Optimal. Leaf size=158 \[ a^3 A x+\frac{1}{2} a^2 x^2 (a B+3 A b)+\frac{1}{2} c x^6 \left (a B c+A b c+b^2 B\right )+a x^3 \left (A \left (a c+b^2\right )+a b B\right )+\frac{1}{5} x^5 \left (3 a A c^2+6 a b B c+3 A b^2 c+b^3 B\right )+\frac{1}{4} x^4 \left (A \left (6 a b c+b^3\right )+3 a B \left (a c+b^2\right )\right )+\frac{1}{7} c^2 x^7 (A c+3 b B)+\frac{1}{8} B c^3 x^8 \]
[Out]
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Rubi [A] time = 0.427088, antiderivative size = 158, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 1, integrand size = 18, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.056 \[ a^3 A x+\frac{1}{2} a^2 x^2 (a B+3 A b)+\frac{1}{2} c x^6 \left (a B c+A b c+b^2 B\right )+a x^3 \left (A \left (a c+b^2\right )+a b B\right )+\frac{1}{5} x^5 \left (3 a A c^2+6 a b B c+3 A b^2 c+b^3 B\right )+\frac{1}{4} x^4 \left (A \left (6 a b c+b^3\right )+3 a B \left (a c+b^2\right )\right )+\frac{1}{7} c^2 x^7 (A c+3 b B)+\frac{1}{8} B c^3 x^8 \]
Antiderivative was successfully verified.
[In] Int[(A + B*x)*(a + b*x + c*x^2)^3,x]
[Out]
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Rubi in Sympy [F] time = 0., size = 0, normalized size = 0. \[ \frac{B c^{3} x^{8}}{8} + a^{3} \int A\, dx + a^{2} \left (3 A b + B a\right ) \int x\, dx + a x^{3} \left (A a c + A b^{2} + B a b\right ) + \frac{c^{2} x^{7} \left (A c + 3 B b\right )}{7} + \frac{c x^{6} \left (A b c + B a c + B b^{2}\right )}{2} + x^{5} \left (\frac{3 A a c^{2}}{5} + \frac{3 A b^{2} c}{5} + \frac{6 B a b c}{5} + \frac{B b^{3}}{5}\right ) + x^{4} \left (\frac{3 A a b c}{2} + \frac{A b^{3}}{4} + \frac{3 B a^{2} c}{4} + \frac{3 B a b^{2}}{4}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate((B*x+A)*(c*x**2+b*x+a)**3,x)
[Out]
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Mathematica [A] time = 0.0853155, size = 158, normalized size = 1. \[ a^3 A x+\frac{1}{2} a^2 x^2 (a B+3 A b)+\frac{1}{2} c x^6 \left (a B c+A b c+b^2 B\right )+a x^3 \left (A \left (a c+b^2\right )+a b B\right )+\frac{1}{5} x^5 \left (3 a A c^2+6 a b B c+3 A b^2 c+b^3 B\right )+\frac{1}{4} x^4 \left (A \left (6 a b c+b^3\right )+3 a B \left (a c+b^2\right )\right )+\frac{1}{7} c^2 x^7 (A c+3 b B)+\frac{1}{8} B c^3 x^8 \]
Antiderivative was successfully verified.
[In] Integrate[(A + B*x)*(a + b*x + c*x^2)^3,x]
[Out]
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Maple [A] time = 0.002, size = 223, normalized size = 1.4 \[{\frac{B{c}^{3}{x}^{8}}{8}}+{\frac{ \left ( A{c}^{3}+3\,Bb{c}^{2} \right ){x}^{7}}{7}}+{\frac{ \left ( 3\,Ab{c}^{2}+B \left ( a{c}^{2}+2\,{b}^{2}c+c \left ( 2\,ac+{b}^{2} \right ) \right ) \right ){x}^{6}}{6}}+{\frac{ \left ( A \left ( a{c}^{2}+2\,{b}^{2}c+c \left ( 2\,ac+{b}^{2} \right ) \right ) +B \left ( 4\,abc+b \left ( 2\,ac+{b}^{2} \right ) \right ) \right ){x}^{5}}{5}}+{\frac{ \left ( A \left ( 4\,abc+b \left ( 2\,ac+{b}^{2} \right ) \right ) +B \left ( a \left ( 2\,ac+{b}^{2} \right ) +2\,{b}^{2}a+c{a}^{2} \right ) \right ){x}^{4}}{4}}+{\frac{ \left ( A \left ( a \left ( 2\,ac+{b}^{2} \right ) +2\,{b}^{2}a+c{a}^{2} \right ) +3\,B{a}^{2}b \right ){x}^{3}}{3}}+{\frac{ \left ( 3\,A{a}^{2}b+B{a}^{3} \right ){x}^{2}}{2}}+{a}^{3}Ax \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int((B*x+A)*(c*x^2+b*x+a)^3,x)
