\(\int \frac {-50 x^6-800 x^7-150 x^8+e^{\frac {3+2 e^4 x}{x}} (-32 x^3-6 x^4)+e^{\frac {3+2 e^4 x}{2 x}} (-3 x^3+6 x^4+320 x^5+60 x^6)}{2 x^2+80 x^3+820 x^4+400 x^5+50 x^6+e^{\frac {3+2 e^4 x}{x}} (32+16 x+2 x^2)+e^{\frac {3+2 e^4 x}{2 x}} (-16 x-324 x^2-160 x^3-20 x^4)} \, dx\) [5360]

Optimal result

Integrand size = 172, antiderivative size = 37 \[ \int \frac {-50 x^6-800 x^7-150 x^8+e^{\frac {3+2 e^4 x}{x}} \left (-32 x^3-6 x^4\right )+e^{\frac {3+2 e^4 x}{2 x}} \left (-3 x^3+6 x^4+320 x^5+60 x^6\right )}{2 x^2+80 x^3+820 x^4+400 x^5+50 x^6+e^{\frac {3+2 e^4 x}{x}} \left (32+16 x+2 x^2\right )+e^{\frac {3+2 e^4 x}{2 x}} \left (-16 x-324 x^2-160 x^3-20 x^4\right )} \, dx=3-\frac {x^4}{4+x+\frac {x}{-e^{e^4+\frac {3}{2 x}}+5 x^2}} \]

[Out]

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

Rubi [F]

\[ \int \frac {-50 x^6-800 x^7-150 x^8+e^{\frac {3+2 e^4 x}{x}} \left (-32 x^3-6 x^4\right )+e^{\frac {3+2 e^4 x}{2 x}} \left (-3 x^3+6 x^4+320 x^5+60 x^6\right )}{2 x^2+80 x^3+820 x^4+400 x^5+50 x^6+e^{\frac {3+2 e^4 x}{x}} \left (32+16 x+2 x^2\right )+e^{\frac {3+2 e^4 x}{2 x}} \left (-16 x-324 x^2-160 x^3-20 x^4\right )} \, dx=\int \frac {-50 x^6-800 x^7-150 x^8+e^{\frac {3+2 e^4 x}{x}} \left (-32 x^3-6 x^4\right )+e^{\frac {3+2 e^4 x}{2 x}} \left (-3 x^3+6 x^4+320 x^5+60 x^6\right )}{2 x^2+80 x^3+820 x^4+400 x^5+50 x^6+e^{\frac {3+2 e^4 x}{x}} \left (32+16 x+2 x^2\right )+e^{\frac {3+2 e^4 x}{2 x}} \left (-16 x-324 x^2-160 x^3-20 x^4\right )} \, dx \]

[In]

Int[(-50*x^6 - 800*x^7 - 150*x^8 + E^((3 + 2*E^4*x)/x)*(-32*x^3 - 6*x^4) + E^((3 + 2*E^4*x)/(2*x))*(-3*x^3 + 6
*x^4 + 320*x^5 + 60*x^6))/(2*x^2 + 80*x^3 + 820*x^4 + 400*x^5 + 50*x^6 + E^((3 + 2*E^4*x)/x)*(32 + 16*x + 2*x^
2) + E^((3 + 2*E^4*x)/(2*x))*(-16*x - 324*x^2 - 160*x^3 - 20*x^4)),x]

[Out]

