Optimal. Leaf size=287 \[ \frac{x^9 \left (c-\frac{a \left (a^2 f-a b e+b^2 d\right )}{b^3}\right )}{4 a \left (a+b x^2\right )^2}-\frac{x^5 \left (17 a^2 b e-29 a^3 f-9 a b^2 d+5 b^3 c\right )}{20 a b^5}+\frac{x^3 \left (15 a^2 b e-23 a^3 f-9 a b^2 d+5 b^3 c\right )}{6 b^6}-\frac{a^2 x \left (13 a^2 b e-17 a^3 f-9 a b^2 d+5 b^3 c\right )}{8 b^7 \left (a+b x^2\right )}-\frac{a x \left (43 a^2 b e-63 a^3 f-27 a b^2 d+15 b^3 c\right )}{4 b^7}+\frac{a^{3/2} \tan ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right ) \left (99 a^2 b e-143 a^3 f-63 a b^2 d+35 b^3 c\right )}{8 b^{15/2}}+\frac{x^7 (b e-3 a f)}{7 b^4}+\frac{f x^9}{9 b^3} \]
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Rubi [A] time = 0.492374, antiderivative size = 287, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 5, integrand size = 30, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.167, Rules used = {1804, 1585, 1257, 1810, 205} \[ \frac{x^9 \left (c-\frac{a \left (a^2 f-a b e+b^2 d\right )}{b^3}\right )}{4 a \left (a+b x^2\right )^2}-\frac{x^5 \left (17 a^2 b e-29 a^3 f-9 a b^2 d+5 b^3 c\right )}{20 a b^5}+\frac{x^3 \left (15 a^2 b e-23 a^3 f-9 a b^2 d+5 b^3 c\right )}{6 b^6}-\frac{a^2 x \left (13 a^2 b e-17 a^3 f-9 a b^2 d+5 b^3 c\right )}{8 b^7 \left (a+b x^2\right )}-\frac{a x \left (43 a^2 b e-63 a^3 f-27 a b^2 d+15 b^3 c\right )}{4 b^7}+\frac{a^{3/2} \tan ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right ) \left (99 a^2 b e-143 a^3 f-63 a b^2 d+35 b^3 c\right )}{8 b^{15/2}}+\frac{x^7 (b e-3 a f)}{7 b^4}+\frac{f x^9}{9 b^3} \]
Antiderivative was successfully verified.
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Rule 1804
Rule 1585
Rule 1257
Rule 1810
Rule 205
Rubi steps
\begin{align*} \int \frac{x^8 \left (c+d x^2+e x^4+f x^6\right )}{\left (a+b x^2\right )^3} \, dx &=\frac{\left (c-\frac{a \left (b^2 d-a b e+a^2 f\right )}{b^3}\right ) x^9}{4 a \left (a+b x^2\right )^2}-\frac{\int \frac{x^7 \left (\left (5 b c-9 a d+\frac{9 a^2 e}{b}-\frac{9 a^3 f}{b^2}\right ) x-4 a \left (e-\frac{a f}{b}\right ) x^3-4 a f x^5\right )}{\left (a+b x^2\right )^2} \, dx}{4 a b}\\ &=\frac{\left (c-\frac{a \left (b^2 d-a b e+a^2 f\right )}{b^3}\right ) x^9}{4 a \left (a+b x^2\right )^2}-\frac{\int \frac{x^8 \left (5 b c-9 a d+\frac{9 a^2 e}{b}-\frac{9 a^3 f}{b^2}-4 a \left (e-\frac{a f}{b}\right ) x^2-4 a f