3.1584 \(\int \frac{(3+5 x)^3}{(1-2 x)^2 (2+3 x)^5} \, dx\)

Optimal. Leaf size=76 \[ \frac{2662}{16807 (1-2 x)}-\frac{3267}{16807 (3 x+2)}+\frac{363}{4802 (3 x+2)^2}-\frac{101}{9261 (3 x+2)^3}+\frac{1}{1764 (3 x+2)^4}-\frac{14520 \log (1-2 x)}{117649}+\frac{14520 \log (3 x+2)}{117649} \]

[Out]

2662/(16807*(1 - 2*x)) + 1/(1764*(2 + 3*x)^4) - 101/(9261*(2 + 3*x)^3) + 363/(4802*(2 + 3*x)^2) - 3267/(16807*
(2 + 3*x)) - (14520*Log[1 - 2*x])/117649 + (14520*Log[2 + 3*x])/117649

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Rubi [A]  time = 0.034965, 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} \[ \frac{2662}{16807 (1-2 x)}-\frac{3267}{16807 (3 x+2)}+\frac{363}{4802 (3 x+2)^2}-\frac{101}{9261 (3 x+2)^3}+\frac{1}{1764 (3 x+2)^4}-\frac{14520 \log (1-2 x)}{117649}+\frac{14520 \log (3 x+2)}{117649} \]

Antiderivative was successfully verified.

[In]

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

[Out]

2662/(16807*(1 - 2*x)) + 1/(1764*(2 + 3*x)^4) - 101/(9261*(2 + 3*x)^3) + 363/(4802*(2 + 3*x)^2) - 3267/(16807*
(2 + 3*x)) - (14520*Log[1 - 2*x])/117649 + (14520*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)^2 (2+3 x)^5} \, dx &=\int \left (\frac{5324}{16807 (-1+2 x)^2}-\frac{29040}{117649 (-1+2 x)}-\frac{1}{147 (2+3 x)^5}+\frac{101}{1029 (2+3 x)^4}-\frac{1089}{2401 (2+3 x)^3}+\frac{9801}{16807 (2+3 x)^2}+\frac{43560}{117649 (2+3 x)}\right ) \, dx\\ &=\frac{2662}{16807 (1-2 x)}+\frac{1}{1764 (2+3 x)^4}-\frac{101}{9261 (2+3 x)^3}+\frac{363}{4802 (2+3 x)^2}-\frac{3267}{16807 (2+3 x)}-\frac{14520 \log (1-2 x)}{117649}+\frac{14520 \log (2+3 x)}{117649}\\ \end{align*}

Mathematica [A]  time = 0.044298, size = 59, normalized size = 0.78 \[ \frac{2 \left (-\frac{7 \left (42340320 x^4+88209000 x^3+66510750 x^2+21109490 x+2287541\right )}{8 (2 x-1) (3 x+2)^4}-196020 \log (1-2 x)+196020 \log (6 x+4)\right )}{3176523} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(2*((-7*(2287541 + 21109490*x + 66510750*x^2 + 88209000*x^3 + 42340320*x^4))/(8*(-1 + 2*x)*(2 + 3*x)^4) - 1960
20*Log[1 - 2*x] + 196020*Log[4 + 6*x]))/3176523

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Maple [A]  time = 0.01, size = 63, normalized size = 0.8 \begin{align*} -{\frac{2662}{33614\,x-16807}}-{\frac{14520\,\ln \left ( 2\,x-1 \right ) }{117649}}+{\frac{1}{1764\, \left ( 2+3\,x \right ) ^{4}}}-{\frac{101}{9261\, \left ( 2+3\,x \right ) ^{3}}}+{\frac{363}{4802\, \left ( 2+3\,x \right ) ^{2}}}-{\frac{3267}{33614+50421\,x}}+{\frac{14520\,\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)^2/(2+3*x)^5,x)

[Out]

-2662/16807/(2*x-1)-14520/117649*ln(2*x-1)+1/1764/(2+3*x)^4-101/9261/(2+3*x)^3+363/4802/(2+3*x)^2-3267/16807/(
2+3*x)+14520/117649*ln(2+3*x)

