3.23.14 \(\int \frac {2 x^4+e^{16 x+4 x^2} (-2+16 x+8 x^2)+e^{12 x+3 x^2} (-4 x+48 x^2+24 x^3)+e^{8 x+2 x^2} (48 x^3+24 x^4)+e^{4 x+x^2} (4 x^3+16 x^4+8 x^5)}{e^{24 x+6 x^2}+6 e^{20 x+5 x^2} x+x^3+3 x^4+3 x^5+x^6+e^{16 x+4 x^2} (3 x+15 x^2)+e^{12 x+3 x^2} (12 x^2+20 x^3)+e^{8 x+2 x^2} (3 x^2+18 x^3+15 x^4)+e^{4 x+x^2} (6 x^3+12 x^4+6 x^5)} \, dx\)

Optimal. Leaf size=21 \[ \left (-1+\frac {x}{x+\left (e^{x (4+x)}+x\right )^2}\right )^2 \]

________________________________________________________________________________________

Rubi [F]  time = 7.53, 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 {2 x^4+e^{16 x+4 x^2} \left (-2+16 x+8 x^2\right )+e^{12 x+3 x^2} \left (-4 x+48 x^2+24 x^3\right )+e^{8 x+2 x^2} \left (48 x^3+24 x^4\right )+e^{4 x+x^2} \left (4 x^3+16 x^4+8 x^5\right )}{e^{24 x+6 x^2}+6 e^{20 x+5 x^2} x+x^3+3 x^4+3 x^5+x^6+e^{16 x+4 x^2} \left (3 x+15 x^2\right )+e^{12 x+3 x^2} \left (12 x^2+20 x^3\right )+e^{8 x+2 x^2} \left (3 x^2+18 x^3+15 x^4\right )+e^{4 x+x^2} \left (6 x^3+12 x^4+6 x^5\right )} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

Int[(2*x^4 + E^(16*x + 4*x^2)*(-2 + 16*x + 8*x^2) + E^(12*x + 3*x^2)*(-4*x + 48*x^2 + 24*x^3) + E^(8*x + 2*x^2
)*(48*x^3 + 24*x^4) + E^(4*x + x^2)*(4*x^3 + 16*x^4 + 8*x^5))/(E^(24*x + 6*x^2) + 6*E^(20*x + 5*x^2)*x + x^3 +
3*x^4 + 3*x^5 + x^6 + E^(16*x + 4*x^2)*(3*x + 15*x^2) + E^(12*x + 3*x^2)*(12*x^2 + 20*x^3) + E^(8*x + 2*x^2)*
(3*x^2 + 18*x^3 + 15*x^4) + E^(4*x + x^2)*(6*x^3 + 12*x^4 + 6*x^5)),x]

[Out]

