Optimal. Leaf size=112 \[ -\frac {c^3 \tanh ^{-1}\left (\frac {\sqrt {b} x}{\sqrt {b x^2+c x^4}}\right )}{16 b^{5/2}}+\frac {c^2 \sqrt {b x^2+c x^4}}{16 b^2 x^3}-\frac {\sqrt {b x^2+c x^4}}{6 x^7}-\frac {c \sqrt {b x^2+c x^4}}{24 b x^5} \]
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Rubi [A] time = 0.14, antiderivative size = 112, 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} \[ \frac {c^2 \sqrt {b x^2+c x^4}}{16 b^2 x^3}-\frac {c^3 \tanh ^{-1}\left (\frac {\sqrt {b} x}{\sqrt {b x^2+c x^4}}\right )}{16 b^{5/2}}-\frac {c \sqrt {b x^2+c x^4}}{24 b x^5}-\frac {\sqrt {b x^2+c x^4}}{6 x^7} \]
Antiderivative was successfully verified.
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Rule 206
Rule 2008
Rule 2020
Rule 2025
Rubi steps
\begin {align*} \int \frac {\sqrt {b x^2+c x^4}}{x^8} \, dx &=-\frac {\sqrt {b x^2+c x^4}}{6 x^7}+\frac {1}{6} c \int \frac {1}{x^4 \sqrt {b x^2+c x^4}} \, dx\\ &=-\frac {\sqrt {b x^2+c x^4}}{6 x^7}-\frac {c \sqrt {b x^2+c x^4}}{24 b x^5}-\frac {c^2 \int \frac {1}{x^2 \sqrt {b x^2+c x^4}} \, dx}{8 b}\\ &=-\frac {\sqrt {b x^2+c x^4}}{6 x^7}-\frac {c \sqrt {b x^2+c x^4}}{24 b x^5}+\frac {c^2 \sqrt {b x^2+c x^4}}{16 b^2 x^3}+\frac {c^3 \int \frac {1}{\sqrt {b x^2+c x^4}} \, dx}{16 b^2}\\ &=-\frac {\sqrt {b x^2+c x^4}}{6 x^7}-\frac {c \sqrt {b x^2+c x^4}}{24 b x^5}+\frac {c^2 \sqrt {b x^2+c x^4}}{16 b^2 x^3}-\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^2}\\ &=-\frac {\sqrt {b x^2+c x^4}}{6 x^7}-\frac {c \sqrt {b x^2+c x^4}}{24 b x^5}+\frac {c^2 \sqrt {b x^2+c x^4}}{16 b^2 x^3}-\frac {c^3 \tanh ^{-1}\left (\frac {\sqrt {b} x}{\sqrt {b x^2+c x^4}}\right )}{16 b^{5/2}}\\ \end {align*}
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Mathematica [C] time = 0.01, size = 46, normalized size = 0.41 \[ \frac {c^3 \left (x^2 \left (b+c x^2\right )\right )^{3/2} \, _2F_1\left (\frac {3}{2},4;\frac {5}{2};\frac {c x^2}{b}+1\right )}{3 b^4 x^3} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.68, size = 185, normalized size = 1.65 \[ \left [\frac {3 \, \sqrt {b} c^{3} x^{7} \log \left (-\frac {c x^{3} + 2 \, b x - 2 \, \sqrt {c x^{4} + b x^{2}} \sqrt {b}}{x^{3}}\right ) + 2 \, {\left (3 \, b c^{2} x^{4} - 2 \, b^{2} c x^{2} - 8 \, b^{3}\right )} \sqrt {c x^{4} + b x^{2}}}{96 \, b^{3} x^{7}}, \frac {3 \, \sqrt {-b} c^{3} x^{7} \arctan \left (\frac {\sqrt {c x^{4} + b x^{2}} \sqrt {-b}}{c x^{3} + b x}\right ) + {\left (3 \, b c^{2} x^{4} - 2 \, b^{2} c x^{2} - 8 \, b^{3}\right )} \sqrt {c x^{4} + b x^{2}}}{48 \, b^{3} x^{7}}\right ] \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.27, size = 100, normalized size = 0.89 \[ \frac {\frac {3 \, c^{4} \arctan \left (\frac {\sqrt {c x^{2} + b}}{\sqrt {-b}}\right ) \mathrm {sgn}\relax (x)}{\sqrt {-b} b^{2}} + \frac {3 \, {\left (c x^{2} + b\right )}^{\frac {5}{2}} c^{4} \mathrm {sgn}\relax (x) - 8 \, {\left (c x^{2} + b\right )}^{\frac {3}{2}} b c^{4} \mathrm {sgn}\relax (x) - 3 \, \sqrt {c x^{2} + b} b^{2} c^{4} \mathrm {sgn}\relax (x)}{b^{2} c^{3} x^{6}}}{48 \, c} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.01, size = 128, normalized size = 1.14 \[ -\frac {\sqrt {c \,x^{4}+b \,x^{2}}\, \left (3 \sqrt {b}\, c^{3} x^{6} \ln \left (\frac {2 b +2 \sqrt {c \,x^{2}+b}\, \sqrt {b}}{x}\right )-3 \sqrt {c \,x^{2}+b}\, c^{3} x^{6}+3 \left (c \,x^{2}+b \right )^{\frac {3}{2}} c^{2} x^{4}-6 \left (c \,x^{2}+b \right )^{\frac {3}{2}} b c \,x^{2}+8 \left (c \,x^{2}+b \right )^{\frac {3}{2}} b^{2}\right )}{48 \sqrt {c \,x^{2}+b}\, b^{3} x^{7}} \]
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 \[ \int \frac {\sqrt {c x^{4} + b x^{2}}}{x^{8}}\,{d x} \]
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 \[ \int \frac {\sqrt {c\,x^4+b\,x^2}}{x^8} \,d x \]
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 \[ \int \frac {\sqrt {x^{2} \left (b + c x^{2}\right )}}{x^{8}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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