Optimal. Leaf size=133 \[ a^4 x+2 a^3 b x^2+\frac{2}{3} a^2 x^3 \left (2 a c+3 b^2\right )+\frac{1}{5} x^5 \left (6 a^2 c^2+12 a b^2 c+b^4\right )+\frac{2}{7} c^2 x^7 \left (2 a c+3 b^2\right )+\frac{2}{3} b c x^6 \left (3 a c+b^2\right )+a b x^4 \left (3 a c+b^2\right )+\frac{1}{2} b c^3 x^8+\frac{c^4 x^9}{9} \]
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Rubi [A] time = 0.244018, antiderivative size = 133, 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 \[ a^4 x+2 a^3 b x^2+\frac{2}{3} a^2 x^3 \left (2 a c+3 b^2\right )+\frac{1}{5} x^5 \left (6 a^2 c^2+12 a b^2 c+b^4\right )+\frac{2}{7} c^2 x^7 \left (2 a c+3 b^2\right )+\frac{2}{3} b c x^6 \left (3 a c+b^2\right )+a b x^4 \left (3 a c+b^2\right )+\frac{1}{2} b c^3 x^8+\frac{c^4 x^9}{9} \]
Antiderivative was successfully verified.
[In] Int[(a + b*x + c*x^2)^4,x]
[Out]
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Rubi in Sympy [F] time = 0., size = 0, normalized size = 0. \[ 4 a^{3} b \int x\, dx + \frac{2 a^{2} x^{3} \left (2 a c + 3 b^{2}\right )}{3} + a b x^{4} \left (3 a c + b^{2}\right ) + \frac{b c^{3} x^{8}}{2} + \frac{2 b c x^{6} \left (3 a c + b^{2}\right )}{3} + \frac{c^{4} x^{9}}{9} + \frac{2 c^{2} x^{7} \left (2 a c + 3 b^{2}\right )}{7} + x^{5} \left (\frac{6 a^{2} c^{2}}{5} + \frac{12 a b^{2} c}{5} + \frac{b^{4}}{5}\right ) + \int a^{4}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate((c*x**2+b*x+a)**4,x)
[Out]
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Mathematica [A] time = 0.0364397, size = 133, normalized size = 1. \[ a^4 x+2 a^3 b x^2+\frac{2}{3} a^2 x^3 \left (2 a c+3 b^2\right )+\frac{1}{5} x^5 \left (6 a^2 c^2+12 a b^2 c+b^4\right )+\frac{2}{7} c^2 x^7 \left (2 a c+3 b^2\right )+\frac{2}{3} b c x^6 \left (3 a c+b^2\right )+a b x^4 \left (3 a c+b^2\right )+\frac{1}{2} b c^3 x^8+\frac{c^4 x^9}{9} \]
Antiderivative was successfully verified.
