### 3.2137 $$\int \frac{(a+b x+c x^2)^3}{(d+e x)^2} \, dx$$

Optimal. Leaf size=256 $\frac{c (d+e x)^3 \left (-c e (5 b d-a e)+b^2 e^2+5 c^2 d^2\right )}{e^7}-\frac{(d+e x)^2 (2 c d-b e) \left (-2 c e (5 b d-3 a e)+b^2 e^2+10 c^2 d^2\right )}{2 e^7}+\frac{3 x \left (a e^2-b d e+c d^2\right ) \left (-c e (5 b d-a e)+b^2 e^2+5 c^2 d^2\right )}{e^6}-\frac{\left (a e^2-b d e+c d^2\right )^3}{e^7 (d+e x)}-\frac{3 (2 c d-b e) \log (d+e x) \left (a e^2-b d e+c d^2\right )^2}{e^7}-\frac{3 c^2 (d+e x)^4 (2 c d-b e)}{4 e^7}+\frac{c^3 (d+e x)^5}{5 e^7}$

[Out]

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

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Rubi [A]  time = 0.320798, antiderivative size = 256, 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} $\frac{c (d+e x)^3 \left (-c e (5 b d-a e)+b^2 e^2+5 c^2 d^2\right )}{e^7}-\frac{(d+e x)^2 (2 c d-b e) \left (-2 c e (5 b d-3 a e)+b^2 e^2+10 c^2 d^2\right )}{2 e^7}+\frac{3 x \left (a e^2-b d e+c d^2\right ) \left (-c e (5 b d-a e)+b^2 e^2+5 c^2 d^2\right )}{e^6}-\frac{\left (a e^2-b d e+c d^2\right )^3}{e^7 (d+e x)}-\frac{3 (2 c d-b e) \log (d+e x) \left (a e^2-b d e+c d^2\right )^2}{e^7}-\frac{3 c^2 (d+e x)^4 (2 c d-b e)}{4 e^7}+\frac{c^3 (d+e x)^5}{5 e^7}$

Antiderivative was successfully veriﬁed.

[In]

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

[Out]

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

Mathematica [A]  time = 0.103753, size = 255, normalized size = 1. $\frac{20 e x \left (3 c e^2 \left (a^2 e^2-4 a b d e+3 b^2 d^2\right )+b^2 e^3 (3 a e-2 b d)+3 c^2 d^2 e (3 a e-4 b d)+5 c^3 d^4\right )+20 c e^3 x^3 \left (c e (a e-2 b d)+b^2 e^2+c^2 d^2\right )+10 e^2 x^2 (b e-c d) \left (c e (6 a e-5 b d)+b^2 e^2+4 c^2 d^2\right )-\frac{20 \left (e (a e-b d)+c d^2\right )^3}{d+e x}-60 (2 c d-b e) \log (d+e x) \left (e (a e-b d)+c d^2\right )^2+5 c^2 e^4 x^4 (3 b e-2 c d)+4 c^3 e^5 x^5}{20 e^7}$

Antiderivative was successfully veriﬁed.

[In]

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

[Out]

