Optimal. Leaf size=111 \[ \frac{3 c^2 d^2 \left (c d^2-a e^2\right )}{5 e^4 (d+e x)^5}-\frac{c d \left (c d^2-a e^2\right )^2}{2 e^4 (d+e x)^6}+\frac{\left (c d^2-a e^2\right )^3}{7 e^4 (d+e x)^7}-\frac{c^3 d^3}{4 e^4 (d+e x)^4} \]
[Out]
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Rubi [A] time = 0.166494, antiderivative size = 111, 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 \[ \frac{3 c^2 d^2 \left (c d^2-a e^2\right )}{5 e^4 (d+e x)^5}-\frac{c d \left (c d^2-a e^2\right )^2}{2 e^4 (d+e x)^6}+\frac{\left (c d^2-a e^2\right )^3}{7 e^4 (d+e x)^7}-\frac{c^3 d^3}{4 e^4 (d+e x)^4} \]
Antiderivative was successfully verified.
[In] Int[(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^3/(d + e*x)^11,x]
[Out]
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Rubi in Sympy [A] time = 44.5606, size = 100, normalized size = 0.9 \[ - \frac{c^{3} d^{3}}{4 e^{4} \left (d + e x\right )^{4}} - \frac{3 c^{2} d^{2} \left (a e^{2} - c d^{2}\right )}{5 e^{4} \left (d + e x\right )^{5}} - \frac{c d \left (a e^{2} - c d^{2}\right )^{2}}{2 e^{4} \left (d + e x\right )^{6}} - \frac{\left (a e^{2} - c d^{2}\right )^{3}}{7 e^{4} \left (d + e x\right )^{7}} \]
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)**3/(e*x+d)**11,x)
[Out]
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Mathematica [A] time = 0.0672588, size = 103, normalized size = 0.93 \[ -\frac{20 a^3 e^6+10 a^2 c d e^4 (d+7 e x)+4 a c^2 d^2 e^2 \left (d^2+7 d e x+21 e^2 x^2\right )+c^3 d^3 \left (d^3+7 d^2 e x+21 d e^2 x^2+35 e^3 x^3\right )}{140 e^4 (d+e x)^7} \]
Antiderivative was successfully verified.
[In] Integrate[(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^3/(d + e*x)^11,x]
[Out]
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Maple [A] time = 0.009, size = 141, normalized size = 1.3 \[ -{\frac{cd \left ({a}^{2}{e}^{4}-2\,ac{d}^{2}{e}^{2}+{c}^{2}{d}^{4} \right ) }{2\,{e}^{4} \left ( ex+d \right ) ^{6}}}-{\frac{{a}^{3}{e}^{6}-3\,{a}^{2}c{d}^{2}{e}^{4}+3\,{c}^{2}{d}^{4}a{e}^{2}-{c}^{3}{d}^{6}}{7\,{e}^{4} \left ( ex+d \right ) ^{7}}}-{\frac{{c}^{3}{d}^{3}}{4\,{e}^{4} \left ( ex+d \right ) ^{4}}}-{\frac{3\,{c}^{2}{d}^{2} \left ( a{e}^{2}-c{d}^{2} \right ) }{5\,{e}^{4} \left ( ex+d \right ) ^{5}}} \]
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)^3/(e*x+d)^11,x)
[Out]
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Maxima [A] time = 0.739169, size = 266, normalized size = 2.4 \[ -\frac{35 \, c^{3} d^{3} e^{3} x^{3} + c^{3} d^{6} + 4 \, a c^{2} d^{4} e^{2} + 10 \, a^{2} c d^{2} e^{4} + 20 \, a^{3} e^{6} + 21 \,{\left (c^{3} d^{4} e^{2} + 4 \, a c^{2} d^{2} e^{4}\right )} x^{2} + 7 \,{\left (c^{3} d^{5} e + 4 \, a c^{2} d^{3} e^{3} + 10 \, a^{2} c d e^{5}\right )} x}{140 \,{\left (e^{11} x^{7} + 7 \, d e^{10} x^{6} + 21 \, d^{2} e^{9} x^{5} + 35 \, d^{3} e^{8} x^{4} + 35 \, d^{4} e^{7} x^{3} + 21 \, d^{5} e^{6} x^{2} + 7 \, d^{6} e^{5} x + d^{7} e^{4}\right )}} \]
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)^3/(e*x + d)^11,x, algorithm="maxima")
[Out]
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Fricas [A] time = 0.215821, size = 266, normalized size = 2.4 \[ -\frac{35 \, c^{3} d^{3} e^{3} x^{3} + c^{3} d^{6} + 4 \, a c^{2} d^{4} e^{2} + 10 \, a^{2} c d^{2} e^{4} + 20 \, a^{3} e^{6} + 21 \,{\left (c^{3} d^{4} e^{2} + 4 \, a c^{2} d^{2} e^{4}\right )} x^{2} + 7 \,{\left (c^{3} d^{5} e + 4 \, a c^{2} d^{3} e^{3} + 10 \, a^{2} c d e^{5}\right )} x}{140 \,{\left (e^{11} x^{7} + 7 \, d e^{10} x^{6} + 21 \, d^{2} e^{9} x^{5} + 35 \, d^{3} e^{8} x^{4} + 35 \, d^{4} e^{7} x^{3} + 21 \, d^{5} e^{6} x^{2} + 7 \, d^{6} e^{5} x + d^{7} e^{4}\right )}} \]
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)^3/(e*x + d)^11,x, algorithm="fricas")
[Out]
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Sympy [F(-1)] time = 0., size = 0, normalized size = 0. \[ \text{Timed out} \]
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)**3/(e*x+d)**11,x)
[Out]
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GIAC/XCAS [A] time = 0.212123, size = 378, normalized size = 3.41 \[ -\frac{{\left (35 \, c^{3} d^{3} x^{6} e^{6} + 126 \, c^{3} d^{4} x^{5} e^{5} + 175 \, c^{3} d^{5} x^{4} e^{4} + 120 \, c^{3} d^{6} x^{3} e^{3} + 45 \, c^{3} d^{7} x^{2} e^{2} + 10 \, c^{3} d^{8} x e + c^{3} d^{9} + 84 \, a c^{2} d^{2} x^{5} e^{7} + 280 \, a c^{2} d^{3} x^{4} e^{6} + 340 \, a c^{2} d^{4} x^{3} e^{5} + 180 \, a c^{2} d^{5} x^{2} e^{4} + 40 \, a c^{2} d^{6} x e^{3} + 4 \, a c^{2} d^{7} e^{2} + 70 \, a^{2} c d x^{4} e^{8} + 220 \, a^{2} c d^{2} x^{3} e^{7} + 240 \, a^{2} c d^{3} x^{2} e^{6} + 100 \, a^{2} c d^{4} x e^{5} + 10 \, a^{2} c d^{5} e^{4} + 20 \, a^{3} x^{3} e^{9} + 60 \, a^{3} d x^{2} e^{8} + 60 \, a^{3} d^{2} x e^{7} + 20 \, a^{3} d^{3} e^{6}\right )} e^{\left (-4\right )}}{140 \,{\left (x e + d\right )}^{10}} \]
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)^3/(e*x + d)^11,x, algorithm="giac")
[Out]