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)-16/41503*ln(1-2*x)+34371/343*ln(2+3*x)-12125/121*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} \[ -\frac {648}{49 (3 x+2)}-\frac {125}{11 (5 x+3)}-\frac {9}{14 (3 x+2)^2}-\frac {16 \log (1-2 x)}{41503}+\frac {34371}{343} \log (3 x+2)-\frac {12125}{121} \log (5 x+3) \]

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*}

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Mathematica [A]  time = 0.04, 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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fricas [A]  time = 0.90, size = 98, normalized size = 1.53 \[ -\frac {24954930 \, x^{2} + 8317750 \, {\left (45 \, x^{3} + 87 \, x^{2} + 56 \, x + 12\right )} \log \left (5 \, x + 3\right ) - 8317782 \, {\left (45 \, x^{3} + 87 \, x^{2} + 56 \, x + 12\right )} \log \left (3 \, x + 2\right ) + 32 \, {\left (45 \, x^{3} + 87 \, x^{2} + 56 \, x + 12\right )} \log \left (2 \, x - 1\right ) + 32442333 \, x + 10519355}{83006 \, {\left (45 \, x^{3} + 87 \, x^{2} + 56 \, x + 12\right )}} \]

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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giac [A]  time = 0.97, size = 64, normalized size = 1.00 \[ -\frac {125}{11 \, {\left (5 \, x + 3\right )}} + \frac {135 \, {\left (\frac {214}{5 \, x + 3} + 537\right )}}{98 \, {\left (\frac {1}{5 \, x + 3} + 3\right )}^{2}} + \frac {34371}{343} \, \log \left ({\left | -\frac {1}{5 \, x + 3} - 3 \right |}\right ) - \frac {16}{41503} \, \log \left ({\left | -\frac {11}{5 \, x + 3} + 2 \right |}\right ) \]

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))

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maple [A]  time = 0.01, size = 53, normalized size = 0.83 \[ -\frac {16 \ln \left (2 x -1\right )}{41503}+\frac {34371 \ln \left (3 x +2\right )}{343}-\frac {12125 \ln \left (5 x +3\right )}{121}-\frac {125}{11 \left (5 x +3\right )}-\frac {9}{14 \left (3 x +2\right )^{2}}-\frac {648}{49 \left (3 x +2\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maxima [A]  time = 0.58, size = 54, normalized size = 0.84 \[ -\frac {324090 \, x^{2} + 421329 \, x + 136615}{1078 \, {\left (45 \, x^{3} + 87 \, x^{2} + 56 \, x + 12\right )}} - \frac {12125}{121} \, \log \left (5 \, x + 3\right ) + \frac {34371}{343} \, \log \left (3 \, x + 2\right ) - \frac {16}{41503} \, \log \left (2 \, x - 1\right ) \]

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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mupad [B]  time = 0.04, size = 46, normalized size = 0.72 \[ \frac {34371\,\ln \left (x+\frac {2}{3}\right )}{343}-\frac {16\,\ln \left (x-\frac {1}{2}\right )}{41503}-\frac {12125\,\ln \left (x+\frac {3}{5}\right )}{121}-\frac {\frac {3601\,x^2}{539}+\frac {140443\,x}{16170}+\frac {27323}{9702}}{x^3+\frac {29\,x^2}{15}+\frac {56\,x}{45}+\frac {4}{15}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

(34371*log(x + 2/3))/343 - (16*log(x - 1/2))/41503 - (12125*log(x + 3/5))/121 - ((140443*x)/16170 + (3601*x^2)
/539 + 27323/9702)/((56*x)/45 + (29*x^2)/15 + x^3 + 4/15)

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sympy [A]  time = 0.22, size = 54, normalized size = 0.84 \[ - \frac {324090 x^{2} + 421329 x + 136615}{48510 x^{3} + 93786 x^{2} + 60368 x + 12936} - \frac {16 \log {\left (x - \frac {1}{2} \right )}}{41503} - \frac {12125 \log {\left (x + \frac {3}{5} \right )}}{121} + \frac {34371 \log {\left (x + \frac {2}{3} \right )}}{343} \]

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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