Optimal. Leaf size=122 \[ \frac {63 a^5 \tanh ^{-1}\left (\frac {\sqrt {b} x}{\sqrt {a+b x^2}}\right )}{256 \sqrt {b}}+\frac {63}{256} a^4 x \sqrt {a+b x^2}+\frac {21}{128} a^3 x \left (a+b x^2\right )^{3/2}+\frac {21}{160} a^2 x \left (a+b x^2\right )^{5/2}+\frac {9}{80} a x \left (a+b x^2\right )^{7/2}+\frac {1}{10} x \left (a+b x^2\right )^{9/2} \]
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Rubi [A] time = 0.04, antiderivative size = 122, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 3, integrand size = 11, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.273, Rules used = {195, 217, 206} \[ \frac {63}{256} a^4 x \sqrt {a+b x^2}+\frac {21}{128} a^3 x \left (a+b x^2\right )^{3/2}+\frac {21}{160} a^2 x \left (a+b x^2\right )^{5/2}+\frac {63 a^5 \tanh ^{-1}\left (\frac {\sqrt {b} x}{\sqrt {a+b x^2}}\right )}{256 \sqrt {b}}+\frac {9}{80} a x \left (a+b x^2\right )^{7/2}+\frac {1}{10} x \left (a+b x^2\right )^{9/2} \]
Antiderivative was successfully verified.
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Rule 195
Rule 206
Rule 217
Rubi steps
\begin {align*} \int \left (a+b x^2\right )^{9/2} \, dx &=\frac {1}{10} x \left (a+b x^2\right )^{9/2}+\frac {1}{10} (9 a) \int \left (a+b x^2\right )^{7/2} \, dx\\ &=\frac {9}{80} a x \left (a+b x^2\right )^{7/2}+\frac {1}{10} x \left (a+b x^2\right )^{9/2}+\frac {1}{80} \left (63 a^2\right ) \int \left (a+b x^2\right )^{5/2} \, dx\\ &=\frac {21}{160} a^2 x \left (a+b x^2\right )^{5/2}+\frac {9}{80} a x \left (a+b x^2\right )^{7/2}+\frac {1}{10} x \left (a+b x^2\right )^{9/2}+\frac {1}{32} \left (21 a^3\right ) \int \left (a+b x^2\right )^{3/2} \, dx\\ &=\frac {21}{128} a^3 x \left (a+b x^2\right )^{3/2}+\frac {21}{160} a^2 x \left (a+b x^2\right )^{5/2}+\frac {9}{80} a x \left (a+b x^2\right )^{7/2}+\frac {1}{10} x \left (a+b x^2\right )^{9/2}+\frac {1}{128} \left (63 a^4\right ) \int \sqrt {a+b x^2} \, dx\\ &=\frac {63}{256} a^4 x \sqrt {a+b x^2}+\frac {21}{128} a^3 x \left (a+b x^2\right )^{3/2}+\frac {21}{160} a^2 x \left (a+b x^2\right )^{5/2}+\frac {9}{80} a x \left (a+b x^2\right )^{7/2}+\frac {1}{10} x \left (a+b x^2\right )^{9/2}+\frac {1}{256} \left (63 a^5\right ) \int \frac {1}{\sqrt {a+b x^2}} \, dx\\ &=\frac {63}{256} a^4 x \sqrt {a+b x^2}+\frac {21}{128} a^3 x \left (a+b x^2\right )^{3/2}+\frac {21}{160} a^2 x \left (a+b x^2\right )^{5/2}+\frac {9}{80} a x \left (a+b x^2\right )^{7/2}+\frac {1}{10} x \left (a+b x^2\right )^{9/2}+\frac {1}{256} \left (63 a^5\right ) \operatorname {Subst}\left (\int \frac {1}{1-b x^2} \, dx,x,\frac {x}{\sqrt {a+b x^2}}\right )\\ &=\frac {63}{256} a^4 x \sqrt {a+b x^2}+\frac {21}{128} a^3 x \left (a+b x^2\right )^{3/2}+\frac {21}{160} a^2 x \left (a+b x^2\right )^{5/2}+\frac {9}{80} a x \left (a+b x^2\right )^{7/2}+\frac {1}{10} x \left (a+b x^2\right )^{9/2}+\frac {63 a^5 \tanh ^{-1}\left (\frac {\sqrt {b} x}{\sqrt {a+b x^2}}\right )}{256 \sqrt {b}}\\ \end {align*}
