3.47.3 \(\int \frac {e^{\frac {1}{3} (5-\log (6 x^2-x^3))} (-4+x)-18 x+3 x^2+(6 x-x^2) \log (20)}{-6 x+x^2} \, dx\)

Optimal. Leaf size=31 \[ -e^{\frac {1}{3} \left (5-\log \left ((6-x) x^2\right )\right )}+3 x-x \log (20) \]

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Rubi [A]  time = 0.92, antiderivative size = 31, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, integrand size = 57, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.053, Rules used = {1593, 6742, 1588} \begin {gather*} x (3-\log (20))-\frac {e^{5/3}}{\sqrt [3]{6 x^2-x^3}} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(E^((5 - Log[6*x^2 - x^3])/3)*(-4 + x) - 18*x + 3*x^2 + (6*x - x^2)*Log[20])/(-6*x + x^2),x]

[Out]

-(E^(5/3)/(6*x^2 - x^3)^(1/3)) + x*(3 - Log[20])

Rule 1588

Int[(Pp_)*(Qq_)^(m_.), x_Symbol] :> With[{p = Expon[Pp, x], q = Expon[Qq, x]}, Simp[(Coeff[Pp, x, p]*x^(p - q
+ 1)*Qq^(m + 1))/((p + m*q + 1)*Coeff[Qq, x, q]), x] /; NeQ[p + m*q + 1, 0] && EqQ[(p + m*q + 1)*Coeff[Qq, x,
q]*Pp, Coeff[Pp, x, p]*x^(p - q)*((p - q + 1)*Qq + (m + 1)*x*D[Qq, x])]] /; FreeQ[m, x] && PolyQ[Pp, x] && Pol
yQ[Qq, x] && NeQ[m, -1]

Rule 1593

Int[(u_.)*((a_.)*(x_)^(p_.) + (b_.)*(x_)^(q_.))^(n_.), x_Symbol] :> Int[u*x^(n*p)*(a + b*x^(q - p))^n, x] /; F
reeQ[{a, b, p, q}, x] && IntegerQ[n] && PosQ[q - p]

Rule 6742

Int[u_, x_Symbol] :> With[{v = ExpandIntegrand[u, x]}, Int[v, x] /; SumQ[v]]

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\int \frac {e^{\frac {1}{3} \left (5-\log \left (6 x^2-x^3\right )\right )} (-4+x)-18 x+3 x^2+\left (6 x-x^2\right ) \log (20)}{(-6+x) x} \, dx\\ &=\int \left (\frac {e^{5/3} (4-x) x}{\left (6 x^2-x^3\right )^{4/3}}+3 \left (1-\frac {\log (20)}{3}\right )\right ) \, dx\\ &=x (3-\log (20))+e^{5/3} \int \frac {(4-x) x}{\left (6 x^2-x^3\right )^{4/3}} \, dx\\ &=-\frac {e^{5/3}}{\sqrt [3]{6 x^2-x^3}}+x (3-\log (20))\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.18, size = 27, normalized size = 0.87 \begin {gather*} -\frac {e^{5/3}}{\sqrt [3]{-\left ((-6+x) x^2\right )}}-x (-3+\log (20)) \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(E^((5 - Log[6*x^2 - x^3])/3)*(-4 + x) - 18*x + 3*x^2 + (6*x - x^2)*Log[20])/(-6*x + x^2),x]

[Out]

-(E^(5/3)/(-((-6 + x)*x^2))^(1/3)) - x*(-3 + Log[20])

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fricas [B]  time = 0.73, size = 52, normalized size = 1.68 \begin {gather*} \frac {3 \, x^{4} - 18 \, x^{3} - {\left (x^{4} - 6 \, x^{3}\right )} \log \left (20\right ) + {\left (-x^{3} + 6 \, x^{2}\right )}^{\frac {2}{3}} e^{\frac {5}{3}}}{x^{3} - 6 \, x^{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((x-4)*exp(-1/3*log(-x^3+6*x^2)+5/3)+(-x^2+6*x)*log(20)+3*x^2-18*x)/(x^2-6*x),x, algorithm="fricas")

[Out]

(3*x^4 - 18*x^3 - (x^4 - 6*x^3)*log(20) + (-x^3 + 6*x^2)^(2/3)*e^(5/3))/(x^3 - 6*x^2)

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giac [A]  time = 0.24, size = 27, normalized size = 0.87 \begin {gather*} -x {\left (\log \left (20\right ) - 3\right )} - e^{\left (-\frac {1}{3} \, \log \left (-x^{3} + 6 \, x^{2}\right ) + \frac {5}{3}\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((x-4)*exp(-1/3*log(-x^3+6*x^2)+5/3)+(-x^2+6*x)*log(20)+3*x^2-18*x)/(x^2-6*x),x, algorithm="giac")

[Out]

-x*(log(20) - 3) - e^(-1/3*log(-x^3 + 6*x^2) + 5/3)

