3.3.36 \(\int \frac {1}{\frac {1}{\sqrt [4]{x}}+\sqrt {x}} \, dx\) [236]

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

[Out]

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

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Rubi [A]
time = 0.02, antiderivative size = 62, normalized size of antiderivative = 1.00, 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 = {1607, 348, 327, 298, 31, 648, 632, 210, 642} \begin {gather*} 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}} \end {gather*}

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 31

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

Rule 210

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

Rule 298

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 327

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 348

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 632

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 642

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]

Rule 648

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 1607

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]

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

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Mathematica [A]
time = 0.04, size = 62, normalized size = 1.00 \begin {gather*} \frac {2}{3} \left (3 \sqrt {x}+2 \sqrt {3} \tan ^{-1}\left (\frac {1-2 \sqrt [4]{x}}{\sqrt {3}}\right )+2 \log \left (1+\sqrt [4]{x}\right )-\log \left (1-\sqrt [4]{x}+\sqrt {x}\right )\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Mathics [A]
time = 2.15, size = 47, normalized size = 0.76 \begin {gather*} 2 \sqrt {x}-\frac {4 \sqrt {3} \text {ArcTan}\left [\frac {\sqrt {3} \left (-1+2 x^{\frac {1}{4}}\right )}{3}\right ]}{3}-\frac {2 \text {Log}\left [4-4 x^{\frac {1}{4}}+4 \sqrt {x}\right ]}{3}+\frac {4 \text {Log}\left [1+x^{\frac {1}{4}}\right ]}{3} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Maple [A]
time = 0.04, size = 46, normalized size = 0.74

method	result	size
derivativedivides	\(2 \sqrt {x}+\frac {4 \ln \left (1+x^{\frac {1}{4}}\right )}{3}-\frac {2 \ln \left (1-x^{\frac {1}{4}}+\sqrt {x}\right )}{3}-\frac {4 \sqrt {3}\, \arctan \left (\frac {\left (2 x^{\frac {1}{4}}-1\right ) \sqrt {3}}{3}\right )}{3}\)	\(46\)
default	\(2 \sqrt {x}+\frac {4 \ln \left (1+x^{\frac {1}{4}}\right )}{3}-\frac {2 \ln \left (1-x^{\frac {1}{4}}+\sqrt {x}\right )}{3}-\frac {4 \sqrt {3}\, \arctan \left (\frac {\left (2 x^{\frac {1}{4}}-1\right ) \sqrt {3}}{3}\right )}{3}\)	\(46\)
meijerg	\(2 \sqrt {x}-\frac {4 \sqrt {x}\, \left (-\frac {\ln \left (1+x^{\frac {1}{4}}\right )}{\sqrt {x}}+\frac {\ln \left (1-x^{\frac {1}{4}}+\sqrt {x}\right )}{2 \sqrt {x}}+\frac {\sqrt {3}\, \arctan \left (\frac {\sqrt {3}\, x^{\frac {1}{4}}}{2-x^{\frac {1}{4}}}\right )}{\sqrt {x}}\right )}{3}\)	\(65\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(1/x^(1/4)+x^(1/2)),x,method=_RETURNVERBOSE)

[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 = 0.34, size = 45, normalized size = 0.73 \begin {gather*} -\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 {gather*}

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 = 0.35, size = 47, normalized size = 0.76 \begin {gather*} -\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 {gather*}

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 = 0.19, size = 68, normalized size = 1.10 \begin {gather*} 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 {gather*}

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 = 0.00, size = 65, normalized size = 1.05 \begin {gather*} 4 \left (\frac {\ln \left (x^{\frac {1}{4}}+1\right )}{3}-\frac {\ln \left (\sqrt {x}-x^{\frac {1}{4}}+1\right )}{6}-\frac {\arctan \left (\frac {2 x^{\frac {1}{4}}-1}{\sqrt {3}}\right )}{\sqrt {3}}+\frac {\sqrt {x}}{2}\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[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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Mupad [B]
time = 0.17, size = 73, normalized size = 1.18 \begin {gather*} \frac {4\,\ln \left (16\,x^{1/4}+16\right )}{3}+\ln \left (9\,{\left (-\frac {2}{3}+\frac {\sqrt {3}\,2{}\mathrm {i}}{3}\right )}^2+16\,x^{1/4}\right )\,\left (-\frac {2}{3}+\frac {\sqrt {3}\,2{}\mathrm {i}}{3}\right )-\ln \left (9\,{\left (\frac {2}{3}+\frac {\sqrt {3}\,2{}\mathrm {i}}{3}\right )}^2+16\,x^{1/4}\right )\,\left (\frac {2}{3}+\frac {\sqrt {3}\,2{}\mathrm {i}}{3}\right )+2\,\sqrt {x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(x^(1/2) + 1/x^(1/4)),x)

[Out]

(4*log(16*x^(1/4) + 16))/3 + log(9*((3^(1/2)*2i)/3 - 2/3)^2 + 16*x^(1/4))*((3^(1/2)*2i)/3 - 2/3) - log(9*((3^(
1/2)*2i)/3 + 2/3)^2 + 16*x^(1/4))*((3^(1/2)*2i)/3 + 2/3) + 2*x^(1/2)

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