3.1591 \(\int \frac{(2+3 x)^5}{(1-2 x)^2 (3+5 x)} \, dx\)

Optimal. Leaf size=51 \[ \frac{81 x^3}{20}+\frac{567 x^2}{25}+\frac{152793 x}{2000}+\frac{16807}{352 (1-2 x)}+\frac{156065 \log (1-2 x)}{1936}+\frac{\log (5 x+3)}{75625} \]

[Out]

16807/(352*(1 - 2*x)) + (152793*x)/2000 + (567*x^2)/25 + (81*x^3)/20 + (156065*Log[1 - 2*x])/1936 + Log[3 + 5*
x]/75625

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Rubi [A]  time = 0.0247458, antiderivative size = 51, 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} \[ \frac{81 x^3}{20}+\frac{567 x^2}{25}+\frac{152793 x}{2000}+\frac{16807}{352 (1-2 x)}+\frac{156065 \log (1-2 x)}{1936}+\frac{\log (5 x+3)}{75625} \]

Antiderivative was successfully verified.

[In]

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

[Out]

16807/(352*(1 - 2*x)) + (152793*x)/2000 + (567*x^2)/25 + (81*x^3)/20 + (156065*Log[1 - 2*x])/1936 + Log[3 + 5*
x]/75625

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)^5}{(1-2 x)^2 (3+5 x)} \, dx &=\int \left (\frac{152793}{2000}+\frac{1134 x}{25}+\frac{243 x^2}{20}+\frac{16807}{176 (-1+2 x)^2}+\frac{156065}{968 (-1+2 x)}+\frac{1}{15125 (3+5 x)}\right ) \, dx\\ &=\frac{16807}{352 (1-2 x)}+\frac{152793 x}{2000}+\frac{567 x^2}{25}+\frac{81 x^3}{20}+\frac{156065 \log (1-2 x)}{1936}+\frac{\log (3+5 x)}{75625}\\ \end{align*}

Mathematica [A]  time = 0.0238067, size = 52, normalized size = 1.02 \[ \frac{81 x^3}{20}+\frac{567 x^2}{25}+\frac{152793 x}{2000}+\frac{16807}{352-704 x}+\frac{156065 \log (5-10 x)}{1936}+\frac{\log (5 x+3)}{75625}+\frac{385479}{10000} \]

Antiderivative was successfully verified.

[In]

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

[Out]

385479/10000 + 16807/(352 - 704*x) + (152793*x)/2000 + (567*x^2)/25 + (81*x^3)/20 + (156065*Log[5 - 10*x])/193
6 + Log[3 + 5*x]/75625

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Maple [A]  time = 0.006, size = 40, normalized size = 0.8 \begin{align*}{\frac{81\,{x}^{3}}{20}}+{\frac{567\,{x}^{2}}{25}}+{\frac{152793\,x}{2000}}-{\frac{16807}{704\,x-352}}+{\frac{156065\,\ln \left ( 2\,x-1 \right ) }{1936}}+{\frac{\ln \left ( 3+5\,x \right ) }{75625}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

81/20*x^3+567/25*x^2+152793/2000*x-16807/352/(2*x-1)+156065/1936*ln(2*x-1)+1/75625*ln(3+5*x)

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Maxima [A]  time = 1.33902, size = 53, normalized size = 1.04 \begin{align*} \frac{81}{20} \, x^{3} + \frac{567}{25} \, x^{2} + \frac{152793}{2000} \, x - \frac{16807}{352 \,{\left (2 \, x - 1\right )}} + \frac{1}{75625} \, \log \left (5 \, x + 3\right ) + \frac{156065}{1936} \, \log \left (2 \, x - 1\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

81/20*x^3 + 567/25*x^2 + 152793/2000*x - 16807/352/(2*x - 1) + 1/75625*log(5*x + 3) + 156065/1936*log(2*x - 1)

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Fricas [A]  time = 1.32327, size = 211, normalized size = 4.14 \begin{align*} \frac{19602000 \, x^{4} + 99970200 \, x^{3} + 314873460 \, x^{2} + 32 \,{\left (2 \, x - 1\right )} \log \left (5 \, x + 3\right ) + 195081250 \,{\left (2 \, x - 1\right )} \log \left (2 \, x - 1\right ) - 184879530 \, x - 115548125}{2420000 \,{\left (2 \, x - 1\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/2420000*(19602000*x^4 + 99970200*x^3 + 314873460*x^2 + 32*(2*x - 1)*log(5*x + 3) + 195081250*(2*x - 1)*log(2
*x - 1) - 184879530*x - 115548125)/(2*x - 1)

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Sympy [A]  time = 0.134243, size = 42, normalized size = 0.82 \begin{align*} \frac{81 x^{3}}{20} + \frac{567 x^{2}}{25} + \frac{152793 x}{2000} + \frac{156065 \log{\left (x - \frac{1}{2} \right )}}{1936} + \frac{\log{\left (x + \frac{3}{5} \right )}}{75625} - \frac{16807}{704 x - 352} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

81*x**3/20 + 567*x**2/25 + 152793*x/2000 + 156065*log(x - 1/2)/1936 + log(x + 3/5)/75625 - 16807/(704*x - 352)

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Giac [A]  time = 1.24423, size = 97, normalized size = 1.9 \begin{align*} \frac{27}{4000} \,{\left (2 \, x - 1\right )}^{3}{\left (\frac{1065}{2 \, x - 1} + \frac{7564}{{\left (2 \, x - 1\right )}^{2}} + 75\right )} - \frac{16807}{352 \,{\left (2 \, x - 1\right )}} - \frac{806121}{10000} \, \log \left (\frac{{\left | 2 \, x - 1 \right |}}{2 \,{\left (2 \, x - 1\right )}^{2}}\right ) + \frac{1}{75625} \, \log \left ({\left | -\frac{11}{2 \, x - 1} - 5 \right |}\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

27/4000*(2*x - 1)^3*(1065/(2*x - 1) + 7564/(2*x - 1)^2 + 75) - 16807/352/(2*x - 1) - 806121/10000*log(1/2*abs(
2*x - 1)/(2*x - 1)^2) + 1/75625*log(abs(-11/(2*x - 1) - 5))