Optimal. Leaf size=101 \[ \frac{5 a^2 \sqrt{x} \sqrt{a+b x}}{8 b^3}-\frac{5 a^3 \tanh ^{-1}\left (\frac{\sqrt{b} \sqrt{x}}{\sqrt{a+b x}}\right )}{8 b^{7/2}}-\frac{5 a x^{3/2} \sqrt{a+b x}}{12 b^2}+\frac{x^{5/2} \sqrt{a+b x}}{3 b} \]
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Rubi [A] time = 0.0304051, antiderivative size = 101, 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} \[ \frac{5 a^2 \sqrt{x} \sqrt{a+b x}}{8 b^3}-\frac{5 a^3 \tanh ^{-1}\left (\frac{\sqrt{b} \sqrt{x}}{\sqrt{a+b x}}\right )}{8 b^{7/2}}-\frac{5 a x^{3/2} \sqrt{a+b x}}{12 b^2}+\frac{x^{5/2} \sqrt{a+b x}}{3 b} \]
Antiderivative was successfully verified.
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Rule 50
Rule 63
Rule 217
Rule 206
Rubi steps
\begin{align*} \int \frac{x^{5/2}}{\sqrt{a+b x}} \, dx &=\frac{x^{5/2} \sqrt{a+b x}}{3 b}-\frac{(5 a) \int \frac{x^{3/2}}{\sqrt{a+b x}} \, dx}{6 b}\\ &=-\frac{5 a x^{3/2} \sqrt{a+b x}}{12 b^2}+\frac{x^{5/2} \sqrt{a+b x}}{3 b}+\frac{\left (5 a^2\right ) \int \frac{\sqrt{x}}{\sqrt{a+b x}} \, dx}{8 b^2}\\ &=\frac{5 a^2 \sqrt{x} \sqrt{a+b x}}{8 b^3}-\frac{5 a x^{3/2} \sqrt{a+b x}}{12 b^2}+\frac{x^{5/2} \sqrt{a+b x}}{3 b}-\frac{\left (5 a^3\right ) \int \frac{1}{\sqrt{x} \sqrt{a+b x}} \, dx}{16 b^3}\\ &=\frac{5 a^2 \sqrt{x} \sqrt{a+b x}}{8 b^3}-\frac{5 a x^{3/2} \sqrt{a+b x}}{12 b^2}+\frac{x^{5/2} \sqrt{a+b x}}{3 b}-\frac{\left (5 a^3\right ) \operatorname{Subst}\left (\int \frac{1}{\sqrt{a+b x^2}} \, dx,x,\sqrt{x}\right )}{8 b^3}\\ &=\frac{5 a^2 \sqrt{x} \sqrt{a+b x}}{8 b^3}-\frac{5 a x^{3/2} \sqrt{a+b x}}{12 b^2}+\frac{x^{5/2} \sqrt{a+b x}}{3 b}-\frac{\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 )}{8 b^3}\\ &=\frac{5 a^2 \sqrt{x} \sqrt{a+b x}}{8 b^3}-\frac{5 a x^{3/2} \sqrt{a+b x}}{12 b^2}+\frac{x^{5/2} \sqrt{a+b x}}{3 b}-\frac{5 a^3 \tanh ^{-1}\left (\frac{\sqrt{b} \sqrt{x}}{\sqrt{a+b x}}\right )}{8 b^{7/2}}\\ \end{align*}
Mathematica [A] time = 0.170838, size = 85, normalized size = 0.84 \[ \frac{\sqrt{a+b x} \left (\sqrt{b} \sqrt{x} \left (15 a^2-10 a b x+8 b^2 x^2\right )-\frac{15 a^{5/2} \sinh ^{-1}\left (\frac{\sqrt{b} \sqrt{x}}{\sqrt{a}}\right )}{\sqrt{\frac{b x}{a}+1}}\right )}{24 b^{7/2}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.004, size = 102, normalized size = 1. \begin{align*}{\frac{1}{3\,b}{x}^{{\frac{5}{2}}}\sqrt{bx+a}}-{\frac{5\,a}{12\,{b}^{2}}{x}^{{\frac{3}{2}}}\sqrt{bx+a}}+{\frac{5\,{a}^{2}}{8\,{b}^{3}}\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 ){b}^{-{\frac{7}{2}}}{\frac{1}{\sqrt{x}}}{\frac{1}{\sqrt{bx+a}}}} \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 = 1.94789, size = 369, normalized size = 3.65 \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} - 10 \, a b^{2} x + 15 \, a^{2} b\right )} \sqrt{b x + a} \sqrt{x}}{48 \, b^{4}}, \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} - 10 \, a b^{2} x + 15 \, a^{2} b\right )} \sqrt{b x + a} \sqrt{x}}{24 \, b^{4}}\right ] \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 13.0539, size = 128, normalized size = 1.27 \begin{align*} \frac{5 a^{\frac{5}{2}} \sqrt{x}}{8 b^{3} \sqrt{1 + \frac{b x}{a}}} + \frac{5 a^{\frac{3}{2}} x^{\frac{3}{2}}}{24 b^{2} \sqrt{1 + \frac{b x}{a}}} - \frac{\sqrt{a} x^{\frac{5}{2}}}{12 b \sqrt{1 + \frac{b x}{a}}} - \frac{5 a^{3} \operatorname{asinh}{\left (\frac{\sqrt{b} \sqrt{x}}{\sqrt{a}} \right )}}{8 b^{\frac{7}{2}}} + \frac{x^{\frac{7}{2}}}{3 \sqrt{a} \sqrt{1 + \frac{b x}{a}}} \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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