Optimal. Leaf size=105 \[ -\frac{\sqrt{b x+c x^2} (4 b B-3 A c)}{4 b^2 x^{3/2}}+\frac{c (4 b B-3 A c) \tanh ^{-1}\left (\frac{\sqrt{b x+c x^2}}{\sqrt{b} \sqrt{x}}\right )}{4 b^{5/2}}-\frac{A \sqrt{b x+c x^2}}{2 b x^{5/2}} \]
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Rubi [A] time = 0.0861174, antiderivative size = 105, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.167, Rules used = {792, 672, 660, 207} \[ -\frac{\sqrt{b x+c x^2} (4 b B-3 A c)}{4 b^2 x^{3/2}}+\frac{c (4 b B-3 A c) \tanh ^{-1}\left (\frac{\sqrt{b x+c x^2}}{\sqrt{b} \sqrt{x}}\right )}{4 b^{5/2}}-\frac{A \sqrt{b x+c x^2}}{2 b x^{5/2}} \]
Antiderivative was successfully verified.
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Rule 792
Rule 672
Rule 660
Rule 207
Rubi steps
\begin{align*} \int \frac{A+B x}{x^{5/2} \sqrt{b x+c x^2}} \, dx &=-\frac{A \sqrt{b x+c x^2}}{2 b x^{5/2}}+\frac{\left (-\frac{5}{2} (-b B+A c)+\frac{1}{2} (-b B+2 A c)\right ) \int \frac{1}{x^{3/2} \sqrt{b x+c x^2}} \, dx}{2 b}\\ &=-\frac{A \sqrt{b x+c x^2}}{2 b x^{5/2}}-\frac{(4 b B-3 A c) \sqrt{b x+c x^2}}{4 b^2 x^{3/2}}-\frac{(c (4 b B-3 A c)) \int \frac{1}{\sqrt{x} \sqrt{b x+c x^2}} \, dx}{8 b^2}\\ &=-\frac{A \sqrt{b x+c x^2}}{2 b x^{5/2}}-\frac{(4 b B-3 A c) \sqrt{b x+c x^2}}{4 b^2 x^{3/2}}-\frac{(c (4 b B-3 A c)) \operatorname{Subst}\left (\int \frac{1}{-b+x^2} \, dx,x,\frac{\sqrt{b x+c x^2}}{\sqrt{x}}\right )}{4 b^2}\\ &=-\frac{A \sqrt{b x+c x^2}}{2 b x^{5/2}}-\frac{(4 b B-3 A c) \sqrt{b x+c x^2}}{4 b^2 x^{3/2}}+\frac{c (4 b B-3 A c) \tanh ^{-1}\left (\frac{\sqrt{b x+c x^2}}{\sqrt{b} \sqrt{x}}\right )}{4 b^{5/2}}\\ \end{align*}
Mathematica [A] time = 0.099285, size = 92, normalized size = 0.88 \[ \frac{\sqrt{x (b+c x)} \left (c x^2 (4 b B-3 A c) \tanh ^{-1}\left (\sqrt{\frac{c x}{b}+1}\right )+b \sqrt{\frac{c x}{b}+1} (-2 A b+3 A c x-4 b B x)\right )}{4 b^3 x^{5/2} \sqrt{\frac{c x}{b}+1}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.017, size = 109, normalized size = 1. \begin{align*} -{\frac{1}{4}\sqrt{x \left ( cx+b \right ) } \left ( 3\,A{\it Artanh} \left ({\frac{\sqrt{cx+b}}{\sqrt{b}}} \right ){x}^{2}{c}^{2}-4\,B{\it Artanh} \left ({\frac{\sqrt{cx+b}}{\sqrt{b}}} \right ){x}^{2}bc-3\,Axc\sqrt{cx+b}\sqrt{b}+4\,Bx{b}^{3/2}\sqrt{cx+b}+2\,A{b}^{3/2}\sqrt{cx+b} \right ){b}^{-{\frac{5}{2}}}{x}^{-{\frac{5}{2}}}{\frac{1}{\sqrt{cx+b}}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{B x + A}{\sqrt{c x^{2} + b x} x^{\frac{5}{2}}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.84778, size = 456, normalized size = 4.34 \begin{align*} \left [-\frac{{\left (4 \, B b c - 3 \, A c^{2}\right )} \sqrt{b} x^{3} \log \left (-\frac{c x^{2} + 2 \, b x - 2 \, \sqrt{c x^{2} + b x} \sqrt{b} \sqrt{x}}{x^{2}}\right ) + 2 \,{\left (2 \, A b^{2} +{\left (4 \, B b^{2} - 3 \, A b c\right )} x\right )} \sqrt{c x^{2} + b x} \sqrt{x}}{8 \, b^{3} x^{3}}, -\frac{{\left (4 \, B b c - 3 \, A c^{2}\right )} \sqrt{-b} x^{3} \arctan \left (\frac{\sqrt{-b} \sqrt{x}}{\sqrt{c x^{2} + b x}}\right ) +{\left (2 \, A b^{2} +{\left (4 \, B b^{2} - 3 \, A b c\right )} x\right )} \sqrt{c x^{2} + b x} \sqrt{x}}{4 \, b^{3} x^{3}}\right ] \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{A + B x}{x^{\frac{5}{2}} \sqrt{x \left (b + c x\right )}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.21563, size = 150, normalized size = 1.43 \begin{align*} -\frac{\frac{{\left (4 \, B b c^{2} - 3 \, A c^{3}\right )} \arctan \left (\frac{\sqrt{c x + b}}{\sqrt{-b}}\right )}{\sqrt{-b} b^{2}} + \frac{4 \,{\left (c x + b\right )}^{\frac{3}{2}} B b c^{2} - 4 \, \sqrt{c x + b} B b^{2} c^{2} - 3 \,{\left (c x + b\right )}^{\frac{3}{2}} A c^{3} + 5 \, \sqrt{c x + b} A b c^{3}}{b^{2} c^{2} x^{2}}}{4 \, c} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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