\(\int \frac {6 e^{\frac {x^2}{3}} x \log (\frac {-2+x}{5 x})+e^{\frac {x^2}{3}} (-6 x+3 x^2-2 x^3+x^4) \log ^2(\frac {-2+x}{5 x})}{-12+6 x} \, dx\) [6563]

Optimal result

Integrand size = 72, antiderivative size = 29 \[ \int \frac {6 e^{\frac {x^2}{3}} x \log \left (\frac {-2+x}{5 x}\right )+e^{\frac {x^2}{3}} \left (-6 x+3 x^2-2 x^3+x^4\right ) \log ^2\left (\frac {-2+x}{5 x}\right )}{-12+6 x} \, dx=\frac {1}{4} e^{\frac {x^2}{3}} x^2 \log ^2\left (\frac {-2+x}{5 x}\right ) \]

[Out]

1/4*ln(1/5*(-2+x)/x)^2*x^2*exp(1/3*x^2)

Rubi [F]

\[ \int \frac {6 e^{\frac {x^2}{3}} x \log \left (\frac {-2+x}{5 x}\right )+e^{\frac {x^2}{3}} \left (-6 x+3 x^2-2 x^3+x^4\right ) \log ^2\left (\frac {-2+x}{5 x}\right )}{-12+6 x} \, dx=\int \frac {6 e^{\frac {x^2}{3}} x \log \left (\frac {-2+x}{5 x}\right )+e^{\frac {x^2}{3}} \left (-6 x+3 x^2-2 x^3+x^4\right ) \log ^2\left (\frac {-2+x}{5 x}\right )}{-12+6 x} \, dx \]

[In]

Int[(6*E^(x^2/3)*x*Log[(-2 + x)/(5*x)] + E^(x^2/3)*(-6*x + 3*x^2 - 2*x^3 + x^4)*Log[(-2 + x)/(5*x)]^2)/(-12 +
6*x),x]

[Out]

x*HypergeometricPFQ[{1/2, 1/2}, {3/2, 3/2}, x^2/3] + (Sqrt[3*Pi]*Erfi[x/Sqrt[3]]*Log[1/5 - 2/(5*x)])/2 + 2*Log
[1/5 - 2/(5*x)]*Defer[Int][E^(x^2/3)/(-2 + x), x] - (Sqrt[3*Pi]*Defer[Int][Erfi[x/Sqrt[3]]/(-2 + x), x])/2 + (
14*Defer[Int][(E^(x^2/3)*Log[1/5 - 2/(5*x)]^2)/(2 - x), x])/3 + (14*Defer[Int][(E^(x^2/3)*Log[1/5 - 2/(5*x)]^2
)/(-2 + x), x])/3 + Defer[Int][E^(x^2/3)*x*Log[1/5 - 2/(5*x)]^2, x]/2 + Defer[Int][E^(x^2/3)*x^3*Log[1/5 - 2/(
5*x)]^2, x]/6 - 2*Defer[Int][Defer[Int][E^(x^2/3)/(-2 + x), x]/(-2 + x), x] + 2*Defer[Int][Defer[Int][E^(x^2/3
)/(-2 + x), x]/x, x]

