3.1443 \(\int \frac{3+5 x}{(1-2 x) (2+3 x)} \, dx\)

Optimal. Leaf size=21 \[ -\frac{11}{14} \log (1-2 x)-\frac{1}{21} \log (3 x+2) \]

[Out]

(-11*Log[1 - 2*x])/14 - Log[2 + 3*x]/21

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Rubi [A]  time = 0.0098538, antiderivative size = 21, 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 = {72} \[ -\frac{11}{14} \log (1-2 x)-\frac{1}{21} \log (3 x+2) \]

Antiderivative was successfully verified.

[In]

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

[Out]

(-11*Log[1 - 2*x])/14 - Log[2 + 3*x]/21

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}{(1-2 x) (2+3 x)} \, dx &=\int \left (-\frac{11}{7 (-1+2 x)}-\frac{1}{7 (2+3 x)}\right ) \, dx\\ &=-\frac{11}{14} \log (1-2 x)-\frac{1}{21} \log (2+3 x)\\ \end{align*}

Mathematica [A]  time = 0.0043201, size = 21, normalized size = 1. \[ -\frac{11}{14} \log (1-2 x)-\frac{1}{21} \log (3 x+2) \]

Antiderivative was successfully verified.

[In]

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

[Out]

(-11*Log[1 - 2*x])/14 - Log[2 + 3*x]/21

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Maple [A]  time = 0.004, size = 18, normalized size = 0.9 \begin{align*} -{\frac{11\,\ln \left ( 2\,x-1 \right ) }{14}}-{\frac{\ln \left ( 2+3\,x \right ) }{21}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-11/14*ln(2*x-1)-1/21*ln(2+3*x)

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Maxima [A]  time = 1.83039, size = 23, normalized size = 1.1 \begin{align*} -\frac{1}{21} \, \log \left (3 \, x + 2\right ) - \frac{11}{14} \, \log \left (2 \, x - 1\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/21*log(3*x + 2) - 11/14*log(2*x - 1)

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Fricas [A]  time = 1.17874, size = 55, normalized size = 2.62 \begin{align*} -\frac{1}{21} \, \log \left (3 \, x + 2\right ) - \frac{11}{14} \, \log \left (2 \, x - 1\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/21*log(3*x + 2) - 11/14*log(2*x - 1)

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Sympy [A]  time = 0.106297, size = 19, normalized size = 0.9 \begin{align*} - \frac{11 \log{\left (x - \frac{1}{2} \right )}}{14} - \frac{\log{\left (x + \frac{2}{3} \right )}}{21} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-11*log(x - 1/2)/14 - log(x + 2/3)/21

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Giac [A]  time = 1.82513, size = 26, normalized size = 1.24 \begin{align*} -\frac{1}{21} \, \log \left ({\left | 3 \, x + 2 \right |}\right ) - \frac{11}{14} \, \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)/(1-2*x)/(2+3*x),x, algorithm="giac")

[Out]

-1/21*log(abs(3*x + 2)) - 11/14*log(abs(2*x - 1))