3.1325 \(\int \frac{(A+B x) \left (a+c x^2\right )^3}{(d+e x)^7} \, dx\)

Optimal. Leaf size=320 \[ \frac{c \left (4 A c d e \left (3 a e^2+5 c d^2\right )-B \left (3 a^2 e^4+30 a c d^2 e^2+35 c^2 d^4\right )\right )}{3 e^8 (d+e x)^3}-\frac{3 c^2 \left (a B e^2-2 A c d e+7 B c d^2\right )}{e^8 (d+e x)}+\frac{c^2 \left (-3 a A e^3+15 a B d e^2-15 A c d^2 e+35 B c d^3\right )}{2 e^8 (d+e x)^2}-\frac{\left (a e^2+c d^2\right )^2 \left (a B e^2-6 A c d e+7 B c d^2\right )}{5 e^8 (d+e x)^5}+\frac{\left (a e^2+c d^2\right )^3 (B d-A e)}{6 e^8 (d+e x)^6}+\frac{3 c \left (a e^2+c d^2\right ) \left (-a A e^3+3 a B d e^2-5 A c d^2 e+7 B c d^3\right )}{4 e^8 (d+e x)^4}-\frac{c^3 (7 B d-A e) \log (d+e x)}{e^8}+\frac{B c^3 x}{e^7} \]

[Out]

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

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Rubi [A]  time = 0.970963, antiderivative size = 320, 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 \[ -\frac{c \left (3 a^2 B e^4-12 a A c d e^3+30 a B c d^2 e^2-20 A c^2 d^3 e+35 B c^2 d^4\right )}{3 e^8 (d+e x)^3}-\frac{3 c^2 \left (a B e^2-2 A c d e+7 B c d^2\right )}{e^8 (d+e x)}+\frac{c^2 \left (-3 a A e^3+15 a B d e^2-15 A c d^2 e+35 B c d^3\right )}{2 e^8 (d+e x)^2}-\frac{\left (a e^2+c d^2\right )^2 \left (a B e^2-6 A c d e+7 B c d^2\right )}{5 e^8 (d+e x)^5}+\frac{\left (a e^2+c d^2\right )^3 (B d-A e)}{6 e^8 (d+e x)^6}+\frac{3 c \left (a e^2+c d^2\right ) \left (-a A e^3+3 a B d e^2-5 A c d^2 e+7 B c d^3\right )}{4 e^8 (d+e x)^4}-\frac{c^3 (7 B d-A e) \log (d+e x)}{e^8}+\frac{B c^3 x}{e^7} \]

Antiderivative was successfully verified.

[In]  Int[((A + B*x)*(a + c*x^2)^3)/(d + e*x)^7,x]

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((B*x+A)*(c*x**2+a)**3/(e*x+d)**7,x)

[Out]

c**3*Integral(B, x)/e**7 + c**3*(A*e - 7*B*d)*log(d + e*x)/e**8 - 3*c**2*(-2*A*c
*d*e + B*a*e**2 + 7*B*c*d**2)/(e**8*(d + e*x)) - c**2*(3*A*a*e**3 + 15*A*c*d**2*
e - 15*B*a*d*e**2 - 35*B*c*d**3)/(2*e**8*(d + e*x)**2) - c*(-12*A*a*c*d*e**3 - 2
0*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)/(3*e**8*(
d + e*x)**3) - 3*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)/(4*e**8*(d + e*x)**4) - (a*e**2 + c*d**2)**2*(-6*A*c*d*e + B*a*e**2 +
7*B*c*d**2)/(5*e**8*(d + e*x)**5) - (A*e - B*d)*(a*e**2 + c*d**2)**3/(6*e**8*(d
+ e*x)**6)

