3.1605 \(\int \frac{(2+3 x)^6}{(1-2 x)^2 (3+5 x)^2} \, dx\)

Optimal. Leaf size=62 \[ \frac{243 x^3}{100}+\frac{13851 x^2}{1000}+\frac{473607 x}{10000}+\frac{117649}{3872 (1-2 x)}-\frac{1}{378125 (5 x+3)}+\frac{67228 \log (1-2 x)}{1331}+\frac{202 \log (5 x+3)}{4159375} \]

[Out]

117649/(3872*(1 - 2*x)) + (473607*x)/10000 + (13851*x^2)/1000 + (243*x^3)/100 - 1/(378125*(3 + 5*x)) + (67228*
Log[1 - 2*x])/1331 + (202*Log[3 + 5*x])/4159375

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Rubi [A]  time = 0.0298386, antiderivative size = 62, 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} \[ \frac{243 x^3}{100}+\frac{13851 x^2}{1000}+\frac{473607 x}{10000}+\frac{117649}{3872 (1-2 x)}-\frac{1}{378125 (5 x+3)}+\frac{67228 \log (1-2 x)}{1331}+\frac{202 \log (5 x+3)}{4159375} \]

Antiderivative was successfully verified.

[In]

Int[(2 + 3*x)^6/((1 - 2*x)^2*(3 + 5*x)^2),x]

[Out]

117649/(3872*(1 - 2*x)) + (473607*x)/10000 + (13851*x^2)/1000 + (243*x^3)/100 - 1/(378125*(3 + 5*x)) + (67228*
Log[1 - 2*x])/1331 + (202*Log[3 + 5*x])/4159375

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)^6}{(1-2 x)^2 (3+5 x)^2} \, dx &=\int \left (\frac{473607}{10000}+\frac{13851 x}{500}+\frac{729 x^2}{100}+\frac{117649}{1936 (-1+2 x)^2}+\frac{134456}{1331 (-1+2 x)}+\frac{1}{75625 (3+5 x)^2}+\frac{202}{831875 (3+5 x)}\right ) \, dx\\ &=\frac{117649}{3872 (1-2 x)}+\frac{473607 x}{10000}+\frac{13851 x^2}{1000}+\frac{243 x^3}{100}-\frac{1}{378125 (3+5 x)}+\frac{67228 \log (1-2 x)}{1331}+\frac{202 \log (3+5 x)}{4159375}\\ \end{align*}

Mathematica [A]  time = 0.0319095, size = 65, normalized size = 1.05 \[ \frac{-\frac{11 (1838265689 x+1102959343)}{10 x^2+x-3}[phone] (3 x+2)^3+132966900 (3 x+2)^2+1425620790 (3 x+2)[phone] \log (3-6 x)+6464 \log (-3 (5 x+3))}{133100000} \]

Antiderivative was successfully verified.

[In]

Integrate[(2 + 3*x)^6/((1 - 2*x)^2*(3 + 5*x)^2),x]

[Out]

(1425620790*(2 + 3*x) + 132966900*(2 + 3*x)^2 + 11979000*(2 + 3*x)^3 - (11*(1102959343 + 1838265689*x))/(-3 +
x + 10*x^2) + 6722800000*Log[3 - 6*x] + 6464*Log[-3*(3 + 5*x)])/133100000

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Maple [A]  time = 0.009, size = 49, normalized size = 0.8 \begin{align*}{\frac{243\,{x}^{3}}{100}}+{\frac{13851\,{x}^{2}}{1000}}+{\frac{473607\,x}{10000}}-{\frac{117649}{7744\,x-3872}}+{\frac{67228\,\ln \left ( 2\,x-1 \right ) }{1331}}-{\frac{1}{1134375+1890625\,x}}+{\frac{202\,\ln \left ( 3+5\,x \right ) }{4159375}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((2+3*x)^6/(1-2*x)^2/(3+5*x)^2,x)

[Out]

243/100*x^3+13851/1000*x^2+473607/10000*x-117649/3872/(2*x-1)+67228/1331*ln(2*x-1)-1/378125/(3+5*x)+202/415937
5*ln(3+5*x)

