Optimal. Leaf size=138 \[ \frac{105 b^3}{8 a^5 \sqrt{a+\frac{b}{x}}}+\frac{35 b^3}{8 a^4 \left (a+\frac{b}{x}\right )^{3/2}}+\frac{21 b^2 x}{8 a^3 \left (a+\frac{b}{x}\right )^{3/2}}-\frac{105 b^3 \tanh ^{-1}\left (\frac{\sqrt{a+\frac{b}{x}}}{\sqrt{a}}\right )}{8 a^{11/2}}-\frac{3 b x^2}{4 a^2 \left (a+\frac{b}{x}\right )^{3/2}}+\frac{x^3}{3 a \left (a+\frac{b}{x}\right )^{3/2}} \]
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Rubi [A] time = 0.0663102, antiderivative size = 134, normalized size of antiderivative = 0.97, number of steps used = 8, number of rules used = 4, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.267, Rules used = {266, 51, 63, 208} \[ \frac{105 b^2 x \sqrt{a+\frac{b}{x}}}{8 a^5}-\frac{105 b^3 \tanh ^{-1}\left (\frac{\sqrt{a+\frac{b}{x}}}{\sqrt{a}}\right )}{8 a^{11/2}}-\frac{35 b x^2 \sqrt{a+\frac{b}{x}}}{4 a^4}+\frac{7 x^3 \sqrt{a+\frac{b}{x}}}{a^3}-\frac{6 x^3}{a^2 \sqrt{a+\frac{b}{x}}}-\frac{2 x^3}{3 a \left (a+\frac{b}{x}\right )^{3/2}} \]
Antiderivative was successfully verified.
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Rule 266
Rule 51
Rule 63
Rule 208
Rubi steps
\begin{align*} \int \frac{x^2}{\left (a+\frac{b}{x}\right )^{5/2}} \, dx &=-\operatorname{Subst}\left (\int \frac{1}{x^4 (a+b x)^{5/2}} \, dx,x,\frac{1}{x}\right )\\ &=-\frac{2 x^3}{3 a \left (a+\frac{b}{x}\right )^{3/2}}-\frac{3 \operatorname{Subst}\left (\int \frac{1}{x^4 (a+b x)^{3/2}} \, dx,x,\frac{1}{x}\right )}{a}\\ &=-\frac{2 x^3}{3 a \left (a+\frac{b}{x}\right )^{3/2}}-\frac{6 x^3}{a^2 \sqrt{a+\frac{b}{x}}}-\frac{21 \operatorname{Subst}\left (\int \frac{1}{x^4 \sqrt{a+b x}} \, dx,x,\frac{1}{x}\right )}{a^2}\\ &=-\frac{2 x^3}{3 a \left (a+\frac{b}{x}\right )^{3/2}}-\frac{6 x^3}{a^2 \sqrt{a+\frac{b}{x}}}+\frac{7 \sqrt{a+\frac{b}{x}} x^3}{a^3}+\frac{(35 b) \operatorname{Subst}\left (\int \frac{1}{x^3 \sqrt{a+b x}} \, dx,x,\frac{1}{x}\right )}{2 a^3}\\ &=-\frac{35 b \sqrt{a+\frac{b}{x}} x^2}{4 a^4}-\frac{2 x^3}{3 a \left (a+\frac{b}{x}\right )^{3/2}}-\frac{6 x^3}{a^2 \sqrt{a+\frac{b}{x}}}+\frac{7 \sqrt{a+\frac{b}{x}} x^3}{a^3}-\frac{\left (105 b^2\right ) \operatorname{Subst}\left (\int \frac{1}{x^2 \sqrt{a+b x}} \, dx,x,\frac{1}{x}\right )}{8 a^4}\\ &=\frac{105 b^2 \sqrt{a+\frac{b}{x}} x}{8 a^5}-\frac{35 b \sqrt{a+\frac{b}{x}} x^2}{4 a^4}-\frac{2 x^3}{3 a \left (a+\frac{b}{x}\right )^{3/2}}-\frac{6 x^3}{a^2 \sqrt{a+\frac{b}{x}}}+\frac{7 \sqrt{a+\frac{b}{x}} x^3}{a^3}+\frac{\left (105 b^3\right ) \operatorname{Subst}\left (\int \frac{1}{x \sqrt{a+b x}} \, dx,x,\frac{1}{x}\right )}{16 a^5}\\ &=\frac{105 b^2 \sqrt{a+\frac{b}{x}} x}{8 a^5}-\frac{35 b \sqrt{a+\frac{b}{x}} x^2}{4 a^4}-\frac{2 x^3}{3 a \left (a+\frac{b}{x}\right )^{3/2}}-\frac{6 x^3}{a^2 \sqrt{a+\frac{b}{x}}}+\frac{7 \sqrt{a+\frac{b}{x}} x^3}{a^3}+\frac{\left (105 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^5}\\ &=\frac{105 b^2 \sqrt{a+\frac{b}{x}} x}{8 a^5}-\frac{35 b \sqrt{a+\frac{b}{x}} x^2}{4 a^4}-\frac{2 x^3}{3 a \left (a+\frac{b}{x}\right )^{3/2}}-\frac{6 x^3}{a^2 \sqrt{a+\frac{b}{x}}}+\frac{7 \sqrt{a+\frac{b}{x}} x^3}{a^3}-\frac{105 b^3 \tanh ^{-1}\left (\frac{\sqrt{a+\frac{b}{x}}}{\sqrt{a}}\right )}{8 a^{11/2}}\\ \end{align*}
Mathematica [C] time = 0.0146011, size = 39, normalized size = 0.28 \[ \frac{2 b^3 \, _2F_1\left (-\frac{3}{2},4;-\frac{1}{2};\frac{b}{a x}+1\right )}{3 a^4 \left (a+\frac{b}{x}\right )^{3/2}} \]
Antiderivative was successfully verified.
