3.930 \(\int \sqrt{\sqrt [4]{x}+x} \, dx\)

Optimal. Leaf size=59 \[ \frac{2}{3} \sqrt{x+\sqrt [4]{x}} x+\frac{1}{3} \sqrt{x+\sqrt [4]{x}} \sqrt [4]{x}-\frac{1}{3} \tanh ^{-1}\left (\frac{\sqrt{x}}{\sqrt{x+\sqrt [4]{x}}}\right ) \]

[Out]

(x^(1/4)*Sqrt[x^(1/4) + x])/3 + (2*x*Sqrt[x^(1/4) + x])/3 - ArcTanh[Sqrt[x]/Sqrt[x^(1/4) + x]]/3

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Rubi [A]  time = 0.0685488, antiderivative size = 59, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 5, integrand size = 11, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.454, Rules used = {2004, 2018, 2024, 2029, 206} \[ \frac{2}{3} \sqrt{x+\sqrt [4]{x}} x+\frac{1}{3} \sqrt{x+\sqrt [4]{x}} \sqrt [4]{x}-\frac{1}{3} \tanh ^{-1}\left (\frac{\sqrt{x}}{\sqrt{x+\sqrt [4]{x}}}\right ) \]

Antiderivative was successfully verified.

[In]

Int[Sqrt[x^(1/4) + x],x]

[Out]

(x^(1/4)*Sqrt[x^(1/4) + x])/3 + (2*x*Sqrt[x^(1/4) + x])/3 - ArcTanh[Sqrt[x]/Sqrt[x^(1/4) + x]]/3

Rule 2004

Int[((a_.)*(x_)^(j_.) + (b_.)*(x_)^(n_.))^(p_), x_Symbol] :> Simp[(x*(a*x^j + b*x^n)^p)/(n*p + 1), x] + Dist[(
a*(n - j)*p)/(n*p + 1), Int[x^j*(a*x^j + b*x^n)^(p - 1), x], x] /; FreeQ[{a, b}, x] &&  !IntegerQ[p] && LtQ[0,
j, n] && GtQ[p, 0] && NeQ[n*p + 1, 0]

Rule 2018

Int[(x_)^(m_.)*((a_.)*(x_)^(j_.) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Dist[1/n, Subst[Int[x^(Simplify[(m + 1)
/n] - 1)*(a*x^Simplify[j/n] + b*x)^p, x], x, x^n], x] /; FreeQ[{a, b, j, m, n, p}, x] &&  !IntegerQ[p] && NeQ[
n, j] && IntegerQ[Simplify[j/n]] && IntegerQ[Simplify[(m + 1)/n]] && NeQ[n^2, 1]

Rule 2024

Int[((c_.)*(x_))^(m_.)*((a_.)*(x_)^(j_.) + (b_.)*(x_)^(n_.))^(p_), x_Symbol] :> Simp[(c^(n - 1)*(c*x)^(m - n +
1)*(a*x^j + b*x^n)^(p + 1))/(b*(m + n*p + 1)), x] - Dist[(a*c^(n - j)*(m + j*p - n + j + 1))/(b*(m + n*p + 1)
), Int[(c*x)^(m - (n - j))*(a*x^j + b*x^n)^p, x], x] /; FreeQ[{a, b, c, m, p}, x] &&  !IntegerQ[p] && LtQ[0, j
, n] && (IntegersQ[j, n] || GtQ[c, 0]) && GtQ[m + j*p + 1 - n + j, 0] && NeQ[m + n*p + 1, 0]

Rule 2029

Int[(x_)^(m_.)/Sqrt[(a_.)*(x_)^(j_.) + (b_.)*(x_)^(n_.)], x_Symbol] :> Dist[-2/(n - j), Subst[Int[1/(1 - a*x^2
), x], x, x^(j/2)/Sqrt[a*x^j + b*x^n]], x] /; FreeQ[{a, b, j, n}, x] && EqQ[m, j/2 - 1] && NeQ[n, j]

Rule 206

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1*ArcTanh[(Rt[-b, 2]*x)/Rt[a, 2]])/(Rt[a, 2]*Rt[-b, 2]), x]
/; FreeQ[{a, b}, x] && NegQ[a/b] && (GtQ[a, 0] || LtQ[b, 0])

