Optimal. Leaf size=68 \[ -\frac{2 a^2}{b^3 \sqrt{x}}-\frac{2 a^{5/2} \tan ^{-1}\left (\frac{\sqrt{a} \sqrt{x}}{\sqrt{b}}\right )}{b^{7/2}}+\frac{2 a}{3 b^2 x^{3/2}}-\frac{2}{5 b x^{5/2}} \]
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Rubi [A] time = 0.0253934, antiderivative size = 68, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 4, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.267, Rules used = {263, 51, 63, 205} \[ -\frac{2 a^2}{b^3 \sqrt{x}}-\frac{2 a^{5/2} \tan ^{-1}\left (\frac{\sqrt{a} \sqrt{x}}{\sqrt{b}}\right )}{b^{7/2}}+\frac{2 a}{3 b^2 x^{3/2}}-\frac{2}{5 b x^{5/2}} \]
Antiderivative was successfully verified.
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Rule 263
Rule 51
Rule 63
Rule 205
Rubi steps
\begin{align*} \int \frac{1}{\left (a+\frac{b}{x}\right ) x^{9/2}} \, dx &=\int \frac{1}{x^{7/2} (b+a x)} \, dx\\ &=-\frac{2}{5 b x^{5/2}}-\frac{a \int \frac{1}{x^{5/2} (b+a x)} \, dx}{b}\\ &=-\frac{2}{5 b x^{5/2}}+\frac{2 a}{3 b^2 x^{3/2}}+\frac{a^2 \int \frac{1}{x^{3/2} (b+a x)} \, dx}{b^2}\\ &=-\frac{2}{5 b x^{5/2}}+\frac{2 a}{3 b^2 x^{3/2}}-\frac{2 a^2}{b^3 \sqrt{x}}-\frac{a^3 \int \frac{1}{\sqrt{x} (b+a x)} \, dx}{b^3}\\ &=-\frac{2}{5 b x^{5/2}}+\frac{2 a}{3 b^2 x^{3/2}}-\frac{2 a^2}{b^3 \sqrt{x}}-\frac{\left (2 a^3\right ) \operatorname{Subst}\left (\int \frac{1}{b+a x^2} \, dx,x,\sqrt{x}\right )}{b^3}\\ &=-\frac{2}{5 b x^{5/2}}+\frac{2 a}{3 b^2 x^{3/2}}-\frac{2 a^2}{b^3 \sqrt{x}}-\frac{2 a^{5/2} \tan ^{-1}\left (\frac{\sqrt{a} \sqrt{x}}{\sqrt{b}}\right )}{b^{7/2}}\\ \end{align*}
Mathematica [C] time = 0.0049082, size = 27, normalized size = 0.4 \[ -\frac{2 \, _2F_1\left (-\frac{5}{2},1;-\frac{3}{2};-\frac{a x}{b}\right )}{5 b x^{5/2}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.008, size = 54, normalized size = 0.8 \begin{align*} -2\,{\frac{{a}^{3}}{{b}^{3}\sqrt{ab}}\arctan \left ({\frac{a\sqrt{x}}{\sqrt{ab}}} \right ) }-{\frac{2}{5\,b}{x}^{-{\frac{5}{2}}}}-2\,{\frac{{a}^{2}}{{b}^{3}\sqrt{x}}}+{\frac{2\,a}{3\,{b}^{2}}{x}^{-{\frac{3}{2}}}} \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 = 1.75661, size = 336, normalized size = 4.94 \begin{align*} \left [\frac{15 \, a^{2} x^{3} \sqrt{-\frac{a}{b}} \log \left (\frac{a x - 2 \, b \sqrt{x} \sqrt{-\frac{a}{b}} - b}{a x + b}\right ) - 2 \,{\left (15 \, a^{2} x^{2} - 5 \, a b x + 3 \, b^{2}\right )} \sqrt{x}}{15 \, b^{3} x^{3}}, \frac{2 \,{\left (15 \, a^{2} x^{3} \sqrt{\frac{a}{b}} \arctan \left (\frac{b \sqrt{\frac{a}{b}}}{a \sqrt{x}}\right ) -{\left (15 \, a^{2} x^{2} - 5 \, a b x + 3 \, b^{2}\right )} \sqrt{x}\right )}}{15 \, b^{3} x^{3}}\right ] \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 103.053, size = 139, normalized size = 2.04 \begin{align*} \begin{cases} \frac{\tilde{\infty }}{x^{\frac{5}{2}}} & \text{for}\: a = 0 \wedge b = 0 \\- \frac{2}{5 b x^{\frac{5}{2}}} & \text{for}\: a = 0 \\- \frac{2}{7 a x^{\frac{7}{2}}} & \text{for}\: b = 0 \\- \frac{2 a^{2}}{b^{3} \sqrt{x}} + \frac{i a^{2} \log{\left (- i \sqrt{b} \sqrt{\frac{1}{a}} + \sqrt{x} \right )}}{b^{\frac{7}{2}} \sqrt{\frac{1}{a}}} - \frac{i a^{2} \log{\left (i \sqrt{b} \sqrt{\frac{1}{a}} + \sqrt{x} \right )}}{b^{\frac{7}{2}} \sqrt{\frac{1}{a}}} + \frac{2 a}{3 b^{2} x^{\frac{3}{2}}} - \frac{2}{5 b x^{\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 = 1.11328, size = 70, normalized size = 1.03 \begin{align*} -\frac{2 \, a^{3} \arctan \left (\frac{a \sqrt{x}}{\sqrt{a b}}\right )}{\sqrt{a b} b^{3}} - \frac{2 \,{\left (15 \, a^{2} x^{2} - 5 \, a b x + 3 \, b^{2}\right )}}{15 \, b^{3} x^{\frac{5}{2}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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