-(x^4/(4 + x)) - 1376*Defer[Int][1/((4 + x)*(-4*E^(E^4 + 3/(2*x)) + x - E^(E^4 + 3/(2*x))*x + 20*x^2 + 5*x^3))
, x] + 1120*Defer[Int][(E^(E^4 + 3/(2*x))*(4 + x) - x*(1 + 20*x + 5*x^2))^(-2), x] - 216*Defer[Int][x/(E^(E^4
+ 3/(2*x))*(4 + x) - x*(1 + 20*x + 5*x^2))^2, x] + 38*Defer[Int][x^2/(E^(E^4 + 3/(2*x))*(4 + x) - x*(1 + 20*x
+ 5*x^2))^2, x] - (11*Defer[Int][x^3/(E^(E^4 + 3/(2*x))*(4 + x) - x*(1 + 20*x + 5*x^2))^2, x])/2 - (15*Defer[I
nt][x^5/(E^(E^4 + 3/(2*x))*(4 + x) - x*(1 + 20*x + 5*x^2))^2, x])/2 - 10*Defer[Int][x^6/(E^(E^4 + 3/(2*x))*(4
+ x) - x*(1 + 20*x + 5*x^2))^2, x] + 4096*Defer[Int][1/((4 + x)^2*(E^(E^4 + 3/(2*x))*(4 + x) - x*(1 + 20*x + 5
*x^2))^2), x] - 5504*Defer[Int][1/((4 + x)*(E^(E^4 + 3/(2*x))*(4 + x) - x*(1 + 20*x + 5*x^2))^2), x] - 216*Def
er[Int][(E^(E^4 + 3/(2*x))*(4 + x) - x*(1 + 20*x + 5*x^2))^(-1), x] - 22*Defer[Int][x/(-(E^(E^4 + 3/(2*x))*(4
+ x)) + x*(1 + 20*x + 5*x^2)), x] - (5*Defer[Int][x^2/(-(E^(E^4 + 3/(2*x))*(4 + x)) + x*(1 + 20*x + 5*x^2)), x
])/2 + 3*Defer[Int][x^3/(-(E^(E^4 + 3/(2*x))*(4 + x)) + x*(1 + 20*x + 5*x^2)), x] + 2048*Defer[Int][1/((4 + x)
^2*(-(E^(E^4 + 3/(2*x))*(4 + x)) + x*(1 + 20*x + 5*x^2))), x]

