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

Optimal. Leaf size=189 \[ \frac{a^2 (e x)^{m+3} (a A d+a B c+3 A b c)}{e^3 (m+3)}+\frac{a^3 A c (e x)^{m+1}}{e (m+1)}+\frac{b^2 (e x)^{m+9} (3 a B d+A b d+b B c)}{e^9 (m+9)}+\frac{a (e x)^{m+5} (3 A b (a d+b c)+a B (a d+3 b c))}{e^5 (m+5)}+\frac{b (e x)^{m+7} (A b (3 a d+b c)+3 a B (a d+b c))}{e^7 (m+7)}+\frac{b^3 B d (e x)^{m+11}}{e^{11} (m+11)} \]

[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))

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Rubi [A]  time = 0.227067, antiderivative size = 189, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 1, integrand size = 29, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.034, Rules used = {570} \[ \frac{a^2 (e x)^{m+3} (a A d+a B c+3 A b c)}{e^3 (m+3)}+\frac{a^3 A c (e x)^{m+1}}{e (m+1)}+\frac{b^2 (e x)^{m+9} (3 a B d+A b d+b B c)}{e^9 (m+9)}+\frac{a (e x)^{m+5} (3 A b (a d+b c)+a B (a d+3 b c))}{e^5 (m+5)}+\frac{b (e x)^{m+7} (A b (3 a d+b c)+3 a B (a d+b c))}{e^7 (m+7)}+\frac{b^3 B d (e x)^{m+11}}{e^{11} (m+11)} \]

Antiderivative was successfully verified.

[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 570

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*} \int (e x)^m \left (a+b x^2\right )^3 \left (A+B x^2\right ) \left (c+d x^2\right ) \, dx &=\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]  time = 0.253054, size = 151, normalized size = 0.8 \[ x (e x)^m \left (\frac{a^2 x^2 (a A d+a B c+3 A b c)}{m+3}+\frac{a^3 A c}{m+1}+\frac{b^2 x^8 (3 a B d+A b d+b B c)}{m+9}+\frac{b x^6 (A b (3 a d+b c)+3 a B (a d+b c))}{m+7}+\frac{a x^4 (3 A b (a d+b c)+a B (a d+3 b c))}{m+5}+\frac{b^3 B d x^{10}}{m+11}\right ) \]

Antiderivative was successfully verified.

[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))

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Maple [B]  time = 0.008, size = 1229, normalized size = 6.5 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[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)

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Maxima [F(-2)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: ValueError} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Exception raised: ValueError

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Fricas [B]  time = 1.64714, size = 2070, normalized size = 10.95 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[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)

