3.1324 \(\int \frac{(A+B x) \left (a+c x^2\right )^3}{(d+e x)^6} \, dx\)

Optimal. Leaf size=313 \[ \frac{c \left (4 A c d e \left (3 a e^2+5 c d^2\right )-B \left (3 a^2 e^4+30 a c d^2 e^2+35 c^2 d^4\right )\right )}{2 e^8 (d+e x)^2}+\frac{3 c^2 \log (d+e x) \left (a B e^2-2 A c d e+7 B c d^2\right )}{e^8}+\frac{c^2 \left (-3 a A e^3+15 a B d e^2-15 A c d^2 e+35 B c d^3\right )}{e^8 (d+e x)}-\frac{\left (a e^2+c d^2\right )^2 \left (a B e^2-6 A c d e+7 B c d^2\right )}{4 e^8 (d+e x)^4}+\frac{\left (a e^2+c d^2\right )^3 (B d-A e)}{5 e^8 (d+e x)^5}+\frac{c \left (a e^2+c d^2\right ) \left (-a A e^3+3 a B d e^2-5 A c d^2 e+7 B c d^3\right )}{e^8 (d+e x)^3}-\frac{c^3 x (6 B d-A e)}{e^7}+\frac{B c^3 x^2}{2 e^6} \]

[Out]

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

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Rubi [A]  time = 0.983089, antiderivative size = 313, 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 \[ -\frac{c \left (3 a^2 B e^4-12 a A c d e^3+30 a B c d^2 e^2-20 A c^2 d^3 e+35 B c^2 d^4\right )}{2 e^8 (d+e x)^2}+\frac{3 c^2 \log (d+e x) \left (a B e^2-2 A c d e+7 B c d^2\right )}{e^8}+\frac{c^2 \left (-3 a A e^3+15 a B d e^2-15 A c d^2 e+35 B c d^3\right )}{e^8 (d+e x)}-\frac{\left (a e^2+c d^2\right )^2 \left (a B e^2-6 A c d e+7 B c d^2\right )}{4 e^8 (d+e x)^4}+\frac{\left (a e^2+c d^2\right )^3 (B d-A e)}{5 e^8 (d+e x)^5}+\frac{c \left (a e^2+c d^2\right ) \left (-a A e^3+3 a B d e^2-5 A c d^2 e+7 B c d^3\right )}{e^8 (d+e x)^3}-\frac{c^3 x (6 B d-A e)}{e^7}+\frac{B c^3 x^2}{2 e^6} \]

Antiderivative was successfully verified.

[In]  Int[((A + B*x)*(a + c*x^2)^3)/(d + e*x)^6,x]

[Out]

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

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Rubi in Sympy [F]  time = 0., size = 0, normalized size = 0. \[ \frac{B c^{3} \int x\, dx}{e^{6}} + \frac{3 c^{2} \left (- 2 A c d e + B a e^{2} + 7 B c d^{2}\right ) \log{\left (d + e x \right )}}{e^{8}} - \frac{c^{2} \left (3 A a e^{3} + 15 A c d^{2} e - 15 B a d e^{2} - 35 B c d^{3}\right )}{e^{8} \left (d + e x\right )} - \frac{c \left (- 12 A a c d e^{3} - 20 A c^{2} d^{3} e + 3 B a^{2} e^{4} + 30 B a c d^{2} e^{2} + 35 B c^{2} d^{4}\right )}{2 e^{8} \left (d + e x\right )^{2}} - \frac{c \left (a e^{2} + c d^{2}\right ) \left (A a e^{3} + 5 A c d^{2} e - 3 B a d e^{2} - 7 B c d^{3}\right )}{e^{8} \left (d + e x\right )^{3}} + \frac{\left (A e - 6 B d\right ) \int c^{3}\, dx}{e^{7}} - \frac{\left (a e^{2} + c d^{2}\right )^{2} \left (- 6 A c d e + B a e^{2} + 7 B c d^{2}\right )}{4 e^{8} \left (d + e x\right )^{4}} - \frac{\left (A e - B d\right ) \left (a e^{2} + c d^{2}\right )^{3}}{5 e^{8} \left (d + e x\right )^{5}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((B*x+A)*(c*x**2+a)**3/(e*x+d)**6,x)

[Out]

