Optimal. Leaf size=184 \[ -\frac{5 b^2 (7 b B-6 A c) \tanh ^{-1}\left (\frac{\sqrt{c} x^2}{\sqrt{b x^2+c x^4}}\right )}{16 c^{9/2}}+\frac{x^4 \sqrt{b x^2+c x^4} (7 b B-6 A c)}{6 b c^2}-\frac{5 x^2 \sqrt{b x^2+c x^4} (7 b B-6 A c)}{24 c^3}+\frac{5 b \sqrt{b x^2+c x^4} (7 b B-6 A c)}{16 c^4}-\frac{x^8 (b B-A c)}{b c \sqrt{b x^2+c x^4}} \]
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Rubi [A] time = 0.33328, antiderivative size = 184, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 6, integrand size = 26, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.231, Rules used = {2034, 788, 670, 640, 620, 206} \[ -\frac{5 b^2 (7 b B-6 A c) \tanh ^{-1}\left (\frac{\sqrt{c} x^2}{\sqrt{b x^2+c x^4}}\right )}{16 c^{9/2}}+\frac{x^4 \sqrt{b x^2+c x^4} (7 b B-6 A c)}{6 b c^2}-\frac{5 x^2 \sqrt{b x^2+c x^4} (7 b B-6 A c)}{24 c^3}+\frac{5 b \sqrt{b x^2+c x^4} (7 b B-6 A c)}{16 c^4}-\frac{x^8 (b B-A c)}{b c \sqrt{b x^2+c x^4}} \]
Antiderivative was successfully verified.
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Rule 2034
Rule 788
Rule 670
Rule 640
Rule 620
Rule 206
Rubi steps
\begin{align*} \int \frac{x^9 \left (A+B x^2\right )}{\left (b x^2+c x^4\right )^{3/2}} \, dx &=\frac{1}{2} \operatorname{Subst}\left (\int \frac{x^4 (A+B x)}{\left (b x+c x^2\right )^{3/2}} \, dx,x,x^2\right )\\ &=-\frac{(b B-A c) x^8}{b c \sqrt{b x^2+c x^4}}+\frac{1}{2} \left (-\frac{6 A}{b}+\frac{7 B}{c}\right ) \operatorname{Subst}\left (\int \frac{x^3}{\sqrt{b x+c x^2}} \, dx,x,x^2\right )\\ &=-\frac{(b B-A c) x^8}{b c \sqrt{b x^2+c x^4}}+\frac{(7 b B-6 A c) x^4 \sqrt{b x^2+c x^4}}{6 b c^2}-\frac{(5 (7 b B-6 A c)) \operatorname{Subst}\left (\int \frac{x^2}{\sqrt{b x+c x^2}} \, dx,x,x^2\right )}{12 c^2}\\ &=-\frac{(b B-A c) x^8}{b c \sqrt{b x^2+c x^4}}-\frac{5 (7 b B-6 A c) x^2 \sqrt{b x^2+c x^4}}{24 c^3}+\frac{(7 b B-6 A c) x^4 \sqrt{b x^2+c x^4}}{6 b c^2}+\frac{(5 b (7 b B-6 A c)) \operatorname{Subst}\left (\int \frac{x}{\sqrt{b x+c x^2}} \, dx,x,x^2\right )}{16 c^3}\\ &=-\frac{(b B-A c) x^8}{b c \sqrt{b x^2+c x^4}}+\frac{5 b (7 b B-6 A c) \sqrt{b x^2+c x^4}}{16 c^4}-\frac{5 (7 b B-6 A c) x^2 \sqrt{b x^2+c x^4}}{24 c^3}+\frac{(7 b B-6 A c) x^4 \sqrt{b x^2+c x^4}}{6 b c^2}-\frac{\left (5 b^2 (7 b B-6 A c)\right ) \operatorname{Subst}\left (\int \frac{1}{\sqrt{b x+c x^2}} \, dx,x,x^2\right )}{32 c^4}\\ &=-\frac{(b B-A c) x^8}{b c \sqrt{b x^2+c x^4}}+\frac{5 b (7 b B-6 A c) \sqrt{b x^2+c x^4}}{16 c^4}-\frac{5 (7 b B-6 A c) x^2 \sqrt{b x^2+c x^4}}{24 c^3}+\frac{(7 b B-6 A c) x^4 \sqrt{b x^2+c x^4}}{6 b c^2}-\frac{\left (5 b^2 (7 b B-6 A c)\right ) \operatorname{Subst}\left (\int \frac{1}{1-c x^2} \, dx,x,\frac{x^2}{\sqrt{b x^2+c x^4}}\right )}{16 c^4}\\ &=-\frac{(b B-A c) x^8}{b c \sqrt{b x^2+c x^4}}+\frac{5 b (7 b B-6 A c) \sqrt{b x^2+c x^4}}{16 c^4}-\frac{5 (7 b B-6 A c) x^2 \sqrt{b x^2+c x^4}}{24 c^3}+\frac{(7 b B-6 A c) x^4 \sqrt{b x^2+c x^4}}{6 b c^2}-\frac{5 b^2 (7 b B-6 A c) \tanh ^{-1}\left (\frac{\sqrt{c} x^2}{\sqrt{b x^2+c x^4}}\right )}{16 c^{9/2}}\\ \end{align*}
