3.975 \(\int \frac{(c x)^{3/2}}{\left (a-b x^2\right )^{3/4}} \, dx\)

Optimal. Leaf size=91 \[ -\frac{c \sqrt{c x} \sqrt [4]{a-b x^2}}{b}-\frac{\sqrt{a} (c x)^{3/2} \left (1-\frac{a}{b x^2}\right )^{3/4} F\left (\left .\frac{1}{2} \csc ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right )\right |2\right )}{\sqrt{b} \left (a-b x^2\right )^{3/4}} \]

[Out]

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

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Rubi [A]  time = 0.208862, antiderivative size = 91, 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 \[ -\frac{c \sqrt{c x} \sqrt [4]{a-b x^2}}{b}-\frac{\sqrt{a} (c x)^{3/2} \left (1-\frac{a}{b x^2}\right )^{3/4} F\left (\left .\frac{1}{2} \csc ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right )\right |2\right )}{\sqrt{b} \left (a-b x^2\right )^{3/4}} \]

Antiderivative was successfully verified.

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

[Out]

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

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Rubi in Sympy [A]  time = 32.4337, size = 76, normalized size = 0.84 \[ - \frac{\sqrt{a} \left (c x\right )^{\frac{3}{2}} \left (- \frac{a}{b x^{2}} + 1\right )^{\frac{3}{4}} F\left (\frac{\operatorname{asin}{\left (\frac{\sqrt{a}}{\sqrt{b} x} \right )}}{2}\middle | 2\right )}{\sqrt{b} \left (a - b x^{2}\right )^{\frac{3}{4}}} - \frac{c \sqrt{c x} \sqrt [4]{a - b x^{2}}}{b} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((c*x)**(3/2)/(-b*x**2+a)**(3/4),x)

[Out]

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

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Mathematica [C]  time = 0.0601226, size = 68, normalized size = 0.75 \[ \frac{c \sqrt{c x} \left (a \left (1-\frac{b x^2}{a}\right )^{3/4} \, _2F_1\left (\frac{1}{4},\frac{3}{4};\frac{5}{4};\frac{b x^2}{a}\right )-a+b x^2\right )}{b \left (a-b x^2\right )^{3/4}} \]

Antiderivative was successfully verified.

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

[Out]

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

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Maple [F]  time = 0.045, size = 0, normalized size = 0. \[ \int{1 \left ( cx \right ) ^{{\frac{3}{2}}} \left ( -b{x}^{2}+a \right ) ^{-{\frac{3}{4}}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((c*x)^(3/2)/(-b*x^2 + a)^(3/4),x, algorithm="maxima")

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((c*x)^(3/2)/(-b*x^2 + a)^(3/4),x, algorithm="fricas")

[Out]

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

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Sympy [A]  time = 13.4456, size = 46, normalized size = 0.51 \[ \frac{c^{\frac{3}{2}} x^{\frac{5}{2}} \Gamma \left (\frac{5}{4}\right ){{}_{2}F_{1}\left (\begin{matrix} \frac{3}{4}, \frac{5}{4} \\ \frac{9}{4} \end{matrix}\middle |{\frac{b x^{2} e^{2 i \pi }}{a}} \right )}}{2 a^{\frac{3}{4}} \Gamma \left (\frac{9}{4}\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((c*x)**(3/2)/(-b*x**2+a)**(3/4),x)

[Out]

c**(3/2)*x**(5/2)*gamma(5/4)*hyper((3/4, 5/4), (9/4,), b*x**2*exp_polar(2*I*pi)/
a)/(2*a**(3/4)*gamma(9/4))

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GIAC/XCAS [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{\left (c x\right )^{\frac{3}{2}}}{{\left (-b x^{2} + a\right )}^{\frac{3}{4}}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((c*x)^(3/2)/(-b*x^2 + a)^(3/4),x, algorithm="giac")

[Out]

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