Optimal. Leaf size=98 \[ -\frac{a^2 A}{3 x^3}-\frac{a^2 B}{2 x^2}+b x (2 a C+A b)-\frac{a (a C+2 A b)}{x}+\frac{1}{2} b x^2 (2 a D+b B)+a \log (x) (a D+2 b B)+\frac{1}{3} b^2 C x^3+\frac{1}{4} b^2 D x^4 \]
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Rubi [A] time = 0.0857923, antiderivative size = 98, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 1, integrand size = 28, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.036, Rules used = {1802} \[ -\frac{a^2 A}{3 x^3}-\frac{a^2 B}{2 x^2}+b x (2 a C+A b)-\frac{a (a C+2 A b)}{x}+\frac{1}{2} b x^2 (2 a D+b B)+a \log (x) (a D+2 b B)+\frac{1}{3} b^2 C x^3+\frac{1}{4} b^2 D x^4 \]
Antiderivative was successfully verified.
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Rule 1802
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^4} \, dx &=\int \left (b (A b+2 a C)+\frac{a^2 A}{x^4}+\frac{a^2 B}{x^3}+\frac{a (2 A b+a C)}{x^2}+\frac{a (2 b B+a D)}{x}+b (b B+2 a D) x+b^2 C x^2+b^2 D x^3\right ) \, dx\\ &=-\frac{a^2 A}{3 x^3}-\frac{a^2 B}{2 x^2}-\frac{a (2 A b+a C)}{x}+b (A b+2 a C) x+\frac{1}{2} b (b B+2 a D) x^2+\frac{1}{3} b^2 C x^3+\frac{1}{4} b^2 D x^4+a (2 b B+a D) \log (x)\\ \end{align*}
Mathematica [A] time = 0.0472645, size = 83, normalized size = 0.85 \[ -\frac{a^2 (2 A+3 x (B+2 C x))}{6 x^3}-\frac{2 a A b}{x}+a \log (x) (a D+2 b B)+a b x (2 C+D x)+\frac{1}{12} b^2 x \left (12 A+x \left (6 B+4 C x+3 D x^2\right )\right ) \]
Antiderivative was successfully verified.
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Maple [A] time = 0.006, size = 97, normalized size = 1. \begin{align*}{\frac{{b}^{2}D{x}^{4}}{4}}+{\frac{{b}^{2}C{x}^{3}}{3}}+{\frac{B{x}^{2}{b}^{2}}{2}}+D{x}^{2}ab+A{b}^{2}x+2\,abCx+2\,B\ln \left ( x \right ) ab+D\ln \left ( x \right ){a}^{2}-{\frac{A{a}^{2}}{3\,{x}^{3}}}-{\frac{B{a}^{2}}{2\,{x}^{2}}}-2\,{\frac{Aab}{x}}-{\frac{{a}^{2}C}{x}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 0.98409, size = 131, normalized size = 1.34 \begin{align*} \frac{1}{4} \, D b^{2} x^{4} + \frac{1}{3} \, C b^{2} x^{3} + \frac{1}{2} \,{\left (2 \, D a b + B b^{2}\right )} x^{2} +{\left (2 \, C a b + A b^{2}\right )} x +{\left (D a^{2} + 2 \, B a b\right )} \log \left (x\right ) - \frac{3 \, B a^{2} x + 2 \, A a^{2} + 6 \,{\left (C a^{2} + 2 \, A a b\right )} x^{2}}{6 \, x^{3}} \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 = 0.974943, size = 99, normalized size = 1.01 \begin{align*} \frac{C b^{2} x^{3}}{3} + \frac{D b^{2} x^{4}}{4} + a \left (2 B b + D a\right ) \log{\left (x \right )} + x^{2} \left (\frac{B b^{2}}{2} + D a b\right ) + x \left (A b^{2} + 2 C a b\right ) - \frac{2 A a^{2} + 3 B a^{2} x + x^{2} \left (12 A a b + 6 C a^{2}\right )}{6 x^{3}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.16514, size = 131, normalized size = 1.34 \begin{align*} \frac{1}{4} \, D b^{2} x^{4} + \frac{1}{3} \, C b^{2} x^{3} + D a b x^{2} + \frac{1}{2} \, B b^{2} x^{2} + 2 \, C a b x + A b^{2} x +{\left (D a^{2} + 2 \, B a b\right )} \log \left ({\left | x \right |}\right ) - \frac{3 \, B a^{2} x + 2 \, A a^{2} + 6 \,{\left (C a^{2} + 2 \, A a b\right )} x^{2}}{6 \, x^{3}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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