Optimal. Leaf size=71 \[ -\frac{16 b^2 (a-b x)^{3/2}}{105 a^3 x^{3/2}}-\frac{8 b (a-b x)^{3/2}}{35 a^2 x^{5/2}}-\frac{2 (a-b x)^{3/2}}{7 a x^{7/2}} \]
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Rubi [A] time = 0.0104142, antiderivative size = 71, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 16, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.125, Rules used = {45, 37} \[ -\frac{16 b^2 (a-b x)^{3/2}}{105 a^3 x^{3/2}}-\frac{8 b (a-b x)^{3/2}}{35 a^2 x^{5/2}}-\frac{2 (a-b x)^{3/2}}{7 a x^{7/2}} \]
Antiderivative was successfully verified.
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Rule 45
Rule 37
Rubi steps
\begin{align*} \int \frac{\sqrt{a-b x}}{x^{9/2}} \, dx &=-\frac{2 (a-b x)^{3/2}}{7 a x^{7/2}}+\frac{(4 b) \int \frac{\sqrt{a-b x}}{x^{7/2}} \, dx}{7 a}\\ &=-\frac{2 (a-b x)^{3/2}}{7 a x^{7/2}}-\frac{8 b (a-b x)^{3/2}}{35 a^2 x^{5/2}}+\frac{\left (8 b^2\right ) \int \frac{\sqrt{a-b x}}{x^{5/2}} \, dx}{35 a^2}\\ &=-\frac{2 (a-b x)^{3/2}}{7 a x^{7/2}}-\frac{8 b (a-b x)^{3/2}}{35 a^2 x^{5/2}}-\frac{16 b^2 (a-b x)^{3/2}}{105 a^3 x^{3/2}}\\ \end{align*}
Mathematica [A] time = 0.0113349, size = 41, normalized size = 0.58 \[ -\frac{2 (a-b x)^{3/2} \left (15 a^2+12 a b x+8 b^2 x^2\right )}{105 a^3 x^{7/2}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.003, size = 36, normalized size = 0.5 \begin{align*} -{\frac{16\,{b}^{2}{x}^{2}+24\,abx+30\,{a}^{2}}{105\,{a}^{3}} \left ( -bx+a \right ) ^{{\frac{3}{2}}}{x}^{-{\frac{7}{2}}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.03152, size = 66, normalized size = 0.93 \begin{align*} -\frac{2 \,{\left (\frac{35 \,{\left (-b x + a\right )}^{\frac{3}{2}} b^{2}}{x^{\frac{3}{2}}} + \frac{42 \,{\left (-b x + a\right )}^{\frac{5}{2}} b}{x^{\frac{5}{2}}} + \frac{15 \,{\left (-b x + a\right )}^{\frac{7}{2}}}{x^{\frac{7}{2}}}\right )}}{105 \, a^{3}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.51523, size = 112, normalized size = 1.58 \begin{align*} \frac{2 \,{\left (8 \, b^{3} x^{3} + 4 \, a b^{2} x^{2} + 3 \, a^{2} b x - 15 \, a^{3}\right )} \sqrt{-b x + a}}{105 \, a^{3} x^{\frac{7}{2}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [B] time = 114.542, size = 711, normalized size = 10.01 \begin{align*} \begin{cases} - \frac{30 a^{5} b^{\frac{9}{2}} \sqrt{\frac{a}{b x} - 1}}{105 a^{5} b^{4} x^{3} - 210 a^{4} b^{5} x^{4} + 105 a^{3} b^{6} x^{5}} + \frac{66 a^{4} b^{\frac{11}{2}} x \sqrt{\frac{a}{b x} - 1}}{105 a^{5} b^{4} x^{3} - 210 a^{4} b^{5} x^{4} + 105 a^{3} b^{6} x^{5}} - \frac{34 a^{3} b^{\frac{13}{2}} x^{2} \sqrt{\frac{a}{b x} - 1}}{105 a^{5} b^{4} x^{3} - 210 a^{4} b^{5} x^{4} + 105 a^{3} b^{6} x^{5}} + \frac{6 a^{2} b^{\frac{15}{2}} x^{3} \sqrt{\frac{a}{b x} - 1}}{105 a^{5} b^{4} x^{3} - 210 a^{4} b^{5} x^{4} + 105 a^{3} b^{6} x^{5}} - \frac{24 a b^{\frac{17}{2}} x^{4} \sqrt{\frac{a}{b x} - 1}}{105 a^{5} b^{4} x^{3} - 210 a^{4} b^{5} x^{4} + 105 a^{3} b^{6} x^{5}} + \frac{16 b^{\frac{19}{2}} x^{5} \sqrt{\frac{a}{b x} - 1}}{105 a^{5} b^{4} x^{3} - 210 a^{4} b^{5} x^{4} + 105 a^{3} b^{6} x^{5}} & \text{for}\: \frac{\left |{a}\right |}{\left |{b}\right | \left |{x}\right |} > 1 \\- \frac{30 i a^{5} b^{\frac{9}{2}} \sqrt{- \frac{a}{b x} + 1}}{105 a^{5} b^{4} x^{3} - 210 a^{4} b^{5} x^{4} + 105 a^{3} b^{6} x^{5}} + \frac{66 i a^{4} b^{\frac{11}{2}} x \sqrt{- \frac{a}{b x} + 1}}{105 a^{5} b^{4} x^{3} - 210 a^{4} b^{5} x^{4} + 105 a^{3} b^{6} x^{5}} - \frac{34 i a^{3} b^{\frac{13}{2}} x^{2} \sqrt{- \frac{a}{b x} + 1}}{105 a^{5} b^{4} x^{3} - 210 a^{4} b^{5} x^{4} + 105 a^{3} b^{6} x^{5}} + \frac{6 i a^{2} b^{\frac{15}{2}} x^{3} \sqrt{- \frac{a}{b x} + 1}}{105 a^{5} b^{4} x^{3} - 210 a^{4} b^{5} x^{4} + 105 a^{3} b^{6} x^{5}} - \frac{24 i a b^{\frac{17}{2}} x^{4} \sqrt{- \frac{a}{b x} + 1}}{105 a^{5} b^{4} x^{3} - 210 a^{4} b^{5} x^{4} + 105 a^{3} b^{6} x^{5}} + \frac{16 i b^{\frac{19}{2}} x^{5} \sqrt{- \frac{a}{b x} + 1}}{105 a^{5} b^{4} x^{3} - 210 a^{4} b^{5} x^{4} + 105 a^{3} b^{6} x^{5}} & \text{otherwise} \end{cases} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.22038, size = 107, normalized size = 1.51 \begin{align*} \frac{{\left (b x - a\right )} \sqrt{-b x + a}{\left (4 \,{\left (b x - a\right )}{\left (\frac{2 \,{\left (b x - a\right )}}{a^{4} b^{5}} + \frac{7}{a^{3} b^{5}}\right )} + \frac{35}{a^{2} b^{5}}\right )} b}{40320 \,{\left ({\left (b x - a\right )} b + a b\right )}^{\frac{7}{2}}{\left | b \right |}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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