Optimal. Leaf size=168 \[ \frac{9 x \left (a-b x^2\right )}{32 a^3 \sqrt{a+b x^2} \sqrt{a^2-b^2 x^4}}+\frac{x \left (a-b x^2\right )}{8 a^2 \left (a+b x^2\right )^{3/2} \sqrt{a^2-b^2 x^4}}+\frac{19 \sqrt{a+b x^2} \sqrt{a-b x^2} \tan ^{-1}\left (\frac{\sqrt{2} \sqrt{b} x}{\sqrt{a-b x^2}}\right )}{32 \sqrt{2} a^3 \sqrt{b} \sqrt{a^2-b^2 x^4}} \]
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Rubi [A] time = 0.0862913, antiderivative size = 168, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 6, integrand size = 28, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.214, Rules used = {1152, 414, 527, 12, 377, 205} \[ \frac{9 x \left (a-b x^2\right )}{32 a^3 \sqrt{a+b x^2} \sqrt{a^2-b^2 x^4}}+\frac{x \left (a-b x^2\right )}{8 a^2 \left (a+b x^2\right )^{3/2} \sqrt{a^2-b^2 x^4}}+\frac{19 \sqrt{a+b x^2} \sqrt{a-b x^2} \tan ^{-1}\left (\frac{\sqrt{2} \sqrt{b} x}{\sqrt{a-b x^2}}\right )}{32 \sqrt{2} a^3 \sqrt{b} \sqrt{a^2-b^2 x^4}} \]
Antiderivative was successfully verified.
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Rule 1152
Rule 414
Rule 527
Rule 12
Rule 377
Rule 205
Rubi steps
\begin{align*} \int \frac{1}{\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{1}{\sqrt{a-b x^2} \left (a+b x^2\right )^3} \, dx}{\sqrt{a^2-b^2 x^4}}\\ &=\frac{x \left (a-b x^2\right )}{8 a^2 \left (a+b x^2\right )^{3/2} \sqrt{a^2-b^2 x^4}}-\frac{\left (\sqrt{a-b x^2} \sqrt{a+b x^2}\right ) \int \frac{-7 a b+2 b^2 x^2}{\sqrt{a-b x^2} \left (a+b x^2\right )^2} \, dx}{8 a^2 b \sqrt{a^2-b^2 x^4}}\\ &=\frac{x \left (a-b x^2\right )}{8 a^2 \left (a+b x^2\right )^{3/2} \sqrt{a^2-b^2 x^4}}+\frac{9 x \left (a-b x^2\right )}{32 a^3 \sqrt{a+b x^2} \sqrt{a^2-b^2 x^4}}+\frac{\left (\sqrt{a-b x^2} \sqrt{a+b x^2}\right ) \int \frac{19 a^2 b^2}{\sqrt{a-b x^2} \left (a+b x^2\right )} \, dx}{32 a^4 b^2 \sqrt{a^2-b^2 x^4}}\\ &=\frac{x \left (a-b x^2\right )}{8 a^2 \left (a+b x^2\right )^{3/2} \sqrt{a^2-b^2 x^4}}+\frac{9 x \left (a-b x^2\right )}{32 a^3 \sqrt{a+b x^2} \sqrt{a^2-b^2 x^4}}+\frac{\left (19 \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}{32 a^2 \sqrt{a^2-b^2 x^4}}\\ &=\frac{x \left (a-b x^2\right )}{8 a^2 \left (a+b x^2\right )^{3/2} \sqrt{a^2-b^2 x^4}}+\frac{9 x \left (a-b x^2\right )}{32 a^3 \sqrt{a+b x^2} \sqrt{a^2-b^2 x^4}}+\frac{\left (19 \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 )}{32 a^2 \sqrt{a^2-b^2 x^4}}\\ &=\frac{x \left (a-b x^2\right )}{8 a^2 \left (a+b x^2\right )^{3/2} \sqrt{a^2-b^2 x^4}}+\frac{9 x \left (a-b x^2\right )}{32 a^3 \sqrt{a+b x^2} \sqrt{a^2-b^2 x^4}}+\frac{19 \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 )}{32 \sqrt{2} a^3 \sqrt{b} \sqrt{a^2-b^2 x^4}}\\ \end{align*}
Mathematica [A] time = 0.108498, size = 123, normalized size = 0.73 \[ \frac{\sqrt{a^2-b^2 x^4} \left (2 \sqrt{b} x \sqrt{a-b x^2} \left (13 a+9 b x^2\right )+19 \sqrt{2} \left (a+b x^2\right )^2 \tan ^{-1}\left (\frac{\sqrt{2} \sqrt{b} x}{\sqrt{a-b x^2}}\right )\right )}{64 a^3 \sqrt{b} \sqrt{a-b x^2} \left (a+b x^2\right )^{5/2}} \]
Antiderivative was successfully verified.
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Maple [B] time = 0.054, size = 711, normalized size = 4.2 \begin{align*} \text{result too large to display} \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{5}{2}}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.87349, size = 801, normalized size = 4.77 \begin{align*} \left [-\frac{19 \, \sqrt{2}{\left (b^{3} x^{6} + 3 \, a b^{2} x^{4} + 3 \, a^{2} b x^{2} + a^{3}\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 ) - 4 \, \sqrt{-b^{2} x^{4} + a^{2}}{\left (9 \, b^{2} x^{3} + 13 \, a b x\right )} \sqrt{b x^{2} + a}}{128 \,{\left (a^{3} b^{4} x^{6} + 3 \, a^{4} b^{3} x^{4} + 3 \, a^{5} b^{2} x^{2} + a^{6} b\right )}}, -\frac{19 \, \sqrt{2}{\left (b^{3} x^{6} + 3 \, a b^{2} x^{4} + 3 \, a^{2} b x^{2} + a^{3}\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 ) - 2 \, \sqrt{-b^{2} x^{4} + a^{2}}{\left (9 \, b^{2} x^{3} + 13 \, a b x\right )} \sqrt{b x^{2} + a}}{64 \,{\left (a^{3} b^{4} x^{6} + 3 \, a^{4} b^{3} x^{4} + 3 \, a^{5} b^{2} x^{2} + a^{6} 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{5}{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{5}{2}}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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