### 3.2136 $$\int \frac{(a+b x+c x^2)^3}{d+e x} \, dx$$

Optimal. Leaf size=260 $\frac{3 c (d+e x)^4 \left (-c e (5 b d-a e)+b^2 e^2+5 c^2 d^2\right )}{4 e^7}-\frac{(d+e x)^3 (2 c d-b e) \left (-2 c e (5 b d-3 a e)+b^2 e^2+10 c^2 d^2\right )}{3 e^7}+\frac{3 (d+e x)^2 \left (a e^2-b d e+c d^2\right ) \left (-c e (5 b d-a e)+b^2 e^2+5 c^2 d^2\right )}{2 e^7}-\frac{3 x (2 c d-b e) \left (a e^2-b d e+c d^2\right )^2}{e^6}+\frac{\log (d+e x) \left (a e^2-b d e+c d^2\right )^3}{e^7}-\frac{3 c^2 (d+e x)^5 (2 c d-b e)}{5 e^7}+\frac{c^3 (d+e x)^6}{6 e^7}$

[Out]

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

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Rubi [A]  time = 0.32271, antiderivative size = 260, 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} $\frac{3 c (d+e x)^4 \left (-c e (5 b d-a e)+b^2 e^2+5 c^2 d^2\right )}{4 e^7}-\frac{(d+e x)^3 (2 c d-b e) \left (-2 c e (5 b d-3 a e)+b^2 e^2+10 c^2 d^2\right )}{3 e^7}+\frac{3 (d+e x)^2 \left (a e^2-b d e+c d^2\right ) \left (-c e (5 b d-a e)+b^2 e^2+5 c^2 d^2\right )}{2 e^7}-\frac{3 x (2 c d-b e) \left (a e^2-b d e+c d^2\right )^2}{e^6}+\frac{\log (d+e x) \left (a e^2-b d e+c d^2\right )^3}{e^7}-\frac{3 c^2 (d+e x)^5 (2 c d-b e)}{5 e^7}+\frac{c^3 (d+e x)^6}{6 e^7}$

Antiderivative was successfully veriﬁed.

[In]

Int[(a + b*x + c*x^2)^3/(d + e*x),x]

[Out]

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

Mathematica [A]  time = 0.159707, size = 308, normalized size = 1.18 $\frac{e x \left (15 c e^2 \left (6 a^2 e^2 (e x-2 d)+4 a b e \left (6 d^2-3 d e x+2 e^2 x^2\right )+b^2 \left (6 d^2 e x-12 d^3-4 d e^2 x^2+3 e^3 x^3\right )\right )+10 b e^3 \left (18 a^2 e^2+9 a b e (e x-2 d)+b^2 \left (6 d^2-3 d e x+2 e^2 x^2\right )\right )+3 c^2 e \left (5 a e \left (6 d^2 e x-12 d^3-4 d e^2 x^2+3 e^3 x^3\right )+b \left (20 d^2 e^2 x^2-30 d^3 e x+60 d^4-15 d e^3 x^3+12 e^4 x^4\right )\right )+c^3 \left (-20 d^3 e^2 x^2+15 d^2 e^3 x^3+30 d^4 e x-60 d^5-12 d e^4 x^4+10 e^5 x^5\right )\right )+60 \log (d+e x) \left (e (a e-b d)+c d^2\right )^3}{60 e^7}$

Antiderivative was successfully veriﬁed.

[In]

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

[Out]

(e*x*(c^3*(-60*d^5 + 30*d^4*e*x - 20*d^3*e^2*x^2 + 15*d^2*e^3*x^3 - 12*d*e^4*x^4 + 10*e^5*x^5) + 10*b*e^3*(18*
a^2*e^2 + 9*a*b*e*(-2*d + e*x) + b^2*(6*d^2 - 3*d*e*x + 2*e^2*x^2)) + 15*c*e^2*(6*a^2*e^2*(-2*d + e*x) + 4*a*b
*e*(6*d^2 - 3*d*e*x + 2*e^2*x^2) + b^2*(-12*d^3 + 6*d^2*e*x - 4*d*e^2*x^2 + 3*e^3*x^3)) + 3*c^2*e*(5*a*e*(-12*
d^3 + 6*d^2*e*x - 4*d*e^2*x^2 + 3*e^3*x^3) + b*(60*d^4 - 30*d^3*e*x + 20*d^2*e^2*x^2 - 15*d*e^3*x^3 + 12*e^4*x
^4))) + 60*(c*d^2 + e*(-(b*d) + a*e))^3*Log[d + e*x])/(60*e^7)

