x^{7} + {\left ({\left (3 \, {\left (B a^{2} b + A a b^{2}\right )} c + {\left (B a^{3} + 3 \, A a^{2} b\right )} d\right )} m^{5} + 31 \, {\left (3 \, {\left (B a^{2} b + A a b^{2}\right )} c + {\left (B a^{3} + 3 \, A a^{2} b\right )} d\right )} m^{4} + 350 \, {\left (3 \, {\left (B a^{2} b + A a b^{2}\right )} c + {\left (B a^{3} + 3 \, A a^{2} b\right )} d\right )} m^{3} + 1730 \, {\left (3 \, {\left (B a^{2} b + A a b^{2}\right )} c + {\left (B a^{3} + 3 \, A a^{2} b\right )} d\right )} m^{2} + 6237 \, {\left (B a^{2} b + A a b^{2}\right )} c + 2079 \, {\left (B a^{3} + 3 \, A a^{2} b\right )} d + 3489 \, {\left (3 \, {\left (B a^{2} b + A a b^{2}\right )} c + {\left (B a^{3} + 3 \, A a^{2} b\right )} d\right )} m\right )} x^{5} + {\left ({\left (A a^{3} d + {\left (B a^{3} + 3 \, A a^{2} b\right )} c\right )} m^{5} + 3465 \, A a^{3} d + 33 \, {\left (A a^{3} d + {\left (B a^{3} + 3 \, A a^{2} b\right )} c\right )} m^{4} + 406 \, {\left (A a^{3} d + {\left (B a^{3} + 3 \, A a^{2} b\right )} c\right )} m^{3} + 2262 \, {\left (A a^{3} d + {\left (B a^{3} + 3 \, A a^{2} b\right )} c\right )} m^{2} + 3465 \, {\left (B a^{3} + 3 \, A a^{2} b\right )} c + 5353 \, {\left (A a^{3} d + {\left (B a^{3} + 3 \, A a^{2} b\right )} c\right )} m\right )} x^{3} + {\left (A a^{3} c m^{5} + 35 \, A a^{3} c m^{4} + 470 \, A a^{3} c m^{3} + 3010 \, A a^{3} c m^{2} + 9129 \, A a^{3} c m + 10395 \, A a^{3} c\right )} x\right )} \left (e x\right )^{m}}{m^{6} + 36 \, m^{5} + 505 \, m^{4} + 3480 \, m^{3} + 12139 \, m^{2} + 19524 \, m + 10395} \]

[In]

integrate((e*x)^m*(b*x^2+a)^3*(B*x^2+A)*(d*x^2+c),x, algorithm="fricas")

[Out]

((B*b^3*d*m^5 + 25*B*b^3*d*m^4 + 230*B*b^3*d*m^3 + 950*B*b^3*d*m^2 + 1689*B*b^3*d*m + 945*B*b^3*d)*x^11 + ((B*
b^3*c + (3*B*a*b^2 + A*b^3)*d)*m^5 + 1155*B*b^3*c + 27*(B*b^3*c + (3*B*a*b^2 + A*b^3)*d)*m^4 + 262*(B*b^3*c +
(3*B*a*b^2 + A*b^3)*d)*m^3 + 1122*(B*b^3*c + (3*B*a*b^2 + A*b^3)*d)*m^2 + 1155*(3*B*a*b^2 + A*b^3)*d + 2041*(B
*b^3*c + (3*B*a*b^2 + A*b^3)*d)*m)*x^9 + (((3*B*a*b^2 + A*b^3)*c + 3*(B*a^2*b + A*a*b^2)*d)*m^5 + 29*((3*B*a*b
^2 + A*b^3)*c + 3*(B*a^2*b + A*a*b^2)*d)*m^4 + 302*((3*B*a*b^2 + A*b^3)*c + 3*(B*a^2*b + A*a*b^2)*d)*m^3 + 136
6*((3*B*a*b^2 + A*b^3)*c + 3*(B*a^2*b + A*a*b^2)*d)*m^2 + 1485*(3*B*a*b^2 + A*b^3)*c + 4455*(B*a^2*b + A*a*b^2
)*d + 2577*((3*B*a*b^2 + A*b^3)*c + 3*(B*a^2*b + A*a*b^2)*d)*m)*x^7 + ((3*(B*a^2*b + A*a*b^2)*c + (B*a^3 + 3*A
*a^2*b)*d)*m^5 + 31*(3*(B*a^2*b + A*a*b^2)*c + (B*a^3 + 3*A*a^2*b)*d)*m^4 + 350*(3*(B*a^2*b + A*a*b^2)*c + (B*
a^3 + 3*A*a^2*b)*d)*m^3 + 1730*(3*(B*a^2*b + A*a*b^2)*c + (B*a^3 + 3*A*a^2*b)*d)*m^2 + 6237*(B*a^2*b + A*a*b^2
)*c + 2079*(B*a^3 + 3*A*a^2*b)*d + 3489*(3*(B*a^2*b + A*a*b^2)*c + (B*a^3 + 3*A*a^2*b)*d)*m)*x^5 + ((A*a^3*d +
(B*a^3 + 3*A*a^2*b)*c)*m^5 + 3465*A*a^3*d + 33*(A*a^3*d + (B*a^3 + 3*A*a^2*b)*c)*m^4 + 406*(A*a^3*d + (B*a^3
+ 3*A*a^2*b)*c)*m^3 + 2262*(A*a^3*d + (B*a^3 + 3*A*a^2*b)*c)*m^2 + 3465*(B*a^3 + 3*A*a^2*b)*c + 5353*(A*a^3*d
+ (B*a^3 + 3*A*a^2*b)*c)*m)*x^3 + (A*a^3*c*m^5 + 35*A*a^3*c*m^4 + 470*A*a^3*c*m^3 + 3010*A*a^3*c*m^2 + 9129*A*
a^3*c*m + 10395*A*a^3*c)*x)*(e*x)^m/(m^6 + 36*m^5 + 505*m^4 + 3480*m^3 + 12139*m^2 + 19524*m + 10395)

Sympy [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 5992 vs. \(2 (184) = 368\).

