Optimal. Leaf size=109 \[ -\frac{4 d^2 (d g+e f)^2 \log (d-e x)}{e^3}-\frac{x^2 \left (4 d^2 g^2+6 d e f g+e^2 f^2\right )}{2 e}-\frac{d x (2 d g+e f) (2 d g+3 e f)}{e^2}-\frac{1}{3} g x^3 (3 d g+2 e f)-\frac{1}{4} e g^2 x^4 \]
[Out]
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Rubi [A] time = 0.242567, antiderivative size = 109, 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 \[ -\frac{4 d^2 (d g+e f)^2 \log (d-e x)}{e^3}-\frac{x^2 \left (4 d^2 g^2+6 d e f g+e^2 f^2\right )}{2 e}-\frac{d x (2 d g+e f) (2 d g+3 e f)}{e^2}-\frac{1}{3} g x^3 (3 d g+2 e f)-\frac{1}{4} e g^2 x^4 \]
Antiderivative was successfully verified.
[In] Int[((d + e*x)^3*(f + g*x)^2)/(d^2 - e^2*x^2),x]
[Out]
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Rubi in Sympy [F] time = 0., size = 0, normalized size = 0. \[ - \frac{4 d^{2} \left (d g + e f\right )^{2} \log{\left (d - e x \right )}}{e^{3}} - \frac{e g^{2} x^{4}}{4} - \frac{g x^{3} \left (3 d g + 2 e f\right )}{3} - \frac{\left (4 d^{2} g^{2} + 6 d e f g + e^{2} f^{2}\right ) \int x\, dx}{e} - \frac{\left (2 d g + e f\right ) \left (2 d g + 3 e f\right ) \int d\, dx}{e^{2}} \]
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),x)
[Out]
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Mathematica [A] time = 0.0952753, size = 103, normalized size = 0.94 \[ -\frac{48 d^2 (d g+e f)^2 \log (d-e x)+e x \left (48 d^3 g^2+24 d^2 e g (4 f+g x)+12 d e^2 \left (3 f^2+3 f g x+g^2 x^2\right )+e^3 x \left (6 f^2+8 f g x+3 g^2 x^2\right )\right )}{12 e^3} \]
Antiderivative was successfully verified.
[In] Integrate[((d + e*x)^3*(f + g*x)^2)/(d^2 - e^2*x^2),x]
[Out]
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Maple [A] time = 0.006, size = 145, normalized size = 1.3 \[ -{\frac{e{g}^{2}{x}^{4}}{4}}-{x}^{3}d{g}^{2}-{\frac{2\,e{x}^{3}fg}{3}}-2\,{\frac{{x}^{2}{d}^{2}{g}^{2}}{e}}-3\,{x}^{2}dfg-{\frac{e{x}^{2}{f}^{2}}{2}}-4\,{\frac{{d}^{3}{g}^{2}x}{{e}^{2}}}-8\,{\frac{{d}^{2}fgx}{e}}-3\,d{f}^{2}x-4\,{\frac{{d}^{4}\ln \left ( ex-d \right ){g}^{2}}{{e}^{3}}}-8\,{\frac{{d}^{3}\ln \left ( ex-d \right ) fg}{{e}^{2}}}-4\,{\frac{{d}^{2}\ln \left ( ex-d \right ){f}^{2}}{e}} \]
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),x)
[Out]
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Maxima [A] time = 0.708116, size = 186, normalized size = 1.71 \[ -\frac{3 \, e^{3} g^{2} x^{4} + 4 \,{\left (2 \, e^{3} f g + 3 \, d e^{2} g^{2}\right )} x^{3} + 6 \,{\left (e^{3} f^{2} + 6 \, d e^{2} f g + 4 \, d^{2} e g^{2}\right )} x^{2} + 12 \,{\left (3 \, d e^{2} f^{2} + 8 \, d^{2} e f g + 4 \, d^{3} g^{2}\right )} x}{12 \, e^{2}} - \frac{4 \,{\left (d^{2} e^{2} f^{2} + 2 \, d^{3} e f g + d^{4} g^{2}\right )} \log \left (e x - d\right )}{e^{3}} \]
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),x, algorithm="maxima")
[Out]
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Fricas [A] time = 0.281944, size = 188, normalized size = 1.72 \[ -\frac{3 \, e^{4} g^{2} x^{4} + 4 \,{\left (2 \, e^{4} f g + 3 \, d e^{3} g^{2}\right )} x^{3} + 6 \,{\left (e^{4} f^{2} + 6 \, d e^{3} f g + 4 \, d^{2} e^{2} g^{2}\right )} x^{2} + 12 \,{\left (3 \, d e^{3} f^{2} + 8 \, d^{2} e^{2} f g + 4 \, d^{3} e g^{2}\right )} x + 48 \,{\left (d^{2} e^{2} f^{2} + 2 \, d^{3} e f g + d^{4} g^{2}\right )} \log \left (e x - d\right )}{12 \, e^{3}} \]
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),x, algorithm="fricas")
[Out]
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Sympy [A] time = 2.2243, size = 116, normalized size = 1.06 \[ - \frac{4 d^{2} \left (d g + e f\right )^{2} \log{\left (- d + e x \right )}}{e^{3}} - \frac{e g^{2} x^{4}}{4} - x^{3} \left (d g^{2} + \frac{2 e f g}{3}\right ) - \frac{x^{2} \left (4 d^{2} g^{2} + 6 d e f g + e^{2} f^{2}\right )}{2 e} - \frac{x \left (4 d^{3} g^{2} + 8 d^{2} e f g + 3 d e^{2} f^{2}\right )}{e^{2}} \]
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),x)
[Out]
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GIAC/XCAS [A] time = 0.273402, size = 285, normalized size = 2.61 \[ -2 \,{\left (d^{4} g^{2} e^{3} + 2 \, d^{3} f g e^{4} + d^{2} f^{2} e^{5}\right )} e^{\left (-6\right )}{\rm ln}\left ({\left | x^{2} e^{2} - d^{2} \right |}\right ) - \frac{1}{12} \,{\left (3 \, g^{2} x^{4} e^{9} + 12 \, d g^{2} x^{3} e^{8} + 24 \, d^{2} g^{2} x^{2} e^{7} + 48 \, d^{3} g^{2} x e^{6} + 8 \, f g x^{3} e^{9} + 36 \, d f g x^{2} e^{8} + 96 \, d^{2} f g x e^{7} + 6 \, f^{2} x^{2} e^{9} + 36 \, d f^{2} x e^{8}\right )} e^{\left (-8\right )} - \frac{2 \,{\left (d^{5} g^{2} e^{2} + 2 \, d^{4} f g e^{3} + d^{3} f^{2} e^{4}\right )} e^{\left (-5\right )}{\rm ln}\left (\frac{{\left | 2 \, x e^{2} - 2 \,{\left | d \right |} e \right |}}{{\left | 2 \, x e^{2} + 2 \,{\left | d \right |} e \right |}}\right )}{{\left | d \right |}} \]
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),x, algorithm="giac")
[Out]