Optimal. Leaf size=61 \[ b^{3/2} \tanh ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a+b x^2}}\right )-\frac{b \sqrt{a+b x^2}}{x}-\frac{\left (a+b x^2\right )^{3/2}}{3 x^3} \]
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Rubi [A] time = 0.0184478, antiderivative size = 61, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.2, Rules used = {277, 217, 206} \[ b^{3/2} \tanh ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a+b x^2}}\right )-\frac{b \sqrt{a+b x^2}}{x}-\frac{\left (a+b x^2\right )^{3/2}}{3 x^3} \]
Antiderivative was successfully verified.
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Rule 277
Rule 217
Rule 206
Rubi steps
\begin{align*} \int \frac{\left (a+b x^2\right )^{3/2}}{x^4} \, dx &=-\frac{\left (a+b x^2\right )^{3/2}}{3 x^3}+b \int \frac{\sqrt{a+b x^2}}{x^2} \, dx\\ &=-\frac{b \sqrt{a+b x^2}}{x}-\frac{\left (a+b x^2\right )^{3/2}}{3 x^3}+b^2 \int \frac{1}{\sqrt{a+b x^2}} \, dx\\ &=-\frac{b \sqrt{a+b x^2}}{x}-\frac{\left (a+b x^2\right )^{3/2}}{3 x^3}+b^2 \operatorname{Subst}\left (\int \frac{1}{1-b x^2} \, dx,x,\frac{x}{\sqrt{a+b x^2}}\right )\\ &=-\frac{b \sqrt{a+b x^2}}{x}-\frac{\left (a+b x^2\right )^{3/2}}{3 x^3}+b^{3/2} \tanh ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a+b x^2}}\right )\\ \end{align*}
Mathematica [C] time = 0.0086008, size = 52, normalized size = 0.85 \[ -\frac{a \sqrt{a+b x^2} \, _2F_1\left (-\frac{3}{2},-\frac{3}{2};-\frac{1}{2};-\frac{b x^2}{a}\right )}{3 x^3 \sqrt{\frac{b x^2}{a}+1}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.005, size = 92, normalized size = 1.5 \begin{align*} -{\frac{1}{3\,a{x}^{3}} \left ( b{x}^{2}+a \right ) ^{{\frac{5}{2}}}}-{\frac{2\,b}{3\,{a}^{2}x} \left ( b{x}^{2}+a \right ) ^{{\frac{5}{2}}}}+{\frac{2\,{b}^{2}x}{3\,{a}^{2}} \left ( b{x}^{2}+a \right ) ^{{\frac{3}{2}}}}+{\frac{{b}^{2}x}{a}\sqrt{b{x}^{2}+a}}+{b}^{{\frac{3}{2}}}\ln \left ( x\sqrt{b}+\sqrt{b{x}^{2}+a} \right ) \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 = 1.60172, size = 279, normalized size = 4.57 \begin{align*} \left [\frac{3 \, b^{\frac{3}{2}} x^{3} \log \left (-2 \, b x^{2} - 2 \, \sqrt{b x^{2} + a} \sqrt{b} x - a\right ) - 2 \,{\left (4 \, b x^{2} + a\right )} \sqrt{b x^{2} + a}}{6 \, x^{3}}, -\frac{3 \, \sqrt{-b} b x^{3} \arctan \left (\frac{\sqrt{-b} x}{\sqrt{b x^{2} + a}}\right ) +{\left (4 \, b x^{2} + a\right )} \sqrt{b x^{2} + a}}{3 \, x^{3}}\right ] \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 1.97584, size = 78, normalized size = 1.28 \begin{align*} - \frac{a \sqrt{b} \sqrt{\frac{a}{b x^{2}} + 1}}{3 x^{2}} - \frac{4 b^{\frac{3}{2}} \sqrt{\frac{a}{b x^{2}} + 1}}{3} - \frac{b^{\frac{3}{2}} \log{\left (\frac{a}{b x^{2}} \right )}}{2} + b^{\frac{3}{2}} \log{\left (\sqrt{\frac{a}{b x^{2}} + 1} + 1 \right )} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [B] time = 2.90562, size = 154, normalized size = 2.52 \begin{align*} -\frac{1}{2} \, b^{\frac{3}{2}} \log \left ({\left (\sqrt{b} x - \sqrt{b x^{2} + a}\right )}^{2}\right ) + \frac{4 \,{\left (3 \,{\left (\sqrt{b} x - \sqrt{b x^{2} + a}\right )}^{4} a b^{\frac{3}{2}} - 3 \,{\left (\sqrt{b} x - \sqrt{b x^{2} + a}\right )}^{2} a^{2} b^{\frac{3}{2}} + 2 \, a^{3} b^{\frac{3}{2}}\right )}}{3 \,{\left ({\left (\sqrt{b} x - \sqrt{b x^{2} + a}\right )}^{2} - a\right )}^{3}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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