Optimal. Leaf size=71 \[ \frac{b^2 \tan ^{-1}\left (\frac{\sqrt{b x-a}}{\sqrt{a}}\right )}{4 a^{3/2}}-\frac{\sqrt{b x-a}}{2 x^2}+\frac{b \sqrt{b x-a}}{4 a x} \]
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Rubi [A] time = 0.015946, antiderivative size = 71, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.267, Rules used = {47, 51, 63, 205} \[ \frac{b^2 \tan ^{-1}\left (\frac{\sqrt{b x-a}}{\sqrt{a}}\right )}{4 a^{3/2}}-\frac{\sqrt{b x-a}}{2 x^2}+\frac{b \sqrt{b x-a}}{4 a x} \]
Antiderivative was successfully verified.
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Rule 47
Rule 51
Rule 63
Rule 205
Rubi steps
\begin{align*} \int \frac{\sqrt{-a+b x}}{x^3} \, dx &=-\frac{\sqrt{-a+b x}}{2 x^2}+\frac{1}{4} b \int \frac{1}{x^2 \sqrt{-a+b x}} \, dx\\ &=-\frac{\sqrt{-a+b x}}{2 x^2}+\frac{b \sqrt{-a+b x}}{4 a x}+\frac{b^2 \int \frac{1}{x \sqrt{-a+b x}} \, dx}{8 a}\\ &=-\frac{\sqrt{-a+b x}}{2 x^2}+\frac{b \sqrt{-a+b x}}{4 a x}+\frac{b \operatorname{Subst}\left (\int \frac{1}{\frac{a}{b}+\frac{x^2}{b}} \, dx,x,\sqrt{-a+b x}\right )}{4 a}\\ &=-\frac{\sqrt{-a+b x}}{2 x^2}+\frac{b \sqrt{-a+b x}}{4 a x}+\frac{b^2 \tan ^{-1}\left (\frac{\sqrt{-a+b x}}{\sqrt{a}}\right )}{4 a^{3/2}}\\ \end{align*}
Mathematica [C] time = 0.0241265, size = 38, normalized size = 0.54 \[ \frac{2 b^2 (b x-a)^{3/2} \, _2F_1\left (\frac{3}{2},3;\frac{5}{2};1-\frac{b x}{a}\right )}{3 a^3} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.011, size = 55, normalized size = 0.8 \begin{align*}{\frac{1}{4\,a{x}^{2}} \left ( bx-a \right ) ^{{\frac{3}{2}}}}-{\frac{1}{4\,{x}^{2}}\sqrt{bx-a}}+{\frac{{b}^{2}}{4}\arctan \left ({\sqrt{bx-a}{\frac{1}{\sqrt{a}}}} \right ){a}^{-{\frac{3}{2}}}} \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 = 1.60857, size = 289, normalized size = 4.07 \begin{align*} \left [-\frac{\sqrt{-a} b^{2} x^{2} \log \left (\frac{b x - 2 \, \sqrt{b x - a} \sqrt{-a} - 2 \, a}{x}\right ) - 2 \,{\left (a b x - 2 \, a^{2}\right )} \sqrt{b x - a}}{8 \, a^{2} x^{2}}, \frac{\sqrt{a} b^{2} x^{2} \arctan \left (\frac{\sqrt{b x - a}}{\sqrt{a}}\right ) +{\left (a b x - 2 \, a^{2}\right )} \sqrt{b x - a}}{4 \, a^{2} x^{2}}\right ] \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 4.88827, size = 211, normalized size = 2.97 \begin{align*} \begin{cases} - \frac{i a}{2 \sqrt{b} x^{\frac{5}{2}} \sqrt{\frac{a}{b x} - 1}} + \frac{3 i \sqrt{b}}{4 x^{\frac{3}{2}} \sqrt{\frac{a}{b x} - 1}} - \frac{i b^{\frac{3}{2}}}{4 a \sqrt{x} \sqrt{\frac{a}{b x} - 1}} + \frac{i b^{2} \operatorname{acosh}{\left (\frac{\sqrt{a}}{\sqrt{b} \sqrt{x}} \right )}}{4 a^{\frac{3}{2}}} & \text{for}\: \frac{\left |{a}\right |}{\left |{b}\right | \left |{x}\right |} > 1 \\\frac{a}{2 \sqrt{b} x^{\frac{5}{2}} \sqrt{- \frac{a}{b x} + 1}} - \frac{3 \sqrt{b}}{4 x^{\frac{3}{2}} \sqrt{- \frac{a}{b x} + 1}} + \frac{b^{\frac{3}{2}}}{4 a \sqrt{x} \sqrt{- \frac{a}{b x} + 1}} - \frac{b^{2} \operatorname{asin}{\left (\frac{\sqrt{a}}{\sqrt{b} \sqrt{x}} \right )}}{4 a^{\frac{3}{2}}} & \text{otherwise} \end{cases} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.21496, size = 89, normalized size = 1.25 \begin{align*} \frac{\frac{b^{3} \arctan \left (\frac{\sqrt{b x - a}}{\sqrt{a}}\right )}{a^{\frac{3}{2}}} + \frac{{\left (b x - a\right )}^{\frac{3}{2}} b^{3} - \sqrt{b x - a} a b^{3}}{a b^{2} x^{2}}}{4 \, b} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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