3.27.1 \(\int \frac {20 e^{2 x}+40 e^x x+20 x^2+e^{\frac {9 x+8 x^3+e^x (4+8 x^2)}{10 e^x+10 x}} (32 e^{2 x} x+32 x^3+e^x (10-10 x+64 x^2))}{5 e^{2 x}+10 e^x x+5 x^2} \, dx\)

Optimal. Leaf size=32 \[ 2+4 \left (3+e^{\frac {1}{5} \left (2+4 x^2+\frac {5 x}{2 \left (e^x+x\right )}\right )}+x\right ) \]

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Rubi [F]  time = 8.34, 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 {20 e^{2 x}+40 e^x x+20 x^2+\exp \left (\frac {9 x+8 x^3+e^x \left (4+8 x^2\right )}{10 e^x+10 x}\right ) \left (32 e^{2 x} x+32 x^3+e^x \left (10-10 x+64 x^2\right )\right )}{5 e^{2 x}+10 e^x x+5 x^2} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

Int[(20*E^(2*x) + 40*E^x*x + 20*x^2 + E^((9*x + 8*x^3 + E^x*(4 + 8*x^2))/(10*E^x + 10*x))*(32*E^(2*x)*x + 32*x
^3 + E^x*(10 - 10*x + 64*x^2)))/(5*E^(2*x) + 10*E^x*x + 5*x^2),x]

[Out]

4*x + (32*Defer[Int][E^((4*E^x + 9*x + 8*E^x*x^2 + 8*x^3)/(10*(E^x + x)))*x, x])/5 - 2*Defer[Int][(E^((4*E^x +
9*x + 8*E^x*x^2 + 8*x^3)/(10*(E^x + x)))*x)/(E^x + x)^2, x] + 2*Defer[Int][(E^((4*E^x + 9*x + 8*E^x*x^2 + 8*x
^3)/(10*(E^x + x)))*x^2)/(E^x + x)^2, x] + 2*Defer[Int][E^((4*E^x + 9*x + 8*E^x*x^2 + 8*x^3)/(10*(E^x + x)))/(
E^x + x), x] - 2*Defer[Int][(E^((4*E^x + 9*x + 8*E^x*x^2 + 8*x^3)/(10*(E^x + x)))*x)/(E^x + x), x]

