3.1431 \(\int \frac{3+5 x}{(1-2 x) (2+3 x)^6} \, dx\)

Optimal. Leaf size=76 \[ -\frac{88}{16807 (3 x+2)}-\frac{22}{2401 (3 x+2)^2}-\frac{22}{1029 (3 x+2)^3}-\frac{11}{196 (3 x+2)^4}+\frac{1}{105 (3 x+2)^5}-\frac{176 \log (1-2 x)}{117649}+\frac{176 \log (3 x+2)}{117649} \]

[Out]

1/(105*(2 + 3*x)^5) - 11/(196*(2 + 3*x)^4) - 22/(1029*(2 + 3*x)^3) - 22/(2401*(2
+ 3*x)^2) - 88/(16807*(2 + 3*x)) - (176*Log[1 - 2*x])/117649 + (176*Log[2 + 3*x
])/117649

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Rubi [A]  time = 0.0702305, antiderivative size = 76, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 1, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.05 \[ -\frac{88}{16807 (3 x+2)}-\frac{22}{2401 (3 x+2)^2}-\frac{22}{1029 (3 x+2)^3}-\frac{11}{196 (3 x+2)^4}+\frac{1}{105 (3 x+2)^5}-\frac{176 \log (1-2 x)}{117649}+\frac{176 \log (3 x+2)}{117649} \]

Antiderivative was successfully verified.

[In]  Int[(3 + 5*x)/((1 - 2*x)*(2 + 3*x)^6),x]

[Out]

1/(105*(2 + 3*x)^5) - 11/(196*(2 + 3*x)^4) - 22/(1029*(2 + 3*x)^3) - 22/(2401*(2
+ 3*x)^2) - 88/(16807*(2 + 3*x)) - (176*Log[1 - 2*x])/117649 + (176*Log[2 + 3*x
])/117649

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Rubi in Sympy [A]  time = 10.8296, size = 66, normalized size = 0.87 \[ - \frac{176 \log{\left (- 2 x + 1 \right )}}{117649} + \frac{176 \log{\left (3 x + 2 \right )}}{117649} - \frac{88}{16807 \left (3 x + 2\right )} - \frac{22}{2401 \left (3 x + 2\right )^{2}} - \frac{22}{1029 \left (3 x + 2\right )^{3}} - \frac{11}{196 \left (3 x + 2\right )^{4}} + \frac{1}{105 \left (3 x + 2\right )^{5}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((3+5*x)/(1-2*x)/(2+3*x)**6,x)

[Out]

-176*log(-2*x + 1)/117649 + 176*log(3*x + 2)/117649 - 88/(16807*(3*x + 2)) - 22/
(2401*(3*x + 2)**2) - 22/(1029*(3*x + 2)**3) - 11/(196*(3*x + 2)**4) + 1/(105*(3
*x + 2)**5)

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Mathematica [A]  time = 0.0581297, size = 50, normalized size = 0.66 \[ \frac{-\frac{7 \left (427680 x^4+1389960 x^3+1833480 x^2+1268025 x+348226\right )}{(3 x+2)^5}-10560 \log (3-6 x)+10560 \log (3 x+2)}{7058940} \]

Antiderivative was successfully verified.

[In]  Integrate[(3 + 5*x)/((1 - 2*x)*(2 + 3*x)^6),x]

[Out]

((-7*(348226 + 1268025*x + 1833480*x^2 + 1389960*x^3 + 427680*x^4))/(2 + 3*x)^5
- 10560*Log[3 - 6*x] + 10560*Log[2 + 3*x])/7058940

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Maple [A]  time = 0.013, size = 63, normalized size = 0.8 \[{\frac{1}{105\, \left ( 2+3\,x \right ) ^{5}}}-{\frac{11}{196\, \left ( 2+3\,x \right ) ^{4}}}-{\frac{22}{1029\, \left ( 2+3\,x \right ) ^{3}}}-{\frac{22}{2401\, \left ( 2+3\,x \right ) ^{2}}}-{\frac{88}{33614+50421\,x}}+{\frac{176\,\ln \left ( 2+3\,x \right ) }{117649}}-{\frac{176\,\ln \left ( -1+2\,x \right ) }{117649}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((3+5*x)/(1-2*x)/(2+3*x)^6,x)

[Out]

1/105/(2+3*x)^5-11/196/(2+3*x)^4-22/1029/(2+3*x)^3-22/2401/(2+3*x)^2-88/16807/(2
+3*x)+176/117649*ln(2+3*x)-176/117649*ln(-1+2*x)

