3.26.92 \(\int \frac {120+360 x^3+90 x^4+e^{x^{x/2}} (30+120 x^3+x^{x/2} (15-15 x-15 x^4+(15-15 x-15 x^4) \log (x)))}{18+2 e^{2 x^{x/2}}+12 x+2 x^2+e^{x^{x/2}} (12+4 x)} \, dx\)

Optimal. Leaf size=24 \[ 5+\frac {15 \left (-1+x+x^4\right )}{3+e^{x^{x/2}}+x} \]

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Rubi [F]  time = 7.32, 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 {120+360 x^3+90 x^4+e^{x^{x/2}} \left (30+120 x^3+x^{x/2} \left (15-15 x-15 x^4+\left (15-15 x-15 x^4\right ) \log (x)\right )\right )}{18+2 e^{2 x^{x/2}}+12 x+2 x^2+e^{x^{x/2}} (12+4 x)} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

Int[(120 + 360*x^3 + 90*x^4 + E^x^(x/2)*(30 + 120*x^3 + x^(x/2)*(15 - 15*x - 15*x^4 + (15 - 15*x - 15*x^4)*Log
[x])))/(18 + 2*E^(2*x^(x/2)) + 12*x + 2*x^2 + E^x^(x/2)*(12 + 4*x)),x]

[Out]

180*Defer[Int][x^3/(3 + E^x^(x/2) + x)^2, x] + 60*Defer[Int][(E^x^(x/2)*x^3)/(3 + E^x^(x/2) + x)^2, x] + 45*De
fer[Int][x^4/(3 + E^x^(x/2) + x)^2, x] + 120*Defer[Subst][Defer[Int][(3 + E^(2^x*x^x) + 2*x)^(-2), x], x, x/2]
+ 30*Defer[Subst][Defer[Int][E^(2^x*x^x)/(3 + E^(2^x*x^x) + 2*x)^2, x], x, x/2] + 15*Defer[Subst][Defer[Int][
(2^x*E^(2^x*x^x)*x^x)/(3 + E^(2^x*x^x) + 2*x)^2, x], x, x/2] + 15*Log[x]*Defer[Subst][Defer[Int][(2^x*E^(2^x*x
^x)*x^x)/(3 + E^(2^x*x^x) + 2*x)^2, x], x, x/2] - 15*Defer[Subst][Defer[Int][(2^(1 + x)*E^(2^x*x^x)*x^(1 + x))
/(3 + E^(2^x*x^x) + 2*x)^2, x], x, x/2] - 15*Log[x]*Defer[Subst][Defer[Int][(2^(1 + x)*E^(2^x*x^x)*x^(1 + x))/
(3 + E^(2^x*x^x) + 2*x)^2, x], x, x/2] - 15*Defer[Subst][Defer[Int][(2^(4 + x)*E^(2^x*x^x)*x^(4 + x))/(3 + E^(
2^x*x^x) + 2*x)^2, x], x, x/2] - 15*Log[x]*Defer[Subst][Defer[Int][(2^(4 + x)*E^(2^x*x^x)*x^(4 + x))/(3 + E^(2
^x*x^x) + 2*x)^2, x], x, x/2] - 15*Defer[Subst][Defer[Int][Defer[Int][(2^x*E^(2^x*x^x)*x^x)/(3 + E^(2^x*x^x) +
2*x)^2, x]/x, x], x, x/2] + 15*Defer[Subst][Defer[Int][Defer[Int][(2^(1 + x)*E^(2^x*x^x)*x^(1 + x))/(3 + E^(2
^x*x^x) + 2*x)^2, x]/x, x], x, x/2] + 15*Defer[Subst][Defer[Int][Defer[Int][(2^(4 + x)*E^(2^x*x^x)*x^(4 + x))/
(3 + E^(2^x*x^x) + 2*x)^2, x]/x, x], x, x/2]

