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

Optimal. Leaf size=127 \[ \frac{2 c \sqrt{c x} \sqrt{a+b x^2}}{3 b}-\frac{a^{3/4} c^{3/2} \left (\sqrt{a}+\sqrt{b} x\right ) \sqrt{\frac{a+b x^2}{\left (\sqrt{a}+\sqrt{b} x\right )^2}} F\left (2 \tan ^{-1}\left (\frac{\sqrt [4]{b} \sqrt{c x}}{\sqrt [4]{a} \sqrt{c}}\right )|\frac{1}{2}\right )}{3 b^{5/4} \sqrt{a+b x^2}} \]

[Out]

(2*c*Sqrt[c*x]*Sqrt[a + b*x^2])/(3*b) - (a^(3/4)*c^(3/2)*(Sqrt[a] + Sqrt[b]*x)*S
qrt[(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*b^(5/4)*Sqrt[a + b*x^2])

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Rubi [A]  time = 0.201396, antiderivative size = 127, 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 \[ \frac{2 c \sqrt{c x} \sqrt{a+b x^2}}{3 b}-\frac{a^{3/4} c^{3/2} \left (\sqrt{a}+\sqrt{b} x\right ) \sqrt{\frac{a+b x^2}{\left (\sqrt{a}+\sqrt{b} x\right )^2}} F\left (2 \tan ^{-1}\left (\frac{\sqrt [4]{b} \sqrt{c x}}{\sqrt [4]{a} \sqrt{c}}\right )|\frac{1}{2}\right )}{3 b^{5/4} \sqrt{a+b x^2}} \]

Antiderivative was successfully verified.

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

[Out]

(2*c*Sqrt[c*x]*Sqrt[a + b*x^2])/(3*b) - (a^(3/4)*c^(3/2)*(Sqrt[a] + Sqrt[b]*x)*S
qrt[(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*b^(5/4)*Sqrt[a + b*x^2])

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Rubi in Sympy [A]  time = 17.58, size = 114, normalized size = 0.9 \[ - \frac{a^{\frac{3}{4}} c^{\frac{3}{2}} \sqrt{\frac{a + b x^{2}}{\left (\sqrt{a} + \sqrt{b} x\right )^{2}}} \left (\sqrt{a} + \sqrt{b} x\right ) F\left (2 \operatorname{atan}{\left (\frac{\sqrt [4]{b} \sqrt{c x}}{\sqrt [4]{a} \sqrt{c}} \right )}\middle | \frac{1}{2}\right )}{3 b^{\frac{5}{4}} \sqrt{a + b x^{2}}} + \frac{2 c \sqrt{c x} \sqrt{a + b x^{2}}}{3 b} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-a**(3/4)*c**(3/2)*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*b**(5/4)*
sqrt(a + b*x**2)) + 2*c*sqrt(c*x)*sqrt(a + b*x**2)/(3*b)

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Mathematica [C]  time = 0.210603, size = 106, normalized size = 0.83 \[ \frac{2 c \sqrt{c x} \left (-\frac{i a \sqrt{x} \sqrt{\frac{a}{b x^2}+1} F\left (\left .i \sinh ^{-1}\left (\frac{\sqrt{\frac{i \sqrt{a}}{\sqrt{b}}}}{\sqrt{x}}\right )\right |-1\right )}{\sqrt{\frac{i \sqrt{a}}{\sqrt{b}}}}+a+b x^2\right )}{3 b \sqrt{a+b x^2}} \]

Antiderivative was successfully verified.

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

[Out]

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

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Maple [A]  time = 0.015, size = 125, normalized size = 1. \[ -{\frac{c}{3\,{b}^{2}x}\sqrt{cx} \left ( \sqrt{-ab}\sqrt{{1 \left ( bx+\sqrt{-ab} \right ){\frac{1}{\sqrt{-ab}}}}}\sqrt{{1 \left ( -bx+\sqrt{-ab} \right ){\frac{1}{\sqrt{-ab}}}}}\sqrt{-{bx{\frac{1}{\sqrt{-ab}}}}}{\it EllipticF} \left ( \sqrt{{1 \left ( bx+\sqrt{-ab} \right ){\frac{1}{\sqrt{-ab}}}}},{\frac{\sqrt{2}}{2}} \right ) \sqrt{2}a-2\,{b}^{2}{x}^{3}-2\,abx \right ){\frac{1}{\sqrt{b{x}^{2}+a}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Sympy [A]  time = 10.994, size = 44, normalized size = 0.35 \[ \frac{c^{\frac{3}{2}} x^{\frac{5}{2}} \Gamma \left (\frac{5}{4}\right ){{}_{2}F_{1}\left (\begin{matrix} \frac{1}{2}, \frac{5}{4} \\ \frac{9}{4} \end{matrix}\middle |{\frac{b x^{2} e^{i \pi }}{a}} \right )}}{2 \sqrt{a} \Gamma \left (\frac{9}{4}\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

c**(3/2)*x**(5/2)*gamma(5/4)*hyper((1/2, 5/4), (9/4,), b*x**2*exp_polar(I*pi)/a)
/(2*sqrt(a)*gamma(9/4))

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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