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Maple [B]  time = 0.044, size = 546, normalized size = 2.1 \begin{align*}{\frac{3\,b{x}^{5}{c}^{2}}{5\,e}}+{\frac{{x}^{2}{c}^{3}{d}^{4}}{2\,{e}^{5}}}+6\,{\frac{abc{d}^{2}x}{{e}^{3}}}-3\,{\frac{ab{x}^{2}cd}{{e}^{2}}}-6\,{\frac{\ln \left ( ex+d \right ) abc{d}^{3}}{{e}^{4}}}+3\,{\frac{b{a}^{2}x}{e}}-{\frac{\ln \left ( ex+d \right ){b}^{3}{d}^{3}}{{e}^{4}}}+{\frac{\ln \left ( ex+d \right ){c}^{3}{d}^{6}}{{e}^{7}}}+{\frac{{c}^{3}{x}^{6}}{6\,e}}+{\frac{3\,{a}^{2}{x}^{2}c}{2\,e}}+{\frac{{x}^{4}{c}^{3}{d}^{2}}{4\,{e}^{3}}}-{\frac{{x}^{2}{b}^{3}d}{2\,{e}^{2}}}+{\frac{3\,{x}^{4}{b}^{2}c}{4\,e}}+{\frac{{b}^{3}{d}^{2}x}{{e}^{3}}}+{\frac{3\,a{b}^{2}{x}^{2}}{2\,e}}-3\,{\frac{\ln \left ( ex+d \right ){a}^{2}bd}{{e}^{2}}}+2\,{\frac{a{x}^{3}bc}{e}}+{\frac{3\,a{x}^{4}{c}^{2}}{4\,e}}+{\frac{\ln \left ( ex+d \right ){a}^{3}}{e}}+{\frac{{x}^{3}{b}^{3}}{3\,e}}-3\,{\frac{a{b}^{2}dx}{{e}^{2}}}-3\,{\frac{a{c}^{2}{d}^{3}x}{{e}^{4}}}-3\,{\frac{{b}^{2}c{d}^{3}x}{{e}^{4}}}-{\frac{3\,b{x}^{2}{c}^{2}{d}^{3}}{2\,{e}^{4}}}-3\,{\frac{cd{a}^{2}x}{{e}^{2}}}-{\frac{{x}^{3}{c}^{3}{d}^{3}}{3\,{e}^{4}}}-{\frac{{c}^{3}{d}^{5}x}{{e}^{6}}}-{\frac{3\,b{x}^{4}{c}^{2}d}{4\,{e}^{2}}}+3\,{\frac{\ln \left ( ex+d \right ){a}^{2}c{d}^{2}}{{e}^{3}}}-{\frac{{b}^{2}c{x}^{3}d}{{e}^{2}}}+{\frac{3\,a{x}^{2}{c}^{2}{d}^{2}}{2\,{e}^{3}}}+{\frac{3\,{b}^{2}{x}^{2}c{d}^{2}}{2\,{e}^{3}}}+3\,{\frac{{d}^{4}b{c}^{2}x}{{e}^{5}}}-{\frac{a{x}^{3}{c}^{2}d}{{e}^{2}}}+{\frac{b{x}^{3}{c}^{2}{d}^{2}}{{e}^{3}}}+3\,{\frac{\ln \left ( ex+d \right ) a{b}^{2}{d}^{2}}{{e}^{3}}}+3\,{\frac{\ln \left ( ex+d \right ) a{c}^{2}{d}^{4}}{{e}^{5}}}+3\,{\frac{\ln \left ( ex+d \right ){b}^{2}c{d}^{4}}{{e}^{5}}}-3\,{\frac{\ln \left ( ex+d \right ) b{c}^{2}{d}^{5}}{{e}^{6}}}-{\frac{{c}^{3}d{x}^{5}}{5\,{e}^{2}}} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

