3.1566 \(\int \frac{(3+5 x)^2}{(1-2 x)^2 (2+3 x)^4} \, dx\)

Optimal. Leaf size=65 \[ \frac{242}{2401 (1-2 x)}-\frac{319}{2401 (3 x+2)}+\frac{11}{343 (3 x+2)^2}-\frac{1}{441 (3 x+2)^3}-\frac{1364 \log (1-2 x)}{16807}+\frac{1364 \log (3 x+2)}{16807} \]

[Out]

242/(2401*(1 - 2*x)) - 1/(441*(2 + 3*x)^3) + 11/(343*(2 + 3*x)^2) - 319/(2401*(2 + 3*x)) - (1364*Log[1 - 2*x])
/16807 + (1364*Log[2 + 3*x])/16807

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Rubi [A]  time = 0.0295785, antiderivative size = 65, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 1, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.045, Rules used = {88} \[ \frac{242}{2401 (1-2 x)}-\frac{319}{2401 (3 x+2)}+\frac{11}{343 (3 x+2)^2}-\frac{1}{441 (3 x+2)^3}-\frac{1364 \log (1-2 x)}{16807}+\frac{1364 \log (3 x+2)}{16807} \]

Antiderivative was successfully verified.

[In]

Int[(3 + 5*x)^2/((1 - 2*x)^2*(2 + 3*x)^4),x]

[Out]

242/(2401*(1 - 2*x)) - 1/(441*(2 + 3*x)^3) + 11/(343*(2 + 3*x)^2) - 319/(2401*(2 + 3*x)) - (1364*Log[1 - 2*x])
/16807 + (1364*Log[2 + 3*x])/16807

Rule 88

Int[((a_.) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_.)*((e_.) + (f_.)*(x_))^(p_.), x_Symbol] :> Int[ExpandI
ntegrand[(a + b*x)^m*(c + d*x)^n*(e + f*x)^p, x], x] /; FreeQ[{a, b, c, d, e, f, p}, x] && IntegersQ[m, n] &&
(IntegerQ[p] || (GtQ[m, 0] && GeQ[n, -1]))

Rubi steps

\begin{align*} \int \frac{(3+5 x)^2}{(1-2 x)^2 (2+3 x)^4} \, dx &=\int \left (\frac{484}{2401 (-1+2 x)^2}-\frac{2728}{16807 (-1+2 x)}+\frac{1}{49 (2+3 x)^4}-\frac{66}{343 (2+3 x)^3}+\frac{957}{2401 (2+3 x)^2}+\frac{4092}{16807 (2+3 x)}\right ) \, dx\\ &=\frac{242}{2401 (1-2 x)}-\frac{1}{441 (2+3 x)^3}+\frac{11}{343 (2+3 x)^2}-\frac{319}{2401 (2+3 x)}-\frac{1364 \log (1-2 x)}{16807}+\frac{1364 \log (2+3 x)}{16807}\\ \end{align*}

Mathematica [A]  time = 0.0372448, size = 54, normalized size = 0.83 \[ \frac{2 \left (-\frac{7 \left (110484 x^3+156519 x^2+66329 x+7277\right )}{2 (2 x-1) (3 x+2)^3}-6138 \log (1-2 x)+6138 \log (6 x+4)\right )}{151263} \]

Antiderivative was successfully verified.

[In]

Integrate[(3 + 5*x)^2/((1 - 2*x)^2*(2 + 3*x)^4),x]

[Out]

(2*((-7*(7277 + 66329*x + 156519*x^2 + 110484*x^3))/(2*(-1 + 2*x)*(2 + 3*x)^3) - 6138*Log[1 - 2*x] + 6138*Log[
4 + 6*x]))/151263

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Maple [A]  time = 0.007, size = 54, normalized size = 0.8 \begin{align*} -{\frac{242}{4802\,x-2401}}-{\frac{1364\,\ln \left ( 2\,x-1 \right ) }{16807}}-{\frac{1}{441\, \left ( 2+3\,x \right ) ^{3}}}+{\frac{11}{343\, \left ( 2+3\,x \right ) ^{2}}}-{\frac{319}{4802+7203\,x}}+{\frac{1364\,\ln \left ( 2+3\,x \right ) }{16807}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((3+5*x)^2/(1-2*x)^2/(2+3*x)^4,x)

