3.52.27 \(\int \frac {e^{\frac {2 x^3+30 \log (x)+15 x \log ^2(x)}{10 \log (x)+5 x \log ^2(x)}} (-4 x^2+(12 x^2-4 x^3) \log (x)+4 x^3 \log ^2(x))}{20 \log ^2(x)+20 x \log ^3(x)+5 x^2 \log ^4(x)} \, dx\)

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

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Rubi [F]  time = 4.28, 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 {\exp \left (\frac {2 x^3+30 \log (x)+15 x \log ^2(x)}{10 \log (x)+5 x \log ^2(x)}\right ) \left (-4 x^2+\left (12 x^2-4 x^3\right ) \log (x)+4 x^3 \log ^2(x)\right )}{20 \log ^2(x)+20 x \log ^3(x)+5 x^2 \log ^4(x)} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

Int[(E^((2*x^3 + 30*Log[x] + 15*x*Log[x]^2)/(10*Log[x] + 5*x*Log[x]^2))*(-4*x^2 + (12*x^2 - 4*x^3)*Log[x] + 4*
x^3*Log[x]^2))/(20*Log[x]^2 + 20*x*Log[x]^3 + 5*x^2*Log[x]^4),x]

[Out]

-1/5*Defer[Int][(E^((2*x^3 + 30*Log[x] + 15*x*Log[x]^2)/(5*Log[x]*(2 + x*Log[x])))*x^2)/Log[x]^2, x] + (3*Defe
r[Int][(E^((2*x^3 + 30*Log[x] + 15*x*Log[x]^2)/(5*Log[x]*(2 + x*Log[x])))*x^2)/Log[x], x])/5 - (2*Defer[Int][(
E^((2*x^3 + 30*Log[x] + 15*x*Log[x]^2)/(5*Log[x]*(2 + x*Log[x])))*x^3)/(2 + x*Log[x])^2, x])/5 + Defer[Int][(E
^((2*x^3 + 30*Log[x] + 15*x*Log[x]^2)/(5*Log[x]*(2 + x*Log[x])))*x^4)/(2 + x*Log[x])^2, x]/5 - (3*Defer[Int][(
E^((2*x^3 + 30*Log[x] + 15*x*Log[x]^2)/(5*Log[x]*(2 + x*Log[x])))*x^3)/(2 + x*Log[x]), x])/5

