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

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

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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 {9 x}{10}-\frac {49}{44} \log (1-2 x)+\frac {1}{275} \log (5 x+3) \end {gather*}

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*}

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Mathematica [A]  time = 0.02, size = 31, normalized size = 1.19 \begin {gather*} -\frac {9 x}{10}-\frac {49}{44} \log (3-6 x)+\frac {1}{275} \log (-3 (5 x+3))-\frac {3}{5} \end {gather*}

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

Verification is not applicable to the result.

[In]

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

[Out]

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

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fricas [A]  time = 1.21, size = 20, normalized size = 0.77 \begin {gather*} -\frac {9}{10} \, x + \frac {1}{275} \, \log \left (5 \, x + 3\right ) - \frac {49}{44} \, \log \left (2 \, x - 1\right ) \end {gather*}

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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giac [A]  time = 0.88, size = 22, normalized size = 0.85 \begin {gather*} -\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 {gather*}

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))

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maple [A]  time = 0.01, size = 21, normalized size = 0.81 \begin {gather*} -\frac {9 x}{10}-\frac {49 \ln \left (2 x -1\right )}{44}+\frac {\ln \left (5 x +3\right )}{275} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maxima [A]  time = 0.48, size = 20, normalized size = 0.77 \begin {gather*} -\frac {9}{10} \, x + \frac {1}{275} \, \log \left (5 \, x + 3\right ) - \frac {49}{44} \, \log \left (2 \, x - 1\right ) \end {gather*}

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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mupad [B]  time = 0.05, size = 16, normalized size = 0.62 \begin {gather*} \frac {\ln \left (x+\frac {3}{5}\right )}{275}-\frac {49\,\ln \left (x-\frac {1}{2}\right )}{44}-\frac {9\,x}{10} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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sympy [A]  time = 0.13, size = 22, normalized size = 0.85 \begin {gather*} - \frac {9 x}{10} - \frac {49 \log {\left (x - \frac {1}{2} \right )}}{44} + \frac {\log {\left (x + \frac {3}{5} \right )}}{275} \end {gather*}

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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