Optimal. Leaf size=28 \[ \frac{(a+b x)^3}{3 (d+e x)^3 (b d-a e)} \]
[Out]
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Rubi [A] time = 0.0246163, antiderivative size = 28, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.083 \[ \frac{(a+b x)^3}{3 (d+e x)^3 (b d-a e)} \]
Antiderivative was successfully verified.
[In] Int[(a^2 + 2*a*b*x + b^2*x^2)/(d + e*x)^4,x]
[Out]
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Rubi in Sympy [A] time = 10.9508, size = 22, normalized size = 0.79 \[ - \frac{\left (a + b x\right )^{3}}{3 \left (d + e x\right )^{3} \left (a e - b d\right )} \]
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)**4,x)
[Out]
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Mathematica [A] time = 0.0469223, size = 53, normalized size = 1.89 \[ -\frac{a^2 e^2+a b e (d+3 e x)+b^2 \left (d^2+3 d e x+3 e^2 x^2\right )}{3 e^3 (d+e x)^3} \]
Antiderivative was successfully verified.
[In] Integrate[(a^2 + 2*a*b*x + b^2*x^2)/(d + e*x)^4,x]
[Out]
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Maple [B] time = 0.008, size = 71, normalized size = 2.5 \[ -{\frac{{a}^{2}{e}^{2}-2\,deab+{b}^{2}{d}^{2}}{3\,{e}^{3} \left ( ex+d \right ) ^{3}}}-{\frac{b \left ( ae-bd \right ) }{{e}^{3} \left ( ex+d \right ) ^{2}}}-{\frac{{b}^{2}}{{e}^{3} \left ( ex+d \right ) }} \]
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)^4,x)
[Out]
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Maxima [A] time = 0.6862, size = 113, normalized size = 4.04 \[ -\frac{3 \, b^{2} e^{2} x^{2} + b^{2} d^{2} + a b d e + a^{2} e^{2} + 3 \,{\left (b^{2} d e + a b e^{2}\right )} x}{3 \,{\left (e^{6} x^{3} + 3 \, d e^{5} x^{2} + 3 \, d^{2} e^{4} x + d^{3} e^{3}\right )}} \]
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)^4,x, algorithm="maxima")
[Out]
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Fricas [A] time = 0.198369, size = 113, normalized size = 4.04 \[ -\frac{3 \, b^{2} e^{2} x^{2} + b^{2} d^{2} + a b d e + a^{2} e^{2} + 3 \,{\left (b^{2} d e + a b e^{2}\right )} x}{3 \,{\left (e^{6} x^{3} + 3 \, d e^{5} x^{2} + 3 \, d^{2} e^{4} x + d^{3} e^{3}\right )}} \]
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)^4,x, algorithm="fricas")
[Out]
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Sympy [A] time = 3.06849, size = 88, normalized size = 3.14 \[ - \frac{a^{2} e^{2} + a b d e + b^{2} d^{2} + 3 b^{2} e^{2} x^{2} + x \left (3 a b e^{2} + 3 b^{2} d e\right )}{3 d^{3} e^{3} + 9 d^{2} e^{4} x + 9 d e^{5} x^{2} + 3 e^{6} x^{3}} \]
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)**4,x)
[Out]
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GIAC/XCAS [A] time = 0.210827, size = 78, normalized size = 2.79 \[ -\frac{{\left (3 \, b^{2} x^{2} e^{2} + 3 \, b^{2} d x e + b^{2} d^{2} + 3 \, a b x e^{2} + a b d e + a^{2} e^{2}\right )} e^{\left (-3\right )}}{3 \,{\left (x e + d\right )}^{3}} \]
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)^4,x, algorithm="giac")
[Out]