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

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

[Out]

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

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Rubi [A]  time = 0.175998, 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} \[ \frac{c^2 (e x)^{m+3} (3 a A d+a B c+A b c)}{e^3 (m+3)}+\frac{d^2 (e x)^{m+9} (a B d+A b d+3 b B c)}{e^9 (m+9)}+\frac{c (e x)^{m+5} (3 a d (A d+B c)+b c (3 A d+B c))}{e^5 (m+5)}+\frac{d (e x)^{m+7} (a d (A d+3 B c)+3 b c (A d+B c))}{e^7 (m+7)}+\frac{a A c^3 (e x)^{m+1}}{e (m+1)}+\frac{b B d^3 (e x)^{m+11}}{e^{11} (m+11)} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

Mathematica [A]  time = 0.246853, size = 151, normalized size = 0.8 \[ x (e x)^m \left (\frac{c^2 x^2 (3 a A d+a B c+A b c)}{m+3}+\frac{d^2 x^8 (a B d+A b d+3 b B c)}{m+9}+\frac{d x^6 (a d (A d+3 B c)+3 b c (A d+B c))}{m+7}+\frac{c x^4 (3 a d (A d+B c)+b c (3 A d+B c))}{m+5}+\frac{a A c^3}{m+1}+\frac{b B d^3 x^{10}}{m+11}\right ) \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Maple [B]  time = 0.006, 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)*(B*x^2+A)*(d*x^2+c)^3,x)

[Out]

x*(B*b*d^3*m^5*x^10+25*B*b*d^3*m^4*x^10+A*b*d^3*m^5*x^8+B*a*d^3*m^5*x^8+3*B*b*c*d^2*m^5*x^8+230*B*b*d^3*m^3*x^
10+27*A*b*d^3*m^4*x^8+27*B*a*d^3*m^4*x^8+81*B*b*c*d^2*m^4*x^8+950*B*b*d^3*m^2*x^10+A*a*d^3*m^5*x^6+3*A*b*c*d^2
*m^5*x^6+262*A*b*d^3*m^3*x^8+3*B*a*c*d^2*m^5*x^6+262*B*a*d^3*m^3*x^8+3*B*b*c^2*d*m^5*x^6+786*B*b*c*d^2*m^3*x^8
+1689*B*b*d^3*m*x^10+29*A*a*d^3*m^4*x^6+87*A*b*c*d^2*m^4*x^6+1122*A*b*d^3*m^2*x^8+87*B*a*c*d^2*m^4*x^6+1122*B*
a*d^3*m^2*x^8+87*B*b*c^2*d*m^4*x^6+3366*B*b*c*d^2*m^2*x^8+945*B*b*d^3*x^10+3*A*a*c*d^2*m^5*x^4+302*A*a*d^3*m^3
*x^6+3*A*b*c^2*d*m^5*x^4+906*A*b*c*d^2*m^3*x^6+2041*A*b*d^3*m*x^8+3*B*a*c^2*d*m^5*x^4+906*B*a*c*d^2*m^3*x^6+20
41*B*a*d^3*m*x^8+B*b*c^3*m^5*x^4+906*B*b*c^2*d*m^3*x^6+6123*B*b*c*d^2*m*x^8+93*A*a*c*d^2*m^4*x^4+1366*A*a*d^3*
m^2*x^6+93*A*b*c^2*d*m^4*x^4+4098*A*b*c*d^2*m^2*x^6+1155*A*b*d^3*x^8+93*B*a*c^2*d*m^4*x^4+4098*B*a*c*d^2*m^2*x
^6+1155*B*a*d^3*x^8+31*B*b*c^3*m^4*x^4+4098*B*b*c^2*d*m^2*x^6+3465*B*b*c*d^2*x^8+3*A*a*c^2*d*m^5*x^2+1050*A*a*
c*d^2*m^3*x^4+2577*A*a*d^3*m*x^6+A*b*c^3*m^5*x^2+1050*A*b*c^2*d*m^3*x^4+7731*A*b*c*d^2*m*x^6+B*a*c^3*m^5*x^2+1
050*B*a*c^2*d*m^3*x^4+7731*B*a*c*d^2*m*x^6+350*B*b*c^3*m^3*x^4+7731*B*b*c^2*d*m*x^6+99*A*a*c^2*d*m^4*x^2+5190*
A*a*c*d^2*m^2*x^4+1485*A*a*d^3*x^6+33*A*b*c^3*m^4*x^2+5190*A*b*c^2*d*m^2*x^4+4455*A*b*c*d^2*x^6+33*B*a*c^3*m^4
*x^2+5190*B*a*c^2*d*m^2*x^4+4455*B*a*c*d^2*x^6+1730*B*b*c^3*m^2*x^4+4455*B*b*c^2*d*x^6+A*a*c^3*m^5+1218*A*a*c^
2*d*m^3*x^2+10467*A*a*c*d^2*m*x^4+406*A*b*c^3*m^3*x^2+10467*A*b*c^2*d*m*x^4+406*B*a*c^3*m^3*x^2+10467*B*a*c^2*
d*m*x^4+3489*B*b*c^3*m*x^4+35*A*a*c^3*m^4+6786*A*a*c^2*d*m^2*x^2+6237*A*a*c*d^2*x^4+2262*A*b*c^3*m^2*x^2+6237*
A*b*c^2*d*x^4+2262*B*a*c^3*m^2*x^2+6237*B*a*c^2*d*x^4+2079*B*b*c^3*x^4+470*A*a*c^3*m^3+16059*A*a*c^2*d*m*x^2+5
353*A*b*c^3*m*x^2+5353*B*a*c^3*m*x^2+3010*A*a*c^3*m^2+10395*A*a*c^2*d*x^2+3465*A*b*c^3*x^2+3465*B*a*c^3*x^2+91
29*A*a*c^3*m+10395*A*a*c^3)*(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)*(B*x^2+A)*(d*x^2+c)^3,x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [B]  time = 1.66352, size = 1976, normalized size = 10.46 \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)*(B*x^2+A)*(d*x^2+c)^3,x, algorithm="fricas")