x^4\right )}{\left (a+b x^2\right )^2} \, dx}{4 a b}\\ &=\frac{\left (c-\frac{a \left (b^2 d-a b e+a^2 f\right )}{b^3}\right ) x^9}{4 a \left (a+b x^2\right )^2}-\frac{a^2 \left (5 b^3 c-9 a b^2 d+13 a^2 b e-17 a^3 f\right ) x}{8 b^7 \left (a+b x^2\right )}+\frac{\int \frac{a^3 \left (5 b^3 c-9 a b^2 d+13 a^2 b e-17 a^3 f\right )-2 a^2 b \left (5 b^3 c-9 a b^2 d+13 a^2 b e-17 a^3 f\right ) x^2+2 a b^2 \left (5 b^3 c-9 a b^2 d+13 a^2 b e-17 a^3 f\right ) x^4-2 b^3 \left (5 b^3 c-9 a b^2 d+13 a^2 b e-17 a^3 f\right ) x^6+8 a b^4 (b e-2 a f) x^8+8 a b^5 f x^{10}}{a+b x^2} \, dx}{8 a b^7}\\ &=\frac{\left (c-\frac{a \left (b^2 d-a b e+a^2 f\right )}{b^3}\right ) x^9}{4 a \left (a+b x^2\right )^2}-\frac{a^2 \left (5 b^3 c-9 a b^2 d+13 a^2 b e-17 a^3 f\right ) x}{8 b^7 \left (a+b x^2\right )}+\frac{\int \left (-2 a^2 \left (15 b^3 c-27 a b^2 d+43 a^2 b e-63 a^3 f\right )+4 a b \left (5 b^3 c-9 a b^2 d+15 a^2 b e-23 a^3 f\right ) x^2-2 b^2 \left (5 b^3 c-9 a b^2 d+17 a^2 b e-29 a^3 f\right ) x^4+8 a b^3 (b e-3 a f) x^6+8 a b^4 f x^8+\frac{35 a^3 b^3 c-63 a^4 b^2 d+99 a^5 b e-143 a^6 f}{a+b x^2}\right ) \, dx}{8 a b^7}\\ &=-\frac{a \left (15 b^3 c-27 a b^2 d+43 a^2 b e-63 a^3 f\right ) x}{4 b^7}+\frac{\left (5 b^3 c-9 a b^2 d+15 a^2 b e-23 a^3 f\right ) x^3}{6 b^6}-\frac{\left (5 b^3 c-9 a b^2 d+17 a^2 b e-29 a^3 f\right ) x^5}{20 a b^5}+\frac{(b e-3 a f) x^7}{7 b^4}+\frac{f x^9}{9 b^3}+\frac{\left (c-\frac{a \left (b^2 d-a b e+a^2 f\right )}{b^3}\right ) x^9}{4 a \left (a+b x^2\right )^2}-\frac{a^2 \left (5 b^3 c-9 a b^2 d+13 a^2 b e-17 a^3 f\right ) x}{8 b^7 \left (a+b x^2\right )}+\frac{\left (a^2 \left (35 b^3 c-63 a b^2 d+99 a^2 b e-143 a^3 f\right )\right ) \int \frac{1}{a+b x^2} \, dx}{8 b^7}\\ &=-\frac{a \left (15 b^3 c-27 a b^2 d+43 a^2 b e-63 a^3 f\right ) x}{4 b^7}+\frac{\left (5 b^3 c-9 a b^2 d+15 a^2 b e-23 a^3 f\right ) x^3}{6 b^6}-\frac{\left (5 b^3 c-9 a b^2 d+17 a^2 b e-29 a^3 f\right ) x^5}{20 a b^5}+\frac{(b e-3 a f) x^7}{7 b^4}+\frac{f x^9}{9 b^3}+\frac{\left (c-\frac{a \left (b^2 d-a b e+a^2 f\right )}{b^3}\right ) x^9}{4 a \left (a+b x^2\right )^2}-\frac{a^2 \left (5 b^3 c-9 a b^2 d+13 a^2 b e-17 a^3 f\right ) x}{8 b^7 \left (a+b x^2\right )}+\frac{a^{3/2} \left (35 b^3 c-63 a b^2 d+99 a^2 b e-143 a^3 f\right ) \tan ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right )}{8 b^{15/2}}\\ \end{align*}
