Optimal. Leaf size=140 \[ -\frac{3 a^4 \sqrt{x} \sqrt{a+b x}}{128 b^2}+\frac{3 a^5 \tanh ^{-1}\left (\frac{\sqrt{b} \sqrt{x}}{\sqrt{a+b x}}\right )}{128 b^{5/2}}+\frac{a^3 x^{3/2} \sqrt{a+b x}}{64 b}+\frac{1}{16} a^2 x^{5/2} \sqrt{a+b x}+\frac{1}{8} a x^{5/2} (a+b x)^{3/2}+\frac{1}{5} x^{5/2} (a+b x)^{5/2} \]
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Rubi [A] time = 0.0476703, antiderivative size = 140, normalized size of antiderivative = 1., number of steps used = 8, number of rules used = 4, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.267, Rules used = {50, 63, 217, 206} \[ -\frac{3 a^4 \sqrt{x} \sqrt{a+b x}}{128 b^2}+\frac{3 a^5 \tanh ^{-1}\left (\frac{\sqrt{b} \sqrt{x}}{\sqrt{a+b x}}\right )}{128 b^{5/2}}+\frac{a^3 x^{3/2} \sqrt{a+b x}}{64 b}+\frac{1}{16} a^2 x^{5/2} \sqrt{a+b x}+\frac{1}{8} a x^{5/2} (a+b x)^{3/2}+\frac{1}{5} x^{5/2} (a+b x)^{5/2} \]
Antiderivative was successfully verified.
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Rule 50
Rule 63
Rule 217
Rule 206
Rubi steps
\begin{align*} \int x^{3/2} (a+b x)^{5/2} \, dx &=\frac{1}{5} x^{5/2} (a+b x)^{5/2}+\frac{1}{2} a \int x^{3/2} (a+b x)^{3/2} \, dx\\ &=\frac{1}{8} a x^{5/2} (a+b x)^{3/2}+\frac{1}{5} x^{5/2} (a+b x)^{5/2}+\frac{1}{16} \left (3 a^2\right ) \int x^{3/2} \sqrt{a+b x} \, dx\\ &=\frac{1}{16} a^2 x^{5/2} \sqrt{a+b x}+\frac{1}{8} a x^{5/2} (a+b x)^{3/2}+\frac{1}{5} x^{5/2} (a+b x)^{5/2}+\frac{1}{32} a^3 \int \frac{x^{3/2}}{\sqrt{a+b x}} \, dx\\ &=\frac{a^3 x^{3/2} \sqrt{a+b x}}{64 b}+\frac{1}{16} a^2 x^{5/2} \sqrt{a+b x}+\frac{1}{8} a x^{5/2} (a+b x)^{3/2}+\frac{1}{5} x^{5/2} (a+b x)^{5/2}-\frac{\left (3 a^4\right ) \int \frac{\sqrt{x}}{\sqrt{a+b x}} \, dx}{128 b}\\ &=-\frac{3 a^4 \sqrt{x} \sqrt{a+b x}}{128 b^2}+\frac{a^3 x^{3/2} \sqrt{a+b x}}{64 b}+\frac{1}{16} a^2 x^{5/2} \sqrt{a+b x}+\frac{1}{8} a x^{5/2} (a+b x)^{3/2}+\frac{1}{5} x^{5/2} (a+b x)^{5/2}+\frac{\left (3 a^5\right ) \int \frac{1}{\sqrt{x} \sqrt{a+b x}} \, dx}{256 b^2}\\ &=-\frac{3 a^4 \sqrt{x} \sqrt{a+b x}}{128 b^2}+\frac{a^3 x^{3/2} \sqrt{a+b x}}{64 b}+\frac{1}{16} a^2 x^{5/2} \sqrt{a+b x}+\frac{1}{8} a x^{5/2} (a+b x)^{3/2}+\frac{1}{5} x^{5/2} (a+b x)^{5/2}+\frac{\left (3 a^5\right ) \operatorname{Subst}\left (\int \frac{1}{\sqrt{a+b x^2}} \, dx,x,\sqrt{x}\right )}{128 b^2}\\ &=-\frac{3 a^4 \sqrt{x} \sqrt{a+b x}}{128 b^2}+\frac{a^3 x^{3/2} \sqrt{a+b x}}{64 b}+\frac{1}{16} a^2 x^{5/2} \sqrt{a+b x}+\frac{1}{8} a x^{5/2} (a+b x)^{3/2}+\frac{1}{5} x^{5/2} (a+b x)^{5/2}+\frac{\left (3 a^5\right ) \operatorname{Subst}\left (\int \frac{1}{1-b x^2} \, dx,x,\frac{\sqrt{x}}{\sqrt{a+b x}}\right )}{128 b^2}\\ &=-\frac{3 a^4 \sqrt{x} \sqrt{a+b x}}{128 b^2}+\frac{a^3 x^{3/2} \sqrt{a+b x}}{64 b}+\frac{1}{16} a^2 x^{5/2} \sqrt{a+b x}+\frac{1}{8} a x^{5/2} (a+b x)^{3/2}+\frac{1}{5} x^{5/2} (a+b x)^{5/2}+\frac{3 a^5 \tanh ^{-1}\left (\frac{\sqrt{b} \sqrt{x}}{\sqrt{a+b x}}\right )}{128 b^{5/2}}\\ \end{align*}
