Optimal. Leaf size=90 \[ -\frac{a^2 A}{x}+a^2 B \log (x)+\frac{1}{3} b x^3 (2 a C+A b)+a x (a C+2 A b)+a b B x^2+\frac{D \left (a+b x^2\right )^3}{6 b}+\frac{1}{4} b^2 B x^4+\frac{1}{5} b^2 C x^5 \]
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Rubi [A] time = 0.197665, antiderivative size = 90, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 28, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.071 \[ -\frac{a^2 A}{x}+a^2 B \log (x)+\frac{1}{3} b x^3 (2 a C+A b)+a x (a C+2 A b)+a b B x^2+\frac{D \left (a+b x^2\right )^3}{6 b}+\frac{1}{4} b^2 B x^4+\frac{1}{5} b^2 C x^5 \]
Antiderivative was successfully verified.
[In] Int[((a + b*x^2)^2*(A + B*x + C*x^2 + D*x^3))/x^2,x]
[Out]
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Rubi in Sympy [F] time = 0., size = 0, normalized size = 0. \[ - \frac{A a^{2}}{x} + B a^{2} \log{\left (x \right )} + \frac{C b^{2} x^{5}}{5} + \frac{D b^{2} x^{6}}{6} + a \left (2 B b + D a\right ) \int x\, dx + \frac{b x^{4} \left (B b + 2 D a\right )}{4} + \frac{b x^{3} \left (A b + 2 C a\right )}{3} + \frac{a \left (2 A b + C a\right ) \int C\, dx}{C} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate((b*x**2+a)**2*(D*x**3+C*x**2+B*x+A)/x**2,x)
[Out]
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Mathematica [A] time = 0.138605, size = 88, normalized size = 0.98 \[ a^2 \left (-\frac{A}{x}+C x+\frac{D x^2}{2}\right )+a^2 B \log (x)+\frac{1}{6} a b x (12 A+x (6 B+x (4 C+3 D x)))+\frac{1}{60} b^2 x^3 (20 A+x (15 B+2 x (6 C+5 D x))) \]
Antiderivative was successfully verified.
[In] Integrate[((a + b*x^2)^2*(A + B*x + C*x^2 + D*x^3))/x^2,x]
[Out]
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Maple [A] time = 0.01, size = 98, normalized size = 1.1 \[{\frac{D{b}^{2}{x}^{6}}{6}}+{\frac{{b}^{2}C{x}^{5}}{5}}+{\frac{{b}^{2}B{x}^{4}}{4}}+{\frac{D{x}^{4}ab}{2}}+{\frac{A{x}^{3}{b}^{2}}{3}}+{\frac{2\,C{x}^{3}ab}{3}}+B{x}^{2}ab+{\frac{D{x}^{2}{a}^{2}}{2}}+2\,Axab+Cx{a}^{2}+{a}^{2}B\ln \left ( x \right ) -{\frac{A{a}^{2}}{x}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int((b*x^2+a)^2*(D*x^3+C*x^2+B*x+A)/x^2,x)
[Out]
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Maxima [A] time = 1.35618, size = 130, normalized size = 1.44 \[ \frac{1}{6} \, D b^{2} x^{6} + \frac{1}{5} \, C b^{2} x^{5} + \frac{1}{4} \,{\left (2 \, D a b + B b^{2}\right )} x^{4} + \frac{1}{3} \,{\left (2 \, C a b + A b^{2}\right )} x^{3} + B a^{2} \log \left (x\right ) + \frac{1}{2} \,{\left (D a^{2} + 2 \, B a b\right )} x^{2} - \frac{A a^{2}}{x} +{\left (C a^{2} + 2 \, A a b\right )} x \]
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)^2/x^2,x, algorithm="maxima")
[Out]
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Fricas [A] time = 0.222919, size = 139, normalized size = 1.54 \[ \frac{10 \, D b^{2} x^{7} + 12 \, C b^{2} x^{6} + 15 \,{\left (2 \, D a b + B b^{2}\right )} x^{5} + 20 \,{\left (2 \, C a b + A b^{2}\right )} x^{4} + 60 \, B a^{2} x \log \left (x\right ) + 30 \,{\left (D a^{2} + 2 \, B a b\right )} x^{3} - 60 \, A a^{2} + 60 \,{\left (C a^{2} + 2 \, A a b\right )} x^{2}}{60 \, x} \]
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)^2/x^2,x, algorithm="fricas")
[Out]
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Sympy [A] time = 0.758291, size = 99, normalized size = 1.1 \[ - \frac{A a^{2}}{x} + B a^{2} \log{\left (x \right )} + \frac{C b^{2} x^{5}}{5} + \frac{D b^{2} x^{6}}{6} + x^{4} \left (\frac{B b^{2}}{4} + \frac{D a b}{2}\right ) + x^{3} \left (\frac{A b^{2}}{3} + \frac{2 C a b}{3}\right ) + x^{2} \left (B a b + \frac{D a^{2}}{2}\right ) + x \left (2 A a b + C a^{2}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((b*x**2+a)**2*(D*x**3+C*x**2+B*x+A)/x**2,x)
[Out]
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GIAC/XCAS [A] time = 0.224227, size = 132, normalized size = 1.47 \[ \frac{1}{6} \, D b^{2} x^{6} + \frac{1}{5} \, C b^{2} x^{5} + \frac{1}{2} \, D a b x^{4} + \frac{1}{4} \, B b^{2} x^{4} + \frac{2}{3} \, C a b x^{3} + \frac{1}{3} \, A b^{2} x^{3} + \frac{1}{2} \, D a^{2} x^{2} + B a b x^{2} + C a^{2} x + 2 \, A a b x + B a^{2}{\rm ln}\left ({\left | x \right |}\right ) - \frac{A a^{2}}{x} \]
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)^2/x^2,x, algorithm="giac")
[Out]