Optimal. Leaf size=250 \[ \frac{x \left (e^2 \left (5 a^2 e^2+2 a b d e+b^2 d^2\right )-2 c d^2 e (11 b d-a e)+29 c^2 d^4\right )}{16 d^3 e^4 \left (d+e x^2\right )}-\frac{\tan ^{-1}\left (\frac{\sqrt{e} x}{\sqrt{d}}\right ) \left (-e^2 \left (5 a^2 e^2+2 a b d e+b^2 d^2\right )-2 c d^2 e (a e+5 b d)+35 c^2 d^4\right )}{16 d^{7/2} e^{9/2}}-\frac{x \left (-5 a e^2-7 b d e+19 c d^2\right ) \left (a e^2-b d e+c d^2\right )}{24 d^2 e^4 \left (d+e x^2\right )^2}+\frac{x \left (a e^2-b d e+c d^2\right )^2}{6 d e^4 \left (d+e x^2\right )^3}+\frac{c^2 x}{e^4} \]
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Rubi [A] time = 0.542902, antiderivative size = 250, 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, 388, 205} \[ \frac{x \left (e^2 \left (5 a^2 e^2+2 a b d e+b^2 d^2\right )-2 c d^2 e (11 b d-a e)+29 c^2 d^4\right )}{16 d^3 e^4 \left (d+e x^2\right )}-\frac{\tan ^{-1}\left (\frac{\sqrt{e} x}{\sqrt{d}}\right ) \left (-e^2 \left (5 a^2 e^2+2 a b d e+b^2 d^2\right )-2 c d^2 e (a e+5 b d)+35 c^2 d^4\right )}{16 d^{7/2} e^{9/2}}-\frac{x \left (-5 a e^2-7 b d e+19 c d^2\right ) \left (a e^2-b d e+c d^2\right )}{24 d^2 e^4 \left (d+e x^2\right )^2}+\frac{x \left (a e^2-b d e+c d^2\right )^2}{6 d e^4 \left (d+e x^2\right )^3}+\frac{c^2 x}{e^4} \]
Antiderivative was successfully verified.
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Rule 1157
Rule 1814
Rule 388
Rule 205
Rubi steps
\begin{align*} \int \frac{\left (a+b x^2+c x^4\right )^2}{\left (d+e x^2\right )^4} \, dx &=\frac{\left (c d^2-b d e+a e^2\right )^2 x}{6 d e^4 \left (d+e x^2\right )^3}-\frac{\int \frac{\frac{c^2 d^4-2 c d^2 e (b d-a e)+e^2 \left (b^2 d^2-2 a b d e-5 a^2 e^2\right )}{e^4}-\frac{6 d \left (c^2 d^2+b^2 e^2-2 c e (b d-a e)\right ) x^2}{e^3}+\frac{6 c d (c d-2 b e) x^4}{e^2}-\frac{6 c^2 d x^6}{e}}{\left (d+e x^2\right )^3} \, dx}{6 d}\\ &=\frac{\left (c d^2-b d e+a e^2\right )^2 x}{6 d e^4 \left (d+e x^2\right )^3}-\frac{\left (19 c d^2-7 b d e-5 a e^2\right ) \left (c d^2-b d e+a e^2\right ) x}{24 d^2 e^4 \left (d+e x^2\right )^2}+\frac{\int \frac{\frac{3 \left (5 c^2 d^4-2 c d^2 e (3 b d-a e)+e^2 \left (b^2 d^2+2 a b d e+5 a^2 e^2\right )\right )}{e^4}-\frac{48 c d^2 (c d-b e) x^2}{e^3}+\frac{24 c^2 d^2 x^4}{e^2}}{\left (d+e x^2\right )^2} \, dx}{24 d^2}\\ &=\frac{\left (c d^2-b d e+a e^2\right )^2 x}{6 d e^4 \left (d+e x^2\right )^3}-\frac{\left (19 c d^2-7 b d e-5 a e^2\right ) \left (c d^2-b d e+a e^2\right ) x}{24 d^2 e^4 \left (d+e x^2\right )^2}+\frac{\left (29 c^2 d^4-2 c d^2 e (11 b d-a e)+e^2 \left (b^2 d^2+2 a b d e+5 a^2 e^2\right )\right ) x}{16 d^3 e^4 \left (d+e x^2\right )}-\frac{\int \frac{\frac{3 \left (19 c^2 d^4-2 c d^2 e (5 b d+a e)-e^2 \left (b^2 d^2+2 a b d e+5 a^2 e^2\right )\right )}{e^4}-\frac{48 c^2 d^3 x^2}{e^3}}{d+e x^2} \, dx}{48 d^3}\\ &=\frac{c^2 