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Mathematica [A]  time = 0.395736, size = 377, normalized size = 1.18 \[ -\frac{A e \left (10 a^3 e^6+3 a^2 c e^4 \left (d^2+6 d e x+15 e^2 x^2\right )+6 a c^2 e^2 \left (d^4+6 d^3 e x+15 d^2 e^2 x^2+20 d e^3 x^3+15 e^4 x^4\right )-c^3 d \left (147 d^5+822 d^4 e x+1875 d^3 e^2 x^2+2200 d^2 e^3 x^3+1350 d e^4 x^4+360 e^5 x^5\right )\right )+B \left (2 a^3 e^6 (d+6 e x)+3 a^2 c e^4 \left (d^3+6 d^2 e x+15 d e^2 x^2+20 e^3 x^3\right )+30 a c^2 e^2 \left (d^5+6 d^4 e x+15 d^3 e^2 x^2+20 d^2 e^3 x^3+15 d e^4 x^4+6 e^5 x^5\right )+c^3 \left (669 d^7+3594 d^6 e x+7725 d^5 e^2 x^2+8200 d^4 e^3 x^3+4050 d^3 e^4 x^4+360 d^2 e^5 x^5-360 d e^6 x^6-60 e^7 x^7\right )\right )+60 c^3 (d+e x)^6 (7 B d-A e) \log (d+e x)}{60 e^8 (d+e x)^6} \]

Antiderivative was successfully verified.

[In]  Integrate[((A + B*x)*(a + c*x^2)^3)/(d + e*x)^7,x]

[Out]

-(A*e*(10*a^3*e^6 + 3*a^2*c*e^4*(d^2 + 6*d*e*x + 15*e^2*x^2) + 6*a*c^2*e^2*(d^4
+ 6*d^3*e*x + 15*d^2*e^2*x^2 + 20*d*e^3*x^3 + 15*e^4*x^4) - c^3*d*(147*d^5 + 822
*d^4*e*x + 1875*d^3*e^2*x^2 + 2200*d^2*e^3*x^3 + 1350*d*e^4*x^4 + 360*e^5*x^5))
+ B*(2*a^3*e^6*(d + 6*e*x) + 3*a^2*c*e^4*(d^3 + 6*d^2*e*x + 15*d*e^2*x^2 + 20*e^
3*x^3) + 30*a*c^2*e^2*(d^5 + 6*d^4*e*x + 15*d^3*e^2*x^2 + 20*d^2*e^3*x^3 + 15*d*
e^4*x^4 + 6*e^5*x^5) + c^3*(669*d^7 + 3594*d^6*e*x + 7725*d^5*e^2*x^2 + 8200*d^4
*e^3*x^3 + 4050*d^3*e^4*x^4 + 360*d^2*e^5*x^5 - 360*d*e^6*x^6 - 60*e^7*x^7)) + 6
0*c^3*(7*B*d - A*e)*(d + e*x)^6*Log[d + e*x])/(60*e^8*(d + e*x)^6)

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Maple [B]  time = 0.018, size = 656, 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)^7,x)

[Out]

