Optimal. Leaf size=152 \[ -\frac{9 a x \sqrt{a-b x^2} \left (a+b x^2\right )}{8 \sqrt{a^2-b^2 x^4}}-\frac{x \left (a-b x^2\right )^{3/2} \left (a+b x^2\right )}{4 \sqrt{a^2-b^2 x^4}}+\frac{19 a^2 \sqrt{a-b x^2} \sqrt{a+b x^2} \tanh ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a+b x^2}}\right )}{8 \sqrt{b} \sqrt{a^2-b^2 x^4}} \]
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Rubi [A] time = 0.0545413, antiderivative size = 152, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 5, integrand size = 29, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.172, Rules used = {1152, 416, 388, 217, 206} \[ -\frac{9 a x \sqrt{a-b x^2} \left (a+b x^2\right )}{8 \sqrt{a^2-b^2 x^4}}-\frac{x \left (a-b x^2\right )^{3/2} \left (a+b x^2\right )}{4 \sqrt{a^2-b^2 x^4}}+\frac{19 a^2 \sqrt{a-b x^2} \sqrt{a+b x^2} \tanh ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a+b x^2}}\right )}{8 \sqrt{b} \sqrt{a^2-b^2 x^4}} \]
Antiderivative was successfully verified.
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Rule 1152
Rule 416
Rule 388
Rule 217
Rule 206
Rubi steps
\begin{align*} \int \frac{\left (a-b x^2\right )^{5/2}}{\sqrt{a^2-b^2 x^4}} \, dx &=\frac{\left (\sqrt{a-b x^2} \sqrt{a+b x^2}\right ) \int \frac{\left (a-b x^2\right )^2}{\sqrt{a+b x^2}} \, dx}{\sqrt{a^2-b^2 x^4}}\\ &=-\frac{x \left (a-b x^2\right )^{3/2} \left (a+b x^2\right )}{4 \sqrt{a^2-b^2 x^4}}+\frac{\left (\sqrt{a-b x^2} \sqrt{a+b x^2}\right ) \int \frac{5 a^2 b-9 a b^2 x^2}{\sqrt{a+b x^2}} \, dx}{4 b \sqrt{a^2-b^2 x^4}}\\ &=-\frac{9 a x \sqrt{a-b x^2} \left (a+b x^2\right )}{8 \sqrt{a^2-b^2 x^4}}-\frac{x \left (a-b x^2\right )^{3/2} \left (a+b x^2\right )}{4 \sqrt{a^2-b^2 x^4}}+\frac{\left (19 a^2 \sqrt{a-b x^2} \sqrt{a+b x^2}\right ) \int \frac{1}{\sqrt{a+b x^2}} \, dx}{8 \sqrt{a^2-b^2 x^4}}\\ &=-\frac{9 a x \sqrt{a-b x^2} \left (a+b x^2\right )}{8 \sqrt{a^2-b^2 x^4}}-\frac{x \left (a-b x^2\right )^{3/2} \left (a+b x^2\right )}{4 \sqrt{a^2-b^2 x^4}}+\frac{\left (19 a^2 \sqrt{a-b x^2} \sqrt{a+b x^2}\right ) \operatorname{Subst}\left (\int \frac{1}{1-b x^2} \, dx,x,\frac{x}{\sqrt{a+b x^2}}\right )}{8 \sqrt{a^2-b^2 x^4}}\\ &=-\frac{9 a x \sqrt{a-b x^2} \left (a+b x^2\right )}{8 \sqrt{a^2-b^2 x^4}}-\frac{x \left (a-b x^2\right )^{3/2} \left (a+b x^2\right )}{4 \sqrt{a^2-b^2 x^4}}+\frac{19 a^2 \sqrt{a-b x^2} \sqrt{a+b x^2} \tanh ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a+b x^2}}\right )}{8 \sqrt{b} \sqrt{a^2-b^2 x^4}}\\ \end{align*}
Mathematica [A] time = 0.200751, size = 123, normalized size = 0.81 \[ \frac{1}{8} \left (\frac{x \left (2 b x^2-11 a\right ) \sqrt{a^2-b^2 x^4}}{\sqrt{a-b x^2}}+\frac{19 a^2 \log \left (\sqrt{b} \sqrt{a-b x^2} \sqrt{a^2-b^2 x^4}+a b x-b^2 x^3\right )}{\sqrt{b}}-\frac{19 a^2 \log \left (b x^2-a\right )}{\sqrt{b}}\right ) \]
Antiderivative was successfully verified.
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Maple [A] time = 0.017, size = 105, normalized size = 0.7 \begin{align*} -{\frac{1}{8\,b{x}^{2}-8\,a}\sqrt{-b{x}^{2}+a}\sqrt{-{b}^{2}{x}^{4}+{a}^{2}} \left ( 2\,{x}^{3}{b}^{3/2}\sqrt{b{x}^{2}+a}-11\,xa\sqrt{b{x}^{2}+a}\sqrt{b}+19\,\ln \left ( x\sqrt{b}+\sqrt{b{x}^{2}+a} \right ){a}^{2} \right ){\frac{1}{\sqrt{b{x}^{2}+a}}}{\frac{1}{\sqrt{b}}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{{\left (-b x^{2} + a\right )}^{\frac{5}{2}}}{\sqrt{-b^{2} x^{4} + a^{2}}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.99549, size = 551, normalized size = 3.62 \begin{align*} \left [\frac{19 \,{\left (a^{2} b x^{2} - a^{3}\right )} \sqrt{b} \log \left (\frac{2 \, b^{2} x^{4} - a b x^{2} - 2 \, \sqrt{-b^{2} x^{4} + a^{2}} \sqrt{-b x^{2} + a} \sqrt{b} x - a^{2}}{b x^{2} - a}\right ) - 2 \, \sqrt{-b^{2} x^{4} + a^{2}}{\left (2 \, b^{2} x^{3} - 11 \, a b x\right )} \sqrt{-b x^{2} + a}}{16 \,{\left (b^{2} x^{2} - a b\right )}}, \frac{19 \,{\left (a^{2} b x^{2} - a^{3}\right )} \sqrt{-b} \arctan \left (\frac{\sqrt{-b^{2} x^{4} + a^{2}} \sqrt{-b x^{2} + a} \sqrt{-b}}{b^{2} x^{3} - a b x}\right ) - \sqrt{-b^{2} x^{4} + a^{2}}{\left (2 \, b^{2} x^{3} - 11 \, a b x\right )} \sqrt{-b x^{2} + a}}{8 \,{\left (b^{2} x^{2} - a b\right )}}\right ] \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\left (a - b x^{2}\right )^{\frac{5}{2}}}{\sqrt{- \left (- a + b x^{2}\right ) \left (a + b x^{2}\right )}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{{\left (-b x^{2} + a\right )}^{\frac{5}{2}}}{\sqrt{-b^{2} x^{4} + a^{2}}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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