Optimal. Leaf size=131 \[ -\frac {63 b^5 \tanh ^{-1}\left (\frac {\sqrt {a+b x^2}}{\sqrt {a}}\right )}{256 \sqrt {a}}-\frac {63 b^4 \sqrt {a+b x^2}}{256 x^2}-\frac {21 b^3 \left (a+b x^2\right )^{3/2}}{128 x^4}-\frac {21 b^2 \left (a+b x^2\right )^{5/2}}{160 x^6}-\frac {\left (a+b x^2\right )^{9/2}}{10 x^{10}}-\frac {9 b \left (a+b x^2\right )^{7/2}}{80 x^8} \]
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Rubi [A] time = 0.08, antiderivative size = 131, normalized size of antiderivative = 1.00, number of steps used = 8, number of rules used = 4, integrand size = 15, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.267, Rules used = {266, 47, 63, 208} \[ -\frac {63 b^4 \sqrt {a+b x^2}}{256 x^2}-\frac {21 b^3 \left (a+b x^2\right )^{3/2}}{128 x^4}-\frac {21 b^2 \left (a+b x^2\right )^{5/2}}{160 x^6}-\frac {63 b^5 \tanh ^{-1}\left (\frac {\sqrt {a+b x^2}}{\sqrt {a}}\right )}{256 \sqrt {a}}-\frac {9 b \left (a+b x^2\right )^{7/2}}{80 x^8}-\frac {\left (a+b x^2\right )^{9/2}}{10 x^{10}} \]
Antiderivative was successfully verified.
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Rule 47
Rule 63
Rule 208
Rule 266
Rubi steps
\begin {align*} \int \frac {\left (a+b x^2\right )^{9/2}}{x^{11}} \, dx &=\frac {1}{2} \operatorname {Subst}\left (\int \frac {(a+b x)^{9/2}}{x^6} \, dx,x,x^2\right )\\ &=-\frac {\left (a+b x^2\right )^{9/2}}{10 x^{10}}+\frac {1}{20} (9 b) \operatorname {Subst}\left (\int \frac {(a+b x)^{7/2}}{x^5} \, dx,x,x^2\right )\\ &=-\frac {9 b \left (a+b x^2\right )^{7/2}}{80 x^8}-\frac {\left (a+b x^2\right )^{9/2}}{10 x^{10}}+\frac {1}{160} \left (63 b^2\right ) \operatorname {Subst}\left (\int \frac {(a+b x)^{5/2}}{x^4} \, dx,x,x^2\right )\\ &=-\frac {21 b^2 \left (a+b x^2\right )^{5/2}}{160 x^6}-\frac {9 b \left (a+b x^2\right )^{7/2}}{80 x^8}-\frac {\left (a+b x^2\right )^{9/2}}{10 x^{10}}+\frac {1}{64} \left (21 b^3\right ) \operatorname {Subst}\left (\int \frac {(a+b x)^{3/2}}{x^3} \, dx,x,x^2\right )\\ &=-\frac {21 b^3 \left (a+b x^2\right )^{3/2}}{128 x^4}-\frac {21 b^2 \left (a+b x^2\right )^{5/2}}{160 x^6}-\frac {9 b \left (a+b x^2\right )^{7/2}}{80 x^8}-\frac {\left (a+b x^2\right )^{9/2}}{10 x^{10}}+\frac {1}{256} \left (63 b^4\right ) \operatorname {Subst}\left (\int \frac {\sqrt {a+b x}}{x^2} \, dx,x,x^2\right )\\ &=-\frac {63 b^4 \sqrt {a+b x^2}}{256 x^2}-\frac {21 b^3 \left (a+b x^2\right )^{3/2}}{128 x^4}-\frac {21 b^2 \left (a+b x^2\right )^{5/2}}{160 x^6}-\frac {9 b \left (a+b x^2\right )^{7/2}}{80 x^8}-\frac {\left (a+b x^2\right )^{9/2}}{10 x^{10}}+\frac {1}{512} \left (63 b^5\right ) \operatorname {Subst}\left (\int \frac {1}{x \sqrt {a+b x}} \, dx,x,x^2\right )\\ &=-\frac {63 b^4 \sqrt {a+b x^2}}{256 x^2}-\frac {21 b^3 \left (a+b x^2\right )^{3/2}}{128 x^4}-\frac {21 b^2 \left (a+b x^2\right )^{5/2}}{160 x^6}-\frac {9 b \left (a+b x^2\right )^{7/2}}{80 x^8}-\frac {\left (a+b x^2\right )^{9/2}}{10 x^{10}}+\frac {1}{256} \left (63 b^4\right ) \operatorname {Subst}\left (\int \frac {1}{-\frac {a}{b}+\frac {x^2}{b}} \, dx,x,\sqrt {a+b x^2}\right )\\ &=-\frac {63 b^4 \sqrt {a+b x^2}}{256 x^2}-\frac {21 b^3 \left (a+b x^2\right )^{3/2}}{128 x^4}-\frac {21 b^2 \left (a+b x^2\right )^{5/2}}{160 x^6}-\frac {9 b \left (a+b x^2\right )^{7/2}}{80 x^8}-\frac {\left (a+b x^2\right )^{9/2}}{10 x^{10}}-\frac {63 b^5 \tanh ^{-1}\left (\frac {\sqrt {a+b x^2}}{\sqrt {a}}\right )}{256 \sqrt {a}}\\ \end {align*}
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Mathematica [A] time = 0.05, size = 109, normalized size = 0.83 \[ -\frac {128 a^5+784 a^4 b x^2+2024 a^3 b^2 x^4+2858 a^2 b^3 x^6+315 b^5 x^{10} \sqrt {\frac {b x^2}{a}+1} \tanh ^{-1}\left (\sqrt {\frac {b x^2}{a}+1}\right )+2455 a b^4 x^8+965 b^5 x^{10}}{1280 x^{10} \sqrt {a+b x^2}} \]
Antiderivative was successfully verified.
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fricas [A] time = 1.05, size = 202, normalized size = 1.54 \[ \left [\frac {315 \, \sqrt {a} b^{5} x^{10} \log \left (-\frac {b x^{2} - 2 \, \sqrt {b x^{2} + a} \sqrt {a} + 2 \, a}{x^{2}}\right ) - 2 \, {\left (965 \, a b^{4} x^{8} + 1490 \, a^{2} b^{3} x^{6} + 1368 \, a^{3} b^{2} x^{4} + 656 \, a^{4} b x^{2} + 128 \, a^{5}\right )} \sqrt {b x^{2} + a}}{2560 \, a x^{10}}, \frac {315 \, \sqrt {-a} b^{5} x^{10} \arctan \left (\frac {\sqrt {-a}}{\sqrt {b x^{2} + a}}\right ) - {\left (965 \, a b^{4} x^{8} + 1490 \, a^{2} b^{3} x^{6} + 1368 \, a^{3} b^{2} x^{4} + 656 \, a^{4} b x^{2} + 128 \, a^{5}\right )} \sqrt {b x^{2} + a}}{1280 \, a x^{10}}\right ] \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 1.09, size = 121, normalized size = 0.92 \[ \frac {\frac {315 \, b^{6} \arctan \left (\frac {\sqrt {b x^{2} + a}}{\sqrt {-a}}\right )}{\sqrt {-a}} - \frac {965 \, {\left (b x^{2} + a\right )}^{\frac {9}{2}} b^{6} - 2370 \, {\left (b x^{2} + a\right )}^{\frac {7}{2}} a b^{6} + 2688 \, {\left (b x^{2} + a\right )}^{\frac {5}{2}} a^{2} b^{6} - 1470 \, {\left (b x^{2} + a\right )}^{\frac {3}{2}} a^{3} b^{6} + 315 \, \sqrt {b x^{2} + a} a^{4} b^{6}}{b^{5} x^{10}}}{1280 \, b} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.05, size = 213, normalized size = 1.63 \[ -\frac {63 b^{5} \ln \left (\frac {2 a +2 \sqrt {b \,x^{2}+a}\, \sqrt {a}}{x}\right )}{256 \sqrt {a}}+\frac {63 \sqrt {b \,x^{2}+a}\, b^{5}}{256 a}+\frac {21 \left (b \,x^{2}+a \right )^{\frac {3}{2}} b^{5}}{256 a^{2}}+\frac {63 \left (b \,x^{2}+a \right )^{\frac {5}{2}} b^{5}}{1280 