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Mathematica [A] time = 0.14, size = 98, normalized size = 0.80 \[ \frac {\sqrt {a+b x^2} \left (\frac {315 a^{9/2} \sinh ^{-1}\left (\frac {\sqrt {b} x}{\sqrt {a}}\right )}{\sqrt {b} \sqrt {\frac {b x^2}{a}+1}}+965 a^4 x+1490 a^3 b x^3+1368 a^2 b^2 x^5+656 a b^3 x^7+128 b^4 x^9\right )}{1280} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.84, size = 190, normalized size = 1.56 \[ \left [\frac {315 \, a^{5} \sqrt {b} \log \left (-2 \, b x^{2} - 2 \, \sqrt {b x^{2} + a} \sqrt {b} x - a\right ) + 2 \, {\left (128 \, b^{5} x^{9} + 656 \, a b^{4} x^{7} + 1368 \, a^{2} b^{3} x^{5} + 1490 \, a^{3} b^{2} x^{3} + 965 \, a^{4} b x\right )} \sqrt {b x^{2} + a}}{2560 \, b}, -\frac {315 \, a^{5} \sqrt {-b} \arctan \left (\frac {\sqrt {-b} x}{\sqrt {b x^{2} + a}}\right ) - {\left (128 \, b^{5} x^{9} + 656 \, a b^{4} x^{7} + 1368 \, a^{2} b^{3} x^{5} + 1490 \, a^{3} b^{2} x^{3} + 965 \, a^{4} b x\right )} \sqrt {b x^{2} + a}}{1280 \, b}\right ] \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 1.12, size = 91, normalized size = 0.75 \[ -\frac {63 \, a^{5} \log \left ({\left | -\sqrt {b} x + \sqrt {b x^{2} + a} \right |}\right )}{256 \, \sqrt {b}} + \frac {1}{1280} \, {\left (965 \, a^{4} + 2 \, {\left (745 \, a^{3} b + 4 \, {\left (171 \, a^{2} b^{2} + 2 \, {\left (8 \, b^{4} x^{2} + 41 \, a b^{3}\right )} x^{2}\right )} x^{2}\right )} x^{2}\right )} \sqrt {b x^{2} + a} x \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.00, size = 96, normalized size = 0.79 \[ \frac {63 a^{5} \ln \left (\sqrt {b}\, x +\sqrt {b \,x^{2}+a}\right )}{256 \sqrt {b}}+\frac {63 \sqrt {b \,x^{2}+a}\, a^{4} x}{256}+\frac {21 \left (b \,x^{2}+a \right )^{\frac {3}{2}} a^{3} x}{128}+\frac {21 \left (b \,x^{2}+a \right )^{\frac {5}{2}} a^{2} x}{160}+\frac {9 \left (b \,x^{2}+a \right )^{\frac {7}{2}} a x}{80}+\frac {\left (b \,x^{2}+a \right )^{\frac {9}{2}} x}{10} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 1.32, size = 88, normalized size = 0.72 \[ \frac {1}{10} \, {\left (b x^{2} + a\right )}^{\frac {9}{2}} x + \frac {9}{80} \, {\left (b x^{2} + a\right )}^{\frac {7}{2}} a x + \frac {21}{160} \, {\left (b x^{2} + a\right )}^{\frac {5}{2}} a^{2} x + \frac {21}{128} \, {\left (b x^{2} + a\right )}^{\frac {3}{2}} a^{3} x + \frac {63}{256} \, \sqrt {b x^{2} + a} a^{4} x + \frac {63 \, a^{5} \operatorname {arsinh}\left (\frac {b x}{\sqrt {a b}}\right )}{256 \, \sqrt {b}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 4.62, size = 37, normalized size = 0.30 \[ \frac {x\,{\left (b\,x^2+a\right )}^{9/2}\,{{}}_2{\mathrm {F}}_1\left (-\frac {9}{2},\frac {1}{2};\ \frac {3}{2};\ -\frac {b\,x^2}{a}\right )}{{\left (\frac {b\,x^2}{a}+1\right )}^{9/2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 8.35, size = 151, normalized size = 1.24 \[ \frac {193 a^{\frac {9}{2}} x \sqrt {1 + \frac {b x^{2}}{a}}}{256} + \frac {149 a^{\frac {7}{2}} b x^{3} \sqrt {1 + \frac {b x^{2}}{a}}}{128} + \frac {171 a^{\frac {5}{2}} b^{2} x^{5} \sqrt {1 + \frac {b x^{2}}{a}}}{160} + \frac {41 a^{\frac {3}{2}} b^{3} x^{7} \sqrt {1 + \frac {b x^{2}}{a}}}{80} + \frac {\sqrt {a} b^{4} x^{9} \sqrt {1 + \frac {b x^{2}}{a}}}{10} + \frac {63 a^{5} \operatorname {asinh}{\left (\frac {\sqrt {b} x}{\sqrt {a}} \right )}}{256 \sqrt {b}} \]
Verification of antiderivative is not currently implemented for this CAS.
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