3.1327 \(\int \frac{(A+B x) \left (a+c x^2\right )^3}{(d+e x)^9} \, dx\)

Optimal. Leaf size=330 \[ \frac{c \left (4 A c d e \left (3 a e^2+5 c d^2\right )-B \left (3 a^2 e^4+30 a c d^2 e^2+35 c^2 d^4\right )\right )}{5 e^8 (d+e x)^5}-\frac{c^2 \left (a B e^2-2 A c d e+7 B c d^2\right )}{e^8 (d+e x)^3}+\frac{c^2 \left (-3 a A e^3+15 a B d e^2-15 A c d^2 e+35 B c d^3\right )}{4 e^8 (d+e x)^4}-\frac{\left (a e^2+c d^2\right )^2 \left (a B e^2-6 A c d e+7 B c d^2\right )}{7 e^8 (d+e x)^7}+\frac{\left (a e^2+c d^2\right )^3 (B d-A e)}{8 e^8 (d+e x)^8}+\frac{c \left (a e^2+c d^2\right ) \left (-a A e^3+3 a B d e^2-5 A c d^2 e+7 B c d^3\right )}{2 e^8 (d+e x)^6}+\frac{c^3 (7 B d-A e)}{2 e^8 (d+e x)^2}-\frac{B c^3}{e^8 (d+e x)} \]

[Out]

((B*d - A*e)*(c*d^2 + a*e^2)^3)/(8*e^8*(d + e*x)^8) - ((c*d^2 + a*e^2)^2*(7*B*c*
d^2 - 6*A*c*d*e + a*B*e^2))/(7*e^8*(d + e*x)^7) + (c*(c*d^2 + a*e^2)*(7*B*c*d^3
- 5*A*c*d^2*e + 3*a*B*d*e^2 - a*A*e^3))/(2*e^8*(d + e*x)^6) + (c*(4*A*c*d*e*(5*c
*d^2 + 3*a*e^2) - B*(35*c^2*d^4 + 30*a*c*d^2*e^2 + 3*a^2*e^4)))/(5*e^8*(d + e*x)
^5) + (c^2*(35*B*c*d^3 - 15*A*c*d^2*e + 15*a*B*d*e^2 - 3*a*A*e^3))/(4*e^8*(d + e
*x)^4) - (c^2*(7*B*c*d^2 - 2*A*c*d*e + a*B*e^2))/(e^8*(d + e*x)^3) + (c^3*(7*B*d
- A*e))/(2*e^8*(d + e*x)^2) - (B*c^3)/(e^8*(d + e*x))

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Rubi [A]  time = 0.960833, antiderivative size = 330, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 1, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.045 \[ -\frac{c \left (3 a^2 B e^4-12 a A c d e^3+30 a B c d^2 e^2-20 A c^2 d^3 e+35 B c^2 d^4\right )}{5 e^8 (d+e x)^5}-\frac{c^2 \left (a B e^2-2 A c d e+7 B c d^2\right )}{e^8 (d+e x)^3}+\frac{c^2 \left (-3 a A e^3+15 a B d e^2-15 A c d^2 e+35 B c d^3\right )}{4 e^8 (d+e x)^4}-\frac{\left (a e^2+c d^2\right )^2 \left (a B e^2-6 A c d e+7 B c d^2\right )}{7 e^8 (d+e x)^7}+\frac{\left (a e^2+c d^2\right )^3 (B d-A e)}{8 e^8 (d+e x)^8}+\frac{c \left (a e^2+c d^2\right ) \left (-a A e^3+3 a B d e^2-5 A c d^2 e+7 B c d^3\right )}{2 e^8 (d+e x)^6}+\frac{c^3 (7 B d-A e)}{2 e^8 (d+e x)^2}-\frac{B c^3}{e^8 (d+e x)} \]

Antiderivative was successfully verified.