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Maxima [A]  time = 1.72267, size = 89, normalized size = 1.17 \begin{align*} -\frac{42340320 \, x^{4} + 88209000 \, x^{3} + 66510750 \, x^{2} + 21109490 \, x + 2287541}{1815156 \,{\left (162 \, x^{5} + 351 \, x^{4} + 216 \, x^{3} - 24 \, x^{2} - 64 \, x - 16\right )}} + \frac{14520}{117649} \, \log \left (3 \, x + 2\right ) - \frac{14520}{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)^2/(2+3*x)^5,x, algorithm="maxima")

[Out]

-1/1815156*(42340320*x^4 + 88209000*x^3 + 66510750*x^2 + 21109490*x + 2287541)/(162*x^5 + 351*x^4 + 216*x^3 -
24*x^2 - 64*x - 16) + 14520/117649*log(3*x + 2) - 14520/117649*log(2*x - 1)

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Fricas [A]  time = 1.30504, size = 385, normalized size = 5.07 \begin{align*} -\frac{296382240 \, x^{4} + 617463000 \, x^{3} + 465575250 \, x^{2} - 1568160 \,{\left (162 \, x^{5} + 351 \, x^{4} + 216 \, x^{3} - 24 \, x^{2} - 64 \, x - 16\right )} \log \left (3 \, x + 2\right ) + 1568160 \,{\left (162 \, x^{5} + 351 \, x^{4} + 216 \, x^{3} - 24 \, x^{2} - 64 \, x - 16\right )} \log \left (2 \, x - 1\right ) + 147766430 \, x + 16012787}{12706092 \,{\left (162 \, x^{5} + 351 \, x^{4} + 216 \, x^{3} - 24 \, x^{2} - 64 \, x - 16\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/12706092*(296382240*x^4 + 617463000*x^3 + 465575250*x^2 - 1568160*(162*x^5 + 351*x^4 + 216*x^3 - 24*x^2 - 6
4*x - 16)*log(3*x + 2) + 1568160*(162*x^5 + 351*x^4 + 216*x^3 - 24*x^2 - 64*x - 16)*log(2*x - 1) + 147766430*x
+ 16012787)/(162*x^5 + 351*x^4 + 216*x^3 - 24*x^2 - 64*x - 16)

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Sympy [A]  time = 0.18156, size = 65, normalized size = 0.86 \begin{align*} - \frac{42340320 x^{4} + 88209000 x^{3} + 66510750 x^{2} + 21109490 x + 2287541}{294055272 x^{5} + 637119756 x^{4} + 392073696 x^{3} - 43563744 x^{2} - 116169984 x - 29042496} - \frac{14520 \log{\left (x - \frac{1}{2} \right )}}{117649} + \frac{14520 \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)**2/(2+3*x)**5,x)

[Out]

-(42340320*x**4 + 88209000*x**3 + 66510750*x**2 + 21109490*x + 2287541)/(294055272*x**5 + 637119756*x**4 + 392
073696*x**3 - 43563744*x**2 - 116169984*x - 29042496) - 14520*log(x - 1/2)/117649 + 14520*log(x + 2/3)/117649

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Giac [A]  time = 2.72989, size = 90, normalized size = 1.18 \begin{align*} -\frac{3267}{16807 \,{\left (3 \, x + 2\right )}} + \frac{15972}{117649 \,{\left (\frac{7}{3 \, x + 2} - 2\right )}} + \frac{363}{4802 \,{\left (3 \, x + 2\right )}^{2}} - \frac{101}{9261 \,{\left (3 \, x + 2\right )}^{3}} + \frac{1}{1764 \,{\left (3 \, x + 2\right )}^{4}} - \frac{14520}{117649} \, \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)^2/(2+3*x)^5,x, algorithm="giac")

[Out]

-3267/16807/(3*x + 2) + 15972/117649/(7/(3*x + 2) - 2) + 363/4802/(3*x + 2)^2 - 101/9261/(3*x + 2)^3 + 1/1764/
(3*x + 2)^4 - 14520/117649*log(abs(-7/(3*x + 2) + 2))