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\int \frac {120+360 x^3+90 x^4+e^{x^{x/2}} \left (30+120 x^3+x^{x/2} \left (15-15 x-15 x^4+\left (15-15 x-15 x^4\right ) \log (x)\right )\right )}{2 \left (3+e^{x^{x/2}}+x\right )^2} \, dx\\ &=\frac {1}{2} \int \frac {120+360 x^3+90 x^4+e^{x^{x/2}} \left (30+120 x^3+x^{x/2} \left (15-15 x-15 x^4+\left (15-15 x-15 x^4\right ) \log (x)\right )\right )}{\left (3+e^{x^{x/2}}+x\right )^2} \, dx\\ &=\frac {1}{2} \int \left (\frac {120}{\left (3+e^{x^{x/2}}+x\right )^2}+\frac {30 e^{x^{x/2}}}{\left (3+e^{x^{x/2}}+x\right )^2}+\frac {360 x^3}{\left (3+e^{x^{x/2}}+x\right )^2}+\frac {120 e^{x^{x/2}} x^3}{\left (3+e^{x^{x/2}}+x\right )^2}+\frac {90 x^4}{\left (3+e^{x^{x/2}}+x\right )^2}-\frac {15 e^{x^{x/2}} x^{x/2} \left (-1+x+x^4\right ) (1+\log (x))}{\left (3+e^{x^{x/2}}+x\right )^2}\right ) \, dx\\ &=-\left (\frac {15}{2} \int \frac {e^{x^{x/2}} x^{x/2} \left (-1+x+x^4\right ) (1+\log (x))}{\left (3+e^{x^{x/2}}+x\right )^2} \, dx\right )+15 \int \frac {e^{x^{x/2}}}{\left (3+e^{x^{x/2}}+x\right )^2} \, dx+45 \int \frac {x^4}{\left (3+e^{x^{x/2}}+x\right )^2} \, dx+60 \int \frac {1}{\left (3+e^{x^{x/2}}+x\right )^2} \, dx+60 \int \frac {e^{x^{x/2}} x^3}{\left (3+e^{x^{x/2}}+x\right )^2} \, dx+180 \int \frac {x^3}{\left (3+e^{x^{x/2}}+x\right )^2} \, dx\\ &=-\left (15 \operatorname {Subst}\left (\int \frac {2^x e^{2^x x^x} x^x \left (-1+2 x+16 x^4\right ) (1+\log (2 x))}{\left (3+e^{2^x x^x}+2 x\right )^2} \, dx,x,\frac {x}{2}\right )\right )+30 \operatorname {Subst}\left (\int \frac {e^{2^x x^x}}{\left (3+e^{2^x x^x}+2 x\right )^2} \, dx,x,\frac {x}{2}\right )+45 \int \frac {x^4}{\left (3+e^{x^{x/2}}+x\right )^2} \, dx+60 \int \frac {e^{x^{x/2}} x^3}{\left (3+e^{x^{x/2}}+x\right )^2} \, dx+120 \operatorname {Subst}\left (\int \frac {1}{\left (3+e^{2^x x^x}+2 x\right )^2} \, dx,x,\frac {x}{2}\right )+180 \int \frac {x^3}{\left (3+e^{x^{x/2}}+x\right )^2} \, dx\\ &=-\left (15 \operatorname {Subst}\left (\int \left (-\frac {2^x e^{2^x x^x} x^x (1+\log (2 x))}{\left (3+e^{2^x x^x}+2 x\right )^2}+\frac {2^{1+x} e^{2^x x^x} x^{1+x} (1+\log (2 x))}{\left (3+e^{2^x x^x}+2 x\right )^2}+\frac {2^{4+x} e^{2^x x^x} x^{4+x} (1+\log (2 x))}{\left (3+e^{2^x x^x}+2 x\right )^2}\right ) \, dx,x,\frac {x}{2}\right )\right )+30 \operatorname {Subst}\left (\int \frac {e^{2^x x^x}}{\left (3+e^{2^x x^x}+2 x\right )^2} \, dx,x,\frac {x}{2}\right )+45 \int \frac {x^4}{\left (3+e^{x^{x/2}}+x\right )^2} \, dx+60 \int \frac {e^{x^{x/2}} x^3}{\left (3+e^{x^{x/2}}+x\right )^2} \, dx+120 \operatorname {Subst}\left (\int \frac {1}{\left (3+e^{2^x x^x}+2 x\right )^2} \, dx,x,\frac {x}{2}\right )+180 \int \frac {x^3}{\left (3+e^{x^{x/2}}+x\right )^2} \, dx\\ &=15 \operatorname {Subst}\left (\int \frac {2^x e^{2^x x^x} x^x (1+\log (2 x))}{\left (3+e^{2^x x^x}+2 x\right )^2} \, dx,x,\frac {x}{2}\right )-15 \operatorname {Subst}\left (\int \frac {2^{1+x} e^{2^x x^x} x^{1+x} (1+\log (2 x))}{\left (3+e^{2^x x^x}+2 x\right )^2} \, dx,x,\frac {x}{2}\right )-15 \operatorname {Subst}\left (\int \frac {2^{4+x} e^{2^x x^x} x^{4+x} (1+\log (2 x))}{\left (3+e^{2^x x^x}+2 x\right )^2} \, dx,x,\frac {x}{2}\right )+30 \operatorname {Subst}\left (\int \frac {e^{2^x x^x}}{\left (3+e^{2^x x^x}+2 x\right )^2} \, dx,x,\frac {x}{2}\right )+45 \int \frac {x^4}{\left (3+e^{x^{x/2}}+x\right )^2} \, dx+60 \int \frac {e^{x^{x/2}} x^3}{\left (3+e^{x^{x/2}}+x\right )^2} \, dx+120 \operatorname {Subst}\left (\int \frac {1}{\left (3+e^{2^x x^x}+2 x\right )^2} \, dx,x,\frac {x}{2}\right )+180 \int \frac {x^3}{\left (3+e^{x^{x/2}}+x\right )^2} \, dx\\ &=15 \operatorname {Subst}\left (\int \left (\frac {2^x e^{2^x x^x} x^x}{\left (3+e^{2^x x^x}+2 x\right )^2}+\frac {2^x e^{2^x x^x} x^x \log (2 x)}{\left (3+e^{2^x x^x}+2 x\right )^2}\right ) \, dx,x,\frac {x}{2}\right )-15 \operatorname {Subst}\left (\int \left (\frac {2^{1+x} e^{2^x x^x} x^{1+x}}{\left (3+e^{2^x x^x}+2 x\right )^2}+\frac {2^{1+x} e^{2^x x^x} x^{1+x} \log (2 x)}{\left (3+e^{2^x x^x}+2 x\right )^2}\right ) \, dx,x,\frac {x}{2}\right )-15 \operatorname {Subst}\left (\int \left (\frac {2^{4+x} e^{2^x x^x} x^{4+x}}{\left (3+e^{2^x x^x}+2 x\right )^2}+\frac {2^{4+x} e^{2^x x^x} x^{4+x} \log (2 x)}{\left (3+e^{2^x x^x}+2 x\right )^2}\right ) \, dx,x,\frac {x}{2}\right )+30 \operatorname {Subst}\left (\int \frac {e^{2^x x^x}}{\left (3+e^{2^x x^x}+2 x\right )^2} \, dx,x,\frac {x}{2}\right )+45 \int \frac {x^4}{\left (3+e^{x^{x/2}}+x\right )^2} \, dx+60 \int \frac {e^{x^{x/2}} x^3}{\left (3+e^{x^{x/2}}+x\right )^2} \, dx+120 \operatorname {Subst}\left (\int \frac {1}{\left (3+e^{2^x x^x}+2 x\right )^2} \, dx,x,\frac {x}{2}\right )+180 \int \frac {x^3}{\left (3+e^{x^{x/2}}+x\right )^2} \, dx\\ &=15 \operatorname {Subst}\left (\int \frac {2^x e^{2^x x^x} x^x}{\left (3+e^{2^x x^x}+2 x\right )^2} \, dx,x,\frac {x}{2}\right )-15 \operatorname {Subst}\left (\int \frac {2^{1+x} e^{2^x x^x} x^{1+x}}{\left (3+e^{2^x x^x}+2 x\right )^2} \, dx,x,\frac {x}{2}\right )-15 \operatorname {Subst}\left (\int \frac {2^{4+x} e^{2^x x^x} x^{4+x}}{\left (3+e^{2^x x^x}+2 x\right )^2} \, dx,x,\frac {x}{2}\right )+15 \operatorname {Subst}\left (\int \frac {2^x e^{2^x x^x} x^x \log (2 x)}{\left (3+e^{2^x x^x}+2 x\right )^2} \, dx,x,\frac {x}{2}\right )-15 \operatorname {Subst}\left (\int \frac {2^{1+x} e^{2^x x^x} x^{1+x} \log (2 x)}{\left (3+e^{2^x x^x}+2 x\right )^2} \, dx,x,\frac {x}{2}\right )-15 \operatorname {Subst}\left (\int \frac {2^{4+x} e^{2^x x^x} x^{4+x} \log (2 x)}{\left (3+e^{2^x x^x}+2 x\right )^2} \, dx,x,\frac {x}{2}\right )+30 \operatorname {Subst}\left (\int \frac {e^{2^x x^x}}{\left (3+e^{2^x x^x}+2 x\right )^2} \, dx,x,\frac {x}{2}\right )+45 \int \frac {x^4}{\left (3+e^{x^{x/2}}+x\right )^2} \, dx+60 \int \frac {e^{x^{x/2}} x^3}{\left (3+e^{x^{x/2}}+x\right )^2} \, dx+120 \operatorname {Subst}\left (\int \frac {1}{\left (3+e^{2^x x^x}+2 x\right )^2} \, dx,x,\frac {x}{2}\right )+180 \int \frac {x^3}{\left (3+e^{x^{x/2}}+x\right )^2} \, dx\\ &=15 \operatorname {Subst}\left (\int \frac {2^x e^{2^x x^x} x^x}{\left (3+e^{2^x x^x}+2 x\right )^2} \, dx,x,\frac {x}{2}\right )-15 \operatorname {Subst}\left (\int \frac {2^{1+x} e^{2^x x^x} x^{1+x}}{\left (3+e^{2^x x^x}+2 x\right )^2} \, dx,x,\frac {x}{2}\right )-15 \operatorname {Subst}\left (\int \frac {2^{4+x} e^{2^x x^x} x^{4+x}}{\left (3+e^{2^x x^x}+2 x\right )^2} \, dx,x,\frac {x}{2}\right )-15 \operatorname {Subst}\left (\int \frac {\int \frac {2^x e^{2^x x^x} x^x}{\left (3+e^{2^x x^x}+2 x\right )^2} \, dx}{x} \, dx,x,\frac {x}{2}\right )+15 \operatorname {Subst}\left (\int \frac {\int \frac {2^{1+x} e^{2^x x^x} x^{1+x}}{\left (3+e^{2^x x^x}+2 x\right )^2} \, dx}{x} \, dx,x,\frac {x}{2}\right )+15 \operatorname {Subst}\left (\int \frac {\int \frac {2^{4+x} e^{2^x x^x} x^{4+x}}{\left (3+e^{2^x x^x}+2 x\right )^2} \, dx}{x} \, dx,x,\frac {x}{2}\right )+30 \operatorname {Subst}\left (\int \frac {e^{2^x x^x}}{\left (3+e^{2^x x^x}+2 x\right )^2} \, dx,x,\frac {x}{2}\right )+45 \int \frac {x^4}{\left (3+e^{x^{x/2}}+x\right )^2} \, dx+60 \int \frac {e^{x^{x/2}} x^3}{\left (3+e^{x^{x/2}}+x\right )^2} \, dx+120 \operatorname {Subst}\left (\int \frac {1}{\left (3+e^{2^x x^x}+2 x\right )^2} \, dx,x,\frac {x}{2}\right )+180 \int \frac {x^3}{\left (3+e^{x^{x/2}}+x\right )^2} \, dx+(15 \log (x)) \operatorname {Subst}\left (\int \frac {2^x e^{2^x x^x} x^x}{\left (3+e^{2^x x^x}+2 x\right )^2} \, dx,x,\frac {x}{2}\right )-(15 \log (x)) \operatorname {Subst}\left (\int \frac {2^{1+x} e^{2^x x^x} x^{1+x}}{\left (3+e^{2^x x^x}+2 x\right )^2} \, dx,x,\frac {x}{2}\right )-(15 \log (x)) \operatorname {Subst}\left (\int \frac {2^{4+x} e^{2^x x^x} x^{4+x}}{\left (3+e^{2^x x^x}+2 x\right )^2} \, dx,x,\frac {x}{2}\right )\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.11, size = 22, normalized size = 0.92 \begin {gather*} \frac {15 \left (-1+x+x^4\right )}{3+e^{x^{x/2}}+x} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(120 + 360*x^3 + 90*x^4 + E^x^(x/2)*(30 + 120*x^3 + x^(x/2)*(15 - 15*x - 15*x^4 + (15 - 15*x - 15*x^
4)*Log[x])))/(18 + 2*E^(2*x^(x/2)) + 12*x + 2*x^2 + E^x^(x/2)*(12 + 4*x)),x]

