\(\int \frac {e^{\frac {x^3}{8 e^3 x^2+e^5 x^3+e^x (8 e^3+e^5 x)}} (128 e^6 x^4+16 e^3 x^5+32 e^8 x^5+2 e^{10} x^6+e^{2 x} (128 e^6+32 e^8 x+2 e^{10} x^2)+e^x (256 e^6 x^2+4 e^{10} x^4+e^3 (48 x^3-16 x^4)+e^5 (64 e^3 x^3+4 x^4-2 x^5)))}{64 e^6 x^4+16 e^8 x^5+e^{10} x^6+e^{2 x} (64 e^6+16 e^8 x+e^{10} x^2)+e^x (128 e^6 x^2+32 e^8 x^3+2 e^{10} x^4)} \, dx\) [74]

Optimal result

Integrand size = 238, antiderivative size = 32 \[ \int \frac {e^{\frac {x^3}{8 e^3 x^2+e^5 x^3+e^x \left (8 e^3+e^5 x\right )}} \left (128 e^6 x^4+16 e^3 x^5+32 e^8 x^5+2 e^{10} x^6+e^{2 x} \left (128 e^6+32 e^8 x+2 e^{10} x^2\right )+e^x \left (256 e^6 x^2+4 e^{10} x^4+e^3 \left (48 x^3-16 x^4\right )+e^5 \left (64 e^3 x^3+4 x^4-2 x^5\right )\right )\right )}{64 e^6 x^4+16 e^8 x^5+e^{10} x^6+e^{2 x} \left (64 e^6+16 e^8 x+e^{10} x^2\right )+e^x \left (128 e^6 x^2+32 e^8 x^3+2 e^{10} x^4\right )} \, dx=2 e^{\frac {x^2}{\left (e^5+\frac {8 e^3}{x}\right ) \left (e^x+x^2\right )}} x \]

[Out]

2*exp(x^2/(exp(5)+8*exp(3)/x)/(x^2+exp(x)))*x

Rubi [F]

\[ \int \frac {e^{\frac {x^3}{8 e^3 x^2+e^5 x^3+e^x \left (8 e^3+e^5 x\right )}} \left (128 e^6 x^4+16 e^3 x^5+32 e^8 x^5+2 e^{10} x^6+e^{2 x} \left (128 e^6+32 e^8 x+2 e^{10} x^2\right )+e^x \left (256 e^6 x^2+4 e^{10} x^4+e^3 \left (48 x^3-16 x^4\right )+e^5 \left (64 e^3 x^3+4 x^4-2 x^5\right )\right )\right )}{64 e^6 x^4+16 e^8 x^5+e^{10} x^6+e^{2 x} \left (64 e^6+16 e^8 x+e^{10} x^2\right )+e^x \left (128 e^6 x^2+32 e^8 x^3+2 e^{10} x^4\right )} \, dx=\int \frac {\exp \left (\frac {x^3}{8 e^3 x^2+e^5 x^3+e^x \left (8 e^3+e^5 x\right )}\right ) \left (128 e^6 x^4+16 e^3 x^5+32 e^8 x^5+2 e^{10} x^6+e^{2 x} \left (128 e^6+32 e^8 x+2 e^{10} x^2\right )+e^x \left (256 e^6 x^2+4 e^{10} x^4+e^3 \left (48 x^3-16 x^4\right )+e^5 \left (64 e^3 x^3+4 x^4-2 x^5\right )\right )\right )}{64 e^6 x^4+16 e^8 x^5+e^{10} x^6+e^{2 x} \left (64 e^6+16 e^8 x+e^{10} x^2\right )+e^x \left (128 e^6 x^2+32 e^8 x^3+2 e^{10} x^4\right )} \, dx \]

[In]

Int[(E^(x^3/(8*E^3*x^2 + E^5*x^3 + E^x*(8*E^3 + E^5*x)))*(128*E^6*x^4 + 16*E^3*x^5 + 32*E^8*x^5 + 2*E^10*x^6 +
E^(2*x)*(128*E^6 + 32*E^8*x + 2*E^10*x^2) + E^x*(256*E^6*x^2 + 4*E^10*x^4 + E^3*(48*x^3 - 16*x^4) + E^5*(64*E
^3*x^3 + 4*x^4 - 2*x^5))))/(64*E^6*x^4 + 16*E^8*x^5 + E^10*x^6 + E^(2*x)*(64*E^6 + 16*E^8*x + E^10*x^2) + E^x*
(128*E^6*x^2 + 32*E^8*x^3 + 2*E^10*x^4)),x]

