3.217 \(\int \frac{\sqrt{a x^3}}{\sqrt{1+x^2}} \, dx\)

Optimal. Leaf size=83 \[ \frac{2 \sqrt{x^2+1} \sqrt{a x^3}}{3 x}-\frac{(x+1) \sqrt{\frac{x^2+1}{(x+1)^2}} \sqrt{a x^3} F\left (2 \tan ^{-1}\left (\sqrt{x}\right )|\frac{1}{2}\right )}{3 x^{3/2} \sqrt{x^2+1}} \]

[Out]

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

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Rubi [A]  time = 0.0605491, antiderivative size = 83, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.21 \[ \frac{2 \sqrt{x^2+1} \sqrt{a x^3}}{3 x}-\frac{(x+1) \sqrt{\frac{x^2+1}{(x+1)^2}} \sqrt{a x^3} F\left (2 \tan ^{-1}\left (\sqrt{x}\right )|\frac{1}{2}\right )}{3 x^{3/2} \sqrt{x^2+1}} \]

Antiderivative was successfully verified.

[In]  Int[Sqrt[a*x^3]/Sqrt[1 + x^2],x]

[Out]

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

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Rubi in Sympy [A]  time = 10.7222, size = 73, normalized size = 0.88 \[ \frac{2 \sqrt{a x^{3}} \sqrt{x^{2} + 1}}{3 x} - \frac{\sqrt{a x^{3}} \sqrt{\frac{x^{2} + 1}{\left (x + 1\right )^{2}}} \left (x + 1\right ) F\left (2 \operatorname{atan}{\left (\sqrt{x} \right )}\middle | \frac{1}{2}\right )}{3 x^{\frac{3}{2}} \sqrt{x^{2} + 1}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

2*sqrt(a*x**3)*sqrt(x**2 + 1)/(3*x) - sqrt(a*x**3)*sqrt((x**2 + 1)/(x + 1)**2)*(
x + 1)*elliptic_f(2*atan(sqrt(x)), 1/2)/(3*x**(3/2)*sqrt(x**2 + 1))

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Mathematica [C]  time = 0.0690683, size = 77, normalized size = 0.93 \[ \frac{2 \sqrt{x^2+1} \sqrt{a x^3} \left (\sqrt{\frac{1}{x^2}+1} x^{3/2}-\sqrt [4]{-1} F\left (\left .i \sinh ^{-1}\left (\frac{\sqrt [4]{-1}}{\sqrt{x}}\right )\right |-1\right )\right )}{3 \sqrt{\frac{1}{x^2}+1} x^{5/2}} \]

Antiderivative was successfully verified.

[In]  Integrate[Sqrt[a*x^3]/Sqrt[1 + x^2],x]

[Out]

(2*Sqrt[a*x^3]*Sqrt[1 + x^2]*(Sqrt[1 + x^(-2)]*x^(3/2) - (-1)^(1/4)*EllipticF[I*
ArcSinh[(-1)^(1/4)/Sqrt[x]], -1]))/(3*Sqrt[1 + x^(-2)]*x^(5/2))

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Maple [C]  time = 0.031, size = 76, normalized size = 0.9 \[ -{\frac{1}{3\,{x}^{2}}\sqrt{a{x}^{3}} \left ( i\sqrt{-i \left ( x+i \right ) }\sqrt{-i \left ( -x+i \right ) }\sqrt{ix}{\it EllipticF} \left ( \sqrt{-i \left ( x+i \right ) },{\frac{\sqrt{2}}{2}} \right ) \sqrt{2}-2\,{x}^{3}-2\,x \right ){\frac{1}{\sqrt{{x}^{2}+1}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Fricas [F]  time = 0., size = 0, normalized size = 0. \[{\rm integral}\left (\frac{\sqrt{a x^{3}}}{\sqrt{x^{2} + 1}}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

integral(sqrt(a*x^3)/sqrt(x^2 + 1), x)

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Sympy [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{\sqrt{a x^{3}}}{\sqrt{x^{2} + 1}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Integral(sqrt(a*x**3)/sqrt(x**2 + 1), x)

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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