Time = 0.95 (sec) , antiderivative size = 5992, normalized size of antiderivative = 31.70 \[ \int (e x)^m \left (a+b x^2\right )^3 \left (A+B x^2\right ) \left (c+d x^2\right ) \, dx=\text {Too large to display} \]

[In]

integrate((e*x)**m*(b*x**2+a)**3*(B*x**2+A)*(d*x**2+c),x)

[Out]

Piecewise(((-A*a**3*c/(10*x**10) - A*a**3*d/(8*x**8) - 3*A*a**2*b*c/(8*x**8) - A*a**2*b*d/(2*x**6) - A*a*b**2*
c/(2*x**6) - 3*A*a*b**2*d/(4*x**4) - A*b**3*c/(4*x**4) - A*b**3*d/(2*x**2) - B*a**3*c/(8*x**8) - B*a**3*d/(6*x
**6) - B*a**2*b*c/(2*x**6) - 3*B*a**2*b*d/(4*x**4) - 3*B*a*b**2*c/(4*x**4) - 3*B*a*b**2*d/(2*x**2) - B*b**3*c/
(2*x**2) + B*b**3*d*log(x))/e**11, Eq(m, -11)), ((-A*a**3*c/(8*x**8) - A*a**3*d/(6*x**6) - A*a**2*b*c/(2*x**6)
- 3*A*a**2*b*d/(4*x**4) - 3*A*a*b**2*c/(4*x**4) - 3*A*a*b**2*d/(2*x**2) - A*b**3*c/(2*x**2) + A*b**3*d*log(x)
- B*a**3*c/(6*x**6) - B*a**3*d/(4*x**4) - 3*B*a**2*b*c/(4*x**4) - 3*B*a**2*b*d/(2*x**2) - 3*B*a*b**2*c/(2*x**
2) + 3*B*a*b**2*d*log(x) + B*b**3*c*log(x) + B*b**3*d*x**2/2)/e**9, Eq(m, -9)), ((-A*a**3*c/(6*x**6) - A*a**3*
d/(4*x**4) - 3*A*a**2*b*c/(4*x**4) - 3*A*a**2*b*d/(2*x**2) - 3*A*a*b**2*c/(2*x**2) + 3*A*a*b**2*d*log(x) + A*b
**3*c*log(x) + A*b**3*d*x**2/2 - B*a**3*c/(4*x**4) - B*a**3*d/(2*x**2) - 3*B*a**2*b*c/(2*x**2) + 3*B*a**2*b*d*
log(x) + 3*B*a*b**2*c*log(x) + 3*B*a*b**2*d*x**2/2 + B*b**3*c*x**2/2 + B*b**3*d*x**4/4)/e**7, Eq(m, -7)), ((-A
*a**3*c/(4*x**4) - A*a**3*d/(2*x**2) - 3*A*a**2*b*c/(2*x**2) + 3*A*a**2*b*d*log(x) + 3*A*a*b**2*c*log(x) + 3*A
*a*b**2*d*x**2/2 + A*b**3*c*x**2/2 + A*b**3*d*x**4/4 - B*a**3*c/(2*x**2) + B*a**3*d*log(x) + 3*B*a**2*b*c*log(
x) + 3*B*a**2*b*d*x**2/2 + 3*B*a*b**2*c*x**2/2 + 3*B*a*b**2*d*x**4/4 + B*b**3*c*x**4/4 + B*b**3*d*x**6/6)/e**5
, Eq(m, -5)), ((-A*a**3*c/(2*x**2) + A*a**3*d*log(x) + 3*A*a**2*b*c*log(x) + 3*A*a**2*b*d*x**2/2 + 3*A*a*b**2*
c*x**2/2 + 3*A*a*b**2*d*x**4/4 + A*b**3*c*x**4/4 + A*b**3*d*x**6/6 + B*a**3*c*log(x) + B*a**3*d*x**2/2 + 3*B*a
**2*b*c*x**2/2 + 3*B*a**2*b*d*x**4/4 + 3*B*a*b**2*c*x**4/4 + B*a*b**2*d*x**6/2 + B*b**3*c*x**6/6 + B*b**3*d*x*
*8/8)/e**3, Eq(m, -3)), ((A*a**3*c*log(x) + A*a**3*d*x**2/2 + 3*A*a**2*b*c*x**2/2 + 3*A*a**2*b*d*x**4/4 + 3*A*
a*b**2*c*x**4/4 + A*a*b**2*d*x**6/2 + A*b**3*c*x**6/6 + A*b**3*d*x**8/8 + B*a**3*c*x**2/2 + B*a**3*d*x**4/4 +
3*B*a**2*b*c*x**4/4 + B*a**2*b*d*x**6/2 + B*a*b**2*c*x**6/2 + 3*B*a*b**2*d*x**8/8 + B*b**3*c*x**8/8 + B*b**3*d
*x**10/10)/e, Eq(m, -1)), (A*a**3*c*m**5*x*(e*x)**m/(m**6 + 36*m**5 + 505*m**4 + 3480*m**3 + 12139*m**2 + 1952
4*m + 10395) + 35*A*a**3*c*m**4*x*(e*x)**m/(m**6 + 36*m**5 + 505*m**4 + 3480*m**3 + 12139*m**2 + 19524*m + 103
95) + 470*A*a**3*c*m**3*x*(e*x)**m/(m**6 + 36*m**5 + 505*m**4 + 3480*m**3 + 12139*m**2 + 19524*m + 10395) + 30
10*A*a**3*c*m**2*x*(e*x)**m/(m**6 + 36*m**5 + 505*m**4 + 3480*m**3 + 12139*m**2 + 19524*m + 10395) + 9129*A*a*
*3*c*m*x*(e*x)**m/(m**6 + 36*m**5 + 505*m**4 + 3480*m**3 + 12139*m**2 + 19524*m + 10395) + 10395*A*a**3*c*x*(e
*x)**m/(m**6 + 36*m**5 + 505*m**4 + 3480*m**3 + 12139*m**2 + 19524*m + 10395) + A*a**3*d*m**5*x**3*(e*x)**m/(m
