3.1495 \(\int \frac{1}{(1-2 x) (2+3 x)^2 (3+5 x)} \, dx\)

Optimal. Leaf size=42 \[ \frac{3}{7 (3 x+2)}-\frac{4}{539} \log (1-2 x)-\frac{111}{49} \log (3 x+2)+\frac{25}{11} \log (5 x+3) \]

[Out]

3/(7*(2 + 3*x)) - (4*Log[1 - 2*x])/539 - (111*Log[2 + 3*x])/49 + (25*Log[3 + 5*x])/11

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Rubi [A]  time = 0.017436, antiderivative size = 42, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 1, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.045, Rules used = {72} \[ \frac{3}{7 (3 x+2)}-\frac{4}{539} \log (1-2 x)-\frac{111}{49} \log (3 x+2)+\frac{25}{11} \log (5 x+3) \]

Antiderivative was successfully verified.

[In]

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

[Out]

3/(7*(2 + 3*x)) - (4*Log[1 - 2*x])/539 - (111*Log[2 + 3*x])/49 + (25*Log[3 + 5*x])/11

Rule 72

Int[((e_.) + (f_.)*(x_))^(p_.)/(((a_.) + (b_.)*(x_))*((c_.) + (d_.)*(x_))), x_Symbol] :> Int[ExpandIntegrand[(
e + f*x)^p/((a + b*x)*(c + d*x)), x], x] /; FreeQ[{a, b, c, d, e, f}, x] && IntegerQ[p]

Rubi steps

\begin{align*} \int \frac{1}{(1-2 x) (2+3 x)^2 (3+5 x)} \, dx &=\int \left (-\frac{8}{539 (-1+2 x)}-\frac{9}{7 (2+3 x)^2}-\frac{333}{49 (2+3 x)}+\frac{125}{11 (3+5 x)}\right ) \, dx\\ &=\frac{3}{7 (2+3 x)}-\frac{4}{539} \log (1-2 x)-\frac{111}{49} \log (2+3 x)+\frac{25}{11} \log (3+5 x)\\ \end{align*}

Mathematica [A]  time = 0.0199891, size = 38, normalized size = 0.9 \[ \frac{1}{539} \left (\frac{231}{3 x+2}-4 \log (1-2 x)-1221 \log (6 x+4)+1225 \log (10 x+6)\right ) \]

Antiderivative was successfully verified.

[In]

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

[Out]

(231/(2 + 3*x) - 4*Log[1 - 2*x] - 1221*Log[4 + 6*x] + 1225*Log[6 + 10*x])/539

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Maple [A]  time = 0.007, size = 35, normalized size = 0.8 \begin{align*} -{\frac{4\,\ln \left ( 2\,x-1 \right ) }{539}}+{\frac{3}{14+21\,x}}-{\frac{111\,\ln \left ( 2+3\,x \right ) }{49}}+{\frac{25\,\ln \left ( 3+5\,x \right ) }{11}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-4/539*ln(2*x-1)+3/7/(2+3*x)-111/49*ln(2+3*x)+25/11*ln(3+5*x)

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Maxima [A]  time = 1.16987, size = 46, normalized size = 1.1 \begin{align*} \frac{3}{7 \,{\left (3 \, x + 2\right )}} + \frac{25}{11} \, \log \left (5 \, x + 3\right ) - \frac{111}{49} \, \log \left (3 \, x + 2\right ) - \frac{4}{539} \, \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*x)^2/(3+5*x),x, algorithm="maxima")

[Out]

3/7/(3*x + 2) + 25/11*log(5*x + 3) - 111/49*log(3*x + 2) - 4/539*log(2*x - 1)

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Fricas [A]  time = 1.40554, size = 149, normalized size = 3.55 \begin{align*} \frac{1225 \,{\left (3 \, x + 2\right )} \log \left (5 \, x + 3\right ) - 1221 \,{\left (3 \, x + 2\right )} \log \left (3 \, x + 2\right ) - 4 \,{\left (3 \, x + 2\right )} \log \left (2 \, x - 1\right ) + 231}{539 \,{\left (3 \, x + 2\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/539*(1225*(3*x + 2)*log(5*x + 3) - 1221*(3*x + 2)*log(3*x + 2) - 4*(3*x + 2)*log(2*x - 1) + 231)/(3*x + 2)

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Sympy [A]  time = 0.159433, size = 36, normalized size = 0.86 \begin{align*} - \frac{4 \log{\left (x - \frac{1}{2} \right )}}{539} + \frac{25 \log{\left (x + \frac{3}{5} \right )}}{11} - \frac{111 \log{\left (x + \frac{2}{3} \right )}}{49} + \frac{3}{21 x + 14} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-4*log(x - 1/2)/539 + 25*log(x + 3/5)/11 - 111*log(x + 2/3)/49 + 3/(21*x + 14)

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Giac [A]  time = 1.45932, size = 54, normalized size = 1.29 \begin{align*} \frac{3}{7 \,{\left (3 \, x + 2\right )}} + \frac{25}{11} \, \log \left ({\left | -\frac{1}{3 \, x + 2} + 5 \right |}\right ) - \frac{4}{539} \, \log \left ({\left | -\frac{7}{3 \, x + 2} + 2 \right |}\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

3/7/(3*x + 2) + 25/11*log(abs(-1/(3*x + 2) + 5)) - 4/539*log(abs(-7/(3*x + 2) + 2))