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*ln(2+3*x)+128634*ln(3+5*x
)

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Rubi [A]  time = 0.03, antiderivative size = 68, 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} \[ \frac {15708}{3 x+2}+\frac {16698}{5 x+3}+\frac {1617}{2 (3 x+2)^2}-\frac {1331}{2 (5 x+3)^2}+\frac {343}{9 (3 x+2)^3}-128634 \log (3 x+2)+128634 \log (5 x+3) \]

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*}

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Mathematica [A]  time = 0.04, 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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fricas [A]  time = 0.77, size = 115, normalized size = 1.69 \[ \frac {104193540 \, x^{4} + 267430086 \, x^{3} + 257165096 \, x^{2} + 2315412 \, {\left (675 \, x^{5} + 2160 \, x^{4} + 2763 \, x^{3} + 1766 \, x^{2} + 564 \, x + 72\right )} \log \left (5 \, x + 3\right ) - 2315412 \, {\left (675 \, x^{5} + 2160 \, x^{4} + 2763 \, x^{3} + 1766 \, x^{2} + 564 \, x + 72\right )} \log \left (3 \, x + 2\right ) + 109804551 \, x + 17564616}{18 \, {\left (675 \, x^{5} + 2160 \, x^{4} + 2763 \, x^{3} + 1766 \, x^{2} + 564 \, x + 72\right )}} \]

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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giac [A]  time = 0.91, size = 55, normalized size = 0.81 \[ \frac {104193540 \, x^{4} + 267430086 \, x^{3} + 257165096 \, x^{2} + 109804551 \, x + 17564616}{18 \, {\left (5 \, x + 3\right )}^{2} {\left (3 \, x + 2\right )}^{3}} + 128634 \, \log \left ({\left | 5 \, x + 3 \right |}\right ) - 128634 \, \log \left ({\left | 3 \, x + 2 \right |}\right ) \]

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))

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maple [A]  time = 0.01, size = 63, normalized size = 0.93 \[ -128634 \ln \left (3 x +2\right )+128634 \ln \left (5 x +3\right )+\frac {343}{9 \left (3 x +2\right )^{3}}+\frac {1617}{2 \left (3 x +2\right )^{2}}+\frac {15708}{3 x +2}-\frac {1331}{2 \left (5 x +3\right )^{2}}+\frac {16698}{5 x +3} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maxima [A]  time = 0.64, size = 66, normalized size = 0.97 \[ \frac {104193540 \, x^{4} + 267430086 \, x^{3} + 257165096 \, x^{2} + 109804551 \, x + 17564616}{18 \, {\left (675 \, x^{5} + 2160 \, x^{4} + 2763 \, x^{3} + 1766 \, x^{2} + 564 \, x + 72\right )}} + 128634 \, \log \left (5 \, x + 3\right ) - 128634 \, \log \left (3 \, x + 2\right ) \]

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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mupad [B]  time = 1.12, size = 55, normalized size = 0.81 \[ \frac {\frac {42878\,x^4}{5}+\frac {1650803\,x^3}{75}+\frac {128582548\,x^2}{6075}+\frac {36601517\,x}{4050}+\frac {975812}{675}}{x^5+\frac {16\,x^4}{5}+\frac {307\,x^3}{75}+\frac {1766\,x^2}{675}+\frac {188\,x}{225}+\frac {8}{75}}-257268\,\mathrm {atanh}\left (30\,x+19\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

((36601517*x)/4050 + (128582548*x^2)/6075 + (1650803*x^3)/75 + (42878*x^4)/5 + 975812/675)/((188*x)/225 + (176
6*x^2)/675 + (307*x^3)/75 + (16*x^4)/5 + x^5 + 8/75) - 257268*atanh(30*x + 19)

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sympy [A]  time = 0.20, size = 63, normalized size = 0.93 \[ - \frac {- 104193540 x^{4} - 267430086 x^{3} - 257165096 x^{2} - 109804551 x - 17564616}{12150 x^{5} + 38880 x^{4} + 49734 x^{3} + 31788 x^{2} + 10152 x + 1296} + 128634 \log {\left (x + \frac {3}{5} \right )} - 128634 \log {\left (x + \frac {2}{3} \right )} \]

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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