[Out]

(15*(-1 + x + x^4))/(3 + E^x^(x/2) + x)

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fricas [A]  time = 0.54, size = 19, normalized size = 0.79 \begin {gather*} \frac {15 \, {\left (x^{4} + x - 1\right )}}{x + e^{\left (x^{\frac {1}{2} \, x}\right )} + 3} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((((-15*x^4-15*x+15)*log(x)-15*x^4-15*x+15)*exp(1/2*x*log(x))+120*x^3+30)*exp(exp(1/2*x*log(x)))+90*
x^4+360*x^3+120)/(2*exp(exp(1/2*x*log(x)))^2+(4*x+12)*exp(exp(1/2*x*log(x)))+2*x^2+12*x+18),x, algorithm="fric
as")

[Out]

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

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giac [F(-1)]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((((-15*x^4-15*x+15)*log(x)-15*x^4-15*x+15)*exp(1/2*x*log(x))+120*x^3+30)*exp(exp(1/2*x*log(x)))+90*
x^4+360*x^3+120)/(2*exp(exp(1/2*x*log(x)))^2+(4*x+12)*exp(exp(1/2*x*log(x)))+2*x^2+12*x+18),x, algorithm="giac
")

[Out]

Timed out

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maple [A]  time = 0.07, size = 20, normalized size = 0.83

method	result	size
risch	\(\frac {15 x^{4}+15 x -15}{x +3+{\mathrm e}^{x^{\frac {x}{2}}}}\)	\(20\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((((-15*x^4-15*x+15)*ln(x)-15*x^4-15*x+15)*exp(1/2*x*ln(x))+120*x^3+30)*exp(exp(1/2*x*ln(x)))+90*x^4+360*x
^3+120)/(2*exp(exp(1/2*x*ln(x)))^2+(4*x+12)*exp(exp(1/2*x*ln(x)))+2*x^2+12*x+18),x,method=_RETURNVERBOSE)

[Out]

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

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maxima [A]  time = 0.47, size = 19, normalized size = 0.79 \begin {gather*} \frac {15 \, {\left (x^{4} + x - 1\right )}}{x + e^{\left (x^{\frac {1}{2} \, x}\right )} + 3} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((((-15*x^4-15*x+15)*log(x)-15*x^4-15*x+15)*exp(1/2*x*log(x))+120*x^3+30)*exp(exp(1/2*x*log(x)))+90*
x^4+360*x^3+120)/(2*exp(exp(1/2*x*log(x)))^2+(4*x+12)*exp(exp(1/2*x*log(x)))+2*x^2+12*x+18),x, algorithm="maxi
ma")

[Out]

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

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mupad [B]  time = 2.03, size = 170, normalized size = 7.08 \begin {gather*} \frac {15\,\left (x^2\,{\mathrm {e}}^{\frac {x\,\ln \relax (x)}{2}}-3\,{\mathrm {e}}^{\frac {x\,\ln \relax (x)}{2}}-2\,x+3\,x^4\,{\mathrm {e}}^{\frac {x\,\ln \relax (x)}{2}}+x^5\,{\mathrm {e}}^{\frac {x\,\ln \relax (x)}{2}}-3\,{\mathrm {e}}^{\frac {x\,\ln \relax (x)}{2}}\,\ln \relax (x)-2\,x^4+2\,x\,{\mathrm {e}}^{\frac {x\,\ln \relax (x)}{2}}+x^2\,{\mathrm {e}}^{\frac {x\,\ln \relax (x)}{2}}\,\ln \relax (x)+3\,x^4\,{\mathrm {e}}^{\frac {x\,\ln \relax (x)}{2}}\,\ln \relax (x)+x^5\,{\mathrm {e}}^{\frac {x\,\ln \relax (x)}{2}}\,\ln \relax (x)+2\,x\,{\mathrm {e}}^{\frac {x\,\ln \relax (x)}{2}}\,\ln \relax (x)+2\right )}{\left (x+{\mathrm {e}}^{{\mathrm {e}}^{\frac {x\,\ln \relax (x)}{2}}}+3\right )\,\left (3\,{\mathrm {e}}^{\frac {x\,\ln \relax (x)}{2}}+3\,{\mathrm {e}}^{\frac {x\,\ln \relax (x)}{2}}\,\ln \relax (x)+x\,{\mathrm {e}}^{\frac {x\,\ln \relax (x)}{2}}+x\,{\mathrm {e}}^{\frac {x\,\ln \relax (x)}{2}}\,\ln \relax (x)-2\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((exp(exp((x*log(x))/2))*(120*x^3 - exp((x*log(x))/2)*(15*x + log(x)*(15*x + 15*x^4 - 15) + 15*x^4 - 15) +
30) + 360*x^3 + 90*x^4 + 120)/(12*x + 2*exp(2*exp((x*log(x))/2)) + exp(exp((x*log(x))/2))*(4*x + 12) + 2*x^2 +
18),x)

[Out]

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

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sympy [A]  time = 0.46, size = 22, normalized size = 0.92 \begin {gather*} \frac {15 x^{4} + 15 x - 15}{x + e^{e^{\frac {x \log {\relax (x )}}{2}}} + 3} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((((-15*x**4-15*x+15)*ln(x)-15*x**4-15*x+15)*exp(1/2*x*ln(x))+120*x**3+30)*exp(exp(1/2*x*ln(x)))+90*
x**4+360*x**3+120)/(2*exp(exp(1/2*x*ln(x)))**2+(4*x+12)*exp(exp(1/2*x*ln(x)))+2*x**2+12*x+18),x)

[Out]

(15*x**4 + 15*x - 15)/(x + exp(exp(x*log(x)/2)) + 3)

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