Optimal. Leaf size=79 \[ -\frac{\left (b x^2+c x^4\right )^{3/2}}{2 x^5}+\frac{3 c \sqrt{b x^2+c x^4}}{2 x}-\frac{3}{2} \sqrt{b} c \tanh ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{b x^2+c x^4}}\right ) \]
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Rubi [A] time = 0.118345, antiderivative size = 79, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.21, Rules used = {2020, 2021, 2008, 206} \[ -\frac{\left (b x^2+c x^4\right )^{3/2}}{2 x^5}+\frac{3 c \sqrt{b x^2+c x^4}}{2 x}-\frac{3}{2} \sqrt{b} c \tanh ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{b x^2+c x^4}}\right ) \]
Antiderivative was successfully verified.
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Rule 2020
Rule 2021
Rule 2008
Rule 206
Rubi steps
\begin{align*} \int \frac{\left (b x^2+c x^4\right )^{3/2}}{x^6} \, dx &=-\frac{\left (b x^2+c x^4\right )^{3/2}}{2 x^5}+\frac{1}{2} (3 c) \int \frac{\sqrt{b x^2+c x^4}}{x^2} \, dx\\ &=\frac{3 c \sqrt{b x^2+c x^4}}{2 x}-\frac{\left (b x^2+c x^4\right )^{3/2}}{2 x^5}+\frac{1}{2} (3 b c) \int \frac{1}{\sqrt{b x^2+c x^4}} \, dx\\ &=\frac{3 c \sqrt{b x^2+c x^4}}{2 x}-\frac{\left (b x^2+c x^4\right )^{3/2}}{2 x^5}-\frac{1}{2} (3 b c) \operatorname{Subst}\left (\int \frac{1}{1-b x^2} \, dx,x,\frac{x}{\sqrt{b x^2+c x^4}}\right )\\ &=\frac{3 c \sqrt{b x^2+c x^4}}{2 x}-\frac{\left (b x^2+c x^4\right )^{3/2}}{2 x^5}-\frac{3}{2} \sqrt{b} c \tanh ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{b x^2+c x^4}}\right )\\ \end{align*}
Mathematica [C] time = 0.0170345, size = 44, normalized size = 0.56 \[ \frac{c \left (x^2 \left (b+c x^2\right )\right )^{5/2} \, _2F_1\left (2,\frac{5}{2};\frac{7}{2};\frac{c x^2}{b}+1\right )}{5 b^2 x^5} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.047, size = 102, normalized size = 1.3 \begin{align*} -{\frac{1}{2\,b{x}^{5}} \left ( c{x}^{4}+b{x}^{2} \right ) ^{{\frac{3}{2}}} \left ( 3\,{b}^{3/2}\ln \left ( 2\,{\frac{\sqrt{b}\sqrt{c{x}^{2}+b}+b}{x}} \right ){x}^{2}c- \left ( c{x}^{2}+b \right ) ^{{\frac{3}{2}}}{x}^{2}c+ \left ( c{x}^{2}+b \right ) ^{{\frac{5}{2}}}-3\,\sqrt{c{x}^{2}+b}{x}^{2}bc \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}}}{x^{6}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.43626, size = 327, normalized size = 4.14 \begin{align*} \left [\frac{3 \, \sqrt{b} c 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 \, \sqrt{c x^{4} + b x^{2}}{\left (2 \, c x^{2} - b\right )}}{4 \, x^{3}}, \frac{3 \, \sqrt{-b} c x^{3} \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 (2 \, c x^{2} - b\right )}}{2 \, 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}}}{x^{6}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.20255, size = 80, normalized size = 1.01 \begin{align*} \frac{1}{2} \,{\left (\frac{3 \, b \arctan \left (\frac{\sqrt{c x^{2} + b}}{\sqrt{-b}}\right )}{\sqrt{-b}} + 2 \, \sqrt{c x^{2} + b} - \frac{\sqrt{c x^{2} + b} b}{c x^{2}}\right )} c \mathrm{sgn}\left (x\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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