3.922 \(\int \frac {\sqrt [6]{x}+\sqrt [5]{x^3}}{\sqrt {x}} \, dx\)

Optimal. Leaf size=26 \[ \frac {3 x^{2/3}}{2}+\frac {10}{11} \sqrt [5]{x^3} \sqrt {x} \]

[Out]

3/2*x^(2/3)+10/11*(x^3)^(1/5)*x^(1/2)

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Rubi [A]  time = 0.01, antiderivative size = 26, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, integrand size = 19, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.158, Rules used = {14, 15, 30} \[ \frac {3 x^{2/3}}{2}+\frac {10}{11} \sqrt [5]{x^3} \sqrt {x} \]

Antiderivative was successfully verified.

[In]

Int[(x^(1/6) + (x^3)^(1/5))/Sqrt[x],x]

[Out]

(3*x^(2/3))/2 + (10*Sqrt[x]*(x^3)^(1/5))/11

Rule 14

Int[(u_)*((c_.)*(x_))^(m_.), x_Symbol] :> Int[ExpandIntegrand[(c*x)^m*u, x], x] /; FreeQ[{c, m}, x] && SumQ[u]
&&  !LinearQ[u, x] &&  !MatchQ[u, (a_) + (b_.)*(v_) /; FreeQ[{a, b}, x] && InverseFunctionQ[v]]

Rule 15

Int[(u_.)*((a_.)*(x_)^(n_))^(m_), x_Symbol] :> Dist[(a^IntPart[m]*(a*x^n)^FracPart[m])/x^(n*FracPart[m]), Int[
u*x^(m*n), x], x] /; FreeQ[{a, m, n}, x] &&  !IntegerQ[m]

Rule 30

Int[(x_)^(m_.), x_Symbol] :> Simp[x^(m + 1)/(m + 1), x] /; FreeQ[m, x] && NeQ[m, -1]

Rubi steps

\begin {align*} \int \frac {\sqrt [6]{x}+\sqrt [5]{x^3}}{\sqrt {x}} \, dx &=\int \left (\frac {1}{\sqrt [3]{x}}+\frac {\sqrt [5]{x^3}}{\sqrt {x}}\right ) \, dx\\ &=\frac {3 x^{2/3}}{2}+\int \frac {\sqrt [5]{x^3}}{\sqrt {x}} \, dx\\ &=\frac {3 x^{2/3}}{2}+\frac {\sqrt [5]{x^3} \int \sqrt [10]{x} \, dx}{x^{3/5}}\\ &=\frac {3 x^{2/3}}{2}+\frac {10}{11} \sqrt {x} \sqrt [5]{x^3}\\ \end {align*}

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Mathematica [A]  time = 0.01, size = 26, normalized size = 1.00 \[ \frac {3 x^{2/3}}{2}+\frac {10}{11} \sqrt [5]{x^3} \sqrt {x} \]

Antiderivative was successfully verified.

[In]

Integrate[(x^(1/6) + (x^3)^(1/5))/Sqrt[x],x]

[Out]

(3*x^(2/3))/2 + (10*Sqrt[x]*(x^3)^(1/5))/11

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fricas [A]  time = 0.85, size = 16, normalized size = 0.62 \[ \frac {10}{11} \, {\left (x^{3}\right )}^{\frac {1}{5}} \sqrt {x} + \frac {3}{2} \, x^{\frac {2}{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((x^(1/6)+(x^3)^(1/5))/x^(1/2),x, algorithm="fricas")

[Out]

10/11*(x^3)^(1/5)*sqrt(x) + 3/2*x^(2/3)

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giac [A]  time = 0.51, size = 11, normalized size = 0.42 \[ \frac {10}{11} \, x^{\frac {11}{10}} + \frac {3}{2} \, x^{\frac {2}{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((x^(1/6)+(x^3)^(1/5))/x^(1/2),x, algorithm="giac")

[Out]

10/11*x^(11/10) + 3/2*x^(2/3)

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maple [A]  time = 0.00, size = 17, normalized size = 0.65 \[ \frac {3 x^{\frac {2}{3}}}{2}+\frac {10 \left (x^{3}\right )^{\frac {1}{5}} \sqrt {x}}{11} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((x^(1/6)+(x^3)^(1/5))/x^(1/2),x)

[Out]

3/2*x^(2/3)+10/11*(x^3)^(1/5)*x^(1/2)

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maxima [A]  time = 0.96, size = 16, normalized size = 0.62 \[ \frac {10}{11} \, {\left (x^{3}\right )}^{\frac {1}{5}} \sqrt {x} + \frac {3}{2} \, x^{\frac {2}{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((x^(1/6)+(x^3)^(1/5))/x^(1/2),x, algorithm="maxima")

[Out]

10/11*(x^3)^(1/5)*sqrt(x) + 3/2*x^(2/3)

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mupad [B]  time = 3.54, size = 16, normalized size = 0.62 \[ \frac {10\,\sqrt {x}\,{\left (x^3\right )}^{1/5}}{11}+\frac {3\,x^{2/3}}{2} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((x^3)^(1/5) + x^(1/6))/x^(1/2),x)

[Out]

(10*x^(1/2)*(x^3)^(1/5))/11 + (3*x^(2/3))/2

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sympy [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((x**(1/6)+(x**3)**(1/5))/x**(1/2),x)

[Out]

Timed out

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