Optimal. Leaf size=135 \[ \frac{d x \left (a e^2-b d e+c d^2\right )}{2 e^4 \left (d+e x^2\right )}+\frac{x \left (3 c d^2-e (2 b d-a e)\right )}{e^4}-\frac{\sqrt{d} \tan ^{-1}\left (\frac{\sqrt{e} x}{\sqrt{d}}\right ) \left (7 c d^2-e (5 b d-3 a e)\right )}{2 e^{9/2}}-\frac{x^3 (2 c d-b e)}{3 e^3}+\frac{c x^5}{5 e^2} \]
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Rubi [A] time = 0.159071, antiderivative size = 135, 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 = {1257, 1810, 205} \[ \frac{d x \left (a e^2-b d e+c d^2\right )}{2 e^4 \left (d+e x^2\right )}+\frac{x \left (3 c d^2-e (2 b d-a e)\right )}{e^4}-\frac{\sqrt{d} \tan ^{-1}\left (\frac{\sqrt{e} x}{\sqrt{d}}\right ) \left (7 c d^2-e (5 b d-3 a e)\right )}{2 e^{9/2}}-\frac{x^3 (2 c d-b e)}{3 e^3}+\frac{c x^5}{5 e^2} \]
Antiderivative was successfully verified.
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Rule 1257
Rule 1810
Rule 205
Rubi steps
\begin{align*} \int \frac{x^4 \left (a+b x^2+c x^4\right )}{\left (d+e x^2\right )^2} \, dx &=\frac{d \left (c d^2-b d e+a e^2\right ) x}{2 e^4 \left (d+e x^2\right )}-\frac{\int \frac{d \left (c d^2-b d e+a e^2\right )-2 e \left (c d^2-b d e+a e^2\right ) x^2+2 e^2 (c d-b e) x^4-2 c e^3 x^6}{d+e x^2} \, dx}{2 e^4}\\ &=\frac{d \left (c d^2-b d e+a e^2\right ) x}{2 e^4 \left (d+e x^2\right )}-\frac{\int \left (-2 \left (3 c d^2-2 b d e+a e^2\right )+2 e (2 c d-b e) x^2-2 c e^2 x^4+\frac{7 c d^3-5 b d^2 e+3 a d e^2}{d+e x^2}\right ) \, dx}{2 e^4}\\ &=\frac{\left (3 c d^2-e (2 b d-a e)\right ) x}{e^4}-\frac{(2 c d-b e) x^3}{3 e^3}+\frac{c x^5}{5 e^2}+\frac{d \left (c d^2-b d e+a e^2\right ) x}{2 e^4 \left (d+e x^2\right )}-\frac{\left (d \left (7 c d^2-e (5 b d-3 a e)\right )\right ) \int \frac{1}{d+e x^2} \, dx}{2 e^4}\\ &=\frac{\left (3 c d^2-e (2 b d-a e)\right ) x}{e^4}-\frac{(2 c d-b e) x^3}{3 e^3}+\frac{c x^5}{5 e^2}+\frac{d \left (c d^2-b d e+a e^2\right ) x}{2 e^4 \left (d+e x^2\right )}-\frac{\sqrt{d} \left (7 c d^2-e (5 b d-3 a e)\right ) \tan ^{-1}\left (\frac{\sqrt{e} x}{\sqrt{d}}\right )}{2 e^{9/2}}\\ \end{align*}
Mathematica [A] time = 0.0789524, size = 133, normalized size = 0.99 \[ \frac{x \left (a d e^2-b d^2 e+c d^3\right )}{2 e^4 \left (d+e x^2\right )}+\frac{x \left (a e^2-2 b d e+3 c d^2\right )}{e^4}-\frac{\sqrt{d} \tan ^{-1}\left (\frac{\sqrt{e} x}{\sqrt{d}}\right ) \left (3 a e^2-5 b d e+7 c d^2\right )}{2 e^{9/2}}+\frac{x^3 (b e-2 c d)}{3 e^3}+\frac{c x^5}{5 e^2} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.012, size = 176, normalized size = 1.3 \begin{align*}{\frac{c{x}^{5}}{5\,{e}^{2}}}+{\frac{{x}^{3}b}{3\,{e}^{2}}}-{\frac{2\,{x}^{3}cd}{3\,{e}^{3}}}+{\frac{ax}{{e}^{2}}}-2\,{\frac{bdx}{{e}^{3}}}+3\,{\frac{c{d}^{2}x}{{e}^{4}}}+{\frac{dxa}{2\,{e}^{2} \left ( e{x}^{2}+d \right ) }}-{\frac{{d}^{2}bx}{2\,{e}^{3} \left ( e{x}^{2}+d \right ) }}+{\frac{{d}^{3}xc}{2\,{e}^{4} \left ( e{x}^{2}+d \right ) }}-{\frac{3\,ad}{2\,{e}^{2}}\arctan \left ({ex{\frac{1}{\sqrt{de}}}} \right ){\frac{1}{\sqrt{de}}}}+{\frac{5\,{d}^{2}b}{2\,{e}^{3}}\arctan \left ({ex{\frac{1}{\sqrt{de}}}} \right ){\frac{1}{\sqrt{de}}}}-{\frac{7\,c{d}^{3}}{2\,{e}^{4}}\arctan \left ({ex{\frac{1}{\sqrt{de}}}} \right ){\frac{1}{\sqrt{de}}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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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.
