Optimal. Leaf size=128 \[ \frac{\left (b x+c x^2\right )^{3/2} (3 A c+2 b B)}{3 b x^{3/2}}+\frac{\sqrt{b x+c x^2} (3 A c+2 b B)}{\sqrt{x}}-\sqrt{b} (3 A c+2 b B) \tanh ^{-1}\left (\frac{\sqrt{b x+c x^2}}{\sqrt{b} \sqrt{x}}\right )-\frac{A \left (b x+c x^2\right )^{5/2}}{b x^{7/2}} \]
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Rubi [A] time = 0.125791, antiderivative size = 128, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 4, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.167, Rules used = {792, 664, 660, 207} \[ \frac{\left (b x+c x^2\right )^{3/2} (3 A c+2 b B)}{3 b x^{3/2}}+\frac{\sqrt{b x+c x^2} (3 A c+2 b B)}{\sqrt{x}}-\sqrt{b} (3 A c+2 b B) \tanh ^{-1}\left (\frac{\sqrt{b x+c x^2}}{\sqrt{b} \sqrt{x}}\right )-\frac{A \left (b x+c x^2\right )^{5/2}}{b x^{7/2}} \]
Antiderivative was successfully verified.
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Rule 792
Rule 664
Rule 660
Rule 207
Rubi steps
\begin{align*} \int \frac{(A+B x) \left (b x+c x^2\right )^{3/2}}{x^{7/2}} \, dx &=-\frac{A \left (b x+c x^2\right )^{5/2}}{b x^{7/2}}+\frac{\left (-\frac{7}{2} (-b B+A c)+\frac{5}{2} (-b B+2 A c)\right ) \int \frac{\left (b x+c x^2\right )^{3/2}}{x^{5/2}} \, dx}{b}\\ &=\frac{(2 b B+3 A c) \left (b x+c x^2\right )^{3/2}}{3 b x^{3/2}}-\frac{A \left (b x+c x^2\right )^{5/2}}{b x^{7/2}}+\frac{1}{2} (2 b B+3 A c) \int \frac{\sqrt{b x+c x^2}}{x^{3/2}} \, dx\\ &=\frac{(2 b B+3 A c) \sqrt{b x+c x^2}}{\sqrt{x}}+\frac{(2 b B+3 A c) \left (b x+c x^2\right )^{3/2}}{3 b x^{3/2}}-\frac{A \left (b x+c x^2\right )^{5/2}}{b x^{7/2}}+\frac{1}{2} (b (2 b B+3 A c)) \int \frac{1}{\sqrt{x} \sqrt{b x+c x^2}} \, dx\\ &=\frac{(2 b B+3 A c) \sqrt{b x+c x^2}}{\sqrt{x}}+\frac{(2 b B+3 A c) \left (b x+c x^2\right )^{3/2}}{3 b x^{3/2}}-\frac{A \left (b x+c x^2\right )^{5/2}}{b x^{7/2}}+(b (2 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 )\\ &=\frac{(2 b B+3 A c) \sqrt{b x+c x^2}}{\sqrt{x}}+\frac{(2 b B+3 A c) \left (b x+c x^2\right )^{3/2}}{3 b x^{3/2}}-\frac{A \left (b x+c x^2\right )^{5/2}}{b x^{7/2}}-\sqrt{b} (2 b B+3 A c) \tanh ^{-1}\left (\frac{\sqrt{b x+c x^2}}{\sqrt{b} \sqrt{x}}\right )\\ \end{align*}
Mathematica [A] time = 0.0601579, size = 94, normalized size = 0.73 \[ \frac{\sqrt{x (b+c x)} \left (\sqrt{b+c x} (2 B x (4 b+c x)-3 A (b-2 c x))-3 \sqrt{b} x (3 A c+2 b B) \tanh ^{-1}\left (\frac{\sqrt{b+c x}}{\sqrt{b}}\right )\right )}{3 x^{3/2} \sqrt{b+c x}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.018, size = 122, normalized size = 1. \begin{align*} -{\frac{1}{3}\sqrt{x \left ( cx+b \right ) } \left ( -2\,B{x}^{2}c\sqrt{b}\sqrt{cx+b}+9\,A{\it Artanh} \left ({\frac{\sqrt{cx+b}}{\sqrt{b}}} \right ) xbc-6\,Axc\sqrt{cx+b}\sqrt{b}+6\,B{\it Artanh} \left ({\frac{\sqrt{cx+b}}{\sqrt{b}}} \right ) x{b}^{2}-8\,Bx{b}^{3/2}\sqrt{cx+b}+3\,A{b}^{3/2}\sqrt{cx+b} \right ){x}^{-{\frac{3}{2}}}{\frac{1}{\sqrt{cx+b}}}{\frac{1}{\sqrt{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*} \frac{2}{3} \,{\left (B c x + B b\right )} \sqrt{c x + b} + \int \frac{{\left (A b +{\left (B b + A c\right )} x\right )} \sqrt{c x + b}}{x^{2}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.60796, size = 454, normalized size = 3.55 \begin{align*} \left [\frac{3 \,{\left (2 \, B b + 3 \, A c\right )} \sqrt{b} x^{2} \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 \, B c x^{2} - 3 \, A b + 2 \,{\left (4 \, B b + 3 \, A c\right )} x\right )} \sqrt{c x^{2} + b x} \sqrt{x}}{6 \, x^{2}}, \frac{3 \,{\left (2 \, B b + 3 \, A c\right )} \sqrt{-b} x^{2} \arctan \left (\frac{\sqrt{-b} \sqrt{x}}{\sqrt{c x^{2} + b x}}\right ) +{\left (2 \, B c x^{2} - 3 \, A b + 2 \,{\left (4 \, B b + 3 \, A c\right )} x\right )} \sqrt{c x^{2} + b x} \sqrt{x}}{3 \, x^{2}}\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{\left (x \left (b + c x\right )\right )^{\frac{3}{2}} \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 = 1.30861, size = 126, normalized size = 0.98 \begin{align*} \frac{2 \,{\left (c x + b\right )}^{\frac{3}{2}} B c + 6 \, \sqrt{c x + b} B b c + 6 \, \sqrt{c x + b} A c^{2} - \frac{3 \, \sqrt{c x + b} A b c}{x} + \frac{3 \,{\left (2 \, B b^{2} c + 3 \, A b c^{2}\right )} \arctan \left (\frac{\sqrt{c x + b}}{\sqrt{-b}}\right )}{\sqrt{-b}}}{3 \, c} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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