Optimal. Leaf size=174 \[ -\frac{5 c \sqrt{x} (4 b B-7 A c)}{4 b^4 \sqrt{b x+c x^2}}-\frac{5 (4 b B-7 A c)}{12 b^3 \sqrt{x} \sqrt{b x+c x^2}}+\frac{\sqrt{x} (4 b B-7 A c)}{6 b^2 \left (b x+c x^2\right )^{3/2}}+\frac{5 c (4 b B-7 A c) \tanh ^{-1}\left (\frac{\sqrt{b x+c x^2}}{\sqrt{b} \sqrt{x}}\right )}{4 b^{9/2}}-\frac{A}{2 b \sqrt{x} \left (b x+c x^2\right )^{3/2}} \]
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Rubi [A] time = 0.146951, antiderivative size = 174, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 5, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.208, Rules used = {792, 666, 672, 660, 207} \[ -\frac{5 c \sqrt{x} (4 b B-7 A c)}{4 b^4 \sqrt{b x+c x^2}}-\frac{5 (4 b B-7 A c)}{12 b^3 \sqrt{x} \sqrt{b x+c x^2}}+\frac{\sqrt{x} (4 b B-7 A c)}{6 b^2 \left (b x+c x^2\right )^{3/2}}+\frac{5 c (4 b B-7 A c) \tanh ^{-1}\left (\frac{\sqrt{b x+c x^2}}{\sqrt{b} \sqrt{x}}\right )}{4 b^{9/2}}-\frac{A}{2 b \sqrt{x} \left (b x+c x^2\right )^{3/2}} \]
Antiderivative was successfully verified.
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Rule 792
Rule 666
Rule 672
Rule 660
Rule 207
Rubi steps
\begin{align*} \int \frac{A+B x}{\sqrt{x} \left (b x+c x^2\right )^{5/2}} \, dx &=-\frac{A}{2 b \sqrt{x} \left (b x+c x^2\right )^{3/2}}+\frac{\left (\frac{1}{2} (b B-A c)-\frac{3}{2} (-b B+2 A c)\right ) \int \frac{\sqrt{x}}{\left (b x+c x^2\right )^{5/2}} \, dx}{2 b}\\ &=-\frac{A}{2 b \sqrt{x} \left (b x+c x^2\right )^{3/2}}+\frac{(4 b B-7 A c) \sqrt{x}}{6 b^2 \left (b x+c x^2\right )^{3/2}}+\frac{(5 (4 b B-7 A c)) \int \frac{1}{\sqrt{x} \left (b x+c x^2\right )^{3/2}} \, dx}{12 b^2}\\ &=-\frac{A}{2 b \sqrt{x} \left (b x+c x^2\right )^{3/2}}+\frac{(4 b B-7 A c) \sqrt{x}}{6 b^2 \left (b x+c x^2\right )^{3/2}}-\frac{5 (4 b B-7 A c)}{12 b^3 \sqrt{x} \sqrt{b x+c x^2}}-\frac{(5 c (4 b B-7 A c)) \int \frac{\sqrt{x}}{\left (b x+c x^2\right )^{3/2}} \, dx}{8 b^3}\\ &=-\frac{A}{2 b \sqrt{x} \left (b x+c x^2\right )^{3/2}}+\frac{(4 b B-7 A c) \sqrt{x}}{6 b^2 \left (b x+c x^2\right )^{3/2}}-\frac{5 (4 b B-7 A c)}{12 b^3 \sqrt{x} \sqrt{b x+c x^2}}-\frac{5 c (4 b B-7 A c) \sqrt{x}}{4 b^4 \sqrt{b x+c x^2}}-\frac{(5 c (4 b B-7 A c)) \int \frac{1}{\sqrt{x} \sqrt{b x+c x^2}} \, dx}{8 b^4}\\ &=-\frac{A}{2 b \sqrt{x} \left (b x+c x^2\right )^{3/2}}+\frac{(4 b B-7 A c) \sqrt{x}}{6 b^2 \left (b x+c x^2\right )^{3/2}}-\frac{5 (4 b B-7 A c)}{12 b^3 \sqrt{x} \sqrt{b x+c x^2}}-\frac{5 c (4 b B-7 A c) \sqrt{x}}{4 b^4 \sqrt{b x+c x^2}}-\frac{(5 c (4 b B-7 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^4}\\ &=-\frac{A}{2 b \sqrt{x} \left (b x+c x^2\right )^{3/2}}+\frac{(4 b B-7 A c) \sqrt{x}}{6 b^2 \left (b x+c x^2\right )^{3/2}}-\frac{5 (4 b B-7 A c)}{12 b^3 \sqrt{x} \sqrt{b x+c x^2}}-\frac{5 c (4 b B-7 A c) \sqrt{x}}{4 b^4 \sqrt{b x+c x^2}}+\frac{5 c (4 b B-7 A c) \tanh ^{-1}\left (\frac{\sqrt{b x+c x^2}}{\sqrt{b} \sqrt{x}}\right )}{4 b^{9/2}}\\ \end{align*}
