3.2449 \(\int \frac{1}{\sqrt{1+x^{4/5}} \sqrt [5]{x}} \, dx\)

Optimal. Leaf size=15 \[ \frac{5}{2} \sqrt{x^{4/5}+1} \]

[Out]

(5*Sqrt[1 + x^(4/5)])/2

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Rubi [A]  time = 0.0030842, antiderivative size = 15, normalized size of antiderivative = 1., number of steps used = 1, number of rules used = 1, integrand size = 17, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.059, Rules used = {261} \[ \frac{5}{2} \sqrt{x^{4/5}+1} \]

Antiderivative was successfully verified.

[In]

Int[1/(Sqrt[1 + x^(4/5)]*x^(1/5)),x]

[Out]

(5*Sqrt[1 + x^(4/5)])/2

Rule 261

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

Rubi steps

\begin{align*} \int \frac{1}{\sqrt{1+x^{4/5}} \sqrt [5]{x}} \, dx &=\frac{5}{2} \sqrt{1+x^{4/5}}\\ \end{align*}

Mathematica [A]  time = 0.00373, size = 15, normalized size = 1. \[ \frac{5}{2} \sqrt{x^{4/5}+1} \]

Antiderivative was successfully verified.

[In]

Integrate[1/(Sqrt[1 + x^(4/5)]*x^(1/5)),x]

[Out]

(5*Sqrt[1 + x^(4/5)])/2

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Maple [A]  time = 0.006, size = 10, normalized size = 0.7 \begin{align*}{\frac{5}{2}\sqrt{1+{x}^{{\frac{4}{5}}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/x^(1/5)/(1+x^(4/5))^(1/2),x)

[Out]

5/2*(1+x^(4/5))^(1/2)

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Maxima [A]  time = 1.00518, size = 12, normalized size = 0.8 \begin{align*} \frac{5}{2} \, \sqrt{x^{\frac{4}{5}} + 1} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/x^(1/5)/(1+x^(4/5))^(1/2),x, algorithm="maxima")

[Out]

5/2*sqrt(x^(4/5) + 1)

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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(1/x^(1/5)/(1+x^(4/5))^(1/2),x, algorithm="fricas")

[Out]

Timed out

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Sympy [A]  time = 0.857914, size = 12, normalized size = 0.8 \begin{align*} \frac{5 \sqrt{x^{\frac{4}{5}} + 1}}{2} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/x**(1/5)/(1+x**(4/5))**(1/2),x)

[Out]

5*sqrt(x**(4/5) + 1)/2

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Giac [A]  time = 1.1832, size = 12, normalized size = 0.8 \begin{align*} \frac{5}{2} \, \sqrt{x^{\frac{4}{5}} + 1} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/x^(1/5)/(1+x^(4/5))^(1/2),x, algorithm="giac")

[Out]

5/2*sqrt(x^(4/5) + 1)