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

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Maple [B]  time = 0.05, size = 585, normalized size = 2.3 \begin{align*} 18\,{\frac{\ln \left ( ex+d \right ) abc{d}^{2}}{{e}^{4}}}-3\,{\frac{a{x}^{2}{c}^{2}d}{{e}^{3}}}-3\,{\frac{{b}^{2}{x}^{2}cd}{{e}^{3}}}+{\frac{{c}^{3}{x}^{5}}{5\,{e}^{2}}}+9\,{\frac{a{c}^{2}{d}^{2}x}{{e}^{4}}}+9\,{\frac{c{b}^{2}{d}^{2}x}{{e}^{4}}}-12\,{\frac{{d}^{3}b{c}^{2}x}{{e}^{5}}}+3\,{\frac{ab{x}^{2}c}{{e}^{2}}}-6\,{\frac{dc\ln \left ( ex+d \right ){a}^{2}}{{e}^{3}}}-6\,{\frac{\ln \left ( ex+d \right ) a{b}^{2}d}{{e}^{3}}}-12\,{\frac{\ln \left ( ex+d \right ) a{c}^{2}{d}^{3}}{{e}^{5}}}-12\,{\frac{\ln \left ( ex+d \right ){b}^{2}c{d}^{3}}{{e}^{5}}}+15\,{\frac{\ln \left ( ex+d \right ) b{c}^{2}{d}^{4}}{{e}^{6}}}+3\,{\frac{{a}^{2}db}{{e}^{2} \left ( ex+d \right ) }}-3\,{\frac{{a}^{2}c{d}^{2}}{{e}^{3} \left ( ex+d \right ) }}-3\,{\frac{a{b}^{2}{d}^{2}}{{e}^{3} \left ( ex+d \right ) }}-3\,{\frac{a{c}^{2}{d}^{4}}{{e}^{5} \left ( ex+d \right ) }}+3\,{\frac{b{c}^{2}{d}^{5}}{{e}^{6} \left ( ex+d \right ) }}-2\,{\frac{b{x}^{3}{c}^{2}d}{{e}^{3}}}-{\frac{{c}^{3}{d}^{6}}{{e}^{7} \left ( ex+d \right ) }}+3\,{\frac{\ln \left ( ex+d \right ){a}^{2}b}{{e}^{2}}}+3\,{\frac{\ln \left ( ex+d \right ){b}^{3}{d}^{2}}{{e}^{4}}}-6\,{\frac{\ln \left ( ex+d \right ){c}^{3}{d}^{5}}{{e}^{7}}}+{\frac{{b}^{3}{d}^{3}}{{e}^{4} \left ( ex+d \right ) }}+3\,{\frac{{b}^{2}ax}{{e}^{2}}}-2\,{\frac{{b}^{3}dx}{{e}^{3}}}+5\,{\frac{{c}^{3}{d}^{4}x}{{e}^{6}}}+{\frac{3\,b{x}^{4}{c}^{2}}{4\,{e}^{2}}}+{\frac{a{x}^{3}{c}^{2}}{{e}^{2}}}+{\frac{{b}^{2}c{x}^{3}}{{e}^{2}}}+{\frac{{x}^{3}{c}^{3}{d}^{2}}{{e}^{4}}}-2\,{\frac{{x}^{2}{c}^{3}{d}^{3}}{{e}^{5}}}+3\,{\frac{{a}^{2}cx}{{e}^{2}}}-{\frac{{a}^{3}}{e \left ( ex+d \right ) }}-3\,{\frac{{b}^{2}c{d}^{4}}{{e}^{5} \left ( ex+d \right ) }}+{\frac{{x}^{2}{b}^{3}}{2\,{e}^{2}}}+{\frac{9\,b{x}^{2}{c}^{2}{d}^{2}}{2\,{e}^{4}}}+6\,{\frac{abc{d}^{3}}{{e}^{4} \left ( ex+d \right ) }}-12\,{\frac{cabdx}{{e}^{3}}}-{\frac{{c}^{3}d{x}^{4}}{2\,{e}^{3}}} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [A]  time = 0.989355, size = 554, normalized size = 2.16 \begin{align*} -\frac{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}}{e^{8} x + d e^{7}} + \frac{4 \, c^{3} e^{4} x^{5} - 5 \,{\left (2 \, c^{3} d e^{3} - 3 \, b c^{2} e^{4}\right )} x^{4} + 20 \,{\left (c^{3} d^{2} e^{2} - 2 \, b c^{2} d e^{3} +{\left (b^{2} c + a c^{2}\right )} e^{4}\right )} x^{3} - 10 \,{\left (4 \, c^{3} d^{3} e - 9 \, b c^{2} d^{2} e^{2} + 6 \,{\left (b^{2} c + a c^{2}\right )} d e^{3} -{\left (b^{3} + 6 \, a b c\right )} e^{4}\right )} x^{2} + 20 \,{\left (5 \, c^{3} d^{4} - 12 \, b c^{2} d^{3} e + 9 \,{\left (b^{2} c + a c^{2}\right )} d^{2} e^{2} - 2 \,{\left (b^{3} + 6 \, a b c\right )} d e^{3} + 3 \,{\left (a b^{2} + a^{2} c\right )} e^{4}\right )} x}{20 \, e^{6}} - \frac{3 \,{\left (2 \, c^{3} d^{5} - 5 \, b c^{2} d^{4} e - a^{2} b e^{5} + 4 \,{\left (b^{2} c + a c^{2}\right )} d^{3} e^{2} -{\left (b^{3} + 6 \, a b c\right )} d^{2} e^{3} + 2 \,{\left (a b^{2} + a^{2} c\right )} d 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)^2,x, algorithm="maxima")

[Out]

