Optimal. Leaf size=57 \[ -\frac{3 \sqrt{a} \tan ^{-1}\left (\frac{\sqrt{b} \sqrt{x}}{\sqrt{a}}\right )}{b^{5/2}}-\frac{x^{3/2}}{b (a+b x)}+\frac{3 \sqrt{x}}{b^2} \]
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Rubi [A] time = 0.0168536, antiderivative size = 57, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 13, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.308, Rules used = {47, 50, 63, 205} \[ -\frac{3 \sqrt{a} \tan ^{-1}\left (\frac{\sqrt{b} \sqrt{x}}{\sqrt{a}}\right )}{b^{5/2}}-\frac{x^{3/2}}{b (a+b x)}+\frac{3 \sqrt{x}}{b^2} \]
Antiderivative was successfully verified.
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Rule 47
Rule 50
Rule 63
Rule 205
Rubi steps
\begin{align*} \int \frac{x^{3/2}}{(a+b x)^2} \, dx &=-\frac{x^{3/2}}{b (a+b x)}+\frac{3 \int \frac{\sqrt{x}}{a+b x} \, dx}{2 b}\\ &=\frac{3 \sqrt{x}}{b^2}-\frac{x^{3/2}}{b (a+b x)}-\frac{(3 a) \int \frac{1}{\sqrt{x} (a+b x)} \, dx}{2 b^2}\\ &=\frac{3 \sqrt{x}}{b^2}-\frac{x^{3/2}}{b (a+b x)}-\frac{(3 a) \operatorname{Subst}\left (\int \frac{1}{a+b x^2} \, dx,x,\sqrt{x}\right )}{b^2}\\ &=\frac{3 \sqrt{x}}{b^2}-\frac{x^{3/2}}{b (a+b x)}-\frac{3 \sqrt{a} \tan ^{-1}\left (\frac{\sqrt{b} \sqrt{x}}{\sqrt{a}}\right )}{b^{5/2}}\\ \end{align*}
Mathematica [C] time = 0.0039348, size = 27, normalized size = 0.47 \[ \frac{2 x^{5/2} \, _2F_1\left (2,\frac{5}{2};\frac{7}{2};-\frac{b x}{a}\right )}{5 a^2} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.009, size = 47, normalized size = 0.8 \begin{align*} 2\,{\frac{\sqrt{x}}{{b}^{2}}}+{\frac{a}{{b}^{2} \left ( bx+a \right ) }\sqrt{x}}-3\,{\frac{a}{{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 = 1.49069, size = 300, normalized size = 5.26 \begin{align*} \left [\frac{3 \,{\left (b x + a\right )} \sqrt{-\frac{a}{b}} \log \left (\frac{b x - 2 \, b \sqrt{x} \sqrt{-\frac{a}{b}} - a}{b x + a}\right ) + 2 \,{\left (2 \, b x + 3 \, a\right )} \sqrt{x}}{2 \,{\left (b^{3} x + a b^{2}\right )}}, -\frac{3 \,{\left (b x + a\right )} \sqrt{\frac{a}{b}} \arctan \left (\frac{b \sqrt{x} \sqrt{\frac{a}{b}}}{a}\right ) -{\left (2 \, b x + 3 \, a\right )} \sqrt{x}}{b^{3} x + a b^{2}}\right ] \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 19.2066, size = 411, normalized size = 7.21 \begin{align*} \begin{cases} \tilde{\infty } \sqrt{x} & \text{for}\: a = 0 \wedge b = 0 \\\frac{2 x^{\frac{5}{2}}}{5 a^{2}} & \text{for}\: b = 0 \\\frac{2 \sqrt{x}}{b^{2}} & \text{for}\: a = 0 \\\frac{6 i a^{\frac{3}{2}} b \sqrt{x} \sqrt{\frac{1}{b}}}{2 i a^{\frac{3}{2}} b^{3} \sqrt{\frac{1}{b}} + 2 i \sqrt{a} b^{4} x \sqrt{\frac{1}{b}}} + \frac{4 i \sqrt{a} b^{2} x^{\frac{3}{2}} \sqrt{\frac{1}{b}}}{2 i a^{\frac{3}{2}} b^{3} \sqrt{\frac{1}{b}} + 2 i \sqrt{a} b^{4} x \sqrt{\frac{1}{b}}} - \frac{3 a^{2} \log{\left (- i \sqrt{a} \sqrt{\frac{1}{b}} + \sqrt{x} \right )}}{2 i a^{\frac{3}{2}} b^{3} \sqrt{\frac{1}{b}} + 2 i \sqrt{a} b^{4} x \sqrt{\frac{1}{b}}} + \frac{3 a^{2} \log{\left (i \sqrt{a} \sqrt{\frac{1}{b}} + \sqrt{x} \right )}}{2 i a^{\frac{3}{2}} b^{3} \sqrt{\frac{1}{b}} + 2 i \sqrt{a} b^{4} x \sqrt{\frac{1}{b}}} - \frac{3 a b x \log{\left (- i \sqrt{a} \sqrt{\frac{1}{b}} + \sqrt{x} \right )}}{2 i a^{\frac{3}{2}} b^{3} \sqrt{\frac{1}{b}} + 2 i \sqrt{a} b^{4} x \sqrt{\frac{1}{b}}} + \frac{3 a b x \log{\left (i \sqrt{a} \sqrt{\frac{1}{b}} + \sqrt{x} \right )}}{2 i a^{\frac{3}{2}} b^{3} \sqrt{\frac{1}{b}} + 2 i \sqrt{a} b^{4} x \sqrt{\frac{1}{b}}} & \text{otherwise} \end{cases} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.19443, size = 62, normalized size = 1.09 \begin{align*} -\frac{3 \, a \arctan \left (\frac{b \sqrt{x}}{\sqrt{a b}}\right )}{\sqrt{a b} b^{2}} + \frac{a \sqrt{x}}{{\left (b x + a\right )} b^{2}} + \frac{2 \, \sqrt{x}}{b^{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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