3.45 \(\int \frac{\left (a^2+2 a b x^3+b^2 x^6\right )^{3/2}}{x^{13}} \, dx\)

Optimal. Leaf size=41 \[ -\frac{\left (a+b x^3\right )^3 \sqrt{a^2+2 a b x^3+b^2 x^6}}{12 a x^{12}} \]

[Out]

-((a + b*x^3)^3*Sqrt[a^2 + 2*a*b*x^3 + b^2*x^6])/(12*a*x^12)

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Rubi [A]  time = 0.0561068, antiderivative size = 41, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 26, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.077 \[ -\frac{\left (a+b x^3\right )^3 \sqrt{a^2+2 a b x^3+b^2 x^6}}{12 a x^{12}} \]

Antiderivative was successfully verified.

[In]  Int[(a^2 + 2*a*b*x^3 + b^2*x^6)^(3/2)/x^13,x]

[Out]

-((a + b*x^3)^3*Sqrt[a^2 + 2*a*b*x^3 + b^2*x^6])/(12*a*x^12)

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Rubi in Sympy [A]  time = 8.42827, size = 39, normalized size = 0.95 \[ - \frac{\left (2 a + 2 b x^{3}\right ) \left (a^{2} + 2 a b x^{3} + b^{2} x^{6}\right )^{\frac{3}{2}}}{24 a x^{12}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((b**2*x**6+2*a*b*x**3+a**2)**(3/2)/x**13,x)

[Out]

-(2*a + 2*b*x**3)*(a**2 + 2*a*b*x**3 + b**2*x**6)**(3/2)/(24*a*x**12)

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Mathematica [A]  time = 0.0263874, size = 59, normalized size = 1.44 \[ -\frac{\sqrt{\left (a+b x^3\right )^2} \left (a^3+4 a^2 b x^3+6 a b^2 x^6+4 b^3 x^9\right )}{12 x^{12} \left (a+b x^3\right )} \]

Antiderivative was successfully verified.

[In]  Integrate[(a^2 + 2*a*b*x^3 + b^2*x^6)^(3/2)/x^13,x]

[Out]

-(Sqrt[(a + b*x^3)^2]*(a^3 + 4*a^2*b*x^3 + 6*a*b^2*x^6 + 4*b^3*x^9))/(12*x^12*(a
+ b*x^3))

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Maple [A]  time = 0.011, size = 56, normalized size = 1.4 \[ -{\frac{4\,{b}^{3}{x}^{9}+6\,a{x}^{6}{b}^{2}+4\,{x}^{3}{a}^{2}b+{a}^{3}}{12\,{x}^{12} \left ( b{x}^{3}+a \right ) ^{3}} \left ( \left ( b{x}^{3}+a \right ) ^{2} \right ) ^{{\frac{3}{2}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((b^2*x^6+2*a*b*x^3+a^2)^(3/2)/x^13,x)

[Out]

-1/12*(4*b^3*x^9+6*a*b^2*x^6+4*a^2*b*x^3+a^3)*((b*x^3+a)^2)^(3/2)/x^12/(b*x^3+a)
^3

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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((b^2*x^6 + 2*a*b*x^3 + a^2)^(3/2)/x^13,x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.264754, size = 47, normalized size = 1.15 \[ -\frac{4 \, b^{3} x^{9} + 6 \, a b^{2} x^{6} + 4 \, a^{2} b x^{3} + a^{3}}{12 \, x^{12}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b^2*x^6 + 2*a*b*x^3 + a^2)^(3/2)/x^13,x, algorithm="fricas")

[Out]

-1/12*(4*b^3*x^9 + 6*a*b^2*x^6 + 4*a^2*b*x^3 + a^3)/x^12

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Sympy [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{\left (\left (a + b x^{3}\right )^{2}\right )^{\frac{3}{2}}}{x^{13}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b**2*x**6+2*a*b*x**3+a**2)**(3/2)/x**13,x)

[Out]

Integral(((a + b*x**3)**2)**(3/2)/x**13, x)

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GIAC/XCAS [A]  time = 0.318631, size = 92, normalized size = 2.24 \[ -\frac{4 \, b^{3} x^{9}{\rm sign}\left (b x^{3} + a\right ) + 6 \, a b^{2} x^{6}{\rm sign}\left (b x^{3} + a\right ) + 4 \, a^{2} b x^{3}{\rm sign}\left (b x^{3} + a\right ) + a^{3}{\rm sign}\left (b x^{3} + a\right )}{12 \, x^{12}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b^2*x^6 + 2*a*b*x^3 + a^2)^(3/2)/x^13,x, algorithm="giac")

[Out]

-1/12*(4*b^3*x^9*sign(b*x^3 + a) + 6*a*b^2*x^6*sign(b*x^3 + a) + 4*a^2*b*x^3*sig
n(b*x^3 + a) + a^3*sign(b*x^3 + a))/x^12