Rubi steps \begin{align*} \text {integral}& = \int \frac {x^3 \left (-e^{2 e^4+\frac {3}{x}} (32+6 x)-50 x^3 \left (1+16 x+3 x^2\right )+e^{e^4+\frac {3}{2 x}} \left (-3+6 x+320 x^2+60 x^3\right )\right )}{2 \left (e^{e^4+\frac {3}{2 x}} (4+x)-x \left (1+20 x+5 x^2\right )\right )^2} \, dx \\ & = \frac {1}{2} \int \frac {x^3 \left (-e^{2 e^4+\frac {3}{x}} (32+6 x)-50 x^3 \left (1+16 x+3 x^2\right )+e^{e^4+\frac {3}{2 x}} \left (-3+6 x+320 x^2+60 x^3\right )\right )}{\left (e^{e^4+\frac {3}{2 x}} (4+x)-x \left (1+20 x+5 x^2\right )\right )^2} \, dx \\ & = \frac {1}{2} \int \left (-\frac {2 x^3 (16+3 x)}{(4+x)^2}+\frac {x^3 \left (12+43 x+6 x^2\right )}{(4+x)^2 \left (-4 e^{e^4+\frac {3}{2 x}}+x-e^{e^4+\frac {3}{2 x}} x+20 x^2+5 x^3\right )}-\frac {x^4 \left (12+251 x+440 x^2+175 x^3+20 x^4\right )}{(4+x)^2 \left (-4 e^{e^4+\frac {3}{2 x}}+x-e^{e^4+\frac {3}{2 x}} x+20 x^2+5 x^3\right )^2}\right ) \, dx \\ & = \frac {1}{2} \int \frac {x^3 \left (12+43 x+6 x^2\right )}{(4+x)^2 \left (-4 e^{e^4+\frac {3}{2 x}}+x-e^{e^4+\frac {3}{2 x}} x+20 x^2+5 x^3\right )} \, dx-\frac {1}{2} \int \frac {x^4 \left (12+251 x+440 x^2+175 x^3+20 x^4\right )}{(4+x)^2 \left (-4 e^{e^4+\frac {3}{2 x}}+x-e^{e^4+\frac {3}{2 x}} x+20 x^2+5 x^3\right )^2} \, dx-\int \frac {x^3 (16+3 x)}{(4+x)^2} \, dx \\ & = -\frac {x^4}{4+x}-\frac {1}{2} \int \frac {x^4 \left (12+251 x+440 x^2+175 x^3+20 x^4\right )}{(4+x)^2 \left (e^{e^4+\frac {3}{2 x}} (4+x)-x \left (1+20 x+5 x^2\right )\right )^2} \, dx+\frac {1}{2} \int \left (-\frac {432}{4 e^{e^4+\frac {3}{2 x}}-x+e^{e^4+\frac {3}{2 x}} x-20 x^2-5 x^3}-\frac {44 x}{-4 e^{e^4+\frac {3}{2 x}}+x-e^{e^4+\frac {3}{2 x}} x+20 x^2+5 x^3}-\frac {5 x^2}{-4 e^{e^4+\frac {3}{2 x}}+x-e^{e^4+\frac {3}{2 x}} x+20 x^2+5 x^3}+\frac {6 x^3}{-4 e^{e^4+\frac {3}{2 x}}+x-e^{e^4+\frac {3}{2 x}} x+20 x^2+5 x^3}+\frac {4096}{(4+x)^2 \left (-4 e^{e^4+\frac {3}{2 x}}+x-e^{e^4+\frac {3}{2 x}} x+20 x^2+5 x^3\right )}-\frac {2752}{(4+x) \left (-4 e^{e^4+\frac {3}{2 x}}+x-e^{e^4+\frac {3}{2 x}} x+20 x^2+5 x^3\right )}\right ) \, dx \\ & = -\frac {x^4}{4+x}-\frac {1}{2} \int \left (-\frac {2240}{\left (4 e^{e^4+\frac {3}{2 x}}-x+e^{e^4+\frac {3}{2 x}} x-20 x^2-5 x^3\right )^2}+\frac {432 x}{\left (-4 e^{e^4+\frac {3}{2 x}}+x-e^{e^4+\frac {3}{2 x}} x+20 x^2+5 x^3\right )^2}-\frac {76 x^2}{\left (-4 e^{e^4+\frac {3}{2 x}}+x-e^{e^4+\frac {3}{2 x}} x+20 x^2+5 x^3\right )^2}+\frac {11 x^3}{\left (-4 e^{e^4+\frac {3}{2 x}}+x-e^{e^4+\frac {3}{2 x}} x+20 x^2+5 x^3\right )^2}+\frac {15 x^5}{\left (-4 e^{e^4+\frac {3}{2 x}}+x-e^{e^4+\frac {3}{2 x}} x+20 x^2+5 x^3\right )^2}+\frac {20 x^6}{\left (-4 e^{e^4+\frac {3}{2 x}}+x-e^{e^4+\frac {3}{2 x}} x+20 x^2+5 x^3\right )^2}-\frac {8192}{(4+x)^2 \left (-4 e^{e^4+\frac {3}{2 x}}+x-e^{e^4+\frac {3}{2 x}} x+20 x^2+5 x^3\right )^2}+\frac {11008}{(4+x) \left (-4 e^{e^4+\frac {3}{2 x}}+x-e^{e^4+\frac {3}{2 x}} x+20 x^2+5 x^3\right )^2}\right ) \, dx-\frac {5}{2} \int \frac {x^2}{-4 e^{e^4+\frac {3}{2 x}}+x-e^{e^4+\frac {3}{2 x}} x+20 x^2+5 x^3} \, dx+3 \int \frac {x^3}{-4 e^{e^4+\frac {3}{2 x}}+x-e^{e^4+\frac {3}{2 x}} x+20 x^2+5 x^3} \, dx-22 \int \frac {x}{-4 e^{e^4+\frac {3}{2 x}}+x-e^{e^4+\frac {3}{2 x}} x+20 x^2+5 x^3} \, dx-216 \int \frac {1}{4 e^{e^4+\frac {3}{2 x}}-x+e^{e^4+\frac {3}{2 x}} x-20 x^2-5 x^3} \, dx-1376 \int \frac {1}{(4+x) \left (-4 e^{e^4+\frac {3}{2 x}}+x-e^{e^4+\frac {3}{2 x}} x+20 x^2+5 x^3\right )} \, dx+2048 \int \frac {1}{(4+x)^2 \left (-4 e^{e^4+\frac {3}{2 x}}+x-e^{e^4+\frac {3}{2 x}} x+20 x^2+5 x^3\right )} \, dx \\ & = -\frac {x^4}{4+x}-\frac {5}{2} \int \frac {x^2}{-e^{e^4+\frac {3}{2 x}} (4+x)+x \left (1+20 x+5 x^2\right )} \, dx+3 \int \frac {x^3}{-e^{e^4+\frac {3}{2 x}} (4+x)+x \left (1+20 x+5 x^2\right )} \, dx-\frac {11}{2} \int \frac {x^3}{\left (-4 e^{e^4+\frac {3}{2 x}}+x-e^{e^4+\frac {3}{2 x}} x+20 x^2+5 x^3\right )^2} \, dx-\frac {15}{2} \int \frac {x^5}{\left (-4 e^{e^4+\frac {3}{2 x}}+x-e^{e^4+\frac {3}{2 x}} x+20 x^2+5 x^3\right )^2} \, dx-10 \int \frac {x^6}{\left (-4 e^{e^4+\frac {3}{2 x}}+x-e^{e^4+\frac {3}{2 x}} x+20 x^2+5 x^3\right )^2} \, dx-22 \int \frac {x}{-e^{e^4+\frac {3}{2 x}} (4+x)+x \left (1+20 x+5 x^2\right )} \, dx+38 \int \frac {x^2}{\left (-4 e^{e^4+\frac {3}{2 x}}+x-e^{e^4+\frac {3}{2 x}} x+20 x^2+5 x^3\right )^2} \, dx-216 \int \frac {x}{\left (-4 e^{e^4+\frac {3}{2 x}}+x-e^{e^4+\frac {3}{2 x}} x+20 x^2+5 x^3\right )^2} \, dx-216 \int \frac {1}{e^{e^4+\frac {3}{2 x}} (4+x)-x \left (1+20 x+5 x^2\right )} \, dx+1120 \int \frac {1}{\left (4 e^{e^4+\frac {3}{2 x}}-x+e^{e^4+\frac {3}{2 x}} x-20 x^2-5 x^3\right )^2} \, dx-1376 \int \frac {1}{(4+x) \left (-4 e^{e^4+\frac {3}{2 x}}+x-e^{e^4+\frac {3}{2 x}} x+20 x^2+5 x^3\right )} \, dx+2048 \int \frac {1}{(4+x)^2 \left (-e^{e^4+\frac {3}{2 x}} (4+x)+x \left (1+20 x+5 x^2\right )\right )} \, dx+4096 \int \frac {1}{(4+x)^2 \left (-4 e^{e^4+\frac {3}{2 x}}+x-e^{e^4+\frac {3}{2 x}} x+20 x^2+5 x^3\right )^2} \, dx-5504 \int \frac {1}{(4+x) \left (-4 e^{e^4+\frac {3}{2 x}}+x-e^{e^4+\frac {3}{2 x}} x+20 x^2+5 x^3\right )^2} \, dx \\ & = -\frac {x^4}{4+x}-\frac {5}{2} \int \frac {x^2}{-e^{e^4+\frac {3}{2 x}} (4+x)+x \left (1+20 x+5 x^2\right )} \, dx+3 \int \frac {x^3}{-e^{e^4+\frac {3}{2 x}} (4+x)+x \left (1+20 x+5 x^2\right )} \, dx-\frac {11}{2} \int \frac {x^3}{\left (e^{e^4+\frac {3}{2 x}} (4+x)-x \left (1+20 x+5 x^2\right )\right )^2} \, dx-\frac {15}{2} \int \frac {x^5}{\left (e^{e^4+\frac {3}{2 x}} (4+x)-x \left (1+20 x+5 x^2\right )\right )^2} \, dx-10 \int \frac {x^6}{\left (e^{e^4+\frac {3}{2 x}} (4+x)-x \left (1+20 x+5 x^2\right )\right )^2} \, dx-22 \int \frac {x}{-e^{e^4+\frac {3}{2 x}} (4+x)+x \left (1+20 x+5 x^2\right )} \, dx+38 \int \frac {x^2}{\left (e^{e^4+\frac {3}{2 x}} (4+x)-x \left (1+20 x+5 x^2\right )\right )^2} \, dx-216 \int \frac {x}{\left (e^{e^4+\frac {3}{2 x}} (4+x)-x \left (1+20 x+5 x^2\right )\right )^2} \, dx-216 \int \frac {1}{e^{e^4+\frac {3}{2 x}} (4+x)-x \left (1+20 x+5 x^2\right )} \, dx+1120 \int \frac {1}{\left (e^{e^4+\frac {3}{2 x}} (4+x)-x \left (1+20 x+5 x^2\right )\right )^2} \, dx-1376 \int \frac {1}{(4+x) \left (-4 e^{e^4+\frac {3}{2 x}}+x-e^{e^4+\frac {3}{2 x}} x+20 x^2+5 x^3\right )} \, dx+2048 \int \frac {1}{(4+x)^2 \left (-e^{e^4+\frac {3}{2 x}} (4+x)+x \left (1+20 x+5 x^2\right )\right )} \, dx+4096 \int \frac {1}{(4+x)^2 \left (e^{e^4+\frac {3}{2 x}} (4+x)-x \left (1+20 x+5 x^2\right )\right )^2} \, dx-5504 \int \frac {1}{(4+x) \left (e^{e^4+\frac {3}{2 x}} (4+x)-x \left (1+20 x+5 x^2\right )\right )^2} \, dx \\ \end{align*}