[Out]
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Maxima [A] time = 0.692459, size = 219, normalized size = 1.39 \[ \frac{1}{8} \, B c^{3} x^{8} + \frac{1}{7} \,{\left (3 \, B b c^{2} + A c^{3}\right )} x^{7} + \frac{1}{2} \,{\left (B b^{2} c +{\left (B a + A b\right )} c^{2}\right )} x^{6} + \frac{1}{5} \,{\left (B b^{3} + 3 \, A a c^{2} + 3 \,{\left (2 \, B a b + A b^{2}\right )} c\right )} x^{5} + A a^{3} x + \frac{1}{4} \,{\left (3 \, B a b^{2} + A b^{3} + 3 \,{\left (B a^{2} + 2 \, A a b\right )} c\right )} x^{4} +{\left (B a^{2} b + A a b^{2} + A a^{2} c\right )} x^{3} + \frac{1}{2} \,{\left (B a^{3} + 3 \, A a^{2} b\right )} x^{2} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((c*x^2 + b*x + a)^3*(B*x + A),x, algorithm="maxima")
[Out]
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Fricas [A] time = 0.240802, size = 1, normalized size = 0.01 \[ \frac{1}{8} x^{8} c^{3} B + \frac{3}{7} x^{7} c^{2} b B + \frac{1}{7} x^{7} c^{3} A + \frac{1}{2} x^{6} c b^{2} B + \frac{1}{2} x^{6} c^{2} a B + \frac{1}{2} x^{6} c^{2} b A + \frac{1}{5} x^{5} b^{3} B + \frac{6}{5} x^{5} c b a B + \frac{3}{5} x^{5} c b^{2} A + \frac{3}{5} x^{5} c^{2} a A + \frac{3}{4} x^{4} b^{2} a B + \frac{3}{4} x^{4} c a^{2} B + \frac{1}{4} x^{4} b^{3} A + \frac{3}{2} x^{4} c b a A + x^{3} b a^{2} B + x^{3} b^{2} a A + x^{3} c a^{2} A + \frac{1}{2} x^{2} a^{3} B + \frac{3}{2} x^{2} b a^{2} A + x a^{3} A \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((c*x^2 + b*x + a)^3*(B*x + A),x, algorithm="fricas")
[Out]
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Sympy [A] time = 0.199783, size = 190, normalized size = 1.2 \[ A a^{3} x + \frac{B c^{3} x^{8}}{8} + x^{7} \left (\frac{A c^{3}}{7} + \frac{3 B b c^{2}}{7}\right ) + x^{6} \left (\frac{A b c^{2}}{2} + \frac{B a c^{2}}{2} + \frac{B b^{2} c}{2}\right ) + x^{5} \left (\frac{3 A a c^{2}}{5} + \frac{3 A b^{2} c}{5} + \frac{6 B a b c}{5} + \frac{B b^{3}}{5}\right ) + x^{4} \left (\frac{3 A a b c}{2} + \frac{A b^{3}}{4} + \frac{3 B a^{2} c}{4} + \frac{3 B a b^{2}}{4}\right ) + x^{3} \left (A a^{2} c + A a b^{2} + B a^{2} b\right ) + x^{2} \left (\frac{3 A a^{2} b}{2} + \frac{B a^{3}}{2}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((B*x+A)*(c*x**2+b*x+a)**3,x)
[Out]
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GIAC/XCAS [A] time = 0.282472, size = 252, normalized size = 1.59 \[ \frac{1}{8} \, B c^{3} x^{8} + \frac{3}{7} \, B b c^{2} x^{7} + \frac{1}{7} \, A c^{3} x^{7} + \frac{1}{2} \, B b^{2} c x^{6} + \frac{1}{2} \, B a c^{2} x^{6} + \frac{1}{2} \, A b c^{2} x^{6} + \frac{1}{5} \, B b^{3} x^{5} + \frac{6}{5} \, B a b c x^{5} + \frac{3}{5} \, A b^{2} c x^{5} + \frac{3}{5} \, A a c^{2} x^{5} + \frac{3}{4} \, B a b^{2} x^{4} + \frac{1}{4} \, A b^{3} x^{4} + \frac{3}{4} \, B a^{2} c x^{4} + \frac{3}{2} \, A a b c x^{4} + B a^{2} b x^{3} + A a b^{2} x^{3} + A a^{2} c x^{3} + \frac{1}{2} \, B a^{3} x^{2} + \frac{3}{2} \, A a^{2} b x^{2} + A a^{3} x \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((c*x^2 + b*x + a)^3*(B*x + A),x, algorithm="giac")
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