Optimal. Leaf size=170 \[ \frac{16 c^3 \left (b x^2+c x^4\right )^{3/2} (11 b B-8 A c)}{3465 b^5 x^6}-\frac{8 c^2 \left (b x^2+c x^4\right )^{3/2} (11 b B-8 A c)}{1155 b^4 x^8}+\frac{2 c \left (b x^2+c x^4\right )^{3/2} (11 b B-8 A c)}{231 b^3 x^{10}}-\frac{\left (b x^2+c x^4\right )^{3/2} (11 b B-8 A c)}{99 b^2 x^{12}}-\frac{A \left (b x^2+c x^4\right )^{3/2}}{11 b x^{14}} \]
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Rubi [A] time = 0.300112, antiderivative size = 170, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 4, integrand size = 26, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.154, Rules used = {2034, 792, 658, 650} \[ \frac{16 c^3 \left (b x^2+c x^4\right )^{3/2} (11 b B-8 A c)}{3465 b^5 x^6}-\frac{8 c^2 \left (b x^2+c x^4\right )^{3/2} (11 b B-8 A c)}{1155 b^4 x^8}+\frac{2 c \left (b x^2+c x^4\right )^{3/2} (11 b B-8 A c)}{231 b^3 x^{10}}-\frac{\left (b x^2+c x^4\right )^{3/2} (11 b B-8 A c)}{99 b^2 x^{12}}-\frac{A \left (b x^2+c x^4\right )^{3/2}}{11 b x^{14}} \]
Antiderivative was successfully verified.
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Rule 2034
Rule 792
Rule 658
Rule 650
Rubi steps
\begin{align*} \int \frac{\left (A+B x^2\right ) \sqrt{b x^2+c x^4}}{x^{13}} \, dx &=\frac{1}{2} \operatorname{Subst}\left (\int \frac{(A+B x) \sqrt{b x+c x^2}}{x^7} \, dx,x,x^2\right )\\ &=-\frac{A \left (b x^2+c x^4\right )^{3/2}}{11 b x^{14}}+\frac{\left (-7 (-b B+A c)+\frac{3}{2} (-b B+2 A c)\right ) \operatorname{Subst}\left (\int \frac{\sqrt{b x+c x^2}}{x^6} \, dx,x,x^2\right )}{11 b}\\ &=-\frac{A \left (b x^2+c x^4\right )^{3/2}}{11 b x^{14}}-\frac{(11 b B-8 A c) \left (b x^2+c x^4\right )^{3/2}}{99 b^2 x^{12}}-\frac{(c (11 b B-8 A c)) \operatorname{Subst}\left (\int \frac{\sqrt{b x+c x^2}}{x^5} \, dx,x,x^2\right )}{33 b^2}\\ &=-\frac{A \left (b x^2+c x^4\right )^{3/2}}{11 b x^{14}}-\frac{(11 b B-8 A c) \left (b x^2+c x^4\right )^{3/2}}{99 b^2 x^{12}}+\frac{2 c (11 b B-8 A c) \left (b x^2+c x^4\right )^{3/2}}{231 b^3 x^{10}}+\frac{\left (4 c^2 (11 b B-8 A c)\right ) \operatorname{Subst}\left (\int \frac{\sqrt{b x+c x^2}}{x^4} \, dx,x,x^2\right )}{231 b^3}\\ &=-\frac{A \left (b x^2+c x^4\right )^{3/2}}{11 b x^{14}}-\frac{(11 b B-8 A c) \left (b x^2+c x^4\right )^{3/2}}{99 b^2 x^{12}}+\frac{2 c (11 b B-8 A c) \left (b x^2+c x^4\right )^{3/2}}{231 b^3 x^{10}}-\frac{8 c^2 (11 b B-8 A c) \left (b x^2+c x^4\right )^{3/2}}{1155 b^4 x^8}-\frac{\left (8 c^3 (11 b B-8 A c)\right ) \operatorname{Subst}\left (\int \frac{\sqrt{b x+c x^2}}{x^3} \, dx,x,x^2\right )}{1155 b^4}\\ &=-\frac{A \left (b x^2+c x^4\right )^{3/2}}{11 b x^{14}}-\frac{(11 b B-8 A c) \left (b x^2+c x^4\right )^{3/2}}{99 b^2 x^{12}}+\frac{2 c (11 b B-8 A c) \left (b x^2+c x^4\right )^{3/2}}{231 b^3 x^{10}}-\frac{8 c^2 (11 b B-8 A c) \left (b x^2+c x^4\right )^{3/2}}{1155 b^4 x^8}+\frac{16 c^3 (11 b B-8 A c) \left (b x^2+c x^4\right )^{3/2}}{3465 b^5 x^6}\\ \end{align*}
