Optimal. Leaf size=72 \[ \frac{3 \sqrt{x}}{4 b^2 (a-b x)}-\frac{3 \tanh ^{-1}\left (\frac{\sqrt{b} \sqrt{x}}{\sqrt{a}}\right )}{4 \sqrt{a} b^{5/2}}-\frac{x^{3/2}}{2 b (a-b x)^2} \]
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Rubi [A] time = 0.0208706, antiderivative size = 72, 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 = {47, 63, 208} \[ \frac{3 \sqrt{x}}{4 b^2 (a-b x)}-\frac{3 \tanh ^{-1}\left (\frac{\sqrt{b} \sqrt{x}}{\sqrt{a}}\right )}{4 \sqrt{a} b^{5/2}}-\frac{x^{3/2}}{2 b (a-b x)^2} \]
Antiderivative was successfully verified.
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Rule 47
Rule 63
Rule 208
Rubi steps
\begin{align*} \int \frac{x^{3/2}}{(-a+b x)^3} \, dx &=-\frac{x^{3/2}}{2 b (a-b x)^2}+\frac{3 \int \frac{\sqrt{x}}{(-a+b x)^2} \, dx}{4 b}\\ &=-\frac{x^{3/2}}{2 b (a-b x)^2}+\frac{3 \sqrt{x}}{4 b^2 (a-b x)}+\frac{3 \int \frac{1}{\sqrt{x} (-a+b x)} \, dx}{8 b^2}\\ &=-\frac{x^{3/2}}{2 b (a-b x)^2}+\frac{3 \sqrt{x}}{4 b^2 (a-b x)}+\frac{3 \operatorname{Subst}\left (\int \frac{1}{-a+b x^2} \, dx,x,\sqrt{x}\right )}{4 b^2}\\ &=-\frac{x^{3/2}}{2 b (a-b x)^2}+\frac{3 \sqrt{x}}{4 b^2 (a-b x)}-\frac{3 \tanh ^{-1}\left (\frac{\sqrt{b} \sqrt{x}}{\sqrt{a}}\right )}{4 \sqrt{a} b^{5/2}}\\ \end{align*}
Mathematica [A] time = 0.0357522, size = 60, normalized size = 0.83 \[ \frac{\sqrt{x} (3 a-5 b x)}{4 b^2 (a-b x)^2}-\frac{3 \tanh ^{-1}\left (\frac{\sqrt{b} \sqrt{x}}{\sqrt{a}}\right )}{4 \sqrt{a} b^{5/2}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.01, size = 52, normalized size = 0.7 \begin{align*} 2\,{\frac{1}{ \left ( bx-a \right ) ^{2}} \left ( -5/8\,{\frac{{x}^{3/2}}{b}}+3/8\,{\frac{a\sqrt{x}}{{b}^{2}}} \right ) }-{\frac{3}{4\,{b}^{2}}{\it Artanh} \left ({b\sqrt{x}{\frac{1}{\sqrt{ab}}}} \right ){\frac{1}{\sqrt{ab}}}} \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 = 1.71914, size = 420, normalized size = 5.83 \begin{align*} \left [\frac{3 \,{\left (b^{2} x^{2} - 2 \, a b x + a^{2}\right )} \sqrt{a b} \log \left (\frac{b x + a - 2 \, \sqrt{a b} \sqrt{x}}{b x - a}\right ) - 2 \,{\left (5 \, a b^{2} x - 3 \, a^{2} b\right )} \sqrt{x}}{8 \,{\left (a b^{5} x^{2} - 2 \, a^{2} b^{4} x + a^{3} b^{3}\right )}}, \frac{3 \,{\left (b^{2} x^{2} - 2 \, a b x + a^{2}\right )} \sqrt{-a b} \arctan \left (\frac{\sqrt{-a b}}{b \sqrt{x}}\right ) -{\left (5 \, a b^{2} x - 3 \, a^{2} b\right )} \sqrt{x}}{4 \,{\left (a b^{5} x^{2} - 2 \, a^{2} b^{4} x + a^{3} b^{3}\right )}}\right ] \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 80.4734, size = 673, normalized size = 9.35 \begin{align*} \begin{cases} \frac{\tilde{\infty }}{\sqrt{x}} & \text{for}\: a = 0 \wedge b = 0 \\- \frac{2}{b^{3} \sqrt{x}} & \text{for}\: a = 0 \\- \frac{2 x^{\frac{5}{2}}}{5 a^{3}} & \text{for}\: b = 0 \\\frac{6 a^{\frac{3}{2}} b \sqrt{x} \sqrt{\frac{1}{b}}}{8 a^{\frac{5}{2}} b^{3} \sqrt{\frac{1}{b}} - 16 a^{\frac{3}{2}} b^{4} x \sqrt{\frac{1}{b}} + 8 \sqrt{a} b^{5} x^{2} \sqrt{\frac{1}{b}}} - \frac{10 \sqrt{a} b^{2} x^{\frac{3}{2}} \sqrt{\frac{1}{b}}}{8 a^{\frac{5}{2}} b^{3} \sqrt{\frac{1}{b}} - 16 a^{\frac{3}{2}} b^{4} x \sqrt{\frac{1}{b}} + 8 \sqrt{a} b^{5} x^{2} \sqrt{\frac{1}{b}}} + \frac{3 a^{2} \log{\left (- \sqrt{a} \sqrt{\frac{1}{b}} + \sqrt{x} \right )}}{8 a^{\frac{5}{2}} b^{3} \sqrt{\frac{1}{b}} - 16 a^{\frac{3}{2}} b^{4} x \sqrt{\frac{1}{b}} + 8 \sqrt{a} b^{5} x^{2} \sqrt{\frac{1}{b}}} - \frac{3 a^{2} \log{\left (\sqrt{a} \sqrt{\frac{1}{b}} + \sqrt{x} \right )}}{8 a^{\frac{5}{2}} b^{3} \sqrt{\frac{1}{b}} - 16 a^{\frac{3}{2}} b^{4} x \sqrt{\frac{1}{b}} + 8 \sqrt{a} b^{5} x^{2} \sqrt{\frac{1}{b}}} - \frac{6 a b x \log{\left (- \sqrt{a} \sqrt{\frac{1}{b}} + \sqrt{x} \right )}}{8 a^{\frac{5}{2}} b^{3} \sqrt{\frac{1}{b}} - 16 a^{\frac{3}{2}} b^{4} x \sqrt{\frac{1}{b}} + 8 \sqrt{a} b^{5} x^{2} \sqrt{\frac{1}{b}}} + \frac{6 a b x \log{\left (\sqrt{a} \sqrt{\frac{1}{b}} + \sqrt{x} \right )}}{8 a^{\frac{5}{2}} b^{3} \sqrt{\frac{1}{b}} - 16 a^{\frac{3}{2}} b^{4} x \sqrt{\frac{1}{b}} + 8 \sqrt{a} b^{5} x^{2} \sqrt{\frac{1}{b}}} + \frac{3 b^{2} x^{2} \log{\left (- \sqrt{a} \sqrt{\frac{1}{b}} + \sqrt{x} \right )}}{8 a^{\frac{5}{2}} b^{3} \sqrt{\frac{1}{b}} - 16 a^{\frac{3}{2}} b^{4} x \sqrt{\frac{1}{b}} + 8 \sqrt{a} b^{5} x^{2} \sqrt{\frac{1}{b}}} - \frac{3 b^{2} x^{2} \log{\left (\sqrt{a} \sqrt{\frac{1}{b}} + \sqrt{x} \right )}}{8 a^{\frac{5}{2}} b^{3} \sqrt{\frac{1}{b}} - 16 a^{\frac{3}{2}} b^{4} x \sqrt{\frac{1}{b}} + 8 \sqrt{a} b^{5} x^{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 = 1.22572, size = 69, normalized size = 0.96 \begin{align*} \frac{3 \, \arctan \left (\frac{b \sqrt{x}}{\sqrt{-a b}}\right )}{4 \, \sqrt{-a b} b^{2}} - \frac{5 \, b x^{\frac{3}{2}} - 3 \, a \sqrt{x}}{4 \,{\left (b x - a\right )}^{2} b^{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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