Optimal. Leaf size=71 \[ 5 a^{3/2} b \tanh ^{-1}\left (\frac{\sqrt{a+\frac{b}{x}}}{\sqrt{a}}\right )+x \left (a+\frac{b}{x}\right )^{5/2}-\frac{5}{3} b \left (a+\frac{b}{x}\right )^{3/2}-5 a b \sqrt{a+\frac{b}{x}} \]
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Rubi [A] time = 0.0343127, antiderivative size = 71, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 5, integrand size = 11, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.454, Rules used = {242, 47, 50, 63, 208} \[ 5 a^{3/2} b \tanh ^{-1}\left (\frac{\sqrt{a+\frac{b}{x}}}{\sqrt{a}}\right )+x \left (a+\frac{b}{x}\right )^{5/2}-\frac{5}{3} b \left (a+\frac{b}{x}\right )^{3/2}-5 a b \sqrt{a+\frac{b}{x}} \]
Antiderivative was successfully verified.
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Rule 242
Rule 47
Rule 50
Rule 63
Rule 208
Rubi steps
\begin{align*} \int \left (a+\frac{b}{x}\right )^{5/2} \, dx &=-\operatorname{Subst}\left (\int \frac{(a+b x)^{5/2}}{x^2} \, dx,x,\frac{1}{x}\right )\\ &=\left (a+\frac{b}{x}\right )^{5/2} x-\frac{1}{2} (5 b) \operatorname{Subst}\left (\int \frac{(a+b x)^{3/2}}{x} \, dx,x,\frac{1}{x}\right )\\ &=-\frac{5}{3} b \left (a+\frac{b}{x}\right )^{3/2}+\left (a+\frac{b}{x}\right )^{5/2} x-\frac{1}{2} (5 a b) \operatorname{Subst}\left (\int \frac{\sqrt{a+b x}}{x} \, dx,x,\frac{1}{x}\right )\\ &=-5 a b \sqrt{a+\frac{b}{x}}-\frac{5}{3} b \left (a+\frac{b}{x}\right )^{3/2}+\left (a+\frac{b}{x}\right )^{5/2} x-\frac{1}{2} \left (5 a^2 b\right ) \operatorname{Subst}\left (\int \frac{1}{x \sqrt{a+b x}} \, dx,x,\frac{1}{x}\right )\\ &=-5 a b \sqrt{a+\frac{b}{x}}-\frac{5}{3} b \left (a+\frac{b}{x}\right )^{3/2}+\left (a+\frac{b}{x}\right )^{5/2} x-\left (5 a^2\right ) \operatorname{Subst}\left (\int \frac{1}{-\frac{a}{b}+\frac{x^2}{b}} \, dx,x,\sqrt{a+\frac{b}{x}}\right )\\ &=-5 a b \sqrt{a+\frac{b}{x}}-\frac{5}{3} b \left (a+\frac{b}{x}\right )^{3/2}+\left (a+\frac{b}{x}\right )^{5/2} x+5 a^{3/2} b \tanh ^{-1}\left (\frac{\sqrt{a+\frac{b}{x}}}{\sqrt{a}}\right )\\ \end{align*}
Mathematica [A] time = 0.0442537, size = 64, normalized size = 0.9 \[ \frac{\sqrt{a+\frac{b}{x}} \left (3 a^2 x^2-14 a b x-2 b^2\right )}{3 x}+5 a^{3/2} b \tanh ^{-1}\left (\frac{\sqrt{a+\frac{b}{x}}}{\sqrt{a}}\right ) \]
Antiderivative was successfully verified.
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Maple [B] time = 0.007, size = 120, normalized size = 1.7 \begin{align*} -{\frac{1}{6\,{x}^{2}}\sqrt{{\frac{ax+b}{x}}} \left ( -30\,\sqrt{a{x}^{2}+bx}{a}^{5/2}{x}^{3}-15\,\ln \left ( 1/2\,{\frac{2\,\sqrt{a{x}^{2}+bx}\sqrt{a}+2\,ax+b}{\sqrt{a}}} \right ){x}^{3}{a}^{2}b+24\, \left ( a{x}^{2}+bx \right ) ^{3/2}{a}^{3/2}x+4\,b \left ( a{x}^{2}+bx \right ) ^{3/2}\sqrt{a} \right ){\frac{1}{\sqrt{ \left ( ax+b \right ) x}}}{\frac{1}{\sqrt{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 = 1.42284, size = 329, normalized size = 4.63 \begin{align*} \left [\frac{15 \, a^{\frac{3}{2}} b x \log \left (2 \, a x + 2 \, \sqrt{a} x \sqrt{\frac{a x + b}{x}} + b\right ) + 2 \,{\left (3 \, a^{2} x^{2} - 14 \, a b x - 2 \, b^{2}\right )} \sqrt{\frac{a x + b}{x}}}{6 \, x}, -\frac{15 \, \sqrt{-a} a b x \arctan \left (\frac{\sqrt{-a} \sqrt{\frac{a x + b}{x}}}{a}\right ) -{\left (3 \, a^{2} x^{2} - 14 \, a b x - 2 \, b^{2}\right )} \sqrt{\frac{a x + b}{x}}}{3 \, x}\right ] \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 3.37097, size = 99, normalized size = 1.39 \begin{align*} a^{\frac{5}{2}} x \sqrt{1 + \frac{b}{a x}} - \frac{14 a^{\frac{3}{2}} b \sqrt{1 + \frac{b}{a x}}}{3} - \frac{5 a^{\frac{3}{2}} b \log{\left (\frac{b}{a x} \right )}}{2} + 5 a^{\frac{3}{2}} b \log{\left (\sqrt{1 + \frac{b}{a x}} + 1 \right )} - \frac{2 \sqrt{a} b^{2} \sqrt{1 + \frac{b}{a x}}}{3 x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F(-2)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: TypeError} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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