**6 + 36*m**5 + 505*m**4 + 3480*m**3 + 12139*m**2 + 19524*m + 10395) + 33*A*a**3*d*m**4*x**3*(e*x)**m/(m**6 +
36*m**5 + 505*m**4 + 3480*m**3 + 12139*m**2 + 19524*m + 10395) + 406*A*a**3*d*m**3*x**3*(e*x)**m/(m**6 + 36*m*
*5 + 505*m**4 + 3480*m**3 + 12139*m**2 + 19524*m + 10395) + 2262*A*a**3*d*m**2*x**3*(e*x)**m/(m**6 + 36*m**5 +
505*m**4 + 3480*m**3 + 12139*m**2 + 19524*m + 10395) + 5353*A*a**3*d*m*x**3*(e*x)**m/(m**6 + 36*m**5 + 505*m*
*4 + 3480*m**3 + 12139*m**2 + 19524*m + 10395) + 3465*A*a**3*d*x**3*(e*x)**m/(m**6 + 36*m**5 + 505*m**4 + 3480
*m**3 + 12139*m**2 + 19524*m + 10395) + 3*A*a**2*b*c*m**5*x**3*(e*x)**m/(m**6 + 36*m**5 + 505*m**4 + 3480*m**3
+ 12139*m**2 + 19524*m + 10395) + 99*A*a**2*b*c*m**4*x**3*(e*x)**m/(m**6 + 36*m**5 + 505*m**4 + 3480*m**3 + 1
2139*m**2 + 19524*m + 10395) + 1218*A*a**2*b*c*m**3*x**3*(e*x)**m/(m**6 + 36*m**5 + 505*m**4 + 3480*m**3 + 121
39*m**2 + 19524*m + 10395) + 6786*A*a**2*b*c*m**2*x**3*(e*x)**m/(m**6 + 36*m**5 + 505*m**4 + 3480*m**3 + 12139
*m**2 + 19524*m + 10395) + 16059*A*a**2*b*c*m*x**3*(e*x)**m/(m**6 + 36*m**5 + 505*m**4 + 3480*m**3 + 12139*m**
2 + 19524*m + 10395) + 10395*A*a**2*b*c*x**3*(e*x)**m/(m**6 + 36*m**5 + 505*m**4 + 3480*m**3 + 12139*m**2 + 19
524*m + 10395) + 3*A*a**2*b*d*m**5*x**5*(e*x)**m/(m**6 + 36*m**5 + 505*m**4 + 3480*m**3 + 12139*m**2 + 19524*m
+ 10395) + 93*A*a**2*b*d*m**4*x**5*(e*x)**m/(m**6 + 36*m**5 + 505*m**4 + 3480*m**3 + 12139*m**2 + 19524*m + 1
0395) + 1050*A*a**2*b*d*m**3*x**5*(e*x)**m/(m**6 + 36*m**5 + 505*m**4 + 3480*m**3 + 12139*m**2 + 19524*m + 103
95) + 5190*A*a**2*b*d*m**2*x**5*(e*x)**m/(m**6 + 36*m**5 + 505*m**4 + 3480*m**3 + 12139*m**2 + 19524*m + 10395
) + 10467*A*a**2*b*d*m*x**5*(e*x)**m/(m**6 + 36*m**5 + 505*m**4 + 3480*m**3 + 12139*m**2 + 19524*m + 10395) +
6237*A*a**2*b*d*x**5*(e*x)**m/(m**6 + 36*m**5 + 505*m**4 + 3480*m**3 + 12139*m**2 + 19524*m + 10395) + 3*A*a*b
**2*c*m**5*x**5*(e*x)**m/(m**6 + 36*m**5 + 505*m**4 + 3480*m**3 + 12139*m**2 + 19524*m + 10395) + 93*A*a*b**2*
c*m**4*x**5*(e*x)**m/(m**6 + 36*m**5 + 505*m**4 + 3480*m**3 + 12139*m**2 + 19524*m + 10395) + 1050*A*a*b**2*c*
m**3*x**5*(e*x)**m/(m**6 + 36*m**5 + 505*m**4 + 3480*m**3 + 12139*m**2 + 19524*m + 10395) + 5190*A*a*b**2*c*m*
*2*x**5*(e*x)**m/(m**6 + 36*m**5 + 505*m**4 + 3480*m**3 + 12139*m**2 + 19524*m + 10395) + 10467*A*a*b**2*c*m*x
**5*(e*x)**m/(m**6 + 36*m**5 + 505*m**4 + 3480*m**3 + 12139*m**2 + 19524*m + 10395) + 6237*A*a*b**2*c*x**5*(e*
x)**m/(m**6 + 36*m**5 + 505*m**4 + 3480*m**3 + 12139*m**2 + 19524*m + 10395) + 3*A*a*b**2*d*m**5*x**7*(e*x)**m
/(m**6 + 36*m**5 + 505*m**4 + 3480*m**3 + 12139*m**2 + 19524*m + 10395) + 87*A*a*b**2*d*m**4*x**7*(e*x)**m/(m*
*6 + 36*m**5 + 505*m**4 + 3480*m**3 + 12139*m**2 + 19524*m + 10395) + 906*A*a*b**2*d*m**3*x**7*(e*x)**m/(m**6
+ 36*m**5 + 505*m**4 + 3480*m**3 + 12139*m**2 + 19524*m + 10395) + 4098*A*a*b**2*d*m**2*x**7*(e*x)**m/(m**6 +
36*m**5 + 505*m**4 + 3480*m**3 + 12139*m**2 + 19524*m + 10395) + 7731*A*a*b**2*d*m*x**7*(e*x)**m/(m**6 + 36*m*
*5 + 505*m**4 + 3480*m**3 + 12139*m**2 + 19524*m + 10395) + 4455*A*a*b**2*d*x**7*(e*x)**m/(m**6 + 36*m**5 + 50
5*m**4 + 3480*m**3 + 12139*m**2 + 19524*m + 10395) + A*b**3*c*m**5*x**7*(e*x)**m/(m**6 + 36*m**5 + 505*m**4 +
