3.1695 \(\int \frac{1}{(1-2 x)^3 (2+3 x)^3 (3+5 x)^2} \, dx\)

Optimal. Leaf size=86 \[ \frac{2704}{3195731 (1-2 x)}-\frac{6156}{2401 (3 x+2)}-\frac{3125}{1331 (5 x+3)}+\frac{8}{41503 (1-2 x)^2}-\frac{81}{686 (3 x+2)^2}-\frac{274224 \log (1-2 x)}{246071287}+\frac{333639 \log (3 x+2)}{16807}-\frac{290625 \log (5 x+3)}{14641} \]

[Out]

8/(41503*(1 - 2*x)^2) + 2704/(3195731*(1 - 2*x)) - 81/(686*(2 + 3*x)^2) - 6156/(2401*(2 + 3*x)) - 3125/(1331*(
3 + 5*x)) - (274224*Log[1 - 2*x])/246071287 + (333639*Log[2 + 3*x])/16807 - (290625*Log[3 + 5*x])/14641

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Rubi [A]  time = 0.0459781, antiderivative size = 86, 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} \[ \frac{2704}{3195731 (1-2 x)}-\frac{6156}{2401 (3 x+2)}-\frac{3125}{1331 (5 x+3)}+\frac{8}{41503 (1-2 x)^2}-\frac{81}{686 (3 x+2)^2}-\frac{274224 \log (1-2 x)}{246071287}+\frac{333639 \log (3 x+2)}{16807}-\frac{290625 \log (5 x+3)}{14641} \]

Antiderivative was successfully verified.

[In]

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

[Out]

8/(41503*(1 - 2*x)^2) + 2704/(3195731*(1 - 2*x)) - 81/(686*(2 + 3*x)^2) - 6156/(2401*(2 + 3*x)) - 3125/(1331*(
3 + 5*x)) - (274224*Log[1 - 2*x])/246071287 + (333639*Log[2 + 3*x])/16807 - (290625*Log[3 + 5*x])/14641

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{1}{(1-2 x)^3 (2+3 x)^3 (3+5 x)^2} \, dx &=\int \left (-\frac{32}{41503 (-1+2 x)^3}+\frac{5408}{3195731 (-1+2 x)^2}-\frac{548448}{246071287 (-1+2 x)}+\frac{243}{343 (2+3 x)^3}+\frac{18468}{2401 (2+3 x)^2}+\frac{1000917}{16807 (2+3 x)}+\frac{15625}{1331 (3+5 x)^2}-\frac{1453125}{14641 (3+5 x)}\right ) \, dx\\ &=\frac{8}{41503 (1-2 x)^2}+\frac{2704}{3195731 (1-2 x)}-\frac{81}{686 (2+3 x)^2}-\frac{6156}{2401 (2+3 x)}-\frac{3125}{1331 (3+5 x)}-\frac{274224 \log (1-2 x)}{246071287}+\frac{333639 \log (2+3 x)}{16807}-\frac{290625 \log (3+5 x)}{14641}\\ \end{align*}

Mathematica [A]  time = 0.0611108, size = 74, normalized size = 0.86 \[ \frac{\frac{41503 (6558 x-3251)}{\left (6 x^2+x-2\right )^2}-\frac{154 (16395384 x-7937593)}{6 x^2+x-2}-\frac{1155481250}{5 x+3}-548448 \log (5-10 x)[phone] \log (5 (3 x+2))-9769068750 \log (5 x+3)}{492142574} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(-1155481250/(3 + 5*x) + (41503*(-3251 + 6558*x))/(-2 + x + 6*x^2)^2 - (154*(-7937593 + 16395384*x))/(-2 + x +
6*x^2) - 548448*Log[5 - 10*x] + 9769617198*Log[5*(2 + 3*x)] - 9769068750*Log[3 + 5*x])/492142574

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Maple [A]  time = 0.011, size = 71, normalized size = 0.8 \begin{align*}{\frac{8}{41503\, \left ( 2\,x-1 \right ) ^{2}}}-{\frac{2704}{6391462\,x-3195731}}-{\frac{274224\,\ln \left ( 2\,x-1 \right ) }{246071287}}-{\frac{81}{686\, \left ( 2+3\,x \right ) ^{2}}}-{\frac{6156}{4802+7203\,x}}+{\frac{333639\,\ln \left ( 2+3\,x \right ) }{16807}}-{\frac{3125}{3993+6655\,x}}-{\frac{290625\,\ln \left ( 3+5\,x \right ) }{14641}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

8/41503/(2*x-1)^2-2704/3195731/(2*x-1)-274224/246071287*ln(2*x-1)-81/686/(2+3*x)^2-6156/2401/(2+3*x)+333639/16
807*ln(2+3*x)-3125/1331/(3+5*x)-290625/14641*ln(3+5*x)

