Optimal. Leaf size=114 \[ -\frac{3 b^3 x \sqrt{a+\frac{b}{x}}}{64 a^2}+\frac{3 b^4 \tanh ^{-1}\left (\frac{\sqrt{a+\frac{b}{x}}}{\sqrt{a}}\right )}{64 a^{5/2}}+\frac{b^2 x^2 \sqrt{a+\frac{b}{x}}}{32 a}+\frac{1}{8} b x^3 \sqrt{a+\frac{b}{x}}+\frac{1}{4} x^4 \left (a+\frac{b}{x}\right )^{3/2} \]
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Rubi [A] time = 0.0518756, antiderivative size = 114, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 5, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.333, Rules used = {266, 47, 51, 63, 208} \[ -\frac{3 b^3 x \sqrt{a+\frac{b}{x}}}{64 a^2}+\frac{3 b^4 \tanh ^{-1}\left (\frac{\sqrt{a+\frac{b}{x}}}{\sqrt{a}}\right )}{64 a^{5/2}}+\frac{b^2 x^2 \sqrt{a+\frac{b}{x}}}{32 a}+\frac{1}{8} b x^3 \sqrt{a+\frac{b}{x}}+\frac{1}{4} x^4 \left (a+\frac{b}{x}\right )^{3/2} \]
Antiderivative was successfully verified.
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Rule 266
Rule 47
Rule 51
Rule 63
Rule 208
Rubi steps
\begin{align*} \int \left (a+\frac{b}{x}\right )^{3/2} x^3 \, dx &=-\operatorname{Subst}\left (\int \frac{(a+b x)^{3/2}}{x^5} \, dx,x,\frac{1}{x}\right )\\ &=\frac{1}{4} \left (a+\frac{b}{x}\right )^{3/2} x^4-\frac{1}{8} (3 b) \operatorname{Subst}\left (\int \frac{\sqrt{a+b x}}{x^4} \, dx,x,\frac{1}{x}\right )\\ &=\frac{1}{8} b \sqrt{a+\frac{b}{x}} x^3+\frac{1}{4} \left (a+\frac{b}{x}\right )^{3/2} x^4-\frac{1}{16} b^2 \operatorname{Subst}\left (\int \frac{1}{x^3 \sqrt{a+b x}} \, dx,x,\frac{1}{x}\right )\\ &=\frac{b^2 \sqrt{a+\frac{b}{x}} x^2}{32 a}+\frac{1}{8} b \sqrt{a+\frac{b}{x}} x^3+\frac{1}{4} \left (a+\frac{b}{x}\right )^{3/2} x^4+\frac{\left (3 b^3\right ) \operatorname{Subst}\left (\int \frac{1}{x^2 \sqrt{a+b x}} \, dx,x,\frac{1}{x}\right )}{64 a}\\ &=-\frac{3 b^3 \sqrt{a+\frac{b}{x}} x}{64 a^2}+\frac{b^2 \sqrt{a+\frac{b}{x}} x^2}{32 a}+\frac{1}{8} b \sqrt{a+\frac{b}{x}} x^3+\frac{1}{4} \left (a+\frac{b}{x}\right )^{3/2} x^4-\frac{\left (3 b^4\right ) \operatorname{Subst}\left (\int \frac{1}{x \sqrt{a+b x}} \, dx,x,\frac{1}{x}\right )}{128 a^2}\\ &=-\frac{3 b^3 \sqrt{a+\frac{b}{x}} x}{64 a^2}+\frac{b^2 \sqrt{a+\frac{b}{x}} x^2}{32 a}+\frac{1}{8} b \sqrt{a+\frac{b}{x}} x^3+\frac{1}{4} \left (a+\frac{b}{x}\right )^{3/2} x^4-\frac{\left (3 b^3\right ) \operatorname{Subst}\left (\int \frac{1}{-\frac{a}{b}+\frac{x^2}{b}} \, dx,x,\sqrt{a+\frac{b}{x}}\right )}{64 a^2}\\ &=-\frac{3 b^3 \sqrt{a+\frac{b}{x}} x}{64 a^2}+\frac{b^2 \sqrt{a+\frac{b}{x}} x^2}{32 a}+\frac{1}{8} b \sqrt{a+\frac{b}{x}} x^3+\frac{1}{4} \left (a+\frac{b}{x}\right )^{3/2} x^4+\frac{3 b^4 \tanh ^{-1}\left (\frac{\sqrt{a+\frac{b}{x}}}{\sqrt{a}}\right )}{64 a^{5/2}}\\ \end{align*}