3480*m**3 + 12139*m**2 + 19524*m + 10395) + 29*A*b**3*c*m**4*x**7*(e*x)**m/(m**6 + 36*m**5 + 505*m**4 + 3480*m
**3 + 12139*m**2 + 19524*m + 10395) + 302*A*b**3*c*m**3*x**7*(e*x)**m/(m**6 + 36*m**5 + 505*m**4 + 3480*m**3 +
12139*m**2 + 19524*m + 10395) + 1366*A*b**3*c*m**2*x**7*(e*x)**m/(m**6 + 36*m**5 + 505*m**4 + 3480*m**3 + 121
39*m**2 + 19524*m + 10395) + 2577*A*b**3*c*m*x**7*(e*x)**m/(m**6 + 36*m**5 + 505*m**4 + 3480*m**3 + 12139*m**2
+ 19524*m + 10395) + 1485*A*b**3*c*x**7*(e*x)**m/(m**6 + 36*m**5 + 505*m**4 + 3480*m**3 + 12139*m**2 + 19524*
m + 10395) + A*b**3*d*m**5*x**9*(e*x)**m/(m**6 + 36*m**5 + 505*m**4 + 3480*m**3 + 12139*m**2 + 19524*m + 10395
) + 27*A*b**3*d*m**4*x**9*(e*x)**m/(m**6 + 36*m**5 + 505*m**4 + 3480*m**3 + 12139*m**2 + 19524*m + 10395) + 26
2*A*b**3*d*m**3*x**9*(e*x)**m/(m**6 + 36*m**5 + 505*m**4 + 3480*m**3 + 12139*m**2 + 19524*m + 10395) + 1122*A*
b**3*d*m**2*x**9*(e*x)**m/(m**6 + 36*m**5 + 505*m**4 + 3480*m**3 + 12139*m**2 + 19524*m + 10395) + 2041*A*b**3
*d*m*x**9*(e*x)**m/(m**6 + 36*m**5 + 505*m**4 + 3480*m**3 + 12139*m**2 + 19524*m + 10395) + 1155*A*b**3*d*x**9
*(e*x)**m/(m**6 + 36*m**5 + 505*m**4 + 3480*m**3 + 12139*m**2 + 19524*m + 10395) + B*a**3*c*m**5*x**3*(e*x)**m
/(m**6 + 36*m**5 + 505*m**4 + 3480*m**3 + 12139*m**2 + 19524*m + 10395) + 33*B*a**3*c*m**4*x**3*(e*x)**m/(m**6
+ 36*m**5 + 505*m**4 + 3480*m**3 + 12139*m**2 + 19524*m + 10395) + 406*B*a**3*c*m**3*x**3*(e*x)**m/(m**6 + 36
*m**5 + 505*m**4 + 3480*m**3 + 12139*m**2 + 19524*m + 10395) + 2262*B*a**3*c*m**2*x**3*(e*x)**m/(m**6 + 36*m**
5 + 505*m**4 + 3480*m**3 + 12139*m**2 + 19524*m + 10395) + 5353*B*a**3*c*m*x**3*(e*x)**m/(m**6 + 36*m**5 + 505
*m**4 + 3480*m**3 + 12139*m**2 + 19524*m + 10395) + 3465*B*a**3*c*x**3*(e*x)**m/(m**6 + 36*m**5 + 505*m**4 + 3
480*m**3 + 12139*m**2 + 19524*m + 10395) + B*a**3*d*m**5*x**5*(e*x)**m/(m**6 + 36*m**5 + 505*m**4 + 3480*m**3
+ 12139*m**2 + 19524*m + 10395) + 31*B*a**3*d*m**4*x**5*(e*x)**m/(m**6 + 36*m**5 + 505*m**4 + 3480*m**3 + 1213
9*m**2 + 19524*m + 10395) + 350*B*a**3*d*m**3*x**5*(e*x)**m/(m**6 + 36*m**5 + 505*m**4 + 3480*m**3 + 12139*m**
2 + 19524*m + 10395) + 1730*B*a**3*d*m**2*x**5*(e*x)**m/(m**6 + 36*m**5 + 505*m**4 + 3480*m**3 + 12139*m**2 +
19524*m + 10395) + 3489*B*a**3*d*m*x**5*(e*x)**m/(m**6 + 36*m**5 + 505*m**4 + 3480*m**3 + 12139*m**2 + 19524*m
+ 10395) + 2079*B*a**3*d*x**5*(e*x)**m/(m**6 + 36*m**5 + 505*m**4 + 3480*m**3 + 12139*m**2 + 19524*m + 10395)
+ 3*B*a**2*b*c*m**5*x**5*(e*x)**m/(m**6 + 36*m**5 + 505*m**4 + 3480*m**3 + 12139*m**2 + 19524*m + 10395) + 93
*B*a**2*b*c*m**4*x**5*(e*x)**m/(m**6 + 36*m**5 + 505*m**4 + 3480*m**3 + 12139*m**2 + 19524*m + 10395) + 1050*B
*a**2*b*c*m**3*x**5*(e*x)**m/(m**6 + 36*m**5 + 505*m**4 + 3480*m**3 + 12139*m**2 + 19524*m + 10395) + 5190*B*a
**2*b*c*m**2*x**5*(e*x)**m/(m**6 + 36*m**5 + 505*m**4 + 3480*m**3 + 12139*m**2 + 19524*m + 10395) + 10467*B*a*
*2*b*c*m*x**5*(e*x)**m/(m**6 + 36*m**5 + 505*m**4 + 3480*m**3 + 12139*m**2 + 19524*m + 10395) + 6237*B*a**2*b*
c*x**5*(e*x)**m/(m**6 + 36*m**5 + 505*m**4 + 3480*m**3 + 12139*m**2 + 19524*m + 10395) + 3*B*a**2*b*d*m**5*x**
