Optimal. Leaf size=68 \[ \frac{2 a^2 \sqrt{x}}{b^3}-\frac{2 a^{5/2} \tan ^{-1}\left (\frac{\sqrt{b} \sqrt{x}}{\sqrt{a}}\right )}{b^{7/2}}-\frac{2 a x^{3/2}}{3 b^2}+\frac{2 x^{5/2}}{5 b} \]
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Rubi [A] time = 0.0319405, antiderivative size = 68, 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 = {50, 63, 205} \[ \frac{2 a^2 \sqrt{x}}{b^3}-\frac{2 a^{5/2} \tan ^{-1}\left (\frac{\sqrt{b} \sqrt{x}}{\sqrt{a}}\right )}{b^{7/2}}-\frac{2 a x^{3/2}}{3 b^2}+\frac{2 x^{5/2}}{5 b} \]
Antiderivative was successfully verified.
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Rule 50
Rule 63
Rule 205
Rubi steps
\begin{align*} \int \frac{x^{5/2}}{a+b x} \, dx &=\frac{2 x^{5/2}}{5 b}-\frac{a \int \frac{x^{3/2}}{a+b x} \, dx}{b}\\ &=-\frac{2 a x^{3/2}}{3 b^2}+\frac{2 x^{5/2}}{5 b}+\frac{a^2 \int \frac{\sqrt{x}}{a+b x} \, dx}{b^2}\\ &=\frac{2 a^2 \sqrt{x}}{b^3}-\frac{2 a x^{3/2}}{3 b^2}+\frac{2 x^{5/2}}{5 b}-\frac{a^3 \int \frac{1}{\sqrt{x} (a+b x)} \, dx}{b^3}\\ &=\frac{2 a^2 \sqrt{x}}{b^3}-\frac{2 a x^{3/2}}{3 b^2}+\frac{2 x^{5/2}}{5 b}-\frac{\left (2 a^3\right ) \operatorname{Subst}\left (\int \frac{1}{a+b x^2} \, dx,x,\sqrt{x}\right )}{b^3}\\ &=\frac{2 a^2 \sqrt{x}}{b^3}-\frac{2 a x^{3/2}}{3 b^2}+\frac{2 x^{5/2}}{5 b}-\frac{2 a^{5/2} \tan ^{-1}\left (\frac{\sqrt{b} \sqrt{x}}{\sqrt{a}}\right )}{b^{7/2}}\\ \end{align*}
Mathematica [A] time = 0.0296076, size = 61, normalized size = 0.9 \[ \frac{2 \sqrt{x} \left (15 a^2-5 a b x+3 b^2 x^2\right )}{15 b^3}-\frac{2 a^{5/2} \tan ^{-1}\left (\frac{\sqrt{b} \sqrt{x}}{\sqrt{a}}\right )}{b^{7/2}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.007, size = 54, normalized size = 0.8 \begin{align*}{\frac{2}{5\,b}{x}^{{\frac{5}{2}}}}-{\frac{2\,a}{3\,{b}^{2}}{x}^{{\frac{3}{2}}}}+2\,{\frac{{a}^{2}\sqrt{x}}{{b}^{3}}}-2\,{\frac{{a}^{3}}{{b}^{3}\sqrt{ab}}\arctan \left ({\frac{b\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 = 1.52884, size = 308, normalized size = 4.53 \begin{align*} \left [\frac{15 \, a^{2} \sqrt{-\frac{a}{b}} \log \left (\frac{b x - 2 \, b \sqrt{x} \sqrt{-\frac{a}{b}} - a}{b x + a}\right ) + 2 \,{\left (3 \, b^{2} x^{2} - 5 \, a b x + 15 \, a^{2}\right )} \sqrt{x}}{15 \, b^{3}}, -\frac{2 \,{\left (15 \, a^{2} \sqrt{\frac{a}{b}} \arctan \left (\frac{b \sqrt{x} \sqrt{\frac{a}{b}}}{a}\right ) -{\left (3 \, b^{2} x^{2} - 5 \, a b x + 15 \, a^{2}\right )} \sqrt{x}\right )}}{15 \, b^{3}}\right ] \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 16.3615, size = 121, normalized size = 1.78 \begin{align*} \begin{cases} \frac{i a^{\frac{5}{2}} \log{\left (- i \sqrt{a} \sqrt{\frac{1}{b}} + \sqrt{x} \right )}}{b^{4} \sqrt{\frac{1}{b}}} - \frac{i a^{\frac{5}{2}} \log{\left (i \sqrt{a} \sqrt{\frac{1}{b}} + \sqrt{x} \right )}}{b^{4} \sqrt{\frac{1}{b}}} + \frac{2 a^{2} \sqrt{x}}{b^{3}} - \frac{2 a x^{\frac{3}{2}}}{3 b^{2}} + \frac{2 x^{\frac{5}{2}}}{5 b} & \text{for}\: b \neq 0 \\\frac{2 x^{\frac{7}{2}}}{7 a} & \text{otherwise} \end{cases} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.19747, size = 80, normalized size = 1.18 \begin{align*} -\frac{2 \, a^{3} \arctan \left (\frac{b \sqrt{x}}{\sqrt{a b}}\right )}{\sqrt{a b} b^{3}} + \frac{2 \,{\left (3 \, b^{4} x^{\frac{5}{2}} - 5 \, a b^{3} x^{\frac{3}{2}} + 15 \, a^{2} b^{2} \sqrt{x}\right )}}{15 \, b^{5}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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