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\int \frac {20 e^{2 x}+40 e^x x+20 x^2+\exp \left (\frac {9 x+8 x^3+e^x \left (4+8 x^2\right )}{10 e^x+10 x}\right ) \left (32 e^{2 x} x+32 x^3+e^x \left (10-10 x+64 x^2\right )\right )}{5 \left (e^x+x\right )^2} \, dx\\ &=\frac {1}{5} \int \frac {20 e^{2 x}+40 e^x x+20 x^2+\exp \left (\frac {9 x+8 x^3+e^x \left (4+8 x^2\right )}{10 e^x+10 x}\right ) \left (32 e^{2 x} x+32 x^3+e^x \left (10-10 x+64 x^2\right )\right )}{\left (e^x+x\right )^2} \, dx\\ &=\frac {1}{5} \int \left (20+\frac {2 \exp \left (\frac {4 e^x+9 x+8 e^x x^2+8 x^3}{10 \left (e^x+x\right )}\right ) \left (5 e^x-5 e^x x+16 e^{2 x} x+32 e^x x^2+16 x^3\right )}{\left (e^x+x\right )^2}\right ) \, dx\\ &=4 x+\frac {2}{5} \int \frac {\exp \left (\frac {4 e^x+9 x+8 e^x x^2+8 x^3}{10 \left (e^x+x\right )}\right ) \left (5 e^x-5 e^x x+16 e^{2 x} x+32 e^x x^2+16 x^3\right )}{\left (e^x+x\right )^2} \, dx\\ &=4 x+\frac {2}{5} \int \left (16 \exp \left (\frac {4 e^x+9 x+8 e^x x^2+8 x^3}{10 \left (e^x+x\right )}\right ) x+\frac {5 \exp \left (\frac {4 e^x+9 x+8 e^x x^2+8 x^3}{10 \left (e^x+x\right )}\right ) (-1+x) x}{\left (e^x+x\right )^2}-\frac {5 \exp \left (\frac {4 e^x+9 x+8 e^x x^2+8 x^3}{10 \left (e^x+x\right )}\right ) (-1+x)}{e^x+x}\right ) \, dx\\ &=4 x+2 \int \frac {\exp \left (\frac {4 e^x+9 x+8 e^x x^2+8 x^3}{10 \left (e^x+x\right )}\right ) (-1+x) x}{\left (e^x+x\right )^2} \, dx-2 \int \frac {\exp \left (\frac {4 e^x+9 x+8 e^x x^2+8 x^3}{10 \left (e^x+x\right )}\right ) (-1+x)}{e^x+x} \, dx+\frac {32}{5} \int \exp \left (\frac {4 e^x+9 x+8 e^x x^2+8 x^3}{10 \left (e^x+x\right )}\right ) x \, dx\\ &=4 x+2 \int \left (-\frac {\exp \left (\frac {4 e^x+9 x+8 e^x x^2+8 x^3}{10 \left (e^x+x\right )}\right ) x}{\left (e^x+x\right )^2}+\frac {\exp \left (\frac {4 e^x+9 x+8 e^x x^2+8 x^3}{10 \left (e^x+x\right )}\right ) x^2}{\left (e^x+x\right )^2}\right ) \, dx-2 \int \left (-\frac {\exp \left (\frac {4 e^x+9 x+8 e^x x^2+8 x^3}{10 \left (e^x+x\right )}\right )}{e^x+x}+\frac {\exp \left (\frac {4 e^x+9 x+8 e^x x^2+8 x^3}{10 \left (e^x+x\right )}\right ) x}{e^x+x}\right ) \, dx+\frac {32}{5} \int \exp \left (\frac {4 e^x+9 x+8 e^x x^2+8 x^3}{10 \left (e^x+x\right )}\right ) x \, dx\\ &=4 x-2 \int \frac {\exp \left (\frac {4 e^x+9 x+8 e^x x^2+8 x^3}{10 \left (e^x+x\right )}\right ) x}{\left (e^x+x\right )^2} \, dx+2 \int \frac {\exp \left (\frac {4 e^x+9 x+8 e^x x^2+8 x^3}{10 \left (e^x+x\right )}\right ) x^2}{\left (e^x+x\right )^2} \, dx+2 \int \frac {\exp \left (\frac {4 e^x+9 x+8 e^x x^2+8 x^3}{10 \left (e^x+x\right )}\right )}{e^x+x} \, dx-2 \int \frac {\exp \left (\frac {4 e^x+9 x+8 e^x x^2+8 x^3}{10 \left (e^x+x\right )}\right ) x}{e^x+x} \, dx+\frac {32}{5} \int \exp \left (\frac {4 e^x+9 x+8 e^x x^2+8 x^3}{10 \left (e^x+x\right )}\right ) x \, dx\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.19, size = 27, normalized size = 0.84 \begin {gather*} 4 \left (e^{\frac {1}{10} \left (4+8 x^2+\frac {5 x}{e^x+x}\right )}+x\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(20*E^(2*x) + 40*E^x*x + 20*x^2 + E^((9*x + 8*x^3 + E^x*(4 + 8*x^2))/(10*E^x + 10*x))*(32*E^(2*x)*x
+ 32*x^3 + E^x*(10 - 10*x + 64*x^2)))/(5*E^(2*x) + 10*E^x*x + 5*x^2),x]

[Out]

4*(E^((4 + 8*x^2 + (5*x)/(E^x + x))/10) + x)

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fricas [A]  time = 0.53, size = 35, normalized size = 1.09 \begin {gather*} 4 \, x + 4 \, e^{\left (\frac {8 \, x^{3} + 4 \, {\left (2 \, x^{2} + 1\right )} e^{x} + 9 \, x}{10 \, {\left (x + e^{x}\right )}}\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((32*x*exp(x)^2+(64*x^2-10*x+10)*exp(x)+32*x^3)*exp(((8*x^2+4)*exp(x)+8*x^3+9*x)/(10*exp(x)+10*x))+2
0*exp(x)^2+40*exp(x)*x+20*x^2)/(5*exp(x)^2+10*exp(x)*x+5*x^2),x, algorithm="fricas")

[Out]

4*x + 4*e^(1/10*(8*x^3 + 4*(2*x^2 + 1)*e^x + 9*x)/(x + e^x))

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giac [F(-2)]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: TypeError} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((32*x*exp(x)^2+(64*x^2-10*x+10)*exp(x)+32*x^3)*exp(((8*x^2+4)*exp(x)+8*x^3+9*x)/(10*exp(x)+10*x))+2
0*exp(x)^2+40*exp(x)*x+20*x^2)/(5*exp(x)^2+10*exp(x)*x+5*x^2),x, algorithm="giac")

[Out]

