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

Optimal. Leaf size=151 \[ \frac{c x^4 \left (3 b^2 e^2-3 b c d e+c^2 d^2\right )}{4 e^3}-\frac{c^2 x^5 (c d-3 b e)}{5 e^2}+\frac{d^3 (c d-b e)^3 \log (d+e x)}{e^7}-\frac{d^2 x (c d-b e)^3}{e^6}+\frac{d x^2 (c d-b e)^3}{2 e^5}-\frac{x^3 (c d-b e)^3}{3 e^4}+\frac{c^3 x^6}{6 e} \]

[Out]

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

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Rubi [A]  time = 0.355798, antiderivative size = 151, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.105 \[ \frac{c x^4 \left (3 b^2 e^2-3 b c d e+c^2 d^2\right )}{4 e^3}-\frac{c^2 x^5 (c d-3 b e)}{5 e^2}+\frac{d^3 (c d-b e)^3 \log (d+e x)}{e^7}-\frac{d^2 x (c d-b e)^3}{e^6}+\frac{d x^2 (c d-b e)^3}{2 e^5}-\frac{x^3 (c d-b e)^3}{3 e^4}+\frac{c^3 x^6}{6 e} \]

Antiderivative was successfully verified.

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

[Out]

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

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Rubi in Sympy [F]  time = 0., size = 0, normalized size = 0. \[ \frac{c^{3} x^{6}}{6 e} + \frac{c^{2} x^{5} \left (3 b e - c d\right )}{5 e^{2}} + \frac{c x^{4} \left (3 b^{2} e^{2} - 3 b c d e + c^{2} d^{2}\right )}{4 e^{3}} - \frac{d^{3} \left (b e - c d\right )^{3} \log{\left (d + e x \right )}}{e^{7}} - \frac{d \left (b e - c d\right )^{3} \int x\, dx}{e^{5}} + \frac{x^{3} \left (b e - c d\right )^{3}}{3 e^{4}} + \frac{\left (b e - c d\right )^{3} \int d^{2}\, dx}{e^{6}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((c*x**2+b*x)**3/(e*x+d),x)

[Out]

c**3*x**6/(6*e) + c**2*x**5*(3*b*e - c*d)/(5*e**2) + c*x**4*(3*b**2*e**2 - 3*b*c
*d*e + c**2*d**2)/(4*e**3) - d**3*(b*e - c*d)**3*log(d + e*x)/e**7 - d*(b*e - c*
d)**3*Integral(x, x)/e**5 + x**3*(b*e - c*d)**3/(3*e**4) + (b*e - c*d)**3*Integr
al(d**2, x)/e**6

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Mathematica [A]  time = 0.142786, size = 144, normalized size = 0.95 \[ \frac{15 c e^4 x^4 \left (3 b^2 e^2-3 b c d e+c^2 d^2\right )-12 c^2 e^5 x^5 (c d-3 b e)+60 d^3 (c d-b e)^3 \log (d+e x)-60 d^2 e x (c d-b e)^3+20 e^3 x^3 (b e-c d)^3+30 d e^2 x^2 (c d-b e)^3+10 c^3 e^6 x^6}{60 e^7} \]

Antiderivative was successfully verified.

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

[Out]

