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

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

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Rubi [A]  time = 0.01, antiderivative size = 26, normalized size of antiderivative = 1.00, 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} \begin {gather*} -\frac {25 x}{6}-\frac {121}{28} \log (1-2 x)+\frac {1}{63} \log (3 x+2) \end {gather*}

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*}

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Mathematica [A]  time = 0.01, size = 32, normalized size = 1.23 \begin {gather*} -\frac {5}{6} (5 x+3)-\frac {121}{28} \log (5-10 x)+\frac {1}{63} \log (5 (3 x+2)) \end {gather*}

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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IntegrateAlgebraic [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {(3+5 x)^2}{(1-2 x) (2+3 x)} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

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

[Out]

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

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fricas [A]  time = 1.04, size = 20, normalized size = 0.77 \begin {gather*} -\frac {25}{6} \, x + \frac {1}{63} \, \log \left (3 \, x + 2\right ) - \frac {121}{28} \, \log \left (2 \, x - 1\right ) \end {gather*}

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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giac [A]  time = 0.92, size = 22, normalized size = 0.85 \begin {gather*} -\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 {gather*}

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))

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maple [A]  time = 0.01, size = 21, normalized size = 0.81 \begin {gather*} -\frac {25 x}{6}-\frac {121 \ln \left (2 x -1\right )}{28}+\frac {\ln \left (3 x +2\right )}{63} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maxima [A]  time = 0.51, size = 20, normalized size = 0.77 \begin {gather*} -\frac {25}{6} \, x + \frac {1}{63} \, \log \left (3 \, x + 2\right ) - \frac {121}{28} \, \log \left (2 \, x - 1\right ) \end {gather*}

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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mupad [B]  time = 1.12, size = 16, normalized size = 0.62 \begin {gather*} \frac {\ln \left (x+\frac {2}{3}\right )}{63}-\frac {121\,\ln \left (x-\frac {1}{2}\right )}{28}-\frac {25\,x}{6} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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sympy [A]  time = 0.13, size = 22, normalized size = 0.85 \begin {gather*} - \frac {25 x}{6} - \frac {121 \log {\left (x - \frac {1}{2} \right )}}{28} + \frac {\log {\left (x + \frac {2}{3} \right )}}{63} \end {gather*}

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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