7*(e*x)**m/(m**6 + 36*m**5 + 505*m**4 + 3480*m**3 + 12139*m**2 + 19524*m + 10395) + 87*B*a**2*b*d*m**4*x**7*(e
*x)**m/(m**6 + 36*m**5 + 505*m**4 + 3480*m**3 + 12139*m**2 + 19524*m + 10395) + 906*B*a**2*b*d*m**3*x**7*(e*x)
**m/(m**6 + 36*m**5 + 505*m**4 + 3480*m**3 + 12139*m**2 + 19524*m + 10395) + 4098*B*a**2*b*d*m**2*x**7*(e*x)**
m/(m**6 + 36*m**5 + 505*m**4 + 3480*m**3 + 12139*m**2 + 19524*m + 10395) + 7731*B*a**2*b*d*m*x**7*(e*x)**m/(m*
*6 + 36*m**5 + 505*m**4 + 3480*m**3 + 12139*m**2 + 19524*m + 10395) + 4455*B*a**2*b*d*x**7*(e*x)**m/(m**6 + 36
*m**5 + 505*m**4 + 3480*m**3 + 12139*m**2 + 19524*m + 10395) + 3*B*a*b**2*c*m**5*x**7*(e*x)**m/(m**6 + 36*m**5
+ 505*m**4 + 3480*m**3 + 12139*m**2 + 19524*m + 10395) + 87*B*a*b**2*c*m**4*x**7*(e*x)**m/(m**6 + 36*m**5 + 5
05*m**4 + 3480*m**3 + 12139*m**2 + 19524*m + 10395) + 906*B*a*b**2*c*m**3*x**7*(e*x)**m/(m**6 + 36*m**5 + 505*
m**4 + 3480*m**3 + 12139*m**2 + 19524*m + 10395) + 4098*B*a*b**2*c*m**2*x**7*(e*x)**m/(m**6 + 36*m**5 + 505*m*
*4 + 3480*m**3 + 12139*m**2 + 19524*m + 10395) + 7731*B*a*b**2*c*m*x**7*(e*x)**m/(m**6 + 36*m**5 + 505*m**4 +
3480*m**3 + 12139*m**2 + 19524*m + 10395) + 4455*B*a*b**2*c*x**7*(e*x)**m/(m**6 + 36*m**5 + 505*m**4 + 3480*m*
*3 + 12139*m**2 + 19524*m + 10395) + 3*B*a*b**2*d*m**5*x**9*(e*x)**m/(m**6 + 36*m**5 + 505*m**4 + 3480*m**3 +
12139*m**2 + 19524*m + 10395) + 81*B*a*b**2*d*m**4*x**9*(e*x)**m/(m**6 + 36*m**5 + 505*m**4 + 3480*m**3 + 1213
9*m**2 + 19524*m + 10395) + 786*B*a*b**2*d*m**3*x**9*(e*x)**m/(m**6 + 36*m**5 + 505*m**4 + 3480*m**3 + 12139*m
**2 + 19524*m + 10395) + 3366*B*a*b**2*d*m**2*x**9*(e*x)**m/(m**6 + 36*m**5 + 505*m**4 + 3480*m**3 + 12139*m**
2 + 19524*m + 10395) + 6123*B*a*b**2*d*m*x**9*(e*x)**m/(m**6 + 36*m**5 + 505*m**4 + 3480*m**3 + 12139*m**2 + 1
9524*m + 10395) + 3465*B*a*b**2*d*x**9*(e*x)**m/(m**6 + 36*m**5 + 505*m**4 + 3480*m**3 + 12139*m**2 + 19524*m
+ 10395) + B*b**3*c*m**5*x**9*(e*x)**m/(m**6 + 36*m**5 + 505*m**4 + 3480*m**3 + 12139*m**2 + 19524*m + 10395)
+ 27*B*b**3*c*m**4*x**9*(e*x)**m/(m**6 + 36*m**5 + 505*m**4 + 3480*m**3 + 12139*m**2 + 19524*m + 10395) + 262*
B*b**3*c*m**3*x**9*(e*x)**m/(m**6 + 36*m**5 + 505*m**4 + 3480*m**3 + 12139*m**2 + 19524*m + 10395) + 1122*B*b*
*3*c*m**2*x**9*(e*x)**m/(m**6 + 36*m**5 + 505*m**4 + 3480*m**3 + 12139*m**2 + 19524*m + 10395) + 2041*B*b**3*c
*m*x**9*(e*x)**m/(m**6 + 36*m**5 + 505*m**4 + 3480*m**3 + 12139*m**2 + 19524*m + 10395) + 1155*B*b**3*c*x**9*(
e*x)**m/(m**6 + 36*m**5 + 505*m**4 + 3480*m**3 + 12139*m**2 + 19524*m + 10395) + B*b**3*d*m**5*x**11*(e*x)**m/
(m**6 + 36*m**5 + 505*m**4 + 3480*m**3 + 12139*m**2 + 19524*m + 10395) + 25*B*b**3*d*m**4*x**11*(e*x)**m/(m**6
+ 36*m**5 + 505*m**4 + 3480*m**3 + 12139*m**2 + 19524*m + 10395) + 230*B*b**3*d*m**3*x**11*(e*x)**m/(m**6 + 3
6*m**5 + 505*m**4 + 3480*m**3 + 12139*m**2 + 19524*m + 10395) + 950*B*b**3*d*m**2*x**11*(e*x)**m/(m**6 + 36*m*
*5 + 505*m**4 + 3480*m**3 + 12139*m**2 + 19524*m + 10395) + 1689*B*b**3*d*m*x**11*(e*x)**m/(m**6 + 36*m**5 + 5
05*m**4 + 3480*m**3 + 12139*m**2 + 19524*m + 10395) + 945*B*b**3*d*x**11*(e*x)**m/(m**6 + 36*m**5 + 505*m**4 +
3480*m**3 + 12139*m**2 + 19524*m + 10395), True))

