3.20 \(\int \frac{\sqrt{a^2+2 a b x^3+b^2 x^6}}{x^9} \, dx\)

Optimal. Leaf size=79 \[ -\frac{a \sqrt{a^2+2 a b x^3+b^2 x^6}}{8 x^8 \left (a+b x^3\right )}-\frac{b \sqrt{a^2+2 a b x^3+b^2 x^6}}{5 x^5 \left (a+b x^3\right )} \]

[Out]

-(a*Sqrt[a^2 + 2*a*b*x^3 + b^2*x^6])/(8*x^8*(a + b*x^3)) - (b*Sqrt[a^2 + 2*a*b*x
^3 + b^2*x^6])/(5*x^5*(a + b*x^3))

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Rubi [A]  time = 0.0629029, 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 \[ -\frac{a \sqrt{a^2+2 a b x^3+b^2 x^6}}{8 x^8 \left (a+b x^3\right )}-\frac{b \sqrt{a^2+2 a b x^3+b^2 x^6}}{5 x^5 \left (a+b x^3\right )} \]

Antiderivative was successfully verified.

[In]  Int[Sqrt[a^2 + 2*a*b*x^3 + b^2*x^6]/x^9,x]

[Out]

-(a*Sqrt[a^2 + 2*a*b*x^3 + b^2*x^6])/(8*x^8*(a + b*x^3)) - (b*Sqrt[a^2 + 2*a*b*x
^3 + b^2*x^6])/(5*x^5*(a + b*x^3))

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Rubi in Sympy [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{\sqrt{\left (a + b x^{3}\right )^{2}}}{x^{9}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(((b*x**3+a)**2)**(1/2)/x**9,x)

[Out]

Integral(sqrt((a + b*x**3)**2)/x**9, x)

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Mathematica [A]  time = 0.0134127, size = 39, normalized size = 0.49 \[ -\frac{\sqrt{\left (a+b x^3\right )^2} \left (5 a+8 b x^3\right )}{40 x^8 \left (a+b x^3\right )} \]

Antiderivative was successfully verified.

[In]  Integrate[Sqrt[a^2 + 2*a*b*x^3 + b^2*x^6]/x^9,x]

[Out]

-(Sqrt[(a + b*x^3)^2]*(5*a + 8*b*x^3))/(40*x^8*(a + b*x^3))

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Maple [A]  time = 0.005, size = 36, normalized size = 0.5 \[ -{\frac{8\,b{x}^{3}+5\,a}{40\,{x}^{8} \left ( b{x}^{3}+a \right ) }\sqrt{ \left ( b{x}^{3}+a \right ) ^{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(((b*x^3+a)^2)^(1/2)/x^9,x)

[Out]

-1/40*(8*b*x^3+5*a)*((b*x^3+a)^2)^(1/2)/x^8/(b*x^3+a)

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Maxima [A]  time = 0.791242, size = 20, normalized size = 0.25 \[ -\frac{8 \, b x^{3} + 5 \, a}{40 \, x^{8}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(sqrt((b*x^3 + a)^2)/x^9,x, algorithm="maxima")

[Out]

-1/40*(8*b*x^3 + 5*a)/x^8

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Fricas [A]  time = 0.27184, size = 20, normalized size = 0.25 \[ -\frac{8 \, b x^{3} + 5 \, a}{40 \, x^{8}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(sqrt((b*x^3 + a)^2)/x^9,x, algorithm="fricas")

[Out]

-1/40*(8*b*x^3 + 5*a)/x^8

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Sympy [A]  time = 1.37279, size = 15, normalized size = 0.19 \[ - \frac{5 a + 8 b x^{3}}{40 x^{8}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(((b*x**3+a)**2)**(1/2)/x**9,x)

[Out]

-(5*a + 8*b*x**3)/(40*x**8)

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GIAC/XCAS [A]  time = 0.286681, size = 42, normalized size = 0.53 \[ -\frac{8 \, b x^{3}{\rm sign}\left (b x^{3} + a\right ) + 5 \, a{\rm sign}\left (b x^{3} + a\right )}{40 \, x^{8}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(sqrt((b*x^3 + a)^2)/x^9,x, algorithm="giac")

[Out]

-1/40*(8*b*x^3*sign(b*x^3 + a) + 5*a*sign(b*x^3 + a))/x^8