3.555 \(\int \frac{\sqrt{a^2+2 a b x^2+b^2 x^4}}{x^6} \, dx\)

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

[Out]

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

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Rubi [A]  time = 0.0701777, 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 \[ -\frac{a \sqrt{a^2+2 a b x^2+b^2 x^4}}{5 x^5 \left (a+b x^2\right )}-\frac{b \sqrt{a^2+2 a b x^2+b^2 x^4}}{3 x^3 \left (a+b x^2\right )} \]

Antiderivative was successfully verified.

[In]  Int[Sqrt[a^2 + 2*a*b*x^2 + b^2*x^4]/x^6,x]

[Out]

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

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Rubi in Sympy [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{\sqrt{\left (a + b x^{2}\right )^{2}}}{x^{6}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(((b*x**2+a)**2)**(1/2)/x**6,x)

[Out]

Integral(sqrt((a + b*x**2)**2)/x**6, x)

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Mathematica [A]  time = 0.0130489, size = 39, normalized size = 0.49 \[ -\frac{\sqrt{\left (a+b x^2\right )^2} \left (3 a+5 b x^2\right )}{15 x^5 \left (a+b x^2\right )} \]

Antiderivative was successfully verified.

[In]  Integrate[Sqrt[a^2 + 2*a*b*x^2 + b^2*x^4]/x^6,x]

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(((b*x^2+a)^2)^(1/2)/x^6,x)

[Out]

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

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Maxima [A]  time = 0.697821, size = 20, normalized size = 0.25 \[ -\frac{5 \, b x^{2} + 3 \, a}{15 \, x^{5}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(sqrt((b*x^2 + a)^2)/x^6,x, algorithm="maxima")

[Out]

-1/15*(5*b*x^2 + 3*a)/x^5

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Fricas [A]  time = 0.263699, size = 20, normalized size = 0.25 \[ -\frac{5 \, b x^{2} + 3 \, a}{15 \, x^{5}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(sqrt((b*x^2 + a)^2)/x^6,x, algorithm="fricas")

[Out]

-1/15*(5*b*x^2 + 3*a)/x^5

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Sympy [A]  time = 1.16266, size = 15, normalized size = 0.19 \[ - \frac{3 a + 5 b x^{2}}{15 x^{5}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(((b*x**2+a)**2)**(1/2)/x**6,x)

[Out]

-(3*a + 5*b*x**2)/(15*x**5)

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GIAC/XCAS [A]  time = 0.27023, size = 42, normalized size = 0.53 \[ -\frac{5 \, b x^{2}{\rm sign}\left (b x^{2} + a\right ) + 3 \, a{\rm sign}\left (b x^{2} + a\right )}{15 \, x^{5}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(sqrt((b*x^2 + a)^2)/x^6,x, algorithm="giac")

[Out]

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