3.1460 \(\int \frac{(3+5 x)^2}{(1-2 x) (2+3 x)} \, dx\)

Optimal. Leaf size=26 \[ -\frac{25 x}{6}-\frac{121}{28} \log (1-2 x)+\frac{1}{63} \log (3 x+2) \]

[Out]

(-25*x)/6 - (121*Log[1 - 2*x])/28 + Log[2 + 3*x]/63

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Rubi [A]  time = 0.0127887, antiderivative size = 26, 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} \[ -\frac{25 x}{6}-\frac{121}{28} \log (1-2 x)+\frac{1}{63} \log (3 x+2) \]

Antiderivative was successfully verified.

[In]

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

[Out]

(-25*x)/6 - (121*Log[1 - 2*x])/28 + Log[2 + 3*x]/63

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{(3+5 x)^2}{(1-2 x) (2+3 x)} \, dx &=\int \left (-\frac{25}{6}-\frac{121}{14 (-1+2 x)}+\frac{1}{21 (2+3 x)}\right ) \, dx\\ &=-\frac{25 x}{6}-\frac{121}{28} \log (1-2 x)+\frac{1}{63} \log (2+3 x)\\ \end{align*}

Mathematica [A]  time = 0.0122791, size = 32, normalized size = 1.23 \[ -\frac{5}{6} (5 x+3)-\frac{121}{28} \log (5-10 x)+\frac{1}{63} \log (5 (3 x+2)) \]

Antiderivative was successfully verified.

[In]

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

[Out]

(-5*(3 + 5*x))/6 - (121*Log[5 - 10*x])/28 + Log[5*(2 + 3*x)]/63

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Maple [A]  time = 0.005, size = 21, normalized size = 0.8 \begin{align*} -{\frac{25\,x}{6}}-{\frac{121\,\ln \left ( 2\,x-1 \right ) }{28}}+{\frac{\ln \left ( 2+3\,x \right ) }{63}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-25/6*x-121/28*ln(2*x-1)+1/63*ln(2+3*x)

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Maxima [A]  time = 2.36591, size = 27, normalized size = 1.04 \begin{align*} -\frac{25}{6} \, x + \frac{1}{63} \, \log \left (3 \, x + 2\right ) - \frac{121}{28} \, \log \left (2 \, x - 1\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-25/6*x + 1/63*log(3*x + 2) - 121/28*log(2*x - 1)

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Fricas [A]  time = 1.29075, size = 69, normalized size = 2.65 \begin{align*} -\frac{25}{6} \, x + \frac{1}{63} \, \log \left (3 \, x + 2\right ) - \frac{121}{28} \, \log \left (2 \, x - 1\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-25/6*x + 1/63*log(3*x + 2) - 121/28*log(2*x - 1)

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Sympy [A]  time = 0.116808, size = 22, normalized size = 0.85 \begin{align*} - \frac{25 x}{6} - \frac{121 \log{\left (x - \frac{1}{2} \right )}}{28} + \frac{\log{\left (x + \frac{2}{3} \right )}}{63} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-25*x/6 - 121*log(x - 1/2)/28 + log(x + 2/3)/63

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Giac [A]  time = 1.50511, size = 30, normalized size = 1.15 \begin{align*} -\frac{25}{6} \, x + \frac{1}{63} \, \log \left ({\left | 3 \, x + 2 \right |}\right ) - \frac{121}{28} \, \log \left ({\left | 2 \, x - 1 \right |}\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-25/6*x + 1/63*log(abs(3*x + 2)) - 121/28*log(abs(2*x - 1))