Optimal. Leaf size=114 \[ -\frac{35 b^2}{4 a^4 \sqrt{a+\frac{b}{x}}}-\frac{35 b^2}{12 a^3 \left (a+\frac{b}{x}\right )^{3/2}}+\frac{35 b^2 \tanh ^{-1}\left (\frac{\sqrt{a+\frac{b}{x}}}{\sqrt{a}}\right )}{4 a^{9/2}}-\frac{7 b x}{4 a^2 \left (a+\frac{b}{x}\right )^{3/2}}+\frac{x^2}{2 a \left (a+\frac{b}{x}\right )^{3/2}} \]
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Rubi [A] time = 0.0502586, antiderivative size = 114, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 4, integrand size = 13, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.308, Rules used = {266, 51, 63, 208} \[ \frac{35 b^2 \tanh ^{-1}\left (\frac{\sqrt{a+\frac{b}{x}}}{\sqrt{a}}\right )}{4 a^{9/2}}+\frac{35 x^2 \sqrt{a+\frac{b}{x}}}{6 a^3}-\frac{14 x^2}{3 a^2 \sqrt{a+\frac{b}{x}}}-\frac{35 b x \sqrt{a+\frac{b}{x}}}{4 a^4}-\frac{2 x^2}{3 a \left (a+\frac{b}{x}\right )^{3/2}} \]
Antiderivative was successfully verified.
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Rule 266
Rule 51
Rule 63
Rule 208
Rubi steps
\begin{align*} \int \frac{x}{\left (a+\frac{b}{x}\right )^{5/2}} \, dx &=-\operatorname{Subst}\left (\int \frac{1}{x^3 (a+b x)^{5/2}} \, dx,x,\frac{1}{x}\right )\\ &=-\frac{2 x^2}{3 a \left (a+\frac{b}{x}\right )^{3/2}}-\frac{7 \operatorname{Subst}\left (\int \frac{1}{x^3 (a+b x)^{3/2}} \, dx,x,\frac{1}{x}\right )}{3 a}\\ &=-\frac{2 x^2}{3 a \left (a+\frac{b}{x}\right )^{3/2}}-\frac{14 x^2}{3 a^2 \sqrt{a+\frac{b}{x}}}-\frac{35 \operatorname{Subst}\left (\int \frac{1}{x^3 \sqrt{a+b x}} \, dx,x,\frac{1}{x}\right )}{3 a^2}\\ &=-\frac{2 x^2}{3 a \left (a+\frac{b}{x}\right )^{3/2}}-\frac{14 x^2}{3 a^2 \sqrt{a+\frac{b}{x}}}+\frac{35 \sqrt{a+\frac{b}{x}} x^2}{6 a^3}+\frac{(35 b) \operatorname{Subst}\left (\int \frac{1}{x^2 \sqrt{a+b x}} \, dx,x,\frac{1}{x}\right )}{4 a^3}\\ &=-\frac{35 b \sqrt{a+\frac{b}{x}} x}{4 a^4}-\frac{2 x^2}{3 a \left (a+\frac{b}{x}\right )^{3/2}}-\frac{14 x^2}{3 a^2 \sqrt{a+\frac{b}{x}}}+\frac{35 \sqrt{a+\frac{b}{x}} x^2}{6 a^3}-\frac{\left (35 b^2\right ) \operatorname{Subst}\left (\int \frac{1}{x \sqrt{a+b x}} \, dx,x,\frac{1}{x}\right )}{8 a^4}\\ &=-\frac{35 b \sqrt{a+\frac{b}{x}} x}{4 a^4}-\frac{2 x^2}{3 a \left (a+\frac{b}{x}\right )^{3/2}}-\frac{14 x^2}{3 a^2 \sqrt{a+\frac{b}{x}}}+\frac{35 \sqrt{a+\frac{b}{x}} x^2}{6 a^3}-\frac{(35 b) \operatorname{Subst}\left (\int \frac{1}{-\frac{a}{b}+\frac{x^2}{b}} \, dx,x,\sqrt{a+\frac{b}{x}}\right )}{4 a^4}\\ &=-\frac{35 b \sqrt{a+\frac{b}{x}} x}{4 a^4}-\frac{2 x^2}{3 a \left (a+\frac{b}{x}\right )^{3/2}}-\frac{14 x^2}{3 a^2 \sqrt{a+\frac{b}{x}}}+\frac{35 \sqrt{a+\frac{b}{x}} x^2}{6 a^3}+\frac{35 b^2 \tanh ^{-1}\left (\frac{\sqrt{a+\frac{b}{x}}}{\sqrt{a}}\right )}{4 a^{9/2}}\\ \end{align*}
Mathematica [C] time = 0.0106617, size = 39, normalized size = 0.34 \[ -\frac{2 b^2 \, _2F_1\left (-\frac{3}{2},3;-\frac{1}{2};\frac{b}{a x}+1\right )}{3 a^3 \left (a+\frac{b}{x}\right )^{3/2}} \]
Antiderivative was successfully verified.
