Optimal. Leaf size=109 \[ \frac {c^3 \tanh ^{-1}\left (\frac {\sqrt {b} x}{\sqrt {b x^2+c x^4}}\right )}{16 b^{3/2}}-\frac {c^2 \sqrt {b x^2+c x^4}}{16 b x^3}-\frac {\left (b x^2+c x^4\right )^{3/2}}{6 x^9}-\frac {c \sqrt {b x^2+c x^4}}{8 x^5} \]
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Rubi [A] time = 0.16, antiderivative size = 109, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 4, integrand size = 19, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.210, 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 206
Rule 2008
Rule 2020
Rule 2025
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*}
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Mathematica [C] time = 0.02, 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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fricas [A] time = 0.90, size = 185, normalized size = 1.70 \[ \left [\frac {3 \, \sqrt {b} c^{3} x^{7} \log \left (-\frac {c x^{3} + 2 \, b x + 2 \, \sqrt {c x^{4} + b x^{2}} \sqrt {b}}{x^{3}}\right ) - 2 \, {\left (3 \, b c^{2} x^{4} + 14 \, b^{2} c x^{2} + 8 \, b^{3}\right )} \sqrt {c x^{4} + b x^{2}}}{96 \, b^{2} x^{7}}, -\frac {3 \, \sqrt {-b} c^{3} x^{7} \arctan \left (\frac {\sqrt {c x^{4} + b x^{2}} \sqrt {-b}}{c x^{3} + b x}\right ) + {\left (3 \, b c^{2} x^{4} + 14 \, b^{2} c x^{2} + 8 \, b^{3}\right )} \sqrt {c x^{4} + b x^{2}}}{48 \, b^{2} x^{7}}\right ] \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.24, size = 100, normalized size = 0.92 \[ -\frac {\frac {3 \, c^{4} \arctan \left (\frac {\sqrt {c x^{2} + b}}{\sqrt {-b}}\right ) \mathrm {sgn}\relax (x)}{\sqrt {-b} b} + \frac {3 \, {\left (c x^{2} + b\right )}^{\frac {5}{2}} c^{4} \mathrm {sgn}\relax (x) + 8 \, {\left (c x^{2} + b\right )}^{\frac {3}{2}} b c^{4} \mathrm {sgn}\relax (x) - 3 \, \sqrt {c x^{2} + b} b^{2} c^{4} \mathrm {sgn}\relax (x)}{b c^{3} x^{6}}}{48 \, c} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.01, size = 145, normalized size = 1.33 \[ \frac {\left (c \,x^{4}+b \,x^{2}\right )^{\frac {3}{2}} \left (3 b^{\frac {3}{2}} c^{3} x^{6} \ln \left (\frac {2 b +2 \sqrt {c \,x^{2}+b}\, \sqrt {b}}{x}\right )-3 \sqrt {c \,x^{2}+b}\, b \,c^{3} x^{6}-\left (c \,x^{2}+b \right )^{\frac {3}{2}} c^{3} x^{6}+\left (c \,x^{2}+b \right )^{\frac {5}{2}} c^{2} x^{4}+2 \left (c \,x^{2}+b \right )^{\frac {5}{2}} b c \,x^{2}-8 \left (c \,x^{2}+b \right )^{\frac {5}{2}} b^{2}\right )}{48 \left (c \,x^{2}+b \right )^{\frac {3}{2}} b^{3} x^{9}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {{\left (c x^{4} + b x^{2}\right )}^{\frac {3}{2}}}{x^{10}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.01 \[ \int \frac {{\left (c\,x^4+b\,x^2\right )}^{3/2}}{x^{10}} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\left (x^{2} \left (b + c x^{2}\right )\right )^{\frac {3}{2}}}{x^{10}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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