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Maxima [A]  time = 1.82156, size = 100, normalized size = 1.16 \begin{align*} -\frac{1523948040 \, x^{4} + 458007084 \, x^{3} - 957482214 \, x^{2} - 147486147 \, x + 160532983}{6391462 \,{\left (180 \, x^{5} + 168 \, x^{4} - 79 \, x^{3} - 89 \, x^{2} + 8 \, x + 12\right )}} - \frac{290625}{14641} \, \log \left (5 \, x + 3\right ) + \frac{333639}{16807} \, \log \left (3 \, x + 2\right ) - \frac{274224}{246071287} \, \log \left (2 \, x - 1\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/6391462*(1523948040*x^4 + 458007084*x^3 - 957482214*x^2 - 147486147*x + 160532983)/(180*x^5 + 168*x^4 - 79*
x^3 - 89*x^2 + 8*x + 12) - 290625/14641*log(5*x + 3) + 333639/16807*log(3*x + 2) - 274224/246071287*log(2*x -
1)

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Fricas [B]  time = 1.57332, size = 498, normalized size = 5.79 \begin{align*} -\frac{117343999080 \, x^{4} + 35266545468 \, x^{3} - 73726130478 \, x^{2} + 9769068750 \,{\left (180 \, x^{5} + 168 \, x^{4} - 79 \, x^{3} - 89 \, x^{2} + 8 \, x + 12\right )} \log \left (5 \, x + 3\right ) - 9769617198 \,{\left (180 \, x^{5} + 168 \, x^{4} - 79 \, x^{3} - 89 \, x^{2} + 8 \, x + 12\right )} \log \left (3 \, x + 2\right ) + 548448 \,{\left (180 \, x^{5} + 168 \, x^{4} - 79 \, x^{3} - 89 \, x^{2} + 8 \, x + 12\right )} \log \left (2 \, x - 1\right ) - 11356433319 \, x + 12361039691}{492142574 \,{\left (180 \, x^{5} + 168 \, x^{4} - 79 \, x^{3} - 89 \, x^{2} + 8 \, x + 12\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/492142574*(117343999080*x^4 + 35266545468*x^3 - 73726130478*x^2 + 9769068750*(180*x^5 + 168*x^4 - 79*x^3 -
89*x^2 + 8*x + 12)*log(5*x + 3) - 9769617198*(180*x^5 + 168*x^4 - 79*x^3 - 89*x^2 + 8*x + 12)*log(3*x + 2) + 5
48448*(180*x^5 + 168*x^4 - 79*x^3 - 89*x^2 + 8*x + 12)*log(2*x - 1) - 11356433319*x + 12361039691)/(180*x^5 +
168*x^4 - 79*x^3 - 89*x^2 + 8*x + 12)

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Sympy [A]  time = 0.2368, size = 75, normalized size = 0.87 \begin{align*} - \frac{1523948040 x^{4} + 458007084 x^{3} - 957482214 x^{2} - 147486147 x + 160532983}{1150463160 x^{5} + 1073765616 x^{4} - 504925498 x^{3} - 568840118 x^{2} + 51131696 x + 76697544} - \frac{274224 \log{\left (x - \frac{1}{2} \right )}}{246071287} - \frac{290625 \log{\left (x + \frac{3}{5} \right )}}{14641} + \frac{333639 \log{\left (x + \frac{2}{3} \right )}}{16807} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-(1523948040*x**4 + 458007084*x**3 - 957482214*x**2 - 147486147*x + 160532983)/(1150463160*x**5 + 1073765616*x
**4 - 504925498*x**3 - 568840118*x**2 + 51131696*x + 76697544) - 274224*log(x - 1/2)/246071287 - 290625*log(x
+ 3/5)/14641 + 333639*log(x + 2/3)/16807

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Giac [A]  time = 2.08651, size = 128, normalized size = 1.49 \begin{align*} -\frac{3125}{1331 \,{\left (5 \, x + 3\right )}} - \frac{5 \,{\left (\frac{84659379036}{5 \, x + 3} - \frac{206753119043}{{\left (5 \, x + 3\right )}^{2}} - \frac{95568773322}{{\left (5 \, x + 3\right )}^{3}} - 7983405324\right )}}{70306082 \,{\left (\frac{11}{5 \, x + 3} - 2\right )}^{2}{\left (\frac{1}{5 \, x + 3} + 3\right )}^{2}} + \frac{333639}{16807} \, \log \left ({\left | -\frac{1}{5 \, x + 3} - 3 \right |}\right ) - \frac{274224}{246071287} \, \log \left ({\left | -\frac{11}{5 \, x + 3} + 2 \right |}\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-3125/1331/(5*x + 3) - 5/70306082*(84659379036/(5*x + 3) - 206753119043/(5*x + 3)^2 - 95568773322/(5*x + 3)^3
- 7983405324)/((11/(5*x + 3) - 2)^2*(1/(5*x + 3) + 3)^2) + 333639/16807*log(abs(-1/(5*x + 3) - 3)) - 274224/24
6071287*log(abs(-11/(5*x + 3) + 2))