Rubi steps

\begin{align*} \int \sqrt{\sqrt [4]{x}+x} \, dx &=\frac{2}{3} x \sqrt{\sqrt [4]{x}+x}+\frac{1}{4} \int \frac{\sqrt [4]{x}}{\sqrt{\sqrt [4]{x}+x}} \, dx\\ &=\frac{2}{3} x \sqrt{\sqrt [4]{x}+x}+\operatorname{Subst}\left (\int \frac{x^4}{\sqrt{x+x^4}} \, dx,x,\sqrt [4]{x}\right )\\ &=\frac{1}{3} \sqrt [4]{x} \sqrt{\sqrt [4]{x}+x}+\frac{2}{3} x \sqrt{\sqrt [4]{x}+x}-\frac{1}{2} \operatorname{Subst}\left (\int \frac{x}{\sqrt{x+x^4}} \, dx,x,\sqrt [4]{x}\right )\\ &=\frac{1}{3} \sqrt [4]{x} \sqrt{\sqrt [4]{x}+x}+\frac{2}{3} x \sqrt{\sqrt [4]{x}+x}-\frac{1}{3} \operatorname{Subst}\left (\int \frac{1}{1-x^2} \, dx,x,\frac{\sqrt{x}}{\sqrt{\sqrt [4]{x}+x}}\right )\\ &=\frac{1}{3} \sqrt [4]{x} \sqrt{\sqrt [4]{x}+x}+\frac{2}{3} x \sqrt{\sqrt [4]{x}+x}-\frac{1}{3} \tanh ^{-1}\left (\frac{\sqrt{x}}{\sqrt{\sqrt [4]{x}+x}}\right )\\ \end{align*}

Mathematica [A]  time = 0.0348624, size = 57, normalized size = 0.97 \[ \frac{2 x^2+3 x^{5/4}-\sqrt{x^{3/4}+1} \sqrt [8]{x} \sinh ^{-1}\left (x^{3/8}\right )+\sqrt{x}}{3 \sqrt{x+\sqrt [4]{x}}} \]

Antiderivative was successfully verified.

[In]

Integrate[Sqrt[x^(1/4) + x],x]

[Out]

(Sqrt[x] + 3*x^(5/4) + 2*x^2 - Sqrt[1 + x^(3/4)]*x^(1/8)*ArcSinh[x^(3/8)])/(3*Sqrt[x^(1/4) + x])

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Maple [C]  time = 0.067, size = 342, normalized size = 5.8 \begin{align*}{\frac{2\,x}{3}\sqrt{\sqrt [4]{x}+x}}+{\frac{1}{3}\sqrt [4]{x}\sqrt{\sqrt [4]{x}+x}}+{\frac{-{\frac{1}{2}}-{\frac{i}{2}}\sqrt{3}}{{\frac{3}{2}}+{\frac{i}{2}}\sqrt{3}}\sqrt{{\frac{{\frac{3}{2}}+{\frac{i}{2}}\sqrt{3}}{{\frac{1}{2}}+{\frac{i}{2}}\sqrt{3}}\sqrt [4]{x} \left ( 1+\sqrt [4]{x} \right ) ^{-1}}} \left ( 1+\sqrt [4]{x} \right ) ^{2}\sqrt{-{\frac{1}{{\frac{1}{2}}-{\frac{i}{2}}\sqrt{3}} \left ( \sqrt [4]{x}-{\frac{1}{2}}+{\frac{i}{2}}\sqrt{3} \right ) \left ( 1+\sqrt [4]{x} \right ) ^{-1}}}\sqrt{-{\frac{1}{{\frac{1}{2}}+{\frac{i}{2}}\sqrt{3}} \left ( \sqrt [4]{x}-{\frac{1}{2}}-{\frac{i}{2}}\sqrt{3} \right ) \left ( 1+\sqrt [4]{x} \right ) ^{-1}}} \left ( -{\it EllipticF} \left ( \sqrt{{\frac{{\frac{3}{2}}+{\frac{i}{2}}\sqrt{3}}{{\frac{1}{2}}+{\frac{i}{2}}\sqrt{3}}\sqrt [4]{x} \left ( 1+\sqrt [4]{x} \right ) ^{-1}}},\sqrt{{\frac{ \left ( -{\frac{3}{2}}+{\frac{i}{2}}\sqrt{3} \right ) \left ( -{\frac{1}{2}}-{\frac{i}{2}}\sqrt{3} \right ) }{ \left ( -{\frac{1}{2}}+{\frac{i}{2}}\sqrt{3} \right ) \left ( -{\frac{3}{2}}-{\frac{i}{2}}\sqrt{3} \right ) }}} \right ) +{\it EllipticPi} \left ( \sqrt{{\frac{{\frac{3}{2}}+{\frac{i}{2}}\sqrt{3}}{{\frac{1}{2}}+{\frac{i}{2}}\sqrt{3}}\sqrt [4]{x} \left ( 1+\sqrt [4]{x} \right ) ^{-1}}},{\frac{{\frac{1}{2}}+{\frac{i}{2}}\sqrt{3}}{{\frac{3}{2}}+{\frac{i}{2}}\sqrt{3}}},\sqrt{{\frac{ \left ( -{\frac{3}{2}}+{\frac{i}{2}}\sqrt{3} \right ) \left ( -{\frac{1}{2}}-{\frac{i}{2}}\sqrt{3} \right ) }{ \left ( -{\frac{1}{2}}+{\frac{i}{2}}\sqrt{3} \right ) \left ( -{\frac{3}{2}}-{\frac{i}{2}}\sqrt{3} \right ) }}} \right ) \right ){\frac{1}{\sqrt{\sqrt [4]{x} \left ( 1+\sqrt [4]{x} \right ) \left ( \sqrt [4]{x}-{\frac{1}{2}}+{\frac{i}{2}}\sqrt{3} \right ) \left ( \sqrt [4]{x}-{\frac{1}{2}}-{\frac{i}{2}}\sqrt{3} \right ) }}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((x^(1/4)+x)^(1/2),x)