[Out]

2*Defer[Int][E^(x^3/(E^3*(8 + E^2*x)*(E^x + x^2))), x] - 16384*(4 + E^2)*Defer[Int][E^(-15 + x^3/(E^3*(8 + E^2
*x)*(E^x + x^2)))/(E^x + x^2)^2, x] + 2048*(4 + E^2)*Defer[Int][(E^(-13 + x^3/(E^3*(8 + E^2*x)*(E^x + x^2)))*x
)/(E^x + x^2)^2, x] - 256*(4 + E^2)*Defer[Int][(E^(-11 + x^3/(E^3*(8 + E^2*x)*(E^x + x^2)))*x^2)/(E^x + x^2)^2
, x] + 32*(4 + E^2)*Defer[Int][(E^(-9 + x^3/(E^3*(8 + E^2*x)*(E^x + x^2)))*x^3)/(E^x + x^2)^2, x] - 4*(4 + E^2
)*Defer[Int][(E^(-7 + x^3/(E^3*(8 + E^2*x)*(E^x + x^2)))*x^4)/(E^x + x^2)^2, x] + 2*Defer[Int][(E^(-5 + x^3/(E
^3*(8 + E^2*x)*(E^x + x^2)))*x^5)/(E^x + x^2)^2, x] + 131072*(4 + E^2)*Defer[Int][E^(-15 + x^3/(E^3*(8 + E^2*x
)*(E^x + x^2)))/((8 + E^2*x)*(E^x + x^2)^2), x] + 1024*Defer[Int][E^(-11 + x^3/(E^3*(8 + E^2*x)*(E^x + x^2)))/
(E^x + x^2), x] - 16*(8 + E^2)*Defer[Int][(E^(-9 + x^3/(E^3*(8 + E^2*x)*(E^x + x^2)))*x)/(E^x + x^2), x] + 4*(
4 + E^2)*Defer[Int][(E^(-7 + x^3/(E^3*(8 + E^2*x)*(E^x + x^2)))*x^2)/(E^x + x^2), x] - 2*Defer[Int][(E^(-5 + x
^3/(E^3*(8 + E^2*x)*(E^x + x^2)))*x^3)/(E^x + x^2), x] - 8192*Defer[Int][E^(-9 + x^3/(E^3*(8 + E^2*x)*(E^x + x
^2)))/((8 + E^2*x)^2*(E^x + x^2)), x] - 1024*(8 - E^2)*Defer[Int][E^(-11 + x^3/(E^3*(8 + E^2*x)*(E^x + x^2)))/
((8 + E^2*x)*(E^x + x^2)), x]

