Optimal. Leaf size=46 \[ -\frac{4 b (a-b x)^{3/2}}{15 a^2 x^{3/2}}-\frac{2 (a-b x)^{3/2}}{5 a x^{5/2}} \]
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Rubi [A] time = 0.0051817, antiderivative size = 46, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 16, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.125, Rules used = {45, 37} \[ -\frac{4 b (a-b x)^{3/2}}{15 a^2 x^{3/2}}-\frac{2 (a-b x)^{3/2}}{5 a x^{5/2}} \]
Antiderivative was successfully verified.
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Rule 45
Rule 37
Rubi steps
\begin{align*} \int \frac{\sqrt{a-b x}}{x^{7/2}} \, dx &=-\frac{2 (a-b x)^{3/2}}{5 a x^{5/2}}+\frac{(2 b) \int \frac{\sqrt{a-b x}}{x^{5/2}} \, dx}{5 a}\\ &=-\frac{2 (a-b x)^{3/2}}{5 a x^{5/2}}-\frac{4 b (a-b x)^{3/2}}{15 a^2 x^{3/2}}\\ \end{align*}
Mathematica [A] time = 0.0086985, size = 30, normalized size = 0.65 \[ -\frac{2 (a-b x)^{3/2} (3 a+2 b x)}{15 a^2 x^{5/2}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.003, size = 25, normalized size = 0.5 \begin{align*} -{\frac{4\,bx+6\,a}{15\,{a}^{2}} \left ( -bx+a \right ) ^{{\frac{3}{2}}}{x}^{-{\frac{5}{2}}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.05495, size = 45, normalized size = 0.98 \begin{align*} -\frac{2 \,{\left (\frac{5 \,{\left (-b x + a\right )}^{\frac{3}{2}} b}{x^{\frac{3}{2}}} + \frac{3 \,{\left (-b x + a\right )}^{\frac{5}{2}}}{x^{\frac{5}{2}}}\right )}}{15 \, a^{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.63756, size = 85, normalized size = 1.85 \begin{align*} \frac{2 \,{\left (2 \, b^{2} x^{2} + a b x - 3 \, a^{2}\right )} \sqrt{-b x + a}}{15 \, a^{2} x^{\frac{5}{2}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 20.0139, size = 245, normalized size = 5.33 \begin{align*} \begin{cases} - \frac{2 \sqrt{b} \sqrt{\frac{a}{b x} - 1}}{5 x^{2}} + \frac{2 b^{\frac{3}{2}} \sqrt{\frac{a}{b x} - 1}}{15 a x} + \frac{4 b^{\frac{5}{2}} \sqrt{\frac{a}{b x} - 1}}{15 a^{2}} & \text{for}\: \frac{\left |{a}\right |}{\left |{b}\right | \left |{x}\right |} > 1 \\\frac{6 i a^{3} b^{\frac{3}{2}} \sqrt{- \frac{a}{b x} + 1}}{x \left (- 15 a^{3} b x + 15 a^{2} b^{2} x^{2}\right )} - \frac{8 i a^{2} b^{\frac{5}{2}} \sqrt{- \frac{a}{b x} + 1}}{- 15 a^{3} b x + 15 a^{2} b^{2} x^{2}} - \frac{2 i a b^{\frac{7}{2}} x \sqrt{- \frac{a}{b x} + 1}}{- 15 a^{3} b x + 15 a^{2} b^{2} x^{2}} + \frac{4 i b^{\frac{9}{2}} x^{2} \sqrt{- \frac{a}{b x} + 1}}{- 15 a^{3} b x + 15 a^{2} b^{2} x^{2}} & \text{otherwise} \end{cases} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.23375, size = 82, normalized size = 1.78 \begin{align*} -\frac{{\left (b x - a\right )} \sqrt{-b x + a} b{\left (\frac{2 \,{\left (b x - a\right )}}{a^{3} b^{4}} + \frac{5}{a^{2} b^{4}}\right )}}{480 \,{\left ({\left (b x - a\right )} b + a b\right )}^{\frac{5}{2}}{\left | b \right |}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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