3.250 \(\int \frac{(b x+c x^2)^3}{(d+e x)^2} \, dx\)

Optimal. Leaf size=166 \[ -\frac{c^2 x^4 (2 c d-3 b e)}{4 e^3}-\frac{d^3 (c d-b e)^3}{e^7 (d+e x)}-\frac{3 d^2 (c d-b e)^2 (2 c d-b e) \log (d+e x)}{e^7}+\frac{c x^3 (c d-b e)^2}{e^4}-\frac{x^2 (c d-b e)^2 (4 c d-b e)}{2 e^5}+\frac{d x (5 c d-2 b e) (c d-b e)^2}{e^6}+\frac{c^3 x^5}{5 e^2} \]

[Out]

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

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Rubi [A]  time = 0.186282, antiderivative size = 166, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 1, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.053, Rules used = {698} \[ -\frac{c^2 x^4 (2 c d-3 b e)}{4 e^3}-\frac{d^3 (c d-b e)^3}{e^7 (d+e x)}-\frac{3 d^2 (c d-b e)^2 (2 c d-b e) \log (d+e x)}{e^7}+\frac{c x^3 (c d-b e)^2}{e^4}-\frac{x^2 (c d-b e)^2 (4 c d-b e)}{2 e^5}+\frac{d x (5 c d-2 b e) (c d-b e)^2}{e^6}+\frac{c^3 x^5}{5 e^2} \]

Antiderivative was successfully verified.

[In]

Int[(b*x + c*x^2)^3/(d + e*x)^2,x]

[Out]

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

Mathematica [A]  time = 0.0618319, size = 160, normalized size = 0.96 \[ \frac{-5 c^2 e^4 x^4 (2 c d-3 b e)-\frac{20 d^3 (c d-b e)^3}{d+e x}-60 d^2 (c d-b e)^2 (2 c d-b e) \log (d+e x)+20 c e^3 x^3 (c d-b e)^2+10 e^2 x^2 (c d-b e)^2 (b e-4 c d)+20 d e x (5 c d-2 b e) (c d-b e)^2+4 c^3 e^5 x^5}{20 e^7} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Maple [A]  time = 0.056, size = 318, normalized size = 1.9 \begin{align*}{\frac{{c}^{3}{x}^{5}}{5\,{e}^{2}}}+{\frac{3\,b{x}^{4}{c}^{2}}{4\,{e}^{2}}}-{\frac{{c}^{3}d{x}^{4}}{2\,{e}^{3}}}+{\frac{{b}^{2}c{x}^{3}}{{e}^{2}}}-2\,{\frac{b{x}^{3}{c}^{2}d}{{e}^{3}}}+{\frac{{x}^{3}{c}^{3}{d}^{2}}{{e}^{4}}}+{\frac{{x}^{2}{b}^{3}}{2\,{e}^{2}}}-3\,{\frac{{b}^{2}{x}^{2}cd}{{e}^{3}}}+{\frac{9\,b{x}^{2}{c}^{2}{d}^{2}}{2\,{e}^{4}}}-2\,{\frac{{x}^{2}{c}^{3}{d}^{3}}{{e}^{5}}}-2\,{\frac{{b}^{3}dx}{{e}^{3}}}+9\,{\frac{{b}^{2}c{d}^{2}x}{{e}^{4}}}-12\,{\frac{b{c}^{2}{d}^{3}x}{{e}^{5}}}+5\,{\frac{{c}^{3}{d}^{4}x}{{e}^{6}}}+3\,{\frac{{d}^{2}\ln \left ( ex+d \right ){b}^{3}}{{e}^{4}}}-12\,{\frac{{d}^{3}\ln \left ( ex+d \right ){b}^{2}c}{{e}^{5}}}+15\,{\frac{{d}^{4}\ln \left ( ex+d \right ) b{c}^{2}}{{e}^{6}}}-6\,{\frac{{d}^{5}\ln \left ( ex+d \right ){c}^{3}}{{e}^{7}}}+{\frac{{d}^{3}{b}^{3}}{{e}^{4} \left ( ex+d \right ) }}-3\,{\frac{{d}^{4}{b}^{2}c}{{e}^{5} \left ( ex+d \right ) }}+3\,{\frac{{d}^{5}b{c}^{2}}{{e}^{6} \left ( ex+d \right ) }}-{\frac{{c}^{3}{d}^{6}}{{e}^{7} \left ( ex+d \right ) }} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [A]  time = 1.0369, size = 369, normalized size = 2.22 \begin{align*} -\frac{c^{3} d^{6} - 3 \, b c^{2} d^{5} e + 3 \, b^{2} c d^{4} e^{2} - b^{3} d^{3} e^{3}}{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} + b^{2} c e^{4}\right )} x^{3} - 10 \,{\left (4 \, c^{3} d^{3} e - 9 \, b c^{2} d^{2} e^{2} + 6 \, b^{2} c d e^{3} - b^{3} e^{4}\right )} x^{2} + 20 \,{\left (5 \, c^{3} d^{4} - 12 \, b c^{2} d^{3} e + 9 \, b^{2} c d^{2} e^{2} - 2 \, b^{3} d e^{3}\right )} x}{20 \, e^{6}} - \frac{3 \,{\left (2 \, c^{3} d^{5} - 5 \, b c^{2} d^{4} e + 4 \, b^{2} c d^{3} e^{2} - b^{3} d^{2} e^{3}\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)^3/(e*x+d)^2,x, algorithm="maxima")

