Optimal. Leaf size=153 \[ -\frac{9 a x \left (a-b x^2\right ) \sqrt{a+b x^2}}{8 \sqrt{a^2-b^2 x^4}}-\frac{x \left (a-b x^2\right ) \left (a+b x^2\right )^{3/2}}{4 \sqrt{a^2-b^2 x^4}}+\frac{19 a^2 \sqrt{a-b x^2} \sqrt{a+b x^2} \tan ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a-b x^2}}\right )}{8 \sqrt{b} \sqrt{a^2-b^2 x^4}} \]
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Rubi [A] time = 0.0547431, antiderivative size = 153, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 5, integrand size = 28, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.179, Rules used = {1152, 416, 388, 217, 203} \[ -\frac{9 a x \left (a-b x^2\right ) \sqrt{a+b x^2}}{8 \sqrt{a^2-b^2 x^4}}-\frac{x \left (a-b x^2\right ) \left (a+b x^2\right )^{3/2}}{4 \sqrt{a^2-b^2 x^4}}+\frac{19 a^2 \sqrt{a-b x^2} \sqrt{a+b x^2} \tan ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a-b x^2}}\right )}{8 \sqrt{b} \sqrt{a^2-b^2 x^4}} \]
Antiderivative was successfully verified.
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Rule 1152
Rule 416
Rule 388
Rule 217
Rule 203
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 ) \left (a+b x^2\right )^{3/2}}{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 \left (a-b x^2\right ) \sqrt{a+b x^2}}{8 \sqrt{a^2-b^2 x^4}}-\frac{x \left (a-b x^2\right ) \left (a+b x^2\right )^{3/2}}{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 \left (a-b x^2\right ) \sqrt{a+b x^2}}{8 \sqrt{a^2-b^2 x^4}}-\frac{x \left (a-b x^2\right ) \left (a+b x^2\right )^{3/2}}{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 \left (a-b x^2\right ) \sqrt{a+b x^2}}{8 \sqrt{a^2-b^2 x^4}}-\frac{x \left (a-b x^2\right ) \left (a+b x^2\right )^{3/2}}{4 \sqrt{a^2-b^2 x^4}}+\frac{19 a^2 \sqrt{a-b x^2} \sqrt{a+b x^2} \tan ^{-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 [C] time = 0.164907, size = 98, normalized size = 0.64 \[ -\frac{\left (11 a x+2 b x^3\right ) \sqrt{a^2-b^2 x^4}}{8 \sqrt{a+b x^2}}+\frac{19 i a^2 \log \left (\frac{2 \sqrt{a^2-b^2 x^4}}{\sqrt{a+b x^2}}-2 i \sqrt{b} x\right )}{8 \sqrt{b}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.089, size = 132, normalized size = 0.9 \begin{align*} -{\frac{1}{8}\sqrt{-{b}^{2}{x}^{4}+{a}^{2}} \left ( 2\,{x}^{3}{b}^{3/2}\sqrt{-b{x}^{2}+a}+11\,\sqrt{b}\sqrt{-b{x}^{2}+a}xa+13\,\arctan \left ({\frac{x\sqrt{b}}{\sqrt{-b{x}^{2}+a}}} \right ){a}^{2}-32\,\arctan \left ({x\sqrt{b}{\frac{1}{\sqrt{{\frac{ \left ( -bx+\sqrt{ab} \right ) \left ( bx+\sqrt{ab} \right ) }{b}}}}}} \right ){a}^{2} \right ){\frac{1}{\sqrt{b{x}^{2}+a}}}{\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 = 2.08634, size = 549, normalized size = 3.59 \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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