Mathematica [C] time = 0.0270372, size = 60, normalized size = 0.34 \[ \frac{c x^2 (7 A c-4 b B) \, _2F_1\left (-\frac{3}{2},2;-\frac{1}{2};\frac{c x}{b}+1\right )-3 A b^2}{6 b^3 \sqrt{x} (x (b+c x))^{3/2}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.025, size = 208, normalized size = 1.2 \begin{align*} -{\frac{1}{12\, \left ( cx+b \right ) ^{2}}\sqrt{x \left ( cx+b \right ) } \left ( 105\,A{\it Artanh} \left ({\frac{\sqrt{cx+b}}{\sqrt{b}}} \right ) \sqrt{cx+b}{x}^{3}{c}^{3}-60\,B{\it Artanh} \left ({\frac{\sqrt{cx+b}}{\sqrt{b}}} \right ) \sqrt{cx+b}{x}^{3}b{c}^{2}+105\,A{\it Artanh} \left ({\frac{\sqrt{cx+b}}{\sqrt{b}}} \right ){x}^{2}b{c}^{2}\sqrt{cx+b}-105\,A\sqrt{b}{x}^{3}{c}^{3}-60\,B{\it Artanh} \left ({\frac{\sqrt{cx+b}}{\sqrt{b}}} \right ){x}^{2}{b}^{2}c\sqrt{cx+b}+60\,B{b}^{3/2}{x}^{3}{c}^{2}-140\,A{b}^{3/2}{x}^{2}{c}^{2}+80\,B{b}^{5/2}{x}^{2}c-21\,A{b}^{5/2}xc+12\,B{b}^{7/2}x+6\,A{b}^{7/2} \right ){x}^{-{\frac{5}{2}}}{b}^{-{\frac{9}{2}}}} \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}{{\left (c x^{2} + b x\right )}^{\frac{5}{2}} \sqrt{x}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.99501, size = 932, normalized size = 5.36 \begin{align*} \left [-\frac{15 \,{\left ({\left (4 \, B b c^{3} - 7 \, A c^{4}\right )} x^{5} + 2 \,{\left (4 \, B b^{2} c^{2} - 7 \, A b c^{3}\right )} x^{4} +{\left (4 \, B b^{3} c - 7 \, A b^{2} c^{2}\right )} x^{3}\right )} \sqrt{b} \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 (6 \, A b^{4} + 15 \,{\left (4 \, B b^{2} c^{2} - 7 \, A b c^{3}\right )} x^{3} + 20 \,{\left (4 \, B b^{3} c - 7 \, A b^{2} c^{2}\right )} x^{2} + 3 \,{\left (4 \, B b^{4} - 7 \, A b^{3} c\right )} x\right )} \sqrt{c x^{2} + b x} \sqrt{x}}{24 \,{\left (b^{5} c^{2} x^{5} + 2 \, b^{6} c x^{4} + b^{7} x^{3}\right )}}, -\frac{15 \,{\left ({\left (4 \, B b c^{3} - 7 \, A c^{4}\right )} x^{5} + 2 \,{\left (4 \, B b^{2} c^{2} - 7 \, A b c^{3}\right )} x^{4} +{\left (4 \, B b^{3} c - 7 \, A b^{2} c^{2}\right )} x^{3}\right )} \sqrt{-b} \arctan \left (\frac{\sqrt{-b} \sqrt{x}}{\sqrt{c x^{2} + b x}}\right ) +{\left (6 \, A b^{4} + 15 \,{\left (4 \, B b^{2} c^{2} - 7 \, A b c^{3}\right )} x^{3} + 20 \,{\left (4 \, B b^{3} c - 7 \, A b^{2} c^{2}\right )} x^{2} + 3 \,{\left (4 \, B b^{4} - 7 \, A b^{3} c\right )} x\right )} \sqrt{c x^{2} + b x} \sqrt{x}}{12 \,{\left (b^{5} c^{2} x^{5} + 2 \, b^{6} c x^{4} + b^{7} x^{3}\right )}}\right ] \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F(-1)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.27845, size = 201, normalized size = 1.16 \begin{align*} -\frac{5 \,{\left (4 \, B b c - 7 \, A c^{2}\right )} \arctan \left (\frac{\sqrt{c x + b}}{\sqrt{-b}}\right )}{4 \, \sqrt{-b} b^{4}} - \frac{2 \,{\left (6 \,{\left (c x + b\right )} B b c + B b^{2} c - 9 \,{\left (c x + b\right )} A c^{2} - A b c^{2}\right )}}{3 \,{\left (c x + b\right )}^{\frac{3}{2}} b^{4}} - \frac{4 \,{\left (c x + b\right )}^{\frac{3}{2}} B b c - 4 \, \sqrt{c x + b} B b^{2} c - 11 \,{\left (c x + b\right )}^{\frac{3}{2}} A c^{2} + 13 \, \sqrt{c x + b} A b c^{2}}{4 \, b^{4} c^{2} x^{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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