Rubi steps \begin{align*} \text {integral}& = \int \left (\frac {e^{\frac {x^2}{3}} x \log \left (\frac {1}{5}-\frac {2}{5 x}\right )}{-2+x}+\frac {e^{\frac {x^2}{3}} x \log ^2\left (\frac {1}{5}-\frac {2}{5 x}\right )}{2-x}+\frac {e^{\frac {x^2}{3}} x^2 \log ^2\left (\frac {1}{5}-\frac {2}{5 x}\right )}{2 (-2+x)}+\frac {e^{\frac {x^2}{3}} x^3 \log ^2\left (\frac {1}{5}-\frac {2}{5 x}\right )}{3 (2-x)}+\frac {e^{\frac {x^2}{3}} x^4 \log ^2\left (\frac {1}{5}-\frac {2}{5 x}\right )}{6 (-2+x)}\right ) \, dx \\ & = \frac {1}{6} \int \frac {e^{\frac {x^2}{3}} x^4 \log ^2\left (\frac {1}{5}-\frac {2}{5 x}\right )}{-2+x} \, dx+\frac {1}{3} \int \frac {e^{\frac {x^2}{3}} x^3 \log ^2\left (\frac {1}{5}-\frac {2}{5 x}\right )}{2-x} \, dx+\frac {1}{2} \int \frac {e^{\frac {x^2}{3}} x^2 \log ^2\left (\frac {1}{5}-\frac {2}{5 x}\right )}{-2+x} \, dx+\int \frac {e^{\frac {x^2}{3}} x \log \left (\frac {1}{5}-\frac {2}{5 x}\right )}{-2+x} \, dx+\int \frac {e^{\frac {x^2}{3}} x \log ^2\left (\frac {1}{5}-\frac {2}{5 x}\right )}{2-x} \, dx \\ & = \frac {1}{2} \sqrt {3 \pi } \text {erfi}\left (\frac {x}{\sqrt {3}}\right ) \log \left (\frac {1}{5}-\frac {2}{5 x}\right )+\frac {1}{6} \int \left (8 e^{\frac {x^2}{3}} \log ^2\left (\frac {1}{5}-\frac {2}{5 x}\right )+\frac {16 e^{\frac {x^2}{3}} \log ^2\left (\frac {1}{5}-\frac {2}{5 x}\right )}{-2+x}+4 e^{\frac {x^2}{3}} x \log ^2\left (\frac {1}{5}-\frac {2}{5 x}\right )+2 e^{\frac {x^2}{3}} x^2 \log ^2\left (\frac {1}{5}-\frac {2}{5 x}\right )+e^{\frac {x^2}{3}} x^3 \log ^2\left (\frac {1}{5}-\frac {2}{5 x}\right )\right ) \, dx+\frac {1}{3} \int \left (-4 e^{\frac {x^2}{3}} \log ^2\left (\frac {1}{5}-\frac {2}{5 x}\right )+\frac {8 e^{\frac {x^2}{3}} \log ^2\left (\frac {1}{5}-\frac {2}{5 x}\right )}{2-x}-2 e^{\frac {x^2}{3}} x \log ^2\left (\frac {1}{5}-\frac {2}{5 x}\right )-e^{\frac {x^2}{3}} x^2 \log ^2\left (\frac {1}{5}-\frac {2}{5 x}\right )\right ) \, dx+\frac {1}{2} \int \left (2 e^{\frac {x^2}{3}} \log ^2\left (\frac {1}{5}-\frac {2}{5 x}\right )+\frac {4 e^{\frac {x^2}{3}} \log ^2\left (\frac {1}{5}-\frac {2}{5 x}\right )}{-2+x}+e^{\frac {x^2}{3}} x \log ^2\left (\frac {1}{5}-\frac {2}{5 x}\right )\right ) \, dx+\left (2 \log \left (\frac {1}{5}-\frac {2}{5 x}\right )\right ) \int \frac {e^{\frac {x^2}{3}}}{-2+x} \, dx+\int \left (-e^{\frac {x^2}{3}} \log ^2\left (\frac {1}{5}-\frac {2}{5 x}\right )+\frac {2 e^{\frac {x^2}{3}} \log ^2\left (\frac {1}{5}-\frac {2}{5 x}\right )}{2-x}\right ) \, dx-\int \frac {-\sqrt {3 \pi } \text {erfi}\left (\frac {x}{\sqrt {3}}\right )-4 \int \frac {e^{\frac {x^2}{3}}}{-2+x} \, dx}{(2-x) x} \, dx \\ & = \frac {1}{2} \sqrt {3 \pi } \text {erfi}\left (\frac {x}{\sqrt {3}}\right ) \log \left (\frac {1}{5}-\frac {2}{5 x}\right )+\frac {1}{6} \int e^{\frac {x^2}{3}} x^3 \log ^2\left (\frac {1}{5}-\frac {2}{5 x}\right ) \, dx+\frac {1}{2} \int e^{\frac {x^2}{3}} x \log ^2\left (\frac {1}{5}-\frac {2}{5 x}\right ) \, dx+2 \int \frac {e^{\frac {x^2}{3}} \log ^2\left (\frac {1}{5}-\frac {2}{5 x}\right )}{2-x} \, dx+2 \int \frac {e^{\frac {x^2}{3}} \log ^2\left (\frac {1}{5}-\frac {2}{5 x}\right )}{-2+x} \, dx+\frac {8}{3} \int \frac {e^{\frac {x^2}{3}} \log ^2\left (\frac {1}{5}-\frac {2}{5 x}\right )}{2-x} \, dx+\frac {8}{3} \int \frac {e^{\frac {x^2}{3}} \log ^2\left (\frac {1}{5}-\frac {2}{5 x}\right )}{-2+x} \, dx+\left (2 \log \left (\frac {1}{5}-\frac {2}{5 x}\right )\right ) \int \frac {e^{\frac {x^2}{3}}}{-2+x} \, dx-\int \left (\frac {\sqrt {3 \pi } \text {erfi}\left (\frac {x}{\sqrt {3}}\right )}{(-2+x) x}+\frac {4 \int \frac {e^{\frac {x^2}{3}}}{-2+x} \, dx}{(-2+x) x}\right ) \, dx \\ & = \frac {1}{2} \sqrt {3 \pi } \text {erfi}\left (\frac {x}{\sqrt {3}}\right ) \log \left (\frac {1}{5}-\frac {2}{5 x}\right )+\frac {1}{6} \int e^{\frac {x^2}{3}} x^3 \log ^2\left (\frac {1}{5}-\frac {2}{5 x}\right ) \, dx+\frac {1}{2} \int e^{\frac {x^2}{3}} x \log ^2\left (\frac {1}{5}-\frac {2}{5 x}\right ) \, dx+2 \int \frac {e^{\frac {x^2}{3}} \log ^2\left (\frac {1}{5}-\frac {2}{5 x}\right )}{2-x} \, dx+2 \int \frac {e^{\frac {x^2}{3}} \log ^2\left (\frac {1}{5}-\frac {2}{5 x}\right )}{-2+x} \, dx+\frac {8}{3} \int \frac {e^{\frac {x^2}{3}} \log ^2\left (\frac {1}{5}-\frac {2}{5 x}\right )}{2-x} \, dx+\frac {8}{3} \int \frac {e^{\frac {x^2}{3}} \log ^2\left (\frac {1}{5}-\frac {2}{5 x}\right )}{-2+x} \, dx-4 \int \frac {\int \frac {e^{\frac {x^2}{3}}}{-2+x} \, dx}{(-2+x) x} \, dx-\sqrt {3 \pi } \int \frac {\text {erfi}\left (\frac {x}{\sqrt {3}}\right )}{(-2+x) x} \, dx+\left (2 \log \left (\frac {1}{5}-\frac {2}{5 x}\right )\right ) \int \frac {e^{\frac {x^2}{3}}}{-2+x} \, dx \\ & = \frac {1}{2} \sqrt {3 \pi } \text {erfi}\left (\frac {x}{\sqrt {3}}\right ) \log \left (\frac {1}{5}-\frac {2}{5 x}\right )+\frac {1}{6} \int e^{\frac {x^2}{3}} x^3 \log ^2\left (\frac {1}{5}-\frac {2}{5 x}\right ) \, dx+\frac {1}{2} \int e^{\frac {x^2}{3}} x \log ^2\left (\frac {1}{5}-\frac {2}{5 x}\right ) \, dx+2 \int \frac {e^{\frac {x^2}{3}} \log ^2\left (\frac {1}{5}-\frac {2}{5 x}\right )}{2-x} \, dx+2 \int \frac {e^{\frac {x^2}{3}} \log ^2\left (\frac {1}{5}-\frac {2}{5 x}\right )}{-2+x} \, dx+\frac {8}{3} \int \frac {e^{\frac {x^2}{3}} \log ^2\left (\frac {1}{5}-\frac {2}{5 x}\right )}{2-x} \, dx+\frac {8}{3} \int \frac {e^{\frac {x^2}{3}} \log ^2\left (\frac {1}{5}-\frac {2}{5 x}\right )}{-2+x} \, dx-4 \int \left (\frac {\int \frac {e^{\frac {x^2}{3}}}{-2+x} \, dx}{2 (-2+x)}-\frac {\int \frac {e^{\frac {x^2}{3}}}{-2+x} \, dx}{2 x}\right ) \, dx-\sqrt {3 \pi } \int \left (\frac {\text {erfi}\left (\frac {x}{\sqrt {3}}\right )}{2 (-2+x)}-\frac {\text {erfi}\left (\frac {x}{\sqrt {3}}\right )}{2 x}\right ) \, dx+\left (2 \log \left (\frac {1}{5}-\frac {2}{5 x}\right )\right ) \int \frac {e^{\frac {x^2}{3}}}{-2+x} \, dx \\ & = \frac {1}{2} \sqrt {3 \pi } \text {erfi}\left (\frac {x}{\sqrt {3}}\right ) \log \left (\frac {1}{5}-\frac {2}{5 x}\right )+\frac {1}{6} \int e^{\frac {x^2}{3}} x^3 \log ^2\left (\frac {1}{5}-\frac {2}{5 x}\right ) \, dx+\frac {1}{2} \int e^{\frac {x^2}{3}} x \log ^2\left (\frac {1}{5}-\frac {2}{5 x}\right ) \, dx+2 \int \frac {e^{\frac {x^2}{3}} \log ^2\left (\frac {1}{5}-\frac {2}{5 x}\right )}{2-x} \, dx+2 \int \frac {e^{\frac {x^2}{3}} \log ^2\left (\frac {1}{5}-\frac {2}{5 x}\right )}{-2+x} \, dx-2 \int \frac {\int \frac {e^{\frac {x^2}{3}}}{-2+x} \, dx}{-2+x} \, dx+2 \int \frac {\int \frac {e^{\frac {x^2}{3}}}{-2+x} \, dx}{x} \, dx+\frac {8}{3} \int \frac {e^{\frac {x^2}{3}} \log ^2\left (\frac {1}{5}-\frac {2}{5 x}\right )}{2-x} \, dx+\frac {8}{3} \int \frac {e^{\frac {x^2}{3}} \log ^2\left (\frac {1}{5}-\frac {2}{5 x}\right )}{-2+x} \, dx-\frac {1}{2} \sqrt {3 \pi } \int \frac {\text {erfi}\left (\frac {x}{\sqrt {3}}\right )}{-2+x} \, dx+\frac {1}{2} \sqrt {3 \pi } \int \frac {\text {erfi}\left (\frac {x}{\sqrt {3}}\right )}{x} \, dx+\left (2 \log \left (\frac {1}{5}-\frac {2}{5 x}\right )\right ) \int \frac {e^{\frac {x^2}{3}}}{-2+x} \, dx \\ & = x \, _2F_2\left (\frac {1}{2},\frac {1}{2};\frac {3}{2},\frac {3}{2};\frac {x^2}{3}\right )+\frac {1}{2} \sqrt {3 \pi } \text {erfi}\left (\frac {x}{\sqrt {3}}\right ) \log \left (\frac {1}{5}-\frac {2}{5 x}\right )+\frac {1}{6} \int e^{\frac {x^2}{3}} x^3 \log ^2\left (\frac {1}{5}-\frac {2}{5 x}\right ) \, dx+\frac {1}{2} \int e^{\frac {x^2}{3}} x \log ^2\left (\frac {1}{5}-\frac {2}{5 x}\right ) \, dx+2 \int \frac {e^{\frac {x^2}{3}} \log ^2\left (\frac {1}{5}-\frac {2}{5 x}\right )}{2-x} \, dx+2 \int \frac {e^{\frac {x^2}{3}} \log ^2\left (\frac {1}{5}-\frac {2}{5 x}\right )}{-2+x} \, dx-2 \int \frac {\int \frac {e^{\frac {x^2}{3}}}{-2+x} \, dx}{-2+x} \, dx+2 \int \frac {\int \frac {e^{\frac {x^2}{3}}}{-2+x} \, dx}{x} \, dx+\frac {8}{3} \int \frac {e^{\frac {x^2}{3}} \log ^2\left (\frac {1}{5}-\frac {2}{5 x}\right )}{2-x} \, dx+\frac {8}{3} \int \frac {e^{\frac {x^2}{3}} \log ^2\left (\frac {1}{5}-\frac {2}{5 x}\right )}{-2+x} \, dx-\frac {1}{2} \sqrt {3 \pi } \int \frac {\text {erfi}\left (\frac {x}{\sqrt {3}}\right )}{-2+x} \, dx+\left (2 \log \left (\frac {1}{5}-\frac {2}{5 x}\right )\right ) \int \frac {e^{\frac {x^2}{3}}}{-2+x} \, dx \\ \end{align*}