Mathematica [A] (verified)

Time = 0.15 (sec) , antiderivative size = 63, normalized size of antiderivative = 1.70 \[ \int \frac {-50 x^6-800 x^7-150 x^8+e^{\frac {3+2 e^4 x}{x}} \left (-32 x^3-6 x^4\right )+e^{\frac {3+2 e^4 x}{2 x}} \left (-3 x^3+6 x^4+320 x^5+60 x^6\right )}{2 x^2+80 x^3+820 x^4+400 x^5+50 x^6+e^{\frac {3+2 e^4 x}{x}} \left (32+16 x+2 x^2\right )+e^{\frac {3+2 e^4 x}{2 x}} \left (-16 x-324 x^2-160 x^3-20 x^4\right )} \, dx=-16 x+4 x^2-x^3-\frac {256}{4+x}+\frac {x^5}{(4+x) \left (-e^{e^4+\frac {3}{2 x}} (4+x)+x \left (1+20 x+5 x^2\right )\right )} \]

[In]

Integrate[(-50*x^6 - 800*x^7 - 150*x^8 + E^((3 + 2*E^4*x)/x)*(-32*x^3 - 6*x^4) + E^((3 + 2*E^4*x)/(2*x))*(-3*x
^3 + 6*x^4 + 320*x^5 + 60*x^6))/(2*x^2 + 80*x^3 + 820*x^4 + 400*x^5 + 50*x^6 + E^((3 + 2*E^4*x)/x)*(32 + 16*x
+ 2*x^2) + E^((3 + 2*E^4*x)/(2*x))*(-16*x - 324*x^2 - 160*x^3 - 20*x^4)),x]

[Out]

-16*x + 4*x^2 - x^3 - 256/(4 + x) + x^5/((4 + x)*(-(E^(E^4 + 3/(2*x))*(4 + x)) + x*(1 + 20*x + 5*x^2)))

Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(69\) vs. \(2(33)=66\).

Time = 0.80 (sec) , antiderivative size = 70, normalized size of antiderivative = 1.89

method	result	size
norman	\(\frac {{\mathrm e}^{\frac {2 x \,{\mathrm e}^{4}+3}{2 x}} x^{4}-5 x^{6}}{5 x^{3}+20 x^{2}-x \,{\mathrm e}^{\frac {2 x \,{\mathrm e}^{4}+3}{2 x}}+x -4 \,{\mathrm e}^{\frac {2 x \,{\mathrm e}^{4}+3}{2 x}}}\)	\(70\)
parallelrisch	\(-\frac {10 x^{6}-2 \,{\mathrm e}^{\frac {2 x \,{\mathrm e}^{4}+3}{2 x}} x^{4}}{2 \left (5 x^{3}+20 x^{2}-x \,{\mathrm e}^{\frac {2 x \,{\mathrm e}^{4}+3}{2 x}}+x -4 \,{\mathrm e}^{\frac {2 x \,{\mathrm e}^{4}+3}{2 x}}\right )}\)	\(72\)
risch	\(-x^{3}+4 x^{2}-16 x -\frac {256}{4+x}+\frac {x^{5}}{\left (4+x \right ) \left (5 x^{3}+20 x^{2}-x \,{\mathrm e}^{\frac {2 x \,{\mathrm e}^{4}+3}{2 x}}+x -4 \,{\mathrm e}^{\frac {2 x \,{\mathrm e}^{4}+3}{2 x}}\right )}\)	\(76\)

