Optimal. Leaf size=39 \[ \frac{1}{5} (d+e x)^5 \left (a-\frac{c d^2}{e^2}\right )+\frac{c d (d+e x)^6}{6 e^2} \]
[Out]
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Rubi [A] time = 0.0752802, 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 \[ \frac{1}{5} (d+e x)^5 \left (a-\frac{c d^2}{e^2}\right )+\frac{c d (d+e x)^6}{6 e^2} \]
Antiderivative was successfully verified.
[In] Int[(d + e*x)^3*(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2),x]
[Out]
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Rubi in Sympy [A] time = 23.6721, size = 36, normalized size = 0.92 \[ \frac{c d \left (d + e x\right )^{6}}{6 e^{2}} + \frac{\left (d + e x\right )^{5} \left (a e^{2} - c d^{2}\right )}{5 e^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate((e*x+d)**3*(a*d*e+(a*e**2+c*d**2)*x+c*d*e*x**2),x)
[Out]
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Mathematica [B] time = 0.0369817, size = 95, normalized size = 2.44 \[ \frac{1}{30} x \left (6 a e \left (5 d^4+10 d^3 e x+10 d^2 e^2 x^2+5 d e^3 x^3+e^4 x^4\right )+c d x \left (15 d^4+40 d^3 e x+45 d^2 e^2 x^2+24 d e^3 x^3+5 e^4 x^4\right )\right ) \]
Antiderivative was successfully verified.
[In] Integrate[(d + e*x)^3*(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2),x]
[Out]
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Maple [B] time = 0.002, size = 155, normalized size = 4. \[{\frac{{e}^{4}dc{x}^{6}}{6}}+{\frac{ \left ( 3\,{d}^{2}{e}^{3}c+{e}^{3} \left ( a{e}^{2}+c{d}^{2} \right ) \right ){x}^{5}}{5}}+{\frac{ \left ( 3\,{d}^{3}{e}^{2}c+3\,d{e}^{2} \left ( a{e}^{2}+c{d}^{2} \right ) +{e}^{4}ad \right ){x}^{4}}{4}}+{\frac{ \left ({d}^{4}ec+3\,{d}^{2}e \left ( a{e}^{2}+c{d}^{2} \right ) +3\,{d}^{2}{e}^{3}a \right ){x}^{3}}{3}}+{\frac{ \left ({d}^{3} \left ( a{e}^{2}+c{d}^{2} \right ) +3\,{d}^{3}{e}^{2}a \right ){x}^{2}}{2}}+{d}^{4}aex \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int((e*x+d)^3*(a*e*d+(a*e^2+c*d^2)*x+c*d*e*x^2),x)
[Out]
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Maxima [A] time = 0.711707, size = 138, normalized size = 3.54 \[ \frac{1}{6} \, c d e^{4} x^{6} + a d^{4} e x + \frac{1}{5} \,{\left (4 \, c d^{2} e^{3} + a e^{5}\right )} x^{5} + \frac{1}{2} \,{\left (3 \, c d^{3} e^{2} + 2 \, a d e^{4}\right )} x^{4} + \frac{2}{3} \,{\left (2 \, c d^{4} e + 3 \, a d^{2} e^{3}\right )} x^{3} + \frac{1}{2} \,{\left (c d^{5} + 4 \, a d^{3} e^{2}\right )} x^{2} \]
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)^3,x, algorithm="maxima")
[Out]
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Fricas [A] time = 0.177223, size = 1, normalized size = 0.03 \[ \frac{1}{6} x^{6} e^{4} d c + \frac{4}{5} x^{5} e^{3} d^{2} c + \frac{1}{5} x^{5} e^{5} a + \frac{3}{2} x^{4} e^{2} d^{3} c + x^{4} e^{4} d a + \frac{4}{3} x^{3} e d^{4} c + 2 x^{3} e^{3} d^{2} a + \frac{1}{2} x^{2} d^{5} c + 2 x^{2} e^{2} d^{3} a + x e d^{4} a \]
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)^3,x, algorithm="fricas")
[Out]
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Sympy [A] time = 0.165399, size = 107, normalized size = 2.74 \[ a d^{4} e x + \frac{c d e^{4} x^{6}}{6} + x^{5} \left (\frac{a e^{5}}{5} + \frac{4 c d^{2} e^{3}}{5}\right ) + x^{4} \left (a d e^{4} + \frac{3 c d^{3} e^{2}}{2}\right ) + x^{3} \left (2 a d^{2} e^{3} + \frac{4 c d^{4} e}{3}\right ) + x^{2} \left (2 a d^{3} e^{2} + \frac{c d^{5}}{2}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((e*x+d)**3*(a*d*e+(a*e**2+c*d**2)*x+c*d*e*x**2),x)
[Out]
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GIAC/XCAS [A] time = 0.208519, size = 132, normalized size = 3.38 \[ \frac{1}{6} \, c d x^{6} e^{4} + \frac{4}{5} \, c d^{2} x^{5} e^{3} + \frac{3}{2} \, c d^{3} x^{4} e^{2} + \frac{4}{3} \, c d^{4} x^{3} e + \frac{1}{2} \, c d^{5} x^{2} + \frac{1}{5} \, a x^{5} e^{5} + a d x^{4} e^{4} + 2 \, a d^{2} x^{3} e^{3} + 2 \, a d^{3} x^{2} e^{2} + a d^{4} x e \]
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)^3,x, algorithm="giac")
[Out]