c^3/e^7*ln(e*x+d)*A-1/5/e^2/(e*x+d)^5*B*a^3-1/6/e/(e*x+d)^6*A*a^3-3/4*c/e^3/(e*x
+d)^4*A*a^2-1/2/e^3/(e*x+d)^6*A*d^2*a^2*c-1/2/e^5/(e*x+d)^6*A*d^4*a*c^2+15/2*c^2
/e^6/(e*x+d)^2*a*B*d-1/6/e^7/(e*x+d)^6*A*d^6*c^3+1/6/e^2/(e*x+d)^6*B*d*a^3+B*c^3
*x/e^7-10/e^6*c^2/(e*x+d)^3*B*a*d^2+1/2/e^6/(e*x+d)^6*B*d^5*a*c^2-9/2*c^2/e^5/(e
*x+d)^4*A*d^2*a+9/4*c/e^4/(e*x+d)^4*B*d*a^2+15/2*c^2/e^6/(e*x+d)^4*a*B*d^3+4/e^5
*c^2/(e*x+d)^3*A*a*d+6/5/e^3/(e*x+d)^5*A*a^2*c*d+12/5/e^5/(e*x+d)^5*A*a*c^2*d^3-
9/5/e^4/(e*x+d)^5*B*a^2*c*d^2-3/e^6/(e*x+d)^5*B*a*c^2*d^4+1/2/e^4/(e*x+d)^6*B*d^
3*a^2*c-15/4*c^3/e^7/(e*x+d)^4*A*d^4+21/4*c^3/e^8/(e*x+d)^4*B*d^5-35/3/e^8*c^3/(
e*x+d)^3*B*d^4-7*c^3/e^8*ln(e*x+d)*B*d+6/5/e^7/(e*x+d)^5*A*c^3*d^5-7/5/e^8/(e*x+
d)^5*B*c^3*d^6+20/3/e^7*c^3/(e*x+d)^3*A*d^3-1/e^4*c/(e*x+d)^3*B*a^2-3*c^2/e^6/(e
*x+d)*a*B-21*c^3/e^8/(e*x+d)*B*d^2+35/2*c^3/e^8/(e*x+d)^2*B*d^3+1/6/e^8/(e*x+d)^
6*B*c^3*d^7+6*c^3/e^7/(e*x+d)*A*d-3/2*c^2/e^5/(e*x+d)^2*a*A-15/2*c^3/e^7/(e*x+d)
^2*A*d^2

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Maxima [A]  time = 0.720988, size = 690, normalized size = 2.16 \[ -\frac{669 \, B c^{3} d^{7} - 147 \, A c^{3} d^{6} e + 30 \, B a c^{2} d^{5} e^{2} + 6 \, A a c^{2} d^{4} e^{3} + 3 \, B a^{2} c d^{3} e^{4} + 3 \, A a^{2} c d^{2} e^{5} + 2 \, B a^{3} d e^{6} + 10 \, A a^{3} e^{7} + 180 \,{\left (7 \, B c^{3} d^{2} e^{5} - 2 \, A c^{3} d e^{6} + B a c^{2} e^{7}\right )} x^{5} + 30 \,{\left (175 \, B c^{3} d^{3} e^{4} - 45 \, A c^{3} d^{2} e^{5} + 15 \, B a c^{2} d e^{6} + 3 \, A a c^{2} e^{7}\right )} x^{4} + 20 \,{\left (455 \, B c^{3} d^{4} e^{3} - 110 \, A c^{3} d^{3} e^{4} + 30 \, B a c^{2} d^{2} e^{5} + 6 \, A a c^{2} d e^{6} + 3 \, B a^{2} c e^{7}\right )} x^{3} + 15 \,{\left (539 \, B c^{3} d^{5} e^{2} - 125 \, A c^{3} d^{4} e^{3} + 30 \, B a c^{2} d^{3} e^{4} + 6 \, A a c^{2} d^{2} e^{5} + 3 \, B a^{2} c d e^{6} + 3 \, A a^{2} c e^{7}\right )} x^{2} + 6 \,{\left (609 \, B c^{3} d^{6} e - 137 \, A c^{3} d^{5} e^{2} + 30 \, B a c^{2} d^{4} e^{3} + 6 \, A a c^{2} d^{3} e^{4} + 3 \, B a^{2} c d^{2} e^{5} + 3 \, A a^{2} c d e^{6} + 2 \, B a^{3} e^{7}\right )} x}{60 \,{\left (e^{14} x^{6} + 6 \, d e^{13} x^{5} + 15 \, d^{2} e^{12} x^{4} + 20 \, d^{3} e^{11} x^{3} + 15 \, d^{4} e^{10} x^{2} + 6 \, d^{5} e^{9} x + d^{6} e^{8}\right )}} + \frac{B c^{3} x}{e^{7}} - \frac{{\left (7 \, B c^{3} d - A c^{3} e\right )} \log \left (e x + d\right )}{e^{8}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((c*x^2 + a)^3*(B*x + A)/(e*x + d)^7,x, algorithm="maxima")

[Out]

