3.38.87 \(\int \frac {1}{2} e^{\frac {1}{6} (4-9 x)} (10 e^{5 x+\frac {1}{6} (-4+9 x)}+e^{e^{\frac {1}{6} (4-9 x)} x} (2-3 x)) \, dx\)

Optimal. Leaf size=28 \[ 25+e^{5 x}+e^{e^{\frac {1}{2} \left (\frac {4}{3}-x\right )-x} x} \]

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Rubi [F]  time = 0.55, 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}{2} e^{\frac {1}{6} (4-9 x)} \left (10 e^{5 x+\frac {1}{6} (-4+9 x)}+e^{e^{\frac {1}{6} (4-9 x)} x} (2-3 x)\right ) \, dx \end {gather*}

Verification is not applicable to the result.

[In]

Int[(E^((4 - 9*x)/6)*(10*E^(5*x + (-4 + 9*x)/6) + E^(E^((4 - 9*x)/6)*x)*(2 - 3*x)))/2,x]

[Out]

E^(5*x) + Defer[Int][E^(2/3 + (-3/2 + E^(2/3 - (3*x)/2))*x), x] - (3*Defer[Int][E^(2/3 + (-3/2 + E^(2/3 - (3*x
)/2))*x)*x, x])/2

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\frac {1}{2} \int e^{\frac {1}{6} (4-9 x)} \left (10 e^{5 x+\frac {1}{6} (-4+9 x)}+e^{e^{\frac {1}{6} (4-9 x)} x} (2-3 x)\right ) \, dx\\ &=\frac {1}{2} \int \left (10 e^{5 x}+e^{\frac {2}{3}+\left (-\frac {3}{2}+e^{\frac {2}{3}-\frac {3 x}{2}}\right ) x} (2-3 x)\right ) \, dx\\ &=\frac {1}{2} \int e^{\frac {2}{3}+\left (-\frac {3}{2}+e^{\frac {2}{3}-\frac {3 x}{2}}\right ) x} (2-3 x) \, dx+5 \int e^{5 x} \, dx\\ &=e^{5 x}+\frac {1}{2} \int \left (2 e^{\frac {2}{3}+\left (-\frac {3}{2}+e^{\frac {2}{3}-\frac {3 x}{2}}\right ) x}-3 e^{\frac {2}{3}+\left (-\frac {3}{2}+e^{\frac {2}{3}-\frac {3 x}{2}}\right ) x} x\right ) \, dx\\ &=e^{5 x}-\frac {3}{2} \int e^{\frac {2}{3}+\left (-\frac {3}{2}+e^{\frac {2}{3}-\frac {3 x}{2}}\right ) x} x \, dx+\int e^{\frac {2}{3}+\left (-\frac {3}{2}+e^{\frac {2}{3}-\frac {3 x}{2}}\right ) x} \, dx\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.23, size = 21, normalized size = 0.75 \begin {gather*} e^{5 x}+e^{e^{\frac {2}{3}-\frac {3 x}{2}} x} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(E^((4 - 9*x)/6)*(10*E^(5*x + (-4 + 9*x)/6) + E^(E^((4 - 9*x)/6)*x)*(2 - 3*x)))/2,x]

[Out]

E^(5*x) + E^(E^(2/3 - (3*x)/2)*x)

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fricas [A]  time = 0.80, size = 24, normalized size = 0.86 \begin {gather*} {\left (e^{\frac {20}{9}} + e^{\left (x e^{\left (-\frac {3}{2} \, x + \frac {2}{3}\right )} - 5 \, x + \frac {20}{9}\right )}\right )} e^{\left (5 \, x - \frac {20}{9}\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/2*((-3*x+2)*exp(x/exp(3/2*x-2/3))+10*exp(3/2*x-2/3)*exp(5*x))/exp(3/2*x-2/3),x, algorithm="fricas"
)

[Out]

(e^(20/9) + e^(x*e^(-3/2*x + 2/3) - 5*x + 20/9))*e^(5*x - 20/9)

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giac [A]  time = 0.19, size = 28, normalized size = 1.00 \begin {gather*} {\left (e^{\left (x e^{\left (-\frac {3}{2} \, x + \frac {2}{3}\right )} - \frac {3}{2} \, x + \frac {2}{3}\right )} + e^{\left (\frac {7}{2} \, x + \frac {2}{3}\right )}\right )} e^{\left (\frac {3}{2} \, x - \frac {2}{3}\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/2*((-3*x+2)*exp(x/exp(3/2*x-2/3))+10*exp(3/2*x-2/3)*exp(5*x))/exp(3/2*x-2/3),x, algorithm="giac")

[Out]

(e^(x*e^(-3/2*x + 2/3) - 3/2*x + 2/3) + e^(7/2*x + 2/3))*e^(3/2*x - 2/3)

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maple [A]  time = 0.09, size = 15, normalized size = 0.54

method	result	size
risch	\({\mathrm e}^{5 x}+{\mathrm e}^{x \,{\mathrm e}^{-\frac {3 x}{2}+\frac {2}{3}}}\)	\(15\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/2*((-3*x+2)*exp(x/exp(3/2*x-2/3))+10*exp(3/2*x-2/3)*exp(5*x))/exp(3/2*x-2/3),x,method=_RETURNVERBOSE)

[Out]

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

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maxima [A]  time = 0.46, size = 14, normalized size = 0.50 \begin {gather*} e^{\left (x e^{\left (-\frac {3}{2} \, x + \frac {2}{3}\right )}\right )} + e^{\left (5 \, x\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/2*((-3*x+2)*exp(x/exp(3/2*x-2/3))+10*exp(3/2*x-2/3)*exp(5*x))/exp(3/2*x-2/3),x, algorithm="maxima"
)

[Out]

e^(x*e^(-3/2*x + 2/3)) + e^(5*x)

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mupad [B]  time = 2.41, size = 14, normalized size = 0.50 \begin {gather*} {\mathrm {e}}^{5\,x}+{\mathrm {e}}^{\frac {x\,{\mathrm {e}}^{2/3}}{{\left ({\mathrm {e}}^x\right )}^{3/2}}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(exp(2/3 - (3*x)/2)*(5*exp(5*x)*exp((3*x)/2 - 2/3) - (exp(x*exp(2/3 - (3*x)/2))*(3*x - 2))/2),x)

[Out]

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

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sympy [A]  time = 0.32, size = 20, normalized size = 0.71 \begin {gather*} e^{5 x} + e^{\frac {x e^{\frac {2}{3}}}{\left (e^{5 x}\right )^{\frac {3}{10}}}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/2*((-3*x+2)*exp(x/exp(3/2*x-2/3))+10*exp(3/2*x-2/3)*exp(5*x))/exp(3/2*x-2/3),x)

[Out]

exp(5*x) + exp(x*exp(2/3)/exp(5*x)**(3/10))

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