Optimal. Leaf size=61 \[ -\frac{\left (b x^2+c x^4\right )^{5/2} (7 b B-2 A c)}{35 b^2 x^{10}}-\frac{A \left (b x^2+c x^4\right )^{5/2}}{7 b x^{12}} \]
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Rubi [A] time = 0.173004, antiderivative size = 61, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 26, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.115, Rules used = {2034, 792, 650} \[ -\frac{\left (b x^2+c x^4\right )^{5/2} (7 b B-2 A c)}{35 b^2 x^{10}}-\frac{A \left (b x^2+c x^4\right )^{5/2}}{7 b x^{12}} \]
Antiderivative was successfully verified.
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Rule 2034
Rule 792
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^{11}} \, dx &=\frac{1}{2} \operatorname{Subst}\left (\int \frac{(A+B x) \left (b x+c x^2\right )^{3/2}}{x^6} \, dx,x,x^2\right )\\ &=-\frac{A \left (b x^2+c x^4\right )^{5/2}}{7 b x^{12}}+\frac{\left (-6 (-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^5} \, dx,x,x^2\right )}{7 b}\\ &=-\frac{A \left (b x^2+c x^4\right )^{5/2}}{7 b x^{12}}-\frac{(7 b B-2 A c) \left (b x^2+c x^4\right )^{5/2}}{35 b^2 x^{10}}\\ \end{align*}
Mathematica [A] time = 0.0238985, size = 44, normalized size = 0.72 \[ -\frac{\left (x^2 \left (b+c x^2\right )\right )^{5/2} \left (5 A b-2 A c x^2+7 b B x^2\right )}{35 b^2 x^{12}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.004, size = 48, normalized size = 0.8 \begin{align*} -{\frac{ \left ( c{x}^{2}+b \right ) \left ( -2\,A{x}^{2}c+7\,B{x}^{2}b+5\,Ab \right ) }{35\,{x}^{10}{b}^{2}} \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 = 1.30742, size = 178, normalized size = 2.92 \begin{align*} -\frac{{\left ({\left (7 \, B b c^{2} - 2 \, A c^{3}\right )} x^{6} +{\left (14 \, B b^{2} c + A b c^{2}\right )} x^{4} + 5 \, A b^{3} +{\left (7 \, B b^{3} + 8 \, A b^{2} c\right )} x^{2}\right )} \sqrt{c x^{4} + b x^{2}}}{35 \, b^{2} x^{8}} \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^{11}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [B] time = 3.53941, size = 500, normalized size = 8.2 \begin{align*} \frac{2 \,{\left (35 \,{\left (\sqrt{c} x - \sqrt{c x^{2} + b}\right )}^{12} B c^{\frac{5}{2}} \mathrm{sgn}\left (x\right ) - 70 \,{\left (\sqrt{c} x - \sqrt{c x^{2} + b}\right )}^{10} B b c^{\frac{5}{2}} \mathrm{sgn}\left (x\right ) + 70 \,{\left (\sqrt{c} x - \sqrt{c x^{2} + b}\right )}^{10} A c^{\frac{7}{2}} \mathrm{sgn}\left (x\right ) + 105 \,{\left (\sqrt{c} x - \sqrt{c x^{2} + b}\right )}^{8} B b^{2} c^{\frac{5}{2}} \mathrm{sgn}\left (x\right ) + 70 \,{\left (\sqrt{c} x - \sqrt{c x^{2} + b}\right )}^{8} A b c^{\frac{7}{2}} \mathrm{sgn}\left (x\right ) - 140 \,{\left (\sqrt{c} x - \sqrt{c x^{2} + b}\right )}^{6} B b^{3} c^{\frac{5}{2}} \mathrm{sgn}\left (x\right ) + 140 \,{\left (\sqrt{c} x - \sqrt{c x^{2} + b}\right )}^{6} A b^{2} c^{\frac{7}{2}} \mathrm{sgn}\left (x\right ) + 77 \,{\left (\sqrt{c} x - \sqrt{c x^{2} + b}\right )}^{4} B b^{4} c^{\frac{5}{2}} \mathrm{sgn}\left (x\right ) + 28 \,{\left (\sqrt{c} x - \sqrt{c x^{2} + b}\right )}^{4} A b^{3} c^{\frac{7}{2}} \mathrm{sgn}\left (x\right ) - 14 \,{\left (\sqrt{c} x - \sqrt{c x^{2} + b}\right )}^{2} B b^{5} c^{\frac{5}{2}} \mathrm{sgn}\left (x\right ) + 14 \,{\left (\sqrt{c} x - \sqrt{c x^{2} + b}\right )}^{2} A b^{4} c^{\frac{7}{2}} \mathrm{sgn}\left (x\right ) + 7 \, B b^{6} c^{\frac{5}{2}} \mathrm{sgn}\left (x\right ) - 2 \, A b^{5} c^{\frac{7}{2}} \mathrm{sgn}\left (x\right )\right )}}{35 \,{\left ({\left (\sqrt{c} x - \sqrt{c x^{2} + b}\right )}^{2} - b\right )}^{7}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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