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

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Maple [B]  time = 0.006, size = 302, normalized size = 2. \[{\frac{{c}^{3}{x}^{6}}{6\,e}}+{\frac{3\,b{x}^{5}{c}^{2}}{5\,e}}-{\frac{d{c}^{3}{x}^{5}}{5\,{e}^{2}}}+{\frac{3\,{b}^{2}{x}^{4}c}{4\,e}}-{\frac{3\,b{x}^{4}{c}^{2}d}{4\,{e}^{2}}}+{\frac{{x}^{4}{c}^{3}{d}^{2}}{4\,{e}^{3}}}+{\frac{{b}^{3}{x}^{3}}{3\,e}}-{\frac{{b}^{2}{x}^{3}cd}{{e}^{2}}}+{\frac{b{x}^{3}{c}^{2}{d}^{2}}{{e}^{3}}}-{\frac{{x}^{3}{c}^{3}{d}^{3}}{3\,{e}^{4}}}-{\frac{{x}^{2}{b}^{3}d}{2\,{e}^{2}}}+{\frac{3\,{b}^{2}{x}^{2}c{d}^{2}}{2\,{e}^{3}}}-{\frac{3\,b{x}^{2}{c}^{2}{d}^{3}}{2\,{e}^{4}}}+{\frac{{x}^{2}{c}^{3}{d}^{4}}{2\,{e}^{5}}}+{\frac{{b}^{3}{d}^{2}x}{{e}^{3}}}-3\,{\frac{{d}^{3}{b}^{2}cx}{{e}^{4}}}+3\,{\frac{{d}^{4}b{c}^{2}x}{{e}^{5}}}-{\frac{{c}^{3}{d}^{5}x}{{e}^{6}}}-{\frac{{d}^{3}\ln \left ( ex+d \right ){b}^{3}}{{e}^{4}}}+3\,{\frac{{d}^{4}\ln \left ( ex+d \right ){b}^{2}c}{{e}^{5}}}-3\,{\frac{{d}^{5}\ln \left ( ex+d \right ) b{c}^{2}}{{e}^{6}}}+{\frac{{d}^{6}\ln \left ( ex+d \right ){c}^{3}}{{e}^{7}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((c*x^2+b*x)^3/(e*x+d),x)

[Out]

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

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Maxima [A]  time = 0.69383, size = 356, normalized size = 2.36 \[ \frac{10 \, c^{3} e^{5} x^{6} - 12 \,{\left (c^{3} d e^{4} - 3 \, b c^{2} e^{5}\right )} x^{5} + 15 \,{\left (c^{3} d^{2} e^{3} - 3 \, b c^{2} d e^{4} + 3 \, b^{2} c e^{5}\right )} x^{4} - 20 \,{\left (c^{3} d^{3} e^{2} - 3 \, b c^{2} d^{2} e^{3} + 3 \, b^{2} c d e^{4} - b^{3} e^{5}\right )} x^{3} + 30 \,{\left (c^{3} d^{4} e - 3 \, b c^{2} d^{3} e^{2} + 3 \, b^{2} c d^{2} e^{3} - b^{3} d e^{4}\right )} x^{2} - 60 \,{\left (c^{3} d^{5} - 3 \, b c^{2} d^{4} e + 3 \, b^{2} c d^{3} e^{2} - b^{3} d^{2} e^{3}\right )} x}{60 \, e^{6}} + \frac{{\left (c^{3} d^{6} - 3 \, b c^{2} d^{5} e + 3 \, b^{2} c d^{4} e^{2} - b^{3} d^{3} e^{3}\right )} \log \left (e x + d\right )}{e^{7}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((c*x^2 + b*x)^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*e^5)*x^4 - 20*(c^3*d^3*e^2 - 3*b*c^2*d^2*e^3 + 3*b^2*c*d*e^4
- b^3*e^5)*x^3 + 30*(c^3*d^4*e - 3*b*c^2*d^3*e^2 + 3*b^2*c*d^2*e^3 - b^3*d*e^4)
*x^2 - 60*(c^3*d^5 - 3*b*c^2*d^4*e + 3*b^2*c*d^3*e^2 - b^3*d^2*e^3)*x)/e^6 + (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)*log(e*x + d)/e^7

