Optimal. Leaf size=152 \[ -\frac{x \left (a^2 b e+a^3 (-f)-a b^2 d+b^3 c\right )}{2 a^4 \left (a+b x^2\right )}-\frac{\tan ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right ) \left (3 a^2 b e+a^3 (-f)-5 a b^2 d+7 b^3 c\right )}{2 a^{9/2} \sqrt{b}}-\frac{a^2 e-2 a b d+3 b^2 c}{a^4 x}+\frac{2 b c-a d}{3 a^3 x^3}-\frac{c}{5 a^2 x^5} \]
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Rubi [A] time = 0.213107, antiderivative size = 152, 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, 1802, 205} \[ -\frac{x \left (a^2 b e+a^3 (-f)-a b^2 d+b^3 c\right )}{2 a^4 \left (a+b x^2\right )}-\frac{\tan ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right ) \left (3 a^2 b e+a^3 (-f)-5 a b^2 d+7 b^3 c\right )}{2 a^{9/2} \sqrt{b}}-\frac{a^2 e-2 a b d+3 b^2 c}{a^4 x}+\frac{2 b c-a d}{3 a^3 x^3}-\frac{c}{5 a^2 x^5} \]
Antiderivative was successfully verified.
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Rule 1805
Rule 1802
Rule 205
Rubi steps
\begin{align*} \int \frac{c+d x^2+e x^4+f x^6}{x^6 \left (a+b x^2\right )^2} \, dx &=-\frac{\left (b^3 c-a b^2 d+a^2 b e-a^3 f\right ) x}{2 a^4 \left (a+b x^2\right )}-\frac{\int \frac{-2 c+2 \left (\frac{b c}{a}-d\right ) x^2-\frac{2 \left (b^2 c-a b d+a^2 e\right ) x^4}{a^2}+\frac{\left (b^3 c-a b^2 d+a^2 b e-a^3 f\right ) x^6}{a^3}}{x^6 \left (a+b x^2\right )} \, dx}{2 a}\\ &=-\frac{\left (b^3 c-a b^2 d+a^2 b e-a^3 f\right ) x}{2 a^4 \left (a+b x^2\right )}-\frac{\int \left (-\frac{2 c}{a x^6}-\frac{2 (-2 b c+a d)}{a^2 x^4}-\frac{2 \left (3 b^2 c-2 a b d+a^2 e\right )}{a^3 x^2}+\frac{7 b^3 c-5 a b^2 d+3 a^2 b e-a^3 f}{a^3 \left (a+b x^2\right )}\right ) \, dx}{2 a}\\ &=-\frac{c}{5 a^2 x^5}+\frac{2 b c-a d}{3 a^3 x^3}-\frac{3 b^2 c-2 a b d+a^2 e}{a^4 x}-\frac{\left (b^3 c-a b^2 d+a^2 b e-a^3 f\right ) x}{2 a^4 \left (a+b x^2\right )}-\frac{\left (7 b^3 c-5 a b^2 d+3 a^2 b e-a^3 f\right ) \int \frac{1}{a+b x^2} \, dx}{2 a^4}\\ &=-\frac{c}{5 a^2 x^5}+\frac{2 b c-a d}{3 a^3 x^3}-\frac{3 b^2 c-2 a b d+a^2 e}{a^4 x}-\frac{\left (b^3 c-a b^2 d+a^2 b e-a^3 f\right ) x}{2 a^4 \left (a+b x^2\right )}-\frac{\left (7 b^3 c-5 a b^2 d+3 a^2 b e-a^3 f\right ) \tan ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right )}{2 a^{9/2} \sqrt{b}}\\ \end{align*}
Mathematica [A] time = 0.0824897, size = 151, normalized size = 0.99 \[ \frac{x \left (-a^2 b e+a^3 f+a b^2 d-b^3 c\right )}{2 a^4 \left (a+b x^2\right )}+\frac{\tan ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right ) \left (-3 a^2 b e+a^3 f+5 a b^2 d-7 b^3 c\right )}{2 a^{9/2} \sqrt{b}}+\frac{a^2 (-e)+2 a b d-3 b^2 c}{a^4 x}+\frac{2 b c-a d}{3 a^3 x^3}-\frac{c}{5 a^2 x^5} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.014, size = 219, normalized size = 1.4 \begin{align*} -{\frac{c}{5\,{x}^{5}{a}^{2}}}-{\frac{d}{3\,{x}^{3}{a}^{2}}}+{\frac{2\,bc}{3\,{a}^{3}{x}^{3}}}-{\frac{e}{{a}^{2}x}}+2\,{\frac{bd}{{a}^{3}x}}-3\,{\frac{{b}^{2}c}{{a}^{4}x}}+{\frac{fx}{2\,a \left ( b{x}^{2}+a \right ) }}-{\frac{bxe}{2\,{a}^{2} \left ( b{x}^{2}+a \right ) }}+{\frac{x{b}^{2}d}{2\,{a}^{3} \left ( b{x}^{2}+a \right ) }}-{\frac{x{b}^{3}c}{2\,{a}^{4} \left ( b{x}^{2}+a \right ) }}+{\frac{f}{2\,a}\arctan \left ({bx{\frac{1}{\sqrt{ab}}}} \right ){\frac{1}{\sqrt{ab}}}}-{\frac{3\,be}{2\,{a}^{2}}\arctan \left ({bx{\frac{1}{\sqrt{ab}}}} \right ){\frac{1}{\sqrt{ab}}}}+{\frac{5\,{b}^{2}d}{2\,{a}^{3}}\arctan \left ({bx{\frac{1}{\sqrt{ab}}}} \right ){\frac{1}{\sqrt{ab}}}}-{\frac{7\,{b}^{3}c}{2\,{a}^{4}}\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 = 1.39414, size = 926, normalized size = 6.09 \begin{align*} \left [-\frac{30 \,{\left (7 \, a b^{4} c - 5 \, a^{2} b^{3} d + 3 \, a^{3} b^{2} e - a^{4} b f\right )} x^{6} + 12 \, a^{4} b c + 20 \,{\left (7 \, a^{2} b^{3} c - 5 \, a^{3} b^{2} d + 3 \, a^{4} b e\right )} x^{4} - 4 \,{\left (7 \, a^{3} b^{2} c - 5 \, a^{4} b d\right )} x^{2} - 15 \,{\left ({\left (7 \, b^{4} c - 5 \, a b^{3} d + 3 \, a^{2} b^{2} e - a^{3} b f\right )} x^{7} +{\left (7 \, a b^{3} c - 5 \, a^{2} b^{2} d + 3 \, a^{3} b e - a^{4} f\right )} x^{5}\right )} \sqrt{-a b} \log \left (\frac{b x^{2} - 2 \, \sqrt{-a b} x - a}{b x^{2} + a}\right )}{60 \,{\left (a^{5} b^{2} x^{7} + a^{6} b x^{5}\right )}}, -\frac{15 \,{\left (7 \, a b^{4} c - 5 \, a^{2} b^{3} d + 3 \, a^{3} b^{2} e - a^{4} b f\right )} x^{6} + 6 \, a^{4} b c + 10 \,{\left (7 \, a^{2} b^{3} c - 5 \, a^{3} b^{2} d + 3 \, a^{4} b e\right )} x^{4} - 2 \,{\left (7 \, a^{3} b^{2} c - 5 \, a^{4} b d\right )} x^{2} + 15 \,{\left ({\left (7 \, b^{4} c - 5 \, a b^{3} d + 3 \, a^{2} b^{2} e - a^{3} b f\right )} x^{7} +{\left (7 \, a b^{3} c - 5 \, a^{2} b^{2} d + 3 \, a^{3} b e - a^{4} f\right )} x^{5}\right )} \sqrt{a b} \arctan \left (\frac{\sqrt{a b} x}{a}\right )}{30 \,{\left (a^{5} b^{2} x^{7} + a^{6} b x^{5}\right )}}\right ] \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 30.5347, size = 226, normalized size = 1.49 \begin{align*} - \frac{\sqrt{- \frac{1}{a^{9} b}} \left (a^{3} f - 3 a^{2} b e + 5 a b^{2} d - 7 b^{3} c\right ) \log{\left (- a^{5} \sqrt{- \frac{1}{a^{9} b}} + x \right )}}{4} + \frac{\sqrt{- \frac{1}{a^{9} b}} \left (a^{3} f - 3 a^{2} b e + 5 a b^{2} d - 7 b^{3} c\right ) \log{\left (a^{5} \sqrt{- \frac{1}{a^{9} b}} + x \right )}}{4} + \frac{- 6 a^{3} c + x^{6} \left (15 a^{3} f - 45 a^{2} b e + 75 a b^{2} d - 105 b^{3} c\right ) + x^{4} \left (- 30 a^{3} e + 50 a^{2} b d - 70 a b^{2} c\right ) + x^{2} \left (- 10 a^{3} d + 14 a^{2} b c\right )}{30 a^{5} x^{5} + 30 a^{4} b x^{7}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.1733, size = 204, normalized size = 1.34 \begin{align*} -\frac{{\left (7 \, b^{3} c - 5 \, a b^{2} d - a^{3} f + 3 \, a^{2} b e\right )} \arctan \left (\frac{b x}{\sqrt{a b}}\right )}{2 \, \sqrt{a b} a^{4}} - \frac{b^{3} c x - a b^{2} d x - a^{3} f x + a^{2} b x e}{2 \,{\left (b x^{2} + a\right )} a^{4}} - \frac{45 \, b^{2} c x^{4} - 30 \, a b d x^{4} + 15 \, a^{2} x^{4} e - 10 \, a b c x^{2} + 5 \, a^{2} d x^{2} + 3 \, a^{2} c}{15 \, a^{4} x^{5}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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