Optimal. Leaf size=81 \[ a^3 x+\frac{3}{2} a^2 b x^2+\frac{3}{5} c x^5 \left (a c+b^2\right )+\frac{1}{4} b x^4 \left (6 a c+b^2\right )+a x^3 \left (a c+b^2\right )+\frac{1}{2} b c^2 x^6+\frac{c^3 x^7}{7} \]
[Out]
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Rubi [A] time = 0.121756, antiderivative size = 81, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 1, integrand size = 12, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.083 \[ a^3 x+\frac{3}{2} a^2 b x^2+\frac{3}{5} c x^5 \left (a c+b^2\right )+\frac{1}{4} b x^4 \left (6 a c+b^2\right )+a x^3 \left (a c+b^2\right )+\frac{1}{2} b c^2 x^6+\frac{c^3 x^7}{7} \]
Antiderivative was successfully verified.
[In] Int[(a + b*x + c*x^2)^3,x]
[Out]
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Rubi in Sympy [F] time = 0., size = 0, normalized size = 0. \[ 3 a^{2} b \int x\, dx + a x^{3} \left (a c + b^{2}\right ) + \frac{b c^{2} x^{6}}{2} + \frac{b x^{4} \left (6 a c + b^{2}\right )}{4} + \frac{c^{3} x^{7}}{7} + \frac{3 c x^{5} \left (a c + b^{2}\right )}{5} + \int a^{3}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate((c*x**2+b*x+a)**3,x)
[Out]
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Mathematica [A] time = 0.0250905, size = 81, normalized size = 1. \[ a^3 x+\frac{3}{2} a^2 b x^2+\frac{3}{5} c x^5 \left (a c+b^2\right )+\frac{1}{4} b x^4 \left (6 a c+b^2\right )+a x^3 \left (a c+b^2\right )+\frac{1}{2} b c^2 x^6+\frac{c^3 x^7}{7} \]
Antiderivative was successfully verified.
[In] Integrate[(a + b*x + c*x^2)^3,x]
[Out]
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Maple [A] time = 0.001, size = 108, normalized size = 1.3 \[{\frac{{c}^{3}{x}^{7}}{7}}+{\frac{b{c}^{2}{x}^{6}}{2}}+{\frac{ \left ( a{c}^{2}+2\,{b}^{2}c+c \left ( 2\,ac+{b}^{2} \right ) \right ){x}^{5}}{5}}+{\frac{ \left ( 4\,abc+b \left ( 2\,ac+{b}^{2} \right ) \right ){x}^{4}}{4}}+{\frac{ \left ( a \left ( 2\,ac+{b}^{2} \right ) +2\,a{b}^{2}+{a}^{2}c \right ){x}^{3}}{3}}+{\frac{3\,{a}^{2}b{x}^{2}}{2}}+{a}^{3}x \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int((c*x^2+b*x+a)^3,x)
[Out]
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Maxima [A] time = 0.797455, size = 115, normalized size = 1.42 \[ \frac{1}{7} \, c^{3} x^{7} + \frac{1}{2} \, b c^{2} x^{6} + \frac{3}{5} \, b^{2} c x^{5} + \frac{1}{4} \, b^{3} x^{4} + a^{3} x + \frac{1}{2} \,{\left (2 \, c x^{3} + 3 \, b x^{2}\right )} a^{2} + \frac{1}{10} \,{\left (6 \, c^{2} x^{5} + 15 \, b c x^{4} + 10 \, b^{2} x^{3}\right )} a \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((c*x^2 + b*x + a)^3,x, algorithm="maxima")
[Out]
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Fricas [A] time = 0.191142, size = 1, normalized size = 0.01 \[ \frac{1}{7} x^{7} c^{3} + \frac{1}{2} x^{6} c^{2} b + \frac{3}{5} x^{5} c b^{2} + \frac{3}{5} x^{5} c^{2} a + \frac{1}{4} x^{4} b^{3} + \frac{3}{2} x^{4} c b a + x^{3} b^{2} a + x^{3} c a^{2} + \frac{3}{2} x^{2} b a^{2} + x a^{3} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((c*x^2 + b*x + a)^3,x, algorithm="fricas")
[Out]
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Sympy [A] time = 0.141302, size = 85, normalized size = 1.05 \[ a^{3} x + \frac{3 a^{2} b x^{2}}{2} + \frac{b c^{2} x^{6}}{2} + \frac{c^{3} x^{7}}{7} + x^{5} \left (\frac{3 a c^{2}}{5} + \frac{3 b^{2} c}{5}\right ) + x^{4} \left (\frac{3 a b c}{2} + \frac{b^{3}}{4}\right ) + x^{3} \left (a^{2} c + a b^{2}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((c*x**2+b*x+a)**3,x)
[Out]
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GIAC/XCAS [A] time = 0.202217, size = 111, normalized size = 1.37 \[ \frac{1}{7} \, c^{3} x^{7} + \frac{1}{2} \, b c^{2} x^{6} + \frac{3}{5} \, b^{2} c x^{5} + \frac{3}{5} \, a c^{2} x^{5} + \frac{1}{4} \, b^{3} x^{4} + \frac{3}{2} \, a b c x^{4} + a b^{2} x^{3} + a^{2} c x^{3} + \frac{3}{2} \, a^{2} b x^{2} + a^{3} x \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((c*x^2 + b*x + a)^3,x, algorithm="giac")
[Out]