3.548 \(\int \frac{\sqrt{a^2+2 a b x^2+b^2 x^4}}{x^9} \, dx\)

Optimal. Leaf size=79 \[ -\frac{a \sqrt{a^2+2 a b x^2+b^2 x^4}}{8 x^8 \left (a+b x^2\right )}-\frac{b \sqrt{a^2+2 a b x^2+b^2 x^4}}{6 x^6 \left (a+b x^2\right )} \]

[Out]

-(a*Sqrt[a^2 + 2*a*b*x^2 + b^2*x^4])/(8*x^8*(a + b*x^2)) - (b*Sqrt[a^2 + 2*a*b*x
^2 + b^2*x^4])/(6*x^6*(a + b*x^2))

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Rubi [A]  time = 0.144664, antiderivative size = 79, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 26, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.115 \[ -\frac{a \sqrt{a^2+2 a b x^2+b^2 x^4}}{8 x^8 \left (a+b x^2\right )}-\frac{b \sqrt{a^2+2 a b x^2+b^2 x^4}}{6 x^6 \left (a+b x^2\right )} \]

Antiderivative was successfully verified.

[In]  Int[Sqrt[a^2 + 2*a*b*x^2 + b^2*x^4]/x^9,x]

[Out]

-(a*Sqrt[a^2 + 2*a*b*x^2 + b^2*x^4])/(8*x^8*(a + b*x^2)) - (b*Sqrt[a^2 + 2*a*b*x
^2 + b^2*x^4])/(6*x^6*(a + b*x^2))

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Rubi in Sympy [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{\sqrt{\left (a + b x^{2}\right )^{2}}}{x^{9}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(((b*x**2+a)**2)**(1/2)/x**9,x)

[Out]

Integral(sqrt((a + b*x**2)**2)/x**9, x)

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Mathematica [A]  time = 0.0137151, size = 39, normalized size = 0.49 \[ -\frac{\sqrt{\left (a+b x^2\right )^2} \left (3 a+4 b x^2\right )}{24 x^8 \left (a+b x^2\right )} \]

Antiderivative was successfully verified.

[In]  Integrate[Sqrt[a^2 + 2*a*b*x^2 + b^2*x^4]/x^9,x]

[Out]

-(Sqrt[(a + b*x^2)^2]*(3*a + 4*b*x^2))/(24*x^8*(a + b*x^2))

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Maple [A]  time = 0.005, size = 36, normalized size = 0.5 \[ -{\frac{4\,b{x}^{2}+3\,a}{24\,{x}^{8} \left ( b{x}^{2}+a \right ) }\sqrt{ \left ( b{x}^{2}+a \right ) ^{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(((b*x^2+a)^2)^(1/2)/x^9,x)

[Out]

-1/24*(4*b*x^2+3*a)*((b*x^2+a)^2)^(1/2)/x^8/(b*x^2+a)

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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^9,x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.255692, size = 20, normalized size = 0.25 \[ -\frac{4 \, b x^{2} + 3 \, a}{24 \, x^{8}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(sqrt((b*x^2 + a)^2)/x^9,x, algorithm="fricas")

[Out]

-1/24*(4*b*x^2 + 3*a)/x^8

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Sympy [A]  time = 1.24565, size = 15, normalized size = 0.19 \[ - \frac{3 a + 4 b x^{2}}{24 x^{8}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(((b*x**2+a)**2)**(1/2)/x**9,x)

[Out]

-(3*a + 4*b*x**2)/(24*x**8)

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GIAC/XCAS [A]  time = 0.269581, size = 42, normalized size = 0.53 \[ -\frac{4 \, b x^{2}{\rm sign}\left (b x^{2} + a\right ) + 3 \, a{\rm sign}\left (b x^{2} + a\right )}{24 \, x^{8}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(sqrt((b*x^2 + a)^2)/x^9,x, algorithm="giac")

[Out]

-1/24*(4*b*x^2*sign(b*x^2 + a) + 3*a*sign(b*x^2 + a))/x^8