3.1680 \(\int \frac{1}{(1-2 x)^3 (2+3 x) (3+5 x)} \, dx\)

Optimal. Leaf size=53 \[ \frac{136}{5929 (1-2 x)}+\frac{1}{77 (1-2 x)^2}-\frac{6938 \log (1-2 x)}{456533}-\frac{27}{343} \log (3 x+2)+\frac{125 \log (5 x+3)}{1331} \]

[Out]

1/(77*(1 - 2*x)^2) + 136/(5929*(1 - 2*x)) - (6938*Log[1 - 2*x])/456533 - (27*Log[2 + 3*x])/343 + (125*Log[3 +
5*x])/1331

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Rubi [A]  time = 0.0237475, antiderivative size = 53, 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 = {72} \[ \frac{136}{5929 (1-2 x)}+\frac{1}{77 (1-2 x)^2}-\frac{6938 \log (1-2 x)}{456533}-\frac{27}{343} \log (3 x+2)+\frac{125 \log (5 x+3)}{1331} \]

Antiderivative was successfully verified.

[In]

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

[Out]

1/(77*(1 - 2*x)^2) + 136/(5929*(1 - 2*x)) - (6938*Log[1 - 2*x])/456533 - (27*Log[2 + 3*x])/343 + (125*Log[3 +
5*x])/1331

Rule 72

Int[((e_.) + (f_.)*(x_))^(p_.)/(((a_.) + (b_.)*(x_))*((c_.) + (d_.)*(x_))), x_Symbol] :> Int[ExpandIntegrand[(
e + f*x)^p/((a + b*x)*(c + d*x)), x], x] /; FreeQ[{a, b, c, d, e, f}, x] && IntegerQ[p]

Rubi steps

\begin{align*} \int \frac{1}{(1-2 x)^3 (2+3 x) (3+5 x)} \, dx &=\int \left (-\frac{4}{77 (-1+2 x)^3}+\frac{272}{5929 (-1+2 x)^2}-\frac{13876}{456533 (-1+2 x)}-\frac{81}{343 (2+3 x)}+\frac{625}{1331 (3+5 x)}\right ) \, dx\\ &=\frac{1}{77 (1-2 x)^2}+\frac{136}{5929 (1-2 x)}-\frac{6938 \log (1-2 x)}{456533}-\frac{27}{343} \log (2+3 x)+\frac{125 \log (3+5 x)}{1331}\\ \end{align*}

Mathematica [A]  time = 0.0250934, size = 52, normalized size = 0.98 \[ \frac{-6938 \log (3-6 x)-35937 \log (3 x+2)+\frac{7 \left (-2992 x+6125 (1-2 x)^2 \log (-3 (5 x+3))+2343\right )}{(1-2 x)^2}}{456533} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(-6938*Log[3 - 6*x] - 35937*Log[2 + 3*x] + (7*(2343 - 2992*x + 6125*(1 - 2*x)^2*Log[-3*(3 + 5*x)]))/(1 - 2*x)^
2)/456533

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Maple [A]  time = 0.007, size = 44, normalized size = 0.8 \begin{align*}{\frac{1}{77\, \left ( 2\,x-1 \right ) ^{2}}}-{\frac{136}{11858\,x-5929}}-{\frac{6938\,\ln \left ( 2\,x-1 \right ) }{456533}}-{\frac{27\,\ln \left ( 2+3\,x \right ) }{343}}+{\frac{125\,\ln \left ( 3+5\,x \right ) }{1331}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/77/(2*x-1)^2-136/5929/(2*x-1)-6938/456533*ln(2*x-1)-27/343*ln(2+3*x)+125/1331*ln(3+5*x)

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Maxima [A]  time = 1.38795, size = 59, normalized size = 1.11 \begin{align*} -\frac{272 \, x - 213}{5929 \,{\left (4 \, x^{2} - 4 \, x + 1\right )}} + \frac{125}{1331} \, \log \left (5 \, x + 3\right ) - \frac{27}{343} \, \log \left (3 \, x + 2\right ) - \frac{6938}{456533} \, \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/(2+3*x)/(3+5*x),x, algorithm="maxima")

[Out]

-1/5929*(272*x - 213)/(4*x^2 - 4*x + 1) + 125/1331*log(5*x + 3) - 27/343*log(3*x + 2) - 6938/456533*log(2*x -
1)

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Fricas [A]  time = 1.53099, size = 219, normalized size = 4.13 \begin{align*} \frac{42875 \,{\left (4 \, x^{2} - 4 \, x + 1\right )} \log \left (5 \, x + 3\right ) - 35937 \,{\left (4 \, x^{2} - 4 \, x + 1\right )} \log \left (3 \, x + 2\right ) - 6938 \,{\left (4 \, x^{2} - 4 \, x + 1\right )} \log \left (2 \, x - 1\right ) - 20944 \, x + 16401}{456533 \,{\left (4 \, x^{2} - 4 \, x + 1\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/456533*(42875*(4*x^2 - 4*x + 1)*log(5*x + 3) - 35937*(4*x^2 - 4*x + 1)*log(3*x + 2) - 6938*(4*x^2 - 4*x + 1)
*log(2*x - 1) - 20944*x + 16401)/(4*x^2 - 4*x + 1)

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Sympy [A]  time = 0.177326, size = 44, normalized size = 0.83 \begin{align*} - \frac{272 x - 213}{23716 x^{2} - 23716 x + 5929} - \frac{6938 \log{\left (x - \frac{1}{2} \right )}}{456533} + \frac{125 \log{\left (x + \frac{3}{5} \right )}}{1331} - \frac{27 \log{\left (x + \frac{2}{3} \right )}}{343} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-(272*x - 213)/(23716*x**2 - 23716*x + 5929) - 6938*log(x - 1/2)/456533 + 125*log(x + 3/5)/1331 - 27*log(x + 2
/3)/343

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Giac [A]  time = 3.00802, size = 57, normalized size = 1.08 \begin{align*} -\frac{272 \, x - 213}{5929 \,{\left (2 \, x - 1\right )}^{2}} + \frac{125}{1331} \, \log \left ({\left | 5 \, x + 3 \right |}\right ) - \frac{27}{343} \, \log \left ({\left | 3 \, x + 2 \right |}\right ) - \frac{6938}{456533} \, \log \left ({\left | 2 \, x - 1 \right |}\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/5929*(272*x - 213)/(2*x - 1)^2 + 125/1331*log(abs(5*x + 3)) - 27/343*log(abs(3*x + 2)) - 6938/456533*log(ab
s(2*x - 1))