\(\int \frac {-x^2+2 x^3+e^{-x+2 e^{\frac {1}{2} (6+8 x+5 \log (x))} x+x^2+e^{6+8 x} x^5} (-1-x+2 x^2+e^{6+8 x} x^5 (5+8 x)+e^{\frac {1}{2} (6+8 x+5 \log (x))} (7 x+8 x^2))}{x^2} \, dx\) [2349]

Optimal result

Integrand size = 105, antiderivative size = 38 \[ \int \frac {-x^2+2 x^3+e^{-x+2 e^{\frac {1}{2} (6+8 x+5 \log (x))} x+x^2+e^{6+8 x} x^5} \left (-1-x+2 x^2+e^{6+8 x} x^5 (5+8 x)+e^{\frac {1}{2} (6+8 x+5 \log (x))} \left (7 x+8 x^2\right )\right )}{x^2} \, dx=\frac {e^{-x+\left (e^{3-x+5 \left (x+\frac {\log (x)}{2}\right )}+x\right )^2}}{x}-x+x^2 \]

[Out]

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

Rubi [F]

\[ \int \frac {-x^2+2 x^3+e^{-x+2 e^{\frac {1}{2} (6+8 x+5 \log (x))} x+x^2+e^{6+8 x} x^5} \left (-1-x+2 x^2+e^{6+8 x} x^5 (5+8 x)+e^{\frac {1}{2} (6+8 x+5 \log (x))} \left (7 x+8 x^2\right )\right )}{x^2} \, dx=\int \frac {-x^2+2 x^3+\exp \left (-x+2 e^{\frac {1}{2} (6+8 x+5 \log (x))} x+x^2+e^{6+8 x} x^5\right ) \left (-1-x+2 x^2+e^{6+8 x} x^5 (5+8 x)+e^{\frac {1}{2} (6+8 x+5 \log (x))} \left (7 x+8 x^2\right )\right )}{x^2} \, dx \]

[In]

Int[(-x^2 + 2*x^3 + E^(-x + 2*E^((6 + 8*x + 5*Log[x])/2)*x + x^2 + E^(6 + 8*x)*x^5)*(-1 - x + 2*x^2 + E^(6 + 8
*x)*x^5*(5 + 8*x) + E^((6 + 8*x + 5*Log[x])/2)*(7*x + 8*x^2)))/x^2,x]

[Out]

-x + x^2 - 2*Defer[Subst][Defer[Int][E^(-x^2 + (x^2 + E^(3 + 4*x^2)*x^5)^2)/x^3, x], x, Sqrt[x]] - 2*Defer[Sub
st][Defer[Int][E^(-x^2 + (x^2 + E^(3 + 4*x^2)*x^5)^2)/x, x], x, Sqrt[x]] + 4*Defer[Subst][Defer[Int][E^(-x^2 +
(x^2 + E^(3 + 4*x^2)*x^5)^2)*x, x], x, Sqrt[x]] + 14*Defer[Subst][Defer[Int][E^(3 + 3*x^2 + x^4 + 2*E^(3 + 4*
x^2)*x^7 + E^(6 + 8*x^2)*x^10)*x^4, x], x, Sqrt[x]] + 16*Defer[Subst][Defer[Int][E^(3 + 3*x^2 + x^4 + 2*E^(3 +
4*x^2)*x^7 + E^(6 + 8*x^2)*x^10)*x^6, x], x, Sqrt[x]] + 10*Defer[Subst][Defer[Int][E^(6 + 7*x^2 + x^4 + 2*E^(
3 + 4*x^2)*x^7 + E^(6 + 8*x^2)*x^10)*x^7, x], x, Sqrt[x]] + 16*Defer[Subst][Defer[Int][E^(6 + 7*x^2 + x^4 + 2*
E^(3 + 4*x^2)*x^7 + E^(6 + 8*x^2)*x^10)*x^9, x], x, Sqrt[x]]

