3.1694 \(\int \frac{1}{(1-2 x)^3 (2+3 x)^2 (3+5 x)^2} \, dx\)

Optimal. Leaf size=75 \[ \frac{1088}{456533 (1-2 x)}-\frac{81}{343 (3 x+2)}-\frac{625}{1331 (5 x+3)}+\frac{4}{5929 (1-2 x)^2}-\frac{92496 \log (1-2 x)}{35153041}+\frac{6156 \log (3 x+2)}{2401}-\frac{37500 \log (5 x+3)}{14641} \]

[Out]

4/(5929*(1 - 2*x)^2) + 1088/(456533*(1 - 2*x)) - 81/(343*(2 + 3*x)) - 625/(1331*(3 + 5*x)) - (92496*Log[1 - 2*
x])/35153041 + (6156*Log[2 + 3*x])/2401 - (37500*Log[3 + 5*x])/14641

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Rubi [A]  time = 0.0382461, antiderivative size = 75, 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} \[ \frac{1088}{456533 (1-2 x)}-\frac{81}{343 (3 x+2)}-\frac{625}{1331 (5 x+3)}+\frac{4}{5929 (1-2 x)^2}-\frac{92496 \log (1-2 x)}{35153041}+\frac{6156 \log (3 x+2)}{2401}-\frac{37500 \log (5 x+3)}{14641} \]

Antiderivative was successfully verified.

[In]

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

[Out]

4/(5929*(1 - 2*x)^2) + 1088/(456533*(1 - 2*x)) - 81/(343*(2 + 3*x)) - 625/(1331*(3 + 5*x)) - (92496*Log[1 - 2*
x])/35153041 + (6156*Log[2 + 3*x])/2401 - (37500*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)^2 (3+5 x)^2} \, dx &=\int \left (-\frac{16}{5929 (-1+2 x)^3}+\frac{2176}{456533 (-1+2 x)^2}-\frac{184992}{35153041 (-1+2 x)}+\frac{243}{343 (2+3 x)^2}+\frac{18468}{2401 (2+3 x)}+\frac{3125}{1331 (3+5 x)^2}-\frac{187500}{14641 (3+5 x)}\right ) \, dx\\ &=\frac{4}{5929 (1-2 x)^2}+\frac{1088}{456533 (1-2 x)}-\frac{81}{343 (2+3 x)}-\frac{625}{1331 (3+5 x)}-\frac{92496 \log (1-2 x)}{35153041}+\frac{6156 \log (2+3 x)}{2401}-\frac{37500 \log (3+5 x)}{14641}\\ \end{align*}

Mathematica [A]  time = 0.0663568, size = 68, normalized size = 0.91 \[ \frac{2 \left (77 \left (-\frac{107811}{6 x+4}-\frac{214375}{10 x+6}+\frac{544}{1-2 x}+\frac{154}{(1-2 x)^2}\right )-46248 \log (1-2 x)[phone] \log (6 x+4)-45018750 \log (10 x+6)\right )}{35153041} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(2*(77*(154/(1 - 2*x)^2 + 544/(1 - 2*x) - 107811/(4 + 6*x) - 214375/(6 + 10*x)) - 46248*Log[1 - 2*x] + 4506499
8*Log[4 + 6*x] - 45018750*Log[6 + 10*x]))/35153041

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Maple [A]  time = 0.01, size = 62, normalized size = 0.8 \begin{align*}{\frac{4}{5929\, \left ( 2\,x-1 \right ) ^{2}}}-{\frac{1088}{913066\,x-456533}}-{\frac{92496\,\ln \left ( 2\,x-1 \right ) }{35153041}}-{\frac{81}{686+1029\,x}}+{\frac{6156\,\ln \left ( 2+3\,x \right ) }{2401}}-{\frac{625}{3993+6655\,x}}-{\frac{37500\,\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)^2/(3+5*x)^2,x)

[Out]

4/5929/(2*x-1)^2-1088/456533/(2*x-1)-92496/35153041*ln(2*x-1)-81/343/(2+3*x)+6156/2401*ln(2+3*x)-625/1331/(3+5
*x)-37500/14641*ln(3+5*x)