Mathematica [C] time = 0.0156935, size = 39, normalized size = 0.34 \[ \frac{2 b^4 \left (a+\frac{b}{x}\right )^{5/2} \, _2F_1\left (\frac{5}{2},5;\frac{7}{2};\frac{b}{a x}+1\right )}{5 a^5} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.004, size = 135, normalized size = 1.2 \begin{align*}{\frac{x}{128}\sqrt{{\frac{ax+b}{x}}} \left ( 32\,x \left ( a{x}^{2}+bx \right ) ^{3/2}{a}^{7/2}+16\,{a}^{5/2} \left ( a{x}^{2}+bx \right ) ^{3/2}b-12\,{a}^{5/2}\sqrt{a{x}^{2}+bx}x{b}^{2}-6\,{a}^{3/2}\sqrt{a{x}^{2}+bx}{b}^{3}+3\,\ln \left ( 1/2\,{\frac{2\,\sqrt{a{x}^{2}+bx}\sqrt{a}+2\,ax+b}{\sqrt{a}}} \right ) a{b}^{4} \right ){a}^{-{\frac{7}{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 = 1.77754, size = 401, normalized size = 3.52 \begin{align*} \left [\frac{3 \, \sqrt{a} b^{4} \log \left (2 \, a x + 2 \, \sqrt{a} x \sqrt{\frac{a x + b}{x}} + b\right ) + 2 \,{\left (16 \, a^{4} x^{4} + 24 \, a^{3} b x^{3} + 2 \, a^{2} b^{2} x^{2} - 3 \, a b^{3} x\right )} \sqrt{\frac{a x + b}{x}}}{128 \, a^{3}}, -\frac{3 \, \sqrt{-a} b^{4} \arctan \left (\frac{\sqrt{-a} \sqrt{\frac{a x + b}{x}}}{a}\right ) -{\left (16 \, a^{4} x^{4} + 24 \, a^{3} b x^{3} + 2 \, a^{2} b^{2} x^{2} - 3 \, a b^{3} x\right )} \sqrt{\frac{a x + b}{x}}}{64 \, a^{3}}\right ] \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 7.98564, size = 153, normalized size = 1.34 \begin{align*} \frac{a^{2} x^{\frac{9}{2}}}{4 \sqrt{b} \sqrt{\frac{a x}{b} + 1}} + \frac{5 a \sqrt{b} x^{\frac{7}{2}}}{8 \sqrt{\frac{a x}{b} + 1}} + \frac{13 b^{\frac{3}{2}} x^{\frac{5}{2}}}{32 \sqrt{\frac{a x}{b} + 1}} - \frac{b^{\frac{5}{2}} x^{\frac{3}{2}}}{64 a \sqrt{\frac{a x}{b} + 1}} - \frac{3 b^{\frac{7}{2}} \sqrt{x}}{64 a^{2} \sqrt{\frac{a x}{b} + 1}} + \frac{3 b^{4} \operatorname{asinh}{\left (\frac{\sqrt{a} \sqrt{x}}{\sqrt{b}} \right )}}{64 a^{\frac{5}{2}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.14136, size = 143, normalized size = 1.25 \begin{align*} -\frac{3 \, b^{4} \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 )}{128 \, a^{\frac{5}{2}}} + \frac{3 \, b^{4} \log \left ({\left | b \right |}\right ) \mathrm{sgn}\left (x\right )}{128 \, a^{\frac{5}{2}}} + \frac{1}{64} \, \sqrt{a x^{2} + b x}{\left (2 \,{\left (4 \,{\left (2 \, a x \mathrm{sgn}\left (x\right ) + 3 \, b \mathrm{sgn}\left (x\right )\right )} x + \frac{b^{2} \mathrm{sgn}\left (x\right )}{a}\right )} x - \frac{3 \, b^{3} \mathrm{sgn}\left (x\right )}{a^{2}}\right )} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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