Optimal. Leaf size=81 \[ -\frac{5 b^2 \sqrt{a+b x}}{8 x}-\frac{5 b^3 \tanh ^{-1}\left (\frac{\sqrt{a+b x}}{\sqrt{a}}\right )}{8 \sqrt{a}}-\frac{5 b (a+b x)^{3/2}}{12 x^2}-\frac{(a+b x)^{5/2}}{3 x^3} \]
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Rubi [A] time = 0.0226134, antiderivative size = 81, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 3, integrand size = 13, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.231, Rules used = {47, 63, 208} \[ -\frac{5 b^2 \sqrt{a+b x}}{8 x}-\frac{5 b^3 \tanh ^{-1}\left (\frac{\sqrt{a+b x}}{\sqrt{a}}\right )}{8 \sqrt{a}}-\frac{5 b (a+b x)^{3/2}}{12 x^2}-\frac{(a+b x)^{5/2}}{3 x^3} \]
Antiderivative was successfully verified.
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Rule 47
Rule 63
Rule 208
Rubi steps
\begin{align*} \int \frac{(a+b x)^{5/2}}{x^4} \, dx &=-\frac{(a+b x)^{5/2}}{3 x^3}+\frac{1}{6} (5 b) \int \frac{(a+b x)^{3/2}}{x^3} \, dx\\ &=-\frac{5 b (a+b x)^{3/2}}{12 x^2}-\frac{(a+b x)^{5/2}}{3 x^3}+\frac{1}{8} \left (5 b^2\right ) \int \frac{\sqrt{a+b x}}{x^2} \, dx\\ &=-\frac{5 b^2 \sqrt{a+b x}}{8 x}-\frac{5 b (a+b x)^{3/2}}{12 x^2}-\frac{(a+b x)^{5/2}}{3 x^3}+\frac{1}{16} \left (5 b^3\right ) \int \frac{1}{x \sqrt{a+b x}} \, dx\\ &=-\frac{5 b^2 \sqrt{a+b x}}{8 x}-\frac{5 b (a+b x)^{3/2}}{12 x^2}-\frac{(a+b x)^{5/2}}{3 x^3}+\frac{1}{8} \left (5 b^2\right ) \operatorname{Subst}\left (\int \frac{1}{-\frac{a}{b}+\frac{x^2}{b}} \, dx,x,\sqrt{a+b x}\right )\\ &=-\frac{5 b^2 \sqrt{a+b x}}{8 x}-\frac{5 b (a+b x)^{3/2}}{12 x^2}-\frac{(a+b x)^{5/2}}{3 x^3}-\frac{5 b^3 \tanh ^{-1}\left (\frac{\sqrt{a+b x}}{\sqrt{a}}\right )}{8 \sqrt{a}}\\ \end{align*}
Mathematica [A] time = 0.0569685, size = 79, normalized size = 0.98 \[ -\frac{34 a^2 b x+8 a^3+59 a b^2 x^2+15 b^3 x^3 \sqrt{\frac{b x}{a}+1} \tanh ^{-1}\left (\sqrt{\frac{b x}{a}+1}\right )+33 b^3 x^3}{24 x^3 \sqrt{a+b x}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.009, size = 63, normalized size = 0.8 \begin{align*} 2\,{b}^{3} \left ({\frac{1}{{b}^{3}{x}^{3}} \left ( -{\frac{11\, \left ( bx+a \right ) ^{5/2}}{16}}+5/6\,a \left ( bx+a \right ) ^{3/2}-{\frac{5\,{a}^{2}\sqrt{bx+a}}{16}} \right ) }-{\frac{5}{16\,\sqrt{a}}{\it Artanh} \left ({\frac{\sqrt{bx+a}}{\sqrt{a}}} \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 = 1.59282, size = 350, normalized size = 4.32 \begin{align*} \left [\frac{15 \, \sqrt{a} b^{3} x^{3} \log \left (\frac{b x - 2 \, \sqrt{b x + a} \sqrt{a} + 2 \, a}{x}\right ) - 2 \,{\left (33 \, a b^{2} x^{2} + 26 \, a^{2} b x + 8 \, a^{3}\right )} \sqrt{b x + a}}{48 \, a x^{3}}, \frac{15 \, \sqrt{-a} b^{3} x^{3} \arctan \left (\frac{\sqrt{b x + a} \sqrt{-a}}{a}\right ) -{\left (33 \, a b^{2} x^{2} + 26 \, a^{2} b x + 8 \, a^{3}\right )} \sqrt{b x + a}}{24 \, a x^{3}}\right ] \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 6.73303, size = 104, normalized size = 1.28 \begin{align*} - \frac{a^{2} \sqrt{b} \sqrt{\frac{a}{b x} + 1}}{3 x^{\frac{5}{2}}} - \frac{13 a b^{\frac{3}{2}} \sqrt{\frac{a}{b x} + 1}}{12 x^{\frac{3}{2}}} - \frac{11 b^{\frac{5}{2}} \sqrt{\frac{a}{b x} + 1}}{8 \sqrt{x}} - \frac{5 b^{3} \operatorname{asinh}{\left (\frac{\sqrt{a}}{\sqrt{b} \sqrt{x}} \right )}}{8 \sqrt{a}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.28311, size = 107, normalized size = 1.32 \begin{align*} \frac{\frac{15 \, b^{4} \arctan \left (\frac{\sqrt{b x + a}}{\sqrt{-a}}\right )}{\sqrt{-a}} - \frac{33 \,{\left (b x + a\right )}^{\frac{5}{2}} b^{4} - 40 \,{\left (b x + a\right )}^{\frac{3}{2}} a b^{4} + 15 \, \sqrt{b x + a} a^{2} b^{4}}{b^{3} x^{3}}}{24 \, b} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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