Mathematica [A] time = 0.154838, size = 272, normalized size = 0.95 \[ \frac{x^3 \left (6 a^2 b e-10 a^3 f-3 a b^2 d+b^3 c\right )}{3 b^6}+\frac{a^2 x \left (-21 a^2 b e+25 a^3 f+17 a b^2 d-13 b^3 c\right )}{8 b^7 \left (a+b x^2\right )}+\frac{a^3 x \left (a^2 b e+a^3 (-f)-a b^2 d+b^3 c\right )}{4 b^7 \left (a+b x^2\right )^2}+\frac{a x \left (-10 a^2 b e+15 a^3 f+6 a b^2 d-3 b^3 c\right )}{b^7}-\frac{a^{3/2} \tan ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right ) \left (-99 a^2 b e+143 a^3 f+63 a b^2 d-35 b^3 c\right )}{8 b^{15/2}}+\frac{x^5 \left (6 a^2 f-3 a b e+b^2 d\right )}{5 b^5}+\frac{x^7 (b e-3 a f)}{7 b^4}+\frac{f x^9}{9 b^3} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.014, size = 394, normalized size = 1.4 \begin{align*}{\frac{{x}^{7}e}{7\,{b}^{3}}}+{\frac{{x}^{5}d}{5\,{b}^{3}}}+{\frac{{x}^{3}c}{3\,{b}^{3}}}+{\frac{15\,{a}^{4}dx}{8\,{b}^{5} \left ( b{x}^{2}+a \right ) ^{2}}}+{\frac{25\,{a}^{5}{x}^{3}f}{8\,{b}^{6} \left ( b{x}^{2}+a \right ) ^{2}}}+{\frac{f{x}^{9}}{9\,{b}^{3}}}-{\frac{11\,{a}^{3}cx}{8\,{b}^{4} \left ( b{x}^{2}+a \right ) ^{2}}}-{\frac{143\,{a}^{5}f}{8\,{b}^{7}}\arctan \left ({bx{\frac{1}{\sqrt{ab}}}} \right ){\frac{1}{\sqrt{ab}}}}+{\frac{99\,{a}^{4}e}{8\,{b}^{6}}\arctan \left ({bx{\frac{1}{\sqrt{ab}}}} \right ){\frac{1}{\sqrt{ab}}}}-{\frac{63\,{a}^{3}d}{8\,{b}^{5}}\arctan \left ({bx{\frac{1}{\sqrt{ab}}}} \right ){\frac{1}{\sqrt{ab}}}}+{\frac{35\,{a}^{2}c}{8\,{b}^{4}}\arctan \left ({bx{\frac{1}{\sqrt{ab}}}} \right ){\frac{1}{\sqrt{ab}}}}-{\frac{21\,{a}^{4}{x}^{3}e}{8\,{b}^{5} \left ( b{x}^{2}+a \right ) ^{2}}}+{\frac{17\,{a}^{3}{x}^{3}d}{8\,{b}^{4} \left ( b{x}^{2}+a \right ) ^{2}}}-{\frac{13\,{x}^{3}{a}^{2}c}{8\,{b}^{3} \left ( b{x}^{2}+a \right ) ^{2}}}+{\frac{23\,{a}^{6}fx}{8\,{b}^{7} \left ( b{x}^{2}+a \right ) ^{2}}}-{\frac{19\,{a}^{5}ex}{8\,{b}^{6} \left ( b{x}^{2}+a \right ) ^{2}}}-{\frac{10\,{a}^{3}{x}^{3}f}{3\,{b}^{6}}}+2\,{\frac{{x}^{3}{a}^{2}e}{{b}^{5}}}-{\frac{a{x}^{3}d}{{b}^{4}}}+15\,{\frac{{a}^{4}fx}{{b}^{7}}}-10\,{\frac{{a}^{3}ex}{{b}^{6}}}+6\,{\frac{{a}^{2}dx}{{b}^{5}}}-3\,{\frac{acx}{{b}^{4}}}-{\frac{3\,{x}^{7}af}{7\,{b}^{4}}}+{\frac{6\,{x}^{5}{a}^{2}f}{5\,{b}^{5}}}-{\frac{3\,{x}^{5}ae}{5\,{b}^{4}}} \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 = 1.25273, size = 1748, normalized size = 6.09 \begin{align*} \text{result