Optimal. Leaf size=74 \[ 5 a^{3/2} b \tan ^{-1}\left (\frac{\sqrt{b x-a}}{\sqrt{a}}\right )-\frac{(b x-a)^{5/2}}{x}+\frac{5}{3} b (b x-a)^{3/2}-5 a b \sqrt{b x-a} \]
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Rubi [A] time = 0.0197625, antiderivative size = 74, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 4, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.267, Rules used = {47, 50, 63, 205} \[ 5 a^{3/2} b \tan ^{-1}\left (\frac{\sqrt{b x-a}}{\sqrt{a}}\right )-\frac{(b x-a)^{5/2}}{x}+\frac{5}{3} b (b x-a)^{3/2}-5 a b \sqrt{b x-a} \]
Antiderivative was successfully verified.
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Rule 47
Rule 50
Rule 63
Rule 205
Rubi steps
\begin{align*} \int \frac{(-a+b x)^{5/2}}{x^2} \, dx &=-\frac{(-a+b x)^{5/2}}{x}+\frac{1}{2} (5 b) \int \frac{(-a+b x)^{3/2}}{x} \, dx\\ &=\frac{5}{3} b (-a+b x)^{3/2}-\frac{(-a+b x)^{5/2}}{x}-\frac{1}{2} (5 a b) \int \frac{\sqrt{-a+b x}}{x} \, dx\\ &=-5 a b \sqrt{-a+b x}+\frac{5}{3} b (-a+b x)^{3/2}-\frac{(-a+b x)^{5/2}}{x}+\frac{1}{2} \left (5 a^2 b\right ) \int \frac{1}{x \sqrt{-a+b x}} \, dx\\ &=-5 a b \sqrt{-a+b x}+\frac{5}{3} b (-a+b x)^{3/2}-\frac{(-a+b x)^{5/2}}{x}+\left (5 a^2\right ) \operatorname{Subst}\left (\int \frac{1}{\frac{a}{b}+\frac{x^2}{b}} \, dx,x,\sqrt{-a+b x}\right )\\ &=-5 a b \sqrt{-a+b x}+\frac{5}{3} b (-a+b x)^{3/2}-\frac{(-a+b x)^{5/2}}{x}+5 a^{3/2} b \tan ^{-1}\left (\frac{\sqrt{-a+b x}}{\sqrt{a}}\right )\\ \end{align*}
Mathematica [C] time = 0.0175202, size = 36, normalized size = 0.49 \[ \frac{2 b (b x-a)^{7/2} \, _2F_1\left (2,\frac{7}{2};\frac{9}{2};1-\frac{b x}{a}\right )}{7 a^2} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.01, size = 64, normalized size = 0.9 \begin{align*}{\frac{2\,b}{3} \left ( bx-a \right ) ^{{\frac{3}{2}}}}-4\,ab\sqrt{bx-a}-{\frac{{a}^{2}}{x}\sqrt{bx-a}}+5\,{a}^{3/2}b\arctan \left ({\frac{\sqrt{bx-a}}{\sqrt{a}}} \right ) \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 = 1.5684, size = 306, normalized size = 4.14 \begin{align*} \left [\frac{15 \, \sqrt{-a} a b x \log \left (\frac{b x + 2 \, \sqrt{b x - a} \sqrt{-a} - 2 \, a}{x}\right ) + 2 \,{\left (2 \, b^{2} x^{2} - 14 \, a b x - 3 \, a^{2}\right )} \sqrt{b x - a}}{6 \, x}, \frac{15 \, a^{\frac{3}{2}} b x \arctan \left (\frac{\sqrt{b x - a}}{\sqrt{a}}\right ) +{\left (2 \, b^{2} x^{2} - 14 \, a b x - 3 \, a^{2}\right )} \sqrt{b x - a}}{3 \, x}\right ] \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [C] time = 4.85502, size = 246, normalized size = 3.32 \begin{align*} \begin{cases} - \frac{a^{\frac{5}{2}} \sqrt{-1 + \frac{b x}{a}}}{x} - \frac{14 a^{\frac{3}{2}} b \sqrt{-1 + \frac{b x}{a}}}{3} - \frac{5 i a^{\frac{3}{2}} b \log{\left (\frac{b x}{a} \right )}}{2} + 5 i a^{\frac{3}{2}} b \log{\left (\frac{\sqrt{b} \sqrt{x}}{\sqrt{a}} \right )} - 5 a^{\frac{3}{2}} b \operatorname{asin}{\left (\frac{\sqrt{a}}{\sqrt{b} \sqrt{x}} \right )} + \frac{2 \sqrt{a} b^{2} x \sqrt{-1 + \frac{b x}{a}}}{3} & \text{for}\: \frac{\left |{b x}\right |}{\left |{a}\right |} > 1 \\- \frac{i a^{\frac{5}{2}} \sqrt{1 - \frac{b x}{a}}}{x} - \frac{14 i a^{\frac{3}{2}} b \sqrt{1 - \frac{b x}{a}}}{3} - \frac{5 i a^{\frac{3}{2}} b \log{\left (\frac{b x}{a} \right )}}{2} + 5 i a^{\frac{3}{2}} b \log{\left (\sqrt{1 - \frac{b x}{a}} + 1 \right )} + \frac{2 i \sqrt{a} b^{2} x \sqrt{1 - \frac{b x}{a}}}{3} & \text{otherwise} \end{cases} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.14342, size = 101, normalized size = 1.36 \begin{align*} \frac{15 \, a^{\frac{3}{2}} b^{2} \arctan \left (\frac{\sqrt{b x - a}}{\sqrt{a}}\right ) + 2 \,{\left (b x - a\right )}^{\frac{3}{2}} b^{2} - 12 \, \sqrt{b x - a} a b^{2} - \frac{3 \, \sqrt{b x - a} a^{2} b}{x}}{3 \, b} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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