3.1490 \(\int \frac{(2+3 x)^3}{(1-2 x) (3+5 x)} \, dx\)

Optimal. Leaf size=33 \[ -\frac{27 x^2}{20}-\frac{513 x}{100}-\frac{343}{88} \log (1-2 x)+\frac{\log (5 x+3)}{1375} \]

[Out]

(-513*x)/100 - (27*x^2)/20 - (343*Log[1 - 2*x])/88 + Log[3 + 5*x]/1375

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Rubi [A]  time = 0.0136325, antiderivative size = 33, 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 = {72} \[ -\frac{27 x^2}{20}-\frac{513 x}{100}-\frac{343}{88} \log (1-2 x)+\frac{\log (5 x+3)}{1375} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(-513*x)/100 - (27*x^2)/20 - (343*Log[1 - 2*x])/88 + Log[3 + 5*x]/1375

Rule 72

Int[((e_.) + (f_.)*(x_))^(p_.)/(((a_.) + (b_.)*(x_))*((c_.) + (d_.)*(x_))), x_Symbol] :> Int[ExpandIntegrand[(
e + f*x)^p/((a + b*x)*(c + d*x)), x], x] /; FreeQ[{a, b, c, d, e, f}, x] && IntegerQ[p]

Rubi steps

\begin{align*} \int \frac{(2+3 x)^3}{(1-2 x) (3+5 x)} \, dx &=\int \left (-\frac{513}{100}-\frac{27 x}{10}-\frac{343}{44 (-1+2 x)}+\frac{1}{275 (3+5 x)}\right ) \, dx\\ &=-\frac{513 x}{100}-\frac{27 x^2}{20}-\frac{343}{88} \log (1-2 x)+\frac{\log (3+5 x)}{1375}\\ \end{align*}

Mathematica [A]  time = 0.0136481, size = 35, normalized size = 1.06 \[ \frac{-330 \left (45 x^2+171 x+94\right )-42875 \log (3-6 x)+8 \log (-3 (5 x+3))}{11000} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(-330*(94 + 171*x + 45*x^2) - 42875*Log[3 - 6*x] + 8*Log[-3*(3 + 5*x)])/11000

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Maple [A]  time = 0.005, size = 26, normalized size = 0.8 \begin{align*} -{\frac{27\,{x}^{2}}{20}}-{\frac{513\,x}{100}}-{\frac{343\,\ln \left ( 2\,x-1 \right ) }{88}}+{\frac{\ln \left ( 3+5\,x \right ) }{1375}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-27/20*x^2-513/100*x-343/88*ln(2*x-1)+1/1375*ln(3+5*x)

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Maxima [A]  time = 1.03344, size = 34, normalized size = 1.03 \begin{align*} -\frac{27}{20} \, x^{2} - \frac{513}{100} \, x + \frac{1}{1375} \, \log \left (5 \, x + 3\right ) - \frac{343}{88} \, \log \left (2 \, x - 1\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-27/20*x^2 - 513/100*x + 1/1375*log(5*x + 3) - 343/88*log(2*x - 1)

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Fricas [A]  time = 1.30842, size = 92, normalized size = 2.79 \begin{align*} -\frac{27}{20} \, x^{2} - \frac{513}{100} \, x + \frac{1}{1375} \, \log \left (5 \, x + 3\right ) - \frac{343}{88} \, \log \left (2 \, x - 1\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-27/20*x^2 - 513/100*x + 1/1375*log(5*x + 3) - 343/88*log(2*x - 1)

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Sympy [A]  time = 0.119686, size = 29, normalized size = 0.88 \begin{align*} - \frac{27 x^{2}}{20} - \frac{513 x}{100} - \frac{343 \log{\left (x - \frac{1}{2} \right )}}{88} + \frac{\log{\left (x + \frac{3}{5} \right )}}{1375} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-27*x**2/20 - 513*x/100 - 343*log(x - 1/2)/88 + log(x + 3/5)/1375

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Giac [A]  time = 3.02467, size = 36, normalized size = 1.09 \begin{align*} -\frac{27}{20} \, x^{2} - \frac{513}{100} \, x + \frac{1}{1375} \, \log \left ({\left | 5 \, x + 3 \right |}\right ) - \frac{343}{88} \, \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)^3/(1-2*x)/(3+5*x),x, algorithm="giac")

[Out]

-27/20*x^2 - 513/100*x + 1/1375*log(abs(5*x + 3)) - 343/88*log(abs(2*x - 1))