Optimal. Leaf size=83 \[ \frac{2 b^2 x^{3/2}}{3 a^3}-\frac{2 b^3 \sqrt{x}}{a^4}+\frac{2 b^{7/2} \tan ^{-1}\left (\frac{\sqrt{a} \sqrt{x}}{\sqrt{b}}\right )}{a^{9/2}}-\frac{2 b x^{5/2}}{5 a^2}+\frac{2 x^{7/2}}{7 a} \]
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Rubi [A] time = 0.0355053, antiderivative size = 83, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 4, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.267, Rules used = {263, 50, 63, 205} \[ \frac{2 b^2 x^{3/2}}{3 a^3}-\frac{2 b^3 \sqrt{x}}{a^4}+\frac{2 b^{7/2} \tan ^{-1}\left (\frac{\sqrt{a} \sqrt{x}}{\sqrt{b}}\right )}{a^{9/2}}-\frac{2 b x^{5/2}}{5 a^2}+\frac{2 x^{7/2}}{7 a} \]
Antiderivative was successfully verified.
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Rule 263
Rule 50
Rule 63
Rule 205
Rubi steps
\begin{align*} \int \frac{x^{5/2}}{a+\frac{b}{x}} \, dx &=\int \frac{x^{7/2}}{b+a x} \, dx\\ &=\frac{2 x^{7/2}}{7 a}-\frac{b \int \frac{x^{5/2}}{b+a x} \, dx}{a}\\ &=-\frac{2 b x^{5/2}}{5 a^2}+\frac{2 x^{7/2}}{7 a}+\frac{b^2 \int \frac{x^{3/2}}{b+a x} \, dx}{a^2}\\ &=\frac{2 b^2 x^{3/2}}{3 a^3}-\frac{2 b x^{5/2}}{5 a^2}+\frac{2 x^{7/2}}{7 a}-\frac{b^3 \int \frac{\sqrt{x}}{b+a x} \, dx}{a^3}\\ &=-\frac{2 b^3 \sqrt{x}}{a^4}+\frac{2 b^2 x^{3/2}}{3 a^3}-\frac{2 b x^{5/2}}{5 a^2}+\frac{2 x^{7/2}}{7 a}+\frac{b^4 \int \frac{1}{\sqrt{x} (b+a x)} \, dx}{a^4}\\ &=-\frac{2 b^3 \sqrt{x}}{a^4}+\frac{2 b^2 x^{3/2}}{3 a^3}-\frac{2 b x^{5/2}}{5 a^2}+\frac{2 x^{7/2}}{7 a}+\frac{\left (2 b^4\right ) \operatorname{Subst}\left (\int \frac{1}{b+a x^2} \, dx,x,\sqrt{x}\right )}{a^4}\\ &=-\frac{2 b^3 \sqrt{x}}{a^4}+\frac{2 b^2 x^{3/2}}{3 a^3}-\frac{2 b x^{5/2}}{5 a^2}+\frac{2 x^{7/2}}{7 a}+\frac{2 b^{7/2} \tan ^{-1}\left (\frac{\sqrt{a} \sqrt{x}}{\sqrt{b}}\right )}{a^{9/2}}\\ \end{align*}
Mathematica [A] time = 0.0286505, size = 72, normalized size = 0.87 \[ \frac{2 \sqrt{x} \left (-21 a^2 b x^2+15 a^3 x^3+35 a b^2 x-105 b^3\right )}{105 a^4}+\frac{2 b^{7/2} \tan ^{-1}\left (\frac{\sqrt{a} \sqrt{x}}{\sqrt{b}}\right )}{a^{9/2}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.008, size = 65, normalized size = 0.8 \begin{align*}{\frac{2}{7\,a}{x}^{{\frac{7}{2}}}}-{\frac{2\,b}{5\,{a}^{2}}{x}^{{\frac{5}{2}}}}+{\frac{2\,{b}^{2}}{3\,{a}^{3}}{x}^{{\frac{3}{2}}}}-2\,{\frac{{b}^{3}\sqrt{x}}{{a}^{4}}}+2\,{\frac{{b}^{4}}{{a}^{4}\sqrt{ab}}\arctan \left ({\frac{a\sqrt{x}}{\sqrt{ab}}} \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 = 1.82599, size = 366, normalized size = 4.41 \begin{align*} \left [\frac{105 \, b^{3} \sqrt{-\frac{b}{a}} \log \left (\frac{a x + 2 \, a \sqrt{x} \sqrt{-\frac{b}{a}} - b}{a x + b}\right ) + 2 \,{\left (15 \, a^{3} x^{3} - 21 \, a^{2} b x^{2} + 35 \, a b^{2} x - 105 \, b^{3}\right )} \sqrt{x}}{105 \, a^{4}}, \frac{2 \,{\left (105 \, b^{3} \sqrt{\frac{b}{a}} \arctan \left (\frac{a \sqrt{x} \sqrt{\frac{b}{a}}}{b}\right ) +{\left (15 \, a^{3} x^{3} - 21 \, a^{2} b x^{2} + 35 \, a b^{2} x - 105 \, b^{3}\right )} \sqrt{x}\right )}}{105 \, a^{4}}\right ] \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 20.7491, size = 136, normalized size = 1.64 \begin{align*} \begin{cases} \frac{2 x^{\frac{7}{2}}}{7 a} - \frac{2 b x^{\frac{5}{2}}}{5 a^{2}} + \frac{2 b^{2} x^{\frac{3}{2}}}{3 a^{3}} - \frac{2 b^{3} \sqrt{x}}{a^{4}} - \frac{i b^{\frac{7}{2}} \log{\left (- i \sqrt{b} \sqrt{\frac{1}{a}} + \sqrt{x} \right )}}{a^{5} \sqrt{\frac{1}{a}}} + \frac{i b^{\frac{7}{2}} \log{\left (i \sqrt{b} \sqrt{\frac{1}{a}} + \sqrt{x} \right )}}{a^{5} \sqrt{\frac{1}{a}}} & \text{for}\: a \neq 0 \\\frac{2 x^{\frac{9}{2}}}{9 b} & \text{otherwise} \end{cases} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.09908, size = 95, normalized size = 1.14 \begin{align*} \frac{2 \, b^{4} \arctan \left (\frac{a \sqrt{x}}{\sqrt{a b}}\right )}{\sqrt{a b} a^{4}} + \frac{2 \,{\left (15 \, a^{6} x^{\frac{7}{2}} - 21 \, a^{5} b x^{\frac{5}{2}} + 35 \, a^{4} b^{2} x^{\frac{3}{2}} - 105 \, a^{3} b^{3} \sqrt{x}\right )}}{105 \, a^{7}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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