3.236 \(\int \frac{1}{\frac{1}{\sqrt [4]{x}}+\sqrt{x}} \, dx\)

Optimal. Leaf size=62 \[ 2 \sqrt{x}+\frac{4}{3} \log \left (\sqrt [4]{x}+1\right )-\frac{2}{3} \log \left (\sqrt{x}-\sqrt [4]{x}+1\right )+\frac{4 \tan ^{-1}\left (\frac{1-2 \sqrt [4]{x}}{\sqrt{3}}\right )}{\sqrt{3}} \]

[Out]

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

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Rubi [A]  time = 0.037231, antiderivative size = 62, normalized size of antiderivative = 1., number of steps used = 9, number of rules used = 9, integrand size = 13, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.692, Rules used = {1593, 341, 321, 292, 31, 634, 618, 204, 628} \[ 2 \sqrt{x}+\frac{4}{3} \log \left (\sqrt [4]{x}+1\right )-\frac{2}{3} \log \left (\sqrt{x}-\sqrt [4]{x}+1\right )+\frac{4 \tan ^{-1}\left (\frac{1-2 \sqrt [4]{x}}{\sqrt{3}}\right )}{\sqrt{3}} \]

Antiderivative was successfully verified.

[In]

Int[(x^(-1/4) + Sqrt[x])^(-1),x]

[Out]

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

Rule 1593

Int[(u_.)*((a_.)*(x_)^(p_.) + (b_.)*(x_)^(q_.))^(n_.), x_Symbol] :> Int[u*x^(n*p)*(a + b*x^(q - p))^n, x] /; F
reeQ[{a, b, p, q}, x] && IntegerQ[n] && PosQ[q - p]

Rule 341

Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> With[{k = Denominator[n]}, Dist[k, Subst[Int[x^(k*(
m + 1) - 1)*(a + b*x^(k*n))^p, x], x, x^(1/k)], x]] /; FreeQ[{a, b, m, p}, x] && FractionQ[n]

Rule 321

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

Rule 292

Int[(x_)/((a_) + (b_.)*(x_)^3), x_Symbol] :> -Dist[(3*Rt[a, 3]*Rt[b, 3])^(-1), Int[1/(Rt[a, 3] + Rt[b, 3]*x),
x], x] + Dist[1/(3*Rt[a, 3]*Rt[b, 3]), Int[(Rt[a, 3] + Rt[b, 3]*x)/(Rt[a, 3]^2 - Rt[a, 3]*Rt[b, 3]*x + Rt[b, 3
]^2*x^2), x], x] /; FreeQ[{a, b}, x]

Rule 31

Int[((a_) + (b_.)*(x_))^(-1), x_Symbol] :> Simp[Log[RemoveContent[a + b*x, x]]/b, x] /; FreeQ[{a, b}, x]

Rule 634

Int[((d_.) + (e_.)*(x_))/((a_) + (b_.)*(x_) + (c_.)*(x_)^2), x_Symbol] :> Dist[(2*c*d - b*e)/(2*c), Int[1/(a +
b*x + c*x^2), x], x] + Dist[e/(2*c), Int[(b + 2*c*x)/(a + b*x + c*x^2), x], x] /; FreeQ[{a, b, c, d, e}, x] &
& NeQ[2*c*d - b*e, 0] && NeQ[b^2 - 4*a*c, 0] &&  !NiceSqrtQ[b^2 - 4*a*c]

Rule 618

Int[((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(-1), x_Symbol] :> Dist[-2, Subst[Int[1/Simp[b^2 - 4*a*c - x^2, x], x]
, x, b + 2*c*x], x] /; FreeQ[{a, b, c}, x] && NeQ[b^2 - 4*a*c, 0]

Rule 204

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

Rule 628

Int[((d_) + (e_.)*(x_))/((a_.) + (b_.)*(x_) + (c_.)*(x_)^2), x_Symbol] :> Simp[(d*Log[RemoveContent[a + b*x +
c*x^2, x]])/b, x] /; FreeQ[{a, b, c, d, e}, x] && EqQ[2*c*d - b*e, 0]

Rubi steps

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

Mathematica [C]  time = 0.0066814, size = 24, normalized size = 0.39 \[ -2 \sqrt{x} \left (\text{Hypergeometric2F1}\left (\frac{2}{3},1,\frac{5}{3},-x^{3/4}\right )-1\right ) \]

Antiderivative was successfully verified.