Mathematica [A] time = 0.131813, size = 107, normalized size = 0.76 \[ \frac{\sqrt{a+b x} \left (\sqrt{b} \sqrt{x} \left (248 a^2 b^2 x^2+10 a^3 b x-15 a^4+336 a b^3 x^3+128 b^4 x^4\right )+\frac{15 a^{9/2} \sinh ^{-1}\left (\frac{\sqrt{b} \sqrt{x}}{\sqrt{a}}\right )}{\sqrt{\frac{b x}{a}+1}}\right )}{640 b^{5/2}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.004, size = 138, normalized size = 1. \begin{align*}{\frac{1}{5\,b}{x}^{{\frac{3}{2}}} \left ( bx+a \right ) ^{{\frac{7}{2}}}}-{\frac{3\,a}{40\,{b}^{2}}\sqrt{x} \left ( bx+a \right ) ^{{\frac{7}{2}}}}+{\frac{{a}^{2}}{80\,{b}^{2}} \left ( bx+a \right ) ^{{\frac{5}{2}}}\sqrt{x}}+{\frac{{a}^{3}}{64\,{b}^{2}} \left ( bx+a \right ) ^{{\frac{3}{2}}}\sqrt{x}}+{\frac{3\,{a}^{4}}{128\,{b}^{2}}\sqrt{x}\sqrt{bx+a}}+{\frac{3\,{a}^{5}}{256}\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 = 1.94763, size = 477, normalized size = 3.41 \begin{align*} \left [\frac{15 \, a^{5} \sqrt{b} \log \left (2 \, b x + 2 \, \sqrt{b x + a} \sqrt{b} \sqrt{x} + a\right ) + 2 \,{\left (128 \, b^{5} x^{4} + 336 \, a b^{4} x^{3} + 248 \, a^{2} b^{3} x^{2} + 10 \, a^{3} b^{2} x - 15 \, a^{4} b\right )} \sqrt{b x + a} \sqrt{x}}{1280 \, b^{3}}, -\frac{15 \, a^{5} \sqrt{-b} \arctan \left (\frac{\sqrt{b x + a} \sqrt{-b}}{b \sqrt{x}}\right ) -{\left (128 \, b^{5} x^{4} + 336 \, a b^{4} x^{3} + 248 \, a^{2} b^{3} x^{2} + 10 \, a^{3} b^{2} x - 15 \, a^{4} b\right )} \sqrt{b x + a} \sqrt{x}}{640 \, b^{3}}\right ] \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 50.7545, size = 180, normalized size = 1.29 \begin{align*} - \frac{3 a^{\frac{9}{2}} \sqrt{x}}{128 b^{2} \sqrt{1 + \frac{b x}{a}}} - \frac{a^{\frac{7}{2}} x^{\frac{3}{2}}}{128 b \sqrt{1 + \frac{b x}{a}}} + \frac{129 a^{\frac{5}{2}} x^{\frac{5}{2}}}{320 \sqrt{1 + \frac{b x}{a}}} + \frac{73 a^{\frac{3}{2}} b x^{\frac{7}{2}}}{80 \sqrt{1 + \frac{b x}{a}}} + \frac{29 \sqrt{a} b^{2} x^{\frac{9}{2}}}{40 \sqrt{1 + \frac{b x}{a}}} + \frac{3 a^{5} \operatorname{asinh}{\left (\frac{\sqrt{b} \sqrt{x}}{\sqrt{a}} \right )}}{128 b^{\frac{5}{2}}} + \frac{b^{3} x^{\frac{11}{2}}}{5 \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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