Optimal. Leaf size=126 \[ \frac{5 a^4 \tanh ^{-1}\left (\frac{\sqrt{b}}{\sqrt{x} \sqrt{a+\frac{b}{x}}}\right )}{64 b^{3/2}}-\frac{5 a^2 \sqrt{a+\frac{b}{x}}}{32 x^{3/2}}-\frac{5 a^3 \sqrt{a+\frac{b}{x}}}{64 b \sqrt{x}}-\frac{5 a \left (a+\frac{b}{x}\right )^{3/2}}{24 x^{3/2}}-\frac{\left (a+\frac{b}{x}\right )^{5/2}}{4 x^{3/2}} \]
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Rubi [A] time = 0.0682302, antiderivative size = 126, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 5, integrand size = 17, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.294, Rules used = {337, 279, 321, 217, 206} \[ \frac{5 a^4 \tanh ^{-1}\left (\frac{\sqrt{b}}{\sqrt{x} \sqrt{a+\frac{b}{x}}}\right )}{64 b^{3/2}}-\frac{5 a^2 \sqrt{a+\frac{b}{x}}}{32 x^{3/2}}-\frac{5 a^3 \sqrt{a+\frac{b}{x}}}{64 b \sqrt{x}}-\frac{5 a \left (a+\frac{b}{x}\right )^{3/2}}{24 x^{3/2}}-\frac{\left (a+\frac{b}{x}\right )^{5/2}}{4 x^{3/2}} \]
Antiderivative was successfully verified.
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Rule 337
Rule 279
Rule 321
Rule 217
Rule 206
Rubi steps
\begin{align*} \int \frac{\left (a+\frac{b}{x}\right )^{5/2}}{x^{5/2}} \, dx &=-\left (2 \operatorname{Subst}\left (\int x^2 \left (a+b x^2\right )^{5/2} \, dx,x,\frac{1}{\sqrt{x}}\right )\right )\\ &=-\frac{\left (a+\frac{b}{x}\right )^{5/2}}{4 x^{3/2}}-\frac{1}{4} (5 a) \operatorname{Subst}\left (\int x^2 \left (a+b x^2\right )^{3/2} \, dx,x,\frac{1}{\sqrt{x}}\right )\\ &=-\frac{5 a \left (a+\frac{b}{x}\right )^{3/2}}{24 x^{3/2}}-\frac{\left (a+\frac{b}{x}\right )^{5/2}}{4 x^{3/2}}-\frac{1}{8} \left (5 a^2\right ) \operatorname{Subst}\left (\int x^2 \sqrt{a+b x^2} \, dx,x,\frac{1}{\sqrt{x}}\right )\\ &=-\frac{5 a^2 \sqrt{a+\frac{b}{x}}}{32 x^{3/2}}-\frac{5 a \left (a+\frac{b}{x}\right )^{3/2}}{24 x^{3/2}}-\frac{\left (a+\frac{b}{x}\right )^{5/2}}{4 x^{3/2}}-\frac{1}{32} \left (5 a^3\right ) \operatorname{Subst}\left (\int \frac{x^2}{\sqrt{a+b x^2}} \, dx,x,\frac{1}{\sqrt{x}}\right )\\ &=-\frac{5 a^2 \sqrt{a+\frac{b}{x}}}{32 x^{3/2}}-\frac{5 a \left (a+\frac{b}{x}\right )^{3/2}}{24 x^{3/2}}-\frac{\left (a+\frac{b}{x}\right )^{5/2}}{4 x^{3/2}}-\frac{5 a^3 \sqrt{a+\frac{b}{x}}}{64 b \sqrt{x}}+\frac{\left (5 a^4\right ) \operatorname{Subst}\left (\int \frac{1}{\sqrt{a+b x^2}} \, dx,x,\frac{1}{\sqrt{x}}\right )}{64 b}\\ &=-\frac{5 a^2 \sqrt{a+\frac{b}{x}}}{32 x^{3/2}}-\frac{5 a \left (a+\frac{b}{x}\right )^{3/2}}{24 x^{3/2}}-\frac{\left (a+\frac{b}{x}\right )^{5/2}}{4 x^{3/2}}-\frac{5 a^3 \sqrt{a+\frac{b}{x}}}{64 b \sqrt{x}}+\frac{\left (5 a^4\right ) \operatorname{Subst}\left (\int \frac{1}{1-b x^2} \, dx,x,\frac{1}{\sqrt{a+\frac{b}{x}} \sqrt{x}}\right )}{64 b}\\ &=-\frac{5 a^2 \sqrt{a+\frac{b}{x}}}{32 x^{3/2}}-\frac{5 a \left (a+\frac{b}{x}\right )^{3/2}}{24 x^{3/2}}-\frac{\left (a+\frac{b}{x}\right )^{5/2}}{4 x^{3/2}}-\frac{5 a^3 \sqrt{a+\frac{b}{x}}}{64 b \sqrt{x}}+\frac{5 a^4 \tanh ^{-1}\left (\frac{\sqrt{b}}{\sqrt{a+\frac{b}{x}} \sqrt{x}}\right )}{64 b^{3/2}}\\ \end{align*}
