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

Optimal. Leaf size=43 \[ \frac {121}{98 (1-2 x)}-\frac {1}{147 (3 x+2)}+\frac {22}{343} \log (1-2 x)-\frac {22}{343} \log (3 x+2) \]

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Rubi [A]  time = 0.02, antiderivative size = 43, 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}{98 (1-2 x)}-\frac {1}{147 (3 x+2)}+\frac {22}{343} \log (1-2 x)-\frac {22}{343} \log (3 x+2) \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

121/(98*(1 - 2*x)) - 1/(147*(2 + 3*x)) + (22*Log[1 - 2*x])/343 - (22*Log[2 + 3*x])/343

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)^2} \, dx &=\int \left (\frac {121}{49 (-1+2 x)^2}+\frac {44}{343 (-1+2 x)}+\frac {1}{49 (2+3 x)^2}-\frac {66}{343 (2+3 x)}\right ) \, dx\\ &=\frac {121}{98 (1-2 x)}-\frac {1}{147 (2+3 x)}+\frac {22}{343} \log (1-2 x)-\frac {22}{343} \log (2+3 x)\\ \end {align*}

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Mathematica [A]  time = 0.04, size = 38, normalized size = 0.88 \begin {gather*} \frac {-\frac {7 (1093 x+724)}{6 x^2+x-2}+132 \log (1-2 x)-132 \log (3 x+2)}{2058} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

((-7*(724 + 1093*x))/(-2 + x + 6*x^2) + 132*Log[1 - 2*x] - 132*Log[2 + 3*x])/2058

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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)^2} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

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

[Out]

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

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fricas [A]  time = 1.46, size = 49, normalized size = 1.14 \begin {gather*} -\frac {132 \, {\left (6 \, x^{2} + x - 2\right )} \log \left (3 \, x + 2\right ) - 132 \, {\left (6 \, x^{2} + x - 2\right )} \log \left (2 \, x - 1\right ) + 7651 \, x + 5068}{2058 \, {\left (6 \, x^{2} + x - 2\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)^2,x, algorithm="fricas")

[Out]

-1/2058*(132*(6*x^2 + x - 2)*log(3*x + 2) - 132*(6*x^2 + x - 2)*log(2*x - 1) + 7651*x + 5068)/(6*x^2 + x - 2)

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giac [A]  time = 1.12, size = 40, normalized size = 0.93 \begin {gather*} -\frac {1}{147 \, {\left (3 \, x + 2\right )}} + \frac {363}{343 \, {\left (\frac {7}{3 \, x + 2} - 2\right )}} + \frac {22}{343} \, \log \left ({\left | -\frac {7}{3 \, x + 2} + 2 \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)^2,x, algorithm="giac")

[Out]

-1/147/(3*x + 2) + 363/343/(7/(3*x + 2) - 2) + 22/343*log(abs(-7/(3*x + 2) + 2))

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maple [A]  time = 0.01, size = 36, normalized size = 0.84 \begin {gather*} \frac {22 \ln \left (2 x -1\right )}{343}-\frac {22 \ln \left (3 x +2\right )}{343}-\frac {1}{147 \left (3 x +2\right )}-\frac {121}{98 \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)^2,x)

[Out]

-1/147/(3*x+2)-22/343*ln(3*x+2)-121/98/(2*x-1)+22/343*ln(2*x-1)

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maxima [A]  time = 0.53, size = 34, normalized size = 0.79 \begin {gather*} -\frac {1093 \, x + 724}{294 \, {\left (6 \, x^{2} + x - 2\right )}} - \frac {22}{343} \, \log \left (3 \, x + 2\right ) + \frac {22}{343} \, \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)^2,x, algorithm="maxima")

[Out]

-1/294*(1093*x + 724)/(6*x^2 + x - 2) - 22/343*log(3*x + 2) + 22/343*log(2*x - 1)

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mupad [B]  time = 0.04, size = 26, normalized size = 0.60 \begin {gather*} -\frac {44\,\mathrm {atanh}\left (\frac {12\,x}{7}+\frac {1}{7}\right )}{343}-\frac {\frac {1093\,x}{1764}+\frac {181}{441}}{x^2+\frac {x}{6}-\frac {1}{3}} \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)^2),x)

[Out]

- (44*atanh((12*x)/7 + 1/7))/343 - ((1093*x)/1764 + 181/441)/(x/6 + x^2 - 1/3)

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sympy [A]  time = 0.14, size = 36, normalized size = 0.84 \begin {gather*} \frac {- 1093 x - 724}{1764 x^{2} + 294 x - 588} + \frac {22 \log {\left (x - \frac {1}{2} \right )}}{343} - \frac {22 \log {\left (x + \frac {2}{3} \right )}}{343} \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)**2,x)

[Out]

(-1093*x - 724)/(1764*x**2 + 294*x - 588) + 22*log(x - 1/2)/343 - 22*log(x + 2/3)/343

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