[In]

int(((-6*x^4-32*x^3)*exp(1/2*(2*x*exp(4)+3)/x)^2+(60*x^6+320*x^5+6*x^4-3*x^3)*exp(1/2*(2*x*exp(4)+3)/x)-150*x^
8-800*x^7-50*x^6)/((2*x^2+16*x+32)*exp(1/2*(2*x*exp(4)+3)/x)^2+(-20*x^4-160*x^3-324*x^2-16*x)*exp(1/2*(2*x*exp
(4)+3)/x)+50*x^6+400*x^5+820*x^4+80*x^3+2*x^2),x,method=_RETURNVERBOSE)

[Out]

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

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 76 vs. \(2 (33) = 66\).

Time = 0.26 (sec) , antiderivative size = 76, normalized size of antiderivative = 2.05 \[ \int \frac {-50 x^6-800 x^7-150 x^8+e^{\frac {3+2 e^4 x}{x}} \left (-32 x^3-6 x^4\right )+e^{\frac {3+2 e^4 x}{2 x}} \left (-3 x^3+6 x^4+320 x^5+60 x^6\right )}{2 x^2+80 x^3+820 x^4+400 x^5+50 x^6+e^{\frac {3+2 e^4 x}{x}} \left (32+16 x+2 x^2\right )+e^{\frac {3+2 e^4 x}{2 x}} \left (-16 x-324 x^2-160 x^3-20 x^4\right )} \, dx=-\frac {5 \, x^{6} + 320 \, x^{3} + 1280 \, x^{2} - {\left (x^{4} + 64 \, x + 256\right )} e^{\left (\frac {2 \, x e^{4} + 3}{2 \, x}\right )} + 64 \, x}{5 \, x^{3} + 20 \, x^{2} - {\left (x + 4\right )} e^{\left (\frac {2 \, x e^{4} + 3}{2 \, x}\right )} + x} \]

[In]

integrate(((-6*x^4-32*x^3)*exp(1/2*(2*x*exp(4)+3)/x)^2+(60*x^6+320*x^5+6*x^4-3*x^3)*exp(1/2*(2*x*exp(4)+3)/x)-
150*x^8-800*x^7-50*x^6)/((2*x^2+16*x+32)*exp(1/2*(2*x*exp(4)+3)/x)^2+(-20*x^4-160*x^3-324*x^2-16*x)*exp(1/2*(2
*x*exp(4)+3)/x)+50*x^6+400*x^5+820*x^4+80*x^3+2*x^2),x, algorithm="fricas")

[Out]

-(5*x^6 + 320*x^3 + 1280*x^2 - (x^4 + 64*x + 256)*e^(1/2*(2*x*e^4 + 3)/x) + 64*x)/(5*x^3 + 20*x^2 - (x + 4)*e^
(1/2*(2*x*e^4 + 3)/x) + x)

Sympy [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 58 vs. \(2 (24) = 48\).

Time = 0.21 (sec) , antiderivative size = 58, normalized size of antiderivative = 1.57 \[ \int \frac {-50 x^6-800 x^7-150 x^8+e^{\frac {3+2 e^4 x}{x}} \left (-32 x^3-6 x^4\right )+e^{\frac {3+2 e^4 x}{2 x}} \left (-3 x^3+6 x^4+320 x^5+60 x^6\right )}{2 x^2+80 x^3+820 x^4+400 x^5+50 x^6+e^{\frac {3+2 e^4 x}{x}} \left (32+16 x+2 x^2\right )+e^{\frac {3+2 e^4 x}{2 x}} \left (-16 x-324 x^2-160 x^3-20 x^4\right )} \, dx=- \frac {x^{5}}{- 5 x^{4} - 40 x^{3} - 81 x^{2} - 4 x + \left (x^{2} + 8 x + 16\right ) e^{\frac {x e^{4} + \frac {3}{2}}{x}}} - x^{3} + 4 x^{2} - 16 x - \frac {256}{x + 4} \]

