3.2128 \(\int \frac{\left (a+b x+c x^2\right )^3}{(d+e x)^4} \, dx\)

Optimal. Leaf size=251 \[ -\frac{3 \left (-c e (5 b d-a e)+b^2 e^2+5 c^2 d^2\right ) \left (a e^2-b d e+c d^2\right )}{e^7 (d+e x)}-\frac{(2 c d-b e) \log (d+e x) \left (-2 c e (5 b d-3 a e)+b^2 e^2+10 c^2 d^2\right )}{e^7}+\frac{c x \left (-3 c e (4 b d-a e)+3 b^2 e^2+10 c^2 d^2\right )}{e^6}-\frac{\left (a e^2-b d e+c d^2\right )^3}{3 e^7 (d+e x)^3}+\frac{3 (2 c d-b e) \left (a e^2-b d e+c d^2\right )^2}{2 e^7 (d+e x)^2}-\frac{c^2 x^2 (4 c d-3 b e)}{2 e^5}+\frac{c^3 x^3}{3 e^4} \]

[Out]

(c*(10*c^2*d^2 + 3*b^2*e^2 - 3*c*e*(4*b*d - a*e))*x)/e^6 - (c^2*(4*c*d - 3*b*e)*
x^2)/(2*e^5) + (c^3*x^3)/(3*e^4) - (c*d^2 - b*d*e + a*e^2)^3/(3*e^7*(d + e*x)^3)
+ (3*(2*c*d - b*e)*(c*d^2 - b*d*e + a*e^2)^2)/(2*e^7*(d + e*x)^2) - (3*(c*d^2 -
b*d*e + a*e^2)*(5*c^2*d^2 + b^2*e^2 - c*e*(5*b*d - a*e)))/(e^7*(d + e*x)) - ((2
*c*d - b*e)*(10*c^2*d^2 + b^2*e^2 - 2*c*e*(5*b*d - 3*a*e))*Log[d + e*x])/e^7

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Rubi [A]  time = 0.818662, antiderivative size = 251, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 1, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.05 \[ -\frac{3 \left (-c e (5 b d-a e)+b^2 e^2+5 c^2 d^2\right ) \left (a e^2-b d e+c d^2\right )}{e^7 (d+e x)}-\frac{(2 c d-b e) \log (d+e x) \left (-2 c e (5 b d-3 a e)+b^2 e^2+10 c^2 d^2\right )}{e^7}+\frac{c x \left (-3 c e (4 b d-a e)+3 b^2 e^2+10 c^2 d^2\right )}{e^6}-\frac{\left (a e^2-b d e+c d^2\right )^3}{3 e^7 (d+e x)^3}+\frac{3 (2 c d-b e) \left (a e^2-b d e+c d^2\right )^2}{2 e^7 (d+e x)^2}-\frac{c^2 x^2 (4 c d-3 b e)}{2 e^5}+\frac{c^3 x^3}{3 e^4} \]

Antiderivative was successfully verified.

[In]  Int[(a + b*x + c*x^2)^3/(d + e*x)^4,x]

[Out]

(c*(10*c^2*d^2 + 3*b^2*e^2 - 3*c*e*(4*b*d - a*e))*x)/e^6 - (c^2*(4*c*d - 3*b*e)*
x^2)/(2*e^5) + (c^3*x^3)/(3*e^4) - (c*d^2 - b*d*e + a*e^2)^3/(3*e^7*(d + e*x)^3)
+ (3*(2*c*d - b*e)*(c*d^2 - b*d*e + a*e^2)^2)/(2*e^7*(d + e*x)^2) - (3*(c*d^2 -
b*d*e + a*e^2)*(5*c^2*d^2 + b^2*e^2 - c*e*(5*b*d - a*e)))/(e^7*(d + e*x)) - ((2
*c*d - b*e)*(10*c^2*d^2 + b^2*e^2 - 2*c*e*(5*b*d - 3*a*e))*Log[d + e*x])/e^7

