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

Optimal. Leaf size=25 \[ 2 \sqrt {x}-4 \sqrt [4]{x}+4 \log \left (\sqrt [4]{x}+1\right ) \]

[Out]

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

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Rubi [A]  time = 0.01, antiderivative size = 25, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, integrand size = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.231, Rules used = {1593, 266, 43} \[ 2 \sqrt {x}-4 \sqrt [4]{x}+4 \log \left (\sqrt [4]{x}+1\right ) \]

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 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])

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 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]

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+x+\frac {1}{1+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*}

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Mathematica [A]  time = 0.01, size = 25, normalized size = 1.00 \[ 2 \sqrt {x}-4 \sqrt [4]{x}+4 \log \left (\sqrt [4]{x}+1\right ) \]

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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fricas [A]  time = 0.42, size = 19, normalized size = 0.76 \[ 2 \, \sqrt {x} - 4 \, x^{\frac {1}{4}} + 4 \, \log \left (x^{\frac {1}{4}} + 1\right ) \]

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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giac [A]  time = 0.34, size = 19, normalized size = 0.76 \[ 2 \, \sqrt {x} - 4 \, x^{\frac {1}{4}} + 4 \, \log \left (x^{\frac {1}{4}} + 1\right ) \]

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(x^(1/4) + 1)

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maple [A]  time = 0.01, size = 20, normalized size = 0.80 \[ 4 \ln \left (x^{\frac {1}{4}}+1\right )+2 \sqrt {x}-4 x^{\frac {1}{4}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maxima [A]  time = 0.69, size = 19, normalized size = 0.76 \[ 2 \, \sqrt {x} - 4 \, x^{\frac {1}{4}} + 4 \, \log \left (x^{\frac {1}{4}} + 1\right ) \]

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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mupad [B]  time = 0.03, size = 19, normalized size = 0.76 \[ 4\,\ln \left (x^{1/4}+1\right )+2\,\sqrt {x}-4\,x^{1/4} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(x^(1/2) + x^(1/4)),x)

[Out]

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

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sympy [A]  time = 0.24, size = 22, normalized size = 0.88 \[ - 4 \sqrt [4]{x} + 2 \sqrt {x} + 4 \log {\left (\sqrt [4]{x} + 1 \right )} \]

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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