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

Optimal. Leaf size=74 \[ \frac{a x \sqrt{a^2+2 a b x^2+b^2 x^4}}{a+b x^2}+\frac{b x^3 \sqrt{a^2+2 a b x^2+b^2 x^4}}{3 \left (a+b x^2\right )} \]

[Out]

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

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Rubi [A]  time = 0.0403233, antiderivative size = 74, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 1, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.045 \[ \frac{a x \sqrt{a^2+2 a b x^2+b^2 x^4}}{a+b x^2}+\frac{b x^3 \sqrt{a^2+2 a b x^2+b^2 x^4}}{3 \left (a+b x^2\right )} \]

Antiderivative was successfully verified.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Integral(sqrt((a + b*x**2)**2), x)

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Mathematica [A]  time = 0.0161419, size = 36, normalized size = 0.49 \[ \frac{\sqrt{\left (a+b x^2\right )^2} \left (3 a x+b x^3\right )}{3 \left (a+b x^2\right )} \]

Antiderivative was successfully verified.

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

[Out]

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

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Maple [A]  time = 0.003, size = 33, normalized size = 0.5 \[{\frac{x \left ( b{x}^{2}+3\,a \right ) }{3\,b{x}^{2}+3\,a}\sqrt{ \left ( b{x}^{2}+a \right ) ^{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(((b*x^2+a)^2)^(1/2),x)

[Out]

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

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Maxima [A]  time = 0.703227, size = 14, normalized size = 0.19 \[ \frac{1}{3} \, b x^{3} + a x \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/3*b*x^3 + a*x

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Fricas [A]  time = 0.254874, size = 14, normalized size = 0.19 \[ \frac{1}{3} \, b x^{3} + a x \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/3*b*x^3 + a*x

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Sympy [A]  time = 0.153532, size = 8, normalized size = 0.11 \[ a x + \frac{b x^{3}}{3} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

a*x + b*x**3/3

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GIAC/XCAS [A]  time = 0.268615, size = 27, normalized size = 0.36 \[ \frac{1}{3} \,{\left (b x^{3} + 3 \, a x\right )}{\rm sign}\left (b x^{2} + a\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/3*(b*x^3 + 3*a*x)*sign(b*x^2 + a)