3.11.66 \(\int \frac {1-2 x}{(2+3 x)^6 (3+5 x)} \, dx\)

Optimal. Leaf size=70 \[ \frac {1375}{3 x+2}+\frac {275}{2 (3 x+2)^2}+\frac {55}{3 (3 x+2)^3}+\frac {11}{4 (3 x+2)^4}+\frac {7}{15 (3 x+2)^5}-6875 \log (3 x+2)+6875 \log (5 x+3) \]

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Rubi [A]  time = 0.03, antiderivative size = 70, 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 {1375}{3 x+2}+\frac {275}{2 (3 x+2)^2}+\frac {55}{3 (3 x+2)^3}+\frac {11}{4 (3 x+2)^4}+\frac {7}{15 (3 x+2)^5}-6875 \log (3 x+2)+6875 \log (5 x+3) \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(1 - 2*x)/((2 + 3*x)^6*(3 + 5*x)),x]

[Out]

7/(15*(2 + 3*x)^5) + 11/(4*(2 + 3*x)^4) + 55/(3*(2 + 3*x)^3) + 275/(2*(2 + 3*x)^2) + 1375/(2 + 3*x) - 6875*Log
[2 + 3*x] + 6875*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)^6 (3+5 x)} \, dx &=\int \left (-\frac {7}{(2+3 x)^6}-\frac {33}{(2+3 x)^5}-\frac {165}{(2+3 x)^4}-\frac {825}{(2+3 x)^3}-\frac {4125}{(2+3 x)^2}-\frac {20625}{2+3 x}+\frac {34375}{3+5 x}\right ) \, dx\\ &=\frac {7}{15 (2+3 x)^5}+\frac {11}{4 (2+3 x)^4}+\frac {55}{3 (2+3 x)^3}+\frac {275}{2 (2+3 x)^2}+\frac {1375}{2+3 x}-6875 \log (2+3 x)+6875 \log (3+5 x)\\ \end {align*}

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Mathematica [A]  time = 0.05, size = 50, normalized size = 0.71 \begin {gather*} \frac {2227500 x^4+6014250 x^3+6091800 x^2+2743565 x+463586}{20 (3 x+2)^5}-6875 \log (3 x+2)+6875 \log (-3 (5 x+3)) \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(1 - 2*x)/((2 + 3*x)^6*(3 + 5*x)),x]

[Out]

(463586 + 2743565*x + 6091800*x^2 + 6014250*x^3 + 2227500*x^4)/(20*(2 + 3*x)^5) - 6875*Log[2 + 3*x] + 6875*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)^6 (3+5 x)} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

IntegrateAlgebraic[(1 - 2*x)/((2 + 3*x)^6*(3 + 5*x)),x]

[Out]

IntegrateAlgebraic[(1 - 2*x)/((2 + 3*x)^6*(3 + 5*x)), x]

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fricas [A]  time = 1.49, size = 115, normalized size = 1.64 \begin {gather*} \frac {2227500 \, x^{4} + 6014250 \, x^{3} + 6091800 \, x^{2} + 137500 \, {\left (243 \, x^{5} + 810 \, x^{4} + 1080 \, x^{3} + 720 \, x^{2} + 240 \, x + 32\right )} \log \left (5 \, x + 3\right ) - 137500 \, {\left (243 \, x^{5} + 810 \, x^{4} + 1080 \, x^{3} + 720 \, x^{2} + 240 \, x + 32\right )} \log \left (3 \, x + 2\right ) + 2743565 \, x + 463586}{20 \, {\left (243 \, x^{5} + 810 \, x^{4} + 1080 \, x^{3} + 720 \, x^{2} + 240 \, x + 32\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((1-2*x)/(2+3*x)^6/(3+5*x),x, algorithm="fricas")

[Out]

1/20*(2227500*x^4 + 6014250*x^3 + 6091800*x^2 + 137500*(243*x^5 + 810*x^4 + 1080*x^3 + 720*x^2 + 240*x + 32)*l
og(5*x + 3) - 137500*(243*x^5 + 810*x^4 + 1080*x^3 + 720*x^2 + 240*x + 32)*log(3*x + 2) + 2743565*x + 463586)/
(243*x^5 + 810*x^4 + 1080*x^3 + 720*x^2 + 240*x + 32)

