3.1512 \(\int \frac{1}{(1-2 x) (2+3 x)^3 (3+5 x)^2} \, dx\)

Optimal. Leaf size=64 \[ -\frac{648}{49 (3 x+2)}-\frac{125}{11 (5 x+3)}-\frac{9}{14 (3 x+2)^2}-\frac{16 \log (1-2 x)}{41503}+\frac{34371}{343} \log (3 x+2)-\frac{12125}{121} \log (5 x+3) \]

[Out]

-9/(14*(2 + 3*x)^2) - 648/(49*(2 + 3*x)) - 125/(11*(3 + 5*x)) - (16*Log[1 - 2*x])/41503 + (34371*Log[2 + 3*x])
/343 - (12125*Log[3 + 5*x])/121

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Rubi [A]  time = 0.0291976, 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} \[ -\frac{648}{49 (3 x+2)}-\frac{125}{11 (5 x+3)}-\frac{9}{14 (3 x+2)^2}-\frac{16 \log (1-2 x)}{41503}+\frac{34371}{343} \log (3 x+2)-\frac{12125}{121} \log (5 x+3) \]

Antiderivative was successfully verified.

[In]

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

[Out]

-9/(14*(2 + 3*x)^2) - 648/(49*(2 + 3*x)) - 125/(11*(3 + 5*x)) - (16*Log[1 - 2*x])/41503 + (34371*Log[2 + 3*x])
/343 - (12125*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+3 x)^3 (3+5 x)^2} \, dx &=\int \left (-\frac{32}{41503 (-1+2 x)}+\frac{27}{7 (2+3 x)^3}+\frac{1944}{49 (2+3 x)^2}+\frac{103113}{343 (2+3 x)}+\frac{625}{11 (3+5 x)^2}-\frac{60625}{121 (3+5 x)}\right ) \, dx\\ &=-\frac{9}{14 (2+3 x)^2}-\frac{648}{49 (2+3 x)}-\frac{125}{11 (3+5 x)}-\frac{16 \log (1-2 x)}{41503}+\frac{34371}{343} \log (2+3 x)-\frac{12125}{121} \log (3+5 x)\\ \end{align*}

Mathematica [A]  time = 0.0249262, size = 60, normalized size = 0.94 \[ -\frac{125}{55 x+33}-\frac{648}{147 x+98}-\frac{9}{14 (3 x+2)^2}-\frac{16 \log (1-2 x)}{41503}+\frac{34371}{343} \log (6 x+4)-\frac{12125}{121} \log (10 x+6) \]

Antiderivative was successfully verified.

[In]

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

[Out]

-9/(14*(2 + 3*x)^2) - 125/(33 + 55*x) - 648/(98 + 147*x) - (16*Log[1 - 2*x])/41503 + (34371*Log[4 + 6*x])/343
- (12125*Log[6 + 10*x])/121

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Maple [A]  time = 0.008, size = 53, normalized size = 0.8 \begin{align*} -{\frac{16\,\ln \left ( 2\,x-1 \right ) }{41503}}-{\frac{9}{14\, \left ( 2+3\,x \right ) ^{2}}}-{\frac{648}{98+147\,x}}+{\frac{34371\,\ln \left ( 2+3\,x \right ) }{343}}-{\frac{125}{33+55\,x}}-{\frac{12125\,\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+3*x)^3/(3+5*x)^2,x)

[Out]

-16/41503*ln(2*x-1)-9/14/(2+3*x)^2-648/49/(2+3*x)+34371/343*ln(2+3*x)-125/11/(3+5*x)-12125/121*ln(3+5*x)

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Maxima [A]  time = 1.28462, size = 73, normalized size = 1.14 \begin{align*} -\frac{324090 \, x^{2} + 421329 \, x + 136615}{1078 \,{\left (45 \, x^{3} + 87 \, x^{2} + 56 \, x + 12\right )}} - \frac{12125}{121} \, \log \left (5 \, x + 3\right ) + \frac{34371}{343} \, \log \left (3 \, x + 2\right ) - \frac{16}{41503} \, \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+3*x)^3/(3+5*x)^2,x, algorithm="maxima")

[Out]

-1/1078*(324090*x^2 + 421329*x + 136615)/(45*x^3 + 87*x^2 + 56*x + 12) - 12125/121*log(5*x + 3) + 34371/343*lo
g(3*x + 2) - 16/41503*log(2*x - 1)

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Fricas [A]  time = 1.62943, size = 315, normalized size = 4.92 \begin{align*} -\frac{24954930 \, x^{2} + 8317750 \,{\left (45 \, x^{3} + 87 \, x^{2} + 56 \, x + 12\right )} \log \left (5 \, x + 3\right ) - 8317782 \,{\left (45 \, x^{3} + 87 \, x^{2} + 56 \, x + 12\right )} \log \left (3 \, x + 2\right ) + 32 \,{\left (45 \, x^{3} + 87 \, x^{2} + 56 \, x + 12\right )} \log \left (2 \, x - 1\right ) + 32442333 \, x + 10519355}{83006 \,{\left (45 \, x^{3} + 87 \, x^{2} + 56 \, x + 12\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/83006*(24954930*x^2 + 8317750*(45*x^3 + 87*x^2 + 56*x + 12)*log(5*x + 3) - 8317782*(45*x^3 + 87*x^2 + 56*x
+ 12)*log(3*x + 2) + 32*(45*x^3 + 87*x^2 + 56*x + 12)*log(2*x - 1) + 32442333*x + 10519355)/(45*x^3 + 87*x^2 +
56*x + 12)

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Sympy [A]  time = 0.200627, size = 54, normalized size = 0.84 \begin{align*} - \frac{324090 x^{2} + 421329 x + 136615}{48510 x^{3} + 93786 x^{2} + 60368 x + 12936} - \frac{16 \log{\left (x - \frac{1}{2} \right )}}{41503} - \frac{12125 \log{\left (x + \frac{3}{5} \right )}}{121} + \frac{34371 \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)/(2+3*x)**3/(3+5*x)**2,x)

[Out]

-(324090*x**2 + 421329*x + 136615)/(48510*x**3 + 93786*x**2 + 60368*x + 12936) - 16*log(x - 1/2)/41503 - 12125
*log(x + 3/5)/121 + 34371*log(x + 2/3)/343

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Giac [A]  time = 2.62337, size = 86, normalized size = 1.34 \begin{align*} -\frac{125}{11 \,{\left (5 \, x + 3\right )}} + \frac{135 \,{\left (\frac{214}{5 \, x + 3} + 537\right )}}{98 \,{\left (\frac{1}{5 \, x + 3} + 3\right )}^{2}} + \frac{34371}{343} \, \log \left ({\left | -\frac{1}{5 \, x + 3} - 3 \right |}\right ) - \frac{16}{41503} \, \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)/(2+3*x)^3/(3+5*x)^2,x, algorithm="giac")

[Out]

-125/11/(5*x + 3) + 135/98*(214/(5*x + 3) + 537)/(1/(5*x + 3) + 3)^2 + 34371/343*log(abs(-1/(5*x + 3) - 3)) -
16/41503*log(abs(-11/(5*x + 3) + 2))