Maxima [A] (verification not implemented)

none

Time = 0.23 (sec) , antiderivative size = 338, normalized size of antiderivative = 1.79 \[ \int (e x)^m \left (a+b x^2\right )^3 \left (A+B x^2\right ) \left (c+d x^2\right ) \, dx=\frac {B b^{3} d e^{m} x^{11} x^{m}}{m + 11} + \frac {B b^{3} c e^{m} x^{9} x^{m}}{m + 9} + \frac {3 \, B a b^{2} d e^{m} x^{9} x^{m}}{m + 9} + \frac {A b^{3} d e^{m} x^{9} x^{m}}{m + 9} + \frac {3 \, B a b^{2} c e^{m} x^{7} x^{m}}{m + 7} + \frac {A b^{3} c e^{m} x^{7} x^{m}}{m + 7} + \frac {3 \, B a^{2} b d e^{m} x^{7} x^{m}}{m + 7} + \frac {3 \, A a b^{2} d e^{m} x^{7} x^{m}}{m + 7} + \frac {3 \, B a^{2} b c e^{m} x^{5} x^{m}}{m + 5} + \frac {3 \, A a b^{2} c e^{m} x^{5} x^{m}}{m + 5} + \frac {B a^{3} d e^{m} x^{5} x^{m}}{m + 5} + \frac {3 \, A a^{2} b d e^{m} x^{5} x^{m}}{m + 5} + \frac {B a^{3} c e^{m} x^{3} x^{m}}{m + 3} + \frac {3 \, A a^{2} b c e^{m} x^{3} x^{m}}{m + 3} + \frac {A a^{3} d e^{m} x^{3} x^{m}}{m + 3} + \frac {\left (e x\right )^{m + 1} A a^{3} c}{e {\left (m + 1\right )}} \]

[In]

integrate((e*x)^m*(b*x^2+a)^3*(B*x^2+A)*(d*x^2+c),x, algorithm="maxima")

[Out]

B*b^3*d*e^m*x^11*x^m/(m + 11) + B*b^3*c*e^m*x^9*x^m/(m + 9) + 3*B*a*b^2*d*e^m*x^9*x^m/(m + 9) + A*b^3*d*e^m*x^
9*x^m/(m + 9) + 3*B*a*b^2*c*e^m*x^7*x^m/(m + 7) + A*b^3*c*e^m*x^7*x^m/(m + 7) + 3*B*a^2*b*d*e^m*x^7*x^m/(m + 7
) + 3*A*a*b^2*d*e^m*x^7*x^m/(m + 7) + 3*B*a^2*b*c*e^m*x^5*x^m/(m + 5) + 3*A*a*b^2*c*e^m*x^5*x^m/(m + 5) + B*a^
3*d*e^m*x^5*x^m/(m + 5) + 3*A*a^2*b*d*e^m*x^5*x^m/(m + 5) + B*a^3*c*e^m*x^3*x^m/(m + 3) + 3*A*a^2*b*c*e^m*x^3*
x^m/(m + 3) + A*a^3*d*e^m*x^3*x^m/(m + 3) + (e*x)^(m + 1)*A*a^3*c/(e*(m + 1))

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 1708 vs. \(2 (189) = 378\).

