3.13 \(\int \frac{\sqrt{a^2+2 a b x^3+b^2 x^6}}{x^2} \, dx\)

Optimal. Leaf size=77 \[ \frac{b x^2 \sqrt{a^2+2 a b x^3+b^2 x^6}}{2 \left (a+b x^3\right )}-\frac{a \sqrt{a^2+2 a b x^3+b^2 x^6}}{x \left (a+b x^3\right )} \]

[Out]

-((a*Sqrt[a^2 + 2*a*b*x^3 + b^2*x^6])/(x*(a + b*x^3))) + (b*x^2*Sqrt[a^2 + 2*a*b
*x^3 + b^2*x^6])/(2*(a + b*x^3))

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Rubi [A]  time = 0.0624719, antiderivative size = 77, 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 \[ \frac{b x^2 \sqrt{a^2+2 a b x^3+b^2 x^6}}{2 \left (a+b x^3\right )}-\frac{a \sqrt{a^2+2 a b x^3+b^2 x^6}}{x \left (a+b x^3\right )} \]

Antiderivative was successfully verified.

[In]  Int[Sqrt[a^2 + 2*a*b*x^3 + b^2*x^6]/x^2,x]

[Out]

-((a*Sqrt[a^2 + 2*a*b*x^3 + b^2*x^6])/(x*(a + b*x^3))) + (b*x^2*Sqrt[a^2 + 2*a*b
*x^3 + b^2*x^6])/(2*(a + b*x^3))

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Rubi in Sympy [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{\sqrt{\left (a + b x^{3}\right )^{2}}}{x^{2}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(((b*x**3+a)**2)**(1/2)/x**2,x)

[Out]

Integral(sqrt((a + b*x**3)**2)/x**2, x)

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Mathematica [A]  time = 0.0185209, size = 38, normalized size = 0.49 \[ \frac{\left (b x^3-2 a\right ) \sqrt{\left (a+b x^3\right )^2}}{2 x \left (a+b x^3\right )} \]

Antiderivative was successfully verified.

[In]  Integrate[Sqrt[a^2 + 2*a*b*x^3 + b^2*x^6]/x^2,x]

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(((b*x^3+a)^2)^(1/2)/x^2,x)

[Out]

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

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Maxima [A]  time = 0.765795, size = 19, normalized size = 0.25 \[ \frac{b x^{3} - 2 \, a}{2 \, x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(sqrt((b*x^3 + a)^2)/x^2,x, algorithm="maxima")

[Out]

1/2*(b*x^3 - 2*a)/x

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Fricas [A]  time = 0.264117, size = 19, normalized size = 0.25 \[ \frac{b x^{3} - 2 \, a}{2 \, x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(sqrt((b*x^3 + a)^2)/x^2,x, algorithm="fricas")

[Out]

1/2*(b*x^3 - 2*a)/x

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Sympy [A]  time = 1.10838, size = 8, normalized size = 0.1 \[ - \frac{a}{x} + \frac{b x^{2}}{2} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(((b*x**3+a)**2)**(1/2)/x**2,x)

[Out]

-a/x + b*x**2/2

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GIAC/XCAS [A]  time = 0.281235, size = 39, normalized size = 0.51 \[ \frac{1}{2} \, b x^{2}{\rm sign}\left (b x^{3} + a\right ) - \frac{a{\rm sign}\left (b x^{3} + a\right )}{x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(sqrt((b*x^3 + a)^2)/x^2,x, algorithm="giac")

[Out]

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