[Out]

((B*b*d^3*m^5 + 25*B*b*d^3*m^4 + 230*B*b*d^3*m^3 + 950*B*b*d^3*m^2 + 1689*B*b*d^3*m + 945*B*b*d^3)*x^11 + ((3*
B*b*c*d^2 + (B*a + A*b)*d^3)*m^5 + 3465*B*b*c*d^2 + 27*(3*B*b*c*d^2 + (B*a + A*b)*d^3)*m^4 + 1155*(B*a + A*b)*
d^3 + 262*(3*B*b*c*d^2 + (B*a + A*b)*d^3)*m^3 + 1122*(3*B*b*c*d^2 + (B*a + A*b)*d^3)*m^2 + 2041*(3*B*b*c*d^2 +
(B*a + A*b)*d^3)*m)*x^9 + ((3*B*b*c^2*d + A*a*d^3 + 3*(B*a + A*b)*c*d^2)*m^5 + 4455*B*b*c^2*d + 1485*A*a*d^3
+ 29*(3*B*b*c^2*d + A*a*d^3 + 3*(B*a + A*b)*c*d^2)*m^4 + 4455*(B*a + A*b)*c*d^2 + 302*(3*B*b*c^2*d + A*a*d^3 +
3*(B*a + A*b)*c*d^2)*m^3 + 1366*(3*B*b*c^2*d + A*a*d^3 + 3*(B*a + A*b)*c*d^2)*m^2 + 2577*(3*B*b*c^2*d + A*a*d
^3 + 3*(B*a + A*b)*c*d^2)*m)*x^7 + ((B*b*c^3 + 3*A*a*c*d^2 + 3*(B*a + A*b)*c^2*d)*m^5 + 2079*B*b*c^3 + 6237*A*
a*c*d^2 + 31*(B*b*c^3 + 3*A*a*c*d^2 + 3*(B*a + A*b)*c^2*d)*m^4 + 6237*(B*a + A*b)*c^2*d + 350*(B*b*c^3 + 3*A*a
*c*d^2 + 3*(B*a + A*b)*c^2*d)*m^3 + 1730*(B*b*c^3 + 3*A*a*c*d^2 + 3*(B*a + A*b)*c^2*d)*m^2 + 3489*(B*b*c^3 + 3
*A*a*c*d^2 + 3*(B*a + A*b)*c^2*d)*m)*x^5 + ((3*A*a*c^2*d + (B*a + A*b)*c^3)*m^5 + 10395*A*a*c^2*d + 33*(3*A*a*
c^2*d + (B*a + A*b)*c^3)*m^4 + 3465*(B*a + A*b)*c^3 + 406*(3*A*a*c^2*d + (B*a + A*b)*c^3)*m^3 + 2262*(3*A*a*c^
2*d + (B*a + A*b)*c^3)*m^2 + 5353*(3*A*a*c^2*d + (B*a + A*b)*c^3)*m)*x^3 + (A*a*c^3*m^5 + 35*A*a*c^3*m^4 + 470
*A*a*c^3*m^3 + 3010*A*a*c^3*m^2 + 9129*A*a*c^3*m + 10395*A*a*c^3)*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 = 6.83399, 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)*(B*x**2+A)*(d*x**2+c)**3,x)

