Optimal. Leaf size=65 \[ -\frac{2 e (b d-a e)}{3 b^3 (a+b x)^3}-\frac{(b d-a e)^2}{4 b^3 (a+b x)^4}-\frac{e^2}{2 b^3 (a+b x)^2} \]
[Out]
_______________________________________________________________________________________
Rubi [A] time = 0.0897226, antiderivative size = 65, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 31, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.065 \[ -\frac{2 e (b d-a e)}{3 b^3 (a+b x)^3}-\frac{(b d-a e)^2}{4 b^3 (a+b x)^4}-\frac{e^2}{2 b^3 (a+b x)^2} \]
Antiderivative was successfully verified.
[In] Int[((a + b*x)*(d + e*x)^2)/(a^2 + 2*a*b*x + b^2*x^2)^3,x]
[Out]
_______________________________________________________________________________________
Rubi in Sympy [A] time = 34.2779, size = 56, normalized size = 0.86 \[ - \frac{e^{2}}{2 b^{3} \left (a + b x\right )^{2}} + \frac{2 e \left (a e - b d\right )}{3 b^{3} \left (a + b x\right )^{3}} - \frac{\left (a e - b d\right )^{2}}{4 b^{3} \left (a + b x\right )^{4}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate((b*x+a)*(e*x+d)**2/(b**2*x**2+2*a*b*x+a**2)**3,x)
[Out]
_______________________________________________________________________________________
Mathematica [A] time = 0.0381164, size = 56, normalized size = 0.86 \[ -\frac{a^2 e^2+2 a b e (d+2 e x)+b^2 \left (3 d^2+8 d e x+6 e^2 x^2\right )}{12 b^3 (a+b x)^4} \]
Antiderivative was successfully verified.
[In] Integrate[((a + b*x)*(d + e*x)^2)/(a^2 + 2*a*b*x + b^2*x^2)^3,x]
[Out]
_______________________________________________________________________________________
Maple [A] time = 0.009, size = 71, normalized size = 1.1 \[ -{\frac{{a}^{2}{e}^{2}-2\,abde+{b}^{2}{d}^{2}}{4\,{b}^{3} \left ( bx+a \right ) ^{4}}}+{\frac{2\,e \left ( ae-bd \right ) }{3\,{b}^{3} \left ( bx+a \right ) ^{3}}}-{\frac{{e}^{2}}{2\,{b}^{3} \left ( bx+a \right ) ^{2}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int((b*x+a)*(e*x+d)^2/(b^2*x^2+2*a*b*x+a^2)^3,x)
[Out]
_______________________________________________________________________________________
Maxima [A] time = 0.718462, size = 132, normalized size = 2.03 \[ -\frac{6 \, b^{2} e^{2} x^{2} + 3 \, b^{2} d^{2} + 2 \, a b d e + a^{2} e^{2} + 4 \,{\left (2 \, b^{2} d e + a b e^{2}\right )} x}{12 \,{\left (b^{7} x^{4} + 4 \, a b^{6} x^{3} + 6 \, a^{2} b^{5} x^{2} + 4 \, a^{3} b^{4} x + a^{4} b^{3}\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((b*x + a)*(e*x + d)^2/(b^2*x^2 + 2*a*b*x + a^2)^3,x, algorithm="maxima")
[Out]
_______________________________________________________________________________________
Fricas [A] time = 0.278517, size = 132, normalized size = 2.03 \[ -\frac{6 \, b^{2} e^{2} x^{2} + 3 \, b^{2} d^{2} + 2 \, a b d e + a^{2} e^{2} + 4 \,{\left (2 \, b^{2} d e + a b e^{2}\right )} x}{12 \,{\left (b^{7} x^{4} + 4 \, a b^{6} x^{3} + 6 \, a^{2} b^{5} x^{2} + 4 \, a^{3} b^{4} x + a^{4} b^{3}\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((b*x + a)*(e*x + d)^2/(b^2*x^2 + 2*a*b*x + a^2)^3,x, algorithm="fricas")
[Out]
_______________________________________________________________________________________
Sympy [A] time = 3.7498, size = 104, normalized size = 1.6 \[ - \frac{a^{2} e^{2} + 2 a b d e + 3 b^{2} d^{2} + 6 b^{2} e^{2} x^{2} + x \left (4 a b e^{2} + 8 b^{2} d e\right )}{12 a^{4} b^{3} + 48 a^{3} b^{4} x + 72 a^{2} b^{5} x^{2} + 48 a b^{6} x^{3} + 12 b^{7} x^{4}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((b*x+a)*(e*x+d)**2/(b**2*x**2+2*a*b*x+a**2)**3,x)
[Out]
_______________________________________________________________________________________
GIAC/XCAS [A] time = 0.292116, size = 81, normalized size = 1.25 \[ -\frac{6 \, b^{2} x^{2} e^{2} + 8 \, b^{2} d x e + 3 \, b^{2} d^{2} + 4 \, a b x e^{2} + 2 \, a b d e + a^{2} e^{2}}{12 \,{\left (b x + a\right )}^{4} b^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((b*x + a)*(e*x + d)^2/(b^2*x^2 + 2*a*b*x + a^2)^3,x, algorithm="giac")
[Out]