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Rubi in Sympy [F]  time = 0., size = 0, normalized size = 0. \[ \frac{c^{3} x^{3}}{3 e^{4}} + \frac{c^{2} \left (3 b e - 4 c d\right ) \int x\, dx}{e^{5}} + \frac{\left (3 a c e^{2} + 3 b^{2} e^{2} - 12 b c d e + 10 c^{2} d^{2}\right ) \int c\, dx}{e^{6}} + \frac{\left (b e - 2 c d\right ) \left (6 a c e^{2} + b^{2} e^{2} - 10 b c d e + 10 c^{2} d^{2}\right ) \log{\left (d + e x \right )}}{e^{7}} - \frac{3 \left (a e^{2} - b d e + c d^{2}\right ) \left (a c e^{2} + b^{2} e^{2} - 5 b c d e + 5 c^{2} d^{2}\right )}{e^{7} \left (d + e x\right )} - \frac{3 \left (b e - 2 c d\right ) \left (a e^{2} - b d e + c d^{2}\right )^{2}}{2 e^{7} \left (d + e x\right )^{2}} - \frac{\left (a e^{2} - b d e + c d^{2}\right )^{3}}{3 e^{7} \left (d + e x\right )^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((c*x**2+b*x+a)**3/(e*x+d)**4,x)

[Out]

c**3*x**3/(3*e**4) + c**2*(3*b*e - 4*c*d)*Integral(x, x)/e**5 + (3*a*c*e**2 + 3*
b**2*e**2 - 12*b*c*d*e + 10*c**2*d**2)*Integral(c, x)/e**6 + (b*e - 2*c*d)*(6*a*
c*e**2 + b**2*e**2 - 10*b*c*d*e + 10*c**2*d**2)*log(d + e*x)/e**7 - 3*(a*e**2 -
b*d*e + c*d**2)*(a*c*e**2 + b**2*e**2 - 5*b*c*d*e + 5*c**2*d**2)/(e**7*(d + e*x)
) - 3*(b*e - 2*c*d)*(a*e**2 - b*d*e + c*d**2)**2/(2*e**7*(d + e*x)**2) - (a*e**2
- b*d*e + c*d**2)**3/(3*e**7*(d + e*x)**3)

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Mathematica [A]  time = 0.228815, size = 260, normalized size = 1.04 \[ \frac{-\frac{18 \left (c e^2 \left (a^2 e^2-6 a b d e+6 b^2 d^2\right )+b^2 e^3 (a e-b d)+2 c^2 d^2 e (3 a e-5 b d)+5 c^3 d^4\right )}{d+e x}+6 c e x \left (3 c e (a e-4 b d)+3 b^2 e^2+10 c^2 d^2\right )-6 (2 c d-b e) \log (d+e x) \left (2 c e (3 a e-5 b d)+b^2 e^2+10 c^2 d^2\right )-\frac{2 \left (e (a e-b d)+c d^2\right )^3}{(d+e x)^3}+\frac{9 (2 c d-b e) \left (e (a e-b d)+c d^2\right )^2}{(d+e x)^2}+3 c^2 e^2 x^2 (3 b e-4 c d)+2 c^3 e^3 x^3}{6 e^7} \]

Antiderivative was successfully verified.

[In]  Integrate[(a + b*x + c*x^2)^3/(d + e*x)^4,x]

[Out]

(6*c*e*(10*c^2*d^2 + 3*b^2*e^2 + 3*c*e*(-4*b*d + a*e))*x + 3*c^2*e^2*(-4*c*d + 3
*b*e)*x^2 + 2*c^3*e^3*x^3 - (2*(c*d^2 + e*(-(b*d) + a*e))^3)/(d + e*x)^3 + (9*(2
*c*d - b*e)*(c*d^2 + e*(-(b*d) + a*e))^2)/(d + e*x)^2 - (18*(5*c^3*d^4 + b^2*e^3
*(-(b*d) + a*e) + 2*c^2*d^2*e*(-5*b*d + 3*a*e) + c*e^2*(6*b^2*d^2 - 6*a*b*d*e +
a^2*e^2)))/(d + e*x) - 6*(2*c*d - b*e)*(10*c^2*d^2 + b^2*e^2 + 2*c*e*(-5*b*d + 3
*a*e))*Log[d + e*x])/(6*e^7)

