Optimal. Leaf size=35 \[ -\frac{(d+e x)^2}{2 \left (c d^2-a e^2\right ) (a e+c d x)^2} \]
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Rubi [A] time = 0.0405761, antiderivative size = 35, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 35, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.057 \[ -\frac{(d+e x)^2}{2 \left (c d^2-a e^2\right ) (a e+c d x)^2} \]
Antiderivative was successfully verified.
[In] Int[(d + e*x)^4/(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^3,x]
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Rubi in Sympy [A] time = 13.26, size = 27, normalized size = 0.77 \[ \frac{\left (d + e x\right )^{2}}{2 \left (a e + c d x\right )^{2} \left (a e^{2} - c d^{2}\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate((e*x+d)**4/(a*d*e+(a*e**2+c*d**2)*x+c*d*e*x**2)**3,x)
[Out]
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Mathematica [A] time = 0.0259218, size = 35, normalized size = 1. \[ -\frac{a e^2+c d (d+2 e x)}{2 c^2 d^2 (a e+c d x)^2} \]
Antiderivative was successfully verified.
[In] Integrate[(d + e*x)^4/(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^3,x]
[Out]
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Maple [A] time = 0.007, size = 51, normalized size = 1.5 \[ -{\frac{e}{{c}^{2}{d}^{2} \left ( cdx+ae \right ) }}-{\frac{-a{e}^{2}+c{d}^{2}}{2\,{c}^{2}{d}^{2} \left ( cdx+ae \right ) ^{2}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int((e*x+d)^4/(a*e*d+(a*e^2+c*d^2)*x+c*d*e*x^2)^3,x)
[Out]
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Maxima [A] time = 0.744066, size = 76, normalized size = 2.17 \[ -\frac{2 \, c d e x + c d^{2} + a e^{2}}{2 \,{\left (c^{4} d^{4} x^{2} + 2 \, a c^{3} d^{3} e x + a^{2} c^{2} d^{2} e^{2}\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((e*x + d)^4/(c*d*e*x^2 + a*d*e + (c*d^2 + a*e^2)*x)^3,x, algorithm="maxima")
[Out]
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Fricas [A] time = 0.203153, size = 76, normalized size = 2.17 \[ -\frac{2 \, c d e x + c d^{2} + a e^{2}}{2 \,{\left (c^{4} d^{4} x^{2} + 2 \, a c^{3} d^{3} e x + a^{2} c^{2} d^{2} e^{2}\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((e*x + d)^4/(c*d*e*x^2 + a*d*e + (c*d^2 + a*e^2)*x)^3,x, algorithm="fricas")
[Out]
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Sympy [A] time = 2.11969, size = 60, normalized size = 1.71 \[ - \frac{a e^{2} + c d^{2} + 2 c d e x}{2 a^{2} c^{2} d^{2} e^{2} + 4 a c^{3} d^{3} e x + 2 c^{4} d^{4} x^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((e*x+d)**4/(a*d*e+(a*e**2+c*d**2)*x+c*d*e*x**2)**3,x)
[Out]
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GIAC/XCAS [A] time = 1.53286, size = 1, normalized size = 0.03 \[ \mathit{Done} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((e*x + d)^4/(c*d*e*x^2 + a*d*e + (c*d^2 + a*e^2)*x)^3,x, algorithm="giac")
[Out]