Optimal. Leaf size=103 \[ \frac {c (4 b B-3 A c) \tanh ^{-1}\left (\frac {\sqrt {b} x}{\sqrt {b x^2+c x^4}}\right )}{8 b^{5/2}}-\frac {\sqrt {b x^2+c x^4} (4 b B-3 A c)}{8 b^2 x^3}-\frac {A \sqrt {b x^2+c x^4}}{4 b x^5} \]
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Rubi [A] time = 0.17, antiderivative size = 103, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, integrand size = 26, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.154, Rules used = {2038, 2025, 2008, 206} \begin {gather*} -\frac {\sqrt {b x^2+c x^4} (4 b B-3 A c)}{8 b^2 x^3}+\frac {c (4 b B-3 A c) \tanh ^{-1}\left (\frac {\sqrt {b} x}{\sqrt {b x^2+c x^4}}\right )}{8 b^{5/2}}-\frac {A \sqrt {b x^2+c x^4}}{4 b x^5} \end {gather*}
Antiderivative was successfully verified.
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Rule 206
Rule 2008
Rule 2025
Rule 2038
Rubi steps
\begin {align*} \int \frac {A+B x^2}{x^4 \sqrt {b x^2+c x^4}} \, dx &=-\frac {A \sqrt {b x^2+c x^4}}{4 b x^5}-\frac {(-4 b B+3 A c) \int \frac {1}{x^2 \sqrt {b x^2+c x^4}} \, dx}{4 b}\\ &=-\frac {A \sqrt {b x^2+c x^4}}{4 b x^5}-\frac {(4 b B-3 A c) \sqrt {b x^2+c x^4}}{8 b^2 x^3}-\frac {(c (4 b B-3 A c)) \int \frac {1}{\sqrt {b x^2+c x^4}} \, dx}{8 b^2}\\ &=-\frac {A \sqrt {b x^2+c x^4}}{4 b x^5}-\frac {(4 b B-3 A c) \sqrt {b x^2+c x^4}}{8 b^2 x^3}+\frac {(c (4 b B-3 A c)) \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 {A \sqrt {b x^2+c x^4}}{4 b x^5}-\frac {(4 b B-3 A c) \sqrt {b x^2+c x^4}}{8 b^2 x^3}+\frac {c (4 b B-3 A c) \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 [A] time = 0.22, size = 104, normalized size = 1.01 \begin {gather*} -\frac {\sqrt {x^2 \left (b+c x^2\right )} \left (b \sqrt {\frac {c x^2}{b}+1} \left (2 A b-3 A c x^2+4 b B x^2\right )+c x^4 (3 A c-4 b B) \tanh ^{-1}\left (\sqrt {\frac {c x^2}{b}+1}\right )\right )}{8 b^3 x^5 \sqrt {\frac {c x^2}{b}+1}} \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [A] time = 0.22, size = 89, normalized size = 0.86 \begin {gather*} \frac {\left (4 b B c-3 A c^2\right ) \tanh ^{-1}\left (\frac {\sqrt {b} x}{\sqrt {b x^2+c x^4}}\right )}{8 b^{5/2}}+\frac {\sqrt {b x^2+c x^4} \left (-2 A b+3 A c x^2-4 b B x^2\right )}{8 b^2 x^5} \end {gather*}
Antiderivative was successfully verified.
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fricas [A] time = 0.44, size = 199, normalized size = 1.93 \begin {gather*} \left [-\frac {{\left (4 \, B b c - 3 \, A c^{2}\right )} \sqrt {b} 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 (2 \, A b^{2} + {\left (4 \, B b^{2} - 3 \, A b c\right )} x^{2}\right )}}{16 \, b^{3} x^{5}}, -\frac {{\left (4 \, B b c - 3 \, A c^{2}\right )} \sqrt {-b} 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 (2 \, A b^{2} + {\left (4 \, B b^{2} - 3 \, A b c\right )} x^{2}\right )}}{8 \, b^{3} x^{5}}\right ] \end {gather*}
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 \begin {gather*} \text {Exception raised: NotImplementedError} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.06, size = 146, normalized size = 1.42 \begin {gather*} -\frac {\sqrt {c \,x^{2}+b}\, \left (3 A b \,c^{2} x^{4} \ln \left (\frac {2 b +2 \sqrt {c \,x^{2}+b}\, \sqrt {b}}{x}\right )-4 B \,b^{2} c \,x^{4} \ln \left (\frac {2 b +2 \sqrt {c \,x^{2}+b}\, \sqrt {b}}{x}\right )-3 \sqrt {c \,x^{2}+b}\, A \,b^{\frac {3}{2}} c \,x^{2}+4 \sqrt {c \,x^{2}+b}\, B \,b^{\frac {5}{2}} x^{2}+2 \sqrt {c \,x^{2}+b}\, A \,b^{\frac {5}{2}}\right )}{8 \sqrt {c \,x^{4}+b \,x^{2}}\, b^{\frac {7}{2}} x^{3}} \end {gather*}
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 \begin {gather*} \int \frac {B x^{2} + A}{\sqrt {c x^{4} + b x^{2}} x^{4}}\,{d x} \end {gather*}
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 \begin {gather*} \int \frac {B\,x^2+A}{x^4\,\sqrt {c\,x^4+b\,x^2}} \,d x \end {gather*}
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 \begin {gather*} \int \frac {A + B x^{2}}{x^{4} \sqrt {x^{2} \left (b + c x^{2}\right )}}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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