3.1702 \(\int \frac{(2+3 x)^4}{(1-2 x)^3 (3+5 x)^3} \, dx\)

Optimal. Leaf size=65 \[ -\frac{9261}{58564 (1-2 x)}-\frac{138}{366025 (5 x+3)}+\frac{2401}{10648 (1-2 x)^2}-\frac{1}{66550 (5 x+3)^2}-\frac{294 \log (1-2 x)}{161051}+\frac{294 \log (5 x+3)}{161051} \]

[Out]

2401/(10648*(1 - 2*x)^2) - 9261/(58564*(1 - 2*x)) - 1/(66550*(3 + 5*x)^2) - 138/(366025*(3 + 5*x)) - (294*Log[
1 - 2*x])/161051 + (294*Log[3 + 5*x])/161051

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Rubi [A]  time = 0.0318931, 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} \[ -\frac{9261}{58564 (1-2 x)}-\frac{138}{366025 (5 x+3)}+\frac{2401}{10648 (1-2 x)^2}-\frac{1}{66550 (5 x+3)^2}-\frac{294 \log (1-2 x)}{161051}+\frac{294 \log (5 x+3)}{161051} \]

Antiderivative was successfully verified.

[In]

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

[Out]

2401/(10648*(1 - 2*x)^2) - 9261/(58564*(1 - 2*x)) - 1/(66550*(3 + 5*x)^2) - 138/(366025*(3 + 5*x)) - (294*Log[
1 - 2*x])/161051 + (294*Log[3 + 5*x])/161051

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)^4}{(1-2 x)^3 (3+5 x)^3} \, dx &=\int \left (-\frac{2401}{2662 (-1+2 x)^3}-\frac{9261}{29282 (-1+2 x)^2}-\frac{588}{161051 (-1+2 x)}+\frac{1}{6655 (3+5 x)^3}+\frac{138}{73205 (3+5 x)^2}+\frac{1470}{161051 (3+5 x)}\right ) \, dx\\ &=\frac{2401}{10648 (1-2 x)^2}-\frac{9261}{58564 (1-2 x)}-\frac{1}{66550 (3+5 x)^2}-\frac{138}{366025 (3+5 x)}-\frac{294 \log (1-2 x)}{161051}+\frac{294 \log (3+5 x)}{161051}\\ \end{align*}

Mathematica [A]  time = 0.0284607, size = 48, normalized size = 0.74 \[ \frac{\frac{11 \left (23130420 x^3+32722281 x^2+14259554 x+1771669\right )}{\left (10 x^2+x-3\right )^2}+58800 \log (-5 x-3)-58800 \log (1-2 x)}{32210200} \]

Antiderivative was successfully verified.

[In]

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

[Out]

((11*(1771669 + 14259554*x + 32722281*x^2 + 23130420*x^3))/(-3 + x + 10*x^2)^2 + 58800*Log[-3 - 5*x] - 58800*L
og[1 - 2*x])/32210200

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Maple [A]  time = 0.009, size = 54, normalized size = 0.8 \begin{align*}{\frac{2401}{10648\, \left ( 2\,x-1 \right ) ^{2}}}+{\frac{9261}{117128\,x-58564}}-{\frac{294\,\ln \left ( 2\,x-1 \right ) }{161051}}-{\frac{1}{66550\, \left ( 3+5\,x \right ) ^{2}}}-{\frac{138}{1098075+1830125\,x}}+{\frac{294\,\ln \left ( 3+5\,x \right ) }{161051}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2401/10648/(2*x-1)^2+9261/58564/(2*x-1)-294/161051*ln(2*x-1)-1/66550/(3+5*x)^2-138/366025/(3+5*x)+294/161051*l
n(3+5*x)

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Maxima [A]  time = 1.05521, size = 76, normalized size = 1.17 \begin{align*} \frac{23130420 \, x^{3} + 32722281 \, x^{2} + 14259554 \, x + 1771669}{2928200 \,{\left (100 \, x^{4} + 20 \, x^{3} - 59 \, x^{2} - 6 \, x + 9\right )}} + \frac{294}{161051} \, \log \left (5 \, x + 3\right ) - \frac{294}{161051} \, \log \left (2 \, x - 1\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/2928200*(23130420*x^3 + 32722281*x^2 + 14259554*x + 1771669)/(100*x^4 + 20*x^3 - 59*x^2 - 6*x + 9) + 294/161
051*log(5*x + 3) - 294/161051*log(2*x - 1)

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Fricas [A]  time = 1.48406, size = 304, normalized size = 4.68 \begin{align*} \frac{254434620 \, x^{3} + 359945091 \, x^{2} + 58800 \,{\left (100 \, x^{4} + 20 \, x^{3} - 59 \, x^{2} - 6 \, x + 9\right )} \log \left (5 \, x + 3\right ) - 58800 \,{\left (100 \, x^{4} + 20 \, x^{3} - 59 \, x^{2} - 6 \, x + 9\right )} \log \left (2 \, x - 1\right ) + 156855094 \, x + 19488359}{32210200 \,{\left (100 \, x^{4} + 20 \, x^{3} - 59 \, x^{2} - 6 \, x + 9\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/32210200*(254434620*x^3 + 359945091*x^2 + 58800*(100*x^4 + 20*x^3 - 59*x^2 - 6*x + 9)*log(5*x + 3) - 58800*(
100*x^4 + 20*x^3 - 59*x^2 - 6*x + 9)*log(2*x - 1) + 156855094*x + 19488359)/(100*x^4 + 20*x^3 - 59*x^2 - 6*x +
9)

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Sympy [A]  time = 0.175378, size = 54, normalized size = 0.83 \begin{align*} \frac{23130420 x^{3} + 32722281 x^{2} + 14259554 x + 1771669}{292820000 x^{4} + 58564000 x^{3} - 172763800 x^{2} - 17569200 x + 26353800} - \frac{294 \log{\left (x - \frac{1}{2} \right )}}{161051} + \frac{294 \log{\left (x + \frac{3}{5} \right )}}{161051} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

(23130420*x**3 + 32722281*x**2 + 14259554*x + 1771669)/(292820000*x**4 + 58564000*x**3 - 172763800*x**2 - 1756
9200*x + 26353800) - 294*log(x - 1/2)/161051 + 294*log(x + 3/5)/161051

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Giac [A]  time = 2.74217, size = 62, normalized size = 0.95 \begin{align*} \frac{23130420 \, x^{3} + 32722281 \, x^{2} + 14259554 \, x + 1771669}{2928200 \,{\left (10 \, x^{2} + x - 3\right )}^{2}} + \frac{294}{161051} \, \log \left ({\left | 5 \, x + 3 \right |}\right ) - \frac{294}{161051} \, \log \left ({\left | 2 \, x - 1 \right |}\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/2928200*(23130420*x^3 + 32722281*x^2 + 14259554*x + 1771669)/(10*x^2 + x - 3)^2 + 294/161051*log(abs(5*x + 3
)) - 294/161051*log(abs(2*x - 1))