Optimal. Leaf size=78 \[ \frac{2 d (d g+e f)^2}{e^3 (d-e x)}+\frac{(5 d g+e f) (d g+e f) \log (d-e x)}{e^3}+\frac{g x (3 d g+2 e f)}{e^2}+\frac{g^2 x^2}{2 e} \]
[Out]
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Rubi [A] time = 0.208224, antiderivative size = 78, 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 \[ \frac{2 d (d g+e f)^2}{e^3 (d-e x)}+\frac{(5 d g+e f) (d g+e f) \log (d-e x)}{e^3}+\frac{g x (3 d g+2 e f)}{e^2}+\frac{g^2 x^2}{2 e} \]
Antiderivative was successfully verified.
[In] Int[((d + e*x)^3*(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. \[ \frac{2 d \left (d g + e f\right )^{2}}{e^{3} \left (d - e x\right )} + \frac{g^{2} \int x\, dx}{e} + \frac{\left (3 d g + 2 e f\right ) \int g\, dx}{e^{2}} + \frac{\left (d g + e f\right ) \left (5 d g + e f\right ) \log{\left (d - e x \right )}}{e^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate((e*x+d)**3*(g*x+f)**2/(-e**2*x**2+d**2)**2,x)
[Out]
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Mathematica [A] time = 0.116785, size = 83, normalized size = 1.06 \[ \frac{2 \left (5 d^2 g^2+6 d e f g+e^2 f^2\right ) \log (d-e x)+\frac{4 d (d g+e f)^2}{d-e x}+2 e g x (3 d g+2 e f)+e^2 g^2 x^2}{2 e^3} \]
Antiderivative was successfully verified.
[In] Integrate[((d + e*x)^3*(f + g*x)^2)/(d^2 - e^2*x^2)^2,x]
[Out]
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Maple [A] time = 0.012, size = 138, normalized size = 1.8 \[{\frac{{g}^{2}{x}^{2}}{2\,e}}+3\,{\frac{d{g}^{2}x}{{e}^{2}}}+2\,{\frac{fgx}{e}}+5\,{\frac{\ln \left ( ex-d \right ){d}^{2}{g}^{2}}{{e}^{3}}}+6\,{\frac{\ln \left ( ex-d \right ) dfg}{{e}^{2}}}+{\frac{\ln \left ( ex-d \right ){f}^{2}}{e}}-2\,{\frac{{d}^{3}{g}^{2}}{{e}^{3} \left ( ex-d \right ) }}-4\,{\frac{{d}^{2}fg}{{e}^{2} \left ( ex-d \right ) }}-2\,{\frac{d{f}^{2}}{e \left ( ex-d \right ) }} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int((e*x+d)^3*(g*x+f)^2/(-e^2*x^2+d^2)^2,x)
[Out]
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Maxima [A] time = 0.687276, size = 140, normalized size = 1.79 \[ -\frac{2 \,{\left (d e^{2} f^{2} + 2 \, d^{2} e f g + d^{3} g^{2}\right )}}{e^{4} x - d e^{3}} + \frac{e g^{2} x^{2} + 2 \,{\left (2 \, e f g + 3 \, d g^{2}\right )} x}{2 \, e^{2}} + \frac{{\left (e^{2} f^{2} + 6 \, d e f g + 5 \, d^{2} g^{2}\right )} \log \left (e x - d\right )}{e^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((e*x + d)^3*(g*x + f)^2/(e^2*x^2 - d^2)^2,x, algorithm="maxima")
[Out]
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Fricas [A] time = 0.275545, size = 212, normalized size = 2.72 \[ \frac{e^{3} g^{2} x^{3} - 4 \, d e^{2} f^{2} - 8 \, d^{2} e f g - 4 \, d^{3} g^{2} +{\left (4 \, e^{3} f g + 5 \, d e^{2} g^{2}\right )} x^{2} - 2 \,{\left (2 \, d e^{2} f g + 3 \, d^{2} e g^{2}\right )} x - 2 \,{\left (d e^{2} f^{2} + 6 \, d^{2} e f g + 5 \, d^{3} g^{2} -{\left (e^{3} f^{2} + 6 \, d e^{2} f g + 5 \, d^{2} e g^{2}\right )} x\right )} \log \left (e x - d\right )}{2 \,{\left (e^{4} x - d e^{3}\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((e*x + d)^3*(g*x + f)^2/(e^2*x^2 - d^2)^2,x, algorithm="fricas")
[Out]
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Sympy [A] time = 2.76637, size = 92, normalized size = 1.18 \[ - \frac{2 d^{3} g^{2} + 4 d^{2} e f g + 2 d e^{2} f^{2}}{- d e^{3} + e^{4} x} + \frac{g^{2} x^{2}}{2 e} + \frac{x \left (3 d g^{2} + 2 e f g\right )}{e^{2}} + \frac{\left (d g + e f\right ) \left (5 d g + e f\right ) \log{\left (- d + e x \right )}}{e^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((e*x+d)**3*(g*x+f)**2/(-e**2*x**2+d**2)**2,x)
[Out]
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GIAC/XCAS [A] time = 0.291678, size = 286, normalized size = 3.67 \[ \frac{1}{2} \,{\left (5 \, d^{2} g^{2} e^{3} + 6 \, d f g e^{4} + f^{2} e^{5}\right )} e^{\left (-6\right )}{\rm ln}\left ({\left | x^{2} e^{2} - d^{2} \right |}\right ) + \frac{1}{2} \,{\left (g^{2} x^{2} e^{7} + 6 \, d g^{2} x e^{6} + 4 \, f g x e^{7}\right )} e^{\left (-8\right )} + \frac{{\left (5 \, d^{3} g^{2} e^{2} + 6 \, d^{2} f g e^{3} + d f^{2} e^{4}\right )} e^{\left (-5\right )}{\rm ln}\left (\frac{{\left | 2 \, x e^{2} - 2 \,{\left | d \right |} e \right |}}{{\left | 2 \, x e^{2} + 2 \,{\left | d \right |} e \right |}}\right )}{2 \,{\left | d \right |}} - \frac{2 \,{\left (d^{4} g^{2} e^{3} + 2 \, d^{3} f g e^{4} + d^{2} f^{2} e^{5} +{\left (d^{3} g^{2} e^{4} + 2 \, d^{2} f g e^{5} + d f^{2} e^{6}\right )} x\right )} e^{\left (-6\right )}}{x^{2} e^{2} - d^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((e*x + d)^3*(g*x + f)^2/(e^2*x^2 - d^2)^2,x, algorithm="giac")
[Out]