Rubi steps \begin{align*} \text {integral}& = \int \frac {\exp \left (\frac {x^3}{8 e^3 x^2+e^5 x^3+e^x \left (8 e^3+e^5 x\right )}\right ) \left (128 e^6 x^4+\left (16 e^3+32 e^8\right ) x^5+2 e^{10} x^6+e^{2 x} \left (128 e^6+32 e^8 x+2 e^{10} x^2\right )+e^x \left (256 e^6 x^2+4 e^{10} x^4+e^3 \left (48 x^3-16 x^4\right )+e^5 \left (64 e^3 x^3+4 x^4-2 x^5\right )\right )\right )}{64 e^6 x^4+16 e^8 x^5+e^{10} x^6+e^{2 x} \left (64 e^6+16 e^8 x+e^{10} x^2\right )+e^x \left (128 e^6 x^2+32 e^8 x^3+2 e^{10} x^4\right )} \, dx \\ & = \int \frac {2 e^{-3+\frac {x^3}{e^3 \left (8+e^2 x\right ) \left (e^x+x^2\right )}} \left (64 e^{3+2 x}+16 e^{5+2 x} x+128 e^{3+x} x^2+e^{7+2 x} x^2+32 e^{5+x} x^3-8 e^x (-3+x) x^3+64 e^3 x^4+2 e^{7+x} x^4-e^{2+x} (-2+x) x^4+8 \left (1+2 e^5\right ) x^5+e^7 x^6\right )}{\left (8+e^2 x\right )^2 \left (e^x+x^2\right )^2} \, dx \\ & = 2 \int \frac {e^{-3+\frac {x^3}{e^3 \left (8+e^2 x\right ) \left (e^x+x^2\right )}} \left (64 e^{3+2 x}+16 e^{5+2 x} x+128 e^{3+x} x^2+e^{7+2 x} x^2+32 e^{5+x} x^3-8 e^x (-3+x) x^3+64 e^3 x^4+2 e^{7+x} x^4-e^{2+x} (-2+x) x^4+8 \left (1+2 e^5\right ) x^5+e^7 x^6\right )}{\left (8+e^2 x\right )^2 \left (e^x+x^2\right )^2} \, dx \\ & = 2 \int \left (e^{\frac {x^3}{e^3 \left (8+e^2 x\right ) \left (e^x+x^2\right )}}+\frac {e^{-3+\frac {x^3}{e^3 \left (8+e^2 x\right ) \left (e^x+x^2\right )}} (-2+x) x^5}{\left (8+e^2 x\right ) \left (e^x+x^2\right )^2}+\frac {e^{-3+\frac {x^3}{e^3 \left (8+e^2 x\right ) \left (e^x+x^2\right )}} x^3 \left (24-2 \left (4-e^2\right ) x-e^2 x^2\right )}{\left (8+e^2 x\right )^2 \left (e^x+x^2\right )}\right ) \, dx \\ & = 2 \int e^{\frac {x^3}{e^3 \left (8+e^2 x\right ) \left (e^x+x^2\right )}} \, dx+2 \int \frac {e^{-3+\frac {x^3}{e^3 \left (8+e^2 x\right ) \left (e^x+x^2\right )}} (-2+x) x^5}{\left (8+e^2 x\right ) \left (e^x+x^2\right )^2} \, dx+2 \int \frac {e^{-3+\frac {x^3}{e^3 \left (8+e^2 x\right ) \left (e^x+x^2\right )}} x^3 \left (24-2 \left (4-e^2\right ) x-e^2 x^2\right )}{\left (8+e^2 x\right )^2 \left (e^x+x^2\right )} \, dx \\ & = 2 \int e^{\frac {x^3}{e^3 \left (8+e^2 x\right ) \left (e^x+x^2\right )}} \, dx+2 \int \left (-\frac {8192 e^{-15+\frac {x^3}{e^3 \left (8+e^2 x\right ) \left (e^x+x^2\right )}} \left (4+e^2\right )}{\left (e^x+x^2\right )^2}+\frac {1024 e^{-13+\frac {x^3}{e^3 \left (8+e^2 x\right ) \left (e^x+x^2\right )}} \left (4+e^2\right ) x}{\left (e^x+x^2\right )^2}-\frac {128 e^{-11+\frac {x^3}{e^3 \left (8+e^2 x\right ) \left (e^x+x^2\right )}} \left (4+e^2\right ) x^2}{\left (e^x+x^2\right )^2}+\frac {16 e^{-9+\frac {x^3}{e^3 \left (8+e^2 x\right ) \left (e^x+x^2\right )}} \left (4+e^2\right ) x^3}{\left (e^x+x^2\right )^2}-\frac {2 e^{-7+\frac {x^3}{e^3 \left (8+e^2 x\right ) \left (e^x+x^2\right )}} \left (4+e^2\right ) x^4}{\left (e^x+x^2\right )^2}+\frac {e^{-5+\frac {x^3}{e^3 \left (8+e^2 x\right ) \left (e^x+x^2\right )}} x^5}{\left (e^x+x^2\right )^2}+\frac {65536 e^{-15+\frac {x^3}{e^3 \left (8+e^2 x\right ) \left (e^x+x^2\right )}} \left (4+e^2\right )}{\left (8+e^2 x\right ) \left (e^x+x^2\right )^2}\right ) \, dx+2 \int \left (\frac {512 e^{-11+\frac {x^3}{e^3 \left (8+e^2 x\right ) \left (e^x+x^2\right )}}}{e^x+x^2}-\frac {8 e^{-9+\frac {x^3}{e^3 \left (8+e^2 x\right ) \left (e^x+x^2\right )}} \left (8+e^2\right ) x}{e^x+x^2}+\frac {2 e^{-7+\frac {x^3}{e^3 \left (8+e^2 x\right ) \left (e^x+x^2\right )}} \left (4+e^2\right ) x^2}{e^x+x^2}-\frac {e^{-5+\frac {x^3}{e^3 \left (8+e^2 x\right ) \left (e^x+x^2\right )}} x^3}{e^x+x^2}-\frac {4096 e^{-9+\frac {x^3}{e^3 \left (8+e^2 x\right ) \left (e^x+x^2\right )}}}{\left (8+e^2 x\right )^2 \left (e^x+x^2\right )}+\frac {512 e^{-11+\frac {x^3}{e^3 \left (8+e^2 x\right ) \left (e^x+x^2\right )}} \left (-8+e^2\right )}{\left (8+e^2 x\right ) \left (e^x+x^2\right )}\right ) \, dx \\ & = 2 \int e^{\frac {x^3}{e^3 \left (8+e^2 x\right ) \left (e^x+x^2\right )}} \, dx+2 \int \frac {e^{-5+\frac {x^3}{e^3 \left (8+e^2 x\right ) \left (e^x+x^2\right )}} x^5}{\left (e^x+x^2\right )^2} \, dx-2 \int \frac {e^{-5+\frac {x^3}{e^3 \left (8+e^2 x\right ) \left (e^x+x^2\right )}} x^3}{e^x+x^2} \, dx+1024 \int \frac {e^{-11+\frac {x^3}{e^3 \left (8+e^2 x\right ) \left (e^x+x^2\right )}}}{e^x+x^2} \, dx-8192 \int \frac {e^{-9+\frac {x^3}{e^3 \left (8+e^2 x\right ) \left (e^x+x^2\right )}}}{\left (8+e^2 x\right )^2 \left (e^x+x^2\right )} \, dx-\left (1024 \left (8-e^2\right )\right ) \int \frac {e^{-11+\frac {x^3}{e^3 \left (8+e^2 x\right ) \left (e^x+x^2\right )}}}{\left (8+e^2 x\right ) \left (e^x+x^2\right )} \, dx-\left (4 \left (4+e^2\right )\right ) \int \frac {e^{-7+\frac {x^3}{e^3 \left (8+e^2 x\right ) \left (e^x+x^2\right )}} x^4}{\left (e^x+x^2\right )^2} \, dx+\left (4 \left (4+e^2\right )\right ) \int \frac {e^{-7+\frac {x^3}{e^3 \left (8+e^2 x\right ) \left (e^x+x^2\right )}} x^2}{e^x+x^2} \, dx+\left (32 \left (4+e^2\right )\right ) \int \frac {e^{-9+\frac {x^3}{e^3 \left (8+e^2 x\right ) \left (e^x+x^2\right )}} x^3}{\left (e^x+x^2\right )^2} \, dx-\left (256 \left (4+e^2\right )\right ) \int \frac {e^{-11+\frac {x^3}{e^3 \left (8+e^2 x\right ) \left (e^x+x^2\right )}} x^2}{\left (e^x+x^2\right )^2} \, dx+\left (2048 \left (4+e^2\right )\right ) \int \frac {e^{-13+\frac {x^3}{e^3 \left (8+e^2 x\right ) \left (e^x+x^2\right )}} x}{\left (e^x+x^2\right )^2} \, dx-\left (16384 \left (4+e^2\right )\right ) \int \frac {e^{-15+\frac {x^3}{e^3 \left (8+e^2 x\right ) \left (e^x+x^2\right )}}}{\left (e^x+x^2\right )^2} \, dx+\left (131072 \left (4+e^2\right )\right ) \int \frac {e^{-15+\frac {x^3}{e^3 \left (8+e^2 x\right ) \left (e^x+x^2\right )}}}{\left (8+e^2 x\right ) \left (e^x+x^2\right )^2} \, dx-\left (16 \left (8+e^2\right )\right ) \int \frac {e^{-9+\frac {x^3}{e^3 \left (8+e^2 x\right ) \left (e^x+x^2\right )}} x}{e^x+x^2} \, dx \\ \end{align*}

