3.1666 \(\int \frac{(3+5 x)^3}{(1-2 x)^3 (2+3 x)} \, dx\)

Optimal. Leaf size=43 \[ -\frac{1089}{49 (1-2 x)}+\frac{1331}{112 (1-2 x)^2}-\frac{14289 \log (1-2 x)}{2744}-\frac{\log (3 x+2)}{1029} \]

[Out]

1331/(112*(1 - 2*x)^2) - 1089/(49*(1 - 2*x)) - (14289*Log[1 - 2*x])/2744 - Log[2 + 3*x]/1029

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Rubi [A]  time = 0.0180265, 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} \[ -\frac{1089}{49 (1-2 x)}+\frac{1331}{112 (1-2 x)^2}-\frac{14289 \log (1-2 x)}{2744}-\frac{\log (3 x+2)}{1029} \]

Antiderivative was successfully verified.

[In]

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

[Out]

1331/(112*(1 - 2*x)^2) - 1089/(49*(1 - 2*x)) - (14289*Log[1 - 2*x])/2744 - Log[2 + 3*x]/1029

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)^3 (2+3 x)} \, dx &=\int \left (-\frac{1331}{28 (-1+2 x)^3}-\frac{2178}{49 (-1+2 x)^2}-\frac{14289}{1372 (-1+2 x)}-\frac{1}{343 (2+3 x)}\right ) \, dx\\ &=\frac{1331}{112 (1-2 x)^2}-\frac{1089}{49 (1-2 x)}-\frac{14289 \log (1-2 x)}{2744}-\frac{\log (2+3 x)}{1029}\\ \end{align*}

Mathematica [A]  time = 0.0217933, size = 35, normalized size = 0.81 \[ \frac{\frac{2541 (288 x-67)}{(1-2 x)^2}-85734 \log (3-6 x)-16 \log (3 x+2)}{16464} \]

Antiderivative was successfully verified.

[In]

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

[Out]

((2541*(-67 + 288*x))/(1 - 2*x)^2 - 85734*Log[3 - 6*x] - 16*Log[2 + 3*x])/16464

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Maple [A]  time = 0.008, size = 36, normalized size = 0.8 \begin{align*}{\frac{1331}{112\, \left ( 2\,x-1 \right ) ^{2}}}+{\frac{1089}{98\,x-49}}-{\frac{14289\,\ln \left ( 2\,x-1 \right ) }{2744}}-{\frac{\ln \left ( 2+3\,x \right ) }{1029}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1331/112/(2*x-1)^2+1089/49/(2*x-1)-14289/2744*ln(2*x-1)-1/1029*ln(2+3*x)

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Maxima [A]  time = 1.05331, size = 49, normalized size = 1.14 \begin{align*} \frac{121 \,{\left (288 \, x - 67\right )}}{784 \,{\left (4 \, x^{2} - 4 \, x + 1\right )}} - \frac{1}{1029} \, \log \left (3 \, x + 2\right ) - \frac{14289}{2744} \, \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)^3/(2+3*x),x, algorithm="maxima")

[Out]

121/784*(288*x - 67)/(4*x^2 - 4*x + 1) - 1/1029*log(3*x + 2) - 14289/2744*log(2*x - 1)

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Fricas [A]  time = 1.57519, size = 166, normalized size = 3.86 \begin{align*} -\frac{16 \,{\left (4 \, x^{2} - 4 \, x + 1\right )} \log \left (3 \, x + 2\right ) + 85734 \,{\left (4 \, x^{2} - 4 \, x + 1\right )} \log \left (2 \, x - 1\right ) - 731808 \, x + 170247}{16464 \,{\left (4 \, x^{2} - 4 \, x + 1\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/16464*(16*(4*x^2 - 4*x + 1)*log(3*x + 2) + 85734*(4*x^2 - 4*x + 1)*log(2*x - 1) - 731808*x + 170247)/(4*x^2
- 4*x + 1)

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Sympy [A]  time = 0.158366, size = 32, normalized size = 0.74 \begin{align*} \frac{34848 x - 8107}{3136 x^{2} - 3136 x + 784} - \frac{14289 \log{\left (x - \frac{1}{2} \right )}}{2744} - \frac{\log{\left (x + \frac{2}{3} \right )}}{1029} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

(34848*x - 8107)/(3136*x**2 - 3136*x + 784) - 14289*log(x - 1/2)/2744 - log(x + 2/3)/1029

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Giac [A]  time = 3.42161, size = 45, normalized size = 1.05 \begin{align*} \frac{121 \,{\left (288 \, x - 67\right )}}{784 \,{\left (2 \, x - 1\right )}^{2}} - \frac{1}{1029} \, \log \left ({\left | 3 \, x + 2 \right |}\right ) - \frac{14289}{2744} \, \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)^3/(2+3*x),x, algorithm="giac")

[Out]

121/784*(288*x - 67)/(2*x - 1)^2 - 1/1029*log(abs(3*x + 2)) - 14289/2744*log(abs(2*x - 1))