Optimal. Leaf size=75 \[ \frac{2}{b^2 \sqrt{x} \sqrt{a+\frac{b}{x}}}-\frac{2 \tanh ^{-1}\left (\frac{\sqrt{b}}{\sqrt{x} \sqrt{a+\frac{b}{x}}}\right )}{b^{5/2}}+\frac{2}{3 b x^{3/2} \left (a+\frac{b}{x}\right )^{3/2}} \]
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Rubi [A] time = 0.0404249, antiderivative size = 75, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 4, integrand size = 17, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.235, Rules used = {337, 288, 217, 206} \[ \frac{2}{b^2 \sqrt{x} \sqrt{a+\frac{b}{x}}}-\frac{2 \tanh ^{-1}\left (\frac{\sqrt{b}}{\sqrt{x} \sqrt{a+\frac{b}{x}}}\right )}{b^{5/2}}+\frac{2}{3 b x^{3/2} \left (a+\frac{b}{x}\right )^{3/2}} \]
Antiderivative was successfully verified.
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Rule 337
Rule 288
Rule 217
Rule 206
Rubi steps
\begin{align*} \int \frac{1}{\left (a+\frac{b}{x}\right )^{5/2} x^{7/2}} \, dx &=-\left (2 \operatorname{Subst}\left (\int \frac{x^4}{\left (a+b x^2\right )^{5/2}} \, dx,x,\frac{1}{\sqrt{x}}\right )\right )\\ &=\frac{2}{3 b \left (a+\frac{b}{x}\right )^{3/2} x^{3/2}}-\frac{2 \operatorname{Subst}\left (\int \frac{x^2}{\left (a+b x^2\right )^{3/2}} \, dx,x,\frac{1}{\sqrt{x}}\right )}{b}\\ &=\frac{2}{3 b \left (a+\frac{b}{x}\right )^{3/2} x^{3/2}}+\frac{2}{b^2 \sqrt{a+\frac{b}{x}} \sqrt{x}}-\frac{2 \operatorname{Subst}\left (\int \frac{1}{\sqrt{a+b x^2}} \, dx,x,\frac{1}{\sqrt{x}}\right )}{b^2}\\ &=\frac{2}{3 b \left (a+\frac{b}{x}\right )^{3/2} x^{3/2}}+\frac{2}{b^2 \sqrt{a+\frac{b}{x}} \sqrt{x}}-\frac{2 \operatorname{Subst}\left (\int \frac{1}{1-b x^2} \, dx,x,\frac{1}{\sqrt{a+\frac{b}{x}} \sqrt{x}}\right )}{b^2}\\ &=\frac{2}{3 b \left (a+\frac{b}{x}\right )^{3/2} x^{3/2}}+\frac{2}{b^2 \sqrt{a+\frac{b}{x}} \sqrt{x}}-\frac{2 \tanh ^{-1}\left (\frac{\sqrt{b}}{\sqrt{a+\frac{b}{x}} \sqrt{x}}\right )}{b^{5/2}}\\ \end{align*}
Mathematica [A] time = 0.16973, size = 94, normalized size = 1.25 \[ \frac{2 \left (\sqrt{b} \sqrt{x} (3 a x+4 b)-3 \sqrt{a} x \sqrt{\frac{b}{a x}+1} (a x+b) \sinh ^{-1}\left (\frac{\sqrt{b}}{\sqrt{a} \sqrt{x}}\right )\right )}{3 b^{5/2} x \sqrt{a+\frac{b}{x}} (a x+b)} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.014, size = 85, normalized size = 1.1 \begin{align*}{\frac{2}{3\, \left ( ax+b \right ) ^{2}}\sqrt{{\frac{ax+b}{x}}}\sqrt{x} \left ( -3\,{\it Artanh} \left ({\frac{\sqrt{ax+b}}{\sqrt{b}}} \right ) \sqrt{ax+b}xa+4\,{b}^{3/2}+3\,ax\sqrt{b}-3\,{\it Artanh} \left ({\frac{\sqrt{ax+b}}{\sqrt{b}}} \right ) b\sqrt{ax+b} \right ){b}^{-{\frac{5}{2}}}} \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 = 1.45265, size = 474, normalized size = 6.32 \begin{align*} \left [\frac{3 \,{\left (a^{2} x^{2} + 2 \, a b x + b^{2}\right )} \sqrt{b} \log \left (\frac{a x - 2 \, \sqrt{b} \sqrt{x} \sqrt{\frac{a x + b}{x}} + 2 \, b}{x}\right ) + 2 \,{\left (3 \, a b x + 4 \, b^{2}\right )} \sqrt{x} \sqrt{\frac{a x + b}{x}}}{3 \,{\left (a^{2} b^{3} x^{2} + 2 \, a b^{4} x + b^{5}\right )}}, \frac{2 \,{\left (3 \,{\left (a^{2} x^{2} + 2 \, a b x + b^{2}\right )} \sqrt{-b} \arctan \left (\frac{\sqrt{-b} \sqrt{x} \sqrt{\frac{a x + b}{x}}}{b}\right ) +{\left (3 \, a b x + 4 \, b^{2}\right )} \sqrt{x} \sqrt{\frac{a x + b}{x}}\right )}}{3 \,{\left (a^{2} b^{3} x^{2} + 2 \, a b^{4} x + b^{5}\right )}}\right ] \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F(-1)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.19328, size = 105, normalized size = 1.4 \begin{align*} \frac{2 \, \arctan \left (\frac{\sqrt{a x + b}}{\sqrt{-b}}\right )}{\sqrt{-b} b^{2}} - \frac{2 \,{\left (3 \, \sqrt{b} \arctan \left (\frac{\sqrt{b}}{\sqrt{-b}}\right ) + 4 \, \sqrt{-b}\right )}}{3 \, \sqrt{-b} b^{\frac{5}{2}}} + \frac{2 \,{\left (3 \, a x + 4 \, b\right )}}{3 \,{\left (a x + b\right )}^{\frac{3}{2}} b^{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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