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Maple [B]  time = 0.019, size = 653, normalized size = 2.6 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((c*x^2+b*x+a)^3/(e*x+d)^4,x)

[Out]

30/e^6*ln(e*x+d)*b*c^2*d^2-1/3/e/(e*x+d)^3*a^3+b^3/e^4*ln(e*x+d)+1/e^2/(e*x+d)^3
*d*a^2*b-1/e^3/(e*x+d)^3*a^2*c*d^2-1/e^3/(e*x+d)^3*d^2*a*b^2-1/e^5/(e*x+d)^3*c^2
*d^4*a-1/e^5/(e*x+d)^3*b^2*c*d^4+1/e^6/(e*x+d)^3*b*c^2*d^5+3/e^3/(e*x+d)^2*a^2*c
*d+3/e^3/(e*x+d)^2*a*b^2*d+6/e^5/(e*x+d)^2*a*c^2*d^3+6/e^5/(e*x+d)^2*b^2*c*d^3+6
/e^4*ln(e*x+d)*a*b*c-12/e^5*ln(e*x+d)*a*c^2*d-12/e^5*ln(e*x+d)*b^2*c*d+30/e^6/(e
*x+d)*d^3*b*c^2-2*c^3*d*x^2/e^5-18/e^5/(e*x+d)*a*c^2*d^2-18/e^5/(e*x+d)*b^2*c*d^
2-12*c^2/e^5*b*d*x-15/2/e^6/(e*x+d)^2*b*c^2*d^4+2/e^4/(e*x+d)^3*d^3*a*b*c-9/e^4/
(e*x+d)^2*a*b*c*d^2+18/e^4/(e*x+d)*a*b*c*d-3/2/e^2/(e*x+d)^2*a^2*b-3/2/e^4/(e*x+
d)^2*b^3*d^2+3/e^7/(e*x+d)^2*c^3*d^5+1/3/e^4/(e*x+d)^3*d^3*b^3-1/3/e^7/(e*x+d)^3
*c^3*d^6-20/e^7*ln(e*x+d)*c^3*d^3-3/e^3/(e*x+d)*a^2*c-3/e^3/(e*x+d)*a*b^2+3/e^4/
(e*x+d)*b^3*d-15/e^7/(e*x+d)*c^3*d^4+3/2*c^2/e^4*x^2*b+3*c^2/e^4*a*x+3*c/e^4*b^2
*x+10*c^3/e^6*d^2*x+1/3*c^3*x^3/e^4

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Maxima [A]  time = 0.80357, size = 581, normalized size = 2.31 \[ -\frac{74 \, c^{3} d^{6} - 141 \, b c^{2} d^{5} e + 3 \, a^{2} b d e^{5} + 2 \, a^{3} e^{6} + 78 \,{\left (b^{2} c + a c^{2}\right )} d^{4} e^{2} - 11 \,{\left (b^{3} + 6 \, a b c\right )} d^{3} e^{3} + 6 \,{\left (a b^{2} + a^{2} c\right )} d^{2} e^{4} + 18 \,{\left (5 \, c^{3} d^{4} e^{2} - 10 \, b c^{2} d^{3} e^{3} + 6 \,{\left (b^{2} c + a c^{2}\right )} d^{2} e^{4} -{\left (b^{3} + 6 \, a b c\right )} d e^{5} +{\left (a b^{2} + a^{2} c\right )} e^{6}\right )} x^{2} + 9 \,{\left (18 \, c^{3} d^{5} e - 35 \, b c^{2} d^{4} e^{2} + a^{2} b e^{6} + 20 \,{\left (b^{2} c + a c^{2}\right )} d^{3} e^{3} - 3 \,{\left (b^{3} + 6 \, a b c\right )} d^{2} e^{4} + 2 \,{\left (a b^{2} + a^{2} c\right )} d e^{5}\right )} x}{6 \,{\left (e^{10} x^{3} + 3 \, d e^{9} x^{2} + 3 \, d^{2} e^{8} x + d^{3} e^{7}\right )}} + \frac{2 \, c^{3} e^{2} x^{3} - 3 \,{\left (4 \, c^{3} d e - 3 \, b c^{2} e^{2}\right )} x^{2} + 6 \,{\left (10 \, c^{3} d^{2} - 12 \, b c^{2} d e + 3 \,{\left (b^{2} c + a c^{2}\right )} e^{2}\right )} x}{6 \, e^{6}} - \frac{{\left (20 \, c^{3} d^{3} - 30 \, b c^{2} d^{2} e + 12 \,{\left (b^{2} c + a c^{2}\right )} d e^{2} -{\left (b^{3} + 6 \, a b c\right )} e^{3}\right )} \log \left (e x + d\right )}{e^{7}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((c*x^2 + b*x + a)^3/(e*x + d)^4,x, algorithm="maxima")

