Optimal. Leaf size=90 \[ -\frac{a^2 A}{2 x^2}+x \left (2 a B c+2 A b c+b^2 B\right )+\log (x) \left (A \left (2 a c+b^2\right )+2 a b B\right )-\frac{a (a B+2 A b)}{x}+\frac{1}{2} c x^2 (A c+2 b B)+\frac{1}{3} B c^2 x^3 \]
[Out]
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Rubi [A] time = 0.171018, antiderivative size = 90, 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 \[ -\frac{a^2 A}{2 x^2}+x \left (2 a B c+2 A b c+b^2 B\right )+\log (x) \left (A \left (2 a c+b^2\right )+2 a b B\right )-\frac{a (a B+2 A b)}{x}+\frac{1}{2} c x^2 (A c+2 b B)+\frac{1}{3} B c^2 x^3 \]
Antiderivative was successfully verified.
[In] Int[((A + B*x)*(a + b*x + c*x^2)^2)/x^3,x]
[Out]
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Rubi in Sympy [F] time = 0., size = 0, normalized size = 0. \[ - \frac{A a^{2}}{2 x^{2}} + \frac{B c^{2} x^{3}}{3} - \frac{a \left (2 A b + B a\right )}{x} + c \left (A c + 2 B b\right ) \int x\, dx + \left (2 A a c + A b^{2} + 2 B a b\right ) \log{\left (x \right )} + \frac{\left (B b^{2} + 2 c \left (A b + B a\right )\right ) \int B\, dx}{B} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate((B*x+A)*(c*x**2+b*x+a)**2/x**3,x)
[Out]
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Mathematica [A] time = 0.102648, size = 86, normalized size = 0.96 \[ -\frac{a^2 (A+2 B x)}{2 x^2}+A \log (x) \left (2 a c+b^2\right )+a \left (2 B c x-\frac{2 A b}{x}\right )+2 a b B \log (x)+b c x (2 A+B x)+\frac{1}{6} c^2 x^2 (3 A+2 B x)+b^2 B x \]
Antiderivative was successfully verified.
[In] Integrate[((A + B*x)*(a + b*x + c*x^2)^2)/x^3,x]
[Out]
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Maple [A] time = 0.01, size = 92, normalized size = 1. \[{\frac{B{c}^{2}{x}^{3}}{3}}+{\frac{A{c}^{2}{x}^{2}}{2}}+B{x}^{2}bc+2\,Abcx+2\,aBcx+{b}^{2}Bx+2\,aAc\ln \left ( x \right ) +A{b}^{2}\ln \left ( x \right ) +2\,B\ln \left ( x \right ) ab-{\frac{A{a}^{2}}{2\,{x}^{2}}}-2\,{\frac{abA}{x}}-{\frac{{a}^{2}B}{x}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int((B*x+A)*(c*x^2+b*x+a)^2/x^3,x)
[Out]
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Maxima [A] time = 0.690932, size = 119, normalized size = 1.32 \[ \frac{1}{3} \, B c^{2} x^{3} + \frac{1}{2} \,{\left (2 \, B b c + A c^{2}\right )} x^{2} +{\left (B b^{2} + 2 \,{\left (B a + A b\right )} c\right )} x +{\left (2 \, B a b + A b^{2} + 2 \, A a c\right )} \log \left (x\right ) - \frac{A a^{2} + 2 \,{\left (B a^{2} + 2 \, A a b\right )} x}{2 \, x^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((c*x^2 + b*x + a)^2*(B*x + A)/x^3,x, algorithm="maxima")
[Out]
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Fricas [A] time = 0.269561, size = 128, normalized size = 1.42 \[ \frac{2 \, B c^{2} x^{5} + 3 \,{\left (2 \, B b c + A c^{2}\right )} x^{4} + 6 \,{\left (B b^{2} + 2 \,{\left (B a + A b\right )} c\right )} x^{3} + 6 \,{\left (2 \, B a b + A b^{2} + 2 \, A a c\right )} x^{2} \log \left (x\right ) - 3 \, A a^{2} - 6 \,{\left (B a^{2} + 2 \, A a b\right )} x}{6 \, x^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((c*x^2 + b*x + a)^2*(B*x + A)/x^3,x, algorithm="fricas")
[Out]
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Sympy [A] time = 2.35564, size = 92, normalized size = 1.02 \[ \frac{B c^{2} x^{3}}{3} + x^{2} \left (\frac{A c^{2}}{2} + B b c\right ) + x \left (2 A b c + 2 B a c + B b^{2}\right ) + \left (2 A a c + A b^{2} + 2 B a b\right ) \log{\left (x \right )} - \frac{A a^{2} + x \left (4 A a b + 2 B a^{2}\right )}{2 x^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((B*x+A)*(c*x**2+b*x+a)**2/x**3,x)
[Out]
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GIAC/XCAS [A] time = 0.267144, size = 120, normalized size = 1.33 \[ \frac{1}{3} \, B c^{2} x^{3} + B b c x^{2} + \frac{1}{2} \, A c^{2} x^{2} + B b^{2} x + 2 \, B a c x + 2 \, A b c x +{\left (2 \, B a b + A b^{2} + 2 \, A a c\right )}{\rm ln}\left ({\left | x \right |}\right ) - \frac{A a^{2} + 2 \,{\left (B a^{2} + 2 \, A a b\right )} x}{2 \, x^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((c*x^2 + b*x + a)^2*(B*x + A)/x^3,x, algorithm="giac")
[Out]