Optimal. Leaf size=81 \[ \frac{2 A \sqrt{b x+c x^2}}{\sqrt{x}}-2 A \sqrt{b} \tanh ^{-1}\left (\frac{\sqrt{b x+c x^2}}{\sqrt{b} \sqrt{x}}\right )+\frac{2 B \left (b x+c x^2\right )^{3/2}}{3 c x^{3/2}} \]
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Rubi [A] time = 0.0648041, antiderivative size = 81, 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 = {794, 664, 660, 207} \[ \frac{2 A \sqrt{b x+c x^2}}{\sqrt{x}}-2 A \sqrt{b} \tanh ^{-1}\left (\frac{\sqrt{b x+c x^2}}{\sqrt{b} \sqrt{x}}\right )+\frac{2 B \left (b x+c x^2\right )^{3/2}}{3 c x^{3/2}} \]
Antiderivative was successfully verified.
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Rule 794
Rule 664
Rule 660
Rule 207
Rubi steps
\begin{align*} \int \frac{(A+B x) \sqrt{b x+c x^2}}{x^{3/2}} \, dx &=\frac{2 B \left (b x+c x^2\right )^{3/2}}{3 c x^{3/2}}+A \int \frac{\sqrt{b x+c x^2}}{x^{3/2}} \, dx\\ &=\frac{2 A \sqrt{b x+c x^2}}{\sqrt{x}}+\frac{2 B \left (b x+c x^2\right )^{3/2}}{3 c x^{3/2}}+(A b) \int \frac{1}{\sqrt{x} \sqrt{b x+c x^2}} \, dx\\ &=\frac{2 A \sqrt{b x+c x^2}}{\sqrt{x}}+\frac{2 B \left (b x+c x^2\right )^{3/2}}{3 c x^{3/2}}+(2 A b) \operatorname{Subst}\left (\int \frac{1}{-b+x^2} \, dx,x,\frac{\sqrt{b x+c x^2}}{\sqrt{x}}\right )\\ &=\frac{2 A \sqrt{b x+c x^2}}{\sqrt{x}}+\frac{2 B \left (b x+c x^2\right )^{3/2}}{3 c x^{3/2}}-2 A \sqrt{b} \tanh ^{-1}\left (\frac{\sqrt{b x+c x^2}}{\sqrt{b} \sqrt{x}}\right )\\ \end{align*}
Mathematica [A] time = 0.0587543, size = 80, normalized size = 0.99 \[ \frac{2 \sqrt{x} \sqrt{b+c x} \left (\sqrt{b+c x} (3 A c+b B+B c x)-3 A \sqrt{b} c \tanh ^{-1}\left (\frac{\sqrt{b+c x}}{\sqrt{b}}\right )\right )}{3 c \sqrt{x (b+c x)}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.057, size = 79, normalized size = 1. \begin{align*} -{\frac{2}{3\,c}\sqrt{x \left ( cx+b \right ) } \left ( 3\,A\sqrt{b}c{\it Artanh} \left ({\frac{\sqrt{cx+b}}{\sqrt{b}}} \right ) -Bxc\sqrt{cx+b}-3\,Ac\sqrt{cx+b}-Bb\sqrt{cx+b} \right ){\frac{1}{\sqrt{x}}}{\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*} A \int \frac{\sqrt{c x + b}}{x}\,{d x} + \frac{2 \,{\left (B c x + B b\right )} \sqrt{c x + b}}{3 \, c} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.53339, size = 367, normalized size = 4.53 \begin{align*} \left [\frac{3 \, A \sqrt{b} c x \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 (B c x + B b + 3 \, A c\right )} \sqrt{c x^{2} + b x} \sqrt{x}}{3 \, c x}, \frac{2 \,{\left (3 \, A \sqrt{-b} c x \arctan \left (\frac{\sqrt{-b} \sqrt{x}}{\sqrt{c x^{2} + b x}}\right ) +{\left (B c x + B b + 3 \, A c\right )} \sqrt{c x^{2} + b x} \sqrt{x}\right )}}{3 \, c x}\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{\sqrt{x \left (b + c x\right )} \left (A + B x\right )}{x^{\frac{3}{2}}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.16953, size = 139, normalized size = 1.72 \begin{align*} \frac{2 \, A b \arctan \left (\frac{\sqrt{c x + b}}{\sqrt{-b}}\right )}{\sqrt{-b}} - \frac{2 \,{\left (3 \, A b c \arctan \left (\frac{\sqrt{b}}{\sqrt{-b}}\right ) + B \sqrt{-b} b^{\frac{3}{2}} + 3 \, A \sqrt{-b} \sqrt{b} c\right )}}{3 \, \sqrt{-b} c} + \frac{2 \,{\left ({\left (c x + b\right )}^{\frac{3}{2}} B c^{2} + 3 \, \sqrt{c x + b} A c^{3}\right )}}{3 \, c^{3}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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