3.106 \(\int \frac{1}{x^4 \left (a^2+2 a b x^3+b^2 x^6\right )^{3/2}} \, dx\)

Optimal. Leaf size=188 \[ -\frac{b}{6 a^2 \left (a+b x^3\right ) \sqrt{a^2+2 a b x^3+b^2 x^6}}-\frac{3 b \log (x) \left (a+b x^3\right )}{a^4 \sqrt{a^2+2 a b x^3+b^2 x^6}}+\frac{b \left (a+b x^3\right ) \log \left (a+b x^3\right )}{a^4 \sqrt{a^2+2 a b x^3+b^2 x^6}}-\frac{2 b}{3 a^3 \sqrt{a^2+2 a b x^3+b^2 x^6}}-\frac{a+b x^3}{3 a^3 x^3 \sqrt{a^2+2 a b x^3+b^2 x^6}} \]

[Out]

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

_______________________________________________________________________________________

Rubi [A]  time = 0.214629, antiderivative size = 188, 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 \[ -\frac{b}{6 a^2 \left (a+b x^3\right ) \sqrt{a^2+2 a b x^3+b^2 x^6}}-\frac{3 b \log (x) \left (a+b x^3\right )}{a^4 \sqrt{a^2+2 a b x^3+b^2 x^6}}+\frac{b \left (a+b x^3\right ) \log \left (a+b x^3\right )}{a^4 \sqrt{a^2+2 a b x^3+b^2 x^6}}-\frac{2 b}{3 a^3 \sqrt{a^2+2 a b x^3+b^2 x^6}}-\frac{a+b x^3}{3 a^3 x^3 \sqrt{a^2+2 a b x^3+b^2 x^6}} \]

Antiderivative was successfully verified.

[In]  Int[1/(x^4*(a^2 + 2*a*b*x^3 + b^2*x^6)^(3/2)),x]

[Out]

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

_______________________________________________________________________________________

Rubi in Sympy [A]  time = 35.7636, size = 180, normalized size = 0.96 \[ \frac{2 a + 2 b x^{3}}{12 a x^{3} \left (a^{2} + 2 a b x^{3} + b^{2} x^{6}\right )^{\frac{3}{2}}} + \frac{1}{2 a^{2} x^{3} \sqrt{a^{2} + 2 a b x^{3} + b^{2} x^{6}}} - \frac{b \sqrt{a^{2} + 2 a b x^{3} + b^{2} x^{6}} \log{\left (x^{3} \right )}}{a^{4} \left (a + b x^{3}\right )} + \frac{b \sqrt{a^{2} + 2 a b x^{3} + b^{2} x^{6}} \log{\left (a + b x^{3} \right )}}{a^{4} \left (a + b x^{3}\right )} - \frac{\sqrt{a^{2} + 2 a b x^{3} + b^{2} x^{6}}}{a^{4} x^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(1/x**4/(b**2*x**6+2*a*b*x**3+a**2)**(3/2),x)

[Out]

(2*a + 2*b*x**3)/(12*a*x**3*(a**2 + 2*a*b*x**3 + b**2*x**6)**(3/2)) + 1/(2*a**2*
x**3*sqrt(a**2 + 2*a*b*x**3 + b**2*x**6)) - b*sqrt(a**2 + 2*a*b*x**3 + b**2*x**6
)*log(x**3)/(a**4*(a + b*x**3)) + b*sqrt(a**2 + 2*a*b*x**3 + b**2*x**6)*log(a +
b*x**3)/(a**4*(a + b*x**3)) - sqrt(a**2 + 2*a*b*x**3 + b**2*x**6)/(a**4*x**3)

_______________________________________________________________________________________

Mathematica [A]  time = 0.0613676, size = 97, normalized size = 0.52 \[ \frac{-a \left (2 a^2+9 a b x^3+6 b^2 x^6\right )-18 b x^3 \log (x) \left (a+b x^3\right )^2+6 b x^3 \left (a+b x^3\right )^2 \log \left (a+b x^3\right )}{6 a^4 x^3 \left (a+b x^3\right ) \sqrt{\left (a+b x^3\right )^2}} \]

Antiderivative was successfully verified.

