Optimal. Leaf size=119 \[ -\frac{2 A \left (b x+c x^2\right )^{7/2}}{7 b x^7}-\frac{2 B c^2 \sqrt{b x+c x^2}}{x}+2 B c^{5/2} \tanh ^{-1}\left (\frac{\sqrt{c} x}{\sqrt{b x+c x^2}}\right )-\frac{2 B \left (b x+c x^2\right )^{5/2}}{5 x^5}-\frac{2 B c \left (b x+c x^2\right )^{3/2}}{3 x^3} \]
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Rubi [A] time = 0.13935, antiderivative size = 119, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 4, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.182, Rules used = {792, 662, 620, 206} \[ -\frac{2 A \left (b x+c x^2\right )^{7/2}}{7 b x^7}-\frac{2 B c^2 \sqrt{b x+c x^2}}{x}+2 B c^{5/2} \tanh ^{-1}\left (\frac{\sqrt{c} x}{\sqrt{b x+c x^2}}\right )-\frac{2 B \left (b x+c x^2\right )^{5/2}}{5 x^5}-\frac{2 B c \left (b x+c x^2\right )^{3/2}}{3 x^3} \]
Antiderivative was successfully verified.
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Rule 792
Rule 662
Rule 620
Rule 206
Rubi steps
\begin{align*} \int \frac{(A+B x) \left (b x+c x^2\right )^{5/2}}{x^7} \, dx &=-\frac{2 A \left (b x+c x^2\right )^{7/2}}{7 b x^7}+B \int \frac{\left (b x+c x^2\right )^{5/2}}{x^6} \, dx\\ &=-\frac{2 B \left (b x+c x^2\right )^{5/2}}{5 x^5}-\frac{2 A \left (b x+c x^2\right )^{7/2}}{7 b x^7}+(B c) \int \frac{\left (b x+c x^2\right )^{3/2}}{x^4} \, dx\\ &=-\frac{2 B c \left (b x+c x^2\right )^{3/2}}{3 x^3}-\frac{2 B \left (b x+c x^2\right )^{5/2}}{5 x^5}-\frac{2 A \left (b x+c x^2\right )^{7/2}}{7 b x^7}+\left (B c^2\right ) \int \frac{\sqrt{b x+c x^2}}{x^2} \, dx\\ &=-\frac{2 B c^2 \sqrt{b x+c x^2}}{x}-\frac{2 B c \left (b x+c x^2\right )^{3/2}}{3 x^3}-\frac{2 B \left (b x+c x^2\right )^{5/2}}{5 x^5}-\frac{2 A \left (b x+c x^2\right )^{7/2}}{7 b x^7}+\left (B c^3\right ) \int \frac{1}{\sqrt{b x+c x^2}} \, dx\\ &=-\frac{2 B c^2 \sqrt{b x+c x^2}}{x}-\frac{2 B c \left (b x+c x^2\right )^{3/2}}{3 x^3}-\frac{2 B \left (b x+c x^2\right )^{5/2}}{5 x^5}-\frac{2 A \left (b x+c x^2\right )^{7/2}}{7 b x^7}+\left (2 B c^3\right ) \operatorname{Subst}\left (\int \frac{1}{1-c x^2} \, dx,x,\frac{x}{\sqrt{b x+c x^2}}\right )\\ &=-\frac{2 B c^2 \sqrt{b x+c x^2}}{x}-\frac{2 B c \left (b x+c x^2\right )^{3/2}}{3 x^3}-\frac{2 B \left (b x+c x^2\right )^{5/2}}{5 x^5}-\frac{2 A \left (b x+c x^2\right )^{7/2}}{7 b x^7}+2 B c^{5/2} \tanh ^{-1}\left (\frac{\sqrt{c} x}{\sqrt{b x+c x^2}}\right )\\ \end{align*}
Mathematica [C] time = 0.0643442, size = 83, normalized size = 0.7 \[ \frac{2 (x (b+c x))^{5/2} \left ((b+c x)^3 (b B-A c)-\frac{b^4 B \, _2F_1\left (-\frac{7}{2},-\frac{7}{2};-\frac{5}{2};-\frac{c x}{b}\right )}{\sqrt{\frac{c x}{b}+1}}\right )}{7 b c x^6 (b+c x)^2} \]
Antiderivative was successfully verified.
