3.7 \(\int \frac{-x^2+x^3}{(-6+x) (3+5 x)^3} \, dx\)

Optimal. Leaf size=43 \[ \frac{201}{15125 (5 x+3)}-\frac{12}{1375 (5 x+3)^2}+\frac{20 \log (6-x)}{3993}+\frac{1493 \log (5 x+3)}{499125} \]

[Out]

-12/(1375*(3 + 5*x)^2) + 201/(15125*(3 + 5*x)) + (20*Log[6 - x])/3993 + (1493*Lo
g[3 + 5*x])/499125

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Rubi [A]  time = 0.0721782, antiderivative size = 43, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.091 \[ \frac{201}{15125 (5 x+3)}-\frac{12}{1375 (5 x+3)^2}+\frac{20 \log (6-x)}{3993}+\frac{1493 \log (5 x+3)}{499125} \]

Antiderivative was successfully verified.

[In]  Int[(-x^2 + x^3)/((-6 + x)*(3 + 5*x)^3),x]

[Out]

-12/(1375*(3 + 5*x)^2) + 201/(15125*(3 + 5*x)) + (20*Log[6 - x])/3993 + (1493*Lo
g[3 + 5*x])/499125

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Rubi in Sympy [A]  time = 9.14792, size = 34, normalized size = 0.79 \[ \frac{20 \log{\left (- x + 6 \right )}}{3993} + \frac{1493 \log{\left (5 x + 3 \right )}}{499125} + \frac{201}{15125 \left (5 x + 3\right )} - \frac{12}{1375 \left (5 x + 3\right )^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((x**3-x**2)/(-6+x)/(3+5*x)**3,x)

[Out]

20*log(-x + 6)/3993 + 1493*log(5*x + 3)/499125 + 201/(15125*(5*x + 3)) - 12/(137
5*(5*x + 3)**2)

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Mathematica [A]  time = 0.0388779, size = 33, normalized size = 0.77 \[ \frac{\frac{99 (335 x+157)}{(5 x+3)^2}+2500 \log (x-6)+1493 \log (5 x+3)}{499125} \]

Antiderivative was successfully verified.

[In]  Integrate[(-x^2 + x^3)/((-6 + x)*(3 + 5*x)^3),x]

[Out]

((99*(157 + 335*x))/(3 + 5*x)^2 + 2500*Log[-6 + x] + 1493*Log[3 + 5*x])/499125

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Maple [A]  time = 0.015, size = 34, normalized size = 0.8 \[{\frac{20\,\ln \left ( -6+x \right ) }{3993}}-{\frac{12}{1375\, \left ( 3+5\,x \right ) ^{2}}}+{\frac{201}{45375+75625\,x}}+{\frac{1493\,\ln \left ( 3+5\,x \right ) }{499125}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((x^3-x^2)/(-6+x)/(3+5*x)^3,x)

[Out]

20/3993*ln(-6+x)-12/1375/(3+5*x)^2+201/15125/(3+5*x)+1493/499125*ln(3+5*x)

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Maxima [A]  time = 1.3653, size = 46, normalized size = 1.07 \[ \frac{3 \,{\left (335 \, x + 157\right )}}{15125 \,{\left (25 \, x^{2} + 30 \, x + 9\right )}} + \frac{1493}{499125} \, \log \left (5 \, x + 3\right ) + \frac{20}{3993} \, \log \left (x - 6\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((x^3 - x^2)/((5*x + 3)^3*(x - 6)),x, algorithm="maxima")

[Out]

3/15125*(335*x + 157)/(25*x^2 + 30*x + 9) + 1493/499125*log(5*x + 3) + 20/3993*l
og(x - 6)

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Fricas [A]  time = 0.245539, size = 72, normalized size = 1.67 \[ \frac{1493 \,{\left (25 \, x^{2} + 30 \, x + 9\right )} \log \left (5 \, x + 3\right ) + 2500 \,{\left (25 \, x^{2} + 30 \, x + 9\right )} \log \left (x - 6\right ) + 33165 \, x + 15543}{499125 \,{\left (25 \, x^{2} + 30 \, x + 9\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((x^3 - x^2)/((5*x + 3)^3*(x - 6)),x, algorithm="fricas")

[Out]

1/499125*(1493*(25*x^2 + 30*x + 9)*log(5*x + 3) + 2500*(25*x^2 + 30*x + 9)*log(x
- 6) + 33165*x + 15543)/(25*x^2 + 30*x + 9)

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Sympy [A]  time = 0.402441, size = 32, normalized size = 0.74 \[ \frac{1005 x + 471}{378125 x^{2} + 453750 x + 136125} + \frac{20 \log{\left (x - 6 \right )}}{3993} + \frac{1493 \log{\left (x + \frac{3}{5} \right )}}{499125} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((x**3-x**2)/(-6+x)/(3+5*x)**3,x)

[Out]

(1005*x + 471)/(378125*x**2 + 453750*x + 136125) + 20*log(x - 6)/3993 + 1493*log
(x + 3/5)/499125

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GIAC/XCAS [A]  time = 0.212962, size = 42, normalized size = 0.98 \[ \frac{3 \,{\left (335 \, x + 157\right )}}{15125 \,{\left (5 \, x + 3\right )}^{2}} + \frac{1493}{499125} \,{\rm ln}\left ({\left | 5 \, x + 3 \right |}\right ) + \frac{20}{3993} \,{\rm ln}\left ({\left | x - 6 \right |}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((x^3 - x^2)/((5*x + 3)^3*(x - 6)),x, algorithm="giac")

[Out]

3/15125*(335*x + 157)/(5*x + 3)^2 + 1493/499125*ln(abs(5*x + 3)) + 20/3993*ln(ab
s(x - 6))