Mathematica [A] (verified)

Time = 0.12 (sec) , antiderivative size = 29, normalized size of antiderivative = 1.00 \[ \int \frac {6 e^{\frac {x^2}{3}} x \log \left (\frac {-2+x}{5 x}\right )+e^{\frac {x^2}{3}} \left (-6 x+3 x^2-2 x^3+x^4\right ) \log ^2\left (\frac {-2+x}{5 x}\right )}{-12+6 x} \, dx=\frac {1}{4} e^{\frac {x^2}{3}} x^2 \log ^2\left (\frac {-2+x}{5 x}\right ) \]

[In]

Integrate[(6*E^(x^2/3)*x*Log[(-2 + x)/(5*x)] + E^(x^2/3)*(-6*x + 3*x^2 - 2*x^3 + x^4)*Log[(-2 + x)/(5*x)]^2)/(
-12 + 6*x),x]

[Out]

(E^(x^2/3)*x^2*Log[(-2 + x)/(5*x)]^2)/4

Maple [A] (verified)

Time = 0.69 (sec) , antiderivative size = 23, normalized size of antiderivative = 0.79

method	result	size
parallelrisch	\(\frac {\ln \left (\frac {-2+x}{5 x}\right )^{2} x^{2} {\mathrm e}^{\frac {x^{2}}{3}}}{4}\)	\(23\)
risch	\(\text {Expression too large to display}\)	\(820\)

[In]

int(((x^4-2*x^3+3*x^2-6*x)*exp(1/3*x^2)*ln(1/5*(-2+x)/x)^2+6*x*exp(1/3*x^2)*ln(1/5*(-2+x)/x))/(6*x-12),x,metho
d=_RETURNVERBOSE)

[Out]

1/4*ln(1/5*(-2+x)/x)^2*x^2*exp(1/3*x^2)

Fricas [A] (verification not implemented)

none

Time = 0.25 (sec) , antiderivative size = 22, normalized size of antiderivative = 0.76 \[ \int \frac {6 e^{\frac {x^2}{3}} x \log \left (\frac {-2+x}{5 x}\right )+e^{\frac {x^2}{3}} \left (-6 x+3 x^2-2 x^3+x^4\right ) \log ^2\left (\frac {-2+x}{5 x}\right )}{-12+6 x} \, dx=\frac {1}{4} \, x^{2} e^{\left (\frac {1}{3} \, x^{2}\right )} \log \left (\frac {x - 2}{5 \, x}\right )^{2} \]

[In]

integrate(((x^4-2*x^3+3*x^2-6*x)*exp(1/3*x^2)*log(1/5*(-2+x)/x)^2+6*x*exp(1/3*x^2)*log(1/5*(-2+x)/x))/(6*x-12)
,x, algorithm="fricas")

[Out]

1/4*x^2*e^(1/3*x^2)*log(1/5*(x - 2)/x)^2

Sympy [A] (verification not implemented)

