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

Optimal. Leaf size=64 \[ \frac {27}{7 (3 x+2)}+\frac {1600}{121 (5 x+3)}-\frac {25}{22 (5 x+3)^2}-\frac {16 \log (1-2 x)}{65219}-\frac {2889}{49} \log (3 x+2)+\frac {78475 \log (5 x+3)}{1331} \]

________________________________________________________________________________________

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} \begin {gather*} \frac {27}{7 (3 x+2)}+\frac {1600}{121 (5 x+3)}-\frac {25}{22 (5 x+3)^2}-\frac {16 \log (1-2 x)}{65219}-\frac {2889}{49} \log (3 x+2)+\frac {78475 \log (5 x+3)}{1331} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

27/(7*(2 + 3*x)) - 25/(22*(3 + 5*x)^2) + 1600/(121*(3 + 5*x)) - (16*Log[1 - 2*x])/65219 - (2889*Log[2 + 3*x])/
49 + (78475*Log[3 + 5*x])/1331

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)^2 (3+5 x)^3} \, dx &=\int \left (-\frac {32}{65219 (-1+2 x)}-\frac {81}{7 (2+3 x)^2}-\frac {8667}{49 (2+3 x)}+\frac {125}{11 (3+5 x)^3}-\frac {8000}{121 (3+5 x)^2}+\frac {392375}{1331 (3+5 x)}\right ) \, dx\\ &=\frac {27}{7 (2+3 x)}-\frac {25}{22 (3+5 x)^2}+\frac {1600}{121 (3+5 x)}-\frac {16 \log (1-2 x)}{65219}-\frac {2889}{49} \log (2+3 x)+\frac {78475 \log (3+5 x)}{1331}\\ \end {align*}

________________________________________________________________________________________

Mathematica [A]  time = 0.03, size = 60, normalized size = 0.94 \begin {gather*} \frac {27}{21 x+14}+\frac {1600}{605 x+363}-\frac {25}{22 (5 x+3)^2}-\frac {16 \log (1-2 x)}{65219}-\frac {2889}{49} \log (6 x+4)+\frac {78475 \log (10 x+6)}{1331} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

-25/(22*(3 + 5*x)^2) + 27/(14 + 21*x) + 1600/(363 + 605*x) - (16*Log[1 - 2*x])/65219 - (2889*Log[4 + 6*x])/49
+ (78475*Log[6 + 10*x])/1331

________________________________________________________________________________________

IntegrateAlgebraic [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {1}{(1-2 x) (2+3 x)^2 (3+5 x)^3} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

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

[Out]

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

________________________________________________________________________________________

fricas [A]  time = 1.32, size = 98, normalized size = 1.53 \begin {gather*} \frac {38449950 \, x^{2} + 7690550 \, {\left (75 \, x^{3} + 140 \, x^{2} + 87 \, x + 18\right )} \log \left (5 \, x + 3\right ) - 7690518 \, {\left (75 \, x^{3} + 140 \, x^{2} + 87 \, x + 18\right )} \log \left (3 \, x + 2\right ) - 32 \, {\left (75 \, x^{3} + 140 \, x^{2} + 87 \, x + 18\right )} \log \left (2 \, x - 1\right ) + 47420065 \, x + 14580412}{130438 \, {\left (75 \, x^{3} + 140 \, x^{2} + 87 \, x + 18\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/130438*(38449950*x^2 + 7690550*(75*x^3 + 140*x^2 + 87*x + 18)*log(5*x + 3) - 7690518*(75*x^3 + 140*x^2 + 87*
x + 18)*log(3*x + 2) - 32*(75*x^3 + 140*x^2 + 87*x + 18)*log(2*x - 1) + 47420065*x + 14580412)/(75*x^3 + 140*x
^2 + 87*x + 18)

