3.81.34 \(\int \frac {e^{1+(3-5 e^{\frac {3 x^3}{2}}+x) \log (x)} (6-10 e^{\frac {3 x^3}{2}}+2 x+(2 x-45 e^{\frac {3 x^3}{2}} x^3) \log (x))}{2 x} \, dx\)

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

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Rubi [F]  time = 2.34, 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 {e^{1+\left (3-5 e^{\frac {3 x^3}{2}}+x\right ) \log (x)} \left (6-10 e^{\frac {3 x^3}{2}}+2 x+\left (2 x-45 e^{\frac {3 x^3}{2}} x^3\right ) \log (x)\right )}{2 x} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

Int[(E^(1 + (3 - 5*E^((3*x^3)/2) + x)*Log[x])*(6 - 10*E^((3*x^3)/2) + 2*x + (2*x - 45*E^((3*x^3)/2)*x^3)*Log[x
]))/(2*x),x]

[Out]

-5*Defer[Int][E^(1 + (3*x^3)/2 + (3 - 5*E^((3*x^3)/2) + x)*Log[x])/x, x] + 3*E*Defer[Int][x^(2 - 5*E^((3*x^3)/
2) + x), x] + E*Defer[Int][x^(3 - 5*E^((3*x^3)/2) + x), x] + E*Log[x]*Defer[Int][x^(3 - 5*E^((3*x^3)/2) + x),
x] - (45*Defer[Int][E^(1 + (3*x^3)/2 + (3 - 5*E^((3*x^3)/2) + x)*Log[x])*x^2*Log[x], x])/2 - E*Defer[Int][Defe
r[Int][x^(3 - 5*E^((3*x^3)/2) + x), x]/x, x]

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\frac {1}{2} \int \frac {e^{1+\left (3-5 e^{\frac {3 x^3}{2}}+x\right ) \log (x)} \left (6-10 e^{\frac {3 x^3}{2}}+2 x+\left (2 x-45 e^{\frac {3 x^3}{2}} x^3\right ) \log (x)\right )}{x} \, dx\\ &=\frac {1}{2} \int \left (\frac {2 e^{1+\left (3-5 e^{\frac {3 x^3}{2}}+x\right ) \log (x)} (3+x+x \log (x))}{x}-\frac {5 e^{1+\frac {3 x^3}{2}+\left (3-5 e^{\frac {3 x^3}{2}}+x\right ) \log (x)} \left (2+9 x^3 \log (x)\right )}{x}\right ) \, dx\\ &=-\left (\frac {5}{2} \int \frac {e^{1+\frac {3 x^3}{2}+\left (3-5 e^{\frac {3 x^3}{2}}+x\right ) \log (x)} \left (2+9 x^3 \log (x)\right )}{x} \, dx\right )+\int \frac {e^{1+\left (3-5 e^{\frac {3 x^3}{2}}+x\right ) \log (x)} (3+x+x \log (x))}{x} \, dx\\ &=-\left (\frac {5}{2} \int \left (\frac {2 e^{1+\frac {3 x^3}{2}+\left (3-5 e^{\frac {3 x^3}{2}}+x\right ) \log (x)}}{x}+9 e^{1+\frac {3 x^3}{2}+\left (3-5 e^{\frac {3 x^3}{2}}+x\right ) \log (x)} x^2 \log (x)\right ) \, dx\right )+\int e x^{2-5 e^{\frac {3 x^3}{2}}+x} (3+x+x \log (x)) \, dx\\ &=-\left (5 \int \frac {e^{1+\frac {3 x^3}{2}+\left (3-5 e^{\frac {3 x^3}{2}}+x\right ) \log (x)}}{x} \, dx\right )-\frac {45}{2} \int e^{1+\frac {3 x^3}{2}+\left (3-5 e^{\frac {3 x^3}{2}}+x\right ) \log (x)} x^2 \log (x) \, dx+e \int x^{2-5 e^{\frac {3 x^3}{2}}+x} (3+x+x \log (x)) \, dx\\ &=-\left (5 \int \frac {e^{1+\frac {3 x^3}{2}+\left (3-5 e^{\frac {3 x^3}{2}}+x\right ) \log (x)}}{x} \, dx\right )-\frac {45}{2} \int e^{1+\frac {3 x^3}{2}+\left (3-5 e^{\frac {3 x^3}{2}}+x\right ) \log (x)} x^2 \log (x) \, dx+e \int \left (3 x^{2-5 e^{\frac {3 x^3}{2}}+x}+x^{3-5 e^{\frac {3 x^3}{2}}+x}+x^{3-5 e^{\frac {3 x^3}{2}}+x} \log (x)\right ) \, dx\\ &=-\left (5 \int \frac {e^{1+\frac {3 x^3}{2}+\left (3-5 e^{\frac {3 x^3}{2}}+x\right ) \log (x)}}{x} \, dx\right )-\frac {45}{2} \int e^{1+\frac {3 x^3}{2}+\left (3-5 e^{\frac {3 x^3}{2}}+x\right ) \log (x)} x^2 \log (x) \, dx+e \int x^{3-5 e^{\frac {3 x^3}{2}}+x} \, dx+e \int x^{3-5 e^{\frac {3 x^3}{2}}+x} \log (x) \, dx+(3 e) \int x^{2-5 e^{\frac {3 x^3}{2}}+x} \, dx\\ &=-\left (5 \int \frac {e^{1+\frac {3 x^3}{2}+\left (3-5 e^{\frac {3 x^3}{2}}+x\right ) \log (x)}}{x} \, dx\right )-\frac {45}{2} \int e^{1+\frac {3 x^3}{2}+\left (3-5 e^{\frac {3 x^3}{2}}+x\right ) \log (x)} x^2 \log (x) \, dx+e \int x^{3-5 e^{\frac {3 x^3}{2}}+x} \, dx-e \int \frac {\int x^{3-5 e^{\frac {3 x^3}{2}}+x} \, dx}{x} \, dx+(3 e) \int x^{2-5 e^{\frac {3 x^3}{2}}+x} \, dx+(e \log (x)) \int x^{3-5 e^{\frac {3 x^3}{2}}+x} \, dx\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.20, size = 18, normalized size = 0.86 \begin {gather*} e x^{3-5 e^{\frac {3 x^3}{2}}+x} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(E^(1 + (3 - 5*E^((3*x^3)/2) + x)*Log[x])*(6 - 10*E^((3*x^3)/2) + 2*x + (2*x - 45*E^((3*x^3)/2)*x^3)
*Log[x]))/(2*x),x]