[In]  Int[((A + B*x)*(a + c*x^2)^3)/(d + e*x)^9,x]

[Out]

((B*d - A*e)*(c*d^2 + a*e^2)^3)/(8*e^8*(d + e*x)^8) - ((c*d^2 + a*e^2)^2*(7*B*c*
d^2 - 6*A*c*d*e + a*B*e^2))/(7*e^8*(d + e*x)^7) + (c*(c*d^2 + a*e^2)*(7*B*c*d^3
- 5*A*c*d^2*e + 3*a*B*d*e^2 - a*A*e^3))/(2*e^8*(d + e*x)^6) - (c*(35*B*c^2*d^4 -
20*A*c^2*d^3*e + 30*a*B*c*d^2*e^2 - 12*a*A*c*d*e^3 + 3*a^2*B*e^4))/(5*e^8*(d +
e*x)^5) + (c^2*(35*B*c*d^3 - 15*A*c*d^2*e + 15*a*B*d*e^2 - 3*a*A*e^3))/(4*e^8*(d
+ e*x)^4) - (c^2*(7*B*c*d^2 - 2*A*c*d*e + a*B*e^2))/(e^8*(d + e*x)^3) + (c^3*(7
*B*d - A*e))/(2*e^8*(d + e*x)^2) - (B*c^3)/(e^8*(d + e*x))

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Rubi in Sympy [A]  time = 142.212, size = 337, normalized size = 1.02 \[ - \frac{B c^{3}}{e^{8} \left (d + e x\right )} - \frac{c^{3} \left (A e - 7 B d\right )}{2 e^{8} \left (d + e x\right )^{2}} - \frac{c^{2} \left (- 2 A c d e + B a e^{2} + 7 B c d^{2}\right )}{e^{8} \left (d + e x\right )^{3}} - \frac{c^{2} \left (3 A a e^{3} + 15 A c d^{2} e - 15 B a d e^{2} - 35 B c d^{3}\right )}{4 e^{8} \left (d + e x\right )^{4}} - \frac{c \left (- 12 A a c d e^{3} - 20 A c^{2} d^{3} e + 3 B a^{2} e^{4} + 30 B a c d^{2} e^{2} + 35 B c^{2} d^{4}\right )}{5 e^{8} \left (d + e x\right )^{5}} - \frac{c \left (a e^{2} + c d^{2}\right ) \left (A a e^{3} + 5 A c d^{2} e - 3 B a d e^{2} - 7 B c d^{3}\right )}{2 e^{8} \left (d + e x\right )^{6}} - \frac{\left (a e^{2} + c d^{2}\right )^{2} \left (- 6 A c d e + B a e^{2} + 7 B c d^{2}\right )}{7 e^{8} \left (d + e x\right )^{7}} - \frac{\left (A e - B d\right ) \left (a e^{2} + c d^{2}\right )^{3}}{8 e^{8} \left (d + e x\right )^{8}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((B*x+A)*(c*x**2+a)**3/(e*x+d)**9,x)

[Out]

-B*c**3/(e**8*(d + e*x)) - c**3*(A*e - 7*B*d)/(2*e**8*(d + e*x)**2) - c**2*(-2*A
*c*d*e + B*a*e**2 + 7*B*c*d**2)/(e**8*(d + e*x)**3) - c**2*(3*A*a*e**3 + 15*A*c*
d**2*e - 15*B*a*d*e**2 - 35*B*c*d**3)/(4*e**8*(d + e*x)**4) - c*(-12*A*a*c*d*e**
3 - 20*A*c**2*d**3*e + 3*B*a**2*e**4 + 30*B*a*c*d**2*e**2 + 35*B*c**2*d**4)/(5*e
**8*(d + e*x)**5) - c*(a*e**2 + c*d**2)*(A*a*e**3 + 5*A*c*d**2*e - 3*B*a*d*e**2
- 7*B*c*d**3)/(2*e**8*(d + e*x)**6) - (a*e**2 + c*d**2)**2*(-6*A*c*d*e + B*a*e**
2 + 7*B*c*d**2)/(7*e**8*(d + e*x)**7) - (A*e - B*d)*(a*e**2 + c*d**2)**3/(8*e**8
*(d + e*x)**8)