4*Defer[Int][(E^(4*x + x^2)*x^2)/(-E^(2*x*(4 + x)) - x - 2*E^(x*(4 + x))*x - x^2)^3, x] - 2*Defer[Int][x^2/(E^
(2*x*(4 + x)) + x + 2*E^(x*(4 + x))*x + x^2)^3, x] + 12*Defer[Int][x^3/(E^(2*x*(4 + x)) + x + 2*E^(x*(4 + x))*
x + x^2)^3, x] + 16*Defer[Int][(E^(4*x + x^2)*x^3)/(E^(2*x*(4 + x)) + x + 2*E^(x*(4 + x))*x + x^2)^3, x] + 24*
Defer[Int][x^4/(E^(2*x*(4 + x)) + x + 2*E^(x*(4 + x))*x + x^2)^3, x] + 8*Defer[Int][(E^(4*x + x^2)*x^4)/(E^(2*
x*(4 + x)) + x + 2*E^(x*(4 + x))*x + x^2)^3, x] + 8*Defer[Int][x^5/(E^(2*x*(4 + x)) + x + 2*E^(x*(4 + x))*x +
x^2)^3, x] + 4*Defer[Int][x/(E^(2*x*(4 + x)) + x + 2*E^(x*(4 + x))*x + x^2)^2, x] + 4*Defer[Int][(E^(4*x + x^2
)*x)/(E^(2*x*(4 + x)) + x + 2*E^(x*(4 + x))*x + x^2)^2, x] - 28*Defer[Int][x^2/(E^(2*x*(4 + x)) + x + 2*E^(x*(
4 + x))*x + x^2)^2, x] - 16*Defer[Int][(E^(4*x + x^2)*x^2)/(E^(2*x*(4 + x)) + x + 2*E^(x*(4 + x))*x + x^2)^2,
x] - 32*Defer[Int][x^3/(E^(2*x*(4 + x)) + x + 2*E^(x*(4 + x))*x + x^2)^2, x] - 8*Defer[Int][(E^(4*x + x^2)*x^3
)/(E^(2*x*(4 + x)) + x + 2*E^(x*(4 + x))*x + x^2)^2, x] - 8*Defer[Int][x^4/(E^(2*x*(4 + x)) + x + 2*E^(x*(4 +
x))*x + x^2)^2, x] - 2*Defer[Int][(E^(2*x*(4 + x)) + x + 2*E^(x*(4 + x))*x + x^2)^(-1), x] + 16*Defer[Int][x/(
E^(2*x*(4 + x)) + x + 2*E^(x*(4 + x))*x + x^2), x] + 8*Defer[Int][x^2/(E^(2*x*(4 + x)) + x + 2*E^(x*(4 + x))*x
+ x^2), x]

