3.545 \(\int \frac{\sqrt{a^2+2 a b x^2+b^2 x^4}}{x^3} \, dx\)

Optimal. Leaf size=75 \[ \frac{b \log (x) \sqrt{a^2+2 a b x^2+b^2 x^4}}{a+b x^2}-\frac{a \sqrt{a^2+2 a b x^2+b^2 x^4}}{2 x^2 \left (a+b x^2\right )} \]

[Out]

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

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Rubi [A]  time = 0.0682473, antiderivative size = 75, 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 \[ \frac{b \log (x) \sqrt{a^2+2 a b x^2+b^2 x^4}}{a+b x^2}-\frac{a \sqrt{a^2+2 a b x^2+b^2 x^4}}{2 x^2 \left (a+b x^2\right )} \]

Antiderivative was successfully verified.

[In]  Int[Sqrt[a^2 + 2*a*b*x^2 + b^2*x^4]/x^3,x]

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [A]  time = 0.0161505, size = 39, normalized size = 0.52 \[ -\frac{\sqrt{\left (a+b x^2\right )^2} \left (a-2 b x^2 \log (x)\right )}{2 x^2 \left (a+b x^2\right )} \]

Antiderivative was successfully verified.

[In]  Integrate[Sqrt[a^2 + 2*a*b*x^2 + b^2*x^4]/x^3,x]

[Out]

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

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Maple [A]  time = 0.012, size = 38, normalized size = 0.5 \[{\frac{2\,b\ln \left ( x \right ){x}^{2}-a}{ \left ( 2\,b{x}^{2}+2\,a \right ){x}^{2}}\sqrt{ \left ( b{x}^{2}+a \right ) ^{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(((b*x^2+a)^2)^(1/2)/x^3,x)

[Out]

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

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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^3,x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.259938, size = 23, normalized size = 0.31 \[ \frac{2 \, b x^{2} \log \left (x\right ) - a}{2 \, x^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/2*(2*b*x^2*log(x) - a)/x^2

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Sympy [A]  time = 1.0711, size = 10, normalized size = 0.13 \[ - \frac{a}{2 x^{2}} + b \log{\left (x \right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-a/(2*x**2) + b*log(x)

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GIAC/XCAS [A]  time = 0.269956, size = 61, normalized size = 0.81 \[ \frac{1}{2} \, b{\rm ln}\left (x^{2}\right ){\rm sign}\left (b x^{2} + a\right ) - \frac{b x^{2}{\rm sign}\left (b x^{2} + a\right ) + a{\rm sign}\left (b x^{2} + a\right )}{2 \, x^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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