Optimal. Leaf size=109 \[ \frac{x^{3/2} (5 b B-3 A c)}{3 b c^2}-\frac{\sqrt{x} (5 b B-3 A c)}{c^3}+\frac{\sqrt{b} (5 b B-3 A c) \tan ^{-1}\left (\frac{\sqrt{c} \sqrt{x}}{\sqrt{b}}\right )}{c^{7/2}}-\frac{x^{5/2} (b B-A c)}{b c (b+c x)} \]
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Rubi [A] time = 0.0557021, antiderivative size = 109, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 5, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.227, Rules used = {781, 78, 50, 63, 205} \[ \frac{x^{3/2} (5 b B-3 A c)}{3 b c^2}-\frac{\sqrt{x} (5 b B-3 A c)}{c^3}+\frac{\sqrt{b} (5 b B-3 A c) \tan ^{-1}\left (\frac{\sqrt{c} \sqrt{x}}{\sqrt{b}}\right )}{c^{7/2}}-\frac{x^{5/2} (b B-A c)}{b c (b+c x)} \]
Antiderivative was successfully verified.
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Rule 781
Rule 78
Rule 50
Rule 63
Rule 205
Rubi steps
\begin{align*} \int \frac{x^{7/2} (A+B x)}{\left (b x+c x^2\right )^2} \, dx &=\int \frac{x^{3/2} (A+B x)}{(b+c x)^2} \, dx\\ &=-\frac{(b B-A c) x^{5/2}}{b c (b+c x)}-\frac{\left (-\frac{5 b B}{2}+\frac{3 A c}{2}\right ) \int \frac{x^{3/2}}{b+c x} \, dx}{b c}\\ &=\frac{(5 b B-3 A c) x^{3/2}}{3 b c^2}-\frac{(b B-A c) x^{5/2}}{b c (b+c x)}-\frac{(5 b B-3 A c) \int \frac{\sqrt{x}}{b+c x} \, dx}{2 c^2}\\ &=-\frac{(5 b B-3 A c) \sqrt{x}}{c^3}+\frac{(5 b B-3 A c) x^{3/2}}{3 b c^2}-\frac{(b B-A c) x^{5/2}}{b c (b+c x)}+\frac{(b (5 b B-3 A c)) \int \frac{1}{\sqrt{x} (b+c x)} \, dx}{2 c^3}\\ &=-\frac{(5 b B-3 A c) \sqrt{x}}{c^3}+\frac{(5 b B-3 A c) x^{3/2}}{3 b c^2}-\frac{(b B-A c) x^{5/2}}{b c (b+c x)}+\frac{(b (5 b B-3 A c)) \operatorname{Subst}\left (\int \frac{1}{b+c x^2} \, dx,x,\sqrt{x}\right )}{c^3}\\ &=-\frac{(5 b B-3 A c) \sqrt{x}}{c^3}+\frac{(5 b B-3 A c) x^{3/2}}{3 b c^2}-\frac{(b B-A c) x^{5/2}}{b c (b+c x)}+\frac{\sqrt{b} (5 b B-3 A c) \tan ^{-1}\left (\frac{\sqrt{c} \sqrt{x}}{\sqrt{b}}\right )}{c^{7/2}}\\ \end{align*}
Mathematica [A] time = 0.0606876, size = 88, normalized size = 0.81 \[ \frac{\sqrt{x} \left (b c (9 A-10 B x)+2 c^2 x (3 A+B x)-15 b^2 B\right )}{3 c^3 (b+c x)}+\frac{\sqrt{b} (5 b B-3 A c) \tan ^{-1}\left (\frac{\sqrt{c} \sqrt{x}}{\sqrt{b}}\right )}{c^{7/2}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.013, size = 113, normalized size = 1. \begin{align*}{\frac{2\,B}{3\,{c}^{2}}{x}^{{\frac{3}{2}}}}+2\,{\frac{A\sqrt{x}}{{c}^{2}}}-4\,{\frac{bB\sqrt{x}}{{c}^{3}}}+{\frac{Ab}{{c}^{2} \left ( cx+b \right ) }\sqrt{x}}-{\frac{{b}^{2}B}{{c}^{3} \left ( cx+b \right ) }\sqrt{x}}-3\,{\frac{Ab}{{c}^{2}\sqrt{bc}}\arctan \left ({\frac{\sqrt{x}c}{\sqrt{bc}}} \right ) }+5\,{\frac{{b}^{2}B}{{c}^{3}\sqrt{bc}}\arctan \left ({\frac{\sqrt{x}c}{\sqrt{bc}}} \right ) } \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 = 1.59858, size = 524, normalized size = 4.81 \begin{align*} \left [-\frac{3 \,{\left (5 \, B b^{2} - 3 \, A b c +{\left (5 \, B b c - 3 \, A c^{2}\right )} x\right )} \sqrt{-\frac{b}{c}} \log \left (\frac{c x - 2 \, c \sqrt{x} \sqrt{-\frac{b}{c}} - b}{c x + b}\right ) - 2 \,{\left (2 \, B c^{2} x^{2} - 15 \, B b^{2} + 9 \, A b c - 2 \,{\left (5 \, B b c - 3 \, A c^{2}\right )} x\right )} \sqrt{x}}{6 \,{\left (c^{4} x + b c^{3}\right )}}, \frac{3 \,{\left (5 \, B b^{2} - 3 \, A b c +{\left (5 \, B b c - 3 \, A c^{2}\right )} x\right )} \sqrt{\frac{b}{c}} \arctan \left (\frac{c \sqrt{x} \sqrt{\frac{b}{c}}}{b}\right ) +{\left (2 \, B c^{2} x^{2} - 15 \, B b^{2} + 9 \, A b c - 2 \,{\left (5 \, B b c - 3 \, A c^{2}\right )} x\right )} \sqrt{x}}{3 \,{\left (c^{4} x + b c^{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 = 1.12049, size = 128, normalized size = 1.17 \begin{align*} \frac{{\left (5 \, B b^{2} - 3 \, A b c\right )} \arctan \left (\frac{c \sqrt{x}}{\sqrt{b c}}\right )}{\sqrt{b c} c^{3}} - \frac{B b^{2} \sqrt{x} - A b c \sqrt{x}}{{\left (c x + b\right )} c^{3}} + \frac{2 \,{\left (B c^{4} x^{\frac{3}{2}} - 6 \, B b c^{3} \sqrt{x} + 3 \, A c^{4} \sqrt{x}\right )}}{3 \, c^{6}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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