### 3.2138 $$\int \frac{(a+b x+c x^2)^3}{(d+e x)^3} \, dx$$

Optimal. Leaf size=255 $\frac{3 c x^2 \left (-c e (3 b d-a e)+b^2 e^2+2 c^2 d^2\right )}{2 e^5}-\frac{x \left (-9 c^2 d e (2 b d-a e)+3 b c e^2 (3 b d-2 a e)-b^3 e^3+10 c^3 d^3\right )}{e^6}+\frac{3 \log (d+e x) \left (a e^2-b d e+c d^2\right ) \left (-c e (5 b d-a e)+b^2 e^2+5 c^2 d^2\right )}{e^7}+\frac{3 (2 c d-b e) \left (a e^2-b d e+c d^2\right )^2}{e^7 (d+e x)}-\frac{\left (a e^2-b d e+c d^2\right )^3}{2 e^7 (d+e x)^2}-\frac{c^2 x^3 (c d-b e)}{e^4}+\frac{c^3 x^4}{4 e^3}$

[Out]

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

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Rubi [A]  time = 0.343087, antiderivative size = 255, 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, Rules used = {698} $\frac{3 c x^2 \left (-c e (3 b d-a e)+b^2 e^2+2 c^2 d^2\right )}{2 e^5}-\frac{x \left (-9 c^2 d e (2 b d-a e)+3 b c e^2 (3 b d-2 a e)-b^3 e^3+10 c^3 d^3\right )}{e^6}+\frac{3 \log (d+e x) \left (a e^2-b d e+c d^2\right ) \left (-c e (5 b d-a e)+b^2 e^2+5 c^2 d^2\right )}{e^7}+\frac{3 (2 c d-b e) \left (a e^2-b d e+c d^2\right )^2}{e^7 (d+e x)}-\frac{\left (a e^2-b d e+c d^2\right )^3}{2 e^7 (d+e x)^2}-\frac{c^2 x^3 (c d-b e)}{e^4}+\frac{c^3 x^4}{4 e^3}$

Antiderivative was successfully veriﬁed.

[In]

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

[Out]

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

Rule 698

Int[((d_.) + (e_.)*(x_))^(m_)*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> Int[ExpandIntegrand[(d +
e*x)^m*(a + b*x + c*x^2)^p, x], x] /; FreeQ[{a, b, c, d, e, m}, x] && NeQ[b^2 - 4*a*c, 0] && NeQ[c*d^2 - b*d*
e + a*e^2, 0] && NeQ[2*c*d - b*e, 0] && IntegerQ[p] && (GtQ[p, 0] || (EqQ[a, 0] && IntegerQ[m]))

Rubi steps

\begin{align*} \int \frac{\left (a+b x+c x^2\right )^3}{(d+e x)^3} \, dx &=\int \left (\frac{-10 c^3 d^3+b^3 e^3-3 b c e^2 (3 b d-2 a e)+9 c^2 d e (2 b d-a e)}{e^6}+\frac{3 c \left (2 c^2 d^2+b^2 e^2-c e (3 b d-a e)\right ) x}{e^5}-\frac{3 c^2 (c d-b e) x^2}{e^4}+\frac{c^3 x^3}{e^3}+\frac{\left (c d^2-b d e+a e^2\right )^3}{e^6 (d+e x)^3}+\frac{3 (-2 c d+b e) \left (c d^2-b d e+a e^2\right )^2}{e^6 (d+e x)^2}+\frac{3 \left (c d^2-b d e+a e^2\right ) \left (5 c^2 d^2-5 b c d e+b^2 e^2+a c e^2\right )}{e^6 (d+e x)}\right ) \, dx\\ &=-\frac{\left (10 c^3 d^3-b^3 e^3+3 b c e^2 (3 b d-2 a e)-9 c^2 d e (2 b d-a e)\right ) x}{e^6}+\frac{3 c \left (2 c^2 d^2+b^2 e^2-c e (3 b d-a e)\right ) x^2}{2 e^5}-\frac{c^2 (c d-b e) x^3}{e^4}+\frac{c^3 x^4}{4 e^3}-\frac{\left (c d^2-b d e+a e^2\right )^3}{2 e^7 (d+e x)^2}+\frac{3 (2 c d-b e) \left (c d^2-b d e+a e^2\right )^2}{e^7 (d+e x)}+\frac{3 \left (c d^2-b d e+a e^2\right ) \left (5 c^2 d^2+b^2 e^2-c e (5 b d-a e)\right ) \log (d+e x)}{e^7}\\ \end{align*}