Mathematica [A] (verified)

Time = 0.13 (sec) , antiderivative size = 30, normalized size of antiderivative = 0.94 \[ \int \frac {e^{\frac {x^3}{8 e^3 x^2+e^5 x^3+e^x \left (8 e^3+e^5 x\right )}} \left (128 e^6 x^4+16 e^3 x^5+32 e^8 x^5+2 e^{10} x^6+e^{2 x} \left (128 e^6+32 e^8 x+2 e^{10} x^2\right )+e^x \left (256 e^6 x^2+4 e^{10} x^4+e^3 \left (48 x^3-16 x^4\right )+e^5 \left (64 e^3 x^3+4 x^4-2 x^5\right )\right )\right )}{64 e^6 x^4+16 e^8 x^5+e^{10} x^6+e^{2 x} \left (64 e^6+16 e^8 x+e^{10} x^2\right )+e^x \left (128 e^6 x^2+32 e^8 x^3+2 e^{10} x^4\right )} \, dx=2 e^{\frac {x^3}{e^3 \left (8+e^2 x\right ) \left (e^x+x^2\right )}} x \]

[In]

Integrate[(E^(x^3/(8*E^3*x^2 + E^5*x^3 + E^x*(8*E^3 + E^5*x)))*(128*E^6*x^4 + 16*E^3*x^5 + 32*E^8*x^5 + 2*E^10
*x^6 + E^(2*x)*(128*E^6 + 32*E^8*x + 2*E^10*x^2) + E^x*(256*E^6*x^2 + 4*E^10*x^4 + E^3*(48*x^3 - 16*x^4) + E^5
*(64*E^3*x^3 + 4*x^4 - 2*x^5))))/(64*E^6*x^4 + 16*E^8*x^5 + E^10*x^6 + E^(2*x)*(64*E^6 + 16*E^8*x + E^10*x^2)
+ E^x*(128*E^6*x^2 + 32*E^8*x^3 + 2*E^10*x^4)),x]

[Out]

2*E^(x^3/(E^3*(8 + E^2*x)*(E^x + x^2)))*x

Maple [A] (verified)

Time = 34.75 (sec) , antiderivative size = 37, normalized size of antiderivative = 1.16

method	result	size
risch	\(2 x \,{\mathrm e}^{\frac {x^{3}}{x^{3} {\mathrm e}^{5}+x \,{\mathrm e}^{5+x}+8 x^{2} {\mathrm e}^{3}+8 \,{\mathrm e}^{3+x}}}\)	\(37\)
parallelrisch	\(2 x \,{\mathrm e}^{\frac {x^{3}}{x^{3} {\mathrm e}^{5}+x \,{\mathrm e}^{5} {\mathrm e}^{x}+8 x^{2} {\mathrm e}^{3}+8 \,{\mathrm e}^{x} {\mathrm e}^{3}}}\)	\(37\)

[In]

int(((2*x^2*exp(5)^2+32*x*exp(3)*exp(5)+128*exp(3)^2)*exp(x)^2+(4*x^4*exp(5)^2+(64*x^3*exp(3)-2*x^5+4*x^4)*exp
(5)+256*x^2*exp(3)^2+(-16*x^4+48*x^3)*exp(3))*exp(x)+2*x^6*exp(5)^2+32*x^5*exp(3)*exp(5)+128*x^4*exp(3)^2+16*x
^5*exp(3))*exp(x^3/((x*exp(5)+8*exp(3))*exp(x)+x^3*exp(5)+8*x^2*exp(3)))/((x^2*exp(5)^2+16*x*exp(3)*exp(5)+64*
exp(3)^2)*exp(x)^2+(2*x^4*exp(5)^2+32*x^3*exp(3)*exp(5)+128*x^2*exp(3)^2)*exp(x)+x^6*exp(5)^2+16*x^5*exp(3)*ex
p(5)+64*x^4*exp(3)^2),x,method=_RETURNVERBOSE)

[Out]

2*x*exp(x^3/(x^3*exp(5)+x*exp(5+x)+8*x^2*exp(3)+8*exp(3+x)))

Fricas [A] (verification not implemented)

none

Time = 0.24 (sec) , antiderivative size = 36, normalized size of antiderivative = 1.12 \[ \int \frac {e^{\frac {x^3}{8 e^3 x^2+e^5 x^3+e^x \left (8 e^3+e^5 x\right )}} \left (128 e^6 x^4+16 e^3 x^5+32 e^8 x^5+2 e^{10} x^6+e^{2 x} \left (128 e^6+32 e^8 x+2 e^{10} x^2\right )+e^x \left (256 e^6 x^2+4 e^{10} x^4+e^3 \left (48 x^3-16 x^4\right )+e^5 \left (64 e^3 x^3+4 x^4-2 x^5\right )\right )\right )}{64 e^6 x^4+16 e^8 x^5+e^{10} x^6+e^{2 x} \left (64 e^6+16 e^8 x+e^{10} x^2\right )+e^x \left (128 e^6 x^2+32 e^8 x^3+2 e^{10} x^4\right )} \, dx=2 \, x e^{\left (\frac {x^{3}}{x^{3} e^{5} + 8 \, x^{2} e^{3} + {\left (x e^{5} + 8 \, e^{3}\right )} e^{x}}\right )} \]

