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

Optimal. Leaf size=142 \[ \frac{b \tanh ^{-1}\left (\frac{2 a+b x^3}{2 \sqrt{a} \sqrt{a+b x^3+c x^6}}\right )}{2 a^{5/2}}-\frac{\left (3 b^2-8 a c\right ) \sqrt{a+b x^3+c x^6}}{3 a^2 x^3 \left (b^2-4 a c\right )}+\frac{2 \left (-2 a c+b^2+b c x^3\right )}{3 a x^3 \left (b^2-4 a c\right ) \sqrt{a+b x^3+c x^6}} \]

[Out]

(2*(b^2 - 2*a*c + b*c*x^3))/(3*a*(b^2 - 4*a*c)*x^3*Sqrt[a + b*x^3 + c*x^6]) - ((
3*b^2 - 8*a*c)*Sqrt[a + b*x^3 + c*x^6])/(3*a^2*(b^2 - 4*a*c)*x^3) + (b*ArcTanh[(
2*a + b*x^3)/(2*Sqrt[a]*Sqrt[a + b*x^3 + c*x^6])])/(2*a^(5/2))

_______________________________________________________________________________________

Rubi [A]  time = 0.306353, antiderivative size = 142, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 5, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.25 \[ \frac{b \tanh ^{-1}\left (\frac{2 a+b x^3}{2 \sqrt{a} \sqrt{a+b x^3+c x^6}}\right )}{2 a^{5/2}}-\frac{\left (3 b^2-8 a c\right ) \sqrt{a+b x^3+c x^6}}{3 a^2 x^3 \left (b^2-4 a c\right )}+\frac{2 \left (-2 a c+b^2+b c x^3\right )}{3 a x^3 \left (b^2-4 a c\right ) \sqrt{a+b x^3+c x^6}} \]

Antiderivative was successfully verified.

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

[Out]

(2*(b^2 - 2*a*c + b*c*x^3))/(3*a*(b^2 - 4*a*c)*x^3*Sqrt[a + b*x^3 + c*x^6]) - ((
3*b^2 - 8*a*c)*Sqrt[a + b*x^3 + c*x^6])/(3*a^2*(b^2 - 4*a*c)*x^3) + (b*ArcTanh[(
2*a + b*x^3)/(2*Sqrt[a]*Sqrt[a + b*x^3 + c*x^6])])/(2*a^(5/2))

_______________________________________________________________________________________

Rubi in Sympy [A]  time = 37.5344, size = 131, normalized size = 0.92 \[ \frac{2 \left (- 2 a c + b^{2} + b c x^{3}\right )}{3 a x^{3} \left (- 4 a c + b^{2}\right ) \sqrt{a + b x^{3} + c x^{6}}} - \frac{\left (- 8 a c + 3 b^{2}\right ) \sqrt{a + b x^{3} + c x^{6}}}{3 a^{2} x^{3} \left (- 4 a c + b^{2}\right )} + \frac{b \operatorname{atanh}{\left (\frac{2 a + b x^{3}}{2 \sqrt{a} \sqrt{a + b x^{3} + c x^{6}}} \right )}}{2 a^{\frac{5}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

2*(-2*a*c + b**2 + b*c*x**3)/(3*a*x**3*(-4*a*c + b**2)*sqrt(a + b*x**3 + c*x**6)
) - (-8*a*c + 3*b**2)*sqrt(a + b*x**3 + c*x**6)/(3*a**2*x**3*(-4*a*c + b**2)) +
b*atanh((2*a + b*x**3)/(2*sqrt(a)*sqrt(a + b*x**3 + c*x**6)))/(2*a**(5/2))

_______________________________________________________________________________________

Mathematica [A]  time = 0.220457, size = 132, normalized size = 0.93 \[ \frac{b \left (\log \left (2 \sqrt{a} \sqrt{a+b x^3+c x^6}+2 a+b x^3\right )-\log \left (x^3\right )\right )}{2 a^{5/2}}+\frac{-4 a^2 c+a \left (b^2-10 b c x^3-8 c^2 x^6\right )+3 b^2 x^3 \left (b+c x^3\right )}{3 a^2 x^3 \left (4 a c-b^2\right ) \sqrt{a+b x^3+c x^6}} \]

Antiderivative was successfully verified.

