3.75.11 \(\int \frac {e^{\frac {3 x^3-e^{x^2} x^3+x^4-x^3 \log (x)}{-3+3 x}} (-8 x^2+x^3+3 x^4+e^{x^2} (3 x^2-2 x^3+2 x^4-2 x^5)+(3 x^2-2 x^3) \log (x))}{15-30 x+15 x^2} \, dx\)

Optimal. Leaf size=33 \[ \frac {1}{5} e^{\frac {x^2 \left (3-e^{x^2}+x-\log (x)\right )}{3-\frac {3}{x}}} \]

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Rubi [F]  time = 16.62, antiderivative size = 0, normalized size of antiderivative = 0.00, number of steps used = 0, number of rules used = 0, integrand size = 0, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.000, Rules used = {} \begin {gather*} \int \frac {\exp \left (\frac {3 x^3-e^{x^2} x^3+x^4-x^3 \log (x)}{-3+3 x}\right ) \left (-8 x^2+x^3+3 x^4+e^{x^2} \left (3 x^2-2 x^3+2 x^4-2 x^5\right )+\left (3 x^2-2 x^3\right ) \log (x)\right )}{15-30 x+15 x^2} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

Int[(E^((3*x^3 - E^x^2*x^3 + x^4 - x^3*Log[x])/(-3 + 3*x))*(-8*x^2 + x^3 + 3*x^4 + E^x^2*(3*x^2 - 2*x^3 + 2*x^
4 - 2*x^5) + (3*x^2 - 2*x^3)*Log[x]))/(15 - 30*x + 15*x^2),x]

[Out]

-1/5*Defer[Int][E^((x^2*(-3 + 6*x - E^x^2*x + x^2 - x*Log[x]))/(3*(-1 + x))), x] + Defer[Int][E^((3*x^3 - E^x^
2*x^3 + x^4 - x^3*Log[x])/(-3 + 3*x)), x]/5 + Defer[Int][E^((x^2*(-3 + 6*x - E^x^2*x + x^2 - x*Log[x]))/(3*(-1
+ x)))/(-1 + x)^2, x]/15 - (4*Defer[Int][E^((3*x^3 - E^x^2*x^3 + x^4 - x^3*Log[x])/(-3 + 3*x))/(-1 + x)^2, x]
)/15 - (2*Defer[Int][E^((x^2*(-3 + 6*x - E^x^2*x + x^2 - x*Log[x]))/(3*(-1 + x)))/(-1 + x), x])/15 - Defer[Int
][E^((3*x^3 - E^x^2*x^3 + x^4 - x^3*Log[x])/(-3 + 3*x))/(-1 + x), x]/15 - (4*Defer[Int][E^((x^2*(-3 + 6*x - E^
x^2*x + x^2 - x*Log[x]))/(3*(-1 + x)))*x, x])/15 + (7*Defer[Int][E^((3*x^3 - E^x^2*x^3 + x^4 - x^3*Log[x])/(-3
+ 3*x))*x, x])/15 - (2*Defer[Int][E^((x^2*(-3 + 6*x - E^x^2*x + x^2 - x*Log[x]))/(3*(-1 + x)))*x^2, x])/15 +
Defer[Int][E^((3*x^3 - E^x^2*x^3 + x^4 - x^3*Log[x])/(-3 + 3*x))*x^2, x]/5 - (2*Defer[Int][E^((x^2*(-3 + 6*x -
E^x^2*x + x^2 - x*Log[x]))/(3*(-1 + x)))*x^3, x])/15 - Defer[Int][E^((3*x^3 - E^x^2*x^3 + x^4 - x^3*Log[x])/(
-3 + 3*x))*Log[x], x]/15 + Defer[Int][(E^((3*x^3 - E^x^2*x^3 + x^4 - x^3*Log[x])/(-3 + 3*x))*Log[x])/(-1 + x)^
2, x]/15 - (2*Defer[Int][E^((3*x^3 - E^x^2*x^3 + x^4 - x^3*Log[x])/(-3 + 3*x))*x*Log[x], x])/15

