Optimal. Leaf size=98 \[ -\frac{a^2 \sqrt{x} \sqrt{a+b x}}{8 b^2}+\frac{a^3 \tanh ^{-1}\left (\frac{\sqrt{b} \sqrt{x}}{\sqrt{a+b x}}\right )}{8 b^{5/2}}+\frac{a x^{3/2} \sqrt{a+b x}}{12 b}+\frac{1}{3} x^{5/2} \sqrt{a+b x} \]
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Rubi [A] time = 0.0307894, antiderivative size = 98, 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} \[ -\frac{a^2 \sqrt{x} \sqrt{a+b x}}{8 b^2}+\frac{a^3 \tanh ^{-1}\left (\frac{\sqrt{b} \sqrt{x}}{\sqrt{a+b x}}\right )}{8 b^{5/2}}+\frac{a x^{3/2} \sqrt{a+b x}}{12 b}+\frac{1}{3} x^{5/2} \sqrt{a+b x} \]
Antiderivative was successfully verified.
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Rule 50
Rule 63
Rule 217
Rule 206
Rubi steps
\begin{align*} \int x^{3/2} \sqrt{a+b x} \, dx &=\frac{1}{3} x^{5/2} \sqrt{a+b x}+\frac{1}{6} a \int \frac{x^{3/2}}{\sqrt{a+b x}} \, dx\\ &=\frac{a x^{3/2} \sqrt{a+b x}}{12 b}+\frac{1}{3} x^{5/2} \sqrt{a+b x}-\frac{a^2 \int \frac{\sqrt{x}}{\sqrt{a+b x}} \, dx}{8 b}\\ &=-\frac{a^2 \sqrt{x} \sqrt{a+b x}}{8 b^2}+\frac{a x^{3/2} \sqrt{a+b x}}{12 b}+\frac{1}{3} x^{5/2} \sqrt{a+b x}+\frac{a^3 \int \frac{1}{\sqrt{x} \sqrt{a+b x}} \, dx}{16 b^2}\\ &=-\frac{a^2 \sqrt{x} \sqrt{a+b x}}{8 b^2}+\frac{a x^{3/2} \sqrt{a+b x}}{12 b}+\frac{1}{3} x^{5/2} \sqrt{a+b x}+\frac{a^3 \operatorname{Subst}\left (\int \frac{1}{\sqrt{a+b x^2}} \, dx,x,\sqrt{x}\right )}{8 b^2}\\ &=-\frac{a^2 \sqrt{x} \sqrt{a+b x}}{8 b^2}+\frac{a x^{3/2} \sqrt{a+b x}}{12 b}+\frac{1}{3} x^{5/2} \sqrt{a+b x}+\frac{a^3 \operatorname{Subst}\left (\int \frac{1}{1-b x^2} \, dx,x,\frac{\sqrt{x}}{\sqrt{a+b x}}\right )}{8 b^2}\\ &=-\frac{a^2 \sqrt{x} \sqrt{a+b x}}{8 b^2}+\frac{a x^{3/2} \sqrt{a+b x}}{12 b}+\frac{1}{3} x^{5/2} \sqrt{a+b x}+\frac{a^3 \tanh ^{-1}\left (\frac{\sqrt{b} \sqrt{x}}{\sqrt{a+b x}}\right )}{8 b^{5/2}}\\ \end{align*}
Mathematica [A] time = 0.105517, size = 85, normalized size = 0.87 \[ \frac{\sqrt{a+b x} \left (\sqrt{b} \sqrt{x} \left (-3 a^2+2 a b x+8 b^2 x^2\right )+\frac{3 a^{5/2} \sinh ^{-1}\left (\frac{\sqrt{b} \sqrt{x}}{\sqrt{a}}\right )}{\sqrt{\frac{b x}{a}+1}}\right )}{24 b^{5/2}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.003, size = 102, normalized size = 1. \begin{align*}{\frac{1}{3\,b}{x}^{{\frac{3}{2}}} \left ( bx+a \right ) ^{{\frac{3}{2}}}}-{\frac{a}{4\,{b}^{2}} \left ( bx+a \right ) ^{{\frac{3}{2}}}\sqrt{x}}+{\frac{{a}^{2}}{8\,{b}^{2}}\sqrt{x}\sqrt{bx+a}}+{\frac{{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{5}{2}}}{\frac{1}{\sqrt{bx+a}}}{\frac{1}{\sqrt{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 = 1.69255, size = 362, normalized size = 3.69 \begin{align*} \left [\frac{3 \, 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} + 2 \, a b^{2} x - 3 \, a^{2} b\right )} \sqrt{b x + a} \sqrt{x}}{48 \, b^{3}}, -\frac{3 \, a^{3} \sqrt{-b} \arctan \left (\frac{\sqrt{b x + a} \sqrt{-b}}{b \sqrt{x}}\right ) -{\left (8 \, b^{3} x^{2} + 2 \, a b^{2} x - 3 \, a^{2} b\right )} \sqrt{b x + a} \sqrt{x}}{24 \, b^{3}}\right ] \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 7.25147, size = 122, normalized size = 1.24 \begin{align*} - \frac{a^{\frac{5}{2}} \sqrt{x}}{8 b^{2} \sqrt{1 + \frac{b x}{a}}} - \frac{a^{\frac{3}{2}} x^{\frac{3}{2}}}{24 b \sqrt{1 + \frac{b x}{a}}} + \frac{5 \sqrt{a} x^{\frac{5}{2}}}{12 \sqrt{1 + \frac{b x}{a}}} + \frac{a^{3} \operatorname{asinh}{\left (\frac{\sqrt{b} \sqrt{x}}{\sqrt{a}} \right )}}{8 b^{\frac{5}{2}}} + \frac{b 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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