3.633 \(\int \frac{1}{\sqrt{c x} \left (a+b x^2\right )^{5/2}} \, dx\)

Optimal. Leaf size=157 \[ \frac{5 \left (\sqrt{a}+\sqrt{b} x\right ) \sqrt{\frac{a+b x^2}{\left (\sqrt{a}+\sqrt{b} x\right )^2}} F\left (2 \tan ^{-1}\left (\frac{\sqrt [4]{b} \sqrt{c x}}{\sqrt [4]{a} \sqrt{c}}\right )|\frac{1}{2}\right )}{12 a^{9/4} \sqrt [4]{b} \sqrt{c} \sqrt{a+b x^2}}+\frac{5 \sqrt{c x}}{6 a^2 c \sqrt{a+b x^2}}+\frac{\sqrt{c x}}{3 a c \left (a+b x^2\right )^{3/2}} \]

[Out]

Sqrt[c*x]/(3*a*c*(a + b*x^2)^(3/2)) + (5*Sqrt[c*x])/(6*a^2*c*Sqrt[a + b*x^2]) +
(5*(Sqrt[a] + Sqrt[b]*x)*Sqrt[(a + b*x^2)/(Sqrt[a] + Sqrt[b]*x)^2]*EllipticF[2*A
rcTan[(b^(1/4)*Sqrt[c*x])/(a^(1/4)*Sqrt[c])], 1/2])/(12*a^(9/4)*b^(1/4)*Sqrt[c]*
Sqrt[a + b*x^2])

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Rubi [A]  time = 0.257674, antiderivative size = 157, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.158 \[ \frac{5 \left (\sqrt{a}+\sqrt{b} x\right ) \sqrt{\frac{a+b x^2}{\left (\sqrt{a}+\sqrt{b} x\right )^2}} F\left (2 \tan ^{-1}\left (\frac{\sqrt [4]{b} \sqrt{c x}}{\sqrt [4]{a} \sqrt{c}}\right )|\frac{1}{2}\right )}{12 a^{9/4} \sqrt [4]{b} \sqrt{c} \sqrt{a+b x^2}}+\frac{5 \sqrt{c x}}{6 a^2 c \sqrt{a+b x^2}}+\frac{\sqrt{c x}}{3 a c \left (a+b x^2\right )^{3/2}} \]

Antiderivative was successfully verified.

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

[Out]

Sqrt[c*x]/(3*a*c*(a + b*x^2)^(3/2)) + (5*Sqrt[c*x])/(6*a^2*c*Sqrt[a + b*x^2]) +
(5*(Sqrt[a] + Sqrt[b]*x)*Sqrt[(a + b*x^2)/(Sqrt[a] + Sqrt[b]*x)^2]*EllipticF[2*A
rcTan[(b^(1/4)*Sqrt[c*x])/(a^(1/4)*Sqrt[c])], 1/2])/(12*a^(9/4)*b^(1/4)*Sqrt[c]*
Sqrt[a + b*x^2])

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Rubi in Sympy [A]  time = 23.5794, size = 139, normalized size = 0.89 \[ \frac{\sqrt{c x}}{3 a c \left (a + b x^{2}\right )^{\frac{3}{2}}} + \frac{5 \sqrt{c x}}{6 a^{2} c \sqrt{a + b x^{2}}} + \frac{5 \sqrt{\frac{a + b x^{2}}{\left (\sqrt{a} + \sqrt{b} x\right )^{2}}} \left (\sqrt{a} + \sqrt{b} x\right ) F\left (2 \operatorname{atan}{\left (\frac{\sqrt [4]{b} \sqrt{c x}}{\sqrt [4]{a} \sqrt{c}} \right )}\middle | \frac{1}{2}\right )}{12 a^{\frac{9}{4}} \sqrt [4]{b} \sqrt{c} \sqrt{a + b x^{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

sqrt(c*x)/(3*a*c*(a + b*x**2)**(3/2)) + 5*sqrt(c*x)/(6*a**2*c*sqrt(a + b*x**2))
+ 5*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)/(12*a**(9/4)*b**(1/4)*sqrt(
c)*sqrt(a + b*x**2))

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Mathematica [C]  time = 0.333311, size = 115, normalized size = 0.73 \[ \frac{x \left (\frac{5 i \sqrt{x} \sqrt{\frac{a}{b x^2}+1} \left (a+b x^2\right ) F\left (\left .i \sinh ^{-1}\left (\frac{\sqrt{\frac{i \sqrt{a}}{\sqrt{b}}}}{\sqrt{x}}\right )\right |-1\right )}{\sqrt{\frac{i \sqrt{a}}{\sqrt{b}}}}+7 a+5 b x^2\right )}{6 a^2 \sqrt{c x} \left (a+b x^2\right )^{3/2}} \]

Antiderivative was successfully verified.

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

[Out]

(x*(7*a + 5*b*x^2 + ((5*I)*Sqrt[1 + a/(b*x^2)]*Sqrt[x]*(a + b*x^2)*EllipticF[I*A
rcSinh[Sqrt[(I*Sqrt[a])/Sqrt[b]]/Sqrt[x]], -1])/Sqrt[(I*Sqrt[a])/Sqrt[b]]))/(6*a
^2*Sqrt[c*x]*(a + b*x^2)^(3/2))

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Maple [A]  time = 0.028, size = 216, normalized size = 1.4 \[{\frac{1}{12\,{a}^{2}b} \left ( 5\,\sqrt{{\frac{bx+\sqrt{-ab}}{\sqrt{-ab}}}}\sqrt{{\frac{-bx+\sqrt{-ab}}{\sqrt{-ab}}}}\sqrt{-{\frac{bx}{\sqrt{-ab}}}}{\it EllipticF} \left ( \sqrt{{\frac{bx+\sqrt{-ab}}{\sqrt{-ab}}}},1/2\,\sqrt{2} \right ) \sqrt{-ab}\sqrt{2}{x}^{2}b+5\,\sqrt{-ab}\sqrt{{\frac{bx+\sqrt{-ab}}{\sqrt{-ab}}}}\sqrt{{\frac{-bx+\sqrt{-ab}}{\sqrt{-ab}}}}\sqrt{-{\frac{bx}{\sqrt{-ab}}}}{\it EllipticF} \left ( \sqrt{{\frac{bx+\sqrt{-ab}}{\sqrt{-ab}}}},1/2\,\sqrt{2} \right ) \sqrt{2}a+10\,{b}^{2}{x}^{3}+14\,abx \right ){\frac{1}{\sqrt{cx}}} \left ( b{x}^{2}+a \right ) ^{-{\frac{3}{2}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/12*(5*((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^2*b+5*(-a*b)^(1/2)*((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))*2^(1/2)*a+
10*b^2*x^3+14*a*b*x)/(c*x)^(1/2)/a^2/b/(b*x^2+a)^(3/2)

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Sympy [A]  time = 54.9977, size = 44, normalized size = 0.28 \[ \frac{\sqrt{x} \Gamma \left (\frac{1}{4}\right ){{}_{2}F_{1}\left (\begin{matrix} \frac{1}{4}, \frac{5}{2} \\ \frac{5}{4} \end{matrix}\middle |{\frac{b x^{2} e^{i \pi }}{a}} \right )}}{2 a^{\frac{5}{2}} \sqrt{c} \Gamma \left (\frac{5}{4}\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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