3.2450 \(\int (a+\frac{b}{x^{3/5}})^{2/3} \, dx\)

Optimal. Leaf size=18 \[ \frac{x \left (a+\frac{b}{x^{3/5}}\right )^{5/3}}{a} \]

[Out]

((a + b/x^(3/5))^(5/3)*x)/a

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Rubi [A]  time = 0.0025717, antiderivative size = 18, normalized size of antiderivative = 1., number of steps used = 1, number of rules used = 1, integrand size = 13, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.077, Rules used = {191} \[ \frac{x \left (a+\frac{b}{x^{3/5}}\right )^{5/3}}{a} \]

Antiderivative was successfully verified.

[In]

Int[(a + b/x^(3/5))^(2/3),x]

[Out]

((a + b/x^(3/5))^(5/3)*x)/a

Rule 191

Int[((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[(x*(a + b*x^n)^(p + 1))/a, x] /; FreeQ[{a, b, n, p}, x] &
& EqQ[1/n + p + 1, 0]

Rubi steps

\begin{align*} \int \left (a+\frac{b}{x^{3/5}}\right )^{2/3} \, dx &=\frac{\left (a+\frac{b}{x^{3/5}}\right )^{5/3} x}{a}\\ \end{align*}

Mathematica [A]  time = 0.0172939, size = 18, normalized size = 1. \[ \frac{x \left (a+\frac{b}{x^{3/5}}\right )^{5/3}}{a} \]

Antiderivative was successfully verified.

[In]

Integrate[(a + b/x^(3/5))^(2/3),x]

[Out]

((a + b/x^(3/5))^(5/3)*x)/a

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Maple [F]  time = 0.018, size = 0, normalized size = 0. \begin{align*} \int \left ( a+{b{x}^{-{\frac{3}{5}}}} \right ) ^{{\frac{2}{3}}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a+b/x^(3/5))^(2/3),x)

[Out]

int((a+b/x^(3/5))^(2/3),x)

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Maxima [A]  time = 0.982942, size = 19, normalized size = 1.06 \begin{align*} \frac{{\left (a + \frac{b}{x^{\frac{3}{5}}}\right )}^{\frac{5}{3}} x}{a} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b/x^(3/5))^(2/3),x, algorithm="maxima")

[Out]

(a + b/x^(3/5))^(5/3)*x/a

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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((a+b/x^(3/5))^(2/3),x, algorithm="fricas")

[Out]

Timed out

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Sympy [B]  time = 59.8269, size = 78, normalized size = 4.33 \begin{align*} - \frac{5 b^{\frac{2}{3}} x^{\frac{3}{5}} \left (\frac{a x^{\frac{3}{5}}}{b} + 1\right )^{\frac{2}{3}} \Gamma \left (- \frac{5}{3}\right )}{3 \Gamma \left (- \frac{2}{3}\right )} - \frac{5 b^{\frac{5}{3}} \left (\frac{a x^{\frac{3}{5}}}{b} + 1\right )^{\frac{2}{3}} \Gamma \left (- \frac{5}{3}\right )}{3 a \Gamma \left (- \frac{2}{3}\right )} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b/x**(3/5))**(2/3),x)

[Out]

-5*b**(2/3)*x**(3/5)*(a*x**(3/5)/b + 1)**(2/3)*gamma(-5/3)/(3*gamma(-2/3)) - 5*b**(5/3)*(a*x**(3/5)/b + 1)**(2
/3)*gamma(-5/3)/(3*a*gamma(-2/3))

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int{\left (a + \frac{b}{x^{\frac{3}{5}}}\right )}^{\frac{2}{3}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b/x^(3/5))^(2/3),x, algorithm="giac")

[Out]

integrate((a + b/x^(3/5))^(2/3), x)