Optimal. Leaf size=177 \[ -\frac{c^2 \sqrt{b x^2+c x^4} (8 b B-3 A c)}{128 b^2 x^3}+\frac{c^3 (8 b B-3 A c) \tanh ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{b x^2+c x^4}}\right )}{128 b^{5/2}}-\frac{c \sqrt{b x^2+c x^4} (8 b B-3 A c)}{64 b x^5}-\frac{\left (b x^2+c x^4\right )^{3/2} (8 b B-3 A c)}{48 b x^9}-\frac{A \left (b x^2+c x^4\right )^{5/2}}{8 b x^{13}} \]
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Rubi [A] time = 0.281209, antiderivative size = 177, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 5, integrand size = 26, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.192, Rules used = {2038, 2020, 2025, 2008, 206} \[ -\frac{c^2 \sqrt{b x^2+c x^4} (8 b B-3 A c)}{128 b^2 x^3}+\frac{c^3 (8 b B-3 A c) \tanh ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{b x^2+c x^4}}\right )}{128 b^{5/2}}-\frac{c \sqrt{b x^2+c x^4} (8 b B-3 A c)}{64 b x^5}-\frac{\left (b x^2+c x^4\right )^{3/2} (8 b B-3 A c)}{48 b x^9}-\frac{A \left (b x^2+c x^4\right )^{5/2}}{8 b x^{13}} \]
Antiderivative was successfully verified.
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Rule 2038
Rule 2020
Rule 2025
Rule 2008
Rule 206
Rubi steps
\begin{align*} \int \frac{\left (A+B x^2\right ) \left (b x^2+c x^4\right )^{3/2}}{x^{12}} \, dx &=-\frac{A \left (b x^2+c x^4\right )^{5/2}}{8 b x^{13}}-\frac{(-8 b B+3 A c) \int \frac{\left (b x^2+c x^4\right )^{3/2}}{x^{10}} \, dx}{8 b}\\ &=-\frac{(8 b B-3 A c) \left (b x^2+c x^4\right )^{3/2}}{48 b x^9}-\frac{A \left (b x^2+c x^4\right )^{5/2}}{8 b x^{13}}+\frac{(c (8 b B-3 A c)) \int \frac{\sqrt{b x^2+c x^4}}{x^6} \, dx}{16 b}\\ &=-\frac{c (8 b B-3 A c) \sqrt{b x^2+c x^4}}{64 b x^5}-\frac{(8 b B-3 A c) \left (b x^2+c x^4\right )^{3/2}}{48 b x^9}-\frac{A \left (b x^2+c x^4\right )^{5/2}}{8 b x^{13}}+\frac{\left (c^2 (8 b B-3 A c)\right ) \int \frac{1}{x^2 \sqrt{b x^2+c x^4}} \, dx}{64 b}\\ &=-\frac{c (8 b B-3 A c) \sqrt{b x^2+c x^4}}{64 b x^5}-\frac{c^2 (8 b B-3 A c) \sqrt{b x^2+c x^4}}{128 b^2 x^3}-\frac{(8 b B-3 A c) \left (b x^2+c x^4\right )^{3/2}}{48 b x^9}-\frac{A \left (b x^2+c x^4\right )^{5/2}}{8 b x^{13}}-\frac{\left (c^3 (8 b B-3 A c)\right ) \int \frac{1}{\sqrt{b x^2+c x^4}} \, dx}{128 b^2}\\ &=-\frac{c (8 b B-3 A c) \sqrt{b x^2+c x^4}}{64 b x^5}-\frac{c^2 (8 b B-3 A c) \sqrt{b x^2+c x^4}}{128 b^2 x^3}-\frac{(8 b B-3 A c) \left (b x^2+c x^4\right )^{3/2}}{48 b x^9}-\frac{A \left (b x^2+c x^4\right )^{5/2}}{8 b x^{13}}+\frac{\left (c^3 (8 b B-3 A c)\right ) \operatorname{Subst}\left (\int \frac{1}{1-b x^2} \, dx,x,\frac{x}{\sqrt{b x^2+c x^4}}\right )}{128 b^2}\\ &=-\frac{c (8 b B-3 A c) \sqrt{b x^2+c x^4}}{64 b x^5}-\frac{c^2 (8 b B-3 A c) \sqrt{b x^2+c x^4}}{128 b^2 x^3}-\frac{(8 b B-3 A c) \left (b x^2+c x^4\right )^{3/2}}{48 b x^9}-\frac{A \left (b x^2+c x^4\right )^{5/2}}{8 b x^{13}}+\frac{c^3 (8 b B-3 A c) \tanh ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{b x^2+c x^4}}\right )}{128 b^{5/2}}\\ \end{align*}
