3.1491 \(\int \frac{(2+3 x)^2}{(1-2 x) (3+5 x)} \, dx\)

Optimal. Leaf size=26 \[ -\frac{9 x}{10}-\frac{49}{44} \log (1-2 x)+\frac{1}{275} \log (5 x+3) \]

[Out]

(-9*x)/10 - (49*Log[1 - 2*x])/44 + Log[3 + 5*x]/275

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Rubi [A]  time = 0.0127338, 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} \[ -\frac{9 x}{10}-\frac{49}{44} \log (1-2 x)+\frac{1}{275} \log (5 x+3) \]

Antiderivative was successfully verified.

[In]

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

[Out]

(-9*x)/10 - (49*Log[1 - 2*x])/44 + Log[3 + 5*x]/275

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{(2+3 x)^2}{(1-2 x) (3+5 x)} \, dx &=\int \left (-\frac{9}{10}-\frac{49}{22 (-1+2 x)}+\frac{1}{55 (3+5 x)}\right ) \, dx\\ &=-\frac{9 x}{10}-\frac{49}{44} \log (1-2 x)+\frac{1}{275} \log (3+5 x)\\ \end{align*}

Mathematica [A]  time = 0.0117347, size = 31, normalized size = 1.19 \[ -\frac{9 x}{10}-\frac{49}{44} \log (3-6 x)+\frac{1}{275} \log (-3 (5 x+3))-\frac{3}{5} \]

Antiderivative was successfully verified.

[In]

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

[Out]

-3/5 - (9*x)/10 - (49*Log[3 - 6*x])/44 + Log[-3*(3 + 5*x)]/275

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Maple [A]  time = 0.005, size = 21, normalized size = 0.8 \begin{align*} -{\frac{9\,x}{10}}-{\frac{49\,\ln \left ( 2\,x-1 \right ) }{44}}+{\frac{\ln \left ( 3+5\,x \right ) }{275}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-9/10*x-49/44*ln(2*x-1)+1/275*ln(3+5*x)

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Maxima [A]  time = 1.04088, size = 27, normalized size = 1.04 \begin{align*} -\frac{9}{10} \, x + \frac{1}{275} \, \log \left (5 \, x + 3\right ) - \frac{49}{44} \, \log \left (2 \, x - 1\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-9/10*x + 1/275*log(5*x + 3) - 49/44*log(2*x - 1)

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Fricas [A]  time = 1.3051, size = 69, normalized size = 2.65 \begin{align*} -\frac{9}{10} \, x + \frac{1}{275} \, \log \left (5 \, x + 3\right ) - \frac{49}{44} \, \log \left (2 \, x - 1\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-9/10*x + 1/275*log(5*x + 3) - 49/44*log(2*x - 1)

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Sympy [A]  time = 0.115384, size = 22, normalized size = 0.85 \begin{align*} - \frac{9 x}{10} - \frac{49 \log{\left (x - \frac{1}{2} \right )}}{44} + \frac{\log{\left (x + \frac{3}{5} \right )}}{275} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-9*x/10 - 49*log(x - 1/2)/44 + log(x + 3/5)/275

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Giac [A]  time = 1.77081, size = 30, normalized size = 1.15 \begin{align*} -\frac{9}{10} \, x + \frac{1}{275} \, \log \left ({\left | 5 \, x + 3 \right |}\right ) - \frac{49}{44} \, \log \left ({\left | 2 \, x - 1 \right |}\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-9/10*x + 1/275*log(abs(5*x + 3)) - 49/44*log(abs(2*x - 1))