Optimal. Leaf size=125 \[ \frac{x \left (a-b x^2\right )}{4 a^2 \sqrt{a+b x^2} \sqrt{a^2-b^2 x^4}}+\frac{3 \sqrt{a+b x^2} \sqrt{a-b x^2} \tan ^{-1}\left (\frac{\sqrt{2} \sqrt{b} x}{\sqrt{a-b x^2}}\right )}{4 \sqrt{2} a^2 \sqrt{b} \sqrt{a^2-b^2 x^4}} \]
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Rubi [A] time = 0.0511505, antiderivative size = 125, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 28, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.143, Rules used = {1152, 382, 377, 205} \[ \frac{x \left (a-b x^2\right )}{4 a^2 \sqrt{a+b x^2} \sqrt{a^2-b^2 x^4}}+\frac{3 \sqrt{a+b x^2} \sqrt{a-b x^2} \tan ^{-1}\left (\frac{\sqrt{2} \sqrt{b} x}{\sqrt{a-b x^2}}\right )}{4 \sqrt{2} a^2 \sqrt{b} \sqrt{a^2-b^2 x^4}} \]
Antiderivative was successfully verified.
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Rule 1152
Rule 382
Rule 377
Rule 205
Rubi steps
\begin{align*} \int \frac{1}{\left (a+b x^2\right )^{3/2} \sqrt{a^2-b^2 x^4}} \, dx &=\frac{\left (\sqrt{a-b x^2} \sqrt{a+b x^2}\right ) \int \frac{1}{\sqrt{a-b x^2} \left (a+b x^2\right )^2} \, dx}{\sqrt{a^2-b^2 x^4}}\\ &=\frac{x \left (a-b x^2\right )}{4 a^2 \sqrt{a+b x^2} \sqrt{a^2-b^2 x^4}}+\frac{\left (3 \sqrt{a-b x^2} \sqrt{a+b x^2}\right ) \int \frac{1}{\sqrt{a-b x^2} \left (a+b x^2\right )} \, dx}{4 a \sqrt{a^2-b^2 x^4}}\\ &=\frac{x \left (a-b x^2\right )}{4 a^2 \sqrt{a+b x^2} \sqrt{a^2-b^2 x^4}}+\frac{\left (3 \sqrt{a-b x^2} \sqrt{a+b x^2}\right ) \operatorname{Subst}\left (\int \frac{1}{a+2 a b x^2} \, dx,x,\frac{x}{\sqrt{a-b x^2}}\right )}{4 a \sqrt{a^2-b^2 x^4}}\\ &=\frac{x \left (a-b x^2\right )}{4 a^2 \sqrt{a+b x^2} \sqrt{a^2-b^2 x^4}}+\frac{3 \sqrt{a-b x^2} \sqrt{a+b x^2} \tan ^{-1}\left (\frac{\sqrt{2} \sqrt{b} x}{\sqrt{a-b x^2}}\right )}{4 \sqrt{2} a^2 \sqrt{b} \sqrt{a^2-b^2 x^4}}\\ \end{align*}
Mathematica [A] time = 0.0842704, size = 111, normalized size = 0.89 \[ \frac{\sqrt{a^2-b^2 x^4} \left (2 \sqrt{b} x \sqrt{a-b x^2}+3 \sqrt{2} \left (a+b x^2\right ) \tan ^{-1}\left (\frac{\sqrt{2} \sqrt{b} x}{\sqrt{a-b x^2}}\right )\right )}{8 a^2 \sqrt{b} \sqrt{a-b x^2} \left (a+b x^2\right )^{3/2}} \]
Antiderivative was successfully verified.
