Optimal. Leaf size=56 \[ -\frac{3 \sqrt{b} \tan ^{-1}\left (\frac{\sqrt{b} \sqrt{x}}{\sqrt{a}}\right )}{a^{5/2}}-\frac{3}{a^2 \sqrt{x}}+\frac{1}{a \sqrt{x} (a+b x)} \]
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Rubi [A] time = 0.0168896, antiderivative size = 56, 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 = {51, 63, 205} \[ -\frac{3 \sqrt{b} \tan ^{-1}\left (\frac{\sqrt{b} \sqrt{x}}{\sqrt{a}}\right )}{a^{5/2}}-\frac{3}{a^2 \sqrt{x}}+\frac{1}{a \sqrt{x} (a+b x)} \]
Antiderivative was successfully verified.
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Rule 51
Rule 63
Rule 205
Rubi steps
\begin{align*} \int \frac{1}{x^{3/2} (a+b x)^2} \, dx &=\frac{1}{a \sqrt{x} (a+b x)}+\frac{3 \int \frac{1}{x^{3/2} (a+b x)} \, dx}{2 a}\\ &=-\frac{3}{a^2 \sqrt{x}}+\frac{1}{a \sqrt{x} (a+b x)}-\frac{(3 b) \int \frac{1}{\sqrt{x} (a+b x)} \, dx}{2 a^2}\\ &=-\frac{3}{a^2 \sqrt{x}}+\frac{1}{a \sqrt{x} (a+b x)}-\frac{(3 b) \operatorname{Subst}\left (\int \frac{1}{a+b x^2} \, dx,x,\sqrt{x}\right )}{a^2}\\ &=-\frac{3}{a^2 \sqrt{x}}+\frac{1}{a \sqrt{x} (a+b x)}-\frac{3 \sqrt{b} \tan ^{-1}\left (\frac{\sqrt{b} \sqrt{x}}{\sqrt{a}}\right )}{a^{5/2}}\\ \end{align*}
Mathematica [C] time = 0.0047055, size = 25, normalized size = 0.45 \[ -\frac{2 \, _2F_1\left (-\frac{1}{2},2;\frac{1}{2};-\frac{b x}{a}\right )}{a^2 \sqrt{x}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.01, size = 48, normalized size = 0.9 \begin{align*} -2\,{\frac{1}{{a}^{2}\sqrt{x}}}-{\frac{b}{{a}^{2} \left ( bx+a \right ) }\sqrt{x}}-3\,{\frac{b}{{a}^{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 = 1.56498, size = 323, normalized size = 5.77 \begin{align*} \left [\frac{3 \,{\left (b x^{2} + a x\right )} \sqrt{-\frac{b}{a}} \log \left (\frac{b x - 2 \, a \sqrt{x} \sqrt{-\frac{b}{a}} - a}{b x + a}\right ) - 2 \,{\left (3 \, b x + 2 \, a\right )} \sqrt{x}}{2 \,{\left (a^{2} b x^{2} + a^{3} x\right )}}, \frac{3 \,{\left (b x^{2} + a x\right )} \sqrt{\frac{b}{a}} \arctan \left (\frac{a \sqrt{\frac{b}{a}}}{b \sqrt{x}}\right ) -{\left (3 \, b x + 2 \, a\right )} \sqrt{x}}{a^{2} b x^{2} + a^{3} x}\right ] \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 38.7742, size = 434, normalized size = 7.75 \begin{align*} \begin{cases} \frac{\tilde{\infty }}{x^{\frac{5}{2}}} & \text{for}\: a = 0 \wedge b = 0 \\- \frac{2}{5 b^{2} x^{\frac{5}{2}}} & \text{for}\: a = 0 \\- \frac{2}{a^{2} \sqrt{x}} & \text{for}\: b = 0 \\- \frac{4 i a^{\frac{3}{2}} \sqrt{\frac{1}{b}}}{2 i a^{\frac{7}{2}} \sqrt{x} \sqrt{\frac{1}{b}} + 2 i a^{\frac{5}{2}} b x^{\frac{3}{2}} \sqrt{\frac{1}{b}}} - \frac{6 i \sqrt{a} b x \sqrt{\frac{1}{b}}}{2 i a^{\frac{7}{2}} \sqrt{x} \sqrt{\frac{1}{b}} + 2 i a^{\frac{5}{2}} b x^{\frac{3}{2}} \sqrt{\frac{1}{b}}} - \frac{3 a \sqrt{x} \log{\left (- i \sqrt{a} \sqrt{\frac{1}{b}} + \sqrt{x} \right )}}{2 i a^{\frac{7}{2}} \sqrt{x} \sqrt{\frac{1}{b}} + 2 i a^{\frac{5}{2}} b x^{\frac{3}{2}} \sqrt{\frac{1}{b}}} + \frac{3 a \sqrt{x} \log{\left (i \sqrt{a} \sqrt{\frac{1}{b}} + \sqrt{x} \right )}}{2 i a^{\frac{7}{2}} \sqrt{x} \sqrt{\frac{1}{b}} + 2 i a^{\frac{5}{2}} b x^{\frac{3}{2}} \sqrt{\frac{1}{b}}} - \frac{3 b x^{\frac{3}{2}} \log{\left (- i \sqrt{a} \sqrt{\frac{1}{b}} + \sqrt{x} \right )}}{2 i a^{\frac{7}{2}} \sqrt{x} \sqrt{\frac{1}{b}} + 2 i a^{\frac{5}{2}} b x^{\frac{3}{2}} \sqrt{\frac{1}{b}}} + \frac{3 b x^{\frac{3}{2}} \log{\left (i \sqrt{a} \sqrt{\frac{1}{b}} + \sqrt{x} \right )}}{2 i a^{\frac{7}{2}} \sqrt{x} \sqrt{\frac{1}{b}} + 2 i a^{\frac{5}{2}} b x^{\frac{3}{2}} \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 = 1.24275, size = 66, normalized size = 1.18 \begin{align*} -\frac{3 \, b \arctan \left (\frac{b \sqrt{x}}{\sqrt{a b}}\right )}{\sqrt{a b} a^{2}} - \frac{3 \, b x + 2 \, a}{{\left (b x^{\frac{3}{2}} + a \sqrt{x}\right )} a^{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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