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\int \frac {2 \left (e^{x (4+x)}+x\right )^3 \left (x+e^{x (4+x)} \left (-1+8 x+4 x^2\right )\right )}{\left (e^{2 x (4+x)}+x+2 e^{x (4+x)} x+x^2\right )^3} \, dx\\ &=2 \int \frac {\left (e^{x (4+x)}+x\right )^3 \left (x+e^{x (4+x)} \left (-1+8 x+4 x^2\right )\right )}{\left (e^{2 x (4+x)}+x+2 e^{x (4+x)} x+x^2\right )^3} \, dx\\ &=2 \int \left (\frac {-1+8 x+4 x^2}{e^{2 x (4+x)}+x+2 e^{x (4+x)} x+x^2}-\frac {2 x \left (-1-e^{x (4+x)}+7 x+4 e^{x (4+x)} x+8 x^2+2 e^{x (4+x)} x^2+2 x^3\right )}{\left (e^{2 x (4+x)}+x+2 e^{x (4+x)} x+x^2\right )^2}+\frac {x^2 \left (-1-2 e^{x (4+x)}+6 x+8 e^{x (4+x)} x+12 x^2+4 e^{x (4+x)} x^2+4 x^3\right )}{\left (e^{2 x (4+x)}+x+2 e^{x (4+x)} x+x^2\right )^3}\right ) \, dx\\ &=2 \int \frac {-1+8 x+4 x^2}{e^{2 x (4+x)}+x+2 e^{x (4+x)} x+x^2} \, dx+2 \int \frac {x^2 \left (-1-2 e^{x (4+x)}+6 x+8 e^{x (4+x)} x+12 x^2+4 e^{x (4+x)} x^2+4 x^3\right )}{\left (e^{2 x (4+x)}+x+2 e^{x (4+x)} x+x^2\right )^3} \, dx-4 \int \frac {x \left (-1-e^{x (4+x)}+7 x+4 e^{x (4+x)} x+8 x^2+2 e^{x (4+x)} x^2+2 x^3\right )}{\left (e^{2 x (4+x)}+x+2 e^{x (4+x)} x+x^2\right )^2} \, dx\\ &=2 \int \left (-\frac {1}{e^{2 x (4+x)}+x+2 e^{x (4+x)} x+x^2}+\frac {8 x}{e^{2 x (4+x)}+x+2 e^{x (4+x)} x+x^2}+\frac {4 x^2}{e^{2 x (4+x)}+x+2 e^{x (4+x)} x+x^2}\right ) \, dx+2 \int \frac {x^2 \left (-1+6 x+12 x^2+4 x^3+e^{x (4+x)} \left (-2+8 x+4 x^2\right )\right )}{\left (e^{2 x (4+x)}+x+2 e^{x (4+x)} x+x^2\right )^3} \, dx-4 \int \frac {x \left (-1+7 x+8 x^2+2 x^3+e^{x (4+x)} \left (-1+4 x+2 x^2\right )\right )}{\left (e^{2 x (4+x)}+x+2 e^{x (4+x)} x+x^2\right )^2} \, dx\\ &=-\left (2 \int \frac {1}{e^{2 x (4+x)}+x+2 e^{x (4+x)} x+x^2} \, dx\right )+2 \int \left (\frac {2 e^{4 x+x^2} x^2}{\left (-e^{2 x (4+x)}-x-2 e^{x (4+x)} x-x^2\right )^3}-\frac {x^2}{\left (e^{2 x (4+x)}+x+2 e^{x (4+x)} x+x^2\right )^3}+\frac {6 x^3}{\left (e^{2 x (4+x)}+x+2 e^{x (4+x)} x+x^2\right )^3}+\frac {8 e^{4 x+x^2} x^3}{\left (e^{2 x (4+x)}+x+2 e^{x (4+x)} x+x^2\right )^3}+\frac {12 x^4}{\left (e^{2 x (4+x)}+x+2 e^{x (4+x)} x+x^2\right )^3}+\frac {4 e^{4 x+x^2} x^4}{\left (e^{2 x (4+x)}+x+2 e^{x (4+x)} x+x^2\right )^3}+\frac {4 x^5}{\left (e^{2 x (4+x)}+x+2 e^{x (4+x)} x+x^2\right )^3}\right ) \, dx-4 \int \left (-\frac {x}{\left (e^{2 x (4+x)}+x+2 e^{x (4+x)} x+x^2\right )^2}-\frac {e^{4 x+x^2} x}{\left (e^{2 x (4+x)}+x+2 e^{x (4+x)} x+x^2\right )^2}+\frac {7 x^2}{\left (e^{2 x (4+x)}+x+2 e^{x (4+x)} x+x^2\right )^2}+\frac {4 e^{4 x+x^2} x^2}{\left (e^{2 x (4+x)}+x+2 e^{x (4+x)} x+x^2\right )^2}+\frac {8 x^3}{\left (e^{2 x (4+x)}+x+2 e^{x (4+x)} x+x^2\right )^2}+\frac {2 e^{4 x+x^2} x^3}{\left (e^{2 x (4+x)}+x+2 e^{x (4+x)} x+x^2\right )^2}+\frac {2 x^4}{\left (e^{2 x (4+x)}+x+2 e^{x (4+x)} x+x^2\right )^2}\right ) \, dx+8 \int \frac {x^2}{e^{2 x (4+x)}+x+2 e^{x (4+x)} x+x^2} \, dx+16 \int \frac {x}{e^{2 x (4+x)}+x+2 e^{x (4+x)} x+x^2} \, dx\\ &=-\left (2 \int \frac {x^2}{\left (e^{2 x (4+x)}+x+2 e^{x (4+x)} x+x^2\right )^3} \, dx\right )-2 \int \frac {1}{e^{2 x (4+x)}+x+2 e^{x (4+x)} x+x^2} \, dx+4 \int \frac {e^{4 x+x^2} x^2}{\left (-e^{2 x (4+x)}-x-2 e^{x (4+x)} x-x^2\right )^3} \, dx+4 \int \frac {x}{\left (e^{2 x (4+x)}+x+2 e^{x (4+x)} x+x^2\right )^2} \, dx+4 \int \frac {e^{4 x+x^2} x}{\left (e^{2 x (4+x)}+x+2 e^{x (4+x)} x+x^2\right )^2} \, dx+8 \int \frac {e^{4 x+x^2} x^4}{\left (e^{2 x (4+x)}+x+2 e^{x (4+x)} x+x^2\right )^3} \, dx+8 \int \frac {x^5}{\left (e^{2 x (4+x)}+x+2 e^{x (4+x)} x+x^2\right )^3} \, dx-8 \int \frac {e^{4 x+x^2} x^3}{\left (e^{2 x (4+x)}+x+2 e^{x (4+x)} x+x^2\right )^2} \, dx-8 \int \frac {x^4}{\left (e^{2 x (4+x)}+x+2 e^{x (4+x)} x+x^2\right )^2} \, dx+8 \int \frac {x^2}{e^{2 x (4+x)}+x+2 e^{x (4+x)} x+x^2} \, dx+12 \int \frac {x^3}{\left (e^{2 x (4+x)}+x+2 e^{x (4+x)} x+x^2\right )^3} \, dx+16 \int \frac {e^{4 x+x^2} x^3}{\left (e^{2 x (4+x)}+x+2 e^{x (4+x)} x+x^2\right )^3} \, dx-16 \int \frac {e^{4 x+x^2} x^2}{\left (e^{2 x (4+x)}+x+2 e^{x (4+x)} x+x^2\right )^2} \, dx+16 \int \frac {x}{e^{2 x (4+x)}+x+2 e^{x (4+x)} x+x^2} \, dx+24 \int \frac {x^4}{\left (e^{2 x (4+x)}+x+2 e^{x (4+x)} x+x^2\right )^3} \, dx-28 \int \frac {x^2}{\left (e^{2 x (4+x)}+x+2 e^{x (4+x)} x+x^2\right )^2} \, dx-32 \int \frac {x^3}{\left (e^{2 x (4+x)}+x+2 e^{x (4+x)} x+x^2\right )^2} \, dx\\ \end {aligned} \end {gather*}

