3.2396 \(\int \frac{(5-x) (3+2 x)^4}{\left (2+5 x+3 x^2\right )^3} \, dx\)

Optimal. Leaf size=59 \[ -\frac{12083 x+11597}{162 \left (3 x^2+5 x+2\right )^2}+\frac{7 (20298 x+16651)}{162 \left (3 x^2+5 x+2\right )}-883 \log (x+1)+\frac{23825}{27} \log (3 x+2) \]

[Out]

-(11597 + 12083*x)/(162*(2 + 5*x + 3*x^2)^2) + (7*(16651 + 20298*x))/(162*(2 + 5
*x + 3*x^2)) - 883*Log[1 + x] + (23825*Log[2 + 3*x])/27

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Rubi [A]  time = 0.099492, antiderivative size = 59, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 4, integrand size = 25, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.16 \[ -\frac{12083 x+11597}{162 \left (3 x^2+5 x+2\right )^2}+\frac{7 (20298 x+16651)}{162 \left (3 x^2+5 x+2\right )}-883 \log (x+1)+\frac{23825}{27} \log (3 x+2) \]

Antiderivative was successfully verified.

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

[Out]

-(11597 + 12083*x)/(162*(2 + 5*x + 3*x^2)^2) + (7*(16651 + 20298*x))/(162*(2 + 5
*x + 3*x^2)) - 883*Log[1 + x] + (23825*Log[2 + 3*x])/27

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Rubi in Sympy [A]  time = 22.1285, size = 61, normalized size = 1.03 \[ - \frac{\left (2 x + 3\right )^{3} \left (139 x + 121\right )}{6 \left (3 x^{2} + 5 x + 2\right )^{2}} + \frac{\left (2 x + 3\right ) \left (12736 x + 11149\right )}{18 \left (3 x^{2} + 5 x + 2\right )} - 883 \log{\left (x + 1 \right )} + \frac{23825 \log{\left (3 x + 2 \right )}}{27} \]

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)**3,x)

[Out]

-(2*x + 3)**3*(139*x + 121)/(6*(3*x**2 + 5*x + 2)**2) + (2*x + 3)*(12736*x + 111
49)/(18*(3*x**2 + 5*x + 2)) - 883*log(x + 1) + 23825*log(3*x + 2)/27

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Mathematica [A]  time = 0.0646151, size = 62, normalized size = 1.05 \[ \frac{1}{54} \left (47650 \log (-6 x-4)-\frac{3 \left (-47362 x^3-117789 x^2+15894 \left (3 x^2+5 x+2\right )^2 \log (-2 (x+1))-94986 x-24613\right )}{\left (3 x^2+5 x+2\right )^2}\right ) \]

Antiderivative was successfully verified.

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

[Out]

(47650*Log[-4 - 6*x] - (3*(-24613 - 94986*x - 117789*x^2 - 47362*x^3 + 15894*(2
+ 5*x + 3*x^2)^2*Log[-2*(1 + x)]))/(2 + 5*x + 3*x^2)^2)/54

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Maple [A]  time = 0.016, size = 48, normalized size = 0.8 \[ -{\frac{10625}{54\, \left ( 2+3\,x \right ) ^{2}}}+{\frac{15500}{54+81\,x}}+{\frac{23825\,\ln \left ( 2+3\,x \right ) }{27}}+3\, \left ( 1+x \right ) ^{-2}+101\, \left ( 1+x \right ) ^{-1}-883\,\ln \left ( 1+x \right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-10625/54/(2+3*x)^2+15500/27/(2+3*x)+23825/27*ln(2+3*x)+3/(1+x)^2+101/(1+x)-883*
ln(1+x)

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Maxima [A]  time = 0.690656, size = 73, normalized size = 1.24 \[ \frac{47362 \, x^{3} + 117789 \, x^{2} + 94986 \, x + 24613}{18 \,{\left (9 \, x^{4} + 30 \, x^{3} + 37 \, x^{2} + 20 \, x + 4\right )}} + \frac{23825}{27} \, \log \left (3 \, x + 2\right ) - 883 \, \log \left (x + 1\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/18*(47362*x^3 + 117789*x^2 + 94986*x + 24613)/(9*x^4 + 30*x^3 + 37*x^2 + 20*x
+ 4) + 23825/27*log(3*x + 2) - 883*log(x + 1)

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Fricas [A]  time = 0.261854, size = 126, normalized size = 2.14 \[ \frac{142086 \, x^{3} + 353367 \, x^{2} + 47650 \,{\left (9 \, x^{4} + 30 \, x^{3} + 37 \, x^{2} + 20 \, x + 4\right )} \log \left (3 \, x + 2\right ) - 47682 \,{\left (9 \, x^{4} + 30 \, x^{3} + 37 \, x^{2} + 20 \, x + 4\right )} \log \left (x + 1\right ) + 284958 \, x + 73839}{54 \,{\left (9 \, x^{4} + 30 \, x^{3} + 37 \, x^{2} + 20 \, x + 4\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/54*(142086*x^3 + 353367*x^2 + 47650*(9*x^4 + 30*x^3 + 37*x^2 + 20*x + 4)*log(3
*x + 2) - 47682*(9*x^4 + 30*x^3 + 37*x^2 + 20*x + 4)*log(x + 1) + 284958*x + 738
39)/(9*x^4 + 30*x^3 + 37*x^2 + 20*x + 4)

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Sympy [A]  time = 0.510305, size = 51, normalized size = 0.86 \[ \frac{47362 x^{3} + 117789 x^{2} + 94986 x + 24613}{162 x^{4} + 540 x^{3} + 666 x^{2} + 360 x + 72} + \frac{23825 \log{\left (x + \frac{2}{3} \right )}}{27} - 883 \log{\left (x + 1 \right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

(47362*x**3 + 117789*x**2 + 94986*x + 24613)/(162*x**4 + 540*x**3 + 666*x**2 + 3
60*x + 72) + 23825*log(x + 2/3)/27 - 883*log(x + 1)

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GIAC/XCAS [A]  time = 0.292497, size = 62, normalized size = 1.05 \[ \frac{47362 \, x^{3} + 117789 \, x^{2} + 94986 \, x + 24613}{18 \,{\left (3 \, x + 2\right )}^{2}{\left (x + 1\right )}^{2}} + \frac{23825}{27} \,{\rm ln}\left ({\left | 3 \, x + 2 \right |}\right ) - 883 \,{\rm ln}\left ({\left | x + 1 \right |}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/18*(47362*x^3 + 117789*x^2 + 94986*x + 24613)/((3*x + 2)^2*(x + 1)^2) + 23825/
27*ln(abs(3*x + 2)) - 883*ln(abs(x + 1))