3.1599 \(\int \frac{1}{(1-2 x)^2 (2+3 x)^3 (3+5 x)} \, dx\)

Optimal. Leaf size=64 \[ \frac{8}{3773 (1-2 x)}+\frac{351}{343 (3 x+2)}+\frac{9}{98 (3 x+2)^2}-\frac{1072 \log (1-2 x)}{290521}-\frac{12393 \log (3 x+2)}{2401}+\frac{625}{121} \log (5 x+3) \]

[Out]

8/(3773*(1 - 2*x)) + 9/(98*(2 + 3*x)^2) + 351/(343*(2 + 3*x)) - (1072*Log[1 - 2*x])/290521 - (12393*Log[2 + 3*
x])/2401 + (625*Log[3 + 5*x])/121

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Rubi [A]  time = 0.0323773, 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} \[ \frac{8}{3773 (1-2 x)}+\frac{351}{343 (3 x+2)}+\frac{9}{98 (3 x+2)^2}-\frac{1072 \log (1-2 x)}{290521}-\frac{12393 \log (3 x+2)}{2401}+\frac{625}{121} \log (5 x+3) \]

Antiderivative was successfully verified.

[In]

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

[Out]

8/(3773*(1 - 2*x)) + 9/(98*(2 + 3*x)^2) + 351/(343*(2 + 3*x)) - (1072*Log[1 - 2*x])/290521 - (12393*Log[2 + 3*
x])/2401 + (625*Log[3 + 5*x])/121

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)^2 (2+3 x)^3 (3+5 x)} \, dx &=\int \left (\frac{16}{3773 (-1+2 x)^2}-\frac{2144}{290521 (-1+2 x)}-\frac{27}{49 (2+3 x)^3}-\frac{1053}{343 (2+3 x)^2}-\frac{37179}{2401 (2+3 x)}+\frac{3125}{121 (3+5 x)}\right ) \, dx\\ &=\frac{8}{3773 (1-2 x)}+\frac{9}{98 (2+3 x)^2}+\frac{351}{343 (2+3 x)}-\frac{1072 \log (1-2 x)}{290521}-\frac{12393 \log (2+3 x)}{2401}+\frac{625}{121} \log (3+5 x)\\ \end{align*}

Mathematica [A]  time = 0.0447408, size = 61, normalized size = 0.95 \[ \frac{-2144 \log (5-10 x)-2999106 \log (5 (3 x+2))+7 \left (\frac{176}{1-2 x}+\frac{84942}{3 x+2}+\frac{7623}{(3 x+2)^2}+428750 \log (5 x+3)\right )}{581042} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(-2144*Log[5 - 10*x] - 2999106*Log[5*(2 + 3*x)] + 7*(176/(1 - 2*x) + 7623/(2 + 3*x)^2 + 84942/(2 + 3*x) + 4287
50*Log[3 + 5*x]))/581042

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Maple [A]  time = 0.01, size = 53, normalized size = 0.8 \begin{align*} -{\frac{8}{7546\,x-3773}}-{\frac{1072\,\ln \left ( 2\,x-1 \right ) }{290521}}+{\frac{9}{98\, \left ( 2+3\,x \right ) ^{2}}}+{\frac{351}{686+1029\,x}}-{\frac{12393\,\ln \left ( 2+3\,x \right ) }{2401}}+{\frac{625\,\ln \left ( 3+5\,x \right ) }{121}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-8/3773/(2*x-1)-1072/290521*ln(2*x-1)+9/98/(2+3*x)^2+351/343/(2+3*x)-12393/2401*ln(2+3*x)+625/121*ln(3+5*x)

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Maxima [A]  time = 1.04119, size = 73, normalized size = 1.14 \begin{align*} \frac{46188 \, x^{2} + 8916 \, x - 16201}{7546 \,{\left (18 \, x^{3} + 15 \, x^{2} - 4 \, x - 4\right )}} + \frac{625}{121} \, \log \left (5 \, x + 3\right ) - \frac{12393}{2401} \, \log \left (3 \, x + 2\right ) - \frac{1072}{290521} \, \log \left (2 \, x - 1\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/7546*(46188*x^2 + 8916*x - 16201)/(18*x^3 + 15*x^2 - 4*x - 4) + 625/121*log(5*x + 3) - 12393/2401*log(3*x +
2) - 1072/290521*log(2*x - 1)

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Fricas [A]  time = 1.19493, size = 301, normalized size = 4.7 \begin{align*} \frac{3556476 \, x^{2} + 3001250 \,{\left (18 \, x^{3} + 15 \, x^{2} - 4 \, x - 4\right )} \log \left (5 \, x + 3\right ) - 2999106 \,{\left (18 \, x^{3} + 15 \, x^{2} - 4 \, x - 4\right )} \log \left (3 \, x + 2\right ) - 2144 \,{\left (18 \, x^{3} + 15 \, x^{2} - 4 \, x - 4\right )} \log \left (2 \, x - 1\right ) + 686532 \, x - 1247477}{581042 \,{\left (18 \, x^{3} + 15 \, x^{2} - 4 \, x - 4\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/581042*(3556476*x^2 + 3001250*(18*x^3 + 15*x^2 - 4*x - 4)*log(5*x + 3) - 2999106*(18*x^3 + 15*x^2 - 4*x - 4)
*log(3*x + 2) - 2144*(18*x^3 + 15*x^2 - 4*x - 4)*log(2*x - 1) + 686532*x - 1247477)/(18*x^3 + 15*x^2 - 4*x - 4
)

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Sympy [A]  time = 0.195513, size = 54, normalized size = 0.84 \begin{align*} \frac{46188 x^{2} + 8916 x - 16201}{135828 x^{3} + 113190 x^{2} - 30184 x - 30184} - \frac{1072 \log{\left (x - \frac{1}{2} \right )}}{290521} + \frac{625 \log{\left (x + \frac{3}{5} \right )}}{121} - \frac{12393 \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)**2/(2+3*x)**3/(3+5*x),x)

[Out]

(46188*x**2 + 8916*x - 16201)/(135828*x**3 + 113190*x**2 - 30184*x - 30184) - 1072*log(x - 1/2)/290521 + 625*l
og(x + 3/5)/121 - 12393*log(x + 2/3)/2401

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Giac [A]  time = 1.46821, size = 89, normalized size = 1.39 \begin{align*} -\frac{8}{3773 \,{\left (2 \, x - 1\right )}} - \frac{54 \,{\left (\frac{287}{2 \, x - 1} + 120\right )}}{2401 \,{\left (\frac{7}{2 \, x - 1} + 3\right )}^{2}} - \frac{12393}{2401} \, \log \left ({\left | -\frac{7}{2 \, x - 1} - 3 \right |}\right ) + \frac{625}{121} \, \log \left ({\left | -\frac{11}{2 \, x - 1} - 5 \right |}\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-8/3773/(2*x - 1) - 54/2401*(287/(2*x - 1) + 120)/(7/(2*x - 1) + 3)^2 - 12393/2401*log(abs(-7/(2*x - 1) - 3))
+ 625/121*log(abs(-11/(2*x - 1) - 5))