________________________________________________________________________________________

Mathematica [B]  time = 1.23, size = 55, normalized size = 2.62 \begin {gather*} -\frac {x \left (2 e^{2 x (4+x)}+x+4 e^{x (4+x)} x+2 x^2\right )}{\left (e^{2 x (4+x)}+x+2 e^{x (4+x)} x+x^2\right )^2} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(2*x^4 + E^(16*x + 4*x^2)*(-2 + 16*x + 8*x^2) + E^(12*x + 3*x^2)*(-4*x + 48*x^2 + 24*x^3) + E^(8*x +
2*x^2)*(48*x^3 + 24*x^4) + E^(4*x + x^2)*(4*x^3 + 16*x^4 + 8*x^5))/(E^(24*x + 6*x^2) + 6*E^(20*x + 5*x^2)*x +
x^3 + 3*x^4 + 3*x^5 + x^6 + E^(16*x + 4*x^2)*(3*x + 15*x^2) + E^(12*x + 3*x^2)*(12*x^2 + 20*x^3) + E^(8*x + 2
*x^2)*(3*x^2 + 18*x^3 + 15*x^4) + E^(4*x + x^2)*(6*x^3 + 12*x^4 + 6*x^5)),x]

[Out]

-((x*(2*E^(2*x*(4 + x)) + x + 4*E^(x*(4 + x))*x + 2*x^2))/(E^(2*x*(4 + x)) + x + 2*E^(x*(4 + x))*x + x^2)^2)

