3.934 \(\int \frac{\sqrt [4]{a-b x^2}}{(c x)^{5/2}} \, dx\)

Optimal. Leaf size=97 \[ \frac{2 b^{3/2} (c x)^{3/2} \left (1-\frac{a}{b x^2}\right )^{3/4} F\left (\left .\frac{1}{2} \csc ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right )\right |2\right )}{3 \sqrt{a} c^4 \left (a-b x^2\right )^{3/4}}-\frac{2 \sqrt [4]{a-b x^2}}{3 c (c x)^{3/2}} \]

[Out]

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

_______________________________________________________________________________________

Rubi [A]  time = 0.215662, antiderivative size = 97, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 6, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.3 \[ \frac{2 b^{3/2} (c x)^{3/2} \left (1-\frac{a}{b x^2}\right )^{3/4} F\left (\left .\frac{1}{2} \csc ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right )\right |2\right )}{3 \sqrt{a} c^4 \left (a-b x^2\right )^{3/4}}-\frac{2 \sqrt [4]{a-b x^2}}{3 c (c x)^{3/2}} \]

Antiderivative was successfully verified.

[In]  Int[(a - b*x^2)^(1/4)/(c*x)^(5/2),x]

[Out]

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

_______________________________________________________________________________________

Rubi in Sympy [A]  time = 32.6221, size = 83, normalized size = 0.86 \[ - \frac{2 \sqrt [4]{a - b x^{2}}}{3 c \left (c x\right )^{\frac{3}{2}}} + \frac{2 b^{\frac{3}{2}} \left (c x\right )^{\frac{3}{2}} \left (- \frac{a}{b x^{2}} + 1\right )^{\frac{3}{4}} F\left (\frac{\operatorname{asin}{\left (\frac{\sqrt{a}}{\sqrt{b} x} \right )}}{2}\middle | 2\right )}{3 \sqrt{a} c^{4} \left (a - b x^{2}\right )^{\frac{3}{4}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((-b*x**2+a)**(1/4)/(c*x)**(5/2),x)

[Out]

-2*(a - b*x**2)**(1/4)/(3*c*(c*x)**(3/2)) + 2*b**(3/2)*(c*x)**(3/2)*(-a/(b*x**2)
+ 1)**(3/4)*elliptic_f(asin(sqrt(a)/(sqrt(b)*x))/2, 2)/(3*sqrt(a)*c**4*(a - b*x
**2)**(3/4))

_______________________________________________________________________________________

Mathematica [C]  time = 0.0561065, size = 70, normalized size = 0.72 \[ -\frac{2 x \left (b x^2 \left (1-\frac{b x^2}{a}\right )^{3/4} \, _2F_1\left (\frac{1}{4},\frac{3}{4};\frac{5}{4};\frac{b x^2}{a}\right )+a-b x^2\right )}{3 (c x)^{5/2} \left (a-b x^2\right )^{3/4}} \]

Antiderivative was successfully verified.

[In]  Integrate[(a - b*x^2)^(1/4)/(c*x)^(5/2),x]

[Out]

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

_______________________________________________________________________________________

Maple [F]  time = 0.053, size = 0, normalized size = 0. \[ \int{1\sqrt [4]{-b{x}^{2}+a} \left ( cx \right ) ^{-{\frac{5}{2}}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((-b*x^2+a)^(1/4)/(c*x)^(5/2),x)

[Out]

int((-b*x^2+a)^(1/4)/(c*x)^(5/2),x)

_______________________________________________________________________________________

Maxima [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{{\left (-b x^{2} + a\right )}^{\frac{1}{4}}}{\left (c x\right )^{\frac{5}{2}}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((-b*x^2 + a)^(1/4)/(c*x)^(5/2),x, algorithm="maxima")

[Out]

integrate((-b*x^2 + a)^(1/4)/(c*x)^(5/2), x)

_______________________________________________________________________________________

Fricas [F]  time = 0., size = 0, normalized size = 0. \[{\rm integral}\left (\frac{{\left (-b x^{2} + a\right )}^{\frac{1}{4}}}{\sqrt{c x} c^{2} x^{2}}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((-b*x^2 + a)^(1/4)/(c*x)^(5/2),x, algorithm="fricas")

[Out]

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

_______________________________________________________________________________________

Sympy [A]  time = 58.3374, size = 37, normalized size = 0.38 \[ - \frac{i \sqrt [4]{b} e^{\frac{7 i \pi }{4}}{{}_{2}F_{1}\left (\begin{matrix} - \frac{1}{4}, \frac{1}{2} \\ \frac{3}{2} \end{matrix}\middle |{\frac{a}{b x^{2}}} \right )}}{c^{\frac{5}{2}} x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((-b*x**2+a)**(1/4)/(c*x)**(5/2),x)

[Out]

-I*b**(1/4)*exp(7*I*pi/4)*hyper((-1/4, 1/2), (3/2,), a/(b*x**2))/(c**(5/2)*x)

_______________________________________________________________________________________

GIAC/XCAS [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{{\left (-b x^{2} + a\right )}^{\frac{1}{4}}}{\left (c x\right )^{\frac{5}{2}}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((-b*x^2 + a)^(1/4)/(c*x)^(5/2),x, algorithm="giac")

[Out]

integrate((-b*x^2 + a)^(1/4)/(c*x)^(5/2), x)