B*c**3*Integral(x, x)/e**6 + 3*c**2*(-2*A*c*d*e + B*a*e**2 + 7*B*c*d**2)*log(d +
e*x)/e**8 - 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*(d + e*x)) - 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)/(2*e**8*(d + e*x)**2) - 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)/(e**8*(d + e*x)**3) + (A*e - 6*B
*d)*Integral(c**3, x)/e**7 - (a*e**2 + c*d**2)**2*(-6*A*c*d*e + B*a*e**2 + 7*B*c
*d**2)/(4*e**8*(d + e*x)**4) - (A*e - B*d)*(a*e**2 + c*d**2)**3/(5*e**8*(d + e*x
)**5)

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Mathematica [A]  time = 0.387494, size = 388, normalized size = 1.24 \[ \frac{-2 A e \left (2 a^3 e^6+a^2 c e^4 \left (d^2+5 d e x+10 e^2 x^2\right )+6 a c^2 e^2 \left (d^4+5 d^3 e x+10 d^2 e^2 x^2+10 d e^3 x^3+5 e^4 x^4\right )+c^3 \left (87 d^6+375 d^5 e x+600 d^4 e^2 x^2+400 d^3 e^3 x^3+50 d^2 e^4 x^4-50 d e^5 x^5-10 e^6 x^6\right )\right )+B \left (-a^3 e^6 (d+5 e x)-3 a^2 c e^4 \left (d^3+5 d^2 e x+10 d e^2 x^2+10 e^3 x^3\right )+a c^2 d e^2 \left (137 d^4+625 d^3 e x+1100 d^2 e^2 x^2+900 d e^3 x^3+300 e^4 x^4\right )+c^3 \left (459 d^7+1875 d^6 e x+2700 d^5 e^2 x^2+1300 d^4 e^3 x^3-400 d^3 e^4 x^4-500 d^2 e^5 x^5-70 d e^6 x^6+10 e^7 x^7\right )\right )+60 c^2 (d+e x)^5 \log (d+e x) \left (a B e^2-2 A c d e+7 B c d^2\right )}{20 e^8 (d+e x)^5} \]

Antiderivative was successfully verified.

[In]  Integrate[((A + B*x)*(a + c*x^2)^3)/(d + e*x)^6,x]

[Out]

(-2*A*e*(2*a^3*e^6 + a^2*c*e^4*(d^2 + 5*d*e*x + 10*e^2*x^2) + 6*a*c^2*e^2*(d^4 +
5*d^3*e*x + 10*d^2*e^2*x^2 + 10*d*e^3*x^3 + 5*e^4*x^4) + c^3*(87*d^6 + 375*d^5*
e*x + 600*d^4*e^2*x^2 + 400*d^3*e^3*x^3 + 50*d^2*e^4*x^4 - 50*d*e^5*x^5 - 10*e^6
*x^6)) + B*(-(a^3*e^6*(d + 5*e*x)) - 3*a^2*c*e^4*(d^3 + 5*d^2*e*x + 10*d*e^2*x^2
+ 10*e^3*x^3) + a*c^2*d*e^2*(137*d^4 + 625*d^3*e*x + 1100*d^2*e^2*x^2 + 900*d*e
^3*x^3 + 300*e^4*x^4) + c^3*(459*d^7 + 1875*d^6*e*x + 2700*d^5*e^2*x^2 + 1300*d^
4*e^3*x^3 - 400*d^3*e^4*x^4 - 500*d^2*e^5*x^5 - 70*d*e^6*x^6 + 10*e^7*x^7)) + 60
*c^2*(7*B*c*d^2 - 2*A*c*d*e + a*B*e^2)*(d + e*x)^5*Log[d + e*x])/(20*e^8*(d + e*
x)^5)

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Maple [B]  time = 0.02, size = 646, normalized size = 2.1 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((B*x+A)*(c*x^2+a)^3/(e*x+d)^6,x)

[Out]

