Optimal. Leaf size=87 \[ -\frac {3 c^2 \tanh ^{-1}\left (\frac {\sqrt {b} x}{\sqrt {b x^2+c x^4}}\right )}{8 b^{5/2}}+\frac {3 c \sqrt {b x^2+c x^4}}{8 b^2 x^3}-\frac {\sqrt {b x^2+c x^4}}{4 b x^5} \]
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Rubi [A] time = 0.10, antiderivative size = 87, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 4, integrand size = 28, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.143, Rules used = {3, 2025, 2008, 206} \[ -\frac {3 c^2 \tanh ^{-1}\left (\frac {\sqrt {b} x}{\sqrt {b x^2+c x^4}}\right )}{8 b^{5/2}}+\frac {3 c \sqrt {b x^2+c x^4}}{8 b^2 x^3}-\frac {\sqrt {b x^2+c x^4}}{4 b x^5} \]
Antiderivative was successfully verified.
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Rule 3
Rule 206
Rule 2008
Rule 2025
Rubi steps
\begin {align*} \int \frac {1}{x^4 \sqrt {2+2 a-2 (1+a)+b x^2+c x^4}} \, dx &=\int \frac {1}{x^4 \sqrt {b x^2+c x^4}} \, dx\\ &=-\frac {\sqrt {b x^2+c x^4}}{4 b x^5}-\frac {(3 c) \int \frac {1}{x^2 \sqrt {b x^2+c x^4}} \, dx}{4 b}\\ &=-\frac {\sqrt {b x^2+c x^4}}{4 b x^5}+\frac {3 c \sqrt {b x^2+c x^4}}{8 b^2 x^3}+\frac {\left (3 c^2\right ) \int \frac {1}{\sqrt {b x^2+c x^4}} \, dx}{8 b^2}\\ &=-\frac {\sqrt {b x^2+c x^4}}{4 b x^5}+\frac {3 c \sqrt {b x^2+c x^4}}{8 b^2 x^3}-\frac {\left (3 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^2}\\ &=-\frac {\sqrt {b x^2+c x^4}}{4 b x^5}+\frac {3 c \sqrt {b x^2+c x^4}}{8 b^2 x^3}-\frac {3 c^2 \tanh ^{-1}\left (\frac {\sqrt {b} x}{\sqrt {b x^2+c x^4}}\right )}{8 b^{5/2}}\\ \end {align*}
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Mathematica [C] time = 0.01, size = 44, normalized size = 0.51 \[ -\frac {c^2 \sqrt {x^2 \left (b+c x^2\right )} \, _2F_1\left (\frac {1}{2},3;\frac {3}{2};\frac {c x^2}{b}+1\right )}{b^3 x} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.83, size = 163, normalized size = 1.87 \[ \left [\frac {3 \, \sqrt {b} c^{2} x^{5} \log \left (-\frac {c x^{3} + 2 \, b x - 2 \, \sqrt {c x^{4} + b x^{2}} \sqrt {b}}{x^{3}}\right ) + 2 \, \sqrt {c x^{4} + b x^{2}} {\left (3 \, b c x^{2} - 2 \, b^{2}\right )}}{16 \, b^{3} x^{5}}, \frac {3 \, \sqrt {-b} c^{2} x^{5} \arctan \left (\frac {\sqrt {c x^{4} + b x^{2}} \sqrt {-b}}{c x^{3} + b x}\right ) + \sqrt {c x^{4} + b x^{2}} {\left (3 \, b c x^{2} - 2 \, b^{2}\right )}}{8 \, b^{3} x^{5}}\right ] \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [F(-2)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Exception raised: NotImplementedError} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.01, size = 94, normalized size = 1.08 \[ -\frac {\sqrt {c \,x^{2}+b}\, \left (3 b \,c^{2} x^{4} \ln \left (\frac {2 b +2 \sqrt {c \,x^{2}+b}\, \sqrt {b}}{x}\right )-3 \sqrt {c \,x^{2}+b}\, b^{\frac {3}{2}} c \,x^{2}+2 \sqrt {c \,x^{2}+b}\, b^{\frac {5}{2}}\right )}{8 \sqrt {c \,x^{4}+b \,x^{2}}\, b^{\frac {7}{2}} x^{3}} \]
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 \[ \int \frac {1}{\sqrt {c x^{4} + b x^{2}} x^{4}}\,{d x} \]
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 \[ \int \frac {1}{x^4\,\sqrt {c\,x^4+b\,x^2}} \,d x \]
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 \[ \int \frac {1}{x^{4} \sqrt {x^{2} \left (b + c x^{2}\right )}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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