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Maxima [A]  time = 1.05718, size = 63, normalized size = 1.02 \begin{align*} \frac{243}{100} \, x^{3} + \frac{13851}{1000} \, x^{2} + \frac{473607}{10000} \, x - \frac{1838265689 \, x + 1102959343}{12100000 \,{\left (10 \, x^{2} + x - 3\right )}} + \frac{202}{4159375} \, \log \left (5 \, x + 3\right ) + \frac{67228}{1331} \, \log \left (2 \, x - 1\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((2+3*x)^6/(1-2*x)^2/(3+5*x)^2,x, algorithm="maxima")

[Out]

243/100*x^3 + 13851/1000*x^2 + 473607/10000*x - 1/12100000*(1838265689*x + 1102959343)/(10*x^2 + x - 3) + 202/
4159375*log(5*x + 3) + 67228/1331*log(2*x - 1)

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Fricas [A]  time = 1.32732, size = 282, normalized size = 4.55 \begin{align*} \frac{3234330000 \, x^{5} + 18759114000 \, x^{4} + 63910360800 \, x^{3} + 773004870 \, x^{2} + 6464 \,{\left (10 \, x^{2} + x - 3\right )} \log \left (5 \, x + 3\right ) + 6722800000 \,{\left (10 \, x^{2} + x - 3\right )} \log \left (2 \, x - 1\right ) - 39132050089 \, x - 12132552773}{133100000 \,{\left (10 \, x^{2} + x - 3\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((2+3*x)^6/(1-2*x)^2/(3+5*x)^2,x, algorithm="fricas")

[Out]

1/133100000*(3234330000*x^5 + 18759114000*x^4 + 63910360800*x^3 + 773004870*x^2 + 6464*(10*x^2 + x - 3)*log(5*
x + 3) + 6722800000*(10*x^2 + x - 3)*log(2*x - 1) - 39132050089*x - 12132552773)/(10*x^2 + x - 3)

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Sympy [A]  time = 0.161792, size = 53, normalized size = 0.85 \begin{align*} \frac{243 x^{3}}{100} + \frac{13851 x^{2}}{1000} + \frac{473607 x}{10000} - \frac{1838265689 x + 1102959343}{121000000 x^{2} + 12100000 x - 36300000} + \frac{67228 \log{\left (x - \frac{1}{2} \right )}}{1331} + \frac{202 \log{\left (x + \frac{3}{5} \right )}}{4159375} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((2+3*x)**6/(1-2*x)**2/(3+5*x)**2,x)

[Out]

243*x**3/100 + 13851*x**2/1000 + 473607*x/10000 - (1838265689*x + 1102959343)/(121000000*x**2 + 12100000*x - 3
6300000) + 67228*log(x - 1/2)/1331 + 202*log(x + 3/5)/4159375

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Giac [A]  time = 1.95496, size = 127, normalized size = 2.05 \begin{align*} -\frac{{\left (5 \, x + 3\right )}^{3}{\left (\frac{4528062}{5 \, x + 3} + \frac{76330188}{{\left (5 \, x + 3\right )}^{2}} - \frac{840384278}{{\left (5 \, x + 3\right )}^{3}} + 323433\right )}}{8318750 \,{\left (\frac{11}{5 \, x + 3} - 2\right )}} - \frac{1}{378125 \,{\left (5 \, x + 3\right )}} - \frac{157842}{3125} \, \log \left (\frac{{\left | 5 \, x + 3 \right |}}{5 \,{\left (5 \, x + 3\right )}^{2}}\right ) + \frac{67228}{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)^6/(1-2*x)^2/(3+5*x)^2,x, algorithm="giac")

[Out]

-1/8318750*(5*x + 3)^3*(4528062/(5*x + 3) + 76330188/(5*x + 3)^2 - 840384278/(5*x + 3)^3 + 323433)/(11/(5*x +
3) - 2) - 1/378125/(5*x + 3) - 157842/3125*log(1/5*abs(5*x + 3)/(5*x + 3)^2) + 67228/1331*log(abs(-11/(5*x + 3
) + 2))