Optimal. Leaf size=170 \[ \frac{16 c^3 \sqrt{b x^2+c x^4} (9 b B-8 A c)}{315 b^5 x^2}-\frac{8 c^2 \sqrt{b x^2+c x^4} (9 b B-8 A c)}{315 b^4 x^4}+\frac{2 c \sqrt{b x^2+c x^4} (9 b B-8 A c)}{105 b^3 x^6}-\frac{\sqrt{b x^2+c x^4} (9 b B-8 A c)}{63 b^2 x^8}-\frac{A \sqrt{b x^2+c x^4}}{9 b x^{10}} \]
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Rubi [A] time = 0.30266, 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} \[ \frac{16 c^3 \sqrt{b x^2+c x^4} (9 b B-8 A c)}{315 b^5 x^2}-\frac{8 c^2 \sqrt{b x^2+c x^4} (9 b B-8 A c)}{315 b^4 x^4}+\frac{2 c \sqrt{b x^2+c x^4} (9 b B-8 A c)}{105 b^3 x^6}-\frac{\sqrt{b x^2+c x^4} (9 b B-8 A c)}{63 b^2 x^8}-\frac{A \sqrt{b x^2+c x^4}}{9 b x^{10}} \]
Antiderivative was successfully verified.
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Rule 2034
Rule 792
Rule 658
Rule 650
Rubi steps
\begin{align*} \int \frac{A+B x^2}{x^9 \sqrt{b x^2+c x^4}} \, dx &=\frac{1}{2} \operatorname{Subst}\left (\int \frac{A+B x}{x^5 \sqrt{b x+c x^2}} \, dx,x,x^2\right )\\ &=-\frac{A \sqrt{b x^2+c x^4}}{9 b x^{10}}+\frac{\left (-5 (-b B+A c)+\frac{1}{2} (-b B+2 A c)\right ) \operatorname{Subst}\left (\int \frac{1}{x^4 \sqrt{b x+c x^2}} \, dx,x,x^2\right )}{9 b}\\ &=-\frac{A \sqrt{b x^2+c x^4}}{9 b x^{10}}-\frac{(9 b B-8 A c) \sqrt{b x^2+c x^4}}{63 b^2 x^8}-\frac{(c (9 b B-8 A c)) \operatorname{Subst}\left (\int \frac{1}{x^3 \sqrt{b x+c x^2}} \, dx,x,x^2\right )}{21 b^2}\\ &=-\frac{A \sqrt{b x^2+c x^4}}{9 b x^{10}}-\frac{(9 b B-8 A c) \sqrt{b x^2+c x^4}}{63 b^2 x^8}+\frac{2 c (9 b B-8 A c) \sqrt{b x^2+c x^4}}{105 b^3 x^6}+\frac{\left (4 c^2 (9 b B-8 A c)\right ) \operatorname{Subst}\left (\int \frac{1}{x^2 \sqrt{b x+c x^2}} \, dx,x,x^2\right )}{105 b^3}\\ &=-\frac{A \sqrt{b x^2+c x^4}}{9 b x^{10}}-\frac{(9 b B-8 A c) \sqrt{b x^2+c x^4}}{63 b^2 x^8}+\frac{2 c (9 b B-8 A c) \sqrt{b x^2+c x^4}}{105 b^3 x^6}-\frac{8 c^2 (9 b B-8 A c) \sqrt{b x^2+c x^4}}{315 b^4 x^4}-\frac{\left (8 c^3 (9 b B-8 A c)\right ) \operatorname{Subst}\left (\int \frac{1}{x \sqrt{b x+c x^2}} \, dx,x,x^2\right )}{315 b^4}\\ &=-\frac{A \sqrt{b x^2+c x^4}}{9 b x^{10}}-\frac{(9 b B-8 A c) \sqrt{b x^2+c x^4}}{63 b^2 x^8}+\frac{2 c (9 b B-8 A c) \sqrt{b x^2+c x^4}}{105 b^3 x^6}-\frac{8 c^2 (9 b B-8 A c) \sqrt{b x^2+c x^4}}{315 b^4 x^4}+\frac{16 c^3 (9 b B-8 A c) \sqrt{b x^2+c x^4}}{315 b^5 x^2}\\ \end{align*}
Mathematica [A] time = 0.0576179, size = 94, normalized size = 0.55 \[ \frac{x^2 \left (\frac{c x^2}{b}+1\right ) \left (-6 b^2 c x^2+5 b^3+8 b c^2 x^4-16 c^3 x^6\right ) (8 A c-9 b B)-35 A b^3 \left (b+c x^2\right )}{315 b^4 x^8 \sqrt{x^2 \left (b+c x^2\right )}} \]
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}-144\,Bb{c}^{3}{x}^{8}-64\,Ab{c}^{3}{x}^{6}+72\,B{b}^{2}{c}^{2}{x}^{6}+48\,A{b}^{2}{c}^{2}{x}^{4}-54\,B{b}^{3}c{x}^{4}-40\,A{b}^{3}c{x}^{2}+45\,B{b}^{4}{x}^{2}+35\,A{b}^{4} \right ) }{315\,{x}^{8}{b}^{5}}{\frac{1}{\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 = 1.26573, size = 240, normalized size = 1.41 \begin{align*} \frac{{\left (16 \,{\left (9 \, B b c^{3} - 8 \, A c^{4}\right )} x^{8} - 8 \,{\left (9 \, B b^{2} c^{2} - 8 \, A b c^{3}\right )} x^{6} - 35 \, A b^{4} + 6 \,{\left (9 \, B b^{3} c - 8 \, A b^{2} c^{2}\right )} x^{4} - 5 \,{\left (9 \, B b^{4} - 8 \, A b^{3} c\right )} x^{2}\right )} \sqrt{c x^{4} + b x^{2}}}{315 \, b^{5} 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{A + B x^{2}}{x^{9} \sqrt{x^{2} \left (b + c x^{2}\right )}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.18785, size = 182, normalized size = 1.07 \begin{align*} -\frac{45 \, B b{\left (c + \frac{b}{x^{2}}\right )}^{\frac{7}{2}} + 35 \, A{\left (c + \frac{b}{x^{2}}\right )}^{\frac{9}{2}} - 189 \, B b{\left (c + \frac{b}{x^{2}}\right )}^{\frac{5}{2}} c - 180 \, A{\left (c + \frac{b}{x^{2}}\right )}^{\frac{7}{2}} c + 315 \, B b{\left (c + \frac{b}{x^{2}}\right )}^{\frac{3}{2}} c^{2} + 378 \, A{\left (c + \frac{b}{x^{2}}\right )}^{\frac{5}{2}} c^{2} - 315 \, B b \sqrt{c + \frac{b}{x^{2}}} c^{3} - 420 \, A{\left (c + \frac{b}{x^{2}}\right )}^{\frac{3}{2}} c^{3} + 315 \, A \sqrt{c + \frac{b}{x^{2}}} c^{4}}{315 \, b^{5}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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