Optimal. Leaf size=109 \[ \frac{c^3 \tanh ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{b x^2+c x^4}}\right )}{16 b^{3/2}}-\frac{c^2 \sqrt{b x^2+c x^4}}{16 b x^3}-\frac{c \sqrt{b x^2+c x^4}}{8 x^5}-\frac{\left (b x^2+c x^4\right )^{3/2}}{6 x^9} \]
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Rubi [A] time = 0.164261, 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 = {2020, 2025, 2008, 206} \[ \frac{c^3 \tanh ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{b x^2+c x^4}}\right )}{16 b^{3/2}}-\frac{c^2 \sqrt{b x^2+c x^4}}{16 b x^3}-\frac{c \sqrt{b x^2+c x^4}}{8 x^5}-\frac{\left (b x^2+c x^4\right )^{3/2}}{6 x^9} \]
Antiderivative was successfully verified.
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Rule 2020
Rule 2025
Rule 2008
Rule 206
Rubi steps
\begin{align*} \int \frac{\left (b x^2+c x^4\right )^{3/2}}{x^{10}} \, dx &=-\frac{\left (b x^2+c x^4\right )^{3/2}}{6 x^9}+\frac{1}{2} c \int \frac{\sqrt{b x^2+c x^4}}{x^6} \, dx\\ &=-\frac{c \sqrt{b x^2+c x^4}}{8 x^5}-\frac{\left (b x^2+c x^4\right )^{3/2}}{6 x^9}+\frac{1}{8} c^2 \int \frac{1}{x^2 \sqrt{b x^2+c x^4}} \, dx\\ &=-\frac{c \sqrt{b x^2+c x^4}}{8 x^5}-\frac{c^2 \sqrt{b x^2+c x^4}}{16 b x^3}-\frac{\left (b x^2+c x^4\right )^{3/2}}{6 x^9}-\frac{c^3 \int \frac{1}{\sqrt{b x^2+c x^4}} \, dx}{16 b}\\ &=-\frac{c \sqrt{b x^2+c x^4}}{8 x^5}-\frac{c^2 \sqrt{b x^2+c x^4}}{16 b x^3}-\frac{\left (b x^2+c x^4\right )^{3/2}}{6 x^9}+\frac{c^3 \operatorname{Subst}\left (\int \frac{1}{1-b x^2} \, dx,x,\frac{x}{\sqrt{b x^2+c x^4}}\right )}{16 b}\\ &=-\frac{c \sqrt{b x^2+c x^4}}{8 x^5}-\frac{c^2 \sqrt{b x^2+c x^4}}{16 b x^3}-\frac{\left (b x^2+c x^4\right )^{3/2}}{6 x^9}+\frac{c^3 \tanh ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{b x^2+c x^4}}\right )}{16 b^{3/2}}\\ \end{align*}
Mathematica [C] time = 0.0193717, size = 46, normalized size = 0.42 \[ \frac{c^3 \left (x^2 \left (b+c x^2\right )\right )^{5/2} \, _2F_1\left (\frac{5}{2},4;\frac{7}{2};\frac{c x^2}{b}+1\right )}{5 b^4 x^5} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.05, size = 145, normalized size = 1.3 \begin{align*}{\frac{1}{48\,{x}^{9}{b}^{3}} \left ( c{x}^{4}+b{x}^{2} \right ) ^{{\frac{3}{2}}} \left ( - \left ( c{x}^{2}+b \right ) ^{{\frac{3}{2}}}{x}^{6}{c}^{3}+3\,{b}^{3/2}\ln \left ( 2\,{\frac{\sqrt{b}\sqrt{c{x}^{2}+b}+b}{x}} \right ){x}^{6}{c}^{3}+ \left ( c{x}^{2}+b \right ) ^{{\frac{5}{2}}}{x}^{4}{c}^{2}-3\,\sqrt{c{x}^{2}+b}{x}^{6}b{c}^{3}+2\, \left ( c{x}^{2}+b \right ) ^{5/2}{x}^{2}bc-8\, \left ( c{x}^{2}+b \right ) ^{5/2}{b}^{2} \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^{10}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.29058, size = 414, normalized size = 3.8 \begin{align*} \left [\frac{3 \, \sqrt{b} c^{3} x^{7} \log \left (-\frac{c x^{3} + 2 \, b x + 2 \, \sqrt{c x^{4} + b x^{2}} \sqrt{b}}{x^{3}}\right ) - 2 \,{\left (3 \, b c^{2} x^{4} + 14 \, b^{2} c x^{2} + 8 \, b^{3}\right )} \sqrt{c x^{4} + b x^{2}}}{96 \, b^{2} x^{7}}, -\frac{3 \, \sqrt{-b} c^{3} x^{7} \arctan \left (\frac{\sqrt{c x^{4} + b x^{2}} \sqrt{-b}}{c x^{3} + b x}\right ) +{\left (3 \, b c^{2} x^{4} + 14 \, b^{2} c x^{2} + 8 \, b^{3}\right )} \sqrt{c x^{4} + b x^{2}}}{48 \, b^{2} x^{7}}\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^{10}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.18825, size = 111, normalized size = 1.02 \begin{align*} -\frac{1}{48} \, c^{3}{\left (\frac{3 \, \arctan \left (\frac{\sqrt{c x^{2} + b}}{\sqrt{-b}}\right )}{\sqrt{-b} b} + \frac{3 \,{\left (c x^{2} + b\right )}^{\frac{5}{2}} + 8 \,{\left (c x^{2} + b\right )}^{\frac{3}{2}} b - 3 \, \sqrt{c x^{2} + b} b^{2}}{b c^{3} x^{6}}\right )} \mathrm{sgn}\left (x\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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