[Out]

E*x^(3 - 5*E^((3*x^3)/2) + x)

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fricas [A]  time = 0.82, size = 17, normalized size = 0.81 \begin {gather*} e^{\left ({\left (x - 5 \, e^{\left (\frac {3}{2} \, x^{3}\right )} + 3\right )} \log \relax (x) + 1\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/2*((-45*x^3*exp(3/2*x^3)+2*x)*log(x)-10*exp(3/2*x^3)+2*x+6)*exp((-5*exp(3/2*x^3)+3+x)*log(x)+1)/x,
x, algorithm="fricas")

[Out]

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

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giac [A]  time = 0.29, size = 21, normalized size = 1.00 \begin {gather*} e^{\left (x \log \relax (x) - 5 \, e^{\left (\frac {3}{2} \, x^{3}\right )} \log \relax (x) + 3 \, \log \relax (x) + 1\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/2*((-45*x^3*exp(3/2*x^3)+2*x)*log(x)-10*exp(3/2*x^3)+2*x+6)*exp((-5*exp(3/2*x^3)+3+x)*log(x)+1)/x,
x, algorithm="giac")

[Out]

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

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maple [A]  time = 0.09, size = 17, normalized size = 0.81

method	result	size
risch	\(x^{-5 \,{\mathrm e}^{\frac {3 x^{3}}{2}}+3+x} {\mathrm e}\)	\(17\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/2*((-45*x^3*exp(3/2*x^3)+2*x)*ln(x)-10*exp(3/2*x^3)+2*x+6)*exp((-5*exp(3/2*x^3)+3+x)*ln(x)+1)/x,x,method
=_RETURNVERBOSE)

[Out]

x^(-5*exp(3/2*x^3)+3+x)*exp(1)

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maxima [A]  time = 0.45, size = 21, normalized size = 1.00 \begin {gather*} x^{3} e^{\left (x \log \relax (x) - 5 \, e^{\left (\frac {3}{2} \, x^{3}\right )} \log \relax (x) + 1\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/2*((-45*x^3*exp(3/2*x^3)+2*x)*log(x)-10*exp(3/2*x^3)+2*x+6)*exp((-5*exp(3/2*x^3)+3+x)*log(x)+1)/x,
x, algorithm="maxima")

[Out]

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

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mupad [B]  time = 5.96, size = 21, normalized size = 1.00 \begin {gather*} \frac {x^x\,x^3\,\mathrm {e}}{x^{5\,{\mathrm {e}}^{\frac {3\,x^3}{2}}}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((exp(log(x)*(x - 5*exp((3*x^3)/2) + 3) + 1)*(2*x - 10*exp((3*x^3)/2) + log(x)*(2*x - 45*x^3*exp((3*x^3)/2)
) + 6))/(2*x),x)

[Out]

(x^x*x^3*exp(1))/x^(5*exp((3*x^3)/2))

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sympy [A]  time = 0.59, size = 19, normalized size = 0.90 \begin {gather*} e^{\left (x - 5 e^{\frac {3 x^{3}}{2}} + 3\right ) \log {\relax (x )} + 1} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/2*((-45*x**3*exp(3/2*x**3)+2*x)*ln(x)-10*exp(3/2*x**3)+2*x+6)*exp((-5*exp(3/2*x**3)+3+x)*ln(x)+1)/
x,x)

[Out]

exp((x - 5*exp(3*x**3/2) + 3)*log(x) + 1)

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