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Maxima [A]  time = 1.33531, size = 89, normalized size = 1.17 \[ -\frac{427680 \, x^{4} + 1389960 \, x^{3} + 1833480 \, x^{2} + 1268025 \, x + 348226}{1008420 \,{\left (243 \, x^{5} + 810 \, x^{4} + 1080 \, x^{3} + 720 \, x^{2} + 240 \, x + 32\right )}} + \frac{176}{117649} \, \log \left (3 \, x + 2\right ) - \frac{176}{117649} \, \log \left (2 \, x - 1\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(-(5*x + 3)/((3*x + 2)^6*(2*x - 1)),x, algorithm="maxima")

[Out]

-1/1008420*(427680*x^4 + 1389960*x^3 + 1833480*x^2 + 1268025*x + 348226)/(243*x^
5 + 810*x^4 + 1080*x^3 + 720*x^2 + 240*x + 32) + 176/117649*log(3*x + 2) - 176/1
17649*log(2*x - 1)

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Fricas [A]  time = 0.224891, size = 155, normalized size = 2.04 \[ -\frac{2993760 \, x^{4} + 9729720 \, x^{3} + 12834360 \, x^{2} - 10560 \,{\left (243 \, x^{5} + 810 \, x^{4} + 1080 \, x^{3} + 720 \, x^{2} + 240 \, x + 32\right )} \log \left (3 \, x + 2\right ) + 10560 \,{\left (243 \, x^{5} + 810 \, x^{4} + 1080 \, x^{3} + 720 \, x^{2} + 240 \, x + 32\right )} \log \left (2 \, x - 1\right ) + 8876175 \, x + 2437582}{7058940 \,{\left (243 \, x^{5} + 810 \, x^{4} + 1080 \, x^{3} + 720 \, x^{2} + 240 \, x + 32\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(-(5*x + 3)/((3*x + 2)^6*(2*x - 1)),x, algorithm="fricas")

[Out]

-1/7058940*(2993760*x^4 + 9729720*x^3 + 12834360*x^2 - 10560*(243*x^5 + 810*x^4
+ 1080*x^3 + 720*x^2 + 240*x + 32)*log(3*x + 2) + 10560*(243*x^5 + 810*x^4 + 108
0*x^3 + 720*x^2 + 240*x + 32)*log(2*x - 1) + 8876175*x + 2437582)/(243*x^5 + 810
*x^4 + 1080*x^3 + 720*x^2 + 240*x + 32)

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Sympy [A]  time = 0.486902, size = 65, normalized size = 0.86 \[ - \frac{427680 x^{4} + 1389960 x^{3} + 1833480 x^{2} + 1268025 x + 348226}{245046060 x^{5} + 816820200 x^{4} + 1089093600 x^{3} + 726062400 x^{2} + 242020800 x + 32269440} - \frac{176 \log{\left (x - \frac{1}{2} \right )}}{117649} + \frac{176 \log{\left (x + \frac{2}{3} \right )}}{117649} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((3+5*x)/(1-2*x)/(2+3*x)**6,x)

[Out]

-(427680*x**4 + 1389960*x**3 + 1833480*x**2 + 1268025*x + 348226)/(245046060*x**
5 + 816820200*x**4 + 1089093600*x**3 + 726062400*x**2 + 242020800*x + 32269440)
- 176*log(x - 1/2)/117649 + 176*log(x + 2/3)/117649

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GIAC/XCAS [A]  time = 0.210746, size = 65, normalized size = 0.86 \[ -\frac{427680 \, x^{4} + 1389960 \, x^{3} + 1833480 \, x^{2} + 1268025 \, x + 348226}{1008420 \,{\left (3 \, x + 2\right )}^{5}} + \frac{176}{117649} \,{\rm ln}\left ({\left | 3 \, x + 2 \right |}\right ) - \frac{176}{117649} \,{\rm ln}\left ({\left | 2 \, x - 1 \right |}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(-(5*x + 3)/((3*x + 2)^6*(2*x - 1)),x, algorithm="giac")

[Out]

-1/1008420*(427680*x^4 + 1389960*x^3 + 1833480*x^2 + 1268025*x + 348226)/(3*x +
2)^5 + 176/117649*ln(abs(3*x + 2)) - 176/117649*ln(abs(2*x - 1))