3.1667 \(\int \frac{(3+5 x)^3}{(1-2 x)^3 (2+3 x)^2} \, dx\)

Optimal. Leaf size=54 \[ -\frac{4719}{1372 (1-2 x)}+\frac{1}{1029 (3 x+2)}+\frac{1331}{392 (1-2 x)^2}-\frac{33 \log (1-2 x)}{2401}+\frac{33 \log (3 x+2)}{2401} \]

[Out]

1331/(392*(1 - 2*x)^2) - 4719/(1372*(1 - 2*x)) + 1/(1029*(2 + 3*x)) - (33*Log[1 - 2*x])/2401 + (33*Log[2 + 3*x
])/2401

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Rubi [A]  time = 0.0239914, 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} \[ -\frac{4719}{1372 (1-2 x)}+\frac{1}{1029 (3 x+2)}+\frac{1331}{392 (1-2 x)^2}-\frac{33 \log (1-2 x)}{2401}+\frac{33 \log (3 x+2)}{2401} \]

Antiderivative was successfully verified.

[In]

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

[Out]

1331/(392*(1 - 2*x)^2) - 4719/(1372*(1 - 2*x)) + 1/(1029*(2 + 3*x)) - (33*Log[1 - 2*x])/2401 + (33*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)^3 (2+3 x)^2} \, dx &=\int \left (-\frac{1331}{98 (-1+2 x)^3}-\frac{4719}{686 (-1+2 x)^2}-\frac{66}{2401 (-1+2 x)}-\frac{1}{343 (2+3 x)^2}+\frac{99}{2401 (2+3 x)}\right ) \, dx\\ &=\frac{1331}{392 (1-2 x)^2}-\frac{4719}{1372 (1-2 x)}+\frac{1}{1029 (2+3 x)}-\frac{33 \log (1-2 x)}{2401}+\frac{33 \log (2+3 x)}{2401}\\ \end{align*}

Mathematica [A]  time = 0.0334147, size = 47, normalized size = 0.87 \[ \frac{\frac{7 \left (169916 x^2+112135 x-718\right )}{(1-2 x)^2 (3 x+2)}-792 \log (1-2 x)+792 \log (6 x+4)}{57624} \]

Antiderivative was successfully verified.

[In]

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

[Out]

((7*(-718 + 112135*x + 169916*x^2))/((1 - 2*x)^2*(2 + 3*x)) - 792*Log[1 - 2*x] + 792*Log[4 + 6*x])/57624

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Maple [A]  time = 0.008, size = 45, normalized size = 0.8 \begin{align*}{\frac{1331}{392\, \left ( 2\,x-1 \right ) ^{2}}}+{\frac{4719}{2744\,x-1372}}-{\frac{33\,\ln \left ( 2\,x-1 \right ) }{2401}}+{\frac{1}{2058+3087\,x}}+{\frac{33\,\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)^3/(2+3*x)^2,x)

[Out]

1331/392/(2*x-1)^2+4719/1372/(2*x-1)-33/2401*ln(2*x-1)+1/1029/(2+3*x)+33/2401*ln(2+3*x)

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Maxima [A]  time = 2.47075, size = 62, normalized size = 1.15 \begin{align*} \frac{169916 \, x^{2} + 112135 \, x - 718}{8232 \,{\left (12 \, x^{3} - 4 \, x^{2} - 5 \, x + 2\right )}} + \frac{33}{2401} \, \log \left (3 \, x + 2\right ) - \frac{33}{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)^3/(2+3*x)^2,x, algorithm="maxima")

[Out]

1/8232*(169916*x^2 + 112135*x - 718)/(12*x^3 - 4*x^2 - 5*x + 2) + 33/2401*log(3*x + 2) - 33/2401*log(2*x - 1)

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Fricas [A]  time = 1.49134, size = 216, normalized size = 4. \begin{align*} \frac{1189412 \, x^{2} + 792 \,{\left (12 \, x^{3} - 4 \, x^{2} - 5 \, x + 2\right )} \log \left (3 \, x + 2\right ) - 792 \,{\left (12 \, x^{3} - 4 \, x^{2} - 5 \, x + 2\right )} \log \left (2 \, x - 1\right ) + 784945 \, x - 5026}{57624 \,{\left (12 \, x^{3} - 4 \, x^{2} - 5 \, x + 2\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/57624*(1189412*x^2 + 792*(12*x^3 - 4*x^2 - 5*x + 2)*log(3*x + 2) - 792*(12*x^3 - 4*x^2 - 5*x + 2)*log(2*x -
1) + 784945*x - 5026)/(12*x^3 - 4*x^2 - 5*x + 2)

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Sympy [A]  time = 0.162877, size = 44, normalized size = 0.81 \begin{align*} \frac{169916 x^{2} + 112135 x - 718}{98784 x^{3} - 32928 x^{2} - 41160 x + 16464} - \frac{33 \log{\left (x - \frac{1}{2} \right )}}{2401} + \frac{33 \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)**3/(2+3*x)**2,x)

[Out]

(169916*x**2 + 112135*x - 718)/(98784*x**3 - 32928*x**2 - 41160*x + 16464) - 33*log(x - 1/2)/2401 + 33*log(x +
2/3)/2401

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Giac [A]  time = 2.81683, size = 69, normalized size = 1.28 \begin{align*} \frac{1}{1029 \,{\left (3 \, x + 2\right )}} - \frac{1089 \,{\left (\frac{14}{3 \, x + 2} - 15\right )}}{4802 \,{\left (\frac{7}{3 \, x + 2} - 2\right )}^{2}} - \frac{33}{2401} \, \log \left ({\left | -\frac{7}{3 \, x + 2} + 2 \right |}\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/1029/(3*x + 2) - 1089/4802*(14/(3*x + 2) - 15)/(7/(3*x + 2) - 2)^2 - 33/2401*log(abs(-7/(3*x + 2) + 2))