3.1479 \(\int \frac{(3+5 x)^3}{(1-2 x) (2+3 x)^4} \, dx\)

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

[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

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Rubi [A]  time = 0.0227337, antiderivative size = 54, normalized size of antiderivative = 1., 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} \[ \frac{3469}{9261 (3 x+2)}-\frac{103}{2646 (3 x+2)^2}+\frac{1}{567 (3 x+2)^3}-\frac{1331 \log (1-2 x)}{2401}+\frac{1331 \log (3 x+2)}{2401} \]

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*}

Mathematica [A]  time = 0.0247894, size = 40, normalized size = 0.74 \[ \frac{\frac{7 \left (187326 x^2+243279 x+79028\right )}{(3 x+2)^3}-215622 \log (1-2 x)+215622 \log (6 x+4)}{388962} \]

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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Maple [A]  time = 0.007, size = 45, normalized size = 0.8 \begin{align*} -{\frac{1331\,\ln \left ( 2\,x-1 \right ) }{2401}}+{\frac{1}{567\, \left ( 2+3\,x \right ) ^{3}}}-{\frac{103}{2646\, \left ( 2+3\,x \right ) ^{2}}}+{\frac{3469}{18522+27783\,x}}+{\frac{1331\,\ln \left ( 2+3\,x \right ) }{2401}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [A]  time = 1.17799, size = 62, normalized size = 1.15 \begin{align*} \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{align*}

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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Fricas [A]  time = 1.34804, size = 238, normalized size = 4.41 \begin{align*} \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{align*}

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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Sympy [A]  time = 0.162575, size = 44, normalized size = 0.81 \begin{align*} \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{align*}

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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Giac [A]  time = 2.64981, size = 51, normalized size = 0.94 \begin{align*} \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{align*}

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)
)