-3*c^2/e^5/(e*x+d)*a*A+3/2/e^3/(e*x+d)^4*A*a^2*c*d+3/e^5/(e*x+d)^4*A*a*c^2*d^3+1
0*c^2/e^6/(e*x+d)^3*a*B*d^3+6/e^5*c^2/(e*x+d)^2*A*d*a-15/e^6*c^2/(e*x+d)^2*a*B*d
^2-3/5/e^3/(e*x+d)^5*A*d^2*a^2*c-3/5/e^5/(e*x+d)^5*A*d^4*a*c^2+3/5/e^4/(e*x+d)^5
*B*d^3*a^2*c-1/5/e/(e*x+d)^5*A*a^3+c^3/e^6*A*x-1/4/e^2/(e*x+d)^4*B*a^3-3/2/e^4*c
/(e*x+d)^2*B*a^2-35/2/e^8*c^3/(e*x+d)^2*B*d^4-6*c^3/e^7*ln(e*x+d)*A*d+3*c^2/e^6*
ln(e*x+d)*a*B+21*c^3/e^8*ln(e*x+d)*B*d^2-6*c^3/e^7*B*d*x+3*c/e^4/(e*x+d)^3*B*d*a
^2+3/5/e^6/(e*x+d)^5*B*d^5*a*c^2+15*c^2/e^6/(e*x+d)*a*B*d-6*c^2/e^5/(e*x+d)^3*A*
d^2*a-1/5/e^7/(e*x+d)^5*A*c^3*d^6+1/5/e^2/(e*x+d)^5*B*d*a^3-15/4/e^6/(e*x+d)^4*B
*a*c^2*d^4-9/4/e^4/(e*x+d)^4*B*a^2*c*d^2+1/2*B*c^3*x^2/e^6+1/5/e^8/(e*x+d)^5*B*c
^3*d^7-15*c^3/e^7/(e*x+d)*A*d^2+35*c^3/e^8/(e*x+d)*B*d^3+3/2/e^7/(e*x+d)^4*A*c^3
*d^5-7/4/e^8/(e*x+d)^4*B*c^3*d^6-c/e^3/(e*x+d)^3*A*a^2-5*c^3/e^7/(e*x+d)^3*A*d^4
+7*c^3/e^8/(e*x+d)^3*B*d^5+10/e^7*c^3/(e*x+d)^2*A*d^3

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Maxima [A]  time = 0.721432, size = 674, normalized size = 2.15 \[ \frac{459 \, B c^{3} d^{7} - 174 \, A c^{3} d^{6} e + 137 \, B a c^{2} d^{5} e^{2} - 12 \, A a c^{2} d^{4} e^{3} - 3 \, B a^{2} c d^{3} e^{4} - 2 \, A a^{2} c d^{2} e^{5} - B a^{3} d e^{6} - 4 \, A a^{3} e^{7} + 20 \,{\left (35 \, B c^{3} d^{3} e^{4} - 15 \, A c^{3} d^{2} e^{5} + 15 \, B a c^{2} d e^{6} - 3 \, A a c^{2} e^{7}\right )} x^{4} + 10 \,{\left (245 \, B c^{3} d^{4} e^{3} - 100 \, A c^{3} d^{3} e^{4} + 90 \, B a c^{2} d^{2} e^{5} - 12 \, A a c^{2} d e^{6} - 3 \, B a^{2} c e^{7}\right )} x^{3} + 10 \,{\left (329 \, B c^{3} d^{5} e^{2} - 130 \, A c^{3} d^{4} e^{3} + 110 \, B a c^{2} d^{3} e^{4} - 12 \, A a c^{2} d^{2} e^{5} - 3 \, B a^{2} c d e^{6} - 2 \, A a^{2} c e^{7}\right )} x^{2} + 5 \,{\left (399 \, B c^{3} d^{6} e - 154 \, A c^{3} d^{5} e^{2} + 125 \, B a c^{2} d^{4} e^{3} - 12 \, A a c^{2} d^{3} e^{4} - 3 \, B a^{2} c d^{2} e^{5} - 2 \, A a^{2} c d e^{6} - B a^{3} e^{7}\right )} x}{20 \,{\left (e^{13} x^{5} + 5 \, d e^{12} x^{4} + 10 \, d^{2} e^{11} x^{3} + 10 \, d^{3} e^{10} x^{2} + 5 \, d^{4} e^{9} x + d^{5} e^{8}\right )}} + \frac{B c^{3} e x^{2} - 2 \,{\left (6 \, B c^{3} d - A c^{3} e\right )} x}{2 \, e^{7}} + \frac{3 \,{\left (7 \, B c^{3} d^{2} - 2 \, A c^{3} d e + B a c^{2} e^{2}\right )} \log \left (e x + d\right )}{e^{8}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((c*x^2 + a)^3*(B*x + A)/(e*x + d)^6,x, algorithm="maxima")