[Out]

-1/6*(74*c^3*d^6 - 141*b*c^2*d^5*e + 3*a^2*b*d*e^5 + 2*a^3*e^6 + 78*(b^2*c + a*c
^2)*d^4*e^2 - 11*(b^3 + 6*a*b*c)*d^3*e^3 + 6*(a*b^2 + a^2*c)*d^2*e^4 + 18*(5*c^3
*d^4*e^2 - 10*b*c^2*d^3*e^3 + 6*(b^2*c + a*c^2)*d^2*e^4 - (b^3 + 6*a*b*c)*d*e^5
+ (a*b^2 + a^2*c)*e^6)*x^2 + 9*(18*c^3*d^5*e - 35*b*c^2*d^4*e^2 + a^2*b*e^6 + 20
*(b^2*c + a*c^2)*d^3*e^3 - 3*(b^3 + 6*a*b*c)*d^2*e^4 + 2*(a*b^2 + a^2*c)*d*e^5)*
x)/(e^10*x^3 + 3*d*e^9*x^2 + 3*d^2*e^8*x + d^3*e^7) + 1/6*(2*c^3*e^2*x^3 - 3*(4*
c^3*d*e - 3*b*c^2*e^2)*x^2 + 6*(10*c^3*d^2 - 12*b*c^2*d*e + 3*(b^2*c + a*c^2)*e^
2)*x)/e^6 - (20*c^3*d^3 - 30*b*c^2*d^2*e + 12*(b^2*c + a*c^2)*d*e^2 - (b^3 + 6*a
*b*c)*e^3)*log(e*x + d)/e^7

