Optimal. Leaf size=149 \[ -\frac{a^3 x^{3/2} \sqrt{a-b x}}{64 b^2}-\frac{3 a^4 \sqrt{x} \sqrt{a-b x}}{128 b^3}+\frac{3 a^5 \tan ^{-1}\left (\frac{\sqrt{b} \sqrt{x}}{\sqrt{a-b x}}\right )}{128 b^{7/2}}-\frac{a^2 x^{5/2} \sqrt{a-b x}}{80 b}+\frac{3}{40} a x^{7/2} \sqrt{a-b x}+\frac{1}{5} x^{7/2} (a-b x)^{3/2} \]
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Rubi [A] time = 0.0532699, antiderivative size = 149, normalized size of antiderivative = 1., number of steps used = 8, number of rules used = 4, integrand size = 16, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.25, Rules used = {50, 63, 217, 203} \[ -\frac{a^3 x^{3/2} \sqrt{a-b x}}{64 b^2}-\frac{3 a^4 \sqrt{x} \sqrt{a-b x}}{128 b^3}+\frac{3 a^5 \tan ^{-1}\left (\frac{\sqrt{b} \sqrt{x}}{\sqrt{a-b x}}\right )}{128 b^{7/2}}-\frac{a^2 x^{5/2} \sqrt{a-b x}}{80 b}+\frac{3}{40} a x^{7/2} \sqrt{a-b x}+\frac{1}{5} x^{7/2} (a-b x)^{3/2} \]
Antiderivative was successfully verified.
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Rule 50
Rule 63
Rule 217
Rule 203
Rubi steps
\begin{align*} \int x^{5/2} (a-b x)^{3/2} \, dx &=\frac{1}{5} x^{7/2} (a-b x)^{3/2}+\frac{1}{10} (3 a) \int x^{5/2} \sqrt{a-b x} \, dx\\ &=\frac{3}{40} a x^{7/2} \sqrt{a-b x}+\frac{1}{5} x^{7/2} (a-b x)^{3/2}+\frac{1}{80} \left (3 a^2\right ) \int \frac{x^{5/2}}{\sqrt{a-b x}} \, dx\\ &=-\frac{a^2 x^{5/2} \sqrt{a-b x}}{80 b}+\frac{3}{40} a x^{7/2} \sqrt{a-b x}+\frac{1}{5} x^{7/2} (a-b x)^{3/2}+\frac{a^3 \int \frac{x^{3/2}}{\sqrt{a-b x}} \, dx}{32 b}\\ &=-\frac{a^3 x^{3/2} \sqrt{a-b x}}{64 b^2}-\frac{a^2 x^{5/2} \sqrt{a-b x}}{80 b}+\frac{3}{40} a x^{7/2} \sqrt{a-b x}+\frac{1}{5} x^{7/2} (a-b x)^{3/2}+\frac{\left (3 a^4\right ) \int \frac{\sqrt{x}}{\sqrt{a-b x}} \, dx}{128 b^2}\\ &=-\frac{3 a^4 \sqrt{x} \sqrt{a-b x}}{128 b^3}-\frac{a^3 x^{3/2} \sqrt{a-b x}}{64 b^2}-\frac{a^2 x^{5/2} \sqrt{a-b x}}{80 b}+\frac{3}{40} a x^{7/2} \sqrt{a-b x}+\frac{1}{5} x^{7/2} (a-b x)^{3/2}+\frac{\left (3 a^5\right ) \int \frac{1}{\sqrt{x} \sqrt{a-b x}} \, dx}{256 b^3}\\ &=-\frac{3 a^4 \sqrt{x} \sqrt{a-b x}}{128 b^3}-\frac{a^3 x^{3/2} \sqrt{a-b x}}{64 b^2}-\frac{a^2 x^{5/2} \sqrt{a-b x}}{80 b}+\frac{3}{40} a x^{7/2} \sqrt{a-b x}+\frac{1}{5} x^{7/2} (a-b x)^{3/2}+\frac{\left (3 a^5\right ) \operatorname{Subst}\left (\int \frac{1}{\sqrt{a-b x^2}} \, dx,x,\sqrt{x}\right )}{128 b^3}\\ &=-\frac{3 a^4 \sqrt{x} \sqrt{a-b x}}{128 b^3}-\frac{a^3 x^{3/2} \sqrt{a-b x}}{64 b^2}-\frac{a^2 x^{5/2} \sqrt{a-b x}}{80 b}+\frac{3}{40} a x^{7/2} \sqrt{a-b x}+\frac{1}{5} x^{7/2} (a-b x)^{3/2}+\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^3}\\ &=-\frac{3 a^4 \sqrt{x} \sqrt{a-b x}}{128 b^3}-\frac{a^3 x^{3/2} \sqrt{a-b x}}{64 b^2}-\frac{a^2 x^{5/2} \sqrt{a-b x}}{80 b}+\frac{3}{40} a x^{7/2} \sqrt{a-b x}+\frac{1}{5} x^{7/2} (a-b x)^{3/2}+\frac{3 a^5 \tan ^{-1}\left (\frac{\sqrt{b} \sqrt{x}}{\sqrt{a-b x}}\right )}{128 b^{7/2}}\\ \end{align*}
