Optimal. Leaf size=71 \[ \frac{b^2 \tanh ^{-1}\left (\frac{\sqrt{a+b x^2}}{\sqrt{a}}\right )}{8 a^{3/2}}-\frac{b \sqrt{a+b x^2}}{8 a x^2}-\frac{\sqrt{a+b x^2}}{4 x^4} \]
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Rubi [A] time = 0.0390252, antiderivative size = 71, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 5, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.333, Rules used = {266, 47, 51, 63, 208} \[ \frac{b^2 \tanh ^{-1}\left (\frac{\sqrt{a+b x^2}}{\sqrt{a}}\right )}{8 a^{3/2}}-\frac{b \sqrt{a+b x^2}}{8 a x^2}-\frac{\sqrt{a+b x^2}}{4 x^4} \]
Antiderivative was successfully verified.
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Rule 266
Rule 47
Rule 51
Rule 63
Rule 208
Rubi steps
\begin{align*} \int \frac{\sqrt{a+b x^2}}{x^5} \, dx &=\frac{1}{2} \operatorname{Subst}\left (\int \frac{\sqrt{a+b x}}{x^3} \, dx,x,x^2\right )\\ &=-\frac{\sqrt{a+b x^2}}{4 x^4}+\frac{1}{8} b \operatorname{Subst}\left (\int \frac{1}{x^2 \sqrt{a+b x}} \, dx,x,x^2\right )\\ &=-\frac{\sqrt{a+b x^2}}{4 x^4}-\frac{b \sqrt{a+b x^2}}{8 a x^2}-\frac{b^2 \operatorname{Subst}\left (\int \frac{1}{x \sqrt{a+b x}} \, dx,x,x^2\right )}{16 a}\\ &=-\frac{\sqrt{a+b x^2}}{4 x^4}-\frac{b \sqrt{a+b x^2}}{8 a x^2}-\frac{b \operatorname{Subst}\left (\int \frac{1}{-\frac{a}{b}+\frac{x^2}{b}} \, dx,x,\sqrt{a+b x^2}\right )}{8 a}\\ &=-\frac{\sqrt{a+b x^2}}{4 x^4}-\frac{b \sqrt{a+b x^2}}{8 a x^2}+\frac{b^2 \tanh ^{-1}\left (\frac{\sqrt{a+b x^2}}{\sqrt{a}}\right )}{8 a^{3/2}}\\ \end{align*}
Mathematica [C] time = 0.008291, size = 39, normalized size = 0.55 \[ -\frac{b^2 \left (a+b x^2\right )^{3/2} \, _2F_1\left (\frac{3}{2},3;\frac{5}{2};\frac{b x^2}{a}+1\right )}{3 a^3} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.005, size = 85, normalized size = 1.2 \begin{align*} -{\frac{1}{4\,a{x}^{4}} \left ( b{x}^{2}+a \right ) ^{{\frac{3}{2}}}}+{\frac{b}{8\,{a}^{2}{x}^{2}} \left ( b{x}^{2}+a \right ) ^{{\frac{3}{2}}}}+{\frac{{b}^{2}}{8}\ln \left ({\frac{1}{x} \left ( 2\,a+2\,\sqrt{a}\sqrt{b{x}^{2}+a} \right ) } \right ){a}^{-{\frac{3}{2}}}}-{\frac{{b}^{2}}{8\,{a}^{2}}\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 = 1.6081, size = 313, normalized size = 4.41 \begin{align*} \left [\frac{\sqrt{a} b^{2} x^{4} \log \left (-\frac{b x^{2} + 2 \, \sqrt{b x^{2} + a} \sqrt{a} + 2 \, a}{x^{2}}\right ) - 2 \,{\left (a b x^{2} + 2 \, a^{2}\right )} \sqrt{b x^{2} + a}}{16 \, a^{2} x^{4}}, -\frac{\sqrt{-a} b^{2} x^{4} \arctan \left (\frac{\sqrt{-a}}{\sqrt{b x^{2} + a}}\right ) +{\left (a b x^{2} + 2 \, a^{2}\right )} \sqrt{b x^{2} + a}}{8 \, a^{2} x^{4}}\right ] \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 3.62681, size = 92, normalized size = 1.3 \begin{align*} - \frac{a}{4 \sqrt{b} x^{5} \sqrt{\frac{a}{b x^{2}} + 1}} - \frac{3 \sqrt{b}}{8 x^{3} \sqrt{\frac{a}{b x^{2}} + 1}} - \frac{b^{\frac{3}{2}}}{8 a x \sqrt{\frac{a}{b x^{2}} + 1}} + \frac{b^{2} \operatorname{asinh}{\left (\frac{\sqrt{a}}{\sqrt{b} x} \right )}}{8 a^{\frac{3}{2}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 2.66669, size = 84, normalized size = 1.18 \begin{align*} -\frac{1}{8} \, b^{2}{\left (\frac{\arctan \left (\frac{\sqrt{b x^{2} + a}}{\sqrt{-a}}\right )}{\sqrt{-a} a} + \frac{{\left (b x^{2} + a\right )}^{\frac{3}{2}} + \sqrt{b x^{2} + a} a}{a b^{2} x^{4}}\right )} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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