[In]

integrate(((-6*x**4-32*x**3)*exp(1/2*(2*x*exp(4)+3)/x)**2+(60*x**6+320*x**5+6*x**4-3*x**3)*exp(1/2*(2*x*exp(4)
+3)/x)-150*x**8-800*x**7-50*x**6)/((2*x**2+16*x+32)*exp(1/2*(2*x*exp(4)+3)/x)**2+(-20*x**4-160*x**3-324*x**2-1
6*x)*exp(1/2*(2*x*exp(4)+3)/x)+50*x**6+400*x**5+820*x**4+80*x**3+2*x**2),x)

[Out]

-x**5/(-5*x**4 - 40*x**3 - 81*x**2 - 4*x + (x**2 + 8*x + 16)*exp((x*exp(4) + 3/2)/x)) - x**3 + 4*x**2 - 16*x -
256/(x + 4)

Maxima [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 81 vs. \(2 (33) = 66\).

Time = 0.23 (sec) , antiderivative size = 81, normalized size of antiderivative = 2.19 \[ \int \frac {-50 x^6-800 x^7-150 x^8+e^{\frac {3+2 e^4 x}{x}} \left (-32 x^3-6 x^4\right )+e^{\frac {3+2 e^4 x}{2 x}} \left (-3 x^3+6 x^4+320 x^5+60 x^6\right )}{2 x^2+80 x^3+820 x^4+400 x^5+50 x^6+e^{\frac {3+2 e^4 x}{x}} \left (32+16 x+2 x^2\right )+e^{\frac {3+2 e^4 x}{2 x}} \left (-16 x-324 x^2-160 x^3-20 x^4\right )} \, dx=-\frac {5 \, x^{6} + 320 \, x^{3} + 1280 \, x^{2} - {\left (x^{4} e^{\left (e^{4}\right )} + 64 \, x e^{\left (e^{4}\right )} + 256 \, e^{\left (e^{4}\right )}\right )} e^{\left (\frac {3}{2 \, x}\right )} + 64 \, x}{5 \, x^{3} + 20 \, x^{2} - {\left (x e^{\left (e^{4}\right )} + 4 \, e^{\left (e^{4}\right )}\right )} e^{\left (\frac {3}{2 \, x}\right )} + x} \]

[In]

integrate(((-6*x^4-32*x^3)*exp(1/2*(2*x*exp(4)+3)/x)^2+(60*x^6+320*x^5+6*x^4-3*x^3)*exp(1/2*(2*x*exp(4)+3)/x)-
150*x^8-800*x^7-50*x^6)/((2*x^2+16*x+32)*exp(1/2*(2*x*exp(4)+3)/x)^2+(-20*x^4-160*x^3-324*x^2-16*x)*exp(1/2*(2
*x*exp(4)+3)/x)+50*x^6+400*x^5+820*x^4+80*x^3+2*x^2),x, algorithm="maxima")

[Out]

-(5*x^6 + 320*x^3 + 1280*x^2 - (x^4*e^(e^4) + 64*x*e^(e^4) + 256*e^(e^4))*e^(3/2/x) + 64*x)/(5*x^3 + 20*x^2 -
(x*e^(e^4) + 4*e^(e^4))*e^(3/2/x) + x)

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 115 vs. \(2 (33) = 66\).

Time = 0.34 (sec) , antiderivative size = 115, normalized size of antiderivative = 3.11 \[ \int \frac {-50 x^6-800 x^7-150 x^8+e^{\frac {3+2 e^4 x}{x}} \left (-32 x^3-6 x^4\right )+e^{\frac {3+2 e^4 x}{2 x}} \left (-3 x^3+6 x^4+320 x^5+60 x^6\right )}{2 x^2+80 x^3+820 x^4+400 x^5+50 x^6+e^{\frac {3+2 e^4 x}{x}} \left (32+16 x+2 x^2\right )+e^{\frac {3+2 e^4 x}{2 x}} \left (-16 x-324 x^2-160 x^3-20 x^4\right )} \, dx=-\frac {5 \, x^{6} - x^{4} e^{\left (\frac {2 \, x e^{4} + 3}{2 \, x}\right )} + 320 \, x^{3} + 1280 \, x^{2} - 64 \, x e^{\left (\frac {2 \, x e^{4} + 3}{2 \, x}\right )} + 64 \, x - 256 \, e^{\left (\frac {2 \, x e^{4} + 3}{2 \, x}\right )}}{5 \, x^{3} + 20 \, x^{2} - x e^{\left (\frac {2 \, x e^{4} + 3}{2 \, x}\right )} + x - 4 \, e^{\left (\frac {2 \, x e^{4} + 3}{2 \, x}\right )}} \]

