Optimal. Leaf size=147 \[ -\frac{x^{5/2} (7 b B-3 A c)}{4 b c^2 (b+c x)}+\frac{5 x^{3/2} (7 b B-3 A c)}{12 b c^3}-\frac{5 \sqrt{x} (7 b B-3 A c)}{4 c^4}+\frac{5 \sqrt{b} (7 b B-3 A c) \tan ^{-1}\left (\frac{\sqrt{c} \sqrt{x}}{\sqrt{b}}\right )}{4 c^{9/2}}-\frac{x^{7/2} (b B-A c)}{2 b c (b+c x)^2} \]
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Rubi [A] time = 0.070526, antiderivative size = 147, normalized size of antiderivative = 1., number of steps used = 7, 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} \[ -\frac{x^{5/2} (7 b B-3 A c)}{4 b c^2 (b+c x)}+\frac{5 x^{3/2} (7 b B-3 A c)}{12 b c^3}-\frac{5 \sqrt{x} (7 b B-3 A c)}{4 c^4}+\frac{5 \sqrt{b} (7 b B-3 A c) \tan ^{-1}\left (\frac{\sqrt{c} \sqrt{x}}{\sqrt{b}}\right )}{4 c^{9/2}}-\frac{x^{7/2} (b B-A c)}{2 b c (b+c x)^2} \]
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^{11/2} (A+B x)}{\left (b x+c x^2\right )^3} \, dx &=\int \frac{x^{5/2} (A+B x)}{(b+c x)^3} \, dx\\ &=-\frac{(b B-A c) x^{7/2}}{2 b c (b+c x)^2}-\frac{\left (-\frac{7 b B}{2}+\frac{3 A c}{2}\right ) \int \frac{x^{5/2}}{(b+c x)^2} \, dx}{2 b c}\\ &=-\frac{(b B-A c) x^{7/2}}{2 b c (b+c x)^2}-\frac{(7 b B-3 A c) x^{5/2}}{4 b c^2 (b+c x)}+\frac{(5 (7 b B-3 A c)) \int \frac{x^{3/2}}{b+c x} \, dx}{8 b c^2}\\ &=\frac{5 (7 b B-3 A c) x^{3/2}}{12 b c^3}-\frac{(b B-A c) x^{7/2}}{2 b c (b+c x)^2}-\frac{(7 b B-3 A c) x^{5/2}}{4 b c^2 (b+c x)}-\frac{(5 (7 b B-3 A c)) \int \frac{\sqrt{x}}{b+c x} \, dx}{8 c^3}\\ &=-\frac{5 (7 b B-3 A c) \sqrt{x}}{4 c^4}+\frac{5 (7 b B-3 A c) x^{3/2}}{12 b c^3}-\frac{(b B-A c) x^{7/2}}{2 b c (b+c x)^2}-\frac{(7 b B-3 A c) x^{5/2}}{4 b c^2 (b+c x)}+\frac{(5 b (7 b B-3 A c)) \int \frac{1}{\sqrt{x} (b+c x)} \, dx}{8 c^4}\\ &=-\frac{5 (7 b B-3 A c) \sqrt{x}}{4 c^4}+\frac{5 (7 b B-3 A c) x^{3/2}}{12 b c^3}-\frac{(b B-A c) x^{7/2}}{2 b c (b+c x)^2}-\frac{(7 b B-3 A c) x^{5/2}}{4 b c^2 (b+c x)}+\frac{(5 b (7 b B-3 A c)) \operatorname{Subst}\left (\int \frac{1}{b+c x^2} \, dx,x,\sqrt{x}\right )}{4 c^4}\\ &=-\frac{5 (7 b B-3 A c) \sqrt{x}}{4 c^4}+\frac{5 (7 b B-3 A c) x^{3/2}}{12 b c^3}-\frac{(b B-A c) x^{7/2}}{2 b c (b+c x)^2}-\frac{(7 b B-3 A c) x^{5/2}}{4 b c^2 (b+c x)}+\frac{5 \sqrt{b} (7 b B-3 A c) \tan ^{-1}\left (\frac{\sqrt{c} \sqrt{x}}{\sqrt{b}}\right )}{4 c^{9/2}}\\ \end{align*}