________________________________________________________________________________________

fricas [B]  time = 0.98, size = 110, normalized size = 5.24 \begin {gather*} -\frac {2 \, x^{3} + 4 \, x^{2} e^{\left (x^{2} + 4 \, x\right )} + x^{2} + 2 \, x e^{\left (2 \, x^{2} + 8 \, x\right )}}{x^{4} + 2 \, x^{3} + x^{2} + 4 \, x e^{\left (3 \, x^{2} + 12 \, x\right )} + 2 \, {\left (3 \, x^{2} + x\right )} e^{\left (2 \, x^{2} + 8 \, x\right )} + 4 \, {\left (x^{3} + x^{2}\right )} e^{\left (x^{2} + 4 \, x\right )} + e^{\left (4 \, x^{2} + 16 \, x\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((8*x^2+16*x-2)*exp(x^2+4*x)^4+(24*x^3+48*x^2-4*x)*exp(x^2+4*x)^3+(24*x^4+48*x^3)*exp(x^2+4*x)^2+(8*
x^5+16*x^4+4*x^3)*exp(x^2+4*x)+2*x^4)/(exp(x^2+4*x)^6+6*x*exp(x^2+4*x)^5+(15*x^2+3*x)*exp(x^2+4*x)^4+(20*x^3+1
2*x^2)*exp(x^2+4*x)^3+(15*x^4+18*x^3+3*x^2)*exp(x^2+4*x)^2+(6*x^5+12*x^4+6*x^3)*exp(x^2+4*x)+x^6+3*x^5+3*x^4+x
^3),x, algorithm="fricas")

[Out]

-(2*x^3 + 4*x^2*e^(x^2 + 4*x) + x^2 + 2*x*e^(2*x^2 + 8*x))/(x^4 + 2*x^3 + x^2 + 4*x*e^(3*x^2 + 12*x) + 2*(3*x^
2 + x)*e^(2*x^2 + 8*x) + 4*(x^3 + x^2)*e^(x^2 + 4*x) + e^(4*x^2 + 16*x))

________________________________________________________________________________________

giac [B]  time = 5.77, size = 128, normalized size = 6.10 \begin {gather*} -\frac {2 \, {\left (2 \, x^{3} + 4 \, x^{2} e^{\left (x^{2} + 4 \, x\right )} + x^{2} + 2 \, x e^{\left (2 \, x^{2} + 8 \, x\right )}\right )}}{x^{4} + 4 \, x^{3} e^{\left (x^{2} + 4 \, x\right )} + 2 \, x^{3} + 6 \, x^{2} e^{\left (2 \, x^{2} + 8 \, x\right )} + 4 \, x^{2} e^{\left (x^{2} + 4 \, x\right )} + x^{2} + 4 \, x e^{\left (3 \, x^{2} + 12 \, x\right )} + 2 \, x e^{\left (2 \, x^{2} + 8 \, x\right )} + e^{\left (4 \, x^{2} + 16 \, x\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((8*x^2+16*x-2)*exp(x^2+4*x)^4+(24*x^3+48*x^2-4*x)*exp(x^2+4*x)^3+(24*x^4+48*x^3)*exp(x^2+4*x)^2+(8*
x^5+16*x^4+4*x^3)*exp(x^2+4*x)+2*x^4)/(exp(x^2+4*x)^6+6*x*exp(x^2+4*x)^5+(15*x^2+3*x)*exp(x^2+4*x)^4+(20*x^3+1
2*x^2)*exp(x^2+4*x)^3+(15*x^4+18*x^3+3*x^2)*exp(x^2+4*x)^2+(6*x^5+12*x^4+6*x^3)*exp(x^2+4*x)+x^6+3*x^5+3*x^4+x
^3),x, algorithm="giac")

[Out]

-2*(2*x^3 + 4*x^2*e^(x^2 + 4*x) + x^2 + 2*x*e^(2*x^2 + 8*x))/(x^4 + 4*x^3*e^(x^2 + 4*x) + 2*x^3 + 6*x^2*e^(2*x
^2 + 8*x) + 4*x^2*e^(x^2 + 4*x) + x^2 + 4*x*e^(3*x^2 + 12*x) + 2*x*e^(2*x^2 + 8*x) + e^(4*x^2 + 16*x))

