3.10 \(\int x \left (2+3 x^2\right ) \sqrt{5+x^4} \, dx\)

Optimal. Leaf size=44 \[ \frac{1}{2} \left (x^4+5\right )^{3/2}+\frac{5}{2} \sinh ^{-1}\left (\frac{x^2}{\sqrt{5}}\right )+\frac{1}{2} \sqrt{x^4+5} x^2 \]

[Out]

(x^2*Sqrt[5 + x^4])/2 + (5 + x^4)^(3/2)/2 + (5*ArcSinh[x^2/Sqrt[5]])/2

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Rubi [A]  time = 0.0585607, antiderivative size = 44, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 18, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.222 \[ \frac{1}{2} \left (x^4+5\right )^{3/2}+\frac{5}{2} \sinh ^{-1}\left (\frac{x^2}{\sqrt{5}}\right )+\frac{1}{2} \sqrt{x^4+5} x^2 \]

Antiderivative was successfully verified.

[In]  Int[x*(2 + 3*x^2)*Sqrt[5 + x^4],x]

[Out]

(x^2*Sqrt[5 + x^4])/2 + (5 + x^4)^(3/2)/2 + (5*ArcSinh[x^2/Sqrt[5]])/2

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Rubi in Sympy [A]  time = 7.07213, size = 37, normalized size = 0.84 \[ \frac{x^{2} \sqrt{x^{4} + 5}}{2} + \frac{\left (x^{4} + 5\right )^{\frac{3}{2}}}{2} + \frac{5 \operatorname{asinh}{\left (\frac{\sqrt{5} x^{2}}{5} \right )}}{2} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(x*(3*x**2+2)*(x**4+5)**(1/2),x)

[Out]

x**2*sqrt(x**4 + 5)/2 + (x**4 + 5)**(3/2)/2 + 5*asinh(sqrt(5)*x**2/5)/2

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Mathematica [A]  time = 0.0233472, size = 36, normalized size = 0.82 \[ \frac{5}{2} \sinh ^{-1}\left (\frac{x^2}{\sqrt{5}}\right )+\frac{1}{2} \sqrt{x^4+5} \left (x^4+x^2+5\right ) \]

Antiderivative was successfully verified.

[In]  Integrate[x*(2 + 3*x^2)*Sqrt[5 + x^4],x]

[Out]

(Sqrt[5 + x^4]*(5 + x^2 + x^4))/2 + (5*ArcSinh[x^2/Sqrt[5]])/2

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Maple [A]  time = 0.013, size = 34, normalized size = 0.8 \[{\frac{1}{2} \left ({x}^{4}+5 \right ) ^{{\frac{3}{2}}}}+{\frac{5}{2}{\it Arcsinh} \left ({\frac{\sqrt{5}{x}^{2}}{5}} \right ) }+{\frac{{x}^{2}}{2}\sqrt{{x}^{4}+5}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x*(3*x^2+2)*(x^4+5)^(1/2),x)

[Out]

1/2*(x^4+5)^(3/2)+5/2*arcsinh(1/5*5^(1/2)*x^2)+1/2*x^2*(x^4+5)^(1/2)

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Maxima [A]  time = 0.785638, size = 90, normalized size = 2.05 \[ \frac{1}{2} \,{\left (x^{4} + 5\right )}^{\frac{3}{2}} + \frac{5 \, \sqrt{x^{4} + 5}}{2 \, x^{2}{\left (\frac{x^{4} + 5}{x^{4}} - 1\right )}} + \frac{5}{4} \, \log \left (\frac{\sqrt{x^{4} + 5}}{x^{2}} + 1\right ) - \frac{5}{4} \, \log \left (\frac{\sqrt{x^{4} + 5}}{x^{2}} - 1\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(sqrt(x^4 + 5)*(3*x^2 + 2)*x,x, algorithm="maxima")

[Out]

1/2*(x^4 + 5)^(3/2) + 5/2*sqrt(x^4 + 5)/(x^2*((x^4 + 5)/x^4 - 1)) + 5/4*log(sqrt
(x^4 + 5)/x^2 + 1) - 5/4*log(sqrt(x^4 + 5)/x^2 - 1)

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Fricas [A]  time = 0.277566, size = 190, normalized size = 4.32 \[ -\frac{4 \, x^{12} + 4 \, x^{10} + 45 \, x^{8} + 25 \, x^{6} + 150 \, x^{4} + 25 \, x^{2} + 5 \,{\left (4 \, x^{6} + 15 \, x^{2} -{\left (4 \, x^{4} + 5\right )} \sqrt{x^{4} + 5}\right )} \log \left (-x^{2} + \sqrt{x^{4} + 5}\right ) -{\left (4 \, x^{10} + 4 \, x^{8} + 35 \, x^{6} + 15 \, x^{4} + 75 \, x^{2}\right )} \sqrt{x^{4} + 5} + 125}{2 \,{\left (4 \, x^{6} + 15 \, x^{2} -{\left (4 \, x^{4} + 5\right )} \sqrt{x^{4} + 5}\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(sqrt(x^4 + 5)*(3*x^2 + 2)*x,x, algorithm="fricas")

[Out]

-1/2*(4*x^12 + 4*x^10 + 45*x^8 + 25*x^6 + 150*x^4 + 25*x^2 + 5*(4*x^6 + 15*x^2 -
(4*x^4 + 5)*sqrt(x^4 + 5))*log(-x^2 + sqrt(x^4 + 5)) - (4*x^10 + 4*x^8 + 35*x^6
+ 15*x^4 + 75*x^2)*sqrt(x^4 + 5) + 125)/(4*x^6 + 15*x^2 - (4*x^4 + 5)*sqrt(x^4
+ 5))

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Sympy [A]  time = 6.5713, size = 53, normalized size = 1.2 \[ \frac{x^{6}}{2 \sqrt{x^{4} + 5}} + \frac{5 x^{2}}{2 \sqrt{x^{4} + 5}} + \frac{\left (x^{4} + 5\right )^{\frac{3}{2}}}{2} + \frac{5 \operatorname{asinh}{\left (\frac{\sqrt{5} x^{2}}{5} \right )}}{2} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x*(3*x**2+2)*(x**4+5)**(1/2),x)

[Out]

x**6/(2*sqrt(x**4 + 5)) + 5*x**2/(2*sqrt(x**4 + 5)) + (x**4 + 5)**(3/2)/2 + 5*as
inh(sqrt(5)*x**2/5)/2

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GIAC/XCAS [A]  time = 0.264549, size = 50, normalized size = 1.14 \[ \frac{1}{2} \, \sqrt{x^{4} + 5}{\left ({\left (x^{2} + 1\right )} x^{2} + 5\right )} - \frac{5}{2} \,{\rm ln}\left (-x^{2} + \sqrt{x^{4} + 5}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(sqrt(x^4 + 5)*(3*x^2 + 2)*x,x, algorithm="giac")

[Out]

1/2*sqrt(x^4 + 5)*((x^2 + 1)*x^2 + 5) - 5/2*ln(-x^2 + sqrt(x^4 + 5))