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Sympy [A]  time = 6.37624, size = 6156, normalized size = 32.57 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[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*e**m*m**5*x*x**m/(m**6 + 36*m**5 + 505*m**4 + 3480*m**3 + 12139*m**2 + 195
24*m + 10395) + 35*A*a**3*c*e**m*m**4*x*x**m/(m**6 + 36*m**5 + 505*m**4 + 3480*m**3 + 12139*m**2 + 19524*m + 1
0395) + 470*A*a**3*c*e**m*m**3*x*x**m/(m**6 + 36*m**5 + 505*m**4 + 3480*m**3 + 12139*m**2 + 19524*m + 10395) +
3010*A*a**3*c*e**m*m**2*x*x**m/(m**6 + 36*m**5 + 505*m**4 + 3480*m**3 + 12139*m**2 + 19524*m + 10395) + 9129*
A*a**3*c*e**m*m*x*x**m/(m**6 + 36*m**5 + 505*m**4 + 3480*m**3 + 12139*m**2 + 19524*m + 10395) + 10395*A*a**3*c
*e**m*x*x**m/(m**6 + 36*m**5 + 505*m**4 + 3480*m**3 + 12139*m**2 + 19524*m + 10395) + A*a**3*d*e**m*m**5*x**3*
x**m/(m**6 + 36*m**5 + 505*m**4 + 3480*m**3 + 12139*m**2 + 19524*m + 10395) + 33*A*a**3*d*e**m*m**4*x**3*x**m/
(m**6 + 36*m**5 + 505*m**4 + 3480*m**3 + 12139*m**2 + 19524*m + 10395) + 406*A*a**3*d*e**m*m**3*x**3*x**m/(m**
6 + 36*m**5 + 505*m**4 + 3480*m**3 + 12139*m**2 + 19524*m + 10395) + 2262*A*a**3*d*e**m*m**2*x**3*x**m/(m**6 +
36*m**5 + 505*m**4 + 3480*m**3 + 12139*m**2 + 19524*m + 10395) + 5353*A*a**3*d*e**m*m*x**3*x**m/(m**6 + 36*m*
*5 + 505*m**4 + 3480*m**3 + 12139*m**2 + 19524*m + 10395) + 3465*A*a**3*d*e**m*x**3*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*e**m*m**5*x**3*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*e**m*m**4*x**3*x**m/(m**6 + 36*m**5 + 505*m**4 +
3480*m**3 + 12139*m**2 + 19524*m + 10395) + 1218*A*a**2*b*c*e**m*m**3*x**3*x**m/(m**6 + 36*m**5 + 505*m**4 +
3480*m**3 + 12139*m**2 + 19524*m + 10395) + 6786*A*a**2*b*c*e**m*m**2*x**3*x**m/(m**6 + 36*m**5 + 505*m**4 + 3
480*m**3 + 12139*m**2 + 19524*m + 10395) + 16059*A*a**2*b*c*e**m*m*x**3*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*e**m*x**3*x**m/(m**6 + 36*m**5 + 505*m**4 + 3480*m**3
+ 12139*m**2 + 19524*m + 10395) + 3*A*a**2*b*d*e**m*m**5*x**5*x**m/(m**6 + 36*m**5 + 505*m**4 + 3480*m**3 + 1
2139*m**2 + 19524*m + 10395) + 93*A*a**2*b*d*e**m*m**4*x**5*x**m/(m**6 + 36*m**5 + 505*m**4 + 3480*m**3 + 1213
9*m**2 + 19524*m + 10395) + 1050*A*a**2*b*d*e**m*m**3*x**5*x**m/(m**6 + 36*m**5 + 505*m**4 + 3480*m**3 + 12139
*m**2 + 19524*m + 10395) + 5190*A*a**2*b*d*e**m*m**2*x**5*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*e**m*m*x**5*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*e**m*x**5*x**m/(m**6 + 36*m**5 + 505*m**4 + 3480*m**3 + 12139*m**2 + 19
524*m + 10395) + 3*A*a*b**2*c*e**m*m**5*x**5*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*e**m*m**4*x**5*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*e**m*m**3*x**5*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*e**m*m**2*x**5*x**m/(m**6 + 36*m**5 + 505*m**4 + 3480*m**3 + 12139*m**2 + 19524*m + 1
0395) + 10467*A*a*b**2*c*e**m*m*x**5*x**m/(m**6 + 36*m**5 + 505*m**4 + 3480*m**3 + 12139*m**2 + 19524*m + 1039