Mathematica [A]  time = 0.12979, size = 265, normalized size = 1.04 $\frac{12 \log (d+e x) \left (c e^2 \left (a^2 e^2-6 a b d e+6 b^2 d^2\right )+b^2 e^3 (a e-b d)+2 c^2 d^2 e (3 a e-5 b d)+5 c^3 d^4\right )+6 c e^2 x^2 \left (c e (a e-3 b d)+b^2 e^2+2 c^2 d^2\right )+4 e x \left (9 c^2 d e (2 b d-a e)+3 b c e^2 (2 a e-3 b d)+b^3 e^3-10 c^3 d^3\right )+\frac{12 (2 c d-b e) \left (e (a e-b d)+c d^2\right )^2}{d+e x}-\frac{2 \left (e (a e-b d)+c d^2\right )^3}{(d+e x)^2}+4 c^2 e^3 x^3 (b e-c d)+c^3 e^4 x^4}{4 e^7}$

Antiderivative was successfully veriﬁed.

[In]

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

[Out]

(4*e*(-10*c^3*d^3 + b^3*e^3 + 9*c^2*d*e*(2*b*d - a*e) + 3*b*c*e^2*(-3*b*d + 2*a*e))*x + 6*c*e^2*(2*c^2*d^2 + b
^2*e^2 + c*e*(-3*b*d + a*e))*x^2 + 4*c^2*e^3*(-(c*d) + b*e)*x^3 + c^3*e^4*x^4 - (2*(c*d^2 + e*(-(b*d) + a*e))^
3)/(d + e*x)^2 + (12*(2*c*d - b*e)*(c*d^2 + e*(-(b*d) + a*e))^2)/(d + e*x) + 12*(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))*Log[d + e*x])/(4*e^7)

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Maple [B]  time = 0.053, size = 624, normalized size = 2.5 \begin{align*} \text{result too large to display} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [A]  time = 1.05871, size = 563, normalized size = 2.21 \begin{align*} \frac{11 \, c^{3} d^{6} - 27 \, b c^{2} d^{5} e - 3 \, a^{2} b d e^{5} - a^{3} e^{6} + 21 \,{\left (b^{2} c + a c^{2}\right )} d^{4} e^{2} - 5 \,{\left (b^{3} + 6 \, a b c\right )} d^{3} e^{3} + 9 \,{\left (a b^{2} + a^{2} c\right )} d^{2} e^{4} + 6 \,{\left (2 \, c^{3} d^{5} e - 5 \, b c^{2} d^{4} e^{2} - a^{2} b e^{6} + 4 \,{\left (b^{2} c + a c^{2}\right )} d^{3} e^{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}{2 \,{\left (e^{9} x^{2} + 2 \, d e^{8} x + d^{2} e^{7}\right )}} + \frac{c^{3} e^{3} x^{4} - 4 \,{\left (c^{3} d e^{2} - b c^{2} e^{3}\right )} x^{3} + 6 \,{\left (2 \, c^{3} d^{2} e - 3 \, b c^{2} d e^{2} +{\left (b^{2} c + a c^{2}\right )} e^{3}\right )} x^{2} - 4 \,{\left (10 \, c^{3} d^{3} - 18 \, b c^{2} d^{2} e + 9 \,{\left (b^{2} c + a c^{2}\right )} d e^{2} -{\left (b^{3} + 6 \, a b c\right )} e^{3}\right )} x}{4 \, e^{6}} + \frac{3 \,{\left (5 \, c^{3} d^{4} - 10 \, b c^{2} d^{3} e + 6 \,{\left (b^{2} c + a c^{2}\right )} d^{2} e^{2} -{\left (b^{3} + 6 \, a b c\right )} d e^{3} +{\left (a b^{2} + a^{2} c\right )} e^{4}\right )} \log \left (e x + d\right )}{e^{7}} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/2*(11*c^3*d^6 - 27*b*c^2*d^5*e - 3*a^2*b*d*e^5 - a^3*e^6 + 21*(b^2*c + a*c^2)*d^4*e^2 - 5*(b^3 + 6*a*b*c)*d^
3*e^3 + 9*(a*b^2 + a^2*c)*d^2*e^4 + 6*(2*c^3*d^5*e - 5*b*c^2*d^4*e^2 - a^2*b*e^6 + 4*(b^2*c + a*c^2)*d^3*e^3 -
(b^3 + 6*a*b*c)*d^2*e^4 + 2*(a*b^2 + a^2*c)*d*e^5)*x)/(e^9*x^2 + 2*d*e^8*x + d^2*e^7) + 1/4*(c^3*e^3*x^4 - 4*
(c^3*d*e^2 - b*c^2*e^3)*x^3 + 6*(2*c^3*d^2*e - 3*b*c^2*d*e^2 + (b^2*c + a*c^2)*e^3)*x^2 - 4*(10*c^3*d^3 - 18*b
*c^2*d^2*e + 9*(b^2*c + a*c^2)*d*e^2 - (b^3 + 6*a*b*c)*e^3)*x)/e^6 + 3*(5*c^3*d^4 - 10*b*c^2*d^3*e + 6*(b^2*c
+ a*c^2)*d^2*e^2 - (b^3 + 6*a*b*c)*d*e^3 + (a*b^2 + a^2*c)*e^4)*log(e*x + d)/e^7

