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

Optimal. Leaf size=32 \[ \frac {121}{28 (1-2 x)}+\frac {407}{196} \log (1-2 x)+\frac {1}{147} \log (3 x+2) \]

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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 {121}{28 (1-2 x)}+\frac {407}{196} \log (1-2 x)+\frac {1}{147} \log (3 x+2) \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

121/(28*(1 - 2*x)) + (407*Log[1 - 2*x])/196 + Log[2 + 3*x]/147

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 {(3+5 x)^2}{(1-2 x)^2 (2+3 x)} \, dx &=\int \left (\frac {121}{14 (-1+2 x)^2}+\frac {407}{98 (-1+2 x)}+\frac {1}{49 (2+3 x)}\right ) \, dx\\ &=\frac {121}{28 (1-2 x)}+\frac {407}{196} \log (1-2 x)+\frac {1}{147} \log (2+3 x)\\ \end {align*}

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Mathematica [A]  time = 0.03, size = 40, normalized size = 1.25 \begin {gather*} -\frac {363}{28 (2 (3 x+2)-7)}+\frac {1}{147} \log (3 x+2)+\frac {407}{196} \log (7-2 (3 x+2)) \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

-363/(28*(-7 + 2*(2 + 3*x))) + Log[2 + 3*x]/147 + (407*Log[7 - 2*(2 + 3*x)])/196

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

Verification is not applicable to the result.

[In]

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

[Out]

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

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fricas [A]  time = 1.57, size = 37, normalized size = 1.16 \begin {gather*} \frac {4 \, {\left (2 \, x - 1\right )} \log \left (3 \, x + 2\right ) + 1221 \, {\left (2 \, x - 1\right )} \log \left (2 \, x - 1\right ) - 2541}{588 \, {\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/(2+3*x),x, algorithm="fricas")

[Out]

1/588*(4*(2*x - 1)*log(3*x + 2) + 1221*(2*x - 1)*log(2*x - 1) - 2541)/(2*x - 1)

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giac [A]  time = 0.89, size = 43, normalized size = 1.34 \begin {gather*} -\frac {121}{28 \, {\left (2 \, x - 1\right )}} - \frac {25}{12} \, \log \left (\frac {{\left | 2 \, x - 1 \right |}}{2 \, {\left (2 \, x - 1\right )}^{2}}\right ) + \frac {1}{147} \, \log \left ({\left | -\frac {7}{2 \, x - 1} - 3 \right |}\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-121/28/(2*x - 1) - 25/12*log(1/2*abs(2*x - 1)/(2*x - 1)^2) + 1/147*log(abs(-7/(2*x - 1) - 3))

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maple [A]  time = 0.01, size = 27, normalized size = 0.84 \begin {gather*} \frac {407 \ln \left (2 x -1\right )}{196}+\frac {\ln \left (3 x +2\right )}{147}-\frac {121}{28 \left (2 x -1\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/147*ln(3*x+2)-121/28/(2*x-1)+407/196*ln(2*x-1)

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maxima [A]  time = 0.48, size = 26, normalized size = 0.81 \begin {gather*} -\frac {121}{28 \, {\left (2 \, x - 1\right )}} + \frac {1}{147} \, \log \left (3 \, x + 2\right ) + \frac {407}{196} \, \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/(2+3*x),x, algorithm="maxima")

[Out]

-121/28/(2*x - 1) + 1/147*log(3*x + 2) + 407/196*log(2*x - 1)

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mupad [B]  time = 0.05, size = 22, normalized size = 0.69 \begin {gather*} \frac {407\,\ln \left (x-\frac {1}{2}\right )}{196}+\frac {\ln \left (x+\frac {2}{3}\right )}{147}-\frac {121}{56\,\left (x-\frac {1}{2}\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

(407*log(x - 1/2))/196 + log(x + 2/3)/147 - 121/(56*(x - 1/2))

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sympy [A]  time = 0.14, size = 24, normalized size = 0.75 \begin {gather*} \frac {407 \log {\left (x - \frac {1}{2} \right )}}{196} + \frac {\log {\left (x + \frac {2}{3} \right )}}{147} - \frac {121}{56 x - 28} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

407*log(x - 1/2)/196 + log(x + 2/3)/147 - 121/(56*x - 28)

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