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

Optimal. Leaf size=27 \[ 2 \sqrt{x}+4 \sqrt [4]{x}+4 \log \left (1-\sqrt [4]{x}\right ) \]

[Out]

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

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Rubi [A]  time = 0.0116985, antiderivative size = 27, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.2, Rules used = {1593, 266, 43} \[ 2 \sqrt{x}+4 \sqrt [4]{x}+4 \log \left (1-\sqrt [4]{x}\right ) \]

Antiderivative was successfully verified.

[In]

Int[(-x^(1/4) + Sqrt[x])^(-1),x]

[Out]

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

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 266

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

Rule 43

Int[((a_.) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_.), x_Symbol] :> Int[ExpandIntegrand[(a + b*x)^m*(c + d
*x)^n, x], x] /; FreeQ[{a, b, c, d, n}, x] && NeQ[b*c - a*d, 0] && IGtQ[m, 0] && ( !IntegerQ[n] || (EqQ[c, 0]
&& LeQ[7*m + 4*n + 4, 0]) || LtQ[9*m + 5*(n + 1), 0] || GtQ[m + n + 2, 0])

Rubi steps

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

Mathematica [A]  time = 0.0100669, size = 27, normalized size = 1. \[ 2 \sqrt{x}+4 \sqrt [4]{x}+4 \log \left (1-\sqrt [4]{x}\right ) \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Maple [A]  time = 0.007, size = 20, normalized size = 0.7 \begin{align*} 2\,\sqrt{x}+4\,\sqrt [4]{x}+4\,\ln \left ( \sqrt [4]{x}-1 \right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [A]  time = 1.31642, size = 26, normalized size = 0.96 \begin{align*} 2 \, \sqrt{x} + 4 \, x^{\frac{1}{4}} + 4 \, \log \left (x^{\frac{1}{4}} - 1\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Fricas [A]  time = 1.17699, size = 59, normalized size = 2.19 \begin{align*} 2 \, \sqrt{x} + 4 \, x^{\frac{1}{4}} + 4 \, \log \left (x^{\frac{1}{4}} - 1\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Sympy [A]  time = 0.212634, size = 22, normalized size = 0.81 \begin{align*} 4 \sqrt [4]{x} + 2 \sqrt{x} + 4 \log{\left (\sqrt [4]{x} - 1 \right )} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Giac [A]  time = 1.13031, size = 27, normalized size = 1. \begin{align*} 2 \, \sqrt{x} + 4 \, x^{\frac{1}{4}} + 4 \, \log \left ({\left | x^{\frac{1}{4}} - 1 \right |}\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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