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Fricas [A]  time = 0.23181, size = 896, normalized size = 3.57 \[ \frac{2 \, c^{3} e^{6} x^{6} - 74 \, c^{3} d^{6} + 141 \, b c^{2} d^{5} e - 3 \, a^{2} b d e^{5} - 2 \, a^{3} e^{6} - 78 \,{\left (b^{2} c + a c^{2}\right )} d^{4} e^{2} + 11 \,{\left (b^{3} + 6 \, a b c\right )} d^{3} e^{3} - 6 \,{\left (a b^{2} + a^{2} c\right )} d^{2} e^{4} - 3 \,{\left (2 \, c^{3} d e^{5} - 3 \, b c^{2} e^{6}\right )} x^{5} + 3 \,{\left (10 \, c^{3} d^{2} e^{4} - 15 \, b c^{2} d e^{5} + 6 \,{\left (b^{2} c + a c^{2}\right )} e^{6}\right )} x^{4} +{\left (146 \, c^{3} d^{3} e^{3} - 189 \, b c^{2} d^{2} e^{4} + 54 \,{\left (b^{2} c + a c^{2}\right )} d e^{5}\right )} x^{3} + 3 \,{\left (26 \, c^{3} d^{4} e^{2} - 9 \, b c^{2} d^{3} e^{3} - 18 \,{\left (b^{2} c + a c^{2}\right )} d^{2} e^{4} + 6 \,{\left (b^{3} + 6 \, a b c\right )} d e^{5} - 6 \,{\left (a b^{2} + a^{2} c\right )} e^{6}\right )} x^{2} - 3 \,{\left (34 \, c^{3} d^{5} e - 81 \, b c^{2} d^{4} e^{2} + 3 \, a^{2} b e^{6} + 54 \,{\left (b^{2} c + a c^{2}\right )} d^{3} e^{3} - 9 \,{\left (b^{3} + 6 \, a b c\right )} d^{2} e^{4} + 6 \,{\left (a b^{2} + a^{2} c\right )} d e^{5}\right )} x - 6 \,{\left (20 \, c^{3} d^{6} - 30 \, b c^{2} d^{5} e + 12 \,{\left (b^{2} c + a c^{2}\right )} d^{4} e^{2} -{\left (b^{3} + 6 \, a b c\right )} d^{3} e^{3} +{\left (20 \, c^{3} d^{3} e^{3} - 30 \, b c^{2} d^{2} e^{4} + 12 \,{\left (b^{2} c + a c^{2}\right )} d e^{5} -{\left (b^{3} + 6 \, a b c\right )} e^{6}\right )} x^{3} + 3 \,{\left (20 \, c^{3} d^{4} e^{2} - 30 \, b c^{2} d^{3} e^{3} + 12 \,{\left (b^{2} c + a c^{2}\right )} d^{2} e^{4} -{\left (b^{3} + 6 \, a b c\right )} d e^{5}\right )} x^{2} + 3 \,{\left (20 \, c^{3} d^{5} e - 30 \, b c^{2} d^{4} e^{2} + 12 \,{\left (b^{2} c + a c^{2}\right )} d^{3} e^{3} -{\left (b^{3} + 6 \, a b c\right )} d^{2} e^{4}\right )} x\right )} \log \left (e x + d\right )}{6 \,{\left (e^{10} x^{3} + 3 \, d e^{9} x^{2} + 3 \, d^{2} e^{8} x + d^{3} e^{7}\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((c*x^2 + b*x + a)^3/(e*x + d)^4,x, algorithm="fricas")

[Out]

1/6*(2*c^3*e^6*x^6 - 74*c^3*d^6 + 141*b*c^2*d^5*e - 3*a^2*b*d*e^5 - 2*a^3*e^6 -
78*(b^2*c + a*c^2)*d^4*e^2 + 11*(b^3 + 6*a*b*c)*d^3*e^3 - 6*(a*b^2 + a^2*c)*d^2*
e^4 - 3*(2*c^3*d*e^5 - 3*b*c^2*e^6)*x^5 + 3*(10*c^3*d^2*e^4 - 15*b*c^2*d*e^5 + 6
*(b^2*c + a*c^2)*e^6)*x^4 + (146*c^3*d^3*e^3 - 189*b*c^2*d^2*e^4 + 54*(b^2*c + a
*c^2)*d*e^5)*x^3 + 3*(26*c^3*d^4*e^2 - 9*b*c^2*d^3*e^3 - 18*(b^2*c + a*c^2)*d^2*
e^4 + 6*(b^3 + 6*a*b*c)*d*e^5 - 6*(a*b^2 + a^2*c)*e^6)*x^2 - 3*(34*c^3*d^5*e - 8
1*b*c^2*d^4*e^2 + 3*a^2*b*e^6 + 54*(b^2*c + a*c^2)*d^3*e^3 - 9*(b^3 + 6*a*b*c)*d
^2*e^4 + 6*(a*b^2 + a^2*c)*d*e^5)*x - 6*(20*c^3*d^6 - 30*b*c^2*d^5*e + 12*(b^2*c
+ a*c^2)*d^4*e^2 - (b^3 + 6*a*b*c)*d^3*e^3 + (20*c^3*d^3*e^3 - 30*b*c^2*d^2*e^4
+ 12*(b^2*c + a*c^2)*d*e^5 - (b^3 + 6*a*b*c)*e^6)*x^3 + 3*(20*c^3*d^4*e^2 - 30*
b*c^2*d^3*e^3 + 12*(b^2*c + a*c^2)*d^2*e^4 - (b^3 + 6*a*b*c)*d*e^5)*x^2 + 3*(20*
c^3*d^5*e - 30*b*c^2*d^4*e^2 + 12*(b^2*c + a*c^2)*d^3*e^3 - (b^3 + 6*a*b*c)*d^2*
e^4)*x)*log(e*x + d))/(e^10*x^3 + 3*d*e^9*x^2 + 3*d^2*e^8*x + d^3*e^7)

