Optimal. Leaf size=47 \[ \frac{\log (a e+c d x)}{c d^2-a e^2}-\frac{\log (d+e x)}{c d^2-a e^2} \]
[Out]
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Rubi [A] time = 0.0454907, antiderivative size = 47, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 27, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.074 \[ \frac{\log (a e+c d x)}{c d^2-a e^2}-\frac{\log (d+e x)}{c d^2-a e^2} \]
Antiderivative was successfully verified.
[In] Int[(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^(-1),x]
[Out]
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Rubi in Sympy [A] time = 6.43775, size = 42, normalized size = 0.89 \[ - \frac{2 \operatorname{atanh}{\left (\frac{a e^{2} + c d^{2} + 2 c d e x}{a e^{2} - c d^{2}} \right )}}{a e^{2} - c d^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate(1/(a*d*e+(a*e**2+c*d**2)*x+c*d*e*x**2),x)
[Out]
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Mathematica [A] time = 0.026676, size = 33, normalized size = 0.7 \[ \frac{\log (a e+c d x)-\log (d+e x)}{c d^2-a e^2} \]
Antiderivative was successfully verified.
[In] Integrate[(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^(-1),x]
[Out]
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Maple [A] time = 0.009, size = 48, normalized size = 1. \[{\frac{\ln \left ( ex+d \right ) }{a{e}^{2}-c{d}^{2}}}-{\frac{\ln \left ( cdx+ae \right ) }{a{e}^{2}-c{d}^{2}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int(1/(a*e*d+(a*e^2+c*d^2)*x+c*d*e*x^2),x)
[Out]
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Maxima [A] time = 0.72516, size = 63, normalized size = 1.34 \[ \frac{\log \left (c d x + a e\right )}{c d^{2} - a e^{2}} - \frac{\log \left (e x + d\right )}{c d^{2} - a e^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(1/(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 = 0.212506, size = 45, normalized size = 0.96 \[ \frac{\log \left (c d x + a e\right ) - \log \left (e x + d\right )}{c d^{2} - a e^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(1/(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 = 1.06362, size = 172, normalized size = 3.66 \[ \frac{\log{\left (x + \frac{- \frac{a^{2} e^{4}}{a e^{2} - c d^{2}} + \frac{2 a c d^{2} e^{2}}{a e^{2} - c d^{2}} + a e^{2} - \frac{c^{2} d^{4}}{a e^{2} - c d^{2}} + c d^{2}}{2 c d e} \right )}}{a e^{2} - c d^{2}} - \frac{\log{\left (x + \frac{\frac{a^{2} e^{4}}{a e^{2} - c d^{2}} - \frac{2 a c d^{2} e^{2}}{a e^{2} - c d^{2}} + a e^{2} + \frac{c^{2} d^{4}}{a e^{2} - c d^{2}} + c d^{2}}{2 c d e} \right )}}{a e^{2} - c d^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(1/(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 = 0.212871, size = 101, normalized size = 2.15 \[ \frac{2 \, \arctan \left (\frac{2 \, c d x e + c d^{2} + a e^{2}}{\sqrt{-c^{2} d^{4} + 2 \, a c d^{2} e^{2} - a^{2} e^{4}}}\right )}{\sqrt{-c^{2} d^{4} + 2 \, a c d^{2} e^{2} - a^{2} e^{4}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(1/(c*d*e*x^2 + a*d*e + (c*d^2 + a*e^2)*x),x, algorithm="giac")
[Out]