Optimal. Leaf size=108 \[ a^4 \sqrt{a+b x^2}+\frac{1}{3} a^3 \left (a+b x^2\right )^{3/2}+\frac{1}{5} a^2 \left (a+b x^2\right )^{5/2}+a^{9/2} \left (-\tanh ^{-1}\left (\frac{\sqrt{a+b x^2}}{\sqrt{a}}\right )\right )+\frac{1}{7} a \left (a+b x^2\right )^{7/2}+\frac{1}{9} \left (a+b x^2\right )^{9/2} \]
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Rubi [A] time = 0.0702031, antiderivative size = 108, normalized size of antiderivative = 1., 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, 50, 63, 208} \[ a^4 \sqrt{a+b x^2}+\frac{1}{3} a^3 \left (a+b x^2\right )^{3/2}+\frac{1}{5} a^2 \left (a+b x^2\right )^{5/2}+a^{9/2} \left (-\tanh ^{-1}\left (\frac{\sqrt{a+b x^2}}{\sqrt{a}}\right )\right )+\frac{1}{7} a \left (a+b x^2\right )^{7/2}+\frac{1}{9} \left (a+b x^2\right )^{9/2} \]
Antiderivative was successfully verified.
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Rule 266
Rule 50
Rule 63
Rule 208
Rubi steps
\begin{align*} \int \frac{\left (a+b x^2\right )^{9/2}}{x} \, dx &=\frac{1}{2} \operatorname{Subst}\left (\int \frac{(a+b x)^{9/2}}{x} \, dx,x,x^2\right )\\ &=\frac{1}{9} \left (a+b x^2\right )^{9/2}+\frac{1}{2} a \operatorname{Subst}\left (\int \frac{(a+b x)^{7/2}}{x} \, dx,x,x^2\right )\\ &=\frac{1}{7} a \left (a+b x^2\right )^{7/2}+\frac{1}{9} \left (a+b x^2\right )^{9/2}+\frac{1}{2} a^2 \operatorname{Subst}\left (\int \frac{(a+b x)^{5/2}}{x} \, dx,x,x^2\right )\\ &=\frac{1}{5} a^2 \left (a+b x^2\right )^{5/2}+\frac{1}{7} a \left (a+b x^2\right )^{7/2}+\frac{1}{9} \left (a+b x^2\right )^{9/2}+\frac{1}{2} a^3 \operatorname{Subst}\left (\int \frac{(a+b x)^{3/2}}{x} \, dx,x,x^2\right )\\ &=\frac{1}{3} a^3 \left (a+b x^2\right )^{3/2}+\frac{1}{5} a^2 \left (a+b x^2\right )^{5/2}+\frac{1}{7} a \left (a+b x^2\right )^{7/2}+\frac{1}{9} \left (a+b x^2\right )^{9/2}+\frac{1}{2} a^4 \operatorname{Subst}\left (\int \frac{\sqrt{a+b x}}{x} \, dx,x,x^2\right )\\ &=a^4 \sqrt{a+b x^2}+\frac{1}{3} a^3 \left (a+b x^2\right )^{3/2}+\frac{1}{5} a^2 \left (a+b x^2\right )^{5/2}+\frac{1}{7} a \left (a+b x^2\right )^{7/2}+\frac{1}{9} \left (a+b x^2\right )^{9/2}+\frac{1}{2} a^5 \operatorname{Subst}\left (\int \frac{1}{x \sqrt{a+b x}} \, dx,x,x^2\right )\\ &=a^4 \sqrt{a+b x^2}+\frac{1}{3} a^3 \left (a+b x^2\right )^{3/2}+\frac{1}{5} a^2 \left (a+b x^2\right )^{5/2}+\frac{1}{7} a \left (a+b x^2\right )^{7/2}+\frac{1}{9} \left (a+b x^2\right )^{9/2}+\frac{a^5 \operatorname{Subst}\left (\int \frac{1}{-\frac{a}{b}+\frac{x^2}{b}} \, dx,x,\sqrt{a+b x^2}\right )}{b}\\ &=a^4 \sqrt{a+b x^2}+\frac{1}{3} a^3 \left (a+b x^2\right )^{3/2}+\frac{1}{5} a^2 \left (a+b x^2\right )^{5/2}+\frac{1}{7} a \left (a+b x^2\right )^{7/2}+\frac{1}{9} \left (a+b x^2\right )^{9/2}-a^{9/2} \tanh ^{-1}\left (\frac{\sqrt{a+b x^2}}{\sqrt{a}}\right )\\ \end{align*}
Mathematica [A] time = 0.0393282, size = 84, normalized size = 0.78 \[ \frac{1}{315} \sqrt{a+b x^2} \left (408 a^2 b^2 x^4+506 a^3 b x^2+563 a^4+185 a b^3 x^6+35 b^4 x^8\right )-a^{9/2} \tanh ^{-1}\left (\frac{\sqrt{a+b x^2}}{\sqrt{a}}\right ) \]
Antiderivative was successfully verified.