Mathematica [C] time = 0.0242449, size = 61, normalized size = 0.41 \[ \frac{x^{7/2} \left (\frac{7 b^2 (A c-b B)}{(b+c x)^2}+(7 b B-3 A c) \, _2F_1\left (2,\frac{7}{2};\frac{9}{2};-\frac{c x}{b}\right )\right )}{14 b^3 c} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.016, size = 152, normalized size = 1. \begin{align*}{\frac{2\,B}{3\,{c}^{3}}{x}^{{\frac{3}{2}}}}+2\,{\frac{A\sqrt{x}}{{c}^{3}}}-6\,{\frac{bB\sqrt{x}}{{c}^{4}}}+{\frac{9\,Ab}{4\,{c}^{2} \left ( cx+b \right ) ^{2}}{x}^{{\frac{3}{2}}}}-{\frac{13\,{b}^{2}B}{4\,{c}^{3} \left ( cx+b \right ) ^{2}}{x}^{{\frac{3}{2}}}}+{\frac{7\,A{b}^{2}}{4\,{c}^{3} \left ( cx+b \right ) ^{2}}\sqrt{x}}-{\frac{11\,{b}^{3}B}{4\,{c}^{4} \left ( cx+b \right ) ^{2}}\sqrt{x}}-{\frac{15\,Ab}{4\,{c}^{3}}\arctan \left ({c\sqrt{x}{\frac{1}{\sqrt{bc}}}} \right ){\frac{1}{\sqrt{bc}}}}+{\frac{35\,{b}^{2}B}{4\,{c}^{4}}\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 = 1.97242, size = 775, normalized size = 5.27 \begin{align*} \left [-\frac{15 \,{\left (7 \, B b^{3} - 3 \, A b^{2} c +{\left (7 \, B b c^{2} - 3 \, A c^{3}\right )} x^{2} + 2 \,{\left (7 \, B b^{2} c - 3 \, A b 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 (8 \, B c^{3} x^{3} - 105 \, B b^{3} + 45 \, A b^{2} c - 8 \,{\left (7 \, B b c^{2} - 3 \, A c^{3}\right )} x^{2} - 25 \,{\left (7 \, B b^{2} c - 3 \, A b c^{2}\right )} x\right )} \sqrt{x}}{24 \,{\left (c^{6} x^{2} + 2 \, b c^{5} x + b^{2} c^{4}\right )}}, \frac{15 \,{\left (7 \, B b^{3} - 3 \, A b^{2} c +{\left (7 \, B b c^{2} - 3 \, A c^{3}\right )} x^{2} + 2 \,{\left (7 \, B b^{2} c - 3 \, A b c^{2}\right )} x\right )} \sqrt{\frac{b}{c}} \arctan \left (\frac{c \sqrt{x} \sqrt{\frac{b}{c}}}{b}\right ) +{\left (8 \, B c^{3} x^{3} - 105 \, B b^{3} + 45 \, A b^{2} c - 8 \,{\left (7 \, B b c^{2} - 3 \, A c^{3}\right )} x^{2} - 25 \,{\left (7 \, B b^{2} c - 3 \, A b c^{2}\right )} x\right )} \sqrt{x}}{12 \,{\left (c^{6} x^{2} + 2 \, b c^{5} x + b^{2} 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 = 1.1444, size = 161, normalized size = 1.1 \begin{align*} \frac{5 \,{\left (7 \, B b^{2} - 3 \, A b c\right )} \arctan \left (\frac{c \sqrt{x}}{\sqrt{b c}}\right )}{4 \, \sqrt{b c} c^{4}} - \frac{13 \, B b^{2} c x^{\frac{3}{2}} - 9 \, A b c^{2} x^{\frac{3}{2}} + 11 \, B b^{3} \sqrt{x} - 7 \, A b^{2} c \sqrt{x}}{4 \,{\left (c x + b\right )}^{2} c^{4}} + \frac{2 \,{\left (B c^{6} x^{\frac{3}{2}} - 9 \, B b c^{5} \sqrt{x} + 3 \, A c^{6} \sqrt{x}\right )}}{3 \, c^{9}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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