Rubi steps \begin{align*} \text {integral}& = \int \left (\exp \left (3+3 x+x^2+2 e^{3+4 x} x^{7/2}+e^{6+8 x} x^5\right ) x^{3/2} (7+8 x)+\frac {e^{-x} \left (-e^{\left (x+e^{3+4 x} x^{5/2}\right )^2}-e^{\left (x+e^{3+4 x} x^{5/2}\right )^2} x-e^x x^2+2 e^{\left (x+e^{3+4 x} x^{5/2}\right )^2} x^2+2 e^x x^3+5 \exp \left (6+8 x+x^2+2 e^{3+4 x} x^{7/2}+e^{6+8 x} x^5\right ) x^5+8 \exp \left (6+8 x+x^2+2 e^{3+4 x} x^{7/2}+e^{6+8 x} x^5\right ) x^6\right )}{x^2}\right ) \, dx \\ & = \int \exp \left (3+3 x+x^2+2 e^{3+4 x} x^{7/2}+e^{6+8 x} x^5\right ) x^{3/2} (7+8 x) \, dx+\int \frac {e^{-x} \left (-e^{\left (x+e^{3+4 x} x^{5/2}\right )^2}-e^{\left (x+e^{3+4 x} x^{5/2}\right )^2} x-e^x x^2+2 e^{\left (x+e^{3+4 x} x^{5/2}\right )^2} x^2+2 e^x x^3+5 \exp \left (6+8 x+x^2+2 e^{3+4 x} x^{7/2}+e^{6+8 x} x^5\right ) x^5+8 \exp \left (6+8 x+x^2+2 e^{3+4 x} x^{7/2}+e^{6+8 x} x^5\right ) x^6\right )}{x^2} \, dx \\ & = 2 \text {Subst}\left (\int \exp \left (3+3 x^2+x^4+2 e^{3+4 x^2} x^7+e^{6+8 x^2} x^{10}\right ) x^4 \left (7+8 x^2\right ) \, dx,x,\sqrt {x}\right )+\int \left (-1+2 x+\exp \left (6+7 x+x^2+2 e^{3+4 x} x^{7/2}+e^{6+8 x} x^5\right ) x^3 (5+8 x)+\frac {e^{-x+\left (x+e^{3+4 x} x^{5/2}\right )^2} \left (-1-x+2 x^2\right )}{x^2}\right ) \, dx \\ & = -x+x^2+2 \text {Subst}\left (\int \left (7 \exp \left (3+3 x^2+x^4+2 e^{3+4 x^2} x^7+e^{6+8 x^2} x^{10}\right ) x^4+8 \exp \left (3+3 x^2+x^4+2 e^{3+4 x^2} x^7+e^{6+8 x^2} x^{10}\right ) x^6\right ) \, dx,x,\sqrt {x}\right )+\int \exp \left (6+7 x+x^2+2 e^{3+4 x} x^{7/2}+e^{6+8 x} x^5\right ) x^3 (5+8 x) \, dx+\int \frac {e^{-x+\left (x+e^{3+4 x} x^{5/2}\right )^2} \left (-1-x+2 x^2\right )}{x^2} \, dx \\ & = -x+x^2+2 \text {Subst}\left (\int \exp \left (6+7 x^2+x^4+2 e^{3+4 x^2} x^7+e^{6+8 x^2} x^{10}\right ) x^7 \left (5+8 x^2\right ) \, dx,x,\sqrt {x}\right )+2 \text {Subst}\left (\int \frac {e^{-x^2+\left (x^2+e^{3+4 x^2} x^5\right )^2} \left (-1-x^2+2 x^4\right )}{x^3} \, dx,x,\sqrt {x}\right )+14 \text {Subst}\left (\int \exp \left (3+3 x^2+x^4+2 e^{3+4 x^2} x^7+e^{6+8 x^2} x^{10}\right ) x^4 \, dx,x,\sqrt {x}\right )+16 \text {Subst}\left (\int \exp \left (3+3 x^2+x^4+2 e^{3+4 x^2} x^7+e^{6+8 x^2} x^{10}\right ) x^6 \, dx,x,\sqrt {x}\right ) \\ & = -x+x^2+2 \text {Subst}\left (\int \left (-\frac {e^{-x^2+\left (x^2+e^{3+4 x^2} x^5\right )^2}}{x^3}-\frac {e^{-x^2+\left (x^2+e^{3+4 x^2} x^5\right )^2}}{x}+2 e^{-x^2+\left (x^2+e^{3+4 x^2} x^5\right )^2} x\right ) \, dx,x,\sqrt {x}\right )+2 \text {Subst}\left (\int \left (5 \exp \left (6+7 x^2+x^4+2 e^{3+4 x^2} x^7+e^{6+8 x^2} x^{10}\right ) x^7+8 \exp \left (6+7 x^2+x^4+2 e^{3+4 x^2} x^7+e^{6+8 x^2} x^{10}\right ) x^9\right ) \, dx,x,\sqrt {x}\right )+14 \text {Subst}\left (\int \exp \left (3+3 x^2+x^4+2 e^{3+4 x^2} x^7+e^{6+8 x^2} x^{10}\right ) x^4 \, dx,x,\sqrt {x}\right )+16 \text {Subst}\left (\int \exp \left (3+3 x^2+x^4+2 e^{3+4 x^2} x^7+e^{6+8 x^2} x^{10}\right ) x^6 \, dx,x,\sqrt {x}\right ) \\ & = -x+x^2-2 \text {Subst}\left (\int \frac {e^{-x^2+\left (x^2+e^{3+4 x^2} x^5\right )^2}}{x^3} \, dx,x,\sqrt {x}\right )-2 \text {Subst}\left (\int \frac {e^{-x^2+\left (x^2+e^{3+4 x^2} x^5\right )^2}}{x} \, dx,x,\sqrt {x}\right )+4 \text {Subst}\left (\int e^{-x^2+\left (x^2+e^{3+4 x^2} x^5\right )^2} x \, dx,x,\sqrt {x}\right )+10 \text {Subst}\left (\int \exp \left (6+7 x^2+x^4+2 e^{3+4 x^2} x^7+e^{6+8 x^2} x^{10}\right ) x^7 \, dx,x,\sqrt {x}\right )+14 \text {Subst}\left (\int \exp \left (3+3 x^2+x^4+2 e^{3+4 x^2} x^7+e^{6+8 x^2} x^{10}\right ) x^4 \, dx,x,\sqrt {x}\right )+16 \text {Subst}\left (\int \exp \left (3+3 x^2+x^4+2 e^{3+4 x^2} x^7+e^{6+8 x^2} x^{10}\right ) x^6 \, dx,x,\sqrt {x}\right )+16 \text {Subst}\left (\int \exp \left (6+7 x^2+x^4+2 e^{3+4 x^2} x^7+e^{6+8 x^2} x^{10}\right ) x^9 \, dx,x,\sqrt {x}\right ) \\ \end{align*}

