Optimal. Leaf size=92 \[ \frac{5}{8} a^2 \sqrt{x} \sqrt{a+b x}+\frac{5 a^3 \tanh ^{-1}\left (\frac{\sqrt{b} \sqrt{x}}{\sqrt{a+b x}}\right )}{8 \sqrt{b}}+\frac{5}{12} a \sqrt{x} (a+b x)^{3/2}+\frac{1}{3} \sqrt{x} (a+b x)^{5/2} \]
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Rubi [A] time = 0.0277851, antiderivative size = 92, 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 = {50, 63, 217, 206} \[ \frac{5}{8} a^2 \sqrt{x} \sqrt{a+b x}+\frac{5 a^3 \tanh ^{-1}\left (\frac{\sqrt{b} \sqrt{x}}{\sqrt{a+b x}}\right )}{8 \sqrt{b}}+\frac{5}{12} a \sqrt{x} (a+b x)^{3/2}+\frac{1}{3} \sqrt{x} (a+b x)^{5/2} \]
Antiderivative was successfully verified.
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Rule 50
Rule 63
Rule 217
Rule 206
Rubi steps
\begin{align*} \int \frac{(a+b x)^{5/2}}{\sqrt{x}} \, dx &=\frac{1}{3} \sqrt{x} (a+b x)^{5/2}+\frac{1}{6} (5 a) \int \frac{(a+b x)^{3/2}}{\sqrt{x}} \, dx\\ &=\frac{5}{12} a \sqrt{x} (a+b x)^{3/2}+\frac{1}{3} \sqrt{x} (a+b x)^{5/2}+\frac{1}{8} \left (5 a^2\right ) \int \frac{\sqrt{a+b x}}{\sqrt{x}} \, dx\\ &=\frac{5}{8} a^2 \sqrt{x} \sqrt{a+b x}+\frac{5}{12} a \sqrt{x} (a+b x)^{3/2}+\frac{1}{3} \sqrt{x} (a+b x)^{5/2}+\frac{1}{16} \left (5 a^3\right ) \int \frac{1}{\sqrt{x} \sqrt{a+b x}} \, dx\\ &=\frac{5}{8} a^2 \sqrt{x} \sqrt{a+b x}+\frac{5}{12} a \sqrt{x} (a+b x)^{3/2}+\frac{1}{3} \sqrt{x} (a+b x)^{5/2}+\frac{1}{8} \left (5 a^3\right ) \operatorname{Subst}\left (\int \frac{1}{\sqrt{a+b x^2}} \, dx,x,\sqrt{x}\right )\\ &=\frac{5}{8} a^2 \sqrt{x} \sqrt{a+b x}+\frac{5}{12} a \sqrt{x} (a+b x)^{3/2}+\frac{1}{3} \sqrt{x} (a+b x)^{5/2}+\frac{1}{8} \left (5 a^3\right ) \operatorname{Subst}\left (\int \frac{1}{1-b x^2} \, dx,x,\frac{\sqrt{x}}{\sqrt{a+b x}}\right )\\ &=\frac{5}{8} a^2 \sqrt{x} \sqrt{a+b x}+\frac{5}{12} a \sqrt{x} (a+b x)^{3/2}+\frac{1}{3} \sqrt{x} (a+b x)^{5/2}+\frac{5 a^3 \tanh ^{-1}\left (\frac{\sqrt{b} \sqrt{x}}{\sqrt{a+b x}}\right )}{8 \sqrt{b}}\\ \end{align*}
Mathematica [A] time = 0.110568, size = 80, normalized size = 0.87 \[ \frac{1}{24} \sqrt{a+b x} \left (\sqrt{x} \left (33 a^2+26 a b x+8 b^2 x^2\right )+\frac{15 a^{5/2} \sinh ^{-1}\left (\frac{\sqrt{b} \sqrt{x}}{\sqrt{a}}\right )}{\sqrt{b} \sqrt{\frac{b x}{a}+1}}\right ) \]
Antiderivative was successfully verified.
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Maple [A] time = 0.004, size = 93, normalized size = 1. \begin{align*}{\frac{1}{3} \left ( bx+a \right ) ^{{\frac{5}{2}}}\sqrt{x}}+{\frac{5\,a}{12} \left ( bx+a \right ) ^{{\frac{3}{2}}}\sqrt{x}}+{\frac{5\,{a}^{2}}{8}\sqrt{x}\sqrt{bx+a}}+{\frac{5\,{a}^{3}}{16}\sqrt{x \left ( bx+a \right ) }\ln \left ({ \left ({\frac{a}{2}}+bx \right ){\frac{1}{\sqrt{b}}}}+\sqrt{b{x}^{2}+ax} \right ){\frac{1}{\sqrt{x}}}{\frac{1}{\sqrt{bx+a}}}{\frac{1}{\sqrt{b}}}} \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 = 1.85233, size = 365, normalized size = 3.97 \begin{align*} \left [\frac{15 \, a^{3} \sqrt{b} \log \left (2 \, b x + 2 \, \sqrt{b x + a} \sqrt{b} \sqrt{x} + a\right ) + 2 \,{\left (8 \, b^{3} x^{2} + 26 \, a b^{2} x + 33 \, a^{2} b\right )} \sqrt{b x + a} \sqrt{x}}{48 \, b}, -\frac{15 \, a^{3} \sqrt{-b} \arctan \left (\frac{\sqrt{b x + a} \sqrt{-b}}{b \sqrt{x}}\right ) -{\left (8 \, b^{3} x^{2} + 26 \, a b^{2} x + 33 \, a^{2} b\right )} \sqrt{b x + a} \sqrt{x}}{24 \, b}\right ] \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 11.7799, size = 102, normalized size = 1.11 \begin{align*} \frac{11 a^{\frac{5}{2}} \sqrt{x} \sqrt{1 + \frac{b x}{a}}}{8} + \frac{13 a^{\frac{3}{2}} b x^{\frac{3}{2}} \sqrt{1 + \frac{b x}{a}}}{12} + \frac{\sqrt{a} b^{2} x^{\frac{5}{2}} \sqrt{1 + \frac{b x}{a}}}{3} + \frac{5 a^{3} \operatorname{asinh}{\left (\frac{\sqrt{b} \sqrt{x}}{\sqrt{a}} \right )}}{8 \sqrt{b}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [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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