Optimal. Leaf size=81 \[ -\frac {a^{9/2} \tan ^{-1}\left (\frac {\sqrt {b} x}{\sqrt {a}}\right )}{b^{11/2}}+\frac {a^4 x}{b^5}-\frac {a^3 x^3}{3 b^4}+\frac {a^2 x^5}{5 b^3}-\frac {a x^7}{7 b^2}+\frac {x^9}{9 b} \]
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Rubi [A] time = 0.04, antiderivative size = 81, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, integrand size = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.154, Rules used = {302, 205} \[ \frac {a^2 x^5}{5 b^3}-\frac {a^3 x^3}{3 b^4}+\frac {a^4 x}{b^5}-\frac {a^{9/2} \tan ^{-1}\left (\frac {\sqrt {b} x}{\sqrt {a}}\right )}{b^{11/2}}-\frac {a x^7}{7 b^2}+\frac {x^9}{9 b} \]
Antiderivative was successfully verified.
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Rule 205
Rule 302
Rubi steps
\begin {align*} \int \frac {x^{10}}{a+b x^2} \, dx &=\int \left (\frac {a^4}{b^5}-\frac {a^3 x^2}{b^4}+\frac {a^2 x^4}{b^3}-\frac {a x^6}{b^2}+\frac {x^8}{b}-\frac {a^5}{b^5 \left (a+b x^2\right )}\right ) \, dx\\ &=\frac {a^4 x}{b^5}-\frac {a^3 x^3}{3 b^4}+\frac {a^2 x^5}{5 b^3}-\frac {a x^7}{7 b^2}+\frac {x^9}{9 b}-\frac {a^5 \int \frac {1}{a+b x^2} \, dx}{b^5}\\ &=\frac {a^4 x}{b^5}-\frac {a^3 x^3}{3 b^4}+\frac {a^2 x^5}{5 b^3}-\frac {a x^7}{7 b^2}+\frac {x^9}{9 b}-\frac {a^{9/2} \tan ^{-1}\left (\frac {\sqrt {b} x}{\sqrt {a}}\right )}{b^{11/2}}\\ \end {align*}
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Mathematica [A] time = 0.03, size = 81, normalized size = 1.00 \[ -\frac {a^{9/2} \tan ^{-1}\left (\frac {\sqrt {b} x}{\sqrt {a}}\right )}{b^{11/2}}+\frac {a^4 x}{b^5}-\frac {a^3 x^3}{3 b^4}+\frac {a^2 x^5}{5 b^3}-\frac {a x^7}{7 b^2}+\frac {x^9}{9 b} \]
Antiderivative was successfully verified.
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fricas [A] time = 1.01, size = 170, normalized size = 2.10 \[ \left [\frac {70 \, b^{4} x^{9} - 90 \, a b^{3} x^{7} + 126 \, a^{2} b^{2} x^{5} - 210 \, a^{3} b x^{3} + 315 \, a^{4} \sqrt {-\frac {a}{b}} \log \left (\frac {b x^{2} - 2 \, b x \sqrt {-\frac {a}{b}} - a}{b x^{2} + a}\right ) + 630 \, a^{4} x}{630 \, b^{5}}, \frac {35 \, b^{4} x^{9} - 45 \, a b^{3} x^{7} + 63 \, a^{2} b^{2} x^{5} - 105 \, a^{3} b x^{3} - 315 \, a^{4} \sqrt {\frac {a}{b}} \arctan \left (\frac {b x \sqrt {\frac {a}{b}}}{a}\right ) + 315 \, a^{4} x}{315 \, b^{5}}\right ] \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 1.08, size = 77, normalized size = 0.95 \[ -\frac {a^{5} \arctan \left (\frac {b x}{\sqrt {a b}}\right )}{\sqrt {a b} b^{5}} + \frac {35 \, b^{8} x^{9} - 45 \, a b^{7} x^{7} + 63 \, a^{2} b^{6} x^{5} - 105 \, a^{3} b^{5} x^{3} + 315 \, a^{4} b^{4} x}{315 \, b^{9}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.00, size = 71, normalized size = 0.88 \[ \frac {x^{9}}{9 b}-\frac {a \,x^{7}}{7 b^{2}}+\frac {a^{2} x^{5}}{5 b^{3}}-\frac {a^{3} x^{3}}{3 b^{4}}-\frac {a^{5} \arctan \left (\frac {b x}{\sqrt {a b}}\right )}{\sqrt {a b}\, b^{5}}+\frac {a^{4} x}{b^{5}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 2.92, size = 72, normalized size = 0.89 \[ -\frac {a^{5} \arctan \left (\frac {b x}{\sqrt {a b}}\right )}{\sqrt {a b} b^{5}} + \frac {35 \, b^{4} x^{9} - 45 \, a b^{3} x^{7} + 63 \, a^{2} b^{2} x^{5} - 105 \, a^{3} b x^{3} + 315 \, a^{4} x}{315 \, b^{5}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.06, size = 65, normalized size = 0.80 \[ \frac {x^9}{9\,b}-\frac {a\,x^7}{7\,b^2}+\frac {a^4\,x}{b^5}-\frac {a^{9/2}\,\mathrm {atan}\left (\frac {\sqrt {b}\,x}{\sqrt {a}}\right )}{b^{11/2}}+\frac {a^2\,x^5}{5\,b^3}-\frac {a^3\,x^3}{3\,b^4} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.20, size = 119, normalized size = 1.47 \[ \frac {a^{4} x}{b^{5}} - \frac {a^{3} x^{3}}{3 b^{4}} + \frac {a^{2} x^{5}}{5 b^{3}} - \frac {a x^{7}}{7 b^{2}} + \frac {\sqrt {- \frac {a^{9}}{b^{11}}} \log {\left (x - \frac {b^{5} \sqrt {- \frac {a^{9}}{b^{11}}}}{a^{4}} \right )}}{2} - \frac {\sqrt {- \frac {a^{9}}{b^{11}}} \log {\left (x + \frac {b^{5} \sqrt {- \frac {a^{9}}{b^{11}}}}{a^{4}} \right )}}{2} + \frac {x^{9}}{9 b} \]
Verification of antiderivative is not currently implemented for this CAS.
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