Mathematica [A] (verified)

Time = 0.17 (sec) , antiderivative size = 45, normalized size of antiderivative = 1.18 \[ \int \frac {-x^2+2 x^3+e^{-x+2 e^{\frac {1}{2} (6+8 x+5 \log (x))} x+x^2+e^{6+8 x} x^5} \left (-1-x+2 x^2+e^{6+8 x} x^5 (5+8 x)+e^{\frac {1}{2} (6+8 x+5 \log (x))} \left (7 x+8 x^2\right )\right )}{x^2} \, dx=\frac {e^{-x+x^2+2 e^{3+4 x} x^{7/2}+e^{6+8 x} x^5}}{x}-x+x^2 \]

[In]

Integrate[(-x^2 + 2*x^3 + E^(-x + 2*E^((6 + 8*x + 5*Log[x])/2)*x + x^2 + E^(6 + 8*x)*x^5)*(-1 - x + 2*x^2 + E^
(6 + 8*x)*x^5*(5 + 8*x) + E^((6 + 8*x + 5*Log[x])/2)*(7*x + 8*x^2)))/x^2,x]

[Out]

E^(-x + x^2 + 2*E^(3 + 4*x)*x^(7/2) + E^(6 + 8*x)*x^5)/x - x + x^2

Maple [A] (verified)