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

[Out]

(-4*a^2*c + 3*b^2*x^3*(b + c*x^3) + a*(b^2 - 10*b*c*x^3 - 8*c^2*x^6))/(3*a^2*(-b
^2 + 4*a*c)*x^3*Sqrt[a + b*x^3 + c*x^6]) + (b*(-Log[x^3] + Log[2*a + b*x^3 + 2*S
qrt[a]*Sqrt[a + b*x^3 + c*x^6]]))/(2*a^(5/2))

_______________________________________________________________________________________

Maple [F]  time = 0.062, size = 0, normalized size = 0. \[ \int{\frac{1}{{x}^{4}} \left ( c{x}^{6}+b{x}^{3}+a \right ) ^{-{\frac{3}{2}}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

_______________________________________________________________________________________

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

[Out]

Exception raised: ValueError

_______________________________________________________________________________________

Fricas [A]  time = 0.302067, size = 1, normalized size = 0.01 \[ \left [-\frac{4 \,{\left ({\left (3 \, b^{2} c - 8 \, a c^{2}\right )} x^{6} +{\left (3 \, b^{3} - 10 \, a b c\right )} x^{3} + a b^{2} - 4 \, a^{2} c\right )} \sqrt{c x^{6} + b x^{3} + a} \sqrt{a} - 3 \,{\left ({\left (b^{3} c - 4 \, a b c^{2}\right )} x^{9} +{\left (b^{4} - 4 \, a b^{2} c\right )} x^{6} +{\left (a b^{3} - 4 \, a^{2} b c\right )} x^{3}\right )} \log \left (-\frac{4 \, \sqrt{c x^{6} + b x^{3} + a}{\left (a b x^{3} + 2 \, a^{2}\right )} +{\left ({\left (b^{2} + 4 \, a c\right )} x^{6} + 8 \, a b x^{3} + 8 \, a^{2}\right )} \sqrt{a}}{x^{6}}\right )}{12 \,{\left ({\left (a^{2} b^{2} c - 4 \, a^{3} c^{2}\right )} x^{9} +{\left (a^{2} b^{3} - 4 \, a^{3} b c\right )} x^{6} +{\left (a^{3} b^{2} - 4 \, a^{4} c\right )} x^{3}\right )} \sqrt{a}}, -\frac{2 \,{\left ({\left (3 \, b^{2} c - 8 \, a c^{2}\right )} x^{6} +{\left (3 \, b^{3} - 10 \, a b c\right )} x^{3} + a b^{2} - 4 \, a^{2} c\right )} \sqrt{c x^{6} + b x^{3} + a} \sqrt{-a} - 3 \,{\left ({\left (b^{3} c - 4 \, a b c^{2}\right )} x^{9} +{\left (b^{4} - 4 \, a b^{2} c\right )} x^{6} +{\left (a b^{3} - 4 \, a^{2} b c\right )} x^{3}\right )} \arctan \left (\frac{{\left (b x^{3} + 2 \, a\right )} \sqrt{-a}}{2 \, \sqrt{c x^{6} + b x^{3} + a} a}\right )}{6 \,{\left ({\left (a^{2} b^{2} c - 4 \, a^{3} c^{2}\right )} x^{9} +{\left (a^{2} b^{3} - 4 \, a^{3} b c\right )} x^{6} +{\left (a^{3} b^{2} - 4 \, a^{4} c\right )} x^{3}\right )} \sqrt{-a}}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

[-1/12*(4*((3*b^2*c - 8*a*c^2)*x^6 + (3*b^3 - 10*a*b*c)*x^3 + a*b^2 - 4*a^2*c)*s
qrt(c*x^6 + b*x^3 + a)*sqrt(a) - 3*((b^3*c - 4*a*b*c^2)*x^9 + (b^4 - 4*a*b^2*c)*
x^6 + (a*b^3 - 4*a^2*b*c)*x^3)*log(-(4*sqrt(c*x^6 + b*x^3 + a)*(a*b*x^3 + 2*a^2)
+ ((b^2 + 4*a*c)*x^6 + 8*a*b*x^3 + 8*a^2)*sqrt(a))/x^6))/(((a^2*b^2*c - 4*a^3*c
^2)*x^9 + (a^2*b^3 - 4*a^3*b*c)*x^6 + (a^3*b^2 - 4*a^4*c)*x^3)*sqrt(a)), -1/6*(2
*((3*b^2*c - 8*a*c^2)*x^6 + (3*b^3 - 10*a*b*c)*x^3 + a*b^2 - 4*a^2*c)*sqrt(c*x^6
+ b*x^3 + a)*sqrt(-a) - 3*((b^3*c - 4*a*b*c^2)*x^9 + (b^4 - 4*a*b^2*c)*x^6 + (a
*b^3 - 4*a^2*b*c)*x^3)*arctan(1/2*(b*x^3 + 2*a)*sqrt(-a)/(sqrt(c*x^6 + b*x^3 + a
)*a)))/(((a^2*b^2*c - 4*a^3*c^2)*x^9 + (a^2*b^3 - 4*a^3*b*c)*x^6 + (a^3*b^2 - 4*
a^4*c)*x^3)*sqrt(-a))]

_______________________________________________________________________________________

Sympy [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{1}{x^{4} \left (a + b x^{3} + c x^{6}\right )^{\frac{3}{2}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Integral(1/(x**4*(a + b*x**3 + c*x**6)**(3/2)), x)

_______________________________________________________________________________________

GIAC/XCAS [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{1}{{\left (c x^{6} + b x^{3} + a\right )}^{\frac{3}{2}} x^{4}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

integrate(1/((c*x^6 + b*x^3 + a)^(3/2)*x^4), x)