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)-1072/290521*ln(1-2*x)-12393/2401*ln(2+3*x)+625/121*ln(3+5*x)

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Rubi [A]  time = 0.03, antiderivative size = 64, normalized size of antiderivative = 1.00, 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*}

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Mathematica [A]  time = 0.07, 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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fricas [A]  time = 0.65, size = 98, normalized size = 1.53 \[ \frac {3556476 \, x^{2} + 3001250 \, {\left (18 \, x^{3} + 15 \, x^{2} - 4 \, x - 4\right )} \log \left (5 \, x + 3\right ) - 2999106 \, {\left (18 \, x^{3} + 15 \, x^{2} - 4 \, x - 4\right )} \log \left (3 \, x + 2\right ) - 2144 \, {\left (18 \, x^{3} + 15 \, x^{2} - 4 \, x - 4\right )} \log \left (2 \, x - 1\right ) + 686532 \, x - 1247477}{581042 \, {\left (18 \, x^{3} + 15 \, x^{2} - 4 \, x - 4\right )}} \]

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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giac [A]  time = 1.17, size = 66, normalized size = 1.03 \[ -\frac {8}{3773 \, {\left (2 \, x - 1\right )}} - \frac {54 \, {\left (\frac {287}{2 \, x - 1} + 120\right )}}{2401 \, {\left (\frac {7}{2 \, x - 1} + 3\right )}^{2}} - \frac {12393}{2401} \, \log \left ({\left | -\frac {7}{2 \, x - 1} - 3 \right |}\right ) + \frac {625}{121} \, \log \left ({\left | -\frac {11}{2 \, x - 1} - 5 \right |}\right ) \]

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))

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maple [A]  time = 0.01, size = 53, normalized size = 0.83 \[ -\frac {1072 \ln \left (2 x -1\right )}{290521}-\frac {12393 \ln \left (3 x +2\right )}{2401}+\frac {625 \ln \left (5 x +3\right )}{121}+\frac {9}{98 \left (3 x +2\right )^{2}}+\frac {351}{343 \left (3 x +2\right )}-\frac {8}{3773 \left (2 x -1\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maxima [A]  time = 0.58, size = 54, normalized size = 0.84 \[ \frac {46188 \, x^{2} + 8916 \, x - 16201}{7546 \, {\left (18 \, x^{3} + 15 \, x^{2} - 4 \, x - 4\right )}} + \frac {625}{121} \, \log \left (5 \, x + 3\right ) - \frac {12393}{2401} \, \log \left (3 \, x + 2\right ) - \frac {1072}{290521} \, \log \left (2 \, x - 1\right ) \]

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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mupad [B]  time = 1.07, size = 48, normalized size = 0.75 \[ \frac {625\,\ln \left (x+\frac {3}{5}\right )}{121}-\frac {12393\,\ln \left (x+\frac {2}{3}\right )}{2401}-\frac {1072\,\ln \left (x-\frac {1}{2}\right )}{290521}-\frac {\frac {1283\,x^2}{3773}+\frac {743\,x}{11319}-\frac {16201}{135828}}{-x^3-\frac {5\,x^2}{6}+\frac {2\,x}{9}+\frac {2}{9}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

(625*log(x + 3/5))/121 - (12393*log(x + 2/3))/2401 - (1072*log(x - 1/2))/290521 - ((743*x)/11319 + (1283*x^2)/
3773 - 16201/135828)/((2*x)/9 - (5*x^2)/6 - x^3 + 2/9)

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sympy [A]  time = 0.22, size = 54, normalized size = 0.84 \[ \frac {46188 x^{2} + 8916 x - 16201}{135828 x^{3} + 113190 x^{2} - 30184 x - 30184} - \frac {1072 \log {\left (x - \frac {1}{2} \right )}}{290521} + \frac {625 \log {\left (x + \frac {3}{5} \right )}}{121} - \frac {12393 \log {\left (x + \frac {2}{3} \right )}}{2401} \]

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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