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Mathematica [A]  time = 0.353226, size = 357, normalized size = 1.08 \[ -\frac{A e \left (35 a^3 e^6+5 a^2 c e^4 \left (d^2+8 d e x+28 e^2 x^2\right )+3 a c^2 e^2 \left (d^4+8 d^3 e x+28 d^2 e^2 x^2+56 d e^3 x^3+70 e^4 x^4\right )+5 c^3 \left (d^6+8 d^5 e x+28 d^4 e^2 x^2+56 d^3 e^3 x^3+70 d^2 e^4 x^4+56 d e^5 x^5+28 e^6 x^6\right )\right )+B \left (5 a^3 e^6 (d+8 e x)+3 a^2 c e^4 \left (d^3+8 d^2 e x+28 d e^2 x^2+56 e^3 x^3\right )+5 a c^2 e^2 \left (d^5+8 d^4 e x+28 d^3 e^2 x^2+56 d^2 e^3 x^3+70 d e^4 x^4+56 e^5 x^5\right )+35 c^3 \left (d^7+8 d^6 e x+28 d^5 e^2 x^2+56 d^4 e^3 x^3+70 d^3 e^4 x^4+56 d^2 e^5 x^5+28 d e^6 x^6+8 e^7 x^7\right )\right )}{280 e^8 (d+e x)^8} \]

Antiderivative was successfully verified.

[In]  Integrate[((A + B*x)*(a + c*x^2)^3)/(d + e*x)^9,x]

[Out]

-(A*e*(35*a^3*e^6 + 5*a^2*c*e^4*(d^2 + 8*d*e*x + 28*e^2*x^2) + 3*a*c^2*e^2*(d^4
+ 8*d^3*e*x + 28*d^2*e^2*x^2 + 56*d*e^3*x^3 + 70*e^4*x^4) + 5*c^3*(d^6 + 8*d^5*e
*x + 28*d^4*e^2*x^2 + 56*d^3*e^3*x^3 + 70*d^2*e^4*x^4 + 56*d*e^5*x^5 + 28*e^6*x^
6)) + B*(5*a^3*e^6*(d + 8*e*x) + 3*a^2*c*e^4*(d^3 + 8*d^2*e*x + 28*d*e^2*x^2 + 5
6*e^3*x^3) + 5*a*c^2*e^2*(d^5 + 8*d^4*e*x + 28*d^3*e^2*x^2 + 56*d^2*e^3*x^3 + 70
*d*e^4*x^4 + 56*e^5*x^5) + 35*c^3*(d^7 + 8*d^6*e*x + 28*d^5*e^2*x^2 + 56*d^4*e^3
*x^3 + 70*d^3*e^4*x^4 + 56*d^2*e^5*x^5 + 28*d*e^6*x^6 + 8*e^7*x^7)))/(280*e^8*(d
+ e*x)^8)

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Maple [A]  time = 0.013, size = 448, normalized size = 1.4 \[ -{\frac{c \left ( A{a}^{2}{e}^{5}+6\,A{d}^{2}ac{e}^{3}+5\,A{d}^{4}{c}^{2}e-3\,Bd{a}^{2}{e}^{4}-10\,aBc{d}^{3}{e}^{2}-7\,B{c}^{2}{d}^{5} \right ) }{2\,{e}^{8} \left ( ex+d \right ) ^{6}}}-{\frac{{c}^{2} \left ( 3\,aA{e}^{3}+15\,Ac{d}^{2}e-15\,aBd{e}^{2}-35\,Bc{d}^{3} \right ) }{4\,{e}^{8} \left ( ex+d \right ) ^{4}}}+{\frac{{c}^{2} \left ( 2\,Acde-aB{e}^{2}-7\,Bc{d}^{2} \right ) }{{e}^{8} \left ( ex+d \right ) ^{3}}}-{\frac{{c}^{3} \left ( Ae-7\,Bd \right ) }{2\,{e}^{8} \left ( ex+d \right ) ^{2}}}+{\frac{c \left ( 12\,Aacd{e}^{3}+20\,A{c}^{2}{d}^{3}e-3\,B{e}^{4}{a}^{2}-30\,Bac{d}^{2}{e}^{2}-35\,B{c}^{2}{d}^{4} \right ) }{5\,{e}^{8} \left ( ex+d \right ) ^{5}}}-{\frac{A{a}^{3}{e}^{7}+3\,A{d}^{2}{a}^{2}c{e}^{5}+3\,A{d}^{4}a{c}^{2}{e}^{3}+A{d}^{6}{c}^{3}e-B{a}^{3}d{e}^{6}-3\,B{a}^{2}c{d}^{3}{e}^{4}-3\,Ba{c}^{2}{d}^{5}{e}^{2}-B{c}^{3}{d}^{7}}{8\,{e}^{8} \left ( ex+d \right ) ^{8}}}-{\frac{-6\,Ad{a}^{2}c{e}^{5}-12\,A{d}^{3}a{c}^{2}{e}^{3}-6\,A{d}^{5}{c}^{3}e+B{a}^{3}{e}^{6}+9\,B{a}^{2}c{d}^{2}{e}^{4}+15\,Ba{c}^{2}{d}^{4}{e}^{2}+7\,B{c}^{3}{d}^{6}}{7\,{e}^{8} \left ( ex+d \right ) ^{7}}}-{\frac{B{c}^{3}}{{e}^{8} \left ( ex+d \right ) }} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((B*x+A)*(c*x^2+a)^3/(e*x+d)^9,x)

