Optimal. Leaf size=44 \[ -\frac{A b^2}{x}+c x (A c+2 b B)+b \log (x) (2 A c+b B)+\frac{1}{2} B c^2 x^2 \]
[Out]
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Rubi [A] time = 0.0889057, antiderivative size = 44, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.1 \[ -\frac{A b^2}{x}+c x (A c+2 b B)+b \log (x) (2 A c+b B)+\frac{1}{2} B c^2 x^2 \]
Antiderivative was successfully verified.
[In] Int[((A + B*x)*(b*x + c*x^2)^2)/x^4,x]
[Out]
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Rubi in Sympy [F] time = 0., size = 0, normalized size = 0. \[ - \frac{A b^{2}}{x} + B c^{2} \int x\, dx + b \left (2 A c + B b\right ) \log{\left (x \right )} + \frac{c \left (A c + 2 B b\right ) \int A\, dx}{A} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate((B*x+A)*(c*x**2+b*x)**2/x**4,x)
[Out]
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Mathematica [A] time = 0.0629413, size = 43, normalized size = 0.98 \[ A \left (c^2 x-\frac{b^2}{x}\right )+b \log (x) (2 A c+b B)+\frac{1}{2} B c x (4 b+c x) \]
Antiderivative was successfully verified.
[In] Integrate[((A + B*x)*(b*x + c*x^2)^2)/x^4,x]
[Out]
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Maple [A] time = 0.009, size = 46, normalized size = 1.1 \[{\frac{B{c}^{2}{x}^{2}}{2}}+Ax{c}^{2}+2\,Bxbc+2\,A\ln \left ( x \right ) bc+{b}^{2}B\ln \left ( x \right ) -{\frac{{b}^{2}A}{x}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int((B*x+A)*(c*x^2+b*x)^2/x^4,x)
[Out]
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Maxima [A] time = 0.692852, size = 62, normalized size = 1.41 \[ \frac{1}{2} \, B c^{2} x^{2} - \frac{A b^{2}}{x} +{\left (2 \, B b c + A c^{2}\right )} x +{\left (B b^{2} + 2 \, A b c\right )} \log \left (x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((c*x^2 + b*x)^2*(B*x + A)/x^4,x, algorithm="maxima")
[Out]
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Fricas [A] time = 0.270737, size = 70, normalized size = 1.59 \[ \frac{B c^{2} x^{3} - 2 \, A b^{2} + 2 \,{\left (2 \, B b c + A c^{2}\right )} x^{2} + 2 \,{\left (B b^{2} + 2 \, A b c\right )} x \log \left (x\right )}{2 \, x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((c*x^2 + b*x)^2*(B*x + A)/x^4,x, algorithm="fricas")
[Out]
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Sympy [A] time = 1.40745, size = 42, normalized size = 0.95 \[ - \frac{A b^{2}}{x} + \frac{B c^{2} x^{2}}{2} + b \left (2 A c + B b\right ) \log{\left (x \right )} + x \left (A c^{2} + 2 B b c\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((B*x+A)*(c*x**2+b*x)**2/x**4,x)
[Out]
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GIAC/XCAS [A] time = 0.268348, size = 62, normalized size = 1.41 \[ \frac{1}{2} \, B c^{2} x^{2} + 2 \, B b c x + A c^{2} x - \frac{A b^{2}}{x} +{\left (B b^{2} + 2 \, A b c\right )}{\rm ln}\left ({\left | x \right |}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((c*x^2 + b*x)^2*(B*x + A)/x^4,x, algorithm="giac")
[Out]