Optimal. Leaf size=99 \[ \frac{a^3 \tan ^{-1}\left (\frac{\sqrt{b} \sqrt{x}}{\sqrt{a-b x}}\right )}{8 b^{3/2}}-\frac{a^2 \sqrt{x} \sqrt{a-b x}}{8 b}+\frac{1}{4} a x^{3/2} \sqrt{a-b x}+\frac{1}{3} x^{3/2} (a-b x)^{3/2} \]
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Rubi [A] time = 0.0309863, antiderivative size = 99, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 4, integrand size = 16, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.25, Rules used = {50, 63, 217, 203} \[ \frac{a^3 \tan ^{-1}\left (\frac{\sqrt{b} \sqrt{x}}{\sqrt{a-b x}}\right )}{8 b^{3/2}}-\frac{a^2 \sqrt{x} \sqrt{a-b x}}{8 b}+\frac{1}{4} a x^{3/2} \sqrt{a-b x}+\frac{1}{3} x^{3/2} (a-b x)^{3/2} \]
Antiderivative was successfully verified.
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Rule 50
Rule 63
Rule 217
Rule 203
Rubi steps
\begin{align*} \int \sqrt{x} (a-b x)^{3/2} \, dx &=\frac{1}{3} x^{3/2} (a-b x)^{3/2}+\frac{1}{2} a \int \sqrt{x} \sqrt{a-b x} \, dx\\ &=\frac{1}{4} a x^{3/2} \sqrt{a-b x}+\frac{1}{3} x^{3/2} (a-b x)^{3/2}+\frac{1}{8} a^2 \int \frac{\sqrt{x}}{\sqrt{a-b x}} \, dx\\ &=-\frac{a^2 \sqrt{x} \sqrt{a-b x}}{8 b}+\frac{1}{4} a x^{3/2} \sqrt{a-b x}+\frac{1}{3} x^{3/2} (a-b x)^{3/2}+\frac{a^3 \int \frac{1}{\sqrt{x} \sqrt{a-b x}} \, dx}{16 b}\\ &=-\frac{a^2 \sqrt{x} \sqrt{a-b x}}{8 b}+\frac{1}{4} a x^{3/2} \sqrt{a-b x}+\frac{1}{3} x^{3/2} (a-b x)^{3/2}+\frac{a^3 \operatorname{Subst}\left (\int \frac{1}{\sqrt{a-b x^2}} \, dx,x,\sqrt{x}\right )}{8 b}\\ &=-\frac{a^2 \sqrt{x} \sqrt{a-b x}}{8 b}+\frac{1}{4} a x^{3/2} \sqrt{a-b x}+\frac{1}{3} x^{3/2} (a-b x)^{3/2}+\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}\\ &=-\frac{a^2 \sqrt{x} \sqrt{a-b x}}{8 b}+\frac{1}{4} a x^{3/2} \sqrt{a-b x}+\frac{1}{3} x^{3/2} (a-b x)^{3/2}+\frac{a^3 \tan ^{-1}\left (\frac{\sqrt{b} \sqrt{x}}{\sqrt{a-b x}}\right )}{8 b^{3/2}}\\ \end{align*}
Mathematica [A] time = 0.114232, size = 87, normalized size = 0.88 \[ \frac{\sqrt{a-b x} \left (\sqrt{b} \sqrt{x} \left (-3 a^2+14 a b x-8 b^2 x^2\right )+\frac{3 a^{5/2} \sin ^{-1}\left (\frac{\sqrt{b} \sqrt{x}}{\sqrt{a}}\right )}{\sqrt{1-\frac{b x}{a}}}\right )}{24 b^{3/2}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.004, size = 102, normalized size = 1. \begin{align*}{\frac{1}{3}{x}^{{\frac{3}{2}}} \left ( -bx+a \right ) ^{{\frac{3}{2}}}}+{\frac{a}{4}{x}^{{\frac{3}{2}}}\sqrt{-bx+a}}-{\frac{{a}^{2}}{8\,b}\sqrt{x}\sqrt{-bx+a}}+{\frac{{a}^{3}}{16}\sqrt{x \left ( -bx+a \right ) }\arctan \left ({\sqrt{b} \left ( x-{\frac{a}{2\,b}} \right ){\frac{1}{\sqrt{-b{x}^{2}+ax}}}} \right ){b}^{-{\frac{3}{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 = 1.97079, size = 370, normalized size = 3.74 \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} - 14 \, a b^{2} x + 3 \, a^{2} b\right )} \sqrt{-b x + a} \sqrt{x}}{48 \, b^{2}}, -\frac{3 \, a^{3} \sqrt{b} \arctan \left (\frac{\sqrt{-b x + a}}{\sqrt{b} \sqrt{x}}\right ) +{\left (8 \, b^{3} x^{2} - 14 \, a b^{2} x + 3 \, a^{2} b\right )} \sqrt{-b x + a} \sqrt{x}}{24 \, b^{2}}\right ] \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 6.47694, size = 265, normalized size = 2.68 \begin{align*} \begin{cases} \frac{i a^{\frac{5}{2}} \sqrt{x}}{8 b \sqrt{-1 + \frac{b x}{a}}} - \frac{17 i a^{\frac{3}{2}} x^{\frac{3}{2}}}{24 \sqrt{-1 + \frac{b x}{a}}} + \frac{11 i \sqrt{a} b x^{\frac{5}{2}}}{12 \sqrt{-1 + \frac{b x}{a}}} - \frac{i a^{3} \operatorname{acosh}{\left (\frac{\sqrt{b} \sqrt{x}}{\sqrt{a}} \right )}}{8 b^{\frac{3}{2}}} - \frac{i b^{2} x^{\frac{7}{2}}}{3 \sqrt{a} \sqrt{-1 + \frac{b x}{a}}} & \text{for}\: \frac{\left |{b x}\right |}{\left |{a}\right |} > 1 \\- \frac{a^{\frac{5}{2}} \sqrt{x}}{8 b \sqrt{1 - \frac{b x}{a}}} + \frac{17 a^{\frac{3}{2}} x^{\frac{3}{2}}}{24 \sqrt{1 - \frac{b x}{a}}} - \frac{11 \sqrt{a} b x^{\frac{5}{2}}}{12 \sqrt{1 - \frac{b x}{a}}} + \frac{a^{3} \operatorname{asin}{\left (\frac{\sqrt{b} \sqrt{x}}{\sqrt{a}} \right )}}{8 b^{\frac{3}{2}}} + \frac{b^{2} x^{\frac{7}{2}}}{3 \sqrt{a} \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 [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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