________________________________________________________________________________________

maple [B]  time = 0.07, size = 52, normalized size = 2.48

method	result	size
risch	\(-\frac {\left (2 x^{2}+4 x \,{\mathrm e}^{\left (4+x \right ) x}+2 \,{\mathrm e}^{2 \left (4+x \right ) x}+x \right ) x}{\left ({\mathrm e}^{2 \left (4+x \right ) x}+2 x \,{\mathrm e}^{\left (4+x \right ) x}+x^{2}+x \right )^{2}}\)	\(52\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((8*x^2+16*x-2)*exp(x^2+4*x)^4+(24*x^3+48*x^2-4*x)*exp(x^2+4*x)^3+(24*x^4+48*x^3)*exp(x^2+4*x)^2+(8*x^5+16
*x^4+4*x^3)*exp(x^2+4*x)+2*x^4)/(exp(x^2+4*x)^6+6*x*exp(x^2+4*x)^5+(15*x^2+3*x)*exp(x^2+4*x)^4+(20*x^3+12*x^2)
*exp(x^2+4*x)^3+(15*x^4+18*x^3+3*x^2)*exp(x^2+4*x)^2+(6*x^5+12*x^4+6*x^3)*exp(x^2+4*x)+x^6+3*x^5+3*x^4+x^3),x,
method=_RETURNVERBOSE)

[Out]

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

________________________________________________________________________________________

maxima [B]  time = 0.51, size = 110, normalized size = 5.24 \begin {gather*} -\frac {2 \, x^{3} + 4 \, x^{2} e^{\left (x^{2} + 4 \, x\right )} + x^{2} + 2 \, x e^{\left (2 \, x^{2} + 8 \, x\right )}}{x^{4} + 2 \, x^{3} + x^{2} + 4 \, x e^{\left (3 \, x^{2} + 12 \, x\right )} + 2 \, {\left (3 \, x^{2} + x\right )} e^{\left (2 \, x^{2} + 8 \, x\right )} + 4 \, {\left (x^{3} + x^{2}\right )} e^{\left (x^{2} + 4 \, x\right )} + e^{\left (4 \, x^{2} + 16 \, x\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((8*x^2+16*x-2)*exp(x^2+4*x)^4+(24*x^3+48*x^2-4*x)*exp(x^2+4*x)^3+(24*x^4+48*x^3)*exp(x^2+4*x)^2+(8*
x^5+16*x^4+4*x^3)*exp(x^2+4*x)+2*x^4)/(exp(x^2+4*x)^6+6*x*exp(x^2+4*x)^5+(15*x^2+3*x)*exp(x^2+4*x)^4+(20*x^3+1
2*x^2)*exp(x^2+4*x)^3+(15*x^4+18*x^3+3*x^2)*exp(x^2+4*x)^2+(6*x^5+12*x^4+6*x^3)*exp(x^2+4*x)+x^6+3*x^5+3*x^4+x
^3),x, algorithm="maxima")

[Out]

-(2*x^3 + 4*x^2*e^(x^2 + 4*x) + x^2 + 2*x*e^(2*x^2 + 8*x))/(x^4 + 2*x^3 + x^2 + 4*x*e^(3*x^2 + 12*x) + 2*(3*x^
2 + x)*e^(2*x^2 + 8*x) + 4*(x^3 + x^2)*e^(x^2 + 4*x) + e^(4*x^2 + 16*x))