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Fricas [A] time = 1.82637, size = 756, normalized size = 5.6 \begin{align*} \left [\frac{12 \, c e^{3} x^{7} - 4 \,{\left (7 \, c d e^{2} - 5 \, b e^{3}\right )} x^{5} + 20 \,{\left (7 \, c d^{2} e - 5 \, b d e^{2} + 3 \, a e^{3}\right )} x^{3} + 15 \,{\left (7 \, c d^{3} - 5 \, b d^{2} e + 3 \, a d e^{2} +{\left (7 \, c d^{2} e - 5 \, b d e^{2} + 3 \, a e^{3}\right )} x^{2}\right )} \sqrt{-\frac{d}{e}} \log \left (\frac{e x^{2} - 2 \, e x \sqrt{-\frac{d}{e}} - d}{e x^{2} + d}\right ) + 30 \,{\left (7 \, c d^{3} - 5 \, b d^{2} e + 3 \, a d e^{2}\right )} x}{60 \,{\left (e^{5} x^{2} + d e^{4}\right )}}, \frac{6 \, c e^{3} x^{7} - 2 \,{\left (7 \, c d e^{2} - 5 \, b e^{3}\right )} x^{5} + 10 \,{\left (7 \, c d^{2} e - 5 \, b d e^{2} + 3 \, a e^{3}\right )} x^{3} - 15 \,{\left (7 \, c d^{3} - 5 \, b d^{2} e + 3 \, a d e^{2} +{\left (7 \, c d^{2} e - 5 \, b d e^{2} + 3 \, a e^{3}\right )} x^{2}\right )} \sqrt{\frac{d}{e}} \arctan \left (\frac{e x \sqrt{\frac{d}{e}}}{d}\right ) + 15 \,{\left (7 \, c d^{3} - 5 \, b d^{2} e + 3 \, a d e^{2}\right )} x}{30 \,{\left (e^{5} x^{2} + d e^{4}\right )}}\right ] \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 1.40457, size = 184, normalized size = 1.36 \begin{align*} \frac{c x^{5}}{5 e^{2}} + \frac{x \left (a d e^{2} - b d^{2} e + c d^{3}\right )}{2 d e^{4} + 2 e^{5} x^{2}} + \frac{\sqrt{- \frac{d}{e^{9}}} \left (3 a e^{2} - 5 b d e + 7 c d^{2}\right ) \log{\left (- e^{4} \sqrt{- \frac{d}{e^{9}}} + x \right )}}{4} - \frac{\sqrt{- \frac{d}{e^{9}}} \left (3 a e^{2} - 5 b d e + 7 c d^{2}\right ) \log{\left (e^{4} \sqrt{- \frac{d}{e^{9}}} + x \right )}}{4} + \frac{x^{3} \left (b e - 2 c d\right )}{3 e^{3}} + \frac{x \left (a e^{2} - 2 b d e + 3 c d^{2}\right )}{e^{4}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.11209, size = 169, normalized size = 1.25 \begin{align*} -\frac{{\left (7 \, c d^{3} - 5 \, b d^{2} e + 3 \, a d e^{2}\right )} \arctan \left (\frac{x e^{\frac{1}{2}}}{\sqrt{d}}\right ) e^{\left (-\frac{9}{2}\right )}}{2 \, \sqrt{d}} + \frac{1}{15} \,{\left (3 \, c x^{5} e^{8} - 10 \, c d x^{3} e^{7} + 5 \, b x^{3} e^{8} + 45 \, c d^{2} x e^{6} - 30 \, b d x e^{7} + 15 \, a x e^{8}\right )} e^{\left (-10\right )} + \frac{{\left (c d^{3} x - b d^{2} x e + a d x e^{2}\right )} e^{\left (-4\right )}}{2 \,{\left (x^{2} e + d\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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