[Out]

-242/2401/(2*x-1)-1364/16807*ln(2*x-1)-1/441/(2+3*x)^3+11/343/(2+3*x)^2-319/2401/(2+3*x)+1364/16807*ln(2+3*x)

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Maxima [A]  time = 2.10071, size = 76, normalized size = 1.17 \begin{align*} -\frac{110484 \, x^{3} + 156519 \, x^{2} + 66329 \, x + 7277}{21609 \,{\left (54 \, x^{4} + 81 \, x^{3} + 18 \, x^{2} - 20 \, x - 8\right )}} + \frac{1364}{16807} \, \log \left (3 \, x + 2\right ) - \frac{1364}{16807} \, \log \left (2 \, x - 1\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((3+5*x)^2/(1-2*x)^2/(2+3*x)^4,x, algorithm="maxima")

[Out]

-1/21609*(110484*x^3 + 156519*x^2 + 66329*x + 7277)/(54*x^4 + 81*x^3 + 18*x^2 - 20*x - 8) + 1364/16807*log(3*x
+ 2) - 1364/16807*log(2*x - 1)

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Fricas [A]  time = 1.25264, size = 288, normalized size = 4.43 \begin{align*} -\frac{773388 \, x^{3} + 1095633 \, x^{2} - 12276 \,{\left (54 \, x^{4} + 81 \, x^{3} + 18 \, x^{2} - 20 \, x - 8\right )} \log \left (3 \, x + 2\right ) + 12276 \,{\left (54 \, x^{4} + 81 \, x^{3} + 18 \, x^{2} - 20 \, x - 8\right )} \log \left (2 \, x - 1\right ) + 464303 \, x + 50939}{151263 \,{\left (54 \, x^{4} + 81 \, x^{3} + 18 \, x^{2} - 20 \, x - 8\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((3+5*x)^2/(1-2*x)^2/(2+3*x)^4,x, algorithm="fricas")

[Out]

-1/151263*(773388*x^3 + 1095633*x^2 - 12276*(54*x^4 + 81*x^3 + 18*x^2 - 20*x - 8)*log(3*x + 2) + 12276*(54*x^4
+ 81*x^3 + 18*x^2 - 20*x - 8)*log(2*x - 1) + 464303*x + 50939)/(54*x^4 + 81*x^3 + 18*x^2 - 20*x - 8)

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Sympy [A]  time = 0.162151, size = 54, normalized size = 0.83 \begin{align*} - \frac{110484 x^{3} + 156519 x^{2} + 66329 x + 7277}{1166886 x^{4} + 1750329 x^{3} + 388962 x^{2} - 432180 x - 172872} - \frac{1364 \log{\left (x - \frac{1}{2} \right )}}{16807} + \frac{1364 \log{\left (x + \frac{2}{3} \right )}}{16807} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((3+5*x)**2/(1-2*x)**2/(2+3*x)**4,x)

[Out]

-(110484*x**3 + 156519*x**2 + 66329*x + 7277)/(1166886*x**4 + 1750329*x**3 + 388962*x**2 - 432180*x - 172872)
- 1364*log(x - 1/2)/16807 + 1364*log(x + 2/3)/16807

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Giac [A]  time = 2.56663, size = 81, normalized size = 1.25 \begin{align*} -\frac{242}{2401 \,{\left (2 \, x - 1\right )}} + \frac{2 \,{\left (\frac{36120}{2 \, x - 1} + \frac{40621}{{\left (2 \, x - 1\right )}^{2}} + 8031\right )}}{16807 \,{\left (\frac{7}{2 \, x - 1} + 3\right )}^{3}} + \frac{1364}{16807} \, \log \left ({\left | -\frac{7}{2 \, x - 1} - 3 \right |}\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((3+5*x)^2/(1-2*x)^2/(2+3*x)^4,x, algorithm="giac")

[Out]

-242/2401/(2*x - 1) + 2/16807*(36120/(2*x - 1) + 40621/(2*x - 1)^2 + 8031)/(7/(2*x - 1) + 3)^3 + 1364/16807*lo
g(abs(-7/(2*x - 1) - 3))