3.77.27 \(\int \frac {1}{5} e^{-6+\frac {4 e^{2 x} x+e^{3+x} (20 x+4 x^2)+e^6 (25 x+10 x^2+x^3)}{5 e^6}} (e^{2 x} (8+16 x)+e^6 (50+40 x+6 x^2)+e^{3+x} (40+56 x+8 x^2)) \, dx\)

Optimal. Leaf size=21 \[ 2 e^{\frac {1}{5} x \left (5+2 e^{-3+x}+x\right )^2} \]

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Rubi [F]  time = 4.20, 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 {1}{5} \exp \left (-6+\frac {4 e^{2 x} x+e^{3+x} \left (20 x+4 x^2\right )+e^6 \left (25 x+10 x^2+x^3\right )}{5 e^6}\right ) \left (e^{2 x} (8+16 x)+e^6 \left (50+40 x+6 x^2\right )+e^{3+x} \left (40+56 x+8 x^2\right )\right ) \, dx \end {gather*}

Verification is not applicable to the result.

[In]

Int[(E^(-6 + (4*E^(2*x)*x + E^(3 + x)*(20*x + 4*x^2) + E^6*(25*x + 10*x^2 + x^3))/(5*E^6))*(E^(2*x)*(8 + 16*x)
+ E^6*(50 + 40*x + 6*x^2) + E^(3 + x)*(40 + 56*x + 8*x^2)))/5,x]

[Out]

10*Defer[Int][E^((x*(2*E^x + E^3*(5 + x))^2)/(5*E^6)), x] + (8*Defer[Int][E^((-30 + 35*x + 4*E^(-6 + 2*x)*x +
10*x^2 + x^3 + 4*E^(-3 + x)*x*(5 + x))/5), x])/5 + 8*Defer[Int][E^(-3 + x + (4*E^(2*x)*x + E^(3 + x)*(20*x + 4
*x^2) + E^6*(25*x + 10*x^2 + x^3))/(5*E^6)), x] + 8*Defer[Int][E^((x*(2*E^x + E^3*(5 + x))^2)/(5*E^6))*x, x] +
(16*Defer[Int][E^((-30 + 35*x + 4*E^(-6 + 2*x)*x + 10*x^2 + x^3 + 4*E^(-3 + x)*x*(5 + x))/5)*x, x])/5 + (56*D
efer[Int][E^(-3 + x + (4*E^(2*x)*x + E^(3 + x)*(20*x + 4*x^2) + E^6*(25*x + 10*x^2 + x^3))/(5*E^6))*x, x])/5 +
(6*Defer[Int][E^((x*(2*E^x + E^3*(5 + x))^2)/(5*E^6))*x^2, x])/5 + (8*Defer[Int][E^(-3 + x + (4*E^(2*x)*x + E
^(3 + x)*(20*x + 4*x^2) + E^6*(25*x + 10*x^2 + x^3))/(5*E^6))*x^2, x])/5

