Optimal. Leaf size=142 \[ -\frac {(2 b B-3 A c) \tanh ^{-1}\left (\frac {\sqrt {b} x}{\sqrt {b x^2+c x^4}}\right )}{2 b^{5/2}}+\frac {\sqrt {b x^2+c x^4} (2 b B-3 A c)}{2 b^2 c x^3}-\frac {2 b B-3 A c}{3 b c x \sqrt {b x^2+c x^4}}-\frac {B}{3 c x \sqrt {b x^2+c x^4}} \]
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Rubi [A] time = 0.09, antiderivative size = 142, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 5, integrand size = 23, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.217, Rules used = {1145, 2006, 2025, 2008, 206} \begin {gather*} \frac {\sqrt {b x^2+c x^4} (2 b B-3 A c)}{2 b^2 c x^3}-\frac {(2 b B-3 A c) \tanh ^{-1}\left (\frac {\sqrt {b} x}{\sqrt {b x^2+c x^4}}\right )}{2 b^{5/2}}-\frac {2 b B-3 A c}{3 b c x \sqrt {b x^2+c x^4}}-\frac {B}{3 c x \sqrt {b x^2+c x^4}} \end {gather*}
Antiderivative was successfully verified.
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Rule 206
Rule 1145
Rule 2006
Rule 2008
Rule 2025
Rubi steps
\begin {align*} \int \frac {A+B x^2}{\left (b x^2+c x^4\right )^{3/2}} \, dx &=-\frac {B}{3 c x \sqrt {b x^2+c x^4}}+\frac {(-2 b B+3 A c) \int \frac {1}{\left (b x^2+c x^4\right )^{3/2}} \, dx}{3 c}\\ &=-\frac {B}{3 c x \sqrt {b x^2+c x^4}}-\frac {2 b B-3 A c}{3 b c x \sqrt {b x^2+c x^4}}+\frac {(-2 b B+3 A c) \int \frac {1}{x^2 \sqrt {b x^2+c x^4}} \, dx}{b c}\\ &=-\frac {B}{3 c x \sqrt {b x^2+c x^4}}-\frac {2 b B-3 A c}{3 b c x \sqrt {b x^2+c x^4}}+\frac {(2 b B-3 A c) \sqrt {b x^2+c x^4}}{2 b^2 c x^3}+\frac {(2 b B-3 A c) \int \frac {1}{\sqrt {b x^2+c x^4}} \, dx}{2 b^2}\\ &=-\frac {B}{3 c x \sqrt {b x^2+c x^4}}-\frac {2 b B-3 A c}{3 b c x \sqrt {b x^2+c x^4}}+\frac {(2 b B-3 A c) \sqrt {b x^2+c x^4}}{2 b^2 c x^3}-\frac {(2 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 )}{2 b^2}\\ &=-\frac {B}{3 c x \sqrt {b x^2+c x^4}}-\frac {2 b B-3 A c}{3 b c x \sqrt {b x^2+c x^4}}+\frac {(2 b B-3 A c) \sqrt {b x^2+c x^4}}{2 b^2 c x^3}-\frac {(2 b B-3 A c) \tanh ^{-1}\left (\frac {\sqrt {b} x}{\sqrt {b x^2+c x^4}}\right )}{2 b^{5/2}}\\ \end {align*}
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Mathematica [C] time = 0.03, size = 61, normalized size = 0.43 \begin {gather*} \frac {x^2 (2 b B-3 A c) \, _2F_1\left (-\frac {1}{2},1;\frac {1}{2};\frac {c x^2}{b}+1\right )-A b}{2 b^2 x \sqrt {x^2 \left (b+c x^2\right )}} \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [A] time = 0.88, size = 95, normalized size = 0.67 \begin {gather*} \frac {(3 A c-2 b B) \tanh ^{-1}\left (\frac {\sqrt {b} x}{\sqrt {b x^2+c x^4}}\right )}{2 b^{5/2}}+\frac {\sqrt {b x^2+c x^4} \left (-A b-3 A c x^2+2 b B x^2\right )}{2 b^2 x^3 \left (b+c x^2\right )} \end {gather*}
Antiderivative was successfully verified.
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fricas [A] time = 0.44, size = 260, normalized size = 1.83 \begin {gather*} \left [-\frac {{\left ({\left (2 \, B b c - 3 \, A c^{2}\right )} x^{5} + {\left (2 \, B b^{2} - 3 \, A b c\right )} x^{3}\right )} \sqrt {b} \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 (A b^{2} - {\left (2 \, B b^{2} - 3 \, A b c\right )} x^{2}\right )}}{4 \, {\left (b^{3} c x^{5} + b^{4} x^{3}\right )}}, \frac {{\left ({\left (2 \, B b c - 3 \, A c^{2}\right )} x^{5} + {\left (2 \, B b^{2} - 3 \, A b c\right )} x^{3}\right )} \sqrt {-b} \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 (A b^{2} - {\left (2 \, B b^{2} - 3 \, A b c\right )} x^{2}\right )}}{2 \, {\left (b^{3} c x^{5} + b^{4} x^{3}\right )}}\right ] \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [F] time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \mathit {sage}_{0} x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.06, size = 129, normalized size = 0.91 \begin {gather*} -\frac {\left (c \,x^{2}+b \right ) \left (-3 \sqrt {c \,x^{2}+b}\, A b c \,x^{2} \ln \left (\frac {2 b +2 \sqrt {c \,x^{2}+b}\, \sqrt {b}}{x}\right )+2 \sqrt {c \,x^{2}+b}\, B \,b^{2} x^{2} \ln \left (\frac {2 b +2 \sqrt {c \,x^{2}+b}\, \sqrt {b}}{x}\right )+3 A \,b^{\frac {3}{2}} c \,x^{2}-2 B \,b^{\frac {5}{2}} x^{2}+A \,b^{\frac {5}{2}}\right ) x}{2 \left (c \,x^{4}+b \,x^{2}\right )^{\frac {3}{2}} b^{\frac {7}{2}}} \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}{{\left (c x^{4} + b x^{2}\right )}^{\frac {3}{2}}}\,{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}{{\left (c\,x^4+b\,x^2\right )}^{3/2}} \,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}}{\left (x^{2} \left (b + c x^{2}\right )\right )^{\frac {3}{2}}}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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