3.2377 \(\int \frac{(5-x) (3+2 x)^4}{2+5 x+3 x^2} \, dx\)

Optimal. Leaf size=43 \[ -\frac{4 x^4}{3}+\frac{32 x^3}{27}+\frac{1156 x^2}{27}+\frac{11576 x}{81}-6 \log (x+1)+\frac{10625}{243} \log (3 x+2) \]

[Out]

(11576*x)/81 + (1156*x^2)/27 + (32*x^3)/27 - (4*x^4)/3 - 6*Log[1 + x] + (10625*L
og[2 + 3*x])/243

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Rubi [A]  time = 0.0621279, antiderivative size = 43, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 3, integrand size = 25, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.12 \[ -\frac{4 x^4}{3}+\frac{32 x^3}{27}+\frac{1156 x^2}{27}+\frac{11576 x}{81}-6 \log (x+1)+\frac{10625}{243} \log (3 x+2) \]

Antiderivative was successfully verified.

[In]  Int[((5 - x)*(3 + 2*x)^4)/(2 + 5*x + 3*x^2),x]

[Out]

(11576*x)/81 + (1156*x^2)/27 + (32*x^3)/27 - (4*x^4)/3 - 6*Log[1 + x] + (10625*L
og[2 + 3*x])/243

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Rubi in Sympy [F]  time = 0., size = 0, normalized size = 0. \[ - \frac{4 x^{4}}{3} + \frac{32 x^{3}}{27} - 6 \log{\left (x + 1 \right )} + \frac{10625 \log{\left (3 x + 2 \right )}}{243} + \int \frac{11576}{81}\, dx + \frac{2312 \int x\, dx}{27} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((5-x)*(3+2*x)**4/(3*x**2+5*x+2),x)

[Out]

-4*x**4/3 + 32*x**3/27 - 6*log(x + 1) + 10625*log(3*x + 2)/243 + Integral(11576/
81, x) + 2312*Integral(x, x)/27

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Mathematica [A]  time = 0.0404638, size = 43, normalized size = 1. \[ \frac{1}{972} \left (42500 \log (-6 x-4)-3 \left (432 x^4-384 x^3-13872 x^2-46304 x+1944 \log (-2 (x+1))-41727\right )\right ) \]

Antiderivative was successfully verified.

[In]  Integrate[((5 - x)*(3 + 2*x)^4)/(2 + 5*x + 3*x^2),x]

[Out]

(42500*Log[-4 - 6*x] - 3*(-41727 - 46304*x - 13872*x^2 - 384*x^3 + 432*x^4 + 194
4*Log[-2*(1 + x)]))/972

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Maple [A]  time = 0.01, size = 34, normalized size = 0.8 \[{\frac{11576\,x}{81}}+{\frac{1156\,{x}^{2}}{27}}+{\frac{32\,{x}^{3}}{27}}-{\frac{4\,{x}^{4}}{3}}-6\,\ln \left ( 1+x \right ) +{\frac{10625\,\ln \left ( 2+3\,x \right ) }{243}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((5-x)*(3+2*x)^4/(3*x^2+5*x+2),x)

[Out]

11576/81*x+1156/27*x^2+32/27*x^3-4/3*x^4-6*ln(1+x)+10625/243*ln(2+3*x)

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Maxima [A]  time = 0.691223, size = 45, normalized size = 1.05 \[ -\frac{4}{3} \, x^{4} + \frac{32}{27} \, x^{3} + \frac{1156}{27} \, x^{2} + \frac{11576}{81} \, x + \frac{10625}{243} \, \log \left (3 \, x + 2\right ) - 6 \, \log \left (x + 1\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(-(2*x + 3)^4*(x - 5)/(3*x^2 + 5*x + 2),x, algorithm="maxima")

[Out]

-4/3*x^4 + 32/27*x^3 + 1156/27*x^2 + 11576/81*x + 10625/243*log(3*x + 2) - 6*log
(x + 1)

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Fricas [A]  time = 0.27859, size = 45, normalized size = 1.05 \[ -\frac{4}{3} \, x^{4} + \frac{32}{27} \, x^{3} + \frac{1156}{27} \, x^{2} + \frac{11576}{81} \, x + \frac{10625}{243} \, \log \left (3 \, x + 2\right ) - 6 \, \log \left (x + 1\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(-(2*x + 3)^4*(x - 5)/(3*x^2 + 5*x + 2),x, algorithm="fricas")

[Out]

-4/3*x^4 + 32/27*x^3 + 1156/27*x^2 + 11576/81*x + 10625/243*log(3*x + 2) - 6*log
(x + 1)

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Sympy [A]  time = 0.276917, size = 41, normalized size = 0.95 \[ - \frac{4 x^{4}}{3} + \frac{32 x^{3}}{27} + \frac{1156 x^{2}}{27} + \frac{11576 x}{81} + \frac{10625 \log{\left (x + \frac{2}{3} \right )}}{243} - 6 \log{\left (x + 1 \right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((5-x)*(3+2*x)**4/(3*x**2+5*x+2),x)

[Out]

-4*x**4/3 + 32*x**3/27 + 1156*x**2/27 + 11576*x/81 + 10625*log(x + 2/3)/243 - 6*
log(x + 1)

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GIAC/XCAS [A]  time = 0.301347, size = 47, normalized size = 1.09 \[ -\frac{4}{3} \, x^{4} + \frac{32}{27} \, x^{3} + \frac{1156}{27} \, x^{2} + \frac{11576}{81} \, x + \frac{10625}{243} \,{\rm ln}\left ({\left | 3 \, x + 2 \right |}\right ) - 6 \,{\rm ln}\left ({\left | x + 1 \right |}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(-(2*x + 3)^4*(x - 5)/(3*x^2 + 5*x + 2),x, algorithm="giac")

[Out]

-4/3*x^4 + 32/27*x^3 + 1156/27*x^2 + 11576/81*x + 10625/243*ln(abs(3*x + 2)) - 6
*ln(abs(x + 1))