3.1692 \(\int \frac{1}{(1-2 x)^3 (3+5 x)^2} \, dx\)

Optimal. Leaf size=54 \[ \frac{20}{1331 (1-2 x)}-\frac{25}{1331 (5 x+3)}+\frac{1}{121 (1-2 x)^2}-\frac{150 \log (1-2 x)}{14641}+\frac{150 \log (5 x+3)}{14641} \]

[Out]

1/(121*(1 - 2*x)^2) + 20/(1331*(1 - 2*x)) - 25/(1331*(3 + 5*x)) - (150*Log[1 - 2*x])/14641 + (150*Log[3 + 5*x]
)/14641

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Rubi [A]  time = 0.0212377, antiderivative size = 54, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 1, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.067, Rules used = {44} \[ \frac{20}{1331 (1-2 x)}-\frac{25}{1331 (5 x+3)}+\frac{1}{121 (1-2 x)^2}-\frac{150 \log (1-2 x)}{14641}+\frac{150 \log (5 x+3)}{14641} \]

Antiderivative was successfully verified.

[In]

Int[1/((1 - 2*x)^3*(3 + 5*x)^2),x]

[Out]

1/(121*(1 - 2*x)^2) + 20/(1331*(1 - 2*x)) - 25/(1331*(3 + 5*x)) - (150*Log[1 - 2*x])/14641 + (150*Log[3 + 5*x]
)/14641

Rule 44

Int[((a_) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_.), x_Symbol] :> Int[ExpandIntegrand[(a + b*x)^m*(c + d*
x)^n, x], x] /; FreeQ[{a, b, c, d}, x] && NeQ[b*c - a*d, 0] && ILtQ[m, 0] && IntegerQ[n] &&  !(IGtQ[n, 0] && L
tQ[m + n + 2, 0])

Rubi steps

\begin{align*} \int \frac{1}{(1-2 x)^3 (3+5 x)^2} \, dx &=\int \left (-\frac{4}{121 (-1+2 x)^3}+\frac{40}{1331 (-1+2 x)^2}-\frac{300}{14641 (-1+2 x)}+\frac{125}{1331 (3+5 x)^2}+\frac{750}{14641 (3+5 x)}\right ) \, dx\\ &=\frac{1}{121 (1-2 x)^2}+\frac{20}{1331 (1-2 x)}-\frac{25}{1331 (3+5 x)}-\frac{150 \log (1-2 x)}{14641}+\frac{150 \log (3+5 x)}{14641}\\ \end{align*}

Mathematica [A]  time = 0.0158241, size = 62, normalized size = 1.15 \[ \frac{50}{1331 (5 (1-2 x)-11)}+\frac{20}{1331 (1-2 x)}+\frac{1}{121 (1-2 x)^2}+\frac{150 \log (11-5 (1-2 x))}{14641}-\frac{150 \log (1-2 x)}{14641} \]

Antiderivative was successfully verified.

[In]

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

[Out]

50/(1331*(-11 + 5*(1 - 2*x))) + 1/(121*(1 - 2*x)^2) + 20/(1331*(1 - 2*x)) + (150*Log[11 - 5*(1 - 2*x)])/14641
- (150*Log[1 - 2*x])/14641

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Maple [A]  time = 0.008, size = 45, normalized size = 0.8 \begin{align*}{\frac{1}{121\, \left ( 2\,x-1 \right ) ^{2}}}-{\frac{20}{2662\,x-1331}}-{\frac{150\,\ln \left ( 2\,x-1 \right ) }{14641}}-{\frac{25}{3993+6655\,x}}+{\frac{150\,\ln \left ( 3+5\,x \right ) }{14641}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/121/(2*x-1)^2-20/1331/(2*x-1)-150/14641*ln(2*x-1)-25/1331/(3+5*x)+150/14641*ln(3+5*x)

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Maxima [A]  time = 2.73909, size = 62, normalized size = 1.15 \begin{align*} -\frac{300 \, x^{2} - 135 \, x - 68}{1331 \,{\left (20 \, x^{3} - 8 \, x^{2} - 7 \, x + 3\right )}} + \frac{150}{14641} \, \log \left (5 \, x + 3\right ) - \frac{150}{14641} \, \log \left (2 \, x - 1\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/1331*(300*x^2 - 135*x - 68)/(20*x^3 - 8*x^2 - 7*x + 3) + 150/14641*log(5*x + 3) - 150/14641*log(2*x - 1)

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Fricas [A]  time = 1.53873, size = 209, normalized size = 3.87 \begin{align*} -\frac{3300 \, x^{2} - 150 \,{\left (20 \, x^{3} - 8 \, x^{2} - 7 \, x + 3\right )} \log \left (5 \, x + 3\right ) + 150 \,{\left (20 \, x^{3} - 8 \, x^{2} - 7 \, x + 3\right )} \log \left (2 \, x - 1\right ) - 1485 \, x - 748}{14641 \,{\left (20 \, x^{3} - 8 \, x^{2} - 7 \, x + 3\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/14641*(3300*x^2 - 150*(20*x^3 - 8*x^2 - 7*x + 3)*log(5*x + 3) + 150*(20*x^3 - 8*x^2 - 7*x + 3)*log(2*x - 1)
- 1485*x - 748)/(20*x^3 - 8*x^2 - 7*x + 3)

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Sympy [A]  time = 0.152926, size = 44, normalized size = 0.81 \begin{align*} - \frac{300 x^{2} - 135 x - 68}{26620 x^{3} - 10648 x^{2} - 9317 x + 3993} - \frac{150 \log{\left (x - \frac{1}{2} \right )}}{14641} + \frac{150 \log{\left (x + \frac{3}{5} \right )}}{14641} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-(300*x**2 - 135*x - 68)/(26620*x**3 - 10648*x**2 - 9317*x + 3993) - 150*log(x - 1/2)/14641 + 150*log(x + 3/5)
/14641

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Giac [A]  time = 3.2107, size = 69, normalized size = 1.28 \begin{align*} -\frac{25}{1331 \,{\left (5 \, x + 3\right )}} + \frac{100 \,{\left (\frac{33}{5 \, x + 3} - 5\right )}}{14641 \,{\left (\frac{11}{5 \, x + 3} - 2\right )}^{2}} - \frac{150}{14641} \, \log \left ({\left | -\frac{11}{5 \, x + 3} + 2 \right |}\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-25/1331/(5*x + 3) + 100/14641*(33/(5*x + 3) - 5)/(11/(5*x + 3) - 2)^2 - 150/14641*log(abs(-11/(5*x + 3) + 2))