[Out]

-1/2*c*(A*a^2*e^5+6*A*a*c*d^2*e^3+5*A*c^2*d^4*e-3*B*a^2*d*e^4-10*B*a*c*d^3*e^2-7
*B*c^2*d^5)/e^8/(e*x+d)^6-1/4*c^2*(3*A*a*e^3+15*A*c*d^2*e-15*B*a*d*e^2-35*B*c*d^
3)/e^8/(e*x+d)^4+c^2*(2*A*c*d*e-B*a*e^2-7*B*c*d^2)/e^8/(e*x+d)^3-1/2*c^3*(A*e-7*
B*d)/e^8/(e*x+d)^2+1/5*c*(12*A*a*c*d*e^3+20*A*c^2*d^3*e-3*B*a^2*e^4-30*B*a*c*d^2
*e^2-35*B*c^2*d^4)/e^8/(e*x+d)^5-1/8*(A*a^3*e^7+3*A*a^2*c*d^2*e^5+3*A*a*c^2*d^4*
e^3+A*c^3*d^6*e-B*a^3*d*e^6-3*B*a^2*c*d^3*e^4-3*B*a*c^2*d^5*e^2-B*c^3*d^7)/e^8/(
e*x+d)^8-1/7*(-6*A*a^2*c*d*e^5-12*A*a*c^2*d^3*e^3-6*A*c^3*d^5*e+B*a^3*e^6+9*B*a^
2*c*d^2*e^4+15*B*a*c^2*d^4*e^2+7*B*c^3*d^6)/e^8/(e*x+d)^7-B*c^3/e^8/(e*x+d)

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Maxima [A]  time = 0.731137, size = 718, normalized size = 2.18 \[ -\frac{280 \, B c^{3} e^{7} x^{7} + 35 \, B c^{3} d^{7} + 5 \, A c^{3} d^{6} e + 5 \, B a c^{2} d^{5} e^{2} + 3 \, A a c^{2} d^{4} e^{3} + 3 \, B a^{2} c d^{3} e^{4} + 5 \, A a^{2} c d^{2} e^{5} + 5 \, B a^{3} d e^{6} + 35 \, A a^{3} e^{7} + 140 \,{\left (7 \, B c^{3} d e^{6} + A c^{3} e^{7}\right )} x^{6} + 280 \,{\left (7 \, B c^{3} d^{2} e^{5} + A c^{3} d e^{6} + B a c^{2} e^{7}\right )} x^{5} + 70 \,{\left (35 \, B c^{3} d^{3} e^{4} + 5 \, A c^{3} d^{2} e^{5} + 5 \, B a c^{2} d e^{6} + 3 \, A a c^{2} e^{7}\right )} x^{4} + 56 \,{\left (35 \, B c^{3} d^{4} e^{3} + 5 \, A c^{3} d^{3} e^{4} + 5 \, B a c^{2} d^{2} e^{5} + 3 \, A a c^{2} d e^{6} + 3 \, B a^{2} c e^{7}\right )} x^{3} + 28 \,{\left (35 \, B c^{3} d^{5} e^{2} + 5 \, A c^{3} d^{4} e^{3} + 5 \, B a c^{2} d^{3} e^{4} + 3 \, A a c^{2} d^{2} e^{5} + 3 \, B a^{2} c d e^{6} + 5 \, A a^{2} c e^{7}\right )} x^{2} + 8 \,{\left (35 \, B c^{3} d^{6} e + 5 \, A c^{3} d^{5} e^{2} + 5 \, B a c^{2} d^{4} e^{3} + 3 \, A a c^{2} d^{3} e^{4} + 3 \, B a^{2} c d^{2} e^{5} + 5 \, A a^{2} c d e^{6} + 5 \, B a^{3} e^{7}\right )} x}{280 \,{\left (e^{16} x^{8} + 8 \, d e^{15} x^{7} + 28 \, d^{2} e^{14} x^{6} + 56 \, d^{3} e^{13} x^{5} + 70 \, d^{4} e^{12} x^{4} + 56 \, d^{5} e^{11} x^{3} + 28 \, d^{6} e^{10} x^{2} + 8 \, d^{7} e^{9} x + d^{8} e^{8}\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((c*x^2 + a)^3*(B*x + A)/(e*x + d)^9,x, algorithm="maxima")

