3.1600 \(\int \frac{1}{(1-2 x)^2 (2+3 x)^4 (3+5 x)} \, dx\)

Optimal. Leaf size=75 \[ \frac{16}{26411 (1-2 x)}+\frac{12393}{2401 (3 x+2)}+\frac{351}{686 (3 x+2)^2}+\frac{3}{49 (3 x+2)^3}-\frac{2672 \log (1-2 x)}{2033647}-\frac{434043 \log (3 x+2)}{16807}+\frac{3125}{121} \log (5 x+3) \]

[Out]

16/(26411*(1 - 2*x)) + 3/(49*(2 + 3*x)^3) + 351/(686*(2 + 3*x)^2) + 12393/(2401*(2 + 3*x)) - (2672*Log[1 - 2*x
])/2033647 - (434043*Log[2 + 3*x])/16807 + (3125*Log[3 + 5*x])/121

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Rubi [A]  time = 0.0376873, antiderivative size = 75, 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} \[ \frac{16}{26411 (1-2 x)}+\frac{12393}{2401 (3 x+2)}+\frac{351}{686 (3 x+2)^2}+\frac{3}{49 (3 x+2)^3}-\frac{2672 \log (1-2 x)}{2033647}-\frac{434043 \log (3 x+2)}{16807}+\frac{3125}{121} \log (5 x+3) \]

Antiderivative was successfully verified.

[In]

Int[1/((1 - 2*x)^2*(2 + 3*x)^4*(3 + 5*x)),x]

[Out]

16/(26411*(1 - 2*x)) + 3/(49*(2 + 3*x)^3) + 351/(686*(2 + 3*x)^2) + 12393/(2401*(2 + 3*x)) - (2672*Log[1 - 2*x
])/2033647 - (434043*Log[2 + 3*x])/16807 + (3125*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)^4 (3+5 x)} \, dx &=\int \left (\frac{32}{26411 (-1+2 x)^2}-\frac{5344}{2033647 (-1+2 x)}-\frac{27}{49 (2+3 x)^4}-\frac{1053}{343 (2+3 x)^3}-\frac{37179}{2401 (2+3 x)^2}-\frac{1302129}{16807 (2+3 x)}+\frac{15625}{121 (3+5 x)}\right ) \, dx\\ &=\frac{16}{26411 (1-2 x)}+\frac{3}{49 (2+3 x)^3}+\frac{351}{686 (2+3 x)^2}+\frac{12393}{2401 (2+3 x)}-\frac{2672 \log (1-2 x)}{2033647}-\frac{434043 \log (2+3 x)}{16807}+\frac{3125}{121} \log (3+5 x)\\ \end{align*}

Mathematica [A]  time = 0.0573208, size = 70, normalized size = 0.93 \[ \frac{77 \left (\frac{272646}{3 x+2}+\frac{27027}{(3 x+2)^2}+\frac{3234}{(3 x+2)^3}+\frac{32}{1-2 x}\right )-5344 \log (5-10 x)-105038406 \log (5 (3 x+2))[phone] \log (5 x+3)}{4067294} \]

Antiderivative was successfully verified.

[In]

Integrate[1/((1 - 2*x)^2*(2 + 3*x)^4*(3 + 5*x)),x]

[Out]

(77*(32/(1 - 2*x) + 3234/(2 + 3*x)^3 + 27027/(2 + 3*x)^2 + 272646/(2 + 3*x)) - 5344*Log[5 - 10*x] - 105038406*
Log[5*(2 + 3*x)] + 105043750*Log[3 + 5*x])/4067294

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Maple [A]  time = 0.01, size = 62, normalized size = 0.8 \begin{align*} -{\frac{16}{52822\,x-26411}}-{\frac{2672\,\ln \left ( 2\,x-1 \right ) }{2033647}}+{\frac{3}{49\, \left ( 2+3\,x \right ) ^{3}}}+{\frac{351}{686\, \left ( 2+3\,x \right ) ^{2}}}+{\frac{12393}{4802+7203\,x}}-{\frac{434043\,\ln \left ( 2+3\,x \right ) }{16807}}+{\frac{3125\,\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)^4/(3+5*x),x)

[Out]

-16/26411/(2*x-1)-2672/2033647*ln(2*x-1)+3/49/(2+3*x)^3+351/686/(2+3*x)^2+12393/2401/(2+3*x)-434043/16807*ln(2
+3*x)+3125/121*ln(3+5*x)