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Fricas [B]  time = 2.06215, size = 1318, normalized size = 5.17 \begin{align*} \frac{c^{3} e^{6} x^{6} + 22 \, c^{3} d^{6} - 54 \, b c^{2} d^{5} e - 6 \, a^{2} b d e^{5} - 2 \, a^{3} e^{6} + 42 \,{\left (b^{2} c + a c^{2}\right )} d^{4} e^{2} - 10 \,{\left (b^{3} + 6 \, a b c\right )} d^{3} e^{3} + 18 \,{\left (a b^{2} + a^{2} c\right )} d^{2} e^{4} - 2 \,{\left (c^{3} d e^{5} - 2 \, b c^{2} e^{6}\right )} x^{5} +{\left (5 \, c^{3} d^{2} e^{4} - 10 \, b c^{2} d e^{5} + 6 \,{\left (b^{2} c + a c^{2}\right )} e^{6}\right )} x^{4} - 4 \,{\left (5 \, c^{3} d^{3} e^{3} - 10 \, b c^{2} d^{2} e^{4} + 6 \,{\left (b^{2} c + a c^{2}\right )} d e^{5} -{\left (b^{3} + 6 \, a b c\right )} e^{6}\right )} x^{3} - 2 \,{\left (34 \, c^{3} d^{4} e^{2} - 63 \, b c^{2} d^{3} e^{3} + 33 \,{\left (b^{2} c + a c^{2}\right )} d^{2} e^{4} - 4 \,{\left (b^{3} + 6 \, a b c\right )} d e^{5}\right )} x^{2} - 4 \,{\left (4 \, c^{3} d^{5} e - 3 \, b c^{2} d^{4} e^{2} + 3 \, a^{2} b e^{6} - 3 \,{\left (b^{2} c + a c^{2}\right )} d^{3} e^{3} + 2 \,{\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 + 12 \,{\left (5 \, c^{3} d^{6} - 10 \, b c^{2} d^{5} e + 6 \,{\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 (a b^{2} + a^{2} c\right )} d^{2} e^{4} +{\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} + 2 \,{\left (5 \, c^{3} d^{5} e - 10 \, b c^{2} d^{4} e^{2} + 6 \,{\left (b^{2} c + a c^{2}\right )} d^{3} e^{3} -{\left (b^{3} + 6 \, a b c\right )} d^{2} e^{4} +{\left (a b^{2} + a^{2} c\right )} d e^{5}\right )} x\right )} \log \left (e x + d\right )}{4 \,{\left (e^{9} x^{2} + 2 \, d e^{8} x + d^{2} e^{7}\right )}} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/4*(c^3*e^6*x^6 + 22*c^3*d^6 - 54*b*c^2*d^5*e - 6*a^2*b*d*e^5 - 2*a^3*e^6 + 42*(b^2*c + a*c^2)*d^4*e^2 - 10*(
b^3 + 6*a*b*c)*d^3*e^3 + 18*(a*b^2 + a^2*c)*d^2*e^4 - 2*(c^3*d*e^5 - 2*b*c^2*e^6)*x^5 + (5*c^3*d^2*e^4 - 10*b*
c^2*d*e^5 + 6*(b^2*c + a*c^2)*e^6)*x^4 - 4*(5*c^3*d^3*e^3 - 10*b*c^2*d^2*e^4 + 6*(b^2*c + a*c^2)*d*e^5 - (b^3
+ 6*a*b*c)*e^6)*x^3 - 2*(34*c^3*d^4*e^2 - 63*b*c^2*d^3*e^3 + 33*(b^2*c + a*c^2)*d^2*e^4 - 4*(b^3 + 6*a*b*c)*d*
e^5)*x^2 - 4*(4*c^3*d^5*e - 3*b*c^2*d^4*e^2 + 3*a^2*b*e^6 - 3*(b^2*c + a*c^2)*d^3*e^3 + 2*(b^3 + 6*a*b*c)*d^2*
e^4 - 6*(a*b^2 + a^2*c)*d*e^5)*x + 12*(5*c^3*d^6 - 10*b*c^2*d^5*e + 6*(b^2*c + a*c^2)*d^4*e^2 - (b^3 + 6*a*b*c
)*d^3*e^3 + (a*b^2 + a^2*c)*d^2*e^4 + (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 + 2*(5*c^3*d^5*e - 10*b*c^2*d^4*e^2 + 6*(b^2*c + a*c^2)*d^3*e^3 - (b^
3 + 6*a*b*c)*d^2*e^4 + (a*b^2 + a^2*c)*d*e^5)*x)*log(e*x + d))/(e^9*x^2 + 2*d*e^8*x + d^2*e^7)