Mathematica [A] time = 0.160506, size = 110, normalized size = 0.74 \[ \frac{\sqrt{a-b x} \left (\frac{15 a^{9/2} \sin ^{-1}\left (\frac{\sqrt{b} \sqrt{x}}{\sqrt{a}}\right )}{\sqrt{1-\frac{b x}{a}}}-\sqrt{b} \sqrt{x} \left (8 a^2 b^2 x^2+10 a^3 b x+15 a^4-176 a b^3 x^3+128 b^4 x^4\right )\right )}{640 b^{7/2}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.005, size = 146, normalized size = 1. \begin{align*} -{\frac{1}{5\,b}{x}^{{\frac{5}{2}}} \left ( -bx+a \right ) ^{{\frac{5}{2}}}}-{\frac{a}{8\,{b}^{2}}{x}^{{\frac{3}{2}}} \left ( -bx+a \right ) ^{{\frac{5}{2}}}}-{\frac{{a}^{2}}{16\,{b}^{3}} \left ( -bx+a \right ) ^{{\frac{5}{2}}}\sqrt{x}}+{\frac{{a}^{3}}{64\,{b}^{3}} \left ( -bx+a \right ) ^{{\frac{3}{2}}}\sqrt{x}}+{\frac{3\,{a}^{4}}{128\,{b}^{3}}\sqrt{x}\sqrt{-bx+a}}+{\frac{3\,{a}^{5}}{256}\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{7}{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 = 1.9582, size = 477, normalized size = 3.2 \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} - 176 \, a b^{4} x^{3} + 8 \, 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^{4}}, -\frac{15 \, a^{5} \sqrt{b} \arctan \left (\frac{\sqrt{-b x + a}}{\sqrt{b} \sqrt{x}}\right ) +{\left (128 \, b^{5} x^{4} - 176 \, a b^{4} x^{3} + 8 \, 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^{4}}\right ] \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 56.4197, size = 377, normalized size = 2.53 \begin{align*} \begin{cases} \frac{3 i a^{\frac{9}{2}} \sqrt{x}}{128 b^{3} \sqrt{-1 + \frac{b x}{a}}} - \frac{i a^{\frac{7}{2}} x^{\frac{3}{2}}}{128 b^{2} \sqrt{-1 + \frac{b x}{a}}} - \frac{i a^{\frac{5}{2}} x^{\frac{5}{2}}}{320 b \sqrt{-1 + \frac{b x}{a}}} - \frac{23 i a^{\frac{3}{2}} x^{\frac{7}{2}}}{80 \sqrt{-1 + \frac{b x}{a}}} + \frac{19 i \sqrt{a} b x^{\frac{9}{2}}}{40 \sqrt{-1 + \frac{b x}{a}}} - \frac{3 i a^{5} \operatorname{acosh}{\left (\frac{\sqrt{b} \sqrt{x}}{\sqrt{a}} \right )}}{128 b^{\frac{7}{2}}} - \frac{i b^{2} x^{\frac{11}{2}}}{5 \sqrt{a} \sqrt{-1 + \frac{b x}{a}}} & \text{for}\: \frac{\left |{b x}\right |}{\left |{a}\right |} > 1 \\- \frac{3 a^{\frac{9}{2}} \sqrt{x}}{128 b^{3} \sqrt{1 - \frac{b x}{a}}} + \frac{a^{\frac{7}{2}} x^{\frac{3}{2}}}{128 b^{2} \sqrt{1 - \frac{b x}{a}}} + \frac{a^{\frac{5}{2}} x^{\frac{5}{2}}}{320 b \sqrt{1 - \frac{b x}{a}}} + \frac{23 a^{\frac{3}{2}} x^{\frac{7}{2}}}{80 \sqrt{1 - \frac{b x}{a}}} - \frac{19 \sqrt{a} b x^{\frac{9}{2}}}{40 \sqrt{1 - \frac{b x}{a}}} + \frac{3 a^{5} \operatorname{asin}{\left (\frac{\sqrt{b} \sqrt{x}}{\sqrt{a}} \right )}}{128 b^{\frac{7}{2}}} + \frac{b^{2} x^{\frac{11}{2}}}{5 \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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