3/5/e*x^5*b*c^2+1/2/e^5*x^2*c^3*d^4+6/e^3*a*b*c*d^2*x-3/e^2*x^2*a*b*c*d-6/e^4*ln(e*x+d)*a*b*c*d^3+3/e*b*a^2*x-
1/e^4*ln(e*x+d)*b^3*d^3+1/e^7*ln(e*x+d)*c^3*d^6+1/6*c^3*x^6/e+3/2/e*x^2*a^2*c+1/4/e^3*x^4*c^3*d^2-1/2/e^2*x^2*
b^3*d+3/4/e*x^4*b^2*c+1/e^3*b^3*d^2*x+3/2/e*x^2*a*b^2-3/e^2*ln(e*x+d)*a^2*b*d+2/e*x^3*a*b*c+3/4/e*x^4*a*c^2+1/
e*ln(e*x+d)*a^3+1/3/e*x^3*b^3-3/e^2*a*b^2*d*x-3/e^4*a*c^2*d^3*x-3/e^4*b^2*c*d^3*x-3/2/e^4*x^2*b*c^2*d^3-3/e^2*
a^2*c*d*x-1/3/e^4*x^3*c^3*d^3-1/e^6*c^3*d^5*x-3/4/e^2*x^4*b*c^2*d+3/e^3*ln(e*x+d)*a^2*c*d^2-1/e^2*x^3*b^2*c*d+
3/2/e^3*x^2*a*c^2*d^2+3/2/e^3*x^2*b^2*c*d^2+3/e^5*b*c^2*d^4*x-1/e^2*x^3*a*c^2*d+1/e^3*x^3*b*c^2*d^2+3/e^3*ln(e
*x+d)*a*b^2*d^2+3/e^5*ln(e*x+d)*a*c^2*d^4+3/e^5*ln(e*x+d)*b^2*c*d^4-3/e^6*ln(e*x+d)*b*c^2*d^5-1/5*c^3*d*x^5/e^
2

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Maxima [A]  time = 0.968411, size = 541, normalized size = 2.08 \begin{align*} \frac{10 \, c^{3} e^{5} x^{6} - 12 \,{\left (c^{3} d e^{4} - 3 \, b c^{2} e^{5}\right )} x^{5} + 15 \,{\left (c^{3} d^{2} e^{3} - 3 \, b c^{2} d e^{4} + 3 \,{\left (b^{2} c + a c^{2}\right )} e^{5}\right )} x^{4} - 20 \,{\left (c^{3} d^{3} e^{2} - 3 \, b c^{2} d^{2} e^{3} + 3 \,{\left (b^{2} c + a c^{2}\right )} d e^{4} -{\left (b^{3} + 6 \, a b c\right )} e^{5}\right )} x^{3} + 30 \,{\left (c^{3} d^{4} e - 3 \, b c^{2} d^{3} e^{2} + 3 \,{\left (b^{2} c + a c^{2}\right )} d^{2} e^{3} -{\left (b^{3} + 6 \, a b c\right )} d e^{4} + 3 \,{\left (a b^{2} + a^{2} c\right )} e^{5}\right )} x^{2} - 60 \,{\left (c^{3} d^{5} - 3 \, b c^{2} d^{4} e - 3 \, a^{2} b e^{5} + 3 \,{\left (b^{2} c + a c^{2}\right )} d^{3} e^{2} -{\left (b^{3} + 6 \, a b c\right )} d^{2} e^{3} + 3 \,{\left (a b^{2} + a^{2} c\right )} d e^{4}\right )} x}{60 \, e^{6}} + \frac{{\left (c^{3} d^{6} - 3 \, b c^{2} d^{5} e - 3 \, a^{2} b d e^{5} + a^{3} e^{6} + 3 \,{\left (b^{2} c + a c^{2}\right )} d^{4} e^{2} -{\left (b^{3} + 6 \, a b c\right )} d^{3} e^{3} + 3 \,{\left (a b^{2} + a^{2} c\right )} d^{2} 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),x, algorithm="maxima")

[Out]

1/60*(10*c^3*e^5*x^6 - 12*(c^3*d*e^4 - 3*b*c^2*e^5)*x^5 + 15*(c^3*d^2*e^3 - 3*b*c^2*d*e^4 + 3*(b^2*c + a*c^2)*
e^5)*x^4 - 20*(c^3*d^3*e^2 - 3*b*c^2*d^2*e^3 + 3*(b^2*c + a*c^2)*d*e^4 - (b^3 + 6*a*b*c)*e^5)*x^3 + 30*(c^3*d^
4*e - 3*b*c^2*d^3*e^2 + 3*(b^2*c + a*c^2)*d^2*e^3 - (b^3 + 6*a*b*c)*d*e^4 + 3*(a*b^2 + a^2*c)*e^5)*x^2 - 60*(c
^3*d^5 - 3*b*c^2*d^4*e - 3*a^2*b*e^5 + 3*(b^2*c + a*c^2)*d^3*e^2 - (b^3 + 6*a*b*c)*d^2*e^3 + 3*(a*b^2 + a^2*c)
*d*e^4)*x)/e^6 + (c^3*d^6 - 3*b*c^2*d^5*e - 3*a^2*b*d*e^5 + a^3*e^6 + 3*(b^2*c + a*c^2)*d^4*e^2 - (b^3 + 6*a*b
*c)*d^3*e^3 + 3*(a*b^2 + a^2*c)*d^2*e^4)*log(e*x + d)/e^7