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Maple [B] time = 0.063, size = 488, normalized size = 3.9 \begin{align*} -{\frac{1}{4}\sqrt{-{b}^{2}{x}^{4}+{a}^{2}}{b}^{{\frac{5}{2}}} \left ( 3\,\ln \left ( 2\,{\frac{b \left ( \sqrt{2}\sqrt{a}\sqrt{-b{x}^{2}+a}-\sqrt{-ab}x+a \right ) }{bx-\sqrt{-ab}}} \right ) \sqrt{2}{x}^{2}{b}^{3/2}\sqrt{a}-3\,\ln \left ( 2\,{\frac{b \left ( \sqrt{2}\sqrt{a}\sqrt{-b{x}^{2}+a}+\sqrt{-ab}x+a \right ) }{bx+\sqrt{-ab}}} \right ) \sqrt{2}{x}^{2}{b}^{3/2}\sqrt{a}+3\,\ln \left ( 2\,{\frac{b \left ( \sqrt{2}\sqrt{a}\sqrt{-b{x}^{2}+a}-\sqrt{-ab}x+a \right ) }{bx-\sqrt{-ab}}} \right ) \sqrt{2}{a}^{3/2}\sqrt{b}-3\,\ln \left ( 2\,{\frac{b \left ( \sqrt{2}\sqrt{a}\sqrt{-b{x}^{2}+a}+\sqrt{-ab}x+a \right ) }{bx+\sqrt{-ab}}} \right ) \sqrt{2}{a}^{3/2}\sqrt{b}+4\,\arctan \left ({\frac{x\sqrt{b}}{\sqrt{-b{x}^{2}+a}}} \right ){x}^{2}b\sqrt{-ab}-4\,\arctan \left ({x\sqrt{b}{\frac{1}{\sqrt{{\frac{ \left ( -bx+\sqrt{ab} \right ) \left ( bx+\sqrt{ab} \right ) }{b}}}}}} \right ){x}^{2}b\sqrt{-ab}-4\,\sqrt{b}\sqrt{-ab}\sqrt{-b{x}^{2}+a}x+4\,\arctan \left ({\frac{x\sqrt{b}}{\sqrt{-b{x}^{2}+a}}} \right ) a\sqrt{-ab}-4\,\arctan \left ({x\sqrt{b}{\frac{1}{\sqrt{{\frac{ \left ( -bx+\sqrt{ab} \right ) \left ( bx+\sqrt{ab} \right ) }{b}}}}}} \right ) a\sqrt{-ab} \right ){\frac{1}{\sqrt{b{x}^{2}+a}}}{\frac{1}{\sqrt{-b{x}^{2}+a}}} \left ( \sqrt{-ab}+\sqrt{ab} \right ) ^{-2} \left ( -\sqrt{-ab}+\sqrt{ab} \right ) ^{-2}{\frac{1}{\sqrt{-ab}}} \left ( bx+\sqrt{-ab} \right ) ^{-1} \left ( bx-\sqrt{-ab} \right ) ^{-1}} \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{1}{\sqrt{-b^{2} x^{4} + a^{2}}{\left (b x^{2} + a\right )}^{\frac{3}{2}}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 2.23803, size = 655, normalized size = 5.24 \begin{align*} \left [\frac{4 \, \sqrt{-b^{2} x^{4} + a^{2}} \sqrt{b x^{2} + a} b x - 3 \, \sqrt{2}{\left (b^{2} x^{4} + 2 \, a b x^{2} + a^{2}\right )} \sqrt{-b} \log \left (-\frac{3 \, b^{2} x^{4} + 2 \, a b x^{2} - 2 \, \sqrt{2} \sqrt{-b^{2} x^{4} + a^{2}} \sqrt{b x^{2} + a} \sqrt{-b} x - a^{2}}{b^{2} x^{4} + 2 \, a b x^{2} + a^{2}}\right )}{16 \,{\left (a^{2} b^{3} x^{4} + 2 \, a^{3} b^{2} x^{2} + a^{4} b\right )}}, \frac{2 \, \sqrt{-b^{2} x^{4} + a^{2}} \sqrt{b x^{2} + a} b x - 3 \, \sqrt{2}{\left (b^{2} x^{4} + 2 \, a b x^{2} + a^{2}\right )} \sqrt{b} \arctan \left (\frac{\sqrt{2} \sqrt{-b^{2} x^{4} + a^{2}} \sqrt{b x^{2} + a} \sqrt{b}}{2 \,{\left (b^{2} x^{3} + a b x\right )}}\right )}{8 \,{\left (a^{2} b^{3} x^{4} + 2 \, a^{3} b^{2} x^{2} + a^{4} 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{1}{\sqrt{- \left (- a + b x^{2}\right ) \left (a + b x^{2}\right )} \left (a + b x^{2}\right )^{\frac{3}{2}}}\, 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{1}{\sqrt{-b^{2} x^{4} + a^{2}}{\left (b x^{2} + a\right )}^{\frac{3}{2}}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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