Exception raised: TypeError >> An error occurred running a Giac command:INPUT:sage2:=int(sage0,sageVARx):;OUTP
UT:Evaluation time: 0.41Unable to divide, perhaps due to rounding error%%%{-80000000,[1,12]%%%}+%%%{220000000,
[1,11]%%%}+

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maple [A]  time = 0.10, size = 36, normalized size = 1.12

method	result	size
risch	\(4 x +4 \,{\mathrm e}^{\frac {8 \,{\mathrm e}^{x} x^{2}+8 x^{3}+4 \,{\mathrm e}^{x}+9 x}{10 \,{\mathrm e}^{x}+10 x}}\)	\(36\)
norman	\(\frac {4 x^{2}+4 x \,{\mathrm e}^{\frac {\left (8 x^{2}+4\right ) {\mathrm e}^{x}+8 x^{3}+9 x}{10 \,{\mathrm e}^{x}+10 x}}+4 \,{\mathrm e}^{x} x +4 \,{\mathrm e}^{x} {\mathrm e}^{\frac {\left (8 x^{2}+4\right ) {\mathrm e}^{x}+8 x^{3}+9 x}{10 \,{\mathrm e}^{x}+10 x}}}{{\mathrm e}^{x}+x}\)	\(88\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((32*x*exp(x)^2+(64*x^2-10*x+10)*exp(x)+32*x^3)*exp(((8*x^2+4)*exp(x)+8*x^3+9*x)/(10*exp(x)+10*x))+20*exp(
x)^2+40*exp(x)*x+20*x^2)/(5*exp(x)^2+10*exp(x)*x+5*x^2),x,method=_RETURNVERBOSE)

[Out]

4*x+4*exp(1/10*(8*exp(x)*x^2+8*x^3+4*exp(x)+9*x)/(exp(x)+x))

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maxima [A]  time = 0.74, size = 24, normalized size = 0.75 \begin {gather*} 4 \, x + 4 \, e^{\left (\frac {4}{5} \, x^{2} - \frac {e^{x}}{2 \, {\left (x + e^{x}\right )}} + \frac {9}{10}\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((32*x*exp(x)^2+(64*x^2-10*x+10)*exp(x)+32*x^3)*exp(((8*x^2+4)*exp(x)+8*x^3+9*x)/(10*exp(x)+10*x))+2
0*exp(x)^2+40*exp(x)*x+20*x^2)/(5*exp(x)^2+10*exp(x)*x+5*x^2),x, algorithm="maxima")

[Out]

4*x + 4*e^(4/5*x^2 - 1/2*e^x/(x + e^x) + 9/10)

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mupad [B]  time = 1.79, size = 69, normalized size = 2.16 \begin {gather*} 4\,x+4\,{\mathrm {e}}^{\frac {9\,x}{10\,x+10\,{\mathrm {e}}^x}}\,{\mathrm {e}}^{\frac {8\,x^2\,{\mathrm {e}}^x}{10\,x+10\,{\mathrm {e}}^x}}\,{\mathrm {e}}^{\frac {8\,x^3}{10\,x+10\,{\mathrm {e}}^x}}\,{\mathrm {e}}^{\frac {4\,{\mathrm {e}}^x}{10\,x+10\,{\mathrm {e}}^x}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((20*exp(2*x) + exp((9*x + exp(x)*(8*x^2 + 4) + 8*x^3)/(10*x + 10*exp(x)))*(32*x*exp(2*x) + exp(x)*(64*x^2
- 10*x + 10) + 32*x^3) + 40*x*exp(x) + 20*x^2)/(5*exp(2*x) + 10*x*exp(x) + 5*x^2),x)

[Out]

4*x + 4*exp((9*x)/(10*x + 10*exp(x)))*exp((8*x^2*exp(x))/(10*x + 10*exp(x)))*exp((8*x^3)/(10*x + 10*exp(x)))*e
xp((4*exp(x))/(10*x + 10*exp(x)))

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sympy [A]  time = 0.33, size = 32, normalized size = 1.00 \begin {gather*} 4 x + 4 e^{\frac {8 x^{3} + 9 x + \left (8 x^{2} + 4\right ) e^{x}}{10 x + 10 e^{x}}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((32*x*exp(x)**2+(64*x**2-10*x+10)*exp(x)+32*x**3)*exp(((8*x**2+4)*exp(x)+8*x**3+9*x)/(10*exp(x)+10*
x))+20*exp(x)**2+40*exp(x)*x+20*x**2)/(5*exp(x)**2+10*exp(x)*x+5*x**2),x)

[Out]

4*x + 4*exp((8*x**3 + 9*x + (8*x**2 + 4)*exp(x))/(10*x + 10*exp(x)))

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