3.785 \(\int \frac {1}{\sqrt [3]{c x} (a+b x^2)^{2/3}} \, dx\)

Optimal. Leaf size=58 \[ \frac {3 (c x)^{2/3} \left (\frac {b x^2}{a}+1\right )^{2/3} \, _2F_1\left (\frac {1}{3},\frac {2}{3};\frac {4}{3};-\frac {b x^2}{a}\right )}{2 c \left (a+b x^2\right )^{2/3}} \]

[Out]

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

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Rubi [A]  time = 0.02, antiderivative size = 58, normalized size of antiderivative = 1.00, 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} \[ \frac {3 (c x)^{2/3} \left (\frac {b x^2}{a}+1\right )^{2/3} \, _2F_1\left (\frac {1}{3},\frac {2}{3};\frac {4}{3};-\frac {b x^2}{a}\right )}{2 c \left (a+b x^2\right )^{2/3}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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])

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])

Rubi steps

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

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Mathematica [A]  time = 0.01, size = 56, normalized size = 0.97 \[ \frac {3 x \left (\frac {b x^2}{a}+1\right )^{2/3} \, _2F_1\left (\frac {1}{3},\frac {2}{3};\frac {4}{3};-\frac {b x^2}{a}\right )}{2 \sqrt [3]{c x} \left (a+b x^2\right )^{2/3}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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fricas [F]  time = 0.85, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {{\left (b x^{2} + a\right )}^{\frac {1}{3}} \left (c x\right )^{\frac {2}{3}}}{b c x^{3} + a c x}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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giac [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {1}{{\left (b x^{2} + a\right )}^{\frac {2}{3}} \left (c x\right )^{\frac {1}{3}}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maple [F]  time = 0.27, size = 0, normalized size = 0.00 \[ \int \frac {1}{\left (c x \right )^{\frac {1}{3}} \left (b \,x^{2}+a \right )^{\frac {2}{3}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maxima [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {1}{{\left (b x^{2} + a\right )}^{\frac {2}{3}} \left (c x\right )^{\frac {1}{3}}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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mupad [F]  time = 0.00, size = -1, normalized size = -0.02 \[ \int \frac {1}{{\left (c\,x\right )}^{1/3}\,{\left (b\,x^2+a\right )}^{2/3}} \,d x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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sympy [C]  time = 1.38, size = 46, normalized size = 0.79 \[ \frac {\Gamma \left (- \frac {1}{3}\right ) {{}_{2}F_{1}\left (\begin {matrix} \frac {1}{3}, \frac {2}{3} \\ \frac {4}{3} \end {matrix}\middle | {\frac {a e^{i \pi }}{b x^{2}}} \right )}}{2 b^{\frac {2}{3}} \sqrt [3]{c} x^{\frac {2}{3}} \Gamma \left (\frac {2}{3}\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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