-1/60*(669*B*c^3*d^7 - 147*A*c^3*d^6*e + 30*B*a*c^2*d^5*e^2 + 6*A*a*c^2*d^4*e^3
+ 3*B*a^2*c*d^3*e^4 + 3*A*a^2*c*d^2*e^5 + 2*B*a^3*d*e^6 + 10*A*a^3*e^7 + 180*(7*
B*c^3*d^2*e^5 - 2*A*c^3*d*e^6 + B*a*c^2*e^7)*x^5 + 30*(175*B*c^3*d^3*e^4 - 45*A*
c^3*d^2*e^5 + 15*B*a*c^2*d*e^6 + 3*A*a*c^2*e^7)*x^4 + 20*(455*B*c^3*d^4*e^3 - 11
0*A*c^3*d^3*e^4 + 30*B*a*c^2*d^2*e^5 + 6*A*a*c^2*d*e^6 + 3*B*a^2*c*e^7)*x^3 + 15
*(539*B*c^3*d^5*e^2 - 125*A*c^3*d^4*e^3 + 30*B*a*c^2*d^3*e^4 + 6*A*a*c^2*d^2*e^5
+ 3*B*a^2*c*d*e^6 + 3*A*a^2*c*e^7)*x^2 + 6*(609*B*c^3*d^6*e - 137*A*c^3*d^5*e^2
+ 30*B*a*c^2*d^4*e^3 + 6*A*a*c^2*d^3*e^4 + 3*B*a^2*c*d^2*e^5 + 3*A*a^2*c*d*e^6
+ 2*B*a^3*e^7)*x)/(e^14*x^6 + 6*d*e^13*x^5 + 15*d^2*e^12*x^4 + 20*d^3*e^11*x^3 +
15*d^4*e^10*x^2 + 6*d^5*e^9*x + d^6*e^8) + B*c^3*x/e^7 - (7*B*c^3*d - A*c^3*e)*
log(e*x + d)/e^8

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Fricas [A]  time = 0.275221, size = 938, normalized size = 2.93 \[ \frac{60 \, B c^{3} e^{7} x^{7} + 360 \, B c^{3} d e^{6} x^{6} - 669 \, B c^{3} d^{7} + 147 \, A c^{3} d^{6} e - 30 \, B a c^{2} d^{5} e^{2} - 6 \, A a c^{2} d^{4} e^{3} - 3 \, B a^{2} c d^{3} e^{4} - 3 \, A a^{2} c d^{2} e^{5} - 2 \, B a^{3} d e^{6} - 10 \, A a^{3} e^{7} - 180 \,{\left (2 \, B c^{3} d^{2} e^{5} - 2 \, A c^{3} d e^{6} + B a c^{2} e^{7}\right )} x^{5} - 90 \,{\left (45 \, B c^{3} d^{3} e^{4} - 15 \, A c^{3} d^{2} e^{5} + 5 \, B a c^{2} d e^{6} + A a c^{2} e^{7}\right )} x^{4} - 20 \,{\left (410 \, B c^{3} d^{4} e^{3} - 110 \, A c^{3} d^{3} e^{4} + 30 \, B a c^{2} d^{2} e^{5} + 6 \, A a c^{2} d e^{6} + 3 \, B a^{2} c e^{7}\right )} x^{3} - 15 \,{\left (515 \, B c^{3} d^{5} e^{2} - 125 \, A c^{3} d^{4} e^{3} + 30 \, B a c^{2} d^{3} e^{4} + 6 \, A a c^{2} d^{2} e^{5} + 3 \, B a^{2} c d e^{6} + 3 \, A a^{2} c e^{7}\right )} x^{2} - 6 \,{\left (599 \, B c^{3} d^{6} e - 137 \, A c^{3} d^{5} e^{2} + 30 \, B a c^{2} d^{4} e^{3} + 6 \, A a c^{2} d^{3} e^{4} + 3 \, B a^{2} c d^{2} e^{5} + 3 \, A a^{2} c d e^{6} + 2 \, B a^{3} e^{7}\right )} x - 60 \,{\left (7 \, B c^{3} d^{7} - A c^{3} d^{6} e +{\left (7 \, B c^{3} d e^{6} - A c^{3} e^{7}\right )} x^{6} + 6 \,{\left (7 \, B c^{3} d^{2} e^{5} - A c^{3} d e^{6}\right )} x^{5} + 15 \,{\left (7 \, B c^{3} d^{3} e^{4} - A c^{3} d^{2} e^{5}\right )} x^{4} + 20 \,{\left (7 \, B c^{3} d^{4} e^{3} - A c^{3} d^{3} e^{4}\right )} x^{3} + 15 \,{\left (7 \, B c^{3} d^{5} e^{2} - A c^{3} d^{4} e^{3}\right )} x^{2} + 6 \,{\left (7 \, B c^{3} d^{6} e - A c^{3} d^{5} e^{2}\right )} x\right )} \log \left (e x + d\right )}{60 \,{\left (e^{14} x^{6} + 6 \, d e^{13} x^{5} + 15 \, d^{2} e^{12} x^{4} + 20 \, d^{3} e^{11} x^{3} + 15 \, d^{4} e^{10} x^{2} + 6 \, d^{5} e^{9} x + d^{6} e^{8}\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((c*x^2 + a)^3*(B*x + A)/(e*x + d)^7,x, algorithm="fricas")

