3.1669 \(\int \frac{(3+5 x)^3}{(1-2 x)^3 (2+3 x)^4} \, dx\)

Optimal. Leaf size=76 \[ \frac{3267}{16807 (1-2 x)}+\frac{1023}{16807 (3 x+2)}+\frac{1331}{4802 (1-2 x)^2}-\frac{33}{4802 (3 x+2)^2}+\frac{1}{3087 (3 x+2)^3}-\frac{7755 \log (1-2 x)}{117649}+\frac{7755 \log (3 x+2)}{117649} \]

[Out]

1331/(4802*(1 - 2*x)^2) + 3267/(16807*(1 - 2*x)) + 1/(3087*(2 + 3*x)^3) - 33/(4802*(2 + 3*x)^2) + 1023/(16807*
(2 + 3*x)) - (7755*Log[1 - 2*x])/117649 + (7755*Log[2 + 3*x])/117649

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Rubi [A]  time = 0.0356651, antiderivative size = 76, 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} \[ \frac{3267}{16807 (1-2 x)}+\frac{1023}{16807 (3 x+2)}+\frac{1331}{4802 (1-2 x)^2}-\frac{33}{4802 (3 x+2)^2}+\frac{1}{3087 (3 x+2)^3}-\frac{7755 \log (1-2 x)}{117649}+\frac{7755 \log (3 x+2)}{117649} \]

Antiderivative was successfully verified.

[In]

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

[Out]

1331/(4802*(1 - 2*x)^2) + 3267/(16807*(1 - 2*x)) + 1/(3087*(2 + 3*x)^3) - 33/(4802*(2 + 3*x)^2) + 1023/(16807*
(2 + 3*x)) - (7755*Log[1 - 2*x])/117649 + (7755*Log[2 + 3*x])/117649

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)^4} \, dx &=\int \left (-\frac{2662}{2401 (-1+2 x)^3}+\frac{6534}{16807 (-1+2 x)^2}-\frac{15510}{117649 (-1+2 x)}-\frac{1}{343 (2+3 x)^4}+\frac{99}{2401 (2+3 x)^3}-\frac{3069}{16807 (2+3 x)^2}+\frac{23265}{117649 (2+3 x)}\right ) \, dx\\ &=\frac{1331}{4802 (1-2 x)^2}+\frac{3267}{16807 (1-2 x)}+\frac{1}{3087 (2+3 x)^3}-\frac{33}{4802 (2+3 x)^2}+\frac{1023}{16807 (2+3 x)}-\frac{7755 \log (1-2 x)}{117649}+\frac{7755 \log (2+3 x)}{117649}\\ \end{align*}

Mathematica [A]  time = 0.0515793, size = 57, normalized size = 0.75 \[ \frac{\frac{7 \left (-2512620 x^4-2303235 x^3+3054740 x^2+4131175 x+1210868\right )}{(1-2 x)^2 (3 x+2)^3}-139590 \log (1-2 x)+139590 \log (6 x+4)}{2117682} \]

Antiderivative was successfully verified.

[In]

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

[Out]

((7*(1210868 + 4131175*x + 3054740*x^2 - 2303235*x^3 - 2512620*x^4))/((1 - 2*x)^2*(2 + 3*x)^3) - 139590*Log[1
- 2*x] + 139590*Log[4 + 6*x])/2117682

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Maple [A]  time = 0.008, size = 63, normalized size = 0.8 \begin{align*}{\frac{1331}{4802\, \left ( 2\,x-1 \right ) ^{2}}}-{\frac{3267}{33614\,x-16807}}-{\frac{7755\,\ln \left ( 2\,x-1 \right ) }{117649}}+{\frac{1}{3087\, \left ( 2+3\,x \right ) ^{3}}}-{\frac{33}{4802\, \left ( 2+3\,x \right ) ^{2}}}+{\frac{1023}{33614+50421\,x}}+{\frac{7755\,\ln \left ( 2+3\,x \right ) }{117649}} \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)^4,x)

[Out]

1331/4802/(2*x-1)^2-3267/16807/(2*x-1)-7755/117649*ln(2*x-1)+1/3087/(2+3*x)^3-33/4802/(2+3*x)^2+1023/16807/(2+
3*x)+7755/117649*ln(2+3*x)

