Optimal. Leaf size=105 \[ -\frac{5 a^2 \sqrt{x} \sqrt{a-b x}}{8 b^3}+\frac{5 a^3 \tan ^{-1}\left (\frac{\sqrt{b} \sqrt{x}}{\sqrt{a-b x}}\right )}{8 b^{7/2}}-\frac{5 a x^{3/2} \sqrt{a-b x}}{12 b^2}-\frac{x^{5/2} \sqrt{a-b x}}{3 b} \]
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Rubi [A] time = 0.0303057, antiderivative size = 105, 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} \[ -\frac{5 a^2 \sqrt{x} \sqrt{a-b x}}{8 b^3}+\frac{5 a^3 \tan ^{-1}\left (\frac{\sqrt{b} \sqrt{x}}{\sqrt{a-b x}}\right )}{8 b^{7/2}}-\frac{5 a x^{3/2} \sqrt{a-b x}}{12 b^2}-\frac{x^{5/2} \sqrt{a-b x}}{3 b} \]
Antiderivative was successfully verified.
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Rule 50
Rule 63
Rule 217
Rule 203
Rubi steps
\begin{align*} \int \frac{x^{5/2}}{\sqrt{a-b x}} \, dx &=-\frac{x^{5/2} \sqrt{a-b x}}{3 b}+\frac{(5 a) \int \frac{x^{3/2}}{\sqrt{a-b x}} \, dx}{6 b}\\ &=-\frac{5 a x^{3/2} \sqrt{a-b x}}{12 b^2}-\frac{x^{5/2} \sqrt{a-b x}}{3 b}+\frac{\left (5 a^2\right ) \int \frac{\sqrt{x}}{\sqrt{a-b x}} \, dx}{8 b^2}\\ &=-\frac{5 a^2 \sqrt{x} \sqrt{a-b x}}{8 b^3}-\frac{5 a x^{3/2} \sqrt{a-b x}}{12 b^2}-\frac{x^{5/2} \sqrt{a-b x}}{3 b}+\frac{\left (5 a^3\right ) \int \frac{1}{\sqrt{x} \sqrt{a-b x}} \, dx}{16 b^3}\\ &=-\frac{5 a^2 \sqrt{x} \sqrt{a-b x}}{8 b^3}-\frac{5 a x^{3/2} \sqrt{a-b x}}{12 b^2}-\frac{x^{5/2} \sqrt{a-b x}}{3 b}+\frac{\left (5 a^3\right ) \operatorname{Subst}\left (\int \frac{1}{\sqrt{a-b x^2}} \, dx,x,\sqrt{x}\right )}{8 b^3}\\ &=-\frac{5 a^2 \sqrt{x} \sqrt{a-b x}}{8 b^3}-\frac{5 a x^{3/2} \sqrt{a-b x}}{12 b^2}-\frac{x^{5/2} \sqrt{a-b x}}{3 b}+\frac{\left (5 a^3\right ) \operatorname{Subst}\left (\int \frac{1}{1+b x^2} \, dx,x,\frac{\sqrt{x}}{\sqrt{a-b x}}\right )}{8 b^3}\\ &=-\frac{5 a^2 \sqrt{x} \sqrt{a-b x}}{8 b^3}-\frac{5 a x^{3/2} \sqrt{a-b x}}{12 b^2}-\frac{x^{5/2} \sqrt{a-b x}}{3 b}+\frac{5 a^3 \tan ^{-1}\left (\frac{\sqrt{b} \sqrt{x}}{\sqrt{a-b x}}\right )}{8 b^{7/2}}\\ \end{align*}
Mathematica [A] time = 0.173854, size = 88, normalized size = 0.84 \[ \frac{\sqrt{a-b x} \left (\frac{15 a^{5/2} \sin ^{-1}\left (\frac{\sqrt{b} \sqrt{x}}{\sqrt{a}}\right )}{\sqrt{1-\frac{b x}{a}}}-\sqrt{b} \sqrt{x} \left (15 a^2+10 a b x+8 b^2 x^2\right )\right )}{24 b^{7/2}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.003, size = 108, normalized size = 1. \begin{align*} -{\frac{1}{3\,b}{x}^{{\frac{5}{2}}}\sqrt{-bx+a}}-{\frac{5\,a}{12\,{b}^{2}}{x}^{{\frac{3}{2}}}\sqrt{-bx+a}}-{\frac{5\,{a}^{2}}{8\,{b}^{3}}\sqrt{x}\sqrt{-bx+a}}+{\frac{5\,{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{7}{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 = 2.15587, size = 375, normalized size = 3.57 \begin{align*} \left [-\frac{15 \, 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} + 10 \, a b^{2} x + 15 \, a^{2} b\right )} \sqrt{-b x + a} \sqrt{x}}{48 \, b^{4}}, -\frac{15 \, a^{3} \sqrt{b} \arctan \left (\frac{\sqrt{-b x + a}}{\sqrt{b} \sqrt{x}}\right ) +{\left (8 \, b^{3} x^{2} + 10 \, a b^{2} x + 15 \, a^{2} b\right )} \sqrt{-b x + a} \sqrt{x}}{24 \, b^{4}}\right ] \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 15.0901, size = 272, normalized size = 2.59 \begin{align*} \begin{cases} \frac{5 i a^{\frac{5}{2}} \sqrt{x}}{8 b^{3} \sqrt{-1 + \frac{b x}{a}}} - \frac{5 i a^{\frac{3}{2}} x^{\frac{3}{2}}}{24 b^{2} \sqrt{-1 + \frac{b x}{a}}} - \frac{i \sqrt{a} x^{\frac{5}{2}}}{12 b \sqrt{-1 + \frac{b x}{a}}} - \frac{5 i a^{3} \operatorname{acosh}{\left (\frac{\sqrt{b} \sqrt{x}}{\sqrt{a}} \right )}}{8 b^{\frac{7}{2}}} - \frac{i 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{5 a^{\frac{5}{2}} \sqrt{x}}{8 b^{3} \sqrt{1 - \frac{b x}{a}}} + \frac{5 a^{\frac{3}{2}} x^{\frac{3}{2}}}{24 b^{2} \sqrt{1 - \frac{b x}{a}}} + \frac{\sqrt{a} x^{\frac{5}{2}}}{12 b \sqrt{1 - \frac{b x}{a}}} + \frac{5 a^{3} \operatorname{asin}{\left (\frac{\sqrt{b} \sqrt{x}}{\sqrt{a}} \right )}}{8 b^{\frac{7}{2}}} + \frac{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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