Optimal. Leaf size=87 \[ \frac{5}{8} b^2 x \sqrt{a+\frac{b}{x}}+\frac{5 b^3 \tanh ^{-1}\left (\frac{\sqrt{a+\frac{b}{x}}}{\sqrt{a}}\right )}{8 \sqrt{a}}+\frac{5}{12} b x^2 \left (a+\frac{b}{x}\right )^{3/2}+\frac{1}{3} x^3 \left (a+\frac{b}{x}\right )^{5/2} \]
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Rubi [A] time = 0.0375767, antiderivative size = 87, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 4, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.267, Rules used = {266, 47, 63, 208} \[ \frac{5}{8} b^2 x \sqrt{a+\frac{b}{x}}+\frac{5 b^3 \tanh ^{-1}\left (\frac{\sqrt{a+\frac{b}{x}}}{\sqrt{a}}\right )}{8 \sqrt{a}}+\frac{5}{12} b x^2 \left (a+\frac{b}{x}\right )^{3/2}+\frac{1}{3} x^3 \left (a+\frac{b}{x}\right )^{5/2} \]
Antiderivative was successfully verified.
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Rule 266
Rule 47
Rule 63
Rule 208
Rubi steps
\begin{align*} \int \left (a+\frac{b}{x}\right )^{5/2} x^2 \, dx &=-\operatorname{Subst}\left (\int \frac{(a+b x)^{5/2}}{x^4} \, dx,x,\frac{1}{x}\right )\\ &=\frac{1}{3} \left (a+\frac{b}{x}\right )^{5/2} x^3-\frac{1}{6} (5 b) \operatorname{Subst}\left (\int \frac{(a+b x)^{3/2}}{x^3} \, dx,x,\frac{1}{x}\right )\\ &=\frac{5}{12} b \left (a+\frac{b}{x}\right )^{3/2} x^2+\frac{1}{3} \left (a+\frac{b}{x}\right )^{5/2} x^3-\frac{1}{8} \left (5 b^2\right ) \operatorname{Subst}\left (\int \frac{\sqrt{a+b x}}{x^2} \, dx,x,\frac{1}{x}\right )\\ &=\frac{5}{8} b^2 \sqrt{a+\frac{b}{x}} x+\frac{5}{12} b \left (a+\frac{b}{x}\right )^{3/2} x^2+\frac{1}{3} \left (a+\frac{b}{x}\right )^{5/2} x^3-\frac{1}{16} \left (5 b^3\right ) \operatorname{Subst}\left (\int \frac{1}{x \sqrt{a+b x}} \, dx,x,\frac{1}{x}\right )\\ &=\frac{5}{8} b^2 \sqrt{a+\frac{b}{x}} x+\frac{5}{12} b \left (a+\frac{b}{x}\right )^{3/2} x^2+\frac{1}{3} \left (a+\frac{b}{x}\right )^{5/2} x^3-\frac{1}{8} \left (5 b^2\right ) \operatorname{Subst}\left (\int \frac{1}{-\frac{a}{b}+\frac{x^2}{b}} \, dx,x,\sqrt{a+\frac{b}{x}}\right )\\ &=\frac{5}{8} b^2 \sqrt{a+\frac{b}{x}} x+\frac{5}{12} b \left (a+\frac{b}{x}\right )^{3/2} x^2+\frac{1}{3} \left (a+\frac{b}{x}\right )^{5/2} x^3+\frac{5 b^3 \tanh ^{-1}\left (\frac{\sqrt{a+\frac{b}{x}}}{\sqrt{a}}\right )}{8 \sqrt{a}}\\ \end{align*}
Mathematica [A] time = 0.0388304, size = 87, normalized size = 1. \[ \frac{x \sqrt{a+\frac{b}{x}} \left (34 a^2 b x^2+8 a^3 x^3+59 a b^2 x+15 b^3 \sqrt{\frac{b}{a x}+1} \tanh ^{-1}\left (\sqrt{\frac{b}{a x}+1}\right )+33 b^3\right )}{24 (a x+b)} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.007, size = 115, normalized size = 1.3 \begin{align*}{\frac{x}{48}\sqrt{{\frac{ax+b}{x}}} \left ( 16\, \left ( a{x}^{2}+bx \right ) ^{3/2}{a}^{5/2}+36\,\sqrt{a{x}^{2}+bx}{a}^{5/2}xb+66\,\sqrt{a{x}^{2}+bx}{a}^{3/2}{b}^{2}+15\,\ln \left ( 1/2\,{\frac{2\,\sqrt{a{x}^{2}+bx}\sqrt{a}+2\,ax+b}{\sqrt{a}}} \right ) a{b}^{3} \right ){\frac{1}{\sqrt{ \left ( ax+b \right ) x}}}{a}^{-{\frac{3}{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 = 1.76343, size = 354, normalized size = 4.07 \begin{align*} \left [\frac{15 \, \sqrt{a} b^{3} \log \left (2 \, a x + 2 \, \sqrt{a} x \sqrt{\frac{a x + b}{x}} + b\right ) + 2 \,{\left (8 \, a^{3} x^{3} + 26 \, a^{2} b x^{2} + 33 \, a b^{2} x\right )} \sqrt{\frac{a x + b}{x}}}{48 \, a}, -\frac{15 \, \sqrt{-a} b^{3} \arctan \left (\frac{\sqrt{-a} \sqrt{\frac{a x + b}{x}}}{a}\right ) -{\left (8 \, a^{3} x^{3} + 26 \, a^{2} b x^{2} + 33 \, a b^{2} x\right )} \sqrt{\frac{a x + b}{x}}}{24 \, a}\right ] \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 6.14317, size = 102, normalized size = 1.17 \begin{align*} \frac{a^{2} \sqrt{b} x^{\frac{5}{2}} \sqrt{\frac{a x}{b} + 1}}{3} + \frac{13 a b^{\frac{3}{2}} x^{\frac{3}{2}} \sqrt{\frac{a x}{b} + 1}}{12} + \frac{11 b^{\frac{5}{2}} \sqrt{x} \sqrt{\frac{a x}{b} + 1}}{8} + \frac{5 b^{3} \operatorname{asinh}{\left (\frac{\sqrt{a} \sqrt{x}}{\sqrt{b}} \right )}}{8 \sqrt{a}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.13095, size = 126, normalized size = 1.45 \begin{align*} -\frac{5 \, b^{3} \log \left ({\left | -2 \,{\left (\sqrt{a} x - \sqrt{a x^{2} + b x}\right )} \sqrt{a} - b \right |}\right ) \mathrm{sgn}\left (x\right )}{16 \, \sqrt{a}} + \frac{5 \, b^{3} \log \left ({\left | b \right |}\right ) \mathrm{sgn}\left (x\right )}{16 \, \sqrt{a}} + \frac{1}{24} \, \sqrt{a x^{2} + b x}{\left (33 \, b^{2} \mathrm{sgn}\left (x\right ) + 2 \,{\left (4 \, a^{2} x \mathrm{sgn}\left (x\right ) + 13 \, a b \mathrm{sgn}\left (x\right )\right )} x\right )} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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