-(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)/(e^8*x + d*e^7) + 1/20*(4*c^3*e^4*x^5 - 5*(2*c^3*d*e^3 - 3*b*c^2*e^4)*x^4 + 20*(c^3*d
^2*e^2 - 2*b*c^2*d*e^3 + (b^2*c + a*c^2)*e^4)*x^3 - 10*(4*c^3*d^3*e - 9*b*c^2*d^2*e^2 + 6*(b^2*c + a*c^2)*d*e^
3 - (b^3 + 6*a*b*c)*e^4)*x^2 + 20*(5*c^3*d^4 - 12*b*c^2*d^3*e + 9*(b^2*c + a*c^2)*d^2*e^2 - 2*(b^3 + 6*a*b*c)*
d*e^3 + 3*(a*b^2 + a^2*c)*e^4)*x)/e^6 - 3*(2*c^3*d^5 - 5*b*c^2*d^4*e - a^2*b*e^5 + 4*(b^2*c + a*c^2)*d^3*e^2 -
(b^3 + 6*a*b*c)*d^2*e^3 + 2*(a*b^2 + a^2*c)*d*e^4)*log(e*x + d)/e^7

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Fricas [B]  time = 2.11092, size = 1197, normalized size = 4.68 \begin{align*} \frac{4 \, c^{3} e^{6} x^{6} - 20 \, c^{3} d^{6} + 60 \, b c^{2} d^{5} e + 60 \, a^{2} b d e^{5} - 20 \, a^{3} e^{6} - 60 \,{\left (b^{2} c + a c^{2}\right )} d^{4} e^{2} + 20 \,{\left (b^{3} + 6 \, a b c\right )} d^{3} e^{3} - 60 \,{\left (a b^{2} + a^{2} c\right )} d^{2} e^{4} - 3 \,{\left (2 \, c^{3} d e^{5} - 5 \, b c^{2} e^{6}\right )} x^{5} + 5 \,{\left (2 \, c^{3} d^{2} e^{4} - 5 \, b c^{2} d e^{5} + 4 \,{\left (b^{2} c + a c^{2}\right )} e^{6}\right )} x^{4} - 10 \,{\left (2 \, c^{3} d^{3} e^{3} - 5 \, b c^{2} d^{2} e^{4} + 4 \,{\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 (2 \, c^{3} d^{4} e^{2} - 5 \, b c^{2} d^{3} e^{3} + 4 \,{\left (b^{2} c + a c^{2}\right )} d^{2} e^{4} -{\left (b^{3} + 6 \, a b c\right )} d e^{5} + 2 \,{\left (a b^{2} + a^{2} c\right )} e^{6}\right )} x^{2} + 20 \,{\left (5 \, c^{3} d^{5} e - 12 \, b c^{2} d^{4} e^{2} + 9 \,{\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} + 3 \,{\left (a b^{2} + a^{2} c\right )} d e^{5}\right )} x - 60 \,{\left (2 \, c^{3} d^{6} - 5 \, b c^{2} d^{5} e - a^{2} b d e^{5} + 4 \,{\left (b^{2} c + a c^{2}\right )} d^{4} e^{2} -{\left (b^{3} + 6 \, a b c\right )} d^{3} e^{3} + 2 \,{\left (a b^{2} + a^{2} c\right )} d^{2} e^{4} +{\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\right )} \log \left (e x + d\right )}{20 \,{\left (e^{8} x + d 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)^2,x, algorithm="fricas")

[Out]

1/20*(4*c^3*e^6*x^6 - 20*c^3*d^6 + 60*b*c^2*d^5*e + 60*a^2*b*d*e^5 - 20*a^3*e^6 - 60*(b^2*c + a*c^2)*d^4*e^2 +
20*(b^3 + 6*a*b*c)*d^3*e^3 - 60*(a*b^2 + a^2*c)*d^2*e^4 - 3*(2*c^3*d*e^5 - 5*b*c^2*e^6)*x^5 + 5*(2*c^3*d^2*e^
4 - 5*b*c^2*d*e^5 + 4*(b^2*c + a*c^2)*e^6)*x^4 - 10*(2*c^3*d^3*e^3 - 5*b*c^2*d^2*e^4 + 4*(b^2*c + a*c^2)*d*e^5
- (b^3 + 6*a*b*c)*e^6)*x^3 + 30*(2*c^3*d^4*e^2 - 5*b*c^2*d^3*e^3 + 4*(b^2*c + a*c^2)*d^2*e^4 - (b^3 + 6*a*b*c
)*d*e^5 + 2*(a*b^2 + a^2*c)*e^6)*x^2 + 20*(5*c^3*d^5*e - 12*b*c^2*d^4*e^2 + 9*(b^2*c + a*c^2)*d^3*e^3 - 2*(b^3
+ 6*a*b*c)*d^2*e^4 + 3*(a*b^2 + a^2*c)*d*e^5)*x - 60*(2*c^3*d^6 - 5*b*c^2*d^5*e - a^2*b*d*e^5 + 4*(b^2*c + a*
c^2)*d^4*e^2 - (b^3 + 6*a*b*c)*d^3*e^3 + 2*(a*b^2 + a^2*c)*d^2*e^4 + (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)*log(e*x + d))/(e^8*x + d
*e^7)