[Out]

1/20*(459*B*c^3*d^7 - 174*A*c^3*d^6*e + 137*B*a*c^2*d^5*e^2 - 12*A*a*c^2*d^4*e^3
- 3*B*a^2*c*d^3*e^4 - 2*A*a^2*c*d^2*e^5 - B*a^3*d*e^6 - 4*A*a^3*e^7 + 20*(35*B*
c^3*d^3*e^4 - 15*A*c^3*d^2*e^5 + 15*B*a*c^2*d*e^6 - 3*A*a*c^2*e^7)*x^4 + 10*(245
*B*c^3*d^4*e^3 - 100*A*c^3*d^3*e^4 + 90*B*a*c^2*d^2*e^5 - 12*A*a*c^2*d*e^6 - 3*B
*a^2*c*e^7)*x^3 + 10*(329*B*c^3*d^5*e^2 - 130*A*c^3*d^4*e^3 + 110*B*a*c^2*d^3*e^
4 - 12*A*a*c^2*d^2*e^5 - 3*B*a^2*c*d*e^6 - 2*A*a^2*c*e^7)*x^2 + 5*(399*B*c^3*d^6
*e - 154*A*c^3*d^5*e^2 + 125*B*a*c^2*d^4*e^3 - 12*A*a*c^2*d^3*e^4 - 3*B*a^2*c*d^
2*e^5 - 2*A*a^2*c*d*e^6 - B*a^3*e^7)*x)/(e^13*x^5 + 5*d*e^12*x^4 + 10*d^2*e^11*x
^3 + 10*d^3*e^10*x^2 + 5*d^4*e^9*x + d^5*e^8) + 1/2*(B*c^3*e*x^2 - 2*(6*B*c^3*d
- A*c^3*e)*x)/e^7 + 3*(7*B*c^3*d^2 - 2*A*c^3*d*e + B*a*c^2*e^2)*log(e*x + d)/e^8

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Fricas [A]  time = 0.275621, size = 986, normalized size = 3.15 \[ \frac{10 \, B c^{3} e^{7} x^{7} + 459 \, B c^{3} d^{7} - 174 \, A c^{3} d^{6} e + 137 \, B a c^{2} d^{5} e^{2} - 12 \, A a c^{2} d^{4} e^{3} - 3 \, B a^{2} c d^{3} e^{4} - 2 \, A a^{2} c d^{2} e^{5} - B a^{3} d e^{6} - 4 \, A a^{3} e^{7} - 10 \,{\left (7 \, B c^{3} d e^{6} - 2 \, A c^{3} e^{7}\right )} x^{6} - 100 \,{\left (5 \, B c^{3} d^{2} e^{5} - A c^{3} d e^{6}\right )} x^{5} - 20 \,{\left (20 \, B c^{3} d^{3} e^{4} + 5 \, A c^{3} d^{2} e^{5} - 15 \, B a c^{2} d e^{6} + 3 \, A a c^{2} e^{7}\right )} x^{4} + 10 \,{\left (130 \, B c^{3} d^{4} e^{3} - 80 \, A c^{3} d^{3} e^{4} + 90 \, B a c^{2} d^{2} e^{5} - 12 \, A a c^{2} d e^{6} - 3 \, B a^{2} c e^{7}\right )} x^{3} + 10 \,{\left (270 \, B c^{3} d^{5} e^{2} - 120 \, A c^{3} d^{4} e^{3} + 110 \, B a c^{2} d^{3} e^{4} - 12 \, A a c^{2} d^{2} e^{5} - 3 \, B a^{2} c d e^{6} - 2 \, A a^{2} c e^{7}\right )} x^{2} + 5 \,{\left (375 \, B c^{3} d^{6} e - 150 \, A c^{3} d^{5} e^{2} + 125 \, B a c^{2} d^{4} e^{3} - 12 \, A a c^{2} d^{3} e^{4} - 3 \, B a^{2} c d^{2} e^{5} - 2 \, A a^{2} c d e^{6} - B a^{3} e^{7}\right )} x + 60 \,{\left (7 \, B c^{3} d^{7} - 2 \, A c^{3} d^{6} e + B a c^{2} d^{5} e^{2} +{\left (7 \, B c^{3} d^{2} e^{5} - 2 \, A c^{3} d e^{6} + B a c^{2} e^{7}\right )} x^{5} + 5 \,{\left (7 \, B c^{3} d^{3} e^{4} - 2 \, A c^{3} d^{2} e^{5} + B a c^{2} d e^{6}\right )} x^{4} + 10 \,{\left (7 \, B c^{3} d^{4} e^{3} - 2 \, A c^{3} d^{3} e^{4} + B a c^{2} d^{2} e^{5}\right )} x^{3} + 10 \,{\left (7 \, B c^{3} d^{5} e^{2} - 2 \, A c^{3} d^{4} e^{3} + B a c^{2} d^{3} e^{4}\right )} x^{2} + 5 \,{\left (7 \, B c^{3} d^{6} e - 2 \, A c^{3} d^{5} e^{2} + B a c^{2} d^{4} e^{3}\right )} x\right )} \log \left (e x + d\right )}{20 \,{\left (e^{13} x^{5} + 5 \, d e^{12} x^{4} + 10 \, d^{2} e^{11} x^{3} + 10 \, d^{3} e^{10} x^{2} + 5 \, d^{4} e^{9} x + d^{5} e^{8}\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((c*x^2 + a)^3*(B*x + A)/(e*x + d)^6,x, algorithm="fricas")