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Sympy [A]  time = 9.01999, size = 457, normalized size = 1.79 \begin{align*} \frac{c^{3} x^{4}}{4 e^{3}} - \frac{a^{3} e^{6} + 3 a^{2} b d e^{5} - 9 a^{2} c d^{2} e^{4} - 9 a b^{2} d^{2} e^{4} + 30 a b c d^{3} e^{3} - 21 a c^{2} d^{4} e^{2} + 5 b^{3} d^{3} e^{3} - 21 b^{2} c d^{4} e^{2} + 27 b c^{2} d^{5} e - 11 c^{3} d^{6} + x \left (6 a^{2} b e^{6} - 12 a^{2} c d e^{5} - 12 a b^{2} d e^{5} + 36 a b c d^{2} e^{4} - 24 a c^{2} d^{3} e^{3} + 6 b^{3} d^{2} e^{4} - 24 b^{2} c d^{3} e^{3} + 30 b c^{2} d^{4} e^{2} - 12 c^{3} d^{5} e\right )}{2 d^{2} e^{7} + 4 d e^{8} x + 2 e^{9} x^{2}} + \frac{x^{3} \left (b c^{2} e - c^{3} d\right )}{e^{4}} + \frac{x^{2} \left (3 a c^{2} e^{2} + 3 b^{2} c e^{2} - 9 b c^{2} d e + 6 c^{3} d^{2}\right )}{2 e^{5}} + \frac{x \left (6 a b c e^{3} - 9 a c^{2} d e^{2} + b^{3} e^{3} - 9 b^{2} c d e^{2} + 18 b c^{2} d^{2} e - 10 c^{3} d^{3}\right )}{e^{6}} + \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 ) \log{\left (d + e x \right )}}{e^{7}} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

integrate((c*x**2+b*x+a)**3/(e*x+d)**3,x)

[Out]

c**3*x**4/(4*e**3) - (a**3*e**6 + 3*a**2*b*d*e**5 - 9*a**2*c*d**2*e**4 - 9*a*b**2*d**2*e**4 + 30*a*b*c*d**3*e*
*3 - 21*a*c**2*d**4*e**2 + 5*b**3*d**3*e**3 - 21*b**2*c*d**4*e**2 + 27*b*c**2*d**5*e - 11*c**3*d**6 + x*(6*a**
2*b*e**6 - 12*a**2*c*d*e**5 - 12*a*b**2*d*e**5 + 36*a*b*c*d**2*e**4 - 24*a*c**2*d**3*e**3 + 6*b**3*d**2*e**4 -
24*b**2*c*d**3*e**3 + 30*b*c**2*d**4*e**2 - 12*c**3*d**5*e))/(2*d**2*e**7 + 4*d*e**8*x + 2*e**9*x**2) + x**3*
(b*c**2*e - c**3*d)/e**4 + x**2*(3*a*c**2*e**2 + 3*b**2*c*e**2 - 9*b*c**2*d*e + 6*c**3*d**2)/(2*e**5) + x*(6*a
*b*c*e**3 - 9*a*c**2*d*e**2 + b**3*e**3 - 9*b**2*c*d*e**2 + 18*b*c**2*d**2*e - 10*c**3*d**3)/e**6 + 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)*log(d + e*x)/e**7

