Optimal. Leaf size=59 \[ \frac{2 b (b d-a e)}{e^3 (d+e x)}-\frac{(b d-a e)^2}{2 e^3 (d+e x)^2}+\frac{b^2 \log (d+e x)}{e^3} \]
[Out]
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Rubi [A] time = 0.0991435, antiderivative size = 59, 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 \[ \frac{2 b (b d-a e)}{e^3 (d+e x)}-\frac{(b d-a e)^2}{2 e^3 (d+e x)^2}+\frac{b^2 \log (d+e x)}{e^3} \]
Antiderivative was successfully verified.
[In] Int[(a^2 + 2*a*b*x + b^2*x^2)/(d + e*x)^3,x]
[Out]
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Rubi in Sympy [A] time = 30.7813, size = 51, normalized size = 0.86 \[ \frac{b^{2} \log{\left (d + e x \right )}}{e^{3}} - \frac{2 b \left (a e - b d\right )}{e^{3} \left (d + e x\right )} - \frac{\left (a e - b d\right )^{2}}{2 e^{3} \left (d + e x\right )^{2}} \]
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)**3,x)
[Out]
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Mathematica [A] time = 0.045716, size = 48, normalized size = 0.81 \[ \frac{\frac{(b d-a e) (a e+3 b d+4 b e x)}{(d+e x)^2}+2 b^2 \log (d+e x)}{2 e^3} \]
Antiderivative was successfully verified.
[In] Integrate[(a^2 + 2*a*b*x + b^2*x^2)/(d + e*x)^3,x]
[Out]
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Maple [A] time = 0.009, size = 92, normalized size = 1.6 \[{\frac{{b}^{2}\ln \left ( ex+d \right ) }{{e}^{3}}}-{\frac{{a}^{2}}{2\,e \left ( ex+d \right ) ^{2}}}+{\frac{bda}{{e}^{2} \left ( ex+d \right ) ^{2}}}-{\frac{{b}^{2}{d}^{2}}{2\,{e}^{3} \left ( ex+d \right ) ^{2}}}-2\,{\frac{ab}{{e}^{2} \left ( ex+d \right ) }}+2\,{\frac{{b}^{2}d}{{e}^{3} \left ( ex+d \right ) }} \]
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)^3,x)
[Out]
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Maxima [A] time = 0.684348, size = 108, normalized size = 1.83 \[ \frac{3 \, b^{2} d^{2} - 2 \, a b d e - a^{2} e^{2} + 4 \,{\left (b^{2} d e - a b e^{2}\right )} x}{2 \,{\left (e^{5} x^{2} + 2 \, d e^{4} x + d^{2} e^{3}\right )}} + \frac{b^{2} \log \left (e x + d\right )}{e^{3}} \]
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)^3,x, algorithm="maxima")
[Out]
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Fricas [A] time = 0.201958, size = 135, normalized size = 2.29 \[ \frac{3 \, b^{2} d^{2} - 2 \, a b d e - a^{2} e^{2} + 4 \,{\left (b^{2} d e - a b e^{2}\right )} x + 2 \,{\left (b^{2} e^{2} x^{2} + 2 \, b^{2} d e x + b^{2} d^{2}\right )} \log \left (e x + d\right )}{2 \,{\left (e^{5} x^{2} + 2 \, d e^{4} x + d^{2} e^{3}\right )}} \]
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)^3,x, algorithm="fricas")
[Out]
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Sympy [A] time = 2.52957, size = 80, normalized size = 1.36 \[ \frac{b^{2} \log{\left (d + e x \right )}}{e^{3}} - \frac{a^{2} e^{2} + 2 a b d e - 3 b^{2} d^{2} + x \left (4 a b e^{2} - 4 b^{2} d e\right )}{2 d^{2} e^{3} + 4 d e^{4} x + 2 e^{5} x^{2}} \]
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)**3,x)
[Out]
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GIAC/XCAS [A] time = 0.210649, size = 93, normalized size = 1.58 \[ b^{2} e^{\left (-3\right )}{\rm ln}\left ({\left | x e + d \right |}\right ) + \frac{{\left (4 \,{\left (b^{2} d - a b e\right )} x +{\left (3 \, b^{2} d^{2} - 2 \, a b d e - a^{2} e^{2}\right )} e^{\left (-1\right )}\right )} e^{\left (-2\right )}}{2 \,{\left (x e + d\right )}^{2}} \]
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)^3,x, algorithm="giac")
[Out]