[In]

integrate(((2*x^2*exp(5)^2+32*x*exp(3)*exp(5)+128*exp(3)^2)*exp(x)^2+(4*x^4*exp(5)^2+(64*x^3*exp(3)-2*x^5+4*x^
4)*exp(5)+256*x^2*exp(3)^2+(-16*x^4+48*x^3)*exp(3))*exp(x)+2*x^6*exp(5)^2+32*x^5*exp(3)*exp(5)+128*x^4*exp(3)^
2+16*x^5*exp(3))*exp(x^3/((x*exp(5)+8*exp(3))*exp(x)+x^3*exp(5)+8*x^2*exp(3)))/((x^2*exp(5)^2+16*x*exp(3)*exp(
5)+64*exp(3)^2)*exp(x)^2+(2*x^4*exp(5)^2+32*x^3*exp(3)*exp(5)+128*x^2*exp(3)^2)*exp(x)+x^6*exp(5)^2+16*x^5*exp
(3)*exp(5)+64*x^4*exp(3)^2),x, algorithm="fricas")

[Out]

2*x*e^(x^3/(x^3*e^5 + 8*x^2*e^3 + (x*e^5 + 8*e^3)*e^x))

Sympy [A] (verification not implemented)

Time = 2.58 (sec) , antiderivative size = 36, normalized size of antiderivative = 1.12 \[ \int \frac {e^{\frac {x^3}{8 e^3 x^2+e^5 x^3+e^x \left (8 e^3+e^5 x\right )}} \left (128 e^6 x^4+16 e^3 x^5+32 e^8 x^5+2 e^{10} x^6+e^{2 x} \left (128 e^6+32 e^8 x+2 e^{10} x^2\right )+e^x \left (256 e^6 x^2+4 e^{10} x^4+e^3 \left (48 x^3-16 x^4\right )+e^5 \left (64 e^3 x^3+4 x^4-2 x^5\right )\right )\right )}{64 e^6 x^4+16 e^8 x^5+e^{10} x^6+e^{2 x} \left (64 e^6+16 e^8 x+e^{10} x^2\right )+e^x \left (128 e^6 x^2+32 e^8 x^3+2 e^{10} x^4\right )} \, dx=2 x e^{\frac {x^{3}}{x^{3} e^{5} + 8 x^{2} e^{3} + \left (x e^{5} + 8 e^{3}\right ) e^{x}}} \]

[In]

integrate(((2*x**2*exp(5)**2+32*x*exp(3)*exp(5)+128*exp(3)**2)*exp(x)**2+(4*x**4*exp(5)**2+(64*x**3*exp(3)-2*x
**5+4*x**4)*exp(5)+256*x**2*exp(3)**2+(-16*x**4+48*x**3)*exp(3))*exp(x)+2*x**6*exp(5)**2+32*x**5*exp(3)*exp(5)
+128*x**4*exp(3)**2+16*x**5*exp(3))*exp(x**3/((x*exp(5)+8*exp(3))*exp(x)+x**3*exp(5)+8*x**2*exp(3)))/((x**2*ex
p(5)**2+16*x*exp(3)*exp(5)+64*exp(3)**2)*exp(x)**2+(2*x**4*exp(5)**2+32*x**3*exp(3)*exp(5)+128*x**2*exp(3)**2)
*exp(x)+x**6*exp(5)**2+16*x**5*exp(3)*exp(5)+64*x**4*exp(3)**2),x)

[Out]

2*x*exp(x**3/(x**3*exp(5) + 8*x**2*exp(3) + (x*exp(5) + 8*exp(3))*exp(x)))

