Optimal. Leaf size=112 \[ -\frac{x \left (-\frac{a^2 f}{b^2}+\frac{b c}{a}+\frac{a e}{b}-d\right )}{2 a \left (a+b x^2\right )}-\frac{\tan ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right ) \left (-a^2 b e+3 a^3 f-a b^2 d+3 b^3 c\right )}{2 a^{5/2} b^{5/2}}-\frac{c}{a^2 x}+\frac{f x}{b^2} \]
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Rubi [A] time = 0.131882, antiderivative size = 112, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 30, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.1, Rules used = {1805, 1261, 205} \[ -\frac{x \left (-\frac{a^2 f}{b^2}+\frac{b c}{a}+\frac{a e}{b}-d\right )}{2 a \left (a+b x^2\right )}-\frac{\tan ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right ) \left (-a^2 b e+3 a^3 f-a b^2 d+3 b^3 c\right )}{2 a^{5/2} b^{5/2}}-\frac{c}{a^2 x}+\frac{f x}{b^2} \]
Antiderivative was successfully verified.
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Rule 1805
Rule 1261
Rule 205
Rubi steps
\begin{align*} \int \frac{c+d x^2+e x^4+f x^6}{x^2 \left (a+b x^2\right )^2} \, dx &=-\frac{\left (\frac{b c}{a}-d+\frac{a e}{b}-\frac{a^2 f}{b^2}\right ) x}{2 a \left (a+b x^2\right )}-\frac{\int \frac{-2 c+\left (\frac{b c}{a}-d-\frac{a e}{b}+\frac{a^2 f}{b^2}\right ) x^2-\frac{2 a f x^4}{b}}{x^2 \left (a+b x^2\right )} \, dx}{2 a}\\ &=-\frac{\left (\frac{b c}{a}-d+\frac{a e}{b}-\frac{a^2 f}{b^2}\right ) x}{2 a \left (a+b x^2\right )}-\frac{\int \left (-\frac{2 a f}{b^2}-\frac{2 c}{a x^2}+\frac{3 b^3 c-a b^2 d-a^2 b e+3 a^3 f}{a b^2 \left (a+b x^2\right )}\right ) \, dx}{2 a}\\ &=-\frac{c}{a^2 x}+\frac{f x}{b^2}-\frac{\left (\frac{b c}{a}-d+\frac{a e}{b}-\frac{a^2 f}{b^2}\right ) x}{2 a \left (a+b x^2\right )}-\frac{\left (3 b^3 c-a b^2 d-a^2 b e+3 a^3 f\right ) \int \frac{1}{a+b x^2} \, dx}{2 a^2 b^2}\\ &=-\frac{c}{a^2 x}+\frac{f x}{b^2}-\frac{\left (\frac{b c}{a}-d+\frac{a e}{b}-\frac{a^2 f}{b^2}\right ) x}{2 a \left (a+b x^2\right )}-\frac{\left (3 b^3 c-a b^2 d-a^2 b e+3 a^3 f\right ) \tan ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right )}{2 a^{5/2} b^{5/2}}\\ \end{align*}
Mathematica [A] time = 0.0617208, size = 115, normalized size = 1.03 \[ \frac{x \left (-a^2 b e+a^3 f+a b^2 d-b^3 c\right )}{2 a^2 b^2 \left (a+b x^2\right )}-\frac{\tan ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right ) \left (-a^2 b e+3 a^3 f-a b^2 d+3 b^3 c\right )}{2 a^{5/2} b^{5/2}}-\frac{c}{a^2 x}+\frac{f x}{b^2} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.013, size = 165, normalized size = 1.5 \begin{align*}{\frac{fx}{{b}^{2}}}-{\frac{c}{{a}^{2}x}}+{\frac{axf}{2\,{b}^{2} \left ( b{x}^{2}+a \right ) }}-{\frac{ex}{2\,b \left ( b{x}^{2}+a \right ) }}+{\frac{dx}{2\,a \left ( b{x}^{2}+a \right ) }}-{\frac{bcx}{2\,{a}^{2} \left ( b{x}^{2}+a \right ) }}-{\frac{3\,af}{2\,{b}^{2}}\arctan \left ({bx{\frac{1}{\sqrt{ab}}}} \right ){\frac{1}{\sqrt{ab}}}}+{\frac{e}{2\,b}\arctan \left ({bx{\frac{1}{\sqrt{ab}}}} \right ){\frac{1}{\sqrt{ab}}}}+{\frac{d}{2\,a}\arctan \left ({bx{\frac{1}{\sqrt{ab}}}} \right ){\frac{1}{\sqrt{ab}}}}-{\frac{3\,bc}{2\,{a}^{2}}\arctan \left ({bx{\frac{1}{\sqrt{ab}}}} \right ){\frac{1}{\sqrt{ab}}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F(-2)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: ValueError} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.83527, size = 714, normalized size = 6.38 \begin{align*} \left [\frac{4 \, a^{3} b^{2} f x^{4} - 4 \, a^{2} b^{3} c - 2 \,{\left (3 \, a b^{4} c - a^{2} b^{3} d + a^{3} b^{2} e - 3 \, a^{4} b f\right )} x^{2} -{\left ({\left (3 \, b^{4} c - a b^{3} d - a^{2} b^{2} e + 3 \, a^{3} b f\right )} x^{3} +{\left (3 \, a b^{3} c - a^{2} b^{2} d - a^{3} b e + 3 \, a^{4} f\right )} x\right )} \sqrt{-a b} \log \left (\frac{b x^{2} + 2 \, \sqrt{-a b} x - a}{b x^{2} + a}\right )}{4 \,{\left (a^{3} b^{4} x^{3} + a^{4} b^{3} x\right )}}, \frac{2 \, a^{3} b^{2} f x^{4} - 2 \, a^{2} b^{3} c -{\left (3 \, a b^{4} c - a^{2} b^{3} d + a^{3} b^{2} e - 3 \, a^{4} b f\right )} x^{2} -{\left ({\left (3 \, b^{4} c - a b^{3} d - a^{2} b^{2} e + 3 \, a^{3} b f\right )} x^{3} +{\left (3 \, a b^{3} c - a^{2} b^{2} d - a^{3} b e + 3 \, a^{4} f\right )} x\right )} \sqrt{a b} \arctan \left (\frac{\sqrt{a b} x}{a}\right )}{2 \,{\left (a^{3} b^{4} x^{3} + a^{4} b^{3} x\right )}}\right ] \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 4.6105, size = 197, normalized size = 1.76 \begin{align*} \frac{\sqrt{- \frac{1}{a^{5} b^{5}}} \left (3 a^{3} f - a^{2} b e - a b^{2} d + 3 b^{3} c\right ) \log{\left (- a^{3} b^{2} \sqrt{- \frac{1}{a^{5} b^{5}}} + x \right )}}{4} - \frac{\sqrt{- \frac{1}{a^{5} b^{5}}} \left (3 a^{3} f - a^{2} b e - a b^{2} d + 3 b^{3} c\right ) \log{\left (a^{3} b^{2} \sqrt{- \frac{1}{a^{5} b^{5}}} + x \right )}}{4} + \frac{- 2 a b^{2} c + x^{2} \left (a^{3} f - a^{2} b e + a b^{2} d - 3 b^{3} c\right )}{2 a^{3} b^{2} x + 2 a^{2} b^{3} x^{3}} + \frac{f x}{b^{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.1664, size = 165, normalized size = 1.47 \begin{align*} \frac{f x}{b^{2}} - \frac{{\left (3 \, b^{3} c - a b^{2} d + 3 \, a^{3} f - a^{2} b e\right )} \arctan \left (\frac{b x}{\sqrt{a b}}\right )}{2 \, \sqrt{a b} a^{2} b^{2}} - \frac{3 \, b^{3} c x^{2} - a b^{2} d x^{2} - a^{3} f x^{2} + a^{2} b x^{2} e + 2 \, a b^{2} c}{2 \,{\left (b x^{3} + a x\right )} a^{2} b^{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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