[Out]

1/60*(60*B*c^3*e^7*x^7 + 360*B*c^3*d*e^6*x^6 - 669*B*c^3*d^7 + 147*A*c^3*d^6*e -
30*B*a*c^2*d^5*e^2 - 6*A*a*c^2*d^4*e^3 - 3*B*a^2*c*d^3*e^4 - 3*A*a^2*c*d^2*e^5
- 2*B*a^3*d*e^6 - 10*A*a^3*e^7 - 180*(2*B*c^3*d^2*e^5 - 2*A*c^3*d*e^6 + B*a*c^2*
e^7)*x^5 - 90*(45*B*c^3*d^3*e^4 - 15*A*c^3*d^2*e^5 + 5*B*a*c^2*d*e^6 + A*a*c^2*e
^7)*x^4 - 20*(410*B*c^3*d^4*e^3 - 110*A*c^3*d^3*e^4 + 30*B*a*c^2*d^2*e^5 + 6*A*a
*c^2*d*e^6 + 3*B*a^2*c*e^7)*x^3 - 15*(515*B*c^3*d^5*e^2 - 125*A*c^3*d^4*e^3 + 30
*B*a*c^2*d^3*e^4 + 6*A*a*c^2*d^2*e^5 + 3*B*a^2*c*d*e^6 + 3*A*a^2*c*e^7)*x^2 - 6*
(599*B*c^3*d^6*e - 137*A*c^3*d^5*e^2 + 30*B*a*c^2*d^4*e^3 + 6*A*a*c^2*d^3*e^4 +
3*B*a^2*c*d^2*e^5 + 3*A*a^2*c*d*e^6 + 2*B*a^3*e^7)*x - 60*(7*B*c^3*d^7 - A*c^3*d
^6*e + (7*B*c^3*d*e^6 - A*c^3*e^7)*x^6 + 6*(7*B*c^3*d^2*e^5 - A*c^3*d*e^6)*x^5 +
15*(7*B*c^3*d^3*e^4 - A*c^3*d^2*e^5)*x^4 + 20*(7*B*c^3*d^4*e^3 - A*c^3*d^3*e^4)
*x^3 + 15*(7*B*c^3*d^5*e^2 - A*c^3*d^4*e^3)*x^2 + 6*(7*B*c^3*d^6*e - A*c^3*d^5*e
^2)*x)*log(e*x + d))/(e^14*x^6 + 6*d*e^13*x^5 + 15*d^2*e^12*x^4 + 20*d^3*e^11*x^
3 + 15*d^4*e^10*x^2 + 6*d^5*e^9*x + d^6*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)**7,x)

[Out]

