Optimal. Leaf size=35 \[ -\frac{(d+e x)^3}{3 \left (c d^2-a e^2\right ) (a e+c d x)^3} \]
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Rubi [A] time = 0.0402574, 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 \[ -\frac{(d+e x)^3}{3 \left (c d^2-a e^2\right ) (a e+c d x)^3} \]
Antiderivative was successfully verified.
[In] Int[(d + e*x)^6/(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^4,x]
[Out]
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Rubi in Sympy [A] time = 13.8446, size = 27, normalized size = 0.77 \[ \frac{\left (d + e x\right )^{3}}{3 \left (a e + c d x\right )^{3} \left (a e^{2} - c d^{2}\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate((e*x+d)**6/(a*d*e+(a*e**2+c*d**2)*x+c*d*e*x**2)**4,x)
[Out]
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Mathematica [A] time = 0.0618198, size = 65, normalized size = 1.86 \[ -\frac{a^2 e^4+a c d e^2 (d+3 e x)+c^2 d^2 \left (d^2+3 d e x+3 e^2 x^2\right )}{3 c^3 d^3 (a e+c d x)^3} \]
Antiderivative was successfully verified.
[In] Integrate[(d + e*x)^6/(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^4,x]
[Out]
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Maple [B] time = 0.009, size = 96, normalized size = 2.7 \[ -{\frac{{e}^{2}}{{c}^{3}{d}^{3} \left ( cdx+ae \right ) }}+{\frac{e \left ( a{e}^{2}-c{d}^{2} \right ) }{{c}^{3}{d}^{3} \left ( cdx+ae \right ) ^{2}}}-{\frac{{a}^{2}{e}^{4}-2\,ac{d}^{2}{e}^{2}+{c}^{2}{d}^{4}}{3\,{c}^{3}{d}^{3} \left ( cdx+ae \right ) ^{3}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int((e*x+d)^6/(a*e*d+(a*e^2+c*d^2)*x+c*d*e*x^2)^4,x)
[Out]
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Maxima [A] time = 0.741168, size = 153, normalized size = 4.37 \[ -\frac{3 \, c^{2} d^{2} e^{2} x^{2} + c^{2} d^{4} + a c d^{2} e^{2} + a^{2} e^{4} + 3 \,{\left (c^{2} d^{3} e + a c d e^{3}\right )} x}{3 \,{\left (c^{6} d^{6} x^{3} + 3 \, a c^{5} d^{5} e x^{2} + 3 \, a^{2} c^{4} d^{4} e^{2} x + a^{3} c^{3} d^{3} e^{3}\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((e*x + d)^6/(c*d*e*x^2 + a*d*e + (c*d^2 + a*e^2)*x)^4,x, algorithm="maxima")
[Out]
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Fricas [A] time = 0.203471, size = 153, normalized size = 4.37 \[ -\frac{3 \, c^{2} d^{2} e^{2} x^{2} + c^{2} d^{4} + a c d^{2} e^{2} + a^{2} e^{4} + 3 \,{\left (c^{2} d^{3} e + a c d e^{3}\right )} x}{3 \,{\left (c^{6} d^{6} x^{3} + 3 \, a c^{5} d^{5} e x^{2} + 3 \, a^{2} c^{4} d^{4} e^{2} x + a^{3} c^{3} d^{3} e^{3}\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((e*x + d)^6/(c*d*e*x^2 + a*d*e + (c*d^2 + a*e^2)*x)^4,x, algorithm="fricas")
[Out]
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Sympy [A] time = 3.99797, size = 121, normalized size = 3.46 \[ - \frac{a^{2} e^{4} + a c d^{2} e^{2} + c^{2} d^{4} + 3 c^{2} d^{2} e^{2} x^{2} + x \left (3 a c d e^{3} + 3 c^{2} d^{3} e\right )}{3 a^{3} c^{3} d^{3} e^{3} + 9 a^{2} c^{4} d^{4} e^{2} x + 9 a c^{5} d^{5} e x^{2} + 3 c^{6} d^{6} x^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((e*x+d)**6/(a*d*e+(a*e**2+c*d**2)*x+c*d*e*x**2)**4,x)
[Out]
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GIAC/XCAS [A] time = 57.9054, size = 1, normalized size = 0.03 \[ \mathit{Done} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((e*x + d)^6/(c*d*e*x^2 + a*d*e + (c*d^2 + a*e^2)*x)^4,x, algorithm="giac")
[Out]