Mathematica [C] time = 0.0347221, size = 66, normalized size = 0.37 \[ \frac{\left (x^2 \left (b+c x^2\right )\right )^{5/2} \left (c^3 x^8 (8 b B-3 A c) \, _2F_1\left (\frac{5}{2},4;\frac{7}{2};\frac{c x^2}{b}+1\right )-5 A b^4\right )}{40 b^5 x^{13}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.017, size = 302, normalized size = 1.7 \begin{align*} -{\frac{1}{384\,{x}^{11}{b}^{4}} \left ( c{x}^{4}+b{x}^{2} \right ) ^{{\frac{3}{2}}} \left ( 9\,A{b}^{3/2}\ln \left ( 2\,{\frac{\sqrt{b}\sqrt{c{x}^{2}+b}+b}{x}} \right ){x}^{8}{c}^{4}-3\,A \left ( c{x}^{2}+b \right ) ^{3/2}{x}^{8}{c}^{4}-24\,B{b}^{5/2}\ln \left ( 2\,{\frac{\sqrt{b}\sqrt{c{x}^{2}+b}+b}{x}} \right ){x}^{8}{c}^{3}+8\,B \left ( c{x}^{2}+b \right ) ^{3/2}{x}^{8}b{c}^{3}+3\,A \left ( c{x}^{2}+b \right ) ^{5/2}{x}^{6}{c}^{3}-9\,A\sqrt{c{x}^{2}+b}{x}^{8}b{c}^{4}-8\,B \left ( c{x}^{2}+b \right ) ^{5/2}{x}^{6}b{c}^{2}+24\,B\sqrt{c{x}^{2}+b}{x}^{8}{b}^{2}{c}^{3}+6\,A \left ( c{x}^{2}+b \right ) ^{5/2}{x}^{4}b{c}^{2}-16\,B \left ( c{x}^{2}+b \right ) ^{5/2}{x}^{4}{b}^{2}c-24\,A \left ( c{x}^{2}+b \right ) ^{5/2}{x}^{2}{b}^{2}c+64\,B \left ( c{x}^{2}+b \right ) ^{5/2}{x}^{2}{b}^{3}+48\,A \left ( c{x}^{2}+b \right ) ^{5/2}{b}^{3} \right ) \left ( c{x}^{2}+b \right ) ^{-{\frac{3}{2}}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{{\left (c x^{4} + b x^{2}\right )}^{\frac{3}{2}}{\left (B x^{2} + A\right )}}{x^{12}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.23321, size = 664, normalized size = 3.75 \begin{align*} \left [-\frac{3 \,{\left (8 \, B b c^{3} - 3 \, A c^{4}\right )} \sqrt{b} x^{9} \log \left (-\frac{c x^{3} + 2 \, b x - 2 \, \sqrt{c x^{4} + b x^{2}} \sqrt{b}}{x^{3}}\right ) + 2 \,{\left (3 \,{\left (8 \, B b^{2} c^{2} - 3 \, A b c^{3}\right )} x^{6} + 48 \, A b^{4} + 2 \,{\left (56 \, B b^{3} c + 3 \, A b^{2} c^{2}\right )} x^{4} + 8 \,{\left (8 \, B b^{4} + 9 \, A b^{3} c\right )} x^{2}\right )} \sqrt{c x^{4} + b x^{2}}}{768 \, b^{3} x^{9}}, -\frac{3 \,{\left (8 \, B b c^{3} - 3 \, A c^{4}\right )} \sqrt{-b} x^{9} \arctan \left (\frac{\sqrt{c x^{4} + b x^{2}} \sqrt{-b}}{c x^{3} + b x}\right ) +{\left (3 \,{\left (8 \, B b^{2} c^{2} - 3 \, A b c^{3}\right )} x^{6} + 48 \, A b^{4} + 2 \,{\left (56 \, B b^{3} c + 3 \, A b^{2} c^{2}\right )} x^{4} + 8 \,{\left (8 \, B b^{4} + 9 \, A b^{3} c\right )} x^{2}\right )} \sqrt{c x^{4} + b x^{2}}}{384 \, b^{3} x^{9}}\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^{2} \left (b + c x^{2}\right )\right )^{\frac{3}{2}} \left (A + B x^{2}\right )}{x^{12}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.29051, size = 289, normalized size = 1.63 \begin{align*} -\frac{\frac{3 \,{\left (8 \, B b c^{4} \mathrm{sgn}\left (x\right ) - 3 \, A c^{5} \mathrm{sgn}\left (x\right )\right )} \arctan \left (\frac{\sqrt{c x^{2} + b}}{\sqrt{-b}}\right )}{\sqrt{-b} b^{2}} + \frac{24 \,{\left (c x^{2} + b\right )}^{\frac{7}{2}} B b c^{4} \mathrm{sgn}\left (x\right ) + 40 \,{\left (c x^{2} + b\right )}^{\frac{5}{2}} B b^{2} c^{4} \mathrm{sgn}\left (x\right ) - 88 \,{\left (c x^{2} + b\right )}^{\frac{3}{2}} B b^{3} c^{4} \mathrm{sgn}\left (x\right ) + 24 \, \sqrt{c x^{2} + b} B b^{4} c^{4} \mathrm{sgn}\left (x\right ) - 9 \,{\left (c x^{2} + b\right )}^{\frac{7}{2}} A c^{5} \mathrm{sgn}\left (x\right ) + 33 \,{\left (c x^{2} + b\right )}^{\frac{5}{2}} A b c^{5} \mathrm{sgn}\left (x\right ) + 33 \,{\left (c x^{2} + b\right )}^{\frac{3}{2}} A b^{2} c^{5} \mathrm{sgn}\left (x\right ) - 9 \, \sqrt{c x^{2} + b} A b^{3} c^{5} \mathrm{sgn}\left (x\right )}{b^{2} c^{4} x^{8}}}{384 \, c} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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