Time = 0.32 (sec) , antiderivative size = 1708, normalized size of antiderivative = 9.04 \[ \int (e x)^m \left (a+b x^2\right )^3 \left (A+B x^2\right ) \left (c+d x^2\right ) \, dx=\text {Too large to display} \]

[In]

integrate((e*x)^m*(b*x^2+a)^3*(B*x^2+A)*(d*x^2+c),x, algorithm="giac")

[Out]

((e*x)^m*B*b^3*d*m^5*x^11 + 25*(e*x)^m*B*b^3*d*m^4*x^11 + (e*x)^m*B*b^3*c*m^5*x^9 + 3*(e*x)^m*B*a*b^2*d*m^5*x^
9 + (e*x)^m*A*b^3*d*m^5*x^9 + 230*(e*x)^m*B*b^3*d*m^3*x^11 + 27*(e*x)^m*B*b^3*c*m^4*x^9 + 81*(e*x)^m*B*a*b^2*d
*m^4*x^9 + 27*(e*x)^m*A*b^3*d*m^4*x^9 + 950*(e*x)^m*B*b^3*d*m^2*x^11 + 3*(e*x)^m*B*a*b^2*c*m^5*x^7 + (e*x)^m*A
*b^3*c*m^5*x^7 + 3*(e*x)^m*B*a^2*b*d*m^5*x^7 + 3*(e*x)^m*A*a*b^2*d*m^5*x^7 + 262*(e*x)^m*B*b^3*c*m^3*x^9 + 786
*(e*x)^m*B*a*b^2*d*m^3*x^9 + 262*(e*x)^m*A*b^3*d*m^3*x^9 + 1689*(e*x)^m*B*b^3*d*m*x^11 + 87*(e*x)^m*B*a*b^2*c*
m^4*x^7 + 29*(e*x)^m*A*b^3*c*m^4*x^7 + 87*(e*x)^m*B*a^2*b*d*m^4*x^7 + 87*(e*x)^m*A*a*b^2*d*m^4*x^7 + 1122*(e*x
)^m*B*b^3*c*m^2*x^9 + 3366*(e*x)^m*B*a*b^2*d*m^2*x^9 + 1122*(e*x)^m*A*b^3*d*m^2*x^9 + 945*(e*x)^m*B*b^3*d*x^11
+ 3*(e*x)^m*B*a^2*b*c*m^5*x^5 + 3*(e*x)^m*A*a*b^2*c*m^5*x^5 + (e*x)^m*B*a^3*d*m^5*x^5 + 3*(e*x)^m*A*a^2*b*d*m
^5*x^5 + 906*(e*x)^m*B*a*b^2*c*m^3*x^7 + 302*(e*x)^m*A*b^3*c*m^3*x^7 + 906*(e*x)^m*B*a^2*b*d*m^3*x^7 + 906*(e*
x)^m*A*a*b^2*d*m^3*x^7 + 2041*(e*x)^m*B*b^3*c*m*x^9 + 6123*(e*x)^m*B*a*b^2*d*m*x^9 + 2041*(e*x)^m*A*b^3*d*m*x^
9 + 93*(e*x)^m*B*a^2*b*c*m^4*x^5 + 93*(e*x)^m*A*a*b^2*c*m^4*x^5 + 31*(e*x)^m*B*a^3*d*m^4*x^5 + 93*(e*x)^m*A*a^
2*b*d*m^4*x^5 + 4098*(e*x)^m*B*a*b^2*c*m^2*x^7 + 1366*(e*x)^m*A*b^3*c*m^2*x^7 + 4098*(e*x)^m*B*a^2*b*d*m^2*x^7
+ 4098*(e*x)^m*A*a*b^2*d*m^2*x^7 + 1155*(e*x)^m*B*b^3*c*x^9 + 3465*(e*x)^m*B*a*b^2*d*x^9 + 1155*(e*x)^m*A*b^3
*d*x^9 + (e*x)^m*B*a^3*c*m^5*x^3 + 3*(e*x)^m*A*a^2*b*c*m^5*x^3 + (e*x)^m*A*a^3*d*m^5*x^3 + 1050*(e*x)^m*B*a^2*
b*c*m^3*x^5 + 1050*(e*x)^m*A*a*b^2*c*m^3*x^5 + 350*(e*x)^m*B*a^3*d*m^3*x^5 + 1050*(e*x)^m*A*a^2*b*d*m^3*x^5 +
7731*(e*x)^m*B*a*b^2*c*m*x^7 + 2577*(e*x)^m*A*b^3*c*m*x^7 + 7731*(e*x)^m*B*a^2*b*d*m*x^7 + 7731*(e*x)^m*A*a*b^
2*d*m*x^7 + 33*(e*x)^m*B*a^3*c*m^4*x^3 + 99*(e*x)^m*A*a^2*b*c*m^4*x^3 + 33*(e*x)^m*A*a^3*d*m^4*x^3 + 5190*(e*x
)^m*B*a^2*b*c*m^2*x^5 + 5190*(e*x)^m*A*a*b^2*c*m^2*x^5 + 1730*(e*x)^m*B*a^3*d*m^2*x^5 + 5190*(e*x)^m*A*a^2*b*d
*m^2*x^5 + 4455*(e*x)^m*B*a*b^2*c*x^7 + 1485*(e*x)^m*A*b^3*c*x^7 + 4455*(e*x)^m*B*a^2*b*d*x^7 + 4455*(e*x)^m*A
*a*b^2*d*x^7 + (e*x)^m*A*a^3*c*m^5*x + 406*(e*x)^m*B*a^3*c*m^3*x^3 + 1218*(e*x)^m*A*a^2*b*c*m^3*x^3 + 406*(e*x
)^m*A*a^3*d*m^3*x^3 + 10467*(e*x)^m*B*a^2*b*c*m*x^5 + 10467*(e*x)^m*A*a*b^2*c*m*x^5 + 3489*(e*x)^m*B*a^3*d*m*x
^5 + 10467*(e*x)^m*A*a^2*b*d*m*x^5 + 35*(e*x)^m*A*a^3*c*m^4*x + 2262*(e*x)^m*B*a^3*c*m^2*x^3 + 6786*(e*x)^m*A*
a^2*b*c*m^2*x^3 + 2262*(e*x)^m*A*a^3*d*m^2*x^3 + 6237*(e*x)^m*B*a^2*b*c*x^5 + 6237*(e*x)^m*A*a*b^2*c*x^5 + 207
9*(e*x)^m*B*a^3*d*x^5 + 6237*(e*x)^m*A*a^2*b*d*x^5 + 470*(e*x)^m*A*a^3*c*m^3*x + 5353*(e*x)^m*B*a^3*c*m*x^3 +
16059*(e*x)^m*A*a^2*b*c*m*x^3 + 5353*(e*x)^m*A*a^3*d*m*x^3 + 3010*(e*x)^m*A*a^3*c*m^2*x + 3465*(e*x)^m*B*a^3*c
*x^3 + 10395*(e*x)^m*A*a^2*b*c*x^3 + 3465*(e*x)^m*A*a^3*d*x^3 + 9129*(e*x)^m*A*a^3*c*m*x + 10395*(e*x)^m*A*a^3
*c*x)/(m^6 + 36*m^5 + 505*m^4 + 3480*m^3 + 12139*m^2 + 19524*m + 10395)

