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

Optimal. Leaf size=66 \[ \frac {3060}{3 x+2}+\frac {3350}{5 x+3}+\frac {309}{2 (3 x+2)^2}-\frac {275}{2 (5 x+3)^2}+\frac {7}{(3 x+2)^3}-25350 \log (3 x+2)+25350 \log (5 x+3) \]

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Rubi [A]  time = 0.03, antiderivative size = 66, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 1, integrand size = 20, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.050, Rules used = {77} \begin {gather*} \frac {3060}{3 x+2}+\frac {3350}{5 x+3}+\frac {309}{2 (3 x+2)^2}-\frac {275}{2 (5 x+3)^2}+\frac {7}{(3 x+2)^3}-25350 \log (3 x+2)+25350 \log (5 x+3) \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

7/(2 + 3*x)^3 + 309/(2*(2 + 3*x)^2) + 3060/(2 + 3*x) - 275/(2*(3 + 5*x)^2) + 3350/(3 + 5*x) - 25350*Log[2 + 3*
x] + 25350*Log[3 + 5*x]

Rule 77

Int[((a_.) + (b_.)*(x_))*((c_) + (d_.)*(x_))^(n_.)*((e_.) + (f_.)*(x_))^(p_.), x_Symbol] :> Int[ExpandIntegran
d[(a + b*x)*(c + d*x)^n*(e + f*x)^p, x], x] /; FreeQ[{a, b, c, d, e, f, n}, x] && NeQ[b*c - a*d, 0] && ((ILtQ[
n, 0] && ILtQ[p, 0]) || EqQ[p, 1] || (IGtQ[p, 0] && ( !IntegerQ[n] || LeQ[9*p + 5*(n + 2), 0] || GeQ[n + p + 1
, 0] || (GeQ[n + p + 2, 0] && RationalQ[a, b, c, d, e, f]))))

Rubi steps

\begin {align*} \int \frac {1-2 x}{(2+3 x)^4 (3+5 x)^3} \, dx &=\int \left (-\frac {63}{(2+3 x)^4}-\frac {927}{(2+3 x)^3}-\frac {9180}{(2+3 x)^2}-\frac {76050}{2+3 x}+\frac {1375}{(3+5 x)^3}-\frac {16750}{(3+5 x)^2}+\frac {126750}{3+5 x}\right ) \, dx\\ &=\frac {7}{(2+3 x)^3}+\frac {309}{2 (2+3 x)^2}+\frac {3060}{2+3 x}-\frac {275}{2 (3+5 x)^2}+\frac {3350}{3+5 x}-25350 \log (2+3 x)+25350 \log (3+5 x)\\ \end {align*}

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Mathematica [A]  time = 0.05, size = 68, normalized size = 1.03 \begin {gather*} \frac {3060}{3 x+2}+\frac {3350}{5 x+3}+\frac {309}{2 (3 x+2)^2}-\frac {275}{2 (5 x+3)^2}+\frac {7}{(3 x+2)^3}-25350 \log (3 x+2)+25350 \log (-3 (5 x+3)) \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

7/(2 + 3*x)^3 + 309/(2*(2 + 3*x)^2) + 3060/(2 + 3*x) - 275/(2*(3 + 5*x)^2) + 3350/(3 + 5*x) - 25350*Log[2 + 3*
x] + 25350*Log[-3*(3 + 5*x)]

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IntegrateAlgebraic [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {1-2 x}{(2+3 x)^4 (3+5 x)^3} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

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

[Out]

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

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fricas [A]  time = 1.31, size = 115, normalized size = 1.74 \begin {gather*} \frac {2281500 \, x^{4} + 5855850 \, x^{3} + 5631080 \, x^{2} + 50700 \, {\left (675 \, x^{5} + 2160 \, x^{4} + 2763 \, x^{3} + 1766 \, x^{2} + 564 \, x + 72\right )} \log \left (5 \, x + 3\right ) - 50700 \, {\left (675 \, x^{5} + 2160 \, x^{4} + 2763 \, x^{3} + 1766 \, x^{2} + 564 \, x + 72\right )} \log \left (3 \, x + 2\right ) + 2404363 \, x + 384608}{2 \, {\left (675 \, x^{5} + 2160 \, x^{4} + 2763 \, x^{3} + 1766 \, x^{2} + 564 \, x + 72\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/2*(2281500*x^4 + 5855850*x^3 + 5631080*x^2 + 50700*(675*x^5 + 2160*x^4 + 2763*x^3 + 1766*x^2 + 564*x + 72)*l
og(5*x + 3) - 50700*(675*x^5 + 2160*x^4 + 2763*x^3 + 1766*x^2 + 564*x + 72)*log(3*x + 2) + 2404363*x + 384608)
/(675*x^5 + 2160*x^4 + 2763*x^3 + 1766*x^2 + 564*x + 72)