Time = 0.39 (sec) , antiderivative size = 41, normalized size of antiderivative = 1.08

method	result	size
risch	\(x^{2}-x +\frac {{\mathrm e}^{x^{5} {\mathrm e}^{8 x +6}+2 x^{\frac {7}{2}} {\mathrm e}^{3+4 x}+x^{2}-x}}{x}\)	\(41\)
parallelrisch	\(-\frac {-x^{3}+x^{2}-{\mathrm e}^{{\mathrm e}^{5 \ln \left (x \right )+8 x +6}+2 x \,{\mathrm e}^{\frac {5 \ln \left (x \right )}{2}+4 x +3}+x^{2}-x}}{x}\)	\(50\)

[In]

int((((8*x+5)*exp(5/2*ln(x)+4*x+3)^2+(8*x^2+7*x)*exp(5/2*ln(x)+4*x+3)+2*x^2-x-1)*exp(exp(5/2*ln(x)+4*x+3)^2+2*
x*exp(5/2*ln(x)+4*x+3)+x^2-x)+2*x^3-x^2)/x^2,x,method=_RETURNVERBOSE)

[Out]

x^2-x+1/x*exp(x^5*exp(8*x+6)+2*x^(7/2)*exp(3+4*x)+x^2-x)

Fricas [A] (verification not implemented)

none

Time = 0.26 (sec) , antiderivative size = 44, normalized size of antiderivative = 1.16 \[ \int \frac {-x^2+2 x^3+e^{-x+2 e^{\frac {1}{2} (6+8 x+5 \log (x))} x+x^2+e^{6+8 x} x^5} \left (-1-x+2 x^2+e^{6+8 x} x^5 (5+8 x)+e^{\frac {1}{2} (6+8 x+5 \log (x))} \left (7 x+8 x^2\right )\right )}{x^2} \, dx=\frac {x^{3} - x^{2} + e^{\left (x^{2} + 2 \, x e^{\left (4 \, x + \frac {5}{2} \, \log \left (x\right ) + 3\right )} - x + e^{\left (8 \, x + 5 \, \log \left (x\right ) + 6\right )}\right )}}{x} \]

[In]

integrate((((8*x+5)*exp(5/2*log(x)+4*x+3)^2+(8*x^2+7*x)*exp(5/2*log(x)+4*x+3)+2*x^2-x-1)*exp(exp(5/2*log(x)+4*
x+3)^2+2*x*exp(5/2*log(x)+4*x+3)+x^2-x)+2*x^3-x^2)/x^2,x, algorithm="fricas")

[Out]

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

Sympy [F(-1)]

Timed out. \[ \int \frac {-x^2+2 x^3+e^{-x+2 e^{\frac {1}{2} (6+8 x+5 \log (x))} x+x^2+e^{6+8 x} x^5} \left (-1-x+2 x^2+e^{6+8 x} x^5 (5+8 x)+e^{\frac {1}{2} (6+8 x+5 \log (x))} \left (7 x+8 x^2\right )\right )}{x^2} \, dx=\text {Timed out} \]

[In]

integrate((((8*x+5)*exp(5/2*ln(x)+4*x+3)**2+(8*x**2+7*x)*exp(5/2*ln(x)+4*x+3)+2*x**2-x-1)*exp(exp(5/2*ln(x)+4*
x+3)**2+2*x*exp(5/2*ln(x)+4*x+3)+x**2-x)+2*x**3-x**2)/x**2,x)

[Out]

Timed out

Maxima [A] (verification not implemented)

none

Time = 0.32 (sec) , antiderivative size = 40, normalized size of antiderivative = 1.05 \[ \int \frac {-x^2+2 x^3+e^{-x+2 e^{\frac {1}{2} (6+8 x+5 \log (x))} x+x^2+e^{6+8 x} x^5} \left (-1-x+2 x^2+e^{6+8 x} x^5 (5+8 x)+e^{\frac {1}{2} (6+8 x+5 \log (x))} \left (7 x+8 x^2\right )\right )}{x^2} \, dx=x^{2} - x + \frac {e^{\left (x^{5} e^{\left (8 \, x + 6\right )} + 2 \, x^{\frac {7}{2}} e^{\left (4 \, x + 3\right )} + x^{2} - x\right )}}{x} \]