Mupad [B] (verification not implemented)

Time = 5.94 (sec) , antiderivative size = 469, normalized size of antiderivative = 2.48 \[ \int (e x)^m \left (a+b x^2\right )^3 \left (A+B x^2\right ) \left (c+d x^2\right ) \, dx=\frac {a^2\,x^3\,{\left (e\,x\right )}^m\,\left (A\,a\,d+3\,A\,b\,c+B\,a\,c\right )\,\left (m^5+33\,m^4+406\,m^3+2262\,m^2+5353\,m+3465\right )}{m^6+36\,m^5+505\,m^4+3480\,m^3+12139\,m^2+19524\,m+10395}+\frac {b^2\,x^9\,{\left (e\,x\right )}^m\,\left (A\,b\,d+3\,B\,a\,d+B\,b\,c\right )\,\left (m^5+27\,m^4+262\,m^3+1122\,m^2+2041\,m+1155\right )}{m^6+36\,m^5+505\,m^4+3480\,m^3+12139\,m^2+19524\,m+10395}+\frac {a\,x^5\,{\left (e\,x\right )}^m\,\left (3\,A\,b^2\,c+B\,a^2\,d+3\,A\,a\,b\,d+3\,B\,a\,b\,c\right )\,\left (m^5+31\,m^4+350\,m^3+1730\,m^2+3489\,m+2079\right )}{m^6+36\,m^5+505\,m^4+3480\,m^3+12139\,m^2+19524\,m+10395}+\frac {b\,x^7\,{\left (e\,x\right )}^m\,\left (A\,b^2\,c+3\,B\,a^2\,d+3\,A\,a\,b\,d+3\,B\,a\,b\,c\right )\,\left (m^5+29\,m^4+302\,m^3+1366\,m^2+2577\,m+1485\right )}{m^6+36\,m^5+505\,m^4+3480\,m^3+12139\,m^2+19524\,m+10395}+\frac {B\,b^3\,d\,x^{11}\,{\left (e\,x\right )}^m\,\left (m^5+25\,m^4+230\,m^3+950\,m^2+1689\,m+945\right )}{m^6+36\,m^5+505\,m^4+3480\,m^3+12139\,m^2+19524\,m+10395}+\frac {A\,a^3\,c\,x\,{\left (e\,x\right )}^m\,\left (m^5+35\,m^4+470\,m^3+3010\,m^2+9129\,m+10395\right )}{m^6+36\,m^5+505\,m^4+3480\,m^3+12139\,m^2+19524\,m+10395} \]

[In]

int((A + B*x^2)*(e*x)^m*(a + b*x^2)^3*(c + d*x^2),x)

[Out]

(a^2*x^3*(e*x)^m*(A*a*d + 3*A*b*c + B*a*c)*(5353*m + 2262*m^2 + 406*m^3 + 33*m^4 + m^5 + 3465))/(19524*m + 121
39*m^2 + 3480*m^3 + 505*m^4 + 36*m^5 + m^6 + 10395) + (b^2*x^9*(e*x)^m*(A*b*d + 3*B*a*d + B*b*c)*(2041*m + 112
2*m^2 + 262*m^3 + 27*m^4 + m^5 + 1155))/(19524*m + 12139*m^2 + 3480*m^3 + 505*m^4 + 36*m^5 + m^6 + 10395) + (a
*x^5*(e*x)^m*(3*A*b^2*c + B*a^2*d + 3*A*a*b*d + 3*B*a*b*c)*(3489*m + 1730*m^2 + 350*m^3 + 31*m^4 + m^5 + 2079)
)/(19524*m + 12139*m^2 + 3480*m^3 + 505*m^4 + 36*m^5 + m^6 + 10395) + (b*x^7*(e*x)^m*(A*b^2*c + 3*B*a^2*d + 3*
A*a*b*d + 3*B*a*b*c)*(2577*m + 1366*m^2 + 302*m^3 + 29*m^4 + m^5 + 1485))/(19524*m + 12139*m^2 + 3480*m^3 + 50
5*m^4 + 36*m^5 + m^6 + 10395) + (B*b^3*d*x^11*(e*x)^m*(1689*m + 950*m^2 + 230*m^3 + 25*m^4 + m^5 + 945))/(1952
4*m + 12139*m^2 + 3480*m^3 + 505*m^4 + 36*m^5 + m^6 + 10395) + (A*a^3*c*x*(e*x)^m*(9129*m + 3010*m^2 + 470*m^3
+ 35*m^4 + m^5 + 10395))/(19524*m + 12139*m^2 + 3480*m^3 + 505*m^4 + 36*m^5 + m^6 + 10395)