3.1478 \(\int \frac{(3+5 x)^3}{(1-2 x) (2+3 x)^3} \, dx\)

Optimal. Leaf size=43 \[ -\frac{103}{1323 (3 x+2)}+\frac{1}{378 (3 x+2)^2}-\frac{1331}{686} \log (1-2 x)-\frac{3469 \log (3 x+2)}{9261} \]

[Out]

1/(378*(2 + 3*x)^2) - 103/(1323*(2 + 3*x)) - (1331*Log[1 - 2*x])/686 - (3469*Log[2 + 3*x])/9261

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Rubi [A]  time = 0.0173471, antiderivative size = 43, 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} \[ -\frac{103}{1323 (3 x+2)}+\frac{1}{378 (3 x+2)^2}-\frac{1331}{686} \log (1-2 x)-\frac{3469 \log (3 x+2)}{9261} \]

Antiderivative was successfully verified.

[In]

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

[Out]

1/(378*(2 + 3*x)^2) - 103/(1323*(2 + 3*x)) - (1331*Log[1 - 2*x])/686 - (3469*Log[2 + 3*x])/9261

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)^3} \, dx &=\int \left (-\frac{1331}{343 (-1+2 x)}-\frac{1}{63 (2+3 x)^3}+\frac{103}{441 (2+3 x)^2}-\frac{3469}{3087 (2+3 x)}\right ) \, dx\\ &=\frac{1}{378 (2+3 x)^2}-\frac{103}{1323 (2+3 x)}-\frac{1331}{686} \log (1-2 x)-\frac{3469 \log (2+3 x)}{9261}\\ \end{align*}

Mathematica [A]  time = 0.0220743, size = 35, normalized size = 0.81 \[ \frac{-\frac{21 (206 x+135)}{(3 x+2)^2}-35937 \log (1-2 x)-6938 \log (6 x+4)}{18522} \]

Antiderivative was successfully verified.

[In]

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

[Out]

((-21*(135 + 206*x))/(2 + 3*x)^2 - 35937*Log[1 - 2*x] - 6938*Log[4 + 6*x])/18522

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Maple [A]  time = 0.007, size = 36, normalized size = 0.8 \begin{align*} -{\frac{1331\,\ln \left ( 2\,x-1 \right ) }{686}}+{\frac{1}{378\, \left ( 2+3\,x \right ) ^{2}}}-{\frac{103}{2646+3969\,x}}-{\frac{3469\,\ln \left ( 2+3\,x \right ) }{9261}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1331/686*ln(2*x-1)+1/378/(2+3*x)^2-103/1323/(2+3*x)-3469/9261*ln(2+3*x)

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Maxima [A]  time = 1.25147, size = 49, normalized size = 1.14 \begin{align*} -\frac{206 \, x + 135}{882 \,{\left (9 \, x^{2} + 12 \, x + 4\right )}} - \frac{3469}{9261} \, \log \left (3 \, x + 2\right ) - \frac{1331}{686} \, \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)^3,x, algorithm="maxima")

[Out]

-1/882*(206*x + 135)/(9*x^2 + 12*x + 4) - 3469/9261*log(3*x + 2) - 1331/686*log(2*x - 1)

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Fricas [A]  time = 1.21979, size = 167, normalized size = 3.88 \begin{align*} -\frac{6938 \,{\left (9 \, x^{2} + 12 \, x + 4\right )} \log \left (3 \, x + 2\right ) + 35937 \,{\left (9 \, x^{2} + 12 \, x + 4\right )} \log \left (2 \, x - 1\right ) + 4326 \, x + 2835}{18522 \,{\left (9 \, x^{2} + 12 \, x + 4\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)^3,x, algorithm="fricas")

[Out]

-1/18522*(6938*(9*x^2 + 12*x + 4)*log(3*x + 2) + 35937*(9*x^2 + 12*x + 4)*log(2*x - 1) + 4326*x + 2835)/(9*x^2
+ 12*x + 4)

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Sympy [A]  time = 0.160984, size = 36, normalized size = 0.84 \begin{align*} - \frac{206 x + 135}{7938 x^{2} + 10584 x + 3528} - \frac{1331 \log{\left (x - \frac{1}{2} \right )}}{686} - \frac{3469 \log{\left (x + \frac{2}{3} \right )}}{9261} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-(206*x + 135)/(7938*x**2 + 10584*x + 3528) - 1331*log(x - 1/2)/686 - 3469*log(x + 2/3)/9261

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Giac [A]  time = 2.05867, size = 45, normalized size = 1.05 \begin{align*} -\frac{206 \, x + 135}{882 \,{\left (3 \, x + 2\right )}^{2}} - \frac{3469}{9261} \, \log \left ({\left | 3 \, x + 2 \right |}\right ) - \frac{1331}{686} \, \log \left ({\left | 2 \, x - 1 \right |}\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)^3,x, algorithm="giac")

[Out]

-1/882*(206*x + 135)/(3*x + 2)^2 - 3469/9261*log(abs(3*x + 2)) - 1331/686*log(abs(2*x - 1))