Optimal. Leaf size=240 \[ \frac{x^7 \left (c-\frac{a \left (a^2 f-a b e+b^2 d\right )}{b^3}\right )}{2 a \left (a+b x^2\right )}-\frac{x^5 \left (9 a^2 b e-11 a^3 f-7 a b^2 d+5 b^3 c\right )}{10 a b^4}+\frac{x^3 \left (9 a^2 b e-11 a^3 f-7 a b^2 d+5 b^3 c\right )}{6 b^5}-\frac{a x \left (9 a^2 b e-11 a^3 f-7 a b^2 d+5 b^3 c\right )}{2 b^6}+\frac{a^{3/2} \tan ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right ) \left (9 a^2 b e-11 a^3 f-7 a b^2 d+5 b^3 c\right )}{2 b^{13/2}}+\frac{x^7 (b e-2 a f)}{7 b^3}+\frac{f x^9}{9 b^2} \]
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Rubi [A] time = 0.294288, antiderivative size = 240, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 4, integrand size = 30, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.133, Rules used = {1804, 1585, 1261, 205} \[ \frac{x^7 \left (c-\frac{a \left (a^2 f-a b e+b^2 d\right )}{b^3}\right )}{2 a \left (a+b x^2\right )}-\frac{x^5 \left (9 a^2 b e-11 a^3 f-7 a b^2 d+5 b^3 c\right )}{10 a b^4}+\frac{x^3 \left (9 a^2 b e-11 a^3 f-7 a b^2 d+5 b^3 c\right )}{6 b^5}-\frac{a x \left (9 a^2 b e-11 a^3 f-7 a b^2 d+5 b^3 c\right )}{2 b^6}+\frac{a^{3/2} \tan ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right ) \left (9 a^2 b e-11 a^3 f-7 a b^2 d+5 b^3 c\right )}{2 b^{13/2}}+\frac{x^7 (b e-2 a f)}{7 b^3}+\frac{f x^9}{9 b^2} \]
Antiderivative was successfully verified.
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Rule 1804
Rule 1585
Rule 1261
Rule 205
Rubi steps
\begin{align*} \int \frac{x^6 \left (c+d x^2+e x^4+f x^6\right )}{\left (a+b x^2\right )^2} \, dx &=\frac{\left (c-\frac{a \left (b^2 d-a b e+a^2 f\right )}{b^3}\right ) x^7}{2 a \left (a+b x^2\right )}-\frac{\int \frac{x^5 \left (\left (5 b c-7 a d+\frac{7 a^2 e}{b}-\frac{7 a^3 f}{b^2}\right ) x-2 a \left (e-\frac{a f}{b}\right ) x^3-2 a f x^5\right )}{a+b x^2} \, dx}{2 a b}\\ &=\frac{\left (c-\frac{a \left (b^2 d-a b e+a^2 f\right )}{b^3}\right ) x^7}{2 a \left (a+b x^2\right )}-\frac{\int \frac{x^6 \left (5 b c-7 a d+\frac{7 a^2 e}{b}-\frac{7 a^3 f}{b^2}-2 a \left (e-\frac{a f}{b}\right ) x^2-2 a f x^4\right )}{a+b x^2} \, dx}{2 a b}\\ &=\frac{\left (c-\frac{a \left (b^2 d-a b e+a^2 f\right )}{b^3}\right ) x^7}{2 a \left (a+b x^2\right )}-\frac{\int \left (\frac{a^2 \left (5 b^3 c-7 a b^2 d+9 a^2 b e-11 a^3 f\right )}{b^5}-\frac{a \left (5 b^3 c-7 a b^2 d+9 a^2 b e-11 a^3 f\right ) x^2}{b^4}+\frac{\left (5 b^3 c-7 a b^2 d+9 a^2 b e-11 a^3 f\right ) x^4}{b^3}-\frac{2 a (b e-2 a f) x^6}{b^2}-\frac{2 a f x^8}{b}+\frac{-5 a^3 b^3 c+7 a^4 b^2 d-9 a^5 b e+11 a^6 f}{b^5 \left (a+b x^2\right )}\right ) \, dx}{2 a b}\\ &=-\frac{a \left (5 b^3 c-7 a b^2 d+9 a^2 b e-11 a^3 f\right ) x}{2 b^6}+\frac{\left (5 b^3 c-7 a b^2 d+9 a^2 b e-11 a^3 f\right ) x^3}{6 b^5}-\frac{\left (5 b^3 c-7 a b^2 d+9 a^2 b e-11 a^3 f\right ) x^5}{10 a b^4}+\frac{(b e-2 a f) x^7}{7 b^3}+\frac{f x^9}{9 b^2}+\frac{\left (c-\frac{a \left (b^2 d-a b e+a^2 f\right )}{b^3}\right ) x^7}{2 a \left (a+b x^2\right )}+\frac{\left (a^2 \left (5 b^3 c-7 a b^2 d+9 a^2 b