3.22 \(\int \frac{\sqrt{a^2+2 a b x^3+b^2 x^6}}{x^{11}} \, dx\)

Optimal. Leaf size=79 \[ -\frac{a \sqrt{a^2+2 a b x^3+b^2 x^6}}{10 x^{10} \left (a+b x^3\right )}-\frac{b \sqrt{a^2+2 a b x^3+b^2 x^6}}{7 x^7 \left (a+b x^3\right )} \]

[Out]

-(a*Sqrt[a^2 + 2*a*b*x^3 + b^2*x^6])/(10*x^10*(a + b*x^3)) - (b*Sqrt[a^2 + 2*a*b
*x^3 + b^2*x^6])/(7*x^7*(a + b*x^3))

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Rubi [A]  time = 0.0642167, 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 \[ -\frac{a \sqrt{a^2+2 a b x^3+b^2 x^6}}{10 x^{10} \left (a+b x^3\right )}-\frac{b \sqrt{a^2+2 a b x^3+b^2 x^6}}{7 x^7 \left (a+b x^3\right )} \]

Antiderivative was successfully verified.

[In]  Int[Sqrt[a^2 + 2*a*b*x^3 + b^2*x^6]/x^11,x]

[Out]

-(a*Sqrt[a^2 + 2*a*b*x^3 + b^2*x^6])/(10*x^10*(a + b*x^3)) - (b*Sqrt[a^2 + 2*a*b
*x^3 + b^2*x^6])/(7*x^7*(a + b*x^3))

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Rubi in Sympy [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{\sqrt{\left (a + b x^{3}\right )^{2}}}{x^{11}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(((b*x**3+a)**2)**(1/2)/x**11,x)

[Out]

Integral(sqrt((a + b*x**3)**2)/x**11, x)

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Mathematica [A]  time = 0.0137263, size = 39, normalized size = 0.49 \[ -\frac{\sqrt{\left (a+b x^3\right )^2} \left (7 a+10 b x^3\right )}{70 x^{10} \left (a+b x^3\right )} \]

Antiderivative was successfully verified.

[In]  Integrate[Sqrt[a^2 + 2*a*b*x^3 + b^2*x^6]/x^11,x]

[Out]

-(Sqrt[(a + b*x^3)^2]*(7*a + 10*b*x^3))/(70*x^10*(a + b*x^3))

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Maple [A]  time = 0.006, size = 36, normalized size = 0.5 \[ -{\frac{10\,b{x}^{3}+7\,a}{70\,{x}^{10} \left ( b{x}^{3}+a \right ) }\sqrt{ \left ( b{x}^{3}+a \right ) ^{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(((b*x^3+a)^2)^(1/2)/x^11,x)

[Out]

-1/70*(10*b*x^3+7*a)*((b*x^3+a)^2)^(1/2)/x^10/(b*x^3+a)

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Maxima [A]  time = 0.775506, size = 20, normalized size = 0.25 \[ -\frac{10 \, b x^{3} + 7 \, a}{70 \, x^{10}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(sqrt((b*x^3 + a)^2)/x^11,x, algorithm="maxima")

[Out]

-1/70*(10*b*x^3 + 7*a)/x^10

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Fricas [A]  time = 0.274876, size = 20, normalized size = 0.25 \[ -\frac{10 \, b x^{3} + 7 \, a}{70 \, x^{10}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(sqrt((b*x^3 + a)^2)/x^11,x, algorithm="fricas")

[Out]

-1/70*(10*b*x^3 + 7*a)/x^10

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Sympy [A]  time = 1.413, size = 15, normalized size = 0.19 \[ - \frac{7 a + 10 b x^{3}}{70 x^{10}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(((b*x**3+a)**2)**(1/2)/x**11,x)

[Out]

-(7*a + 10*b*x**3)/(70*x**10)

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GIAC/XCAS [A]  time = 0.286571, size = 42, normalized size = 0.53 \[ -\frac{10 \, b x^{3}{\rm sign}\left (b x^{3} + a\right ) + 7 \, a{\rm sign}\left (b x^{3} + a\right )}{70 \, x^{10}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(sqrt((b*x^3 + a)^2)/x^11,x, algorithm="giac")

[Out]

-1/70*(10*b*x^3*sign(b*x^3 + a) + 7*a*sign(b*x^3 + a))/x^10