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

Optimal. Leaf size=92 \[ \frac{\sqrt{c x} \sqrt [4]{a-b x^2}}{c}-\frac{\sqrt{a} \sqrt{b} (c x)^{3/2} \left (1-\frac{a}{b x^2}\right )^{3/4} F\left (\left .\frac{1}{2} \csc ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right )\right |2\right )}{c^2 \left (a-b x^2\right )^{3/4}} \]

[Out]

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

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Rubi [A]  time = 0.215925, antiderivative size = 92, 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 \[ \frac{\sqrt{c x} \sqrt [4]{a-b x^2}}{c}-\frac{\sqrt{a} \sqrt{b} (c x)^{3/2} \left (1-\frac{a}{b x^2}\right )^{3/4} F\left (\left .\frac{1}{2} \csc ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right )\right |2\right )}{c^2 \left (a-b x^2\right )^{3/4}} \]

Antiderivative was successfully verified.

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

[Out]

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

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Rubi in Sympy [A]  time = 32.3137, size = 76, normalized size = 0.83 \[ - \frac{\sqrt{a} \sqrt{b} \left (c x\right )^{\frac{3}{2}} \left (- \frac{a}{b x^{2}} + 1\right )^{\frac{3}{4}} F\left (\frac{\operatorname{asin}{\left (\frac{\sqrt{a}}{\sqrt{b} x} \right )}}{2}\middle | 2\right )}{c^{2} \left (a - b x^{2}\right )^{\frac{3}{4}}} + \frac{\sqrt{c x} \sqrt [4]{a - b x^{2}}}{c} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [C]  time = 0.0468663, size = 66, normalized size = 0.72 \[ \frac{a x \left (1-\frac{b x^2}{a}\right )^{3/4} \, _2F_1\left (\frac{1}{4},\frac{3}{4};\frac{5}{4};\frac{b x^2}{a}\right )+a x-b x^3}{\sqrt{c x} \left (a-b x^2\right )^{3/4}} \]

Antiderivative was successfully verified.

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

[Out]

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

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Maple [F]  time = 0.047, size = 0, normalized size = 0. \[ \int{1\sqrt [4]{-b{x}^{2}+a}{\frac{1}{\sqrt{cx}}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Sympy [A]  time = 5.77888, size = 39, normalized size = 0.42 \[ - \frac{i \sqrt [4]{b} x e^{\frac{3 i \pi }{4}}{{}_{2}F_{1}\left (\begin{matrix} - \frac{1}{2}, - \frac{1}{4} \\ \frac{1}{2} \end{matrix}\middle |{\frac{a}{b x^{2}}} \right )}}{\sqrt{c}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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