Optimal. Leaf size=133 \[ \frac{\left (b x^2+c x^4\right )^{3/2} (3 A c+2 b B)}{6 b x^3}+\frac{\sqrt{b x^2+c x^4} (3 A c+2 b B)}{2 x}-\frac{1}{2} \sqrt{b} (3 A c+2 b B) \tanh ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{b x^2+c x^4}}\right )-\frac{A \left (b x^2+c x^4\right )^{5/2}}{2 b x^7} \]
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Rubi [A] time = 0.219648, antiderivative size = 133, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 4, integrand size = 26, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.154, Rules used = {2038, 2021, 2008, 206} \[ \frac{\left (b x^2+c x^4\right )^{3/2} (3 A c+2 b B)}{6 b x^3}+\frac{\sqrt{b x^2+c x^4} (3 A c+2 b B)}{2 x}-\frac{1}{2} \sqrt{b} (3 A c+2 b B) \tanh ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{b x^2+c x^4}}\right )-\frac{A \left (b x^2+c x^4\right )^{5/2}}{2 b x^7} \]
Antiderivative was successfully verified.
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Rule 2038
Rule 2021
Rule 2008
Rule 206
Rubi steps
\begin{align*} \int \frac{\left (A+B x^2\right ) \left (b x^2+c x^4\right )^{3/2}}{x^6} \, dx &=-\frac{A \left (b x^2+c x^4\right )^{5/2}}{2 b x^7}-\frac{(-2 b B-3 A c) \int \frac{\left (b x^2+c x^4\right )^{3/2}}{x^4} \, dx}{2 b}\\ &=\frac{(2 b B+3 A c) \left (b x^2+c x^4\right )^{3/2}}{6 b x^3}-\frac{A \left (b x^2+c x^4\right )^{5/2}}{2 b x^7}-\frac{1}{2} (-2 b B-3 A c) \int \frac{\sqrt{b x^2+c x^4}}{x^2} \, dx\\ &=\frac{(2 b B+3 A c) \sqrt{b x^2+c x^4}}{2 x}+\frac{(2 b B+3 A c) \left (b x^2+c x^4\right )^{3/2}}{6 b x^3}-\frac{A \left (b x^2+c x^4\right )^{5/2}}{2 b x^7}+\frac{1}{2} (b (2 b B+3 A c)) \int \frac{1}{\sqrt{b x^2+c x^4}} \, dx\\ &=\frac{(2 b B+3 A c) \sqrt{b x^2+c x^4}}{2 x}+\frac{(2 b B+3 A c) \left (b x^2+c x^4\right )^{3/2}}{6 b x^3}-\frac{A \left (b x^2+c x^4\right )^{5/2}}{2 b x^7}-\frac{1}{2} (b (2 b B+3 A c)) \operatorname{Subst}\left (\int \frac{1}{1-b x^2} \, dx,x,\frac{x}{\sqrt{b x^2+c x^4}}\right )\\ &=\frac{(2 b B+3 A c) \sqrt{b x^2+c x^4}}{2 x}+\frac{(2 b B+3 A c) \left (b x^2+c x^4\right )^{3/2}}{6 b x^3}-\frac{A \left (b x^2+c x^4\right )^{5/2}}{2 b x^7}-\frac{1}{2} \sqrt{b} (2 b B+3 A c) \tanh ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{b x^2+c x^4}}\right )\\ \end{align*}
Mathematica [A] time = 0.0628259, size = 109, normalized size = 0.82 \[ \frac{\sqrt{x^2 \left (b+c x^2\right )} \left (\sqrt{b+c x^2} \left (-3 A b+6 A c x^2+8 b B x^2+2 B c x^4\right )-3 \sqrt{b} x^2 (3 A c+2 b B) \tanh ^{-1}\left (\frac{\sqrt{b+c x^2}}{\sqrt{b}}\right )\right )}{6 x^3 \sqrt{b+c x^2}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.009, size = 172, normalized size = 1.3 \begin{align*} -{\frac{1}{6\,b{x}^{5}} \left ( c{x}^{4}+b{x}^{2} \right ) ^{{\frac{3}{2}}} \left ( 9\,A{b}^{3/2}\ln \left ( 2\,{\frac{\sqrt{b}\sqrt{c{x}^{2}+b}+b}{x}} \right ){x}^{2}c-3\,A \left ( c{x}^{2}+b \right ) ^{3/2}{x}^{2}c+6\,B{b}^{5/2}\ln \left ( 2\,{\frac{\sqrt{b}\sqrt{c{x}^{2}+b}+b}{x}} \right ){x}^{2}-2\,B \left ( c{x}^{2}+b \right ) ^{3/2}{x}^{2}b+3\,A \left ( c{x}^{2}+b \right ) ^{5/2}-9\,A\sqrt{c{x}^{2}+b}{x}^{2}bc-6\,B\sqrt{c{x}^{2}+b}{x}^{2}{b}^{2} \right ) \left ( c{x}^{2}+b \right ) ^{-{\frac{3}{2}}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{{\left (c x^{4} + b x^{2}\right )}^{\frac{3}{2}}{\left (B x^{2} + A\right )}}{x^{6}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.11061, size = 447, normalized size = 3.36 \begin{align*} \left [\frac{3 \,{\left (2 \, B b + 3 \, A c\right )} \sqrt{b} x^{3} \log \left (-\frac{c x^{3} + 2 \, b x - 2 \, \sqrt{c x^{4} + b x^{2}} \sqrt{b}}{x^{3}}\right ) + 2 \,{\left (2 \, B c x^{4} + 2 \,{\left (4 \, B b + 3 \, A c\right )} x^{2} - 3 \, A b\right )} \sqrt{c x^{4} + b x^{2}}}{12 \, x^{3}}, \frac{3 \,{\left (2 \, B b + 3 \, A c\right )} \sqrt{-b} x^{3} \arctan \left (\frac{\sqrt{c x^{4} + b x^{2}} \sqrt{-b}}{c x^{3} + b x}\right ) +{\left (2 \, B c x^{4} + 2 \,{\left (4 \, B b + 3 \, A c\right )} x^{2} - 3 \, A b\right )} \sqrt{c x^{4} + b x^{2}}}{6 \, x^{3}}\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{\left (x^{2} \left (b + c x^{2}\right )\right )^{\frac{3}{2}} \left (A + B x^{2}\right )}{x^{6}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.35209, size = 155, normalized size = 1.17 \begin{align*} \frac{2 \,{\left (c x^{2} + b\right )}^{\frac{3}{2}} B c \mathrm{sgn}\left (x\right ) + 6 \, \sqrt{c x^{2} + b} B b c \mathrm{sgn}\left (x\right ) + 6 \, \sqrt{c x^{2} + b} A c^{2} \mathrm{sgn}\left (x\right ) - \frac{3 \, \sqrt{c x^{2} + b} A b c \mathrm{sgn}\left (x\right )}{x^{2}} + \frac{3 \,{\left (2 \, B b^{2} c \mathrm{sgn}\left (x\right ) + 3 \, A b c^{2} \mathrm{sgn}\left (x\right )\right )} \arctan \left (\frac{\sqrt{c x^{2} + b}}{\sqrt{-b}}\right )}{\sqrt{-b}}}{6 \, c} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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