3.1504 \(\int \frac{(2+3 x)^5}{(1-2 x) (3+5 x)^2} \, dx\)

Optimal. Leaf size=51 \[ -\frac{81 x^3}{50}-\frac{6399 x^2}{1000}-\frac{69039 x}{5000}-\frac{1}{34375 (5 x+3)}-\frac{16807 \log (1-2 x)}{1936}+\frac{167 \log (5 x+3)}{378125} \]

[Out]

(-69039*x)/5000 - (6399*x^2)/1000 - (81*x^3)/50 - 1/(34375*(3 + 5*x)) - (16807*Log[1 - 2*x])/1936 + (167*Log[3
+ 5*x])/378125

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Rubi [A]  time = 0.0219625, 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} \[ -\frac{81 x^3}{50}-\frac{6399 x^2}{1000}-\frac{69039 x}{5000}-\frac{1}{34375 (5 x+3)}-\frac{16807 \log (1-2 x)}{1936}+\frac{167 \log (5 x+3)}{378125} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(-69039*x)/5000 - (6399*x^2)/1000 - (81*x^3)/50 - 1/(34375*(3 + 5*x)) - (16807*Log[1 - 2*x])/1936 + (167*Log[3
+ 5*x])/378125

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) (3+5 x)^2} \, dx &=\int \left (-\frac{69039}{5000}-\frac{6399 x}{500}-\frac{243 x^2}{50}-\frac{16807}{968 (-1+2 x)}+\frac{1}{6875 (3+5 x)^2}+\frac{167}{75625 (3+5 x)}\right ) \, dx\\ &=-\frac{69039 x}{5000}-\frac{6399 x^2}{1000}-\frac{81 x^3}{50}-\frac{1}{34375 (3+5 x)}-\frac{16807 \log (1-2 x)}{1936}+\frac{167 \log (3+5 x)}{378125}\\ \end{align*}

Mathematica [A]  time = 0.0231665, size = 50, normalized size = 0.98 \[ \frac{-\frac{11 \left (8910000 x^4+40540500 x^3+97059600 x^2-2318085 x-28730263\right )}{5 x+3}-105043750 \log (1-2 x)+5344 \log (10 x+6)}{12100000} \]

Antiderivative was successfully verified.

[In]

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

[Out]

((-11*(-28730263 - 2318085*x + 97059600*x^2 + 40540500*x^3 + 8910000*x^4))/(3 + 5*x) - 105043750*Log[1 - 2*x]
+ 5344*Log[6 + 10*x])/12100000

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Maple [A]  time = 0.007, size = 40, normalized size = 0.8 \begin{align*} -{\frac{81\,{x}^{3}}{50}}-{\frac{6399\,{x}^{2}}{1000}}-{\frac{69039\,x}{5000}}-{\frac{16807\,\ln \left ( 2\,x-1 \right ) }{1936}}-{\frac{1}{103125+171875\,x}}+{\frac{167\,\ln \left ( 3+5\,x \right ) }{378125}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-81/50*x^3-6399/1000*x^2-69039/5000*x-16807/1936*ln(2*x-1)-1/34375/(3+5*x)+167/378125*ln(3+5*x)

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Maxima [A]  time = 1.05813, size = 53, normalized size = 1.04 \begin{align*} -\frac{81}{50} \, x^{3} - \frac{6399}{1000} \, x^{2} - \frac{69039}{5000} \, x - \frac{1}{34375 \,{\left (5 \, x + 3\right )}} + \frac{167}{378125} \, \log \left (5 \, x + 3\right ) - \frac{16807}{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)/(3+5*x)^2,x, algorithm="maxima")

[Out]

-81/50*x^3 - 6399/1000*x^2 - 69039/5000*x - 1/34375/(5*x + 3) + 167/378125*log(5*x + 3) - 16807/1936*log(2*x -
1)

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Fricas [A]  time = 1.39857, size = 207, normalized size = 4.06 \begin{align*} -\frac{49005000 \, x^{4} + 222972750 \, x^{3} + 533827800 \, x^{2} - 2672 \,{\left (5 \, x + 3\right )} \log \left (5 \, x + 3\right ) + 52521875 \,{\left (5 \, x + 3\right )} \log \left (2 \, x - 1\right ) + 250611570 \, x + 176}{6050000 \,{\left (5 \, x + 3\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/6050000*(49005000*x^4 + 222972750*x^3 + 533827800*x^2 - 2672*(5*x + 3)*log(5*x + 3) + 52521875*(5*x + 3)*lo
g(2*x - 1) + 250611570*x + 176)/(5*x + 3)

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Sympy [A]  time = 0.145629, size = 44, normalized size = 0.86 \begin{align*} - \frac{81 x^{3}}{50} - \frac{6399 x^{2}}{1000} - \frac{69039 x}{5000} - \frac{16807 \log{\left (x - \frac{1}{2} \right )}}{1936} + \frac{167 \log{\left (x + \frac{3}{5} \right )}}{378125} - \frac{1}{171875 x + 103125} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-81*x**3/50 - 6399*x**2/1000 - 69039*x/5000 - 16807*log(x - 1/2)/1936 + 167*log(x + 3/5)/378125 - 1/(171875*x
+ 103125)

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Giac [A]  time = 2.00822, size = 97, normalized size = 1.9 \begin{align*} -\frac{27}{25000} \,{\left (5 \, x + 3\right )}^{3}{\left (\frac{129}{5 \, x + 3} + \frac{1459}{{\left (5 \, x + 3\right )}^{2}} + 12\right )} - \frac{1}{34375 \,{\left (5 \, x + 3\right )}} + \frac{434043}{50000} \, \log \left (\frac{{\left | 5 \, x + 3 \right |}}{5 \,{\left (5 \, x + 3\right )}^{2}}\right ) - \frac{16807}{1936} \, \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)^5/(1-2*x)/(3+5*x)^2,x, algorithm="giac")

[Out]

-27/25000*(5*x + 3)^3*(129/(5*x + 3) + 1459/(5*x + 3)^2 + 12) - 1/34375/(5*x + 3) + 434043/50000*log(1/5*abs(5
*x + 3)/(5*x + 3)^2) - 16807/1936*log(abs(-11/(5*x + 3) + 2))