Optimal. Leaf size=177 \[ \frac{16 d^4 (d g+e f)^2}{e^3 (d-e x)}+\frac{32 d^3 (d g+e f) (2 d g+e f) \log (d-e x)}{e^3}+\frac{1}{3} x^3 \left (17 d^2 g^2+12 d e f g+e^2 f^2\right )+\frac{d x^2 \left (16 d^2 g^2+17 d e f g+3 e^2 f^2\right )}{e}+\frac{d^2 x \left (48 d^2 g^2+64 d e f g+17 e^2 f^2\right )}{e^2}+\frac{1}{2} e g x^4 (3 d g+e f)+\frac{1}{5} e^2 g^2 x^5 \]
[Out]
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Rubi [A] time = 0.458731, antiderivative size = 177, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 29, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.069 \[ \frac{16 d^4 (d g+e f)^2}{e^3 (d-e x)}+\frac{32 d^3 (d g+e f) (2 d g+e f) \log (d-e x)}{e^3}+\frac{1}{3} x^3 \left (17 d^2 g^2+12 d e f g+e^2 f^2\right )+\frac{d x^2 \left (16 d^2 g^2+17 d e f g+3 e^2 f^2\right )}{e}+\frac{d^2 x \left (48 d^2 g^2+64 d e f g+17 e^2 f^2\right )}{e^2}+\frac{1}{2} e g x^4 (3 d g+e f)+\frac{1}{5} e^2 g^2 x^5 \]
Antiderivative was successfully verified.
[In] Int[((d + e*x)^6*(f + g*x)^2)/(d^2 - e^2*x^2)^2,x]
[Out]
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Rubi in Sympy [F] time = 0., size = 0, normalized size = 0. \[ \frac{16 d^{4} \left (d g + e f\right )^{2}}{e^{3} \left (d - e x\right )} + \frac{32 d^{3} \left (d g + e f\right ) \left (2 d g + e f\right ) \log{\left (d - e x \right )}}{e^{3}} + \frac{2 d \left (16 d^{2} g^{2} + 17 d e f g + 3 e^{2} f^{2}\right ) \int x\, dx}{e} + \frac{e^{2} g^{2} x^{5}}{5} + \frac{e g x^{4} \left (3 d g + e f\right )}{2} + x^{3} \left (\frac{17 d^{2} g^{2}}{3} + 4 d e f g + \frac{e^{2} f^{2}}{3}\right ) + \frac{\left (48 d^{2} g^{2} + 64 d e f g + 17 e^{2} f^{2}\right ) \int d^{2}\, dx}{e^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate((e*x+d)**6*(g*x+f)**2/(-e**2*x**2+d**2)**2,x)
[Out]
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Mathematica [A] time = 0.278589, size = 185, normalized size = 1.05 \[ -\frac{16 d^4 (d g+e f)^2}{e^3 (e x-d)}+\frac{1}{3} x^3 \left (17 d^2 g^2+12 d e f g+e^2 f^2\right )+\frac{d x^2 \left (16 d^2 g^2+17 d e f g+3 e^2 f^2\right )}{e}+\frac{d^2 x \left (48 d^2 g^2+64 d e f g+17 e^2 f^2\right )}{e^2}+\frac{32 d^3 \left (2 d^2 g^2+3 d e f g+e^2 f^2\right ) \log (d-e x)}{e^3}+\frac{1}{2} e g x^4 (3 d g+e f)+\frac{1}{5} e^2 g^2 x^5 \]
Antiderivative was successfully verified.
[In] Integrate[((d + e*x)^6*(f + g*x)^2)/(d^2 - e^2*x^2)^2,x]
[Out]
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Maple [A] time = 0.013, size = 245, normalized size = 1.4 \[{\frac{{e}^{2}{g}^{2}{x}^{5}}{5}}+{\frac{3\,e{x}^{4}d{g}^{2}}{2}}+{\frac{{e}^{2}{x}^{4}fg}{2}}+{\frac{17\,{x}^{3}{d}^{2}{g}^{2}}{3}}+4\,e{x}^{3}dfg+{\frac{{e}^{2}{x}^{3}{f}^{2}}{3}}+16\,{\frac{{x}^{2}{d}^{3}{g}^{2}}{e}}+17\,{x}^{2}{d}^{2}fg+3\,e{x}^{2}d{f}^{2}+48\,{\frac{{d}^{4}{g}^{2}x}{{e}^{2}}}+64\,{\frac{{d}^{3}fgx}{e}}+17\,{d}^{2}{f}^{2}x+64\,{\frac{{d}^{5}\ln \left ( ex-d \right ){g}^{2}}{{e}^{3}}}+96\,{\frac{{d}^{4}\ln \left ( ex-d \right ) fg}{{e}^{2}}}+32\,{\frac{{d}^{3}\ln \left ( ex-d \right ){f}^{2}}{e}}-16\,{\frac{{d}^{6}{g}^{2}}{{e}^{3} \left ( ex-d \right ) }}-32\,{\frac{{d}^{5}fg}{{e}^{2} \left ( ex-d \right ) }}-16\,{\frac{{d}^{4}{f}^{2}}{e \left ( ex-d \right ) }} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int((e*x+d)^6*(g*x+f)^2/(-e^2*x^2+d^2)^2,x)
[Out]