e-11 a^3 f\right )\right ) \int \frac{1}{a+b x^2} \, dx}{2 b^6}\\ &=-\frac{a \left (5 b^3 c-7 a b^2 d+9 a^2 b e-11 a^3 f\right ) x}{2 b^6}+\frac{\left (5 b^3 c-7 a b^2 d+9 a^2 b e-11 a^3 f\right ) x^3}{6 b^5}-\frac{\left (5 b^3 c-7 a b^2 d+9 a^2 b e-11 a^3 f\right ) x^5}{10 a b^4}+\frac{(b e-2 a f) x^7}{7 b^3}+\frac{f x^9}{9 b^2}+\frac{\left (c-\frac{a \left (b^2 d-a b e+a^2 f\right )}{b^3}\right ) x^7}{2 a \left (a+b x^2\right )}+\frac{a^{3/2} \left (5 b^3 c-7 a b^2 d+9 a^2 b e-11 a^3 f\right ) \tan ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right )}{2 b^{13/2}}\\ \end{align*}
Mathematica [A] time = 0.123684, size = 227, normalized size = 0.95 \[ \frac{x^3 \left (3 a^2 b e-4 a^3 f-2 a b^2 d+b^3 c\right )}{3 b^5}-\frac{x \left (a^2 b^3 c-a^3 b^2 d+a^4 b e+a^5 (-f)\right )}{2 b^6 \left (a+b x^2\right )}+\frac{a x \left (-4 a^2 b e+5 a^3 f+3 a b^2 d-2 b^3 c\right )}{b^6}-\frac{a^{3/2} \tan ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right ) \left (-9 a^2 b e+11 a^3 f+7 a b^2 d-5 b^3 c\right )}{2 b^{13/2}}+\frac{x^5 \left (3 a^2 f-2 a b e+b^2 d\right )}{5 b^4}+\frac{x^7 (b e-2 a f)}{7 b^3}+\frac{f x^9}{9 b^2} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.011, size = 309, normalized size = 1.3 \begin{align*}{\frac{f{x}^{9}}{9\,{b}^{2}}}-{\frac{2\,{x}^{7}af}{7\,{b}^{3}}}+{\frac{{x}^{7}e}{7\,{b}^{2}}}+{\frac{3\,{x}^{5}{a}^{2}f}{5\,{b}^{4}}}-{\frac{2\,{x}^{5}ae}{5\,{b}^{3}}}+{\frac{{x}^{5}d}{5\,{b}^{2}}}-{\frac{4\,{x}^{3}{a}^{3}f}{3\,{b}^{5}}}+{\frac{{x}^{3}{a}^{2}e}{{b}^{4}}}-{\frac{2\,a{x}^{3}d}{3\,{b}^{3}}}+{\frac{{x}^{3}c}{3\,{b}^{2}}}+5\,{\frac{{a}^{4}fx}{{b}^{6}}}-4\,{\frac{{a}^{3}ex}{{b}^{5}}}+3\,{\frac{{a}^{2}dx}{{b}^{4}}}-2\,{\frac{acx}{{b}^{3}}}+{\frac{{a}^{5}xf}{2\,{b}^{6} \left ( b{x}^{2}+a \right ) }}-{\frac{{a}^{4}xe}{2\,{b}^{5} \left ( b{x}^{2}+a \right ) }}+{\frac{{a}^{3}xd}{2\,{b}^{4} \left ( b{x}^{2}+a \right ) }}-{\frac{{a}^{2}xc}{2\,{b}^{3} \left ( b{x}^{2}+a \right ) }}-{\frac{11\,{a}^{5}f}{2\,{b}^{6}}\arctan \left ({bx{\frac{1}{\sqrt{ab}}}} \right ){\frac{1}{\sqrt{ab}}}}+{\frac{9\,{a}^{4}e}{2\,{b}^{5}}\arctan \left ({bx{\frac{1}{\sqrt{ab}}}} \right ){\frac{1}{\sqrt{ab}}}}-{\frac{7\,{a}^{3}d}{2\,{b}^{4}}\arctan \left ({bx{\frac{1}{\sqrt{ab}}}} \right ){\frac{1}{\sqrt{ab}}}}+{\frac{5\,{a}^{2}c}{2\,{b}^{3}}\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 = 1.58722, size = 1260, normalized size = 5.25 \begin{align*} \left [\frac{140 \, b^{5} f x^{11} + 20 \,{\left (9 \, b^{5} e - 11 \, a b^{4} f\right )} x^{9} + 36 \,{\left (7 \, b^{5} d - 9 \, a b^{4} e + 11 \, a^{2} b^{3} f\right )} x^{7} + 84 \,{\left (5 \, b^{5} c - 7 \, a b^{4} d + 9 \, a^{2} b^{3} e - 11 \, a^{3} b^{2} f\right )} x^{5} - 420 \,{\left (5 \, a b^{4} c - 7 \, a^{2} b^{3} d + 9 \, a^{3} b^{2} e - 11 \, a^{4} b f\right )} x^{3} - 315 \,{\left (5 \, a^{2} b^{3} c - 7 \, a^{3} b^{2} d + 9 \, a^{4} b e - 11 \, a^{5} f +{\left (5 \, a b^{4} c - 7 \, a^{2} b^{3} d + 9 \, a^{3} b^{2} e - 11 \, a^{4} b f\right )} x^{2}\right )} \sqrt{-\frac{a}{b}} \log \left (\frac{b