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Fricas [A]  time = 2.02334, size = 829, normalized size = 3.19 \begin{align*} \frac{10 \, c^{3} e^{6} x^{6} - 12 \,{\left (c^{3} d e^{5} - 3 \, b c^{2} e^{6}\right )} x^{5} + 15 \,{\left (c^{3} d^{2} e^{4} - 3 \, b c^{2} d e^{5} + 3 \,{\left (b^{2} c + a c^{2}\right )} e^{6}\right )} x^{4} - 20 \,{\left (c^{3} d^{3} e^{3} - 3 \, b c^{2} d^{2} e^{4} + 3 \,{\left (b^{2} c + a c^{2}\right )} d e^{5} -{\left (b^{3} + 6 \, a b c\right )} e^{6}\right )} x^{3} + 30 \,{\left (c^{3} d^{4} e^{2} - 3 \, b c^{2} d^{3} e^{3} + 3 \,{\left (b^{2} c + a c^{2}\right )} d^{2} e^{4} -{\left (b^{3} + 6 \, a b c\right )} d e^{5} + 3 \,{\left (a b^{2} + a^{2} c\right )} e^{6}\right )} x^{2} - 60 \,{\left (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} -{\left (b^{3} + 6 \, a b c\right )} d^{2} e^{4} + 3 \,{\left (a b^{2} + a^{2} c\right )} d e^{5}\right )} x + 60 \,{\left (c^{3} d^{6} - 3 \, b c^{2} d^{5} e - 3 \, a^{2} b d e^{5} + a^{3} e^{6} + 3 \,{\left (b^{2} c + a c^{2}\right )} d^{4} e^{2} -{\left (b^{3} + 6 \, a b c\right )} d^{3} e^{3} + 3 \,{\left (a b^{2} + a^{2} c\right )} d^{2} e^{4}\right )} \log \left (e x + d\right )}{60 \, 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),x, algorithm="fricas")

[Out]

1/60*(10*c^3*e^6*x^6 - 12*(c^3*d*e^5 - 3*b*c^2*e^6)*x^5 + 15*(c^3*d^2*e^4 - 3*b*c^2*d*e^5 + 3*(b^2*c + a*c^2)*
e^6)*x^4 - 20*(c^3*d^3*e^3 - 3*b*c^2*d^2*e^4 + 3*(b^2*c + a*c^2)*d*e^5 - (b^3 + 6*a*b*c)*e^6)*x^3 + 30*(c^3*d^
4*e^2 - 3*b*c^2*d^3*e^3 + 3*(b^2*c + a*c^2)*d^2*e^4 - (b^3 + 6*a*b*c)*d*e^5 + 3*(a*b^2 + a^2*c)*e^6)*x^2 - 60*
(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 - (b^3 + 6*a*b*c)*d^2*e^4 + 3*(a*b^2 +
a^2*c)*d*e^5)*x + 60*(c^3*d^6 - 3*b*c^2*d^5*e - 3*a^2*b*d*e^5 + a^3*e^6 + 3*(b^2*c + a*c^2)*d^4*e^2 - (b^3 + 6
*a*b*c)*d^3*e^3 + 3*(a*b^2 + a^2*c)*d^2*e^4)*log(e*x + d))/e^7

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Sympy [A]  time = 1.1198, size = 376, normalized size = 1.45 \begin{align*} \frac{c^{3} x^{6}}{6 e} + \frac{x^{5} \left (3 b c^{2} e - c^{3} d\right )}{5 e^{2}} + \frac{x^{4} \left (3 a c^{2} e^{2} + 3 b^{2} c e^{2} - 3 b c^{2} d e + c^{3} d^{2}\right )}{4 e^{3}} + \frac{x^{3} \left (6 a b c e^{3} - 3 a c^{2} d e^{2} + b^{3} e^{3} - 3 b^{2} c d e^{2} + 3 b c^{2} d^{2} e - c^{3} d^{3}\right )}{3 e^{4}} + \frac{x^{2} \left (3 a^{2} c e^{4} + 3 a b^{2} e^{4} - 6 a b c d e^{3} + 3 a c^{2} d^{2} e^{2} - b^{3} d e^{3} + 3 b^{2} c d^{2} e^{2} - 3 b c^{2} d^{3} e + c^{3} d^{4}\right )}{2 e^{5}} + \frac{x \left (3 a^{2} b e^{5} - 3 a^{2} c d e^{4} - 3 a b^{2} d e^{4} + 6 a b c d^{2} e^{3} - 3 a c^{2} d^{3} e^{2} + b^{3} d^{2} e^{3} - 3 b^{2} c d^{3} e^{2} + 3 b c^{2} d^{4} e - c^{3} d^{5}\right )}{e^{6}} + \frac{\left (a e^{2} - b d e + c d^{2}\right )^{3} \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),x)

