Optimal. Leaf size=201 \[ \frac{\tan ^{-1}\left (\frac{\sqrt{e} x}{\sqrt{d}}\right ) \left (e^2 \left (3 a^2 e^2+2 a b d e+3 b^2 d^2\right )-6 c d^2 e (5 b d-a e)+35 c^2 d^4\right )}{8 d^{5/2} e^{9/2}}-\frac{x \left (-3 a e^2-5 b d e+13 c d^2\right ) \left (a e^2-b d e+c d^2\right )}{8 d^2 e^4 \left (d+e x^2\right )}+\frac{x \left (a e^2-b d e+c d^2\right )^2}{4 d e^4 \left (d+e x^2\right )^2}-\frac{c x (3 c d-2 b e)}{e^4}+\frac{c^2 x^3}{3 e^3} \]
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Rubi [A] time = 0.418775, antiderivative size = 201, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 4, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.167, Rules used = {1157, 1814, 1153, 205} \[ \frac{\tan ^{-1}\left (\frac{\sqrt{e} x}{\sqrt{d}}\right ) \left (e^2 \left (3 a^2 e^2+2 a b d e+3 b^2 d^2\right )-6 c d^2 e (5 b d-a e)+35 c^2 d^4\right )}{8 d^{5/2} e^{9/2}}-\frac{x \left (-3 a e^2-5 b d e+13 c d^2\right ) \left (a e^2-b d e+c d^2\right )}{8 d^2 e^4 \left (d+e x^2\right )}+\frac{x \left (a e^2-b d e+c d^2\right )^2}{4 d e^4 \left (d+e x^2\right )^2}-\frac{c x (3 c d-2 b e)}{e^4}+\frac{c^2 x^3}{3 e^3} \]
Antiderivative was successfully verified.
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Rule 1157
Rule 1814
Rule 1153
Rule 205
Rubi steps
\begin{align*} \int \frac{\left (a+b x^2+c x^4\right )^2}{\left (d+e x^2\right )^3} \, dx &=\frac{\left (c d^2-b d e+a e^2\right )^2 x}{4 d e^4 \left (d+e x^2\right )^2}-\frac{\int \frac{\frac{\left (c d^2-b d e-a e^2\right ) \left (c d^2-b d e+3 a e^2\right )}{e^4}-\frac{4 d \left (c^2 d^2+b^2 e^2-2 c e (b d-a e)\right ) x^2}{e^3}+\frac{4 c d (c d-2 b e) x^4}{e^2}-\frac{4 c^2 d x^6}{e}}{\left (d+e x^2\right )^2} \, dx}{4 d}\\ &=\frac{\left (c d^2-b d e+a e^2\right )^2 x}{4 d e^4 \left (d+e x^2\right )^2}-\frac{\left (13 c d^2-5 b d e-3 a e^2\right ) \left (c d^2-b d e+a e^2\right ) x}{8 d^2 e^4 \left (d+e x^2\right )}+\frac{\int \frac{\frac{11 c^2 d^4-2 c d^2 e (7 b d-3 a e)+e^2 \left (3 b^2 d^2+2 a b d e+3 a^2 e^2\right )}{e^4}-\frac{16 c d^2 (c d-b e) x^2}{e^3}+\frac{8 c^2 d^2 x^4}{e^2}}{d+e x^2} \, dx}{8 d^2}\\ &=\frac{\left (c d^2-b d e+a e^2\right )^2 x}{4 d e^4 \left (d+e x^2\right )^2}-\frac{\left (13 c d^2-5 b d e-3 a e^2\right ) \left (c d^2-b d e+a e^2\right ) x}{8 d^2 e^4 \left (d+e x^2\right )}+\frac{\int \left (-\frac{8 c d^2 (3 c d-2 b e)}{e^4}+\frac{8 c^2 d^2 x^2}{e^3}+\frac{35 c^2 d^4-30 b c d^3 e+3 b^2 d^2 e^2+6 a c d^2 e^2+2 a b d e^3+3 a^2 e^4}{e^4 \left (d+e x^2\right )}\right ) \, dx}{8 d^2}\\ &=-\frac{c (3 c d-2 b e) x}{e^4}+\frac{c^2 x^3}{3 e^3}+\frac{\left (c d^2-b d e+a e^2\right )^2 x}{4 d e^4 \left (d+e x^2\right )^2}-\frac{\left (13 c d^2-5 b d e-3 a e^2\right ) \left (c d^2-b d e+a e^2\right ) x}{8 d^2 e^4 \left (d+e x^2\right )}+\frac{\left (35 c^2 d^4-6 c d^2 e (5 b d-a e)+e^2 \left (3 b^2 d^2+2 a b d e+3 a^2 e^2\right )\right ) \int \frac{1}{d+e x^2} \, dx}{8 d^2 e^4}\\ &=-\frac{c (3 c d-2 b e) x}{e^4}+\frac{c^2 x^3}{3 e^3}+\frac{\left (c d^2-b d e+a e^2\right )^2 x}{4 d e^4 \left (d+e x^2\right )^2}-\frac{\left (13 c d^2-5 b d e-3 a e^2\right ) \left (c d^2-b d e+a e^2\right ) x}{8 d^2 e^4 \left (d+e x^2\right )}+\frac{\left (35 c^2 d^4-6 c d^2 e (5 b d-a e)+e^2 \left (3 b^2 d^2+2 a b d e+3 a^2 e^2\right )\right ) \tan ^{-1}\left (\frac{\sqrt{e} x}{\sqrt{d}}\right )}{8 d^{5/2} e^{9/2}}\\ \end{align*}
Mathematica [A] time = 0.114594, size = 217, normalized size = 1.08 \[ -\frac{x \left (e^2 \left (-3 a^2 e^2-2 a b d e+5 b^2 d^2\right )-2 c d^2 e (9 b d-5 a e)+13 c^2 d^4\right )}{8 d^2 e^4 \left (d+e x^2\right )}+\frac{\tan ^{-1}\left (\frac{\sqrt{e} x}{\sqrt{d}}\right ) \left (e^2 \left (3 a^2 e^2+2 a b d e+3 b^2 d^2\right )+6 c d^2 e (a e-5 b d)+35 c^2 d^4\right )}{8 d^{5/2} e^{9/2}}+\frac{x \left (e (a e-b d)+c d^2\right )^2}{4 d e^4 \left (d+e x^2\right )^2}+\frac{c x (2 b e-3 c d)}{e^4}+\frac{c^2 x^3}{3 e^3} \]
Antiderivative was successfully verified.
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Maple [B] time = 0.012, size = 402, normalized size = 2. \begin{align*}{\frac{{c}^{2}{x}^{3}}{3\,{e}^{3}}}+2\,{\frac{bcx}{{e}^{3}}}-3\,{\frac{d{c}^{2}x}{{e}^{4}}}+{\frac{3\,{a}^{2}e{x}^{3}}{8\, \left ( e{x}^{2}+d \right ) ^{2}{d}^{2}}}+{\frac{ab{x}^{3}}{4\, \left ( e{x}^{2}+d \right ) ^{2}d}}-{\frac{5\,{x}^{3}ac}{4\,e \left ( e{x}^{2}+d \right ) ^{2}}}-{\frac{5\,{x}^{3}{b}^{2}}{8\,e \left ( e{x}^{2}+d \right ) ^{2}}}+{\frac{9\,{x}^{3}bcd}{4\,{e}^{2} \left ( e{x}^{2}+d \right ) ^{2}}}-{\frac{13\,{c}^{2}{x}^{3}{d}^{2}}{8\,{e}^{3} \left ( e{x}^{2}+d \right ) ^{2}}}+{\frac{5\,{a}^{2}x}{8\, \left ( e{x}^{2}+d \right ) ^{2}d}}-{\frac{abx}{4\,e \left ( e{x}^{2}+d \right ) ^{2}}}-{\frac{3\,acdx}{4\,{e}^{2} \left ( e{x}^{2}+d \right ) ^{2}}}-{\frac{3\,{b}^{2}dx}{8\,{e}^{2} \left ( e{x}^{2}+d \right ) ^{2}}}+{\frac{7\,bc{d}^{2}x}{4\,{e}^{3} \left ( e{x}^{2}+d \right ) ^{2}}}-{\frac{11\,{c}^{2}{d}^{3}x}{8\,{e}^{4} \left ( e{x}^{2}+d \right ) ^{2}}}+{\frac{3\,{a}^{2}}{8\,{d}^{2}}\arctan \left ({ex{\frac{1}{\sqrt{de}}}} \right ){\frac{1}{\sqrt{de}}}}+{\frac{ab}{4\,de}\arctan \left ({ex{\frac{1}{\sqrt{de}}}} \right ){\frac{1}{\sqrt{de}}}}+{\frac{3\,ac}{4\,{e}^{2}}\arctan \left ({ex{\frac{1}{\sqrt{de}}}} \right ){\frac{1}{\sqrt{de}}}}+{\frac{3\,{b}^{2}}{8\,{e}^{2}}\arctan \left ({ex{\frac{1}{\sqrt{de}}}} \right ){\frac{1}{\sqrt{de}}}}-{\frac{15\,bcd}{4\,{e}^{3}}\arctan \left ({ex{\frac{1}{\sqrt{de}}}} \right ){\frac{1}{\sqrt{de}}}}+{\frac{35\,{c}^{2}{d}^{2}}{8\,{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 [B] time = 1.72383, size = 1694, normalized size = 8.43 \begin{align*} \text{result too large to display} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 17.1049, size = 396, normalized size = 1.97 \begin{align*} \frac{c^{2} x^{3}}{3 e^{3}} - \frac{\sqrt{- \frac{1}{d^{5} e^{9}}} \left (3 a^{2} e^{4} + 2 a b d e^{3} + 6 a c d^{2} e^{2} + 3 b^{2} d^{2} e^{2} - 30 b c d^{3} e + 35 c^{2} d^{4}\right ) \log{\left (- d^{3} e^{4} \sqrt{- \frac{1}{d^{5} e^{9}}} + x \right )}}{16} + \frac{\sqrt{- \frac{1}{d^{5} e^{9}}} \left (3 a^{2} e^{4} + 2 a b d e^{3} + 6 a c d^{2} e^{2} + 3 b^{2} d^{2} e^{2} - 30 b c d^{3} e + 35 c^{2} d^{4}\right ) \log{\left (d^{3} e^{4} \sqrt{- \frac{1}{d^{5} e^{9}}} + x \right )}}{16} + \frac{x^{3} \left (3 a^{2} e^{5} + 2 a b d e^{4} - 10 a c d^{2} e^{3} - 5 b^{2} d^{2} e^{3} + 18 b c d^{3} e^{2} - 13 c^{2} d^{4} e\right ) + x \left (5 a^{2} d e^{4} - 2 a b d^{2} e^{3} - 6 a c d^{3} e^{2} - 3 b^{2} d^{3} e^{2} + 14 b c d^{4} e - 11 c^{2} d^{5}\right )}{8 d^{4} e^{4} + 16 d^{3} e^{5} x^{2} + 8 d^{2} e^{6} x^{4}} + \frac{x \left (2 b c e - 3 c^{2} d\right )}{e^{4}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.1325, size = 329, normalized size = 1.64 \begin{align*} \frac{1}{3} \,{\left (c^{2} x^{3} e^{6} - 9 \, c^{2} d x e^{5} + 6 \, b c x e^{6}\right )} e^{\left (-9\right )} + \frac{{\left (35 \, c^{2} d^{4} - 30 \, b c d^{3} e + 3 \, b^{2} d^{2} e^{2} + 6 \, a c d^{2} e^{2} + 2 \, a b d e^{3} + 3 \, a^{2} e^{4}\right )} \arctan \left (\frac{x e^{\frac{1}{2}}}{\sqrt{d}}\right ) e^{\left (-\frac{9}{2}\right )}}{8 \, d^{\frac{5}{2}}} - \frac{{\left (13 \, c^{2} d^{4} x^{3} e - 18 \, b c d^{3} x^{3} e^{2} + 11 \, c^{2} d^{5} x + 5 \, b^{2} d^{2} x^{3} e^{3} + 10 \, a c d^{2} x^{3} e^{3} - 14 \, b c d^{4} x e - 2 \, a b d x^{3} e^{4} + 3 \, b^{2} d^{3} x e^{2} + 6 \, a c d^{3} x e^{2} - 3 \, a^{2} x^{3} e^{5} + 2 \, a b d^{2} x e^{3} - 5 \, a^{2} d x e^{4}\right )} e^{\left (-4\right )}}{8 \,{\left (x^{2} e + d\right )}^{2} d^{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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