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Maxima [A] time = 0.701551, size = 294, normalized size = 1.66 \[ -\frac{16 \,{\left (d^{4} e^{2} f^{2} + 2 \, d^{5} e f g + d^{6} g^{2}\right )}}{e^{4} x - d e^{3}} + \frac{6 \, e^{4} g^{2} x^{5} + 15 \,{\left (e^{4} f g + 3 \, d e^{3} g^{2}\right )} x^{4} + 10 \,{\left (e^{4} f^{2} + 12 \, d e^{3} f g + 17 \, d^{2} e^{2} g^{2}\right )} x^{3} + 30 \,{\left (3 \, d e^{3} f^{2} + 17 \, d^{2} e^{2} f g + 16 \, d^{3} e g^{2}\right )} x^{2} + 30 \,{\left (17 \, d^{2} e^{2} f^{2} + 64 \, d^{3} e f g + 48 \, d^{4} g^{2}\right )} x}{30 \, e^{2}} + \frac{32 \,{\left (d^{3} e^{2} f^{2} + 3 \, d^{4} e f g + 2 \, d^{5} g^{2}\right )} \log \left (e x - d\right )}{e^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((e*x + d)^6*(g*x + f)^2/(e^2*x^2 - d^2)^2,x, algorithm="maxima")
[Out]
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Fricas [A] time = 0.275103, size = 389, normalized size = 2.2 \[ \frac{6 \, e^{6} g^{2} x^{6} - 480 \, d^{4} e^{2} f^{2} - 960 \, d^{5} e f g - 480 \, d^{6} g^{2} + 3 \,{\left (5 \, e^{6} f g + 13 \, d e^{5} g^{2}\right )} x^{5} + 5 \,{\left (2 \, e^{6} f^{2} + 21 \, d e^{5} f g + 25 \, d^{2} e^{4} g^{2}\right )} x^{4} + 10 \,{\left (8 \, d e^{5} f^{2} + 39 \, d^{2} e^{4} f g + 31 \, d^{3} e^{3} g^{2}\right )} x^{3} + 30 \,{\left (14 \, d^{2} e^{4} f^{2} + 47 \, d^{3} e^{3} f g + 32 \, d^{4} e^{2} g^{2}\right )} x^{2} - 30 \,{\left (17 \, d^{3} e^{3} f^{2} + 64 \, d^{4} e^{2} f g + 48 \, d^{5} e g^{2}\right )} x - 960 \,{\left (d^{4} e^{2} f^{2} + 3 \, d^{5} e f g + 2 \, d^{6} g^{2} -{\left (d^{3} e^{3} f^{2} + 3 \, d^{4} e^{2} f g + 2 \, d^{5} e g^{2}\right )} x\right )} \log \left (e x - d\right )}{30 \,{\left (e^{4} x - d e^{3}\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((e*x + d)^6*(g*x + f)^2/(e^2*x^2 - d^2)^2,x, algorithm="fricas")
[Out]
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Sympy [A] time = 4.09398, size = 204, normalized size = 1.15 \[ \frac{32 d^{3} \left (d g + e f\right ) \left (2 d g + e f\right ) \log{\left (- d + e x \right )}}{e^{3}} + \frac{e^{2} g^{2} x^{5}}{5} + x^{4} \left (\frac{3 d e g^{2}}{2} + \frac{e^{2} f g}{2}\right ) + x^{3} \left (\frac{17 d^{2} g^{2}}{3} + 4 d e f g + \frac{e^{2} f^{2}}{3}\right ) - \frac{16 d^{6} g^{2} + 32 d^{5} e f g + 16 d^{4} e^{2} f^{2}}{- d e^{3} + e^{4} x} + \frac{x^{2} \left (16 d^{3} g^{2} + 17 d^{2} e f g + 3 d e^{2} f^{2}\right )}{e} + \frac{x \left (48 d^{4} g^{2} + 64 d^{3} e f g + 17 d^{2} e^{2} f^{2}\right )}{e^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((e*x+d)**6*(g*x+f)**2/(-e**2*x**2+d**2)**2,x)
[Out]
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GIAC/XCAS [A] time = 0.303288, size = 441, normalized size = 2.49 \[ 16 \,{\left (2 \, d^{5} g^{2} e^{5} + 3 \, d^{4} f g e^{6} + d^{3} f^{2} e^{7}\right )} e^{\left (-8\right )}{\rm ln}\left ({\left | x^{2} e^{2} - d^{2} \right |}\right ) + \frac{1}{30} \,{\left (6 \, g^{2} x^{5} e^{22} + 45 \, d g^{2} x^{4} e^{21} + 170 \, d^{2} g^{2} x^{3} e^{20} + 480 \, d^{3} g^{2} x^{2} e^{19} + 1440 \, d^{4} g^{2} x e^{18} + 15 \, f g x^{4} e^{22} + 120 \, d f g x^{3} e^{21} + 510 \, d^{2} f g x^{2} e^{20} + 1920 \, d^{3} f g x e^{19} + 10 \, f^{2} x^{3} e^{22} + 90 \, d f^{2} x^{2} e^{21} + 510 \, d^{2} f^{2} x e^{20}\right )} e^{\left (-20\right )} + \frac{16 \,{\left (2 \, d^{6} g^{2} e^{6} + 3 \, d^{5} f g e^{7} + d^{4} f^{2} e^{8}\right )} e^{\left (-9\right )}{\rm ln}\left (\frac{{\left | 2 \, x e^{2} - 2 \,{\left | d \right |} e \right |}}{{\left | 2 \, x e^{2} + 2 \,{\left | d \right |} e \right |}}\right )}{{\left | d \right |}} - \frac{16 \,{\left (d^{7} g^{2} e^{5} + 2 \, d^{6} f g e^{6} + d^{5} f^{2} e^{7} +{\left (d^{6} g^{2} e^{6} + 2 \, d^{5} f g e^{7} + d^{4} f^{2} e^{8}\right )} x\right )} e^{\left (-8\right )}}{x^{2} e^{2} - d^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((e*x + d)^6*(g*x + f)^2/(e^2*x^2 - d^2)^2,x, algorithm="giac")
[Out]