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)-7189/10648*ln(1-2*x)+1/6655*ln(3+5*x)

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Rubi [A]  time = 0.02, antiderivative size = 43, normalized size of antiderivative = 1.00, 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*}

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Mathematica [A]  time = 0.03, 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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fricas [A]  time = 0.75, size = 55, normalized size = 1.28 \[ \frac {16 \, {\left (4 \, x^{2} - 4 \, x + 1\right )} \log \left (5 \, x + 3\right ) - 71890 \, {\left (4 \, x^{2} - 4 \, x + 1\right )} \log \left (2 \, x - 1\right ) + 689920 \, x - 137445}{106480 \, {\left (4 \, x^{2} - 4 \, x + 1\right )}} \]

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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giac [A]  time = 1.15, size = 33, normalized size = 0.77 \[ \frac {49 \, {\left (256 \, x - 51\right )}}{1936 \, {\left (2 \, x - 1\right )}^{2}} + \frac {1}{6655} \, \log \left ({\left | 5 \, x + 3 \right |}\right ) - \frac {7189}{10648} \, \log \left ({\left | 2 \, x - 1 \right |}\right ) \]

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))

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maple [A]  time = 0.01, size = 36, normalized size = 0.84 \[ -\frac {7189 \ln \left (2 x -1\right )}{10648}+\frac {\ln \left (5 x +3\right )}{6655}+\frac {343}{176 \left (2 x -1\right )^{2}}+\frac {392}{121 \left (2 x -1\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maxima [A]  time = 0.50, size = 36, normalized size = 0.84 \[ \frac {49 \, {\left (256 \, x - 51\right )}}{1936 \, {\left (4 \, x^{2} - 4 \, x + 1\right )}} + \frac {1}{6655} \, \log \left (5 \, x + 3\right ) - \frac {7189}{10648} \, \log \left (2 \, x - 1\right ) \]

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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mupad [B]  time = 0.08, size = 29, normalized size = 0.67 \[ \frac {\ln \left (x+\frac {3}{5}\right )}{6655}-\frac {7189\,\ln \left (x-\frac {1}{2}\right )}{10648}+\frac {\frac {196\,x}{121}-\frac {2499}{7744}}{x^2-x+\frac {1}{4}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

log(x + 3/5)/6655 - (7189*log(x - 1/2))/10648 + ((196*x)/121 - 2499/7744)/(x^2 - x + 1/4)

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sympy [A]  time = 0.17, size = 32, normalized size = 0.74 \[ - \frac {2499 - 12544 x}{7744 x^{2} - 7744 x + 1936} - \frac {7189 \log {\left (x - \frac {1}{2} \right )}}{10648} + \frac {\log {\left (x + \frac {3}{5} \right )}}{6655} \]

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]

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

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