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maple [F]  time = 0.03, size = 0, normalized size = 0.00 \[\int \frac {\left (x -4\right ) {\mathrm e}^{-\frac {\ln \left (-x^{3}+6 x^{2}\right )}{3}+\frac {5}{3}}+\left (-x^{2}+6 x \right ) \ln \left (20\right )+3 x^{2}-18 x}{x^{2}-6 x}\, dx\]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((x-4)*exp(-1/3*ln(-x^3+6*x^2)+5/3)+(-x^2+6*x)*ln(20)+3*x^2-18*x)/(x^2-6*x),x)

[Out]

int(((x-4)*exp(-1/3*ln(-x^3+6*x^2)+5/3)+(-x^2+6*x)*ln(20)+3*x^2-18*x)/(x^2-6*x),x)

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maxima [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} -x {\left (\log \relax (5) + 2 \, \log \relax (2) - 3\right )} + \int \frac {x e^{\frac {5}{3}} - 4 \, e^{\frac {5}{3}}}{{\left (x^{\frac {8}{3}} - 6 \, x^{\frac {5}{3}}\right )} {\left (-x + 6\right )}^{\frac {1}{3}}}\,{d x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((x-4)*exp(-1/3*log(-x^3+6*x^2)+5/3)+(-x^2+6*x)*log(20)+3*x^2-18*x)/(x^2-6*x),x, algorithm="maxima")

[Out]

-x*(log(5) + 2*log(2) - 3) + integrate((x*e^(5/3) - 4*e^(5/3))/((x^(8/3) - 6*x^(5/3))*(-x + 6)^(1/3)), x)

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mupad [F]  time = 0.00, size = -1, normalized size = -0.03 \begin {gather*} -\int \frac {\ln \left (20\right )\,\left (6\,x-x^2\right )-18\,x+{\mathrm {e}}^{\frac {5}{3}-\frac {\ln \left (6\,x^2-x^3\right )}{3}}\,\left (x-4\right )+3\,x^2}{6\,x-x^2} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(-(log(20)*(6*x - x^2) - 18*x + exp(5/3 - log(6*x^2 - x^3)/3)*(x - 4) + 3*x^2)/(6*x - x^2),x)

[Out]

-int((log(20)*(6*x - x^2) - 18*x + exp(5/3 - log(6*x^2 - x^3)/3)*(x - 4) + 3*x^2)/(6*x - x^2), x)

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sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} - \int \frac {4 e^{\frac {5}{3}}}{x^{2} \sqrt [3]{- x^{3} + 6 x^{2}} - 6 x \sqrt [3]{- x^{3} + 6 x^{2}}}\, dx - \int \frac {18 x \sqrt [3]{- x^{3} + 6 x^{2}}}{x^{2} \sqrt [3]{- x^{3} + 6 x^{2}} - 6 x \sqrt [3]{- x^{3} + 6 x^{2}}}\, dx - \int \left (- \frac {x e^{\frac {5}{3}}}{x^{2} \sqrt [3]{- x^{3} + 6 x^{2}} - 6 x \sqrt [3]{- x^{3} + 6 x^{2}}}\right )\, dx - \int \left (- \frac {3 x^{2} \sqrt [3]{- x^{3} + 6 x^{2}}}{x^{2} \sqrt [3]{- x^{3} + 6 x^{2}} - 6 x \sqrt [3]{- x^{3} + 6 x^{2}}}\right )\, dx - \int \left (- \frac {6 x \sqrt [3]{- x^{3} + 6 x^{2}} \log {\left (20 \right )}}{x^{2} \sqrt [3]{- x^{3} + 6 x^{2}} - 6 x \sqrt [3]{- x^{3} + 6 x^{2}}}\right )\, dx - \int \frac {x^{2} \sqrt [3]{- x^{3} + 6 x^{2}} \log {\left (20 \right )}}{x^{2} \sqrt [3]{- x^{3} + 6 x^{2}} - 6 x \sqrt [3]{- x^{3} + 6 x^{2}}}\, dx \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((x-4)*exp(-1/3*ln(-x**3+6*x**2)+5/3)+(-x**2+6*x)*ln(20)+3*x**2-18*x)/(x**2-6*x),x)

[Out]

-Integral(4*exp(5/3)/(x**2*(-x**3 + 6*x**2)**(1/3) - 6*x*(-x**3 + 6*x**2)**(1/3)), x) - Integral(18*x*(-x**3 +
6*x**2)**(1/3)/(x**2*(-x**3 + 6*x**2)**(1/3) - 6*x*(-x**3 + 6*x**2)**(1/3)), x) - Integral(-x*exp(5/3)/(x**2*
(-x**3 + 6*x**2)**(1/3) - 6*x*(-x**3 + 6*x**2)**(1/3)), x) - Integral(-3*x**2*(-x**3 + 6*x**2)**(1/3)/(x**2*(-
x**3 + 6*x**2)**(1/3) - 6*x*(-x**3 + 6*x**2)**(1/3)), x) - Integral(-6*x*(-x**3 + 6*x**2)**(1/3)*log(20)/(x**2
*(-x**3 + 6*x**2)**(1/3) - 6*x*(-x**3 + 6*x**2)**(1/3)), x) - Integral(x**2*(-x**3 + 6*x**2)**(1/3)*log(20)/(x
**2*(-x**3 + 6*x**2)**(1/3) - 6*x*(-x**3 + 6*x**2)**(1/3)), x)

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