Optimal. Leaf size=119 \[ -\frac{3 a^3 \sqrt{x} \sqrt{a+b x}}{64 b^2}+\frac{3 a^4 \tanh ^{-1}\left (\frac{\sqrt{b} \sqrt{x}}{\sqrt{a+b x}}\right )}{64 b^{5/2}}+\frac{a^2 x^{3/2} \sqrt{a+b x}}{32 b}+\frac{1}{8} a x^{5/2} \sqrt{a+b x}+\frac{1}{4} x^{5/2} (a+b x)^{3/2} \]
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Rubi [A] time = 0.0379329, antiderivative size = 119, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 4, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.267, Rules used = {50, 63, 217, 206} \[ -\frac{3 a^3 \sqrt{x} \sqrt{a+b x}}{64 b^2}+\frac{3 a^4 \tanh ^{-1}\left (\frac{\sqrt{b} \sqrt{x}}{\sqrt{a+b x}}\right )}{64 b^{5/2}}+\frac{a^2 x^{3/2} \sqrt{a+b x}}{32 b}+\frac{1}{8} a x^{5/2} \sqrt{a+b x}+\frac{1}{4} x^{5/2} (a+b x)^{3/2} \]
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)^{3/2} \, dx &=\frac{1}{4} x^{5/2} (a+b x)^{3/2}+\frac{1}{8} (3 a) \int x^{3/2} \sqrt{a+b x} \, dx\\ &=\frac{1}{8} a x^{5/2} \sqrt{a+b x}+\frac{1}{4} x^{5/2} (a+b x)^{3/2}+\frac{1}{16} a^2 \int \frac{x^{3/2}}{\sqrt{a+b x}} \, dx\\ &=\frac{a^2 x^{3/2} \sqrt{a+b x}}{32 b}+\frac{1}{8} a x^{5/2} \sqrt{a+b x}+\frac{1}{4} x^{5/2} (a+b x)^{3/2}-\frac{\left (3 a^3\right ) \int \frac{\sqrt{x}}{\sqrt{a+b x}} \, dx}{64 b}\\ &=-\frac{3 a^3 \sqrt{x} \sqrt{a+b x}}{64 b^2}+\frac{a^2 x^{3/2} \sqrt{a+b x}}{32 b}+\frac{1}{8} a x^{5/2} \sqrt{a+b x}+\frac{1}{4} x^{5/2} (a+b x)^{3/2}+\frac{\left (3 a^4\right ) \int \frac{1}{\sqrt{x} \sqrt{a+b x}} \, dx}{128 b^2}\\ &=-\frac{3 a^3 \sqrt{x} \sqrt{a+b x}}{64 b^2}+\frac{a^2 x^{3/2} \sqrt{a+b x}}{32 b}+\frac{1}{8} a x^{5/2} \sqrt{a+b x}+\frac{1}{4} x^{5/2} (a+b x)^{3/2}+\frac{\left (3 a^4\right ) \operatorname{Subst}\left (\int \frac{1}{\sqrt{a+b x^2}} \, dx,x,\sqrt{x}\right )}{64 b^2}\\ &=-\frac{3 a^3 \sqrt{x} \sqrt{a+b x}}{64 b^2}+\frac{a^2 x^{3/2} \sqrt{a+b x}}{32 b}+\frac{1}{8} a x^{5/2} \sqrt{a+b x}+\frac{1}{4} x^{5/2} (a+b x)^{3/2}+\frac{\left (3 a^4\right ) \operatorname{Subst}\left (\int \frac{1}{1-b x^2} \, dx,x,\frac{\sqrt{x}}{\sqrt{a+b x}}\right )}{64 b^2}\\ &=-\frac{3 a^3 \sqrt{x} \sqrt{a+b x}}{64 b^2}+\frac{a^2 x^{3/2} \sqrt{a+b x}}{32 b}+\frac{1}{8} a x^{5/2} \sqrt{a+b x}+\frac{1}{4} x^{5/2} (a+b x)^{3/2}+\frac{3 a^4 \tanh ^{-1}\left (\frac{\sqrt{b} \sqrt{x}}{\sqrt{a+b x}}\right )}{64 b^{5/2}}\\ \end{align*}
Mathematica [A] time = 0.123042, size = 96, normalized size = 0.81 \[ \frac{\sqrt{a+b x} \left (\sqrt{b} \sqrt{x} \left (2 a^2 b x-3 a^3+24 a b^2 x^2+16 b^3 x^3\right )+\frac{3 a^{7/2} \sinh ^{-1}\left (\frac{\sqrt{b} \sqrt{x}}{\sqrt{a}}\right )}{\sqrt{\frac{b x}{a}+1}}\right )}{64 b^{5/2}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.004, size = 120, normalized size = 1. \begin{align*}{\frac{1}{4\,b}{x}^{{\frac{3}{2}}} \left ( bx+a \right ) ^{{\frac{5}{2}}}}-{\frac{a}{8\,{b}^{2}} \left ( bx+a \right ) ^{{\frac{5}{2}}}\sqrt{x}}+{\frac{{a}^{2}}{32\,{b}^{2}} \left ( bx+a \right ) ^{{\frac{3}{2}}}\sqrt{x}}+{\frac{3\,{a}^{3}}{64\,{b}^{2}}\sqrt{x}\sqrt{bx+a}}+{\frac{3\,{a}^{4}}{128}\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 = 1.60805, size = 412, normalized size = 3.46 \begin{align*} \left [\frac{3 \, a^{4} \sqrt{b} \log \left (2 \, b x + 2 \, \sqrt{b x + a} \sqrt{b} \sqrt{x} + a\right ) + 2 \,{\left (16 \, b^{4} x^{3} + 24 \, a b^{3} x^{2} + 2 \, a^{2} b^{2} x - 3 \, a^{3} b\right )} \sqrt{b x + a} \sqrt{x}}{128 \, b^{3}}, -\frac{3 \, a^{4} \sqrt{-b} \arctan \left (\frac{\sqrt{b x + a} \sqrt{-b}}{b \sqrt{x}}\right ) -{\left (16 \, b^{4} x^{3} + 24 \, a b^{3} x^{2} + 2 \, a^{2} b^{2} x - 3 \, a^{3} b\right )} \sqrt{b x + a} \sqrt{x}}{64 \, b^{3}}\right ] \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 11.2612, size = 153, normalized size = 1.29 \begin{align*} - \frac{3 a^{\frac{7}{2}} \sqrt{x}}{64 b^{2} \sqrt{1 + \frac{b x}{a}}} - \frac{a^{\frac{5}{2}} x^{\frac{3}{2}}}{64 b \sqrt{1 + \frac{b x}{a}}} + \frac{13 a^{\frac{3}{2}} x^{\frac{5}{2}}}{32 \sqrt{1 + \frac{b x}{a}}} + \frac{5 \sqrt{a} b x^{\frac{7}{2}}}{8 \sqrt{1 + \frac{b x}{a}}} + \frac{3 a^{4} \operatorname{asinh}{\left (\frac{\sqrt{b} \sqrt{x}}{\sqrt{a}} \right )}}{64 b^{\frac{5}{2}}} + \frac{b^{2} x^{\frac{9}{2}}}{4 \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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