3.44 \(\int \frac{\left (a^2+2 a b x^3+b^2 x^6\right )^{3/2}}{x^{12}} \, dx\)

Optimal. Leaf size=167 \[ -\frac{3 a^2 b \sqrt{a^2+2 a b x^3+b^2 x^6}}{8 x^8 \left (a+b x^3\right )}-\frac{3 a b^2 \sqrt{a^2+2 a b x^3+b^2 x^6}}{5 x^5 \left (a+b x^3\right )}-\frac{b^3 \sqrt{a^2+2 a b x^3+b^2 x^6}}{2 x^2 \left (a+b x^3\right )}-\frac{a^3 \sqrt{a^2+2 a b x^3+b^2 x^6}}{11 x^{11} \left (a+b x^3\right )} \]

[Out]

-(a^3*Sqrt[a^2 + 2*a*b*x^3 + b^2*x^6])/(11*x^11*(a + b*x^3)) - (3*a^2*b*Sqrt[a^2
+ 2*a*b*x^3 + b^2*x^6])/(8*x^8*(a + b*x^3)) - (3*a*b^2*Sqrt[a^2 + 2*a*b*x^3 + b
^2*x^6])/(5*x^5*(a + b*x^3)) - (b^3*Sqrt[a^2 + 2*a*b*x^3 + b^2*x^6])/(2*x^2*(a +
b*x^3))

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Rubi [A]  time = 0.114206, antiderivative size = 167, 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 \[ -\frac{3 a^2 b \sqrt{a^2+2 a b x^3+b^2 x^6}}{8 x^8 \left (a+b x^3\right )}-\frac{3 a b^2 \sqrt{a^2+2 a b x^3+b^2 x^6}}{5 x^5 \left (a+b x^3\right )}-\frac{b^3 \sqrt{a^2+2 a b x^3+b^2 x^6}}{2 x^2 \left (a+b x^3\right )}-\frac{a^3 \sqrt{a^2+2 a b x^3+b^2 x^6}}{11 x^{11} \left (a+b x^3\right )} \]

Antiderivative was successfully verified.

[In]  Int[(a^2 + 2*a*b*x^3 + b^2*x^6)^(3/2)/x^12,x]

[Out]

-(a^3*Sqrt[a^2 + 2*a*b*x^3 + b^2*x^6])/(11*x^11*(a + b*x^3)) - (3*a^2*b*Sqrt[a^2
+ 2*a*b*x^3 + b^2*x^6])/(8*x^8*(a + b*x^3)) - (3*a*b^2*Sqrt[a^2 + 2*a*b*x^3 + b
^2*x^6])/(5*x^5*(a + b*x^3)) - (b^3*Sqrt[a^2 + 2*a*b*x^3 + b^2*x^6])/(2*x^2*(a +
b*x^3))

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Rubi in Sympy [A]  time = 17.0504, size = 138, normalized size = 0.83 \[ \frac{81 a b^{2} \sqrt{a^{2} + 2 a b x^{3} + b^{2} x^{6}}}{440 x^{5} \left (a + b x^{3}\right )} + \frac{9 a \left (a + b x^{3}\right ) \sqrt{a^{2} + 2 a b x^{3} + b^{2} x^{6}}}{88 x^{11}} - \frac{27 b^{2} \sqrt{a^{2} + 2 a b x^{3} + b^{2} x^{6}}}{88 x^{5}} - \frac{17 \left (a^{2} + 2 a b x^{3} + b^{2} x^{6}\right )^{\frac{3}{2}}}{88 x^{11}} \]

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**12,x)

[Out]

81*a*b**2*sqrt(a**2 + 2*a*b*x**3 + b**2*x**6)/(440*x**5*(a + b*x**3)) + 9*a*(a +
b*x**3)*sqrt(a**2 + 2*a*b*x**3 + b**2*x**6)/(88*x**11) - 27*b**2*sqrt(a**2 + 2*
a*b*x**3 + b**2*x**6)/(88*x**5) - 17*(a**2 + 2*a*b*x**3 + b**2*x**6)**(3/2)/(88*
x**11)

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Mathematica [A]  time = 0.0313977, size = 61, normalized size = 0.37 \[ -\frac{\sqrt{\left (a+b x^3\right )^2} \left (40 a^3+165 a^2 b x^3+264 a b^2 x^6+220 b^3 x^9\right )}{440 x^{11} \left (a+b x^3\right )} \]

Antiderivative was successfully verified.

[In]  Integrate[(a^2 + 2*a*b*x^3 + b^2*x^6)^(3/2)/x^12,x]

[Out]

-(Sqrt[(a + b*x^3)^2]*(40*a^3 + 165*a^2*b*x^3 + 264*a*b^2*x^6 + 220*b^3*x^9))/(4
40*x^11*(a + b*x^3))

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Maple [A]  time = 0.01, size = 58, normalized size = 0.4 \[ -{\frac{220\,{b}^{3}{x}^{9}+264\,a{x}^{6}{b}^{2}+165\,{x}^{3}{a}^{2}b+40\,{a}^{3}}{440\,{x}^{11} \left ( b{x}^{3}+a \right ) ^{3}} \left ( \left ( b{x}^{3}+a \right ) ^{2} \right ) ^{{\frac{3}{2}}}} \]

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^12,x)

[Out]

-1/440*(220*b^3*x^9+264*a*b^2*x^6+165*a^2*b*x^3+40*a^3)*((b*x^3+a)^2)^(3/2)/x^11
/(b*x^3+a)^3

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Maxima [A]  time = 0.758175, size = 50, normalized size = 0.3 \[ -\frac{220 \, b^{3} x^{9} + 264 \, a b^{2} x^{6} + 165 \, a^{2} b x^{3} + 40 \, a^{3}}{440 \, x^{11}} \]

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^12,x, algorithm="maxima")

[Out]

-1/440*(220*b^3*x^9 + 264*a*b^2*x^6 + 165*a^2*b*x^3 + 40*a^3)/x^11

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Fricas [A]  time = 0.261998, size = 50, normalized size = 0.3 \[ -\frac{220 \, b^{3} x^{9} + 264 \, a b^{2} x^{6} + 165 \, a^{2} b x^{3} + 40 \, a^{3}}{440 \, x^{11}} \]

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^12,x, algorithm="fricas")

[Out]

-1/440*(220*b^3*x^9 + 264*a*b^2*x^6 + 165*a^2*b*x^3 + 40*a^3)/x^11

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Sympy [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{\left (\left (a + b x^{3}\right )^{2}\right )^{\frac{3}{2}}}{x^{12}}\, dx \]

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**12,x)

[Out]

Integral(((a + b*x**3)**2)**(3/2)/x**12, x)

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GIAC/XCAS [A]  time = 0.259786, size = 93, normalized size = 0.56 \[ -\frac{220 \, b^{3} x^{9}{\rm sign}\left (b x^{3} + a\right ) + 264 \, a b^{2} x^{6}{\rm sign}\left (b x^{3} + a\right ) + 165 \, a^{2} b x^{3}{\rm sign}\left (b x^{3} + a\right ) + 40 \, a^{3}{\rm sign}\left (b x^{3} + a\right )}{440 \, x^{11}} \]

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^12,x, algorithm="giac")

[Out]

-1/440*(220*b^3*x^9*sign(b*x^3 + a) + 264*a*b^2*x^6*sign(b*x^3 + a) + 165*a^2*b*
x^3*sign(b*x^3 + a) + 40*a^3*sign(b*x^3 + a))/x^11