Optimal. Leaf size=54 \[ -\frac{c d^2-a e^2}{3 c^2 d^2 (a e+c d x)^3}-\frac{e}{2 c^2 d^2 (a e+c d x)^2} \]
[Out]
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Rubi [A] time = 0.0974425, 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 \[ -\frac{c d^2-a e^2}{3 c^2 d^2 (a e+c d x)^3}-\frac{e}{2 c^2 d^2 (a e+c d x)^2} \]
Antiderivative was successfully verified.
[In] Int[(d + e*x)^5/(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 = 23.5338, size = 48, normalized size = 0.89 \[ - \frac{e}{2 c^{2} d^{2} \left (a e + c d x\right )^{2}} + \frac{a e^{2} - c d^{2}}{3 c^{2} d^{2} \left (a e + c d x\right )^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate((e*x+d)**5/(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 = 0.0301683, size = 37, normalized size = 0.69 \[ -\frac{a e^2+c d (2 d+3 e x)}{6 c^2 d^2 (a e+c d x)^3} \]
Antiderivative was successfully verified.
[In] Integrate[(d + e*x)^5/(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^4,x]
[Out]
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Maple [A] time = 0.008, size = 51, normalized size = 0.9 \[ -{\frac{e}{2\,{c}^{2}{d}^{2} \left ( cdx+ae \right ) ^{2}}}-{\frac{-a{e}^{2}+c{d}^{2}}{3\,{c}^{2}{d}^{2} \left ( cdx+ae \right ) ^{3}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int((e*x+d)^5/(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 = 0.743144, size = 100, normalized size = 1.85 \[ -\frac{3 \, c d e x + 2 \, c d^{2} + a e^{2}}{6 \,{\left (c^{5} d^{5} x^{3} + 3 \, a c^{4} d^{4} e x^{2} + 3 \, a^{2} c^{3} d^{3} e^{2} x + a^{3} c^{2} d^{2} e^{3}\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((e*x + d)^5/(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 = 0.207691, size = 100, normalized size = 1.85 \[ -\frac{3 \, c d e x + 2 \, c d^{2} + a e^{2}}{6 \,{\left (c^{5} d^{5} x^{3} + 3 \, a c^{4} d^{4} e x^{2} + 3 \, a^{2} c^{3} d^{3} e^{2} x + a^{3} c^{2} d^{2} e^{3}\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((e*x + d)^5/(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 = 2.63193, size = 80, normalized size = 1.48 \[ - \frac{a e^{2} + 2 c d^{2} + 3 c d e x}{6 a^{3} c^{2} d^{2} e^{3} + 18 a^{2} c^{3} d^{3} e^{2} x + 18 a c^{4} d^{4} e x^{2} + 6 c^{5} d^{5} x^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((e*x+d)**5/(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 = 0.27777, size = 1, normalized size = 0.02 \[ \mathit{Done} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((e*x + d)^5/(c*d*e*x^2 + a*d*e + (c*d^2 + a*e^2)*x)^4,x, algorithm="giac")
[Out]