3.994 \(\int \frac{\sqrt{c x}}{\left (a+b x^2\right )^{5/4}} \, dx\)

Optimal. Leaf size=63 \[ -\frac{2 \sqrt{c x} \sqrt [4]{\frac{a}{b x^2}+1} E\left (\left .\frac{1}{2} \cot ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right )\right |2\right )}{\sqrt{a} \sqrt{b} \sqrt [4]{a+b x^2}} \]

[Out]

(-2*(1 + a/(b*x^2))^(1/4)*Sqrt[c*x]*EllipticE[ArcCot[(Sqrt[b]*x)/Sqrt[a]]/2, 2])
/(Sqrt[a]*Sqrt[b]*(a + b*x^2)^(1/4))

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Rubi [A]  time = 0.0742066, antiderivative size = 63, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.158 \[ -\frac{2 \sqrt{c x} \sqrt [4]{\frac{a}{b x^2}+1} E\left (\left .\frac{1}{2} \cot ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right )\right |2\right )}{\sqrt{a} \sqrt{b} \sqrt [4]{a+b x^2}} \]

Antiderivative was successfully verified.

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

[Out]

(-2*(1 + a/(b*x^2))^(1/4)*Sqrt[c*x]*EllipticE[ArcCot[(Sqrt[b]*x)/Sqrt[a]]/2, 2])
/(Sqrt[a]*Sqrt[b]*(a + b*x^2)^(1/4))

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Rubi in Sympy [F]  time = 0., size = 0, normalized size = 0. \[ \frac{\sqrt{c x} \sqrt [4]{\frac{a}{b x^{2}} + 1} \int ^{\frac{1}{x}} \frac{1}{\sqrt [4]{\frac{a x^{2}}{b} + 1}}\, dx}{b \sqrt [4]{a + b x^{2}}} - \frac{2 \sqrt{c x}}{b x \sqrt [4]{a + b x^{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [C]  time = 0.0503388, size = 63, normalized size = 1. \[ -\frac{2 x \sqrt{c x} \left (2 \sqrt [4]{\frac{b x^2}{a}+1} \, _2F_1\left (\frac{1}{4},\frac{3}{4};\frac{7}{4};-\frac{b x^2}{a}\right )-3\right )}{3 a \sqrt [4]{a+b x^2}} \]

Antiderivative was successfully verified.

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

[Out]

(-2*x*Sqrt[c*x]*(-3 + 2*(1 + (b*x^2)/a)^(1/4)*Hypergeometric2F1[1/4, 3/4, 7/4, -
((b*x^2)/a)]))/(3*a*(a + b*x^2)^(1/4))

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Maple [F]  time = 0.039, size = 0, normalized size = 0. \[ \int{1\sqrt{cx} \left ( b{x}^{2}+a \right ) ^{-{\frac{5}{4}}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Fricas [F]  time = 0., size = 0, normalized size = 0. \[{\rm integral}\left (\frac{\sqrt{c x}}{{\left (b x^{2} + a\right )}^{\frac{5}{4}}}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

integral(sqrt(c*x)/(b*x^2 + a)^(5/4), x)

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Sympy [A]  time = 14.1861, size = 44, normalized size = 0.7 \[ \frac{\sqrt{c} x^{\frac{3}{2}} \Gamma \left (\frac{3}{4}\right ){{}_{2}F_{1}\left (\begin{matrix} \frac{3}{4}, \frac{5}{4} \\ \frac{7}{4} \end{matrix}\middle |{\frac{b x^{2} e^{i \pi }}{a}} \right )}}{2 a^{\frac{5}{4}} \Gamma \left (\frac{7}{4}\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

sqrt(c)*x**(3/2)*gamma(3/4)*hyper((3/4, 5/4), (7/4,), b*x**2*exp_polar(I*pi)/a)/
(2*a**(5/4)*gamma(7/4))

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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