3.46.91 \(\int e^{4 e^{2 x}-e^x (1-4 x)+x^2} (40 e^{2 x}+e^{-4 e^{2 x}+e^x (1-4 x)-x^2}+10 x+e^x (15+20 x)) \, dx\)

Optimal. Leaf size=22 \[ 3+5 e^{-e^x+\left (2 e^x+x\right )^2}+x \]

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Rubi [F]  time = 1.94, 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 e^{4 e^{2 x}-e^x (1-4 x)+x^2} \left (40 e^{2 x}+e^{-4 e^{2 x}+e^x (1-4 x)-x^2}+10 x+e^x (15+20 x)\right ) \, dx \end {gather*}

Verification is not applicable to the result.

[In]

Int[E^(4*E^(2*x) - E^x*(1 - 4*x) + x^2)*(40*E^(2*x) + E^(-4*E^(2*x) + E^x*(1 - 4*x) - x^2) + 10*x + E^x*(15 +
20*x)),x]

[Out]

x + 15*Defer[Int][E^(4*E^(2*x) - E^x*(1 - 4*x) + x + x^2), x] + 40*Defer[Int][E^(4*E^(2*x) - E^x*(1 - 4*x) + 2
*x + x^2), x] + 10*Defer[Int][E^(4*E^(2*x) - E^x*(1 - 4*x) + x^2)*x, x] + 20*Defer[Int][E^(4*E^(2*x) - E^x*(1
- 4*x) + x + x^2)*x, x]

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\int \left (1+40 e^{4 e^{2 x}-e^x (1-4 x)+2 x+x^2}+10 e^{4 e^{2 x}-e^x (1-4 x)+x^2} x+5 e^{4 e^{2 x}-e^x (1-4 x)+x+x^2} (3+4 x)\right ) \, dx\\ &=x+5 \int e^{4 e^{2 x}-e^x (1-4 x)+x+x^2} (3+4 x) \, dx+10 \int e^{4 e^{2 x}-e^x (1-4 x)+x^2} x \, dx+40 \int e^{4 e^{2 x}-e^x (1-4 x)+2 x+x^2} \, dx\\ &=x+5 \int \left (3 e^{4 e^{2 x}-e^x (1-4 x)+x+x^2}+4 e^{4 e^{2 x}-e^x (1-4 x)+x+x^2} x\right ) \, dx+10 \int e^{4 e^{2 x}-e^x (1-4 x)+x^2} x \, dx+40 \int e^{4 e^{2 x}-e^x (1-4 x)+2 x+x^2} \, dx\\ &=x+10 \int e^{4 e^{2 x}-e^x (1-4 x)+x^2} x \, dx+15 \int e^{4 e^{2 x}-e^x (1-4 x)+x+x^2} \, dx+20 \int e^{4 e^{2 x}-e^x (1-4 x)+x+x^2} x \, dx+40 \int e^{4 e^{2 x}-e^x (1-4 x)+2 x+x^2} \, dx\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 1.60, size = 26, normalized size = 1.18 \begin {gather*} 5 e^{4 e^{2 x}+x^2+e^x (-1+4 x)}+x \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[E^(4*E^(2*x) - E^x*(1 - 4*x) + x^2)*(40*E^(2*x) + E^(-4*E^(2*x) + E^x*(1 - 4*x) - x^2) + 10*x + E^x*
(15 + 20*x)),x]

[Out]

5*E^(4*E^(2*x) + x^2 + E^x*(-1 + 4*x)) + x

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fricas [A]  time = 0.56, size = 23, normalized size = 1.05 \begin {gather*} x + 5 \, e^{\left (x^{2} + {\left (4 \, x - 1\right )} e^{x} + 4 \, e^{\left (2 \, x\right )}\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((exp(-4*exp(x)^2+(-4*x+1)*exp(x)-x^2)+40*exp(x)^2+(20*x+15)*exp(x)+10*x)/exp(-4*exp(x)^2+(-4*x+1)*ex
p(x)-x^2),x, algorithm="fricas")

[Out]

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

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giac [A]  time = 0.19, size = 24, normalized size = 1.09 \begin {gather*} x + 5 \, e^{\left (x^{2} + 4 \, x e^{x} + 4 \, e^{\left (2 \, x\right )} - e^{x}\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((exp(-4*exp(x)^2+(-4*x+1)*exp(x)-x^2)+40*exp(x)^2+(20*x+15)*exp(x)+10*x)/exp(-4*exp(x)^2+(-4*x+1)*ex
p(x)-x^2),x, algorithm="giac")

[Out]

x + 5*e^(x^2 + 4*x*e^x + 4*e^(2*x) - e^x)

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maple [A]  time = 0.07, size = 25, normalized size = 1.14

method	result	size
risch	\(x +5 \,{\mathrm e}^{4 \,{\mathrm e}^{x} x +x^{2}-{\mathrm e}^{x}+4 \,{\mathrm e}^{2 x}}\)	\(25\)
norman	\(\left (5+x \,{\mathrm e}^{-4 \,{\mathrm e}^{2 x}+\left (-4 x +1\right ) {\mathrm e}^{x}-x^{2}}\right ) {\mathrm e}^{4 \,{\mathrm e}^{2 x}-\left (-4 x +1\right ) {\mathrm e}^{x}+x^{2}}\)	\(50\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((exp(-4*exp(x)^2+(-4*x+1)*exp(x)-x^2)+40*exp(x)^2+(20*x+15)*exp(x)+10*x)/exp(-4*exp(x)^2+(-4*x+1)*exp(x)-x
^2),x,method=_RETURNVERBOSE)

[Out]

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

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maxima [A]  time = 0.47, size = 24, normalized size = 1.09 \begin {gather*} x + 5 \, e^{\left (x^{2} + 4 \, x e^{x} + 4 \, e^{\left (2 \, x\right )} - e^{x}\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((exp(-4*exp(x)^2+(-4*x+1)*exp(x)-x^2)+40*exp(x)^2+(20*x+15)*exp(x)+10*x)/exp(-4*exp(x)^2+(-4*x+1)*ex
p(x)-x^2),x, algorithm="maxima")

[Out]

x + 5*e^(x^2 + 4*x*e^x + 4*e^(2*x) - e^x)

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mupad [B]  time = 0.11, size = 26, normalized size = 1.18 \begin {gather*} x+5\,{\mathrm {e}}^{4\,{\mathrm {e}}^{2\,x}}\,{\mathrm {e}}^{4\,x\,{\mathrm {e}}^x}\,{\mathrm {e}}^{x^2}\,{\mathrm {e}}^{-{\mathrm {e}}^x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(exp(4*exp(2*x) + exp(x)*(4*x - 1) + x^2)*(10*x + exp(- 4*exp(2*x) - exp(x)*(4*x - 1) - x^2) + 40*exp(2*x)
+ exp(x)*(20*x + 15)),x)

[Out]

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

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sympy [A]  time = 0.21, size = 22, normalized size = 1.00 \begin {gather*} x + 5 e^{x^{2} - \left (1 - 4 x\right ) e^{x} + 4 e^{2 x}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((exp(-4*exp(x)**2+(-4*x+1)*exp(x)-x**2)+40*exp(x)**2+(20*x+15)*exp(x)+10*x)/exp(-4*exp(x)**2+(-4*x+1
)*exp(x)-x**2),x)

[Out]

x + 5*exp(x**2 - (1 - 4*x)*exp(x) + 4*exp(2*x))

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