3.1430 \(\int \frac{(1-2 x)^3}{(2+3 x)^5 (3+5 x)^3} \, dx\)

Optimal. Leaf size=75 \[ \frac{128634}{3 x+2}+\frac{103455}{5 x+3}+\frac{7854}{(3 x+2)^2}-\frac{6655}{2 (5 x+3)^2}+\frac{539}{(3 x+2)^3}+\frac{343}{12 (3 x+2)^4}-953535 \log (3 x+2)+953535 \log (5 x+3) \]

[Out]

343/(12*(2 + 3*x)^4) + 539/(2 + 3*x)^3 + 7854/(2 + 3*x)^2 + 128634/(2 + 3*x) - 6655/(2*(3 + 5*x)^2) + 103455/(
3 + 5*x) - 953535*Log[2 + 3*x] + 953535*Log[3 + 5*x]

________________________________________________________________________________________

Rubi [A]  time = 0.0409486, 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} \[ \frac{128634}{3 x+2}+\frac{103455}{5 x+3}+\frac{7854}{(3 x+2)^2}-\frac{6655}{2 (5 x+3)^2}+\frac{539}{(3 x+2)^3}+\frac{343}{12 (3 x+2)^4}-953535 \log (3 x+2)+953535 \log (5 x+3) \]

Antiderivative was successfully verified.

[In]

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

[Out]

343/(12*(2 + 3*x)^4) + 539/(2 + 3*x)^3 + 7854/(2 + 3*x)^2 + 128634/(2 + 3*x) - 6655/(2*(3 + 5*x)^2) + 103455/(
3 + 5*x) - 953535*Log[2 + 3*x] + 953535*Log[3 + 5*x]

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{(1-2 x)^3}{(2+3 x)^5 (3+5 x)^3} \, dx &=\int \left (-\frac{343}{(2+3 x)^5}-\frac{4851}{(2+3 x)^4}-\frac{47124}{(2+3 x)^3}-\frac{385902}{(2+3 x)^2}-\frac{2860605}{2+3 x}+\frac{33275}{(3+5 x)^3}-\frac{517275}{(3+5 x)^2}+\frac{4767675}{3+5 x}\right ) \, dx\\ &=\frac{343}{12 (2+3 x)^4}+\frac{539}{(2+3 x)^3}+\frac{7854}{(2+3 x)^2}+\frac{128634}{2+3 x}-\frac{6655}{2 (3+5 x)^2}+\frac{103455}{3+5 x}-953535 \log (2+3 x)+953535 \log (3+5 x)\\ \end{align*}

Mathematica [A]  time = 0.03414, size = 77, normalized size = 1.03 \[ \frac{128634}{3 x+2}+\frac{103455}{5 x+3}+\frac{7854}{(3 x+2)^2}-\frac{6655}{2 (5 x+3)^2}+\frac{539}{(3 x+2)^3}+\frac{343}{12 (3 x+2)^4}-953535 \log (5 (3 x+2))+953535 \log (5 x+3) \]

Antiderivative was successfully verified.

[In]

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

[Out]

343/(12*(2 + 3*x)^4) + 539/(2 + 3*x)^3 + 7854/(2 + 3*x)^2 + 128634/(2 + 3*x) - 6655/(2*(3 + 5*x)^2) + 103455/(
3 + 5*x) - 953535*Log[5*(2 + 3*x)] + 953535*Log[3 + 5*x]

________________________________________________________________________________________

Maple [A]  time = 0.007, size = 72, normalized size = 1. \begin{align*}{\frac{343}{12\, \left ( 2+3\,x \right ) ^{4}}}+539\, \left ( 2+3\,x \right ) ^{-3}+7854\, \left ( 2+3\,x \right ) ^{-2}+128634\, \left ( 2+3\,x \right ) ^{-1}-{\frac{6655}{2\, \left ( 3+5\,x \right ) ^{2}}}+103455\, \left ( 3+5\,x \right ) ^{-1}-953535\,\ln \left ( 2+3\,x \right ) +953535\,\ln \left ( 3+5\,x \right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

343/12/(2+3*x)^4+539/(2+3*x)^3+7854/(2+3*x)^2+128634/(2+3*x)-6655/2/(3+5*x)^2+103455/(3+5*x)-953535*ln(2+3*x)+
953535*ln(3+5*x)