[Out]

-1/280*(280*B*c^3*e^7*x^7 + 35*B*c^3*d^7 + 5*A*c^3*d^6*e + 5*B*a*c^2*d^5*e^2 + 3
*A*a*c^2*d^4*e^3 + 3*B*a^2*c*d^3*e^4 + 5*A*a^2*c*d^2*e^5 + 5*B*a^3*d*e^6 + 35*A*
a^3*e^7 + 140*(7*B*c^3*d*e^6 + A*c^3*e^7)*x^6 + 280*(7*B*c^3*d^2*e^5 + A*c^3*d*e
^6 + B*a*c^2*e^7)*x^5 + 70*(35*B*c^3*d^3*e^4 + 5*A*c^3*d^2*e^5 + 5*B*a*c^2*d*e^6
+ 3*A*a*c^2*e^7)*x^4 + 56*(35*B*c^3*d^4*e^3 + 5*A*c^3*d^3*e^4 + 5*B*a*c^2*d^2*e
^5 + 3*A*a*c^2*d*e^6 + 3*B*a^2*c*e^7)*x^3 + 28*(35*B*c^3*d^5*e^2 + 5*A*c^3*d^4*e
^3 + 5*B*a*c^2*d^3*e^4 + 3*A*a*c^2*d^2*e^5 + 3*B*a^2*c*d*e^6 + 5*A*a^2*c*e^7)*x^
2 + 8*(35*B*c^3*d^6*e + 5*A*c^3*d^5*e^2 + 5*B*a*c^2*d^4*e^3 + 3*A*a*c^2*d^3*e^4
+ 3*B*a^2*c*d^2*e^5 + 5*A*a^2*c*d*e^6 + 5*B*a^3*e^7)*x)/(e^16*x^8 + 8*d*e^15*x^7
+ 28*d^2*e^14*x^6 + 56*d^3*e^13*x^5 + 70*d^4*e^12*x^4 + 56*d^5*e^11*x^3 + 28*d^
6*e^10*x^2 + 8*d^7*e^9*x + d^8*e^8)