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\int \frac {4 \exp \left (\frac {2 x^3+30 \log (x)+15 x \log ^2(x)}{5 \log (x) (2+x \log (x))}\right ) x^2 \left (-1+3 \log (x)-x \log (x)+x \log ^2(x)\right )}{5 \log ^2(x) (2+x \log (x))^2} \, dx\\ &=\frac {4}{5} \int \frac {\exp \left (\frac {2 x^3+30 \log (x)+15 x \log ^2(x)}{5 \log (x) (2+x \log (x))}\right ) x^2 \left (-1+3 \log (x)-x \log (x)+x \log ^2(x)\right )}{\log ^2(x) (2+x \log (x))^2} \, dx\\ &=\frac {4}{5} \int \left (-\frac {\exp \left (\frac {2 x^3+30 \log (x)+15 x \log ^2(x)}{5 \log (x) (2+x \log (x))}\right ) x^2}{4 \log ^2(x)}+\frac {3 \exp \left (\frac {2 x^3+30 \log (x)+15 x \log ^2(x)}{5 \log (x) (2+x \log (x))}\right ) x^2}{4 \log (x)}+\frac {\exp \left (\frac {2 x^3+30 \log (x)+15 x \log ^2(x)}{5 \log (x) (2+x \log (x))}\right ) (-2+x) x^3}{4 (2+x \log (x))^2}-\frac {3 \exp \left (\frac {2 x^3+30 \log (x)+15 x \log ^2(x)}{5 \log (x) (2+x \log (x))}\right ) x^3}{4 (2+x \log (x))}\right ) \, dx\\ &=-\left (\frac {1}{5} \int \frac {\exp \left (\frac {2 x^3+30 \log (x)+15 x \log ^2(x)}{5 \log (x) (2+x \log (x))}\right ) x^2}{\log ^2(x)} \, dx\right )+\frac {1}{5} \int \frac {\exp \left (\frac {2 x^3+30 \log (x)+15 x \log ^2(x)}{5 \log (x) (2+x \log (x))}\right ) (-2+x) x^3}{(2+x \log (x))^2} \, dx+\frac {3}{5} \int \frac {\exp \left (\frac {2 x^3+30 \log (x)+15 x \log ^2(x)}{5 \log (x) (2+x \log (x))}\right ) x^2}{\log (x)} \, dx-\frac {3}{5} \int \frac {\exp \left (\frac {2 x^3+30 \log (x)+15 x \log ^2(x)}{5 \log (x) (2+x \log (x))}\right ) x^3}{2+x \log (x)} \, dx\\ &=-\left (\frac {1}{5} \int \frac {\exp \left (\frac {2 x^3+30 \log (x)+15 x \log ^2(x)}{5 \log (x) (2+x \log (x))}\right ) x^2}{\log ^2(x)} \, dx\right )+\frac {1}{5} \int \left (-\frac {2 \exp \left (\frac {2 x^3+30 \log (x)+15 x \log ^2(x)}{5 \log (x) (2+x \log (x))}\right ) x^3}{(2+x \log (x))^2}+\frac {\exp \left (\frac {2 x^3+30 \log (x)+15 x \log ^2(x)}{5 \log (x) (2+x \log (x))}\right ) x^4}{(2+x \log (x))^2}\right ) \, dx+\frac {3}{5} \int \frac {\exp \left (\frac {2 x^3+30 \log (x)+15 x \log ^2(x)}{5 \log (x) (2+x \log (x))}\right ) x^2}{\log (x)} \, dx-\frac {3}{5} \int \frac {\exp \left (\frac {2 x^3+30 \log (x)+15 x \log ^2(x)}{5 \log (x) (2+x \log (x))}\right ) x^3}{2+x \log (x)} \, dx\\ &=-\left (\frac {1}{5} \int \frac {\exp \left (\frac {2 x^3+30 \log (x)+15 x \log ^2(x)}{5 \log (x) (2+x \log (x))}\right ) x^2}{\log ^2(x)} \, dx\right )+\frac {1}{5} \int \frac {\exp \left (\frac {2 x^3+30 \log (x)+15 x \log ^2(x)}{5 \log (x) (2+x \log (x))}\right ) x^4}{(2+x \log (x))^2} \, dx-\frac {2}{5} \int \frac {\exp \left (\frac {2 x^3+30 \log (x)+15 x \log ^2(x)}{5 \log (x) (2+x \log (x))}\right ) x^3}{(2+x \log (x))^2} \, dx+\frac {3}{5} \int \frac {\exp \left (\frac {2 x^3+30 \log (x)+15 x \log ^2(x)}{5 \log (x) (2+x \log (x))}\right ) x^2}{\log (x)} \, dx-\frac {3}{5} \int \frac {\exp \left (\frac {2 x^3+30 \log (x)+15 x \log ^2(x)}{5 \log (x) (2+x \log (x))}\right ) x^3}{2+x \log (x)} \, dx\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 1.18, size = 34, normalized size = 1.36 \begin {gather*} e^{\frac {2 x^3+30 \log (x)+15 x \log ^2(x)}{10 \log (x)+5 x \log ^2(x)}} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(E^((2*x^3 + 30*Log[x] + 15*x*Log[x]^2)/(10*Log[x] + 5*x*Log[x]^2))*(-4*x^2 + (12*x^2 - 4*x^3)*Log[x
] + 4*x^3*Log[x]^2))/(20*Log[x]^2 + 20*x*Log[x]^3 + 5*x^2*Log[x]^4),x]

[Out]

E^((2*x^3 + 30*Log[x] + 15*x*Log[x]^2)/(10*Log[x] + 5*x*Log[x]^2))

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fricas [A]  time = 0.61, size = 33, normalized size = 1.32 \begin {gather*} e^{\left (\frac {2 \, x^{3} + 15 \, x \log \relax (x)^{2} + 30 \, \log \relax (x)}{5 \, {\left (x \log \relax (x)^{2} + 2 \, \log \relax (x)\right )}}\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((4*x^3*log(x)^2+(-4*x^3+12*x^2)*log(x)-4*x^2)*exp((15*x*log(x)^2+30*log(x)+2*x^3)/(5*x*log(x)^2+10*l
og(x)))/(5*x^2*log(x)^4+20*x*log(x)^3+20*log(x)^2),x, algorithm="fricas")

[Out]

e^(1/5*(2*x^3 + 15*x*log(x)^2 + 30*log(x))/(x*log(x)^2 + 2*log(x)))

