Optimal. Leaf size=211 \[ -\frac{b \left (a^2 b e+a^3 (-f)-a b^2 d+b^3 c\right )}{3 a^5 x^3}+\frac{a^2 b e+a^3 (-f)-a b^2 d+b^3 c}{5 a^4 x^5}+\frac{b^2 \left (a^2 b e+a^3 (-f)-a b^2 d+b^3 c\right )}{a^6 x}+\frac{b^{5/2} \tan ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right ) \left (a^2 b e+a^3 (-f)-a b^2 d+b^3 c\right )}{a^{13/2}}-\frac{a^2 e-a b d+b^2 c}{7 a^3 x^7}+\frac{b c-a d}{9 a^2 x^9}-\frac{c}{11 a x^{11}} \]
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Rubi [A] time = 0.175188, antiderivative size = 211, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 30, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.067, Rules used = {1802, 205} \[ -\frac{b \left (a^2 b e+a^3 (-f)-a b^2 d+b^3 c\right )}{3 a^5 x^3}+\frac{a^2 b e+a^3 (-f)-a b^2 d+b^3 c}{5 a^4 x^5}+\frac{b^2 \left (a^2 b e+a^3 (-f)-a b^2 d+b^3 c\right )}{a^6 x}+\frac{b^{5/2} \tan ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right ) \left (a^2 b e+a^3 (-f)-a b^2 d+b^3 c\right )}{a^{13/2}}-\frac{a^2 e-a b d+b^2 c}{7 a^3 x^7}+\frac{b c-a d}{9 a^2 x^9}-\frac{c}{11 a x^{11}} \]
Antiderivative was successfully verified.
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Rule 1802
Rule 205
Rubi steps
\begin{align*} \int \frac{c+d x^2+e x^4+f x^6}{x^{12} \left (a+b x^2\right )} \, dx &=\int \left (\frac{c}{a x^{12}}+\frac{-b c+a d}{a^2 x^{10}}+\frac{b^2 c-a b d+a^2 e}{a^3 x^8}+\frac{-b^3 c+a b^2 d-a^2 b e+a^3 f}{a^4 x^6}-\frac{b \left (-b^3 c+a b^2 d-a^2 b e+a^3 f\right )}{a^5 x^4}+\frac{b^2 \left (-b^3 c+a b^2 d-a^2 b e+a^3 f\right )}{a^6 x^2}-\frac{b^3 \left (-b^3 c+a b^2 d-a^2 b e+a^3 f\right )}{a^6 \left (a+b x^2\right )}\right ) \, dx\\ &=-\frac{c}{11 a x^{11}}+\frac{b c-a d}{9 a^2 x^9}-\frac{b^2 c-a b d+a^2 e}{7 a^3 x^7}+\frac{b^3 c-a b^2 d+a^2 b e-a^3 f}{5 a^4 x^5}-\frac{b \left (b^3 c-a b^2 d+a^2 b e-a^3 f\right )}{3 a^5 x^3}+\frac{b^2 \left (b^3 c-a b^2 d+a^2 b e-a^3 f\right )}{a^6 x}+\frac{\left (b^3 \left (b^3 c-a b^2 d+a^2 b e-a^3 f\right )\right ) \int \frac{1}{a+b x^2} \, dx}{a^6}\\ &=-\frac{c}{11 a x^{11}}+\frac{b c-a d}{9 a^2 x^9}-\frac{b^2 c-a b d+a^2 e}{7 a^3 x^7}+\frac{b^3 c-a b^2 d+a^2 b e-a^3 f}{5 a^4 x^5}-\frac{b \left (b^3 c-a b^2 d+a^2 b e-a^3 f\right )}{3 a^5 x^3}+\frac{b^2 \left (b^3 c-a b^2 d+a^2 b e-a^3 f\right )}{a^6 x}+\frac{b^{5/2} \left (b^3 c-a b^2 d+a^2 b e-a^3 f\right ) \tan ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right )}{a^{13/2}}\\ \end{align*}