a^{3}}+\frac {9 \left (b \,x^{2}+a \right )^{\frac {7}{2}} b^{5}}{256 a^{4}}+\frac {7 \left (b \,x^{2}+a \right )^{\frac {9}{2}} b^{5}}{256 a^{5}}-\frac {7 \left (b \,x^{2}+a \right )^{\frac {11}{2}} b^{4}}{256 a^{5} x^{2}}-\frac {\left (b \,x^{2}+a \right )^{\frac {11}{2}} b^{3}}{128 a^{4} x^{4}}-\frac {\left (b \,x^{2}+a \right )^{\frac {11}{2}} b^{2}}{160 a^{3} x^{6}}-\frac {\left (b \,x^{2}+a \right )^{\frac {11}{2}} b}{80 a^{2} x^{8}}-\frac {\left (b \,x^{2}+a \right )^{\frac {11}{2}}}{10 a \,x^{10}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 1.48, size = 201, normalized size = 1.53 \[ -\frac {63 \, b^{5} \operatorname {arsinh}\left (\frac {a}{\sqrt {a b} {\left | x \right |}}\right )}{256 \, \sqrt {a}} + \frac {7 \, {\left (b x^{2} + a\right )}^{\frac {9}{2}} b^{5}}{256 \, a^{5}} + \frac {9 \, {\left (b x^{2} + a\right )}^{\frac {7}{2}} b^{5}}{256 \, a^{4}} + \frac {63 \, {\left (b x^{2} + a\right )}^{\frac {5}{2}} b^{5}}{1280 \, a^{3}} + \frac {21 \, {\left (b x^{2} + a\right )}^{\frac {3}{2}} b^{5}}{256 \, a^{2}} + \frac {63 \, \sqrt {b x^{2} + a} b^{5}}{256 \, a} - \frac {7 \, {\left (b x^{2} + a\right )}^{\frac {11}{2}} b^{4}}{256 \, a^{5} x^{2}} - \frac {{\left (b x^{2} + a\right )}^{\frac {11}{2}} b^{3}}{128 \, a^{4} x^{4}} - \frac {{\left (b x^{2} + a\right )}^{\frac {11}{2}} b^{2}}{160 \, a^{3} x^{6}} - \frac {{\left (b x^{2} + a\right )}^{\frac {11}{2}} b}{80 \, a^{2} x^{8}} - \frac {{\left (b x^{2} + a\right )}^{\frac {11}{2}}}{10 \, a x^{10}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 6.62, size = 106, normalized size = 0.81 \[ \frac {237\,a\,{\left (b\,x^2+a\right )}^{7/2}}{128\,x^{10}}-\frac {193\,{\left (b\,x^2+a\right )}^{9/2}}{256\,x^{10}}-\frac {63\,a^4\,\sqrt {b\,x^2+a}}{256\,x^{10}}+\frac {147\,a^3\,{\left (b\,x^2+a\right )}^{3/2}}{128\,x^{10}}-\frac {21\,a^2\,{\left (b\,x^2+a\right )}^{5/2}}{10\,x^{10}}+\frac {b^5\,\mathrm {atan}\left (\frac {\sqrt {b\,x^2+a}\,1{}\mathrm {i}}{\sqrt {a}}\right )\,63{}\mathrm {i}}{256\,\sqrt {a}} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 9.31, size = 153, normalized size = 1.17 \[ - \frac {a^{4} \sqrt {b} \sqrt {\frac {a}{b x^{2}} + 1}}{10 x^{9}} - \frac {41 a^{3} b^{\frac {3}{2}} \sqrt {\frac {a}{b x^{2}} + 1}}{80 x^{7}} - \frac {171 a^{2} b^{\frac {5}{2}} \sqrt {\frac {a}{b x^{2}} + 1}}{160 x^{5}} - \frac {149 a b^{\frac {7}{2}} \sqrt {\frac {a}{b x^{2}} + 1}}{128 x^{3}} - \frac {193 b^{\frac {9}{2}} \sqrt {\frac {a}{b x^{2}} + 1}}{256 x} - \frac {63 b^{5} \operatorname {asinh}{\left (\frac {\sqrt {a}}{\sqrt {b} x} \right )}}{256 \sqrt {a}} \]
Verification of antiderivative is not currently implemented for this CAS.
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