3.14.40 \(\int \frac {(3+5 x)^3}{(1-2 x) (2+3 x)^4} \, dx\)

Optimal. Leaf size=54 \[ \frac {3469}{9261 (3 x+2)}-\frac {103}{2646 (3 x+2)^2}+\frac {1}{567 (3 x+2)^3}-\frac {1331 \log (1-2 x)}{2401}+\frac {1331 \log (3 x+2)}{2401} \]

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Rubi [A]  time = 0.02, antiderivative size = 54, 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 {3469}{9261 (3 x+2)}-\frac {103}{2646 (3 x+2)^2}+\frac {1}{567 (3 x+2)^3}-\frac {1331 \log (1-2 x)}{2401}+\frac {1331 \log (3 x+2)}{2401} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(3 + 5*x)^3/((1 - 2*x)*(2 + 3*x)^4),x]

[Out]

1/(567*(2 + 3*x)^3) - 103/(2646*(2 + 3*x)^2) + 3469/(9261*(2 + 3*x)) - (1331*Log[1 - 2*x])/2401 + (1331*Log[2
+ 3*x])/2401

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)^3}{(1-2 x) (2+3 x)^4} \, dx &=\int \left (-\frac {2662}{2401 (-1+2 x)}-\frac {1}{63 (2+3 x)^4}+\frac {103}{441 (2+3 x)^3}-\frac {3469}{3087 (2+3 x)^2}+\frac {3993}{2401 (2+3 x)}\right ) \, dx\\ &=\frac {1}{567 (2+3 x)^3}-\frac {103}{2646 (2+3 x)^2}+\frac {3469}{9261 (2+3 x)}-\frac {1331 \log (1-2 x)}{2401}+\frac {1331 \log (2+3 x)}{2401}\\ \end {align*}

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Mathematica [A]  time = 0.04, size = 40, normalized size = 0.74 \begin {gather*} \frac {\frac {7 \left (187326 x^2+243279 x+79028\right )}{(3 x+2)^3}-215622 \log (1-2 x)+215622 \log (6 x+4)}{388962} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(3 + 5*x)^3/((1 - 2*x)*(2 + 3*x)^4),x]

[Out]

((7*(79028 + 243279*x + 187326*x^2))/(2 + 3*x)^3 - 215622*Log[1 - 2*x] + 215622*Log[4 + 6*x])/388962

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IntegrateAlgebraic [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {(3+5 x)^3}{(1-2 x) (2+3 x)^4} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

IntegrateAlgebraic[(3 + 5*x)^3/((1 - 2*x)*(2 + 3*x)^4),x]

[Out]

IntegrateAlgebraic[(3 + 5*x)^3/((1 - 2*x)*(2 + 3*x)^4), x]

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fricas [A]  time = 0.83, size = 75, normalized size = 1.39 \begin {gather*} \frac {1311282 \, x^{2} + 215622 \, {\left (27 \, x^{3} + 54 \, x^{2} + 36 \, x + 8\right )} \log \left (3 \, x + 2\right ) - 215622 \, {\left (27 \, x^{3} + 54 \, x^{2} + 36 \, x + 8\right )} \log \left (2 \, x - 1\right ) + 1702953 \, x + 553196}{388962 \, {\left (27 \, x^{3} + 54 \, x^{2} + 36 \, x + 8\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((3+5*x)^3/(1-2*x)/(2+3*x)^4,x, algorithm="fricas")

[Out]

1/388962*(1311282*x^2 + 215622*(27*x^3 + 54*x^2 + 36*x + 8)*log(3*x + 2) - 215622*(27*x^3 + 54*x^2 + 36*x + 8)
*log(2*x - 1) + 1702953*x + 553196)/(27*x^3 + 54*x^2 + 36*x + 8)

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giac [A]  time = 0.91, size = 38, normalized size = 0.70 \begin {gather*} \frac {187326 \, x^{2} + 243279 \, x + 79028}{55566 \, {\left (3 \, x + 2\right )}^{3}} + \frac {1331}{2401} \, \log \left ({\left | 3 \, x + 2 \right |}\right ) - \frac {1331}{2401} \, \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)^3/(1-2*x)/(2+3*x)^4,x, algorithm="giac")

[Out]

1/55566*(187326*x^2 + 243279*x + 79028)/(3*x + 2)^3 + 1331/2401*log(abs(3*x + 2)) - 1331/2401*log(abs(2*x - 1)
)

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maple [A]  time = 0.01, size = 45, normalized size = 0.83 \begin {gather*} -\frac {1331 \ln \left (2 x -1\right )}{2401}+\frac {1331 \ln \left (3 x +2\right )}{2401}+\frac {1}{567 \left (3 x +2\right )^{3}}-\frac {103}{2646 \left (3 x +2\right )^{2}}+\frac {3469}{9261 \left (3 x +2\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((5*x+3)^3/(1-2*x)/(3*x+2)^4,x)

[Out]

1/567/(3*x+2)^3-103/2646/(3*x+2)^2+3469/9261/(3*x+2)+1331/2401*ln(3*x+2)-1331/2401*ln(2*x-1)

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maxima [A]  time = 0.60, size = 46, normalized size = 0.85 \begin {gather*} \frac {187326 \, x^{2} + 243279 \, x + 79028}{55566 \, {\left (27 \, x^{3} + 54 \, x^{2} + 36 \, x + 8\right )}} + \frac {1331}{2401} \, \log \left (3 \, x + 2\right ) - \frac {1331}{2401} \, \log \left (2 \, x - 1\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((3+5*x)^3/(1-2*x)/(2+3*x)^4,x, algorithm="maxima")

[Out]

1/55566*(187326*x^2 + 243279*x + 79028)/(27*x^3 + 54*x^2 + 36*x + 8) + 1331/2401*log(3*x + 2) - 1331/2401*log(
2*x - 1)

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mupad [B]  time = 1.12, size = 35, normalized size = 0.65 \begin {gather*} \frac {2662\,\mathrm {atanh}\left (\frac {12\,x}{7}+\frac {1}{7}\right )}{2401}+\frac {\frac {3469\,x^2}{27783}+\frac {27031\,x}{166698}+\frac {39514}{750141}}{x^3+2\,x^2+\frac {4\,x}{3}+\frac {8}{27}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(-(5*x + 3)^3/((2*x - 1)*(3*x + 2)^4),x)

[Out]

(2662*atanh((12*x)/7 + 1/7))/2401 + ((27031*x)/166698 + (3469*x^2)/27783 + 39514/750141)/((4*x)/3 + 2*x^2 + x^
3 + 8/27)

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sympy [A]  time = 0.17, size = 46, normalized size = 0.85 \begin {gather*} - \frac {- 187326 x^{2} - 243279 x - 79028}{1500282 x^{3} + 3000564 x^{2} + 2000376 x + 444528} - \frac {1331 \log {\left (x - \frac {1}{2} \right )}}{2401} + \frac {1331 \log {\left (x + \frac {2}{3} \right )}}{2401} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((3+5*x)**3/(1-2*x)/(2+3*x)**4,x)

[Out]

-(-187326*x**2 - 243279*x - 79028)/(1500282*x**3 + 3000564*x**2 + 2000376*x + 444528) - 1331*log(x - 1/2)/2401
+ 1331*log(x + 2/3)/2401

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