[Out]

1/20*(10*B*c^3*e^7*x^7 + 459*B*c^3*d^7 - 174*A*c^3*d^6*e + 137*B*a*c^2*d^5*e^2 -
12*A*a*c^2*d^4*e^3 - 3*B*a^2*c*d^3*e^4 - 2*A*a^2*c*d^2*e^5 - B*a^3*d*e^6 - 4*A*
a^3*e^7 - 10*(7*B*c^3*d*e^6 - 2*A*c^3*e^7)*x^6 - 100*(5*B*c^3*d^2*e^5 - A*c^3*d*
e^6)*x^5 - 20*(20*B*c^3*d^3*e^4 + 5*A*c^3*d^2*e^5 - 15*B*a*c^2*d*e^6 + 3*A*a*c^2
*e^7)*x^4 + 10*(130*B*c^3*d^4*e^3 - 80*A*c^3*d^3*e^4 + 90*B*a*c^2*d^2*e^5 - 12*A
*a*c^2*d*e^6 - 3*B*a^2*c*e^7)*x^3 + 10*(270*B*c^3*d^5*e^2 - 120*A*c^3*d^4*e^3 +
110*B*a*c^2*d^3*e^4 - 12*A*a*c^2*d^2*e^5 - 3*B*a^2*c*d*e^6 - 2*A*a^2*c*e^7)*x^2
+ 5*(375*B*c^3*d^6*e - 150*A*c^3*d^5*e^2 + 125*B*a*c^2*d^4*e^3 - 12*A*a*c^2*d^3*
e^4 - 3*B*a^2*c*d^2*e^5 - 2*A*a^2*c*d*e^6 - B*a^3*e^7)*x + 60*(7*B*c^3*d^7 - 2*A
*c^3*d^6*e + B*a*c^2*d^5*e^2 + (7*B*c^3*d^2*e^5 - 2*A*c^3*d*e^6 + B*a*c^2*e^7)*x
^5 + 5*(7*B*c^3*d^3*e^4 - 2*A*c^3*d^2*e^5 + B*a*c^2*d*e^6)*x^4 + 10*(7*B*c^3*d^4
*e^3 - 2*A*c^3*d^3*e^4 + B*a*c^2*d^2*e^5)*x^3 + 10*(7*B*c^3*d^5*e^2 - 2*A*c^3*d^
4*e^3 + B*a*c^2*d^3*e^4)*x^2 + 5*(7*B*c^3*d^6*e - 2*A*c^3*d^5*e^2 + B*a*c^2*d^4*
e^3)*x)*log(e*x + d))/(e^13*x^5 + 5*d*e^12*x^4 + 10*d^2*e^11*x^3 + 10*d^3*e^10*x
^2 + 5*d^4*e^9*x + d^5*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)**6,x)

[Out]

