Optimal. Leaf size=51 \[ -\frac{(b d-a e)^2}{e^3 (d+e x)}-\frac{2 b (b d-a e) \log (d+e x)}{e^3}+\frac{b^2 x}{e^2} \]
[Out]
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Rubi [A] time = 0.0954285, antiderivative size = 51, 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 \[ -\frac{(b d-a e)^2}{e^3 (d+e x)}-\frac{2 b (b d-a e) \log (d+e x)}{e^3}+\frac{b^2 x}{e^2} \]
Antiderivative was successfully verified.
[In] Int[(a^2 + 2*a*b*x + b^2*x^2)/(d + e*x)^2,x]
[Out]
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Rubi in Sympy [A] time = 28.4495, size = 44, normalized size = 0.86 \[ \frac{b^{2} x}{e^{2}} + \frac{2 b \left (a e - b d\right ) \log{\left (d + e x \right )}}{e^{3}} - \frac{\left (a e - b d\right )^{2}}{e^{3} \left (d + e x\right )} \]
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)**2,x)
[Out]
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Mathematica [A] time = 0.0687701, size = 47, normalized size = 0.92 \[ \frac{-\frac{(b d-a e)^2}{d+e x}+2 b (a e-b d) \log (d+e x)+b^2 e x}{e^3} \]
Antiderivative was successfully verified.
[In] Integrate[(a^2 + 2*a*b*x + b^2*x^2)/(d + e*x)^2,x]
[Out]
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Maple [A] time = 0.011, size = 86, normalized size = 1.7 \[{\frac{{b}^{2}x}{{e}^{2}}}+2\,{\frac{b\ln \left ( ex+d \right ) a}{{e}^{2}}}-2\,{\frac{d\ln \left ( ex+d \right ){b}^{2}}{{e}^{3}}}-{\frac{{a}^{2}}{e \left ( ex+d \right ) }}+2\,{\frac{bda}{{e}^{2} \left ( ex+d \right ) }}-{\frac{{b}^{2}{d}^{2}}{{e}^{3} \left ( ex+d \right ) }} \]
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)^2,x)
[Out]
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Maxima [A] time = 0.682597, size = 90, normalized size = 1.76 \[ \frac{b^{2} x}{e^{2}} - \frac{b^{2} d^{2} - 2 \, a b d e + a^{2} e^{2}}{e^{4} x + d e^{3}} - \frac{2 \,{\left (b^{2} d - a b e\right )} \log \left (e x + d\right )}{e^{3}} \]
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)^2,x, algorithm="maxima")
[Out]
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Fricas [A] time = 0.199387, size = 124, normalized size = 2.43 \[ \frac{b^{2} e^{2} x^{2} + b^{2} d e x - b^{2} d^{2} + 2 \, a b d e - a^{2} e^{2} - 2 \,{\left (b^{2} d^{2} - a b d e +{\left (b^{2} d e - a b e^{2}\right )} x\right )} \log \left (e x + d\right )}{e^{4} x + d e^{3}} \]
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)^2,x, algorithm="fricas")
[Out]
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Sympy [A] time = 2.10348, size = 60, normalized size = 1.18 \[ \frac{b^{2} x}{e^{2}} + \frac{2 b \left (a e - b d\right ) \log{\left (d + e x \right )}}{e^{3}} - \frac{a^{2} e^{2} - 2 a b d e + b^{2} d^{2}}{d e^{3} + e^{4} x} \]
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)**2,x)
[Out]
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GIAC/XCAS [A] time = 0.213481, size = 150, normalized size = 2.94 \[ -2 \,{\left (e^{\left (-1\right )}{\rm ln}\left (\frac{{\left | x e + d \right |} e^{\left (-1\right )}}{{\left (x e + d\right )}^{2}}\right ) - \frac{d e^{\left (-1\right )}}{x e + d}\right )} a b e^{\left (-1\right )} +{\left (2 \, d e^{\left (-3\right )}{\rm ln}\left (\frac{{\left | x e + d \right |} e^{\left (-1\right )}}{{\left (x e + d\right )}^{2}}\right ) +{\left (x e + d\right )} e^{\left (-3\right )} - \frac{d^{2} e^{\left (-3\right )}}{x e + d}\right )} b^{2} - \frac{a^{2} e^{\left (-1\right )}}{x e + d} \]
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)^2,x, algorithm="giac")
[Out]