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

Optimal. Leaf size=163 \[ \frac{3 a b^2 x \sqrt{a^2+2 a b x^3+b^2 x^6}}{a+b x^3}-\frac{3 a^2 b \sqrt{a^2+2 a b x^3+b^2 x^6}}{2 x^2 \left (a+b x^3\right )}+\frac{b^3 x^4 \sqrt{a^2+2 a b x^3+b^2 x^6}}{4 \left (a+b x^3\right )}-\frac{a^3 \sqrt{a^2+2 a b x^3+b^2 x^6}}{5 x^5 \left (a+b x^3\right )} \]

[Out]

-(a^3*Sqrt[a^2 + 2*a*b*x^3 + b^2*x^6])/(5*x^5*(a + b*x^3)) - (3*a^2*b*Sqrt[a^2 +
2*a*b*x^3 + b^2*x^6])/(2*x^2*(a + b*x^3)) + (3*a*b^2*x*Sqrt[a^2 + 2*a*b*x^3 + b
^2*x^6])/(a + b*x^3) + (b^3*x^4*Sqrt[a^2 + 2*a*b*x^3 + b^2*x^6])/(4*(a + b*x^3))

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Rubi [A]  time = 0.112273, antiderivative size = 163, 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 \[ \frac{3 a b^2 x \sqrt{a^2+2 a b x^3+b^2 x^6}}{a+b x^3}-\frac{3 a^2 b \sqrt{a^2+2 a b x^3+b^2 x^6}}{2 x^2 \left (a+b x^3\right )}+\frac{b^3 x^4 \sqrt{a^2+2 a b x^3+b^2 x^6}}{4 \left (a+b x^3\right )}-\frac{a^3 \sqrt{a^2+2 a b x^3+b^2 x^6}}{5 x^5 \left (a+b x^3\right )} \]

Antiderivative was successfully verified.

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

[Out]

-(a^3*Sqrt[a^2 + 2*a*b*x^3 + b^2*x^6])/(5*x^5*(a + b*x^3)) - (3*a^2*b*Sqrt[a^2 +
2*a*b*x^3 + b^2*x^6])/(2*x^2*(a + b*x^3)) + (3*a*b^2*x*Sqrt[a^2 + 2*a*b*x^3 + b
^2*x^6])/(a + b*x^3) + (b^3*x^4*Sqrt[a^2 + 2*a*b*x^3 + b^2*x^6])/(4*(a + b*x^3))

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Rubi in Sympy [A]  time = 11.524, size = 136, normalized size = 0.83 \[ \frac{81 a b^{2} x \sqrt{a^{2} + 2 a b x^{3} + b^{2} x^{6}}}{20 \left (a + b x^{3}\right )} + \frac{9 a \left (a + b x^{3}\right ) \sqrt{a^{2} + 2 a b x^{3} + b^{2} x^{6}}}{10 x^{5}} + \frac{27 b^{2} x \sqrt{a^{2} + 2 a b x^{3} + b^{2} x^{6}}}{20} - \frac{11 \left (a^{2} + 2 a b x^{3} + b^{2} x^{6}\right )^{\frac{3}{2}}}{10 x^{5}} \]

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**6,x)

[Out]

81*a*b**2*x*sqrt(a**2 + 2*a*b*x**3 + b**2*x**6)/(20*(a + b*x**3)) + 9*a*(a + b*x
**3)*sqrt(a**2 + 2*a*b*x**3 + b**2*x**6)/(10*x**5) + 27*b**2*x*sqrt(a**2 + 2*a*b
*x**3 + b**2*x**6)/20 - 11*(a**2 + 2*a*b*x**3 + b**2*x**6)**(3/2)/(10*x**5)

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Mathematica [A]  time = 0.0334181, size = 61, normalized size = 0.37 \[ \frac{\sqrt{\left (a+b x^3\right )^2} \left (-4 a^3-30 a^2 b x^3+60 a b^2 x^6+5 b^3 x^9\right )}{20 x^5 \left (a+b x^3\right )} \]

Antiderivative was successfully verified.

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

[Out]

(Sqrt[(a + b*x^3)^2]*(-4*a^3 - 30*a^2*b*x^3 + 60*a*b^2*x^6 + 5*b^3*x^9))/(20*x^5
*(a + b*x^3))

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Maple [A]  time = 0.009, size = 58, normalized size = 0.4 \[ -{\frac{-5\,{b}^{3}{x}^{9}-60\,a{x}^{6}{b}^{2}+30\,{x}^{3}{a}^{2}b+4\,{a}^{3}}{20\,{x}^{5} \left ( b{x}^{3}+a \right ) ^{3}} \left ( \left ( b{x}^{3}+a \right ) ^{2} \right ) ^{{\frac{3}{2}}}} \]

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^6,x)

[Out]

-1/20*(-5*b^3*x^9-60*a*b^2*x^6+30*a^2*b*x^3+4*a^3)*((b*x^3+a)^2)^(3/2)/x^5/(b*x^
3+a)^3

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Maxima [A]  time = 0.790617, size = 50, normalized size = 0.31 \[ \frac{5 \, b^{3} x^{9} + 60 \, a b^{2} x^{6} - 30 \, a^{2} b x^{3} - 4 \, a^{3}}{20 \, x^{5}} \]

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^6,x, algorithm="maxima")

[Out]

1/20*(5*b^3*x^9 + 60*a*b^2*x^6 - 30*a^2*b*x^3 - 4*a^3)/x^5

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Fricas [A]  time = 0.263934, size = 50, normalized size = 0.31 \[ \frac{5 \, b^{3} x^{9} + 60 \, a b^{2} x^{6} - 30 \, a^{2} b x^{3} - 4 \, a^{3}}{20 \, x^{5}} \]

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^6,x, algorithm="fricas")

[Out]

1/20*(5*b^3*x^9 + 60*a*b^2*x^6 - 30*a^2*b*x^3 - 4*a^3)/x^5

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Sympy [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{\left (\left (a + b x^{3}\right )^{2}\right )^{\frac{3}{2}}}{x^{6}}\, dx \]

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**6,x)

[Out]

Integral(((a + b*x**3)**2)**(3/2)/x**6, x)

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GIAC/XCAS [A]  time = 0.29151, size = 92, normalized size = 0.56 \[ \frac{1}{4} \, b^{3} x^{4}{\rm sign}\left (b x^{3} + a\right ) + 3 \, a b^{2} x{\rm sign}\left (b x^{3} + a\right ) - \frac{15 \, a^{2} b x^{3}{\rm sign}\left (b x^{3} + a\right ) + 2 \, a^{3}{\rm sign}\left (b x^{3} + a\right )}{10 \, x^{5}} \]

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^6,x, algorithm="giac")

[Out]

1/4*b^3*x^4*sign(b*x^3 + a) + 3*a*b^2*x*sign(b*x^3 + a) - 1/10*(15*a^2*b*x^3*sig
n(b*x^3 + a) + 2*a^3*sign(b*x^3 + a))/x^5