Optimal. Leaf size=101 \[ \frac{4 c \left (b+2 c x^2\right ) (5 b B-6 A c)}{15 b^4 \sqrt{b x^2+c x^4}}-\frac{5 b B-6 A c}{15 b^2 x^2 \sqrt{b x^2+c x^4}}-\frac{A}{5 b x^4 \sqrt{b x^2+c x^4}} \]
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Rubi [A] time = 0.220201, antiderivative size = 101, 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, 613} \[ \frac{4 c \left (b+2 c x^2\right ) (5 b B-6 A c)}{15 b^4 \sqrt{b x^2+c x^4}}-\frac{5 b B-6 A c}{15 b^2 x^2 \sqrt{b x^2+c x^4}}-\frac{A}{5 b x^4 \sqrt{b x^2+c x^4}} \]
Antiderivative was successfully verified.
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Rule 2034
Rule 792
Rule 658
Rule 613
Rubi steps
\begin{align*} \int \frac{A+B x^2}{x^3 \left (b x^2+c x^4\right )^{3/2}} \, dx &=\frac{1}{2} \operatorname{Subst}\left (\int \frac{A+B x}{x^2 \left (b x+c x^2\right )^{3/2}} \, dx,x,x^2\right )\\ &=-\frac{A}{5 b x^4 \sqrt{b x^2+c x^4}}+\frac{\left (\frac{1}{2} (b B-2 A c)-2 (-b B+A c)\right ) \operatorname{Subst}\left (\int \frac{1}{x \left (b x+c x^2\right )^{3/2}} \, dx,x,x^2\right )}{5 b}\\ &=-\frac{A}{5 b x^4 \sqrt{b x^2+c x^4}}-\frac{5 b B-6 A c}{15 b^2 x^2 \sqrt{b x^2+c x^4}}-\frac{(2 c (5 b B-6 A c)) \operatorname{Subst}\left (\int \frac{1}{\left (b x+c x^2\right )^{3/2}} \, dx,x,x^2\right )}{15 b^2}\\ &=-\frac{A}{5 b x^4 \sqrt{b x^2+c x^4}}-\frac{5 b B-6 A c}{15 b^2 x^2 \sqrt{b x^2+c x^4}}+\frac{4 c (5 b B-6 A c) \left (b+2 c x^2\right )}{15 b^4 \sqrt{b x^2+c x^4}}\\ \end{align*}
Mathematica [A] time = 0.0268884, size = 85, normalized size = 0.84 \[ \frac{-3 A \left (-2 b^2 c x^2+b^3+8 b c^2 x^4+16 c^3 x^6\right )-5 b B x^2 \left (b^2-4 b c x^2-8 c^2 x^4\right )}{15 b^4 x^4 \sqrt{x^2 \left (b+c x^2\right )}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.006, size = 94, normalized size = 0.9 \begin{align*} -{\frac{ \left ( c{x}^{2}+b \right ) \left ( 48\,A{c}^{3}{x}^{6}-40\,B{x}^{6}b{c}^{2}+24\,Ab{c}^{2}{x}^{4}-20\,B{x}^{4}{b}^{2}c-6\,A{b}^{2}c{x}^{2}+5\,B{x}^{2}{b}^{3}+3\,A{b}^{3} \right ) }{15\,{x}^{2}{b}^{4}} \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 = 1.3184, size = 200, normalized size = 1.98 \begin{align*} \frac{{\left (8 \,{\left (5 \, B b c^{2} - 6 \, A c^{3}\right )} x^{6} + 4 \,{\left (5 \, B b^{2} c - 6 \, A b c^{2}\right )} x^{4} - 3 \, A b^{3} -{\left (5 \, B b^{3} - 6 \, A b^{2} c\right )} x^{2}\right )} \sqrt{c x^{4} + b x^{2}}}{15 \,{\left (b^{4} c x^{8} + b^{5} x^{6}\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{A + B x^{2}}{x^{3} \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{B x^{2} + A}{{\left (c x^{4} + b x^{2}\right )}^{\frac{3}{2}} x^{3}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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