3.41 \(\int \frac{\left (a^2+2 a b x^3+b^2 x^6\right )^{3/2}}{x^9} \, dx\)

Optimal. Leaf size=162 \[ -\frac{3 a^2 b \sqrt{a^2+2 a b x^3+b^2 x^6}}{5 x^5 \left (a+b x^3\right )}-\frac{3 a b^2 \sqrt{a^2+2 a b x^3+b^2 x^6}}{2 x^2 \left (a+b x^3\right )}+\frac{b^3 x \sqrt{a^2+2 a b x^3+b^2 x^6}}{a+b x^3}-\frac{a^3 \sqrt{a^2+2 a b x^3+b^2 x^6}}{8 x^8 \left (a+b x^3\right )} \]

[Out]

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

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Rubi [A]  time = 0.110234, antiderivative size = 162, 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 \[ -\frac{3 a^2 b \sqrt{a^2+2 a b x^3+b^2 x^6}}{5 x^5 \left (a+b x^3\right )}-\frac{3 a b^2 \sqrt{a^2+2 a b x^3+b^2 x^6}}{2 x^2 \left (a+b x^3\right )}+\frac{b^3 x \sqrt{a^2+2 a b x^3+b^2 x^6}}{a+b x^3}-\frac{a^3 \sqrt{a^2+2 a b x^3+b^2 x^6}}{8 x^8 \left (a+b x^3\right )} \]

Antiderivative was successfully verified.

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

[Out]

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

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Rubi in Sympy [A]  time = 16.9236, size = 138, normalized size = 0.85 \[ - \frac{81 a b^{2} \sqrt{a^{2} + 2 a b x^{3} + b^{2} x^{6}}}{40 x^{2} \left (a + b x^{3}\right )} + \frac{9 a \left (a + b x^{3}\right ) \sqrt{a^{2} + 2 a b x^{3} + b^{2} x^{6}}}{40 x^{8}} + \frac{27 b^{2} \sqrt{a^{2} + 2 a b x^{3} + b^{2} x^{6}}}{20 x^{2}} - \frac{7 \left (a^{2} + 2 a b x^{3} + b^{2} x^{6}\right )^{\frac{3}{2}}}{20 x^{8}} \]

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**9,x)

[Out]

-81*a*b**2*sqrt(a**2 + 2*a*b*x**3 + b**2*x**6)/(40*x**2*(a + b*x**3)) + 9*a*(a +
b*x**3)*sqrt(a**2 + 2*a*b*x**3 + b**2*x**6)/(40*x**8) + 27*b**2*sqrt(a**2 + 2*a
*b*x**3 + b**2*x**6)/(20*x**2) - 7*(a**2 + 2*a*b*x**3 + b**2*x**6)**(3/2)/(20*x*
*8)

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Mathematica [A]  time = 0.0243241, size = 61, normalized size = 0.38 \[ -\frac{\sqrt{\left (a+b x^3\right )^2} \left (5 a^3+24 a^2 b x^3+60 a b^2 x^6-40 b^3 x^9\right )}{40 x^8 \left (a+b x^3\right )} \]

Antiderivative was successfully verified.

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

[Out]

-(Sqrt[(a + b*x^3)^2]*(5*a^3 + 24*a^2*b*x^3 + 60*a*b^2*x^6 - 40*b^3*x^9))/(40*x^
8*(a + b*x^3))

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Maple [A]  time = 0.009, size = 58, normalized size = 0.4 \[ -{\frac{-40\,{b}^{3}{x}^{9}+60\,a{x}^{6}{b}^{2}+24\,{x}^{3}{a}^{2}b+5\,{a}^{3}}{40\,{x}^{8} \left ( b{x}^{3}+a \right ) ^{3}} \left ( \left ( b{x}^{3}+a \right ) ^{2} \right ) ^{{\frac{3}{2}}}} \]

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^9,x)

[Out]

-1/40*(-40*b^3*x^9+60*a*b^2*x^6+24*a^2*b*x^3+5*a^3)*((b*x^3+a)^2)^(3/2)/x^8/(b*x
^3+a)^3

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Maxima [A]  time = 0.777922, size = 50, normalized size = 0.31 \[ \frac{40 \, b^{3} x^{9} - 60 \, a b^{2} x^{6} - 24 \, a^{2} b x^{3} - 5 \, a^{3}}{40 \, x^{8}} \]

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^9,x, algorithm="maxima")

[Out]

1/40*(40*b^3*x^9 - 60*a*b^2*x^6 - 24*a^2*b*x^3 - 5*a^3)/x^8

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Fricas [A]  time = 0.255093, size = 50, normalized size = 0.31 \[ \frac{40 \, b^{3} x^{9} - 60 \, a b^{2} x^{6} - 24 \, a^{2} b x^{3} - 5 \, a^{3}}{40 \, x^{8}} \]

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^9,x, algorithm="fricas")

[Out]

1/40*(40*b^3*x^9 - 60*a*b^2*x^6 - 24*a^2*b*x^3 - 5*a^3)/x^8

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

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**9,x)

[Out]

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

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GIAC/XCAS [A]  time = 0.295689, size = 90, normalized size = 0.56 \[ b^{3} x{\rm sign}\left (b x^{3} + a\right ) - \frac{60 \, a b^{2} x^{6}{\rm sign}\left (b x^{3} + a\right ) + 24 \, a^{2} b x^{3}{\rm sign}\left (b x^{3} + a\right ) + 5 \, a^{3}{\rm sign}\left (b x^{3} + a\right )}{40 \, x^{8}} \]

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^9,x, algorithm="giac")

[Out]

b^3*x*sign(b*x^3 + a) - 1/40*(60*a*b^2*x^6*sign(b*x^3 + a) + 24*a^2*b*x^3*sign(b
*x^3 + a) + 5*a^3*sign(b*x^3 + a))/x^8