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Maxima [A]  time = 1.08033, size = 86, normalized size = 1.15 \begin{align*} -\frac{4761360 \, x^{3} - 1699584 \, x^{2} - 1840020 \, x + 743807}{456533 \,{\left (60 \, x^{4} + 16 \, x^{3} - 37 \, x^{2} - 5 \, x + 6\right )}} - \frac{37500}{14641} \, \log \left (5 \, x + 3\right ) + \frac{6156}{2401} \, \log \left (3 \, x + 2\right ) - \frac{92496}{35153041} \, \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)^2/(3+5*x)^2,x, algorithm="maxima")

[Out]

-1/456533*(4761360*x^3 - 1699584*x^2 - 1840020*x + 743807)/(60*x^4 + 16*x^3 - 37*x^2 - 5*x + 6) - 37500/14641*
log(5*x + 3) + 6156/2401*log(3*x + 2) - 92496/35153041*log(2*x - 1)

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Fricas [B]  time = 1.49716, size = 387, normalized size = 5.16 \begin{align*} -\frac{366624720 \, x^{3} - 130867968 \, x^{2} + 90037500 \,{\left (60 \, x^{4} + 16 \, x^{3} - 37 \, x^{2} - 5 \, x + 6\right )} \log \left (5 \, x + 3\right ) - 90129996 \,{\left (60 \, x^{4} + 16 \, x^{3} - 37 \, x^{2} - 5 \, x + 6\right )} \log \left (3 \, x + 2\right ) + 92496 \,{\left (60 \, x^{4} + 16 \, x^{3} - 37 \, x^{2} - 5 \, x + 6\right )} \log \left (2 \, x - 1\right ) - 141681540 \, x + 57273139}{35153041 \,{\left (60 \, x^{4} + 16 \, x^{3} - 37 \, x^{2} - 5 \, x + 6\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/35153041*(366624720*x^3 - 130867968*x^2 + 90037500*(60*x^4 + 16*x^3 - 37*x^2 - 5*x + 6)*log(5*x + 3) - 9012
9996*(60*x^4 + 16*x^3 - 37*x^2 - 5*x + 6)*log(3*x + 2) + 92496*(60*x^4 + 16*x^3 - 37*x^2 - 5*x + 6)*log(2*x -
1) - 141681540*x + 57273139)/(60*x^4 + 16*x^3 - 37*x^2 - 5*x + 6)

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Sympy [A]  time = 0.215032, size = 65, normalized size = 0.87 \begin{align*} - \frac{4761360 x^{3} - 1699584 x^{2} - 1840020 x + 743807}{27391980 x^{4} + 7304528 x^{3} - 16891721 x^{2} - 2282665 x + 2739198} - \frac{92496 \log{\left (x - \frac{1}{2} \right )}}{35153041} - \frac{37500 \log{\left (x + \frac{3}{5} \right )}}{14641} + \frac{6156 \log{\left (x + \frac{2}{3} \right )}}{2401} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-(4761360*x**3 - 1699584*x**2 - 1840020*x + 743807)/(27391980*x**4 + 7304528*x**3 - 16891721*x**2 - 2282665*x
+ 2739198) - 92496*log(x - 1/2)/35153041 - 37500*log(x + 3/5)/14641 + 6156*log(x + 2/3)/2401

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Giac [A]  time = 3.01521, size = 116, normalized size = 1.55 \begin{align*} -\frac{625}{1331 \,{\left (5 \, x + 3\right )}} - \frac{5 \,{\left (\frac{156456196}{5 \, x + 3} - \frac{430519419}{{\left (5 \, x + 3\right )}^{2}} - 14216316\right )}}{5021863 \,{\left (\frac{11}{5 \, x + 3} - 2\right )}^{2}{\left (\frac{1}{5 \, x + 3} + 3\right )}} + \frac{6156}{2401} \, \log \left ({\left | -\frac{1}{5 \, x + 3} - 3 \right |}\right ) - \frac{92496}{35153041} \, \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)^2/(3+5*x)^2,x, algorithm="giac")

[Out]

-625/1331/(5*x + 3) - 5/5021863*(156456196/(5*x + 3) - 430519419/(5*x + 3)^2 - 14216316)/((11/(5*x + 3) - 2)^2
*(1/(5*x + 3) + 3)) + 6156/2401*log(abs(-1/(5*x + 3) - 3)) - 92496/35153041*log(abs(-11/(5*x + 3) + 2))