Optimal. Leaf size=142 \[ -\frac{c^2 (6 b B-5 A c) \tanh ^{-1}\left (\frac{\sqrt{b x+c x^2}}{\sqrt{b} \sqrt{x}}\right )}{8 b^{7/2}}+\frac{c \sqrt{b x+c x^2} (6 b B-5 A c)}{8 b^3 x^{3/2}}-\frac{\sqrt{b x+c x^2} (6 b B-5 A c)}{12 b^2 x^{5/2}}-\frac{A \sqrt{b x+c x^2}}{3 b x^{7/2}} \]
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Rubi [A] time = 0.120299, antiderivative size = 142, 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, 672, 660, 207} \[ -\frac{c^2 (6 b B-5 A c) \tanh ^{-1}\left (\frac{\sqrt{b x+c x^2}}{\sqrt{b} \sqrt{x}}\right )}{8 b^{7/2}}+\frac{c \sqrt{b x+c x^2} (6 b B-5 A c)}{8 b^3 x^{3/2}}-\frac{\sqrt{b x+c x^2} (6 b B-5 A c)}{12 b^2 x^{5/2}}-\frac{A \sqrt{b x+c x^2}}{3 b x^{7/2}} \]
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^{7/2} \sqrt{b x+c x^2}} \, dx &=-\frac{A \sqrt{b x+c x^2}}{3 b x^{7/2}}+\frac{\left (-\frac{7}{2} (-b B+A c)+\frac{1}{2} (-b B+2 A c)\right ) \int \frac{1}{x^{5/2} \sqrt{b x+c x^2}} \, dx}{3 b}\\ &=-\frac{A \sqrt{b x+c x^2}}{3 b x^{7/2}}-\frac{(6 b B-5 A c) \sqrt{b x+c x^2}}{12 b^2 x^{5/2}}-\frac{(c (6 b B-5 A c)) \int \frac{1}{x^{3/2} \sqrt{b x+c x^2}} \, dx}{8 b^2}\\ &=-\frac{A \sqrt{b x+c x^2}}{3 b x^{7/2}}-\frac{(6 b B-5 A c) \sqrt{b x+c x^2}}{12 b^2 x^{5/2}}+\frac{c (6 b B-5 A c) \sqrt{b x+c x^2}}{8 b^3 x^{3/2}}+\frac{\left (c^2 (6 b B-5 A c)\right ) \int \frac{1}{\sqrt{x} \sqrt{b x+c x^2}} \, dx}{16 b^3}\\ &=-\frac{A \sqrt{b x+c x^2}}{3 b x^{7/2}}-\frac{(6 b B-5 A c) \sqrt{b x+c x^2}}{12 b^2 x^{5/2}}+\frac{c (6 b B-5 A c) \sqrt{b x+c x^2}}{8 b^3 x^{3/2}}+\frac{\left (c^2 (6 b B-5 A c)\right ) \operatorname{Subst}\left (\int \frac{1}{-b+x^2} \, dx,x,\frac{\sqrt{b x+c x^2}}{\sqrt{x}}\right )}{8 b^3}\\ &=-\frac{A \sqrt{b x+c x^2}}{3 b x^{7/2}}-\frac{(6 b B-5 A c) \sqrt{b x+c x^2}}{12 b^2 x^{5/2}}+\frac{c (6 b B-5 A c) \sqrt{b x+c x^2}}{8 b^3 x^{3/2}}-\frac{c^2 (6 b B-5 A c) \tanh ^{-1}\left (\frac{\sqrt{b x+c x^2}}{\sqrt{b} \sqrt{x}}\right )}{8 b^{7/2}}\\ \end{align*}
Mathematica [C] time = 0.0276484, size = 61, normalized size = 0.43 \[ -\frac{\sqrt{x (b+c x)} \left (A b^3+c^2 x^3 (6 b B-5 A c) \, _2F_1\left (\frac{1}{2},3;\frac{3}{2};\frac{c x}{b}+1\right )\right )}{3 b^4 x^{7/2}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.017, size = 147, normalized size = 1. \begin{align*}{\frac{1}{24}\sqrt{x \left ( cx+b \right ) } \left ( 15\,A{\it Artanh} \left ({\frac{\sqrt{cx+b}}{\sqrt{b}}} \right ){x}^{3}{c}^{3}-18\,B{\it Artanh} \left ({\frac{\sqrt{cx+b}}{\sqrt{b}}} \right ){x}^{3}b{c}^{2}-15\,A{x}^{2}{c}^{2}\sqrt{b}\sqrt{cx+b}+18\,B{x}^{2}{b}^{3/2}c\sqrt{cx+b}+10\,Ax{b}^{3/2}c\sqrt{cx+b}-12\,Bx{b}^{5/2}\sqrt{cx+b}-8\,A{b}^{5/2}\sqrt{cx+b} \right ){b}^{-{\frac{7}{2}}}{x}^{-{\frac{7}{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{7}{2}}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.98262, size = 566, normalized size = 3.99 \begin{align*} \left [-\frac{3 \,{\left (6 \, B b c^{2} - 5 \, A c^{3}\right )} \sqrt{b} x^{4} \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 (8 \, A b^{3} - 3 \,{\left (6 \, B b^{2} c - 5 \, A b c^{2}\right )} x^{2} + 2 \,{\left (6 \, B b^{3} - 5 \, A b^{2} c\right )} x\right )} \sqrt{c x^{2} + b x} \sqrt{x}}{48 \, b^{4} x^{4}}, \frac{3 \,{\left (6 \, B b c^{2} - 5 \, A c^{3}\right )} \sqrt{-b} x^{4} \arctan \left (\frac{\sqrt{-b} \sqrt{x}}{\sqrt{c x^{2} + b x}}\right ) -{\left (8 \, A b^{3} - 3 \,{\left (6 \, B b^{2} c - 5 \, A b c^{2}\right )} x^{2} + 2 \,{\left (6 \, B b^{3} - 5 \, A b^{2} c\right )} x\right )} \sqrt{c x^{2} + b x} \sqrt{x}}{24 \, b^{4} x^{4}}\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{7}{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 = 1.23999, size = 194, normalized size = 1.37 \begin{align*} \frac{\frac{3 \,{\left (6 \, B b c^{3} - 5 \, A c^{4}\right )} \arctan \left (\frac{\sqrt{c x + b}}{\sqrt{-b}}\right )}{\sqrt{-b} b^{3}} + \frac{18 \,{\left (c x + b\right )}^{\frac{5}{2}} B b c^{3} - 48 \,{\left (c x + b\right )}^{\frac{3}{2}} B b^{2} c^{3} + 30 \, \sqrt{c x + b} B b^{3} c^{3} - 15 \,{\left (c x + b\right )}^{\frac{5}{2}} A c^{4} + 40 \,{\left (c x + b\right )}^{\frac{3}{2}} A b c^{4} - 33 \, \sqrt{c x + b} A b^{2} c^{4}}{b^{3} c^{3} x^{3}}}{24 \, c} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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