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\frac {1}{5} \int \exp \left (-6+\frac {4 e^{2 x} x+e^{3+x} \left (20 x+4 x^2\right )+e^6 \left (25 x+10 x^2+x^3\right )}{5 e^6}\right ) \left (e^{2 x} (8+16 x)+e^6 \left (50+40 x+6 x^2\right )+e^{3+x} \left (40+56 x+8 x^2\right )\right ) \, dx\\ &=\frac {1}{5} \int \left (8 \exp \left (-6+2 x+\frac {4 e^{2 x} x+e^{3+x} \left (20 x+4 x^2\right )+e^6 \left (25 x+10 x^2+x^3\right )}{5 e^6}\right ) (1+2 x)+8 \exp \left (-3+x+\frac {4 e^{2 x} x+e^{3+x} \left (20 x+4 x^2\right )+e^6 \left (25 x+10 x^2+x^3\right )}{5 e^6}\right ) \left (5+7 x+x^2\right )+2 \exp \left (\frac {4 e^{2 x} x+e^{3+x} \left (20 x+4 x^2\right )+e^6 \left (25 x+10 x^2+x^3\right )}{5 e^6}\right ) \left (25+20 x+3 x^2\right )\right ) \, dx\\ &=\frac {2}{5} \int \exp \left (\frac {4 e^{2 x} x+e^{3+x} \left (20 x+4 x^2\right )+e^6 \left (25 x+10 x^2+x^3\right )}{5 e^6}\right ) \left (25+20 x+3 x^2\right ) \, dx+\frac {8}{5} \int \exp \left (-6+2 x+\frac {4 e^{2 x} x+e^{3+x} \left (20 x+4 x^2\right )+e^6 \left (25 x+10 x^2+x^3\right )}{5 e^6}\right ) (1+2 x) \, dx+\frac {8}{5} \int \exp \left (-3+x+\frac {4 e^{2 x} x+e^{3+x} \left (20 x+4 x^2\right )+e^6 \left (25 x+10 x^2+x^3\right )}{5 e^6}\right ) \left (5+7 x+x^2\right ) \, dx\\ &=\frac {2}{5} \int e^{\frac {x \left (2 e^x+e^3 (5+x)\right )^2}{5 e^6}} \left (25+20 x+3 x^2\right ) \, dx+\frac {8}{5} \int \exp \left (\frac {1}{5} \left (-30+35 x+4 e^{-6+2 x} x+10 x^2+x^3+4 e^{-3+x} x (5+x)\right )\right ) (1+2 x) \, dx+\frac {8}{5} \int \left (5 \exp \left (-3+x+\frac {4 e^{2 x} x+e^{3+x} \left (20 x+4 x^2\right )+e^6 \left (25 x+10 x^2+x^3\right )}{5 e^6}\right )+7 \exp \left (-3+x+\frac {4 e^{2 x} x+e^{3+x} \left (20 x+4 x^2\right )+e^6 \left (25 x+10 x^2+x^3\right )}{5 e^6}\right ) x+\exp \left (-3+x+\frac {4 e^{2 x} x+e^{3+x} \left (20 x+4 x^2\right )+e^6 \left (25 x+10 x^2+x^3\right )}{5 e^6}\right ) x^2\right ) \, dx\\ &=\frac {2}{5} \int \left (25 e^{\frac {x \left (2 e^x+e^3 (5+x)\right )^2}{5 e^6}}+20 e^{\frac {x \left (2 e^x+e^3 (5+x)\right )^2}{5 e^6}} x+3 e^{\frac {x \left (2 e^x+e^3 (5+x)\right )^2}{5 e^6}} x^2\right ) \, dx+\frac {8}{5} \int \exp \left (-3+x+\frac {4 e^{2 x} x+e^{3+x} \left (20 x+4 x^2\right )+e^6 \left (25 x+10 x^2+x^3\right )}{5 e^6}\right ) x^2 \, dx+\frac {8}{5} \int \left (\exp \left (\frac {1}{5} \left (-30+35 x+4 e^{-6+2 x} x+10 x^2+x^3+4 e^{-3+x} x (5+x)\right )\right )+2 \exp \left (\frac {1}{5} \left (-30+35 x+4 e^{-6+2 x} x+10 x^2+x^3+4 e^{-3+x} x (5+x)\right )\right ) x\right ) \, dx+8 \int \exp \left (-3+x+\frac {4 e^{2 x} x+e^{3+x} \left (20 x+4 x^2\right )+e^6 \left (25 x+10 x^2+x^3\right )}{5 e^6}\right ) \, dx+\frac {56}{5} \int \exp \left (-3+x+\frac {4 e^{2 x} x+e^{3+x} \left (20 x+4 x^2\right )+e^6 \left (25 x+10 x^2+x^3\right )}{5 e^6}\right ) x \, dx\\ &=\frac {6}{5} \int e^{\frac {x \left (2 e^x+e^3 (5+x)\right )^2}{5 e^6}} x^2 \, dx+\frac {8}{5} \int \exp \left (\frac {1}{5} \left (-30+35 x+4 e^{-6+2 x} x+10 x^2+x^3+4 e^{-3+x} x (5+x)\right )\right ) \, dx+\frac {8}{5} \int \exp \left (-3+x+\frac {4 e^{2 x} x+e^{3+x} \left (20 x+4 x^2\right )+e^6 \left (25 x+10 x^2+x^3\right )}{5 e^6}\right ) x^2 \, dx+\frac {16}{5} \int \exp \left (\frac {1}{5} \left (-30+35 x+4 e^{-6+2 x} x+10 x^2+x^3+4 e^{-3+x} x (5+x)\right )\right ) x \, dx+8 \int \exp \left (-3+x+\frac {4 e^{2 x} x+e^{3+x} \left (20 x+4 x^2\right )+e^6 \left (25 x+10 x^2+x^3\right )}{5 e^6}\right ) \, dx+8 \int e^{\frac {x \left (2 e^x+e^3 (5+x)\right )^2}{5 e^6}} x \, dx+10 \int e^{\frac {x \left (2 e^x+e^3 (5+x)\right )^2}{5 e^6}} \, dx+\frac {56}{5} \int \exp \left (-3+x+\frac {4 e^{2 x} x+e^{3+x} \left (20 x+4 x^2\right )+e^6 \left (25 x+10 x^2+x^3\right )}{5 e^6}\right ) x \, dx\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.16, size = 27, normalized size = 1.29 \begin {gather*} 2 e^{\frac {x \left (2 e^x+e^3 (5+x)\right )^2}{5 e^6}} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(E^(-6 + (4*E^(2*x)*x + E^(3 + x)*(20*x + 4*x^2) + E^6*(25*x + 10*x^2 + x^3))/(5*E^6))*(E^(2*x)*(8 +
16*x) + E^6*(50 + 40*x + 6*x^2) + E^(3 + x)*(40 + 56*x + 8*x^2)))/5,x]

