Optimal. Leaf size=92 \[ a^2 A \log (x)+\frac{1}{3} x^3 \left (2 a B c+2 A b c+b^2 B\right )+\frac{1}{2} x^2 \left (A \left (2 a c+b^2\right )+2 a b B\right )+a x (a B+2 A b)+\frac{1}{4} c x^4 (A c+2 b B)+\frac{1}{5} B c^2 x^5 \]
[Out]
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Rubi [A] time = 0.122062, antiderivative size = 92, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 1, integrand size = 21, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.048 \[ a^2 A \log (x)+\frac{1}{3} x^3 \left (2 a B c+2 A b c+b^2 B\right )+\frac{1}{2} x^2 \left (A \left (2 a c+b^2\right )+2 a b B\right )+a x (a B+2 A b)+\frac{1}{4} c x^4 (A c+2 b B)+\frac{1}{5} B c^2 x^5 \]
Antiderivative was successfully verified.
[In] Int[((A + B*x)*(a + b*x + c*x^2)^2)/x,x]
[Out]
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Rubi in Sympy [F] time = 0., size = 0, normalized size = 0. \[ A a^{2} \log{\left (x \right )} + \frac{B c^{2} x^{5}}{5} + \frac{c x^{4} \left (A c + 2 B b\right )}{4} + x^{3} \left (\frac{2 A b c}{3} + \frac{2 B a c}{3} + \frac{B b^{2}}{3}\right ) + \left (2 A b + B a\right ) \int a\, dx + \left (2 A a c + A b^{2} + 2 B a b\right ) \int x\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate((B*x+A)*(c*x**2+b*x+a)**2/x,x)
[Out]
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Mathematica [A] time = 0.0660736, size = 92, normalized size = 1. \[ a^2 A \log (x)+\frac{1}{3} x^3 \left (2 a B c+2 A b c+b^2 B\right )+\frac{1}{2} x^2 \left (2 a A c+2 a b B+A b^2\right )+a x (a B+2 A b)+\frac{1}{4} c x^4 (A c+2 b B)+\frac{1}{5} B c^2 x^5 \]
Antiderivative was successfully verified.
[In] Integrate[((A + B*x)*(a + b*x + c*x^2)^2)/x,x]
[Out]
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Maple [A] time = 0.004, size = 95, normalized size = 1. \[{\frac{B{c}^{2}{x}^{5}}{5}}+{\frac{A{c}^{2}{x}^{4}}{4}}+{\frac{B{x}^{4}bc}{2}}+{\frac{2\,A{x}^{3}bc}{3}}+{\frac{2\,aBc{x}^{3}}{3}}+{\frac{B{b}^{2}{x}^{3}}{3}}+aAc{x}^{2}+{\frac{A{b}^{2}{x}^{2}}{2}}+B{x}^{2}ab+2\,aAbx+{a}^{2}Bx+{a}^{2}A\ln \left ( x \right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int((B*x+A)*(c*x^2+b*x+a)^2/x,x)
[Out]
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Maxima [A] time = 0.689928, size = 119, normalized size = 1.29 \[ \frac{1}{5} \, B c^{2} x^{5} + \frac{1}{4} \,{\left (2 \, B b c + A c^{2}\right )} x^{4} + \frac{1}{3} \,{\left (B b^{2} + 2 \,{\left (B a + A b\right )} c\right )} x^{3} + A a^{2} \log \left (x\right ) + \frac{1}{2} \,{\left (2 \, B a b + A b^{2} + 2 \, A a c\right )} x^{2} +{\left (B a^{2} + 2 \, A a b\right )} x \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((c*x^2 + b*x + a)^2*(B*x + A)/x,x, algorithm="maxima")
[Out]
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Fricas [A] time = 0.287325, size = 119, normalized size = 1.29 \[ \frac{1}{5} \, B c^{2} x^{5} + \frac{1}{4} \,{\left (2 \, B b c + A c^{2}\right )} x^{4} + \frac{1}{3} \,{\left (B b^{2} + 2 \,{\left (B a + A b\right )} c\right )} x^{3} + A a^{2} \log \left (x\right ) + \frac{1}{2} \,{\left (2 \, B a b + A b^{2} + 2 \, A a c\right )} x^{2} +{\left (B a^{2} + 2 \, A a b\right )} x \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((c*x^2 + b*x + a)^2*(B*x + A)/x,x, algorithm="fricas")
[Out]
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Sympy [A] time = 1.39288, size = 95, normalized size = 1.03 \[ A a^{2} \log{\left (x \right )} + \frac{B c^{2} x^{5}}{5} + x^{4} \left (\frac{A c^{2}}{4} + \frac{B b c}{2}\right ) + x^{3} \left (\frac{2 A b c}{3} + \frac{2 B a c}{3} + \frac{B b^{2}}{3}\right ) + x^{2} \left (A a c + \frac{A b^{2}}{2} + B a b\right ) + x \left (2 A a b + B a^{2}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((B*x+A)*(c*x**2+b*x+a)**2/x,x)
[Out]
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GIAC/XCAS [A] time = 0.272252, size = 128, normalized size = 1.39 \[ \frac{1}{5} \, B c^{2} x^{5} + \frac{1}{2} \, B b c x^{4} + \frac{1}{4} \, A c^{2} x^{4} + \frac{1}{3} \, B b^{2} x^{3} + \frac{2}{3} \, B a c x^{3} + \frac{2}{3} \, A b c x^{3} + B a b x^{2} + \frac{1}{2} \, A b^{2} x^{2} + A a c x^{2} + B a^{2} x + 2 \, A a b x + A a^{2}{\rm ln}\left ({\left | x \right |}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((c*x^2 + b*x + a)^2*(B*x + A)/x,x, algorithm="giac")
[Out]