3.1693 \(\int \frac{1}{(1-2 x)^3 (2+3 x) (3+5 x)^2} \, dx\)

Optimal. Leaf size=64 \[ \frac{412}{65219 (1-2 x)}-\frac{125}{1331 (5 x+3)}+\frac{2}{847 (1-2 x)^2}-\frac{28296 \log (1-2 x)}{5021863}+\frac{81}{343} \log (3 x+2)-\frac{3375 \log (5 x+3)}{14641} \]

[Out]

2/(847*(1 - 2*x)^2) + 412/(65219*(1 - 2*x)) - 125/(1331*(3 + 5*x)) - (28296*Log[1 - 2*x])/5021863 + (81*Log[2
+ 3*x])/343 - (3375*Log[3 + 5*x])/14641

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Rubi [A]  time = 0.0305904, antiderivative size = 64, 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} \[ \frac{412}{65219 (1-2 x)}-\frac{125}{1331 (5 x+3)}+\frac{2}{847 (1-2 x)^2}-\frac{28296 \log (1-2 x)}{5021863}+\frac{81}{343} \log (3 x+2)-\frac{3375 \log (5 x+3)}{14641} \]

Antiderivative was successfully verified.

[In]

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

[Out]

2/(847*(1 - 2*x)^2) + 412/(65219*(1 - 2*x)) - 125/(1331*(3 + 5*x)) - (28296*Log[1 - 2*x])/5021863 + (81*Log[2
+ 3*x])/343 - (3375*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+5 x)^2} \, dx &=\int \left (-\frac{8}{847 (-1+2 x)^3}+\frac{824}{65219 (-1+2 x)^2}-\frac{56592}{5021863 (-1+2 x)}+\frac{243}{343 (2+3 x)}+\frac{625}{1331 (3+5 x)^2}-\frac{16875}{14641 (3+5 x)}\right ) \, dx\\ &=\frac{2}{847 (1-2 x)^2}+\frac{412}{65219 (1-2 x)}-\frac{125}{1331 (3+5 x)}-\frac{28296 \log (1-2 x)}{5021863}+\frac{81}{343} \log (2+3 x)-\frac{3375 \log (3+5 x)}{14641}\\ \end{align*}

Mathematica [A]  time = 0.0288789, size = 60, normalized size = 0.94 \[ \frac{3 \left (\frac{31724}{3-6 x}-\frac{471625}{15 x+9}+\frac{11858}{3 (1-2 x)^2}-9432 \log (3-6 x)+395307 \log (3 x+2)-385875 \log (-3 (5 x+3))\right )}{5021863} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(3*(31724/(3 - 6*x) + 11858/(3*(1 - 2*x)^2) - 471625/(9 + 15*x) - 9432*Log[3 - 6*x] + 395307*Log[2 + 3*x] - 38
5875*Log[-3*(3 + 5*x)]))/5021863

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Maple [A]  time = 0.009, size = 53, normalized size = 0.8 \begin{align*}{\frac{2}{847\, \left ( 2\,x-1 \right ) ^{2}}}-{\frac{412}{130438\,x-65219}}-{\frac{28296\,\ln \left ( 2\,x-1 \right ) }{5021863}}+{\frac{81\,\ln \left ( 2+3\,x \right ) }{343}}-{\frac{125}{3993+6655\,x}}-{\frac{3375\,\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+5*x)^2,x)

[Out]

2/847/(2*x-1)^2-412/65219/(2*x-1)-28296/5021863*ln(2*x-1)+81/343*ln(2+3*x)-125/1331/(3+5*x)-3375/14641*ln(3+5*
x)

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Maxima [A]  time = 1.08233, size = 73, normalized size = 1.14 \begin{align*} -\frac{28620 \, x^{2} - 24858 \, x + 4427}{65219 \,{\left (20 \, x^{3} - 8 \, x^{2} - 7 \, x + 3\right )}} - \frac{3375}{14641} \, \log \left (5 \, x + 3\right ) + \frac{81}{343} \, \log \left (3 \, x + 2\right ) - \frac{28296}{5021863} \, \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+5*x)^2,x, algorithm="maxima")

[Out]

-1/65219*(28620*x^2 - 24858*x + 4427)/(20*x^3 - 8*x^2 - 7*x + 3) - 3375/14641*log(5*x + 3) + 81/343*log(3*x +
2) - 28296/5021863*log(2*x - 1)

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Fricas [A]  time = 1.58652, size = 300, normalized size = 4.69 \begin{align*} -\frac{2203740 \, x^{2} + 1157625 \,{\left (20 \, x^{3} - 8 \, x^{2} - 7 \, x + 3\right )} \log \left (5 \, x + 3\right ) - 1185921 \,{\left (20 \, x^{3} - 8 \, x^{2} - 7 \, x + 3\right )} \log \left (3 \, x + 2\right ) + 28296 \,{\left (20 \, x^{3} - 8 \, x^{2} - 7 \, x + 3\right )} \log \left (2 \, x - 1\right ) - 1914066 \, x + 340879}{5021863 \,{\left (20 \, x^{3} - 8 \, x^{2} - 7 \, x + 3\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/5021863*(2203740*x^2 + 1157625*(20*x^3 - 8*x^2 - 7*x + 3)*log(5*x + 3) - 1185921*(20*x^3 - 8*x^2 - 7*x + 3)
*log(3*x + 2) + 28296*(20*x^3 - 8*x^2 - 7*x + 3)*log(2*x - 1) - 1914066*x + 340879)/(20*x^3 - 8*x^2 - 7*x + 3)

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Sympy [A]  time = 0.199354, size = 54, normalized size = 0.84 \begin{align*} - \frac{28620 x^{2} - 24858 x + 4427}{1304380 x^{3} - 521752 x^{2} - 456533 x + 195657} - \frac{28296 \log{\left (x - \frac{1}{2} \right )}}{5021863} - \frac{3375 \log{\left (x + \frac{3}{5} \right )}}{14641} + \frac{81 \log{\left (x + \frac{2}{3} \right )}}{343} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-(28620*x**2 - 24858*x + 4427)/(1304380*x**3 - 521752*x**2 - 456533*x + 195657) - 28296*log(x - 1/2)/5021863 -
3375*log(x + 3/5)/14641 + 81*log(x + 2/3)/343

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Giac [A]  time = 2.96062, size = 89, normalized size = 1.39 \begin{align*} -\frac{125}{1331 \,{\left (5 \, x + 3\right )}} + \frac{40 \,{\left (\frac{1518}{5 \, x + 3} - 241\right )}}{717409 \,{\left (\frac{11}{5 \, x + 3} - 2\right )}^{2}} + \frac{81}{343} \, \log \left ({\left | -\frac{1}{5 \, x + 3} - 3 \right |}\right ) - \frac{28296}{5021863} \, \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+5*x)^2,x, algorithm="giac")

[Out]

-125/1331/(5*x + 3) + 40/717409*(1518/(5*x + 3) - 241)/(11/(5*x + 3) - 2)^2 + 81/343*log(abs(-1/(5*x + 3) - 3)
) - 28296/5021863*log(abs(-11/(5*x + 3) + 2))