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Maple [B] time = 0.009, size = 531, normalized size = 4.7 \begin{align*}{\frac{x}{24\, \left ( ax+b \right ) ^{3}}\sqrt{{\frac{ax+b}{x}}} \left ( 12\,\sqrt{a{x}^{2}+bx}{a}^{11/2}{x}^{4}-216\,\sqrt{ \left ( ax+b \right ) x}{a}^{9/2}{x}^{3}b+42\,\sqrt{a{x}^{2}+bx}{a}^{9/2}{x}^{3}b+108\,{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}^{2}+144\, \left ( \left ( ax+b \right ) x \right ) ^{3/2}{a}^{7/2}xb-648\,\sqrt{ \left ( ax+b \right ) x}{a}^{7/2}{x}^{2}{b}^{2}+54\,\sqrt{a{x}^{2}+bx}{a}^{7/2}{x}^{2}{b}^{2}+324\,{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}^{3}-3\,\ln \left ( 1/2\,{\frac{2\,\sqrt{a{x}^{2}+bx}\sqrt{a}+2\,ax+b}{\sqrt{a}}} \right ){x}^{3}{a}^{4}{b}^{2}+128\, \left ( \left ( ax+b \right ) x \right ) ^{3/2}{a}^{5/2}{b}^{2}-648\,\sqrt{ \left ( ax+b \right ) x}{a}^{5/2}x{b}^{3}+30\,\sqrt{a{x}^{2}+bx}{a}^{5/2}x{b}^{3}+324\,{a}^{2}\ln \left ( 1/2\,{\frac{2\,\sqrt{ \left ( ax+b \right ) x}\sqrt{a}+2\,ax+b}{\sqrt{a}}} \right ) x{b}^{4}-9\,\ln \left ( 1/2\,{\frac{2\,\sqrt{a{x}^{2}+bx}\sqrt{a}+2\,ax+b}{\sqrt{a}}} \right ){x}^{2}{a}^{3}{b}^{3}-216\,\sqrt{ \left ( ax+b \right ) x}{a}^{3/2}{b}^{4}+6\,\sqrt{a{x}^{2}+bx}{a}^{3/2}{b}^{4}+108\,a\ln \left ( 1/2\,{\frac{2\,\sqrt{ \left ( ax+b \right ) x}\sqrt{a}+2\,ax+b}{\sqrt{a}}} \right ){b}^{5}-9\,\ln \left ( 1/2\,{\frac{2\,\sqrt{a{x}^{2}+bx}\sqrt{a}+2\,ax+b}{\sqrt{a}}} \right ) x{a}^{2}{b}^{4}-3\,\ln \left ( 1/2\,{\frac{2\,\sqrt{a{x}^{2}+bx}\sqrt{a}+2\,ax+b}{\sqrt{a}}} \right ) a{b}^{5} \right ){a}^{-{\frac{11}{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 = 1.76382, size = 564, normalized size = 4.95 \begin{align*} \left [\frac{105 \,{\left (a^{2} b^{2} x^{2} + 2 \, a b^{3} x + b^{4}\right )} \sqrt{a} \log \left (2 \, a x + 2 \, \sqrt{a} x \sqrt{\frac{a x + b}{x}} + b\right ) + 2 \,{\left (6 \, a^{4} x^{4} - 21 \, a^{3} b x^{3} - 140 \, a^{2} b^{2} x^{2} - 105 \, a b^{3} x\right )} \sqrt{\frac{a x + b}{x}}}{24 \,{\left (a^{7} x^{2} + 2 \, a^{6} b x + a^{5} b^{2}\right )}}, -\frac{105 \,{\left (a^{2} b^{2} x^{2} + 2 \, a b^{3} x + b^{4}\right )} \sqrt{-a} \arctan \left (\frac{\sqrt{-a} \sqrt{\frac{a x + b}{x}}}{a}\right ) -{\left (6 \, a^{4} x^{4} - 21 \, a^{3} b x^{3} - 140 \, a^{2} b^{2} x^{2} - 105 \, a b^{3} x\right )} \sqrt{\frac{a x + b}{x}}}{12 \,{\left (a^{7} x^{2} + 2 \, a^{6} b x + a^{5} b^{2}\right )}}\right ] \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [B] time = 7.36674, size = 464, normalized size = 4.07 \begin{align*} \frac{6 a^{\frac{89}{2}} b^{75} x^{49}}{12 a^{\frac{93}{2}} b^{\frac{151}{2}} x^{\frac{93}{2}} \sqrt{\frac{a x}{b} + 1} + 12 a^{\frac{91}{2}} b^{\frac{153}{2}} x^{\frac{91}{2}} \sqrt{\frac{a x}{b} + 1}} - \frac{21 a^{\frac{87}{2}} b^{76} x^{48}}{12 a^{\frac{93}{2}} b^{\frac{151}{2}} x^{\frac{93}{2}} \sqrt{\frac{a x}{b} + 1} + 12 a^{\frac{91}{2}} b^{\frac{153}{2}} x^{\frac{91}{2}} \sqrt{\frac{a x}{b} + 1}} - \frac{140 a^{\frac{85}{2}} b^{77} x^{47}}{12 a^{\frac{93}{2}} b^{\frac{151}{2}} x^{\frac{93}{2}} \sqrt{\frac{a x}{b} + 1} + 12 a^{\frac{91}{2}} b^{\frac{153}{2}} x^{\frac{91}{2}} \sqrt{\frac{a x}{b} + 1}} - \frac{105 a^{\frac{83}{2}} b^{78} x^{46}}{12 a^{\frac{93}{2}} b^{\frac{151}{2}} x^{\frac{93}{2}} \sqrt{\frac{a x}{b} + 1} + 12 a^{\frac{91}{2}} b^{\frac{153}{2}} x^{\frac{91}{2}} \sqrt{\frac{a x}{b} + 1}} + \frac{105 a^{42} b^{\frac{155}{2}} x^{\frac{93}{2}} \sqrt{\frac{a x}{b} + 1} \operatorname{asinh}{\left (\frac{\sqrt{a} \sqrt{x}}{\sqrt{b}} \right )}}{12 a^{\frac{93}{2}} b^{\frac{151}{2}} x^{\frac{93}{2}} \sqrt{\frac{a x}{b} + 1} + 12 a^{\frac{91}{2}} b^{\frac{153}{2}} x^{\frac{91}{2}} \sqrt{\frac{a x}{b} + 1}} + \frac{105 a^{41} b^{\frac{157}{2}} x^{\frac{91}{2}} \sqrt{\frac{a x}{b} + 1} \operatorname{asinh}{\left (\frac{\sqrt{a} \sqrt{x}}{\sqrt{b}} \right )}}{12 a^{\frac{93}{2}} b^{\frac{151}{2}} x^{\frac{93}{2}} \sqrt{\frac{a x}{b} + 1} + 12 a^{\frac{91}{2}} b^{\frac{153}{2}} x^{\frac{91}{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 = 1.27404, size = 169, normalized size = 1.48 \begin{align*} -\frac{1}{12} \, b^{2}{\left (\frac{8 \,{\left (a + \frac{9 \,{\left (a x + b\right )}}{x}\right )} x}{{\left (a x + b\right )} a^{4} \sqrt{\frac{a x + b}{x}}} + \frac{105 \, \arctan \left (\frac{\sqrt{\frac{a x + b}{x}}}{\sqrt{-a}}\right )}{\sqrt{-a} a^{4}} - \frac{3 \,{\left (13 \, a \sqrt{\frac{a x + b}{x}} - \frac{11 \,{\left (a x + b\right )} \sqrt{\frac{a x + b}{x}}}{x}\right )}}{{\left (a - \frac{a x + b}{x}\right )}^{2} a^{4}}\right )} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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