Optimal. Leaf size=66 \[ -5 a^{3/2} b \tanh ^{-1}\left (\frac{\sqrt{a+b x}}{\sqrt{a}}\right )-\frac{(a+b x)^{5/2}}{x}+\frac{5}{3} b (a+b x)^{3/2}+5 a b \sqrt{a+b x} \]
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Rubi [A] time = 0.0204844, antiderivative size = 66, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 4, integrand size = 13, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.308, Rules used = {47, 50, 63, 208} \[ -5 a^{3/2} b \tanh ^{-1}\left (\frac{\sqrt{a+b x}}{\sqrt{a}}\right )-\frac{(a+b x)^{5/2}}{x}+\frac{5}{3} b (a+b x)^{3/2}+5 a b \sqrt{a+b x} \]
Antiderivative was successfully verified.
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Rule 47
Rule 50
Rule 63
Rule 208
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 \tanh ^{-1}\left (\frac{\sqrt{a+b x}}{\sqrt{a}}\right )\\ \end{align*}
Mathematica [C] time = 0.012207, size = 33, normalized size = 0.5 \[ \frac{2 b (a+b x)^{7/2} \, _2F_1\left (2,\frac{7}{2};\frac{9}{2};\frac{b x}{a}+1\right )}{7 a^2} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.009, size = 61, normalized size = 0.9 \begin{align*} 2\,b \left ( 1/3\, \left ( bx+a \right ) ^{3/2}+2\,a\sqrt{bx+a}+{a}^{2} \left ( -1/2\,{\frac{\sqrt{bx+a}}{bx}}-5/2\,{\frac{1}{\sqrt{a}}{\it Artanh} \left ({\frac{\sqrt{bx+a}}{\sqrt{a}}} \right ) } \right ) \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 = 1.57004, size = 309, normalized size = 4.68 \begin{align*} \left [\frac{15 \, a^{\frac{3}{2}} 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 \, \sqrt{-a} a b x \arctan \left (\frac{\sqrt{b x + a} \sqrt{-a}}{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 [A] time = 4.71397, size = 99, normalized size = 1.5 \begin{align*} - \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 a^{\frac{3}{2}} b \log{\left (\frac{b x}{a} \right )}}{2} - 5 a^{\frac{3}{2}} b \log{\left (\sqrt{1 + \frac{b x}{a}} + 1 \right )} + \frac{2 \sqrt{a} b^{2} x \sqrt{1 + \frac{b x}{a}}}{3} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.20062, size = 100, normalized size = 1.52 \begin{align*} \frac{\frac{15 \, a^{2} b^{2} \arctan \left (\frac{\sqrt{b x + a}}{\sqrt{-a}}\right )}{\sqrt{-a}} + 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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