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\int \frac {\exp \left (\frac {3 x^3-e^{x^2} x^3+x^4-x^3 \log (x)}{-3+3 x}\right ) \left (-8 x^2+x^3+3 x^4+e^{x^2} \left (3 x^2-2 x^3+2 x^4-2 x^5\right )+\left (3 x^2-2 x^3\right ) \log (x)\right )}{15 (-1+x)^2} \, dx\\ &=\frac {1}{15} \int \frac {\exp \left (\frac {3 x^3-e^{x^2} x^3+x^4-x^3 \log (x)}{-3+3 x}\right ) \left (-8 x^2+x^3+3 x^4+e^{x^2} \left (3 x^2-2 x^3+2 x^4-2 x^5\right )+\left (3 x^2-2 x^3\right ) \log (x)\right )}{(-1+x)^2} \, dx\\ &=\frac {1}{15} \int \left (-\frac {8 \exp \left (\frac {3 x^3-e^{x^2} x^3+x^4-x^3 \log (x)}{-3+3 x}\right ) x^2}{(-1+x)^2}+\frac {\exp \left (\frac {3 x^3-e^{x^2} x^3+x^4-x^3 \log (x)}{-3+3 x}\right ) x^3}{(-1+x)^2}+\frac {3 \exp \left (\frac {3 x^3-e^{x^2} x^3+x^4-x^3 \log (x)}{-3+3 x}\right ) x^4}{(-1+x)^2}-\frac {\exp \left (x^2+\frac {3 x^3-e^{x^2} x^3+x^4-x^3 \log (x)}{-3+3 x}\right ) x^2 \left (-3+2 x-2 x^2+2 x^3\right )}{(-1+x)^2}-\frac {\exp \left (\frac {3 x^3-e^{x^2} x^3+x^4-x^3 \log (x)}{-3+3 x}\right ) x^2 (-3+2 x) \log (x)}{(-1+x)^2}\right ) \, dx\\ &=\frac {1}{15} \int \frac {\exp \left (\frac {3 x^3-e^{x^2} x^3+x^4-x^3 \log (x)}{-3+3 x}\right ) x^3}{(-1+x)^2} \, dx-\frac {1}{15} \int \frac {\exp \left (x^2+\frac {3 x^3-e^{x^2} x^3+x^4-x^3 \log (x)}{-3+3 x}\right ) x^2 \left (-3+2 x-2 x^2+2 x^3\right )}{(-1+x)^2} \, dx-\frac {1}{15} \int \frac {\exp \left (\frac {3 x^3-e^{x^2} x^3+x^4-x^3 \log (x)}{-3+3 x}\right ) x^2 (-3+2 x) \log (x)}{(-1+x)^2} \, dx+\frac {1}{5} \int \frac {\exp \left (\frac {3 x^3-e^{x^2} x^3+x^4-x^3 \log (x)}{-3+3 x}\right ) x^4}{(-1+x)^2} \, dx-\frac {8}{15} \int \frac {\exp \left (\frac {3 x^3-e^{x^2} x^3+x^4-x^3 \log (x)}{-3+3 x}\right ) x^2}{(-1+x)^2} \, dx\\ &=\frac {1}{15} \int \left (2 \exp \left (\frac {3 x^3-e^{x^2} x^3+x^4-x^3 \log (x)}{-3+3 x}\right )+\frac {\exp \left (\frac {3 x^3-e^{x^2} x^3+x^4-x^3 \log (x)}{-3+3 x}\right )}{(-1+x)^2}+\frac {3 \exp \left (\frac {3 x^3-e^{x^2} x^3+x^4-x^3 \log (x)}{-3+3 x}\right )}{-1+x}+\exp \left (\frac {3 x^3-e^{x^2} x^3+x^4-x^3 \log (x)}{-3+3 x}\right ) x\right ) \, dx-\frac {1}{15} \int \frac {\exp \left (\frac {x^2 \left (-3+6 x-e^{x^2} x+x^2-x \log (x)\right )}{3 (-1+x)}\right ) x^2 \left (-3+2 x-2 x^2+2 x^3\right )}{(1-x)^2} \, dx-\frac {1}{15} \int \left (\exp \left (\frac {3 x^3-e^{x^2} x^3+x^4-x^3 \log (x)}{-3+3 x}\right ) \log (x)-\frac {\exp \left (\frac {3 x^3-e^{x^2} x^3+x^4-x^3 \log (x)}{-3+3 x}\right ) \log (x)}{(-1+x)^2}+2 \exp \left (\frac {3 x^3-e^{x^2} x^3+x^4-x^3 \log (x)}{-3+3 