Optimal. Leaf size=39 \[ -\frac{a-\frac{c d^2}{e^2}}{4 (d+e x)^4}-\frac{c d}{3 e^2 (d+e x)^3} \]
[Out]
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Rubi [A] time = 0.07208, antiderivative size = 39, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 33, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.061 \[ -\frac{a-\frac{c d^2}{e^2}}{4 (d+e x)^4}-\frac{c d}{3 e^2 (d+e x)^3} \]
Antiderivative was successfully verified.
[In] Int[(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)/(d + e*x)^6,x]
[Out]
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Rubi in Sympy [A] time = 16.2963, size = 37, normalized size = 0.95 \[ - \frac{c d}{3 e^{2} \left (d + e x\right )^{3}} - \frac{a e^{2} - c d^{2}}{4 e^{2} \left (d + e x\right )^{4}} \]
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)/(e*x+d)**6,x)
[Out]
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Mathematica [A] time = 0.0234266, size = 30, normalized size = 0.77 \[ -\frac{3 a e^2+c d (d+4 e x)}{12 e^2 (d+e x)^4} \]
Antiderivative was successfully verified.
[In] Integrate[(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)/(d + e*x)^6,x]
[Out]
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Maple [A] time = 0.008, size = 40, normalized size = 1. \[ -{\frac{a{e}^{2}-c{d}^{2}}{4\,{e}^{2} \left ( ex+d \right ) ^{4}}}-{\frac{cd}{3\,{e}^{2} \left ( ex+d \right ) ^{3}}} \]
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)/(e*x+d)^6,x)
[Out]
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Maxima [A] time = 0.732291, size = 89, normalized size = 2.28 \[ -\frac{4 \, c d e x + c d^{2} + 3 \, a e^{2}}{12 \,{\left (e^{6} x^{4} + 4 \, d e^{5} x^{3} + 6 \, d^{2} e^{4} x^{2} + 4 \, d^{3} e^{3} x + d^{4} e^{2}\right )}} \]
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)/(e*x + d)^6,x, algorithm="maxima")
[Out]
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Fricas [A] time = 0.21923, size = 89, normalized size = 2.28 \[ -\frac{4 \, c d e x + c d^{2} + 3 \, a e^{2}}{12 \,{\left (e^{6} x^{4} + 4 \, d e^{5} x^{3} + 6 \, d^{2} e^{4} x^{2} + 4 \, d^{3} e^{3} x + d^{4} e^{2}\right )}} \]
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)/(e*x + d)^6,x, algorithm="fricas")
[Out]
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Sympy [A] time = 2.41742, size = 70, normalized size = 1.79 \[ - \frac{3 a e^{2} + c d^{2} + 4 c d e x}{12 d^{4} e^{2} + 48 d^{3} e^{3} x + 72 d^{2} e^{4} x^{2} + 48 d e^{5} x^{3} + 12 e^{6} x^{4}} \]
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)/(e*x+d)**6,x)
[Out]
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GIAC/XCAS [A] time = 0.207763, size = 65, normalized size = 1.67 \[ -\frac{{\left (4 \, c d x^{2} e^{2} + 5 \, c d^{2} x e + c d^{3} + 3 \, a x e^{3} + 3 \, a d e^{2}\right )} e^{\left (-2\right )}}{12 \,{\left (x e + d\right )}^{5}} \]
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)/(e*x + d)^6,x, algorithm="giac")
[Out]