[Out]

2/3*x*(x^(1/4)+x)^(1/2)+1/3*x^(1/4)*(x^(1/4)+x)^(1/2)+(-1/2-1/2*I*3^(1/2))*((3/2+1/2*I*3^(1/2))*x^(1/4)/(1/2+1
/2*I*3^(1/2))/(1+x^(1/4)))^(1/2)*(1+x^(1/4))^2*(-(x^(1/4)-1/2+1/2*I*3^(1/2))/(1/2-1/2*I*3^(1/2))/(1+x^(1/4)))^
(1/2)*(-(x^(1/4)-1/2-1/2*I*3^(1/2))/(1/2+1/2*I*3^(1/2))/(1+x^(1/4)))^(1/2)/(3/2+1/2*I*3^(1/2))/(x^(1/4)*(1+x^(
1/4))*(x^(1/4)-1/2+1/2*I*3^(1/2))*(x^(1/4)-1/2-1/2*I*3^(1/2)))^(1/2)*(-EllipticF(((3/2+1/2*I*3^(1/2))*x^(1/4)/
(1/2+1/2*I*3^(1/2))/(1+x^(1/4)))^(1/2),((-3/2+1/2*I*3^(1/2))*(-1/2-1/2*I*3^(1/2))/(-1/2+1/2*I*3^(1/2))/(-3/2-1
/2*I*3^(1/2)))^(1/2))+EllipticPi(((3/2+1/2*I*3^(1/2))*x^(1/4)/(1/2+1/2*I*3^(1/2))/(1+x^(1/4)))^(1/2),(1/2+1/2*
I*3^(1/2))/(3/2+1/2*I*3^(1/2)),((-3/2+1/2*I*3^(1/2))*(-1/2-1/2*I*3^(1/2))/(-1/2+1/2*I*3^(1/2))/(-3/2-1/2*I*3^(
1/2)))^(1/2)))

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Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \sqrt{x + x^{\frac{1}{4}}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((x^(1/4)+x)^(1/2),x, algorithm="maxima")

[Out]

integrate(sqrt(x + x^(1/4)), x)

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Fricas [F(-1)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((x^(1/4)+x)^(1/2),x, algorithm="fricas")

[Out]

Timed out

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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \sqrt{\sqrt [4]{x} + x}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((x**(1/4)+x)**(1/2),x)

[Out]

Integral(sqrt(x**(1/4) + x), x)

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Giac [F(-2)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: NotImplementedError} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((x^(1/4)+x)^(1/2),x, algorithm="giac")

[Out]

Exception raised: NotImplementedError