Optimal. Leaf size=93 \[ \frac{c \tan ^{-1}\left (\frac{x^3 \sqrt{b c-a d}}{\sqrt{a} \sqrt{c+d x^6}}\right )}{6 \sqrt{a} (b c-a d)^{3/2}}-\frac{x^3 \sqrt{c+d x^6}}{6 \left (a+b x^6\right ) (b c-a d)} \]
[Out]
_______________________________________________________________________________________
Rubi [A] time = 0.257695, antiderivative size = 93, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 5, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.208 \[ \frac{c \tan ^{-1}\left (\frac{x^3 \sqrt{b c-a d}}{\sqrt{a} \sqrt{c+d x^6}}\right )}{6 \sqrt{a} (b c-a d)^{3/2}}-\frac{x^3 \sqrt{c+d x^6}}{6 \left (a+b x^6\right ) (b c-a d)} \]
Antiderivative was successfully verified.
[In] Int[x^8/((a + b*x^6)^2*Sqrt[c + d*x^6]),x]
[Out]
_______________________________________________________________________________________
Rubi in Sympy [A] time = 29.5023, size = 76, normalized size = 0.82 \[ \frac{x^{3} \sqrt{c + d x^{6}}}{6 \left (a + b x^{6}\right ) \left (a d - b c\right )} - \frac{c \operatorname{atanh}{\left (\frac{x^{3} \sqrt{a d - b c}}{\sqrt{a} \sqrt{c + d x^{6}}} \right )}}{6 \sqrt{a} \left (a d - b c\right )^{\frac{3}{2}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate(x**8/(b*x**6+a)**2/(d*x**6+c)**(1/2),x)
[Out]
_______________________________________________________________________________________
Mathematica [A] time = 0.174096, size = 90, normalized size = 0.97 \[ \frac{\frac{x^3 \sqrt{c+d x^6}}{a+b x^6}-\frac{c \tan ^{-1}\left (\frac{x^3 \sqrt{b c-a d}}{\sqrt{a} \sqrt{c+d x^6}}\right )}{\sqrt{a} \sqrt{b c-a d}}}{6 a d-6 b c} \]
Antiderivative was successfully verified.
[In] Integrate[x^8/((a + b*x^6)^2*Sqrt[c + d*x^6]),x]
[Out]
_______________________________________________________________________________________
Maple [F] time = 0.074, size = 0, normalized size = 0. \[ \int{\frac{{x}^{8}}{ \left ( b{x}^{6}+a \right ) ^{2}}{\frac{1}{\sqrt{d{x}^{6}+c}}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int(x^8/(b*x^6+a)^2/(d*x^6+c)^(1/2),x)
[Out]
_______________________________________________________________________________________
Maxima [F] time = 0., size = 0, normalized size = 0. \[ \int \frac{x^{8}}{{\left (b x^{6} + a\right )}^{2} \sqrt{d x^{6} + c}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(x^8/((b*x^6 + a)^2*sqrt(d*x^6 + c)),x, algorithm="maxima")
[Out]
_______________________________________________________________________________________
Fricas [A] time = 0.321661, size = 1, normalized size = 0.01 \[ \left [-\frac{4 \, \sqrt{d x^{6} + c} \sqrt{-a b c + a^{2} d} x^{3} +{\left (b c x^{6} + a c\right )} \log \left (-\frac{4 \,{\left ({\left (a b^{2} c^{2} - 3 \, a^{2} b c d + 2 \, a^{3} d^{2}\right )} x^{9} -{\left (a^{2} b c^{2} - a^{3} c d\right )} x^{3}\right )} \sqrt{d x^{6} + c} -{\left ({\left (b^{2} c^{2} - 8 \, a b c d + 8 \, a^{2} d^{2}\right )} x^{12} - 2 \,{\left (3 \, a b c^{2} - 4 \, a^{2} c d\right )} x^{6} + a^{2} c^{2}\right )} \sqrt{-a b c + a^{2} d}}{b^{2} x^{12} + 2 \, a b x^{6} + a^{2}}\right )}{24 \,{\left ({\left (b^{2} c - a b d\right )} x^{6} + a b c - a^{2} d\right )} \sqrt{-a b c + a^{2} d}}, -\frac{2 \, \sqrt{d x^{6} + c} \sqrt{a b c - a^{2} d} x^{3} -{\left (b c x^{6} + a c\right )} \arctan \left (\frac{{\left (b c - 2 \, a d\right )} x^{6} - a c}{2 \, \sqrt{d x^{6} + c} \sqrt{a b c - a^{2} d} x^{3}}\right )}{12 \,{\left ({\left (b^{2} c - a b d\right )} x^{6} + a b c - a^{2} d\right )} \sqrt{a b c - a^{2} d}}\right ] \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(x^8/((b*x^6 + a)^2*sqrt(d*x^6 + c)),x, algorithm="fricas")
[Out]
_______________________________________________________________________________________
Sympy [F(-1)] time = 0., size = 0, normalized size = 0. \[ \text{Timed out} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(x**8/(b*x**6+a)**2/(d*x**6+c)**(1/2),x)
[Out]
_______________________________________________________________________________________
GIAC/XCAS [A] time = 0.277513, size = 248, normalized size = 2.67 \[ -\frac{1}{6} \, c{\left (\frac{\arctan \left (\frac{a \sqrt{d + \frac{c}{x^{6}}}}{\sqrt{a b c - a^{2} d}}\right )}{\sqrt{a b c - a^{2} d}{\left (b c{\rm sign}\left (x\right ) - a d{\rm sign}\left (x\right )\right )}} + \frac{\sqrt{d + \frac{c}{x^{6}}}}{{\left (b c{\rm sign}\left (x\right ) - a d{\rm sign}\left (x\right )\right )}{\left (b c + a{\left (d + \frac{c}{x^{6}}\right )} - a d\right )}}\right )} + \frac{{\left (b c \arctan \left (\frac{a \sqrt{d}}{\sqrt{a b c - a^{2} d}}\right ) + \sqrt{a b c - a^{2} d} \sqrt{d}\right )}{\rm sign}\left (x\right )}{6 \,{\left (\sqrt{a b c - a^{2} d} b^{2} c - \sqrt{a b c - a^{2} d} a b d\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(x^8/((b*x^6 + a)^2*sqrt(d*x^6 + c)),x, algorithm="giac")
[Out]