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 = 0.080234, 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, Rules used = {1583, 1628} \[ -\frac{a^2 A}{x}+a^2 B \log (x)+\frac{1}{3} b x^3 (2 a C+A b)+a x (a C+2 A b)+a b B x^2+\frac{D \left (a+b x^2\right )^3}{6 b}+\frac{1}{4} b^2 B x^4+\frac{1}{5} b^2 C x^5 \]
Antiderivative was successfully verified.
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Rule 1583
Rule 1628
Rubi steps
\begin{align*} \int \frac{\left (a+b x^2\right )^2 \left (A+B x+C x^2+D x^3\right )}{x^2} \, dx &=\frac{D \left (a+b x^2\right )^3}{6 b}+\int \frac{\left (a+b x^2\right )^2 \left (A+B x+C x^2\right )}{x^2} \, dx\\ &=\frac{D \left (a+b x^2\right )^3}{6 b}+\int \left (a (2 A b+a C)+\frac{a^2 A}{x^2}+\frac{a^2 B}{x}+2 a b B x+b (A b+2 a C) x^2+b^2 B x^3+b^2 C x^4\right ) \, dx\\ &=-\frac{a^2 A}{x}+a (2 A b+a C) x+a b B x^2+\frac{1}{3} b (A b+2 a C) x^3+\frac{1}{4} b^2 B x^4+\frac{1}{5} b^2 C x^5+\frac{D \left (a+b x^2\right )^3}{6 b}+a^2 B \log (x)\\ \end{align*}
Mathematica [A] time = 0.0503056, 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.
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Maple [A] time = 0.006, size = 98, normalized size = 1.1 \begin{align*}{\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\,Aabx+{a}^{2}Cx+{a}^{2}B\ln \left ( x \right ) -{\frac{A{a}^{2}}{x}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 0.994366, size = 130, normalized size = 1.44 \begin{align*} \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 \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [F(-2)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: UnboundLocalError} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 0.37863, size = 99, normalized size = 1.1 \begin{align*} - \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 ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.22434, size = 132, normalized size = 1.47 \begin{align*} \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} \log \left ({\left | x \right |}\right ) - \frac{A a^{2}}{x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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