Optimal. Leaf size=73 \[ \frac{c d (a e+c d x)^4}{20 (d+e x)^4 \left (c d^2-a e^2\right )^2}+\frac{(a e+c d x)^4}{5 (d+e x)^5 \left (c d^2-a e^2\right )} \]
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Rubi [A] time = 0.0772554, antiderivative size = 73, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 35, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.086 \[ \frac{c d (a e+c d x)^4}{20 (d+e x)^4 \left (c d^2-a e^2\right )^2}+\frac{(a e+c d x)^4}{5 (d+e x)^5 \left (c d^2-a e^2\right )} \]
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)^9,x]
[Out]
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Rubi in Sympy [A] time = 20.8233, size = 61, normalized size = 0.84 \[ \frac{c d \left (a e + c d x\right )^{4}}{20 \left (d + e x\right )^{4} \left (a e^{2} - c d^{2}\right )^{2}} - \frac{\left (a e + c d x\right )^{4}}{5 \left (d + e x\right )^{5} \left (a e^{2} - c d^{2}\right )} \]
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)**9,x)
[Out]
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Mathematica [A] time = 0.0759064, size = 103, normalized size = 1.41 \[ -\frac{4 a^3 e^6+3 a^2 c d e^4 (d+5 e x)+2 a c^2 d^2 e^2 \left (d^2+5 d e x+10 e^2 x^2\right )+c^3 d^3 \left (d^3+5 d^2 e x+10 d e^2 x^2+10 e^3 x^3\right )}{20 e^4 (d+e x)^5} \]
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)^9,x]
[Out]
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Maple [B] time = 0.01, size = 141, normalized size = 1.9 \[ -{\frac{3\,cd \left ({a}^{2}{e}^{4}-2\,ac{d}^{2}{e}^{2}+{c}^{2}{d}^{4} \right ) }{4\,{e}^{4} \left ( ex+d \right ) ^{4}}}-{\frac{{c}^{2}{d}^{2} \left ( a{e}^{2}-c{d}^{2} \right ) }{{e}^{4} \left ( ex+d \right ) ^{3}}}-{\frac{{c}^{3}{d}^{3}}{2\,{e}^{4} \left ( ex+d \right ) ^{2}}}-{\frac{{a}^{3}{e}^{6}-3\,{a}^{2}c{d}^{2}{e}^{4}+3\,{c}^{2}{d}^{4}a{e}^{2}-{c}^{3}{d}^{6}}{5\,{e}^{4} \left ( ex+d \right ) ^{5}}} \]
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)^9,x)
[Out]
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Maxima [A] time = 0.728006, size = 236, normalized size = 3.23 \[ -\frac{10 \, c^{3} d^{3} e^{3} x^{3} + c^{3} d^{6} + 2 \, a c^{2} d^{4} e^{2} + 3 \, a^{2} c d^{2} e^{4} + 4 \, a^{3} e^{6} + 10 \,{\left (c^{3} d^{4} e^{2} + 2 \, a c^{2} d^{2} e^{4}\right )} x^{2} + 5 \,{\left (c^{3} d^{5} e + 2 \, a c^{2} d^{3} e^{3} + 3 \, a^{2} c d e^{5}\right )} x}{20 \,{\left (e^{9} x^{5} + 5 \, d e^{8} x^{4} + 10 \, d^{2} e^{7} x^{3} + 10 \, d^{3} e^{6} x^{2} + 5 \, d^{4} e^{5} x + d^{5} e^{4}\right )}} \]
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)^9,x, algorithm="maxima")
[Out]
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Fricas [A] time = 0.213125, size = 236, normalized size = 3.23 \[ -\frac{10 \, c^{3} d^{3} e^{3} x^{3} + c^{3} d^{6} + 2 \, a c^{2} d^{4} e^{2} + 3 \, a^{2} c d^{2} e^{4} + 4 \, a^{3} e^{6} + 10 \,{\left (c^{3} d^{4} e^{2} + 2 \, a c^{2} d^{2} e^{4}\right )} x^{2} + 5 \,{\left (c^{3} d^{5} e + 2 \, a c^{2} d^{3} e^{3} + 3 \, a^{2} c d e^{5}\right )} x}{20 \,{\left (e^{9} x^{5} + 5 \, d e^{8} x^{4} + 10 \, d^{2} e^{7} x^{3} + 10 \, d^{3} e^{6} x^{2} + 5 \, d^{4} e^{5} x + d^{5} e^{4}\right )}} \]
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)^9,x, algorithm="fricas")
[Out]
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Sympy [A] time = 37.4677, size = 185, normalized size = 2.53 \[ - \frac{4 a^{3} e^{6} + 3 a^{2} c d^{2} e^{4} + 2 a c^{2} d^{4} e^{2} + c^{3} d^{6} + 10 c^{3} d^{3} e^{3} x^{3} + x^{2} \left (20 a c^{2} d^{2} e^{4} + 10 c^{3} d^{4} e^{2}\right ) + x \left (15 a^{2} c d e^{5} + 10 a c^{2} d^{3} e^{3} + 5 c^{3} d^{5} e\right )}{20 d^{5} e^{4} + 100 d^{4} e^{5} x + 200 d^{3} e^{6} x^{2} + 200 d^{2} e^{7} x^{3} + 100 d e^{8} x^{4} + 20 e^{9} x^{5}} \]
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)**9,x)
[Out]
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GIAC/XCAS [A] time = 0.211544, size = 378, normalized size = 5.18 \[ -\frac{{\left (10 \, c^{3} d^{3} x^{6} e^{6} + 40 \, c^{3} d^{4} x^{5} e^{5} + 65 \, c^{3} d^{5} x^{4} e^{4} + 56 \, c^{3} d^{6} x^{3} e^{3} + 28 \, c^{3} d^{7} x^{2} e^{2} + 8 \, c^{3} d^{8} x e + c^{3} d^{9} + 20 \, a c^{2} d^{2} x^{5} e^{7} + 70 \, a c^{2} d^{3} x^{4} e^{6} + 92 \, a c^{2} d^{4} x^{3} e^{5} + 56 \, a c^{2} d^{5} x^{2} e^{4} + 16 \, a c^{2} d^{6} x e^{3} + 2 \, a c^{2} d^{7} e^{2} + 15 \, a^{2} c d x^{4} e^{8} + 48 \, a^{2} c d^{2} x^{3} e^{7} + 54 \, a^{2} c d^{3} x^{2} e^{6} + 24 \, a^{2} c d^{4} x e^{5} + 3 \, a^{2} c d^{5} e^{4} + 4 \, a^{3} x^{3} e^{9} + 12 \, a^{3} d x^{2} e^{8} + 12 \, a^{3} d^{2} x e^{7} + 4 \, a^{3} d^{3} e^{6}\right )} e^{\left (-4\right )}}{20 \,{\left (x e + d\right )}^{8}} \]
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)^3/(e*x + d)^9,x, algorithm="giac")
[Out]