3.1676 \(\int \frac{(2+3 x)^3}{(1-2 x)^3 (3+5 x)} \, dx\)

Optimal. Leaf size=43 \[ -\frac{392}{121 (1-2 x)}+\frac{343}{176 (1-2 x)^2}-\frac{7189 \log (1-2 x)}{10648}+\frac{\log (5 x+3)}{6655} \]

[Out]

343/(176*(1 - 2*x)^2) - 392/(121*(1 - 2*x)) - (7189*Log[1 - 2*x])/10648 + Log[3 + 5*x]/6655

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Rubi [A]  time = 0.0185592, 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} \[ -\frac{392}{121 (1-2 x)}+\frac{343}{176 (1-2 x)^2}-\frac{7189 \log (1-2 x)}{10648}+\frac{\log (5 x+3)}{6655} \]

Antiderivative was successfully verified.

[In]

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

[Out]

343/(176*(1 - 2*x)^2) - 392/(121*(1 - 2*x)) - (7189*Log[1 - 2*x])/10648 + Log[3 + 5*x]/6655

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)^3}{(1-2 x)^3 (3+5 x)} \, dx &=\int \left (-\frac{343}{44 (-1+2 x)^3}-\frac{784}{121 (-1+2 x)^2}-\frac{7189}{5324 (-1+2 x)}+\frac{1}{1331 (3+5 x)}\right ) \, dx\\ &=\frac{343}{176 (1-2 x)^2}-\frac{392}{121 (1-2 x)}-\frac{7189 \log (1-2 x)}{10648}+\frac{\log (3+5 x)}{6655}\\ \end{align*}

Mathematica [A]  time = 0.0208941, size = 35, normalized size = 0.81 \[ \frac{\frac{2695 (256 x-51)}{(1-2 x)^2}-71890 \log (5-10 x)+16 \log (5 x+3)}{106480} \]

Antiderivative was successfully verified.

[In]

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

[Out]

((2695*(-51 + 256*x))/(1 - 2*x)^2 - 71890*Log[5 - 10*x] + 16*Log[3 + 5*x])/106480

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Maple [A]  time = 0.007, size = 36, normalized size = 0.8 \begin{align*}{\frac{343}{176\, \left ( 2\,x-1 \right ) ^{2}}}+{\frac{392}{242\,x-121}}-{\frac{7189\,\ln \left ( 2\,x-1 \right ) }{10648}}+{\frac{\ln \left ( 3+5\,x \right ) }{6655}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

343/176/(2*x-1)^2+392/121/(2*x-1)-7189/10648*ln(2*x-1)+1/6655*ln(3+5*x)

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Maxima [A]  time = 1.78873, size = 49, normalized size = 1.14 \begin{align*} \frac{49 \,{\left (256 \, x - 51\right )}}{1936 \,{\left (4 \, x^{2} - 4 \, x + 1\right )}} + \frac{1}{6655} \, \log \left (5 \, x + 3\right ) - \frac{7189}{10648} \, \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/(3+5*x),x, algorithm="maxima")

[Out]

49/1936*(256*x - 51)/(4*x^2 - 4*x + 1) + 1/6655*log(5*x + 3) - 7189/10648*log(2*x - 1)

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Fricas [A]  time = 1.58721, size = 166, normalized size = 3.86 \begin{align*} \frac{16 \,{\left (4 \, x^{2} - 4 \, x + 1\right )} \log \left (5 \, x + 3\right ) - 71890 \,{\left (4 \, x^{2} - 4 \, x + 1\right )} \log \left (2 \, x - 1\right ) + 689920 \, x - 137445}{106480 \,{\left (4 \, x^{2} - 4 \, 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/(3+5*x),x, algorithm="fricas")

[Out]

1/106480*(16*(4*x^2 - 4*x + 1)*log(5*x + 3) - 71890*(4*x^2 - 4*x + 1)*log(2*x - 1) + 689920*x - 137445)/(4*x^2
- 4*x + 1)

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Sympy [A]  time = 0.154243, size = 32, normalized size = 0.74 \begin{align*} \frac{12544 x - 2499}{7744 x^{2} - 7744 x + 1936} - \frac{7189 \log{\left (x - \frac{1}{2} \right )}}{10648} + \frac{\log{\left (x + \frac{3}{5} \right )}}{6655} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

(12544*x - 2499)/(7744*x**2 - 7744*x + 1936) - 7189*log(x - 1/2)/10648 + log(x + 3/5)/6655

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Giac [A]  time = 2.98014, size = 45, normalized size = 1.05 \begin{align*} \frac{49 \,{\left (256 \, x - 51\right )}}{1936 \,{\left (2 \, x - 1\right )}^{2}} + \frac{1}{6655} \, \log \left ({\left | 5 \, x + 3 \right |}\right ) - \frac{7189}{10648} \, \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/(3+5*x),x, algorithm="giac")

[Out]

49/1936*(256*x - 51)/(2*x - 1)^2 + 1/6655*log(abs(5*x + 3)) - 7189/10648*log(abs(2*x - 1))