3.216 \(\int \frac{\sqrt{a x^4}}{\sqrt{1+x^2}} \, dx\)

Optimal. Leaf size=44 \[ \frac{\sqrt{x^2+1} \sqrt{a x^4}}{2 x}-\frac{\sqrt{a x^4} \sinh ^{-1}(x)}{2 x^2} \]

[Out]

(Sqrt[a*x^4]*Sqrt[1 + x^2])/(2*x) - (Sqrt[a*x^4]*ArcSinh[x])/(2*x^2)

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Rubi [A]  time = 0.0230602, antiderivative size = 44, 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 \[ \frac{\sqrt{x^2+1} \sqrt{a x^4}}{2 x}-\frac{\sqrt{a x^4} \sinh ^{-1}(x)}{2 x^2} \]

Antiderivative was successfully verified.

[In]  Int[Sqrt[a*x^4]/Sqrt[1 + x^2],x]

[Out]

(Sqrt[a*x^4]*Sqrt[1 + x^2])/(2*x) - (Sqrt[a*x^4]*ArcSinh[x])/(2*x^2)

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Rubi in Sympy [A]  time = 8.76722, size = 36, normalized size = 0.82 \[ \frac{\sqrt{a x^{4}} \sqrt{x^{2} + 1}}{2 x} - \frac{\sqrt{a x^{4}} \operatorname{asinh}{\left (x \right )}}{2 x^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((a*x**4)**(1/2)/(x**2+1)**(1/2),x)

[Out]

sqrt(a*x**4)*sqrt(x**2 + 1)/(2*x) - sqrt(a*x**4)*asinh(x)/(2*x**2)

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Mathematica [A]  time = 0.017481, size = 32, normalized size = 0.73 \[ \frac{\sqrt{a x^4} \left (x \sqrt{x^2+1}-\sinh ^{-1}(x)\right )}{2 x^2} \]

Antiderivative was successfully verified.

[In]  Integrate[Sqrt[a*x^4]/Sqrt[1 + x^2],x]

[Out]

(Sqrt[a*x^4]*(x*Sqrt[1 + x^2] - ArcSinh[x]))/(2*x^2)

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Maple [A]  time = 0.01, size = 26, normalized size = 0.6 \[ -{\frac{1}{2\,{x}^{2}}\sqrt{a{x}^{4}} \left ( -x\sqrt{{x}^{2}+1}+{\it Arcsinh} \left ( x \right ) \right ) } \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((a*x^4)^(1/2)/(x^2+1)^(1/2),x)

[Out]

-1/2*(a*x^4)^(1/2)*(-x*(x^2+1)^(1/2)+arcsinh(x))/x^2

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Maxima [A]  time = 0.786085, size = 26, normalized size = 0.59 \[ \frac{1}{2} \,{\left (\sqrt{x^{2} + 1} x - \operatorname{arsinh}\left (x\right )\right )} \sqrt{a} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(sqrt(a*x^4)/sqrt(x^2 + 1),x, algorithm="maxima")

[Out]

1/2*(sqrt(x^2 + 1)*x - arcsinh(x))*sqrt(a)

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Fricas [A]  time = 0.299278, size = 177, normalized size = 4.02 \[ \frac{\sqrt{a x^{4}}{\left (8 \, x^{4} + 8 \, x^{2} - 4 \,{\left (2 \, x^{3} + x\right )} \sqrt{x^{2} + 1} + 1\right )} \log \left (-x + \sqrt{x^{2} + 1}\right ) -{\left (8 \, x^{6} + 12 \, x^{4} + 4 \, x^{2} -{\left (8 \, x^{5} + 8 \, x^{3} + x\right )} \sqrt{x^{2} + 1}\right )} \sqrt{a x^{4}}}{2 \,{\left (8 \, x^{6} + 8 \, x^{4} + x^{2} - 4 \,{\left (2 \, x^{5} + x^{3}\right )} \sqrt{x^{2} + 1}\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(sqrt(a*x^4)/sqrt(x^2 + 1),x, algorithm="fricas")

[Out]

1/2*(sqrt(a*x^4)*(8*x^4 + 8*x^2 - 4*(2*x^3 + x)*sqrt(x^2 + 1) + 1)*log(-x + sqrt
(x^2 + 1)) - (8*x^6 + 12*x^4 + 4*x^2 - (8*x^5 + 8*x^3 + x)*sqrt(x^2 + 1))*sqrt(a
*x^4))/(8*x^6 + 8*x^4 + x^2 - 4*(2*x^5 + x^3)*sqrt(x^2 + 1))

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Sympy [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{\sqrt{a x^{4}}}{\sqrt{x^{2} + 1}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((a*x**4)**(1/2)/(x**2+1)**(1/2),x)

[Out]

Integral(sqrt(a*x**4)/sqrt(x**2 + 1), x)

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GIAC/XCAS [A]  time = 0.260717, size = 36, normalized size = 0.82 \[ \frac{1}{2} \,{\left (\sqrt{x^{2} + 1} x +{\rm ln}\left (-x + \sqrt{x^{2} + 1}\right )\right )} \sqrt{a} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(sqrt(a*x^4)/sqrt(x^2 + 1),x, algorithm="giac")

[Out]

1/2*(sqrt(x^2 + 1)*x + ln(-x + sqrt(x^2 + 1)))*sqrt(a)