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Sympy [A]  time = 105.651, size = 498, normalized size = 1.98 \[ \frac{c^{3} x^{3}}{3 e^{4}} - \frac{2 a^{3} e^{6} + 3 a^{2} b d e^{5} + 6 a^{2} c d^{2} e^{4} + 6 a b^{2} d^{2} e^{4} - 66 a b c d^{3} e^{3} + 78 a c^{2} d^{4} e^{2} - 11 b^{3} d^{3} e^{3} + 78 b^{2} c d^{4} e^{2} - 141 b c^{2} d^{5} e + 74 c^{3} d^{6} + x^{2} \left (18 a^{2} c e^{6} + 18 a b^{2} e^{6} - 108 a b c d e^{5} + 108 a c^{2} d^{2} e^{4} - 18 b^{3} d e^{5} + 108 b^{2} c d^{2} e^{4} - 180 b c^{2} d^{3} e^{3} + 90 c^{3} d^{4} e^{2}\right ) + x \left (9 a^{2} b e^{6} + 18 a^{2} c d e^{5} + 18 a b^{2} d e^{5} - 162 a b c d^{2} e^{4} + 180 a c^{2} d^{3} e^{3} - 27 b^{3} d^{2} e^{4} + 180 b^{2} c d^{3} e^{3} - 315 b c^{2} d^{4} e^{2} + 162 c^{3} d^{5} e\right )}{6 d^{3} e^{7} + 18 d^{2} e^{8} x + 18 d e^{9} x^{2} + 6 e^{10} x^{3}} + \frac{x^{2} \left (3 b c^{2} e - 4 c^{3} d\right )}{2 e^{5}} + \frac{x \left (3 a c^{2} e^{2} + 3 b^{2} c e^{2} - 12 b c^{2} d e + 10 c^{3} d^{2}\right )}{e^{6}} + \frac{\left (b e - 2 c d\right ) \left (6 a c e^{2} + b^{2} e^{2} - 10 b c d e + 10 c^{2} d^{2}\right ) \log{\left (d + e x \right )}}{e^{7}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((c*x**2+b*x+a)**3/(e*x+d)**4,x)

[Out]

c**3*x**3/(3*e**4) - (2*a**3*e**6 + 3*a**2*b*d*e**5 + 6*a**2*c*d**2*e**4 + 6*a*b
**2*d**2*e**4 - 66*a*b*c*d**3*e**3 + 78*a*c**2*d**4*e**2 - 11*b**3*d**3*e**3 + 7
8*b**2*c*d**4*e**2 - 141*b*c**2*d**5*e + 74*c**3*d**6 + x**2*(18*a**2*c*e**6 + 1
8*a*b**2*e**6 - 108*a*b*c*d*e**5 + 108*a*c**2*d**2*e**4 - 18*b**3*d*e**5 + 108*b
**2*c*d**2*e**4 - 180*b*c**2*d**3*e**3 + 90*c**3*d**4*e**2) + x*(9*a**2*b*e**6 +
18*a**2*c*d*e**5 + 18*a*b**2*d*e**5 - 162*a*b*c*d**2*e**4 + 180*a*c**2*d**3*e**
3 - 27*b**3*d**2*e**4 + 180*b**2*c*d**3*e**3 - 315*b*c**2*d**4*e**2 + 162*c**3*d
**5*e))/(6*d**3*e**7 + 18*d**2*e**8*x + 18*d*e**9*x**2 + 6*e**10*x**3) + x**2*(3
*b*c**2*e - 4*c**3*d)/(2*e**5) + x*(3*a*c**2*e**2 + 3*b**2*c*e**2 - 12*b*c**2*d*
e + 10*c**3*d**2)/e**6 + (b*e - 2*c*d)*(6*a*c*e**2 + b**2*e**2 - 10*b*c*d*e + 10
*c**2*d**2)*log(d + e*x)/e**7

