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

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Rubi [A]  time = 0.02, antiderivative size = 53, 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 = {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*}

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Mathematica [A]  time = 0.03, 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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fricas [A]  time = 0.55, size = 73, normalized size = 1.38 \[ \frac {42875 \, {\left (4 \, x^{2} - 4 \, x + 1\right )} \log \left (5 \, x + 3\right ) - 35937 \, {\left (4 \, x^{2} - 4 \, x + 1\right )} \log \left (3 \, x + 2\right ) - 6938 \, {\left (4 \, x^{2} - 4 \, x + 1\right )} \log \left (2 \, x - 1\right ) - 20944 \, x + 16401}{456533 \, {\left (4 \, x^{2} - 4 \, x + 1\right )}} \]

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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giac [A]  time = 1.15, size = 42, normalized size = 0.79 \[ -\frac {272 \, x - 213}{5929 \, {\left (2 \, x - 1\right )}^{2}} + \frac {125}{1331} \, \log \left ({\left | 5 \, x + 3 \right |}\right ) - \frac {27}{343} \, \log \left ({\left | 3 \, x + 2 \right |}\right ) - \frac {6938}{456533} \, \log \left ({\left | 2 \, x - 1 \right |}\right ) \]

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))

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maple [A]  time = 0.01, size = 44, normalized size = 0.83 \[ -\frac {6938 \ln \left (2 x -1\right )}{456533}-\frac {27 \ln \left (3 x +2\right )}{343}+\frac {125 \ln \left (5 x +3\right )}{1331}+\frac {1}{77 \left (2 x -1\right )^{2}}-\frac {136}{5929 \left (2 x -1\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maxima [A]  time = 0.48, size = 44, normalized size = 0.83 \[ -\frac {272 \, x - 213}{5929 \, {\left (4 \, x^{2} - 4 \, x + 1\right )}} + \frac {125}{1331} \, \log \left (5 \, x + 3\right ) - \frac {27}{343} \, \log \left (3 \, x + 2\right ) - \frac {6938}{456533} \, \log \left (2 \, x - 1\right ) \]

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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mupad [B]  time = 1.07, size = 36, normalized size = 0.68 \[ \frac {125\,\ln \left (x+\frac {3}{5}\right )}{1331}-\frac {27\,\ln \left (x+\frac {2}{3}\right )}{343}-\frac {6938\,\ln \left (x-\frac {1}{2}\right )}{456533}-\frac {\frac {68\,x}{5929}-\frac {213}{23716}}{x^2-x+\frac {1}{4}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

(125*log(x + 3/5))/1331 - (27*log(x + 2/3))/343 - (6938*log(x - 1/2))/456533 - ((68*x)/5929 - 213/23716)/(x^2
- x + 1/4)

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sympy [A]  time = 0.22, size = 44, normalized size = 0.83 \[ - \frac {272 x - 213}{23716 x^{2} - 23716 x + 5929} - \frac {6938 \log {\left (x - \frac {1}{2} \right )}}{456533} + \frac {125 \log {\left (x + \frac {3}{5} \right )}}{1331} - \frac {27 \log {\left (x + \frac {2}{3} \right )}}{343} \]

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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