3.769 \(\int \frac{(a+b x^2)^{4/3}}{\sqrt [3]{c x}} \, dx\)

Optimal. Leaf size=59 \[ \frac{3 a (c x)^{2/3} \sqrt [3]{a+b x^2} \, _2F_1\left (-\frac{4}{3},\frac{1}{3};\frac{4}{3};-\frac{b x^2}{a}\right )}{2 c \sqrt [3]{\frac{b x^2}{a}+1}} \]

[Out]

(3*a*(c*x)^(2/3)*(a + b*x^2)^(1/3)*Hypergeometric2F1[-4/3, 1/3, 4/3, -((b*x^2)/a)])/(2*c*(1 + (b*x^2)/a)^(1/3)
)

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Rubi [A]  time = 0.0185268, antiderivative size = 59, 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} \[ \frac{3 a (c x)^{2/3} \sqrt [3]{a+b x^2} \, _2F_1\left (-\frac{4}{3},\frac{1}{3};\frac{4}{3};-\frac{b x^2}{a}\right )}{2 c \sqrt [3]{\frac{b x^2}{a}+1}} \]

Antiderivative was successfully verified.

[In]

Int[(a + b*x^2)^(4/3)/(c*x)^(1/3),x]

[Out]

(3*a*(c*x)^(2/3)*(a + b*x^2)^(1/3)*Hypergeometric2F1[-4/3, 1/3, 4/3, -((b*x^2)/a)])/(2*c*(1 + (b*x^2)/a)^(1/3)
)

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{\left (a+b x^2\right )^{4/3}}{\sqrt [3]{c x}} \, dx &=\frac{\left (a \sqrt [3]{a+b x^2}\right ) \int \frac{\left (1+\frac{b x^2}{a}\right )^{4/3}}{\sqrt [3]{c x}} \, dx}{\sqrt [3]{1+\frac{b x^2}{a}}}\\ &=\frac{3 a (c x)^{2/3} \sqrt [3]{a+b x^2} \, _2F_1\left (-\frac{4}{3},\frac{1}{3};\frac{4}{3};-\frac{b x^2}{a}\right )}{2 c \sqrt [3]{1+\frac{b x^2}{a}}}\\ \end{align*}

Mathematica [A]  time = 0.0119165, size = 57, normalized size = 0.97 \[ \frac{3 a x \sqrt [3]{a+b x^2} \, _2F_1\left (-\frac{4}{3},\frac{1}{3};\frac{4}{3};-\frac{b x^2}{a}\right )}{2 \sqrt [3]{c x} \sqrt [3]{\frac{b x^2}{a}+1}} \]

Antiderivative was successfully verified.

[In]

Integrate[(a + b*x^2)^(4/3)/(c*x)^(1/3),x]

[Out]

(3*a*x*(a + b*x^2)^(1/3)*Hypergeometric2F1[-4/3, 1/3, 4/3, -((b*x^2)/a)])/(2*(c*x)^(1/3)*(1 + (b*x^2)/a)^(1/3)
)

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Maple [F]  time = 0.017, size = 0, normalized size = 0. \begin{align*} \int{ \left ( b{x}^{2}+a \right ) ^{{\frac{4}{3}}}{\frac{1}{\sqrt [3]{cx}}}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((b*x^2+a)^(4/3)/(c*x)^(1/3),x)

[Out]

int((b*x^2+a)^(4/3)/(c*x)^(1/3),x)

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Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{{\left (b x^{2} + a\right )}^{\frac{4}{3}}}{\left (c x\right )^{\frac{1}{3}}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x^2+a)^(4/3)/(c*x)^(1/3),x, algorithm="maxima")

[Out]

integrate((b*x^2 + a)^(4/3)/(c*x)^(1/3), x)

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Fricas [F]  time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (\frac{{\left (b x^{2} + a\right )}^{\frac{4}{3}} \left (c x\right )^{\frac{2}{3}}}{c x}, x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x^2+a)^(4/3)/(c*x)^(1/3),x, algorithm="fricas")

[Out]

integral((b*x^2 + a)^(4/3)*(c*x)^(2/3)/(c*x), x)

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Sympy [C]  time = 11.6606, size = 46, normalized size = 0.78 \begin{align*} \frac{a^{\frac{4}{3}} x^{\frac{2}{3}} \Gamma \left (\frac{1}{3}\right ){{}_{2}F_{1}\left (\begin{matrix} - \frac{4}{3}, \frac{1}{3} \\ \frac{4}{3} \end{matrix}\middle |{\frac{b x^{2} e^{i \pi }}{a}} \right )}}{2 \sqrt [3]{c} \Gamma \left (\frac{4}{3}\right )} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x**2+a)**(4/3)/(c*x)**(1/3),x)

[Out]

a**(4/3)*x**(2/3)*gamma(1/3)*hyper((-4/3, 1/3), (4/3,), b*x**2*exp_polar(I*pi)/a)/(2*c**(1/3)*gamma(4/3))

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{{\left (b x^{2} + a\right )}^{\frac{4}{3}}}{\left (c x\right )^{\frac{1}{3}}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x^2+a)^(4/3)/(c*x)^(1/3),x, algorithm="giac")

[Out]

integrate((b*x^2 + a)^(4/3)/(c*x)^(1/3), x)