3.59.85 \(\int \frac {-8 e^{2 x^4} x^5+e^{\frac {3+x+x \log (5)+(3+x+x \log (5)) \log (x)}{x}} (x+x \log (5)-3 \log (x))}{x^2} \, dx\)

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

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Rubi [F]  time = 0.35, 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 {-8 e^{2 x^4} x^5+e^{\frac {3+x+x \log (5)+(3+x+x \log (5)) \log (x)}{x}} (x+x \log (5)-3 \log (x))}{x^2} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

Int[(-8*E^(2*x^4)*x^5 + E^((3 + x + x*Log[5] + (3 + x + x*Log[5])*Log[x])/x)*(x + x*Log[5] - 3*Log[x]))/x^2,x]

[Out]

-E^(2*x^4) - 15*Log[x]*Defer[Int][E^(1 + 3/x)*x^(-1 + 3/x + Log[5]), x] + 5*(1 + Log[5])*Defer[Int][E^(1 + 3/x
)*x^(3/x + Log[5]), x] + 15*Defer[Int][Defer[Int][E^(1 + 3/x)*x^(-1 + 3/x + Log[5]), x]/x, x]

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\int \left (-8 e^{2 x^4} x^3+5 e^{1+\frac {3}{x}} x^{-1+\frac {3}{x}+\log (5)} (x (1+\log (5))-3 \log (x))\right ) \, dx\\ &=5 \int e^{1+\frac {3}{x}} x^{-1+\frac {3}{x}+\log (5)} (x (1+\log (5))-3 \log (x)) \, dx-8 \int e^{2 x^4} x^3 \, dx\\ &=-e^{2 x^4}+5 \int \left (e^{1+\frac {3}{x}} x^{\frac {3}{x}+\log (5)} (1+\log (5))-3 e^{1+\frac {3}{x}} x^{-1+\frac {3}{x}+\log (5)} \log (x)\right ) \, dx\\ &=-e^{2 x^4}-15 \int e^{1+\frac {3}{x}} x^{-1+\frac {3}{x}+\log (5)} \log (x) \, dx+(5 (1+\log (5))) \int e^{1+\frac {3}{x}} x^{\frac {3}{x}+\log (5)} \, dx\\ &=-e^{2 x^4}+15 \int \frac {\int e^{1+\frac {3}{x}} x^{-1+\frac {3}{x}+\log (5)} \, dx}{x} \, dx+(5 (1+\log (5))) \int e^{1+\frac {3}{x}} x^{\frac {3}{x}+\log (5)} \, dx-(15 \log (x)) \int e^{1+\frac {3}{x}} x^{-1+\frac {3}{x}+\log (5)} \, dx\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.20, size = 35, normalized size = 1.25 \begin {gather*} -e^{2 x^4}+5 e^{1+\frac {3}{x}} x^{1+\frac {3+x \log (5)}{x}} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(-8*E^(2*x^4)*x^5 + E^((3 + x + x*Log[5] + (3 + x + x*Log[5])*Log[x])/x)*(x + x*Log[5] - 3*Log[x]))/
x^2,x]

[Out]

-E^(2*x^4) + 5*E^(1 + 3/x)*x^(1 + (3 + x*Log[5])/x)

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fricas [A]  time = 0.59, size = 31, normalized size = 1.11 \begin {gather*} -e^{\left (2 \, x^{4}\right )} + e^{\left (\frac {x \log \relax (5) + {\left (x \log \relax (5) + x + 3\right )} \log \relax (x) + x + 3}{x}\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-3*log(x)+x*log(5)+x)*exp(((x*log(5)+3+x)*log(x)+x*log(5)+3+x)/x)-8*x^5*exp(x^4)^2)/x^2,x, algorit
hm="fricas")

[Out]

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

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giac [A]  time = 0.22, size = 35, normalized size = 1.25 \begin {gather*} -e^{\left (2 \, x^{4}\right )} + e^{\left (\frac {x \log \relax (5) \log \relax (x) + x \log \relax (5) + x \log \relax (x) + x + 3 \, \log \relax (x) + 3}{x}\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-3*log(x)+x*log(5)+x)*exp(((x*log(5)+3+x)*log(x)+x*log(5)+3+x)/x)-8*x^5*exp(x^4)^2)/x^2,x, algorit
hm="giac")

[Out]

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

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maple [A]  time = 0.13, size = 26, normalized size = 0.93

method	result	size
risch	\({\mathrm e}^{\frac {\left (\ln \relax (x )+1\right ) \left (x \ln \relax (5)+3+x \right )}{x}}-{\mathrm e}^{2 x^{4}}\)	\(26\)
default	\({\mathrm e}^{\frac {\left (x \ln \relax (5)+3+x \right ) \ln \relax (x )+x \ln \relax (5)+3+x}{x}}-{\mathrm e}^{2 x^{4}}\)	\(32\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((-3*ln(x)+x*ln(5)+x)*exp(((x*ln(5)+3+x)*ln(x)+x*ln(5)+3+x)/x)-8*x^5*exp(x^4)^2)/x^2,x,method=_RETURNVERBO
SE)

[Out]

exp((ln(x)+1)*(x*ln(5)+3+x)/x)-exp(2*x^4)

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maxima [A]  time = 0.53, size = 32, normalized size = 1.14 \begin {gather*} 5 \, x e^{\left (\log \relax (5) \log \relax (x) + \frac {3 \, \log \relax (x)}{x} + \frac {3}{x} + 1\right )} - e^{\left (2 \, x^{4}\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-3*log(x)+x*log(5)+x)*exp(((x*log(5)+3+x)*log(x)+x*log(5)+3+x)/x)-8*x^5*exp(x^4)^2)/x^2,x, algorit
hm="maxima")

[Out]

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

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mupad [B]  time = 4.09, size = 31, normalized size = 1.11 \begin {gather*} 5\,x\,x^{3/x}\,x^{\ln \relax (5)}\,\mathrm {e}\,{\mathrm {e}}^{3/x}-{\mathrm {e}}^{2\,x^4} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(-(8*x^5*exp(2*x^4) - exp((x + x*log(5) + log(x)*(x + x*log(5) + 3) + 3)/x)*(x - 3*log(x) + x*log(5)))/x^2,
x)

[Out]

5*x*x^(3/x)*x^log(5)*exp(1)*exp(3/x) - exp(2*x^4)

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sympy [A]  time = 0.49, size = 29, normalized size = 1.04 \begin {gather*} - e^{2 x^{4}} + e^{\frac {x + x \log {\relax (5 )} + \left (x + x \log {\relax (5 )} + 3\right ) \log {\relax (x )} + 3}{x}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-3*ln(x)+x*ln(5)+x)*exp(((x*ln(5)+3+x)*ln(x)+x*ln(5)+3+x)/x)-8*x**5*exp(x**4)**2)/x**2,x)

[Out]

-exp(2*x**4) + exp((x + x*log(5) + (x + x*log(5) + 3)*log(x) + 3)/x)

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