Optimal. Leaf size=42 \[ \frac{\left (a+b x^2\right )^3 (b c-a d)}{6 b^2}+\frac{d \left (a+b x^2\right )^4}{8 b^2} \]
[Out]
_______________________________________________________________________________________
Rubi [A] time = 0.151151, antiderivative size = 42, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 18, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.111 \[ \frac{\left (a+b x^2\right )^3 (b c-a d)}{6 b^2}+\frac{d \left (a+b x^2\right )^4}{8 b^2} \]
Antiderivative was successfully verified.
[In] Int[x*(a + b*x^2)^2*(c + d*x^2),x]
[Out]
_______________________________________________________________________________________
Rubi in Sympy [A] time = 15.6108, size = 34, normalized size = 0.81 \[ \frac{d \left (a + b x^{2}\right )^{4}}{8 b^{2}} - \frac{\left (a + b x^{2}\right )^{3} \left (a d - b c\right )}{6 b^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate(x*(b*x**2+a)**2*(d*x**2+c),x)
[Out]
_______________________________________________________________________________________
Mathematica [A] time = 0.0207506, size = 51, normalized size = 1.21 \[ \frac{1}{24} x^2 \left (12 a^2 c+4 b x^4 (2 a d+b c)+6 a x^2 (a d+2 b c)+3 b^2 d x^6\right ) \]
Antiderivative was successfully verified.
[In] Integrate[x*(a + b*x^2)^2*(c + d*x^2),x]
[Out]
_______________________________________________________________________________________
Maple [A] time = 0.001, size = 52, normalized size = 1.2 \[{\frac{{b}^{2}d{x}^{8}}{8}}+{\frac{ \left ( 2\,abd+{b}^{2}c \right ){x}^{6}}{6}}+{\frac{ \left ({a}^{2}d+2\,abc \right ){x}^{4}}{4}}+{\frac{{a}^{2}c{x}^{2}}{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int(x*(b*x^2+a)^2*(d*x^2+c),x)
[Out]
_______________________________________________________________________________________
Maxima [A] time = 1.32345, size = 69, normalized size = 1.64 \[ \frac{1}{8} \, b^{2} d x^{8} + \frac{1}{6} \,{\left (b^{2} c + 2 \, a b d\right )} x^{6} + \frac{1}{2} \, a^{2} c x^{2} + \frac{1}{4} \,{\left (2 \, a b c + a^{2} d\right )} x^{4} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((b*x^2 + a)^2*(d*x^2 + c)*x,x, algorithm="maxima")
[Out]
_______________________________________________________________________________________
Fricas [A] time = 0.194868, size = 1, normalized size = 0.02 \[ \frac{1}{8} x^{8} d b^{2} + \frac{1}{6} x^{6} c b^{2} + \frac{1}{3} x^{6} d b a + \frac{1}{2} x^{4} c b a + \frac{1}{4} x^{4} d a^{2} + \frac{1}{2} x^{2} c a^{2} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((b*x^2 + a)^2*(d*x^2 + c)*x,x, algorithm="fricas")
[Out]
_______________________________________________________________________________________
Sympy [A] time = 0.115064, size = 53, normalized size = 1.26 \[ \frac{a^{2} c x^{2}}{2} + \frac{b^{2} d x^{8}}{8} + x^{6} \left (\frac{a b d}{3} + \frac{b^{2} c}{6}\right ) + x^{4} \left (\frac{a^{2} d}{4} + \frac{a b c}{2}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(x*(b*x**2+a)**2*(d*x**2+c),x)
[Out]
_______________________________________________________________________________________
GIAC/XCAS [A] time = 0.220245, size = 72, normalized size = 1.71 \[ \frac{1}{8} \, b^{2} d x^{8} + \frac{1}{6} \, b^{2} c x^{6} + \frac{1}{3} \, a b d x^{6} + \frac{1}{2} \, a b c x^{4} + \frac{1}{4} \, a^{2} d x^{4} + \frac{1}{2} \, a^{2} c x^{2} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((b*x^2 + a)^2*(d*x^2 + c)*x,x, algorithm="giac")
[Out]