5) + 6237*A*a*b**2*c*e**m*x**5*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*e**m*m**5*x**7*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*e**m*m**4*x**7*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*e**m*m**3*x**7*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*e**m*m**2*x**7*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*e**m*m*x**7*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
*e**m*x**7*x**m/(m**6 + 36*m**5 + 505*m**4 + 3480*m**3 + 12139*m**2 + 19524*m + 10395) + A*b**3*c*e**m*m**5*x*
*7*x**m/(m**6 + 36*m**5 + 505*m**4 + 3480*m**3 + 12139*m**2 + 19524*m + 10395) + 29*A*b**3*c*e**m*m**4*x**7*x*
*m/(m**6 + 36*m**5 + 505*m**4 + 3480*m**3 + 12139*m**2 + 19524*m + 10395) + 302*A*b**3*c*e**m*m**3*x**7*x**m/(
m**6 + 36*m**5 + 505*m**4 + 3480*m**3 + 12139*m**2 + 19524*m + 10395) + 1366*A*b**3*c*e**m*m**2*x**7*x**m/(m**
6 + 36*m**5 + 505*m**4 + 3480*m**3 + 12139*m**2 + 19524*m + 10395) + 2577*A*b**3*c*e**m*m*x**7*x**m/(m**6 + 36
*m**5 + 505*m**4 + 3480*m**3 + 12139*m**2 + 19524*m + 10395) + 1485*A*b**3*c*e**m*x**7*x**m/(m**6 + 36*m**5 +
505*m**4 + 3480*m**3 + 12139*m**2 + 19524*m + 10395) + A*b**3*d*e**m*m**5*x**9*x**m/(m**6 + 36*m**5 + 505*m**4
+ 3480*m**3 + 12139*m**2 + 19524*m + 10395) + 27*A*b**3*d*e**m*m**4*x**9*x**m/(m**6 + 36*m**5 + 505*m**4 + 34
80*m**3 + 12139*m**2 + 19524*m + 10395) + 262*A*b**3*d*e**m*m**3*x**9*x**m/(m**6 + 36*m**5 + 505*m**4 + 3480*m
**3 + 12139*m**2 + 19524*m + 10395) + 1122*A*b**3*d*e**m*m**2*x**9*x**m/(m**6 + 36*m**5 + 505*m**4 + 3480*m**3
+ 12139*m**2 + 19524*m + 10395) + 2041*A*b**3*d*e**m*m*x**9*x**m/(m**6 + 36*m**5 + 505*m**4 + 3480*m**3 + 121
39*m**2 + 19524*m + 10395) + 1155*A*b**3*d*e**m*x**9*x**m/(m**6 + 36*m**5 + 505*m**4 + 3480*m**3 + 12139*m**2
+ 19524*m + 10395) + B*a**3*c*e**m*m**5*x**3*x**m/(m**6 + 36*m**5 + 505*m**4 + 3480*m**3 + 12139*m**2 + 19524*
m + 10395) + 33*B*a**3*c*e**m*m**4*x**3*x**m/(m**6 + 36*m**5 + 505*m**4 + 3480*m**3 + 12139*m**2 + 19524*m + 1
0395) + 406*B*a**3*c*e**m*m**3*x**3*x**m/(m**6 + 36*m**5 + 505*m**4 + 3480*m**3 + 12139*m**2 + 19524*m + 10395
) + 2262*B*a**3*c*e**m*m**2*x**3*x**m/(m**6 + 36*m**5 + 505*m**4 + 3480*m**3 + 12139*m**2 + 19524*m + 10395) +
5353*B*a**3*c*e**m*m*x**3*x**m/(m**6 + 36*m**5 + 505*m**4 + 3480*m**3 + 12139*m**2 + 19524*m + 10395) + 3465*
B*a**3*c*e**m*x**3*x**m/(m**6 + 36*m**5 + 505*m**4 + 3480*m**3 + 12139*m**2 + 19524*m + 10395) + B*a**3*d*e**m
*m**5*x**5*x**m/(m**6 + 36*m**5 + 505*m**4 + 3480*m**3 + 12139*m**2 + 19524*m + 10395) + 31*B*a**3*d*e**m*m**4
*x**5*x**m/(m**6 + 36*m**5 + 505*m**4 + 3480*m**3 + 12139*m**2 + 19524*m + 10395) + 350*B*a**3*d*e**m*m**3*x**
5*x**m/(m**6 + 36*m**5 + 505*m**4 + 3480*m**3 + 12139*m**2 + 19524*m + 10395) + 1730*B*a**3*d*e**m*m**2*x**5*x
**m/(m**6 + 36*m**5 + 505*m**4 + 3480*m**3 + 12139*m**2 + 19524*m + 10395) + 3489*B*a**3*d*e**m*m*x**5*x**m/(m
**6 + 36*m**5 + 505*m**4 + 3480*m**3 + 12139*m**2 + 19524*m + 10395) + 2079*B*a**3*d*e**m*x**5*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*e**m*m**5*x**5*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*e**m*m**4*x**5*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*e**m*m**3*x**5*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*e**m*m**2*x**5*x**m/(m**6 + 36*m**5 + 5
05*m**4 + 3480*m**3 + 12139*m**2 + 19524*m + 10395) + 10467*B*a**2*b*c*e**m*m*x**5*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*e**m*x**5*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*e**m*m**5*x**7*x**m/(m**6 + 36*m**5 + 505*m**4 + 348