Timed out

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GIAC/XCAS [A]  time = 0.272263, size = 575, normalized size = 1.8 \[ B c^{3} x e^{\left (-7\right )} -{\left (7 \, B c^{3} d - A c^{3} e\right )} e^{\left (-8\right )}{\rm ln}\left ({\left | x e + d \right |}\right ) - \frac{{\left (669 \, B c^{3} d^{7} - 147 \, A c^{3} d^{6} e + 30 \, B a c^{2} d^{5} e^{2} + 6 \, A a c^{2} d^{4} e^{3} + 3 \, B a^{2} c d^{3} e^{4} + 3 \, A a^{2} c d^{2} e^{5} + 180 \,{\left (7 \, B c^{3} d^{2} e^{5} - 2 \, A c^{3} d e^{6} + B a c^{2} e^{7}\right )} x^{5} + 2 \, B a^{3} d e^{6} + 30 \,{\left (175 \, B c^{3} d^{3} e^{4} - 45 \, A c^{3} d^{2} e^{5} + 15 \, B a c^{2} d e^{6} + 3 \, A a c^{2} e^{7}\right )} x^{4} + 10 \, A a^{3} e^{7} + 20 \,{\left (455 \, B c^{3} d^{4} e^{3} - 110 \, A c^{3} d^{3} e^{4} + 30 \, B a c^{2} d^{2} e^{5} + 6 \, A a c^{2} d e^{6} + 3 \, B a^{2} c e^{7}\right )} x^{3} + 15 \,{\left (539 \, B c^{3} d^{5} e^{2} - 125 \, A c^{3} d^{4} e^{3} + 30 \, B a c^{2} d^{3} e^{4} + 6 \, A a c^{2} d^{2} e^{5} + 3 \, B a^{2} c d e^{6} + 3 \, A a^{2} c e^{7}\right )} x^{2} + 6 \,{\left (609 \, B c^{3} d^{6} e - 137 \, A c^{3} d^{5} e^{2} + 30 \, B a c^{2} d^{4} e^{3} + 6 \, A a c^{2} d^{3} e^{4} + 3 \, B a^{2} c d^{2} e^{5} + 3 \, A a^{2} c d e^{6} + 2 \, B a^{3} e^{7}\right )} x\right )} e^{\left (-8\right )}}{60 \,{\left (x e + d\right )}^{6}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((c*x^2 + a)^3*(B*x + A)/(e*x + d)^7,x, algorithm="giac")

[Out]

B*c^3*x*e^(-7) - (7*B*c^3*d - A*c^3*e)*e^(-8)*ln(abs(x*e + d)) - 1/60*(669*B*c^3
*d^7 - 147*A*c^3*d^6*e + 30*B*a*c^2*d^5*e^2 + 6*A*a*c^2*d^4*e^3 + 3*B*a^2*c*d^3*
e^4 + 3*A*a^2*c*d^2*e^5 + 180*(7*B*c^3*d^2*e^5 - 2*A*c^3*d*e^6 + B*a*c^2*e^7)*x^
5 + 2*B*a^3*d*e^6 + 30*(175*B*c^3*d^3*e^4 - 45*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*A*a^3*e^7 + 20*(455*B*c^3*d^4*e^3 - 110*A*c^3*d^3*e^4
+ 30*B*a*c^2*d^2*e^5 + 6*A*a*c^2*d*e^6 + 3*B*a^2*c*e^7)*x^3 + 15*(539*B*c^3*d^5*
e^2 - 125*A*c^3*d^4*e^3 + 30*B*a*c^2*d^3*e^4 + 6*A*a*c^2*d^2*e^5 + 3*B*a^2*c*d*e
^6 + 3*A*a^2*c*e^7)*x^2 + 6*(609*B*c^3*d^6*e - 137*A*c^3*d^5*e^2 + 30*B*a*c^2*d^
4*e^3 + 6*A*a*c^2*d^3*e^4 + 3*B*a^2*c*d^2*e^5 + 3*A*a^2*c*d*e^6 + 2*B*a^3*e^7)*x
)*e^(-8)/(x*e + d)^6