________________________________________________________________________________________

giac [A]  time = 1.02, size = 64, normalized size = 1.00 \begin {gather*} \frac {27}{7 \, {\left (3 \, x + 2\right )}} - \frac {375 \, {\left (\frac {194}{3 \, x + 2} - 805\right )}}{242 \, {\left (\frac {1}{3 \, x + 2} - 5\right )}^{2}} + \frac {78475}{1331} \, \log \left ({\left | -\frac {1}{3 \, x + 2} + 5 \right |}\right ) - \frac {16}{65219} \, \log \left ({\left | -\frac {7}{3 \, x + 2} + 2 \right |}\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

27/7/(3*x + 2) - 375/242*(194/(3*x + 2) - 805)/(1/(3*x + 2) - 5)^2 + 78475/1331*log(abs(-1/(3*x + 2) + 5)) - 1
6/65219*log(abs(-7/(3*x + 2) + 2))

________________________________________________________________________________________

maple [A]  time = 0.01, size = 53, normalized size = 0.83 \begin {gather*} -\frac {16 \ln \left (2 x -1\right )}{65219}-\frac {2889 \ln \left (3 x +2\right )}{49}+\frac {78475 \ln \left (5 x +3\right )}{1331}-\frac {25}{22 \left (5 x +3\right )^{2}}+\frac {1600}{121 \left (5 x +3\right )}+\frac {27}{7 \left (3 x +2\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-25/22/(5*x+3)^2+1600/121/(5*x+3)+78475/1331*ln(5*x+3)+27/7/(3*x+2)-2889/49*ln(3*x+2)-16/65219*ln(2*x-1)

________________________________________________________________________________________

maxima [A]  time = 0.60, size = 54, normalized size = 0.84 \begin {gather*} \frac {499350 \, x^{2} + 615845 \, x + 189356}{1694 \, {\left (75 \, x^{3} + 140 \, x^{2} + 87 \, x + 18\right )}} + \frac {78475}{1331} \, \log \left (5 \, x + 3\right ) - \frac {2889}{49} \, \log \left (3 \, x + 2\right ) - \frac {16}{65219} \, \log \left (2 \, x - 1\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/1694*(499350*x^2 + 615845*x + 189356)/(75*x^3 + 140*x^2 + 87*x + 18) + 78475/1331*log(5*x + 3) - 2889/49*log
(3*x + 2) - 16/65219*log(2*x - 1)

________________________________________________________________________________________

mupad [B]  time = 1.10, size = 45, normalized size = 0.70 \begin {gather*} \frac {78475\,\ln \left (x+\frac {3}{5}\right )}{1331}-\frac {2889\,\ln \left (x+\frac {2}{3}\right )}{49}-\frac {16\,\ln \left (x-\frac {1}{2}\right )}{65219}+\frac {\frac {3329\,x^2}{847}+\frac {123169\,x}{25410}+\frac {94678}{63525}}{x^3+\frac {28\,x^2}{15}+\frac {29\,x}{25}+\frac {6}{25}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

(78475*log(x + 3/5))/1331 - (2889*log(x + 2/3))/49 - (16*log(x - 1/2))/65219 + ((123169*x)/25410 + (3329*x^2)/
847 + 94678/63525)/((29*x)/25 + (28*x^2)/15 + x^3 + 6/25)

________________________________________________________________________________________

sympy [A]  time = 0.22, size = 56, normalized size = 0.88 \begin {gather*} - \frac {- 499350 x^{2} - 615845 x - 189356}{127050 x^{3} + 237160 x^{2} + 147378 x + 30492} - \frac {16 \log {\left (x - \frac {1}{2} \right )}}{65219} + \frac {78475 \log {\left (x + \frac {3}{5} \right )}}{1331} - \frac {2889 \log {\left (x + \frac {2}{3} \right )}}{49} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-(-499350*x**2 - 615845*x - 189356)/(127050*x**3 + 237160*x**2 + 147378*x + 30492) - 16*log(x - 1/2)/65219 + 7
8475*log(x + 3/5)/1331 - 2889*log(x + 2/3)/49

________________________________________________________________________________________