Optimal. Leaf size=74 \[ \frac{3 b^2 \tan ^{-1}\left (\frac{\sqrt{b x-a}}{\sqrt{a}}\right )}{4 a^{5/2}}+\frac{3 b \sqrt{b x-a}}{4 a^2 x}+\frac{\sqrt{b x-a}}{2 a x^2} \]
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Rubi [A] time = 0.0170277, antiderivative size = 74, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.2, Rules used = {51, 63, 205} \[ \frac{3 b^2 \tan ^{-1}\left (\frac{\sqrt{b x-a}}{\sqrt{a}}\right )}{4 a^{5/2}}+\frac{3 b \sqrt{b x-a}}{4 a^2 x}+\frac{\sqrt{b x-a}}{2 a x^2} \]
Antiderivative was successfully verified.
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Rule 51
Rule 63
Rule 205
Rubi steps
\begin{align*} \int \frac{1}{x^3 \sqrt{-a+b x}} \, dx &=\frac{\sqrt{-a+b x}}{2 a x^2}+\frac{(3 b) \int \frac{1}{x^2 \sqrt{-a+b x}} \, dx}{4 a}\\ &=\frac{\sqrt{-a+b x}}{2 a x^2}+\frac{3 b \sqrt{-a+b x}}{4 a^2 x}+\frac{\left (3 b^2\right ) \int \frac{1}{x \sqrt{-a+b x}} \, dx}{8 a^2}\\ &=\frac{\sqrt{-a+b x}}{2 a x^2}+\frac{3 b \sqrt{-a+b x}}{4 a^2 x}+\frac{(3 b) \operatorname{Subst}\left (\int \frac{1}{\frac{a}{b}+\frac{x^2}{b}} \, dx,x,\sqrt{-a+b x}\right )}{4 a^2}\\ &=\frac{\sqrt{-a+b x}}{2 a x^2}+\frac{3 b \sqrt{-a+b x}}{4 a^2 x}+\frac{3 b^2 \tan ^{-1}\left (\frac{\sqrt{-a+b x}}{\sqrt{a}}\right )}{4 a^{5/2}}\\ \end{align*}
Mathematica [C] time = 0.0061519, size = 36, normalized size = 0.49 \[ \frac{2 b^2 \sqrt{b x-a} \, _2F_1\left (\frac{1}{2},3;\frac{3}{2};1-\frac{b x}{a}\right )}{a^3} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.006, size = 59, normalized size = 0.8 \begin{align*}{\frac{3\,{b}^{2}}{4}\arctan \left ({\sqrt{bx-a}{\frac{1}{\sqrt{a}}}} \right ){a}^{-{\frac{5}{2}}}}+{\frac{1}{2\,a{x}^{2}}\sqrt{bx-a}}+{\frac{3\,b}{4\,{a}^{2}x}\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 = 1.54129, size = 300, normalized size = 4.05 \begin{align*} \left [-\frac{3 \, \sqrt{-a} b^{2} x^{2} \log \left (\frac{b x - 2 \, \sqrt{b x - a} \sqrt{-a} - 2 \, a}{x}\right ) - 2 \,{\left (3 \, a b x + 2 \, a^{2}\right )} \sqrt{b x - a}}{8 \, a^{3} x^{2}}, \frac{3 \, \sqrt{a} b^{2} x^{2} \arctan \left (\frac{\sqrt{b x - a}}{\sqrt{a}}\right ) +{\left (3 \, a b x + 2 \, a^{2}\right )} \sqrt{b x - a}}{4 \, a^{3} x^{2}}\right ] \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 5.75695, size = 219, normalized size = 2.96 \begin{align*} \begin{cases} \frac{i}{2 \sqrt{b} x^{\frac{5}{2}} \sqrt{\frac{a}{b x} - 1}} + \frac{i \sqrt{b}}{4 a x^{\frac{3}{2}} \sqrt{\frac{a}{b x} - 1}} - \frac{3 i b^{\frac{3}{2}}}{4 a^{2} \sqrt{x} \sqrt{\frac{a}{b x} - 1}} + \frac{3 i b^{2} \operatorname{acosh}{\left (\frac{\sqrt{a}}{\sqrt{b} \sqrt{x}} \right )}}{4 a^{\frac{5}{2}}} & \text{for}\: \frac{\left |{a}\right |}{\left |{b}\right | \left |{x}\right |} > 1 \\- \frac{1}{2 \sqrt{b} x^{\frac{5}{2}} \sqrt{- \frac{a}{b x} + 1}} - \frac{\sqrt{b}}{4 a x^{\frac{3}{2}} \sqrt{- \frac{a}{b x} + 1}} + \frac{3 b^{\frac{3}{2}}}{4 a^{2} \sqrt{x} \sqrt{- \frac{a}{b x} + 1}} - \frac{3 b^{2} \operatorname{asin}{\left (\frac{\sqrt{a}}{\sqrt{b} \sqrt{x}} \right )}}{4 a^{\frac{5}{2}}} & \text{otherwise} \end{cases} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.18143, size = 92, normalized size = 1.24 \begin{align*} \frac{\frac{3 \, b^{3} \arctan \left (\frac{\sqrt{b x - a}}{\sqrt{a}}\right )}{a^{\frac{5}{2}}} + \frac{3 \,{\left (b x - a\right )}^{\frac{3}{2}} b^{3} + 5 \, \sqrt{b x - a} a b^{3}}{a^{2} b^{2} x^{2}}}{4 \, b} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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