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

Optimal. Leaf size=32 \[ \frac {49}{44 (1-2 x)}+\frac {217}{484} \log (1-2 x)+\frac {1}{605} \log (5 x+3) \]

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Rubi [A]  time = 0.01, antiderivative size = 32, 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 = {88} \begin {gather*} \frac {49}{44 (1-2 x)}+\frac {217}{484} \log (1-2 x)+\frac {1}{605} \log (5 x+3) \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

49/(44*(1 - 2*x)) + (217*Log[1 - 2*x])/484 + Log[3 + 5*x]/605

Rule 88

Int[((a_.) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_.)*((e_.) + (f_.)*(x_))^(p_.), x_Symbol] :> Int[ExpandI
ntegrand[(a + b*x)^m*(c + d*x)^n*(e + f*x)^p, x], x] /; FreeQ[{a, b, c, d, e, f, p}, x] && IntegersQ[m, n] &&
(IntegerQ[p] || (GtQ[m, 0] && GeQ[n, -1]))

Rubi steps

\begin {align*} \int \frac {(2+3 x)^2}{(1-2 x)^2 (3+5 x)} \, dx &=\int \left (\frac {49}{22 (-1+2 x)^2}+\frac {217}{242 (-1+2 x)}+\frac {1}{121 (3+5 x)}\right ) \, dx\\ &=\frac {49}{44 (1-2 x)}+\frac {217}{484} \log (1-2 x)+\frac {1}{605} \log (3+5 x)\\ \end {align*}

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Mathematica [A]  time = 0.02, size = 40, normalized size = 1.25 \begin {gather*} -\frac {245}{44 (2 (5 x+3)-11)}+\frac {1}{605} \log (5 x+3)+\frac {217}{484} \log (11-2 (5 x+3)) \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

-245/(44*(-11 + 2*(3 + 5*x))) + Log[3 + 5*x]/605 + (217*Log[11 - 2*(3 + 5*x)])/484

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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)^2 (3+5 x)} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

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

[Out]

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

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fricas [A]  time = 1.33, size = 37, normalized size = 1.16 \begin {gather*} \frac {4 \, {\left (2 \, x - 1\right )} \log \left (5 \, x + 3\right ) + 1085 \, {\left (2 \, x - 1\right )} \log \left (2 \, x - 1\right ) - 2695}{2420 \, {\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)^2/(3+5*x),x, algorithm="fricas")

[Out]

1/2420*(4*(2*x - 1)*log(5*x + 3) + 1085*(2*x - 1)*log(2*x - 1) - 2695)/(2*x - 1)

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giac [A]  time = 1.19, size = 43, normalized size = 1.34 \begin {gather*} -\frac {49}{44 \, {\left (2 \, x - 1\right )}} - \frac {9}{20} \, \log \left (\frac {{\left | 2 \, x - 1 \right |}}{2 \, {\left (2 \, x - 1\right )}^{2}}\right ) + \frac {1}{605} \, \log \left ({\left | -\frac {11}{2 \, x - 1} - 5 \right |}\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-49/44/(2*x - 1) - 9/20*log(1/2*abs(2*x - 1)/(2*x - 1)^2) + 1/605*log(abs(-11/(2*x - 1) - 5))

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maple [A]  time = 0.01, size = 27, normalized size = 0.84 \begin {gather*} \frac {217 \ln \left (2 x -1\right )}{484}+\frac {\ln \left (5 x +3\right )}{605}-\frac {49}{44 \left (2 x -1\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/605*ln(5*x+3)-49/44/(2*x-1)+217/484*ln(2*x-1)

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maxima [A]  time = 0.48, size = 26, normalized size = 0.81 \begin {gather*} -\frac {49}{44 \, {\left (2 \, x - 1\right )}} + \frac {1}{605} \, \log \left (5 \, x + 3\right ) + \frac {217}{484} \, \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)^2/(3+5*x),x, algorithm="maxima")

[Out]

-49/44/(2*x - 1) + 1/605*log(5*x + 3) + 217/484*log(2*x - 1)

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mupad [B]  time = 0.08, size = 22, normalized size = 0.69 \begin {gather*} \frac {217\,\ln \left (x-\frac {1}{2}\right )}{484}+\frac {\ln \left (x+\frac {3}{5}\right )}{605}-\frac {49}{88\,\left (x-\frac {1}{2}\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

(217*log(x - 1/2))/484 + log(x + 3/5)/605 - 49/(88*(x - 1/2))

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sympy [A]  time = 0.14, size = 24, normalized size = 0.75 \begin {gather*} \frac {217 \log {\left (x - \frac {1}{2} \right )}}{484} + \frac {\log {\left (x + \frac {3}{5} \right )}}{605} - \frac {49}{88 x - 44} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

217*log(x - 1/2)/484 + log(x + 3/5)/605 - 49/(88*x - 44)

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