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Maxima [A]  time = 2.67133, size = 86, normalized size = 1.15 \begin{align*} \frac{4906764 \, x^{3} + 4250124 \, x^{2} - 1058241 \, x - 1148128}{52822 \,{\left (54 \, x^{4} + 81 \, x^{3} + 18 \, x^{2} - 20 \, x - 8\right )}} + \frac{3125}{121} \, \log \left (5 \, x + 3\right ) - \frac{434043}{16807} \, \log \left (3 \, x + 2\right ) - \frac{2672}{2033647} \, \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)^4/(3+5*x),x, algorithm="maxima")

[Out]

1/52822*(4906764*x^3 + 4250124*x^2 - 1058241*x - 1148128)/(54*x^4 + 81*x^3 + 18*x^2 - 20*x - 8) + 3125/121*log
(5*x + 3) - 434043/16807*log(3*x + 2) - 2672/2033647*log(2*x - 1)

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Fricas [B]  time = 1.28293, size = 390, normalized size = 5.2 \begin{align*} \frac{377820828 \, x^{3} + 327259548 \, x^{2} + 105043750 \,{\left (54 \, x^{4} + 81 \, x^{3} + 18 \, x^{2} - 20 \, x - 8\right )} \log \left (5 \, x + 3\right ) - 105038406 \,{\left (54 \, x^{4} + 81 \, x^{3} + 18 \, x^{2} - 20 \, x - 8\right )} \log \left (3 \, x + 2\right ) - 5344 \,{\left (54 \, x^{4} + 81 \, x^{3} + 18 \, x^{2} - 20 \, x - 8\right )} \log \left (2 \, x - 1\right ) - 81484557 \, x - 88405856}{4067294 \,{\left (54 \, x^{4} + 81 \, x^{3} + 18 \, x^{2} - 20 \, x - 8\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(1-2*x)^2/(2+3*x)^4/(3+5*x),x, algorithm="fricas")

[Out]

1/4067294*(377820828*x^3 + 327259548*x^2 + 105043750*(54*x^4 + 81*x^3 + 18*x^2 - 20*x - 8)*log(5*x + 3) - 1050
38406*(54*x^4 + 81*x^3 + 18*x^2 - 20*x - 8)*log(3*x + 2) - 5344*(54*x^4 + 81*x^3 + 18*x^2 - 20*x - 8)*log(2*x
- 1) - 81484557*x - 88405856)/(54*x^4 + 81*x^3 + 18*x^2 - 20*x - 8)

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Sympy [A]  time = 0.219769, size = 65, normalized size = 0.87 \begin{align*} \frac{4906764 x^{3} + 4250124 x^{2} - 1058241 x - 1148128}{2852388 x^{4} + 4278582 x^{3} + 950796 x^{2} - 1056440 x - 422576} - \frac{2672 \log{\left (x - \frac{1}{2} \right )}}{2033647} + \frac{3125 \log{\left (x + \frac{3}{5} \right )}}{121} - \frac{434043 \log{\left (x + \frac{2}{3} \right )}}{16807} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(1-2*x)**2/(2+3*x)**4/(3+5*x),x)

[Out]

(4906764*x**3 + 4250124*x**2 - 1058241*x - 1148128)/(2852388*x**4 + 4278582*x**3 + 950796*x**2 - 1056440*x - 4
22576) - 2672*log(x - 1/2)/2033647 + 3125*log(x + 3/5)/121 - 434043*log(x + 2/3)/16807

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Giac [A]  time = 2.52951, size = 101, normalized size = 1.35 \begin{align*} -\frac{16}{26411 \,{\left (2 \, x - 1\right )}} - \frac{54 \,{\left (\frac{60375}{2 \, x - 1} + \frac{71491}{{\left (2 \, x - 1\right )}^{2}} + 12756\right )}}{16807 \,{\left (\frac{7}{2 \, x - 1} + 3\right )}^{3}} - \frac{434043}{16807} \, \log \left ({\left | -\frac{7}{2 \, x - 1} - 3 \right |}\right ) + \frac{3125}{121} \, \log \left ({\left | -\frac{11}{2 \, x - 1} - 5 \right |}\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(1-2*x)^2/(2+3*x)^4/(3+5*x),x, algorithm="giac")

[Out]

-16/26411/(2*x - 1) - 54/16807*(60375/(2*x - 1) + 71491/(2*x - 1)^2 + 12756)/(7/(2*x - 1) + 3)^3 - 434043/1680
7*log(abs(-7/(2*x - 1) - 3)) + 3125/121*log(abs(-11/(2*x - 1) - 5))