Optimal. Leaf size=157 \[ -\frac{2 \left (b x+c x^2\right )^{5/2} (4 A c+3 b B)}{3 b x^3}+\frac{5 c \left (b x+c x^2\right )^{3/2} (4 A c+3 b B)}{6 b x}+\frac{5}{4} c \sqrt{b x+c x^2} (4 A c+3 b B)+\frac{5}{4} b \sqrt{c} (4 A c+3 b B) \tanh ^{-1}\left (\frac{\sqrt{c} x}{\sqrt{b x+c x^2}}\right )-\frac{2 A \left (b x+c x^2\right )^{7/2}}{3 b x^5} \]
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Rubi [A] time = 0.163561, antiderivative size = 157, 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 = {792, 662, 664, 620, 206} \[ -\frac{2 \left (b x+c x^2\right )^{5/2} (4 A c+3 b B)}{3 b x^3}+\frac{5 c \left (b x+c x^2\right )^{3/2} (4 A c+3 b B)}{6 b x}+\frac{5}{4} c \sqrt{b x+c x^2} (4 A c+3 b B)+\frac{5}{4} b \sqrt{c} (4 A c+3 b B) \tanh ^{-1}\left (\frac{\sqrt{c} x}{\sqrt{b x+c x^2}}\right )-\frac{2 A \left (b x+c x^2\right )^{7/2}}{3 b x^5} \]
Antiderivative was successfully verified.
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Rule 792
Rule 662
Rule 664
Rule 620
Rule 206
Rubi steps
\begin{align*} \int \frac{(A+B x) \left (b x+c x^2\right )^{5/2}}{x^5} \, dx &=-\frac{2 A \left (b x+c x^2\right )^{7/2}}{3 b x^5}+\frac{\left (2 \left (-5 (-b B+A c)+\frac{7}{2} (-b B+2 A c)\right )\right ) \int \frac{\left (b x+c x^2\right )^{5/2}}{x^4} \, dx}{3 b}\\ &=-\frac{2 (3 b B+4 A c) \left (b x+c x^2\right )^{5/2}}{3 b x^3}-\frac{2 A \left (b x+c x^2\right )^{7/2}}{3 b x^5}+\frac{(5 c (3 b B+4 A c)) \int \frac{\left (b x+c x^2\right )^{3/2}}{x^2} \, dx}{3 b}\\ &=\frac{5 c (3 b B+4 A c) \left (b x+c x^2\right )^{3/2}}{6 b x}-\frac{2 (3 b B+4 A c) \left (b x+c x^2\right )^{5/2}}{3 b x^3}-\frac{2 A \left (b x+c x^2\right )^{7/2}}{3 b x^5}+\frac{1}{4} (5 c (3 b B+4 A c)) \int \frac{\sqrt{b x+c x^2}}{x} \, dx\\ &=\frac{5}{4} c (3 b B+4 A c) \sqrt{b x+c x^2}+\frac{5 c (3 b B+4 A c) \left (b x+c x^2\right )^{3/2}}{6 b x}-\frac{2 (3 b B+4 A c) \left (b x+c x^2\right )^{5/2}}{3 b x^3}-\frac{2 A \left (b x+c x^2\right )^{7/2}}{3 b x^5}+\frac{1}{8} (5 b c (3 b B+4 A c)) \int \frac{1}{\sqrt{b x+c x^2}} \, dx\\ &=\frac{5}{4} c (3 b B+4 A c) \sqrt{b x+c x^2}+\frac{5 c (3 b B+4 A c) \left (b x+c x^2\right )^{3/2}}{6 b x}-\frac{2 (3 b B+4 A c) \left (b x+c x^2\right )^{5/2}}{3 b x^3}-\frac{2 A \left (b x+c x^2\right )^{7/2}}{3 b x^5}+\frac{1}{4} (5 b c (3 b B+4 A c)) \operatorname{Subst}\left (\int \frac{1}{1-c x^2} \, dx,x,\frac{x}{\sqrt{b x+c x^2}}\right )\\ &=\frac{5}{4} c (3 b B+4 A c) \sqrt{b x+c x^2}+\frac{5 c (3 b B+4 A c) \left (b x+c x^2\right )^{3/2}}{6 b x}-\frac{2 (3 b B+4 A c) \left (b x+c x^2\right )^{5/2}}{3 b x^3}-\frac{2 A \left (b x+c x^2\right )^{7/2}}{3 b x^5}+\frac{5}{4} b \sqrt{c} (3 b B+4 A c) \tanh ^{-1}\left (\frac{\sqrt{c} x}{\sqrt{b x+c x^2}}\right )\\ \end{align*}
Mathematica [C] time = 0.0530951, size = 86, normalized size = 0.55 \[ -\frac{2 \sqrt{x (b+c x)} \left (b^2 x (4 A c+3 b B) \, _2F_1\left (-\frac{5}{2},-\frac{1}{2};\frac{1}{2};-\frac{c x}{b}\right )+A \sqrt{\frac{c x}{b}+1} (b+c x)^3\right )}{3 b x^2 \sqrt{\frac{c x}{b}+1}} \]
Antiderivative was successfully verified.