________________________________________________________________________________________

Maxima [A]  time = 2.53941, size = 103, normalized size = 1.37 \begin{align*} \frac{1544726700 \, x^{5} + 4994616330 \, x^{4} + 6455813364 \, x^{3} + 4169655991 \, x^{2} + 1345680462 \, x + 173603415}{12 \,{\left (2025 \, x^{6} + 7830 \, x^{5} + 12609 \, x^{4} + 10824 \, x^{3} + 5224 \, x^{2} + 1344 \, x + 144\right )}} + 953535 \, \log \left (5 \, x + 3\right ) - 953535 \, \log \left (3 \, x + 2\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/12*(1544726700*x^5 + 4994616330*x^4 + 6455813364*x^3 + 4169655991*x^2 + 1345680462*x + 173603415)/(2025*x^6
+ 7830*x^5 + 12609*x^4 + 10824*x^3 + 5224*x^2 + 1344*x + 144) + 953535*log(5*x + 3) - 953535*log(3*x + 2)

________________________________________________________________________________________

Fricas [A]  time = 1.29895, size = 493, normalized size = 6.57 \begin{align*} \frac{1544726700 \, x^{5} + 4994616330 \, x^{4} + 6455813364 \, x^{3} + 4169655991 \, x^{2} + 11442420 \,{\left (2025 \, x^{6} + 7830 \, x^{5} + 12609 \, x^{4} + 10824 \, x^{3} + 5224 \, x^{2} + 1344 \, x + 144\right )} \log \left (5 \, x + 3\right ) - 11442420 \,{\left (2025 \, x^{6} + 7830 \, x^{5} + 12609 \, x^{4} + 10824 \, x^{3} + 5224 \, x^{2} + 1344 \, x + 144\right )} \log \left (3 \, x + 2\right ) + 1345680462 \, x + 173603415}{12 \,{\left (2025 \, x^{6} + 7830 \, x^{5} + 12609 \, x^{4} + 10824 \, x^{3} + 5224 \, x^{2} + 1344 \, x + 144\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/12*(1544726700*x^5 + 4994616330*x^4 + 6455813364*x^3 + 4169655991*x^2 + 11442420*(2025*x^6 + 7830*x^5 + 1260
9*x^4 + 10824*x^3 + 5224*x^2 + 1344*x + 144)*log(5*x + 3) - 11442420*(2025*x^6 + 7830*x^5 + 12609*x^4 + 10824*
x^3 + 5224*x^2 + 1344*x + 144)*log(3*x + 2) + 1345680462*x + 173603415)/(2025*x^6 + 7830*x^5 + 12609*x^4 + 108
24*x^3 + 5224*x^2 + 1344*x + 144)

________________________________________________________________________________________

Sympy [A]  time = 0.201134, size = 71, normalized size = 0.95 \begin{align*} \frac{1544726700 x^{5} + 4994616330 x^{4} + 6455813364 x^{3} + 4169655991 x^{2} + 1345680462 x + 173603415}{24300 x^{6} + 93960 x^{5} + 151308 x^{4} + 129888 x^{3} + 62688 x^{2} + 16128 x + 1728} + 953535 \log{\left (x + \frac{3}{5} \right )} - 953535 \log{\left (x + \frac{2}{3} \right )} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

(1544726700*x**5 + 4994616330*x**4 + 6455813364*x**3 + 4169655991*x**2 + 1345680462*x + 173603415)/(24300*x**6
+ 93960*x**5 + 151308*x**4 + 129888*x**3 + 62688*x**2 + 16128*x + 1728) + 953535*log(x + 3/5) - 953535*log(x
+ 2/3)

________________________________________________________________________________________

Giac [A]  time = 2.57269, size = 103, normalized size = 1.37 \begin{align*} \frac{128634}{3 \, x + 2} - \frac{27225 \,{\left (\frac{136}{3 \, x + 2} - 625\right )}}{2 \,{\left (\frac{1}{3 \, x + 2} - 5\right )}^{2}} + \frac{7854}{{\left (3 \, x + 2\right )}^{2}} + \frac{539}{{\left (3 \, x + 2\right )}^{3}} + \frac{343}{12 \,{\left (3 \, x + 2\right )}^{4}} + 953535 \, \log \left ({\left | -\frac{1}{3 \, x + 2} + 5 \right |}\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

128634/(3*x + 2) - 27225/2*(136/(3*x + 2) - 625)/(1/(3*x + 2) - 5)^2 + 7854/(3*x + 2)^2 + 539/(3*x + 2)^3 + 34
3/12/(3*x + 2)^4 + 953535*log(abs(-1/(3*x + 2) + 5))