Optimal. Leaf size=80 \[ \frac{(A b-2 a B) \tanh ^{-1}\left (\frac{2 a+b x^2}{2 \sqrt{a} \sqrt{a+b x^2+c x^4}}\right )}{4 a^{3/2}}-\frac{A \sqrt{a+b x^2+c x^4}}{2 a x^2} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.0828435, antiderivative size = 80, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 27, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.148, Rules used = {1251, 806, 724, 206} \[ \frac{(A b-2 a B) \tanh ^{-1}\left (\frac{2 a+b x^2}{2 \sqrt{a} \sqrt{a+b x^2+c x^4}}\right )}{4 a^{3/2}}-\frac{A \sqrt{a+b x^2+c x^4}}{2 a x^2} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 1251
Rule 806
Rule 724
Rule 206
Rubi steps
\begin{align*} \int \frac{A+B x^2}{x^3 \sqrt{a+b x^2+c x^4}} \, dx &=\frac{1}{2} \operatorname{Subst}\left (\int \frac{A+B x}{x^2 \sqrt{a+b x+c x^2}} \, dx,x,x^2\right )\\ &=-\frac{A \sqrt{a+b x^2+c x^4}}{2 a x^2}-\frac{(A b-2 a B) \operatorname{Subst}\left (\int \frac{1}{x \sqrt{a+b x+c x^2}} \, dx,x,x^2\right )}{4 a}\\ &=-\frac{A \sqrt{a+b x^2+c x^4}}{2 a x^2}+\frac{(A b-2 a B) \operatorname{Subst}\left (\int \frac{1}{4 a-x^2} \, dx,x,\frac{2 a+b x^2}{\sqrt{a+b x^2+c x^4}}\right )}{2 a}\\ &=-\frac{A \sqrt{a+b x^2+c x^4}}{2 a x^2}+\frac{(A b-2 a B) \tanh ^{-1}\left (\frac{2 a+b x^2}{2 \sqrt{a} \sqrt{a+b x^2+c x^4}}\right )}{4 a^{3/2}}\\ \end{align*}
Mathematica [A] time = 0.0364715, size = 82, normalized size = 1.02 \[ \frac{1}{2} \left (\frac{(A b-2 a B) \tanh ^{-1}\left (\frac{2 a+b x^2}{2 \sqrt{a} \sqrt{a+b x^2+c x^4}}\right )}{2 a^{3/2}}-\frac{A \sqrt{a+b x^2+c x^4}}{a x^2}\right ) \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [A] time = 0.016, size = 104, normalized size = 1.3 \begin{align*} -{\frac{B}{2}\ln \left ({\frac{1}{{x}^{2}} \left ( 2\,a+b{x}^{2}+2\,\sqrt{a}\sqrt{c{x}^{4}+b{x}^{2}+a} \right ) } \right ){\frac{1}{\sqrt{a}}}}-{\frac{A}{2\,{x}^{2}a}\sqrt{c{x}^{4}+b{x}^{2}+a}}+{\frac{Ab}{4}\ln \left ({\frac{1}{{x}^{2}} \left ( 2\,a+b{x}^{2}+2\,\sqrt{a}\sqrt{c{x}^{4}+b{x}^{2}+a} \right ) } \right ){a}^{-{\frac{3}{2}}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Maxima [F(-2)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: ValueError} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Fricas [A] time = 1.77008, size = 460, normalized size = 5.75 \begin{align*} \left [-\frac{{\left (2 \, B a - A b\right )} \sqrt{a} x^{2} \log \left (-\frac{{\left (b^{2} + 4 \, a c\right )} x^{4} + 8 \, a b x^{2} + 4 \, \sqrt{c x^{4} + b x^{2} + a}{\left (b x^{2} + 2 \, a\right )} \sqrt{a} + 8 \, a^{2}}{x^{4}}\right ) + 4 \, \sqrt{c x^{4} + b x^{2} + a} A a}{8 \, a^{2} x^{2}}, \frac{{\left (2 \, B a - A b\right )} \sqrt{-a} x^{2} \arctan \left (\frac{\sqrt{c x^{4} + b x^{2} + a}{\left (b x^{2} + 2 \, a\right )} \sqrt{-a}}{2 \,{\left (a c x^{4} + a b x^{2} + a^{2}\right )}}\right ) - 2 \, \sqrt{c x^{4} + b x^{2} + a} A a}{4 \, a^{2} x^{2}}\right ] \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Sympy [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{A + B x^{2}}{x^{3} \sqrt{a + b x^{2} + c x^{4}}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Giac [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{B x^{2} + A}{\sqrt{c x^{4} + b x^{2} + a} x^{3}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]