[Out]

2*E^((x*(2*E^x + E^3*(5 + x))^2)/(5*E^6))

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fricas [B]  time = 2.55, size = 48, normalized size = 2.29 \begin {gather*} 2 \, e^{\left (\frac {1}{5} \, {\left ({\left (x^{3} + 10 \, x^{2} + 25 \, x - 30\right )} e^{12} + 4 \, x e^{\left (2 \, x + 6\right )} + 4 \, {\left (x^{2} + 5 \, x\right )} e^{\left (x + 9\right )}\right )} e^{\left (-12\right )} + 6\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/5*((16*x+8)*exp(x)^2+(8*x^2+56*x+40)*exp(3)*exp(x)+(6*x^2+40*x+50)*exp(3)^2)*exp(1/5*(4*x*exp(x)^2
+(4*x^2+20*x)*exp(3)*exp(x)+(x^3+10*x^2+25*x)*exp(3)^2)/exp(3)^2)/exp(3)^2,x, algorithm="fricas")

[Out]

2*e^(1/5*((x^3 + 10*x^2 + 25*x - 30)*e^12 + 4*x*e^(2*x + 6) + 4*(x^2 + 5*x)*e^(x + 9))*e^(-12) + 6)

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giac [B]  time = 0.26, size = 42, normalized size = 2.00 \begin {gather*} 2 \, e^{\left (\frac {1}{5} \, x^{3} + \frac {4}{5} \, x^{2} e^{\left (x - 3\right )} + 2 \, x^{2} + \frac {4}{5} \, x e^{\left (2 \, x - 6\right )} + 4 \, x e^{\left (x - 3\right )} + 5 \, x\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/5*((16*x+8)*exp(x)^2+(8*x^2+56*x+40)*exp(3)*exp(x)+(6*x^2+40*x+50)*exp(3)^2)*exp(1/5*(4*x*exp(x)^2
+(4*x^2+20*x)*exp(3)*exp(x)+(x^3+10*x^2+25*x)*exp(3)^2)/exp(3)^2)/exp(3)^2,x, algorithm="giac")

[Out]

