[next] [prev] [prev-tail] [tail] [up]

3.1601 \(\int \frac{1}{(1-2 x)^2 (2+3 x)^5 (3+5 x)} \, dx\)

Optimal. Leaf size=86 \[ \frac{32}{184877 (1-2 x)}+\frac{434043}{16807 (3 x+2)}+\frac{12393}{4802 (3 x+2)^2}+\frac{117}{343 (3 x+2)^3}+\frac{9}{196 (3 x+2)^4}-\frac{6400 \log (1-2 x)}{14235529}-\frac{15192225 \log (3 x+2)}{117649}+\frac{15625}{121} \log (5 x+3) \]

[Out]

32/(184877*(1 - 2*x)) + 9/(196*(2 + 3*x)^4) + 117/(343*(2 + 3*x)^3) + 12393/(4802*(2 + 3*x)^2) + 434043/(16807
*(2 + 3*x)) - (6400*Log[1 - 2*x])/14235529 - (15192225*Log[2 + 3*x])/117649 + (15625*Log[3 + 5*x])/121

________________________________________________________________________________________

Rubi [A]  time = 0.0460715, antiderivative size = 86, 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} \[ \frac{32}{184877 (1-2 x)}+\frac{434043}{16807 (3 x+2)}+\frac{12393}{4802 (3 x+2)^2}+\frac{117}{343 (3 x+2)^3}+\frac{9}{196 (3 x+2)^4}-\frac{6400 \log (1-2 x)}{14235529}-\frac{15192225 \log (3 x+2)}{117649}+\frac{15625}{121} \log (5 x+3) \]

Antiderivative was successfully verified.

[In]

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

[Out]

32/(184877*(1 - 2*x)) + 9/(196*(2 + 3*x)^4) + 117/(343*(2 + 3*x)^3) + 12393/(4802*(2 + 3*x)^2) + 434043/(16807
*(2 + 3*x)) - (6400*Log[1 - 2*x])/14235529 - (15192225*Log[2 + 3*x])/117649 + (15625*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)^5 (3+5 x)} \, dx &=\int \left (\frac{64}{184877 (-1+2 x)^2}-\frac{12800}{14235529 (-1+2 x)}-\frac{27}{49 (2+3 x)^5}-\frac{1053}{343 (2+3 x)^4}-\frac{37179}{2401 (2+3 x)^3}-\frac{1302129}{16807 (2+3 x)^2}-\frac{45576675}{117649 (2+3 x)}+\frac{78125}{121 (3+5 x)}\right ) \, dx\\ &=\frac{32}{184877 (1-2 x)}+\frac{9}{196 (2+3 x)^4}+\frac{117}{343 (2+3 x)^3}+\frac{12393}{4802 (2+3 x)^2}+\frac{434043}{16807 (2+3 x)}-\frac{6400 \log (1-2 x)}{14235529}-\frac{15192225 \log (2+3 x)}{117649}+\frac{15625}{121} \log (3+5 x)\\ \end{align*}

Mathematica [A]  time = 0.089692, size = 81, normalized size = 0.94 \[ \frac{5 \left (\frac{77}{5} \left (\frac{19097892}{3 x+2}+\frac{1908522}{(3 x+2)^2}+\frac{252252}{(3 x+2)^3}+\frac{33957}{(3 x+2)^4}+\frac{128}{1-2 x}\right )-5120 \log (5-10 x)-1470607380 \log (5 (3 x+2))+1470612500 \log (5 x+3)\right )}{56942116} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(5*((77*(128/(1 - 2*x) + 33957/(2 + 3*x)^4 + 252252/(2 + 3*x)^3 + 1908522/(2 + 3*x)^2 + 19097892/(2 + 3*x)))/5
 - 5120*Log[5 - 10*x] - 1470607380*Log[5*(2 + 3*x)] + 1470612500*Log[3 + 5*x]))/56942116

________________________________________________________________________________________

Maple [A]  time = 0.009, size = 71, normalized size = 0.8 \begin{align*} -{\frac{32}{369754\,x-184877}}-{\frac{6400\,\ln \left ( 2\,x-1 \right ) }{14235529}}+{\frac{9}{196\, \left ( 2+3\,x \right ) ^{4}}}+{\frac{117}{343\, \left ( 2+3\,x \right ) ^{3}}}+{\frac{12393}{4802\, \left ( 2+3\,x \right ) ^{2}}}+{\frac{434043}{33614+50421\,x}}-{\frac{15192225\,\ln \left ( 2+3\,x \right ) }{117649}}+{\frac{15625\,\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/(2+3*x)^5/(3+5*x),x)

[Out]

-32/184877/(2*x-1)-6400/14235529*ln(2*x-1)+9/196/(2+3*x)^4+117/343/(2+3*x)^3+12393/4802/(2+3*x)^2+434043/16807
/(2+3*x)-15192225/117649*ln(2+3*x)+15625/121*ln(3+5*x)

