3.1524 \(\int \frac{1}{(1-2 x) (3+5 x)^3} \, dx\)

Optimal. Leaf size=43 \[ -\frac{2}{121 (5 x+3)}-\frac{1}{22 (5 x+3)^2}-\frac{4 \log (1-2 x)}{1331}+\frac{4 \log (5 x+3)}{1331} \]

[Out]

-1/(22*(3 + 5*x)^2) - 2/(121*(3 + 5*x)) - (4*Log[1 - 2*x])/1331 + (4*Log[3 + 5*x])/1331

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Rubi [A]  time = 0.014529, antiderivative size = 43, 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} \[ -\frac{2}{121 (5 x+3)}-\frac{1}{22 (5 x+3)^2}-\frac{4 \log (1-2 x)}{1331}+\frac{4 \log (5 x+3)}{1331} \]

Antiderivative was successfully verified.

[In]

Int[1/((1 - 2*x)*(3 + 5*x)^3),x]

[Out]

-1/(22*(3 + 5*x)^2) - 2/(121*(3 + 5*x)) - (4*Log[1 - 2*x])/1331 + (4*Log[3 + 5*x])/1331

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)^3} \, dx &=\int \left (-\frac{8}{1331 (-1+2 x)}+\frac{5}{11 (3+5 x)^3}+\frac{10}{121 (3+5 x)^2}+\frac{20}{1331 (3+5 x)}\right ) \, dx\\ &=-\frac{1}{22 (3+5 x)^2}-\frac{2}{121 (3+5 x)}-\frac{4 \log (1-2 x)}{1331}+\frac{4 \log (3+5 x)}{1331}\\ \end{align*}

Mathematica [A]  time = 0.0146901, size = 35, normalized size = 0.81 \[ \frac{-\frac{11 (20 x+23)}{(5 x+3)^2}-8 \log (5-10 x)+8 \log (5 x+3)}{2662} \]

Antiderivative was successfully verified.

[In]

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

[Out]

((-11*(23 + 20*x))/(3 + 5*x)^2 - 8*Log[5 - 10*x] + 8*Log[3 + 5*x])/2662

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Maple [A]  time = 0.006, size = 36, normalized size = 0.8 \begin{align*} -{\frac{4\,\ln \left ( 2\,x-1 \right ) }{1331}}-{\frac{1}{22\, \left ( 3+5\,x \right ) ^{2}}}-{\frac{2}{363+605\,x}}+{\frac{4\,\ln \left ( 3+5\,x \right ) }{1331}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-4/1331*ln(2*x-1)-1/22/(3+5*x)^2-2/121/(3+5*x)+4/1331*ln(3+5*x)

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Maxima [A]  time = 1.03923, size = 49, normalized size = 1.14 \begin{align*} -\frac{20 \, x + 23}{242 \,{\left (25 \, x^{2} + 30 \, x + 9\right )}} + \frac{4}{1331} \, \log \left (5 \, x + 3\right ) - \frac{4}{1331} \, \log \left (2 \, x - 1\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/242*(20*x + 23)/(25*x^2 + 30*x + 9) + 4/1331*log(5*x + 3) - 4/1331*log(2*x - 1)

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Fricas [A]  time = 1.31193, size = 157, normalized size = 3.65 \begin{align*} \frac{8 \,{\left (25 \, x^{2} + 30 \, x + 9\right )} \log \left (5 \, x + 3\right ) - 8 \,{\left (25 \, x^{2} + 30 \, x + 9\right )} \log \left (2 \, x - 1\right ) - 220 \, x - 253}{2662 \,{\left (25 \, x^{2} + 30 \, x + 9\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/2662*(8*(25*x^2 + 30*x + 9)*log(5*x + 3) - 8*(25*x^2 + 30*x + 9)*log(2*x - 1) - 220*x - 253)/(25*x^2 + 30*x
+ 9)

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Sympy [A]  time = 0.135371, size = 34, normalized size = 0.79 \begin{align*} - \frac{20 x + 23}{6050 x^{2} + 7260 x + 2178} - \frac{4 \log{\left (x - \frac{1}{2} \right )}}{1331} + \frac{4 \log{\left (x + \frac{3}{5} \right )}}{1331} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-(20*x + 23)/(6050*x**2 + 7260*x + 2178) - 4*log(x - 1/2)/1331 + 4*log(x + 3/5)/1331

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Giac [A]  time = 1.24637, size = 45, normalized size = 1.05 \begin{align*} -\frac{20 \, x + 23}{242 \,{\left (5 \, x + 3\right )}^{2}} + \frac{4}{1331} \, \log \left ({\left | 5 \, x + 3 \right |}\right ) - \frac{4}{1331} \, \log \left ({\left | 2 \, x - 1 \right |}\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/242*(20*x + 23)/(5*x + 3)^2 + 4/1331*log(abs(5*x + 3)) - 4/1331*log(abs(2*x - 1))