Optimal. Leaf size=38 \[ \frac{\left (c d^2-a e^2\right ) \log (a e+c d x)}{c^2 d^2}+\frac{e x}{c d} \]
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Rubi [A] time = 0.0804165, antiderivative size = 38, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 35, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.057 \[ \frac{\left (c d^2-a e^2\right ) \log (a e+c d x)}{c^2 d^2}+\frac{e x}{c d} \]
Antiderivative was successfully verified.
[In] Int[(d + e*x)^2/(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2),x]
[Out]
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Rubi in Sympy [F] time = 0., size = 0, normalized size = 0. \[ \frac{\int e\, dx}{c d} - \frac{\left (a e^{2} - c d^{2}\right ) \log{\left (a e + c d x \right )}}{c^{2} d^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate((e*x+d)**2/(a*d*e+(a*e**2+c*d**2)*x+c*d*e*x**2),x)
[Out]
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Mathematica [A] time = 0.0180704, size = 35, normalized size = 0.92 \[ \frac{\left (c d^2-a e^2\right ) \log (a e+c d x)+c d e x}{c^2 d^2} \]
Antiderivative was successfully verified.
[In] Integrate[(d + e*x)^2/(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2),x]
[Out]
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Maple [A] time = 0.003, size = 45, normalized size = 1.2 \[{\frac{ex}{cd}}-{\frac{\ln \left ( cdx+ae \right ) a{e}^{2}}{{c}^{2}{d}^{2}}}+{\frac{\ln \left ( cdx+ae \right ) }{c}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int((e*x+d)^2/(a*e*d+(a*e^2+c*d^2)*x+c*d*e*x^2),x)
[Out]
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Maxima [A] time = 0.718649, size = 51, normalized size = 1.34 \[ \frac{e x}{c d} + \frac{{\left (c d^{2} - a e^{2}\right )} \log \left (c d x + a e\right )}{c^{2} d^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((e*x + d)^2/(c*d*e*x^2 + a*d*e + (c*d^2 + a*e^2)*x),x, algorithm="maxima")
[Out]
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Fricas [A] time = 0.212928, size = 47, normalized size = 1.24 \[ \frac{c d e x +{\left (c d^{2} - a e^{2}\right )} \log \left (c d x + a e\right )}{c^{2} d^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((e*x + d)^2/(c*d*e*x^2 + a*d*e + (c*d^2 + a*e^2)*x),x, algorithm="fricas")
[Out]
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Sympy [A] time = 1.3361, size = 32, normalized size = 0.84 \[ \frac{e x}{c d} - \frac{\left (a e^{2} - c d^{2}\right ) \log{\left (a e + c d x \right )}}{c^{2} d^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((e*x+d)**2/(a*d*e+(a*e**2+c*d**2)*x+c*d*e*x**2),x)
[Out]
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GIAC/XCAS [A] time = 0.219023, size = 215, normalized size = 5.66 \[ \frac{x e}{c d} + \frac{{\left (c d^{2} - a e^{2}\right )}{\rm ln}\left (c d x^{2} e + c d^{2} x + a x e^{2} + a d e\right )}{2 \, c^{2} d^{2}} + \frac{{\left (c^{2} d^{4} - 2 \, a c d^{2} e^{2} + a^{2} e^{4}\right )} \arctan \left (\frac{2 \, c d x e + c d^{2} + a e^{2}}{\sqrt{-c^{2} d^{4} + 2 \, a c d^{2} e^{2} - a^{2} e^{4}}}\right )}{\sqrt{-c^{2} d^{4} + 2 \, a c d^{2} e^{2} - a^{2} e^{4}} c^{2} d^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((e*x + d)^2/(c*d*e*x^2 + a*d*e + (c*d^2 + a*e^2)*x),x, algorithm="giac")
[Out]