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Maple [B] time = 0.012, size = 411, normalized size = 2.6 \begin{align*} -2\,{\frac{B \left ( c{x}^{2}+bx \right ) ^{7/2}}{b{x}^{4}}}+12\,{\frac{Bc \left ( c{x}^{2}+bx \right ) ^{7/2}}{{b}^{2}{x}^{3}}}-32\,{\frac{B{c}^{2} \left ( c{x}^{2}+bx \right ) ^{7/2}}{{b}^{3}{x}^{2}}}+32\,{\frac{B{c}^{3} \left ( c{x}^{2}+bx \right ) ^{5/2}}{{b}^{3}}}+20\,{\frac{B{c}^{3} \left ( c{x}^{2}+bx \right ) ^{3/2}x}{{b}^{2}}}+10\,{\frac{B{c}^{2} \left ( c{x}^{2}+bx \right ) ^{3/2}}{b}}-{\frac{15\,B{c}^{2}x}{2}\sqrt{c{x}^{2}+bx}}-{\frac{15\,bBc}{4}\sqrt{c{x}^{2}+bx}}+{\frac{15\,{b}^{2}B}{8}\sqrt{c}\ln \left ({ \left ({\frac{b}{2}}+cx \right ){\frac{1}{\sqrt{c}}}}+\sqrt{c{x}^{2}+bx} \right ) }-{\frac{2\,A}{3\,b{x}^{5}} \left ( c{x}^{2}+bx \right ) ^{{\frac{7}{2}}}}-{\frac{8\,Ac}{3\,{b}^{2}{x}^{4}} \left ( c{x}^{2}+bx \right ) ^{{\frac{7}{2}}}}+16\,{\frac{A{c}^{2} \left ( c{x}^{2}+bx \right ) ^{7/2}}{{b}^{3}{x}^{3}}}-{\frac{128\,A{c}^{3}}{3\,{b}^{4}{x}^{2}} \left ( c{x}^{2}+bx \right ) ^{{\frac{7}{2}}}}+{\frac{128\,A{c}^{4}}{3\,{b}^{4}} \left ( c{x}^{2}+bx \right ) ^{{\frac{5}{2}}}}+{\frac{80\,A{c}^{4}x}{3\,{b}^{3}} \left ( c{x}^{2}+bx \right ) ^{{\frac{3}{2}}}}+{\frac{40\,A{c}^{3}}{3\,{b}^{2}} \left ( c{x}^{2}+bx \right ) ^{{\frac{3}{2}}}}-10\,{\frac{A{c}^{3}\sqrt{c{x}^{2}+bx}x}{b}}-5\,A{c}^{2}\sqrt{c{x}^{2}+bx}+{\frac{5\,Ab}{2}{c}^{{\frac{3}{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 = 2.00727, size = 518, normalized size = 3.3 \begin{align*} \left [\frac{15 \,{\left (3 \, B b^{2} + 4 \, A b c\right )} \sqrt{c} x^{2} \log \left (2 \, c x + b + 2 \, \sqrt{c x^{2} + b x} \sqrt{c}\right ) + 2 \,{\left (6 \, B c^{2} x^{3} - 8 \, A b^{2} + 3 \,{\left (9 \, B b c + 4 \, A c^{2}\right )} x^{2} - 8 \,{\left (3 \, B b^{2} + 7 \, A b c\right )} x\right )} \sqrt{c x^{2} + b x}}{24 \, x^{2}}, -\frac{15 \,{\left (3 \, B b^{2} + 4 \, A b c\right )} \sqrt{-c} x^{2} \arctan \left (\frac{\sqrt{c x^{2} + b x} \sqrt{-c}}{c x}\right ) -{\left (6 \, B c^{2} x^{3} - 8 \, A b^{2} + 3 \,{\left (9 \, B b c + 4 \, A c^{2}\right )} x^{2} - 8 \,{\left (3 \, B b^{2} + 7 \, A b c\right )} x\right )} \sqrt{c x^{2} + b x}}{12 \, x^{2}}\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^{5}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.18783, size = 285, normalized size = 1.82 \begin{align*} \frac{1}{4} \,{\left (2 \, B c^{2} x + \frac{9 \, B b c^{2} + 4 \, A c^{3}}{c}\right )} \sqrt{c x^{2} + b x} - \frac{5 \,{\left (3 \, B b^{2} c + 4 \, A b c^{2}\right )} \log \left ({\left | -2 \,{\left (\sqrt{c} x - \sqrt{c x^{2} + b x}\right )} \sqrt{c} - b \right |}\right )}{8 \, \sqrt{c}} + \frac{2 \,{\left (3 \,{\left (\sqrt{c} x - \sqrt{c x^{2} + b x}\right )}^{2} B b^{3} \sqrt{c} + 9 \,{\left (\sqrt{c} x - \sqrt{c x^{2} + b x}\right )}^{2} A b^{2} c^{\frac{3}{2}} + 3 \,{\left (\sqrt{c} x - \sqrt{c x^{2} + b x}\right )} A b^{3} c + A b^{4} \sqrt{c}\right )}}{3 \,{\left (\sqrt{c} x - \sqrt{c x^{2} + b x}\right )}^{3} \sqrt{c}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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