Optimal. Leaf size=178 \[ \frac{(d g+e f) (d g+5 e f) \tanh ^{-1}\left (\frac{e x}{d}\right )}{32 d^6 e^3}+\frac{(d g+e f)^2}{32 d^5 e^3 (d-e x)}-\frac{f (d g+e f)}{8 d^5 e^2 (d+e x)}-\frac{(3 e f-d g) (d g+e f)}{32 d^4 e^3 (d+e x)^2}-\frac{(e f-d g)^2}{16 d^2 e^3 (d+e x)^4}-\frac{e^2 f^2-d^2 g^2}{12 d^3 e^3 (d+e x)^3} \]
[Out]
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Rubi [A] time = 0.429567, antiderivative size = 178, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 29, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.103 \[ \frac{(d g+e f) (d g+5 e f) \tanh ^{-1}\left (\frac{e x}{d}\right )}{32 d^6 e^3}+\frac{(d g+e f)^2}{32 d^5 e^3 (d-e x)}-\frac{f (d g+e f)}{8 d^5 e^2 (d+e x)}-\frac{(3 e f-d g) (d g+e f)}{32 d^4 e^3 (d+e x)^2}-\frac{(e f-d g)^2}{16 d^2 e^3 (d+e x)^4}-\frac{e^2 f^2-d^2 g^2}{12 d^3 e^3 (d+e x)^3} \]
Antiderivative was successfully verified.
[In] Int[(f + g*x)^2/((d + e*x)^3*(d^2 - e^2*x^2)^2),x]
[Out]
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Rubi in Sympy [A] time = 64.3836, size = 187, normalized size = 1.05 \[ - \frac{\left (d g - e f\right )^{2}}{16 d^{2} e^{3} \left (d + e x\right )^{4}} + \frac{\left (d g - e f\right ) \left (d g + e f\right )}{12 d^{3} e^{3} \left (d + e x\right )^{3}} + \frac{\left (d g - 3 e f\right ) \left (d g + e f\right )}{32 d^{4} e^{3} \left (d + e x\right )^{2}} - \frac{f \left (d g + e f\right )}{8 d^{5} e^{2} \left (d + e x\right )} + \frac{\left (d g + e f\right )^{2}}{32 d^{5} e^{3} \left (d - e x\right )} - \frac{\left (d g + e f\right ) \left (d g + 5 e f\right ) \log{\left (d - e x \right )}}{64 d^{6} e^{3}} + \frac{\left (d g + e f\right ) \left (d g + 5 e f\right ) \log{\left (d + e x \right )}}{64 d^{6} e^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate((g*x+f)**2/(e*x+d)**3/(-e**2*x**2+d**2)**2,x)
[Out]
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Mathematica [A] time = 0.247167, size = 195, normalized size = 1.1 \[ \frac{-\frac{12 d^4 (e f-d g)^2}{(d+e x)^4}+\frac{6 d^2 \left (d^2 g^2-2 d e f g-3 e^2 f^2\right )}{(d+e x)^2}-3 \left (d^2 g^2+6 d e f g+5 e^2 f^2\right ) \log (d-e x)+3 \left (d^2 g^2+6 d e f g+5 e^2 f^2\right ) \log (d+e x)+\frac{16 d^3 \left (d^2 g^2-e^2 f^2\right )}{(d+e x)^3}+\frac{6 d (d g+e f)^2}{d-e x}-\frac{24 d e f (d g+e f)}{d+e x}}{192 d^6 e^3} \]
Antiderivative was successfully verified.
