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

Optimal. Leaf size=58 \[ \frac{3 (c x)^{5/3} \left (\frac{b x^2}{a}+1\right )^{2/3} \, _2F_1\left (\frac{2}{3},\frac{5}{6};\frac{11}{6};-\frac{b x^2}{a}\right )}{5 c \left (a+b x^2\right )^{2/3}} \]

[Out]

(3*(c*x)^(5/3)*(1 + (b*x^2)/a)^(2/3)*Hypergeometric2F1[2/3, 5/6, 11/6, -((b*x^2)/a)])/(5*c*(a + b*x^2)^(2/3))

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Rubi [A]  time = 0.0183455, antiderivative size = 58, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.105, Rules used = {365, 364} \[ \frac{3 (c x)^{5/3} \left (\frac{b x^2}{a}+1\right )^{2/3} \, _2F_1\left (\frac{2}{3},\frac{5}{6};\frac{11}{6};-\frac{b x^2}{a}\right )}{5 c \left (a+b x^2\right )^{2/3}} \]

Antiderivative was successfully verified.

[In]

Int[(c*x)^(2/3)/(a + b*x^2)^(2/3),x]

[Out]

(3*(c*x)^(5/3)*(1 + (b*x^2)/a)^(2/3)*Hypergeometric2F1[2/3, 5/6, 11/6, -((b*x^2)/a)])/(5*c*(a + b*x^2)^(2/3))

Rule 365

Int[((c_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Dist[(a^IntPart[p]*(a + b*x^n)^FracPart[p])
/(1 + (b*x^n)/a)^FracPart[p], Int[(c*x)^m*(1 + (b*x^n)/a)^p, x], x] /; FreeQ[{a, b, c, m, n, p}, x] &&  !IGtQ[
p, 0] &&  !(ILtQ[p, 0] || GtQ[a, 0])

Rule 364

Int[((c_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[(a^p*(c*x)^(m + 1)*Hypergeometric2F1[-
p, (m + 1)/n, (m + 1)/n + 1, -((b*x^n)/a)])/(c*(m + 1)), x] /; FreeQ[{a, b, c, m, n, p}, x] &&  !IGtQ[p, 0] &&
(ILtQ[p, 0] || GtQ[a, 0])

Rubi steps

\begin{align*} \int \frac{(c x)^{2/3}}{\left (a+b x^2\right )^{2/3}} \, dx &=\frac{\left (1+\frac{b x^2}{a}\right )^{2/3} \int \frac{(c x)^{2/3}}{\left (1+\frac{b x^2}{a}\right )^{2/3}} \, dx}{\left (a+b x^2\right )^{2/3}}\\ &=\frac{3 (c x)^{5/3} \left (1+\frac{b x^2}{a}\right )^{2/3} \, _2F_1\left (\frac{2}{3},\frac{5}{6};\frac{11}{6};-\frac{b x^2}{a}\right )}{5 c \left (a+b x^2\right )^{2/3}}\\ \end{align*}

Mathematica [A]  time = 0.0118511, size = 56, normalized size = 0.97 \[ \frac{3 x (c x)^{2/3} \left (\frac{b x^2}{a}+1\right )^{2/3} \, _2F_1\left (\frac{2}{3},\frac{5}{6};\frac{11}{6};-\frac{b x^2}{a}\right )}{5 \left (a+b x^2\right )^{2/3}} \]

Antiderivative was successfully verified.

[In]

Integrate[(c*x)^(2/3)/(a + b*x^2)^(2/3),x]

[Out]

(3*x*(c*x)^(2/3)*(1 + (b*x^2)/a)^(2/3)*Hypergeometric2F1[2/3, 5/6, 11/6, -((b*x^2)/a)])/(5*(a + b*x^2)^(2/3))

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Maple [F]  time = 0.022, size = 0, normalized size = 0. \begin{align*} \int{ \left ( cx \right ) ^{{\frac{2}{3}}} \left ( b{x}^{2}+a \right ) ^{-{\frac{2}{3}}}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Fricas [F]  time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (\frac{\left (c x\right )^{\frac{2}{3}}}{{\left (b x^{2} + a\right )}^{\frac{2}{3}}}, x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((c*x)^(2/3)/(b*x^2+a)^(2/3),x, algorithm="fricas")

[Out]

integral((c*x)^(2/3)/(b*x^2 + a)^(2/3), x)

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Sympy [C]  time = 2.03337, size = 44, normalized size = 0.76 \begin{align*} \frac{c^{\frac{2}{3}} x^{\frac{5}{3}} \Gamma \left (\frac{5}{6}\right ){{}_{2}F_{1}\left (\begin{matrix} \frac{2}{3}, \frac{5}{6} \\ \frac{11}{6} \end{matrix}\middle |{\frac{b x^{2} e^{i \pi }}{a}} \right )}}{2 a^{\frac{2}{3}} \Gamma \left (\frac{11}{6}\right )} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

c**(2/3)*x**(5/3)*gamma(5/6)*hyper((2/3, 5/6), (11/6,), b*x**2*exp_polar(I*pi)/a)/(2*a**(2/3)*gamma(11/6))

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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