Time = 24.69 (sec) , antiderivative size = 22, normalized size of antiderivative = 0.76 \[ \int \frac {6 e^{\frac {x^2}{3}} x \log \left (\frac {-2+x}{5 x}\right )+e^{\frac {x^2}{3}} \left (-6 x+3 x^2-2 x^3+x^4\right ) \log ^2\left (\frac {-2+x}{5 x}\right )}{-12+6 x} \, dx=\frac {x^{2} e^{\frac {x^{2}}{3}} \log {\left (\frac {\frac {x}{5} - \frac {2}{5}}{x} \right )}^{2}}{4} \]

[In]

integrate(((x**4-2*x**3+3*x**2-6*x)*exp(1/3*x**2)*ln(1/5*(-2+x)/x)**2+6*x*exp(1/3*x**2)*ln(1/5*(-2+x)/x))/(6*x
-12),x)

[Out]

x**2*exp(x**2/3)*log((x/5 - 2/5)/x)**2/4

Maxima [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 63 vs. \(2 (22) = 44\).

Time = 0.32 (sec) , antiderivative size = 63, normalized size of antiderivative = 2.17 \[ \int \frac {6 e^{\frac {x^2}{3}} x \log \left (\frac {-2+x}{5 x}\right )+e^{\frac {x^2}{3}} \left (-6 x+3 x^2-2 x^3+x^4\right ) \log ^2\left (\frac {-2+x}{5 x}\right )}{-12+6 x} \, dx=\frac {1}{4} \, {\left (x^{2} \log \left (5\right )^{2} + x^{2} \log \left (x - 2\right )^{2} + 2 \, x^{2} \log \left (5\right ) \log \left (x\right ) + x^{2} \log \left (x\right )^{2} - 2 \, {\left (x^{2} \log \left (5\right ) + x^{2} \log \left (x\right )\right )} \log \left (x - 2\right )\right )} e^{\left (\frac {1}{3} \, x^{2}\right )} \]

[In]

integrate(((x^4-2*x^3+3*x^2-6*x)*exp(1/3*x^2)*log(1/5*(-2+x)/x)^2+6*x*exp(1/3*x^2)*log(1/5*(-2+x)/x))/(6*x-12)
,x, algorithm="maxima")

[Out]

1/4*(x^2*log(5)^2 + x^2*log(x - 2)^2 + 2*x^2*log(5)*log(x) + x^2*log(x)^2 - 2*(x^2*log(5) + x^2*log(x))*log(x
- 2))*e^(1/3*x^2)

Giac [A] (verification not implemented)

none

Time = 0.32 (sec) , antiderivative size = 22, normalized size of antiderivative = 0.76 \[ \int \frac {6 e^{\frac {x^2}{3}} x \log \left (\frac {-2+x}{5 x}\right )+e^{\frac {x^2}{3}} \left (-6 x+3 x^2-2 x^3+x^4\right ) \log ^2\left (\frac {-2+x}{5 x}\right )}{-12+6 x} \, dx=\frac {1}{4} \, x^{2} e^{\left (\frac {1}{3} \, x^{2}\right )} \log \left (\frac {x - 2}{5 \, x}\right )^{2} \]

[In]

integrate(((x^4-2*x^3+3*x^2-6*x)*exp(1/3*x^2)*log(1/5*(-2+x)/x)^2+6*x*exp(1/3*x^2)*log(1/5*(-2+x)/x))/(6*x-12)
,x, algorithm="giac")

[Out]

1/4*x^2*e^(1/3*x^2)*log(1/5*(x - 2)/x)^2

Mupad [B] (verification not implemented)

Time = 11.93 (sec) , antiderivative size = 22, normalized size of antiderivative = 0.76 \[ \int \frac {6 e^{\frac {x^2}{3}} x \log \left (\frac {-2+x}{5 x}\right )+e^{\frac {x^2}{3}} \left (-6 x+3 x^2-2 x^3+x^4\right ) \log ^2\left (\frac {-2+x}{5 x}\right )}{-12+6 x} \, dx=\frac {x^2\,{\mathrm {e}}^{\frac {x^2}{3}}\,{\ln \left (\frac {x-2}{5\,x}\right )}^2}{4} \]

[In]

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

[Out]

(x^2*exp(x^2/3)*log((x - 2)/(5*x))^2)/4