Optimal. Leaf size=104 \[ -\frac{15 a^2 \tanh ^{-1}\left (\frac{\sqrt{b}}{\sqrt{x} \sqrt{a+\frac{b}{x}}}\right )}{4 b^{7/2}}-\frac{5 \sqrt{a+\frac{b}{x}}}{2 b^2 x^{3/2}}+\frac{15 a \sqrt{a+\frac{b}{x}}}{4 b^3 \sqrt{x}}+\frac{2}{b x^{5/2} \sqrt{a+\frac{b}{x}}} \]
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Rubi [A] time = 0.0541592, antiderivative size = 104, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 5, integrand size = 17, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.294, Rules used = {337, 288, 321, 217, 206} \[ -\frac{15 a^2 \tanh ^{-1}\left (\frac{\sqrt{b}}{\sqrt{x} \sqrt{a+\frac{b}{x}}}\right )}{4 b^{7/2}}-\frac{5 \sqrt{a+\frac{b}{x}}}{2 b^2 x^{3/2}}+\frac{15 a \sqrt{a+\frac{b}{x}}}{4 b^3 \sqrt{x}}+\frac{2}{b x^{5/2} \sqrt{a+\frac{b}{x}}} \]
Antiderivative was successfully verified.
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Rule 337
Rule 288
Rule 321
Rule 217
Rule 206
Rubi steps
\begin{align*} \int \frac{1}{\left (a+\frac{b}{x}\right )^{3/2} x^{9/2}} \, dx &=-\left (2 \operatorname{Subst}\left (\int \frac{x^6}{\left (a+b x^2\right )^{3/2}} \, dx,x,\frac{1}{\sqrt{x}}\right )\right )\\ &=\frac{2}{b \sqrt{a+\frac{b}{x}} x^{5/2}}-\frac{10 \operatorname{Subst}\left (\int \frac{x^4}{\sqrt{a+b x^2}} \, dx,x,\frac{1}{\sqrt{x}}\right )}{b}\\ &=\frac{2}{b \sqrt{a+\frac{b}{x}} x^{5/2}}-\frac{5 \sqrt{a+\frac{b}{x}}}{2 b^2 x^{3/2}}+\frac{(15 a) \operatorname{Subst}\left (\int \frac{x^2}{\sqrt{a+b x^2}} \, dx,x,\frac{1}{\sqrt{x}}\right )}{2 b^2}\\ &=\frac{2}{b \sqrt{a+\frac{b}{x}} x^{5/2}}-\frac{5 \sqrt{a+\frac{b}{x}}}{2 b^2 x^{3/2}}+\frac{15 a \sqrt{a+\frac{b}{x}}}{4 b^3 \sqrt{x}}-\frac{\left (15 a^2\right ) \operatorname{Subst}\left (\int \frac{1}{\sqrt{a+b x^2}} \, dx,x,\frac{1}{\sqrt{x}}\right )}{4 b^3}\\ &=\frac{2}{b \sqrt{a+\frac{b}{x}} x^{5/2}}-\frac{5 \sqrt{a+\frac{b}{x}}}{2 b^2 x^{3/2}}+\frac{15 a \sqrt{a+\frac{b}{x}}}{4 b^3 \sqrt{x}}-\frac{\left (15 a^2\right ) \operatorname{Subst}\left (\int \frac{1}{1-b x^2} \, dx,x,\frac{1}{\sqrt{a+\frac{b}{x}} \sqrt{x}}\right )}{4 b^3}\\ &=\frac{2}{b \sqrt{a+\frac{b}{x}} x^{5/2}}-\frac{5 \sqrt{a+\frac{b}{x}}}{2 b^2 x^{3/2}}+\frac{15 a \sqrt{a+\frac{b}{x}}}{4 b^3 \sqrt{x}}-\frac{15 a^2 \tanh ^{-1}\left (\frac{\sqrt{b}}{\sqrt{a+\frac{b}{x}} \sqrt{x}}\right )}{4 b^{7/2}}\\ \end{align*}
Mathematica [C] time = 0.0164082, size = 56, normalized size = 0.54 \[ -\frac{2 \sqrt{\frac{b}{a x}+1} \, _2F_1\left (\frac{3}{2},\frac{7}{2};\frac{9}{2};-\frac{b}{a x}\right )}{7 a x^{7/2} \sqrt{a+\frac{b}{x}}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.016, size = 78, normalized size = 0.8 \begin{align*} -{\frac{1}{4\,ax+4\,b}\sqrt{{\frac{ax+b}{x}}} \left ( 15\,{\it Artanh} \left ({\frac{\sqrt{ax+b}}{\sqrt{b}}} \right ) \sqrt{ax+b}{x}^{2}{a}^{2}-5\,{b}^{3/2}xa-15\,{a}^{2}{x}^{2}\sqrt{b}+2\,{b}^{5/2} \right ){x}^{-{\frac{3}{2}}}{b}^{-{\frac{7}{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 = 1.55056, size = 485, normalized size = 4.66 \begin{align*} \left [\frac{15 \,{\left (a^{3} x^{3} + a^{2} b x^{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 (15 \, a^{2} b x^{2} + 5 \, a b^{2} x - 2 \, b^{3}\right )} \sqrt{x} \sqrt{\frac{a x + b}{x}}}{8 \,{\left (a b^{4} x^{3} + b^{5} x^{2}\right )}}, \frac{15 \,{\left (a^{3} x^{3} + a^{2} b x^{2}\right )} \sqrt{-b} \arctan \left (\frac{\sqrt{-b} \sqrt{x} \sqrt{\frac{a x + b}{x}}}{b}\right ) +{\left (15 \, a^{2} b x^{2} + 5 \, a b^{2} x - 2 \, b^{3}\right )} \sqrt{x} \sqrt{\frac{a x + b}{x}}}{4 \,{\left (a b^{4} x^{3} + b^{5} x^{2}\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 = 1.25446, size = 97, normalized size = 0.93 \begin{align*} \frac{1}{4} \, a^{2}{\left (\frac{15 \, \arctan \left (\frac{\sqrt{a x + b}}{\sqrt{-b}}\right )}{\sqrt{-b} b^{3}} + \frac{8}{\sqrt{a x + b} b^{3}} + \frac{7 \,{\left (a x + b\right )}^{\frac{3}{2}} - 9 \, \sqrt{a x + b} b}{a^{2} b^{3} x^{2}}\right )} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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