[Out]

Piecewise(((-A*a*c**3/(10*x**10) - 3*A*a*c**2*d/(8*x**8) - A*a*c*d**2/(2*x**6) - A*a*d**3/(4*x**4) - A*b*c**3/
(8*x**8) - A*b*c**2*d/(2*x**6) - 3*A*b*c*d**2/(4*x**4) - A*b*d**3/(2*x**2) - B*a*c**3/(8*x**8) - B*a*c**2*d/(2
*x**6) - 3*B*a*c*d**2/(4*x**4) - B*a*d**3/(2*x**2) - B*b*c**3/(6*x**6) - 3*B*b*c**2*d/(4*x**4) - 3*B*b*c*d**2/
(2*x**2) + B*b*d**3*log(x))/e**11, Eq(m, -11)), ((-A*a*c**3/(8*x**8) - A*a*c**2*d/(2*x**6) - 3*A*a*c*d**2/(4*x
**4) - A*a*d**3/(2*x**2) - A*b*c**3/(6*x**6) - 3*A*b*c**2*d/(4*x**4) - 3*A*b*c*d**2/(2*x**2) + A*b*d**3*log(x)
- B*a*c**3/(6*x**6) - 3*B*a*c**2*d/(4*x**4) - 3*B*a*c*d**2/(2*x**2) + B*a*d**3*log(x) - B*b*c**3/(4*x**4) - 3
*B*b*c**2*d/(2*x**2) + 3*B*b*c*d**2*log(x) + B*b*d**3*x**2/2)/e**9, Eq(m, -9)), ((-A*a*c**3/(6*x**6) - 3*A*a*c
**2*d/(4*x**4) - 3*A*a*c*d**2/(2*x**2) + A*a*d**3*log(x) - A*b*c**3/(4*x**4) - 3*A*b*c**2*d/(2*x**2) + 3*A*b*c
*d**2*log(x) + A*b*d**3*x**2/2 - B*a*c**3/(4*x**4) - 3*B*a*c**2*d/(2*x**2) + 3*B*a*c*d**2*log(x) + B*a*d**3*x*
*2/2 - B*b*c**3/(2*x**2) + 3*B*b*c**2*d*log(x) + 3*B*b*c*d**2*x**2/2 + B*b*d**3*x**4/4)/e**7, Eq(m, -7)), ((-A
*a*c**3/(4*x**4) - 3*A*a*c**2*d/(2*x**2) + 3*A*a*c*d**2*log(x) + A*a*d**3*x**2/2 - A*b*c**3/(2*x**2) + 3*A*b*c
**2*d*log(x) + 3*A*b*c*d**2*x**2/2 + A*b*d**3*x**4/4 - B*a*c**3/(2*x**2) + 3*B*a*c**2*d*log(x) + 3*B*a*c*d**2*
x**2/2 + B*a*d**3*x**4/4 + B*b*c**3*log(x) + 3*B*b*c**2*d*x**2/2 + 3*B*b*c*d**2*x**4/4 + B*b*d**3*x**6/6)/e**5
, Eq(m, -5)), ((-A*a*c**3/(2*x**2) + 3*A*a*c**2*d*log(x) + 3*A*a*c*d**2*x**2/2 + A*a*d**3*x**4/4 + A*b*c**3*lo
g(x) + 3*A*b*c**2*d*x**2/2 + 3*A*b*c*d**2*x**4/4 + A*b*d**3*x**6/6 + B*a*c**3*log(x) + 3*B*a*c**2*d*x**2/2 + 3
*B*a*c*d**2*x**4/4 + B*a*d**3*x**6/6 + B*b*c**3*x**2/2 + 3*B*b*c**2*d*x**4/4 + B*b*c*d**2*x**6/2 + B*b*d**3*x*
*8/8)/e**3, Eq(m, -3)), ((A*a*c**3*log(x) + 3*A*a*c**2*d*x**2/2 + 3*A*a*c*d**2*x**4/4 + A*a*d**3*x**6/6 + A*b*
c**3*x**2/2 + 3*A*b*c**2*d*x**4/4 + A*b*c*d**2*x**6/2 + A*b*d**3*x**8/8 + B*a*c**3*x**2/2 + 3*B*a*c**2*d*x**4/
4 + B*a*c*d**2*x**6/2 + B*a*d**3*x**8/8 + B*b*c**3*x**4/4 + B*b*c**2*d*x**6/2 + 3*B*b*c*d**2*x**8/8 + B*b*d**3
*x**10/10)/e, Eq(m, -1)), (A*a*c**3*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*c**3*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*c**3*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*c**3*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*c**3*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*c**3
*e**m*x*x**m/(m**6 + 36*m**5 + 505*m**4 + 3480*m**3 + 12139*m**2 + 19524*m + 10395) + 3*A*a*c**2*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) + 99*A*a*c**2*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) + 1218*A*a*c**2*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) + 6786*A*a*c**2*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) + 16059*A*a*c**2*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) + 10395*A*a*c**2*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*c*d**2*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*c*d**2*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*c*d**2*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*c*d**2*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*c*d**2*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*c*d**2*e**m*x**5*x**m/(m**6 + 36*m**5 + 505*
m**4 + 3480*m**3 + 12139*m**2 + 19524*m + 10395) + A*a*d**3*e**m*m**5*x**7*x**m/(m**6 + 36*m**5 + 505*m**4 + 3
480*m**3 + 12139*m**2 + 19524*m + 10395) + 29*A*a*d**3*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*a*d**3*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*a*d**3*e**m*m**2*x**7*x**m/(m**6 + 36*m**5 + 505*m**4 + 3480*m**3 + 1
2139*m**2 + 19524*m + 10395) + 2577*A*a*d**3*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*a*d**3*e**m*x**7*x**m/(m**6 + 36*m**5 + 505*m**4 + 3480*m**3 + 12139*m**2 + 19
524*m + 10395) + A*b*c**3*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*b*c**3*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*b*c**3*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*b*c**3*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) + 535
3*A*b*c**3*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*b*
