\(\int \frac {x^2-48 x^3+e^{\frac {2 (5+x+x^3)}{x}} (10-4 x^3)+e^{\frac {5+x+x^3}{x}} (50 x-10 x^2-20 x^4)}{e^{\frac {2 (5+x+x^3)}{x}} x^2-x^3+10 e^{\frac {5+x+x^3}{x}} x^3+24 x^4} \, dx\) [7997]

Optimal result

Integrand size = 105, antiderivative size = 31 \[ \int \frac {x^2-48 x^3+e^{\frac {2 \left (5+x+x^3\right )}{x}} \left (10-4 x^3\right )+e^{\frac {5+x+x^3}{x}} \left (50 x-10 x^2-20 x^4\right )}{e^{\frac {2 \left (5+x+x^3\right )}{x}} x^2-x^3+10 e^{\frac {5+x+x^3}{x}} x^3+24 x^4} \, dx=\log \left (\frac {3}{x+x^2-\left (e^{x^2+\frac {5+x}{x}}+5 x\right )^2}\right ) \]

[Out]

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

Rubi [F]

\[ \int \frac {x^2-48 x^3+e^{\frac {2 \left (5+x+x^3\right )}{x}} \left (10-4 x^3\right )+e^{\frac {5+x+x^3}{x}} \left (50 x-10 x^2-20 x^4\right )}{e^{\frac {2 \left (5+x+x^3\right )}{x}} x^2-x^3+10 e^{\frac {5+x+x^3}{x}} x^3+24 x^4} \, dx=\int \frac {x^2-48 x^3+e^{\frac {2 \left (5+x+x^3\right )}{x}} \left (10-4 x^3\right )+e^{\frac {5+x+x^3}{x}} \left (50 x-10 x^2-20 x^4\right )}{e^{\frac {2 \left (5+x+x^3\right )}{x}} x^2-x^3+10 e^{\frac {5+x+x^3}{x}} x^3+24 x^4} \, dx \]

[In]

Int[(x^2 - 48*x^3 + E^((2*(5 + x + x^3))/x)*(10 - 4*x^3) + E^((5 + x + x^3)/x)*(50*x - 10*x^2 - 20*x^4))/(E^((
2*(5 + x + x^3))/x)*x^2 - x^3 + 10*E^((5 + x + x^3)/x)*x^3 + 24*x^4),x]

[Out]

-10/x - 2*x^2 - 239*Defer[Int][(E^(2 + 10/x + 2*x^2) - x + 10*E^(1 + 5/x + x^2)*x + 24*x^2)^(-1), x] - 10*Defe
r[Int][E^(1 + 5/x + x^2)/(E^(2 + 10/x + 2*x^2) - x + 10*E^(1 + 5/x + x^2)*x + 24*x^2), x] + 10*Defer[Int][1/(x
*(E^(2 + 10/x + 2*x^2) - x + 10*E^(1 + 5/x + x^2)*x + 24*x^2)), x] - 50*Defer[Int][E^(1 + 5/x + x^2)/(x*(E^(2
+ 10/x + 2*x^2) - x + 10*E^(1 + 5/x + x^2)*x + 24*x^2)), x] - 48*Defer[Int][x/(E^(2 + 10/x + 2*x^2) - x + 10*E
^(1 + 5/x + x^2)*x + 24*x^2), x] - 4*Defer[Int][x^2/(E^(2 + 10/x + 2*x^2) - x + 10*E^(1 + 5/x + x^2)*x + 24*x^
2), x] + 20*Defer[Int][(E^(1 + 5/x + x^2)*x^2)/(E^(2 + 10/x + 2*x^2) - x + 10*E^(1 + 5/x + x^2)*x + 24*x^2), x
] + 96*Defer[Int][x^3/(E^(2 + 10/x + 2*x^2) - x + 10*E^(1 + 5/x + x^2)*x + 24*x^2), x]

