3.1429 \(\int \frac{(1-2 x)^3}{(2+3 x)^4 (3+5 x)^3} \, dx\)

Optimal. Leaf size=68 \[ \frac{15708}{3 x+2}+\frac{16698}{5 x+3}+\frac{1617}{2 (3 x+2)^2}-\frac{1331}{2 (5 x+3)^2}+\frac{343}{9 (3 x+2)^3}-128634 \log (3 x+2)+128634 \log (5 x+3) \]

[Out]

343/(9*(2 + 3*x)^3) + 1617/(2*(2 + 3*x)^2) + 15708/(2 + 3*x) - 1331/(2*(3 + 5*x)^2) + 16698/(3 + 5*x) - 128634
*Log[2 + 3*x] + 128634*Log[3 + 5*x]

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Rubi [A]  time = 0.0336473, antiderivative size = 68, 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} \[ \frac{15708}{3 x+2}+\frac{16698}{5 x+3}+\frac{1617}{2 (3 x+2)^2}-\frac{1331}{2 (5 x+3)^2}+\frac{343}{9 (3 x+2)^3}-128634 \log (3 x+2)+128634 \log (5 x+3) \]

Antiderivative was successfully verified.

[In]

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

[Out]

343/(9*(2 + 3*x)^3) + 1617/(2*(2 + 3*x)^2) + 15708/(2 + 3*x) - 1331/(2*(3 + 5*x)^2) + 16698/(3 + 5*x) - 128634
*Log[2 + 3*x] + 128634*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)^4 (3+5 x)^3} \, dx &=\int \left (-\frac{343}{(2+3 x)^4}-\frac{4851}{(2+3 x)^3}-\frac{47124}{(2+3 x)^2}-\frac{385902}{2+3 x}+\frac{6655}{(3+5 x)^3}-\frac{83490}{(3+5 x)^2}+\frac{643170}{3+5 x}\right ) \, dx\\ &=\frac{343}{9 (2+3 x)^3}+\frac{1617}{2 (2+3 x)^2}+\frac{15708}{2+3 x}-\frac{1331}{2 (3+5 x)^2}+\frac{16698}{3+5 x}-128634 \log (2+3 x)+128634 \log (3+5 x)\\ \end{align*}

Mathematica [A]  time = 0.0313085, size = 70, normalized size = 1.03 \[ \frac{15708}{3 x+2}+\frac{16698}{5 x+3}+\frac{1617}{2 (3 x+2)^2}-\frac{1331}{2 (5 x+3)^2}+\frac{343}{9 (3 x+2)^3}-128634 \log (5 (3 x+2))+128634 \log (5 x+3) \]

Antiderivative was successfully verified.

[In]

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

[Out]

343/(9*(2 + 3*x)^3) + 1617/(2*(2 + 3*x)^2) + 15708/(2 + 3*x) - 1331/(2*(3 + 5*x)^2) + 16698/(3 + 5*x) - 128634
*Log[5*(2 + 3*x)] + 128634*Log[3 + 5*x]

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Maple [A]  time = 0.009, size = 63, normalized size = 0.9 \begin{align*}{\frac{343}{9\, \left ( 2+3\,x \right ) ^{3}}}+{\frac{1617}{2\, \left ( 2+3\,x \right ) ^{2}}}+15708\, \left ( 2+3\,x \right ) ^{-1}-{\frac{1331}{2\, \left ( 3+5\,x \right ) ^{2}}}+16698\, \left ( 3+5\,x \right ) ^{-1}-128634\,\ln \left ( 2+3\,x \right ) +128634\,\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)^4/(3+5*x)^3,x)

[Out]

343/9/(2+3*x)^3+1617/2/(2+3*x)^2+15708/(2+3*x)-1331/2/(3+5*x)^2+16698/(3+5*x)-128634*ln(2+3*x)+128634*ln(3+5*x
)

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Maxima [A]  time = 1.07935, size = 89, normalized size = 1.31 \begin{align*} \frac{104193540 \, x^{4} + 267430086 \, x^{3} + 257165096 \, x^{2} + 109804551 \, x + 17564616}{18 \,{\left (675 \, x^{5} + 2160 \, x^{4} + 2763 \, x^{3} + 1766 \, x^{2} + 564 \, x + 72\right )}} + 128634 \, \log \left (5 \, x + 3\right ) - 128634 \, \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)^4/(3+5*x)^3,x, algorithm="maxima")

[Out]

1/18*(104193540*x^4 + 267430086*x^3 + 257165096*x^2 + 109804551*x + 17564616)/(675*x^5 + 2160*x^4 + 2763*x^3 +
1766*x^2 + 564*x + 72) + 128634*log(5*x + 3) - 128634*log(3*x + 2)

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Fricas [A]  time = 1.3138, size = 396, normalized size = 5.82 \begin{align*} \frac{104193540 \, x^{4} + 267430086 \, x^{3} + 257165096 \, x^{2} + 2315412 \,{\left (675 \, x^{5} + 2160 \, x^{4} + 2763 \, x^{3} + 1766 \, x^{2} + 564 \, x + 72\right )} \log \left (5 \, x + 3\right ) - 2315412 \,{\left (675 \, x^{5} + 2160 \, x^{4} + 2763 \, x^{3} + 1766 \, x^{2} + 564 \, x + 72\right )} \log \left (3 \, x + 2\right ) + 109804551 \, x + 17564616}{18 \,{\left (675 \, x^{5} + 2160 \, x^{4} + 2763 \, x^{3} + 1766 \, x^{2} + 564 \, x + 72\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/18*(104193540*x^4 + 267430086*x^3 + 257165096*x^2 + 2315412*(675*x^5 + 2160*x^4 + 2763*x^3 + 1766*x^2 + 564*
x + 72)*log(5*x + 3) - 2315412*(675*x^5 + 2160*x^4 + 2763*x^3 + 1766*x^2 + 564*x + 72)*log(3*x + 2) + 10980455
1*x + 17564616)/(675*x^5 + 2160*x^4 + 2763*x^3 + 1766*x^2 + 564*x + 72)

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Sympy [A]  time = 0.183443, size = 61, normalized size = 0.9 \begin{align*} \frac{104193540 x^{4} + 267430086 x^{3} + 257165096 x^{2} + 109804551 x + 17564616}{12150 x^{5} + 38880 x^{4} + 49734 x^{3} + 31788 x^{2} + 10152 x + 1296} + 128634 \log{\left (x + \frac{3}{5} \right )} - 128634 \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)**4/(3+5*x)**3,x)

[Out]

(104193540*x**4 + 267430086*x**3 + 257165096*x**2 + 109804551*x + 17564616)/(12150*x**5 + 38880*x**4 + 49734*x
**3 + 31788*x**2 + 10152*x + 1296) + 128634*log(x + 3/5) - 128634*log(x + 2/3)

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Giac [A]  time = 1.7212, size = 74, normalized size = 1.09 \begin{align*} \frac{104193540 \, x^{4} + 267430086 \, x^{3} + 257165096 \, x^{2} + 109804551 \, x + 17564616}{18 \,{\left (5 \, x + 3\right )}^{2}{\left (3 \, x + 2\right )}^{3}} + 128634 \, \log \left ({\left | 5 \, x + 3 \right |}\right ) - 128634 \, \log \left ({\left | 3 \, x + 2 \right |}\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/18*(104193540*x^4 + 267430086*x^3 + 257165096*x^2 + 109804551*x + 17564616)/((5*x + 3)^2*(3*x + 2)^3) + 1286
34*log(abs(5*x + 3)) - 128634*log(abs(3*x + 2))