Optimal. Leaf size=69 \[ -\frac{2 (b B-A c)}{b^2 \sqrt{x}}-\frac{2 \sqrt{c} (b B-A c) \tan ^{-1}\left (\frac{\sqrt{c} \sqrt{x}}{\sqrt{b}}\right )}{b^{5/2}}-\frac{2 A}{3 b x^{3/2}} \]
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Rubi [A] time = 0.0417975, antiderivative size = 69, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 5, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.227, Rules used = {781, 78, 51, 63, 205} \[ -\frac{2 (b B-A c)}{b^2 \sqrt{x}}-\frac{2 \sqrt{c} (b B-A c) \tan ^{-1}\left (\frac{\sqrt{c} \sqrt{x}}{\sqrt{b}}\right )}{b^{5/2}}-\frac{2 A}{3 b x^{3/2}} \]
Antiderivative was successfully verified.
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Rule 781
Rule 78
Rule 51
Rule 63
Rule 205
Rubi steps
\begin{align*} \int \frac{A+B x}{x^{3/2} \left (b x+c x^2\right )} \, dx &=\int \frac{A+B x}{x^{5/2} (b+c x)} \, dx\\ &=-\frac{2 A}{3 b x^{3/2}}+\frac{\left (2 \left (\frac{3 b B}{2}-\frac{3 A c}{2}\right )\right ) \int \frac{1}{x^{3/2} (b+c x)} \, dx}{3 b}\\ &=-\frac{2 A}{3 b x^{3/2}}-\frac{2 (b B-A c)}{b^2 \sqrt{x}}-\frac{(c (b B-A c)) \int \frac{1}{\sqrt{x} (b+c x)} \, dx}{b^2}\\ &=-\frac{2 A}{3 b x^{3/2}}-\frac{2 (b B-A c)}{b^2 \sqrt{x}}-\frac{(2 c (b B-A c)) \operatorname{Subst}\left (\int \frac{1}{b+c x^2} \, dx,x,\sqrt{x}\right )}{b^2}\\ &=-\frac{2 A}{3 b x^{3/2}}-\frac{2 (b B-A c)}{b^2 \sqrt{x}}-\frac{2 \sqrt{c} (b B-A c) \tan ^{-1}\left (\frac{\sqrt{c} \sqrt{x}}{\sqrt{b}}\right )}{b^{5/2}}\\ \end{align*}
Mathematica [C] time = 0.0134692, size = 44, normalized size = 0.64 \[ \frac{\, _2F_1\left (-\frac{1}{2},1;\frac{1}{2};-\frac{c x}{b}\right ) (6 A c x-6 b B x)-2 A b}{3 b^2 x^{3/2}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.012, size = 78, normalized size = 1.1 \begin{align*} -{\frac{2\,A}{3\,b}{x}^{-{\frac{3}{2}}}}+2\,{\frac{Ac}{{b}^{2}\sqrt{x}}}-2\,{\frac{B}{b\sqrt{x}}}+2\,{\frac{A{c}^{2}}{{b}^{2}\sqrt{bc}}\arctan \left ({\frac{\sqrt{x}c}{\sqrt{bc}}} \right ) }-2\,{\frac{Bc}{b\sqrt{bc}}\arctan \left ({\frac{\sqrt{x}c}{\sqrt{bc}}} \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 = 1.78254, size = 335, normalized size = 4.86 \begin{align*} \left [-\frac{3 \,{\left (B b - A c\right )} x^{2} \sqrt{-\frac{c}{b}} \log \left (\frac{c x + 2 \, b \sqrt{x} \sqrt{-\frac{c}{b}} - b}{c x + b}\right ) + 2 \,{\left (A b + 3 \,{\left (B b - A c\right )} x\right )} \sqrt{x}}{3 \, b^{2} x^{2}}, \frac{2 \,{\left (3 \,{\left (B b - A c\right )} x^{2} \sqrt{\frac{c}{b}} \arctan \left (\frac{b \sqrt{\frac{c}{b}}}{c \sqrt{x}}\right ) -{\left (A b + 3 \,{\left (B b - A c\right )} x\right )} \sqrt{x}\right )}}{3 \, b^{2} x^{2}}\right ] \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 15.2149, size = 248, normalized size = 3.59 \begin{align*} \begin{cases} \tilde{\infty } \left (- \frac{2 A}{5 x^{\frac{5}{2}}} - \frac{2 B}{3 x^{\frac{3}{2}}}\right ) & \text{for}\: b = 0 \wedge c = 0 \\\frac{- \frac{2 A}{5 x^{\frac{5}{2}}} - \frac{2 B}{3 x^{\frac{3}{2}}}}{c} & \text{for}\: b = 0 \\\frac{- \frac{2 A}{3 x^{\frac{3}{2}}} - \frac{2 B}{\sqrt{x}}}{b} & \text{for}\: c = 0 \\- \frac{2 A}{3 b x^{\frac{3}{2}}} + \frac{2 A c}{b^{2} \sqrt{x}} - \frac{i A c \log{\left (- i \sqrt{b} \sqrt{\frac{1}{c}} + \sqrt{x} \right )}}{b^{\frac{5}{2}} \sqrt{\frac{1}{c}}} + \frac{i A c \log{\left (i \sqrt{b} \sqrt{\frac{1}{c}} + \sqrt{x} \right )}}{b^{\frac{5}{2}} \sqrt{\frac{1}{c}}} - \frac{2 B}{b \sqrt{x}} + \frac{i B \log{\left (- i \sqrt{b} \sqrt{\frac{1}{c}} + \sqrt{x} \right )}}{b^{\frac{3}{2}} \sqrt{\frac{1}{c}}} - \frac{i B \log{\left (i \sqrt{b} \sqrt{\frac{1}{c}} + \sqrt{x} \right )}}{b^{\frac{3}{2}} \sqrt{\frac{1}{c}}} & \text{otherwise} \end{cases} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.10225, size = 74, normalized size = 1.07 \begin{align*} -\frac{2 \,{\left (B b c - A c^{2}\right )} \arctan \left (\frac{c \sqrt{x}}{\sqrt{b c}}\right )}{\sqrt{b c} b^{2}} - \frac{2 \,{\left (3 \, B b x - 3 \, A c x + A b\right )}}{3 \, b^{2} x^{\frac{3}{2}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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