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

________________________________________________________________________________________

Rubi [A]  time = 0.0063937, antiderivative size = 26, normalized size of antiderivative = 1., 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*}

Mathematica [A]  time = 0.0117846, size = 26, normalized size = 1. \[ \frac{3 x^{2/3}}{2}+\frac{10}{11} \sqrt [5]{x^3} \sqrt{x} \]

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

________________________________________________________________________________________

Maple [A]  time = 0.002, size = 17, normalized size = 0.7 \begin{align*}{\frac{3}{2}{x}^{{\frac{2}{3}}}}+{\frac{10}{11}\sqrt [5]{{x}^{3}}\sqrt{x}} \end{align*}

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)

________________________________________________________________________________________

Maxima [A]  time = 1.05393, size = 22, normalized size = 0.85 \begin{align*} \frac{10}{11} \,{\left (x^{3}\right )}^{\frac{1}{5}} \sqrt{x} + \frac{3}{2} \, x^{\frac{2}{3}} \end{align*}

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)

________________________________________________________________________________________

Fricas [A]  time = 1.89933, size = 55, normalized size = 2.12 \begin{align*} \frac{10}{11} \,{\left (x^{3}\right )}^{\frac{1}{5}} \sqrt{x} + \frac{3}{2} \, x^{\frac{2}{3}} \end{align*}

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)

________________________________________________________________________________________

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

[Out]

Timed out

________________________________________________________________________________________

Giac [A]  time = 1.06577, size = 15, normalized size = 0.58 \begin{align*} \frac{10}{11} \, x^{\frac{11}{10}} + \frac{3}{2} \, x^{\frac{2}{3}} \end{align*}

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)