3.1610 \(\int \frac{2+3 x}{(1-2 x)^2 (3+5 x)^2} \, dx\)

Optimal. Leaf size=43 \[ \frac{7}{121 (1-2 x)}-\frac{1}{121 (5 x+3)}-\frac{37 \log (1-2 x)}{1331}+\frac{37 \log (5 x+3)}{1331} \]

[Out]

7/(121*(1 - 2*x)) - 1/(121*(3 + 5*x)) - (37*Log[1 - 2*x])/1331 + (37*Log[3 + 5*x])/1331

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Rubi [A]  time = 0.0171225, antiderivative size = 43, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 1, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.05, Rules used = {77} \[ \frac{7}{121 (1-2 x)}-\frac{1}{121 (5 x+3)}-\frac{37 \log (1-2 x)}{1331}+\frac{37 \log (5 x+3)}{1331} \]

Antiderivative was successfully verified.

[In]

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

[Out]

7/(121*(1 - 2*x)) - 1/(121*(3 + 5*x)) - (37*Log[1 - 2*x])/1331 + (37*Log[3 + 5*x])/1331

Rule 77

Int[((a_.) + (b_.)*(x_))*((c_) + (d_.)*(x_))^(n_.)*((e_.) + (f_.)*(x_))^(p_.), x_Symbol] :> Int[ExpandIntegran
d[(a + b*x)*(c + d*x)^n*(e + f*x)^p, x], x] /; FreeQ[{a, b, c, d, e, f, n}, x] && NeQ[b*c - a*d, 0] && ((ILtQ[
n, 0] && ILtQ[p, 0]) || EqQ[p, 1] || (IGtQ[p, 0] && ( !IntegerQ[n] || LeQ[9*p + 5*(n + 2), 0] || GeQ[n + p + 1
, 0] || (GeQ[n + p + 2, 0] && RationalQ[a, b, c, d, e, f]))))

Rubi steps

\begin{align*} \int \frac{2+3 x}{(1-2 x)^2 (3+5 x)^2} \, dx &=\int \left (\frac{14}{121 (-1+2 x)^2}-\frac{74}{1331 (-1+2 x)}+\frac{5}{121 (3+5 x)^2}+\frac{185}{1331 (3+5 x)}\right ) \, dx\\ &=\frac{7}{121 (1-2 x)}-\frac{1}{121 (3+5 x)}-\frac{37 \log (1-2 x)}{1331}+\frac{37 \log (3+5 x)}{1331}\\ \end{align*}

Mathematica [A]  time = 0.0150774, size = 40, normalized size = 0.93 \[ \frac{-37 x-20}{121 \left (10 x^2+x-3\right )}-\frac{37 \log (1-2 x)}{1331}+\frac{37 \log (5 x+3)}{1331} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(-20 - 37*x)/(121*(-3 + x + 10*x^2)) - (37*Log[1 - 2*x])/1331 + (37*Log[3 + 5*x])/1331

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Maple [A]  time = 0.008, size = 36, normalized size = 0.8 \begin{align*} -{\frac{7}{242\,x-121}}-{\frac{37\,\ln \left ( 2\,x-1 \right ) }{1331}}-{\frac{1}{363+605\,x}}+{\frac{37\,\ln \left ( 3+5\,x \right ) }{1331}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-7/121/(2*x-1)-37/1331*ln(2*x-1)-1/121/(3+5*x)+37/1331*ln(3+5*x)

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Maxima [A]  time = 1.0772, size = 46, normalized size = 1.07 \begin{align*} -\frac{37 \, x + 20}{121 \,{\left (10 \, x^{2} + x - 3\right )}} + \frac{37}{1331} \, \log \left (5 \, x + 3\right ) - \frac{37}{1331} \, \log \left (2 \, x - 1\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/121*(37*x + 20)/(10*x^2 + x - 3) + 37/1331*log(5*x + 3) - 37/1331*log(2*x - 1)

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Fricas [A]  time = 1.31441, size = 147, normalized size = 3.42 \begin{align*} \frac{37 \,{\left (10 \, x^{2} + x - 3\right )} \log \left (5 \, x + 3\right ) - 37 \,{\left (10 \, x^{2} + x - 3\right )} \log \left (2 \, x - 1\right ) - 407 \, x - 220}{1331 \,{\left (10 \, x^{2} + x - 3\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/1331*(37*(10*x^2 + x - 3)*log(5*x + 3) - 37*(10*x^2 + x - 3)*log(2*x - 1) - 407*x - 220)/(10*x^2 + x - 3)

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Sympy [A]  time = 0.132012, size = 34, normalized size = 0.79 \begin{align*} - \frac{37 x + 20}{1210 x^{2} + 121 x - 363} - \frac{37 \log{\left (x - \frac{1}{2} \right )}}{1331} + \frac{37 \log{\left (x + \frac{3}{5} \right )}}{1331} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-(37*x + 20)/(1210*x**2 + 121*x - 363) - 37*log(x - 1/2)/1331 + 37*log(x + 3/5)/1331

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Giac [A]  time = 3.15209, size = 54, normalized size = 1.26 \begin{align*} -\frac{1}{121 \,{\left (5 \, x + 3\right )}} + \frac{70}{1331 \,{\left (\frac{11}{5 \, x + 3} - 2\right )}} - \frac{37}{1331} \, \log \left ({\left | -\frac{11}{5 \, x + 3} + 2 \right |}\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/121/(5*x + 3) + 70/1331/(11/(5*x + 3) - 2) - 37/1331*log(abs(-11/(5*x + 3) + 2))