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

Optimal. Leaf size=74 \[ \frac{b x \sqrt{a^2+2 a b x^3+b^2 x^6}}{a+b x^3}-\frac{a \sqrt{a^2+2 a b x^3+b^2 x^6}}{2 x^2 \left (a+b x^3\right )} \]

[Out]

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

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Rubi [A]  time = 0.0618748, antiderivative size = 74, 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 \[ \frac{b x \sqrt{a^2+2 a b x^3+b^2 x^6}}{a+b x^3}-\frac{a \sqrt{a^2+2 a b x^3+b^2 x^6}}{2 x^2 \left (a+b x^3\right )} \]

Antiderivative was successfully verified.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [A]  time = 0.0127244, size = 37, normalized size = 0.5 \[ -\frac{\left (a-2 b x^3\right ) \sqrt{\left (a+b x^3\right )^2}}{2 x^2 \left (a+b x^3\right )} \]

Antiderivative was successfully verified.

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

[Out]

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

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Maple [A]  time = 0.004, size = 34, normalized size = 0.5 \[ -{\frac{-2\,b{x}^{3}+a}{2\,{x}^{2} \left ( b{x}^{3}+a \right ) }\sqrt{ \left ( b{x}^{3}+a \right ) ^{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(((b*x^3+a)^2)^(1/2)/x^3,x)

[Out]

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

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Maxima [A]  time = 0.756498, size = 20, normalized size = 0.27 \[ \frac{2 \, b x^{3} - a}{2 \, x^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/2*(2*b*x^3 - a)/x^2

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Fricas [A]  time = 0.247426, size = 20, normalized size = 0.27 \[ \frac{2 \, b x^{3} - a}{2 \, x^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/2*(2*b*x^3 - a)/x^2

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Sympy [A]  time = 1.21255, size = 8, normalized size = 0.11 \[ - \frac{a}{2 x^{2}} + b x \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-a/(2*x**2) + b*x

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GIAC/XCAS [A]  time = 0.257151, size = 35, normalized size = 0.47 \[ b x{\rm sign}\left (b x^{3} + a\right ) - \frac{a{\rm sign}\left (b x^{3} + a\right )}{2 \, x^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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