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

Optimal. Leaf size=86 \[ \frac {2704}{3195731 (1-2 x)}-\frac {6156}{2401 (3 x+2)}-\frac {3125}{1331 (5 x+3)}+\frac {8}{41503 (1-2 x)^2}-\frac {81}{686 (3 x+2)^2}-\frac {274224 \log (1-2 x)}{246071287}+\frac {333639 \log (3 x+2)}{16807}-\frac {290625 \log (5 x+3)}{14641} \]

________________________________________________________________________________________

Rubi [A]  time = 0.05, antiderivative size = 86, 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 {2704}{3195731 (1-2 x)}-\frac {6156}{2401 (3 x+2)}-\frac {3125}{1331 (5 x+3)}+\frac {8}{41503 (1-2 x)^2}-\frac {81}{686 (3 x+2)^2}-\frac {274224 \log (1-2 x)}{246071287}+\frac {333639 \log (3 x+2)}{16807}-\frac {290625 \log (5 x+3)}{14641} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

8/(41503*(1 - 2*x)^2) + 2704/(3195731*(1 - 2*x)) - 81/(686*(2 + 3*x)^2) - 6156/(2401*(2 + 3*x)) - 3125/(1331*(
3 + 5*x)) - (274224*Log[1 - 2*x])/246071287 + (333639*Log[2 + 3*x])/16807 - (290625*Log[3 + 5*x])/14641

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)^3 (2+3 x)^3 (3+5 x)^2} \, dx &=\int \left (-\frac {32}{41503 (-1+2 x)^3}+\frac {5408}{3195731 (-1+2 x)^2}-\frac {548448}{246071287 (-1+2 x)}+\frac {243}{343 (2+3 x)^3}+\frac {18468}{2401 (2+3 x)^2}+\frac {1000917}{16807 (2+3 x)}+\frac {15625}{1331 (3+5 x)^2}-\frac {1453125}{14641 (3+5 x)}\right ) \, dx\\ &=\frac {8}{41503 (1-2 x)^2}+\frac {2704}{3195731 (1-2 x)}-\frac {81}{686 (2+3 x)^2}-\frac {6156}{2401 (2+3 x)}-\frac {3125}{1331 (3+5 x)}-\frac {274224 \log (1-2 x)}{246071287}+\frac {333639 \log (2+3 x)}{16807}-\frac {290625 \log (3+5 x)}{14641}\\ \end {align*}

________________________________________________________________________________________

Mathematica [A]  time = 0.08, size = 74, normalized size = 0.86 \begin {gather*} \frac {\frac {41503 (6558 x-3251)}{\left (6 x^2+x-2\right )^2}-\frac {154 (16395384 x-7937593)}{6 x^2+x-2}-\frac {1155481250}{5 x+3}-548448 \log (5-10 x)[phone] \log (5 (3 x+2))-9769068750 \log (5 x+3)}{492142574} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(-1155481250/(3 + 5*x) + (41503*(-3251 + 6558*x))/(-2 + x + 6*x^2)^2 - (154*(-7937593 + 16395384*x))/(-2 + x +
6*x^2) - 548448*Log[5 - 10*x] + 9769617198*Log[5*(2 + 3*x)] - 9769068750*Log[3 + 5*x])/492142574

________________________________________________________________________________________

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

Verification is not applicable to the result.

[In]

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

[Out]

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

________________________________________________________________________________________

fricas [B]  time = 0.83, size = 148, normalized size = 1.72 \begin {gather*} -\frac {117343999080 \, x^{4} + 35266545468 \, x^{3} - 73726130478 \, x^{2} + 9769068750 \, {\left (180 \, x^{5} + 168 \, x^{4} - 79 \, x^{3} - 89 \, x^{2} + 8 \, x + 12\right )} \log \left (5 \, x + 3\right ) - 9769617198 \, {\left (180 \, x^{5} + 168 \, x^{4} - 79 \, x^{3} - 89 \, x^{2} + 8 \, x + 12\right )} \log \left (3 \, x + 2\right ) + 548448 \, {\left (180 \, x^{5} + 168 \, x^{4} - 79 \, x^{3} - 89 \, x^{2} + 8 \, x + 12\right )} \log \left (2 \, x - 1\right ) - 11356433319 \, x + 12361039691}{492142574 \, {\left (180 \, x^{5} + 168 \, x^{4} - 79 \, x^{3} - 89 \, x^{2} + 8 \, x + 12\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/492142574*(117343999080*x^4 + 35266545468*x^3 - 73726130478*x^2 + 9769068750*(180*x^5 + 168*x^4 - 79*x^3 -
89*x^2 + 8*x + 12)*log(5*x + 3) - 9769617198*(180*x^5 + 168*x^4 - 79*x^3 - 89*x^2 + 8*x + 12)*log(3*x + 2) + 5
48448*(180*x^5 + 168*x^4 - 79*x^3 - 89*x^2 + 8*x + 12)*log(2*x - 1) - 11356433319*x + 12361039691)/(180*x^5 +
168*x^4 - 79*x^3 - 89*x^2 + 8*x + 12)

