Optimal. Leaf size=90 \[ -\frac{a^2 A}{4 x^4}-\frac{A \left (2 a c+b^2\right )+2 a b B}{2 x^2}-\frac{2 a B c+2 A b c+b^2 B}{x}-\frac{a (a B+2 A b)}{3 x^3}+c \log (x) (A c+2 b B)+B c^2 x \]
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Rubi [A] time = 0.143204, 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 \[ -\frac{a^2 A}{4 x^4}-\frac{A \left (2 a c+b^2\right )+2 a b B}{2 x^2}-\frac{2 a B c+2 A b c+b^2 B}{x}-\frac{a (a B+2 A b)}{3 x^3}+c \log (x) (A c+2 b B)+B c^2 x \]
Antiderivative was successfully verified.
[In] Int[((A + B*x)*(a + b*x + c*x^2)^2)/x^5,x]
[Out]
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Rubi in Sympy [F] time = 0., size = 0, normalized size = 0. \[ - \frac{A a^{2}}{4 x^{4}} - \frac{a \left (2 A b + B a\right )}{3 x^{3}} + c^{2} \int B\, dx + c \left (A c + 2 B b\right ) \log{\left (x \right )} - \frac{2 A b c + 2 B a c + B b^{2}}{x} - \frac{A a c + \frac{A b^{2}}{2} + B a b}{x^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate((B*x+A)*(c*x**2+b*x+a)**2/x**5,x)
[Out]
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Mathematica [A] time = 0.0989163, size = 92, normalized size = 1.02 \[ -\frac{a^2 (3 A+4 B x)+4 a x (A (2 b+3 c x)+3 B x (b+2 c x))+6 x^2 \left (A b (b+4 c x)+2 B x \left (b^2-c^2 x^2\right )\right )-12 c x^4 \log (x) (A c+2 b B)}{12 x^4} \]
Antiderivative was successfully verified.
[In] Integrate[((A + B*x)*(a + b*x + c*x^2)^2)/x^5,x]
[Out]
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Maple [A] time = 0.01, size = 98, normalized size = 1.1 \[ B{c}^{2}x+A\ln \left ( x \right ){c}^{2}+2\,B\ln \left ( x \right ) bc-{\frac{A{a}^{2}}{4\,{x}^{4}}}-{\frac{2\,abA}{3\,{x}^{3}}}-{\frac{{a}^{2}B}{3\,{x}^{3}}}-{\frac{aAc}{{x}^{2}}}-{\frac{{b}^{2}A}{2\,{x}^{2}}}-{\frac{abB}{{x}^{2}}}-2\,{\frac{Abc}{x}}-2\,{\frac{aBc}{x}}-{\frac{{b}^{2}B}{x}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int((B*x+A)*(c*x^2+b*x+a)^2/x^5,x)
[Out]
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Maxima [A] time = 0.692161, size = 120, normalized size = 1.33 \[ B c^{2} x +{\left (2 \, B b c + A c^{2}\right )} \log \left (x\right ) - \frac{12 \,{\left (B b^{2} + 2 \,{\left (B a + A b\right )} c\right )} x^{3} + 3 \, A a^{2} + 6 \,{\left (2 \, B a b + A b^{2} + 2 \, A a c\right )} x^{2} + 4 \,{\left (B a^{2} + 2 \, A a b\right )} x}{12 \, x^{4}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((c*x^2 + b*x + a)^2*(B*x + A)/x^5,x, algorithm="maxima")
[Out]
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Fricas [A] time = 0.276341, size = 128, normalized size = 1.42 \[ \frac{12 \, B c^{2} x^{5} + 12 \,{\left (2 \, B b c + A c^{2}\right )} x^{4} \log \left (x\right ) - 12 \,{\left (B b^{2} + 2 \,{\left (B a + A b\right )} c\right )} x^{3} - 3 \, A a^{2} - 6 \,{\left (2 \, B a b + A b^{2} + 2 \, A a c\right )} x^{2} - 4 \,{\left (B a^{2} + 2 \, A a b\right )} x}{12 \, x^{4}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((c*x^2 + b*x + a)^2*(B*x + A)/x^5,x, algorithm="fricas")
[Out]
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Sympy [A] time = 12.1623, size = 94, normalized size = 1.04 \[ B c^{2} x + c \left (A c + 2 B b\right ) \log{\left (x \right )} - \frac{3 A a^{2} + x^{3} \left (24 A b c + 24 B a c + 12 B b^{2}\right ) + x^{2} \left (12 A a c + 6 A b^{2} + 12 B a b\right ) + x \left (8 A a b + 4 B a^{2}\right )}{12 x^{4}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((B*x+A)*(c*x**2+b*x+a)**2/x**5,x)
[Out]
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GIAC/XCAS [A] time = 0.272903, size = 122, normalized size = 1.36 \[ B c^{2} x +{\left (2 \, B b c + A c^{2}\right )}{\rm ln}\left ({\left | x \right |}\right ) - \frac{12 \,{\left (B b^{2} + 2 \, B a c + 2 \, A b c\right )} x^{3} + 3 \, A a^{2} + 6 \,{\left (2 \, B a b + A b^{2} + 2 \, A a c\right )} x^{2} + 4 \,{\left (B a^{2} + 2 \, A a b\right )} x}{12 \, x^{4}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((c*x^2 + b*x + a)^2*(B*x + A)/x^5,x, algorithm="giac")
[Out]