Optimal. Leaf size=65 \[ \frac{2 b (b d-a e)}{3 e^3 (d+e x)^3}-\frac{(b d-a e)^2}{4 e^3 (d+e x)^4}-\frac{b^2}{2 e^3 (d+e x)^2} \]
[Out]
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Rubi [A] time = 0.0996398, antiderivative size = 65, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.083 \[ \frac{2 b (b d-a e)}{3 e^3 (d+e x)^3}-\frac{(b d-a e)^2}{4 e^3 (d+e x)^4}-\frac{b^2}{2 e^3 (d+e x)^2} \]
Antiderivative was successfully verified.
[In] Int[(a^2 + 2*a*b*x + b^2*x^2)/(d + e*x)^5,x]
[Out]
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Rubi in Sympy [A] time = 31.3469, size = 58, normalized size = 0.89 \[ - \frac{b^{2}}{2 e^{3} \left (d + e x\right )^{2}} - \frac{2 b \left (a e - b d\right )}{3 e^{3} \left (d + e x\right )^{3}} - \frac{\left (a e - b d\right )^{2}}{4 e^{3} \left (d + e x\right )^{4}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate((b**2*x**2+2*a*b*x+a**2)/(e*x+d)**5,x)
[Out]
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Mathematica [A] time = 0.0397531, size = 55, normalized size = 0.85 \[ -\frac{3 a^2 e^2+2 a b e (d+4 e x)+b^2 \left (d^2+4 d e x+6 e^2 x^2\right )}{12 e^3 (d+e x)^4} \]
Antiderivative was successfully verified.
[In] Integrate[(a^2 + 2*a*b*x + b^2*x^2)/(d + e*x)^5,x]
[Out]
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Maple [A] time = 0.008, size = 71, normalized size = 1.1 \[ -{\frac{2\,b \left ( ae-bd \right ) }{3\,{e}^{3} \left ( ex+d \right ) ^{3}}}-{\frac{{a}^{2}{e}^{2}-2\,deab+{b}^{2}{d}^{2}}{4\,{e}^{3} \left ( ex+d \right ) ^{4}}}-{\frac{{b}^{2}}{2\,{e}^{3} \left ( ex+d \right ) ^{2}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int((b^2*x^2+2*a*b*x+a^2)/(e*x+d)^5,x)
[Out]
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Maxima [A] time = 0.695402, size = 132, normalized size = 2.03 \[ -\frac{6 \, b^{2} e^{2} x^{2} + b^{2} d^{2} + 2 \, a b d e + 3 \, a^{2} e^{2} + 4 \,{\left (b^{2} d e + 2 \, a b e^{2}\right )} x}{12 \,{\left (e^{7} x^{4} + 4 \, d e^{6} x^{3} + 6 \, d^{2} e^{5} x^{2} + 4 \, d^{3} e^{4} x + d^{4} e^{3}\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((b^2*x^2 + 2*a*b*x + a^2)/(e*x + d)^5,x, algorithm="maxima")
[Out]
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Fricas [A] time = 0.197234, size = 132, normalized size = 2.03 \[ -\frac{6 \, b^{2} e^{2} x^{2} + b^{2} d^{2} + 2 \, a b d e + 3 \, a^{2} e^{2} + 4 \,{\left (b^{2} d e + 2 \, a b e^{2}\right )} x}{12 \,{\left (e^{7} x^{4} + 4 \, d e^{6} x^{3} + 6 \, d^{2} e^{5} x^{2} + 4 \, d^{3} e^{4} x + d^{4} e^{3}\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((b^2*x^2 + 2*a*b*x + a^2)/(e*x + d)^5,x, algorithm="fricas")
[Out]
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Sympy [A] time = 3.76947, size = 104, normalized size = 1.6 \[ - \frac{3 a^{2} e^{2} + 2 a b d e + b^{2} d^{2} + 6 b^{2} e^{2} x^{2} + x \left (8 a b e^{2} + 4 b^{2} d e\right )}{12 d^{4} e^{3} + 48 d^{3} e^{4} x + 72 d^{2} e^{5} x^{2} + 48 d e^{6} x^{3} + 12 e^{7} x^{4}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((b**2*x**2+2*a*b*x+a**2)/(e*x+d)**5,x)
[Out]
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GIAC/XCAS [A] time = 0.210628, size = 132, normalized size = 2.03 \[ -\frac{1}{12} \,{\left (\frac{6 \, b^{2} e}{{\left (x e + d\right )}^{2}} - \frac{8 \, b^{2} d e}{{\left (x e + d\right )}^{3}} + \frac{3 \, b^{2} d^{2} e}{{\left (x e + d\right )}^{4}} + \frac{8 \, a b e^{2}}{{\left (x e + d\right )}^{3}} - \frac{6 \, a b d e^{2}}{{\left (x e + d\right )}^{4}} + \frac{3 \, a^{2} e^{3}}{{\left (x e + d\right )}^{4}}\right )} e^{\left (-4\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((b^2*x^2 + 2*a*b*x + a^2)/(e*x + d)^5,x, algorithm="giac")
[Out]