3.176 \(\int \frac{x^8}{b x^2+c x^4} \, dx\)

Optimal. Leaf size=55 \[ -\frac{b^{5/2} \tan ^{-1}\left (\frac{\sqrt{c} x}{\sqrt{b}}\right )}{c^{7/2}}+\frac{b^2 x}{c^3}-\frac{b x^3}{3 c^2}+\frac{x^5}{5 c} \]

[Out]

(b^2*x)/c^3 - (b*x^3)/(3*c^2) + x^5/(5*c) - (b^(5/2)*ArcTan[(Sqrt[c]*x)/Sqrt[b]]
)/c^(7/2)

_______________________________________________________________________________________

Rubi [A]  time = 0.0725561, antiderivative size = 55, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 17, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.176 \[ -\frac{b^{5/2} \tan ^{-1}\left (\frac{\sqrt{c} x}{\sqrt{b}}\right )}{c^{7/2}}+\frac{b^2 x}{c^3}-\frac{b x^3}{3 c^2}+\frac{x^5}{5 c} \]

Antiderivative was successfully verified.

[In]  Int[x^8/(b*x^2 + c*x^4),x]

[Out]

(b^2*x)/c^3 - (b*x^3)/(3*c^2) + x^5/(5*c) - (b^(5/2)*ArcTan[(Sqrt[c]*x)/Sqrt[b]]
)/c^(7/2)

_______________________________________________________________________________________

Rubi in Sympy [F]  time = 0., size = 0, normalized size = 0. \[ - \frac{b^{\frac{5}{2}} \operatorname{atan}{\left (\frac{\sqrt{c} x}{\sqrt{b}} \right )}}{c^{\frac{7}{2}}} - \frac{b x^{3}}{3 c^{2}} + \frac{x^{5}}{5 c} + \frac{\int b^{2}\, dx}{c^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(x**8/(c*x**4+b*x**2),x)

[Out]

-b**(5/2)*atan(sqrt(c)*x/sqrt(b))/c**(7/2) - b*x**3/(3*c**2) + x**5/(5*c) + Inte
gral(b**2, x)/c**3

_______________________________________________________________________________________

Mathematica [A]  time = 0.0432015, size = 55, normalized size = 1. \[ -\frac{b^{5/2} \tan ^{-1}\left (\frac{\sqrt{c} x}{\sqrt{b}}\right )}{c^{7/2}}+\frac{b^2 x}{c^3}-\frac{b x^3}{3 c^2}+\frac{x^5}{5 c} \]

Antiderivative was successfully verified.

[In]  Integrate[x^8/(b*x^2 + c*x^4),x]

[Out]

(b^2*x)/c^3 - (b*x^3)/(3*c^2) + x^5/(5*c) - (b^(5/2)*ArcTan[(Sqrt[c]*x)/Sqrt[b]]
)/c^(7/2)

_______________________________________________________________________________________

Maple [A]  time = 0.004, size = 49, normalized size = 0.9 \[{\frac{{x}^{5}}{5\,c}}-{\frac{b{x}^{3}}{3\,{c}^{2}}}+{\frac{{b}^{2}x}{{c}^{3}}}-{\frac{{b}^{3}}{{c}^{3}}\arctan \left ({cx{\frac{1}{\sqrt{bc}}}} \right ){\frac{1}{\sqrt{bc}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x^8/(c*x^4+b*x^2),x)

[Out]

1/5*x^5/c-1/3*b*x^3/c^2+b^2*x/c^3-b^3/c^3/(b*c)^(1/2)*arctan(c*x/(b*c)^(1/2))

_______________________________________________________________________________________

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(x^8/(c*x^4 + b*x^2),x, algorithm="maxima")

[Out]

Exception raised: ValueError

_______________________________________________________________________________________

Fricas [A]  time = 0.263321, size = 1, normalized size = 0.02 \[ \left [\frac{6 \, c^{2} x^{5} - 10 \, b c x^{3} + 15 \, b^{2} \sqrt{-\frac{b}{c}} \log \left (\frac{c x^{2} - 2 \, c x \sqrt{-\frac{b}{c}} - b}{c x^{2} + b}\right ) + 30 \, b^{2} x}{30 \, c^{3}}, \frac{3 \, c^{2} x^{5} - 5 \, b c x^{3} - 15 \, b^{2} \sqrt{\frac{b}{c}} \arctan \left (\frac{x}{\sqrt{\frac{b}{c}}}\right ) + 15 \, b^{2} x}{15 \, c^{3}}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x^8/(c*x^4 + b*x^2),x, algorithm="fricas")

[Out]

[1/30*(6*c^2*x^5 - 10*b*c*x^3 + 15*b^2*sqrt(-b/c)*log((c*x^2 - 2*c*x*sqrt(-b/c)
- b)/(c*x^2 + b)) + 30*b^2*x)/c^3, 1/15*(3*c^2*x^5 - 5*b*c*x^3 - 15*b^2*sqrt(b/c
)*arctan(x/sqrt(b/c)) + 15*b^2*x)/c^3]

_______________________________________________________________________________________

Sympy [A]  time = 1.30234, size = 95, normalized size = 1.73 \[ \frac{b^{2} x}{c^{3}} - \frac{b x^{3}}{3 c^{2}} + \frac{\sqrt{- \frac{b^{5}}{c^{7}}} \log{\left (x - \frac{c^{3} \sqrt{- \frac{b^{5}}{c^{7}}}}{b^{2}} \right )}}{2} - \frac{\sqrt{- \frac{b^{5}}{c^{7}}} \log{\left (x + \frac{c^{3} \sqrt{- \frac{b^{5}}{c^{7}}}}{b^{2}} \right )}}{2} + \frac{x^{5}}{5 c} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x**8/(c*x**4+b*x**2),x)

[Out]

b**2*x/c**3 - b*x**3/(3*c**2) + sqrt(-b**5/c**7)*log(x - c**3*sqrt(-b**5/c**7)/b
**2)/2 - sqrt(-b**5/c**7)*log(x + c**3*sqrt(-b**5/c**7)/b**2)/2 + x**5/(5*c)

_______________________________________________________________________________________

GIAC/XCAS [A]  time = 0.268974, size = 74, normalized size = 1.35 \[ -\frac{b^{3} \arctan \left (\frac{c x}{\sqrt{b c}}\right )}{\sqrt{b c} c^{3}} + \frac{3 \, c^{4} x^{5} - 5 \, b c^{3} x^{3} + 15 \, b^{2} c^{2} x}{15 \, c^{5}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x^8/(c*x^4 + b*x^2),x, algorithm="giac")

[Out]

-b^3*arctan(c*x/sqrt(b*c))/(sqrt(b*c)*c^3) + 1/15*(3*c^4*x^5 - 5*b*c^3*x^3 + 15*
b^2*c^2*x)/c^5