3.42 \(\int \frac{\left (a^2+2 a b x^3+b^2 x^6\right )^{3/2}}{x^{10}} \, dx\)

Optimal. Leaf size=161 \[ -\frac{a^2 b \sqrt{a^2+2 a b x^3+b^2 x^6}}{2 x^6 \left (a+b x^3\right )}-\frac{a b^2 \sqrt{a^2+2 a b x^3+b^2 x^6}}{x^3 \left (a+b x^3\right )}+\frac{b^3 \log (x) \sqrt{a^2+2 a b x^3+b^2 x^6}}{a+b x^3}-\frac{a^3 \sqrt{a^2+2 a b x^3+b^2 x^6}}{9 x^9 \left (a+b x^3\right )} \]

[Out]

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

_______________________________________________________________________________________

Rubi [A]  time = 0.124858, antiderivative size = 161, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 26, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.115 \[ -\frac{a^2 b \sqrt{a^2+2 a b x^3+b^2 x^6}}{2 x^6 \left (a+b x^3\right )}-\frac{a b^2 \sqrt{a^2+2 a b x^3+b^2 x^6}}{x^3 \left (a+b x^3\right )}+\frac{b^3 \log (x) \sqrt{a^2+2 a b x^3+b^2 x^6}}{a+b x^3}-\frac{a^3 \sqrt{a^2+2 a b x^3+b^2 x^6}}{9 x^9 \left (a+b x^3\right )} \]

Antiderivative was successfully verified.

[In]  Int[(a^2 + 2*a*b*x^3 + b^2*x^6)^(3/2)/x^10,x]

[Out]

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

_______________________________________________________________________________________

Rubi in Sympy [A]  time = 22.4744, size = 138, normalized size = 0.86 \[ - \frac{a b^{2} \sqrt{a^{2} + 2 a b x^{3} + b^{2} x^{6}}}{3 x^{3} \left (a + b x^{3}\right )} + \frac{a \left (a + b x^{3}\right ) \sqrt{a^{2} + 2 a b x^{3} + b^{2} x^{6}}}{6 x^{9}} + \frac{b^{3} \sqrt{a^{2} + 2 a b x^{3} + b^{2} x^{6}} \log{\left (x \right )}}{a + b x^{3}} - \frac{5 \left (a^{2} + 2 a b x^{3} + b^{2} x^{6}\right )^{\frac{3}{2}}}{18 x^{9}} \]

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**10,x)

[Out]

-a*b**2*sqrt(a**2 + 2*a*b*x**3 + b**2*x**6)/(3*x**3*(a + b*x**3)) + a*(a + b*x**
3)*sqrt(a**2 + 2*a*b*x**3 + b**2*x**6)/(6*x**9) + b**3*sqrt(a**2 + 2*a*b*x**3 +
b**2*x**6)*log(x)/(a + b*x**3) - 5*(a**2 + 2*a*b*x**3 + b**2*x**6)**(3/2)/(18*x*
*9)

_______________________________________________________________________________________

Mathematica [A]  time = 0.0425568, size = 63, normalized size = 0.39 \[ -\frac{\sqrt{\left (a+b x^3\right )^2} \left (a \left (2 a^2+9 a b x^3+18 b^2 x^6\right )-18 b^3 x^9 \log (x)\right )}{18 x^9 \left (a+b x^3\right )} \]

Antiderivative was successfully verified.

[In]  Integrate[(a^2 + 2*a*b*x^3 + b^2*x^6)^(3/2)/x^10,x]

[Out]

-(Sqrt[(a + b*x^3)^2]*(a*(2*a^2 + 9*a*b*x^3 + 18*b^2*x^6) - 18*b^3*x^9*Log[x]))/
(18*x^9*(a + b*x^3))

_______________________________________________________________________________________

Maple [A]  time = 0.018, size = 60, normalized size = 0.4 \[{\frac{18\,{b}^{3}\ln \left ( x \right ){x}^{9}-18\,a{x}^{6}{b}^{2}-9\,{x}^{3}{a}^{2}b-2\,{a}^{3}}{18\, \left ( b{x}^{3}+a \right ) ^{3}{x}^{9}} \left ( \left ( b{x}^{3}+a \right ) ^{2} \right ) ^{{\frac{3}{2}}}} \]

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^10,x)

[Out]

1/18*((b*x^3+a)^2)^(3/2)*(18*b^3*ln(x)*x^9-18*a*x^6*b^2-9*x^3*a^2*b-2*a^3)/(b*x^
3+a)^3/x^9

_______________________________________________________________________________________

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^10,x, algorithm="maxima")

[Out]

Exception raised: ValueError

_______________________________________________________________________________________

Fricas [A]  time = 0.256578, size = 53, normalized size = 0.33 \[ \frac{18 \, b^{3} x^{9} \log \left (x\right ) - 18 \, a b^{2} x^{6} - 9 \, a^{2} b x^{3} - 2 \, a^{3}}{18 \, x^{9}} \]

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^10,x, algorithm="fricas")

[Out]

1/18*(18*b^3*x^9*log(x) - 18*a*b^2*x^6 - 9*a^2*b*x^3 - 2*a^3)/x^9

_______________________________________________________________________________________

Sympy [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{\left (\left (a + b x^{3}\right )^{2}\right )^{\frac{3}{2}}}{x^{10}}\, dx \]

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**10,x)

[Out]

Integral(((a + b*x**3)**2)**(3/2)/x**10, x)

_______________________________________________________________________________________

GIAC/XCAS [A]  time = 0.269732, size = 115, normalized size = 0.71 \[ b^{3}{\rm ln}\left ({\left | x \right |}\right ){\rm sign}\left (b x^{3} + a\right ) - \frac{11 \, b^{3} x^{9}{\rm sign}\left (b x^{3} + a\right ) + 18 \, a b^{2} x^{6}{\rm sign}\left (b x^{3} + a\right ) + 9 \, a^{2} b x^{3}{\rm sign}\left (b x^{3} + a\right ) + 2 \, a^{3}{\rm sign}\left (b x^{3} + a\right )}{18 \, x^{9}} \]

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^10,x, algorithm="giac")

[Out]

b^3*ln(abs(x))*sign(b*x^3 + a) - 1/18*(11*b^3*x^9*sign(b*x^3 + a) + 18*a*b^2*x^6
*sign(b*x^3 + a) + 9*a^2*b*x^3*sign(b*x^3 + a) + 2*a^3*sign(b*x^3 + a))/x^9