Optimal. Leaf size=96 \[ \frac{2 c \left (b x^2+c x^4\right )^{5/2} (9 b B-4 A c)}{315 b^3 x^{10}}-\frac{\left (b x^2+c x^4\right )^{5/2} (9 b B-4 A c)}{63 b^2 x^{12}}-\frac{A \left (b x^2+c x^4\right )^{5/2}}{9 b x^{14}} \]
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Rubi [A] time = 0.233431, antiderivative size = 96, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 26, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.154, Rules used = {2034, 792, 658, 650} \[ \frac{2 c \left (b x^2+c x^4\right )^{5/2} (9 b B-4 A c)}{315 b^3 x^{10}}-\frac{\left (b x^2+c x^4\right )^{5/2} (9 b B-4 A c)}{63 b^2 x^{12}}-\frac{A \left (b x^2+c x^4\right )^{5/2}}{9 b x^{14}} \]
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 ) \left (b x^2+c x^4\right )^{3/2}}{x^{13}} \, dx &=\frac{1}{2} \operatorname{Subst}\left (\int \frac{(A+B x) \left (b x+c x^2\right )^{3/2}}{x^7} \, dx,x,x^2\right )\\ &=-\frac{A \left (b x^2+c x^4\right )^{5/2}}{9 b x^{14}}+\frac{\left (-7 (-b B+A c)+\frac{5}{2} (-b B+2 A c)\right ) \operatorname{Subst}\left (\int \frac{\left (b x+c x^2\right )^{3/2}}{x^6} \, dx,x,x^2\right )}{9 b}\\ &=-\frac{A \left (b x^2+c x^4\right )^{5/2}}{9 b x^{14}}-\frac{(9 b B-4 A c) \left (b x^2+c x^4\right )^{5/2}}{63 b^2 x^{12}}-\frac{(c (9 b B-4 A c)) \operatorname{Subst}\left (\int \frac{\left (b x+c x^2\right )^{3/2}}{x^5} \, dx,x,x^2\right )}{63 b^2}\\ &=-\frac{A \left (b x^2+c x^4\right )^{5/2}}{9 b x^{14}}-\frac{(9 b B-4 A c) \left (b x^2+c x^4\right )^{5/2}}{63 b^2 x^{12}}+\frac{2 c (9 b B-4 A c) \left (b x^2+c x^4\right )^{5/2}}{315 b^3 x^{10}}\\ \end{align*}
Mathematica [A] time = 0.0306484, size = 66, normalized size = 0.69 \[ \frac{\left (x^2 \left (b+c x^2\right )\right )^{5/2} \left (A \left (-35 b^2+20 b c x^2-8 c^2 x^4\right )+9 b B x^2 \left (2 c x^2-5 b\right )\right )}{315 b^3 x^{14}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.005, size = 70, normalized size = 0.7 \begin{align*} -{\frac{ \left ( c{x}^{2}+b \right ) \left ( 8\,A{c}^{2}{x}^{4}-18\,B{x}^{4}bc-20\,Abc{x}^{2}+45\,B{x}^{2}{b}^{2}+35\,A{b}^{2} \right ) }{315\,{x}^{12}{b}^{3}} \left ( c{x}^{4}+b{x}^{2} \right ) ^{{\frac{3}{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 = 1.33211, size = 236, normalized size = 2.46 \begin{align*} \frac{{\left (2 \,{\left (9 \, B b c^{3} - 4 \, A c^{4}\right )} x^{8} -{\left (9 \, B b^{2} c^{2} - 4 \, A b c^{3}\right )} x^{6} - 35 \, A b^{4} - 3 \,{\left (24 \, B b^{3} c + A b^{2} c^{2}\right )} x^{4} - 5 \,{\left (9 \, B b^{4} + 10 \, A b^{3} c\right )} x^{2}\right )} \sqrt{c x^{4} + b x^{2}}}{315 \, b^{3} x^{10}} \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^{13}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [B] time = 4.78231, size = 581, normalized size = 6.05 \begin{align*} \frac{4 \,{\left (315 \,{\left (\sqrt{c} x - \sqrt{c x^{2} + b}\right )}^{14} B c^{\frac{7}{2}} \mathrm{sgn}\left (x\right ) - 315 \,{\left (\sqrt{c} x - \sqrt{c x^{2} + b}\right )}^{12} B b c^{\frac{7}{2}} \mathrm{sgn}\left (x\right ) + 840 \,{\left (\sqrt{c} x - \sqrt{c x^{2} + b}\right )}^{12} A c^{\frac{9}{2}} \mathrm{sgn}\left (x\right ) + 315 \,{\left (\sqrt{c} x - \sqrt{c x^{2} + b}\right )}^{10} B b^{2} c^{\frac{7}{2}} \mathrm{sgn}\left (x\right ) + 1260 \,{\left (\sqrt{c} x - \sqrt{c x^{2} + b}\right )}^{10} A b c^{\frac{9}{2}} \mathrm{sgn}\left (x\right ) - 819 \,{\left (\sqrt{c} x - \sqrt{c x^{2} + b}\right )}^{8} B b^{3} c^{\frac{7}{2}} \mathrm{sgn}\left (x\right ) + 1764 \,{\left (\sqrt{c} x - \sqrt{c x^{2} + b}\right )}^{8} A b^{2} c^{\frac{9}{2}} \mathrm{sgn}\left (x\right ) + 441 \,{\left (\sqrt{c} x - \sqrt{c x^{2} + b}\right )}^{6} B b^{4} c^{\frac{7}{2}} \mathrm{sgn}\left (x\right ) + 504 \,{\left (\sqrt{c} x - \sqrt{c x^{2} + b}\right )}^{6} A b^{3} c^{\frac{9}{2}} \mathrm{sgn}\left (x\right ) - 9 \,{\left (\sqrt{c} x - \sqrt{c x^{2} + b}\right )}^{4} B b^{5} c^{\frac{7}{2}} \mathrm{sgn}\left (x\right ) + 144 \,{\left (\sqrt{c} x - \sqrt{c x^{2} + b}\right )}^{4} A b^{4} c^{\frac{9}{2}} \mathrm{sgn}\left (x\right ) + 81 \,{\left (\sqrt{c} x - \sqrt{c x^{2} + b}\right )}^{2} B b^{6} c^{\frac{7}{2}} \mathrm{sgn}\left (x\right ) - 36 \,{\left (\sqrt{c} x - \sqrt{c x^{2} + b}\right )}^{2} A b^{5} c^{\frac{9}{2}} \mathrm{sgn}\left (x\right ) - 9 \, B b^{7} c^{\frac{7}{2}} \mathrm{sgn}\left (x\right ) + 4 \, A b^{6} c^{\frac{9}{2}} \mathrm{sgn}\left (x\right )\right )}}{315 \,{\left ({\left (\sqrt{c} x - \sqrt{c x^{2} + b}\right )}^{2} - b\right )}^{9}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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