Optimal. Leaf size=43 \[ a^2 c \log (x)+a b c x^2+\frac{d \left (a+b x^2\right )^3}{6 b}+\frac{1}{4} b^2 c x^4 \]
[Out]
_______________________________________________________________________________________
Rubi [A] time = 0.0796131, antiderivative size = 43, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.15 \[ a^2 c \log (x)+a b c x^2+\frac{d \left (a+b x^2\right )^3}{6 b}+\frac{1}{4} b^2 c x^4 \]
Antiderivative was successfully verified.
[In] Int[((a + b*x^2)^2*(c + d*x^2))/x,x]
[Out]
_______________________________________________________________________________________
Rubi in Sympy [F] time = 0., size = 0, normalized size = 0. \[ \frac{a^{2} c \log{\left (x^{2} \right )}}{2} + a b c x^{2} + \frac{b^{2} c \int ^{x^{2}} x\, dx}{2} + \frac{d \left (a + b x^{2}\right )^{3}}{6 b} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate((b*x**2+a)**2*(d*x**2+c)/x,x)
[Out]
_______________________________________________________________________________________
Mathematica [A] time = 0.0238551, size = 51, normalized size = 1.19 \[ a^2 c \log (x)+\frac{1}{4} b x^4 (2 a d+b c)+\frac{1}{2} a x^2 (a d+2 b c)+\frac{1}{6} b^2 d x^6 \]
Antiderivative was successfully verified.
[In] Integrate[((a + b*x^2)^2*(c + d*x^2))/x,x]
[Out]
_______________________________________________________________________________________
Maple [A] time = 0.003, size = 51, normalized size = 1.2 \[{\frac{{b}^{2}d{x}^{6}}{6}}+{\frac{{x}^{4}abd}{2}}+{\frac{{b}^{2}c{x}^{4}}{4}}+{\frac{{x}^{2}{a}^{2}d}{2}}+abc{x}^{2}+{a}^{2}c\ln \left ( x \right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int((b*x^2+a)^2*(d*x^2+c)/x,x)
[Out]
_______________________________________________________________________________________
Maxima [A] time = 1.34861, size = 70, normalized size = 1.63 \[ \frac{1}{6} \, b^{2} d x^{6} + \frac{1}{4} \,{\left (b^{2} c + 2 \, a b d\right )} x^{4} + \frac{1}{2} \, a^{2} c \log \left (x^{2}\right ) + \frac{1}{2} \,{\left (2 \, a b c + a^{2} d\right )} x^{2} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((b*x^2 + a)^2*(d*x^2 + c)/x,x, algorithm="maxima")
[Out]
_______________________________________________________________________________________
Fricas [A] time = 0.218032, size = 66, normalized size = 1.53 \[ \frac{1}{6} \, b^{2} d x^{6} + \frac{1}{4} \,{\left (b^{2} c + 2 \, a b d\right )} x^{4} + a^{2} c \log \left (x\right ) + \frac{1}{2} \,{\left (2 \, a b c + a^{2} d\right )} x^{2} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((b*x^2 + a)^2*(d*x^2 + c)/x,x, algorithm="fricas")
[Out]
_______________________________________________________________________________________
Sympy [A] time = 1.22489, size = 49, normalized size = 1.14 \[ a^{2} c \log{\left (x \right )} + \frac{b^{2} d x^{6}}{6} + x^{4} \left (\frac{a b d}{2} + \frac{b^{2} c}{4}\right ) + x^{2} \left (\frac{a^{2} d}{2} + a b c\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((b*x**2+a)**2*(d*x**2+c)/x,x)
[Out]
_______________________________________________________________________________________
GIAC/XCAS [A] time = 0.231116, size = 72, normalized size = 1.67 \[ \frac{1}{6} \, b^{2} d x^{6} + \frac{1}{4} \, b^{2} c x^{4} + \frac{1}{2} \, a b d x^{4} + a b c x^{2} + \frac{1}{2} \, a^{2} d x^{2} + \frac{1}{2} \, a^{2} c{\rm ln}\left (x^{2}\right ) \]
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]