Optimal. Leaf size=77 \[ \frac{2 c d \left (c d^2-a e^2\right )}{5 e^3 (d+e x)^5}-\frac{\left (c d^2-a e^2\right )^2}{6 e^3 (d+e x)^6}-\frac{c^2 d^2}{4 e^3 (d+e x)^4} \]
[Out]
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Rubi [A] time = 0.12187, antiderivative size = 77, 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 \[ \frac{2 c d \left (c d^2-a e^2\right )}{5 e^3 (d+e x)^5}-\frac{\left (c d^2-a e^2\right )^2}{6 e^3 (d+e x)^6}-\frac{c^2 d^2}{4 e^3 (d+e x)^4} \]
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)^9,x]
[Out]
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Rubi in Sympy [A] time = 31.0433, size = 70, normalized size = 0.91 \[ - \frac{c^{2} d^{2}}{4 e^{3} \left (d + e x\right )^{4}} - \frac{2 c d \left (a e^{2} - c d^{2}\right )}{5 e^{3} \left (d + e x\right )^{5}} - \frac{\left (a e^{2} - c d^{2}\right )^{2}}{6 e^{3} \left (d + e x\right )^{6}} \]
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)**9,x)
[Out]
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Mathematica [A] time = 0.0443151, size = 61, normalized size = 0.79 \[ -\frac{10 a^2 e^4+4 a c d e^2 (d+6 e x)+c^2 d^2 \left (d^2+6 d e x+15 e^2 x^2\right )}{60 e^3 (d+e x)^6} \]
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)^9,x]
[Out]
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Maple [A] time = 0.008, size = 83, normalized size = 1.1 \[ -{\frac{{a}^{2}{e}^{4}-2\,ac{d}^{2}{e}^{2}+{c}^{2}{d}^{4}}{6\,{e}^{3} \left ( ex+d \right ) ^{6}}}-{\frac{{c}^{2}{d}^{2}}{4\,{e}^{3} \left ( ex+d \right ) ^{4}}}-{\frac{2\,cd \left ( a{e}^{2}-c{d}^{2} \right ) }{5\,{e}^{3} \left ( ex+d \right ) ^{5}}} \]
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)^9,x)
[Out]
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Maxima [A] time = 0.734167, size = 176, normalized size = 2.29 \[ -\frac{15 \, c^{2} d^{2} e^{2} x^{2} + c^{2} d^{4} + 4 \, a c d^{2} e^{2} + 10 \, a^{2} e^{4} + 6 \,{\left (c^{2} d^{3} e + 4 \, a c d e^{3}\right )} x}{60 \,{\left (e^{9} x^{6} + 6 \, d e^{8} x^{5} + 15 \, d^{2} e^{7} x^{4} + 20 \, d^{3} e^{6} x^{3} + 15 \, d^{4} e^{5} x^{2} + 6 \, d^{5} e^{4} x + d^{6} e^{3}\right )}} \]
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)^9,x, algorithm="maxima")
[Out]
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Fricas [A] time = 0.205128, size = 176, normalized size = 2.29 \[ -\frac{15 \, c^{2} d^{2} e^{2} x^{2} + c^{2} d^{4} + 4 \, a c d^{2} e^{2} + 10 \, a^{2} e^{4} + 6 \,{\left (c^{2} d^{3} e + 4 \, a c d e^{3}\right )} x}{60 \,{\left (e^{9} x^{6} + 6 \, d e^{8} x^{5} + 15 \, d^{2} e^{7} x^{4} + 20 \, d^{3} e^{6} x^{3} + 15 \, d^{4} e^{5} x^{2} + 6 \, d^{5} e^{4} x + d^{6} e^{3}\right )}} \]
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)^9,x, algorithm="fricas")
[Out]
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Sympy [A] time = 8.20987, size = 138, normalized size = 1.79 \[ - \frac{10 a^{2} e^{4} + 4 a c d^{2} e^{2} + c^{2} d^{4} + 15 c^{2} d^{2} e^{2} x^{2} + x \left (24 a c d e^{3} + 6 c^{2} d^{3} e\right )}{60 d^{6} e^{3} + 360 d^{5} e^{4} x + 900 d^{4} e^{5} x^{2} + 1200 d^{3} e^{6} x^{3} + 900 d^{2} e^{7} x^{4} + 360 d e^{8} x^{5} + 60 e^{9} x^{6}} \]
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)**9,x)
[Out]
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GIAC/XCAS [A] time = 0.209669, size = 189, normalized size = 2.45 \[ -\frac{{\left (15 \, c^{2} d^{2} x^{4} e^{4} + 36 \, c^{2} d^{3} x^{3} e^{3} + 28 \, c^{2} d^{4} x^{2} e^{2} + 8 \, c^{2} d^{5} x e + c^{2} d^{6} + 24 \, a c d x^{3} e^{5} + 52 \, a c d^{2} x^{2} e^{4} + 32 \, a c d^{3} x e^{3} + 4 \, a c d^{4} e^{2} + 10 \, a^{2} x^{2} e^{6} + 20 \, a^{2} d x e^{5} + 10 \, a^{2} d^{2} e^{4}\right )} e^{\left (-3\right )}}{60 \,{\left (x e + d\right )}^{8}} \]
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)^9,x, algorithm="giac")
[Out]