3.1625 \(\int \frac{1}{(1-2 x)^2 (3+5 x)^3} \, dx\)

Optimal. Leaf size=54 \[ \frac{4}{1331 (1-2 x)}-\frac{20}{1331 (5 x+3)}-\frac{5}{242 (5 x+3)^2}-\frac{60 \log (1-2 x)}{14641}+\frac{60 \log (5 x+3)}{14641} \]

[Out]

4/(1331*(1 - 2*x)) - 5/(242*(3 + 5*x)^2) - 20/(1331*(3 + 5*x)) - (60*Log[1 - 2*x])/14641 + (60*Log[3 + 5*x])/1
4641

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Rubi [A]  time = 0.0208356, antiderivative size = 54, normalized size of antiderivative = 1., 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} \[ \frac{4}{1331 (1-2 x)}-\frac{20}{1331 (5 x+3)}-\frac{5}{242 (5 x+3)^2}-\frac{60 \log (1-2 x)}{14641}+\frac{60 \log (5 x+3)}{14641} \]

Antiderivative was successfully verified.

[In]

Int[1/((1 - 2*x)^2*(3 + 5*x)^3),x]

[Out]

4/(1331*(1 - 2*x)) - 5/(242*(3 + 5*x)^2) - 20/(1331*(3 + 5*x)) - (60*Log[1 - 2*x])/14641 + (60*Log[3 + 5*x])/1
4641

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)^3} \, dx &=\int \left (\frac{8}{1331 (-1+2 x)^2}-\frac{120}{14641 (-1+2 x)}+\frac{25}{121 (3+5 x)^3}+\frac{100}{1331 (3+5 x)^2}+\frac{300}{14641 (3+5 x)}\right ) \, dx\\ &=\frac{4}{1331 (1-2 x)}-\frac{5}{242 (3+5 x)^2}-\frac{20}{1331 (3+5 x)}-\frac{60 \log (1-2 x)}{14641}+\frac{60 \log (3+5 x)}{14641}\\ \end{align*}

Mathematica [A]  time = 0.0210305, size = 47, normalized size = 0.87 \[ \frac{-\frac{11 \left (600 x^2+390 x-103\right )}{(2 x-1) (5 x+3)^2}-120 \log (1-2 x)+120 \log (10 x+6)}{29282} \]

Antiderivative was successfully verified.

[In]

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

[Out]

((-11*(-103 + 390*x + 600*x^2))/((-1 + 2*x)*(3 + 5*x)^2) - 120*Log[1 - 2*x] + 120*Log[6 + 10*x])/29282

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Maple [A]  time = 0.008, size = 45, normalized size = 0.8 \begin{align*} -{\frac{4}{2662\,x-1331}}-{\frac{60\,\ln \left ( 2\,x-1 \right ) }{14641}}-{\frac{5}{242\, \left ( 3+5\,x \right ) ^{2}}}-{\frac{20}{3993+6655\,x}}+{\frac{60\,\ln \left ( 3+5\,x \right ) }{14641}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-4/1331/(2*x-1)-60/14641*ln(2*x-1)-5/242/(3+5*x)^2-20/1331/(3+5*x)+60/14641*ln(3+5*x)

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Maxima [A]  time = 1.05222, size = 62, normalized size = 1.15 \begin{align*} -\frac{600 \, x^{2} + 390 \, x - 103}{2662 \,{\left (50 \, x^{3} + 35 \, x^{2} - 12 \, x - 9\right )}} + \frac{60}{14641} \, \log \left (5 \, x + 3\right ) - \frac{60}{14641} \, \log \left (2 \, x - 1\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/2662*(600*x^2 + 390*x - 103)/(50*x^3 + 35*x^2 - 12*x - 9) + 60/14641*log(5*x + 3) - 60/14641*log(2*x - 1)

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Fricas [A]  time = 1.17036, size = 219, normalized size = 4.06 \begin{align*} -\frac{6600 \, x^{2} - 120 \,{\left (50 \, x^{3} + 35 \, x^{2} - 12 \, x - 9\right )} \log \left (5 \, x + 3\right ) + 120 \,{\left (50 \, x^{3} + 35 \, x^{2} - 12 \, x - 9\right )} \log \left (2 \, x - 1\right ) + 4290 \, x - 1133}{29282 \,{\left (50 \, x^{3} + 35 \, x^{2} - 12 \, x - 9\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/29282*(6600*x^2 - 120*(50*x^3 + 35*x^2 - 12*x - 9)*log(5*x + 3) + 120*(50*x^3 + 35*x^2 - 12*x - 9)*log(2*x
- 1) + 4290*x - 1133)/(50*x^3 + 35*x^2 - 12*x - 9)

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Sympy [A]  time = 0.148568, size = 44, normalized size = 0.81 \begin{align*} - \frac{600 x^{2} + 390 x - 103}{133100 x^{3} + 93170 x^{2} - 31944 x - 23958} - \frac{60 \log{\left (x - \frac{1}{2} \right )}}{14641} + \frac{60 \log{\left (x + \frac{3}{5} \right )}}{14641} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-(600*x**2 + 390*x - 103)/(133100*x**3 + 93170*x**2 - 31944*x - 23958) - 60*log(x - 1/2)/14641 + 60*log(x + 3/
5)/14641

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Giac [A]  time = 2.46827, size = 69, normalized size = 1.28 \begin{align*} -\frac{4}{1331 \,{\left (2 \, x - 1\right )}} + \frac{50 \,{\left (\frac{66}{2 \, x - 1} + 25\right )}}{14641 \,{\left (\frac{11}{2 \, x - 1} + 5\right )}^{2}} + \frac{60}{14641} \, \log \left ({\left | -\frac{11}{2 \, x - 1} - 5 \right |}\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-4/1331/(2*x - 1) + 50/14641*(66/(2*x - 1) + 25)/(11/(2*x - 1) + 5)^2 + 60/14641*log(abs(-11/(2*x - 1) - 5))