3.17.11 \(\int \frac {1}{(1-2 x)^2 (3+5 x)^2} \, dx\) [1611]

Optimal. Leaf size=43 \[ \frac {2}{121 (1-2 x)}-\frac {5}{121 (3+5 x)}-\frac {20 \log (1-2 x)}{1331}+\frac {20 \log (3+5 x)}{1331} \]

[Out]

2/121/(1-2*x)-5/121/(3+5*x)-20/1331*ln(1-2*x)+20/1331*ln(3+5*x)

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Rubi [A]
time = 0.01, antiderivative size = 43, 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 = {46} \begin {gather*} \frac {2}{121 (1-2 x)}-\frac {5}{121 (5 x+3)}-\frac {20 \log (1-2 x)}{1331}+\frac {20 \log (5 x+3)}{1331} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[1/((1 - 2*x)^2*(3 + 5*x)^2),x]

[Out]

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

Rule 46

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] && Lt
Q[m + n + 2, 0])

Rubi steps

\begin {align*} \int \frac {1}{(1-2 x)^2 (3+5 x)^2} \, dx &=\int \left (\frac {4}{121 (-1+2 x)^2}-\frac {40}{1331 (-1+2 x)}+\frac {25}{121 (3+5 x)^2}+\frac {100}{1331 (3+5 x)}\right ) \, dx\\ &=\frac {2}{121 (1-2 x)}-\frac {5}{121 (3+5 x)}-\frac {20 \log (1-2 x)}{1331}+\frac {20 \log (3+5 x)}{1331}\\ \end {align*}

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Mathematica [A]
time = 0.01, size = 40, normalized size = 0.93 \begin {gather*} \frac {-1-20 x}{121 \left (-3+x+10 x^2\right )}-\frac {20 \log (1-2 x)}{1331}+\frac {20 \log (3+5 x)}{1331} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(-1 - 20*x)/(121*(-3 + x + 10*x^2)) - (20*Log[1 - 2*x])/1331 + (20*Log[3 + 5*x])/1331

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Maple [A]
time = 0.10, size = 36, normalized size = 0.84

method	result	size
default	\(-\frac {2}{121 \left (-1+2 x \right )}-\frac {20 \ln \left (-1+2 x \right )}{1331}-\frac {5}{121 \left (3+5 x \right )}+\frac {20 \ln \left (3+5 x \right )}{1331}\)	\(36\)
risch	\(\frac {-\frac {20 x}{121}-\frac {1}{121}}{\left (3+5 x \right ) \left (-1+2 x \right )}-\frac {20 \ln \left (-1+2 x \right )}{1331}+\frac {20 \ln \left (3+5 x \right )}{1331}\)	\(39\)
norman	\(\frac {\frac {200 x^{2}}{121}-\frac {61}{121}}{\left (3+5 x \right ) \left (-1+2 x \right )}-\frac {20 \ln \left (-1+2 x \right )}{1331}+\frac {20 \ln \left (3+5 x \right )}{1331}\)	\(40\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-2/121/(-1+2*x)-20/1331*ln(-1+2*x)-5/121/(3+5*x)+20/1331*ln(3+5*x)

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Maxima [A]
time = 0.32, size = 34, normalized size = 0.79 \begin {gather*} -\frac {20 \, x + 1}{121 \, {\left (10 \, x^{2} + x - 3\right )}} + \frac {20}{1331} \, \log \left (5 \, x + 3\right ) - \frac {20}{1331} \, \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)^2,x, algorithm="maxima")

[Out]

-1/121*(20*x + 1)/(10*x^2 + x - 3) + 20/1331*log(5*x + 3) - 20/1331*log(2*x - 1)

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Fricas [A]
time = 0.43, size = 49, normalized size = 1.14 \begin {gather*} \frac {20 \, {\left (10 \, x^{2} + x - 3\right )} \log \left (5 \, x + 3\right ) - 20 \, {\left (10 \, x^{2} + x - 3\right )} \log \left (2 \, x - 1\right ) - 220 \, x - 11}{1331 \, {\left (10 \, x^{2} + x - 3\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/1331*(20*(10*x^2 + x - 3)*log(5*x + 3) - 20*(10*x^2 + x - 3)*log(2*x - 1) - 220*x - 11)/(10*x^2 + x - 3)

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Sympy [A]
time = 0.07, size = 36, normalized size = 0.84 \begin {gather*} \frac {- 20 x - 1}{1210 x^{2} + 121 x - 363} - \frac {20 \log {\left (x - \frac {1}{2} \right )}}{1331} + \frac {20 \log {\left (x + \frac {3}{5} \right )}}{1331} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

(-20*x - 1)/(1210*x**2 + 121*x - 363) - 20*log(x - 1/2)/1331 + 20*log(x + 3/5)/1331

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Giac [A]
time = 0.57, size = 40, normalized size = 0.93 \begin {gather*} -\frac {5}{121 \, {\left (5 \, x + 3\right )}} + \frac {20}{1331 \, {\left (\frac {11}{5 \, x + 3} - 2\right )}} - \frac {20}{1331} \, \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)^2/(3+5*x)^2,x, algorithm="giac")

[Out]

-5/121/(5*x + 3) + 20/1331/(11/(5*x + 3) - 2) - 20/1331*log(abs(-11/(5*x + 3) + 2))

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Mupad [B]
time = 0.05, size = 42, normalized size = 0.98 \begin {gather*} \frac {20\,\ln \left (\frac {5\,x+3}{2\,x-1}\right )}{1331}-\frac {1}{11\,\left (2\,x-1\right )\,\left (5\,x+3\right )}-\frac {10}{121\,\left (5\,x+3\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/((2*x - 1)^2*(5*x + 3)^2),x)

[Out]

(20*log((5*x + 3)/(2*x - 1)))/1331 - 1/(11*(2*x - 1)*(5*x + 3)) - 10/(121*(5*x + 3))

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