Optimal. Leaf size=169 \[ \frac{7 c^{3/2} (5 b B-9 A c) \tan ^{-1}\left (\frac{\sqrt{c} \sqrt{x}}{\sqrt{b}}\right )}{4 b^{11/2}}-\frac{7 (5 b B-9 A c)}{12 b^4 x^{3/2}}-\frac{5 b B-9 A c}{4 b^2 c x^{5/2} (b+c x)}+\frac{7 (5 b B-9 A c)}{20 b^3 c x^{5/2}}+\frac{7 c (5 b B-9 A c)}{4 b^5 \sqrt{x}}-\frac{b B-A c}{2 b c x^{5/2} (b+c x)^2} \]
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Rubi [A] time = 0.0873458, antiderivative size = 169, normalized size of antiderivative = 1., number of steps used = 8, number of rules used = 5, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.227, Rules used = {781, 78, 51, 63, 205} \[ \frac{7 c^{3/2} (5 b B-9 A c) \tan ^{-1}\left (\frac{\sqrt{c} \sqrt{x}}{\sqrt{b}}\right )}{4 b^{11/2}}-\frac{7 (5 b B-9 A c)}{12 b^4 x^{3/2}}-\frac{5 b B-9 A c}{4 b^2 c x^{5/2} (b+c x)}+\frac{7 (5 b B-9 A c)}{20 b^3 c x^{5/2}}+\frac{7 c (5 b B-9 A c)}{4 b^5 \sqrt{x}}-\frac{b B-A c}{2 b c x^{5/2} (b+c x)^2} \]
Antiderivative was successfully verified.
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Rule 781
Rule 78
Rule 51
Rule 63
Rule 205
Rubi steps
\begin{align*} \int \frac{A+B x}{\sqrt{x} \left (b x+c x^2\right )^3} \, dx &=\int \frac{A+B x}{x^{7/2} (b+c x)^3} \, dx\\ &=-\frac{b B-A c}{2 b c x^{5/2} (b+c x)^2}-\frac{\left (\frac{5 b B}{2}-\frac{9 A c}{2}\right ) \int \frac{1}{x^{7/2} (b+c x)^2} \, dx}{2 b c}\\ &=-\frac{b B-A c}{2 b c x^{5/2} (b+c x)^2}-\frac{5 b B-9 A c}{4 b^2 c x^{5/2} (b+c x)}-\frac{(7 (5 b B-9 A c)) \int \frac{1}{x^{7/2} (b+c x)} \, dx}{8 b^2 c}\\ &=\frac{7 (5 b B-9 A c)}{20 b^3 c x^{5/2}}-\frac{b B-A c}{2 b c x^{5/2} (b+c x)^2}-\frac{5 b B-9 A c}{4 b^2 c x^{5/2} (b+c x)}+\frac{(7 (5 b B-9 A c)) \int \frac{1}{x^{5/2} (b+c x)} \, dx}{8 b^3}\\ &=\frac{7 (5 b B-9 A c)}{20 b^3 c x^{5/2}}-\frac{7 (5 b B-9 A c)}{12 b^4 x^{3/2}}-\frac{b B-A c}{2 b c x^{5/2} (b+c x)^2}-\frac{5 b B-9 A c}{4 b^2 c x^{5/2} (b+c x)}-\frac{(7 c (5 b B-9 A c)) \int \frac{1}{x^{3/2} (b+c x)} \, dx}{8 b^4}\\ &=\frac{7 (5 b B-9 A c)}{20 b^3 c x^{5/2}}-\frac{7 (5 b B-9 A c)}{12 b^4 x^{3/2}}+\frac{7 c (5 b B-9 A c)}{4 b^5 \sqrt{x}}-\frac{b B-A c}{2 b c x^{5/2} (b+c x)^2}-\frac{5 b B-9 A c}{4 b^2 c x^{5/2} (b+c x)}+\frac{\left (7 c^2 (5 b B-9 A c)\right ) \int \frac{1}{\sqrt{x} (b+c x)} \, dx}{8 b^5}\\ &=\frac{7 (5 b B-9 A c)}{20 b^3 c x^{5/2}}-\frac{7 (5 b B-9 A c)}{12 b^4 x^{3/2}}+\frac{7 c (5 b B-9 A c)}{4 b^5 \sqrt{x}}-\frac{b B-A c}{2 b c x^{5/2} (b+c x)^2}-\frac{5 b B-9 A c}{4 b^2 c x^{5/2} (b+c x)}+\frac{\left (7 c^2 (5 b B-9 A c)\right ) \operatorname{Subst}\left (\int \frac{1}{b+c x^2} \, dx,x,\sqrt{x}\right )}{4 b^5}\\ &=\frac{7 (5 b B-9 A c)}{20 b^3 c x^{5/2}}-\frac{7 (5 b B-9 A c)}{12 b^4 x^{3/2}}+\frac{7 c (5 b B-9 A c)}{4 b^5 \sqrt{x}}-\frac{b B-A c}{2 b c x^{5/2} (b+c x)^2}-\frac{5 b B-9 A c}{4 b^2 c x^{5/2} (b+c x)}+\frac{7 c^{3/2} (5 b B-9 A c) \tan ^{-1}\left (\frac{\sqrt{c} \sqrt{x}}{\sqrt{b}}\right )}{4 b^{11/2}}\\ \end{align*}