[In]

Integrate[(x^(-1/4) + Sqrt[x])^(-1),x]

[Out]

-2*Sqrt[x]*(-1 + Hypergeometric2F1[2/3, 1, 5/3, -x^(3/4)])

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Maple [A]  time = 0.007, size = 46, normalized size = 0.7 \begin{align*} 2\,\sqrt{x}+{\frac{4}{3}\ln \left ( 1+\sqrt [4]{x} \right ) }-{\frac{2}{3}\ln \left ( 1-\sqrt [4]{x}+\sqrt{x} \right ) }-{\frac{4\,\sqrt{3}}{3}\arctan \left ({\frac{\sqrt{3}}{3} \left ( 2\,\sqrt [4]{x}-1 \right ) } \right ) } \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [A]  time = 1.44284, size = 61, normalized size = 0.98 \begin{align*} -\frac{4}{3} \, \sqrt{3} \arctan \left (\frac{1}{3} \, \sqrt{3}{\left (2 \, x^{\frac{1}{4}} - 1\right )}\right ) + 2 \, \sqrt{x} - \frac{2}{3} \, \log \left (\sqrt{x} - x^{\frac{1}{4}} + 1\right ) + \frac{4}{3} \, \log \left (x^{\frac{1}{4}} + 1\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-4/3*sqrt(3)*arctan(1/3*sqrt(3)*(2*x^(1/4) - 1)) + 2*sqrt(x) - 2/3*log(sqrt(x) - x^(1/4) + 1) + 4/3*log(x^(1/4
) + 1)

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Fricas [A]  time = 2.15339, size = 167, normalized size = 2.69 \begin{align*} -\frac{4}{3} \, \sqrt{3} \arctan \left (\frac{2}{3} \, \sqrt{3} x^{\frac{1}{4}} - \frac{1}{3} \, \sqrt{3}\right ) + 2 \, \sqrt{x} - \frac{2}{3} \, \log \left (\sqrt{x} - x^{\frac{1}{4}} + 1\right ) + \frac{4}{3} \, \log \left (x^{\frac{1}{4}} + 1\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-4/3*sqrt(3)*arctan(2/3*sqrt(3)*x^(1/4) - 1/3*sqrt(3)) + 2*sqrt(x) - 2/3*log(sqrt(x) - x^(1/4) + 1) + 4/3*log(
x^(1/4) + 1)

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Sympy [A]  time = 0.701258, size = 68, normalized size = 1.1 \begin{align*} 2 \sqrt{x} + \frac{4 \log{\left (\sqrt [4]{x} + 1 \right )}}{3} - \frac{2 \log{\left (- 4 \sqrt [4]{x} + 4 \sqrt{x} + 4 \right )}}{3} - \frac{4 \sqrt{3} \operatorname{atan}{\left (\frac{2 \sqrt{3} \sqrt [4]{x}}{3} - \frac{\sqrt{3}}{3} \right )}}{3} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2*sqrt(x) + 4*log(x**(1/4) + 1)/3 - 2*log(-4*x**(1/4) + 4*sqrt(x) + 4)/3 - 4*sqrt(3)*atan(2*sqrt(3)*x**(1/4)/3
- sqrt(3)/3)/3

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Giac [A]  time = 1.05391, size = 61, normalized size = 0.98 \begin{align*} -\frac{4}{3} \, \sqrt{3} \arctan \left (\frac{1}{3} \, \sqrt{3}{\left (2 \, x^{\frac{1}{4}} - 1\right )}\right ) + 2 \, \sqrt{x} - \frac{2}{3} \, \log \left (\sqrt{x} - x^{\frac{1}{4}} + 1\right ) + \frac{4}{3} \, \log \left (x^{\frac{1}{4}} + 1\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-4/3*sqrt(3)*arctan(1/3*sqrt(3)*(2*x^(1/4) - 1)) + 2*sqrt(x) - 2/3*log(sqrt(x) - x^(1/4) + 1) + 4/3*log(x^(1/4
) + 1)