Optimal. Leaf size=66 \[ -\frac{\left (b^2-4 a c\right )^2}{96 c^3 d^4 (b+2 c x)^3}+\frac{b^2-4 a c}{16 c^3 d^4 (b+2 c x)}+\frac{x}{16 c^2 d^4} \]
[Out]
_______________________________________________________________________________________
Rubi [A] time = 0.133583, antiderivative size = 66, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 1, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.042 \[ -\frac{\left (b^2-4 a c\right )^2}{96 c^3 d^4 (b+2 c x)^3}+\frac{b^2-4 a c}{16 c^3 d^4 (b+2 c x)}+\frac{x}{16 c^2 d^4} \]
Antiderivative was successfully verified.
[In] Int[(a + b*x + c*x^2)^2/(b*d + 2*c*d*x)^4,x]
[Out]
_______________________________________________________________________________________
Rubi in Sympy [F] time = 0., size = 0, normalized size = 0. \[ \frac{\int \frac{1}{16}\, dx}{c^{2} d^{4}} + \frac{- 4 a c + b^{2}}{16 c^{3} d^{4} \left (b + 2 c x\right )} - \frac{\left (- 4 a c + b^{2}\right )^{2}}{96 c^{3} d^{4} \left (b + 2 c x\right )^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate((c*x**2+b*x+a)**2/(2*c*d*x+b*d)**4,x)
[Out]
_______________________________________________________________________________________
Mathematica [A] time = 0.151451, size = 53, normalized size = 0.8 \[ \frac{-\frac{\left (b^2-4 a c\right )^2}{(b+2 c x)^3}+\frac{6 \left (b^2-4 a c\right )}{b+2 c x}+6 c x}{96 c^3 d^4} \]
Antiderivative was successfully verified.
[In] Integrate[(a + b*x + c*x^2)^2/(b*d + 2*c*d*x)^4,x]
[Out]
_______________________________________________________________________________________
Maple [A] time = 0.01, size = 67, normalized size = 1. \[{\frac{1}{{d}^{4}} \left ({\frac{x}{16\,{c}^{2}}}-{\frac{16\,{a}^{2}{c}^{2}-8\,ac{b}^{2}+{b}^{4}}{96\,{c}^{3} \left ( 2\,cx+b \right ) ^{3}}}-{\frac{4\,ac-{b}^{2}}{16\,{c}^{3} \left ( 2\,cx+b \right ) }} \right ) } \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int((c*x^2+b*x+a)^2/(2*c*d*x+b*d)^4,x)
[Out]
_______________________________________________________________________________________
Maxima [A] time = 0.682995, size = 157, normalized size = 2.38 \[ \frac{5 \, b^{4} - 16 \, a b^{2} c - 16 \, a^{2} c^{2} + 24 \,{\left (b^{2} c^{2} - 4 \, a c^{3}\right )} x^{2} + 24 \,{\left (b^{3} c - 4 \, a b c^{2}\right )} x}{96 \,{\left (8 \, c^{6} d^{4} x^{3} + 12 \, b c^{5} d^{4} x^{2} + 6 \, b^{2} c^{4} d^{4} x + b^{3} c^{3} d^{4}\right )}} + \frac{x}{16 \, c^{2} d^{4}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((c*x^2 + b*x + a)^2/(2*c*d*x + b*d)^4,x, algorithm="maxima")
[Out]
_______________________________________________________________________________________
Fricas [A] time = 0.203014, size = 169, normalized size = 2.56 \[ \frac{48 \, c^{4} x^{4} + 72 \, b c^{3} x^{3} + 5 \, b^{4} - 16 \, a b^{2} c - 16 \, a^{2} c^{2} + 12 \,{\left (5 \, b^{2} c^{2} - 8 \, a c^{3}\right )} x^{2} + 6 \,{\left (5 \, b^{3} c - 16 \, a b c^{2}\right )} x}{96 \,{\left (8 \, c^{6} d^{4} x^{3} + 12 \, b c^{5} d^{4} x^{2} + 6 \, b^{2} c^{4} d^{4} x + b^{3} c^{3} d^{4}\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((c*x^2 + b*x + a)^2/(2*c*d*x + b*d)^4,x, algorithm="fricas")
[Out]
_______________________________________________________________________________________
Sympy [A] time = 4.21316, size = 117, normalized size = 1.77 \[ - \frac{16 a^{2} c^{2} + 16 a b^{2} c - 5 b^{4} + x^{2} \left (96 a c^{3} - 24 b^{2} c^{2}\right ) + x \left (96 a b c^{2} - 24 b^{3} c\right )}{96 b^{3} c^{3} d^{4} + 576 b^{2} c^{4} d^{4} x + 1152 b c^{5} d^{4} x^{2} + 768 c^{6} d^{4} x^{3}} + \frac{x}{16 c^{2} d^{4}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((c*x**2+b*x+a)**2/(2*c*d*x+b*d)**4,x)
[Out]
_______________________________________________________________________________________
GIAC/XCAS [A] time = 0.213601, size = 111, normalized size = 1.68 \[ \frac{x}{16 \, c^{2} d^{4}} + \frac{24 \, b^{2} c^{2} x^{2} - 96 \, a c^{3} x^{2} + 24 \, b^{3} c x - 96 \, a b c^{2} x + 5 \, b^{4} - 16 \, a b^{2} c - 16 \, a^{2} c^{2}}{96 \,{\left (2 \, c x + b\right )}^{3} c^{3} d^{4}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((c*x^2 + b*x + a)^2/(2*c*d*x + b*d)^4,x, algorithm="giac")
[Out]