Mathematica [C] time = 0.0248452, size = 61, normalized size = 0.36 \[ \frac{\frac{5 b^2 (A c-b B)}{(b+c x)^2}+(5 b B-9 A c) \, _2F_1\left (-\frac{5}{2},2;-\frac{3}{2};-\frac{c x}{b}\right )}{10 b^3 c x^{5/2}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.02, size = 178, normalized size = 1.1 \begin{align*} -{\frac{2\,A}{5\,{b}^{3}}{x}^{-{\frac{5}{2}}}}+2\,{\frac{Ac}{{b}^{4}{x}^{3/2}}}-{\frac{2\,B}{3\,{b}^{3}}{x}^{-{\frac{3}{2}}}}-12\,{\frac{A{c}^{2}}{{b}^{5}\sqrt{x}}}+6\,{\frac{Bc}{{b}^{4}\sqrt{x}}}-{\frac{15\,{c}^{4}A}{4\,{b}^{5} \left ( cx+b \right ) ^{2}}{x}^{{\frac{3}{2}}}}+{\frac{11\,{c}^{3}B}{4\,{b}^{4} \left ( cx+b \right ) ^{2}}{x}^{{\frac{3}{2}}}}-{\frac{17\,{c}^{3}A}{4\,{b}^{4} \left ( cx+b \right ) ^{2}}\sqrt{x}}+{\frac{13\,{c}^{2}B}{4\,{b}^{3} \left ( cx+b \right ) ^{2}}\sqrt{x}}-{\frac{63\,{c}^{3}A}{4\,{b}^{5}}\arctan \left ({c\sqrt{x}{\frac{1}{\sqrt{bc}}}} \right ){\frac{1}{\sqrt{bc}}}}+{\frac{35\,{c}^{2}B}{4\,{b}^{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 = 1.71587, size = 950, normalized size = 5.62 \begin{align*} \left [-\frac{105 \,{\left ({\left (5 \, B b c^{3} - 9 \, A c^{4}\right )} x^{5} + 2 \,{\left (5 \, B b^{2} c^{2} - 9 \, A b c^{3}\right )} x^{4} +{\left (5 \, B b^{3} c - 9 \, A b^{2} c^{2}\right )} x^{3}\right )} \sqrt{-\frac{c}{b}} \log \left (\frac{c x - 2 \, b \sqrt{x} \sqrt{-\frac{c}{b}} - b}{c x + b}\right ) + 2 \,{\left (24 \, A b^{4} - 105 \,{\left (5 \, B b c^{3} - 9 \, A c^{4}\right )} x^{4} - 175 \,{\left (5 \, B b^{2} c^{2} - 9 \, A b c^{3}\right )} x^{3} - 56 \,{\left (5 \, B b^{3} c - 9 \, A b^{2} c^{2}\right )} x^{2} + 8 \,{\left (5 \, B b^{4} - 9 \, A b^{3} c\right )} x\right )} \sqrt{x}}{120 \,{\left (b^{5} c^{2} x^{5} + 2 \, b^{6} c x^{4} + b^{7} x^{3}\right )}}, -\frac{105 \,{\left ({\left (5 \, B b c^{3} - 9 \, A c^{4}\right )} x^{5} + 2 \,{\left (5 \, B b^{2} c^{2} - 9 \, A b c^{3}\right )} x^{4} +{\left (5 \, B b^{3} c - 9 \, A b^{2} c^{2}\right )} x^{3}\right )} \sqrt{\frac{c}{b}} \arctan \left (\frac{b \sqrt{\frac{c}{b}}}{c \sqrt{x}}\right ) +{\left (24 \, A b^{4} - 105 \,{\left (5 \, B b c^{3} - 9 \, A c^{4}\right )} x^{4} - 175 \,{\left (5 \, B b^{2} c^{2} - 9 \, A b c^{3}\right )} x^{3} - 56 \,{\left (5 \, B b^{3} c - 9 \, A b^{2} c^{2}\right )} x^{2} + 8 \,{\left (5 \, B b^{4} - 9 \, A b^{3} c\right )} x\right )} \sqrt{x}}{60 \,{\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 = 1.13233, size = 182, normalized size = 1.08 \begin{align*} \frac{7 \,{\left (5 \, B b c^{2} - 9 \, A c^{3}\right )} \arctan \left (\frac{c \sqrt{x}}{\sqrt{b c}}\right )}{4 \, \sqrt{b c} b^{5}} + \frac{11 \, B b c^{3} x^{\frac{3}{2}} - 15 \, A c^{4} x^{\frac{3}{2}} + 13 \, B b^{2} c^{2} \sqrt{x} - 17 \, A b c^{3} \sqrt{x}}{4 \,{\left (c x + b\right )}^{2} b^{5}} + \frac{2 \,{\left (45 \, B b c x^{2} - 90 \, A c^{2} x^{2} - 5 \, B b^{2} x + 15 \, A b c x - 3 \, A b^{2}\right )}}{15 \, b^{5} x^{\frac{5}{2}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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