Mathematica [A] time = 0.179336, size = 136, normalized size = 0.74 \[ \frac{x \left (\sqrt{c} x \left (b^2 \left (35 B c x^2-90 A c\right )-2 b c^2 x^2 \left (15 A+7 B x^2\right )+4 c^3 x^4 \left (3 A+2 B x^2\right )+105 b^3 B\right )-15 b^{5/2} \sqrt{\frac{c x^2}{b}+1} (7 b B-6 A c) \sinh ^{-1}\left (\frac{\sqrt{c} x}{\sqrt{b}}\right )\right )}{48 c^{9/2} \sqrt{x^2 \left (b+c x^2\right )}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.015, size = 166, normalized size = 0.9 \begin{align*}{\frac{ \left ( c{x}^{2}+b \right ){x}^{3}}{48} \left ( 8\,B{c}^{9/2}{x}^{7}+12\,A{c}^{9/2}{x}^{5}-14\,B{c}^{7/2}{x}^{5}b-30\,A{c}^{7/2}{x}^{3}b+35\,B{c}^{5/2}{x}^{3}{b}^{2}-90\,A{c}^{5/2}x{b}^{2}+105\,B{c}^{3/2}x{b}^{3}+90\,A\ln \left ( x\sqrt{c}+\sqrt{c{x}^{2}+b} \right ) \sqrt{c{x}^{2}+b}{b}^{2}{c}^{2}-105\,B\ln \left ( x\sqrt{c}+\sqrt{c{x}^{2}+b} \right ) \sqrt{c{x}^{2}+b}{b}^{3}c \right ) \left ( c{x}^{4}+b{x}^{2} \right ) ^{-{\frac{3}{2}}}{c}^{-{\frac{11}{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 = 1.53562, size = 738, normalized size = 4.01 \begin{align*} \left [-\frac{15 \,{\left (7 \, B b^{4} - 6 \, A b^{3} c +{\left (7 \, B b^{3} c - 6 \, A b^{2} c^{2}\right )} x^{2}\right )} \sqrt{c} \log \left (-2 \, c x^{2} - b - 2 \, \sqrt{c x^{4} + b x^{2}} \sqrt{c}\right ) - 2 \,{\left (8 \, B c^{4} x^{6} + 105 \, B b^{3} c - 90 \, A b^{2} c^{2} - 2 \,{\left (7 \, B b c^{3} - 6 \, A c^{4}\right )} x^{4} + 5 \,{\left (7 \, B b^{2} c^{2} - 6 \, A b c^{3}\right )} x^{2}\right )} \sqrt{c x^{4} + b x^{2}}}{96 \,{\left (c^{6} x^{2} + b c^{5}\right )}}, \frac{15 \,{\left (7 \, B b^{4} - 6 \, A b^{3} c +{\left (7 \, B b^{3} c - 6 \, A b^{2} c^{2}\right )} x^{2}\right )} \sqrt{-c} \arctan \left (\frac{\sqrt{c x^{4} + b x^{2}} \sqrt{-c}}{c x^{2} + b}\right ) +{\left (8 \, B c^{4} x^{6} + 105 \, B b^{3} c - 90 \, A b^{2} c^{2} - 2 \,{\left (7 \, B b c^{3} - 6 \, A c^{4}\right )} x^{4} + 5 \,{\left (7 \, B b^{2} c^{2} - 6 \, A b c^{3}\right )} x^{2}\right )} \sqrt{c x^{4} + b x^{2}}}{48 \,{\left (c^{6} x^{2} + b c^{5}\right )}}\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{x^{9} \left (A + B x^{2}\right )}{\left (x^{2} \left (b + c x^{2}\right )\right )^{\frac{3}{2}}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{{\left (B x^{2} + A\right )} x^{9}}{{\left (c x^{4} + b x^{2}\right )}^{\frac{3}{2}}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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