3.1690 \(\int \frac{(2+3 x)^2}{(1-2 x)^3 (3+5 x)^2} \, dx\)

Optimal. Leaf size=54 \[ \frac{14}{1331 (1-2 x)}-\frac{1}{1331 (5 x+3)}+\frac{49}{484 (1-2 x)^2}-\frac{72 \log (1-2 x)}{14641}+\frac{72 \log (5 x+3)}{14641} \]

[Out]

49/(484*(1 - 2*x)^2) + 14/(1331*(1 - 2*x)) - 1/(1331*(3 + 5*x)) - (72*Log[1 - 2*x])/14641 + (72*Log[3 + 5*x])/
14641

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Rubi [A]  time = 0.0240991, antiderivative size = 54, 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} \[ \frac{14}{1331 (1-2 x)}-\frac{1}{1331 (5 x+3)}+\frac{49}{484 (1-2 x)^2}-\frac{72 \log (1-2 x)}{14641}+\frac{72 \log (5 x+3)}{14641} \]

Antiderivative was successfully verified.

[In]

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

[Out]

49/(484*(1 - 2*x)^2) + 14/(1331*(1 - 2*x)) - 1/(1331*(3 + 5*x)) - (72*Log[1 - 2*x])/14641 + (72*Log[3 + 5*x])/
14641

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)^3 (3+5 x)^2} \, dx &=\int \left (-\frac{49}{121 (-1+2 x)^3}+\frac{28}{1331 (-1+2 x)^2}-\frac{144}{14641 (-1+2 x)}+\frac{5}{1331 (3+5 x)^2}+\frac{360}{14641 (3+5 x)}\right ) \, dx\\ &=\frac{49}{484 (1-2 x)^2}+\frac{14}{1331 (1-2 x)}-\frac{1}{1331 (3+5 x)}-\frac{72 \log (1-2 x)}{14641}+\frac{72 \log (3+5 x)}{14641}\\ \end{align*}

Mathematica [A]  time = 0.0292258, size = 48, normalized size = 0.89 \[ \frac{\frac{616}{1-2 x}-\frac{44}{5 x+3}+\frac{5929}{(1-2 x)^2}-288 \log (1-2 x)+288 \log (10 x+6)}{58564} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(5929/(1 - 2*x)^2 + 616/(1 - 2*x) - 44/(3 + 5*x) - 288*Log[1 - 2*x] + 288*Log[6 + 10*x])/58564

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Maple [A]  time = 0.008, size = 45, normalized size = 0.8 \begin{align*}{\frac{49}{484\, \left ( 2\,x-1 \right ) ^{2}}}-{\frac{14}{2662\,x-1331}}-{\frac{72\,\ln \left ( 2\,x-1 \right ) }{14641}}-{\frac{1}{3993+6655\,x}}+{\frac{72\,\ln \left ( 3+5\,x \right ) }{14641}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

49/484/(2*x-1)^2-14/1331/(2*x-1)-72/14641*ln(2*x-1)-1/1331/(3+5*x)+72/14641*ln(3+5*x)

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Maxima [A]  time = 1.06428, size = 62, normalized size = 1.15 \begin{align*} -\frac{576 \, x^{2} - 2655 \, x - 1781}{5324 \,{\left (20 \, x^{3} - 8 \, x^{2} - 7 \, x + 3\right )}} + \frac{72}{14641} \, \log \left (5 \, x + 3\right ) - \frac{72}{14641} \, \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)^3/(3+5*x)^2,x, algorithm="maxima")

[Out]

-1/5324*(576*x^2 - 2655*x - 1781)/(20*x^3 - 8*x^2 - 7*x + 3) + 72/14641*log(5*x + 3) - 72/14641*log(2*x - 1)

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Fricas [A]  time = 1.58147, size = 213, normalized size = 3.94 \begin{align*} -\frac{6336 \, x^{2} - 288 \,{\left (20 \, x^{3} - 8 \, x^{2} - 7 \, x + 3\right )} \log \left (5 \, x + 3\right ) + 288 \,{\left (20 \, x^{3} - 8 \, x^{2} - 7 \, x + 3\right )} \log \left (2 \, x - 1\right ) - 29205 \, x - 19591}{58564 \,{\left (20 \, x^{3} - 8 \, x^{2} - 7 \, x + 3\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/58564*(6336*x^2 - 288*(20*x^3 - 8*x^2 - 7*x + 3)*log(5*x + 3) + 288*(20*x^3 - 8*x^2 - 7*x + 3)*log(2*x - 1)
- 29205*x - 19591)/(20*x^3 - 8*x^2 - 7*x + 3)

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Sympy [A]  time = 0.156859, size = 44, normalized size = 0.81 \begin{align*} - \frac{576 x^{2} - 2655 x - 1781}{106480 x^{3} - 42592 x^{2} - 37268 x + 15972} - \frac{72 \log{\left (x - \frac{1}{2} \right )}}{14641} + \frac{72 \log{\left (x + \frac{3}{5} \right )}}{14641} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-(576*x**2 - 2655*x - 1781)/(106480*x**3 - 42592*x**2 - 37268*x + 15972) - 72*log(x - 1/2)/14641 + 72*log(x +
3/5)/14641

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Giac [A]  time = 3.17162, size = 69, normalized size = 1.28 \begin{align*} -\frac{1}{1331 \,{\left (5 \, x + 3\right )}} + \frac{35 \,{\left (\frac{429}{5 \, x + 3} - 43\right )}}{14641 \,{\left (\frac{11}{5 \, x + 3} - 2\right )}^{2}} - \frac{72}{14641} \, \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)^3/(3+5*x)^2,x, algorithm="giac")

[Out]

-1/1331/(5*x + 3) + 35/14641*(429/(5*x + 3) - 43)/(11/(5*x + 3) - 2)^2 - 72/14641*log(abs(-11/(5*x + 3) + 2))