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giac [A]  time = 1.11, size = 55, normalized size = 0.83 \begin {gather*} \frac {2281500 \, x^{4} + 5855850 \, x^{3} + 5631080 \, x^{2} + 2404363 \, x + 384608}{2 \, {\left (5 \, x + 3\right )}^{2} {\left (3 \, x + 2\right )}^{3}} + 25350 \, \log \left ({\left | 5 \, x + 3 \right |}\right ) - 25350 \, \log \left ({\left | 3 \, x + 2 \right |}\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/2*(2281500*x^4 + 5855850*x^3 + 5631080*x^2 + 2404363*x + 384608)/((5*x + 3)^2*(3*x + 2)^3) + 25350*log(abs(5
*x + 3)) - 25350*log(abs(3*x + 2))

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maple [A]  time = 0.01, size = 63, normalized size = 0.95 \begin {gather*} -25350 \ln \left (3 x +2\right )+25350 \ln \left (5 x +3\right )+\frac {7}{\left (3 x +2\right )^{3}}+\frac {309}{2 \left (3 x +2\right )^{2}}+\frac {3060}{3 x +2}-\frac {275}{2 \left (5 x +3\right )^{2}}+\frac {3350}{5 x +3} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

7/(3*x+2)^3+309/2/(3*x+2)^2+3060/(3*x+2)-275/2/(5*x+3)^2+3350/(5*x+3)-25350*ln(3*x+2)+25350*ln(5*x+3)

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maxima [A]  time = 0.60, size = 66, normalized size = 1.00 \begin {gather*} \frac {2281500 \, x^{4} + 5855850 \, x^{3} + 5631080 \, x^{2} + 2404363 \, x + 384608}{2 \, {\left (675 \, x^{5} + 2160 \, x^{4} + 2763 \, x^{3} + 1766 \, x^{2} + 564 \, x + 72\right )}} + 25350 \, \log \left (5 \, x + 3\right ) - 25350 \, \log \left (3 \, x + 2\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/2*(2281500*x^4 + 5855850*x^3 + 5631080*x^2 + 2404363*x + 384608)/(675*x^5 + 2160*x^4 + 2763*x^3 + 1766*x^2 +
564*x + 72) + 25350*log(5*x + 3) - 25350*log(3*x + 2)

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mupad [B]  time = 0.04, size = 55, normalized size = 0.83 \begin {gather*} \frac {1690\,x^4+\frac {13013\,x^3}{3}+\frac {563108\,x^2}{135}+\frac {2404363\,x}{1350}+\frac {192304}{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}}-50700\,\mathrm {atanh}\left (30\,x+19\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

((2404363*x)/1350 + (563108*x^2)/135 + (13013*x^3)/3 + 1690*x^4 + 192304/675)/((188*x)/225 + (1766*x^2)/675 +
(307*x^3)/75 + (16*x^4)/5 + x^5 + 8/75) - 50700*atanh(30*x + 19)

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sympy [A]  time = 0.18, size = 63, normalized size = 0.95 \begin {gather*} - \frac {- 2281500 x^{4} - 5855850 x^{3} - 5631080 x^{2} - 2404363 x - 384608}{1350 x^{5} + 4320 x^{4} + 5526 x^{3} + 3532 x^{2} + 1128 x + 144} + 25350 \log {\left (x + \frac {3}{5} \right )} - 25350 \log {\left (x + \frac {2}{3} \right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-(-2281500*x**4 - 5855850*x**3 - 5631080*x**2 - 2404363*x - 384608)/(1350*x**5 + 4320*x**4 + 5526*x**3 + 3532*
x**2 + 1128*x + 144) + 25350*log(x + 3/5) - 25350*log(x + 2/3)

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