Optimal. Leaf size=104 \[ -\frac{2 b^3 (d+e x)^2 (b d-a e)}{e^5}+\frac{6 b^2 x (b d-a e)^2}{e^4}-\frac{(b d-a e)^4}{e^5 (d+e x)}-\frac{4 b (b d-a e)^3 \log (d+e x)}{e^5}+\frac{b^4 (d+e x)^3}{3 e^5} \]
[Out]
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Rubi [A] time = 0.228918, antiderivative size = 104, 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 \[ -\frac{2 b^3 (d+e x)^2 (b d-a e)}{e^5}+\frac{6 b^2 x (b d-a e)^2}{e^4}-\frac{(b d-a e)^4}{e^5 (d+e x)}-\frac{4 b (b d-a e)^3 \log (d+e x)}{e^5}+\frac{b^4 (d+e x)^3}{3 e^5} \]
Antiderivative was successfully verified.
[In] Int[(a^2 + 2*a*b*x + b^2*x^2)^2/(d + e*x)^2,x]
[Out]
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Rubi in Sympy [A] time = 53.47, size = 94, normalized size = 0.9 \[ \frac{b^{4} \left (d + e x\right )^{3}}{3 e^{5}} + \frac{2 b^{3} \left (d + e x\right )^{2} \left (a e - b d\right )}{e^{5}} + \frac{6 b^{2} x \left (a e - b d\right )^{2}}{e^{4}} + \frac{4 b \left (a e - b d\right )^{3} \log{\left (d + e x \right )}}{e^{5}} - \frac{\left (a e - b d\right )^{4}}{e^{5} \left (d + e x\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate((b**2*x**2+2*a*b*x+a**2)**2/(e*x+d)**2,x)
[Out]
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Mathematica [A] time = 0.123758, size = 165, normalized size = 1.59 \[ \frac{-3 a^4 e^4+12 a^3 b d e^3+18 a^2 b^2 e^2 \left (-d^2+d e x+e^2 x^2\right )+6 a b^3 e \left (2 d^3-4 d^2 e x-3 d e^2 x^2+e^3 x^3\right )-12 b (d+e x) (b d-a e)^3 \log (d+e x)+b^4 \left (-3 d^4+9 d^3 e x+6 d^2 e^2 x^2-2 d e^3 x^3+e^4 x^4\right )}{3 e^5 (d+e x)} \]
Antiderivative was successfully verified.
[In] Integrate[(a^2 + 2*a*b*x + b^2*x^2)^2/(d + e*x)^2,x]
[Out]
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Maple [B] time = 0.012, size = 230, normalized size = 2.2 \[{\frac{{b}^{4}{x}^{3}}{3\,{e}^{2}}}+2\,{\frac{{b}^{3}{x}^{2}a}{{e}^{2}}}-{\frac{{b}^{4}{x}^{2}d}{{e}^{3}}}+6\,{\frac{{a}^{2}{b}^{2}x}{{e}^{2}}}-8\,{\frac{ad{b}^{3}x}{{e}^{3}}}+3\,{\frac{{b}^{4}{d}^{2}x}{{e}^{4}}}+4\,{\frac{b\ln \left ( ex+d \right ){a}^{3}}{{e}^{2}}}-12\,{\frac{{b}^{2}\ln \left ( ex+d \right ){a}^{2}d}{{e}^{3}}}+12\,{\frac{{b}^{3}\ln \left ( ex+d \right ) a{d}^{2}}{{e}^{4}}}-4\,{\frac{{b}^{4}\ln \left ( ex+d \right ){d}^{3}}{{e}^{5}}}-{\frac{{a}^{4}}{e \left ( ex+d \right ) }}+4\,{\frac{{a}^{3}bd}{{e}^{2} \left ( ex+d \right ) }}-6\,{\frac{{d}^{2}{a}^{2}{b}^{2}}{{e}^{3} \left ( ex+d \right ) }}+4\,{\frac{{d}^{3}a{b}^{3}}{{e}^{4} \left ( ex+d \right ) }}-{\frac{{b}^{4}{d}^{4}}{{e}^{5} \left ( ex+d \right ) }} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int((b^2*x^2+2*a*b*x+a^2)^2/(e*x+d)^2,x)
[Out]
