Optimal. Leaf size=126 \[ -\frac{x^{3/2} (5 b B-A c)}{4 b c^2 (b+c x)}+\frac{3 \sqrt{x} (5 b B-A c)}{4 b c^3}-\frac{3 (5 b B-A c) \tan ^{-1}\left (\frac{\sqrt{c} \sqrt{x}}{\sqrt{b}}\right )}{4 \sqrt{b} c^{7/2}}-\frac{x^{5/2} (b B-A c)}{2 b c (b+c x)^2} \]
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Rubi [A] time = 0.0588881, antiderivative size = 126, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 6, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.273, Rules used = {781, 78, 47, 50, 63, 205} \[ -\frac{x^{3/2} (5 b B-A c)}{4 b c^2 (b+c x)}+\frac{3 \sqrt{x} (5 b B-A c)}{4 b c^3}-\frac{3 (5 b B-A c) \tan ^{-1}\left (\frac{\sqrt{c} \sqrt{x}}{\sqrt{b}}\right )}{4 \sqrt{b} c^{7/2}}-\frac{x^{5/2} (b B-A c)}{2 b c (b+c x)^2} \]
Antiderivative was successfully verified.
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Rule 781
Rule 78
Rule 47
Rule 50
Rule 63
Rule 205
Rubi steps
\begin{align*} \int \frac{x^{9/2} (A+B x)}{\left (b x+c x^2\right )^3} \, dx &=\int \frac{x^{3/2} (A+B x)}{(b+c x)^3} \, dx\\ &=-\frac{(b B-A c) x^{5/2}}{2 b c (b+c x)^2}-\frac{\left (-\frac{5 b B}{2}+\frac{A c}{2}\right ) \int \frac{x^{3/2}}{(b+c x)^2} \, dx}{2 b c}\\ &=-\frac{(b B-A c) x^{5/2}}{2 b c (b+c x)^2}-\frac{(5 b B-A c) x^{3/2}}{4 b c^2 (b+c x)}+\frac{(3 (5 b B-A c)) \int \frac{\sqrt{x}}{b+c x} \, dx}{8 b c^2}\\ &=\frac{3 (5 b B-A c) \sqrt{x}}{4 b c^3}-\frac{(b B-A c) x^{5/2}}{2 b c (b+c x)^2}-\frac{(5 b B-A c) x^{3/2}}{4 b c^2 (b+c x)}-\frac{(3 (5 b B-A c)) \int \frac{1}{\sqrt{x} (b+c x)} \, dx}{8 c^3}\\ &=\frac{3 (5 b B-A c) \sqrt{x}}{4 b c^3}-\frac{(b B-A c) x^{5/2}}{2 b c (b+c x)^2}-\frac{(5 b B-A c) x^{3/2}}{4 b c^2 (b+c x)}-\frac{(3 (5 b B-A c)) \operatorname{Subst}\left (\int \frac{1}{b+c x^2} \, dx,x,\sqrt{x}\right )}{4 c^3}\\ &=\frac{3 (5 b B-A c) \sqrt{x}}{4 b c^3}-\frac{(b B-A c) x^{5/2}}{2 b c (b+c x)^2}-\frac{(5 b B-A c) x^{3/2}}{4 b c^2 (b+c x)}-\frac{3 (5 b B-A c) \tan ^{-1}\left (\frac{\sqrt{c} \sqrt{x}}{\sqrt{b}}\right )}{4 \sqrt{b} c^{7/2}}\\ \end{align*}
Mathematica [C] time = 0.0252955, size = 61, normalized size = 0.48 \[ \frac{x^{5/2} \left (\frac{5 b^2 (A c-b B)}{(b+c x)^2}+(5 b B-A c) \, _2F_1\left (2,\frac{5}{2};\frac{7}{2};-\frac{c x}{b}\right )\right )}{10 b^3 c} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.016, size = 125, normalized size = 1. \begin{align*} 2\,{\frac{B\sqrt{x}}{{c}^{3}}}-{\frac{5\,A}{4\,c \left ( cx+b \right ) ^{2}}{x}^{{\frac{3}{2}}}}+{\frac{9\,bB}{4\,{c}^{2} \left ( cx+b \right ) ^{2}}{x}^{{\frac{3}{2}}}}-{\frac{3\,Ab}{4\,{c}^{2} \left ( cx+b \right ) ^{2}}\sqrt{x}}+{\frac{7\,{b}^{2}B}{4\,{c}^{3} \left ( cx+b \right ) ^{2}}\sqrt{x}}+{\frac{3\,A}{4\,{c}^{2}}\arctan \left ({c\sqrt{x}{\frac{1}{\sqrt{bc}}}} \right ){\frac{1}{\sqrt{bc}}}}-{\frac{15\,bB}{4\,{c}^{3}}\arctan \left ({c\sqrt{x}{\frac{1}{\sqrt{bc}}}} \right ){\frac{1}{\sqrt{bc}}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F(-2)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: ValueError} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 2.00096, size = 687, normalized size = 5.45 \begin{align*} \left [\frac{3 \,{\left (5 \, B b^{3} - A b^{2} c +{\left (5 \, B b c^{2} - A c^{3}\right )} x^{2} + 2 \,{\left (5 \, B b^{2} c - A b c^{2}\right )} x\right )} \sqrt{-b c} \log \left (\frac{c x - b - 2 \, \sqrt{-b c} \sqrt{x}}{c x + b}\right ) + 2 \,{\left (8 \, B b c^{3} x^{2} + 15 \, B b^{3} c - 3 \, A b^{2} c^{2} + 5 \,{\left (5 \, B b^{2} c^{2} - A b c^{3}\right )} x\right )} \sqrt{x}}{8 \,{\left (b c^{6} x^{2} + 2 \, b^{2} c^{5} x + b^{3} c^{4}\right )}}, \frac{3 \,{\left (5 \, B b^{3} - A b^{2} c +{\left (5 \, B b c^{2} - A c^{3}\right )} x^{2} + 2 \,{\left (5 \, B b^{2} c - A b c^{2}\right )} x\right )} \sqrt{b c} \arctan \left (\frac{\sqrt{b c}}{c \sqrt{x}}\right ) +{\left (8 \, B b c^{3} x^{2} + 15 \, B b^{3} c - 3 \, A b^{2} c^{2} + 5 \,{\left (5 \, B b^{2} c^{2} - A b c^{3}\right )} x\right )} \sqrt{x}}{4 \,{\left (b c^{6} x^{2} + 2 \, b^{2} c^{5} x + b^{3} c^{4}\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 = 1.16882, size = 117, normalized size = 0.93 \begin{align*} \frac{2 \, B \sqrt{x}}{c^{3}} - \frac{3 \,{\left (5 \, B b - A c\right )} \arctan \left (\frac{c \sqrt{x}}{\sqrt{b c}}\right )}{4 \, \sqrt{b c} c^{3}} + \frac{9 \, B b c x^{\frac{3}{2}} - 5 \, A c^{2} x^{\frac{3}{2}} + 7 \, B b^{2} \sqrt{x} - 3 \, A b c \sqrt{x}}{4 \,{\left (c x + b\right )}^{2} c^{3}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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