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Maple [B] time = 0.015, size = 616, normalized size = 4.5 \begin{align*}{\frac{x}{48\, \left ( ax+b \right ) ^{3}}\sqrt{{\frac{ax+b}{x}}} \left ( 16\, \left ( a{x}^{2}+bx \right ) ^{3/2}{a}^{11/2}{x}^{3}-84\,\sqrt{a{x}^{2}+bx}{a}^{11/2}{x}^{4}b+48\, \left ( a{x}^{2}+bx \right ) ^{3/2}{a}^{9/2}{x}^{2}b-294\,\sqrt{a{x}^{2}+bx}{a}^{9/2}{x}^{3}{b}^{2}+672\,{a}^{9/2}\sqrt{ \left ( ax+b \right ) x}{x}^{3}{b}^{2}-336\,{a}^{4}\ln \left ( 1/2\,{\frac{2\,\sqrt{ \left ( ax+b \right ) x}\sqrt{a}+2\,ax+b}{\sqrt{a}}} \right ){x}^{3}{b}^{3}+48\, \left ( a{x}^{2}+bx \right ) ^{3/2}{a}^{7/2}x{b}^{2}-378\,\sqrt{a{x}^{2}+bx}{a}^{7/2}{x}^{2}{b}^{3}-384\,{a}^{7/2} \left ( \left ( ax+b \right ) x \right ) ^{3/2}x{b}^{2}+2016\,{a}^{7/2}\sqrt{ \left ( ax+b \right ) x}{x}^{2}{b}^{3}-1008\,{a}^{3}\ln \left ( 1/2\,{\frac{2\,\sqrt{ \left ( ax+b \right ) x}\sqrt{a}+2\,ax+b}{\sqrt{a}}} \right ){x}^{2}{b}^{4}+21\,\ln \left ( 1/2\,{\frac{2\,\sqrt{a{x}^{2}+bx}\sqrt{a}+2\,ax+b}{\sqrt{a}}} \right ){x}^{3}{a}^{4}{b}^{3}+16\, \left ( a{x}^{2}+bx \right ) ^{3/2}{a}^{5/2}{b}^{3}-210\,\sqrt{a{x}^{2}+bx}{a}^{5/2}x{b}^{4}-352\,{b}^{3}{a}^{5/2} \left ( \left ( ax+b \right ) x \right ) ^{3/2}+2016\,{a}^{5/2}\sqrt{ \left ( ax+b \right ) x}x{b}^{4}-1008\,{a}^{2}\ln \left ( 1/2\,{\frac{2\,\sqrt{ \left ( ax+b \right ) x}\sqrt{a}+2\,ax+b}{\sqrt{a}}} \right ) x{b}^{5}+63\,\ln \left ( 1/2\,{\frac{2\,\sqrt{a{x}^{2}+bx}\sqrt{a}+2\,ax+b}{\sqrt{a}}} \right ){x}^{2}{a}^{3}{b}^{4}-42\,\sqrt{a{x}^{2}+bx}{a}^{3/2}{b}^{5}+672\,{a}^{3/2}\sqrt{ \left ( ax+b \right ) x}{b}^{5}-336\,a\ln \left ( 1/2\,{\frac{2\,\sqrt{ \left ( ax+b \right ) x}\sqrt{a}+2\,ax+b}{\sqrt{a}}} \right ){b}^{6}+63\,\ln \left ( 1/2\,{\frac{2\,\sqrt{a{x}^{2}+bx}\sqrt{a}+2\,ax+b}{\sqrt{a}}} \right ) x{a}^{2}{b}^{5}+21\,\ln \left ( 1/2\,{\frac{2\,\sqrt{a{x}^{2}+bx}\sqrt{a}+2\,ax+b}{\sqrt{a}}} \right ) a{b}^{6} \right ){a}^{-{\frac{13}{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 = 1.88782, size = 609, normalized size = 4.41 \begin{align*} \left [\frac{315 \,{\left (a^{2} b^{3} x^{2} + 2 \, a b^{4} x + b^{5}\right )} \sqrt{a} \log \left (2 \, a x - 2 \, \sqrt{a} x \sqrt{\frac{a x + b}{x}} + b\right ) + 2 \,{\left (8 \, a^{5} x^{5} - 18 \, a^{4} b x^{4} + 63 \, a^{3} b^{2} x^{3} + 420 \, a^{2} b^{3} x^{2} + 315 \, a b^{4} x\right )} \sqrt{\frac{a x + b}{x}}}{48 \,{\left (a^{8} x^{2} + 2 \, a^{7} b x + a^{6} b^{2}\right )}}, \frac{315 \,{\left (a^{2} b^{3} x^{2} + 2 \, a b^{4} x + b^{5}\right )} \sqrt{-a} \arctan \left (\frac{\sqrt{-a} \sqrt{\frac{a x + b}{x}}}{a}\right ) +{\left (8 \, a^{5} x^{5} - 18 \, a^{4} b x^{4} + 63 \, a^{3} b^{2} x^{3} + 420 \, a^{2} b^{3} x^{2} + 