________________________________________________________________________________________

Maxima [A]  time = 1.11862, size = 100, normalized size = 1.16 \begin{align*} \frac{1031275800 \, x^{4} + 1581255000 \, x^{3} + 373875750 \, x^{2} - 389284050 \, x - 160957733}{739508 \,{\left (162 \, x^{5} + 351 \, x^{4} + 216 \, x^{3} - 24 \, x^{2} - 64 \, x - 16\right )}} + \frac{15625}{121} \, \log \left (5 \, x + 3\right ) - \frac{15192225}{117649} \, \log \left (3 \, x + 2\right ) - \frac{6400}{14235529} \, \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/(2+3*x)^5/(3+5*x),x, algorithm="maxima")

[Out]

1/739508*(1031275800*x^4 + 1581255000*x^3 + 373875750*x^2 - 389284050*x - 160957733)/(162*x^5 + 351*x^4 + 216*
x^3 - 24*x^2 - 64*x - 16) + 15625/121*log(5*x + 3) - 15192225/117649*log(3*x + 2) - 6400/14235529*log(2*x - 1)

________________________________________________________________________________________

Fricas [B]  time = 1.20395, size = 505, normalized size = 5.87 \begin{align*} \frac{79408236600 \, x^{4} + 121756635000 \, x^{3} + 28788432750 \, x^{2} + 7353062500 \,{\left (162 \, x^{5} + 351 \, x^{4} + 216 \, x^{3} - 24 \, x^{2} - 64 \, x - 16\right )} \log \left (5 \, x + 3\right ) - 7353036900 \,{\left (162 \, x^{5} + 351 \, x^{4} + 216 \, x^{3} - 24 \, x^{2} - 64 \, x - 16\right )} \log \left (3 \, x + 2\right ) - 25600 \,{\left (162 \, x^{5} + 351 \, x^{4} + 216 \, x^{3} - 24 \, x^{2} - 64 \, x - 16\right )} \log \left (2 \, x - 1\right ) - 29974871850 \, x - 12393745441}{56942116 \,{\left (162 \, x^{5} + 351 \, x^{4} + 216 \, x^{3} - 24 \, x^{2} - 64 \, x - 16\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/56942116*(79408236600*x^4 + 121756635000*x^3 + 28788432750*x^2 + 7353062500*(162*x^5 + 351*x^4 + 216*x^3 - 2
4*x^2 - 64*x - 16)*log(5*x + 3) - 7353036900*(162*x^5 + 351*x^4 + 216*x^3 - 24*x^2 - 64*x - 16)*log(3*x + 2) -
 25600*(162*x^5 + 351*x^4 + 216*x^3 - 24*x^2 - 64*x - 16)*log(2*x - 1) - 29974871850*x - 12393745441)/(162*x^5
 + 351*x^4 + 216*x^3 - 24*x^2 - 64*x - 16)

________________________________________________________________________________________

Sympy [A]  time = 0.236142, size = 75, normalized size = 0.87 \begin{align*} \frac{1031275800 x^{4} + 1581255000 x^{3} + 373875750 x^{2} - 389284050 x - 160957733}{119800296 x^{5} + 259567308 x^{4} + 159733728 x^{3} - 17748192 x^{2} - 47328512 x - 11832128} - \frac{6400 \log{\left (x - \frac{1}{2} \right )}}{14235529} + \frac{15625 \log{\left (x + \frac{3}{5} \right )}}{121} - \frac{15192225 \log{\left (x + \frac{2}{3} \right )}}{117649} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

(1031275800*x**4 + 1581255000*x**3 + 373875750*x**2 - 389284050*x - 160957733)/(119800296*x**5 + 259567308*x**
4 + 159733728*x**3 - 17748192*x**2 - 47328512*x - 11832128) - 6400*log(x - 1/2)/14235529 + 15625*log(x + 3/5)/
121 - 15192225*log(x + 2/3)/117649

________________________________________________________________________________________

Giac [A]  time = 2.07531, size = 111, normalized size = 1.29 \begin{align*} \frac{434043}{16807 \,{\left (3 \, x + 2\right )}} + \frac{192}{1294139 \,{\left (\frac{7}{3 \, x + 2} - 2\right )}} + \frac{12393}{4802 \,{\left (3 \, x + 2\right )}^{2}} + \frac{117}{343 \,{\left (3 \, x + 2\right )}^{3}} + \frac{9}{196 \,{\left (3 \, x + 2\right )}^{4}} + \frac{15625}{121} \, \log \left ({\left | -\frac{1}{3 \, x + 2} + 5 \right |}\right ) - \frac{6400}{14235529} \, \log \left ({\left | -\frac{7}{3 \, x + 2} + 2 \right |}\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

434043/16807/(3*x + 2) + 192/1294139/(7/(3*x + 2) - 2) + 12393/4802/(3*x + 2)^2 + 117/343/(3*x + 2)^3 + 9/196/
(3*x + 2)^4 + 15625/121*log(abs(-1/(3*x + 2) + 5)) - 6400/14235529*log(abs(-7/(3*x + 2) + 2))

[next] [prev] [prev-tail] [front] [up]