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Fricas [A]  time = 0.216548, size = 359, normalized size = 2.38 \[ \frac{10 \, c^{3} e^{6} x^{6} - 12 \,{\left (c^{3} d e^{5} - 3 \, b c^{2} e^{6}\right )} x^{5} + 15 \,{\left (c^{3} d^{2} e^{4} - 3 \, b c^{2} d e^{5} + 3 \, b^{2} c e^{6}\right )} x^{4} - 20 \,{\left (c^{3} d^{3} e^{3} - 3 \, b c^{2} d^{2} e^{4} + 3 \, b^{2} c d e^{5} - b^{3} e^{6}\right )} x^{3} + 30 \,{\left (c^{3} d^{4} e^{2} - 3 \, b c^{2} d^{3} e^{3} + 3 \, b^{2} c d^{2} e^{4} - b^{3} d e^{5}\right )} x^{2} - 60 \,{\left (c^{3} d^{5} e - 3 \, b c^{2} d^{4} e^{2} + 3 \, b^{2} c d^{3} e^{3} - b^{3} d^{2} e^{4}\right )} x + 60 \,{\left (c^{3} d^{6} - 3 \, b c^{2} d^{5} e + 3 \, b^{2} c d^{4} e^{2} - b^{3} d^{3} e^{3}\right )} \log \left (e x + d\right )}{60 \, e^{7}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((c*x^2 + b*x)^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*e^6)*x^4 - 20*(c^3*d^3*e^3 - 3*b*c^2*d^2*e^4 + 3*b^2*c*d*e^5
- b^3*e^6)*x^3 + 30*(c^3*d^4*e^2 - 3*b*c^2*d^3*e^3 + 3*b^2*c*d^2*e^4 - b^3*d*e^
5)*x^2 - 60*(c^3*d^5*e - 3*b*c^2*d^4*e^2 + 3*b^2*c*d^3*e^3 - b^3*d^2*e^4)*x + 60
*(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)*log(e*x + d))/e^7

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Sympy [A]  time = 2.36704, size = 231, normalized size = 1.53 \[ \frac{c^{3} x^{6}}{6 e} - \frac{d^{3} \left (b e - c d\right )^{3} \log{\left (d + e x \right )}}{e^{7}} + \frac{x^{5} \left (3 b c^{2} e - c^{3} d\right )}{5 e^{2}} + \frac{x^{4} \left (3 b^{2} c e^{2} - 3 b c^{2} d e + c^{3} d^{2}\right )}{4 e^{3}} + \frac{x^{3} \left (b^{3} e^{3} - 3 b^{2} c d e^{2} + 3 b c^{2} d^{2} e - c^{3} d^{3}\right )}{3 e^{4}} - \frac{x^{2} \left (b^{3} d e^{3} - 3 b^{2} c d^{2} e^{2} + 3 b c^{2} d^{3} e - c^{3} d^{4}\right )}{2 e^{5}} + \frac{x \left (b^{3} d^{2} e^{3} - 3 b^{2} c d^{3} e^{2} + 3 b c^{2} d^{4} e - c^{3} d^{5}\right )}{e^{6}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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GIAC/XCAS [A]  time = 0.211869, size = 365, normalized size = 2.42 \[{\left (c^{3} d^{6} - 3 \, b c^{2} d^{5} e + 3 \, b^{2} c d^{4} e^{2} - b^{3} d^{3} e^{3}\right )} e^{\left (-7\right )}{\rm ln}\left ({\left | x e + d \right |}\right ) + \frac{1}{60} \,{\left (10 \, c^{3} x^{6} e^{5} - 12 \, c^{3} d x^{5} e^{4} + 15 \, c^{3} d^{2} x^{4} e^{3} - 20 \, c^{3} d^{3} x^{3} e^{2} + 30 \, c^{3} d^{4} x^{2} e - 60 \, c^{3} d^{5} x + 36 \, b c^{2} x^{5} e^{5} - 45 \, b c^{2} d x^{4} e^{4} + 60 \, b c^{2} d^{2} x^{3} e^{3} - 90 \, b c^{2} d^{3} x^{2} e^{2} + 180 \, b c^{2} d^{4} x e + 45 \, b^{2} c x^{4} e^{5} - 60 \, b^{2} c d x^{3} e^{4} + 90 \, b^{2} c d^{2} x^{2} e^{3} - 180 \, b^{2} c d^{3} x e^{2} + 20 \, b^{3} x^{3} e^{5} - 30 \, b^{3} d x^{2} e^{4} + 60 \, b^{3} d^{2} x e^{3}\right )} e^{\left (-6\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((c*x^2 + b*x)^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 - b^3*d^3*e^3)*e^(-7)*ln(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 - 60*b^2*c*d*x^3*e^4 + 90*b^2*c*d^2*x^2*e^3 - 180*b^2*c*d^3*x*e^2 + 20*b^3*
x^3*e^5 - 30*b^3*d*x^2*e^4 + 60*b^3*d^2*x*e^3)*e^(-6)