[In] Integrate[(f + g*x)^2/((d + e*x)^3*(d^2 - e^2*x^2)^2),x]
[Out]
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Maple [B] time = 0.022, size = 341, normalized size = 1.9 \[ -{\frac{\ln \left ( ex-d \right ){g}^{2}}{64\,{e}^{3}{d}^{4}}}-{\frac{3\,\ln \left ( ex-d \right ) fg}{32\,{e}^{2}{d}^{5}}}-{\frac{5\,\ln \left ( ex-d \right ){f}^{2}}{64\,e{d}^{6}}}-{\frac{{g}^{2}}{32\,{d}^{3}{e}^{3} \left ( ex-d \right ) }}-{\frac{fg}{16\,{e}^{2}{d}^{4} \left ( ex-d \right ) }}-{\frac{{f}^{2}}{32\,e{d}^{5} \left ( ex-d \right ) }}+{\frac{\ln \left ( ex+d \right ){g}^{2}}{64\,{e}^{3}{d}^{4}}}+{\frac{3\,\ln \left ( ex+d \right ) fg}{32\,{e}^{2}{d}^{5}}}+{\frac{5\,\ln \left ( ex+d \right ){f}^{2}}{64\,e{d}^{6}}}+{\frac{{g}^{2}}{12\,{e}^{3}d \left ( ex+d \right ) ^{3}}}-{\frac{{f}^{2}}{12\,e{d}^{3} \left ( ex+d \right ) ^{3}}}+{\frac{{g}^{2}}{32\,{e}^{3}{d}^{2} \left ( ex+d \right ) ^{2}}}-{\frac{fg}{16\,{e}^{2}{d}^{3} \left ( ex+d \right ) ^{2}}}-{\frac{3\,{f}^{2}}{32\,e{d}^{4} \left ( ex+d \right ) ^{2}}}-{\frac{{g}^{2}}{16\,{e}^{3} \left ( ex+d \right ) ^{4}}}+{\frac{fg}{8\,d{e}^{2} \left ( ex+d \right ) ^{4}}}-{\frac{{f}^{2}}{16\,e{d}^{2} \left ( ex+d \right ) ^{4}}}-{\frac{fg}{8\,{e}^{2}{d}^{4} \left ( ex+d \right ) }}-{\frac{{f}^{2}}{8\,e{d}^{5} \left ( ex+d \right ) }} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int((g*x+f)^2/(e*x+d)^3/(-e^2*x^2+d^2)^2,x)
[Out]
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Maxima [A] time = 0.70641, size = 402, normalized size = 2.26 \[ \frac{32 \, d^{4} e^{2} f^{2} - 8 \, d^{6} g^{2} - 3 \,{\left (5 \, e^{6} f^{2} + 6 \, d e^{5} f g + d^{2} e^{4} g^{2}\right )} x^{4} - 9 \,{\left (5 \, d e^{5} f^{2} + 6 \, d^{2} e^{4} f g + d^{3} e^{3} g^{2}\right )} x^{3} - 7 \,{\left (5 \, d^{2} e^{4} f^{2} + 6 \, d^{3} e^{3} f g + d^{4} e^{2} g^{2}\right )} x^{2} + 3 \,{\left (5 \, d^{3} e^{3} f^{2} + 6 \, d^{4} e^{2} f g - 7 \, d^{5} e g^{2}\right )} x}{96 \,{\left (d^{5} e^{8} x^{5} + 3 \, d^{6} e^{7} x^{4} + 2 \, d^{7} e^{6} x^{3} - 2 \, d^{8} e^{5} x^{2} - 3 \, d^{9} e^{4} x - d^{10} e^{3}\right )}} + \frac{{\left (5 \, e^{2} f^{2} + 6 \, d e f g + d^{2} g^{2}\right )} \log \left (e x + d\right )}{64 \, d^{6} e^{3}} - \frac{{\left (5 \, e^{2} f^{2} + 6 \, d e f g + d^{2} g^{2}\right )} \log \left (e x - d\right )}{64 \, d^{6} e^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((g*x + f)^2/((e^2*x^2 - d^2)^2*(e*x + d)^3),x, algorithm="maxima")
[Out]
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Fricas [A] time = 0.27959, size = 875, normalized size = 4.92 \[ \frac{64 \, d^{5} e^{2} f^{2} - 16 \, d^{7} g^{2} - 6 \,{\left (5 \, d e^{6} f^{2} + 6 \, d^{2} e^{5} f g + d^{3} e^{4} g^{2}\right )} x^{4} - 18 \,{\left (5 \, d^{2} e^{5} f^{2} + 6 \, d^{3} e^{4} f g + d^{4} e^{3} g^{2}\right )} x^{3} - 14 \,{\left (5 \, d^{3} e^{4} f^{2} + 6 \, d^{4} e^{3} f g + d^{5} e^{2} g^{2}\right )} x^{2} + 6 \,{\left (5 \, d^{4} e^{3} f^{2} + 6 \, d^{5} e^{2} f g - 7 \, d^{6} e g^{2}\right )} x - 3 \,{\left (5 \, d^{5} e^{2} f^{2} + 6 \, d^{6} e f g + d^{7} g^{2} -{\left (5 \, e^{7} f^{2} + 6 \, d e^{6} f g + d^{2} e^{5} g^{2}\right )} x^{5} - 3 \,{\left (5 \, d e^{6} f^{2} + 6 \, d^{2} e^{5} f g + d^{3} e^{4} g^{2}\right )} x^{4} - 2 \,{\left (5 \, d^{2} e^{5} f^{2} + 6 \, d^{3} e^{4} f g + d^{4} e^{3} g^{2}\right )} x^{3} + 2 \,{\left (5 \, d^{3} e^{4} f^{2} + 6 \, d^{4} e^{3} f g + d^{5} e^{2} g^{2}\right )} x^{2} + 3 \,{\left (5 \, d^{4} e^{3} f^{2} + 6 \, d^{5} e^{2} f g + d^{6} e g^{2}\right )} x\right )} \log \left (e x + d\right ) + 3 \,{\left (5 \, d^{5} e^{2} f^{2} + 6 \, d^{6} e f g + d^{7} g^{2} -{\left (5 \, e^{7} f^{2} + 6 \, d e^{6} f g + d^{2} e^{5} g^{2}\right )} x^{5} - 3 \,{\left (5 \, d e^{6} f^{2} + 6 \, d^{2} e^{5} f g + d^{3} e^{4} g^{2}\right )} x^{4} - 2 \,{\left (5 \, d^{2} e^{5} f^{2} + 6 \, d^{3} e^{4} f g + d^{4} e^{3} g^{2}\right )} x^{3} + 2 \,{\left (5 \, d^{3} e^{4} f^{2} + 6 \, d^{4} e^{3} f g + d^{5} e^{2} g^{2}\right )} x^{2} + 3 \,{\left (5 \, d^{4} e^{3} f^{2} + 6 \, d^{5} e^{2} f g + d^{6} e g^{2}\right )} x\right )} \log \left (e x - d\right )}{192 \,{\left (d^{6} e^{8} x^{5} + 3 \, d^{7} e^{7} x^{4} + 2 \, d^{8} e^{6} x^{3} - 2 \, d^{9} e^{5} x^{2} - 3 \, d^{10} e^{4} x - d^{11} e^{3}\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((g*x + f)^2/((e^2*x^2 - d^2)^2*(e*x + d)^3),x, algorithm="fricas")
[Out]
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Sympy [A] time = 7.46259, size = 371, normalized size = 2.08 \[ - \frac{8 d^{6} g^{2} - 32 d^{4} e^{2} f^{2} + x^{4} \left (3 d^{2} e^{4} g^{2} + 18 d e^{5} f g + 15 e^{6} f^{2}\right ) + x^{3} \left (9 d^{3} e^{3} g^{2} + 54 d^{2} e^{4} f g + 45 d e^{5} f^{2}\right ) + x^{2} \left (7 d^{4} e^{2} g^{2} + 42 d^{3} e^{3} f g + 35 d^{2} e^{4} f^{2}\right ) + x \left (21 d^{5} e g^{2} - 18 d^{4} e^{2} f g - 15 d^{3} e^{3} f^{2}\right )}{- 96 d^{10} e^{3} - 288 d^{9} e^{4} x - 192 d^{8} e^{5} x^{2} + 192 d^{7} e^{6} x^{3} + 288 d^{6} e^{7} x^{4} + 96 d^{5} e^{8} x^{5}} - \frac{\left (d g + e f\right ) \left (d g + 5 e f\right ) \log{\left (- \frac{d \left (d g + e f\right ) \left (d g + 5 e f\right )}{e \left (d^{2} g^{2} + 6 d e f g + 5 e^{2} f^{2}\right )} + x \right )}}{64 d^{6} e^{3}} + \frac{\left (d g + e f\right ) \left (d g + 5 e f\right ) \log{\left (\frac{d \left (d g + e f\right ) \left (d g + 5 e f\right )}{e \left (d^{2} g^{2} + 6 d e f g + 5 e^{2} f^{2}\right )} + x \right )}}{64 d^{6} e^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((g*x+f)**2/(e*x+d)**3/(-e**2*x**2+d**2)**2,x)
[Out]
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GIAC/XCAS [A] time = 0.27226, size = 1, normalized size = 0.01 \[ \mathit{Done} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((g*x + f)^2/((e^2*x^2 - d^2)^2*(e*x + d)^3),x, algorithm="giac")
[Out]