Optimal. Leaf size=100 \[ \frac {\sqrt {b x^2+c x^4} (A c+2 b B)}{2 b x}-\frac {(A c+2 b B) \tanh ^{-1}\left (\frac {\sqrt {b} x}{\sqrt {b x^2+c x^4}}\right )}{2 \sqrt {b}}-\frac {A \left (b x^2+c x^4\right )^{3/2}}{2 b x^5} \]
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Rubi [A] time = 0.16, antiderivative size = 100, 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, 2021, 2008, 206} \[ \frac {\sqrt {b x^2+c x^4} (A c+2 b B)}{2 b x}-\frac {(A c+2 b B) \tanh ^{-1}\left (\frac {\sqrt {b} x}{\sqrt {b x^2+c x^4}}\right )}{2 \sqrt {b}}-\frac {A \left (b x^2+c x^4\right )^{3/2}}{2 b x^5} \]
Antiderivative was successfully verified.
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Rule 206
Rule 2008
Rule 2021
Rule 2038
Rubi steps
\begin {align*} \int \frac {\left (A+B x^2\right ) \sqrt {b x^2+c x^4}}{x^4} \, dx &=-\frac {A \left (b x^2+c x^4\right )^{3/2}}{2 b x^5}-\frac {(-2 b B-A c) \int \frac {\sqrt {b x^2+c x^4}}{x^2} \, dx}{2 b}\\ &=\frac {(2 b B+A c) \sqrt {b x^2+c x^4}}{2 b x}-\frac {A \left (b x^2+c x^4\right )^{3/2}}{2 b x^5}-\frac {1}{2} (-2 b B-A c) \int \frac {1}{\sqrt {b x^2+c x^4}} \, dx\\ &=\frac {(2 b B+A c) \sqrt {b x^2+c x^4}}{2 b x}-\frac {A \left (b x^2+c x^4\right )^{3/2}}{2 b x^5}-\frac {1}{2} (2 b B+A c) \operatorname {Subst}\left (\int \frac {1}{1-b x^2} \, dx,x,\frac {x}{\sqrt {b x^2+c x^4}}\right )\\ &=\frac {(2 b B+A c) \sqrt {b x^2+c x^4}}{2 b x}-\frac {A \left (b x^2+c x^4\right )^{3/2}}{2 b x^5}-\frac {(2 b B+A c) \tanh ^{-1}\left (\frac {\sqrt {b} x}{\sqrt {b x^2+c x^4}}\right )}{2 \sqrt {b}}\\ \end {align*}
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Mathematica [A] time = 0.05, size = 94, normalized size = 0.94 \[ -\frac {\sqrt {x^2 \left (b+c x^2\right )} \left (\sqrt {b} \left (A-2 B x^2\right ) \sqrt {b+c x^2}+x^2 (A c+2 b B) \tanh ^{-1}\left (\frac {\sqrt {b+c x^2}}{\sqrt {b}}\right )\right )}{2 \sqrt {b} x^3 \sqrt {b+c x^2}} \]
Antiderivative was successfully verified.
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fricas [A] time = 1.14, size = 169, normalized size = 1.69 \[ \left [\frac {{\left (2 \, B b + A c\right )} \sqrt {b} x^{3} \log \left (-\frac {c x^{3} + 2 \, b x - 2 \, \sqrt {c x^{4} + b x^{2}} \sqrt {b}}{x^{3}}\right ) + 2 \, \sqrt {c x^{4} + b x^{2}} {\left (2 \, B b x^{2} - A b\right )}}{4 \, b x^{3}}, \frac {{\left (2 \, B b + A c\right )} \sqrt {-b} x^{3} \arctan \left (\frac {\sqrt {c x^{4} + b x^{2}} \sqrt {-b}}{c x^{3} + b x}\right ) + \sqrt {c x^{4} + b x^{2}} {\left (2 \, B b x^{2} - A b\right )}}{2 \, b x^{3}}\right ] \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.22, size = 76, normalized size = 0.76 \[ \frac {2 \, \sqrt {c x^{2} + b} B c \mathrm {sgn}\relax (x) + \frac {{\left (2 \, B b c \mathrm {sgn}\relax (x) + A c^{2} \mathrm {sgn}\relax (x)\right )} \arctan \left (\frac {\sqrt {c x^{2} + b}}{\sqrt {-b}}\right )}{\sqrt {-b}} - \frac {\sqrt {c x^{2} + b} A c \mathrm {sgn}\relax (x)}{x^{2}}}{2 \, c} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.05, size = 135, normalized size = 1.35 \[ -\frac {\sqrt {c \,x^{4}+b \,x^{2}}\, \left (A \sqrt {b}\, c \,x^{2} \ln \left (\frac {2 b +2 \sqrt {c \,x^{2}+b}\, \sqrt {b}}{x}\right )+2 B \,b^{\frac {3}{2}} x^{2} \ln \left (\frac {2 b +2 \sqrt {c \,x^{2}+b}\, \sqrt {b}}{x}\right )-\sqrt {c \,x^{2}+b}\, A c \,x^{2}-2 \sqrt {c \,x^{2}+b}\, B b \,x^{2}+\left (c \,x^{2}+b \right )^{\frac {3}{2}} A \right )}{2 \sqrt {c \,x^{2}+b}\, b \,x^{3}} \]
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 \[ \int \frac {\sqrt {c x^{4} + b x^{2}} {\left (B x^{2} + A\right )}}{x^{4}}\,{d x} \]
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 \[ \int \frac {\left (B\,x^2+A\right )\,\sqrt {c\,x^4+b\,x^2}}{x^4} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\sqrt {x^{2} \left (b + c x^{2}\right )} \left (A + B x^{2}\right )}{x^{4}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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