Mathematica [A] time = 0.0684124, size = 94, normalized size = 0.55 \[ \frac{\sqrt{x^2 \left (b+c x^2\right )} \left (x^2 \left (\frac{c x^2}{b}+1\right ) \left (-30 b^2 c x^2+35 b^3+24 b c^2 x^4-16 c^3 x^6\right ) (8 A c-11 b B)-315 A b^3 \left (b+c x^2\right )\right )}{3465 b^4 x^{12}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.006, size = 118, normalized size = 0.7 \begin{align*} -{\frac{ \left ( c{x}^{2}+b \right ) \left ( 128\,A{c}^{4}{x}^{8}-176\,Bb{c}^{3}{x}^{8}-192\,Ab{c}^{3}{x}^{6}+264\,B{b}^{2}{c}^{2}{x}^{6}+240\,A{b}^{2}{c}^{2}{x}^{4}-330\,B{b}^{3}c{x}^{4}-280\,A{b}^{3}c{x}^{2}+385\,B{b}^{4}{x}^{2}+315\,A{b}^{4} \right ) }{3465\,{x}^{12}{b}^{5}}\sqrt{c{x}^{4}+b{x}^{2}}} \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 = 1.78683, size = 298, normalized size = 1.75 \begin{align*} \frac{{\left (16 \,{\left (11 \, B b c^{4} - 8 \, A c^{5}\right )} x^{10} - 8 \,{\left (11 \, B b^{2} c^{3} - 8 \, A b c^{4}\right )} x^{8} + 6 \,{\left (11 \, B b^{3} c^{2} - 8 \, A b^{2} c^{3}\right )} x^{6} - 315 \, A b^{5} - 5 \,{\left (11 \, B b^{4} c - 8 \, A b^{3} c^{2}\right )} x^{4} - 35 \,{\left (11 \, B b^{5} + A b^{4} c\right )} x^{2}\right )} \sqrt{c x^{4} + b x^{2}}}{3465 \, b^{5} x^{12}} \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{\sqrt{x^{2} \left (b + c x^{2}\right )} \left (A + B x^{2}\right )}{x^{13}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [B] time = 4.42151, size = 581, normalized size = 3.42 \begin{align*} \frac{32 \,{\left (3465 \,{\left (\sqrt{c} x - \sqrt{c x^{2} + b}\right )}^{14} B c^{\frac{9}{2}} \mathrm{sgn}\left (x\right ) - 4851 \,{\left (\sqrt{c} x - \sqrt{c x^{2} + b}\right )}^{12} B b c^{\frac{9}{2}} \mathrm{sgn}\left (x\right ) + 11088 \,{\left (\sqrt{c} x - \sqrt{c x^{2} + b}\right )}^{12} A c^{\frac{11}{2}} \mathrm{sgn}\left (x\right ) + 231 \,{\left (\sqrt{c} x - \sqrt{c x^{2} + b}\right )}^{10} B b^{2} c^{\frac{9}{2}} \mathrm{sgn}\left (x\right ) + 7392 \,{\left (\sqrt{c} x - \sqrt{c x^{2} + b}\right )}^{10} A b c^{\frac{11}{2}} \mathrm{sgn}\left (x\right ) - 165 \,{\left (\sqrt{c} x - \sqrt{c x^{2} + b}\right )}^{8} B b^{3} c^{\frac{9}{2}} \mathrm{sgn}\left (x\right ) + 2640 \,{\left (\sqrt{c} x - \sqrt{c x^{2} + b}\right )}^{8} A b^{2} c^{\frac{11}{2}} \mathrm{sgn}\left (x\right ) + 1815 \,{\left (\sqrt{c} x - \sqrt{c x^{2} + b}\right )}^{6} B b^{4} c^{\frac{9}{2}} \mathrm{sgn}\left (x\right ) - 1320 \,{\left (\sqrt{c} x - \sqrt{c x^{2} + b}\right )}^{6} A b^{3} c^{\frac{11}{2}} \mathrm{sgn}\left (x\right ) - 605 \,{\left (\sqrt{c} x - \sqrt{c x^{2} + b}\right )}^{4} B b^{5} c^{\frac{9}{2}} \mathrm{sgn}\left (x\right ) + 440 \,{\left (\sqrt{c} x - \sqrt{c x^{2} + b}\right )}^{4} A b^{4} c^{\frac{11}{2}} \mathrm{sgn}\left (x\right ) + 121 \,{\left (\sqrt{c} x - \sqrt{c x^{2} + b}\right )}^{2} B b^{6} c^{\frac{9}{2}} \mathrm{sgn}\left (x\right ) - 88 \,{\left (\sqrt{c} x - \sqrt{c x^{2} + b}\right )}^{2} A b^{5} c^{\frac{11}{2}} \mathrm{sgn}\left (x\right ) - 11 \, B b^{7} c^{\frac{9}{2}} \mathrm{sgn}\left (x\right ) + 8 \, A b^{6} c^{\frac{11}{2}} \mathrm{sgn}\left (x\right )\right )}}{3465 \,{\left ({\left (\sqrt{c} x - \sqrt{c x^{2} + b}\right )}^{2} - b\right )}^{11}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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