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Fricas [A]  time = 0.300489, size = 718, normalized size = 2.18 \[ -\frac{280 \, B c^{3} e^{7} x^{7} + 35 \, B c^{3} d^{7} + 5 \, A c^{3} d^{6} e + 5 \, B a c^{2} d^{5} e^{2} + 3 \, A a c^{2} d^{4} e^{3} + 3 \, B a^{2} c d^{3} e^{4} + 5 \, A a^{2} c d^{2} e^{5} + 5 \, B a^{3} d e^{6} + 35 \, A a^{3} e^{7} + 140 \,{\left (7 \, B c^{3} d e^{6} + A c^{3} e^{7}\right )} x^{6} + 280 \,{\left (7 \, B c^{3} d^{2} e^{5} + A c^{3} d e^{6} + B a c^{2} e^{7}\right )} x^{5} + 70 \,{\left (35 \, B c^{3} d^{3} e^{4} + 5 \, A c^{3} d^{2} e^{5} + 5 \, B a c^{2} d e^{6} + 3 \, A a c^{2} e^{7}\right )} x^{4} + 56 \,{\left (35 \, B c^{3} d^{4} e^{3} + 5 \, A c^{3} d^{3} e^{4} + 5 \, B a c^{2} d^{2} e^{5} + 3 \, A a c^{2} d e^{6} + 3 \, B a^{2} c e^{7}\right )} x^{3} + 28 \,{\left (35 \, B c^{3} d^{5} e^{2} + 5 \, A c^{3} d^{4} e^{3} + 5 \, B a c^{2} d^{3} e^{4} + 3 \, A a c^{2} d^{2} e^{5} + 3 \, B a^{2} c d e^{6} + 5 \, A a^{2} c e^{7}\right )} x^{2} + 8 \,{\left (35 \, B c^{3} d^{6} e + 5 \, A c^{3} d^{5} e^{2} + 5 \, B a c^{2} d^{4} e^{3} + 3 \, A a c^{2} d^{3} e^{4} + 3 \, B a^{2} c d^{2} e^{5} + 5 \, A a^{2} c d e^{6} + 5 \, B a^{3} e^{7}\right )} x}{280 \,{\left (e^{16} x^{8} + 8 \, d e^{15} x^{7} + 28 \, d^{2} e^{14} x^{6} + 56 \, d^{3} e^{13} x^{5} + 70 \, d^{4} e^{12} x^{4} + 56 \, d^{5} e^{11} x^{3} + 28 \, d^{6} e^{10} x^{2} + 8 \, d^{7} e^{9} x + d^{8} e^{8}\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((c*x^2 + a)^3*(B*x + A)/(e*x + d)^9,x, algorithm="fricas")

[Out]

-1/280*(280*B*c^3*e^7*x^7 + 35*B*c^3*d^7 + 5*A*c^3*d^6*e + 5*B*a*c^2*d^5*e^2 + 3
*A*a*c^2*d^4*e^3 + 3*B*a^2*c*d^3*e^4 + 5*A*a^2*c*d^2*e^5 + 5*B*a^3*d*e^6 + 35*A*
a^3*e^7 + 140*(7*B*c^3*d*e^6 + A*c^3*e^7)*x^6 + 280*(7*B*c^3*d^2*e^5 + A*c^3*d*e
^6 + B*a*c^2*e^7)*x^5 + 70*(35*B*c^3*d^3*e^4 + 5*A*c^3*d^2*e^5 + 5*B*a*c^2*d*e^6
+ 3*A*a*c^2*e^7)*x^4 + 56*(35*B*c^3*d^4*e^3 + 5*A*c^3*d^3*e^4 + 5*B*a*c^2*d^2*e
^5 + 3*A*a*c^2*d*e^6 + 3*B*a^2*c*e^7)*x^3 + 28*(35*B*c^3*d^5*e^2 + 5*A*c^3*d^4*e
^3 + 5*B*a*c^2*d^3*e^4 + 3*A*a*c^2*d^2*e^5 + 3*B*a^2*c*d*e^6 + 5*A*a^2*c*e^7)*x^
2 + 8*(35*B*c^3*d^6*e + 5*A*c^3*d^5*e^2 + 5*B*a*c^2*d^4*e^3 + 3*A*a*c^2*d^3*e^4
+ 3*B*a^2*c*d^2*e^5 + 5*A*a^2*c*d*e^6 + 5*B*a^3*e^7)*x)/(e^16*x^8 + 8*d*e^15*x^7
+ 28*d^2*e^14*x^6 + 56*d^3*e^13*x^5 + 70*d^4*e^12*x^4 + 56*d^5*e^11*x^3 + 28*d^
6*e^10*x^2 + 8*d^7*e^9*x + d^8*e^8)

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Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \[ \text{Timed out} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((B*x+A)*(c*x**2+a)**3/(e*x+d)**9,x)

[Out]