Maxima [F]

\[ \int \frac {e^{\frac {x^3}{8 e^3 x^2+e^5 x^3+e^x \left (8 e^3+e^5 x\right )}} \left (128 e^6 x^4+16 e^3 x^5+32 e^8 x^5+2 e^{10} x^6+e^{2 x} \left (128 e^6+32 e^8 x+2 e^{10} x^2\right )+e^x \left (256 e^6 x^2+4 e^{10} x^4+e^3 \left (48 x^3-16 x^4\right )+e^5 \left (64 e^3 x^3+4 x^4-2 x^5\right )\right )\right )}{64 e^6 x^4+16 e^8 x^5+e^{10} x^6+e^{2 x} \left (64 e^6+16 e^8 x+e^{10} x^2\right )+e^x \left (128 e^6 x^2+32 e^8 x^3+2 e^{10} x^4\right )} \, dx=\int { \frac {2 \, {\left (x^{6} e^{10} + 16 \, x^{5} e^{8} + 8 \, x^{5} e^{3} + 64 \, x^{4} e^{6} + {\left (x^{2} e^{10} + 16 \, x e^{8} + 64 \, e^{6}\right )} e^{\left (2 \, x\right )} + {\left (2 \, x^{4} e^{10} + 128 \, x^{2} e^{6} - {\left (x^{5} - 2 \, x^{4} - 32 \, x^{3} e^{3}\right )} e^{5} - 8 \, {\left (x^{4} - 3 \, x^{3}\right )} e^{3}\right )} e^{x}\right )} e^{\left (\frac {x^{3}}{x^{3} e^{5} + 8 \, x^{2} e^{3} + {\left (x e^{5} + 8 \, e^{3}\right )} e^{x}}\right )}}{x^{6} e^{10} + 16 \, x^{5} e^{8} + 64 \, x^{4} e^{6} + {\left (x^{2} e^{10} + 16 \, x e^{8} + 64 \, e^{6}\right )} e^{\left (2 \, x\right )} + 2 \, {\left (x^{4} e^{10} + 16 \, x^{3} e^{8} + 64 \, x^{2} e^{6}\right )} e^{x}} \,d x } \]

[In]

integrate(((2*x^2*exp(5)^2+32*x*exp(3)*exp(5)+128*exp(3)^2)*exp(x)^2+(4*x^4*exp(5)^2+(64*x^3*exp(3)-2*x^5+4*x^
4)*exp(5)+256*x^2*exp(3)^2+(-16*x^4+48*x^3)*exp(3))*exp(x)+2*x^6*exp(5)^2+32*x^5*exp(3)*exp(5)+128*x^4*exp(3)^
2+16*x^5*exp(3))*exp(x^3/((x*exp(5)+8*exp(3))*exp(x)+x^3*exp(5)+8*x^2*exp(3)))/((x^2*exp(5)^2+16*x*exp(3)*exp(
5)+64*exp(3)^2)*exp(x)^2+(2*x^4*exp(5)^2+32*x^3*exp(3)*exp(5)+128*x^2*exp(3)^2)*exp(x)+x^6*exp(5)^2+16*x^5*exp
(3)*exp(5)+64*x^4*exp(3)^2),x, algorithm="maxima")

[Out]

2*integrate((x^6*e^10 + 16*x^5*e^8 + 8*x^5*e^3 + 64*x^4*e^6 + (x^2*e^10 + 16*x*e^8 + 64*e^6)*e^(2*x) + (2*x^4*
e^10 + 128*x^2*e^6 - (x^5 - 2*x^4 - 32*x^3*e^3)*e^5 - 8*(x^4 - 3*x^3)*e^3)*e^x)*e^(x^3/(x^3*e^5 + 8*x^2*e^3 +
(x*e^5 + 8*e^3)*e^x))/(x^6*e^10 + 16*x^5*e^8 + 64*x^4*e^6 + (x^2*e^10 + 16*x*e^8 + 64*e^6)*e^(2*x) + 2*(x^4*e^
10 + 16*x^3*e^8 + 64*x^2*e^6)*e^x), x)

Giac [F(-2)]

