Optimal. Leaf size=61 \[ -\frac{b B-A c}{b^2 x}-\frac{\sqrt{c} (b B-A c) \tan ^{-1}\left (\frac{\sqrt{c} x}{\sqrt{b}}\right )}{b^{5/2}}-\frac{A}{3 b x^3} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.0558791, antiderivative size = 61, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.167, Rules used = {1584, 453, 325, 205} \[ -\frac{b B-A c}{b^2 x}-\frac{\sqrt{c} (b B-A c) \tan ^{-1}\left (\frac{\sqrt{c} x}{\sqrt{b}}\right )}{b^{5/2}}-\frac{A}{3 b x^3} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 1584
Rule 453
Rule 325
Rule 205
Rubi steps
\begin{align*} \int \frac{A+B x^2}{x^2 \left (b x^2+c x^4\right )} \, dx &=\int \frac{A+B x^2}{x^4 \left (b+c x^2\right )} \, dx\\ &=-\frac{A}{3 b x^3}-\frac{(-3 b B+3 A c) \int \frac{1}{x^2 \left (b+c x^2\right )} \, dx}{3 b}\\ &=-\frac{A}{3 b x^3}-\frac{b B-A c}{b^2 x}-\frac{(c (b B-A c)) \int \frac{1}{b+c x^2} \, dx}{b^2}\\ &=-\frac{A}{3 b x^3}-\frac{b B-A c}{b^2 x}-\frac{\sqrt{c} (b B-A c) \tan ^{-1}\left (\frac{\sqrt{c} x}{\sqrt{b}}\right )}{b^{5/2}}\\ \end{align*}
Mathematica [A] time = 0.0525597, size = 60, normalized size = 0.98 \[ \frac{A c-b B}{b^2 x}-\frac{\sqrt{c} (b B-A c) \tan ^{-1}\left (\frac{\sqrt{c} x}{\sqrt{b}}\right )}{b^{5/2}}-\frac{A}{3 b x^3} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [A] time = 0.006, size = 72, normalized size = 1.2 \begin{align*} -{\frac{A}{3\,b{x}^{3}}}+{\frac{Ac}{{b}^{2}x}}-{\frac{B}{bx}}+{\frac{A{c}^{2}}{{b}^{2}}\arctan \left ({cx{\frac{1}{\sqrt{bc}}}} \right ){\frac{1}{\sqrt{bc}}}}-{\frac{cB}{b}\arctan \left ({cx{\frac{1}{\sqrt{bc}}}} \right ){\frac{1}{\sqrt{bc}}}} \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 = 0.86797, size = 296, normalized size = 4.85 \begin{align*} \left [-\frac{3 \,{\left (B b - A c\right )} x^{3} \sqrt{-\frac{c}{b}} \log \left (\frac{c x^{2} + 2 \, b x \sqrt{-\frac{c}{b}} - b}{c x^{2} + b}\right ) + 6 \,{\left (B b - A c\right )} x^{2} + 2 \, A b}{6 \, b^{2} x^{3}}, -\frac{3 \,{\left (B b - A c\right )} x^{3} \sqrt{\frac{c}{b}} \arctan \left (x \sqrt{\frac{c}{b}}\right ) + 3 \,{\left (B b - A c\right )} x^{2} + A b}{3 \, b^{2} x^{3}}\right ] \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Sympy [B] time = 0.56351, size = 129, normalized size = 2.11 \begin{align*} \frac{\sqrt{- \frac{c}{b^{5}}} \left (- A c + B b\right ) \log{\left (- \frac{b^{3} \sqrt{- \frac{c}{b^{5}}} \left (- A c + B b\right )}{- A c^{2} + B b c} + x \right )}}{2} - \frac{\sqrt{- \frac{c}{b^{5}}} \left (- A c + B b\right ) \log{\left (\frac{b^{3} \sqrt{- \frac{c}{b^{5}}} \left (- A c + B b\right )}{- A c^{2} + B b c} + x \right )}}{2} - \frac{A b + x^{2} \left (- 3 A c + 3 B b\right )}{3 b^{2} x^{3}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Giac [A] time = 1.27166, size = 77, normalized size = 1.26 \begin{align*} -\frac{{\left (B b c - A c^{2}\right )} \arctan \left (\frac{c x}{\sqrt{b c}}\right )}{\sqrt{b c} b^{2}} - \frac{3 \, B b x^{2} - 3 \, A c x^{2} + A b}{3 \, b^{2} x^{3}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]