[Out]

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

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Giac [A]  time = 1.12113, size = 621, normalized size = 2.39 \begin{align*}{\left (c^{3} d^{6} - 3 \, b c^{2} d^{5} e + 3 \, b^{2} c d^{4} e^{2} + 3 \, a c^{2} d^{4} e^{2} - b^{3} d^{3} e^{3} - 6 \, a b c d^{3} e^{3} + 3 \, a b^{2} d^{2} e^{4} + 3 \, a^{2} c d^{2} e^{4} - 3 \, a^{2} b d e^{5} + a^{3} e^{6}\right )} e^{\left (-7\right )} \log \left ({\left | x e + d \right |}\right ) + \frac{1}{60} \,{\left (10 \, c^{3} x^{6} e^{5} - 12 \, c^{3} d x^{5} e^{4} + 15 \, c^{3} d^{2} x^{4} e^{3} - 20 \, c^{3} d^{3} x^{3} e^{2} + 30 \, c^{3} d^{4} x^{2} e - 60 \, c^{3} d^{5} x + 36 \, b c^{2} x^{5} e^{5} - 45 \, b c^{2} d x^{4} e^{4} + 60 \, b c^{2} d^{2} x^{3} e^{3} - 90 \, b c^{2} d^{3} x^{2} e^{2} + 180 \, b c^{2} d^{4} x e + 45 \, b^{2} c x^{4} e^{5} + 45 \, a c^{2} x^{4} e^{5} - 60 \, b^{2} c d x^{3} e^{4} - 60 \, a c^{2} d x^{3} e^{4} + 90 \, b^{2} c d^{2} x^{2} e^{3} + 90 \, a c^{2} d^{2} x^{2} e^{3} - 180 \, b^{2} c d^{3} x e^{2} - 180 \, a c^{2} d^{3} x e^{2} + 20 \, b^{3} x^{3} e^{5} + 120 \, a b c x^{3} e^{5} - 30 \, b^{3} d x^{2} e^{4} - 180 \, a b c d x^{2} e^{4} + 60 \, b^{3} d^{2} x e^{3} + 360 \, a b c d^{2} x e^{3} + 90 \, a b^{2} x^{2} e^{5} + 90 \, a^{2} c x^{2} e^{5} - 180 \, a b^{2} d x e^{4} - 180 \, a^{2} c d x e^{4} + 180 \, a^{2} b x e^{5}\right )} e^{\left (-6\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),x, algorithm="giac")

[Out]

(c^3*d^6 - 3*b*c^2*d^5*e + 3*b^2*c*d^4*e^2 + 3*a*c^2*d^4*e^2 - b^3*d^3*e^3 - 6*a*b*c*d^3*e^3 + 3*a*b^2*d^2*e^4
+ 3*a^2*c*d^2*e^4 - 3*a^2*b*d*e^5 + a^3*e^6)*e^(-7)*log(abs(x*e + d)) + 1/60*(10*c^3*x^6*e^5 - 12*c^3*d*x^5*e
^4 + 15*c^3*d^2*x^4*e^3 - 20*c^3*d^3*x^3*e^2 + 30*c^3*d^4*x^2*e - 60*c^3*d^5*x + 36*b*c^2*x^5*e^5 - 45*b*c^2*d
*x^4*e^4 + 60*b*c^2*d^2*x^3*e^3 - 90*b*c^2*d^3*x^2*e^2 + 180*b*c^2*d^4*x*e + 45*b^2*c*x^4*e^5 + 45*a*c^2*x^4*e
^5 - 60*b^2*c*d*x^3*e^4 - 60*a*c^2*d*x^3*e^4 + 90*b^2*c*d^2*x^2*e^3 + 90*a*c^2*d^2*x^2*e^3 - 180*b^2*c*d^3*x*e
^2 - 180*a*c^2*d^3*x*e^2 + 20*b^3*x^3*e^5 + 120*a*b*c*x^3*e^5 - 30*b^3*d*x^2*e^4 - 180*a*b*c*d*x^2*e^4 + 60*b^
3*d^2*x*e^3 + 360*a*b*c*d^2*x*e^3 + 90*a*b^2*x^2*e^5 + 90*a^2*c*x^2*e^5 - 180*a*b^2*d*x*e^4 - 180*a^2*c*d*x*e^
4 + 180*a^2*b*x*e^5)*e^(-6)