[In] Integrate[(a + b*x + c*x^2)^4,x]
[Out]
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Maple [A] time = 0.001, size = 168, normalized size = 1.3 \[{\frac{{c}^{4}{x}^{9}}{9}}+{\frac{b{c}^{3}{x}^{8}}{2}}+{\frac{ \left ( 2\,{c}^{2} \left ( 2\,ac+{b}^{2} \right ) +4\,{b}^{2}{c}^{2} \right ){x}^{7}}{7}}+{\frac{ \left ( 4\,ab{c}^{2}+4\,bc \left ( 2\,ac+{b}^{2} \right ) \right ){x}^{6}}{6}}+{\frac{ \left ( 2\,{a}^{2}{c}^{2}+8\,ac{b}^{2}+ \left ( 2\,ac+{b}^{2} \right ) ^{2} \right ){x}^{5}}{5}}+{\frac{ \left ( 4\,{a}^{2}bc+4\,ab \left ( 2\,ac+{b}^{2} \right ) \right ){x}^{4}}{4}}+{\frac{ \left ( 2\,{a}^{2} \left ( 2\,ac+{b}^{2} \right ) +4\,{a}^{2}{b}^{2} \right ){x}^{3}}{3}}+2\,{a}^{3}b{x}^{2}+{a}^{4}x \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int((c*x^2+b*x+a)^4,x)
[Out]
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Maxima [A] time = 0.803426, size = 184, normalized size = 1.38 \[ \frac{1}{9} \, c^{4} x^{9} + \frac{1}{2} \, b c^{3} x^{8} + \frac{6}{7} \, b^{2} c^{2} x^{7} + \frac{2}{3} \, b^{3} c x^{6} + \frac{1}{5} \, b^{4} x^{5} + a^{4} x + \frac{2}{3} \,{\left (2 \, c x^{3} + 3 \, b x^{2}\right )} a^{3} + \frac{1}{5} \,{\left (6 \, c^{2} x^{5} + 15 \, b c x^{4} + 10 \, b^{2} x^{3}\right )} a^{2} + \frac{1}{35} \,{\left (20 \, c^{3} x^{7} + 70 \, b c^{2} x^{6} + 84 \, b^{2} c x^{5} + 35 \, b^{3} x^{4}\right )} a \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((c*x^2 + b*x + a)^4,x, algorithm="maxima")
[Out]
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Fricas [A] time = 0.181729, size = 1, normalized size = 0.01 \[ \frac{1}{9} x^{9} c^{4} + \frac{1}{2} x^{8} c^{3} b + \frac{6}{7} x^{7} c^{2} b^{2} + \frac{4}{7} x^{7} c^{3} a + \frac{2}{3} x^{6} c b^{3} + 2 x^{6} c^{2} b a + \frac{1}{5} x^{5} b^{4} + \frac{12}{5} x^{5} c b^{2} a + \frac{6}{5} x^{5} c^{2} a^{2} + x^{4} b^{3} a + 3 x^{4} c b a^{2} + 2 x^{3} b^{2} a^{2} + \frac{4}{3} x^{3} c a^{3} + 2 x^{2} b a^{3} + x a^{4} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((c*x^2 + b*x + a)^4,x, algorithm="fricas")
[Out]
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Sympy [A] time = 0.172563, size = 141, normalized size = 1.06 \[ a^{4} x + 2 a^{3} b x^{2} + \frac{b c^{3} x^{8}}{2} + \frac{c^{4} x^{9}}{9} + x^{7} \left (\frac{4 a c^{3}}{7} + \frac{6 b^{2} c^{2}}{7}\right ) + x^{6} \left (2 a b c^{2} + \frac{2 b^{3} c}{3}\right ) + x^{5} \left (\frac{6 a^{2} c^{2}}{5} + \frac{12 a b^{2} c}{5} + \frac{b^{4}}{5}\right ) + x^{4} \left (3 a^{2} b c + a b^{3}\right ) + x^{3} \left (\frac{4 a^{3} c}{3} + 2 a^{2} b^{2}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((c*x**2+b*x+a)**4,x)
[Out]
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GIAC/XCAS [A] time = 0.202827, size = 186, normalized size = 1.4 \[ \frac{1}{9} \, c^{4} x^{9} + \frac{1}{2} \, b c^{3} x^{8} + \frac{6}{7} \, b^{2} c^{2} x^{7} + \frac{4}{7} \, a c^{3} x^{7} + \frac{2}{3} \, b^{3} c x^{6} + 2 \, a b c^{2} x^{6} + \frac{1}{5} \, b^{4} x^{5} + \frac{12}{5} \, a b^{2} c x^{5} + \frac{6}{5} \, a^{2} c^{2} x^{5} + a b^{3} x^{4} + 3 \, a^{2} b c x^{4} + 2 \, a^{2} b^{2} x^{3} + \frac{4}{3} \, a^{3} c x^{3} + 2 \, a^{3} b x^{2} + a^{4} x \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((c*x^2 + b*x + a)^4,x, algorithm="giac")
[Out]