3.618 \(\int \frac{1}{(c x)^{3/2} \sqrt{a+b x^2}} \, dx\)

Optimal. Leaf size=268 \[ \frac{\sqrt [4]{b} \left (\sqrt{a}+\sqrt{b} x\right ) \sqrt{\frac{a+b x^2}{\left (\sqrt{a}+\sqrt{b} x\right )^2}} F\left (2 \tan ^{-1}\left (\frac{\sqrt [4]{b} \sqrt{c x}}{\sqrt [4]{a} \sqrt{c}}\right )|\frac{1}{2}\right )}{a^{3/4} c^{3/2} \sqrt{a+b x^2}}-\frac{2 \sqrt [4]{b} \left (\sqrt{a}+\sqrt{b} x\right ) \sqrt{\frac{a+b x^2}{\left (\sqrt{a}+\sqrt{b} x\right )^2}} E\left (2 \tan ^{-1}\left (\frac{\sqrt [4]{b} \sqrt{c x}}{\sqrt [4]{a} \sqrt{c}}\right )|\frac{1}{2}\right )}{a^{3/4} c^{3/2} \sqrt{a+b x^2}}+\frac{2 \sqrt{b} \sqrt{c x} \sqrt{a+b x^2}}{a c^2 \left (\sqrt{a}+\sqrt{b} x\right )}-\frac{2 \sqrt{a+b x^2}}{a c \sqrt{c x}} \]

[Out]

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

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Rubi [A]  time = 0.513263, antiderivative size = 268, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 5, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.263 \[ \frac{\sqrt [4]{b} \left (\sqrt{a}+\sqrt{b} x\right ) \sqrt{\frac{a+b x^2}{\left (\sqrt{a}+\sqrt{b} x\right )^2}} F\left (2 \tan ^{-1}\left (\frac{\sqrt [4]{b} \sqrt{c x}}{\sqrt [4]{a} \sqrt{c}}\right )|\frac{1}{2}\right )}{a^{3/4} c^{3/2} \sqrt{a+b x^2}}-\frac{2 \sqrt [4]{b} \left (\sqrt{a}+\sqrt{b} x\right ) \sqrt{\frac{a+b x^2}{\left (\sqrt{a}+\sqrt{b} x\right )^2}} E\left (2 \tan ^{-1}\left (\frac{\sqrt [4]{b} \sqrt{c x}}{\sqrt [4]{a} \sqrt{c}}\right )|\frac{1}{2}\right )}{a^{3/4} c^{3/2} \sqrt{a+b x^2}}+\frac{2 \sqrt{b} \sqrt{c x} \sqrt{a+b x^2}}{a c^2 \left (\sqrt{a}+\sqrt{b} x\right )}-\frac{2 \sqrt{a+b x^2}}{a c \sqrt{c x}} \]

Antiderivative was successfully verified.

[In]  Int[1/((c*x)^(3/2)*Sqrt[a + b*x^2]),x]

[Out]

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

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Rubi in Sympy [A]  time = 49.8933, size = 243, normalized size = 0.91 \[ \frac{2 \sqrt{b} \sqrt{c x} \sqrt{a + b x^{2}}}{a c^{2} \left (\sqrt{a} + \sqrt{b} x\right )} - \frac{2 \sqrt{a + b x^{2}}}{a c \sqrt{c x}} - \frac{2 \sqrt [4]{b} \sqrt{\frac{a + b x^{2}}{\left (\sqrt{a} + \sqrt{b} x\right )^{2}}} \left (\sqrt{a} + \sqrt{b} x\right ) E\left (2 \operatorname{atan}{\left (\frac{\sqrt [4]{b} \sqrt{c x}}{\sqrt [4]{a} \sqrt{c}} \right )}\middle | \frac{1}{2}\right )}{a^{\frac{3}{4}} c^{\frac{3}{2}} \sqrt{a + b x^{2}}} + \frac{\sqrt [4]{b} \sqrt{\frac{a + b x^{2}}{\left (\sqrt{a} + \sqrt{b} x\right )^{2}}} \left (\sqrt{a} + \sqrt{b} x\right ) F\left (2 \operatorname{atan}{\left (\frac{\sqrt [4]{b} \sqrt{c x}}{\sqrt [4]{a} \sqrt{c}} \right )}\middle | \frac{1}{2}\right )}{a^{\frac{3}{4}} c^{\frac{3}{2}} \sqrt{a + b x^{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(1/(c*x)**(3/2)/(b*x**2+a)**(1/2),x)