Timed out

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GIAC/XCAS [A]  time = 0.275551, size = 617, normalized size = 1.87 \[ -\frac{{\left (280 \, B c^{3} x^{7} e^{7} + 980 \, B c^{3} d x^{6} e^{6} + 1960 \, B c^{3} d^{2} x^{5} e^{5} + 2450 \, B c^{3} d^{3} x^{4} e^{4} + 1960 \, B c^{3} d^{4} x^{3} e^{3} + 980 \, B c^{3} d^{5} x^{2} e^{2} + 280 \, B c^{3} d^{6} x e + 35 \, B c^{3} d^{7} + 140 \, A c^{3} x^{6} e^{7} + 280 \, A c^{3} d x^{5} e^{6} + 350 \, A c^{3} d^{2} x^{4} e^{5} + 280 \, A c^{3} d^{3} x^{3} e^{4} + 140 \, A c^{3} d^{4} x^{2} e^{3} + 40 \, A c^{3} d^{5} x e^{2} + 5 \, A c^{3} d^{6} e + 280 \, B a c^{2} x^{5} e^{7} + 350 \, B a c^{2} d x^{4} e^{6} + 280 \, B a c^{2} d^{2} x^{3} e^{5} + 140 \, B a c^{2} d^{3} x^{2} e^{4} + 40 \, B a c^{2} d^{4} x e^{3} + 5 \, B a c^{2} d^{5} e^{2} + 210 \, A a c^{2} x^{4} e^{7} + 168 \, A a c^{2} d x^{3} e^{6} + 84 \, A a c^{2} d^{2} x^{2} e^{5} + 24 \, A a c^{2} d^{3} x e^{4} + 3 \, A a c^{2} d^{4} e^{3} + 168 \, B a^{2} c x^{3} e^{7} + 84 \, B a^{2} c d x^{2} e^{6} + 24 \, B a^{2} c d^{2} x e^{5} + 3 \, B a^{2} c d^{3} e^{4} + 140 \, A a^{2} c x^{2} e^{7} + 40 \, A a^{2} c d x e^{6} + 5 \, A a^{2} c d^{2} e^{5} + 40 \, B a^{3} x e^{7} + 5 \, B a^{3} d e^{6} + 35 \, A a^{3} e^{7}\right )} e^{\left (-8\right )}}{280 \,{\left (x e + d\right )}^{8}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((c*x^2 + a)^3*(B*x + A)/(e*x + d)^9,x, algorithm="giac")

[Out]

-1/280*(280*B*c^3*x^7*e^7 + 980*B*c^3*d*x^6*e^6 + 1960*B*c^3*d^2*x^5*e^5 + 2450*
B*c^3*d^3*x^4*e^4 + 1960*B*c^3*d^4*x^3*e^3 + 980*B*c^3*d^5*x^2*e^2 + 280*B*c^3*d
^6*x*e + 35*B*c^3*d^7 + 140*A*c^3*x^6*e^7 + 280*A*c^3*d*x^5*e^6 + 350*A*c^3*d^2*
x^4*e^5 + 280*A*c^3*d^3*x^3*e^4 + 140*A*c^3*d^4*x^2*e^3 + 40*A*c^3*d^5*x*e^2 + 5
*A*c^3*d^6*e + 280*B*a*c^2*x^5*e^7 + 350*B*a*c^2*d*x^4*e^6 + 280*B*a*c^2*d^2*x^3
*e^5 + 140*B*a*c^2*d^3*x^2*e^4 + 40*B*a*c^2*d^4*x*e^3 + 5*B*a*c^2*d^5*e^2 + 210*
A*a*c^2*x^4*e^7 + 168*A*a*c^2*d*x^3*e^6 + 84*A*a*c^2*d^2*x^2*e^5 + 24*A*a*c^2*d^
3*x*e^4 + 3*A*a*c^2*d^4*e^3 + 168*B*a^2*c*x^3*e^7 + 84*B*a^2*c*d*x^2*e^6 + 24*B*
a^2*c*d^2*x*e^5 + 3*B*a^2*c*d^3*e^4 + 140*A*a^2*c*x^2*e^7 + 40*A*a^2*c*d*x*e^6 +
5*A*a^2*c*d^2*e^5 + 40*B*a^3*x*e^7 + 5*B*a^3*d*e^6 + 35*A*a^3*e^7)*e^(-8)/(x*e
+ d)^8