[In]  Integrate[1/(x^4*(a^2 + 2*a*b*x^3 + b^2*x^6)^(3/2)),x]

[Out]

(-(a*(2*a^2 + 9*a*b*x^3 + 6*b^2*x^6)) - 18*b*x^3*(a + b*x^3)^2*Log[x] + 6*b*x^3*
(a + b*x^3)^2*Log[a + b*x^3])/(6*a^4*x^3*(a + b*x^3)*Sqrt[(a + b*x^3)^2])

_______________________________________________________________________________________

Maple [A]  time = 0.023, size = 133, normalized size = 0.7 \[{\frac{ \left ( 6\,\ln \left ( b{x}^{3}+a \right ){x}^{9}{b}^{3}-18\,{b}^{3}\ln \left ( x \right ){x}^{9}+12\,\ln \left ( b{x}^{3}+a \right ){x}^{6}a{b}^{2}-36\,a{b}^{2}\ln \left ( x \right ){x}^{6}-6\,a{x}^{6}{b}^{2}+6\,\ln \left ( b{x}^{3}+a \right ){x}^{3}{a}^{2}b-18\,{a}^{2}b\ln \left ( x \right ){x}^{3}-9\,{x}^{3}{a}^{2}b-2\,{a}^{3} \right ) \left ( b{x}^{3}+a \right ) }{6\,{x}^{3}{a}^{4}} \left ( \left ( b{x}^{3}+a \right ) ^{2} \right ) ^{-{\frac{3}{2}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(1/x^4/(b^2*x^6+2*a*b*x^3+a^2)^(3/2),x)

[Out]

1/6*(6*ln(b*x^3+a)*x^9*b^3-18*b^3*ln(x)*x^9+12*ln(b*x^3+a)*x^6*a*b^2-36*a*b^2*ln
(x)*x^6-6*a*x^6*b^2+6*ln(b*x^3+a)*x^3*a^2*b-18*a^2*b*ln(x)*x^3-9*x^3*a^2*b-2*a^3
)*(b*x^3+a)/x^3/a^4/((b*x^3+a)^2)^(3/2)

_______________________________________________________________________________________

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(1/((b^2*x^6 + 2*a*b*x^3 + a^2)^(3/2)*x^4),x, algorithm="maxima")

[Out]

Exception raised: ValueError

_______________________________________________________________________________________

Fricas [A]  time = 0.274706, size = 161, normalized size = 0.86 \[ -\frac{6 \, a b^{2} x^{6} + 9 \, a^{2} b x^{3} + 2 \, a^{3} - 6 \,{\left (b^{3} x^{9} + 2 \, a b^{2} x^{6} + a^{2} b x^{3}\right )} \log \left (b x^{3} + a\right ) + 18 \,{\left (b^{3} x^{9} + 2 \, a b^{2} x^{6} + a^{2} b x^{3}\right )} \log \left (x\right )}{6 \,{\left (a^{4} b^{2} x^{9} + 2 \, a^{5} b x^{6} + a^{6} x^{3}\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/((b^2*x^6 + 2*a*b*x^3 + a^2)^(3/2)*x^4),x, algorithm="fricas")

[Out]

-1/6*(6*a*b^2*x^6 + 9*a^2*b*x^3 + 2*a^3 - 6*(b^3*x^9 + 2*a*b^2*x^6 + a^2*b*x^3)*
log(b*x^3 + a) + 18*(b^3*x^9 + 2*a*b^2*x^6 + a^2*b*x^3)*log(x))/(a^4*b^2*x^9 + 2
*a^5*b*x^6 + a^6*x^3)

_______________________________________________________________________________________

Sympy [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{1}{x^{4} \left (\left (a + b x^{3}\right )^{2}\right )^{\frac{3}{2}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/x**4/(b**2*x**6+2*a*b*x**3+a**2)**(3/2),x)

[Out]

Integral(1/(x**4*((a + b*x**3)**2)**(3/2)), x)

_______________________________________________________________________________________

GIAC/XCAS [A]  time = 0.69738, size = 4, normalized size = 0.02 \[ \mathit{sage}_{0} x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/((b^2*x^6 + 2*a*b*x^3 + a^2)^(3/2)*x^4),x, algorithm="giac")

[Out]

sage0*x