Optimal. Leaf size=96 \[ \frac{5 b^2 x \sqrt{a+\frac{b}{x}}}{8 a^3}-\frac{5 b^3 \tanh ^{-1}\left (\frac{\sqrt{a+\frac{b}{x}}}{\sqrt{a}}\right )}{8 a^{7/2}}-\frac{5 b x^2 \sqrt{a+\frac{b}{x}}}{12 a^2}+\frac{x^3 \sqrt{a+\frac{b}{x}}}{3 a} \]
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Rubi [A] time = 0.039083, antiderivative size = 96, 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, 51, 63, 208} \[ \frac{5 b^2 x \sqrt{a+\frac{b}{x}}}{8 a^3}-\frac{5 b^3 \tanh ^{-1}\left (\frac{\sqrt{a+\frac{b}{x}}}{\sqrt{a}}\right )}{8 a^{7/2}}-\frac{5 b x^2 \sqrt{a+\frac{b}{x}}}{12 a^2}+\frac{x^3 \sqrt{a+\frac{b}{x}}}{3 a} \]
Antiderivative was successfully verified.
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Rule 266
Rule 51
Rule 63
Rule 208
Rubi steps
\begin{align*} \int \frac{x^2}{\sqrt{a+\frac{b}{x}}} \, dx &=-\operatorname{Subst}\left (\int \frac{1}{x^4 \sqrt{a+b x}} \, dx,x,\frac{1}{x}\right )\\ &=\frac{\sqrt{a+\frac{b}{x}} x^3}{3 a}+\frac{(5 b) \operatorname{Subst}\left (\int \frac{1}{x^3 \sqrt{a+b x}} \, dx,x,\frac{1}{x}\right )}{6 a}\\ &=-\frac{5 b \sqrt{a+\frac{b}{x}} x^2}{12 a^2}+\frac{\sqrt{a+\frac{b}{x}} x^3}{3 a}-\frac{\left (5 b^2\right ) \operatorname{Subst}\left (\int \frac{1}{x^2 \sqrt{a+b x}} \, dx,x,\frac{1}{x}\right )}{8 a^2}\\ &=\frac{5 b^2 \sqrt{a+\frac{b}{x}} x}{8 a^3}-\frac{5 b \sqrt{a+\frac{b}{x}} x^2}{12 a^2}+\frac{\sqrt{a+\frac{b}{x}} x^3}{3 a}+\frac{\left (5 b^3\right ) \operatorname{Subst}\left (\int \frac{1}{x \sqrt{a+b x}} \, dx,x,\frac{1}{x}\right )}{16 a^3}\\ &=\frac{5 b^2 \sqrt{a+\frac{b}{x}} x}{8 a^3}-\frac{5 b \sqrt{a+\frac{b}{x}} x^2}{12 a^2}+\frac{\sqrt{a+\frac{b}{x}} x^3}{3 a}+\frac{\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 )}{8 a^3}\\ &=\frac{5 b^2 \sqrt{a+\frac{b}{x}} x}{8 a^3}-\frac{5 b \sqrt{a+\frac{b}{x}} x^2}{12 a^2}+\frac{\sqrt{a+\frac{b}{x}} x^3}{3 a}-\frac{5 b^3 \tanh ^{-1}\left (\frac{\sqrt{a+\frac{b}{x}}}{\sqrt{a}}\right )}{8 a^{7/2}}\\ \end{align*}
Mathematica [C] time = 0.0091067, size = 37, normalized size = 0.39 \[ -\frac{2 b^3 \sqrt{a+\frac{b}{x}} \, _2F_1\left (\frac{1}{2},4;\frac{3}{2};\frac{b}{a x}+1\right )}{a^4} \]
Antiderivative was successfully verified.
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Maple [B] time = 0.008, size = 164, normalized size = 1.7 \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+48\,\sqrt{ \left ( ax+b \right ) x}{a}^{3/2}{b}^{2}-18\,\sqrt{a{x}^{2}+bx}{a}^{3/2}{b}^{2}-24\,a\ln \left ( 1/2\,{\frac{2\,\sqrt{ \left ( ax+b \right ) x}\sqrt{a}+2\,ax+b}{\sqrt{a}}} \right ){b}^{3}+9\,\ln \left ( 1/2\,{\frac{2\,\sqrt{a{x}^{2}+bx}\sqrt{a}+2\,ax+b}{\sqrt{a}}} \right ) a{b}^{3} \right ){a}^{-{\frac{9}{2}}}{\frac{1}{\sqrt{ \left ( ax+b \right ) x}}}} \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 = 1.76019, size = 358, normalized size = 3.73 \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} - 10 \, a^{2} b x^{2} + 15 \, a b^{2} x\right )} \sqrt{\frac{a x + b}{x}}}{48 \, a^{4}}, \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} - 10 \, a^{2} b x^{2} + 15 \, a b^{2} x\right )} \sqrt{\frac{a x + b}{x}}}{24 \, a^{4}}\right ] \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 5.80324, size = 128, normalized size = 1.33 \begin{align*} \frac{x^{\frac{7}{2}}}{3 \sqrt{b} \sqrt{\frac{a x}{b} + 1}} - \frac{\sqrt{b} x^{\frac{5}{2}}}{12 a \sqrt{\frac{a x}{b} + 1}} + \frac{5 b^{\frac{3}{2}} x^{\frac{3}{2}}}{24 a^{2} \sqrt{\frac{a x}{b} + 1}} + \frac{5 b^{\frac{5}{2}} \sqrt{x}}{8 a^{3} \sqrt{\frac{a x}{b} + 1}} - \frac{5 b^{3} \operatorname{asinh}{\left (\frac{\sqrt{a} \sqrt{x}}{\sqrt{b}} \right )}}{8 a^{\frac{7}{2}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.16718, size = 169, normalized size = 1.76 \begin{align*} \frac{1}{24} \, b{\left (\frac{15 \, b^{2} \arctan \left (\frac{\sqrt{\frac{a x + b}{x}}}{\sqrt{-a}}\right )}{\sqrt{-a} a^{3}} - \frac{33 \, a^{2} b^{2} \sqrt{\frac{a x + b}{x}} - \frac{40 \,{\left (a x + b\right )} a b^{2} \sqrt{\frac{a x + b}{x}}}{x} + \frac{15 \,{\left (a x + b\right )}^{2} b^{2} \sqrt{\frac{a x + b}{x}}}{x^{2}}}{{\left (a - \frac{a x + b}{x}\right )}^{3} a^{3}}\right )} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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