0*m**3 + 12139*m**2 + 19524*m + 10395) + 87*B*a**2*b*d*e**m*m**4*x**7*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*e**m*m**3*x**7*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*e**m*m**2*x**7*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*e**m*m*x**7*x**m/(m**6 + 36*m**5 + 505*m**4 + 3480*m**3 + 1
2139*m**2 + 19524*m + 10395) + 4455*B*a**2*b*d*e**m*x**7*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*e**m*m**5*x**7*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*e**m*m**4*x**7*x**m/(m**6 + 36*m**5 + 505*m**4 + 3480*m**3 + 12139*m**2 + 1
9524*m + 10395) + 906*B*a*b**2*c*e**m*m**3*x**7*x**m/(m**6 + 36*m**5 + 505*m**4 + 3480*m**3 + 12139*m**2 + 195
24*m + 10395) + 4098*B*a*b**2*c*e**m*m**2*x**7*x**m/(m**6 + 36*m**5 + 505*m**4 + 3480*m**3 + 12139*m**2 + 1952
4*m + 10395) + 7731*B*a*b**2*c*e**m*m*x**7*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*e**m*x**7*x**m/(m**6 + 36*m**5 + 505*m**4 + 3480*m**3 + 12139*m**2 + 19524*m + 1039
5) + 3*B*a*b**2*d*e**m*m**5*x**9*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*e**m*m**4*x**9*x**m/(m**6 + 36*m**5 + 505*m**4 + 3480*m**3 + 12139*m**2 + 19524*m + 10395) + 78
6*B*a*b**2*d*e**m*m**3*x**9*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*e**m*m**2*x**9*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*e**m*m*x**9*x**m/(m**6 + 36*m**5 + 505*m**4 + 3480*m**3 + 12139*m**2 + 19524*m + 10395) + 3465*B*a*
b**2*d*e**m*x**9*x**m/(m**6 + 36*m**5 + 505*m**4 + 3480*m**3 + 12139*m**2 + 19524*m + 10395) + B*b**3*c*e**m*m
**5*x**9*x**m/(m**6 + 36*m**5 + 505*m**4 + 3480*m**3 + 12139*m**2 + 19524*m + 10395) + 27*B*b**3*c*e**m*m**4*x
**9*x**m/(m**6 + 36*m**5 + 505*m**4 + 3480*m**3 + 12139*m**2 + 19524*m + 10395) + 262*B*b**3*c*e**m*m**3*x**9*
x**m/(m**6 + 36*m**5 + 505*m**4 + 3480*m**3 + 12139*m**2 + 19524*m + 10395) + 1122*B*b**3*c*e**m*m**2*x**9*x**
m/(m**6 + 36*m**5 + 505*m**4 + 3480*m**3 + 12139*m**2 + 19524*m + 10395) + 2041*B*b**3*c*e**m*m*x**9*x**m/(m**
6 + 36*m**5 + 505*m**4 + 3480*m**3 + 12139*m**2 + 19524*m + 10395) + 1155*B*b**3*c*e**m*x**9*x**m/(m**6 + 36*m
**5 + 505*m**4 + 3480*m**3 + 12139*m**2 + 19524*m + 10395) + B*b**3*d*e**m*m**5*x**11*x**m/(m**6 + 36*m**5 + 5
05*m**4 + 3480*m**3 + 12139*m**2 + 19524*m + 10395) + 25*B*b**3*d*e**m*m**4*x**11*x**m/(m**6 + 36*m**5 + 505*m
**4 + 3480*m**3 + 12139*m**2 + 19524*m + 10395) + 230*B*b**3*d*e**m*m**3*x**11*x**m/(m**6 + 36*m**5 + 505*m**4
+ 3480*m**3 + 12139*m**2 + 19524*m + 10395) + 950*B*b**3*d*e**m*m**2*x**11*x**m/(m**6 + 36*m**5 + 505*m**4 +
3480*m**3 + 12139*m**2 + 19524*m + 10395) + 1689*B*b**3*d*e**m*m*x**11*x**m/(m**6 + 36*m**5 + 505*m**4 + 3480*
m**3 + 12139*m**2 + 19524*m + 10395) + 945*B*b**3*d*e**m*x**11*x**m/(m**6 + 36*m**5 + 505*m**4 + 3480*m**3 + 1
2139*m**2 + 19524*m + 10395), True))

________________________________________________________________________________________

Giac [B]  time = 1.23178, size = 2306, normalized size = 12.2 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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