Optimal. Leaf size=167 \[ \frac{3 a b^2 x^7 \sqrt{a^2+2 a b x^2+b^2 x^4}}{7 \left (a+b x^2\right )}+\frac{3 a^2 b x^5 \sqrt{a^2+2 a b x^2+b^2 x^4}}{5 \left (a+b x^2\right )}+\frac{b^3 x^9 \sqrt{a^2+2 a b x^2+b^2 x^4}}{9 \left (a+b x^2\right )}+\frac{a^3 x^3 \sqrt{a^2+2 a b x^2+b^2 x^4}}{3 \left (a+b x^2\right )} \]
[Out]
_______________________________________________________________________________________
Rubi [A] time = 0.131793, 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 \[ \frac{3 a b^2 x^7 \sqrt{a^2+2 a b x^2+b^2 x^4}}{7 \left (a+b x^2\right )}+\frac{3 a^2 b x^5 \sqrt{a^2+2 a b x^2+b^2 x^4}}{5 \left (a+b x^2\right )}+\frac{b^3 x^9 \sqrt{a^2+2 a b x^2+b^2 x^4}}{9 \left (a+b x^2\right )}+\frac{a^3 x^3 \sqrt{a^2+2 a b x^2+b^2 x^4}}{3 \left (a+b x^2\right )} \]
Antiderivative was successfully verified.
[In] Int[x^2*(a^2 + 2*a*b*x^2 + b^2*x^4)^(3/2),x]
[Out]
_______________________________________________________________________________________
Rubi in Sympy [A] time = 17.6938, size = 136, normalized size = 0.81 \[ \frac{16 a^{3} x^{3} \sqrt{a^{2} + 2 a b x^{2} + b^{2} x^{4}}}{315 \left (a + b x^{2}\right )} + \frac{8 a^{2} x^{3} \sqrt{a^{2} + 2 a b x^{2} + b^{2} x^{4}}}{105} + \frac{2 a x^{3} \left (a + b x^{2}\right ) \sqrt{a^{2} + 2 a b x^{2} + b^{2} x^{4}}}{21} + \frac{x^{3} \left (a^{2} + 2 a b x^{2} + b^{2} x^{4}\right )^{\frac{3}{2}}}{9} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate(x**2*(b**2*x**4+2*a*b*x**2+a**2)**(3/2),x)
[Out]
_______________________________________________________________________________________
Mathematica [A] time = 0.026133, size = 61, normalized size = 0.37 \[ \frac{\sqrt{\left (a+b x^2\right )^2} \left (105 a^3 x^3+189 a^2 b x^5+135 a b^2 x^7+35 b^3 x^9\right )}{315 \left (a+b x^2\right )} \]
Antiderivative was successfully verified.
[In] Integrate[x^2*(a^2 + 2*a*b*x^2 + b^2*x^4)^(3/2),x]
[Out]
_______________________________________________________________________________________
Maple [A] time = 0.009, size = 58, normalized size = 0.4 \[{\frac{{x}^{3} \left ( 35\,{b}^{3}{x}^{6}+135\,a{x}^{4}{b}^{2}+189\,{a}^{2}b{x}^{2}+105\,{a}^{3} \right ) }{315\, \left ( b{x}^{2}+a \right ) ^{3}} \left ( \left ( b{x}^{2}+a \right ) ^{2} \right ) ^{{\frac{3}{2}}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int(x^2*(b^2*x^4+2*a*b*x^2+a^2)^(3/2),x)
[Out]
_______________________________________________________________________________________
Maxima [A] time = 0.695685, size = 47, normalized size = 0.28 \[ \frac{1}{9} \, b^{3} x^{9} + \frac{3}{7} \, a b^{2} x^{7} + \frac{3}{5} \, a^{2} b x^{5} + \frac{1}{3} \, a^{3} x^{3} \]
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^2,x, algorithm="maxima")
[Out]
_______________________________________________________________________________________
Fricas [A] time = 0.26612, size = 47, normalized size = 0.28 \[ \frac{1}{9} \, b^{3} x^{9} + \frac{3}{7} \, a b^{2} x^{7} + \frac{3}{5} \, a^{2} b x^{5} + \frac{1}{3} \, a^{3} x^{3} \]
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^2,x, algorithm="fricas")
[Out]
_______________________________________________________________________________________
Sympy [F] time = 0., size = 0, normalized size = 0. \[ \int x^{2} \left (\left (a + b x^{2}\right )^{2}\right )^{\frac{3}{2}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(x**2*(b**2*x**4+2*a*b*x**2+a**2)**(3/2),x)
[Out]
_______________________________________________________________________________________
GIAC/XCAS [A] time = 0.27157, size = 90, normalized size = 0.54 \[ \frac{1}{9} \, b^{3} x^{9}{\rm sign}\left (b x^{2} + a\right ) + \frac{3}{7} \, a b^{2} x^{7}{\rm sign}\left (b x^{2} + a\right ) + \frac{3}{5} \, a^{2} b x^{5}{\rm sign}\left (b x^{2} + a\right ) + \frac{1}{3} \, a^{3} x^{3}{\rm sign}\left (b x^{2} + a\right ) \]
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^2,x, algorithm="giac")
[Out]