3.624 \(\int \frac{\sqrt{c x}}{\left (a+b x^2\right )^{3/2}} \, dx\)

Optimal. Leaf size=266 \[ -\frac{\sqrt{c} \left (\sqrt{a}+\sqrt{b} x\right ) \sqrt{\frac{a+b x^2}{\left (\sqrt{a}+\sqrt{b} x\right )^2}} F\left (2 \tan ^{-1}\left (\frac{\sqrt [4]{b} \sqrt{c x}}{\sqrt [4]{a} \sqrt{c}}\right )|\frac{1}{2}\right )}{2 a^{3/4} b^{3/4} \sqrt{a+b x^2}}+\frac{\sqrt{c} \left (\sqrt{a}+\sqrt{b} x\right ) \sqrt{\frac{a+b x^2}{\left (\sqrt{a}+\sqrt{b} x\right )^2}} E\left (2 \tan ^{-1}\left (\frac{\sqrt [4]{b} \sqrt{c x}}{\sqrt [4]{a} \sqrt{c}}\right )|\frac{1}{2}\right )}{a^{3/4} b^{3/4} \sqrt{a+b x^2}}+\frac{(c x)^{3/2}}{a c \sqrt{a+b x^2}}-\frac{\sqrt{c x} \sqrt{a+b x^2}}{a \sqrt{b} \left (\sqrt{a}+\sqrt{b} x\right )} \]

[Out]

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

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Rubi [A]  time = 0.500484, antiderivative size = 266, 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 \[ -\frac{\sqrt{c} \left (\sqrt{a}+\sqrt{b} x\right ) \sqrt{\frac{a+b x^2}{\left (\sqrt{a}+\sqrt{b} x\right )^2}} F\left (2 \tan ^{-1}\left (\frac{\sqrt [4]{b} \sqrt{c x}}{\sqrt [4]{a} \sqrt{c}}\right )|\frac{1}{2}\right )}{2 a^{3/4} b^{3/4} \sqrt{a+b x^2}}+\frac{\sqrt{c} \left (\sqrt{a}+\sqrt{b} x\right ) \sqrt{\frac{a+b x^2}{\left (\sqrt{a}+\sqrt{b} x\right )^2}} E\left (2 \tan ^{-1}\left (\frac{\sqrt [4]{b} \sqrt{c x}}{\sqrt [4]{a} \sqrt{c}}\right )|\frac{1}{2}\right )}{a^{3/4} b^{3/4} \sqrt{a+b x^2}}+\frac{(c x)^{3/2}}{a c \sqrt{a+b x^2}}-\frac{\sqrt{c x} \sqrt{a+b x^2}}{a \sqrt{b} \left (\sqrt{a}+\sqrt{b} x\right )} \]

Antiderivative was successfully verified.

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

[Out]

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

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Rubi in Sympy [A]  time = 47.9834, size = 236, normalized size = 0.89 \[ \frac{\left (c x\right )^{\frac{3}{2}}}{a c \sqrt{a + b x^{2}}} - \frac{\sqrt{c x} \sqrt{a + b x^{2}}}{a \sqrt{b} \left (\sqrt{a} + \sqrt{b} x\right )} + \frac{\sqrt{c} \sqrt{\frac{a + b x^{2}}{\left (\sqrt{a} + \sqrt{b} x\right )^{2}}} \left (\sqrt{a} + \sqrt{b} x\right ) E\left (2 \operatorname{atan}{\left (\frac{\sqrt [4]{b} \sqrt{c x}}{\sqrt [4]{a} \sqrt{c}} \right )}\middle | \frac{1}{2}\right )}{a^{\frac{3}{4}} b^{\frac{3}{4}} \sqrt{a + b x^{2}}} - \frac{\sqrt{c} \sqrt{\frac{a + b x^{2}}{\left (\sqrt{a} + \sqrt{b} x\right )^{2}}} \left (\sqrt{a} + \sqrt{b} x\right ) F\left (2 \operatorname{atan}{\left (\frac{\sqrt [4]{b} \sqrt{c x}}{\sqrt [4]{a} \sqrt{c}} \right )}\middle | \frac{1}{2}\right )}{2 a^{\frac{3}{4}} b^{\frac{3}{4}} \sqrt{a + b x^{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

(c*x)**(3/2)/(a*c*sqrt(a + b*x**2)) - sqrt(c*x)*sqrt(a + b*x**2)/(a*sqrt(b)*(sqr
t(a) + sqrt(b)*x)) + sqrt(c)*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)*b**(3/4)*sqrt(a + b*x**2)) - sqrt(c)*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)*sq
rt(c))), 1/2)/(2*a**(3/4)*b**(3/4)*sqrt(a + b*x**2))

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Mathematica [C]  time = 0.137594, size = 166, normalized size = 0.62 \[ \frac{\sqrt{c x} \left (\sqrt{a} \sqrt{\frac{b x^2}{a}+1} F\left (\left .i \sinh ^{-1}\left (\sqrt{\frac{i \sqrt{b} x}{\sqrt{a}}}\right )\right |-1\right )-\sqrt{a} \sqrt{\frac{b x^2}{a}+1} E\left (\left .i \sinh ^{-1}\left (\sqrt{\frac{i \sqrt{b} x}{\sqrt{a}}}\right )\right |-1\right )+\sqrt{b} x \sqrt{\frac{i \sqrt{b} x}{\sqrt{a}}}\right )}{a \sqrt{b} \sqrt{\frac{i \sqrt{b} x}{\sqrt{a}}} \sqrt{a+b x^2}} \]

Antiderivative was successfully verified.

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

[Out]

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

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Maple [A]  time = 0.018, size = 197, normalized size = 0.7 \[ -{\frac{1}{2\,abx}\sqrt{cx} \left ( 2\,\sqrt{{\frac{bx+\sqrt{-ab}}{\sqrt{-ab}}}}\sqrt{{\frac{-bx+\sqrt{-ab}}{\sqrt{-ab}}}}\sqrt{-{\frac{bx}{\sqrt{-ab}}}}{\it EllipticE} \left ( \sqrt{{\frac{bx+\sqrt{-ab}}{\sqrt{-ab}}}},1/2\,\sqrt{2} \right ) \sqrt{2}a-\sqrt{{1 \left ( bx+\sqrt{-ab} \right ){\frac{1}{\sqrt{-ab}}}}}\sqrt{{1 \left ( -bx+\sqrt{-ab} \right ){\frac{1}{\sqrt{-ab}}}}}\sqrt{-{bx{\frac{1}{\sqrt{-ab}}}}}{\it EllipticF} \left ( \sqrt{{1 \left ( bx+\sqrt{-ab} \right ){\frac{1}{\sqrt{-ab}}}}},{\frac{\sqrt{2}}{2}} \right ) \sqrt{2}a-2\,b{x}^{2} \right ){\frac{1}{\sqrt{b{x}^{2}+a}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Sympy [A]  time = 5.24471, size = 44, normalized size = 0.17 \[ \frac{\sqrt{c} x^{\frac{3}{2}} \Gamma \left (\frac{3}{4}\right ){{}_{2}F_{1}\left (\begin{matrix} \frac{3}{4}, \frac{3}{2} \\ \frac{7}{4} \end{matrix}\middle |{\frac{b x^{2} e^{i \pi }}{a}} \right )}}{2 a^{\frac{3}{2}} \Gamma \left (\frac{7}{4}\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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