x}\right ) x \log (x)\right ) \, dx+\frac {1}{5} \int \left (3 \exp \left (\frac {3 x^3-e^{x^2} x^3+x^4-x^3 \log (x)}{-3+3 x}\right )+\frac {\exp \left (\frac {3 x^3-e^{x^2} x^3+x^4-x^3 \log (x)}{-3+3 x}\right )}{(-1+x)^2}+\frac {4 \exp \left (\frac {3 x^3-e^{x^2} x^3+x^4-x^3 \log (x)}{-3+3 x}\right )}{-1+x}+2 \exp \left (\frac {3 x^3-e^{x^2} x^3+x^4-x^3 \log (x)}{-3+3 x}\right ) x+\exp \left (\frac {3 x^3-e^{x^2} x^3+x^4-x^3 \log (x)}{-3+3 x}\right ) x^2\right ) \, dx-\frac {8}{15} \int \left (\exp \left (\frac {3 x^3-e^{x^2} x^3+x^4-x^3 \log (x)}{-3+3 x}\right )+\frac {\exp \left (\frac {3 x^3-e^{x^2} x^3+x^4-x^3 \log (x)}{-3+3 x}\right )}{(-1+x)^2}+\frac {2 \exp \left (\frac {3 x^3-e^{x^2} x^3+x^4-x^3 \log (x)}{-3+3 x}\right )}{-1+x}\right ) \, dx\\ &=\frac {1}{15} \int \frac {e^{\frac {3 x^3-e^{x^2} x^3+x^4-x^3 \log (x)}{-3+3 x}}}{(-1+x)^2} \, dx+\frac {1}{15} \int e^{\frac {3 x^3-e^{x^2} x^3+x^4-x^3 \log (x)}{-3+3 x}} x \, dx-\frac {1}{15} \int \left (3 e^{\frac {x^2 \left (-3+6 x-e^{x^2} x+x^2-x \log (x)\right )}{3 (-1+x)}}-\frac {e^{\frac {x^2 \left (-3+6 x-e^{x^2} x+x^2-x \log (x)\right )}{3 (-1+x)}}}{(-1+x)^2}+\frac {2 e^{\frac {x^2 \left (-3+6 x-e^{x^2} x+x^2-x \log (x)\right )}{3 (-1+x)}}}{-1+x}+4 e^{\frac {x^2 \left (-3+6 x-e^{x^2} x+x^2-x \log (x)\right )}{3 (-1+x)}} x+2 e^{\frac {x^2 \left (-3+6 x-e^{x^2} x+x^2-x \log (x)\right )}{3 (-1+x)}} x^2+2 e^{\frac {x^2 \left (-3+6 x-e^{x^2} x+x^2-x \log (x)\right )}{3 (-1+x)}} x^3\right ) \, dx-\frac {1}{15} \int e^{\frac {3 x^3-e^{x^2} x^3+x^4-x^3 \log (x)}{-3+3 x}} \log (x) \, dx+\frac {1}{15} \int \frac {e^{\frac {3 x^3-e^{x^2} x^3+x^4-x^3 \log (x)}{-3+3 x}} \log (x)}{(-1+x)^2} \, dx+\frac {2}{15} \int e^{\frac {3 x^3-e^{x^2} x^3+x^4-x^3 \log (x)}{-3+3 x}} \, dx-\frac {2}{15} \int e^{\frac {3 x^3-e^{x^2} x^3+x^4-x^3 \log (x)}{-3+3 x}} x \log (x) \, dx+\frac {1}{5} \int \frac {e^{\frac {3 x^3-e^{x^2} x^3+x^4-x^3 \log (x)}{-3+3 x}}}{(-1+x)^2} \, dx+\frac {1}{5} \int \frac {e^{\frac {3 x^3-e^{x^2} x^3+x^4-x^3 \log (x)}{-3+3 x}}}{-1+x} \, dx+\frac {1}{5} \int e^{\frac {3 x^3-e^{x^2} x^3+x^4-x^3 \log (x)}{-3+3 x}} x^2 \, dx+\frac {2}{5} \int e^{\frac {3 x^3-e^{x^2} x^3+x^4-x^3 \log (x)}{-3+3 x}} x \, dx-\frac {8}{15} \int e^{\frac {3 x^3-e^{x^2} x^3+x^4-x^3 \log (x)}{-3+3 x}} \, dx-\frac {8}{15} \int \frac {e^{\frac {3 x^3-e^{x^2} x^3+x^4-x^3 \log (x)}{-3+3 x}}}{(-1+x)^2} \, dx+\frac {3}{5} \int e^{\frac {3 x^3-e^{x^2} x^3+x^4-x^3 \log (x)}{-3+3 x}} \, dx+\frac {4}{5} \int \frac {e^{\frac {3 x^3-e^{x^2} x^3+x^4-x^3 \log (x)}{-3+3 x}}}{-1+x} \, dx-\frac {16}{15} \int \frac {e^{\frac {3 x^3-e^{x^2} x^3+x^4-x^3 \log (x)}{-3+3 x}}}{-1+x} \, dx\\ &=\frac {1}{15} \int \frac {e^{\frac {x^2 \left (-3+6 x-e^{x^2} x+x^2-x \log (x)\right )}{3 (-1+x)}}}{(-1+x)^2} \, dx+\frac {1}{15} \int \frac {e^{\frac {3 x^3-e^{x^2} x^3+x^4-x^3 \log (x)}{-3+3 x}}}{(-1+x)^2} \, dx+\frac {1}{15} \int e^{\frac {3 x^3-e^{x^2} x^3+x^4-x^3 \log (x)}{-3+3 x}} x \, dx-\frac {1}{15} \int e^{\frac {3 x^3-e^{x^2} x^3+x^4-x^3 \log (x)}{-3+3 x}} \log (x) \, dx+\frac {1}{15} \int \frac {e^{\frac {3 x^3-e^{x^2} x^3+x^4-x^3 \log (x)}{-3+3 x}} \log (x)}{(-1+x)^2} \, dx+\frac {2}{15} \int e^{\frac {3 x^3-e^{x^2} x^3+x^4-x^3 \log (x)}{-3+3 x}} \, dx-\frac {2}{15} \int \frac {e^{\frac {x^2 \left (-3+6 x-e^{x^2} x+x^2-x \log (x)\right )}{3 (-1+x)}}}{-1+x} \, dx-\frac {2}{15} \int e^{\frac {x^2 \left (-3+6 x-e^{x^2} x+x^2-x \log (x)\right )}{3 (-1+x)}} x^2 \, dx-\frac {2}{15} \int e^{\frac {x^2 \left (-3+6 x-e^{x^2} x+x^2-x \log (x)\right )}{3 (-1+x)}} x^3 \, dx-\frac {2}{15} \int e^{\frac {3 x^3-e^{x^2} x^3+x^4-x^3 \log (x)}{-3+3 x}} x \log (x) \, dx-\frac {1}{5} \int e^{\frac {x^2 \left (-3+6 x-e^{x^2} x+x^2-x \log (x)\right )}{3 (-1+x)}} \, dx+\frac {1}{5} \int \frac {e^{\frac {3 x^3-e^{x^2} x^3+x^4-x^3 \log (x)}{-3+3 x}}}{(-1+x)^2} \, dx+\frac {1}{5} \int \frac {e^{\frac {3 x^3-e^{x^2} x^3+x^4-x^3 \log (x)}{-3+3 x}}}{-1+x} \, dx+\frac {1}{5} \int e^{\frac {3 x^3-e^{x^2} x^3+x^4-x^3 \log (x)}{-3+3 x}} x^2 \, dx-\frac {4}{15} \int e^{\frac {x^2 \left (-3+6 x-e^{x^2} x+x^2-x \log (x)\right )}{3 (-1+x)}} x \, dx+\frac {2}{5} \int e^{\frac {3 x^3-e^{x^2} x^3+x^4-x^3 \log (x)}{-3+3 x}} x \, dx-\frac {8}{15} \int e^{\frac {3 x^3-e^{x^2} x^3+x^4-x^3 \log (x)}{-3+3 x}} \, dx-\frac {8}{15} \int \frac {e^{\frac {3 x^3-e^{x^2} x^3+x^4-x^3 \log (x)}{-3+3 x}}}{(-1+x)^2} \, dx+\frac {3}{5} \int e^{\frac {3 x^3-e^{x^2} x^3+x^4-x^3 \log (x)}{-3+3 x}} \, dx+\frac {4}{5} \int \frac {e^{\frac {3 x^3-e^{x^2} x^3+x^4-x^3 \log (x)}{-3+3 x}}}{-1+x} \, dx-\frac {16}{15} \int \frac {e^{\frac {3 x^3-e^{x^2} x^3+x^4-x^3 \log (x)}{-3+3 x}}}{-1+x} \, dx\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.20, size = 41, normalized size = 1.24 \begin {gather*} \frac {1}{5} e^{\frac {x^3 \left (3-e^{x^2}+x\right )}{3 (-1+x)}} x^{\frac {x^3}{3-3 x}} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(E^((3*x^3 - E^x^2*x^3 + x^4 - x^3*Log[x])/(-3 + 3*x))*(-8*x^2 + x^3 + 3*x^4 + E^x^2*(3*x^2 - 2*x^3
+ 2*x^4 - 2*x^5) + (3*x^2 - 2*x^3)*Log[x]))/(15 - 30*x + 15*x^2),x]

