3.1608 \(\int \frac{(2+3 x)^3}{(1-2 x)^2 (3+5 x)^2} \, dx\)

Optimal. Leaf size=43 \[ \frac{343}{484 (1-2 x)}-\frac{1}{3025 (5 x+3)}+\frac{1421 \log (1-2 x)}{5324}+\frac{103 \log (5 x+3)}{33275} \]

[Out]

343/(484*(1 - 2*x)) - 1/(3025*(3 + 5*x)) + (1421*Log[1 - 2*x])/5324 + (103*Log[3 + 5*x])/33275

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Rubi [A]  time = 0.0210416, antiderivative size = 43, 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} \[ \frac{343}{484 (1-2 x)}-\frac{1}{3025 (5 x+3)}+\frac{1421 \log (1-2 x)}{5324}+\frac{103 \log (5 x+3)}{33275} \]

Antiderivative was successfully verified.

[In]

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

[Out]

343/(484*(1 - 2*x)) - 1/(3025*(3 + 5*x)) + (1421*Log[1 - 2*x])/5324 + (103*Log[3 + 5*x])/33275

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{(2+3 x)^3}{(1-2 x)^2 (3+5 x)^2} \, dx &=\int \left (\frac{343}{242 (-1+2 x)^2}+\frac{1421}{2662 (-1+2 x)}+\frac{1}{605 (3+5 x)^2}+\frac{103}{6655 (3+5 x)}\right ) \, dx\\ &=\frac{343}{484 (1-2 x)}-\frac{1}{3025 (3+5 x)}+\frac{1421 \log (1-2 x)}{5324}+\frac{103 \log (3+5 x)}{33275}\\ \end{align*}

Mathematica [A]  time = 0.0202356, size = 40, normalized size = 0.93 \[ \frac{-\frac{11 (42883 x+25721)}{10 x^2+x-3}+35525 \log (3-6 x)+412 \log (-3 (5 x+3))}{133100} \]

Antiderivative was successfully verified.

[In]

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

[Out]

((-11*(25721 + 42883*x))/(-3 + x + 10*x^2) + 35525*Log[3 - 6*x] + 412*Log[-3*(3 + 5*x)])/133100

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Maple [A]  time = 0.008, size = 36, normalized size = 0.8 \begin{align*} -{\frac{343}{968\,x-484}}+{\frac{1421\,\ln \left ( 2\,x-1 \right ) }{5324}}-{\frac{1}{9075+15125\,x}}+{\frac{103\,\ln \left ( 3+5\,x \right ) }{33275}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-343/484/(2*x-1)+1421/5324*ln(2*x-1)-1/3025/(3+5*x)+103/33275*ln(3+5*x)

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Maxima [A]  time = 1.03717, size = 46, normalized size = 1.07 \begin{align*} -\frac{42883 \, x + 25721}{12100 \,{\left (10 \, x^{2} + x - 3\right )}} + \frac{103}{33275} \, \log \left (5 \, x + 3\right ) + \frac{1421}{5324} \, \log \left (2 \, x - 1\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/12100*(42883*x + 25721)/(10*x^2 + x - 3) + 103/33275*log(5*x + 3) + 1421/5324*log(2*x - 1)

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Fricas [A]  time = 1.16922, size = 163, normalized size = 3.79 \begin{align*} \frac{412 \,{\left (10 \, x^{2} + x - 3\right )} \log \left (5 \, x + 3\right ) + 35525 \,{\left (10 \, x^{2} + x - 3\right )} \log \left (2 \, x - 1\right ) - 471713 \, x - 282931}{133100 \,{\left (10 \, x^{2} + x - 3\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/133100*(412*(10*x^2 + x - 3)*log(5*x + 3) + 35525*(10*x^2 + x - 3)*log(2*x - 1) - 471713*x - 282931)/(10*x^2
+ x - 3)

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Sympy [A]  time = 0.147, size = 34, normalized size = 0.79 \begin{align*} - \frac{42883 x + 25721}{121000 x^{2} + 12100 x - 36300} + \frac{1421 \log{\left (x - \frac{1}{2} \right )}}{5324} + \frac{103 \log{\left (x + \frac{3}{5} \right )}}{33275} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-(42883*x + 25721)/(121000*x**2 + 12100*x - 36300) + 1421*log(x - 1/2)/5324 + 103*log(x + 3/5)/33275

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Giac [A]  time = 3.00674, size = 78, normalized size = 1.81 \begin{align*} -\frac{1}{3025 \,{\left (5 \, x + 3\right )}} + \frac{1715}{2662 \,{\left (\frac{11}{5 \, x + 3} - 2\right )}} - \frac{27}{100} \, \log \left (\frac{{\left | 5 \, x + 3 \right |}}{5 \,{\left (5 \, x + 3\right )}^{2}}\right ) + \frac{1421}{5324} \, \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((2+3*x)^3/(1-2*x)^2/(3+5*x)^2,x, algorithm="giac")

[Out]

-1/3025/(5*x + 3) + 1715/2662/(11/(5*x + 3) - 2) - 27/100*log(1/5*abs(5*x + 3)/(5*x + 3)^2) + 1421/5324*log(ab
s(-11/(5*x + 3) + 2))