3.8.86 \(\int \frac {1}{(c x)^{4/3} (a+b x^2)^{2/3}} \, dx\) [786]

Optimal. Leaf size=56 \[ -\frac {3 \left (1+\frac {b x^2}{a}\right )^{2/3} \, _2F_1\left (-\frac {1}{6},\frac {2}{3};\frac {5}{6};-\frac {b x^2}{a}\right )}{c \sqrt [3]{c x} \left (a+b x^2\right )^{2/3}} \]

[Out]

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

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Rubi [A]
time = 0.01, antiderivative size = 56, 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 = {372, 371} \begin {gather*} -\frac {3 \left (\frac {b x^2}{a}+1\right )^{2/3} \, _2F_1\left (-\frac {1}{6},\frac {2}{3};\frac {5}{6};-\frac {b x^2}{a}\right )}{c \sqrt [3]{c x} \left (a+b x^2\right )^{2/3}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

Rule 371

Int[((c_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[a^p*((c*x)^(m + 1)/(c*(m + 1)))*Hyperg
eometric2F1[-p, (m + 1)/n, (m + 1)/n + 1, (-b)*(x^n/a)], x] /; FreeQ[{a, b, c, m, n, p}, x] &&  !IGtQ[p, 0] &&
(ILtQ[p, 0] || GtQ[a, 0])

Rule 372

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}{(c x)^{4/3} \left (a+b x^2\right )^{2/3}} \, dx &=\frac {\left (1+\frac {b x^2}{a}\right )^{2/3} \int \frac {1}{(c x)^{4/3} \left (1+\frac {b x^2}{a}\right )^{2/3}} \, dx}{\left (a+b x^2\right )^{2/3}}\\ &=-\frac {3 \left (1+\frac {b x^2}{a}\right )^{2/3} \, _2F_1\left (-\frac {1}{6},\frac {2}{3};\frac {5}{6};-\frac {b x^2}{a}\right )}{c \sqrt [3]{c x} \left (a+b x^2\right )^{2/3}}\\ \end {align*}

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Mathematica [A]
time = 10.01, size = 54, normalized size = 0.96 \begin {gather*} -\frac {3 x \left (1+\frac {b x^2}{a}\right )^{2/3} \, _2F_1\left (-\frac {1}{6},\frac {2}{3};\frac {5}{6};-\frac {b x^2}{a}\right )}{(c x)^{4/3} \left (a+b x^2\right )^{2/3}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(-3*x*(1 + (b*x^2)/a)^(2/3)*Hypergeometric2F1[-1/6, 2/3, 5/6, -((b*x^2)/a)])/((c*x)^(4/3)*(a + b*x^2)^(2/3))

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Maple [F]
time = 0.01, size = 0, normalized size = 0.00 \[\int \frac {1}{\left (c x \right )^{\frac {4}{3}} \left (b \,x^{2}+a \right )^{\frac {2}{3}}}\, dx\]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Fricas [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Sympy [C] Result contains complex when optimal does not.
time = 1.73, size = 48, normalized size = 0.86 \begin {gather*} \frac {\Gamma \left (- \frac {1}{6}\right ) {{}_{2}F_{1}\left (\begin {matrix} - \frac {1}{6}, \frac {2}{3} \\ \frac {5}{6} \end {matrix}\middle | {\frac {b x^{2} e^{i \pi }}{a}} \right )}}{2 a^{\frac {2}{3}} c^{\frac {4}{3}} \sqrt [3]{x} \Gamma \left (\frac {5}{6}\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

gamma(-1/6)*hyper((-1/6, 2/3), (5/6,), b*x**2*exp_polar(I*pi)/a)/(2*a**(2/3)*c**(4/3)*x**(1/3)*gamma(5/6))

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Giac [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Mupad [F]
time = 0.00, size = -1, normalized size = -0.02 \begin {gather*} \int \frac {1}{{\left (c\,x\right )}^{4/3}\,{\left (b\,x^2+a\right )}^{2/3}} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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