3.252 \(\int \frac{(b x+c x^2)^3}{(d+e x)^4} \, dx\)

Optimal. Leaf size=213 \[ -\frac{3 d \left (b^2 e^2-5 b c d e+5 c^2 d^2\right ) (c d-b e)}{e^7 (d+e x)}+\frac{c x \left (3 b^2 e^2-12 b c d e+10 c^2 d^2\right )}{e^6}-\frac{(2 c d-b e) \left (b^2 e^2-10 b c d e+10 c^2 d^2\right ) \log (d+e x)}{e^7}-\frac{c^2 x^2 (4 c d-3 b e)}{2 e^5}-\frac{d^3 (c d-b e)^3}{3 e^7 (d+e x)^3}+\frac{3 d^2 (2 c d-b e) (c d-b e)^2}{2 e^7 (d+e x)^2}+\frac{c^3 x^3}{3 e^4} \]

[Out]

(c*(10*c^2*d^2 - 12*b*c*d*e + 3*b^2*e^2)*x)/e^6 - (c^2*(4*c*d - 3*b*e)*x^2)/(2*e^5) + (c^3*x^3)/(3*e^4) - (d^3
*(c*d - b*e)^3)/(3*e^7*(d + e*x)^3) + (3*d^2*(c*d - b*e)^2*(2*c*d - b*e))/(2*e^7*(d + e*x)^2) - (3*d*(c*d - b*
e)*(5*c^2*d^2 - 5*b*c*d*e + b^2*e^2))/(e^7*(d + e*x)) - ((2*c*d - b*e)*(10*c^2*d^2 - 10*b*c*d*e + b^2*e^2)*Log
[d + e*x])/e^7

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Rubi [A]  time = 0.214941, antiderivative size = 213, 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} \[ -\frac{3 d \left (b^2 e^2-5 b c d e+5 c^2 d^2\right ) (c d-b e)}{e^7 (d+e x)}+\frac{c x \left (3 b^2 e^2-12 b c d e+10 c^2 d^2\right )}{e^6}-\frac{(2 c d-b e) \left (b^2 e^2-10 b c d e+10 c^2 d^2\right ) \log (d+e x)}{e^7}-\frac{c^2 x^2 (4 c d-3 b e)}{2 e^5}-\frac{d^3 (c d-b e)^3}{3 e^7 (d+e x)^3}+\frac{3 d^2 (2 c d-b e) (c d-b e)^2}{2 e^7 (d+e x)^2}+\frac{c^3 x^3}{3 e^4} \]

Antiderivative was successfully verified.

[In]

Int[(b*x + c*x^2)^3/(d + e*x)^4,x]

[Out]

(c*(10*c^2*d^2 - 12*b*c*d*e + 3*b^2*e^2)*x)/e^6 - (c^2*(4*c*d - 3*b*e)*x^2)/(2*e^5) + (c^3*x^3)/(3*e^4) - (d^3
*(c*d - b*e)^3)/(3*e^7*(d + e*x)^3) + (3*d^2*(c*d - b*e)^2*(2*c*d - b*e))/(2*e^7*(d + e*x)^2) - (3*d*(c*d - b*
e)*(5*c^2*d^2 - 5*b*c*d*e + b^2*e^2))/(e^7*(d + e*x)) - ((2*c*d - b*e)*(10*c^2*d^2 - 10*b*c*d*e + b^2*e^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 (b x+c x^2\right )^3}{(d+e x)^4} \, dx &=\int \left (\frac{c \left (10 c^2 d^2-12 b c d e+3 b^2 e^2\right )}{e^6}-\frac{c^2 (4 c d-3 b e) x}{e^5}+\frac{c^3 x^2}{e^4}+\frac{d^3 (c d-b e)^3}{e^6 (d+e x)^4}-\frac{3 d^2 (c d-b e)^2 (2 c d-b e)}{e^6 (d+e x)^3}+\frac{3 d (c d-b e) \left (5 c^2 d^2-5 b c d e+b^2 e^2\right )}{e^6 (d+e x)^2}+\frac{(2 c d-b e) \left (-10 c^2 d^2+10 b c d e-b^2 e^2\right )}{e^6 (d+e x)}\right ) \, dx\\ &=\frac{c \left (10 c^2 d^2-12 b c d e+3 b^2 e^2\right ) x}{e^6}-\frac{c^2 (4 c d-3 b e) x^2}{2 e^5}+\frac{c^3 x^3}{3 e^4}-\frac{d^3 (c d-b e)^3}{3 e^7 (d+e x)^3}+\frac{3 d^2 (c d-b e)^2 (2 c d-b e)}{2 e^7 (d+e x)^2}-\frac{3 d (c d-b e) \left (5 c^2 d^2-5 b c d e+b^2 e^2\right )}{e^7 (d+e x)}-\frac{(2 c d-b e) \left (10 c^2 d^2-10 b c d e+b^2 e^2\right ) \log (d+e x)}{e^7}\\ \end{align*}

