Optimal. Leaf size=73 \[ \frac{1}{(d+e x) \left (c d^2-a e^2\right )}+\frac{c d \log (a e+c d x)}{\left (c d^2-a e^2\right )^2}-\frac{c d \log (d+e x)}{\left (c d^2-a e^2\right )^2} \]
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Rubi [A] time = 0.133147, antiderivative size = 73, 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 \[ \frac{1}{(d+e x) \left (c d^2-a e^2\right )}+\frac{c d \log (a e+c d x)}{\left (c d^2-a e^2\right )^2}-\frac{c d \log (d+e x)}{\left (c d^2-a e^2\right )^2} \]
Antiderivative was successfully verified.
[In] Int[1/((d + e*x)*(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)),x]
[Out]
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Rubi in Sympy [A] time = 37.3464, size = 63, normalized size = 0.86 \[ - \frac{c d \log{\left (d + e x \right )}}{\left (a e^{2} - c d^{2}\right )^{2}} + \frac{c d \log{\left (a e + c d x \right )}}{\left (a e^{2} - c d^{2}\right )^{2}} - \frac{1}{\left (d + e x\right ) \left (a e^{2} - c d^{2}\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate(1/(e*x+d)/(a*d*e+(a*e**2+c*d**2)*x+c*d*e*x**2),x)
[Out]
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Mathematica [A] time = 0.0565733, size = 66, normalized size = 0.9 \[ \frac{c d (d+e x) \log (a e+c d x)-a e^2+c d^2-c d (d+e x) \log (d+e x)}{(d+e x) \left (c d^2-a e^2\right )^2} \]
Antiderivative was successfully verified.
[In] Integrate[1/((d + e*x)*(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)),x]
[Out]
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Maple [A] time = 0.025, size = 75, normalized size = 1. \[ -{\frac{1}{ \left ( a{e}^{2}-c{d}^{2} \right ) \left ( ex+d \right ) }}-{\frac{cd\ln \left ( ex+d \right ) }{ \left ( a{e}^{2}-c{d}^{2} \right ) ^{2}}}+{\frac{cd\ln \left ( cdx+ae \right ) }{ \left ( a{e}^{2}-c{d}^{2} \right ) ^{2}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int(1/(e*x+d)/(a*e*d+(a*e^2+c*d^2)*x+c*d*e*x^2),x)
[Out]
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Maxima [A] time = 0.716639, size = 144, normalized size = 1.97 \[ \frac{c d \log \left (c d x + a e\right )}{c^{2} d^{4} - 2 \, a c d^{2} e^{2} + a^{2} e^{4}} - \frac{c d \log \left (e x + d\right )}{c^{2} d^{4} - 2 \, a c d^{2} e^{2} + a^{2} e^{4}} + \frac{1}{c d^{3} - a d e^{2} +{\left (c d^{2} e - a e^{3}\right )} x} \]
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)*(e*x + d)),x, algorithm="maxima")
[Out]
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Fricas [A] time = 0.209312, size = 147, normalized size = 2.01 \[ \frac{c d^{2} - a e^{2} +{\left (c d e x + c d^{2}\right )} \log \left (c d x + a e\right ) -{\left (c d e x + c d^{2}\right )} \log \left (e x + d\right )}{c^{2} d^{5} - 2 \, a c d^{3} e^{2} + a^{2} d e^{4} +{\left (c^{2} d^{4} e - 2 \, a c d^{2} e^{3} + a^{2} e^{5}\right )} x} \]
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)*(e*x + d)),x, algorithm="fricas")
[Out]
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Sympy [A] time = 3.24132, size = 301, normalized size = 4.12 \[ - \frac{c d \log{\left (x + \frac{- \frac{a^{3} c d e^{6}}{\left (a e^{2} - c d^{2}\right )^{2}} + \frac{3 a^{2} c^{2} d^{3} e^{4}}{\left (a e^{2} - c d^{2}\right )^{2}} - \frac{3 a c^{3} d^{5} e^{2}}{\left (a e^{2} - c d^{2}\right )^{2}} + a c d e^{2} + \frac{c^{4} d^{7}}{\left (a e^{2} - c d^{2}\right )^{2}} + c^{2} d^{3}}{2 c^{2} d^{2} e} \right )}}{\left (a e^{2} - c d^{2}\right )^{2}} + \frac{c d \log{\left (x + \frac{\frac{a^{3} c d e^{6}}{\left (a e^{2} - c d^{2}\right )^{2}} - \frac{3 a^{2} c^{2} d^{3} e^{4}}{\left (a e^{2} - c d^{2}\right )^{2}} + \frac{3 a c^{3} d^{5} e^{2}}{\left (a e^{2} - c d^{2}\right )^{2}} + a c d e^{2} - \frac{c^{4} d^{7}}{\left (a e^{2} - c d^{2}\right )^{2}} + c^{2} d^{3}}{2 c^{2} d^{2} e} \right )}}{\left (a e^{2} - c d^{2}\right )^{2}} - \frac{1}{a d e^{2} - c d^{3} + x \left (a e^{3} - c d^{2} e\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(1/(e*x+d)/(a*d*e+(a*e**2+c*d**2)*x+c*d*e*x**2),x)
[Out]
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GIAC/XCAS [F(-2)] time = 0., size = 0, normalized size = 0. \[ \text{Exception raised: NotImplementedError} \]
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)*(e*x + d)),x, algorithm="giac")
[Out]