3.1477 \(\int \frac{(3+5 x)^3}{(1-2 x) (2+3 x)^2} \, dx\)

Optimal. Leaf size=37 \[ -\frac{125 x}{18}+\frac{1}{189 (3 x+2)}-\frac{1331}{196} \log (1-2 x)+\frac{103 \log (3 x+2)}{1323} \]

[Out]

(-125*x)/18 + 1/(189*(2 + 3*x)) - (1331*Log[1 - 2*x])/196 + (103*Log[2 + 3*x])/1323

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Rubi [A]  time = 0.0158067, antiderivative size = 37, 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} \[ -\frac{125 x}{18}+\frac{1}{189 (3 x+2)}-\frac{1331}{196} \log (1-2 x)+\frac{103 \log (3 x+2)}{1323} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(-125*x)/18 + 1/(189*(2 + 3*x)) - (1331*Log[1 - 2*x])/196 + (103*Log[2 + 3*x])/1323

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{(3+5 x)^3}{(1-2 x) (2+3 x)^2} \, dx &=\int \left (-\frac{125}{18}-\frac{1331}{98 (-1+2 x)}-\frac{1}{63 (2+3 x)^2}+\frac{103}{441 (2+3 x)}\right ) \, dx\\ &=-\frac{125 x}{18}+\frac{1}{189 (2+3 x)}-\frac{1331}{196} \log (1-2 x)+\frac{103 \log (2+3 x)}{1323}\\ \end{align*}

Mathematica [A]  time = 0.0234478, size = 37, normalized size = 1. \[ \frac{18375 (1-2 x)+\frac{28}{3 x+2}-35937 \log (1-2 x)+412 \log (6 x+4)}{5292} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(18375*(1 - 2*x) + 28/(2 + 3*x) - 35937*Log[1 - 2*x] + 412*Log[4 + 6*x])/5292

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Maple [A]  time = 0.006, size = 30, normalized size = 0.8 \begin{align*} -{\frac{125\,x}{18}}-{\frac{1331\,\ln \left ( 2\,x-1 \right ) }{196}}+{\frac{1}{378+567\,x}}+{\frac{103\,\ln \left ( 2+3\,x \right ) }{1323}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-125/18*x-1331/196*ln(2*x-1)+1/189/(2+3*x)+103/1323*ln(2+3*x)

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Maxima [A]  time = 1.18441, size = 39, normalized size = 1.05 \begin{align*} -\frac{125}{18} \, x + \frac{1}{189 \,{\left (3 \, x + 2\right )}} + \frac{103}{1323} \, \log \left (3 \, x + 2\right ) - \frac{1331}{196} \, \log \left (2 \, x - 1\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-125/18*x + 1/189/(3*x + 2) + 103/1323*log(3*x + 2) - 1331/196*log(2*x - 1)

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Fricas [A]  time = 1.25858, size = 144, normalized size = 3.89 \begin{align*} -\frac{110250 \, x^{2} - 412 \,{\left (3 \, x + 2\right )} \log \left (3 \, x + 2\right ) + 35937 \,{\left (3 \, x + 2\right )} \log \left (2 \, x - 1\right ) + 73500 \, x - 28}{5292 \,{\left (3 \, x + 2\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/5292*(110250*x^2 - 412*(3*x + 2)*log(3*x + 2) + 35937*(3*x + 2)*log(2*x - 1) + 73500*x - 28)/(3*x + 2)

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Sympy [A]  time = 0.140839, size = 31, normalized size = 0.84 \begin{align*} - \frac{125 x}{18} - \frac{1331 \log{\left (x - \frac{1}{2} \right )}}{196} + \frac{103 \log{\left (x + \frac{2}{3} \right )}}{1323} + \frac{1}{567 x + 378} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-125*x/18 - 1331*log(x - 1/2)/196 + 103*log(x + 2/3)/1323 + 1/(567*x + 378)

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Giac [A]  time = 1.8411, size = 63, normalized size = 1.7 \begin{align*} -\frac{125}{18} \, x + \frac{1}{189 \,{\left (3 \, x + 2\right )}} + \frac{725}{108} \, \log \left (\frac{{\left | 3 \, x + 2 \right |}}{3 \,{\left (3 \, x + 2\right )}^{2}}\right ) - \frac{1331}{196} \, \log \left ({\left | -\frac{7}{3 \, x + 2} + 2 \right |}\right ) - \frac{125}{27} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-125/18*x + 1/189/(3*x + 2) + 725/108*log(1/3*abs(3*x + 2)/(3*x + 2)^2) - 1331/196*log(abs(-7/(3*x + 2) + 2))
- 125/27