Optimal. Leaf size=68 \[ -\frac {(2 b B-A c) \tanh ^{-1}\left (\frac {\sqrt {b} x}{\sqrt {b x^2+c x^4}}\right )}{2 b^{3/2}}-\frac {A \sqrt {b x^2+c x^4}}{2 b x^3} \]
________________________________________________________________________________________
Rubi [A] time = 0.12, antiderivative size = 68, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, integrand size = 26, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.115, Rules used = {2038, 2008, 206} \begin {gather*} -\frac {(2 b B-A c) \tanh ^{-1}\left (\frac {\sqrt {b} x}{\sqrt {b x^2+c x^4}}\right )}{2 b^{3/2}}-\frac {A \sqrt {b x^2+c x^4}}{2 b x^3} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
Rule 206
Rule 2008
Rule 2038
Rubi steps
\begin {align*} \int \frac {A+B x^2}{x^2 \sqrt {b x^2+c x^4}} \, dx &=-\frac {A \sqrt {b x^2+c x^4}}{2 b x^3}-\frac {(-2 b B+A c) \int \frac {1}{\sqrt {b x^2+c x^4}} \, dx}{2 b}\\ &=-\frac {A \sqrt {b x^2+c x^4}}{2 b x^3}+\frac {(-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 )}{2 b}\\ &=-\frac {A \sqrt {b x^2+c x^4}}{2 b x^3}-\frac {(2 b B-A c) \tanh ^{-1}\left (\frac {\sqrt {b} x}{\sqrt {b x^2+c x^4}}\right )}{2 b^{3/2}}\\ \end {align*}
________________________________________________________________________________________
Mathematica [A] time = 0.09, size = 87, normalized size = 1.28 \begin {gather*} \frac {x \sqrt {b+c x^2} \left (-\frac {2 \left (b B-\frac {A c}{2}\right ) \tanh ^{-1}\left (\frac {\sqrt {b+c x^2}}{\sqrt {b}}\right )}{b^{3/2}}-\frac {A \sqrt {b+c x^2}}{b x^2}\right )}{2 \sqrt {x^2 \left (b+c x^2\right )}} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
IntegrateAlgebraic [A] time = 0.16, size = 67, normalized size = 0.99 \begin {gather*} \frac {(A c-2 b B) \tanh ^{-1}\left (\frac {\sqrt {b} x}{\sqrt {b x^2+c x^4}}\right )}{2 b^{3/2}}-\frac {A \sqrt {b x^2+c x^4}}{2 b x^3} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
fricas [A] time = 0.44, size = 152, normalized size = 2.24 \begin {gather*} \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}} A b}{4 \, b^{2} 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}} A b}{2 \, b^{2} x^{3}}\right ] \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
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.
[In]
[Out]
________________________________________________________________________________________
maple [A] time = 0.06, size = 105, normalized size = 1.54 \begin {gather*} -\frac {\sqrt {c \,x^{2}+b}\, \left (-A b c \,x^{2} \ln \left (\frac {2 b +2 \sqrt {c \,x^{2}+b}\, \sqrt {b}}{x}\right )+2 B \,b^{2} x^{2} \ln \left (\frac {2 b +2 \sqrt {c \,x^{2}+b}\, \sqrt {b}}{x}\right )+\sqrt {c \,x^{2}+b}\, A \,b^{\frac {3}{2}}\right )}{2 \sqrt {c \,x^{4}+b \,x^{2}}\, b^{\frac {5}{2}} x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
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^{2}}\,{d x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
mupad [F] time = 0.00, size = -1, normalized size = -0.01 \begin {gather*} \int \frac {B\,x^2+A}{x^2\,\sqrt {c\,x^4+b\,x^2}} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
sympy [F] time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {A + B x^{2}}{x^{2} \sqrt {x^{2} \left (b + c x^{2}\right )}}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________