[Out]

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

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Mathematica [C]  time = 0.19961, size = 176, normalized size = 0.66 \[ -\frac{2 x \left (\sqrt{\frac{i \sqrt{b} x}{\sqrt{a}}} \left (a+b x^2\right )+\sqrt{a} \sqrt{b} x \sqrt{\frac{b x^2}{a}+1} F\left (\left .i \sinh ^{-1}\left (\sqrt{\frac{i \sqrt{b} x}{\sqrt{a}}}\right )\right |-1\right )-\sqrt{a} \sqrt{b} x \sqrt{\frac{b x^2}{a}+1} E\left (\left .i \sinh ^{-1}\left (\sqrt{\frac{i \sqrt{b} x}{\sqrt{a}}}\right )\right |-1\right )\right )}{a (c x)^{3/2} \sqrt{\frac{i \sqrt{b} x}{\sqrt{a}}} \sqrt{a+b x^2}} \]

Antiderivative was successfully verified.

[In]  Integrate[1/((c*x)^(3/2)*Sqrt[a + b*x^2]),x]

[Out]

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

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Maple [A]  time = 0.023, size = 196, normalized size = 0.7 \[{\frac{1}{ac} \left ( 2\,\sqrt{{\frac{bx+\sqrt{-ab}}{\sqrt{-ab}}}}\sqrt{{\frac{-bx+\sqrt{-ab}}{\sqrt{-ab}}}}\sqrt{-{\frac{bx}{\sqrt{-ab}}}}{\it EllipticE} \left ( \sqrt{{\frac{bx+\sqrt{-ab}}{\sqrt{-ab}}}},1/2\,\sqrt{2} \right ) \sqrt{2}a-\sqrt{{1 \left ( bx+\sqrt{-ab} \right ){\frac{1}{\sqrt{-ab}}}}}\sqrt{{1 \left ( -bx+\sqrt{-ab} \right ){\frac{1}{\sqrt{-ab}}}}}\sqrt{-{bx{\frac{1}{\sqrt{-ab}}}}}{\it EllipticF} \left ( \sqrt{{1 \left ( bx+\sqrt{-ab} \right ){\frac{1}{\sqrt{-ab}}}}},{\frac{\sqrt{2}}{2}} \right ) \sqrt{2}a-2\,b{x}^{2}-2\,a \right ){\frac{1}{\sqrt{cx}}}{\frac{1}{\sqrt{b{x}^{2}+a}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(1/(c*x)^(3/2)/(b*x^2+a)^(1/2),x)

[Out]

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

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Maxima [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{1}{\sqrt{b x^{2} + a} \left (c x\right )^{\frac{3}{2}}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/(sqrt(b*x^2 + a)*(c*x)^(3/2)),x, algorithm="maxima")

[Out]

integrate(1/(sqrt(b*x^2 + a)*(c*x)^(3/2)), x)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/(sqrt(b*x^2 + a)*(c*x)^(3/2)),x, algorithm="fricas")

[Out]

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

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Sympy [A]  time = 7.78478, size = 48, normalized size = 0.18 \[ \frac{\Gamma \left (- \frac{1}{4}\right ){{}_{2}F_{1}\left (\begin{matrix} - \frac{1}{4}, \frac{1}{2} \\ \frac{3}{4} \end{matrix}\middle |{\frac{b x^{2} e^{i \pi }}{a}} \right )}}{2 \sqrt{a} c^{\frac{3}{2}} \sqrt{x} \Gamma \left (\frac{3}{4}\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/(c*x)**(3/2)/(b*x**2+a)**(1/2),x)

[Out]

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

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GIAC/XCAS [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{1}{\sqrt{b x^{2} + a} \left (c x\right )^{\frac{3}{2}}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/(sqrt(b*x^2 + a)*(c*x)^(3/2)),x, algorithm="giac")

[Out]

integrate(1/(sqrt(b*x^2 + a)*(c*x)^(3/2)), x)