[Out]

(E^((x^3*(3 - E^x^2 + x))/(3*(-1 + x)))*x^(x^3/(3 - 3*x)))/5

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fricas [A]  time = 0.57, size = 35, normalized size = 1.06 \begin {gather*} \frac {1}{5} \, e^{\left (\frac {x^{4} - x^{3} e^{\left (x^{2}\right )} - x^{3} \log \relax (x) + 3 \, x^{3}}{3 \, {\left (x - 1\right )}}\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-2*x^3+3*x^2)*log(x)+(-2*x^5+2*x^4-2*x^3+3*x^2)*exp(x^2)+3*x^4+x^3-8*x^2)*exp((-x^3*log(x)-x^3*exp
(x^2)+x^4+3*x^3)/(3*x-3))/(15*x^2-30*x+15),x, algorithm="fricas")

[Out]

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

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giac [A]  time = 1.99, size = 49, normalized size = 1.48 \begin {gather*} \frac {1}{5} \, e^{\left (\frac {x^{4}}{3 \, {\left (x - 1\right )}} - \frac {x^{3} e^{\left (x^{2}\right )}}{3 \, {\left (x - 1\right )}} - \frac {x^{3} \log \relax (x)}{3 \, {\left (x - 1\right )}} + \frac {x^{3}}{x - 1}\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-2*x^3+3*x^2)*log(x)+(-2*x^5+2*x^4-2*x^3+3*x^2)*exp(x^2)+3*x^4+x^3-8*x^2)*exp((-x^3*log(x)-x^3*exp
(x^2)+x^4+3*x^3)/(3*x-3))/(15*x^2-30*x+15),x, algorithm="giac")

