Optimal. Leaf size=106 \[ \frac{x \left (a e^2-b d e+c d^2\right )}{2 d^3 \left (d+e x^2\right )}+\frac{\tan ^{-1}\left (\frac{\sqrt{e} x}{\sqrt{d}}\right ) \left (c d^2-e (3 b d-5 a e)\right )}{2 d^{7/2} \sqrt{e}}-\frac{b d-2 a e}{d^3 x}-\frac{a}{3 d^2 x^3} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.137395, antiderivative size = 106, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 25, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.12, Rules used = {1259, 1261, 205} \[ \frac{x \left (a e^2-b d e+c d^2\right )}{2 d^3 \left (d+e x^2\right )}+\frac{\tan ^{-1}\left (\frac{\sqrt{e} x}{\sqrt{d}}\right ) \left (c d^2-e (3 b d-5 a e)\right )}{2 d^{7/2} \sqrt{e}}-\frac{b d-2 a e}{d^3 x}-\frac{a}{3 d^2 x^3} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 1259
Rule 1261
Rule 205
Rubi steps
\begin{align*} \int \frac{a+b x^2+c x^4}{x^4 \left (d+e x^2\right )^2} \, dx &=\frac{\left (c d^2-b d e+a e^2\right ) x}{2 d^3 \left (d+e x^2\right )}+\frac{\int \frac{2 a d^2 e^2+2 d e^2 (b d-a e) x^2+e^2 \left (c d^2-b d e+a e^2\right ) x^4}{x^4 \left (d+e x^2\right )} \, dx}{2 d^3 e^2}\\ &=\frac{\left (c d^2-b d e+a e^2\right ) x}{2 d^3 \left (d+e x^2\right )}+\frac{\int \left (\frac{2 a d e^2}{x^4}-\frac{2 e^2 (-b d+2 a e)}{x^2}+\frac{e^2 \left (c d^2-e (3 b d-5 a e)\right )}{d+e x^2}\right ) \, dx}{2 d^3 e^2}\\ &=-\frac{a}{3 d^2 x^3}-\frac{b d-2 a e}{d^3 x}+\frac{\left (c d^2-b d e+a e^2\right ) x}{2 d^3 \left (d+e x^2\right )}+\frac{\left (c d^2-e (3 b d-5 a e)\right ) \int \frac{1}{d+e x^2} \, dx}{2 d^3}\\ &=-\frac{a}{3 d^2 x^3}-\frac{b d-2 a e}{d^3 x}+\frac{\left (c d^2-b d e+a e^2\right ) x}{2 d^3 \left (d+e x^2\right )}+\frac{\left (c d^2-e (3 b d-5 a e)\right ) \tan ^{-1}\left (\frac{\sqrt{e} x}{\sqrt{d}}\right )}{2 d^{7/2} \sqrt{e}}\\ \end{align*}
Mathematica [A] time = 0.065207, size = 105, normalized size = 0.99 \[ \frac{x \left (a e^2-b d e+c d^2\right )}{2 d^3 \left (d+e x^2\right )}+\frac{\tan ^{-1}\left (\frac{\sqrt{e} x}{\sqrt{d}}\right ) \left (5 a e^2-3 b d e+c d^2\right )}{2 d^{7/2} \sqrt{e}}+\frac{2 a e-b d}{d^3 x}-\frac{a}{3 d^2 x^3} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [A] time = 0.014, size = 146, normalized size = 1.4 \begin{align*} -{\frac{a}{3\,{d}^{2}{x}^{3}}}+2\,{\frac{ae}{{d}^{3}x}}-{\frac{b}{{d}^{2}x}}+{\frac{xa{e}^{2}}{2\,{d}^{3} \left ( e{x}^{2}+d \right ) }}-{\frac{exb}{2\,{d}^{2} \left ( e{x}^{2}+d \right ) }}+{\frac{xc}{2\,d \left ( e{x}^{2}+d \right ) }}+{\frac{5\,a{e}^{2}}{2\,{d}^{3}}\arctan \left ({ex{\frac{1}{\sqrt{de}}}} \right ){\frac{1}{\sqrt{de}}}}-{\frac{3\,be}{2\,{d}^{2}}\arctan \left ({ex{\frac{1}{\sqrt{de}}}} \right ){\frac{1}{\sqrt{de}}}}+{\frac{c}{2\,d}\arctan \left ({ex{\frac{1}{\sqrt{de}}}} \right ){\frac{1}{\sqrt{de}}}} \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 = 1.86962, size = 672, normalized size = 6.34 \begin{align*} \left [-\frac{4 \, a d^{3} e - 6 \,{\left (c d^{3} e - 3 \, b d^{2} e^{2} + 5 \, a d e^{3}\right )} x^{4} + 4 \,{\left (3 \, b d^{3} e - 5 \, a d^{2} e^{2}\right )} x^{2} + 3 \,{\left ({\left (c d^{2} e - 3 \, b d e^{2} + 5 \, a e^{3}\right )} x^{5} +{\left (c d^{3} - 3 \, b d^{2} e + 5 \, a d e^{2}\right )} x^{3}\right )} \sqrt{-d e} \log \left (\frac{e x^{2} - 2 \, \sqrt{-d e} x - d}{e x^{2} + d}\right )}{12 \,{\left (d^{4} e^{2} x^{5} + d^{5} e x^{3}\right )}}, -\frac{2 \, a d^{3} e - 3 \,{\left (c d^{3} e - 3 \, b d^{2} e^{2} + 5 \, a d e^{3}\right )} x^{4} + 2 \,{\left (3 \, b d^{3} e - 5 \, a d^{2} e^{2}\right )} x^{2} - 3 \,{\left ({\left (c d^{2} e - 3 \, b d e^{2} + 5 \, a e^{3}\right )} x^{5} +{\left (c d^{3} - 3 \, b d^{2} e + 5 \, a d e^{2}\right )} x^{3}\right )} \sqrt{d e} \arctan \left (\frac{\sqrt{d e} x}{d}\right )}{6 \,{\left (d^{4} e^{2} x^{5} + d^{5} e x^{3}\right )}}\right ] \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Sympy [A] time = 1.73317, size = 167, normalized size = 1.58 \begin{align*} - \frac{\sqrt{- \frac{1}{d^{7} e}} \left (5 a e^{2} - 3 b d e + c d^{2}\right ) \log{\left (- d^{4} \sqrt{- \frac{1}{d^{7} e}} + x \right )}}{4} + \frac{\sqrt{- \frac{1}{d^{7} e}} \left (5 a e^{2} - 3 b d e + c d^{2}\right ) \log{\left (d^{4} \sqrt{- \frac{1}{d^{7} e}} + x \right )}}{4} + \frac{- 2 a d^{2} + x^{4} \left (15 a e^{2} - 9 b d e + 3 c d^{2}\right ) + x^{2} \left (10 a d e - 6 b d^{2}\right )}{6 d^{4} x^{3} + 6 d^{3} e x^{5}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Giac [A] time = 1.08961, size = 127, normalized size = 1.2 \begin{align*} \frac{{\left (c d^{2} - 3 \, b d e + 5 \, a e^{2}\right )} \arctan \left (\frac{x e^{\frac{1}{2}}}{\sqrt{d}}\right ) e^{\left (-\frac{1}{2}\right )}}{2 \, d^{\frac{7}{2}}} + \frac{c d^{2} x - b d x e + a x e^{2}}{2 \,{\left (x^{2} e + d\right )} d^{3}} - \frac{3 \, b d x^{2} - 6 \, a x^{2} e + a d}{3 \, d^{3} x^{3}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]