Optimal. Leaf size=96 \[ \frac{2 c \sqrt{b x^2+c x^4} (5 b B-4 A c)}{15 b^3 x^2}-\frac{\sqrt{b x^2+c x^4} (5 b B-4 A c)}{15 b^2 x^4}-\frac{A \sqrt{b x^2+c x^4}}{5 b x^6} \]
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Rubi [A] time = 0.209251, 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} \[ \frac{2 c \sqrt{b x^2+c x^4} (5 b B-4 A c)}{15 b^3 x^2}-\frac{\sqrt{b x^2+c x^4} (5 b B-4 A c)}{15 b^2 x^4}-\frac{A \sqrt{b x^2+c x^4}}{5 b x^6} \]
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^5 \sqrt{b x^2+c x^4}} \, dx &=\frac{1}{2} \operatorname{Subst}\left (\int \frac{A+B x}{x^3 \sqrt{b x+c x^2}} \, dx,x,x^2\right )\\ &=-\frac{A \sqrt{b x^2+c x^4}}{5 b x^6}+\frac{\left (-3 (-b B+A c)+\frac{1}{2} (-b B+2 A c)\right ) \operatorname{Subst}\left (\int \frac{1}{x^2 \sqrt{b x+c x^2}} \, dx,x,x^2\right )}{5 b}\\ &=-\frac{A \sqrt{b x^2+c x^4}}{5 b x^6}-\frac{(5 b B-4 A c) \sqrt{b x^2+c x^4}}{15 b^2 x^4}-\frac{(c (5 b B-4 A c)) \operatorname{Subst}\left (\int \frac{1}{x \sqrt{b x+c x^2}} \, dx,x,x^2\right )}{15 b^2}\\ &=-\frac{A \sqrt{b x^2+c x^4}}{5 b x^6}-\frac{(5 b B-4 A c) \sqrt{b x^2+c x^4}}{15 b^2 x^4}+\frac{2 c (5 b B-4 A c) \sqrt{b x^2+c x^4}}{15 b^3 x^2}\\ \end{align*}
Mathematica [A] time = 0.0258534, size = 64, normalized size = 0.67 \[ \frac{\sqrt{x^2 \left (b+c x^2\right )} \left (A \left (-3 b^2+4 b c x^2-8 c^2 x^4\right )-5 b B x^2 \left (b-2 c x^2\right )\right )}{15 b^3 x^6} \]
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}-10\,B{x}^{4}bc-4\,Abc{x}^{2}+5\,B{x}^{2}{b}^{2}+3\,A{b}^{2} \right ) }{15\,{b}^{3}{x}^{4}}{\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 = 1.18404, size = 135, normalized size = 1.41 \begin{align*} \frac{{\left (2 \,{\left (5 \, B b c - 4 \, A c^{2}\right )} x^{4} - 3 \, A b^{2} -{\left (5 \, B b^{2} - 4 \, A b c\right )} x^{2}\right )} \sqrt{c x^{4} + b x^{2}}}{15 \, b^{3} x^{6}} \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^{5} \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 = 1.15571, size = 99, normalized size = 1.03 \begin{align*} -\frac{5 \, B b{\left (c + \frac{b}{x^{2}}\right )}^{\frac{3}{2}} + 3 \, A{\left (c + \frac{b}{x^{2}}\right )}^{\frac{5}{2}} - 15 \, B b \sqrt{c + \frac{b}{x^{2}}} c - 10 \, A{\left (c + \frac{b}{x^{2}}\right )}^{\frac{3}{2}} c + 15 \, A \sqrt{c + \frac{b}{x^{2}}} c^{2}}{15 \, b^{3}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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