Optimal. Leaf size=71 \[ \frac{3 \sqrt{x} \sqrt{a-b x}}{b^2}-\frac{3 a \tan ^{-1}\left (\frac{\sqrt{b} \sqrt{x}}{\sqrt{a-b x}}\right )}{b^{5/2}}+\frac{2 x^{3/2}}{b \sqrt{a-b x}} \]
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Rubi [A] time = 0.0220824, antiderivative size = 71, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 5, integrand size = 16, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.312, Rules used = {47, 50, 63, 217, 203} \[ \frac{3 \sqrt{x} \sqrt{a-b x}}{b^2}-\frac{3 a \tan ^{-1}\left (\frac{\sqrt{b} \sqrt{x}}{\sqrt{a-b x}}\right )}{b^{5/2}}+\frac{2 x^{3/2}}{b \sqrt{a-b x}} \]
Antiderivative was successfully verified.
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Rule 47
Rule 50
Rule 63
Rule 217
Rule 203
Rubi steps
\begin{align*} \int \frac{x^{3/2}}{(a-b x)^{3/2}} \, dx &=\frac{2 x^{3/2}}{b \sqrt{a-b x}}-\frac{3 \int \frac{\sqrt{x}}{\sqrt{a-b x}} \, dx}{b}\\ &=\frac{2 x^{3/2}}{b \sqrt{a-b x}}+\frac{3 \sqrt{x} \sqrt{a-b x}}{b^2}-\frac{(3 a) \int \frac{1}{\sqrt{x} \sqrt{a-b x}} \, dx}{2 b^2}\\ &=\frac{2 x^{3/2}}{b \sqrt{a-b x}}+\frac{3 \sqrt{x} \sqrt{a-b x}}{b^2}-\frac{(3 a) \operatorname{Subst}\left (\int \frac{1}{\sqrt{a-b x^2}} \, dx,x,\sqrt{x}\right )}{b^2}\\ &=\frac{2 x^{3/2}}{b \sqrt{a-b x}}+\frac{3 \sqrt{x} \sqrt{a-b x}}{b^2}-\frac{(3 a) \operatorname{Subst}\left (\int \frac{1}{1+b x^2} \, dx,x,\frac{\sqrt{x}}{\sqrt{a-b x}}\right )}{b^2}\\ &=\frac{2 x^{3/2}}{b \sqrt{a-b x}}+\frac{3 \sqrt{x} \sqrt{a-b x}}{b^2}-\frac{3 a \tan ^{-1}\left (\frac{\sqrt{b} \sqrt{x}}{\sqrt{a-b x}}\right )}{b^{5/2}}\\ \end{align*}
Mathematica [C] time = 0.0107771, size = 51, normalized size = 0.72 \[ \frac{2 x^{5/2} \sqrt{1-\frac{b x}{a}} \, _2F_1\left (\frac{3}{2},\frac{5}{2};\frac{7}{2};\frac{b x}{a}\right )}{5 a \sqrt{a-b x}} \]
Antiderivative was successfully verified.
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Maple [B] time = 0.02, size = 114, normalized size = 1.6 \begin{align*}{\frac{1}{{b}^{2}}\sqrt{x}\sqrt{-bx+a}}+{ \left ( -{\frac{3\,a}{2}\arctan \left ({\sqrt{b} \left ( x-{\frac{a}{2\,b}} \right ){\frac{1}{\sqrt{-b{x}^{2}+ax}}}} \right ){b}^{-{\frac{5}{2}}}}-2\,{\frac{a}{{b}^{3}}\sqrt{-b \left ( x-{\frac{a}{b}} \right ) ^{2}-a \left ( x-{\frac{a}{b}} \right ) } \left ( x-{\frac{a}{b}} \right ) ^{-1}} \right ) \sqrt{x \left ( -bx+a \right ) }{\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 = 1.79657, size = 369, normalized size = 5.2 \begin{align*} \left [-\frac{3 \,{\left (a b x - a^{2}\right )} \sqrt{-b} \log \left (-2 \, b x - 2 \, \sqrt{-b x + a} \sqrt{-b} \sqrt{x} + a\right ) - 2 \,{\left (b^{2} x - 3 \, a b\right )} \sqrt{-b x + a} \sqrt{x}}{2 \,{\left (b^{4} x - a b^{3}\right )}}, \frac{3 \,{\left (a b x - a^{2}\right )} \sqrt{b} \arctan \left (\frac{\sqrt{-b x + a}}{\sqrt{b} \sqrt{x}}\right ) +{\left (b^{2} x - 3 \, a b\right )} \sqrt{-b x + a} \sqrt{x}}{b^{4} x - a b^{3}}\right ] \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 4.24528, size = 156, normalized size = 2.2 \begin{align*} \begin{cases} - \frac{3 i \sqrt{a} \sqrt{x}}{b^{2} \sqrt{-1 + \frac{b x}{a}}} + \frac{3 i a \operatorname{acosh}{\left (\frac{\sqrt{b} \sqrt{x}}{\sqrt{a}} \right )}}{b^{\frac{5}{2}}} + \frac{i x^{\frac{3}{2}}}{\sqrt{a} b \sqrt{-1 + \frac{b x}{a}}} & \text{for}\: \frac{\left |{b x}\right |}{\left |{a}\right |} > 1 \\\frac{3 \sqrt{a} \sqrt{x}}{b^{2} \sqrt{1 - \frac{b x}{a}}} - \frac{3 a \operatorname{asin}{\left (\frac{\sqrt{b} \sqrt{x}}{\sqrt{a}} \right )}}{b^{\frac{5}{2}}} - \frac{x^{\frac{3}{2}}}{\sqrt{a} b \sqrt{1 - \frac{b x}{a}}} & \text{otherwise} \end{cases} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [B] time = 59.3328, size = 176, normalized size = 2.48 \begin{align*} -\frac{{\left (\frac{8 \, a^{2} \sqrt{-b}}{{\left (\sqrt{-b x + a} \sqrt{-b} - \sqrt{{\left (b x - a\right )} b + a b}\right )}^{2} - a b} + \frac{3 \, a \log \left ({\left (\sqrt{-b x + a} \sqrt{-b} - \sqrt{{\left (b x - a\right )} b + a b}\right )}^{2}\right )}{\sqrt{-b}} - \frac{2 \, \sqrt{{\left (b x - a\right )} b + a b} \sqrt{-b x + a}}{b}\right )}{\left | b \right |}}{2 \, b^{3}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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