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Maxima [A] time = 0.686471, size = 247, normalized size = 2.38 \[ -\frac{b^{4} d^{4} - 4 \, a b^{3} d^{3} e + 6 \, a^{2} b^{2} d^{2} e^{2} - 4 \, a^{3} b d e^{3} + a^{4} e^{4}}{e^{6} x + d e^{5}} + \frac{b^{4} e^{2} x^{3} - 3 \,{\left (b^{4} d e - 2 \, a b^{3} e^{2}\right )} x^{2} + 3 \,{\left (3 \, b^{4} d^{2} - 8 \, a b^{3} d e + 6 \, a^{2} b^{2} e^{2}\right )} x}{3 \, e^{4}} - \frac{4 \,{\left (b^{4} d^{3} - 3 \, a b^{3} d^{2} e + 3 \, a^{2} b^{2} d e^{2} - a^{3} b e^{3}\right )} \log \left (e x + d\right )}{e^{5}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((b^2*x^2 + 2*a*b*x + a^2)^2/(e*x + d)^2,x, algorithm="maxima")
[Out]
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Fricas [A] time = 0.20231, size = 360, normalized size = 3.46 \[ \frac{b^{4} e^{4} x^{4} - 3 \, b^{4} d^{4} + 12 \, a b^{3} d^{3} e - 18 \, a^{2} b^{2} d^{2} e^{2} + 12 \, a^{3} b d e^{3} - 3 \, a^{4} e^{4} - 2 \,{\left (b^{4} d e^{3} - 3 \, a b^{3} e^{4}\right )} x^{3} + 6 \,{\left (b^{4} d^{2} e^{2} - 3 \, a b^{3} d e^{3} + 3 \, a^{2} b^{2} e^{4}\right )} x^{2} + 3 \,{\left (3 \, b^{4} d^{3} e - 8 \, a b^{3} d^{2} e^{2} + 6 \, a^{2} b^{2} d e^{3}\right )} x - 12 \,{\left (b^{4} d^{4} - 3 \, a b^{3} d^{3} e + 3 \, a^{2} b^{2} d^{2} e^{2} - a^{3} b d e^{3} +{\left (b^{4} d^{3} e - 3 \, a b^{3} d^{2} e^{2} + 3 \, a^{2} b^{2} d e^{3} - a^{3} b e^{4}\right )} x\right )} \log \left (e x + d\right )}{3 \,{\left (e^{6} x + d e^{5}\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((b^2*x^2 + 2*a*b*x + a^2)^2/(e*x + d)^2,x, algorithm="fricas")
[Out]
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Sympy [A] time = 4.05675, size = 151, normalized size = 1.45 \[ \frac{b^{4} x^{3}}{3 e^{2}} + \frac{4 b \left (a e - b d\right )^{3} \log{\left (d + e x \right )}}{e^{5}} - \frac{a^{4} e^{4} - 4 a^{3} b d e^{3} + 6 a^{2} b^{2} d^{2} e^{2} - 4 a b^{3} d^{3} e + b^{4} d^{4}}{d e^{5} + e^{6} x} + \frac{x^{2} \left (2 a b^{3} e - b^{4} d\right )}{e^{3}} + \frac{x \left (6 a^{2} b^{2} e^{2} - 8 a b^{3} d e + 3 b^{4} d^{2}\right )}{e^{4}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((b**2*x**2+2*a*b*x+a**2)**2/(e*x+d)**2,x)
[Out]
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GIAC/XCAS [A] time = 0.214615, size = 323, normalized size = 3.11 \[ \frac{1}{3} \,{\left (b^{4} - \frac{6 \,{\left (b^{4} d e - a b^{3} e^{2}\right )} e^{\left (-1\right )}}{x e + d} + \frac{18 \,{\left (b^{4} d^{2} e^{2} - 2 \, a b^{3} d e^{3} + a^{2} b^{2} e^{4}\right )} e^{\left (-2\right )}}{{\left (x e + d\right )}^{2}}\right )}{\left (x e + d\right )}^{3} e^{\left (-5\right )} + 4 \,{\left (b^{4} d^{3} - 3 \, a b^{3} d^{2} e + 3 \, a^{2} b^{2} d e^{2} - a^{3} b e^{3}\right )} e^{\left (-5\right )}{\rm ln}\left (\frac{{\left | x e + d \right |} e^{\left (-1\right )}}{{\left (x e + d\right )}^{2}}\right ) -{\left (\frac{b^{4} d^{4} e^{3}}{x e + d} - \frac{4 \, a b^{3} d^{3} e^{4}}{x e + d} + \frac{6 \, a^{2} b^{2} d^{2} e^{5}}{x e + d} - \frac{4 \, a^{3} b d e^{6}}{x e + d} + \frac{a^{4} e^{7}}{x e + d}\right )} e^{\left (-8\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((b^2*x^2 + 2*a*b*x + a^2)^2/(e*x + d)^2,x, algorithm="giac")
[Out]