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Sympy [A]  time = 2.64275, size = 403, normalized size = 1.57 \begin{align*} \frac{c^{3} x^{5}}{5 e^{2}} - \frac{a^{3} e^{6} - 3 a^{2} b d e^{5} + 3 a^{2} c d^{2} e^{4} + 3 a b^{2} d^{2} e^{4} - 6 a b c d^{3} e^{3} + 3 a c^{2} d^{4} e^{2} - b^{3} d^{3} e^{3} + 3 b^{2} c d^{4} e^{2} - 3 b c^{2} d^{5} e + c^{3} d^{6}}{d e^{7} + e^{8} x} + \frac{x^{4} \left (3 b c^{2} e - 2 c^{3} d\right )}{4 e^{3}} + \frac{x^{3} \left (a c^{2} e^{2} + b^{2} c e^{2} - 2 b c^{2} d e + c^{3} d^{2}\right )}{e^{4}} + \frac{x^{2} \left (6 a b c e^{3} - 6 a c^{2} d e^{2} + b^{3} e^{3} - 6 b^{2} c d e^{2} + 9 b c^{2} d^{2} e - 4 c^{3} d^{3}\right )}{2 e^{5}} + \frac{x \left (3 a^{2} c e^{4} + 3 a b^{2} e^{4} - 12 a b c d e^{3} + 9 a c^{2} d^{2} e^{2} - 2 b^{3} d e^{3} + 9 b^{2} c d^{2} e^{2} - 12 b c^{2} d^{3} e + 5 c^{3} d^{4}\right )}{e^{6}} + \frac{3 \left (b e - 2 c d\right ) \left (a e^{2} - b d e + c d^{2}\right )^{2} \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)**2,x)

[Out]

c**3*x**5/(5*e**2) - (a**3*e**6 - 3*a**2*b*d*e**5 + 3*a**2*c*d**2*e**4 + 3*a*b**2*d**2*e**4 - 6*a*b*c*d**3*e**
3 + 3*a*c**2*d**4*e**2 - b**3*d**3*e**3 + 3*b**2*c*d**4*e**2 - 3*b*c**2*d**5*e + c**3*d**6)/(d*e**7 + e**8*x)
+ x**4*(3*b*c**2*e - 2*c**3*d)/(4*e**3) + x**3*(a*c**2*e**2 + b**2*c*e**2 - 2*b*c**2*d*e + c**3*d**2)/e**4 + x
**2*(6*a*b*c*e**3 - 6*a*c**2*d*e**2 + b**3*e**3 - 6*b**2*c*d*e**2 + 9*b*c**2*d**2*e - 4*c**3*d**3)/(2*e**5) +
x*(3*a**2*c*e**4 + 3*a*b**2*e**4 - 12*a*b*c*d*e**3 + 9*a*c**2*d**2*e**2 - 2*b**3*d*e**3 + 9*b**2*c*d**2*e**2 -
12*b*c**2*d**3*e + 5*c**3*d**4)/e**6 + 3*(b*e - 2*c*d)*(a*e**2 - b*d*e + c*d**2)**2*log(d + e*x)/e**7