[Out]

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

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maple [A]  time = 0.20, size = 25, normalized size = 0.76

method	result	size
risch	\(\frac {{\mathrm e}^{-\frac {x^{3} \left (\ln \relax (x )+{\mathrm e}^{x^{2}}-x -3\right )}{3 \left (x -1\right )}}}{5}\)	\(25\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((-2*x^3+3*x^2)*ln(x)+(-2*x^5+2*x^4-2*x^3+3*x^2)*exp(x^2)+3*x^4+x^3-8*x^2)*exp((-x^3*ln(x)-x^3*exp(x^2)+x^
4+3*x^3)/(3*x-3))/(15*x^2-30*x+15),x,method=_RETURNVERBOSE)

[Out]

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

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maxima [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \frac {1}{15} \, \int \frac {{\left (3 \, x^{4} + x^{3} - 8 \, x^{2} - {\left (2 \, x^{5} - 2 \, x^{4} + 2 \, x^{3} - 3 \, x^{2}\right )} e^{\left (x^{2}\right )} - {\left (2 \, x^{3} - 3 \, x^{2}\right )} \log \relax (x)\right )} e^{\left (\frac {x^{4} - x^{3} e^{\left (x^{2}\right )} - x^{3} \log \relax (x) + 3 \, x^{3}}{3 \, {\left (x - 1\right )}}\right )}}{x^{2} - 2 \, x + 1}\,{d x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-2*x^3+3*x^2)*log(x)+(-2*x^5+2*x^4-2*x^3+3*x^2)*exp(x^2)+3*x^4+x^3-8*x^2)*exp((-x^3*log(x)-x^3*exp
(x^2)+x^4+3*x^3)/(3*x-3))/(15*x^2-30*x+15),x, algorithm="maxima")

[Out]

1/15*integrate((3*x^4 + x^3 - 8*x^2 - (2*x^5 - 2*x^4 + 2*x^3 - 3*x^2)*e^(x^2) - (2*x^3 - 3*x^2)*log(x))*e^(1/3
*(x^4 - x^3*e^(x^2) - x^3*log(x) + 3*x^3)/(x - 1))/(x^2 - 2*x + 1), x)

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mupad [B]  time = 5.00, size = 56, normalized size = 1.70 \begin {gather*} \frac {{\mathrm {e}}^{\frac {x^4}{3\,x-3}}\,{\mathrm {e}}^{-\frac {x^3\,{\mathrm {e}}^{x^2}}{3\,x-3}}\,{\mathrm {e}}^{\frac {x^3}{x-1}}}{5\,x^{\frac {x^3}{3\,x-3}}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((exp(-(x^3*log(x) + x^3*exp(x^2) - 3*x^3 - x^4)/(3*x - 3))*(log(x)*(3*x^2 - 2*x^3) + exp(x^2)*(3*x^2 - 2*x
^3 + 2*x^4 - 2*x^5) - 8*x^2 + x^3 + 3*x^4))/(15*x^2 - 30*x + 15),x)

[Out]

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

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sympy [A]  time = 1.12, size = 31, normalized size = 0.94 \begin {gather*} \frac {e^{\frac {x^{4} - x^{3} e^{x^{2}} - x^{3} \log {\relax (x )} + 3 x^{3}}{3 x - 3}}}{5} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-2*x**3+3*x**2)*ln(x)+(-2*x**5+2*x**4-2*x**3+3*x**2)*exp(x**2)+3*x**4+x**3-8*x**2)*exp((-x**3*ln(x
)-x**3*exp(x**2)+x**4+3*x**3)/(3*x-3))/(15*x**2-30*x+15),x)

[Out]

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

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