3.595 \(\int \frac{\sqrt{a+b x^2}}{(c x)^{5/2}} \, dx\)

Optimal. Leaf size=126 \[ \frac{2 b^{3/4} \left (\sqrt{a}+\sqrt{b} x\right ) \sqrt{\frac{a+b x^2}{\left (\sqrt{a}+\sqrt{b} x\right )^2}} F\left (2 \tan ^{-1}\left (\frac{\sqrt [4]{b} \sqrt{c x}}{\sqrt [4]{a} \sqrt{c}}\right )|\frac{1}{2}\right )}{3 \sqrt [4]{a} c^{5/2} \sqrt{a+b x^2}}-\frac{2 \sqrt{a+b x^2}}{3 c (c x)^{3/2}} \]

[Out]

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

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Rubi [A]  time = 0.199549, antiderivative size = 126, 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 \[ \frac{2 b^{3/4} \left (\sqrt{a}+\sqrt{b} x\right ) \sqrt{\frac{a+b x^2}{\left (\sqrt{a}+\sqrt{b} x\right )^2}} F\left (2 \tan ^{-1}\left (\frac{\sqrt [4]{b} \sqrt{c x}}{\sqrt [4]{a} \sqrt{c}}\right )|\frac{1}{2}\right )}{3 \sqrt [4]{a} c^{5/2} \sqrt{a+b x^2}}-\frac{2 \sqrt{a+b x^2}}{3 c (c x)^{3/2}} \]

Antiderivative was successfully verified.

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

[Out]

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

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Rubi in Sympy [A]  time = 18.0486, size = 114, normalized size = 0.9 \[ - \frac{2 \sqrt{a + b x^{2}}}{3 c \left (c x\right )^{\frac{3}{2}}} + \frac{2 b^{\frac{3}{4}} \sqrt{\frac{a + b x^{2}}{\left (\sqrt{a} + \sqrt{b} x\right )^{2}}} \left (\sqrt{a} + \sqrt{b} x\right ) F\left (2 \operatorname{atan}{\left (\frac{\sqrt [4]{b} \sqrt{c x}}{\sqrt [4]{a} \sqrt{c}} \right )}\middle | \frac{1}{2}\right )}{3 \sqrt [4]{a} c^{\frac{5}{2}} \sqrt{a + b x^{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-2*sqrt(a + b*x**2)/(3*c*(c*x)**(3/2)) + 2*b**(3/4)*sqrt((a + b*x**2)/(sqrt(a) +
sqrt(b)*x)**2)*(sqrt(a) + sqrt(b)*x)*elliptic_f(2*atan(b**(1/4)*sqrt(c*x)/(a**(
1/4)*sqrt(c))), 1/2)/(3*a**(1/4)*c**(5/2)*sqrt(a + b*x**2))

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Mathematica [C]  time = 0.280169, size = 106, normalized size = 0.84 \[ \frac{2 x \left (\frac{2 i b x^{5/2} \sqrt{\frac{a}{b x^2}+1} F\left (\left .i \sinh ^{-1}\left (\frac{\sqrt{\frac{i \sqrt{a}}{\sqrt{b}}}}{\sqrt{x}}\right )\right |-1\right )}{\sqrt{\frac{i \sqrt{a}}{\sqrt{b}}}}-a-b x^2\right )}{3 (c x)^{5/2} \sqrt{a+b x^2}} \]

Antiderivative was successfully verified.

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

[Out]

(2*x*(-a - b*x^2 + ((2*I)*b*Sqrt[1 + a/(b*x^2)]*x^(5/2)*EllipticF[I*ArcSinh[Sqrt
[(I*Sqrt[a])/Sqrt[b]]/Sqrt[x]], -1])/Sqrt[(I*Sqrt[a])/Sqrt[b]]))/(3*(c*x)^(5/2)*
Sqrt[a + b*x^2])

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Maple [A]  time = 0.034, size = 120, normalized size = 1. \[{\frac{2}{3\,x{c}^{2}} \left ( \sqrt{{1 \left ( bx+\sqrt{-ab} \right ){\frac{1}{\sqrt{-ab}}}}}\sqrt{{1 \left ( -bx+\sqrt{-ab} \right ){\frac{1}{\sqrt{-ab}}}}}\sqrt{-{bx{\frac{1}{\sqrt{-ab}}}}}{\it EllipticF} \left ( \sqrt{{1 \left ( bx+\sqrt{-ab} \right ){\frac{1}{\sqrt{-ab}}}}},{\frac{\sqrt{2}}{2}} \right ) \sqrt{-ab}\sqrt{2}x-b{x}^{2}-a \right ){\frac{1}{\sqrt{cx}}}{\frac{1}{\sqrt{b{x}^{2}+a}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

2/3/(b*x^2+a)^(1/2)/x*(((b*x+(-a*b)^(1/2))/(-a*b)^(1/2))^(1/2)*((-b*x+(-a*b)^(1/
2))/(-a*b)^(1/2))^(1/2)*(-x*b/(-a*b)^(1/2))^(1/2)*EllipticF(((b*x+(-a*b)^(1/2))/
(-a*b)^(1/2))^(1/2),1/2*2^(1/2))*(-a*b)^(1/2)*2^(1/2)*x-b*x^2-a)/c^2/(c*x)^(1/2)

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Sympy [A]  time = 24.1588, size = 49, normalized size = 0.39 \[ \frac{\sqrt{a} \Gamma \left (- \frac{3}{4}\right ){{}_{2}F_{1}\left (\begin{matrix} - \frac{3}{4}, - \frac{1}{2} \\ \frac{1}{4} \end{matrix}\middle |{\frac{b x^{2} e^{i \pi }}{a}} \right )}}{2 c^{\frac{5}{2}} x^{\frac{3}{2}} \Gamma \left (\frac{1}{4}\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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