3.37 \(\int \frac{\left (a^2+2 a b x^3+b^2 x^6\right )^{3/2}}{x^5} \, dx\)

Optimal. Leaf size=165 \[ -\frac{3 a^2 b \sqrt{a^2+2 a b x^3+b^2 x^6}}{x \left (a+b x^3\right )}+\frac{3 a b^2 x^2 \sqrt{a^2+2 a b x^3+b^2 x^6}}{2 \left (a+b x^3\right )}+\frac{b^3 x^5 \sqrt{a^2+2 a b x^3+b^2 x^6}}{5 \left (a+b x^3\right )}-\frac{a^3 \sqrt{a^2+2 a b x^3+b^2 x^6}}{4 x^4 \left (a+b x^3\right )} \]

[Out]

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

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Rubi [A]  time = 0.113774, antiderivative size = 165, 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 \[ -\frac{3 a^2 b \sqrt{a^2+2 a b x^3+b^2 x^6}}{x \left (a+b x^3\right )}+\frac{3 a b^2 x^2 \sqrt{a^2+2 a b x^3+b^2 x^6}}{2 \left (a+b x^3\right )}+\frac{b^3 x^5 \sqrt{a^2+2 a b x^3+b^2 x^6}}{5 \left (a+b x^3\right )}-\frac{a^3 \sqrt{a^2+2 a b x^3+b^2 x^6}}{4 x^4 \left (a+b x^3\right )} \]

Antiderivative was successfully verified.

[In]  Int[(a^2 + 2*a*b*x^3 + b^2*x^6)^(3/2)/x^5,x]

[Out]

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

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Rubi in Sympy [A]  time = 16.3367, size = 139, normalized size = 0.84 \[ \frac{81 a b^{2} x^{2} \sqrt{a^{2} + 2 a b x^{3} + b^{2} x^{6}}}{20 \left (a + b x^{3}\right )} + \frac{9 a \left (a + b x^{3}\right ) \sqrt{a^{2} + 2 a b x^{3} + b^{2} x^{6}}}{4 x^{4}} + \frac{27 b^{2} x^{2} \sqrt{a^{2} + 2 a b x^{3} + b^{2} x^{6}}}{10} - \frac{5 \left (a^{2} + 2 a b x^{3} + b^{2} x^{6}\right )^{\frac{3}{2}}}{2 x^{4}} \]

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**5,x)

[Out]

81*a*b**2*x**2*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)/(4*x**4) + 27*b**2*x**2*sqrt(a**2 +
2*a*b*x**3 + b**2*x**6)/10 - 5*(a**2 + 2*a*b*x**3 + b**2*x**6)**(3/2)/(2*x**4)

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Mathematica [A]  time = 0.0339256, size = 61, normalized size = 0.37 \[ \frac{\sqrt{\left (a+b x^3\right )^2} \left (-5 a^3-60 a^2 b x^3+30 a b^2 x^6+4 b^3 x^9\right )}{20 x^4 \left (a+b x^3\right )} \]

Antiderivative was successfully verified.

[In]  Integrate[(a^2 + 2*a*b*x^3 + b^2*x^6)^(3/2)/x^5,x]

[Out]

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

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

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^5,x)

[Out]

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

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Maxima [A]  time = 0.803365, size = 50, normalized size = 0.3 \[ \frac{4 \, b^{3} x^{9} + 30 \, a b^{2} x^{6} - 60 \, a^{2} b x^{3} - 5 \, a^{3}}{20 \, x^{4}} \]

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^5,x, algorithm="maxima")

[Out]

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

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Fricas [A]  time = 0.26626, size = 50, normalized size = 0.3 \[ \frac{4 \, b^{3} x^{9} + 30 \, a b^{2} x^{6} - 60 \, a^{2} b x^{3} - 5 \, a^{3}}{20 \, x^{4}} \]

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^5,x, algorithm="fricas")

[Out]

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

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Sympy [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{\left (\left (a + b x^{3}\right )^{2}\right )^{\frac{3}{2}}}{x^{5}}\, dx \]

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**5,x)

[Out]

Integral(((a + b*x**3)**2)**(3/2)/x**5, x)

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GIAC/XCAS [A]  time = 0.275357, size = 93, normalized size = 0.56 \[ \frac{1}{5} \, b^{3} x^{5}{\rm sign}\left (b x^{3} + a\right ) + \frac{3}{2} \, a b^{2} x^{2}{\rm sign}\left (b x^{3} + a\right ) - \frac{12 \, a^{2} b x^{3}{\rm sign}\left (b x^{3} + a\right ) + a^{3}{\rm sign}\left (b x^{3} + a\right )}{4 \, x^{4}} \]

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^5,x, algorithm="giac")

[Out]

1/5*b^3*x^5*sign(b*x^3 + a) + 3/2*a*b^2*x^2*sign(b*x^3 + a) - 1/4*(12*a^2*b*x^3*
sign(b*x^3 + a) + a^3*sign(b*x^3 + a))/x^4