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Giac [B]  time = 1.1272, size = 730, normalized size = 2.85 \begin{align*} \frac{1}{20} \,{\left (4 \, c^{3} - \frac{15 \,{\left (2 \, c^{3} d e - b c^{2} e^{2}\right )} e^{\left (-1\right )}}{x e + d} + \frac{20 \,{\left (5 \, c^{3} d^{2} e^{2} - 5 \, b c^{2} d e^{3} + b^{2} c e^{4} + a c^{2} e^{4}\right )} e^{\left (-2\right )}}{{\left (x e + d\right )}^{2}} - \frac{10 \,{\left (20 \, c^{3} d^{3} e^{3} - 30 \, b c^{2} d^{2} e^{4} + 12 \, b^{2} c d e^{5} + 12 \, a c^{2} d e^{5} - b^{3} e^{6} - 6 \, a b c e^{6}\right )} e^{\left (-3\right )}}{{\left (x e + d\right )}^{3}} + \frac{60 \,{\left (5 \, c^{3} d^{4} e^{4} - 10 \, b c^{2} d^{3} e^{5} + 6 \, b^{2} c d^{2} e^{6} + 6 \, a c^{2} d^{2} e^{6} - b^{3} d e^{7} - 6 \, a b c d e^{7} + a b^{2} e^{8} + a^{2} c e^{8}\right )} e^{\left (-4\right )}}{{\left (x e + d\right )}^{4}}\right )}{\left (x e + d\right )}^{5} e^{\left (-7\right )} + 3 \,{\left (2 \, c^{3} d^{5} - 5 \, b c^{2} d^{4} e + 4 \, b^{2} c d^{3} e^{2} + 4 \, a c^{2} d^{3} e^{2} - b^{3} d^{2} e^{3} - 6 \, a b c d^{2} e^{3} + 2 \, a b^{2} d e^{4} + 2 \, a^{2} c d e^{4} - a^{2} b e^{5}\right )} e^{\left (-7\right )} \log \left (\frac{{\left | x e + d \right |} e^{\left (-1\right )}}{{\left (x e + d\right )}^{2}}\right ) -{\left (\frac{c^{3} d^{6} e^{5}}{x e + d} - \frac{3 \, b c^{2} d^{5} e^{6}}{x e + d} + \frac{3 \, b^{2} c d^{4} e^{7}}{x e + d} + \frac{3 \, a c^{2} d^{4} e^{7}}{x e + d} - \frac{b^{3} d^{3} e^{8}}{x e + d} - \frac{6 \, a b c d^{3} e^{8}}{x e + d} + \frac{3 \, a b^{2} d^{2} e^{9}}{x e + d} + \frac{3 \, a^{2} c d^{2} e^{9}}{x e + d} - \frac{3 \, a^{2} b d e^{10}}{x e + d} + \frac{a^{3} e^{11}}{x e + d}\right )} e^{\left (-12\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)^2,x, algorithm="giac")

[Out]

1/20*(4*c^3 - 15*(2*c^3*d*e - b*c^2*e^2)*e^(-1)/(x*e + d) + 20*(5*c^3*d^2*e^2 - 5*b*c^2*d*e^3 + b^2*c*e^4 + a*
c^2*e^4)*e^(-2)/(x*e + d)^2 - 10*(20*c^3*d^3*e^3 - 30*b*c^2*d^2*e^4 + 12*b^2*c*d*e^5 + 12*a*c^2*d*e^5 - b^3*e^
6 - 6*a*b*c*e^6)*e^(-3)/(x*e + d)^3 + 60*(5*c^3*d^4*e^4 - 10*b*c^2*d^3*e^5 + 6*b^2*c*d^2*e^6 + 6*a*c^2*d^2*e^6
- b^3*d*e^7 - 6*a*b*c*d*e^7 + a*b^2*e^8 + a^2*c*e^8)*e^(-4)/(x*e + d)^4)*(x*e + d)^5*e^(-7) + 3*(2*c^3*d^5 -
5*b*c^2*d^4*e + 4*b^2*c*d^3*e^2 + 4*a*c^2*d^3*e^2 - b^3*d^2*e^3 - 6*a*b*c*d^2*e^3 + 2*a*b^2*d*e^4 + 2*a^2*c*d*
e^4 - a^2*b*e^5)*e^(-7)*log(abs(x*e + d)*e^(-1)/(x*e + d)^2) - (c^3*d^6*e^5/(x*e + d) - 3*b*c^2*d^5*e^6/(x*e +
d) + 3*b^2*c*d^4*e^7/(x*e + d) + 3*a*c^2*d^4*e^7/(x*e + d) - b^3*d^3*e^8/(x*e + d) - 6*a*b*c*d^3*e^8/(x*e + d
) + 3*a*b^2*d^2*e^9/(x*e + d) + 3*a^2*c*d^2*e^9/(x*e + d) - 3*a^2*b*d*e^10/(x*e + d) + a^3*e^11/(x*e + d))*e^(
-12)