Optimal. Leaf size=60 \[ \frac{1}{3} x^3 (a C+A b)+a A x+\frac{1}{4} x^4 (a D+b B)+\frac{1}{2} a B x^2+\frac{1}{5} b C x^5+\frac{1}{6} b D x^6 \]
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Rubi [A] time = 0.0906429, antiderivative size = 60, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 1, integrand size = 23, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.043 \[ \frac{1}{3} x^3 (a C+A b)+a A x+\frac{1}{4} x^4 (a D+b B)+\frac{1}{2} a B x^2+\frac{1}{5} b C x^5+\frac{1}{6} b D x^6 \]
Antiderivative was successfully verified.
[In] Int[(a + b*x^2)*(A + B*x + C*x^2 + D*x^3),x]
[Out]
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Rubi in Sympy [F] time = 0., size = 0, normalized size = 0. \[ B a \int x\, dx + \frac{C b x^{5}}{5} + \frac{D b x^{6}}{6} + a \int A\, dx + x^{4} \left (\frac{B b}{4} + \frac{D a}{4}\right ) + x^{3} \left (\frac{A b}{3} + \frac{C a}{3}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate((b*x**2+a)*(D*x**3+C*x**2+B*x+A),x)
[Out]
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Mathematica [A] time = 0.0167431, size = 60, normalized size = 1. \[ \frac{1}{3} x^3 (a C+A b)+a A x+\frac{1}{4} x^4 (a D+b B)+\frac{1}{2} a B x^2+\frac{1}{5} b C x^5+\frac{1}{6} b D x^6 \]
Antiderivative was successfully verified.
[In] Integrate[(a + b*x^2)*(A + B*x + C*x^2 + D*x^3),x]
[Out]
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Maple [A] time = 0.003, size = 51, normalized size = 0.9 \[ aAx+{\frac{Ba{x}^{2}}{2}}+{\frac{ \left ( Ab+aC \right ){x}^{3}}{3}}+{\frac{ \left ( Bb+aD \right ){x}^{4}}{4}}+{\frac{bC{x}^{5}}{5}}+{\frac{bD{x}^{6}}{6}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int((b*x^2+a)*(D*x^3+C*x^2+B*x+A),x)
[Out]
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Maxima [A] time = 1.34506, size = 68, normalized size = 1.13 \[ \frac{1}{6} \, D b x^{6} + \frac{1}{5} \, C b x^{5} + \frac{1}{4} \,{\left (D a + B b\right )} x^{4} + \frac{1}{2} \, B a x^{2} + \frac{1}{3} \,{\left (C a + A b\right )} x^{3} + A a x \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((D*x^3 + C*x^2 + B*x + A)*(b*x^2 + a),x, algorithm="maxima")
[Out]
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Fricas [A] time = 0.228372, size = 1, normalized size = 0.02 \[ \frac{1}{6} x^{6} b D + \frac{1}{5} x^{5} b C + \frac{1}{4} x^{4} a D + \frac{1}{4} x^{4} b B + \frac{1}{3} x^{3} a C + \frac{1}{3} x^{3} b A + \frac{1}{2} x^{2} a B + x a A \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((D*x^3 + C*x^2 + B*x + A)*(b*x^2 + a),x, algorithm="fricas")
[Out]
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Sympy [A] time = 0.050122, size = 56, normalized size = 0.93 \[ A a x + \frac{B a x^{2}}{2} + \frac{C b x^{5}}{5} + \frac{D b x^{6}}{6} + x^{4} \left (\frac{B b}{4} + \frac{D a}{4}\right ) + x^{3} \left (\frac{A b}{3} + \frac{C a}{3}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((b*x**2+a)*(D*x**3+C*x**2+B*x+A),x)
[Out]
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GIAC/XCAS [A] time = 0.21818, size = 73, normalized size = 1.22 \[ \frac{1}{6} \, D b x^{6} + \frac{1}{5} \, C b x^{5} + \frac{1}{4} \, D a x^{4} + \frac{1}{4} \, B b x^{4} + \frac{1}{3} \, C a x^{3} + \frac{1}{3} \, A b x^{3} + \frac{1}{2} \, B a x^{2} + A a x \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((D*x^3 + C*x^2 + B*x + A)*(b*x^2 + a),x, algorithm="giac")
[Out]