Optimal. Leaf size=42 \[ \frac{b^{3/2} \tan ^{-1}\left (\frac{\sqrt{c} x}{\sqrt{b}}\right )}{c^{5/2}}-\frac{b x}{c^2}+\frac{x^3}{3 c} \]
[Out]
_______________________________________________________________________________________
Rubi [A] time = 0.0619986, antiderivative size = 42, 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 \[ \frac{b^{3/2} \tan ^{-1}\left (\frac{\sqrt{c} x}{\sqrt{b}}\right )}{c^{5/2}}-\frac{b x}{c^2}+\frac{x^3}{3 c} \]
Antiderivative was successfully verified.
[In] Int[x^6/(b*x^2 + c*x^4),x]
[Out]
_______________________________________________________________________________________
Rubi in Sympy [F] time = 0., size = 0, normalized size = 0. \[ \frac{b^{\frac{3}{2}} \operatorname{atan}{\left (\frac{\sqrt{c} x}{\sqrt{b}} \right )}}{c^{\frac{5}{2}}} + \frac{x^{3}}{3 c} - \frac{\int b\, dx}{c^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate(x**6/(c*x**4+b*x**2),x)
[Out]
_______________________________________________________________________________________
Mathematica [A] time = 0.032989, size = 42, normalized size = 1. \[ \frac{b^{3/2} \tan ^{-1}\left (\frac{\sqrt{c} x}{\sqrt{b}}\right )}{c^{5/2}}-\frac{b x}{c^2}+\frac{x^3}{3 c} \]
Antiderivative was successfully verified.
[In] Integrate[x^6/(b*x^2 + c*x^4),x]
[Out]
_______________________________________________________________________________________
Maple [A] time = 0.003, size = 38, normalized size = 0.9 \[{\frac{{x}^{3}}{3\,c}}-{\frac{bx}{{c}^{2}}}+{\frac{{b}^{2}}{{c}^{2}}\arctan \left ({cx{\frac{1}{\sqrt{bc}}}} \right ){\frac{1}{\sqrt{bc}}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int(x^6/(c*x^4+b*x^2),x)
[Out]
_______________________________________________________________________________________
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^6/(c*x^4 + b*x^2),x, algorithm="maxima")
[Out]
_______________________________________________________________________________________
Fricas [A] time = 0.258052, size = 1, normalized size = 0.02 \[ \left [\frac{2 \, c x^{3} + 3 \, b \sqrt{-\frac{b}{c}} \log \left (\frac{c x^{2} + 2 \, c x \sqrt{-\frac{b}{c}} - b}{c x^{2} + b}\right ) - 6 \, b x}{6 \, c^{2}}, \frac{c x^{3} + 3 \, b \sqrt{\frac{b}{c}} \arctan \left (\frac{x}{\sqrt{\frac{b}{c}}}\right ) - 3 \, b x}{3 \, c^{2}}\right ] \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(x^6/(c*x^4 + b*x^2),x, algorithm="fricas")
[Out]
_______________________________________________________________________________________
Sympy [A] time = 1.3132, size = 80, normalized size = 1.9 \[ - \frac{b x}{c^{2}} - \frac{\sqrt{- \frac{b^{3}}{c^{5}}} \log{\left (x - \frac{c^{2} \sqrt{- \frac{b^{3}}{c^{5}}}}{b} \right )}}{2} + \frac{\sqrt{- \frac{b^{3}}{c^{5}}} \log{\left (x + \frac{c^{2} \sqrt{- \frac{b^{3}}{c^{5}}}}{b} \right )}}{2} + \frac{x^{3}}{3 c} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(x**6/(c*x**4+b*x**2),x)
[Out]
_______________________________________________________________________________________
GIAC/XCAS [A] time = 0.269771, size = 54, normalized size = 1.29 \[ \frac{b^{2} \arctan \left (\frac{c x}{\sqrt{b c}}\right )}{\sqrt{b c} c^{2}} + \frac{c^{2} x^{3} - 3 \, b c x}{3 \, c^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(x^6/(c*x^4 + b*x^2),x, algorithm="giac")
[Out]