Optimal. Leaf size=54 \[ \frac{\left (c d^2-a e^2\right ) (a e+c d x)^3}{3 c^2 d^2}+\frac{e (a e+c d x)^4}{4 c^2 d^2} \]
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Rubi [A] time = 0.111041, antiderivative size = 54, 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 \[ \frac{\left (c d^2-a e^2\right ) (a e+c d x)^3}{3 c^2 d^2}+\frac{e (a e+c d x)^4}{4 c^2 d^2} \]
Antiderivative was successfully verified.
[In] Int[(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^2/(d + e*x),x]
[Out]
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Rubi in Sympy [A] time = 24.9488, size = 48, normalized size = 0.89 \[ \frac{e \left (a e + c d x\right )^{4}}{4 c^{2} d^{2}} - \frac{\left (a e + c d x\right )^{3} \left (a e^{2} - c d^{2}\right )}{3 c^{2} d^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate((a*d*e+(a*e**2+c*d**2)*x+c*d*e*x**2)**2/(e*x+d),x)
[Out]
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Mathematica [A] time = 0.0229265, size = 54, normalized size = 1. \[ \frac{1}{12} x \left (6 a^2 e^2 (2 d+e x)+4 a c d e x (3 d+2 e x)+c^2 d^2 x^2 (4 d+3 e x)\right ) \]
Antiderivative was successfully verified.
[In] Integrate[(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^2/(d + e*x),x]
[Out]
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Maple [A] time = 0.002, size = 77, normalized size = 1.4 \[{\frac{{c}^{2}{d}^{2}e{x}^{4}}{4}}+{\frac{ \left ( ad{e}^{2}c+cd \left ( a{e}^{2}+c{d}^{2} \right ) \right ){x}^{3}}{3}}+{\frac{ \left ( ae \left ( a{e}^{2}+c{d}^{2} \right ) +c{d}^{2}ae \right ){x}^{2}}{2}}+{a}^{2}d{e}^{2}x \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int((a*e*d+(a*e^2+c*d^2)*x+c*d*e*x^2)^2/(e*x+d),x)
[Out]
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Maxima [A] time = 0.722565, size = 86, normalized size = 1.59 \[ \frac{1}{4} \, c^{2} d^{2} e x^{4} + a^{2} d e^{2} x + \frac{1}{3} \,{\left (c^{2} d^{3} + 2 \, a c d e^{2}\right )} x^{3} + \frac{1}{2} \,{\left (2 \, a c d^{2} e + a^{2} e^{3}\right )} x^{2} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((c*d*e*x^2 + a*d*e + (c*d^2 + a*e^2)*x)^2/(e*x + d),x, algorithm="maxima")
[Out]
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Fricas [A] time = 0.234023, size = 86, normalized size = 1.59 \[ \frac{1}{4} \, c^{2} d^{2} e x^{4} + a^{2} d e^{2} x + \frac{1}{3} \,{\left (c^{2} d^{3} + 2 \, a c d e^{2}\right )} x^{3} + \frac{1}{2} \,{\left (2 \, a c d^{2} e + a^{2} e^{3}\right )} x^{2} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((c*d*e*x^2 + a*d*e + (c*d^2 + a*e^2)*x)^2/(e*x + d),x, algorithm="fricas")
[Out]
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Sympy [A] time = 0.241112, size = 66, normalized size = 1.22 \[ a^{2} d e^{2} x + \frac{c^{2} d^{2} e x^{4}}{4} + x^{3} \left (\frac{2 a c d e^{2}}{3} + \frac{c^{2} d^{3}}{3}\right ) + x^{2} \left (\frac{a^{2} e^{3}}{2} + a c d^{2} e\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((a*d*e+(a*e**2+c*d**2)*x+c*d*e*x**2)**2/(e*x+d),x)
[Out]
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GIAC/XCAS [A] time = 0.211026, size = 97, normalized size = 1.8 \[ \frac{1}{12} \,{\left (3 \, c^{2} d^{2} x^{4} e^{5} + 4 \, c^{2} d^{3} x^{3} e^{4} + 8 \, a c d x^{3} e^{6} + 12 \, a c d^{2} x^{2} e^{5} + 6 \, a^{2} x^{2} e^{7} + 12 \, a^{2} d x e^{6}\right )} e^{\left (-4\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((c*d*e*x^2 + a*d*e + (c*d^2 + a*e^2)*x)^2/(e*x + d),x, algorithm="giac")
[Out]