c**3*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*b*c**2*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) + 93*A*b*c**2*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) + 1050*A*b*c**2*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*b*c**2*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*b*c**2*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*b*c**2*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*A*b*c*d**2*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*b*c*d**2*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*b*c*d**2*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*b*c*d**2*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*b*c*d**2*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*b*c*d**2*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*d**3*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*d**3*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*d**3*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*d**3*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*d**3*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*d**3*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*c**3*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*c**3*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*c**3*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*c**3*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*c**3*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*c**3*e**m*x**3*x**m/(m**6 + 36*m**5 + 505*m**4 + 3480*m**3 + 12139*m**2 + 19524*m + 10395) + 3*B*a*c**2*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) + 93*B*a*c**2*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) + 1050*B*a*c**2*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*B*a*c**2*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*B*a*c**2*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*B*a*c**2*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*c*d**2*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*c*d**2*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*c*d**2*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*c*d**2*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*c*d**2*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*c*d**2*e**m*x**7*x**m/(m**6 + 36*m**
5 + 505*m**4 + 3480*m**3 + 12139*m**2 + 19524*m + 10395) + B*a*d**3*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*a*d**3*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*a*d**3*e**m*m**3*x**9*x**m/(m**6 + 36*m**5 + 505*m**4 + 34
80*m**3 + 12139*m**2 + 19524*m + 10395) + 1122*B*a*d**3*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*a*d**3*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*a*d**3*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*c**3*e**m*m**5*x**5*x**m/(m**6 + 36*m**5 + 505*m**4 + 3480*m**3 + 12139*m**2 + 19
524*m + 10395) + 31*B*b*c**3*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*b*c**3*e**m*m**3*x**5*x**m/(m**6 + 36*m**5 + 505*m**4 + 3480*m**3 + 12139*m**2 + 19524*m + 1
0395) + 1730*B*b*c**3*e**m*m**2*x**5*x**m/(m**6 + 36*m**5 + 505*m**4 + 3480*m**3 + 12139*m**2 + 19524*m + 1039
5) + 3489*B*b*c**3*e**m*m*x**5*x**m/(m**6 + 36*m**5 + 505*m**4 + 3480*m**3 + 12139*m**2 + 19524*m + 10395) + 2
079*B*b*c**3*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*b*c**
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*B*b*c**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*B*b*c**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*B*b*c**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*B*b*c**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*B*b*c**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) + 3*B*b*c*d**2*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*b*c*d**2*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) + 786*B*b*c*d**2*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*b*c*d**2*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*b*c*d**2*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*b*c*d**2*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*d**3*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*d**3*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*d**3*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*d**3*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*d**3*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*d**3*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 = 1.7254, 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)*(B*x^2+A)*(d*x^2+c)^3,x, algorithm="giac")