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Maple [B] time = 0.011, size = 263, normalized size = 2.2 \begin{align*} -{\frac{2\,A}{7\,b{x}^{7}} \left ( c{x}^{2}+bx \right ) ^{{\frac{7}{2}}}}-{\frac{2\,B}{5\,b{x}^{6}} \left ( c{x}^{2}+bx \right ) ^{{\frac{7}{2}}}}-{\frac{4\,Bc}{15\,{b}^{2}{x}^{5}} \left ( c{x}^{2}+bx \right ) ^{{\frac{7}{2}}}}-{\frac{16\,B{c}^{2}}{15\,{b}^{3}{x}^{4}} \left ( c{x}^{2}+bx \right ) ^{{\frac{7}{2}}}}+{\frac{32\,B{c}^{3}}{5\,{b}^{4}{x}^{3}} \left ( c{x}^{2}+bx \right ) ^{{\frac{7}{2}}}}-{\frac{256\,B{c}^{4}}{15\,{b}^{5}{x}^{2}} \left ( c{x}^{2}+bx \right ) ^{{\frac{7}{2}}}}+{\frac{256\,B{c}^{5}}{15\,{b}^{5}} \left ( c{x}^{2}+bx \right ) ^{{\frac{5}{2}}}}+{\frac{32\,B{c}^{5}x}{3\,{b}^{4}} \left ( c{x}^{2}+bx \right ) ^{{\frac{3}{2}}}}+{\frac{16\,B{c}^{4}}{3\,{b}^{3}} \left ( c{x}^{2}+bx \right ) ^{{\frac{3}{2}}}}-4\,{\frac{B{c}^{4}\sqrt{c{x}^{2}+bx}x}{{b}^{2}}}-2\,{\frac{B{c}^{3}\sqrt{c{x}^{2}+bx}}{b}}+B{c}^{{\frac{5}{2}}}\ln \left ({ \left ({\frac{b}{2}}+cx \right ){\frac{1}{\sqrt{c}}}}+\sqrt{c{x}^{2}+bx} \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 = 1.85464, size = 564, normalized size = 4.74 \begin{align*} \left [\frac{105 \, B b c^{\frac{5}{2}} x^{4} \log \left (2 \, c x + b + 2 \, \sqrt{c x^{2} + b x} \sqrt{c}\right ) - 2 \,{\left (15 \, A b^{3} +{\left (161 \, B b c^{2} + 15 \, A c^{3}\right )} x^{3} +{\left (77 \, B b^{2} c + 45 \, A b c^{2}\right )} x^{2} + 3 \,{\left (7 \, B b^{3} + 15 \, A b^{2} c\right )} x\right )} \sqrt{c x^{2} + b x}}{105 \, b x^{4}}, -\frac{2 \,{\left (105 \, B b \sqrt{-c} c^{2} x^{4} \arctan \left (\frac{\sqrt{c x^{2} + b x} \sqrt{-c}}{c x}\right ) +{\left (15 \, A b^{3} +{\left (161 \, B b c^{2} + 15 \, A c^{3}\right )} x^{3} +{\left (77 \, B b^{2} c + 45 \, A b c^{2}\right )} x^{2} + 3 \,{\left (7 \, B b^{3} + 15 \, A b^{2} c\right )} x\right )} \sqrt{c x^{2} + b x}\right )}}{105 \, b x^{4}}\right ] \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\left (x \left (b + c x\right )\right )^{\frac{5}{2}} \left (A + B x\right )}{x^{7}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [B] time = 1.18853, size = 527, normalized size = 4.43 \begin{align*} -B c^{\frac{5}{2}} \log \left ({\left | -2 \,{\left (\sqrt{c} x - \sqrt{c x^{2} + b x}\right )} \sqrt{c} - b \right |}\right ) + \frac{2 \,{\left (315 \,{\left (\sqrt{c} x - \sqrt{c x^{2} + b x}\right )}^{6} B b c^{\frac{5}{2}} + 105 \,{\left (\sqrt{c} x - \sqrt{c x^{2} + b x}\right )}^{6} A c^{\frac{7}{2}} + 315 \,{\left (\sqrt{c} x - \sqrt{c x^{2} + b x}\right )}^{5} B b^{2} c^{2} + 315 \,{\left (\sqrt{c} x - \sqrt{c x^{2} + b x}\right )}^{5} A b c^{3} + 245 \,{\left (\sqrt{c} x - \sqrt{c x^{2} + b x}\right )}^{4} B b^{3} c^{\frac{3}{2}} + 525 \,{\left (\sqrt{c} x - \sqrt{c x^{2} + b x}\right )}^{4} A b^{2} c^{\frac{5}{2}} + 105 \,{\left (\sqrt{c} x - \sqrt{c x^{2} + b x}\right )}^{3} B b^{4} c + 525 \,{\left (\sqrt{c} x - \sqrt{c x^{2} + b x}\right )}^{3} A b^{3} c^{2} + 21 \,{\left (\sqrt{c} x - \sqrt{c x^{2} + b x}\right )}^{2} B b^{5} \sqrt{c} + 315 \,{\left (\sqrt{c} x - \sqrt{c x^{2} + b x}\right )}^{2} A b^{4} c^{\frac{3}{2}} + 105 \,{\left (\sqrt{c} x - \sqrt{c x^{2} + b x}\right )} A b^{5} c + 15 \, A b^{6} \sqrt{c}\right )}}{105 \,{\left (\sqrt{c} x - \sqrt{c x^{2} + b x}\right )}^{7} \sqrt{c}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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