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

Optimal. Leaf size=32 \[ \frac {1}{11 (1-2 x)}-\frac {5}{121} \log (1-2 x)+\frac {5}{121} \log (5 x+3) \]

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Rubi [A]  time = 0.01, antiderivative size = 32, normalized size of antiderivative = 1.00, 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} \begin {gather*} \frac {1}{11 (1-2 x)}-\frac {5}{121} \log (1-2 x)+\frac {5}{121} \log (5 x+3) \end {gather*}

Antiderivative was successfully verified.

[In]

Int[1/((1 - 2*x)^2*(3 + 5*x)),x]

[Out]

1/(11*(1 - 2*x)) - (5*Log[1 - 2*x])/121 + (5*Log[3 + 5*x])/121

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)^2 (3+5 x)} \, dx &=\int \left (\frac {2}{11 (-1+2 x)^2}-\frac {10}{121 (-1+2 x)}+\frac {25}{121 (3+5 x)}\right ) \, dx\\ &=\frac {1}{11 (1-2 x)}-\frac {5}{121} \log (1-2 x)+\frac {5}{121} \log (3+5 x)\\ \end {align*}

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Mathematica [A]  time = 0.02, size = 38, normalized size = 1.19 \begin {gather*} \frac {(5-10 x) \log (1-2 x)+5 (2 x-1) \log (10 x+6)-11}{121 (2 x-1)} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(-11 + (5 - 10*x)*Log[1 - 2*x] + 5*(-1 + 2*x)*Log[6 + 10*x])/(121*(-1 + 2*x))

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IntegrateAlgebraic [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {1}{(1-2 x)^2 (3+5 x)} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

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

[Out]

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

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fricas [A]  time = 1.55, size = 37, normalized size = 1.16 \begin {gather*} \frac {5 \, {\left (2 \, x - 1\right )} \log \left (5 \, x + 3\right ) - 5 \, {\left (2 \, x - 1\right )} \log \left (2 \, x - 1\right ) - 11}{121 \, {\left (2 \, x - 1\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/121*(5*(2*x - 1)*log(5*x + 3) - 5*(2*x - 1)*log(2*x - 1) - 11)/(2*x - 1)

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giac [A]  time = 1.18, size = 25, normalized size = 0.78 \begin {gather*} -\frac {1}{11 \, {\left (2 \, x - 1\right )}} + \frac {5}{121} \, \log \left ({\left | -\frac {11}{2 \, x - 1} - 5 \right |}\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/11/(2*x - 1) + 5/121*log(abs(-11/(2*x - 1) - 5))

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maple [A]  time = 0.01, size = 27, normalized size = 0.84 \begin {gather*} -\frac {5 \ln \left (2 x -1\right )}{121}+\frac {5 \ln \left (5 x +3\right )}{121}-\frac {1}{11 \left (2 x -1\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

5/121*ln(5*x+3)-1/11/(2*x-1)-5/121*ln(2*x-1)

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maxima [A]  time = 0.59, size = 26, normalized size = 0.81 \begin {gather*} -\frac {1}{11 \, {\left (2 \, x - 1\right )}} + \frac {5}{121} \, \log \left (5 \, x + 3\right ) - \frac {5}{121} \, \log \left (2 \, x - 1\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/11/(2*x - 1) + 5/121*log(5*x + 3) - 5/121*log(2*x - 1)

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mupad [B]  time = 1.09, size = 26, normalized size = 0.81 \begin {gather*} -\frac {5\,\ln \left (\frac {2\,x-1}{5\,x+3}\right )}{121}-\frac {1}{11\,\left (2\,x-1\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/((2*x - 1)^2*(5*x + 3)),x)

[Out]

- (5*log((2*x - 1)/(5*x + 3)))/121 - 1/(11*(2*x - 1))

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sympy [A]  time = 0.13, size = 26, normalized size = 0.81 \begin {gather*} - \frac {5 \log {\left (x - \frac {1}{2} \right )}}{121} + \frac {5 \log {\left (x + \frac {3}{5} \right )}}{121} - \frac {1}{22 x - 11} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-5*log(x - 1/2)/121 + 5*log(x + 3/5)/121 - 1/(22*x - 11)

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