Optimal. Leaf size=105 \[ -\frac{c (4 b B-A c) \tanh ^{-1}\left (\frac{\sqrt{b x+c x^2}}{\sqrt{b} \sqrt{x}}\right )}{4 b^{3/2}}-\frac{\sqrt{b x+c x^2} (4 b B-A c)}{4 b x^{3/2}}-\frac{A \left (b x+c x^2\right )^{3/2}}{2 b x^{7/2}} \]
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Rubi [A] time = 0.0924073, 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, 662, 660, 207} \[ -\frac{c (4 b B-A c) \tanh ^{-1}\left (\frac{\sqrt{b x+c x^2}}{\sqrt{b} \sqrt{x}}\right )}{4 b^{3/2}}-\frac{\sqrt{b x+c x^2} (4 b B-A c)}{4 b x^{3/2}}-\frac{A \left (b x+c x^2\right )^{3/2}}{2 b x^{7/2}} \]
Antiderivative was successfully verified.
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Rule 792
Rule 662
Rule 660
Rule 207
Rubi steps
\begin{align*} \int \frac{(A+B x) \sqrt{b x+c x^2}}{x^{7/2}} \, dx &=-\frac{A \left (b x+c x^2\right )^{3/2}}{2 b x^{7/2}}+\frac{\left (-\frac{7}{2} (-b B+A c)+\frac{3}{2} (-b B+2 A c)\right ) \int \frac{\sqrt{b x+c x^2}}{x^{5/2}} \, dx}{2 b}\\ &=-\frac{(4 b B-A c) \sqrt{b x+c x^2}}{4 b x^{3/2}}-\frac{A \left (b x+c x^2\right )^{3/2}}{2 b x^{7/2}}+\frac{(c (4 b B-A c)) \int \frac{1}{\sqrt{x} \sqrt{b x+c x^2}} \, dx}{8 b}\\ &=-\frac{(4 b B-A c) \sqrt{b x+c x^2}}{4 b x^{3/2}}-\frac{A \left (b x+c x^2\right )^{3/2}}{2 b x^{7/2}}+\frac{(c (4 b B-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}\\ &=-\frac{(4 b B-A c) \sqrt{b x+c x^2}}{4 b x^{3/2}}-\frac{A \left (b x+c x^2\right )^{3/2}}{2 b x^{7/2}}-\frac{c (4 b B-A c) \tanh ^{-1}\left (\frac{\sqrt{b x+c x^2}}{\sqrt{b} \sqrt{x}}\right )}{4 b^{3/2}}\\ \end{align*}
Mathematica [A] time = 0.103393, size = 83, normalized size = 0.79 \[ -\frac{c x^2 \sqrt{\frac{c x}{b}+1} (4 b B-A c) \tanh ^{-1}\left (\sqrt{\frac{c x}{b}+1}\right )+(b+c x) (2 A b+A c x+4 b B x)}{4 b x^{3/2} \sqrt{x (b+c x)}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.017, size = 108, normalized size = 1. \begin{align*}{\frac{1}{4}\sqrt{x \left ( cx+b \right ) } \left ( A{\it Artanh} \left ({\sqrt{cx+b}{\frac{1}{\sqrt{b}}}} \right ){x}^{2}{c}^{2}-4\,B{\it Artanh} \left ({\frac{\sqrt{cx+b}}{\sqrt{b}}} \right ){x}^{2}bc-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{3}{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{\sqrt{c x^{2} + b x}{\left (B x + A\right )}}{x^{\frac{7}{2}}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.58545, size = 444, normalized size = 4.23 \begin{align*} \left [-\frac{{\left (4 \, B b c - 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} + A b c\right )} x\right )} \sqrt{c x^{2} + b x} \sqrt{x}}{8 \, b^{2} x^{3}}, \frac{{\left (4 \, B b c - 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} + A b c\right )} x\right )} \sqrt{c x^{2} + b x} \sqrt{x}}{4 \, b^{2} 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{\sqrt{x \left (b + c x\right )} \left (A + B x\right )}{x^{\frac{7}{2}}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.27428, size = 149, normalized size = 1.42 \begin{align*} \frac{\frac{{\left (4 \, B b c^{2} - A c^{3}\right )} \arctan \left (\frac{\sqrt{c x + b}}{\sqrt{-b}}\right )}{\sqrt{-b} b} - \frac{4 \,{\left (c x + b\right )}^{\frac{3}{2}} B b c^{2} - 4 \, \sqrt{c x + b} B b^{2} c^{2} +{\left (c x + b\right )}^{\frac{3}{2}} A c^{3} + \sqrt{c x + b} A b c^{3}}{b c^{2} x^{2}}}{4 \, c} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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