Exception generated. \[ \int \frac {e^{\frac {x^3}{8 e^3 x^2+e^5 x^3+e^x \left (8 e^3+e^5 x\right )}} \left (128 e^6 x^4+16 e^3 x^5+32 e^8 x^5+2 e^{10} x^6+e^{2 x} \left (128 e^6+32 e^8 x+2 e^{10} x^2\right )+e^x \left (256 e^6 x^2+4 e^{10} x^4+e^3 \left (48 x^3-16 x^4\right )+e^5 \left (64 e^3 x^3+4 x^4-2 x^5\right )\right )\right )}{64 e^6 x^4+16 e^8 x^5+e^{10} x^6+e^{2 x} \left (64 e^6+16 e^8 x+e^{10} x^2\right )+e^x \left (128 e^6 x^2+32 e^8 x^3+2 e^{10} x^4\right )} \, dx=\text {Exception raised: TypeError} \]

[In]

integrate(((2*x^2*exp(5)^2+32*x*exp(3)*exp(5)+128*exp(3)^2)*exp(x)^2+(4*x^4*exp(5)^2+(64*x^3*exp(3)-2*x^5+4*x^
4)*exp(5)+256*x^2*exp(3)^2+(-16*x^4+48*x^3)*exp(3))*exp(x)+2*x^6*exp(5)^2+32*x^5*exp(3)*exp(5)+128*x^4*exp(3)^
2+16*x^5*exp(3))*exp(x^3/((x*exp(5)+8*exp(3))*exp(x)+x^3*exp(5)+8*x^2*exp(3)))/((x^2*exp(5)^2+16*x*exp(3)*exp(
5)+64*exp(3)^2)*exp(x)^2+(2*x^4*exp(5)^2+32*x^3*exp(3)*exp(5)+128*x^2*exp(3)^2)*exp(x)+x^6*exp(5)^2+16*x^5*exp
(3)*exp(5)+64*x^4*exp(3)^2),x, algorithm="giac")

[Out]

Exception raised: TypeError >> an error occurred running a Giac command:INPUT:sage2:=int(sage0,sageVARx):;OUTP
UT:Unable to divide, perhaps due to rounding error%%%{64,[0,8,42,46]%%%}+%%%{-768,[0,8,41,46]%%%}+%%%{7168,[0,
8,41,44]%%%

Mupad [B] (verification not implemented)

Time = 9.48 (sec) , antiderivative size = 36, normalized size of antiderivative = 1.12 \[ \int \frac {e^{\frac {x^3}{8 e^3 x^2+e^5 x^3+e^x \left (8 e^3+e^5 x\right )}} \left (128 e^6 x^4+16 e^3 x^5+32 e^8 x^5+2 e^{10} x^6+e^{2 x} \left (128 e^6+32 e^8 x+2 e^{10} x^2\right )+e^x \left (256 e^6 x^2+4 e^{10} x^4+e^3 \left (48 x^3-16 x^4\right )+e^5 \left (64 e^3 x^3+4 x^4-2 x^5\right )\right )\right )}{64 e^6 x^4+16 e^8 x^5+e^{10} x^6+e^{2 x} \left (64 e^6+16 e^8 x+e^{10} x^2\right )+e^x \left (128 e^6 x^2+32 e^8 x^3+2 e^{10} x^4\right )} \, dx=2\,x\,{\mathrm {e}}^{\frac {x^3}{8\,x^2\,{\mathrm {e}}^3+x^3\,{\mathrm {e}}^5+8\,{\mathrm {e}}^3\,{\mathrm {e}}^x+x\,{\mathrm {e}}^5\,{\mathrm {e}}^x}} \]

[In]

int((exp(x^3/(8*x^2*exp(3) + x^3*exp(5) + exp(x)*(8*exp(3) + x*exp(5))))*(exp(x)*(exp(5)*(64*x^3*exp(3) + 4*x^
4 - 2*x^5) + exp(3)*(48*x^3 - 16*x^4) + 256*x^2*exp(6) + 4*x^4*exp(10)) + exp(2*x)*(128*exp(6) + 32*x*exp(8) +
2*x^2*exp(10)) + 16*x^5*exp(3) + 128*x^4*exp(6) + 32*x^5*exp(8) + 2*x^6*exp(10)))/(exp(x)*(128*x^2*exp(6) + 3
2*x^3*exp(8) + 2*x^4*exp(10)) + exp(2*x)*(64*exp(6) + 16*x*exp(8) + x^2*exp(10)) + 64*x^4*exp(6) + 16*x^5*exp(
8) + x^6*exp(10)),x)

[Out]

2*x*exp(x^3/(8*x^2*exp(3) + x^3*exp(5) + 8*exp(3)*exp(x) + x*exp(5)*exp(x)))