315 \, a b^{4} x\right )} \sqrt{\frac{a x + b}{x}}}{24 \,{\left (a^{8} x^{2} + 2 \, a^{7} b x + a^{6} b^{2}\right )}}\right ] \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [B] time = 12.3749, size = 532, normalized size = 3.86 \begin{align*} \frac{8 a^{\frac{133}{2}} b^{128} x^{72}}{24 a^{\frac{137}{2}} b^{\frac{257}{2}} x^{\frac{137}{2}} \sqrt{\frac{a x}{b} + 1} + 24 a^{\frac{135}{2}} b^{\frac{259}{2}} x^{\frac{135}{2}} \sqrt{\frac{a x}{b} + 1}} - \frac{18 a^{\frac{131}{2}} b^{129} x^{71}}{24 a^{\frac{137}{2}} b^{\frac{257}{2}} x^{\frac{137}{2}} \sqrt{\frac{a x}{b} + 1} + 24 a^{\frac{135}{2}} b^{\frac{259}{2}} x^{\frac{135}{2}} \sqrt{\frac{a x}{b} + 1}} + \frac{63 a^{\frac{129}{2}} b^{130} x^{70}}{24 a^{\frac{137}{2}} b^{\frac{257}{2}} x^{\frac{137}{2}} \sqrt{\frac{a x}{b} + 1} + 24 a^{\frac{135}{2}} b^{\frac{259}{2}} x^{\frac{135}{2}} \sqrt{\frac{a x}{b} + 1}} + \frac{420 a^{\frac{127}{2}} b^{131} x^{69}}{24 a^{\frac{137}{2}} b^{\frac{257}{2}} x^{\frac{137}{2}} \sqrt{\frac{a x}{b} + 1} + 24 a^{\frac{135}{2}} b^{\frac{259}{2}} x^{\frac{135}{2}} \sqrt{\frac{a x}{b} + 1}} + \frac{315 a^{\frac{125}{2}} b^{132} x^{68}}{24 a^{\frac{137}{2}} b^{\frac{257}{2}} x^{\frac{137}{2}} \sqrt{\frac{a x}{b} + 1} + 24 a^{\frac{135}{2}} b^{\frac{259}{2}} x^{\frac{135}{2}} \sqrt{\frac{a x}{b} + 1}} - \frac{315 a^{63} b^{\frac{263}{2}} x^{\frac{137}{2}} \sqrt{\frac{a x}{b} + 1} \operatorname{asinh}{\left (\frac{\sqrt{a} \sqrt{x}}{\sqrt{b}} \right )}}{24 a^{\frac{137}{2}} b^{\frac{257}{2}} x^{\frac{137}{2}} \sqrt{\frac{a x}{b} + 1} + 24 a^{\frac{135}{2}} b^{\frac{259}{2}} x^{\frac{135}{2}} \sqrt{\frac{a x}{b} + 1}} - \frac{315 a^{62} b^{\frac{265}{2}} x^{\frac{135}{2}} \sqrt{\frac{a x}{b} + 1} \operatorname{asinh}{\left (\frac{\sqrt{a} \sqrt{x}}{\sqrt{b}} \right )}}{24 a^{\frac{137}{2}} b^{\frac{257}{2}} x^{\frac{137}{2}} \sqrt{\frac{a x}{b} + 1} + 24 a^{\frac{135}{2}} b^{\frac{259}{2}} x^{\frac{135}{2}} \sqrt{\frac{a x}{b} + 1}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.26409, size = 203, normalized size = 1.47 \begin{align*} \frac{1}{24} \, b{\left (\frac{315 \, b^{2} \arctan \left (\frac{\sqrt{\frac{a x + b}{x}}}{\sqrt{-a}}\right )}{\sqrt{-a} a^{5}} + \frac{16 \, a^{4} b^{2} + \frac{144 \,{\left (a x + b\right )} a^{3} b^{2}}{x} - \frac{693 \,{\left (a x + b\right )}^{2} a^{2} b^{2}}{x^{2}} + \frac{840 \,{\left (a x + b\right )}^{3} a b^{2}}{x^{3}} - \frac{315 \,{\left (a x + b\right )}^{4} b^{2}}{x^{4}}}{{\left (a \sqrt{\frac{a x + b}{x}} - \frac{{\left (a x + b\right )} \sqrt{\frac{a x + b}{x}}}{x}\right )}^{3} a^{5}}\right )} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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