Mathematica [A]  time = 0.107645, size = 210, normalized size = 0.99 \[ \frac{6 c e x \left (3 b^2 e^2-12 b c d e+10 c^2 d^2\right )+\frac{18 d \left (-6 b^2 c d e^2+b^3 e^3+10 b c^2 d^2 e-5 c^3 d^3\right )}{d+e x}+6 \left (-12 b^2 c d e^2+b^3 e^3+30 b c^2 d^2 e-20 c^3 d^3\right ) \log (d+e x)-3 c^2 e^2 x^2 (4 c d-3 b e)-\frac{2 d^3 (c d-b e)^3}{(d+e x)^3}+\frac{9 d^2 (2 c d-b e) (c d-b e)^2}{(d+e x)^2}+2 c^3 e^3 x^3}{6 e^7} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Maple [A]  time = 0.054, size = 353, normalized size = 1.7 \begin{align*}{\frac{{x}^{3}{c}^{3}}{3\,{e}^{4}}}+{\frac{3\,b{x}^{2}{c}^{2}}{2\,{e}^{4}}}-2\,{\frac{{c}^{3}d{x}^{2}}{{e}^{5}}}+3\,{\frac{{b}^{2}xc}{{e}^{4}}}-12\,{\frac{b{c}^{2}dx}{{e}^{5}}}+10\,{\frac{{c}^{3}{d}^{2}x}{{e}^{6}}}-{\frac{3\,{d}^{2}{b}^{3}}{2\,{e}^{4} \left ( ex+d \right ) ^{2}}}+6\,{\frac{{b}^{2}c{d}^{3}}{{e}^{5} \left ( ex+d \right ) ^{2}}}-{\frac{15\,b{c}^{2}{d}^{4}}{2\,{e}^{6} \left ( ex+d \right ) ^{2}}}+3\,{\frac{{c}^{3}{d}^{5}}{{e}^{7} \left ( ex+d \right ) ^{2}}}+{\frac{\ln \left ( ex+d \right ){b}^{3}}{{e}^{4}}}-12\,{\frac{\ln \left ( ex+d \right ){b}^{2}cd}{{e}^{5}}}+30\,{\frac{\ln \left ( ex+d \right ) b{c}^{2}{d}^{2}}{{e}^{6}}}-20\,{\frac{\ln \left ( ex+d \right ){c}^{3}{d}^{3}}{{e}^{7}}}+{\frac{{d}^{3}{b}^{3}}{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 ) ^{3}}}-{\frac{{c}^{3}{d}^{6}}{3\,{e}^{7} \left ( ex+d \right ) ^{3}}}+3\,{\frac{d{b}^{3}}{{e}^{4} \left ( ex+d \right ) }}-18\,{\frac{{b}^{2}c{d}^{2}}{{e}^{5} \left ( ex+d \right ) }}+30\,{\frac{b{c}^{2}{d}^{3}}{{e}^{6} \left ( ex+d \right ) }}-15\,{\frac{{c}^{3}{d}^{4}}{{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)^4,x)

[Out]

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

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Maxima [A]  time = 1.16581, size = 397, normalized size = 1.86 \begin{align*} -\frac{74 \, c^{3} d^{6} - 141 \, b c^{2} d^{5} e + 78 \, b^{2} c d^{4} e^{2} - 11 \, b^{3} d^{3} e^{3} + 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} - b^{3} d e^{5}\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} - 3 \, b^{3} d^{2} e^{4}\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 \, b^{2} c e^{2}\right )} x}{6 \, e^{6}} - \frac{{\left (20 \, c^{3} d^{3} - 30 \, b c^{2} d^{2} e + 12 \, b^{2} c d e^{2} - b^{3} 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)^4,x, algorithm="maxima")

[Out]

