\(\int e^{\sqrt [4]{x}} \, dx\) [88]

Optimal result

Integrand size = 7, antiderivative size = 30 \[ \int e^{\sqrt [4]{x}} \, dx=4 e^{\sqrt [4]{x}} \left (-6+6 \sqrt [4]{x}-3 \sqrt {x}+x^{3/4}\right ) \]

[Out]

4*exp(x^(1/4))*(-6+6*x^(1/4)-3*x^(1/2)+x^(3/4))

Rubi [A] (verified)

Time = 0.04 (sec) , antiderivative size = 52, normalized size of antiderivative = 1.73, number of steps used = 5, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.429, Rules used = {2238, 2207, 2225} \[ \int e^{\sqrt [4]{x}} \, dx=4 e^{\sqrt [4]{x}} x^{3/4}-12 e^{\sqrt [4]{x}} \sqrt {x}+24 e^{\sqrt [4]{x}} \sqrt [4]{x}-24 e^{\sqrt [4]{x}} \]

[In]

Int[E^x^(1/4),x]

[Out]

-24*E^x^(1/4) + 24*E^x^(1/4)*x^(1/4) - 12*E^x^(1/4)*Sqrt[x] + 4*E^x^(1/4)*x^(3/4)

Rule 2207

Int[((b_.)*(F_)^((g_.)*((e_.) + (f_.)*(x_))))^(n_.)*((c_.) + (d_.)*(x_))^(m_.), x_Symbol] :> Simp[(c + d*x)^m*
((b*F^(g*(e + f*x)))^n/(f*g*n*Log[F])), x] - Dist[d*(m/(f*g*n*Log[F])), Int[(c + d*x)^(m - 1)*(b*F^(g*(e + f*x
)))^n, x], x] /; FreeQ[{F, b, c, d, e, f, g, n}, x] && GtQ[m, 0] && IntegerQ[2*m] &&  !TrueQ[$UseGamma]

Rule 2225

Int[((F_)^((c_.)*((a_.) + (b_.)*(x_))))^(n_.), x_Symbol] :> Simp[(F^(c*(a + b*x)))^n/(b*c*n*Log[F]), x] /; Fre
eQ[{F, a, b, c, n}, x]

Rule 2238

Int[(F_)^((a_.) + (b_.)*((c_.) + (d_.)*(x_))^(n_)), x_Symbol] :> With[{k = Denominator[n]}, Dist[k/d, Subst[In
t[x^(k - 1)*F^(a + b*x^(k*n)), x], x, (c + d*x)^(1/k)], x]] /; FreeQ[{F, a, b, c, d}, x] && IntegerQ[2/n] &&
!IntegerQ[n]

Rubi steps \begin{align*} \text {integral}& = 4 \text {Subst}\left (\int e^x x^3 \, dx,x,\sqrt [4]{x}\right ) \\ & = 4 e^{\sqrt [4]{x}} x^{3/4}-12 \text {Subst}\left (\int e^x x^2 \, dx,x,\sqrt [4]{x}\right ) \\ & = -12 e^{\sqrt [4]{x}} \sqrt {x}+4 e^{\sqrt [4]{x}} x^{3/4}+24 \text {Subst}\left (\int e^x x \, dx,x,\sqrt [4]{x}\right ) \\ & = 24 e^{\sqrt [4]{x}} \sqrt [4]{x}-12 e^{\sqrt [4]{x}} \sqrt {x}+4 e^{\sqrt [4]{x}} x^{3/4}-24 \text {Subst}\left (\int e^x \, dx,x,\sqrt [4]{x}\right ) \\ & = -24 e^{\sqrt [4]{x}}+24 e^{\sqrt [4]{x}} \sqrt [4]{x}-12 e^{\sqrt [4]{x}} \sqrt {x}+4 e^{\sqrt [4]{x}} x^{3/4} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.01 (sec) , antiderivative size = 31, normalized size of antiderivative = 1.03 \[ \int e^{\sqrt [4]{x}} \, dx=e^{\sqrt [4]{x}} \left (-24+24 \sqrt [4]{x}-12 \sqrt {x}+4 x^{3/4}\right ) \]

[In]

Integrate[E^x^(1/4),x]

[Out]

E^x^(1/4)*(-24 + 24*x^(1/4) - 12*Sqrt[x] + 4*x^(3/4))

Maple [A] (verified)

