3.18 \(\int \frac{\sqrt{a^2+2 a b x^3+b^2 x^6}}{x^7} \, dx\)

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

[Out]

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

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Rubi [A]  time = 0.0633717, 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 \[ -\frac{a \sqrt{a^2+2 a b x^3+b^2 x^6}}{6 x^6 \left (a+b x^3\right )}-\frac{b \sqrt{a^2+2 a b x^3+b^2 x^6}}{3 x^3 \left (a+b x^3\right )} \]

Antiderivative was successfully verified.

[In]  Int[Sqrt[a^2 + 2*a*b*x^3 + b^2*x^6]/x^7,x]

[Out]

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

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Rubi in Sympy [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{\sqrt{\left (a + b x^{3}\right )^{2}}}{x^{7}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(((b*x**3+a)**2)**(1/2)/x**7,x)

[Out]

Integral(sqrt((a + b*x**3)**2)/x**7, x)

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Mathematica [A]  time = 0.0167524, size = 37, normalized size = 0.47 \[ -\frac{\sqrt{\left (a+b x^3\right )^2} \left (a+2 b x^3\right )}{6 x^6 \left (a+b x^3\right )} \]

Antiderivative was successfully verified.

[In]  Integrate[Sqrt[a^2 + 2*a*b*x^3 + b^2*x^6]/x^7,x]

[Out]

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

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Maple [A]  time = 0.004, size = 34, normalized size = 0.4 \[ -{\frac{2\,b{x}^{3}+a}{6\,{x}^{6} \left ( b{x}^{3}+a \right ) }\sqrt{ \left ( b{x}^{3}+a \right ) ^{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(((b*x^3+a)^2)^(1/2)/x^7,x)

[Out]

-1/6*(2*b*x^3+a)*((b*x^3+a)^2)^(1/2)/x^6/(b*x^3+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^3 + a)^2)/x^7,x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.24892, size = 18, normalized size = 0.23 \[ -\frac{2 \, b x^{3} + a}{6 \, x^{6}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(sqrt((b*x^3 + a)^2)/x^7,x, algorithm="fricas")

[Out]

-1/6*(2*b*x^3 + a)/x^6

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Sympy [A]  time = 1.27151, size = 14, normalized size = 0.18 \[ - \frac{a + 2 b x^{3}}{6 x^{6}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(((b*x**3+a)**2)**(1/2)/x**7,x)

[Out]

-(a + 2*b*x**3)/(6*x**6)

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GIAC/XCAS [A]  time = 0.285397, size = 41, normalized size = 0.52 \[ -\frac{2 \, b x^{3}{\rm sign}\left (b x^{3} + a\right ) + a{\rm sign}\left (b x^{3} + a\right )}{6 \, x^{6}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(sqrt((b*x^3 + a)^2)/x^7,x, algorithm="giac")

[Out]

-1/6*(2*b*x^3*sign(b*x^3 + a) + a*sign(b*x^3 + a))/x^6