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Giac [A]  time = 1.10103, size = 583, normalized size = 2.29 \begin{align*} 3 \,{\left (5 \, c^{3} d^{4} - 10 \, b c^{2} d^{3} e + 6 \, b^{2} c d^{2} e^{2} + 6 \, a c^{2} d^{2} e^{2} - b^{3} d e^{3} - 6 \, a b c d e^{3} + a b^{2} e^{4} + a^{2} c e^{4}\right )} e^{\left (-7\right )} \log \left ({\left | x e + d \right |}\right ) + \frac{1}{4} \,{\left (c^{3} x^{4} e^{9} - 4 \, c^{3} d x^{3} e^{8} + 12 \, c^{3} d^{2} x^{2} e^{7} - 40 \, c^{3} d^{3} x e^{6} + 4 \, b c^{2} x^{3} e^{9} - 18 \, b c^{2} d x^{2} e^{8} + 72 \, b c^{2} d^{2} x e^{7} + 6 \, b^{2} c x^{2} e^{9} + 6 \, a c^{2} x^{2} e^{9} - 36 \, b^{2} c d x e^{8} - 36 \, a c^{2} d x e^{8} + 4 \, b^{3} x e^{9} + 24 \, a b c x e^{9}\right )} e^{\left (-12\right )} + \frac{{\left (11 \, c^{3} d^{6} - 27 \, b c^{2} d^{5} e + 21 \, b^{2} c d^{4} e^{2} + 21 \, a c^{2} d^{4} e^{2} - 5 \, b^{3} d^{3} e^{3} - 30 \, a b c d^{3} e^{3} + 9 \, a b^{2} d^{2} e^{4} + 9 \, a^{2} c d^{2} e^{4} - 3 \, a^{2} b d e^{5} - a^{3} e^{6} + 6 \,{\left (2 \, c^{3} d^{5} e - 5 \, b c^{2} d^{4} e^{2} + 4 \, b^{2} c d^{3} e^{3} + 4 \, a c^{2} d^{3} e^{3} - b^{3} d^{2} e^{4} - 6 \, 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 )}}{2 \,{\left (x e + d\right )}^{2}} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

3*(5*c^3*d^4 - 10*b*c^2*d^3*e + 6*b^2*c*d^2*e^2 + 6*a*c^2*d^2*e^2 - b^3*d*e^3 - 6*a*b*c*d*e^3 + a*b^2*e^4 + a^
2*c*e^4)*e^(-7)*log(abs(x*e + d)) + 1/4*(c^3*x^4*e^9 - 4*c^3*d*x^3*e^8 + 12*c^3*d^2*x^2*e^7 - 40*c^3*d^3*x*e^6
+ 4*b*c^2*x^3*e^9 - 18*b*c^2*d*x^2*e^8 + 72*b*c^2*d^2*x*e^7 + 6*b^2*c*x^2*e^9 + 6*a*c^2*x^2*e^9 - 36*b^2*c*d*
x*e^8 - 36*a*c^2*d*x*e^8 + 4*b^3*x*e^9 + 24*a*b*c*x*e^9)*e^(-12) + 1/2*(11*c^3*d^6 - 27*b*c^2*d^5*e + 21*b^2*c
*d^4*e^2 + 21*a*c^2*d^4*e^2 - 5*b^3*d^3*e^3 - 30*a*b*c*d^3*e^3 + 9*a*b^2*d^2*e^4 + 9*a^2*c*d^2*e^4 - 3*a^2*b*d
*e^5 - a^3*e^6 + 6*(2*c^3*d^5*e - 5*b*c^2*d^4*e^2 + 4*b^2*c*d^3*e^3 + 4*a*c^2*d^3*e^3 - b^3*d^2*e^4 - 6*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)^2