Time = 0.02 (sec) , antiderivative size = 26, normalized size of antiderivative = 0.87

method	result	size
meijerg	\(24-\left (-4 x^{\frac {3}{4}}+12 \sqrt {x}-24 x^{\frac {1}{4}}+24\right ) {\mathrm e}^{x^{\frac {1}{4}}}\)	\(26\)
derivativedivides	\(4 \,{\mathrm e}^{x^{\frac {1}{4}}} x^{\frac {3}{4}}-12 \sqrt {x}\, {\mathrm e}^{x^{\frac {1}{4}}}+24 x^{\frac {1}{4}} {\mathrm e}^{x^{\frac {1}{4}}}-24 \,{\mathrm e}^{x^{\frac {1}{4}}}\)	\(35\)
default	\(4 \,{\mathrm e}^{x^{\frac {1}{4}}} x^{\frac {3}{4}}-12 \sqrt {x}\, {\mathrm e}^{x^{\frac {1}{4}}}+24 x^{\frac {1}{4}} {\mathrm e}^{x^{\frac {1}{4}}}-24 \,{\mathrm e}^{x^{\frac {1}{4}}}\)	\(35\)

[In]

int(exp(x^(1/4)),x,method=_RETURNVERBOSE)

[Out]

24-(-4*x^(3/4)+12*x^(1/2)-24*x^(1/4)+24)*exp(x^(1/4))

Fricas [A] (verification not implemented)

none

Time = 0.24 (sec) , antiderivative size = 21, normalized size of antiderivative = 0.70 \[ \int e^{\sqrt [4]{x}} \, dx=4 \, {\left (x^{\frac {3}{4}} - 3 \, \sqrt {x} + 6 \, x^{\frac {1}{4}} - 6\right )} e^{\left (x^{\frac {1}{4}}\right )} \]

[In]

integrate(exp(x^(1/4)),x, algorithm="fricas")

[Out]

4*(x^(3/4) - 3*sqrt(x) + 6*x^(1/4) - 6)*e^(x^(1/4))

Sympy [A] (verification not implemented)

Time = 0.14 (sec) , antiderivative size = 48, normalized size of antiderivative = 1.60 \[ \int e^{\sqrt [4]{x}} \, dx=4 x^{\frac {3}{4}} e^{\sqrt [4]{x}} + 24 \sqrt [4]{x} e^{\sqrt [4]{x}} - 12 \sqrt {x} e^{\sqrt [4]{x}} - 24 e^{\sqrt [4]{x}} \]

[In]

integrate(exp(x**(1/4)),x)

[Out]

4*x**(3/4)*exp(x**(1/4)) + 24*x**(1/4)*exp(x**(1/4)) - 12*sqrt(x)*exp(x**(1/4)) - 24*exp(x**(1/4))

Maxima [A] (verification not implemented)

none

Time = 0.21 (sec) , antiderivative size = 21, normalized size of antiderivative = 0.70 \[ \int e^{\sqrt [4]{x}} \, dx=4 \, {\left (x^{\frac {3}{4}} - 3 \, \sqrt {x} + 6 \, x^{\frac {1}{4}} - 6\right )} e^{\left (x^{\frac {1}{4}}\right )} \]

[In]

integrate(exp(x^(1/4)),x, algorithm="maxima")

[Out]

4*(x^(3/4) - 3*sqrt(x) + 6*x^(1/4) - 6)*e^(x^(1/4))

Giac [A] (verification not implemented)

none

Time = 0.28 (sec) , antiderivative size = 21, normalized size of antiderivative = 0.70 \[ \int e^{\sqrt [4]{x}} \, dx=4 \, {\left (x^{\frac {3}{4}} - 3 \, \sqrt {x} + 6 \, x^{\frac {1}{4}} - 6\right )} e^{\left (x^{\frac {1}{4}}\right )} \]

[In]

integrate(exp(x^(1/4)),x, algorithm="giac")

[Out]

4*(x^(3/4) - 3*sqrt(x) + 6*x^(1/4) - 6)*e^(x^(1/4))

Mupad [B] (verification not implemented)

Time = 0.03 (sec) , antiderivative size = 28, normalized size of antiderivative = 0.93 \[ \int e^{\sqrt [4]{x}} \, dx=-4\,x\,{\mathrm {e}}^{x^{1/4}}\,\left (\frac {6}{x}+\frac {3}{\sqrt {x}}-\frac {1}{x^{1/4}}-\frac {6}{x^{3/4}}\right ) \]

[In]

int(exp(x^(1/4)),x)

[Out]

-4*x*exp(x^(1/4))*(6/x + 3/x^(1/2) - 1/x^(1/4) - 6/x^(3/4))