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giac [A]  time = 1.15, size = 48, normalized size = 0.69 \begin {gather*} \frac {2227500 \, x^{4} + 6014250 \, x^{3} + 6091800 \, x^{2} + 2743565 \, x + 463586}{20 \, {\left (3 \, x + 2\right )}^{5}} + 6875 \, \log \left ({\left | 5 \, x + 3 \right |}\right ) - 6875 \, \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)^6/(3+5*x),x, algorithm="giac")

[Out]

1/20*(2227500*x^4 + 6014250*x^3 + 6091800*x^2 + 2743565*x + 463586)/(3*x + 2)^5 + 6875*log(abs(5*x + 3)) - 687
5*log(abs(3*x + 2))

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maple [A]  time = 0.01, size = 63, normalized size = 0.90 \begin {gather*} -6875 \ln \left (3 x +2\right )+6875 \ln \left (5 x +3\right )+\frac {7}{15 \left (3 x +2\right )^{5}}+\frac {11}{4 \left (3 x +2\right )^{4}}+\frac {55}{3 \left (3 x +2\right )^{3}}+\frac {275}{2 \left (3 x +2\right )^{2}}+\frac {1375}{3 x +2} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((1-2*x)/(3*x+2)^6/(5*x+3),x)

[Out]

7/15/(3*x+2)^5+11/4/(3*x+2)^4+55/3/(3*x+2)^3+275/2/(3*x+2)^2+1375/(3*x+2)-6875*ln(3*x+2)+6875*ln(5*x+3)

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maxima [A]  time = 0.62, size = 66, normalized size = 0.94 \begin {gather*} \frac {2227500 \, x^{4} + 6014250 \, x^{3} + 6091800 \, x^{2} + 2743565 \, x + 463586}{20 \, {\left (243 \, x^{5} + 810 \, x^{4} + 1080 \, x^{3} + 720 \, x^{2} + 240 \, x + 32\right )}} + 6875 \, \log \left (5 \, x + 3\right ) - 6875 \, \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)^6/(3+5*x),x, algorithm="maxima")

[Out]

1/20*(2227500*x^4 + 6014250*x^3 + 6091800*x^2 + 2743565*x + 463586)/(243*x^5 + 810*x^4 + 1080*x^3 + 720*x^2 +
240*x + 32) + 6875*log(5*x + 3) - 6875*log(3*x + 2)

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mupad [B]  time = 1.09, size = 55, normalized size = 0.79 \begin {gather*} \frac {\frac {1375\,x^4}{3}+\frac {2475\,x^3}{2}+\frac {101530\,x^2}{81}+\frac {548713\,x}{972}+\frac {231793}{2430}}{x^5+\frac {10\,x^4}{3}+\frac {40\,x^3}{9}+\frac {80\,x^2}{27}+\frac {80\,x}{81}+\frac {32}{243}}-13750\,\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)^6*(5*x + 3)),x)

[Out]

((548713*x)/972 + (101530*x^2)/81 + (2475*x^3)/2 + (1375*x^4)/3 + 231793/2430)/((80*x)/81 + (80*x^2)/27 + (40*
x^3)/9 + (10*x^4)/3 + x^5 + 32/243) - 13750*atanh(30*x + 19)

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sympy [A]  time = 0.18, size = 63, normalized size = 0.90 \begin {gather*} - \frac {- 2227500 x^{4} - 6014250 x^{3} - 6091800 x^{2} - 2743565 x - 463586}{4860 x^{5} + 16200 x^{4} + 21600 x^{3} + 14400 x^{2} + 4800 x + 640} + 6875 \log {\left (x + \frac {3}{5} \right )} - 6875 \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)**6/(3+5*x),x)

[Out]

-(-2227500*x**4 - 6014250*x**3 - 6091800*x**2 - 2743565*x - 463586)/(4860*x**5 + 16200*x**4 + 21600*x**3 + 144
00*x**2 + 4800*x + 640) + 6875*log(x + 3/5) - 6875*log(x + 2/3)

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