[In]

integrate(((-6*x^4-32*x^3)*exp(1/2*(2*x*exp(4)+3)/x)^2+(60*x^6+320*x^5+6*x^4-3*x^3)*exp(1/2*(2*x*exp(4)+3)/x)-
150*x^8-800*x^7-50*x^6)/((2*x^2+16*x+32)*exp(1/2*(2*x*exp(4)+3)/x)^2+(-20*x^4-160*x^3-324*x^2-16*x)*exp(1/2*(2
*x*exp(4)+3)/x)+50*x^6+400*x^5+820*x^4+80*x^3+2*x^2),x, algorithm="giac")

[Out]

-(5*x^6 - x^4*e^(1/2*(2*x*e^4 + 3)/x) + 320*x^3 + 1280*x^2 - 64*x*e^(1/2*(2*x*e^4 + 3)/x) + 64*x - 256*e^(1/2*
(2*x*e^4 + 3)/x))/(5*x^3 + 20*x^2 - x*e^(1/2*(2*x*e^4 + 3)/x) + x - 4*e^(1/2*(2*x*e^4 + 3)/x))

Mupad [F(-1)]

Timed out. \[ \int \frac {-50 x^6-800 x^7-150 x^8+e^{\frac {3+2 e^4 x}{x}} \left (-32 x^3-6 x^4\right )+e^{\frac {3+2 e^4 x}{2 x}} \left (-3 x^3+6 x^4+320 x^5+60 x^6\right )}{2 x^2+80 x^3+820 x^4+400 x^5+50 x^6+e^{\frac {3+2 e^4 x}{x}} \left (32+16 x+2 x^2\right )+e^{\frac {3+2 e^4 x}{2 x}} \left (-16 x-324 x^2-160 x^3-20 x^4\right )} \, dx=\int -\frac {{\mathrm {e}}^{\frac {2\,\left (x\,{\mathrm {e}}^4+\frac {3}{2}\right )}{x}}\,\left (6\,x^4+32\,x^3\right )-{\mathrm {e}}^{\frac {x\,{\mathrm {e}}^4+\frac {3}{2}}{x}}\,\left (60\,x^6+320\,x^5+6\,x^4-3\,x^3\right )+50\,x^6+800\,x^7+150\,x^8}{{\mathrm {e}}^{\frac {2\,\left (x\,{\mathrm {e}}^4+\frac {3}{2}\right )}{x}}\,\left (2\,x^2+16\,x+32\right )-{\mathrm {e}}^{\frac {x\,{\mathrm {e}}^4+\frac {3}{2}}{x}}\,\left (20\,x^4+160\,x^3+324\,x^2+16\,x\right )+2\,x^2+80\,x^3+820\,x^4+400\,x^5+50\,x^6} \,d x \]

[In]

int(-(exp((2*(x*exp(4) + 3/2))/x)*(32*x^3 + 6*x^4) - exp((x*exp(4) + 3/2)/x)*(6*x^4 - 3*x^3 + 320*x^5 + 60*x^6
) + 50*x^6 + 800*x^7 + 150*x^8)/(exp((2*(x*exp(4) + 3/2))/x)*(16*x + 2*x^2 + 32) - exp((x*exp(4) + 3/2)/x)*(16
*x + 324*x^2 + 160*x^3 + 20*x^4) + 2*x^2 + 80*x^3 + 820*x^4 + 400*x^5 + 50*x^6),x)

[Out]

int(-(exp((2*(x*exp(4) + 3/2))/x)*(32*x^3 + 6*x^4) - exp((x*exp(4) + 3/2)/x)*(6*x^4 - 3*x^3 + 320*x^5 + 60*x^6
) + 50*x^6 + 800*x^7 + 150*x^8)/(exp((2*(x*exp(4) + 3/2))/x)*(16*x + 2*x^2 + 32) - exp((x*exp(4) + 3/2)/x)*(16
*x + 324*x^2 + 160*x^3 + 20*x^4) + 2*x^2 + 80*x^3 + 820*x^4 + 400*x^5 + 50*x^6), x)