Optimal. Leaf size=189 \[ \frac{b x \left (a^2 b e+a^3 (-f)-a b^2 d+b^3 c\right )}{2 a^5 \left (a+b x^2\right )}+\frac{2 a^2 b e+a^3 (-f)-3 a b^2 d+4 b^3 c}{a^5 x}+\frac{\sqrt{b} \tan ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right ) \left (5 a^2 b e-3 a^3 f-7 a b^2 d+9 b^3 c\right )}{2 a^{11/2}}-\frac{a^2 e-2 a b d+3 b^2 c}{3 a^4 x^3}+\frac{2 b c-a d}{5 a^3 x^5}-\frac{c}{7 a^2 x^7} \]
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Rubi [A] time = 0.293351, antiderivative size = 189, 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} \[ \frac{b x \left (a^2 b e+a^3 (-f)-a b^2 d+b^3 c\right )}{2 a^5 \left (a+b x^2\right )}+\frac{2 a^2 b e+a^3 (-f)-3 a b^2 d+4 b^3 c}{a^5 x}+\frac{\sqrt{b} \tan ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right ) \left (5 a^2 b e-3 a^3 f-7 a b^2 d+9 b^3 c\right )}{2 a^{11/2}}-\frac{a^2 e-2 a b d+3 b^2 c}{3 a^4 x^3}+\frac{2 b c-a d}{5 a^3 x^5}-\frac{c}{7 a^2 x^7} \]
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^8 \left (a+b x^2\right )^2} \, dx &=\frac{b \left (b^3 c-a b^2 d+a^2 b e-a^3 f\right ) x}{2 a^5 \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{2 \left (b^3 c-a b^2 d+a^2 b e-a^3 f\right ) x^6}{a^3}-\frac{b \left (b^3 c-a b^2 d+a^2 b e-a^3 f\right ) x^8}{a^4}}{x^8 \left (a+b x^2\right )} \, dx}{2 a}\\ &=\frac{b \left (b^3 c-a b^2 d+a^2 b e-a^3 f\right ) x}{2 a^5 \left (a+b x^2\right )}-\frac{\int \left (-\frac{2 c}{a x^8}-\frac{2 (-2 b c+a d)}{a^2 x^6}-\frac{2 \left (3 b^2 c-2 a b d+a^2 e\right )}{a^3 x^4}-\frac{2 \left (-4 b^3 c+3 a b^2 d-2 a^2 b e+a^3 f\right )}{a^4 x^2}+\frac{b \left (-9 b^3 c+7 a b^2 d-5 a^2 b e+3 a^3 f\right )}{a^4 \left (a+b x^2\right )}\right ) \, dx}{2 a}\\ &=-\frac{c}{7 a^2 x^7}+\frac{2 b c-a d}{5 a^3 x^5}-\frac{3 b^2 c-2 a b d+a^2 e}{3 a^4 x^3}+\frac{4 b^3 c-3 a b^2 d+2 a^2 b e-a^3 f}{a^5 x}+\frac{b \left (b^3 c-a b^2 d+a^2 b e-a^3 f\right ) x}{2 a^5 \left (a+b x^2\right )}+\frac{\left (b \left (9 b^3 c-7 a b^2 d+5 a^2 b e-3 a^3 f\right )\right ) \int \frac{1}{a+b x^2} \, dx}{2 a^5}\\ &=-\frac{c}{7 a^2 x^7}+\frac{2 b c-a d}{5 a^3 x^5}-\frac{3 b^2 c-2 a b d+a^2 e}{3 a^4 x^3}+\frac{4 b^3 c-3 a b^2 d+2 a^2 b e-a^3 f}{a^5 x}+\frac{b \left (b^3 c-a b^2 d+a^2 b e-a^3 f\right ) x}{2 a^5 \left (a+b x^2\right )}+\frac{\sqrt{b} \left (9 b^3 c-7 a b^2 d+5 a^2 b e-3 a^3 f\right ) \tan ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right )}{2 a^{11/2}}\\ \end{align*}
