Optimal. Leaf size=70 \[ \frac{5 b^{3/2} \tanh ^{-1}\left (\frac{\sqrt{b} \sqrt{x}}{\sqrt{a}}\right )}{a^{7/2}}-\frac{5 b}{a^3 \sqrt{x}}-\frac{5}{3 a^2 x^{3/2}}+\frac{1}{a x^{3/2} (a-b x)} \]
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Rubi [A] time = 0.0224064, antiderivative size = 70, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 3, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.2, Rules used = {51, 63, 208} \[ \frac{5 b^{3/2} \tanh ^{-1}\left (\frac{\sqrt{b} \sqrt{x}}{\sqrt{a}}\right )}{a^{7/2}}-\frac{5 b}{a^3 \sqrt{x}}-\frac{5}{3 a^2 x^{3/2}}+\frac{1}{a x^{3/2} (a-b x)} \]
Antiderivative was successfully verified.
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Rule 51
Rule 63
Rule 208
Rubi steps
\begin{align*} \int \frac{1}{x^{5/2} (-a+b x)^2} \, dx &=\frac{1}{a x^{3/2} (a-b x)}-\frac{5 \int \frac{1}{x^{5/2} (-a+b x)} \, dx}{2 a}\\ &=-\frac{5}{3 a^2 x^{3/2}}+\frac{1}{a x^{3/2} (a-b x)}-\frac{(5 b) \int \frac{1}{x^{3/2} (-a+b x)} \, dx}{2 a^2}\\ &=-\frac{5}{3 a^2 x^{3/2}}-\frac{5 b}{a^3 \sqrt{x}}+\frac{1}{a x^{3/2} (a-b x)}-\frac{\left (5 b^2\right ) \int \frac{1}{\sqrt{x} (-a+b x)} \, dx}{2 a^3}\\ &=-\frac{5}{3 a^2 x^{3/2}}-\frac{5 b}{a^3 \sqrt{x}}+\frac{1}{a x^{3/2} (a-b x)}-\frac{\left (5 b^2\right ) \operatorname{Subst}\left (\int \frac{1}{-a+b x^2} \, dx,x,\sqrt{x}\right )}{a^3}\\ &=-\frac{5}{3 a^2 x^{3/2}}-\frac{5 b}{a^3 \sqrt{x}}+\frac{1}{a x^{3/2} (a-b x)}+\frac{5 b^{3/2} \tanh ^{-1}\left (\frac{\sqrt{b} \sqrt{x}}{\sqrt{a}}\right )}{a^{7/2}}\\ \end{align*}
Mathematica [C] time = 0.0058593, size = 26, normalized size = 0.37 \[ -\frac{2 \, _2F_1\left (-\frac{3}{2},2;-\frac{1}{2};\frac{b x}{a}\right )}{3 a^2 x^{3/2}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.011, size = 60, normalized size = 0.9 \begin{align*} -{\frac{2}{3\,{a}^{2}}{x}^{-{\frac{3}{2}}}}-4\,{\frac{b}{{a}^{3}\sqrt{x}}}-2\,{\frac{{b}^{2}}{{a}^{3}} \left ( 1/2\,{\frac{\sqrt{x}}{bx-a}}-5/2\,{\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 = 1.55767, size = 402, normalized size = 5.74 \begin{align*} \left [\frac{15 \,{\left (b^{2} x^{3} - a b x^{2}\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 (15 \, b^{2} x^{2} - 10 \, a b x - 2 \, a^{2}\right )} \sqrt{x}}{6 \,{\left (a^{3} b x^{3} - a^{4} x^{2}\right )}}, -\frac{15 \,{\left (b^{2} x^{3} - a b x^{2}\right )} \sqrt{-\frac{b}{a}} \arctan \left (\frac{a \sqrt{-\frac{b}{a}}}{b \sqrt{x}}\right ) +{\left (15 \, b^{2} x^{2} - 10 \, a b x - 2 \, a^{2}\right )} \sqrt{x}}{3 \,{\left (a^{3} b x^{3} - a^{4} x^{2}\right )}}\right ] \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 139.005, size = 471, normalized size = 6.73 \begin{align*} \begin{cases} \frac{\tilde{\infty }}{x^{\frac{7}{2}}} & \text{for}\: a = 0 \wedge b = 0 \\- \frac{2}{7 b^{2} x^{\frac{7}{2}}} & \text{for}\: a = 0 \\- \frac{2}{3 a^{2} x^{\frac{3}{2}}} & \text{for}\: b = 0 \\- \frac{4 a^{\frac{5}{2}} \sqrt{\frac{1}{b}}}{6 a^{\frac{9}{2}} x^{\frac{3}{2}} \sqrt{\frac{1}{b}} - 6 a^{\frac{7}{2}} b x^{\frac{5}{2}} \sqrt{\frac{1}{b}}} - \frac{20 a^{\frac{3}{2}} b x \sqrt{\frac{1}{b}}}{6 a^{\frac{9}{2}} x^{\frac{3}{2}} \sqrt{\frac{1}{b}} - 6 a^{\frac{7}{2}} b x^{\frac{5}{2}} \sqrt{\frac{1}{b}}} + \frac{30 \sqrt{a} b^{2} x^{2} \sqrt{\frac{1}{b}}}{6 a^{\frac{9}{2}} x^{\frac{3}{2}} \sqrt{\frac{1}{b}} - 6 a^{\frac{7}{2}} b x^{\frac{5}{2}} \sqrt{\frac{1}{b}}} - \frac{15 a b x^{\frac{3}{2}} \log{\left (- \sqrt{a} \sqrt{\frac{1}{b}} + \sqrt{x} \right )}}{6 a^{\frac{9}{2}} x^{\frac{3}{2}} \sqrt{\frac{1}{b}} - 6 a^{\frac{7}{2}} b x^{\frac{5}{2}} \sqrt{\frac{1}{b}}} + \frac{15 a b x^{\frac{3}{2}} \log{\left (\sqrt{a} \sqrt{\frac{1}{b}} + \sqrt{x} \right )}}{6 a^{\frac{9}{2}} x^{\frac{3}{2}} \sqrt{\frac{1}{b}} - 6 a^{\frac{7}{2}} b x^{\frac{5}{2}} \sqrt{\frac{1}{b}}} + \frac{15 b^{2} x^{\frac{5}{2}} \log{\left (- \sqrt{a} \sqrt{\frac{1}{b}} + \sqrt{x} \right )}}{6 a^{\frac{9}{2}} x^{\frac{3}{2}} \sqrt{\frac{1}{b}} - 6 a^{\frac{7}{2}} b x^{\frac{5}{2}} \sqrt{\frac{1}{b}}} - \frac{15 b^{2} x^{\frac{5}{2}} \log{\left (\sqrt{a} \sqrt{\frac{1}{b}} + \sqrt{x} \right )}}{6 a^{\frac{9}{2}} x^{\frac{3}{2}} \sqrt{\frac{1}{b}} - 6 a^{\frac{7}{2}} b x^{\frac{5}{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 = 1.19526, size = 82, normalized size = 1.17 \begin{align*} -\frac{5 \, b^{2} \arctan \left (\frac{b \sqrt{x}}{\sqrt{-a b}}\right )}{\sqrt{-a b} a^{3}} - \frac{b^{2} \sqrt{x}}{{\left (b x - a\right )} a^{3}} - \frac{2 \,{\left (6 \, b x + a\right )}}{3 \, a^{3} x^{\frac{3}{2}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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