[In]

integrate((((8*x+5)*exp(5/2*log(x)+4*x+3)^2+(8*x^2+7*x)*exp(5/2*log(x)+4*x+3)+2*x^2-x-1)*exp(exp(5/2*log(x)+4*
x+3)^2+2*x*exp(5/2*log(x)+4*x+3)+x^2-x)+2*x^3-x^2)/x^2,x, algorithm="maxima")

[Out]

x^2 - x + e^(x^5*e^(8*x + 6) + 2*x^(7/2)*e^(4*x + 3) + x^2 - x)/x

Giac [F]

\[ \int \frac {-x^2+2 x^3+e^{-x+2 e^{\frac {1}{2} (6+8 x+5 \log (x))} x+x^2+e^{6+8 x} x^5} \left (-1-x+2 x^2+e^{6+8 x} x^5 (5+8 x)+e^{\frac {1}{2} (6+8 x+5 \log (x))} \left (7 x+8 x^2\right )\right )}{x^2} \, dx=\int { \frac {2 \, x^{3} - x^{2} + {\left (2 \, x^{2} + {\left (8 \, x + 5\right )} e^{\left (8 \, x + 5 \, \log \left (x\right ) + 6\right )} + {\left (8 \, x^{2} + 7 \, x\right )} e^{\left (4 \, x + \frac {5}{2} \, \log \left (x\right ) + 3\right )} - x - 1\right )} e^{\left (x^{2} + 2 \, x e^{\left (4 \, x + \frac {5}{2} \, \log \left (x\right ) + 3\right )} - x + e^{\left (8 \, x + 5 \, \log \left (x\right ) + 6\right )}\right )}}{x^{2}} \,d x } \]

[In]

integrate((((8*x+5)*exp(5/2*log(x)+4*x+3)^2+(8*x^2+7*x)*exp(5/2*log(x)+4*x+3)+2*x^2-x-1)*exp(exp(5/2*log(x)+4*
x+3)^2+2*x*exp(5/2*log(x)+4*x+3)+x^2-x)+2*x^3-x^2)/x^2,x, algorithm="giac")

[Out]

integrate((2*x^3 - x^2 + (2*x^2 + (8*x + 5)*e^(8*x + 5*log(x) + 6) + (8*x^2 + 7*x)*e^(4*x + 5/2*log(x) + 3) -
x - 1)*e^(x^2 + 2*x*e^(4*x + 5/2*log(x) + 3) - x + e^(8*x + 5*log(x) + 6)))/x^2, x)

Mupad [B] (verification not implemented)

Time = 8.98 (sec) , antiderivative size = 42, normalized size of antiderivative = 1.11 \[ \int \frac {-x^2+2 x^3+e^{-x+2 e^{\frac {1}{2} (6+8 x+5 \log (x))} x+x^2+e^{6+8 x} x^5} \left (-1-x+2 x^2+e^{6+8 x} x^5 (5+8 x)+e^{\frac {1}{2} (6+8 x+5 \log (x))} \left (7 x+8 x^2\right )\right )}{x^2} \, dx=x^2-x+\frac {{\mathrm {e}}^{-x}\,{\mathrm {e}}^{x^2}\,{\mathrm {e}}^{2\,x^{7/2}\,{\mathrm {e}}^{4\,x}\,{\mathrm {e}}^3}\,{\mathrm {e}}^{x^5\,{\mathrm {e}}^{8\,x}\,{\mathrm {e}}^6}}{x} \]

[In]

int((2*x^3 - x^2 + exp(exp(8*x + 5*log(x) + 6) - x + 2*x*exp(4*x + (5*log(x))/2 + 3) + x^2)*(exp(8*x + 5*log(x
) + 6)*(8*x + 5) - x + exp(4*x + (5*log(x))/2 + 3)*(7*x + 8*x^2) + 2*x^2 - 1))/x^2,x)

[Out]

x^2 - x + (exp(-x)*exp(x^2)*exp(2*x^(7/2)*exp(4*x)*exp(3))*exp(x^5*exp(8*x)*exp(6)))/x