Rubi steps \begin{align*} \text {integral}& = \int \left (-\frac {2 \left (-5+2 x^3\right )}{x^2}+\frac {10-50 e^{1+\frac {5}{x}+x^2}-239 x-10 e^{1+\frac {5}{x}+x^2} x-48 x^2-4 x^3+20 e^{1+\frac {5}{x}+x^2} x^3+96 x^4}{x \left (e^{2+\frac {10}{x}+2 x^2}-x+10 e^{1+\frac {5}{x}+x^2} x+24 x^2\right )}\right ) \, dx \\ & = -\left (2 \int \frac {-5+2 x^3}{x^2} \, dx\right )+\int \frac {10-50 e^{1+\frac {5}{x}+x^2}-239 x-10 e^{1+\frac {5}{x}+x^2} x-48 x^2-4 x^3+20 e^{1+\frac {5}{x}+x^2} x^3+96 x^4}{x \left (e^{2+\frac {10}{x}+2 x^2}-x+10 e^{1+\frac {5}{x}+x^2} x+24 x^2\right )} \, dx \\ & = -\left (2 \int \left (-\frac {5}{x^2}+2 x\right ) \, dx\right )+\int \left (-\frac {239}{e^{2+\frac {10}{x}+2 x^2}-x+10 e^{1+\frac {5}{x}+x^2} x+24 x^2}-\frac {10 e^{1+\frac {5}{x}+x^2}}{e^{2+\frac {10}{x}+2 x^2}-x+10 e^{1+\frac {5}{x}+x^2} x+24 x^2}+\frac {10}{x \left (e^{2+\frac {10}{x}+2 x^2}-x+10 e^{1+\frac {5}{x}+x^2} x+24 x^2\right )}-\frac {50 e^{1+\frac {5}{x}+x^2}}{x \left (e^{2+\frac {10}{x}+2 x^2}-x+10 e^{1+\frac {5}{x}+x^2} x+24 x^2\right )}-\frac {48 x}{e^{2+\frac {10}{x}+2 x^2}-x+10 e^{1+\frac {5}{x}+x^2} x+24 x^2}-\frac {4 x^2}{e^{2+\frac {10}{x}+2 x^2}-x+10 e^{1+\frac {5}{x}+x^2} x+24 x^2}+\frac {20 e^{1+\frac {5}{x}+x^2} x^2}{e^{2+\frac {10}{x}+2 x^2}-x+10 e^{1+\frac {5}{x}+x^2} x+24 x^2}+\frac {96 x^3}{e^{2+\frac {10}{x}+2 x^2}-x+10 e^{1+\frac {5}{x}+x^2} x+24 x^2}\right ) \, dx \\ & = -\frac {10}{x}-2 x^2-4 \int \frac {x^2}{e^{2+\frac {10}{x}+2 x^2}-x+10 e^{1+\frac {5}{x}+x^2} x+24 x^2} \, dx-10 \int \frac {e^{1+\frac {5}{x}+x^2}}{e^{2+\frac {10}{x}+2 x^2}-x+10 e^{1+\frac {5}{x}+x^2} x+24 x^2} \, dx+10 \int \frac {1}{x \left (e^{2+\frac {10}{x}+2 x^2}-x+10 e^{1+\frac {5}{x}+x^2} x+24 x^2\right )} \, dx+20 \int \frac {e^{1+\frac {5}{x}+x^2} x^2}{e^{2+\frac {10}{x}+2 x^2}-x+10 e^{1+\frac {5}{x}+x^2} x+24 x^2} \, dx-48 \int \frac {x}{e^{2+\frac {10}{x}+2 x^2}-x+10 e^{1+\frac {5}{x}+x^2} x+24 x^2} \, dx-50 \int \frac {e^{1+\frac {5}{x}+x^2}}{x \left (e^{2+\frac {10}{x}+2 x^2}-x+10 e^{1+\frac {5}{x}+x^2} x+24 x^2\right )} \, dx+96 \int \frac {x^3}{e^{2+\frac {10}{x}+2 x^2}-x+10 e^{1+\frac {5}{x}+x^2} x+24 x^2} \, dx-239 \int \frac {1}{e^{2+\frac {10}{x}+2 x^2}-x+10 e^{1+\frac {5}{x}+x^2} x+24 x^2} \, dx \\ \end{align*}

Mathematica [F]

\[ \int \frac {x^2-48 x^3+e^{\frac {2 \left (5+x+x^3\right )}{x}} \left (10-4 x^3\right )+e^{\frac {5+x+x^3}{x}} \left (50 x-10 x^2-20 x^4\right )}{e^{\frac {2 \left (5+x+x^3\right )}{x}} x^2-x^3+10 e^{\frac {5+x+x^3}{x}} x^3+24 x^4} \, dx=\int \frac {x^2-48 x^3+e^{\frac {2 \left (5+x+x^3\right )}{x}} \left (10-4 x^3\right )+e^{\frac {5+x+x^3}{x}} \left (50 x-10 x^2-20 x^4\right )}{e^{\frac {2 \left (5+x+x^3\right )}{x}} x^2-x^3+10 e^{\frac {5+x+x^3}{x}} x^3+24 x^4} \, dx \]

[In]

Integrate[(x^2 - 48*x^3 + E^((2*(5 + x + x^3))/x)*(10 - 4*x^3) + E^((5 + x + x^3)/x)*(50*x - 10*x^2 - 20*x^4))
/(E^((2*(5 + x + x^3))/x)*x^2 - x^3 + 10*E^((5 + x + x^3)/x)*x^3 + 24*x^4),x]