x^{2} - 2 \, b x \sqrt{-\frac{a}{b}} - a}{b x^{2} + a}\right ) - 630 \,{\left (5 \, a^{2} b^{3} c - 7 \, a^{3} b^{2} d + 9 \, a^{4} b e - 11 \, a^{5} f\right )} x}{1260 \,{\left (b^{7} x^{2} + a b^{6}\right )}}, \frac{70 \, b^{5} f x^{11} + 10 \,{\left (9 \, b^{5} e - 11 \, a b^{4} f\right )} x^{9} + 18 \,{\left (7 \, b^{5} d - 9 \, a b^{4} e + 11 \, a^{2} b^{3} f\right )} x^{7} + 42 \,{\left (5 \, b^{5} c - 7 \, a b^{4} d + 9 \, a^{2} b^{3} e - 11 \, a^{3} b^{2} f\right )} x^{5} - 210 \,{\left (5 \, a b^{4} c - 7 \, a^{2} b^{3} d + 9 \, a^{3} b^{2} e - 11 \, a^{4} b f\right )} x^{3} + 315 \,{\left (5 \, a^{2} b^{3} c - 7 \, a^{3} b^{2} d + 9 \, a^{4} b e - 11 \, a^{5} f +{\left (5 \, a b^{4} c - 7 \, a^{2} b^{3} d + 9 \, a^{3} b^{2} e - 11 \, a^{4} b f\right )} x^{2}\right )} \sqrt{\frac{a}{b}} \arctan \left (\frac{b x \sqrt{\frac{a}{b}}}{a}\right ) - 315 \,{\left (5 \, a^{2} b^{3} c - 7 \, a^{3} b^{2} d + 9 \, a^{4} b e - 11 \, a^{5} f\right )} x}{630 \,{\left (b^{7} x^{2} + a b^{6}\right )}}\right ] \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 2.44927, size = 430, normalized size = 1.79 \begin{align*} \frac{x \left (a^{5} f - a^{4} b e + a^{3} b^{2} d - a^{2} b^{3} c\right )}{2 a b^{6} + 2 b^{7} x^{2}} + \frac{\sqrt{- \frac{a^{3}}{b^{13}}} \left (11 a^{3} f - 9 a^{2} b e + 7 a b^{2} d - 5 b^{3} c\right ) \log{\left (- \frac{b^{6} \sqrt{- \frac{a^{3}}{b^{13}}} \left (11 a^{3} f - 9 a^{2} b e + 7 a b^{2} d - 5 b^{3} c\right )}{11 a^{4} f - 9 a^{3} b e + 7 a^{2} b^{2} d - 5 a b^{3} c} + x \right )}}{4} - \frac{\sqrt{- \frac{a^{3}}{b^{13}}} \left (11 a^{3} f - 9 a^{2} b e + 7 a b^{2} d - 5 b^{3} c\right ) \log{\left (\frac{b^{6} \sqrt{- \frac{a^{3}}{b^{13}}} \left (11 a^{3} f - 9 a^{2} b e + 7 a b^{2} d - 5 b^{3} c\right )}{11 a^{4} f - 9 a^{3} b e + 7 a^{2} b^{2} d - 5 a b^{3} c} + x \right )}}{4} + \frac{f x^{9}}{9 b^{2}} - \frac{x^{7} \left (2 a f - b e\right )}{7 b^{3}} + \frac{x^{5} \left (3 a^{2} f - 2 a b e + b^{2} d\right )}{5 b^{4}} - \frac{x^{3} \left (4 a^{3} f - 3 a^{2} b e + 2 a b^{2} d - b^{3} c\right )}{3 b^{5}} + \frac{x \left (5 a^{4} f - 4 a^{3} b e + 3 a^{2} b^{2} d - 2 a b^{3} c\right )}{b^{6}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.15514, size = 340, normalized size = 1.42 \begin{align*} \frac{{\left (5 \, a^{2} b^{3} c - 7 \, a^{3} b^{2} d - 11 \, a^{5} f + 9 \, a^{4} b e\right )} \arctan \left (\frac{b x}{\sqrt{a b}}\right )}{2 \, \sqrt{a b} b^{6}} - \frac{a^{2} b^{3} c x - a^{3} b^{2} d x - a^{5} f x + a^{4} b x e}{2 \,{\left (b x^{2} + a\right )} b^{6}} + \frac{35 \, b^{16} f x^{9} - 90 \, a b^{15} f x^{7} + 45 \, b^{16} x^{7} e + 63 \, b^{16} d x^{5} + 189 \, a^{2} b^{14} f x^{5} - 126 \, a b^{15} x^{5} e + 105 \, b^{16} c x^{3} - 210 \, a b^{15} d x^{3} - 420 \, a^{3} b^{13} f x^{3} + 315 \, a^{2} b^{14} x^{3} e - 630 \, a b^{15} c x + 945 \, a^{2} b^{14} d x + 1575 \, a^{4} b^{12} f x - 1260 \, a^{3} b^{13} x e}{315 \, b^{18}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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