Mathematica [A] time = 0.101657, size = 190, normalized size = 1.01 \[ -\frac{b x \left (-a^2 b e+a^3 f+a b^2 d-b^3 c\right )}{2 a^5 \left (a+b x^2\right )}+\frac{2 a^2 b e+a^3 (-f)-3 a b^2 d+4 b^3 c}{a^5 x}-\frac{\sqrt{b} \tan ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right ) \left (-5 a^2 b e+3 a^3 f+7 a b^2 d-9 b^3 c\right )}{2 a^{11/2}}+\frac{a^2 (-e)+2 a b d-3 b^2 c}{3 a^4 x^3}+\frac{2 b c-a d}{5 a^3 x^5}-\frac{c}{7 a^2 x^7} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.016, size = 268, normalized size = 1.4 \begin{align*} -{\frac{c}{7\,{a}^{2}{x}^{7}}}-{\frac{d}{5\,{x}^{5}{a}^{2}}}+{\frac{2\,bc}{5\,{a}^{3}{x}^{5}}}-{\frac{e}{3\,{x}^{3}{a}^{2}}}+{\frac{2\,bd}{3\,{a}^{3}{x}^{3}}}-{\frac{{b}^{2}c}{{a}^{4}{x}^{3}}}-{\frac{f}{{a}^{2}x}}+2\,{\frac{be}{{a}^{3}x}}-3\,{\frac{{b}^{2}d}{{a}^{4}x}}+4\,{\frac{{b}^{3}c}{{a}^{5}x}}-{\frac{bxf}{2\,{a}^{2} \left ( b{x}^{2}+a \right ) }}+{\frac{{b}^{2}xe}{2\,{a}^{3} \left ( b{x}^{2}+a \right ) }}-{\frac{{b}^{3}xd}{2\,{a}^{4} \left ( b{x}^{2}+a \right ) }}+{\frac{{b}^{4}xc}{2\,{a}^{5} \left ( b{x}^{2}+a \right ) }}-{\frac{3\,bf}{2\,{a}^{2}}\arctan \left ({bx{\frac{1}{\sqrt{ab}}}} \right ){\frac{1}{\sqrt{ab}}}}+{\frac{5\,{b}^{2}e}{2\,{a}^{3}}\arctan \left ({bx{\frac{1}{\sqrt{ab}}}} \right ){\frac{1}{\sqrt{ab}}}}-{\frac{7\,{b}^{3}d}{2\,{a}^{4}}\arctan \left ({bx{\frac{1}{\sqrt{ab}}}} \right ){\frac{1}{\sqrt{ab}}}}+{\frac{9\,{b}^{4}c}{2\,{a}^{5}}\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 = 1.37974, size = 1057, normalized size = 5.59 \begin{align*} \left [\frac{210 \,{\left (9 \, b^{4} c - 7 \, a b^{3} d + 5 \, a^{2} b^{2} e - 3 \, a^{3} b f\right )} x^{8} + 140 \,{\left (9 \, a b^{3} c - 7 \, a^{2} b^{2} d + 5 \, a^{3} b e - 3 \, a^{4} f\right )} x^{6} - 60 \, a^{4} c - 28 \,{\left (9 \, a^{2} b^{2} c - 7 \, a^{3} b d + 5 \, a^{4} e\right )} x^{4} + 12 \,{\left (9 \, a^{3} b c - 7 \, a^{4} d\right )} x^{2} - 105 \,{\left ({\left (9 \, b^{4} c - 7 \, a b^{3} d + 5 \, a^{2} b^{2} e - 3 \, a^{3} b f\right )} x^{9} +{\left (9 \, a b^{3} c - 7 \, a^{2} b^{2} d + 5 \, a^{3} b e - 3 \, a^{4} f\right )} x^{7}\right )} \sqrt{-\frac{b}{a}} \log \left (\frac{b x^{2} - 2 \, a x \sqrt{-\frac{b}{a}} - a}{b x^{2} + a}\right )}{420 \,{\left (a^{5} b x^{9} + a^{6} x^{7}\right )}}, \frac{105 \,{\left (9 \, b^{4} c - 7 \, a b^{3} d + 5 \, a^{2} b^{2} e - 3 \, a^{3} b f\right )} x^{8} + 70 \,{\left (9 \, a b^{3} c - 7 \, a^{2} b^{2} d + 5 \, a^{3} b e - 3 \, a^{4} f\right )} x^{6} - 30 \, a^{4} c - 14 \,{\left (9 \, a^{2} b^{2} c - 7 \, a^{3} b d + 5 \, a^{4} e\right )} x^{4} + 6 \,{\left (9 \, a^{3} b c - 7 \, a^{4} d\right )} x^{2} + 105 \,{\left ({\left (9 \, b^{4} c - 7 \, a b^{3} d + 5 \, a^{2} b^{2} e - 3 \, a^{3} b f\right )} x^{9} +{\left (9 \, a b^{3} c - 7 \, a^{2} b^{2} d + 5 \, a^{3} b e - 3 \, a^{4} f\right )} x^{7}\right )} \sqrt{\frac{b}{a}} \arctan \left (x \sqrt{\frac{b}{a}}\right )}{210 \,{\left (a^{5} b x^{9} + a^{6} x^{7}\right )}}\right ] \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [B] time = 76.7361, size = 394, normalized size = 2.08 \begin{align*} \frac{\sqrt{- \frac{b}{a^{11}}} \left (3 a^{3} f - 5 a^{2} b e + 7 a b^{2} d - 9 b^{3} c\right ) \log{\left (- \frac{a^{6} \sqrt{- \frac{b}{a^{11}}} \left (3 a^{3} f - 5 a^{2} b e + 7 a b^{2} d - 9 b^{3} c\right )}{3 a^{3} b f - 5 a^{2} b^{2} e + 7 a b^{3} d - 9 b^{4} c} + x \right )}}{4} - \frac{\sqrt{- \frac{b}{a^{11}}} \left (3 a^{3} f - 5 a^{2} b e + 7 a b^{2} d - 9 b^{3} c\right ) \log{\left (\frac{a^{6} \sqrt{- \frac{b}{a^{11}}} \left (3 a^{3} f - 5 a^{2} b e + 7 a b^{2} d - 9 b^{3} c\right )}{3 a^{3} b f - 5 a^{2} b^{2} e + 7 a b^{3} d - 9 b^{4} c} + x \right )}}{4} - \frac{30 a^{4} c + x^{8} \left (315 a^{3} b f - 525 a^{2} b^{2} e + 735 a b^{3} d - 945 b^{4} c\right ) + x^{6} \left (210 a^{4} f - 350 a^{3} b e + 490 a^{2} b^{2} d - 630 a b^{3} c\right ) + x^{4} \left (70 a^{4} e - 98 a^{3} b d + 126 a^{2} b^{2} c\right ) + x^{2} \left (42 a^{4} d - 54 a^{3} b c\right )}{210 a^{6} x^{7} + 210 a^{5} b x^{9}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.19663, size = 271, normalized size = 1.43 \begin{align*} \frac{{\left (9 \, b^{4} c - 7 \, a b^{3} d - 3 \, a^{3} b f + 5 \, a^{2} b^{2} e\right )} \arctan \left (\frac{b x}{\sqrt{a b}}\right )}{2 \, \sqrt{a b} a^{5}} + \frac{b^{4} c x - a b^{3} d x - a^{3} b f x + a^{2} b^{2} x e}{2 \,{\left (b x^{2} + a\right )} a^{5}} + \frac{420 \, b^{3} c x^{6} - 315 \, a b^{2} d x^{6} - 105 \, a^{3} f x^{6} + 210 \, a^{2} b x^{6} e - 105 \, a b^{2} c x^{4} + 70 \, a^{2} b d x^{4} - 35 \, a^{3} x^{4} e + 42 \, a^{2} b c x^{2} - 21 \, a^{3} d x^{2} - 15 \, a^{3} c}{105 \, a^{5} x^{7}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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