Timed out

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GIAC/XCAS [A]  time = 0.284391, size = 579, normalized size = 1.85 \[ 3 \,{\left (7 \, B c^{3} d^{2} - 2 \, A c^{3} d e + B a c^{2} e^{2}\right )} e^{\left (-8\right )}{\rm ln}\left ({\left | x e + d \right |}\right ) + \frac{1}{2} \,{\left (B c^{3} x^{2} e^{6} - 12 \, B c^{3} d x e^{5} + 2 \, A c^{3} x e^{6}\right )} e^{\left (-12\right )} + \frac{{\left (459 \, B c^{3} d^{7} - 174 \, A c^{3} d^{6} e + 137 \, B a c^{2} d^{5} e^{2} - 12 \, A a c^{2} d^{4} e^{3} - 3 \, B a^{2} c d^{3} e^{4} - 2 \, A a^{2} c d^{2} e^{5} - B a^{3} d e^{6} + 20 \,{\left (35 \, B c^{3} d^{3} e^{4} - 15 \, A c^{3} d^{2} e^{5} + 15 \, B a c^{2} d e^{6} - 3 \, A a c^{2} e^{7}\right )} x^{4} - 4 \, A a^{3} e^{7} + 10 \,{\left (245 \, B c^{3} d^{4} e^{3} - 100 \, A c^{3} d^{3} e^{4} + 90 \, B a c^{2} d^{2} e^{5} - 12 \, A a c^{2} d e^{6} - 3 \, B a^{2} c e^{7}\right )} x^{3} + 10 \,{\left (329 \, B c^{3} d^{5} e^{2} - 130 \, A c^{3} d^{4} e^{3} + 110 \, B a c^{2} d^{3} e^{4} - 12 \, A a c^{2} d^{2} e^{5} - 3 \, B a^{2} c d e^{6} - 2 \, A a^{2} c e^{7}\right )} x^{2} + 5 \,{\left (399 \, B c^{3} d^{6} e - 154 \, A c^{3} d^{5} e^{2} + 125 \, B a c^{2} d^{4} e^{3} - 12 \, A a c^{2} d^{3} e^{4} - 3 \, B a^{2} c d^{2} e^{5} - 2 \, A a^{2} c d e^{6} - B a^{3} e^{7}\right )} x\right )} e^{\left (-8\right )}}{20 \,{\left (x e + d\right )}^{5}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((c*x^2 + a)^3*(B*x + A)/(e*x + d)^6,x, algorithm="giac")

[Out]

3*(7*B*c^3*d^2 - 2*A*c^3*d*e + B*a*c^2*e^2)*e^(-8)*ln(abs(x*e + d)) + 1/2*(B*c^3
*x^2*e^6 - 12*B*c^3*d*x*e^5 + 2*A*c^3*x*e^6)*e^(-12) + 1/20*(459*B*c^3*d^7 - 174
*A*c^3*d^6*e + 137*B*a*c^2*d^5*e^2 - 12*A*a*c^2*d^4*e^3 - 3*B*a^2*c*d^3*e^4 - 2*
A*a^2*c*d^2*e^5 - B*a^3*d*e^6 + 20*(35*B*c^3*d^3*e^4 - 15*A*c^3*d^2*e^5 + 15*B*a
*c^2*d*e^6 - 3*A*a*c^2*e^7)*x^4 - 4*A*a^3*e^7 + 10*(245*B*c^3*d^4*e^3 - 100*A*c^
3*d^3*e^4 + 90*B*a*c^2*d^2*e^5 - 12*A*a*c^2*d*e^6 - 3*B*a^2*c*e^7)*x^3 + 10*(329
*B*c^3*d^5*e^2 - 130*A*c^3*d^4*e^3 + 110*B*a*c^2*d^3*e^4 - 12*A*a*c^2*d^2*e^5 -
3*B*a^2*c*d*e^6 - 2*A*a^2*c*e^7)*x^2 + 5*(399*B*c^3*d^6*e - 154*A*c^3*d^5*e^2 +
125*B*a*c^2*d^4*e^3 - 12*A*a*c^2*d^3*e^4 - 3*B*a^2*c*d^2*e^5 - 2*A*a^2*c*d*e^6 -
B*a^3*e^7)*x)*e^(-8)/(x*e + d)^5