Optimal. Leaf size=53 \[ \frac{2 a^{3/2} \tan ^{-1}\left (\frac{\sqrt{b} \sqrt{x}}{\sqrt{a}}\right )}{b^{5/2}}-\frac{2 a \sqrt{x}}{b^2}+\frac{2 x^{3/2}}{3 b} \]
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Rubi [A] time = 0.0166034, antiderivative size = 53, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 13, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.231, Rules used = {50, 63, 205} \[ \frac{2 a^{3/2} \tan ^{-1}\left (\frac{\sqrt{b} \sqrt{x}}{\sqrt{a}}\right )}{b^{5/2}}-\frac{2 a \sqrt{x}}{b^2}+\frac{2 x^{3/2}}{3 b} \]
Antiderivative was successfully verified.
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Rule 50
Rule 63
Rule 205
Rubi steps
\begin{align*} \int \frac{x^{3/2}}{a+b x} \, dx &=\frac{2 x^{3/2}}{3 b}-\frac{a \int \frac{\sqrt{x}}{a+b x} \, dx}{b}\\ &=-\frac{2 a \sqrt{x}}{b^2}+\frac{2 x^{3/2}}{3 b}+\frac{a^2 \int \frac{1}{\sqrt{x} (a+b x)} \, dx}{b^2}\\ &=-\frac{2 a \sqrt{x}}{b^2}+\frac{2 x^{3/2}}{3 b}+\frac{\left (2 a^2\right ) \operatorname{Subst}\left (\int \frac{1}{a+b x^2} \, dx,x,\sqrt{x}\right )}{b^2}\\ &=-\frac{2 a \sqrt{x}}{b^2}+\frac{2 x^{3/2}}{3 b}+\frac{2 a^{3/2} \tan ^{-1}\left (\frac{\sqrt{b} \sqrt{x}}{\sqrt{a}}\right )}{b^{5/2}}\\ \end{align*}
Mathematica [A] time = 0.018496, size = 49, normalized size = 0.92 \[ \frac{2 a^{3/2} \tan ^{-1}\left (\frac{\sqrt{b} \sqrt{x}}{\sqrt{a}}\right )}{b^{5/2}}+\frac{2 \sqrt{x} (b x-3 a)}{3 b^2} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.003, size = 43, normalized size = 0.8 \begin{align*}{\frac{2}{3\,b}{x}^{{\frac{3}{2}}}}-2\,{\frac{a\sqrt{x}}{{b}^{2}}}+2\,{\frac{{a}^{2}}{{b}^{2}\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 = 1.63256, size = 244, normalized size = 4.6 \begin{align*} \left [\frac{3 \, a \sqrt{-\frac{a}{b}} \log \left (\frac{b x + 2 \, b \sqrt{x} \sqrt{-\frac{a}{b}} - a}{b x + a}\right ) + 2 \,{\left (b x - 3 \, a\right )} \sqrt{x}}{3 \, b^{2}}, \frac{2 \,{\left (3 \, a \sqrt{\frac{a}{b}} \arctan \left (\frac{b \sqrt{x} \sqrt{\frac{a}{b}}}{a}\right ) +{\left (b x - 3 \, a\right )} \sqrt{x}\right )}}{3 \, b^{2}}\right ] \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 3.92548, size = 105, normalized size = 1.98 \begin{align*} \begin{cases} - \frac{i a^{\frac{3}{2}} \log{\left (- i \sqrt{a} \sqrt{\frac{1}{b}} + \sqrt{x} \right )}}{b^{3} \sqrt{\frac{1}{b}}} + \frac{i a^{\frac{3}{2}} \log{\left (i \sqrt{a} \sqrt{\frac{1}{b}} + \sqrt{x} \right )}}{b^{3} \sqrt{\frac{1}{b}}} - \frac{2 a \sqrt{x}}{b^{2}} + \frac{2 x^{\frac{3}{2}}}{3 b} & \text{for}\: b \neq 0 \\\frac{2 x^{\frac{5}{2}}}{5 a} & \text{otherwise} \end{cases} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.16497, size = 61, normalized size = 1.15 \begin{align*} \frac{2 \, a^{2} \arctan \left (\frac{b \sqrt{x}}{\sqrt{a b}}\right )}{\sqrt{a b} b^{2}} + \frac{2 \,{\left (b^{2} x^{\frac{3}{2}} - 3 \, a b \sqrt{x}\right )}}{3 \, b^{3}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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