Optimal. Leaf size=137 \[ -\frac{\left (b x+c x^2\right )^{3/2} (A c+4 b B)}{4 b x^{5/2}}+\frac{3 c \sqrt{b x+c x^2} (A c+4 b B)}{4 b \sqrt{x}}-\frac{3 c (A c+4 b B) \tanh ^{-1}\left (\frac{\sqrt{b x+c x^2}}{\sqrt{b} \sqrt{x}}\right )}{4 \sqrt{b}}-\frac{A \left (b x+c x^2\right )^{5/2}}{2 b x^{9/2}} \]
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Rubi [A] time = 0.127509, antiderivative size = 137, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 5, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.208, Rules used = {792, 662, 664, 660, 207} \[ -\frac{\left (b x+c x^2\right )^{3/2} (A c+4 b B)}{4 b x^{5/2}}+\frac{3 c \sqrt{b x+c x^2} (A c+4 b B)}{4 b \sqrt{x}}-\frac{3 c (A c+4 b B) \tanh ^{-1}\left (\frac{\sqrt{b x+c x^2}}{\sqrt{b} \sqrt{x}}\right )}{4 \sqrt{b}}-\frac{A \left (b x+c x^2\right )^{5/2}}{2 b x^{9/2}} \]
Antiderivative was successfully verified.
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Rule 792
Rule 662
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^{9/2}} \, dx &=-\frac{A \left (b x+c x^2\right )^{5/2}}{2 b x^{9/2}}+\frac{\left (-\frac{9}{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^{7/2}} \, dx}{2 b}\\ &=-\frac{(4 b B+A c) \left (b x+c x^2\right )^{3/2}}{4 b x^{5/2}}-\frac{A \left (b x+c x^2\right )^{5/2}}{2 b x^{9/2}}+\frac{(3 c (4 b B+A c)) \int \frac{\sqrt{b x+c x^2}}{x^{3/2}} \, dx}{8 b}\\ &=\frac{3 c (4 b B+A c) \sqrt{b x+c x^2}}{4 b \sqrt{x}}-\frac{(4 b B+A c) \left (b x+c x^2\right )^{3/2}}{4 b x^{5/2}}-\frac{A \left (b x+c x^2\right )^{5/2}}{2 b x^{9/2}}+\frac{1}{8} (3 c (4 b B+A c)) \int \frac{1}{\sqrt{x} \sqrt{b x+c x^2}} \, dx\\ &=\frac{3 c (4 b B+A c) \sqrt{b x+c x^2}}{4 b \sqrt{x}}-\frac{(4 b B+A c) \left (b x+c x^2\right )^{3/2}}{4 b x^{5/2}}-\frac{A \left (b x+c x^2\right )^{5/2}}{2 b x^{9/2}}+\frac{1}{4} (3 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 )\\ &=\frac{3 c (4 b B+A c) \sqrt{b x+c x^2}}{4 b \sqrt{x}}-\frac{(4 b B+A c) \left (b x+c x^2\right )^{3/2}}{4 b x^{5/2}}-\frac{A \left (b x+c x^2\right )^{5/2}}{2 b x^{9/2}}-\frac{3 c (4 b B+A c) \tanh ^{-1}\left (\frac{\sqrt{b x+c x^2}}{\sqrt{b} \sqrt{x}}\right )}{4 \sqrt{b}}\\ \end{align*}
Mathematica [C] time = 0.0336948, size = 59, normalized size = 0.43 \[ \frac{(x (b+c x))^{5/2} \left (c x^2 (A c+4 b B) \, _2F_1\left (2,\frac{5}{2};\frac{7}{2};\frac{c x}{b}+1\right )-5 A b^2\right )}{10 b^3 x^{9/2}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.02, size = 126, normalized size = 0.9 \begin{align*} -{\frac{1}{4}\sqrt{x \left ( cx+b \right ) } \left ( 3\,A{\it Artanh} \left ({\frac{\sqrt{cx+b}}{\sqrt{b}}} \right ){x}^{2}{c}^{2}+12\,B{\it Artanh} \left ({\frac{\sqrt{cx+b}}{\sqrt{b}}} \right ){x}^{2}bc-8\,B{x}^{2}c\sqrt{b}\sqrt{cx+b}+5\,Axc\sqrt{cx+b}\sqrt{b}+4\,Bx{b}^{3/2}\sqrt{cx+b}+2\,A{b}^{3/2}\sqrt{cx+b} \right ){x}^{-{\frac{5}{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*} \int \frac{{\left (c x^{2} + b x\right )}^{\frac{3}{2}}{\left (B x + A\right )}}{x^{\frac{9}{2}}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.69992, size = 486, normalized size = 3.55 \begin{align*} \left [\frac{3 \,{\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 (8 \, B b c x^{2} - 2 \, A b^{2} -{\left (4 \, B b^{2} + 5 \, A b c\right )} x\right )} \sqrt{c x^{2} + b x} \sqrt{x}}{8 \, b x^{3}}, \frac{3 \,{\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 (8 \, B b c x^{2} - 2 \, A b^{2} -{\left (4 \, B b^{2} + 5 \, A b c\right )} x\right )} \sqrt{c x^{2} + b x} \sqrt{x}}{4 \, b x^{3}}\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 = 1.25535, size = 161, normalized size = 1.18 \begin{align*} \frac{8 \, \sqrt{c x + b} B c^{2} + \frac{3 \,{\left (4 \, B b c^{2} + A c^{3}\right )} \arctan \left (\frac{\sqrt{c x + b}}{\sqrt{-b}}\right )}{\sqrt{-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} + 5 \,{\left (c x + b\right )}^{\frac{3}{2}} A c^{3} - 3 \, \sqrt{c x + b} A b c^{3}}{c^{2} x^{2}}}{4 \, c} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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