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]

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

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Rubi [A]  time = 0.00, antiderivative size = 15, normalized size of antiderivative = 1.00, 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*}

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Mathematica [A]  time = 0.00, size = 15, normalized size = 1.00 \[ \frac {5}{2} \sqrt {x^{4/5}+1} \]

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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fricas [F(-1)]  time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]

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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giac [A]  time = 0.16, size = 9, normalized size = 0.60 \[ \frac {5}{2} \, \sqrt {x^{\frac {4}{5}} + 1} \]

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)

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maple [A]  time = 0.00, size = 10, normalized size = 0.67 \[ \frac {5 \sqrt {x^{\frac {4}{5}}+1}}{2} \]

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 = 0.52, size = 9, normalized size = 0.60 \[ \frac {5}{2} \, \sqrt {x^{\frac {4}{5}} + 1} \]

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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mupad [B]  time = 1.41, size = 9, normalized size = 0.60 \[ \frac {5\,\sqrt {x^{4/5}+1}}{2} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

(5*(x^(4/5) + 1)^(1/2))/2

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sympy [A]  time = 0.64, size = 12, normalized size = 0.80 \[ \frac {5 \sqrt {x^{\frac {4}{5}} + 1}}{2} \]

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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