3.2395 \(\int \frac{(5-x) (3+2 x)^5}{\left (2+5 x+3 x^2\right )^3} \, dx\)

Optimal. Leaf size=64 \[ \frac{502254 x+398585}{486 \left (3 x^2+5 x+2\right )}-\frac{57499 x+56041}{486 \left (3 x^2+5 x+2\right )^2}-\frac{32 x}{27}-1085 \log (x+1)+\frac{29375}{27} \log (3 x+2) \]

[Out]

(-32*x)/27 - (56041 + 57499*x)/(486*(2 + 5*x + 3*x^2)^2) + (398585 + 502254*x)/(
486*(2 + 5*x + 3*x^2)) - 1085*Log[1 + x] + (29375*Log[2 + 3*x])/27

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Rubi [A]  time = 0.141617, antiderivative size = 64, normalized size of antiderivative = 1., number of steps used = 8, number of rules used = 5, integrand size = 25, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.2 \[ \frac{502254 x+398585}{486 \left (3 x^2+5 x+2\right )}-\frac{57499 x+56041}{486 \left (3 x^2+5 x+2\right )^2}-\frac{32 x}{27}-1085 \log (x+1)+\frac{29375}{27} \log (3 x+2) \]

Antiderivative was successfully verified.

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

[Out]

(-32*x)/27 - (56041 + 57499*x)/(486*(2 + 5*x + 3*x^2)^2) + (398585 + 502254*x)/(
486*(2 + 5*x + 3*x^2)) - 1085*Log[1 + x] + (29375*Log[2 + 3*x])/27

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Rubi in Sympy [A]  time = 28.6883, size = 68, normalized size = 1.06 \[ - \frac{7780 x}{9} - \frac{\left (2 x + 3\right )^{4} \left (139 x + 121\right )}{6 \left (3 x^{2} + 5 x + 2\right )^{2}} + \frac{\left (2 x + 3\right )^{2} \left (12210 x + 10515\right )}{18 \left (3 x^{2} + 5 x + 2\right )} - 1085 \log{\left (x + 1 \right )} + \frac{29375 \log{\left (3 x + 2 \right )}}{27} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-7780*x/9 - (2*x + 3)**4*(139*x + 121)/(6*(3*x**2 + 5*x + 2)**2) + (2*x + 3)**2*
(12210*x + 10515)/(18*(3*x**2 + 5*x + 2)) - 1085*log(x + 1) + 29375*log(3*x + 2)
/27

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Mathematica [A]  time = 0.061183, size = 81, normalized size = 1.27 \[ \frac{-1728 x^5-8352 x^4+486510 x^3+1221179 x^2+176250 \left (3 x^2+5 x+2\right )^2 \log (-6 x-4)-175770 \left (3 x^2+5 x+2\right )^2 \log (-2 (x+1))+973450 x+245891}{162 \left (3 x^2+5 x+2\right )^2} \]

Antiderivative was successfully verified.

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

[Out]

(245891 + 973450*x + 1221179*x^2 + 486510*x^3 - 8352*x^4 - 1728*x^5 + 176250*(2
+ 5*x + 3*x^2)^2*Log[-4 - 6*x] - 175770*(2 + 5*x + 3*x^2)^2*Log[-2*(1 + x)])/(16
2*(2 + 5*x + 3*x^2)^2)

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Maple [A]  time = 0.016, size = 51, normalized size = 0.8 \[ -{\frac{32\,x}{27}}-{\frac{53125}{162\, \left ( 2+3\,x \right ) ^{2}}}+{\frac{6250}{18+27\,x}}+{\frac{29375\,\ln \left ( 2+3\,x \right ) }{27}}+3\, \left ( 1+x \right ) ^{-2}+113\, \left ( 1+x \right ) ^{-1}-1085\,\ln \left ( 1+x \right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-32/27*x-53125/162/(2+3*x)^2+6250/9/(2+3*x)+29375/27*ln(2+3*x)+3/(1+x)^2+113/(1+
x)-1085*ln(1+x)

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Maxima [A]  time = 0.688255, size = 77, normalized size = 1.2 \[ -\frac{32}{27} \, x + \frac{502254 \, x^{3} + 1235675 \, x^{2} + 979978 \, x + 247043}{162 \,{\left (9 \, x^{4} + 30 \, x^{3} + 37 \, x^{2} + 20 \, x + 4\right )}} + \frac{29375}{27} \, \log \left (3 \, x + 2\right ) - 1085 \, \log \left (x + 1\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-32/27*x + 1/162*(502254*x^3 + 1235675*x^2 + 979978*x + 247043)/(9*x^4 + 30*x^3
+ 37*x^2 + 20*x + 4) + 29375/27*log(3*x + 2) - 1085*log(x + 1)

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Fricas [A]  time = 0.261693, size = 139, normalized size = 2.17 \[ -\frac{1728 \, x^{5} + 5760 \, x^{4} - 495150 \, x^{3} - 1231835 \, x^{2} - 176250 \,{\left (9 \, x^{4} + 30 \, x^{3} + 37 \, x^{2} + 20 \, x + 4\right )} \log \left (3 \, x + 2\right ) + 175770 \,{\left (9 \, x^{4} + 30 \, x^{3} + 37 \, x^{2} + 20 \, x + 4\right )} \log \left (x + 1\right ) - 979210 \, x - 247043}{162 \,{\left (9 \, x^{4} + 30 \, x^{3} + 37 \, x^{2} + 20 \, x + 4\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/162*(1728*x^5 + 5760*x^4 - 495150*x^3 - 1231835*x^2 - 176250*(9*x^4 + 30*x^3
+ 37*x^2 + 20*x + 4)*log(3*x + 2) + 175770*(9*x^4 + 30*x^3 + 37*x^2 + 20*x + 4)*
log(x + 1) - 979210*x - 247043)/(9*x^4 + 30*x^3 + 37*x^2 + 20*x + 4)

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Sympy [A]  time = 0.527669, size = 56, normalized size = 0.88 \[ - \frac{32 x}{27} + \frac{502254 x^{3} + 1235675 x^{2} + 979978 x + 247043}{1458 x^{4} + 4860 x^{3} + 5994 x^{2} + 3240 x + 648} + \frac{29375 \log{\left (x + \frac{2}{3} \right )}}{27} - 1085 \log{\left (x + 1 \right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-32*x/27 + (502254*x**3 + 1235675*x**2 + 979978*x + 247043)/(1458*x**4 + 4860*x*
*3 + 5994*x**2 + 3240*x + 648) + 29375*log(x + 2/3)/27 - 1085*log(x + 1)

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GIAC/XCAS [A]  time = 0.265084, size = 66, normalized size = 1.03 \[ -\frac{32}{27} \, x + \frac{502254 \, x^{3} + 1235675 \, x^{2} + 979978 \, x + 247043}{162 \,{\left (3 \, x + 2\right )}^{2}{\left (x + 1\right )}^{2}} + \frac{29375}{27} \,{\rm ln}\left ({\left | 3 \, x + 2 \right |}\right ) - 1085 \,{\rm ln}\left ({\left | x + 1 \right |}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-32/27*x + 1/162*(502254*x^3 + 1235675*x^2 + 979978*x + 247043)/((3*x + 2)^2*(x
+ 1)^2) + 29375/27*ln(abs(3*x + 2)) - 1085*ln(abs(x + 1))