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

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Fricas [B]  time = 1.61528, size = 996, normalized size = 4.68 \begin{align*} \frac{2 \, c^{3} e^{6} x^{6} - 74 \, c^{3} d^{6} + 141 \, b c^{2} d^{5} e - 78 \, b^{2} c d^{4} e^{2} + 11 \, b^{3} d^{3} e^{3} - 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 \, b^{2} c e^{6}\right )} x^{4} +{\left (146 \, c^{3} d^{3} e^{3} - 189 \, b c^{2} d^{2} e^{4} + 54 \, b^{2} c d e^{5}\right )} x^{3} + 3 \,{\left (26 \, c^{3} d^{4} e^{2} - 9 \, b c^{2} d^{3} e^{3} - 18 \, b^{2} c d^{2} e^{4} + 6 \, b^{3} d e^{5}\right )} x^{2} - 3 \,{\left (34 \, c^{3} d^{5} e - 81 \, b c^{2} d^{4} e^{2} + 54 \, b^{2} c d^{3} e^{3} - 9 \, b^{3} d^{2} e^{4}\right )} x - 6 \,{\left (20 \, c^{3} d^{6} - 30 \, b c^{2} d^{5} e + 12 \, b^{2} c d^{4} e^{2} - b^{3} d^{3} e^{3} +{\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 )} x^{3} + 3 \,{\left (20 \, c^{3} d^{4} e^{2} - 30 \, b c^{2} d^{3} e^{3} + 12 \, b^{2} c d^{2} e^{4} - b^{3} d e^{5}\right )} x^{2} + 3 \,{\left (20 \, c^{3} d^{5} e - 30 \, b c^{2} d^{4} e^{2} + 12 \, b^{2} c d^{3} e^{3} - b^{3} 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 )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((c*x^2+b*x)^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 - 78*b^2*c*d^4*e^2 + 11*b^3*d^3*e^3 - 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*e^6)*x^4 + (146*c^3*d^3*e^3 - 189*b*c^2*d^2*e^4 + 5
4*b^2*c*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*d^2*e^4 + 6*b^3*d*e^5)*x^2 - 3*(34*c^3*d^5
*e - 81*b*c^2*d^4*e^2 + 54*b^2*c*d^3*e^3 - 9*b^3*d^2*e^4)*x - 6*(20*c^3*d^6 - 30*b*c^2*d^5*e + 12*b^2*c*d^4*e^
2 - b^3*d^3*e^3 + (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)*x^3 + 3*(20*c^3*d^4*e^2 - 30*
b*c^2*d^3*e^3 + 12*b^2*c*d^2*e^4 - b^3*d*e^5)*x^2 + 3*(20*c^3*d^5*e - 30*b*c^2*d^4*e^2 + 12*b^2*c*d^3*e^3 - b^
3*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 = 12.896, size = 298, normalized size = 1.4 \begin{align*} \frac{c^{3} x^{3}}{3 e^{4}} + \frac{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 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 (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 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 (b^{2} e^{2} - 10 b c d e + 10 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)**3/(e*x+d)**4,x)

[Out]

c**3*x**3/(3*e**4) + (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*(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*(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*b**2*c*e**2 - 12*b*c**2*d*e + 10*c**3*d**2)/e**6 + (b*
e - 2*c*d)*(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 [A]  time = 1.31645, size = 352, normalized size = 1.65 \begin{align*} -{\left (20 \, c^{3} d^{3} - 30 \, b c^{2} d^{2} e + 12 \, b^{2} c d e^{2} - b^{3} e^{3}\right )} e^{\left (-7\right )} \log \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}\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} - 11 \, b^{3} d^{3} e^{3} + 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} - b^{3} d e^{5}\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} - 3 \, b^{3} d^{2} e^{4}\right )} x\right )} e^{\left (-7\right )}}{6 \,{\left (x e + d\right )}^{3}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((c*x^2+b*x)^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 - b^3*e^3)*e^(-7)*log(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)*e^(-12) - 1/6*(74*c^3*d
^6 - 141*b*c^2*d^5*e + 78*b^2*c*d^4*e^2 - 11*b^3*d^3*e^3 + 18*(5*c^3*d^4*e^2 - 10*b*c^2*d^3*e^3 + 6*b^2*c*d^2*
e^4 - b^3*d*e^5)*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 - 3*b^3*d^2*e^4)*x)*e^(-7)/(x*e +
d)^3