Optimal. Leaf size=58 \[ -\frac{b^{5/2} \tan ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right )}{a^{7/2}}-\frac{b^2}{a^3 x}+\frac{b}{3 a^2 x^3}-\frac{1}{5 a x^5} \]
[Out]
_______________________________________________________________________________________
Rubi [A] time = 0.0754309, antiderivative size = 58, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 2, integrand size = 13, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.154 \[ -\frac{b^{5/2} \tan ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right )}{a^{7/2}}-\frac{b^2}{a^3 x}+\frac{b}{3 a^2 x^3}-\frac{1}{5 a x^5} \]
Antiderivative was successfully verified.
[In] Int[1/(x^6*(a + b*x^2)),x]
[Out]
_______________________________________________________________________________________
Rubi in Sympy [A] time = 14.989, size = 49, normalized size = 0.84 \[ - \frac{1}{5 a x^{5}} + \frac{b}{3 a^{2} x^{3}} - \frac{b^{2}}{a^{3} x} - \frac{b^{\frac{5}{2}} \operatorname{atan}{\left (\frac{\sqrt{b} x}{\sqrt{a}} \right )}}{a^{\frac{7}{2}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate(1/x**6/(b*x**2+a),x)
[Out]
_______________________________________________________________________________________
Mathematica [A] time = 0.0455841, size = 58, normalized size = 1. \[ -\frac{b^{5/2} \tan ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right )}{a^{7/2}}-\frac{b^2}{a^3 x}+\frac{b}{3 a^2 x^3}-\frac{1}{5 a x^5} \]
Antiderivative was successfully verified.
[In] Integrate[1/(x^6*(a + b*x^2)),x]
[Out]
_______________________________________________________________________________________
Maple [A] time = 0.009, size = 52, normalized size = 0.9 \[ -{\frac{1}{5\,a{x}^{5}}}-{\frac{{b}^{2}}{{a}^{3}x}}+{\frac{b}{3\,{a}^{2}{x}^{3}}}-{\frac{{b}^{3}}{{a}^{3}}\arctan \left ({bx{\frac{1}{\sqrt{ab}}}} \right ){\frac{1}{\sqrt{ab}}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int(1/x^6/(b*x^2+a),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(1/((b*x^2 + a)*x^6),x, algorithm="maxima")
[Out]
_______________________________________________________________________________________
Fricas [A] time = 0.207747, size = 1, normalized size = 0.02 \[ \left [\frac{15 \, b^{2} x^{5} \sqrt{-\frac{b}{a}} \log \left (\frac{b x^{2} - 2 \, a x \sqrt{-\frac{b}{a}} - a}{b x^{2} + a}\right ) - 30 \, b^{2} x^{4} + 10 \, a b x^{2} - 6 \, a^{2}}{30 \, a^{3} x^{5}}, -\frac{15 \, b^{2} x^{5} \sqrt{\frac{b}{a}} \arctan \left (\frac{b x}{a \sqrt{\frac{b}{a}}}\right ) + 15 \, b^{2} x^{4} - 5 \, a b x^{2} + 3 \, a^{2}}{15 \, a^{3} x^{5}}\right ] \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(1/((b*x^2 + a)*x^6),x, algorithm="fricas")
[Out]
_______________________________________________________________________________________
Sympy [A] time = 1.74768, size = 100, normalized size = 1.72 \[ \frac{\sqrt{- \frac{b^{5}}{a^{7}}} \log{\left (- \frac{a^{4} \sqrt{- \frac{b^{5}}{a^{7}}}}{b^{3}} + x \right )}}{2} - \frac{\sqrt{- \frac{b^{5}}{a^{7}}} \log{\left (\frac{a^{4} \sqrt{- \frac{b^{5}}{a^{7}}}}{b^{3}} + x \right )}}{2} - \frac{3 a^{2} - 5 a b x^{2} + 15 b^{2} x^{4}}{15 a^{3} x^{5}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(1/x**6/(b*x**2+a),x)
[Out]
_______________________________________________________________________________________
GIAC/XCAS [A] time = 0.207905, size = 70, normalized size = 1.21 \[ -\frac{b^{3} \arctan \left (\frac{b x}{\sqrt{a b}}\right )}{\sqrt{a b} a^{3}} - \frac{15 \, b^{2} x^{4} - 5 \, a b x^{2} + 3 \, a^{2}}{15 \, a^{3} x^{5}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(1/((b*x^2 + a)*x^6),x, algorithm="giac")
[Out]