________________________________________________________________________________________

mupad [F]  time = 0.00, size = -1, normalized size = -0.05 \begin {gather*} \int \frac {{\mathrm {e}}^{4\,x^2+16\,x}\,\left (8\,x^2+16\,x-2\right )+{\mathrm {e}}^{3\,x^2+12\,x}\,\left (24\,x^3+48\,x^2-4\,x\right )+{\mathrm {e}}^{x^2+4\,x}\,\left (8\,x^5+16\,x^4+4\,x^3\right )+2\,x^4+{\mathrm {e}}^{2\,x^2+8\,x}\,\left (24\,x^4+48\,x^3\right )}{{\mathrm {e}}^{6\,x^2+24\,x}+{\mathrm {e}}^{2\,x^2+8\,x}\,\left (15\,x^4+18\,x^3+3\,x^2\right )+{\mathrm {e}}^{4\,x^2+16\,x}\,\left (15\,x^2+3\,x\right )+6\,x\,{\mathrm {e}}^{5\,x^2+20\,x}+{\mathrm {e}}^{x^2+4\,x}\,\left (6\,x^5+12\,x^4+6\,x^3\right )+x^3+3\,x^4+3\,x^5+x^6+{\mathrm {e}}^{3\,x^2+12\,x}\,\left (20\,x^3+12\,x^2\right )} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((exp(16*x + 4*x^2)*(16*x + 8*x^2 - 2) + exp(12*x + 3*x^2)*(48*x^2 - 4*x + 24*x^3) + exp(4*x + x^2)*(4*x^3
+ 16*x^4 + 8*x^5) + 2*x^4 + exp(8*x + 2*x^2)*(48*x^3 + 24*x^4))/(exp(24*x + 6*x^2) + exp(8*x + 2*x^2)*(3*x^2 +
18*x^3 + 15*x^4) + exp(16*x + 4*x^2)*(3*x + 15*x^2) + 6*x*exp(20*x + 5*x^2) + exp(4*x + x^2)*(6*x^3 + 12*x^4
+ 6*x^5) + x^3 + 3*x^4 + 3*x^5 + x^6 + exp(12*x + 3*x^2)*(12*x^2 + 20*x^3)),x)

[Out]

int((exp(16*x + 4*x^2)*(16*x + 8*x^2 - 2) + exp(12*x + 3*x^2)*(48*x^2 - 4*x + 24*x^3) + exp(4*x + x^2)*(4*x^3
+ 16*x^4 + 8*x^5) + 2*x^4 + exp(8*x + 2*x^2)*(48*x^3 + 24*x^4))/(exp(24*x + 6*x^2) + exp(8*x + 2*x^2)*(3*x^2 +
18*x^3 + 15*x^4) + exp(16*x + 4*x^2)*(3*x + 15*x^2) + 6*x*exp(20*x + 5*x^2) + exp(4*x + x^2)*(6*x^3 + 12*x^4
+ 6*x^5) + x^3 + 3*x^4 + 3*x^5 + x^6 + exp(12*x + 3*x^2)*(12*x^2 + 20*x^3)), x)

________________________________________________________________________________________

sympy [B]  time = 0.34, size = 109, normalized size = 5.19 \begin {gather*} \frac {- 2 x^{3} - 4 x^{2} e^{x^{2} + 4 x} - x^{2} - 2 x e^{2 x^{2} + 8 x}}{x^{4} + 2 x^{3} + x^{2} + 4 x e^{3 x^{2} + 12 x} + \left (6 x^{2} + 2 x\right ) e^{2 x^{2} + 8 x} + \left (4 x^{3} + 4 x^{2}\right ) e^{x^{2} + 4 x} + e^{4 x^{2} + 16 x}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((8*x**2+16*x-2)*exp(x**2+4*x)**4+(24*x**3+48*x**2-4*x)*exp(x**2+4*x)**3+(24*x**4+48*x**3)*exp(x**2+
4*x)**2+(8*x**5+16*x**4+4*x**3)*exp(x**2+4*x)+2*x**4)/(exp(x**2+4*x)**6+6*x*exp(x**2+4*x)**5+(15*x**2+3*x)*exp
(x**2+4*x)**4+(20*x**3+12*x**2)*exp(x**2+4*x)**3+(15*x**4+18*x**3+3*x**2)*exp(x**2+4*x)**2+(6*x**5+12*x**4+6*x
**3)*exp(x**2+4*x)+x**6+3*x**5+3*x**4+x**3),x)

[Out]

(-2*x**3 - 4*x**2*exp(x**2 + 4*x) - x**2 - 2*x*exp(2*x**2 + 8*x))/(x**4 + 2*x**3 + x**2 + 4*x*exp(3*x**2 + 12*
x) + (6*x**2 + 2*x)*exp(2*x**2 + 8*x) + (4*x**3 + 4*x**2)*exp(x**2 + 4*x) + exp(4*x**2 + 16*x))

________________________________________________________________________________________