3.543 \(\int x \sqrt{a^2+2 a b x^2+b^2 x^4} \, dx\)

Optimal. Leaf size=36 \[ \frac{\left (a+b x^2\right ) \sqrt{a^2+2 a b x^2+b^2 x^4}}{4 b} \]

[Out]

((a + b*x^2)*Sqrt[a^2 + 2*a*b*x^2 + b^2*x^4])/(4*b)

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Rubi [A]  time = 0.0656612, antiderivative size = 36, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.083 \[ \frac{\left (a+b x^2\right ) \sqrt{a^2+2 a b x^2+b^2 x^4}}{4 b} \]

Antiderivative was successfully verified.

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

[Out]

((a + b*x^2)*Sqrt[a^2 + 2*a*b*x^2 + b^2*x^4])/(4*b)

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Rubi in Sympy [F]  time = 0., size = 0, normalized size = 0. \[ \frac{\sqrt{\left (a + b x^{2}\right )^{2}} \int ^{a + b x^{2}} x\, dx}{2 b \left (a + b x^{2}\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

sqrt((a + b*x**2)**2)*Integral(x, (x, a + b*x**2))/(2*b*(a + b*x**2))

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Mathematica [A]  time = 0.012167, size = 38, normalized size = 1.06 \[ \frac{\sqrt{\left (a+b x^2\right )^2} \left (2 a x^2+b x^4\right )}{4 \left (a+b x^2\right )} \]

Antiderivative was successfully verified.

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

[Out]

(Sqrt[(a + b*x^2)^2]*(2*a*x^2 + b*x^4))/(4*(a + b*x^2))

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Maple [A]  time = 0.062, size = 35, normalized size = 1. \[{\frac{{x}^{2} \left ( b{x}^{2}+2\,a \right ) }{4\,b{x}^{2}+4\,a}\sqrt{ \left ( b{x}^{2}+a \right ) ^{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x*((b*x^2+a)^2)^(1/2),x)

[Out]

1/4*x^2*(b*x^2+2*a)*((b*x^2+a)^2)^(1/2)/(b*x^2+a)

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Maxima [F]  time = 0., size = 0, normalized size = 0. \[ \text{Exception raised: ValueError} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.258766, size = 18, normalized size = 0.5 \[ \frac{1}{4} \, b x^{4} + \frac{1}{2} \, a x^{2} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/4*b*x^4 + 1/2*a*x^2

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Sympy [A]  time = 0.167063, size = 12, normalized size = 0.33 \[ \frac{a x^{2}}{2} + \frac{b x^{4}}{4} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

a*x**2/2 + b*x**4/4

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GIAC/XCAS [A]  time = 0.269295, size = 30, normalized size = 0.83 \[ \frac{1}{4} \,{\left (b x^{4} + 2 \, a x^{2}\right )}{\rm sign}\left (b x^{2} + a\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/4*(b*x^4 + 2*a*x^2)*sign(b*x^2 + a)