Optimal. Leaf size=168 \[ \frac{x^3 \left (3 c d^2-e (2 b d-a e)\right )}{3 e^4}-\frac{d^2 x \left (a e^2-b d e+c d^2\right )}{2 e^5 \left (d+e x^2\right )}-\frac{d x \left (4 c d^2-e (3 b d-2 a e)\right )}{e^5}+\frac{d^{3/2} \tan ^{-1}\left (\frac{\sqrt{e} x}{\sqrt{d}}\right ) \left (9 c d^2-e (7 b d-5 a e)\right )}{2 e^{11/2}}-\frac{x^5 (2 c d-b e)}{5 e^3}+\frac{c x^7}{7 e^2} \]
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Rubi [A] time = 0.234486, antiderivative size = 168, 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} \[ \frac{x^3 \left (3 c d^2-e (2 b d-a e)\right )}{3 e^4}-\frac{d^2 x \left (a e^2-b d e+c d^2\right )}{2 e^5 \left (d+e x^2\right )}-\frac{d x \left (4 c d^2-e (3 b d-2 a e)\right )}{e^5}+\frac{d^{3/2} \tan ^{-1}\left (\frac{\sqrt{e} x}{\sqrt{d}}\right ) \left (9 c d^2-e (7 b d-5 a e)\right )}{2 e^{11/2}}-\frac{x^5 (2 c d-b e)}{5 e^3}+\frac{c x^7}{7 e^2} \]
Antiderivative was successfully verified.
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Rule 1257
Rule 1810
Rule 205
Rubi steps
\begin{align*} \int \frac{x^6 \left (a+b x^2+c x^4\right )}{\left (d+e x^2\right )^2} \, dx &=-\frac{d^2 \left (c d^2-b d e+a e^2\right ) x}{2 e^5 \left (d+e x^2\right )}-\frac{\int \frac{-d^2 \left (c d^2-b d e+a e^2\right )+2 d e \left (c d^2-b d e+a e^2\right ) x^2-2 e^2 \left (c d^2-b d e+a e^2\right ) x^4+2 e^3 (c d-b e) x^6-2 c e^4 x^8}{d+e x^2} \, dx}{2 e^5}\\ &=-\frac{d^2 \left (c d^2-b d e+a e^2\right ) x}{2 e^5 \left (d+e x^2\right )}-\frac{\int \left (2 d \left (4 c d^2-e (3 b d-2 a e)\right )-2 e \left (3 c d^2-e (2 b d-a e)\right ) x^2+2 e^2 (2 c d-b e) x^4-2 c e^3 x^6+\frac{-9 c d^4+7 b d^3 e-5 a d^2 e^2}{d+e x^2}\right ) \, dx}{2 e^5}\\ &=-\frac{d \left (4 c d^2-e (3 b d-2 a e)\right ) x}{e^5}+\frac{\left (3 c d^2-e (2 b d-a e)\right ) x^3}{3 e^4}-\frac{(2 c d-b e) x^5}{5 e^3}+\frac{c x^7}{7 e^2}-\frac{d^2 \left (c d^2-b d e+a e^2\right ) x}{2 e^5 \left (d+e x^2\right )}+\frac{\left (d^2 \left (9 c d^2-e (7 b d-5 a e)\right )\right ) \int \frac{1}{d+e x^2} \, dx}{2 e^5}\\ &=-\frac{d \left (4 c d^2-e (3 b d-2 a e)\right ) x}{e^5}+\frac{\left (3 c d^2-e (2 b d-a e)\right ) x^3}{3 e^4}-\frac{(2 c d-b e) x^5}{5 e^3}+\frac{c x^7}{7 e^2}-\frac{d^2 \left (c d^2-b d e+a e^2\right ) x}{2 e^5 \left (d+e x^2\right )}+\frac{d^{3/2} \left (9 c d^2-e (7 b d-5 a e)\right ) \tan ^{-1}\left (\frac{\sqrt{e} x}{\sqrt{d}}\right )}{2 e^{11/2}}\\ \end{align*}
Mathematica [A] time = 0.13674, size = 165, normalized size = 0.98 \[ \frac{x^3 \left (a e^2-2 b d e+3 c d^2\right )}{3 e^4}-\frac{x \left (a d^2 e^2-b d^3 e+c d^4\right )}{2 e^5 \left (d+e x^2\right )}-\frac{d x \left (2 a e^2-3 b d e+4 c d^2\right )}{e^5}+\frac{d^{3/2} \tan ^{-1}\left (\frac{\sqrt{e} x}{\sqrt{d}}\right ) \left (5 a e^2-7 b d e+9 c d^2\right )}{2 e^{11/2}}+\frac{x^5 (b e-2 c d)}{5 e^3}+\frac{c x^7}{7 e^2} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.01, size = 214, normalized size = 1.3 \begin{align*}{\frac{c{x}^{7}}{7\,{e}^{2}}}+{\frac{{x}^{5}b}{5\,{e}^{2}}}-{\frac{2\,{x}^{5}cd}{5\,{e}^{3}}}+{\frac{{x}^{3}a}{3\,{e}^{2}}}-{\frac{2\,{x}^{3}bd}{3\,{e}^{3}}}+{\frac{{x}^{3}c{d}^{2}}{{e}^{4}}}-2\,{\frac{adx}{{e}^{3}}}+3\,{\frac{{d}^{2}bx}{{e}^{4}}}-4\,{\frac{c{d}^{3}x}{{e}^{5}}}-{\frac{a{d}^{2}x}{2\,{e}^{3} \left ( e{x}^{2}+d \right ) }}+{\frac{{d}^{3}xb}{2\,{e}^{4} \left ( e{x}^{2}+d \right ) }}-{\frac{{d}^{4}xc}{2\,{e}^{5} \left ( e{x}^{2}+d \right ) }}+{\frac{5\,a{d}^{2}}{2\,{e}^{3}}\arctan \left ({ex{\frac{1}{\sqrt{de}}}} \right ){\frac{1}{\sqrt{de}}}}-{\frac{7\,b{d}^{3}}{2\,{e}^{4}}\arctan \left ({ex{\frac{1}{\sqrt{de}}}} \right ){\frac{1}{\sqrt{de}}}}+{\frac{9\,c{d}^{4}}{2\,{e}^{5}}\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 = 1.9179, size = 922, normalized size = 5.49 \begin{align*} \left [\frac{60 \, c e^{4} x^{9} - 12 \,{\left (9 \, c d e^{3} - 7 \, b e^{4}\right )} x^{7} + 28 \,{\left (9 \, c d^{2} e^{2} - 7 \, b d e^{3} + 5 \, a e^{4}\right )} x^{5} - 140 \,{\left (9 \, c d^{3} e - 7 \, b d^{2} e^{2} + 5 \, a d e^{3}\right )} x^{3} + 105 \,{\left (9 \, c d^{4} - 7 \, b d^{3} e + 5 \, a d^{2} e^{2} +{\left (9 \, c d^{3} e - 7 \, b d^{2} e^{2} + 5 \, a d 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 ) - 210 \,{\left (9 \, c d^{4} - 7 \, b d^{3} e + 5 \, a d^{2} e^{2}\right )} x}{420 \,{\left (e^{6} x^{2} + d e^{5}\right )}}, \frac{30 \, c e^{4} x^{9} - 6 \,{\left (9 \, c d e^{3} - 7 \, b e^{4}\right )} x^{7} + 14 \,{\left (9 \, c d^{2} e^{2} - 7 \, b d e^{3} + 5 \, a e^{4}\right )} x^{5} - 70 \,{\left (9 \, c d^{3} e - 7 \, b d^{2} e^{2} + 5 \, a d e^{3}\right )} x^{3} + 105 \,{\left (9 \, c d^{4} - 7 \, b d^{3} e + 5 \, a d^{2} e^{2} +{\left (9 \, c d^{3} e - 7 \, b d^{2} e^{2} + 5 \, a d e^{3}\right )} x^{2}\right )} \sqrt{\frac{d}{e}} \arctan \left (\frac{e x \sqrt{\frac{d}{e}}}{d}\right ) - 105 \,{\left (9 \, c d^{4} - 7 \, b d^{3} e + 5 \, a d^{2} e^{2}\right )} x}{210 \,{\left (e^{6} x^{2} + d e^{5}\right )}}\right ] \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [B] time = 1.50105, size = 313, normalized size = 1.86 \begin{align*} \frac{c x^{7}}{7 e^{2}} - \frac{x \left (a d^{2} e^{2} - b d^{3} e + c d^{4}\right )}{2 d e^{5} + 2 e^{6} x^{2}} - \frac{\sqrt{- \frac{d^{3}}{e^{11}}} \left (5 a e^{2} - 7 b d e + 9 c d^{2}\right ) \log{\left (- \frac{e^{5} \sqrt{- \frac{d^{3}}{e^{11}}} \left (5 a e^{2} - 7 b d e + 9 c d^{2}\right )}{5 a d e^{2} - 7 b d^{2} e + 9 c d^{3}} + x \right )}}{4} + \frac{\sqrt{- \frac{d^{3}}{e^{11}}} \left (5 a e^{2} - 7 b d e + 9 c d^{2}\right ) \log{\left (\frac{e^{5} \sqrt{- \frac{d^{3}}{e^{11}}} \left (5 a e^{2} - 7 b d e + 9 c d^{2}\right )}{5 a d e^{2} - 7 b d^{2} e + 9 c d^{3}} + x \right )}}{4} + \frac{x^{5} \left (b e - 2 c d\right )}{5 e^{3}} + \frac{x^{3} \left (a e^{2} - 2 b d e + 3 c d^{2}\right )}{3 e^{4}} - \frac{x \left (2 a d e^{2} - 3 b d^{2} e + 4 c d^{3}\right )}{e^{5}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.09757, size = 216, normalized size = 1.29 \begin{align*} \frac{{\left (9 \, c d^{4} - 7 \, b d^{3} e + 5 \, a d^{2} e^{2}\right )} \arctan \left (\frac{x e^{\frac{1}{2}}}{\sqrt{d}}\right ) e^{\left (-\frac{11}{2}\right )}}{2 \, \sqrt{d}} + \frac{1}{105} \,{\left (15 \, c x^{7} e^{12} - 42 \, c d x^{5} e^{11} + 21 \, b x^{5} e^{12} + 105 \, c d^{2} x^{3} e^{10} - 70 \, b d x^{3} e^{11} - 420 \, c d^{3} x e^{9} + 35 \, a x^{3} e^{12} + 315 \, b d^{2} x e^{10} - 210 \, a d x e^{11}\right )} e^{\left (-14\right )} - \frac{{\left (c d^{4} x - b d^{3} x e + a d^{2} x e^{2}\right )} e^{\left (-5\right )}}{2 \,{\left (x^{2} e + d\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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