too large to display} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 16.1898, size = 491, normalized size = 1.71 \begin{align*} \frac{\sqrt{- \frac{a^{3}}{b^{15}}} \left (143 a^{3} f - 99 a^{2} b e + 63 a b^{2} d - 35 b^{3} c\right ) \log{\left (- \frac{b^{7} \sqrt{- \frac{a^{3}}{b^{15}}} \left (143 a^{3} f - 99 a^{2} b e + 63 a b^{2} d - 35 b^{3} c\right )}{143 a^{4} f - 99 a^{3} b e + 63 a^{2} b^{2} d - 35 a b^{3} c} + x \right )}}{16} - \frac{\sqrt{- \frac{a^{3}}{b^{15}}} \left (143 a^{3} f - 99 a^{2} b e + 63 a b^{2} d - 35 b^{3} c\right ) \log{\left (\frac{b^{7} \sqrt{- \frac{a^{3}}{b^{15}}} \left (143 a^{3} f - 99 a^{2} b e + 63 a b^{2} d - 35 b^{3} c\right )}{143 a^{4} f - 99 a^{3} b e + 63 a^{2} b^{2} d - 35 a b^{3} c} + x \right )}}{16} + \frac{x^{3} \left (25 a^{5} b f - 21 a^{4} b^{2} e + 17 a^{3} b^{3} d - 13 a^{2} b^{4} c\right ) + x \left (23 a^{6} f - 19 a^{5} b e + 15 a^{4} b^{2} d - 11 a^{3} b^{3} c\right )}{8 a^{2} b^{7} + 16 a b^{8} x^{2} + 8 b^{9} x^{4}} + \frac{f x^{9}}{9 b^{3}} - \frac{x^{7} \left (3 a f - b e\right )}{7 b^{4}} + \frac{x^{5} \left (6 a^{2} f - 3 a b e + b^{2} d\right )}{5 b^{5}} - \frac{x^{3} \left (10 a^{3} f - 6 a^{2} b e + 3 a b^{2} d - b^{3} c\right )}{3 b^{6}} + \frac{x \left (15 a^{4} f - 10 a^{3} b e + 6 a^{2} b^{2} d - 3 a b^{3} c\right )}{b^{7}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.17982, size = 406, normalized size = 1.41 \begin{align*} \frac{{\left (35 \, a^{2} b^{3} c - 63 \, a^{3} b^{2} d - 143 \, a^{5} f + 99 \, a^{4} b e\right )} \arctan \left (\frac{b x}{\sqrt{a b}}\right )}{8 \, \sqrt{a b} b^{7}} - \frac{13 \, a^{2} b^{4} c x^{3} - 17 \, a^{3} b^{3} d x^{3} - 25 \, a^{5} b f x^{3} + 21 \, a^{4} b^{2} x^{3} e + 11 \, a^{3} b^{3} c x - 15 \, a^{4} b^{2} d x - 23 \, a^{6} f x + 19 \, a^{5} b x e}{8 \,{\left (b x^{2} + a\right )}^{2} b^{7}} + \frac{35 \, b^{24} f x^{9} - 135 \, a b^{23} f x^{7} + 45 \, b^{24} x^{7} e + 63 \, b^{24} d x^{5} + 378 \, a^{2} b^{22} f x^{5} - 189 \, a b^{23} x^{5} e + 105 \, b^{24} c x^{3} - 315 \, a b^{23} d x^{3} - 1050 \, a^{3} b^{21} f x^{3} + 630 \, a^{2} b^{22} x^{3} e - 945 \, a b^{23} c x + 1890 \, a^{2} b^{22} d x + 4725 \, a^{4} b^{20} f x - 3150 \, a^{3} b^{21} x e}{315 \, b^{27}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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