Optimal. Leaf size=57 \[ \frac{3 \sqrt{b} \tanh ^{-1}\left (\frac{\sqrt{b} \sqrt{x}}{\sqrt{a}}\right )}{a^{5/2}}-\frac{3}{a^2 \sqrt{x}}+\frac{1}{a \sqrt{x} (a-b x)} \]
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Rubi [A] time = 0.0179159, antiderivative size = 57, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.2, Rules used = {51, 63, 208} \[ \frac{3 \sqrt{b} \tanh ^{-1}\left (\frac{\sqrt{b} \sqrt{x}}{\sqrt{a}}\right )}{a^{5/2}}-\frac{3}{a^2 \sqrt{x}}+\frac{1}{a \sqrt{x} (a-b x)} \]
Antiderivative was successfully verified.
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Rule 51
Rule 63
Rule 208
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} \tanh ^{-1}\left (\frac{\sqrt{b} \sqrt{x}}{\sqrt{a}}\right )}{a^{5/2}}\\ \end{align*}
Mathematica [C] time = 0.0058549, size = 24, normalized size = 0.42 \[ -\frac{2 \, _2F_1\left (-\frac{1}{2},2;\frac{1}{2};\frac{b x}{a}\right )}{a^2 \sqrt{x}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.01, size = 49, normalized size = 0.9 \begin{align*} -2\,{\frac{1}{{a}^{2}\sqrt{x}}}-2\,{\frac{b}{{a}^{2}} \left ( 1/2\,{\frac{\sqrt{x}}{bx-a}}-3/2\,{\frac{1}{\sqrt{ab}}{\it Artanh} \left ({\frac{b\sqrt{x}}{\sqrt{ab}}} \right ) } \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 = 1.62697, size = 324, normalized size = 5.68 \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 = 42.7898, size = 403, normalized size = 7.07 \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 a^{\frac{3}{2}} \sqrt{\frac{1}{b}}}{2 a^{\frac{7}{2}} \sqrt{x} \sqrt{\frac{1}{b}} - 2 a^{\frac{5}{2}} b x^{\frac{3}{2}} \sqrt{\frac{1}{b}}} + \frac{6 \sqrt{a} b x \sqrt{\frac{1}{b}}}{2 a^{\frac{7}{2}} \sqrt{x} \sqrt{\frac{1}{b}} - 2 a^{\frac{5}{2}} b x^{\frac{3}{2}} \sqrt{\frac{1}{b}}} - \frac{3 a \sqrt{x} \log{\left (- \sqrt{a} \sqrt{\frac{1}{b}} + \sqrt{x} \right )}}{2 a^{\frac{7}{2}} \sqrt{x} \sqrt{\frac{1}{b}} - 2 a^{\frac{5}{2}} b x^{\frac{3}{2}} \sqrt{\frac{1}{b}}} + \frac{3 a \sqrt{x} \log{\left (\sqrt{a} \sqrt{\frac{1}{b}} + \sqrt{x} \right )}}{2 a^{\frac{7}{2}} \sqrt{x} \sqrt{\frac{1}{b}} - 2 a^{\frac{5}{2}} b x^{\frac{3}{2}} \sqrt{\frac{1}{b}}} + \frac{3 b x^{\frac{3}{2}} \log{\left (- \sqrt{a} \sqrt{\frac{1}{b}} + \sqrt{x} \right )}}{2 a^{\frac{7}{2}} \sqrt{x} \sqrt{\frac{1}{b}} - 2 a^{\frac{5}{2}} b x^{\frac{3}{2}} \sqrt{\frac{1}{b}}} - \frac{3 b x^{\frac{3}{2}} \log{\left (\sqrt{a} \sqrt{\frac{1}{b}} + \sqrt{x} \right )}}{2 a^{\frac{7}{2}} \sqrt{x} \sqrt{\frac{1}{b}} - 2 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 = 1.20185, size = 70, normalized size = 1.23 \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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