[Out]

Integrate[(x^2 - 48*x^3 + E^((2*(5 + x + x^3))/x)*(10 - 4*x^3) + E^((5 + x + x^3)/x)*(50*x - 10*x^2 - 20*x^4))
/(E^((2*(5 + x + x^3))/x)*x^2 - x^3 + 10*E^((5 + x + x^3)/x)*x^3 + 24*x^4), x]

Maple [A] (verified)

Time = 0.29 (sec) , antiderivative size = 40, normalized size of antiderivative = 1.29

method	result	size
norman	\(-\ln \left (10 x \,{\mathrm e}^{\frac {x^{3}+x +5}{x}}+24 x^{2}+{\mathrm e}^{\frac {2 x^{3}+2 x +10}{x}}-x \right )\)	\(40\)
parallelrisch	\(-\ln \left (x^{2}+\frac {5 x \,{\mathrm e}^{\frac {x^{3}+x +5}{x}}}{12}+\frac {{\mathrm e}^{\frac {2 x^{3}+2 x +10}{x}}}{24}-\frac {x}{24}\right )\)	\(40\)
risch	\(-2 x^{2}-\frac {10}{x}+\frac {2 x^{3}+2 x +10}{x}-\ln \left (10 x \,{\mathrm e}^{\frac {x^{3}+x +5}{x}}+24 x^{2}+{\mathrm e}^{\frac {2 x^{3}+2 x +10}{x}}-x \right )\)	\(61\)

[In]

int(((-4*x^3+10)*exp((x^3+x+5)/x)^2+(-20*x^4-10*x^2+50*x)*exp((x^3+x+5)/x)-48*x^3+x^2)/(x^2*exp((x^3+x+5)/x)^2
+10*x^3*exp((x^3+x+5)/x)+24*x^4-x^3),x,method=_RETURNVERBOSE)

[Out]

-ln(exp((x^3+x+5)/x)^2+10*x*exp((x^3+x+5)/x)+24*x^2-x)

Fricas [A] (verification not implemented)

none

Time = 0.26 (sec) , antiderivative size = 38, normalized size of antiderivative = 1.23 \[ \int \frac {x^2-48 x^3+e^{\frac {2 \left (5+x+x^3\right )}{x}} \left (10-4 x^3\right )+e^{\frac {5+x+x^3}{x}} \left (50 x-10 x^2-20 x^4\right )}{e^{\frac {2 \left (5+x+x^3\right )}{x}} x^2-x^3+10 e^{\frac {5+x+x^3}{x}} x^3+24 x^4} \, dx=-\log \left (24 \, x^{2} + 10 \, x e^{\left (\frac {x^{3} + x + 5}{x}\right )} - x + e^{\left (\frac {2 \, {\left (x^{3} + x + 5\right )}}{x}\right )}\right ) \]

[In]

integrate(((-4*x^3+10)*exp((x^3+x+5)/x)^2+(-20*x^4-10*x^2+50*x)*exp((x^3+x+5)/x)-48*x^3+x^2)/(x^2*exp((x^3+x+5
)/x)^2+10*x^3*exp((x^3+x+5)/x)+24*x^4-x^3),x, algorithm="fricas")

[Out]

-log(24*x^2 + 10*x*e^((x^3 + x + 5)/x) - x + e^(2*(x^3 + x + 5)/x))

Sympy [A] (verification not implemented)

Time = 0.15 (sec) , antiderivative size = 36, normalized size of antiderivative = 1.16 \[ \int \frac {x^2-48 x^3+e^{\frac {2 \left (5+x+x^3\right )}{x}} \left (10-4 x^3\right )+e^{\frac {5+x+x^3}{x}} \left (50 x-10 x^2-20 x^4\right )}{e^{\frac {2 \left (5+x+x^3\right )}{x}} x^2-x^3+10 e^{\frac {5+x+x^3}{x}} x^3+24 x^4} \, dx=- \log {\left (24 x^{2} + 10 x e^{\frac {x^{3} + x + 5}{x}} - x + e^{\frac {2 \left (x^{3} + x + 5\right )}{x}} \right )} \]

[In]

integrate(((-4*x**3+10)*exp((x**3+x+5)/x)**2+(-20*x**4-10*x**2+50*x)*exp((x**3+x+5)/x)-48*x**3+x**2)/(x**2*exp
((x**3+x+5)/x)**2+10*x**3*exp((x**3+x+5)/x)+24*x**4-x**3),x)

[Out]

-log(24*x**2 + 10*x*exp((x**3 + x + 5)/x) - x + exp(2*(x**3 + x + 5)/x))

