Optimal. Leaf size=109 \[ -\frac{15 c^2 \tanh ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{b x^2+c x^4}}\right )}{8 b^{7/2}}+\frac{15 c \sqrt{b x^2+c x^4}}{8 b^3 x^3}-\frac{5 \sqrt{b x^2+c x^4}}{4 b^2 x^5}+\frac{1}{b x^3 \sqrt{b x^2+c x^4}} \]
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Rubi [A] time = 0.153449, antiderivative size = 109, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 4, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.21, Rules used = {2023, 2025, 2008, 206} \[ -\frac{15 c^2 \tanh ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{b x^2+c x^4}}\right )}{8 b^{7/2}}+\frac{15 c \sqrt{b x^2+c x^4}}{8 b^3 x^3}-\frac{5 \sqrt{b x^2+c x^4}}{4 b^2 x^5}+\frac{1}{b x^3 \sqrt{b x^2+c x^4}} \]
Antiderivative was successfully verified.
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Rule 2023
Rule 2025
Rule 2008
Rule 206
Rubi steps
\begin{align*} \int \frac{1}{x^2 \left (b x^2+c x^4\right )^{3/2}} \, dx &=\frac{1}{b x^3 \sqrt{b x^2+c x^4}}+\frac{5 \int \frac{1}{x^4 \sqrt{b x^2+c x^4}} \, dx}{b}\\ &=\frac{1}{b x^3 \sqrt{b x^2+c x^4}}-\frac{5 \sqrt{b x^2+c x^4}}{4 b^2 x^5}-\frac{(15 c) \int \frac{1}{x^2 \sqrt{b x^2+c x^4}} \, dx}{4 b^2}\\ &=\frac{1}{b x^3 \sqrt{b x^2+c x^4}}-\frac{5 \sqrt{b x^2+c x^4}}{4 b^2 x^5}+\frac{15 c \sqrt{b x^2+c x^4}}{8 b^3 x^3}+\frac{\left (15 c^2\right ) \int \frac{1}{\sqrt{b x^2+c x^4}} \, dx}{8 b^3}\\ &=\frac{1}{b x^3 \sqrt{b x^2+c x^4}}-\frac{5 \sqrt{b x^2+c x^4}}{4 b^2 x^5}+\frac{15 c \sqrt{b x^2+c x^4}}{8 b^3 x^3}-\frac{\left (15 c^2\right ) \operatorname{Subst}\left (\int \frac{1}{1-b x^2} \, dx,x,\frac{x}{\sqrt{b x^2+c x^4}}\right )}{8 b^3}\\ &=\frac{1}{b x^3 \sqrt{b x^2+c x^4}}-\frac{5 \sqrt{b x^2+c x^4}}{4 b^2 x^5}+\frac{15 c \sqrt{b x^2+c x^4}}{8 b^3 x^3}-\frac{15 c^2 \tanh ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{b x^2+c x^4}}\right )}{8 b^{7/2}}\\ \end{align*}
Mathematica [C] time = 0.0111332, size = 41, normalized size = 0.38 \[ \frac{c^2 x \, _2F_1\left (-\frac{1}{2},3;\frac{1}{2};\frac{c x^2}{b}+1\right )}{b^3 \sqrt{x^2 \left (b+c x^2\right )}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.046, size = 94, normalized size = 0.9 \begin{align*} -{\frac{c{x}^{2}+b}{8\,x} \left ( 15\,\sqrt{c{x}^{2}+b}\ln \left ( 2\,{\frac{\sqrt{b}\sqrt{c{x}^{2}+b}+b}{x}} \right ){x}^{4}b{c}^{2}-15\,{b}^{3/2}{x}^{4}{c}^{2}-5\,{b}^{5/2}{x}^{2}c+2\,{b}^{7/2} \right ) \left ( c{x}^{4}+b{x}^{2} \right ) ^{-{\frac{3}{2}}}{b}^{-{\frac{9}{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}} x^{2}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.34802, size = 485, normalized size = 4.45 \begin{align*} \left [\frac{15 \,{\left (c^{3} x^{7} + b c^{2} x^{5}\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 \,{\left (15 \, b c^{2} x^{4} + 5 \, b^{2} c x^{2} - 2 \, b^{3}\right )} \sqrt{c x^{4} + b x^{2}}}{16 \,{\left (b^{4} c x^{7} + b^{5} x^{5}\right )}}, \frac{15 \,{\left (c^{3} x^{7} + b c^{2} x^{5}\right )} \sqrt{-b} \arctan \left (\frac{\sqrt{c x^{4} + b x^{2}} \sqrt{-b}}{c x^{3} + b x}\right ) +{\left (15 \, b c^{2} x^{4} + 5 \, b^{2} c x^{2} - 2 \, b^{3}\right )} \sqrt{c x^{4} + b x^{2}}}{8 \,{\left (b^{4} c x^{7} + b^{5} x^{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{1}{x^{2} \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{1}{{\left (c x^{4} + b x^{2}\right )}^{\frac{3}{2}} x^{2}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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