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

Optimal. Leaf size=79 \[ \frac{b x^5 \sqrt{a^2+2 a b x^2+b^2 x^4}}{5 \left (a+b x^2\right )}+\frac{a x^3 \sqrt{a^2+2 a b x^2+b^2 x^4}}{3 \left (a+b x^2\right )} \]

[Out]

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

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Rubi [A]  time = 0.0741372, antiderivative size = 79, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 26, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.077 \[ \frac{b x^5 \sqrt{a^2+2 a b x^2+b^2 x^4}}{5 \left (a+b x^2\right )}+\frac{a x^3 \sqrt{a^2+2 a b x^2+b^2 x^4}}{3 \left (a+b x^2\right )} \]

Antiderivative was successfully verified.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [A]  time = 0.0122918, size = 39, normalized size = 0.49 \[ \frac{\sqrt{\left (a+b x^2\right )^2} \left (5 a x^3+3 b x^5\right )}{15 \left (a+b x^2\right )} \]

Antiderivative was successfully verified.

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

[Out]

(Sqrt[(a + b*x^2)^2]*(5*a*x^3 + 3*b*x^5))/(15*(a + b*x^2))

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Maple [A]  time = 0.004, size = 36, normalized size = 0.5 \[{\frac{{x}^{3} \left ( 3\,b{x}^{2}+5\,a \right ) }{15\,b{x}^{2}+15\,a}\sqrt{ \left ( b{x}^{2}+a \right ) ^{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x^2*((b*x^2+a)^2)^(1/2),x)

[Out]

1/15*x^3*(3*b*x^2+5*a)*((b*x^2+a)^2)^(1/2)/(b*x^2+a)

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Maxima [A]  time = 0.692213, size = 18, normalized size = 0.23 \[ \frac{1}{5} \, b x^{5} + \frac{1}{3} \, a x^{3} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/5*b*x^5 + 1/3*a*x^3

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Fricas [A]  time = 0.25614, size = 18, normalized size = 0.23 \[ \frac{1}{5} \, b x^{5} + \frac{1}{3} \, a x^{3} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/5*b*x^5 + 1/3*a*x^3

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Sympy [A]  time = 0.165286, size = 12, normalized size = 0.15 \[ \frac{a x^{3}}{3} + \frac{b x^{5}}{5} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

a*x**3/3 + b*x**5/5

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GIAC/XCAS [A]  time = 0.268559, size = 39, normalized size = 0.49 \[ \frac{1}{5} \, b x^{5}{\rm sign}\left (b x^{2} + a\right ) + \frac{1}{3} \, a x^{3}{\rm sign}\left (b x^{2} + a\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/5*b*x^5*sign(b*x^2 + a) + 1/3*a*x^3*sign(b*x^2 + a)