Maxima [A] (verification not implemented)

none

Time = 0.20 (sec) , antiderivative size = 54, normalized size of antiderivative = 1.74 \[ \int \frac {x^2-48 x^3+e^{\frac {2 \left (5+x+x^3\right )}{x}} \left (10-4 x^3\right )+e^{\frac {5+x+x^3}{x}} \left (50 x-10 x^2-20 x^4\right )}{e^{\frac {2 \left (5+x+x^3\right )}{x}} x^2-x^3+10 e^{\frac {5+x+x^3}{x}} x^3+24 x^4} \, dx=-\frac {10}{x} - \log \left ({\left (24 \, x^{2} + 10 \, x e^{\left (x^{2} + \frac {5}{x} + 1\right )} - x + e^{\left (2 \, x^{2} + \frac {10}{x} + 2\right )}\right )} e^{\left (-\frac {10}{x} - 2\right )}\right ) \]

[In]

integrate(((-4*x^3+10)*exp((x^3+x+5)/x)^2+(-20*x^4-10*x^2+50*x)*exp((x^3+x+5)/x)-48*x^3+x^2)/(x^2*exp((x^3+x+5
)/x)^2+10*x^3*exp((x^3+x+5)/x)+24*x^4-x^3),x, algorithm="maxima")

[Out]

-10/x - log((24*x^2 + 10*x*e^(x^2 + 5/x + 1) - x + e^(2*x^2 + 10/x + 2))*e^(-10/x - 2))

Giac [A] (verification not implemented)

none

Time = 0.35 (sec) , antiderivative size = 38, normalized size of antiderivative = 1.23 \[ \int \frac {x^2-48 x^3+e^{\frac {2 \left (5+x+x^3\right )}{x}} \left (10-4 x^3\right )+e^{\frac {5+x+x^3}{x}} \left (50 x-10 x^2-20 x^4\right )}{e^{\frac {2 \left (5+x+x^3\right )}{x}} x^2-x^3+10 e^{\frac {5+x+x^3}{x}} x^3+24 x^4} \, dx=-\log \left (24 \, x^{2} + 10 \, x e^{\left (\frac {x^{3} + x + 5}{x}\right )} - x + e^{\left (\frac {2 \, {\left (x^{3} + x + 5\right )}}{x}\right )}\right ) \]

[In]

integrate(((-4*x^3+10)*exp((x^3+x+5)/x)^2+(-20*x^4-10*x^2+50*x)*exp((x^3+x+5)/x)-48*x^3+x^2)/(x^2*exp((x^3+x+5
)/x)^2+10*x^3*exp((x^3+x+5)/x)+24*x^4-x^3),x, algorithm="giac")

[Out]

-log(24*x^2 + 10*x*e^((x^3 + x + 5)/x) - x + e^(2*(x^3 + x + 5)/x))

Mupad [F(-1)]

Timed out. \[ \int \frac {x^2-48 x^3+e^{\frac {2 \left (5+x+x^3\right )}{x}} \left (10-4 x^3\right )+e^{\frac {5+x+x^3}{x}} \left (50 x-10 x^2-20 x^4\right )}{e^{\frac {2 \left (5+x+x^3\right )}{x}} x^2-x^3+10 e^{\frac {5+x+x^3}{x}} x^3+24 x^4} \, dx=\int -\frac {{\mathrm {e}}^{\frac {2\,\left (x^3+x+5\right )}{x}}\,\left (4\,x^3-10\right )+{\mathrm {e}}^{\frac {x^3+x+5}{x}}\,\left (20\,x^4+10\,x^2-50\,x\right )-x^2+48\,x^3}{10\,x^3\,{\mathrm {e}}^{\frac {x^3+x+5}{x}}+x^2\,{\mathrm {e}}^{\frac {2\,\left (x^3+x+5\right )}{x}}-x^3+24\,x^4} \,d x \]

[In]

int(-(exp((2*(x + x^3 + 5))/x)*(4*x^3 - 10) + exp((x + x^3 + 5)/x)*(10*x^2 - 50*x + 20*x^4) - x^2 + 48*x^3)/(1
0*x^3*exp((x + x^3 + 5)/x) + x^2*exp((2*(x + x^3 + 5))/x) - x^3 + 24*x^4),x)

[Out]

int(-(exp((2*(x + x^3 + 5))/x)*(4*x^3 - 10) + exp((x + x^3 + 5)/x)*(10*x^2 - 50*x + 20*x^4) - x^2 + 48*x^3)/(1
0*x^3*exp((x + x^3 + 5)/x) + x^2*exp((2*(x + x^3 + 5))/x) - x^3 + 24*x^4), x)