3.1609 \(\int \frac{(2+3 x)^2}{(1-2 x)^2 (3+5 x)^2} \, dx\)

Optimal. Leaf size=43 \[ \frac{49}{242 (1-2 x)}-\frac{1}{605 (5 x+3)}-\frac{14 \log (1-2 x)}{1331}+\frac{14 \log (5 x+3)}{1331} \]

[Out]

49/(242*(1 - 2*x)) - 1/(605*(3 + 5*x)) - (14*Log[1 - 2*x])/1331 + (14*Log[3 + 5*x])/1331

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Rubi [A]  time = 0.0184056, 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} \[ \frac{49}{242 (1-2 x)}-\frac{1}{605 (5 x+3)}-\frac{14 \log (1-2 x)}{1331}+\frac{14 \log (5 x+3)}{1331} \]

Antiderivative was successfully verified.

[In]

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

[Out]

49/(242*(1 - 2*x)) - 1/(605*(3 + 5*x)) - (14*Log[1 - 2*x])/1331 + (14*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{(2+3 x)^2}{(1-2 x)^2 (3+5 x)^2} \, dx &=\int \left (\frac{49}{121 (-1+2 x)^2}-\frac{28}{1331 (-1+2 x)}+\frac{1}{121 (3+5 x)^2}+\frac{70}{1331 (3+5 x)}\right ) \, dx\\ &=\frac{49}{242 (1-2 x)}-\frac{1}{605 (3+5 x)}-\frac{14 \log (1-2 x)}{1331}+\frac{14 \log (3+5 x)}{1331}\\ \end{align*}

Mathematica [A]  time = 0.0252977, size = 38, normalized size = 0.88 \[ \frac{-\frac{11 (1229 x+733)}{10 x^2+x-3}+140 \log (-5 x-3)-140 \log (1-2 x)}{13310} \]

Antiderivative was successfully verified.

[In]

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

[Out]

((-11*(733 + 1229*x))/(-3 + x + 10*x^2) + 140*Log[-3 - 5*x] - 140*Log[1 - 2*x])/13310

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Maple [A]  time = 0.008, size = 36, normalized size = 0.8 \begin{align*} -{\frac{49}{484\,x-242}}-{\frac{14\,\ln \left ( 2\,x-1 \right ) }{1331}}-{\frac{1}{1815+3025\,x}}+{\frac{14\,\ln \left ( 3+5\,x \right ) }{1331}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-49/242/(2*x-1)-14/1331*ln(2*x-1)-1/605/(3+5*x)+14/1331*ln(3+5*x)

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Maxima [A]  time = 2.89797, size = 46, normalized size = 1.07 \begin{align*} -\frac{1229 \, x + 733}{1210 \,{\left (10 \, x^{2} + x - 3\right )}} + \frac{14}{1331} \, \log \left (5 \, x + 3\right ) - \frac{14}{1331} \, \log \left (2 \, x - 1\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/1210*(1229*x + 733)/(10*x^2 + x - 3) + 14/1331*log(5*x + 3) - 14/1331*log(2*x - 1)

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Fricas [A]  time = 1.25408, size = 155, normalized size = 3.6 \begin{align*} \frac{140 \,{\left (10 \, x^{2} + x - 3\right )} \log \left (5 \, x + 3\right ) - 140 \,{\left (10 \, x^{2} + x - 3\right )} \log \left (2 \, x - 1\right ) - 13519 \, x - 8063}{13310 \,{\left (10 \, x^{2} + x - 3\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/13310*(140*(10*x^2 + x - 3)*log(5*x + 3) - 140*(10*x^2 + x - 3)*log(2*x - 1) - 13519*x - 8063)/(10*x^2 + x -
3)

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Sympy [A]  time = 0.152339, size = 34, normalized size = 0.79 \begin{align*} - \frac{1229 x + 733}{12100 x^{2} + 1210 x - 3630} - \frac{14 \log{\left (x - \frac{1}{2} \right )}}{1331} + \frac{14 \log{\left (x + \frac{3}{5} \right )}}{1331} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-(1229*x + 733)/(12100*x**2 + 1210*x - 3630) - 14*log(x - 1/2)/1331 + 14*log(x + 3/5)/1331

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Giac [A]  time = 2.78163, size = 54, normalized size = 1.26 \begin{align*} -\frac{1}{605 \,{\left (5 \, x + 3\right )}} + \frac{245}{1331 \,{\left (\frac{11}{5 \, x + 3} - 2\right )}} - \frac{14}{1331} \, \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)^2/(1-2*x)^2/(3+5*x)^2,x, algorithm="giac")

[Out]

-1/605/(5*x + 3) + 245/1331/(11/(5*x + 3) - 2) - 14/1331*log(abs(-11/(5*x + 3) + 2))