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giac [B]  time = 0.31, size = 57, normalized size = 2.28 \begin {gather*} e^{\left (\frac {2 \, x^{3}}{5 \, {\left (x \log \relax (x)^{2} + 2 \, \log \relax (x)\right )}} + \frac {3 \, x \log \relax (x)^{2}}{x \log \relax (x)^{2} + 2 \, \log \relax (x)} + \frac {6 \, \log \relax (x)}{x \log \relax (x)^{2} + 2 \, \log \relax (x)}\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((4*x^3*log(x)^2+(-4*x^3+12*x^2)*log(x)-4*x^2)*exp((15*x*log(x)^2+30*log(x)+2*x^3)/(5*x*log(x)^2+10*l
og(x)))/(5*x^2*log(x)^4+20*x*log(x)^3+20*log(x)^2),x, algorithm="giac")

[Out]

e^(2/5*x^3/(x*log(x)^2 + 2*log(x)) + 3*x*log(x)^2/(x*log(x)^2 + 2*log(x)) + 6*log(x)/(x*log(x)^2 + 2*log(x)))

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maple [A]  time = 0.03, size = 33, normalized size = 1.32

method	result	size
risch	\({\mathrm e}^{\frac {15 x \ln \relax (x )^{2}+30 \ln \relax (x )+2 x^{3}}{5 \ln \relax (x ) \left (x \ln \relax (x )+2\right )}}\)	\(33\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((4*x^3*ln(x)^2+(-4*x^3+12*x^2)*ln(x)-4*x^2)*exp((15*x*ln(x)^2+30*ln(x)+2*x^3)/(5*x*ln(x)^2+10*ln(x)))/(5*x
^2*ln(x)^4+20*x*ln(x)^3+20*ln(x)^2),x,method=_RETURNVERBOSE)

[Out]

exp(1/5*(15*x*ln(x)^2+30*ln(x)+2*x^3)/ln(x)/(x*ln(x)+2))

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maxima [A]  time = 0.52, size = 42, normalized size = 1.68 \begin {gather*} e^{\left (\frac {2 \, x^{2}}{5 \, \log \relax (x)^{2}} - \frac {16}{5 \, {\left (x \log \relax (x)^{5} + 2 \, \log \relax (x)^{4}\right )}} - \frac {4 \, x}{5 \, \log \relax (x)^{3}} + \frac {8}{5 \, \log \relax (x)^{4}} + 3\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((4*x^3*log(x)^2+(-4*x^3+12*x^2)*log(x)-4*x^2)*exp((15*x*log(x)^2+30*log(x)+2*x^3)/(5*x*log(x)^2+10*l
og(x)))/(5*x^2*log(x)^4+20*x*log(x)^3+20*log(x)^2),x, algorithm="maxima")

[Out]

e^(2/5*x^2/log(x)^2 - 16/5/(x*log(x)^5 + 2*log(x)^4) - 4/5*x/log(x)^3 + 8/5/log(x)^4 + 3)

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mupad [B]  time = 3.42, size = 51, normalized size = 2.04 \begin {gather*} x^{\frac {3\,x}{x\,\ln \relax (x)+2}}\,x^{\frac {6}{x\,{\ln \relax (x)}^2+2\,\ln \relax (x)}}\,{\mathrm {e}}^{\frac {2\,x^3}{5\,x\,{\ln \relax (x)}^2+10\,\ln \relax (x)}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((exp((30*log(x) + 15*x*log(x)^2 + 2*x^3)/(10*log(x) + 5*x*log(x)^2))*(log(x)*(12*x^2 - 4*x^3) + 4*x^3*log(
x)^2 - 4*x^2))/(20*x*log(x)^3 + 20*log(x)^2 + 5*x^2*log(x)^4),x)

[Out]

x^((3*x)/(x*log(x) + 2))*x^(6/(2*log(x) + x*log(x)^2))*exp((2*x^3)/(10*log(x) + 5*x*log(x)^2))

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sympy [A]  time = 0.48, size = 32, normalized size = 1.28 \begin {gather*} e^{\frac {2 x^{3} + 15 x \log {\relax (x )}^{2} + 30 \log {\relax (x )}}{5 x \log {\relax (x )}^{2} + 10 \log {\relax (x )}}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((4*x**3*ln(x)**2+(-4*x**3+12*x**2)*ln(x)-4*x**2)*exp((15*x*ln(x)**2+30*ln(x)+2*x**3)/(5*x*ln(x)**2+1
0*ln(x)))/(5*x**2*ln(x)**4+20*x*ln(x)**3+20*ln(x)**2),x)

[Out]

exp((2*x**3 + 15*x*log(x)**2 + 30*log(x))/(5*x*log(x)**2 + 10*log(x)))

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