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

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maple [B]  time = 0.14, size = 44, normalized size = 2.10

method	result	size
risch	\(2 \,{\mathrm e}^{\frac {x \left (x^{2} {\mathrm e}^{6}+4 \,{\mathrm e}^{3+x} x +10 x \,{\mathrm e}^{6}+20 \,{\mathrm e}^{3+x}+25 \,{\mathrm e}^{6}+4 \,{\mathrm e}^{2 x}\right ) {\mathrm e}^{-6}}{5}}\)	\(44\)
norman	\(2 \,{\mathrm e}^{\frac {\left (4 x \,{\mathrm e}^{2 x}+\left (4 x^{2}+20 x \right ) {\mathrm e}^{3} {\mathrm e}^{x}+\left (x^{3}+10 x^{2}+25 x \right ) {\mathrm e}^{6}\right ) {\mathrm e}^{-6}}{5}}\)	\(49\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/5*((16*x+8)*exp(x)^2+(8*x^2+56*x+40)*exp(3)*exp(x)+(6*x^2+40*x+50)*exp(3)^2)*exp(1/5*(4*x*exp(x)^2+(4*x^
2+20*x)*exp(3)*exp(x)+(x^3+10*x^2+25*x)*exp(3)^2)/exp(3)^2)/exp(3)^2,x,method=_RETURNVERBOSE)

[Out]

2*exp(1/5*x*(x^2*exp(6)+4*exp(3+x)*x+10*x*exp(6)+20*exp(3+x)+25*exp(6)+4*exp(2*x))*exp(-6))

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maxima [B]  time = 0.57, size = 42, normalized size = 2.00 \begin {gather*} 2 \, e^{\left (\frac {1}{5} \, x^{3} + \frac {4}{5} \, x^{2} e^{\left (x - 3\right )} + 2 \, x^{2} + \frac {4}{5} \, x e^{\left (2 \, x - 6\right )} + 4 \, x e^{\left (x - 3\right )} + 5 \, x\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/5*((16*x+8)*exp(x)^2+(8*x^2+56*x+40)*exp(3)*exp(x)+(6*x^2+40*x+50)*exp(3)^2)*exp(1/5*(4*x*exp(x)^2
+(4*x^2+20*x)*exp(3)*exp(x)+(x^3+10*x^2+25*x)*exp(3)^2)/exp(3)^2)/exp(3)^2,x, algorithm="maxima")

[Out]

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

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mupad [B]  time = 4.77, size = 46, normalized size = 2.19 \begin {gather*} 2\,{\mathrm {e}}^{5\,x}\,{\mathrm {e}}^{4\,x\,{\mathrm {e}}^{-3}\,{\mathrm {e}}^x}\,{\mathrm {e}}^{2\,x^2}\,{\mathrm {e}}^{\frac {x^3}{5}}\,{\mathrm {e}}^{\frac {4\,x\,{\mathrm {e}}^{2\,x}\,{\mathrm {e}}^{-6}}{5}}\,{\mathrm {e}}^{\frac {4\,x^2\,{\mathrm {e}}^{-3}\,{\mathrm {e}}^x}{5}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((exp(-6)*exp(exp(-6)*((exp(6)*(25*x + 10*x^2 + x^3))/5 + (4*x*exp(2*x))/5 + (exp(3)*exp(x)*(20*x + 4*x^2))
/5))*(exp(6)*(40*x + 6*x^2 + 50) + exp(2*x)*(16*x + 8) + exp(3)*exp(x)*(56*x + 8*x^2 + 40)))/5,x)

[Out]

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

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sympy [B]  time = 0.43, size = 49, normalized size = 2.33 \begin {gather*} 2 e^{\frac {\frac {4 x e^{2 x}}{5} + \frac {\left (4 x^{2} + 20 x\right ) e^{3} e^{x}}{5} + \frac {\left (x^{3} + 10 x^{2} + 25 x\right ) e^{6}}{5}}{e^{6}}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/5*((16*x+8)*exp(x)**2+(8*x**2+56*x+40)*exp(3)*exp(x)+(6*x**2+40*x+50)*exp(3)**2)*exp(1/5*(4*x*exp(
x)**2+(4*x**2+20*x)*exp(3)*exp(x)+(x**3+10*x**2+25*x)*exp(3)**2)/exp(3)**2)/exp(3)**2,x)

[Out]

2*exp((4*x*exp(2*x)/5 + (4*x**2 + 20*x)*exp(3)*exp(x)/5 + (x**3 + 10*x**2 + 25*x)*exp(6)/5)*exp(-6))

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