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Maple [A] time = 0.004, size = 94, normalized size = 0.9 \begin{align*}{\frac{1}{9} \left ( b{x}^{2}+a \right ) ^{{\frac{9}{2}}}}+{\frac{a}{7} \left ( b{x}^{2}+a \right ) ^{{\frac{7}{2}}}}+{\frac{{a}^{2}}{5} \left ( b{x}^{2}+a \right ) ^{{\frac{5}{2}}}}+{\frac{{a}^{3}}{3} \left ( b{x}^{2}+a \right ) ^{{\frac{3}{2}}}}-{a}^{{\frac{9}{2}}}\ln \left ({\frac{1}{x} \left ( 2\,a+2\,\sqrt{a}\sqrt{b{x}^{2}+a} \right ) } \right ) +{a}^{4}\sqrt{b{x}^{2}+a} \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 = 1.69137, size = 419, normalized size = 3.88 \begin{align*} \left [\frac{1}{2} \, a^{\frac{9}{2}} \log \left (-\frac{b x^{2} - 2 \, \sqrt{b x^{2} + a} \sqrt{a} + 2 \, a}{x^{2}}\right ) + \frac{1}{315} \,{\left (35 \, b^{4} x^{8} + 185 \, a b^{3} x^{6} + 408 \, a^{2} b^{2} x^{4} + 506 \, a^{3} b x^{2} + 563 \, a^{4}\right )} \sqrt{b x^{2} + a}, \sqrt{-a} a^{4} \arctan \left (\frac{\sqrt{-a}}{\sqrt{b x^{2} + a}}\right ) + \frac{1}{315} \,{\left (35 \, b^{4} x^{8} + 185 \, a b^{3} x^{6} + 408 \, a^{2} b^{2} x^{4} + 506 \, a^{3} b x^{2} + 563 \, a^{4}\right )} \sqrt{b x^{2} + a}\right ] \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 9.37214, size = 160, normalized size = 1.48 \begin{align*} \frac{563 a^{\frac{9}{2}} \sqrt{1 + \frac{b x^{2}}{a}}}{315} + \frac{a^{\frac{9}{2}} \log{\left (\frac{b x^{2}}{a} \right )}}{2} - a^{\frac{9}{2}} \log{\left (\sqrt{1 + \frac{b x^{2}}{a}} + 1 \right )} + \frac{506 a^{\frac{7}{2}} b x^{2} \sqrt{1 + \frac{b x^{2}}{a}}}{315} + \frac{136 a^{\frac{5}{2}} b^{2} x^{4} \sqrt{1 + \frac{b x^{2}}{a}}}{105} + \frac{37 a^{\frac{3}{2}} b^{3} x^{6} \sqrt{1 + \frac{b x^{2}}{a}}}{63} + \frac{\sqrt{a} b^{4} x^{8} \sqrt{1 + \frac{b x^{2}}{a}}}{9} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.58725, size = 122, normalized size = 1.13 \begin{align*} \frac{a^{5} \arctan \left (\frac{\sqrt{b x^{2} + a}}{\sqrt{-a}}\right )}{\sqrt{-a}} + \frac{1}{9} \,{\left (b x^{2} + a\right )}^{\frac{9}{2}} + \frac{1}{7} \,{\left (b x^{2} + a\right )}^{\frac{7}{2}} a + \frac{1}{5} \,{\left (b x^{2} + a\right )}^{\frac{5}{2}} a^{2} + \frac{1}{3} \,{\left (b x^{2} + a\right )}^{\frac{3}{2}} a^{3} + \sqrt{b x^{2} + a} a^{4} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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