Optimal. Leaf size=98 \[ -\frac {63 a^{5/2} \tan ^{-1}\left (\frac {\sqrt {b} x}{\sqrt {a}}\right )}{8 b^{11/2}}+\frac {63 a^2 x}{8 b^5}-\frac {21 a x^3}{8 b^4}-\frac {9 x^7}{8 b^2 \left (a+b x^2\right )}-\frac {x^9}{4 b \left (a+b x^2\right )^2}+\frac {63 x^5}{40 b^3} \]
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Rubi [A] time = 0.04, antiderivative size = 98, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 3, integrand size = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.231, Rules used = {288, 302, 205} \[ \frac {63 a^2 x}{8 b^5}-\frac {63 a^{5/2} \tan ^{-1}\left (\frac {\sqrt {b} x}{\sqrt {a}}\right )}{8 b^{11/2}}-\frac {9 x^7}{8 b^2 \left (a+b x^2\right )}-\frac {21 a x^3}{8 b^4}-\frac {x^9}{4 b \left (a+b x^2\right )^2}+\frac {63 x^5}{40 b^3} \]
Antiderivative was successfully verified.
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Rule 205
Rule 288
Rule 302
Rubi steps
\begin {align*} \int \frac {x^{10}}{\left (a+b x^2\right )^3} \, dx &=-\frac {x^9}{4 b \left (a+b x^2\right )^2}+\frac {9 \int \frac {x^8}{\left (a+b x^2\right )^2} \, dx}{4 b}\\ &=-\frac {x^9}{4 b \left (a+b x^2\right )^2}-\frac {9 x^7}{8 b^2 \left (a+b x^2\right )}+\frac {63 \int \frac {x^6}{a+b x^2} \, dx}{8 b^2}\\ &=-\frac {x^9}{4 b \left (a+b x^2\right )^2}-\frac {9 x^7}{8 b^2 \left (a+b x^2\right )}+\frac {63 \int \left (\frac {a^2}{b^3}-\frac {a x^2}{b^2}+\frac {x^4}{b}-\frac {a^3}{b^3 \left (a+b x^2\right )}\right ) \, dx}{8 b^2}\\ &=\frac {63 a^2 x}{8 b^5}-\frac {21 a x^3}{8 b^4}+\frac {63 x^5}{40 b^3}-\frac {x^9}{4 b \left (a+b x^2\right )^2}-\frac {9 x^7}{8 b^2 \left (a+b x^2\right )}-\frac {\left (63 a^3\right ) \int \frac {1}{a+b x^2} \, dx}{8 b^5}\\ &=\frac {63 a^2 x}{8 b^5}-\frac {21 a x^3}{8 b^4}+\frac {63 x^5}{40 b^3}-\frac {x^9}{4 b \left (a+b x^2\right )^2}-\frac {9 x^7}{8 b^2 \left (a+b x^2\right )}-\frac {63 a^{5/2} \tan ^{-1}\left (\frac {\sqrt {b} x}{\sqrt {a}}\right )}{8 b^{11/2}}\\ \end {align*}
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Mathematica [A] time = 0.05, size = 88, normalized size = 0.90 \[ \frac {315 a^4 x+525 a^3 b x^3+168 a^2 b^2 x^5-24 a b^3 x^7+8 b^4 x^9}{40 b^5 \left (a+b x^2\right )^2}-\frac {63 a^{5/2} \tan ^{-1}\left (\frac {\sqrt {b} x}{\sqrt {a}}\right )}{8 b^{11/2}} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.93, size = 256, normalized size = 2.61 \[ \left [\frac {16 \, b^{4} x^{9} - 48 \, a b^{3} x^{7} + 336 \, a^{2} b^{2} x^{5} + 1050 \, a^{3} b x^{3} + 630 \, a^{4} x + 315 \, {\left (a^{2} b^{2} x^{4} + 2 \, a^{3} b x^{2} + a^{4}\right )} \sqrt {-\frac {a}{b}} \log \left (\frac {b x^{2} - 2 \, b x \sqrt {-\frac {a}{b}} - a}{b x^{2} + a}\right )}{80 \, {\left (b^{7} x^{4} + 2 \, a b^{6} x^{2} + a^{2} b^{5}\right )}}, \frac {8 \, b^{4} x^{9} - 24 \, a b^{3} x^{7} + 168 \, a^{2} b^{2} x^{5} + 525 \, a^{3} b x^{3} + 315 \, a^{4} x - 315 \, {\left (a^{2} b^{2} x^{4} + 2 \, a^{3} b x^{2} + a^{4}\right )} \sqrt {\frac {a}{b}} \arctan \left (\frac {b x \sqrt {\frac {a}{b}}}{a}\right )}{40 \, {\left (b^{7} x^{4} + 2 \, a b^{6} x^{2} + a^{2} b^{5}\right )}}\right ] \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.64, size = 84, normalized size = 0.86 \[ -\frac {63 \, a^{3} \arctan \left (\frac {b x}{\sqrt {a b}}\right )}{8 \, \sqrt {a b} b^{5}} + \frac {17 \, a^{3} b x^{3} + 15 \, a^{4} x}{8 \, {\left (b x^{2} + a\right )}^{2} b^{5}} + \frac {b^{12} x^{5} - 5 \, a b^{11} x^{3} + 30 \, a^{2} b^{10} x}{5 \, b^{15}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.01, size = 88, normalized size = 0.90 \[ \frac {17 a^{3} x^{3}}{8 \left (b \,x^{2}+a \right )^{2} b^{4}}+\frac {x^{5}}{5 b^{3}}+\frac {15 a^{4} x}{8 \left (b \,x^{2}+a \right )^{2} b^{5}}-\frac {a \,x^{3}}{b^{4}}-\frac {63 a^{3} \arctan \left (\frac {b x}{\sqrt {a b}}\right )}{8 \sqrt {a b}\, b^{5}}+\frac {6 a^{2} x}{b^{5}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 2.86, size = 93, normalized size = 0.95 \[ \frac {17 \, a^{3} b x^{3} + 15 \, a^{4} x}{8 \, {\left (b^{7} x^{4} + 2 \, a b^{6} x^{2} + a^{2} b^{5}\right )}} - \frac {63 \, a^{3} \arctan \left (\frac {b x}{\sqrt {a b}}\right )}{8 \, \sqrt {a b} b^{5}} + \frac {b^{2} x^{5} - 5 \, a b x^{3} + 30 \, a^{2} x}{5 \, b^{5}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.07, size = 87, normalized size = 0.89 \[ \frac {\frac {15\,a^4\,x}{8}+\frac {17\,b\,a^3\,x^3}{8}}{a^2\,b^5+2\,a\,b^6\,x^2+b^7\,x^4}+\frac {x^5}{5\,b^3}-\frac {a\,x^3}{b^4}+\frac {6\,a^2\,x}{b^5}-\frac {63\,a^{5/2}\,\mathrm {atan}\left (\frac {\sqrt {b}\,x}{\sqrt {a}}\right )}{8\,b^{11/2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.47, size = 144, normalized size = 1.47 \[ \frac {6 a^{2} x}{b^{5}} - \frac {a x^{3}}{b^{4}} + \frac {63 \sqrt {- \frac {a^{5}}{b^{11}}} \log {\left (x - \frac {b^{5} \sqrt {- \frac {a^{5}}{b^{11}}}}{a^{2}} \right )}}{16} - \frac {63 \sqrt {- \frac {a^{5}}{b^{11}}} \log {\left (x + \frac {b^{5} \sqrt {- \frac {a^{5}}{b^{11}}}}{a^{2}} \right )}}{16} + \frac {15 a^{4} x + 17 a^{3} b x^{3}}{8 a^{2} b^{5} + 16 a b^{6} x^{2} + 8 b^{7} x^{4}} + \frac {x^{5}}{5 b^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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