Mathematica [A] time = 0.146404, size = 101, normalized size = 0.8 \[ \frac{\sqrt{a+\frac{b}{x}} \left (\frac{15 a^{7/2} \sinh ^{-1}\left (\frac{\sqrt{b}}{\sqrt{a} \sqrt{x}}\right )}{\sqrt{\frac{b}{a x}+1}}-\frac{\sqrt{b} \left (118 a^2 b x^2+15 a^3 x^3+136 a b^2 x+48 b^3\right )}{x^{7/2}}\right )}{192 b^{3/2}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.017, size = 110, normalized size = 0.9 \begin{align*} -{\frac{1}{192}\sqrt{{\frac{ax+b}{x}}} \left ( -15\,{\it Artanh} \left ({\frac{\sqrt{ax+b}}{\sqrt{b}}} \right ){a}^{4}{x}^{4}+48\,{b}^{7/2}\sqrt{ax+b}+136\,xa{b}^{5/2}\sqrt{ax+b}+118\,{x}^{2}{a}^{2}{b}^{3/2}\sqrt{ax+b}+15\,{x}^{3}{a}^{3}\sqrt{ax+b}\sqrt{b} \right ){x}^{-{\frac{7}{2}}}{b}^{-{\frac{3}{2}}}{\frac{1}{\sqrt{ax+b}}}} \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 = 1.53178, size = 478, normalized size = 3.79 \begin{align*} \left [\frac{15 \, a^{4} \sqrt{b} x^{4} \log \left (\frac{a x + 2 \, \sqrt{b} \sqrt{x} \sqrt{\frac{a x + b}{x}} + 2 \, b}{x}\right ) - 2 \,{\left (15 \, a^{3} b x^{3} + 118 \, a^{2} b^{2} x^{2} + 136 \, a b^{3} x + 48 \, b^{4}\right )} \sqrt{x} \sqrt{\frac{a x + b}{x}}}{384 \, b^{2} x^{4}}, -\frac{15 \, a^{4} \sqrt{-b} x^{4} \arctan \left (\frac{\sqrt{-b} \sqrt{x} \sqrt{\frac{a x + b}{x}}}{b}\right ) +{\left (15 \, a^{3} b x^{3} + 118 \, a^{2} b^{2} x^{2} + 136 \, a b^{3} x + 48 \, b^{4}\right )} \sqrt{x} \sqrt{\frac{a x + b}{x}}}{192 \, b^{2} x^{4}}\right ] \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 74.2673, size = 155, normalized size = 1.23 \begin{align*} - \frac{5 a^{\frac{7}{2}}}{64 b \sqrt{x} \sqrt{1 + \frac{b}{a x}}} - \frac{133 a^{\frac{5}{2}}}{192 x^{\frac{3}{2}} \sqrt{1 + \frac{b}{a x}}} - \frac{127 a^{\frac{3}{2}} b}{96 x^{\frac{5}{2}} \sqrt{1 + \frac{b}{a x}}} - \frac{23 \sqrt{a} b^{2}}{24 x^{\frac{7}{2}} \sqrt{1 + \frac{b}{a x}}} + \frac{5 a^{4} \operatorname{asinh}{\left (\frac{\sqrt{b}}{\sqrt{a} \sqrt{x}} \right )}}{64 b^{\frac{3}{2}}} - \frac{b^{3}}{4 \sqrt{a} x^{\frac{9}{2}} \sqrt{1 + \frac{b}{a x}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.354, size = 113, normalized size = 0.9 \begin{align*} -\frac{1}{192} \, a^{4}{\left (\frac{15 \, \arctan \left (\frac{\sqrt{a x + b}}{\sqrt{-b}}\right )}{\sqrt{-b} b} + \frac{15 \,{\left (a x + b\right )}^{\frac{7}{2}} + 73 \,{\left (a x + b\right )}^{\frac{5}{2}} b - 55 \,{\left (a x + b\right )}^{\frac{3}{2}} b^{2} + 15 \, \sqrt{a x + b} b^{3}}{a^{4} b x^{4}}\right )} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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