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

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

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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 (5 x+3)}-\frac {2}{121} \log (1-2 x)+\frac {2}{121} \log (5 x+3) \end {gather*}

Antiderivative was successfully verified.

[In]

Int[1/((1 - 2*x)*(3 + 5*x)^2),x]

[Out]

-1/(11*(3 + 5*x)) - (2*Log[1 - 2*x])/121 + (2*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) (3+5 x)^2} \, dx &=\int \left (-\frac {4}{121 (-1+2 x)}+\frac {5}{11 (3+5 x)^2}+\frac {10}{121 (3+5 x)}\right ) \, dx\\ &=-\frac {1}{11 (3+5 x)}-\frac {2}{121} \log (1-2 x)+\frac {2}{121} \log (3+5 x)\\ \end {align*}

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Mathematica [A]  time = 0.01, size = 30, normalized size = 0.94 \begin {gather*} \frac {1}{121} \left (-\frac {11}{5 x+3}-2 \log (5-10 x)+2 \log (5 x+3)\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(-11/(3 + 5*x) - 2*Log[5 - 10*x] + 2*Log[3 + 5*x])/121

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

Verification is not applicable to the result.

[In]

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

[Out]

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

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fricas [A]  time = 1.48, size = 37, normalized size = 1.16 \begin {gather*} \frac {2 \, {\left (5 \, x + 3\right )} \log \left (5 \, x + 3\right ) - 2 \, {\left (5 \, x + 3\right )} \log \left (2 \, x - 1\right ) - 11}{121 \, {\left (5 \, x + 3\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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giac [A]  time = 0.90, size = 25, normalized size = 0.78 \begin {gather*} -\frac {1}{11 \, {\left (5 \, x + 3\right )}} - \frac {2}{121} \, \log \left ({\left | -\frac {11}{5 \, x + 3} + 2 \right |}\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maxima [A]  time = 0.55, size = 26, normalized size = 0.81 \begin {gather*} -\frac {1}{11 \, {\left (5 \, x + 3\right )}} + \frac {2}{121} \, \log \left (5 \, x + 3\right ) - \frac {2}{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)/(3+5*x)^2,x, algorithm="maxima")

[Out]

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

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mupad [B]  time = 1.11, size = 26, normalized size = 0.81 \begin {gather*} \frac {2\,\ln \left (\frac {5\,x+3}{2\,x-1}\right )}{121}-\frac {1}{11\,\left (5\,x+3\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(-1/((2*x - 1)*(5*x + 3)^2),x)

[Out]

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

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sympy [A]  time = 0.13, size = 26, normalized size = 0.81 \begin {gather*} - \frac {2 \log {\left (x - \frac {1}{2} \right )}}{121} + \frac {2 \log {\left (x + \frac {3}{5} \right )}}{121} - \frac {1}{55 x + 33} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-2*log(x - 1/2)/121 + 2*log(x + 3/5)/121 - 1/(55*x + 33)

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