Optimal. Leaf size=77 \[ -\frac{2 c d (d+e x)^5 \left (c d^2-a e^2\right )}{5 e^3}+\frac{(d+e x)^4 \left (c d^2-a e^2\right )^2}{4 e^3}+\frac{c^2 d^2 (d+e x)^6}{6 e^3} \]
[Out]
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Rubi [A] time = 0.237548, antiderivative size = 77, 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 \[ -\frac{2 c d (d+e x)^5 \left (c d^2-a e^2\right )}{5 e^3}+\frac{(d+e x)^4 \left (c d^2-a e^2\right )^2}{4 e^3}+\frac{c^2 d^2 (d+e x)^6}{6 e^3} \]
Antiderivative was successfully verified.
[In] Int[(d + e*x)*(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^2,x]
[Out]
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Rubi in Sympy [A] time = 36.7809, size = 68, normalized size = 0.88 \[ \frac{c^{2} d^{2} \left (d + e x\right )^{6}}{6 e^{3}} + \frac{2 c d \left (d + e x\right )^{5} \left (a e^{2} - c d^{2}\right )}{5 e^{3}} + \frac{\left (d + e x\right )^{4} \left (a e^{2} - c d^{2}\right )^{2}}{4 e^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate((e*x+d)*(a*d*e+(a*e**2+c*d**2)*x+c*d*e*x**2)**2,x)
[Out]
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Mathematica [A] time = 0.0667513, size = 120, normalized size = 1.56 \[ \frac{1}{60} x \left (15 a^2 e^2 \left (4 d^3+6 d^2 e x+4 d e^2 x^2+e^3 x^3\right )+6 a c d e x \left (10 d^3+20 d^2 e x+15 d e^2 x^2+4 e^3 x^3\right )+c^2 d^2 x^2 \left (20 d^3+45 d^2 e x+36 d e^2 x^2+10 e^3 x^3\right )\right ) \]
Antiderivative was successfully verified.
[In] Integrate[(d + e*x)*(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^2,x]
[Out]
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Maple [B] time = 0.001, size = 195, normalized size = 2.5 \[{\frac{{d}^{2}{e}^{3}{c}^{2}{x}^{6}}{6}}+{\frac{ \left ({d}^{3}{e}^{2}{c}^{2}+2\,{e}^{2} \left ( a{e}^{2}+c{d}^{2} \right ) dc \right ){x}^{5}}{5}}+{\frac{ \left ( 2\,{d}^{2} \left ( a{e}^{2}+c{d}^{2} \right ) ec+e \left ( 2\,ac{d}^{2}{e}^{2}+ \left ( a{e}^{2}+c{d}^{2} \right ) ^{2} \right ) \right ){x}^{4}}{4}}+{\frac{ \left ( d \left ( 2\,ac{d}^{2}{e}^{2}+ \left ( a{e}^{2}+c{d}^{2} \right ) ^{2} \right ) +2\,{e}^{2}ad \left ( a{e}^{2}+c{d}^{2} \right ) \right ){x}^{3}}{3}}+{\frac{ \left ( 2\,{d}^{2}ae \left ( a{e}^{2}+c{d}^{2} \right ) +{a}^{2}{e}^{3}{d}^{2} \right ){x}^{2}}{2}}+{a}^{2}{e}^{2}{d}^{3}x \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int((e*x+d)*(a*e*d+(a*e^2+c*d^2)*x+c*d*e*x^2)^2,x)
[Out]
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Maxima [A] time = 0.723371, size = 189, normalized size = 2.45 \[ \frac{1}{6} \, c^{2} d^{2} e^{3} x^{6} + a^{2} d^{3} e^{2} x + \frac{1}{5} \,{\left (3 \, c^{2} d^{3} e^{2} + 2 \, a c d e^{4}\right )} x^{5} + \frac{1}{4} \,{\left (3 \, c^{2} d^{4} e + 6 \, a c d^{2} e^{3} + a^{2} e^{5}\right )} x^{4} + \frac{1}{3} \,{\left (c^{2} d^{5} + 6 \, a c d^{3} e^{2} + 3 \, a^{2} d e^{4}\right )} x^{3} + \frac{1}{2} \,{\left (2 \, a c d^{4} e + 3 \, a^{2} d^{2} e^{3}\right )} x^{2} \]
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)^2*(e*x + d),x, algorithm="maxima")
[Out]
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Fricas [A] time = 0.212832, size = 1, normalized size = 0.01 \[ \frac{1}{6} x^{6} e^{3} d^{2} c^{2} + \frac{3}{5} x^{5} e^{2} d^{3} c^{2} + \frac{2}{5} x^{5} e^{4} d c a + \frac{3}{4} x^{4} e d^{4} c^{2} + \frac{3}{2} x^{4} e^{3} d^{2} c a + \frac{1}{4} x^{4} e^{5} a^{2} + \frac{1}{3} x^{3} d^{5} c^{2} + 2 x^{3} e^{2} d^{3} c a + x^{3} e^{4} d a^{2} + x^{2} e d^{4} c a + \frac{3}{2} x^{2} e^{3} d^{2} a^{2} + x e^{2} d^{3} a^{2} \]
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)^2*(e*x + d),x, algorithm="fricas")
[Out]
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Sympy [A] time = 0.20705, size = 150, normalized size = 1.95 \[ a^{2} d^{3} e^{2} x + \frac{c^{2} d^{2} e^{3} x^{6}}{6} + x^{5} \left (\frac{2 a c d e^{4}}{5} + \frac{3 c^{2} d^{3} e^{2}}{5}\right ) + x^{4} \left (\frac{a^{2} e^{5}}{4} + \frac{3 a c d^{2} e^{3}}{2} + \frac{3 c^{2} d^{4} e}{4}\right ) + x^{3} \left (a^{2} d e^{4} + 2 a c d^{3} e^{2} + \frac{c^{2} d^{5}}{3}\right ) + x^{2} \left (\frac{3 a^{2} d^{2} e^{3}}{2} + a c d^{4} e\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((e*x+d)*(a*d*e+(a*e**2+c*d**2)*x+c*d*e*x**2)**2,x)
[Out]
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GIAC/XCAS [A] time = 0.20809, size = 188, normalized size = 2.44 \[ \frac{1}{6} \, c^{2} d^{2} x^{6} e^{3} + \frac{3}{5} \, c^{2} d^{3} x^{5} e^{2} + \frac{3}{4} \, c^{2} d^{4} x^{4} e + \frac{1}{3} \, c^{2} d^{5} x^{3} + \frac{2}{5} \, a c d x^{5} e^{4} + \frac{3}{2} \, a c d^{2} x^{4} e^{3} + 2 \, a c d^{3} x^{3} e^{2} + a c d^{4} x^{2} e + \frac{1}{4} \, a^{2} x^{4} e^{5} + a^{2} d x^{3} e^{4} + \frac{3}{2} \, a^{2} d^{2} x^{2} e^{3} + a^{2} d^{3} x e^{2} \]
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)^2*(e*x + d),x, algorithm="giac")
[Out]