[Out]

(B*b*d^3*m^5*x^11*x^m*e^m + 25*B*b*d^3*m^4*x^11*x^m*e^m + 3*B*b*c*d^2*m^5*x^9*x^m*e^m + B*a*d^3*m^5*x^9*x^m*e^
m + A*b*d^3*m^5*x^9*x^m*e^m + 230*B*b*d^3*m^3*x^11*x^m*e^m + 81*B*b*c*d^2*m^4*x^9*x^m*e^m + 27*B*a*d^3*m^4*x^9
*x^m*e^m + 27*A*b*d^3*m^4*x^9*x^m*e^m + 950*B*b*d^3*m^2*x^11*x^m*e^m + 3*B*b*c^2*d*m^5*x^7*x^m*e^m + 3*B*a*c*d
^2*m^5*x^7*x^m*e^m + 3*A*b*c*d^2*m^5*x^7*x^m*e^m + A*a*d^3*m^5*x^7*x^m*e^m + 786*B*b*c*d^2*m^3*x^9*x^m*e^m + 2
62*B*a*d^3*m^3*x^9*x^m*e^m + 262*A*b*d^3*m^3*x^9*x^m*e^m + 1689*B*b*d^3*m*x^11*x^m*e^m + 87*B*b*c^2*d*m^4*x^7*
x^m*e^m + 87*B*a*c*d^2*m^4*x^7*x^m*e^m + 87*A*b*c*d^2*m^4*x^7*x^m*e^m + 29*A*a*d^3*m^4*x^7*x^m*e^m + 3366*B*b*
c*d^2*m^2*x^9*x^m*e^m + 1122*B*a*d^3*m^2*x^9*x^m*e^m + 1122*A*b*d^3*m^2*x^9*x^m*e^m + 945*B*b*d^3*x^11*x^m*e^m
+ B*b*c^3*m^5*x^5*x^m*e^m + 3*B*a*c^2*d*m^5*x^5*x^m*e^m + 3*A*b*c^2*d*m^5*x^5*x^m*e^m + 3*A*a*c*d^2*m^5*x^5*x
^m*e^m + 906*B*b*c^2*d*m^3*x^7*x^m*e^m + 906*B*a*c*d^2*m^3*x^7*x^m*e^m + 906*A*b*c*d^2*m^3*x^7*x^m*e^m + 302*A
*a*d^3*m^3*x^7*x^m*e^m + 6123*B*b*c*d^2*m*x^9*x^m*e^m + 2041*B*a*d^3*m*x^9*x^m*e^m + 2041*A*b*d^3*m*x^9*x^m*e^
m + 31*B*b*c^3*m^4*x^5*x^m*e^m + 93*B*a*c^2*d*m^4*x^5*x^m*e^m + 93*A*b*c^2*d*m^4*x^5*x^m*e^m + 93*A*a*c*d^2*m^
4*x^5*x^m*e^m + 4098*B*b*c^2*d*m^2*x^7*x^m*e^m + 4098*B*a*c*d^2*m^2*x^7*x^m*e^m + 4098*A*b*c*d^2*m^2*x^7*x^m*e
^m + 1366*A*a*d^3*m^2*x^7*x^m*e^m + 3465*B*b*c*d^2*x^9*x^m*e^m + 1155*B*a*d^3*x^9*x^m*e^m + 1155*A*b*d^3*x^9*x
