Optimal. Leaf size=81 \[ -\frac{3 \sqrt{b x^2+c x^4}}{2 b^2 x^3}+\frac{3 c \tanh ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{b x^2+c x^4}}\right )}{2 b^{5/2}}+\frac{1}{b x \sqrt{b x^2+c x^4}} \]
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Rubi [A] time = 0.0646457, antiderivative size = 81, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.267, Rules used = {2006, 2025, 2008, 206} \[ -\frac{3 \sqrt{b x^2+c x^4}}{2 b^2 x^3}+\frac{3 c \tanh ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{b x^2+c x^4}}\right )}{2 b^{5/2}}+\frac{1}{b x \sqrt{b x^2+c x^4}} \]
Antiderivative was successfully verified.
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Rule 2006
Rule 2025
Rule 2008
Rule 206
Rubi steps
\begin{align*} \int \frac{1}{\left (b x^2+c x^4\right )^{3/2}} \, dx &=\frac{1}{b x \sqrt{b x^2+c x^4}}+\frac{3 \int \frac{1}{x^2 \sqrt{b x^2+c x^4}} \, dx}{b}\\ &=\frac{1}{b x \sqrt{b x^2+c x^4}}-\frac{3 \sqrt{b x^2+c x^4}}{2 b^2 x^3}-\frac{(3 c) \int \frac{1}{\sqrt{b x^2+c x^4}} \, dx}{2 b^2}\\ &=\frac{1}{b x \sqrt{b x^2+c x^4}}-\frac{3 \sqrt{b x^2+c x^4}}{2 b^2 x^3}+\frac{(3 c) \operatorname{Subst}\left (\int \frac{1}{1-b x^2} \, dx,x,\frac{x}{\sqrt{b x^2+c x^4}}\right )}{2 b^2}\\ &=\frac{1}{b x \sqrt{b x^2+c x^4}}-\frac{3 \sqrt{b x^2+c x^4}}{2 b^2 x^3}+\frac{3 c \tanh ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{b x^2+c x^4}}\right )}{2 b^{5/2}}\\ \end{align*}
Mathematica [C] time = 0.0081719, size = 40, normalized size = 0.49 \[ -\frac{c x \, _2F_1\left (-\frac{1}{2},2;\frac{1}{2};\frac{c x^2}{b}+1\right )}{b^2 \sqrt{x^2 \left (b+c x^2\right )}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.046, size = 77, normalized size = 1. \begin{align*} -{\frac{x \left ( c{x}^{2}+b \right ) }{2} \left ( 3\,{b}^{3/2}{x}^{2}c-3\,\ln \left ( 2\,{\frac{\sqrt{b}\sqrt{c{x}^{2}+b}+b}{x}} \right ) \sqrt{c{x}^{2}+b}{x}^{2}bc+{b}^{{\frac{5}{2}}} \right ) \left ( c{x}^{4}+b{x}^{2} \right ) ^{-{\frac{3}{2}}}{b}^{-{\frac{7}{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{1}{{\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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Fricas [A] time = 1.41731, size = 425, normalized size = 5.25 \begin{align*} \left [\frac{3 \,{\left (c^{2} x^{5} + b c x^{3}\right )} \sqrt{b} \log \left (-\frac{c x^{3} + 2 \, b x + 2 \, \sqrt{c x^{4} + b x^{2}} \sqrt{b}}{x^{3}}\right ) - 2 \, \sqrt{c x^{4} + b x^{2}}{\left (3 \, b c x^{2} + b^{2}\right )}}{4 \,{\left (b^{3} c x^{5} + b^{4} x^{3}\right )}}, -\frac{3 \,{\left (c^{2} x^{5} + b c x^{3}\right )} \sqrt{-b} \arctan \left (\frac{\sqrt{c x^{4} + b x^{2}} \sqrt{-b}}{c x^{3} + b x}\right ) + \sqrt{c x^{4} + b x^{2}}{\left (3 \, b c x^{2} + b^{2}\right )}}{2 \,{\left (b^{3} c x^{5} + b^{4} x^{3}\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{1}{\left (b x^{2} + c x^{4}\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*} \mathit{sage}_{0} x \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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