Optimal. Leaf size=167 \[ \frac{3 a b^2 x^{11} \sqrt{a^2+2 a b x^2+b^2 x^4}}{11 \left (a+b x^2\right )}+\frac{a^2 b x^9 \sqrt{a^2+2 a b x^2+b^2 x^4}}{3 \left (a+b x^2\right )}+\frac{b^3 x^{13} \sqrt{a^2+2 a b x^2+b^2 x^4}}{13 \left (a+b x^2\right )}+\frac{a^3 x^7 \sqrt{a^2+2 a b x^2+b^2 x^4}}{7 \left (a+b x^2\right )} \]
[Out]
_______________________________________________________________________________________
Rubi [A] time = 0.132912, antiderivative size = 167, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 26, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.077 \[ \frac{3 a b^2 x^{11} \sqrt{a^2+2 a b x^2+b^2 x^4}}{11 \left (a+b x^2\right )}+\frac{a^2 b x^9 \sqrt{a^2+2 a b x^2+b^2 x^4}}{3 \left (a+b x^2\right )}+\frac{b^3 x^{13} \sqrt{a^2+2 a b x^2+b^2 x^4}}{13 \left (a+b x^2\right )}+\frac{a^3 x^7 \sqrt{a^2+2 a b x^2+b^2 x^4}}{7 \left (a+b x^2\right )} \]
Antiderivative was successfully verified.
[In] Int[x^6*(a^2 + 2*a*b*x^2 + b^2*x^4)^(3/2),x]
[Out]
_______________________________________________________________________________________
Rubi in Sympy [A] time = 17.2463, size = 136, normalized size = 0.81 \[ \frac{16 a^{3} x^{7} \sqrt{a^{2} + 2 a b x^{2} + b^{2} x^{4}}}{3003 \left (a + b x^{2}\right )} + \frac{8 a^{2} x^{7} \sqrt{a^{2} + 2 a b x^{2} + b^{2} x^{4}}}{429} + \frac{6 a x^{7} \left (a + b x^{2}\right ) \sqrt{a^{2} + 2 a b x^{2} + b^{2} x^{4}}}{143} + \frac{x^{7} \left (a^{2} + 2 a b x^{2} + b^{2} x^{4}\right )^{\frac{3}{2}}}{13} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate(x**6*(b**2*x**4+2*a*b*x**2+a**2)**(3/2),x)
[Out]
_______________________________________________________________________________________
Mathematica [A] time = 0.0298419, size = 61, normalized size = 0.37 \[ \frac{x^7 \sqrt{\left (a+b x^2\right )^2} \left (429 a^3+1001 a^2 b x^2+819 a b^2 x^4+231 b^3 x^6\right )}{3003 \left (a+b x^2\right )} \]
Antiderivative was successfully verified.
[In] Integrate[x^6*(a^2 + 2*a*b*x^2 + b^2*x^4)^(3/2),x]
[Out]
_______________________________________________________________________________________
Maple [A] time = 0.01, size = 58, normalized size = 0.4 \[{\frac{{x}^{7} \left ( 231\,{b}^{3}{x}^{6}+819\,a{x}^{4}{b}^{2}+1001\,{a}^{2}b{x}^{2}+429\,{a}^{3} \right ) }{3003\, \left ( b{x}^{2}+a \right ) ^{3}} \left ( \left ( b{x}^{2}+a \right ) ^{2} \right ) ^{{\frac{3}{2}}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int(x^6*(b^2*x^4+2*a*b*x^2+a^2)^(3/2),x)
[Out]
_______________________________________________________________________________________
Maxima [A] time = 0.702122, size = 47, normalized size = 0.28 \[ \frac{1}{13} \, b^{3} x^{13} + \frac{3}{11} \, a b^{2} x^{11} + \frac{1}{3} \, a^{2} b x^{9} + \frac{1}{7} \, a^{3} x^{7} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((b^2*x^4 + 2*a*b*x^2 + a^2)^(3/2)*x^6,x, algorithm="maxima")
[Out]
_______________________________________________________________________________________
Fricas [A] time = 0.256437, size = 47, normalized size = 0.28 \[ \frac{1}{13} \, b^{3} x^{13} + \frac{3}{11} \, a b^{2} x^{11} + \frac{1}{3} \, a^{2} b x^{9} + \frac{1}{7} \, a^{3} x^{7} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((b^2*x^4 + 2*a*b*x^2 + a^2)^(3/2)*x^6,x, algorithm="fricas")
[Out]
_______________________________________________________________________________________
Sympy [F] time = 0., size = 0, normalized size = 0. \[ \int x^{6} \left (\left (a + b x^{2}\right )^{2}\right )^{\frac{3}{2}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(x**6*(b**2*x**4+2*a*b*x**2+a**2)**(3/2),x)
[Out]
_______________________________________________________________________________________
GIAC/XCAS [A] time = 0.270074, size = 90, normalized size = 0.54 \[ \frac{1}{13} \, b^{3} x^{13}{\rm sign}\left (b x^{2} + a\right ) + \frac{3}{11} \, a b^{2} x^{11}{\rm sign}\left (b x^{2} + a\right ) + \frac{1}{3} \, a^{2} b x^{9}{\rm sign}\left (b x^{2} + a\right ) + \frac{1}{7} \, a^{3} x^{7}{\rm sign}\left (b x^{2} + a\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((b^2*x^4 + 2*a*b*x^2 + a^2)^(3/2)*x^6,x, algorithm="giac")
[Out]