Optimal. Leaf size=65 \[ -\frac{e (b d-a e)}{2 b^3 (a+b x)^4}-\frac{(b d-a e)^2}{5 b^3 (a+b x)^5}-\frac{e^2}{3 b^3 (a+b x)^3} \]
[Out]
_______________________________________________________________________________________
Rubi [A] time = 0.107336, antiderivative size = 65, 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 \[ -\frac{e (b d-a e)}{2 b^3 (a+b x)^4}-\frac{(b d-a e)^2}{5 b^3 (a+b x)^5}-\frac{e^2}{3 b^3 (a+b x)^3} \]
Antiderivative was successfully verified.
[In] Int[(d + e*x)^2/(a^2 + 2*a*b*x + b^2*x^2)^3,x]
[Out]
_______________________________________________________________________________________
Rubi in Sympy [A] time = 31.5751, size = 54, normalized size = 0.83 \[ - \frac{e^{2}}{3 b^{3} \left (a + b x\right )^{3}} + \frac{e \left (a e - b d\right )}{2 b^{3} \left (a + b x\right )^{4}} - \frac{\left (a e - b d\right )^{2}}{5 b^{3} \left (a + b x\right )^{5}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate((e*x+d)**2/(b**2*x**2+2*a*b*x+a**2)**3,x)
[Out]
_______________________________________________________________________________________
Mathematica [A] time = 0.0467927, size = 57, normalized size = 0.88 \[ -\frac{a^2 e^2+a b e (3 d+5 e x)+b^2 \left (6 d^2+15 d e x+10 e^2 x^2\right )}{30 b^3 (a+b x)^5} \]
Antiderivative was successfully verified.
[In] Integrate[(d + e*x)^2/(a^2 + 2*a*b*x + b^2*x^2)^3,x]
[Out]
_______________________________________________________________________________________
Maple [A] time = 0.008, size = 71, normalized size = 1.1 \[{\frac{e \left ( ae-bd \right ) }{2\,{b}^{3} \left ( bx+a \right ) ^{4}}}-{\frac{{a}^{2}{e}^{2}-2\,aedb+{b}^{2}{d}^{2}}{5\,{b}^{3} \left ( bx+a \right ) ^{5}}}-{\frac{{e}^{2}}{3\,{b}^{3} \left ( bx+a \right ) ^{3}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int((e*x+d)^2/(b^2*x^2+2*a*b*x+a^2)^3,x)
[Out]
_______________________________________________________________________________________
Maxima [A] time = 0.690112, size = 147, normalized size = 2.26 \[ -\frac{10 \, b^{2} e^{2} x^{2} + 6 \, b^{2} d^{2} + 3 \, a b d e + a^{2} e^{2} + 5 \,{\left (3 \, b^{2} d e + a b e^{2}\right )} x}{30 \,{\left (b^{8} x^{5} + 5 \, a b^{7} x^{4} + 10 \, a^{2} b^{6} x^{3} + 10 \, a^{3} b^{5} x^{2} + 5 \, a^{4} b^{4} x + a^{5} b^{3}\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((e*x + d)^2/(b^2*x^2 + 2*a*b*x + a^2)^3,x, algorithm="maxima")
[Out]
_______________________________________________________________________________________
Fricas [A] time = 0.19758, size = 147, normalized size = 2.26 \[ -\frac{10 \, b^{2} e^{2} x^{2} + 6 \, b^{2} d^{2} + 3 \, a b d e + a^{2} e^{2} + 5 \,{\left (3 \, b^{2} d e + a b e^{2}\right )} x}{30 \,{\left (b^{8} x^{5} + 5 \, a b^{7} x^{4} + 10 \, a^{2} b^{6} x^{3} + 10 \, a^{3} b^{5} x^{2} + 5 \, a^{4} b^{4} x + a^{5} b^{3}\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((e*x + d)^2/(b^2*x^2 + 2*a*b*x + a^2)^3,x, algorithm="fricas")
[Out]
_______________________________________________________________________________________
Sympy [A] time = 4.46008, size = 116, normalized size = 1.78 \[ - \frac{a^{2} e^{2} + 3 a b d e + 6 b^{2} d^{2} + 10 b^{2} e^{2} x^{2} + x \left (5 a b e^{2} + 15 b^{2} d e\right )}{30 a^{5} b^{3} + 150 a^{4} b^{4} x + 300 a^{3} b^{5} x^{2} + 300 a^{2} b^{6} x^{3} + 150 a b^{7} x^{4} + 30 b^{8} x^{5}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((e*x+d)**2/(b**2*x**2+2*a*b*x+a**2)**3,x)
[Out]
_______________________________________________________________________________________
GIAC/XCAS [A] time = 0.211503, size = 81, normalized size = 1.25 \[ -\frac{10 \, b^{2} x^{2} e^{2} + 15 \, b^{2} d x e + 6 \, b^{2} d^{2} + 5 \, a b x e^{2} + 3 \, a b d e + a^{2} e^{2}}{30 \,{\left (b x + a\right )}^{5} b^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((e*x + d)^2/(b^2*x^2 + 2*a*b*x + a^2)^3,x, algorithm="giac")
[Out]