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GIAC/XCAS [A]  time = 0.20638, size = 572, normalized size = 2.28 \[ -{\left (20 \, c^{3} d^{3} - 30 \, b c^{2} d^{2} e + 12 \, b^{2} c d e^{2} + 12 \, a c^{2} d e^{2} - b^{3} e^{3} - 6 \, a b c e^{3}\right )} e^{\left (-7\right )}{\rm ln}\left ({\left | x e + d \right |}\right ) + \frac{1}{6} \,{\left (2 \, c^{3} x^{3} e^{8} - 12 \, c^{3} d x^{2} e^{7} + 60 \, c^{3} d^{2} x e^{6} + 9 \, b c^{2} x^{2} e^{8} - 72 \, b c^{2} d x e^{7} + 18 \, b^{2} c x e^{8} + 18 \, a c^{2} x e^{8}\right )} e^{\left (-12\right )} - \frac{{\left (74 \, c^{3} d^{6} - 141 \, b c^{2} d^{5} e + 78 \, b^{2} c d^{4} e^{2} + 78 \, a c^{2} d^{4} e^{2} - 11 \, b^{3} d^{3} e^{3} - 66 \, a b c d^{3} e^{3} + 6 \, a b^{2} d^{2} e^{4} + 6 \, a^{2} c d^{2} e^{4} + 3 \, a^{2} b d e^{5} + 2 \, a^{3} e^{6} + 18 \,{\left (5 \, c^{3} d^{4} e^{2} - 10 \, b c^{2} d^{3} e^{3} + 6 \, b^{2} c d^{2} e^{4} + 6 \, a c^{2} d^{2} e^{4} - b^{3} d e^{5} - 6 \, a b c d e^{5} + a b^{2} e^{6} + a^{2} c e^{6}\right )} x^{2} + 9 \,{\left (18 \, c^{3} d^{5} e - 35 \, b c^{2} d^{4} e^{2} + 20 \, b^{2} c d^{3} e^{3} + 20 \, a c^{2} d^{3} e^{3} - 3 \, b^{3} d^{2} e^{4} - 18 \, a b c d^{2} e^{4} + 2 \, a b^{2} d e^{5} + 2 \, a^{2} c d e^{5} + a^{2} b e^{6}\right )} x\right )} e^{\left (-7\right )}}{6 \,{\left (x e + d\right )}^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((c*x^2 + b*x + a)^3/(e*x + d)^4,x, algorithm="giac")

[Out]

-(20*c^3*d^3 - 30*b*c^2*d^2*e + 12*b^2*c*d*e^2 + 12*a*c^2*d*e^2 - b^3*e^3 - 6*a*
b*c*e^3)*e^(-7)*ln(abs(x*e + d)) + 1/6*(2*c^3*x^3*e^8 - 12*c^3*d*x^2*e^7 + 60*c^
3*d^2*x*e^6 + 9*b*c^2*x^2*e^8 - 72*b*c^2*d*x*e^7 + 18*b^2*c*x*e^8 + 18*a*c^2*x*e
^8)*e^(-12) - 1/6*(74*c^3*d^6 - 141*b*c^2*d^5*e + 78*b^2*c*d^4*e^2 + 78*a*c^2*d^
4*e^2 - 11*b^3*d^3*e^3 - 66*a*b*c*d^3*e^3 + 6*a*b^2*d^2*e^4 + 6*a^2*c*d^2*e^4 +
3*a^2*b*d*e^5 + 2*a^3*e^6 + 18*(5*c^3*d^4*e^2 - 10*b*c^2*d^3*e^3 + 6*b^2*c*d^2*e
^4 + 6*a*c^2*d^2*e^4 - b^3*d*e^5 - 6*a*b*c*d*e^5 + a*b^2*e^6 + a^2*c*e^6)*x^2 +
9*(18*c^3*d^5*e - 35*b*c^2*d^4*e^2 + 20*b^2*c*d^3*e^3 + 20*a*c^2*d^3*e^3 - 3*b^3
*d^2*e^4 - 18*a*b*c*d^2*e^4 + 2*a*b^2*d*e^5 + 2*a^2*c*d*e^5 + a^2*b*e^6)*x)*e^(-
7)/(x*e + d)^3