Optimal. Leaf size=68 \[ \frac{3 b^2 \tan ^{-1}\left (\frac{\sqrt{b x-a}}{\sqrt{a}}\right )}{4 \sqrt{a}}-\frac{(b x-a)^{3/2}}{2 x^2}-\frac{3 b \sqrt{b x-a}}{4 x} \]
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Rubi [A] time = 0.0148347, antiderivative size = 68, 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 = {47, 63, 205} \[ \frac{3 b^2 \tan ^{-1}\left (\frac{\sqrt{b x-a}}{\sqrt{a}}\right )}{4 \sqrt{a}}-\frac{(b x-a)^{3/2}}{2 x^2}-\frac{3 b \sqrt{b x-a}}{4 x} \]
Antiderivative was successfully verified.
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Rule 47
Rule 63
Rule 205
Rubi steps
\begin{align*} \int \frac{(-a+b x)^{3/2}}{x^3} \, dx &=-\frac{(-a+b x)^{3/2}}{2 x^2}+\frac{1}{4} (3 b) \int \frac{\sqrt{-a+b x}}{x^2} \, dx\\ &=-\frac{3 b \sqrt{-a+b x}}{4 x}-\frac{(-a+b x)^{3/2}}{2 x^2}+\frac{1}{8} \left (3 b^2\right ) \int \frac{1}{x \sqrt{-a+b x}} \, dx\\ &=-\frac{3 b \sqrt{-a+b x}}{4 x}-\frac{(-a+b x)^{3/2}}{2 x^2}+\frac{1}{4} (3 b) \operatorname{Subst}\left (\int \frac{1}{\frac{a}{b}+\frac{x^2}{b}} \, dx,x,\sqrt{-a+b x}\right )\\ &=-\frac{3 b \sqrt{-a+b x}}{4 x}-\frac{(-a+b x)^{3/2}}{2 x^2}+\frac{3 b^2 \tan ^{-1}\left (\frac{\sqrt{-a+b x}}{\sqrt{a}}\right )}{4 \sqrt{a}}\\ \end{align*}
Mathematica [A] time = 0.0521995, size = 72, normalized size = 1.06 \[ -\frac{2 a^2+3 b^2 x^2 \sqrt{1-\frac{b x}{a}} \tanh ^{-1}\left (\sqrt{1-\frac{b x}{a}}\right )-7 a b x+5 b^2 x^2}{4 x^2 \sqrt{b x-a}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.01, size = 53, normalized size = 0.8 \begin{align*} -{\frac{5}{4\,{x}^{2}} \left ( bx-a \right ) ^{{\frac{3}{2}}}}-{\frac{3\,a}{4\,{x}^{2}}\sqrt{bx-a}}+{\frac{3\,{b}^{2}}{4}\arctan \left ({\sqrt{bx-a}{\frac{1}{\sqrt{a}}}} \right ){\frac{1}{\sqrt{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 = 1.557, size = 294, normalized size = 4.32 \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 (5 \, a b x - 2 \, a^{2}\right )} \sqrt{b x - a}}{8 \, a x^{2}}, \frac{3 \, \sqrt{a} b^{2} x^{2} \arctan \left (\frac{\sqrt{b x - a}}{\sqrt{a}}\right ) -{\left (5 \, a b x - 2 \, a^{2}\right )} \sqrt{b x - a}}{4 \, a x^{2}}\right ] \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 4.50158, size = 194, normalized size = 2.85 \begin{align*} \begin{cases} \frac{i a^{2}}{2 \sqrt{b} x^{\frac{5}{2}} \sqrt{\frac{a}{b x} - 1}} - \frac{7 i a \sqrt{b}}{4 x^{\frac{3}{2}} \sqrt{\frac{a}{b x} - 1}} + \frac{5 i b^{\frac{3}{2}}}{4 \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 \sqrt{a}} & \text{for}\: \frac{\left |{a}\right |}{\left |{b}\right | \left |{x}\right |} > 1 \\\frac{a \sqrt{b} \sqrt{- \frac{a}{b x} + 1}}{2 x^{\frac{3}{2}}} - \frac{5 b^{\frac{3}{2}} \sqrt{- \frac{a}{b x} + 1}}{4 \sqrt{x}} - \frac{3 b^{2} \operatorname{asin}{\left (\frac{\sqrt{a}}{\sqrt{b} \sqrt{x}} \right )}}{4 \sqrt{a}} & \text{otherwise} \end{cases} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.21901, size = 89, normalized size = 1.31 \begin{align*} \frac{\frac{3 \, b^{3} \arctan \left (\frac{\sqrt{b x - a}}{\sqrt{a}}\right )}{\sqrt{a}} - \frac{5 \,{\left (b x - a\right )}^{\frac{3}{2}} b^{3} + 3 \, \sqrt{b x - a} a b^{3}}{b^{2} x^{2}}}{4 \, b} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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