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Maxima [A]  time = 1.04693, size = 89, normalized size = 1.17 \begin{align*} -\frac{2512620 \, x^{4} + 2303235 \, x^{3} - 3054740 \, x^{2} - 4131175 \, x - 1210868}{302526 \,{\left (108 \, x^{5} + 108 \, x^{4} - 45 \, x^{3} - 58 \, x^{2} + 4 \, x + 8\right )}} + \frac{7755}{117649} \, \log \left (3 \, x + 2\right ) - \frac{7755}{117649} \, \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)^4,x, algorithm="maxima")

[Out]

-1/302526*(2512620*x^4 + 2303235*x^3 - 3054740*x^2 - 4131175*x - 1210868)/(108*x^5 + 108*x^4 - 45*x^3 - 58*x^2
+ 4*x + 8) + 7755/117649*log(3*x + 2) - 7755/117649*log(2*x - 1)

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Fricas [A]  time = 1.48928, size = 362, normalized size = 4.76 \begin{align*} -\frac{17588340 \, x^{4} + 16122645 \, x^{3} - 21383180 \, x^{2} - 139590 \,{\left (108 \, x^{5} + 108 \, x^{4} - 45 \, x^{3} - 58 \, x^{2} + 4 \, x + 8\right )} \log \left (3 \, x + 2\right ) + 139590 \,{\left (108 \, x^{5} + 108 \, x^{4} - 45 \, x^{3} - 58 \, x^{2} + 4 \, x + 8\right )} \log \left (2 \, x - 1\right ) - 28918225 \, x - 8476076}{2117682 \,{\left (108 \, x^{5} + 108 \, x^{4} - 45 \, x^{3} - 58 \, x^{2} + 4 \, x + 8\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)^4,x, algorithm="fricas")

[Out]

-1/2117682*(17588340*x^4 + 16122645*x^3 - 21383180*x^2 - 139590*(108*x^5 + 108*x^4 - 45*x^3 - 58*x^2 + 4*x + 8
)*log(3*x + 2) + 139590*(108*x^5 + 108*x^4 - 45*x^3 - 58*x^2 + 4*x + 8)*log(2*x - 1) - 28918225*x - 8476076)/(
108*x^5 + 108*x^4 - 45*x^3 - 58*x^2 + 4*x + 8)

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Sympy [A]  time = 0.19386, size = 65, normalized size = 0.86 \begin{align*} - \frac{2512620 x^{4} + 2303235 x^{3} - 3054740 x^{2} - 4131175 x - 1210868}{32672808 x^{5} + 32672808 x^{4} - 13613670 x^{3} - 17546508 x^{2} + 1210104 x + 2420208} - \frac{7755 \log{\left (x - \frac{1}{2} \right )}}{117649} + \frac{7755 \log{\left (x + \frac{2}{3} \right )}}{117649} \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)**4,x)

[Out]

-(2512620*x**4 + 2303235*x**3 - 3054740*x**2 - 4131175*x - 1210868)/(32672808*x**5 + 32672808*x**4 - 13613670*
x**3 - 17546508*x**2 + 1210104*x + 2420208) - 7755*log(x - 1/2)/117649 + 7755*log(x + 2/3)/117649

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Giac [A]  time = 2.96708, size = 74, normalized size = 0.97 \begin{align*} -\frac{2512620 \, x^{4} + 2303235 \, x^{3} - 3054740 \, x^{2} - 4131175 \, x - 1210868}{302526 \,{\left (3 \, x + 2\right )}^{3}{\left (2 \, x - 1\right )}^{2}} + \frac{7755}{117649} \, \log \left ({\left | 3 \, x + 2 \right |}\right ) - \frac{7755}{117649} \, \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)^3/(2+3*x)^4,x, algorithm="giac")

[Out]

-1/302526*(2512620*x^4 + 2303235*x^3 - 3054740*x^2 - 4131175*x - 1210868)/((3*x + 2)^3*(2*x - 1)^2) + 7755/117
649*log(abs(3*x + 2)) - 7755/117649*log(abs(2*x - 1))