[Out]

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

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Fricas [B]  time = 1.44909, size = 751, normalized size = 4.52 \begin{align*} \frac{4 \, c^{3} e^{6} x^{6} - 20 \, c^{3} d^{6} + 60 \, b c^{2} d^{5} e - 60 \, b^{2} c d^{4} e^{2} + 20 \, b^{3} d^{3} e^{3} - 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 \, b^{2} c e^{6}\right )} x^{4} - 10 \,{\left (2 \, c^{3} d^{3} e^{3} - 5 \, b c^{2} d^{2} e^{4} + 4 \, b^{2} c d e^{5} - b^{3} e^{6}\right )} x^{3} + 30 \,{\left (2 \, c^{3} d^{4} e^{2} - 5 \, b c^{2} d^{3} e^{3} + 4 \, b^{2} c d^{2} e^{4} - b^{3} d e^{5}\right )} x^{2} + 20 \,{\left (5 \, c^{3} d^{5} e - 12 \, b c^{2} d^{4} e^{2} + 9 \, b^{2} c d^{3} e^{3} - 2 \, b^{3} d^{2} e^{4}\right )} x - 60 \,{\left (2 \, c^{3} d^{6} - 5 \, b c^{2} d^{5} e + 4 \, b^{2} c d^{4} e^{2} - b^{3} d^{3} e^{3} +{\left (2 \, c^{3} d^{5} e - 5 \, b c^{2} d^{4} e^{2} + 4 \, b^{2} c d^{3} e^{3} - b^{3} d^{2} e^{4}\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)^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*b^2*c*d^4*e^2 + 20*b^3*d^3*e^3 - 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*e^6)*x^4 - 10*(2*c^3*d^3*e^3 - 5*b*c^2*d^2*e^4 + 4*b^
2*c*d*e^5 - b^3*e^6)*x^3 + 30*(2*c^3*d^4*e^2 - 5*b*c^2*d^3*e^3 + 4*b^2*c*d^2*e^4 - b^3*d*e^5)*x^2 + 20*(5*c^3*
d^5*e - 12*b*c^2*d^4*e^2 + 9*b^2*c*d^3*e^3 - 2*b^3*d^2*e^4)*x - 60*(2*c^3*d^6 - 5*b*c^2*d^5*e + 4*b^2*c*d^4*e^
2 - b^3*d^3*e^3 + (2*c^3*d^5*e - 5*b*c^2*d^4*e^2 + 4*b^2*c*d^3*e^3 - b^3*d^2*e^4)*x)*log(e*x + d))/(e^8*x + d*
e^7)

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Sympy [A]  time = 3.46736, size = 248, normalized size = 1.49 \begin{align*} \frac{c^{3} x^{5}}{5 e^{2}} + \frac{3 d^{2} \left (b e - 2 c d\right ) \left (b e - c d\right )^{2} \log{\left (d + e x \right )}}{e^{7}} + \frac{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 (b^{2} c e^{2} - 2 b c^{2} d e + c^{3} d^{2}\right )}{e^{4}} + \frac{x^{2} \left (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 (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}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Giac [B]  time = 1.43872, size = 448, normalized size = 2.7 \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}\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} - b^{3} 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} - b^{3} d e^{7}\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} - b^{3} d^{2} e^{3}\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{b^{3} d^{3} e^{8}}{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)^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)*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 - b^3*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 - b^3*d*e^7)*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 - b^3*d^2*e^3)*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) - b^3*d^3*e^8/(x*e + d))*e^(-12)