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

Optimal. Leaf size=48 \[ \frac {55}{3 x+2}+\frac {11}{2 (3 x+2)^2}+\frac {7}{9 (3 x+2)^3}-275 \log (3 x+2)+275 \log (5 x+3) \]

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Rubi [A]  time = 0.02, antiderivative size = 48, 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 {55}{3 x+2}+\frac {11}{2 (3 x+2)^2}+\frac {7}{9 (3 x+2)^3}-275 \log (3 x+2)+275 \log (5 x+3) \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

7/(9*(2 + 3*x)^3) + 11/(2*(2 + 3*x)^2) + 55/(2 + 3*x) - 275*Log[2 + 3*x] + 275*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)} \, dx &=\int \left (-\frac {7}{(2+3 x)^4}-\frac {33}{(2+3 x)^3}-\frac {165}{(2+3 x)^2}-\frac {825}{2+3 x}+\frac {1375}{3+5 x}\right ) \, dx\\ &=\frac {7}{9 (2+3 x)^3}+\frac {11}{2 (2+3 x)^2}+\frac {55}{2+3 x}-275 \log (2+3 x)+275 \log (3+5 x)\\ \end {align*}

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Mathematica [A]  time = 0.04, size = 40, normalized size = 0.83 \begin {gather*} \frac {8910 x^2+12177 x+4172}{18 (3 x+2)^3}-275 \log (3 x+2)+275 \log (-3 (5 x+3)) \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(4172 + 12177*x + 8910*x^2)/(18*(2 + 3*x)^3) - 275*Log[2 + 3*x] + 275*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)} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

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

[Out]

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

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fricas [A]  time = 1.36, size = 75, normalized size = 1.56 \begin {gather*} \frac {8910 \, x^{2} + 4950 \, {\left (27 \, x^{3} + 54 \, x^{2} + 36 \, x + 8\right )} \log \left (5 \, x + 3\right ) - 4950 \, {\left (27 \, x^{3} + 54 \, x^{2} + 36 \, x + 8\right )} \log \left (3 \, x + 2\right ) + 12177 \, x + 4172}{18 \, {\left (27 \, x^{3} + 54 \, x^{2} + 36 \, x + 8\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),x, algorithm="fricas")

[Out]

1/18*(8910*x^2 + 4950*(27*x^3 + 54*x^2 + 36*x + 8)*log(5*x + 3) - 4950*(27*x^3 + 54*x^2 + 36*x + 8)*log(3*x +
2) + 12177*x + 4172)/(27*x^3 + 54*x^2 + 36*x + 8)

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giac [A]  time = 1.22, size = 38, normalized size = 0.79 \begin {gather*} \frac {8910 \, x^{2} + 12177 \, x + 4172}{18 \, {\left (3 \, x + 2\right )}^{3}} + 275 \, \log \left ({\left | 5 \, x + 3 \right |}\right ) - 275 \, \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),x, algorithm="giac")

[Out]

1/18*(8910*x^2 + 12177*x + 4172)/(3*x + 2)^3 + 275*log(abs(5*x + 3)) - 275*log(abs(3*x + 2))

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maple [A]  time = 0.01, size = 45, normalized size = 0.94 \begin {gather*} -275 \ln \left (3 x +2\right )+275 \ln \left (5 x +3\right )+\frac {7}{9 \left (3 x +2\right )^{3}}+\frac {11}{2 \left (3 x +2\right )^{2}}+\frac {55}{3 x +2} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

7/9/(3*x+2)^3+11/2/(3*x+2)^2+55/(3*x+2)-275*ln(3*x+2)+275*ln(5*x+3)

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maxima [A]  time = 0.57, size = 46, normalized size = 0.96 \begin {gather*} \frac {8910 \, x^{2} + 12177 \, x + 4172}{18 \, {\left (27 \, x^{3} + 54 \, x^{2} + 36 \, x + 8\right )}} + 275 \, \log \left (5 \, x + 3\right ) - 275 \, \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),x, algorithm="maxima")

[Out]

1/18*(8910*x^2 + 12177*x + 4172)/(27*x^3 + 54*x^2 + 36*x + 8) + 275*log(5*x + 3) - 275*log(3*x + 2)

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mupad [B]  time = 0.04, size = 35, normalized size = 0.73 \begin {gather*} \frac {\frac {55\,x^2}{3}+\frac {451\,x}{18}+\frac {2086}{243}}{x^3+2\,x^2+\frac {4\,x}{3}+\frac {8}{27}}-550\,\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)),x)

[Out]

((451*x)/18 + (55*x^2)/3 + 2086/243)/((4*x)/3 + 2*x^2 + x^3 + 8/27) - 550*atanh(30*x + 19)

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sympy [A]  time = 0.15, size = 42, normalized size = 0.88 \begin {gather*} - \frac {- 8910 x^{2} - 12177 x - 4172}{486 x^{3} + 972 x^{2} + 648 x + 144} + 275 \log {\left (x + \frac {3}{5} \right )} - 275 \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),x)

[Out]

-(-8910*x**2 - 12177*x - 4172)/(486*x**3 + 972*x**2 + 648*x + 144) + 275*log(x + 3/5) - 275*log(x + 2/3)

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