________________________________________________________________________________________

giac [A]  time = 1.15, size = 95, normalized size = 1.10 \begin {gather*} -\frac {3125}{1331 \, {\left (5 \, x + 3\right )}} - \frac {5 \, {\left (\frac {84659379036}{5 \, x + 3} - \frac {206753119043}{{\left (5 \, x + 3\right )}^{2}} - \frac {95568773322}{{\left (5 \, x + 3\right )}^{3}} - 7983405324\right )}}{70306082 \, {\left (\frac {11}{5 \, x + 3} - 2\right )}^{2} {\left (\frac {1}{5 \, x + 3} + 3\right )}^{2}} + \frac {333639}{16807} \, \log \left ({\left | -\frac {1}{5 \, x + 3} - 3 \right |}\right ) - \frac {274224}{246071287} \, \log \left ({\left | -\frac {11}{5 \, x + 3} + 2 \right |}\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-3125/1331/(5*x + 3) - 5/70306082*(84659379036/(5*x + 3) - 206753119043/(5*x + 3)^2 - 95568773322/(5*x + 3)^3
- 7983405324)/((11/(5*x + 3) - 2)^2*(1/(5*x + 3) + 3)^2) + 333639/16807*log(abs(-1/(5*x + 3) - 3)) - 274224/24
6071287*log(abs(-11/(5*x + 3) + 2))

________________________________________________________________________________________

maple [A]  time = 0.01, size = 71, normalized size = 0.83 \begin {gather*} -\frac {274224 \ln \left (2 x -1\right )}{246071287}+\frac {333639 \ln \left (3 x +2\right )}{16807}-\frac {290625 \ln \left (5 x +3\right )}{14641}-\frac {3125}{1331 \left (5 x +3\right )}-\frac {81}{686 \left (3 x +2\right )^{2}}-\frac {6156}{2401 \left (3 x +2\right )}+\frac {8}{41503 \left (2 x -1\right )^{2}}-\frac {2704}{3195731 \left (2 x -1\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-3125/1331/(5*x+3)-290625/14641*ln(5*x+3)-81/686/(3*x+2)^2-6156/2401/(3*x+2)+333639/16807*ln(3*x+2)+8/41503/(2
*x-1)^2-2704/3195731/(2*x-1)-274224/246071287*ln(2*x-1)

________________________________________________________________________________________

maxima [A]  time = 0.44, size = 74, normalized size = 0.86 \begin {gather*} -\frac {1523948040 \, x^{4} + 458007084 \, x^{3} - 957482214 \, x^{2} - 147486147 \, x + 160532983}{6391462 \, {\left (180 \, x^{5} + 168 \, x^{4} - 79 \, x^{3} - 89 \, x^{2} + 8 \, x + 12\right )}} - \frac {290625}{14641} \, \log \left (5 \, x + 3\right ) + \frac {333639}{16807} \, \log \left (3 \, x + 2\right ) - \frac {274224}{246071287} \, \log \left (2 \, x - 1\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/6391462*(1523948040*x^4 + 458007084*x^3 - 957482214*x^2 - 147486147*x + 160532983)/(180*x^5 + 168*x^4 - 79*
x^3 - 89*x^2 + 8*x + 12) - 290625/14641*log(5*x + 3) + 333639/16807*log(3*x + 2) - 274224/246071287*log(2*x -
1)

________________________________________________________________________________________

mupad [B]  time = 1.10, size = 66, normalized size = 0.77 \begin {gather*} \frac {333639\,\ln \left (x+\frac {2}{3}\right )}{16807}-\frac {274224\,\ln \left (x-\frac {1}{2}\right )}{246071287}-\frac {290625\,\ln \left (x+\frac {3}{5}\right )}{14641}-\frac {\frac {4233189\,x^4}{3195731}+\frac {12722419\,x^3}{31957310}-\frac {159580369\,x^2}{191743860}-\frac {49162049\,x}{383487720}+\frac {160532983}{1150463160}}{x^5+\frac {14\,x^4}{15}-\frac {79\,x^3}{180}-\frac {89\,x^2}{180}+\frac {2\,x}{45}+\frac {1}{15}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

(333639*log(x + 2/3))/16807 - (274224*log(x - 1/2))/246071287 - (290625*log(x + 3/5))/14641 - ((12722419*x^3)/
31957310 - (159580369*x^2)/191743860 - (49162049*x)/383487720 + (4233189*x^4)/3195731 + 160532983/1150463160)/
((2*x)/45 - (89*x^2)/180 - (79*x^3)/180 + (14*x^4)/15 + x^5 + 1/15)

________________________________________________________________________________________

sympy [A]  time = 0.26, size = 75, normalized size = 0.87 \begin {gather*} - \frac {1523948040 x^{4} + 458007084 x^{3} - 957482214 x^{2} - 147486147 x + 160532983}{1150463160 x^{5} + 1073765616 x^{4} - 504925498 x^{3} - 568840118 x^{2} + 51131696 x + 76697544} - \frac {274224 \log {\left (x - \frac {1}{2} \right )}}{246071287} - \frac {290625 \log {\left (x + \frac {3}{5} \right )}}{14641} + \frac {333639 \log {\left (x + \frac {2}{3} \right )}}{16807} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-(1523948040*x**4 + 458007084*x**3 - 957482214*x**2 - 147486147*x + 160532983)/(1150463160*x**5 + 1073765616*x
**4 - 504925498*x**3 - 568840118*x**2 + 51131696*x + 76697544) - 274224*log(x - 1/2)/246071287 - 290625*log(x
+ 3/5)/14641 + 333639*log(x + 2/3)/16807

________________________________________________________________________________________