3.1612 \(\int \frac{1}{(1-2 x)^2 (2+3 x) (3+5 x)^2} \, dx\)

Optimal. Leaf size=53 \[ \frac{4}{847 (1-2 x)}-\frac{25}{121 (5 x+3)}-\frac{412 \log (1-2 x)}{65219}+\frac{27}{49} \log (3 x+2)-\frac{725 \log (5 x+3)}{1331} \]

[Out]

4/(847*(1 - 2*x)) - 25/(121*(3 + 5*x)) - (412*Log[1 - 2*x])/65219 + (27*Log[2 + 3*x])/49 - (725*Log[3 + 5*x])/
1331

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Rubi [A]  time = 0.0252081, antiderivative size = 53, 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} \[ \frac{4}{847 (1-2 x)}-\frac{25}{121 (5 x+3)}-\frac{412 \log (1-2 x)}{65219}+\frac{27}{49} \log (3 x+2)-\frac{725 \log (5 x+3)}{1331} \]

Antiderivative was successfully verified.

[In]

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

[Out]

4/(847*(1 - 2*x)) - 25/(121*(3 + 5*x)) - (412*Log[1 - 2*x])/65219 + (27*Log[2 + 3*x])/49 - (725*Log[3 + 5*x])/
1331

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{1}{(1-2 x)^2 (2+3 x) (3+5 x)^2} \, dx &=\int \left (\frac{8}{847 (-1+2 x)^2}-\frac{824}{65219 (-1+2 x)}+\frac{81}{49 (2+3 x)}+\frac{125}{121 (3+5 x)^2}-\frac{3625}{1331 (3+5 x)}\right ) \, dx\\ &=\frac{4}{847 (1-2 x)}-\frac{25}{121 (3+5 x)}-\frac{412 \log (1-2 x)}{65219}+\frac{27}{49} \log (2+3 x)-\frac{725 \log (3+5 x)}{1331}\\ \end{align*}

Mathematica [A]  time = 0.0318662, size = 48, normalized size = 0.91 \[ \frac{-\frac{77 (370 x-163)}{10 x^2+x-3}-412 \log (3-6 x)+35937 \log (3 x+2)-35525 \log (-3 (5 x+3))}{65219} \]

Antiderivative was successfully verified.

[In]

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

[Out]

((-77*(-163 + 370*x))/(-3 + x + 10*x^2) - 412*Log[3 - 6*x] + 35937*Log[2 + 3*x] - 35525*Log[-3*(3 + 5*x)])/652
19

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Maple [A]  time = 0.01, size = 44, normalized size = 0.8 \begin{align*} -{\frac{4}{1694\,x-847}}-{\frac{412\,\ln \left ( 2\,x-1 \right ) }{65219}}+{\frac{27\,\ln \left ( 2+3\,x \right ) }{49}}-{\frac{25}{363+605\,x}}-{\frac{725\,\ln \left ( 3+5\,x \right ) }{1331}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-4/847/(2*x-1)-412/65219*ln(2*x-1)+27/49*ln(2+3*x)-25/121/(3+5*x)-725/1331*ln(3+5*x)

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Maxima [A]  time = 1.95621, size = 57, normalized size = 1.08 \begin{align*} -\frac{370 \, x - 163}{847 \,{\left (10 \, x^{2} + x - 3\right )}} - \frac{725}{1331} \, \log \left (5 \, x + 3\right ) + \frac{27}{49} \, \log \left (3 \, x + 2\right ) - \frac{412}{65219} \, \log \left (2 \, x - 1\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/847*(370*x - 163)/(10*x^2 + x - 3) - 725/1331*log(5*x + 3) + 27/49*log(3*x + 2) - 412/65219*log(2*x - 1)

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Fricas [A]  time = 1.47198, size = 212, normalized size = 4. \begin{align*} -\frac{35525 \,{\left (10 \, x^{2} + x - 3\right )} \log \left (5 \, x + 3\right ) - 35937 \,{\left (10 \, x^{2} + x - 3\right )} \log \left (3 \, x + 2\right ) + 412 \,{\left (10 \, x^{2} + x - 3\right )} \log \left (2 \, x - 1\right ) + 28490 \, x - 12551}{65219 \,{\left (10 \, x^{2} + x - 3\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/65219*(35525*(10*x^2 + x - 3)*log(5*x + 3) - 35937*(10*x^2 + x - 3)*log(3*x + 2) + 412*(10*x^2 + x - 3)*log
(2*x - 1) + 28490*x - 12551)/(10*x^2 + x - 3)

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Sympy [A]  time = 0.181762, size = 44, normalized size = 0.83 \begin{align*} - \frac{370 x - 163}{8470 x^{2} + 847 x - 2541} - \frac{412 \log{\left (x - \frac{1}{2} \right )}}{65219} - \frac{725 \log{\left (x + \frac{3}{5} \right )}}{1331} + \frac{27 \log{\left (x + \frac{2}{3} \right )}}{49} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-(370*x - 163)/(8470*x**2 + 847*x - 2541) - 412*log(x - 1/2)/65219 - 725*log(x + 3/5)/1331 + 27*log(x + 2/3)/4
9

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Giac [A]  time = 3.28268, size = 74, normalized size = 1.4 \begin{align*} -\frac{25}{121 \,{\left (5 \, x + 3\right )}} + \frac{40}{9317 \,{\left (\frac{11}{5 \, x + 3} - 2\right )}} + \frac{27}{49} \, \log \left ({\left | -\frac{1}{5 \, x + 3} - 3 \right |}\right ) - \frac{412}{65219} \, \log \left ({\left | -\frac{11}{5 \, x + 3} + 2 \right |}\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-25/121/(5*x + 3) + 40/9317/(11/(5*x + 3) - 2) + 27/49*log(abs(-1/(5*x + 3) - 3)) - 412/65219*log(abs(-11/(5*x
+ 3) + 2))