^m*e^m + B*a*c^3*m^5*x^3*x^m*e^m + A*b*c^3*m^5*x^3*x^m*e^m + 3*A*a*c^2*d*m^5*x^3*x^m*e^m + 350*B*b*c^3*m^3*x^5
*x^m*e^m + 1050*B*a*c^2*d*m^3*x^5*x^m*e^m + 1050*A*b*c^2*d*m^3*x^5*x^m*e^m + 1050*A*a*c*d^2*m^3*x^5*x^m*e^m +
7731*B*b*c^2*d*m*x^7*x^m*e^m + 7731*B*a*c*d^2*m*x^7*x^m*e^m + 7731*A*b*c*d^2*m*x^7*x^m*e^m + 2577*A*a*d^3*m*x^
7*x^m*e^m + 33*B*a*c^3*m^4*x^3*x^m*e^m + 33*A*b*c^3*m^4*x^3*x^m*e^m + 99*A*a*c^2*d*m^4*x^3*x^m*e^m + 1730*B*b*
c^3*m^2*x^5*x^m*e^m + 5190*B*a*c^2*d*m^2*x^5*x^m*e^m + 5190*A*b*c^2*d*m^2*x^5*x^m*e^m + 5190*A*a*c*d^2*m^2*x^5
*x^m*e^m + 4455*B*b*c^2*d*x^7*x^m*e^m + 4455*B*a*c*d^2*x^7*x^m*e^m + 4455*A*b*c*d^2*x^7*x^m*e^m + 1485*A*a*d^3
*x^7*x^m*e^m + A*a*c^3*m^5*x*x^m*e^m + 406*B*a*c^3*m^3*x^3*x^m*e^m + 406*A*b*c^3*m^3*x^3*x^m*e^m + 1218*A*a*c^
2*d*m^3*x^3*x^m*e^m + 3489*B*b*c^3*m*x^5*x^m*e^m + 10467*B*a*c^2*d*m*x^5*x^m*e^m + 10467*A*b*c^2*d*m*x^5*x^m*e
^m + 10467*A*a*c*d^2*m*x^5*x^m*e^m + 35*A*a*c^3*m^4*x*x^m*e^m + 2262*B*a*c^3*m^2*x^3*x^m*e^m + 2262*A*b*c^3*m^
2*x^3*x^m*e^m + 6786*A*a*c^2*d*m^2*x^3*x^m*e^m + 2079*B*b*c^3*x^5*x^m*e^m + 6237*B*a*c^2*d*x^5*x^m*e^m + 6237*
A*b*c^2*d*x^5*x^m*e^m + 6237*A*a*c*d^2*x^5*x^m*e^m + 470*A*a*c^3*m^3*x*x^m*e^m + 5353*B*a*c^3*m*x^3*x^m*e^m +
5353*A*b*c^3*m*x^3*x^m*e^m + 16059*A*a*c^2*d*m*x^3*x^m*e^m + 3010*A*a*c^3*m^2*x*x^m*e^m + 3465*B*a*c^3*x^3*x^m
*e^m + 3465*A*b*c^3*x^3*x^m*e^m + 10395*A*a*c^2*d*x^3*x^m*e^m + 9129*A*a*c^3*m*x*x^m*e^m + 10395*A*a*c^3*x*x^m
*e^m)/(m^6 + 36*m^5 + 505*m^4 + 3480*m^3 + 12139*m^2 + 19524*m + 10395)