Mathematica [A] time = 0.160742, size = 211, normalized size = 1. \[ \frac{b \left (-a^2 b e+a^3 f+a b^2 d-b^3 c\right )}{3 a^5 x^3}+\frac{a^2 b e+a^3 (-f)-a b^2 d+b^3 c}{5 a^4 x^5}+\frac{b^2 \left (a^2 b e+a^3 (-f)-a b^2 d+b^3 c\right )}{a^6 x}+\frac{b^{5/2} \tan ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right ) \left (a^2 b e+a^3 (-f)-a b^2 d+b^3 c\right )}{a^{13/2}}-\frac{a^2 e-a b d+b^2 c}{7 a^3 x^7}+\frac{b c-a d}{9 a^2 x^9}-\frac{c}{11 a x^{11}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.011, size = 286, normalized size = 1.4 \begin{align*} -{\frac{c}{11\,a{x}^{11}}}-{\frac{d}{9\,a{x}^{9}}}+{\frac{bc}{9\,{a}^{2}{x}^{9}}}-{\frac{e}{7\,a{x}^{7}}}+{\frac{bd}{7\,{a}^{2}{x}^{7}}}-{\frac{{b}^{2}c}{7\,{a}^{3}{x}^{7}}}-{\frac{f}{5\,a{x}^{5}}}+{\frac{be}{5\,{x}^{5}{a}^{2}}}-{\frac{{b}^{2}d}{5\,{a}^{3}{x}^{5}}}+{\frac{{b}^{3}c}{5\,{a}^{4}{x}^{5}}}-{\frac{{b}^{2}f}{{a}^{3}x}}+{\frac{{b}^{3}e}{{a}^{4}x}}-{\frac{{b}^{4}d}{{a}^{5}x}}+{\frac{{b}^{5}c}{{a}^{6}x}}+{\frac{bf}{3\,{x}^{3}{a}^{2}}}-{\frac{{b}^{2}e}{3\,{a}^{3}{x}^{3}}}+{\frac{{b}^{3}d}{3\,{a}^{4}{x}^{3}}}-{\frac{{b}^{4}c}{3\,{a}^{5}{x}^{3}}}-{\frac{{b}^{3}f}{{a}^{3}}\arctan \left ({bx{\frac{1}{\sqrt{ab}}}} \right ){\frac{1}{\sqrt{ab}}}}+{\frac{{b}^{4}e}{{a}^{4}}\arctan \left ({bx{\frac{1}{\sqrt{ab}}}} \right ){\frac{1}{\sqrt{ab}}}}-{\frac{{b}^{5}d}{{a}^{5}}\arctan \left ({bx{\frac{1}{\sqrt{ab}}}} \right ){\frac{1}{\sqrt{ab}}}}+{\frac{{b}^{6}c}{{a}^{6}}\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 = 1.53972, size = 976, normalized size = 4.63 \begin{align*} \left [-\frac{3465 \,{\left (b^{5} c - a b^{4} d + a^{2} b^{3} e - a^{3} b^{2} f\right )} x^{11} \sqrt{-\frac{b}{a}} \log \left (\frac{b x^{2} - 2 \, a x \sqrt{-\frac{b}{a}} - a}{b x^{2} + a}\right ) - 6930 \,{\left (b^{5} c - a b^{4} d + a^{2} b^{3} e - a^{3} b^{2} f\right )} x^{10} + 2310 \,{\left (a b^{4} c - a^{2} b^{3} d + a^{3} b^{2} e - a^{4} b f\right )} x^{8} - 1386 \,{\left (a^{2} b^{3} c - a^{3} b^{2} d + a^{4} b e - a^{5} f\right )} x^{6} + 630 \, a^{5} c + 990 \,{\left (a^{3} b^{2} c - a^{4} b d + a^{5} e\right )} x^{4} - 770 \,{\left (a^{4} b c - a^{5} d\right )} x^{2}}{6930 \, a^{6} x^{11}}, \frac{3465 \,{\left (b^{5} c - a b^{4} d + a^{2} b^{3} e - a^{3} b^{2} f\right )} x^{11} \sqrt{\frac{b}{a}} \arctan \left (x \sqrt{\frac{b}{a}}\right ) + 3465 \,{\left (b^{5} c - a b^{4} d + a^{2} b^{3} e - a^{3} b^{2} f\right )} x^{10} - 1155 \,{\left (a b^{4} c - a^{2} b^{3} d + a^{3} b^{2} e - a^{4} b f\right )} x^{8} + 693 \,{\left (a^{2} b^{3} c - a^{3} b^{2} d + a^{4} b e - a^{5} f\right )} x^{6} - 315 \, a^{5} c - 495 \,{\left (a^{3} b^{2} c - a^{4} b d + a^{5} e\right )} x^{4} + 385 \,{\left (a^{4} b c - a^{5} d\right )} x^{2}}{3465 \, a^{6} x^{11}}\right ] \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 51.001, size = 398, normalized size = 1.89 \begin{align*} \frac{\sqrt{- \frac{b^{5}}{a^{13}}} \left (a^{3} f - a^{2} b e + a b^{2} d - b^{3} c\right ) \log{\left (- \frac{a^{7} \sqrt{- \frac{b^{5}}{a^{13}}} \left (a^{3} f - a^{2} b e + a b^{2} d - b^{3} c\right )}{a^{3} b^{3} f - a^{2} b^{4} e + a b^{5} d - b^{6} c} + x \right )}}{2} - \frac{\sqrt{- \frac{b^{5}}{a^{13}}} \left (a^{3} f - a^{2} b e + a b^{2} d - b^{3} c\right ) \log{\left (\frac{a^{7} \sqrt{- \frac{b^{5}}{a^{13}}} \left (a^{3} f - a^{2} b e + a b^{2} d - b^{3} c\right )}{a^{3} b^{3} f - a^{2} b^{4} e + a b^{5} d - b^{6} c} + x \right )}}{2} - \frac{315 a^{5} c + x^{10} \left (3465 a^{3} b^{2} f - 3465 a^{2} b^{3} e + 3465 a b^{4} d - 3465 b^{5} c\right ) + x^{8} \left (- 1155 a^{4} b f + 1155 a^{3} b^{2} e - 1155 a^{2} b^{3} d + 1155 a b^{4} c\right ) + x^{6} \left (693 a^{5} f - 693 a^{4} b e + 693 a^{3} b^{2} d - 693 a^{2} b^{3} c\right ) + x^{4} \left (495 a^{5} e - 495 a^{4} b d + 495 a^{3} b^{2} c\right ) + x^{2} \left (385 a^{5} d - 385 a^{4} b c\right )}{3465 a^{6} x^{11}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.17518, size = 336, normalized size = 1.59 \begin{align*} \frac{{\left (b^{6} c - a b^{5} d - a^{3} b^{3} f + a^{2} b^{4} e\right )} \arctan \left (\frac{b x}{\sqrt{a b}}\right )}{\sqrt{a b} a^{6}} + \frac{3465 \, b^{5} c x^{10} - 3465 \, a b^{4} d x^{10} - 3465 \, a^{3} b^{2} f x^{10} + 3465 \, a^{2} b^{3} x^{10} e - 1155 \, a b^{4} c x^{8} + 1155 \, a^{2} b^{3} d x^{8} + 1155 \, a^{4} b f x^{8} - 1155 \, a^{3} b^{2} x^{8} e + 693 \, a^{2} b^{3} c x^{6} - 693 \, a^{3} b^{2} d x^{6} - 693 \, a^{5} f x^{6} + 693 \, a^{4} b x^{6} e - 495 \, a^{3} b^{2} c x^{4} + 495 \, a^{4} b d x^{4} - 495 \, a^{5} x^{4} e + 385 \, a^{4} b c x^{2} - 385 \, a^{5} d x^{2} - 315 \, a^{5} c}{3465 \, a^{6} x^{11}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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