x}{e^4}+\frac{\left (c d^2-b d e+a e^2\right )^2 x}{6 d e^4 \left (d+e x^2\right )^3}-\frac{\left (19 c d^2-7 b d e-5 a e^2\right ) \left (c d^2-b d e+a e^2\right ) x}{24 d^2 e^4 \left (d+e x^2\right )^2}+\frac{\left (29 c^2 d^4-2 c d^2 e (11 b d-a e)+e^2 \left (b^2 d^2+2 a b d e+5 a^2 e^2\right )\right ) x}{16 d^3 e^4 \left (d+e x^2\right )}-\frac{\left (35 c^2 d^4-2 c d^2 e (5 b d+a e)-e^2 \left (b^2 d^2+2 a b d e+5 a^2 e^2\right )\right ) \int \frac{1}{d+e x^2} \, dx}{16 d^3 e^4}\\ &=\frac{c^2 x}{e^4}+\frac{\left (c d^2-b d e+a e^2\right )^2 x}{6 d e^4 \left (d+e x^2\right )^3}-\frac{\left (19 c d^2-7 b d e-5 a e^2\right ) \left (c d^2-b d e+a e^2\right ) x}{24 d^2 e^4 \left (d+e x^2\right )^2}+\frac{\left (29 c^2 d^4-2 c d^2 e (11 b d-a e)+e^2 \left (b^2 d^2+2 a b d e+5 a^2 e^2\right )\right ) x}{16 d^3 e^4 \left (d+e x^2\right )}-\frac{\left (35 c^2 d^4-2 c d^2 e (5 b d+a e)-e^2 \left (b^2 d^2+2 a b d e+5 a^2 e^2\right )\right ) \tan ^{-1}\left (\frac{\sqrt{e} x}{\sqrt{d}}\right )}{16 d^{7/2} e^{9/2}}\\ \end{align*}
Mathematica [A] time = 0.149606, size = 267, normalized size = 1.07 \[ \frac{x \left (e^2 \left (5 a^2 e^2+2 a b d e+b^2 d^2\right )+2 c d^2 e (a e-11 b d)+29 c^2 d^4\right )}{16 d^3 e^4 \left (d+e x^2\right )}-\frac{x \left (e^2 \left (-5 a^2 e^2-2 a b d e+7 b^2 d^2\right )+2 c d^2 e (7 a e-13 b d)+19 c^2 d^4\right )}{24 d^2 e^4 \left (d+e x^2\right )^2}-\frac{\tan ^{-1}\left (\frac{\sqrt{e} x}{\sqrt{d}}\right ) \left (-e^2 \left (5 a^2 e^2+2 a b d e+b^2 d^2\right )-2 c d^2 e (a e+5 b d)+35 c^2 d^4\right )}{16 d^{7/2} e^{9/2}}+\frac{x \left (e (a e-b d)+c d^2\right )^2}{6 d e^4 \left (d+e x^2\right )^3}+\frac{c^2 x}{e^4} \]
Antiderivative was successfully verified.
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Maple [B] time = 0.011, size = 506, normalized size = 2. \begin{align*}{\frac{29\,{x}^{5}{c}^{2}d}{16\,{e}^{2} \left ( e{x}^{2}+d \right ) ^{3}}}+{\frac{5\,{a}^{2}e{x}^{3}}{6\, \left ( e{x}^{2}+d \right ) ^{3}{d}^{2}}}-{\frac{{x}^{3}ac}{3\,e \left ( e{x}^{2}+d \right ) ^{3}}}+{\frac{17\,{x}^{3}{c}^{2}{d}^{2}}{6\,{e}^{3} \left ( e{x}^{2}+d \right ) ^{3}}}-{\frac{abx}{8\,e \left ( e{x}^{2}+d \right ) ^{3}}}-{\frac{{b}^{2}dx}{16\,{e}^{2} \left ( e{x}^{2}+d \right ) ^{3}}}+{\frac{19\,{c}^{2}{d}^{3}x}{16\,{e}^{4} \left ( e{x}^{2}+d \right ) ^{3}}}+{\frac{ac{x}^{5}}{8\, \left ( e{x}^{2}+d \right ) ^{3}d}}+{\frac{ab{x}^{3}}{3\, \left ( e{x}^{2}+d \right ) ^{3}d}}-{\frac{35\,d{c}^{2}}{16\,{e}^{4}}\arctan \left ({ex{\frac{1}{\sqrt{de}}}} \right ){\frac{1}{\sqrt{de}}}}+{\frac{5\,{e}^{2}{x}^{5}{a}^{2}}{16\, \left ( e{x}^{2}+d \right ) ^{3}{d}^{3}}}-{\frac{11\,{x}^{5}bc}{8\,e \left ( e{x}^{2}+d \right ) ^{3}}}+{\frac{{b}^{2}}{16\,d{e}^{2}}\arctan \left ({ex{\frac{1}{\sqrt{de}}}} \right ){\frac{1}{\sqrt{de}}}}+{\frac{5\,bc}{8\,{e}^{3}}\arctan \left ({ex{\frac{1}{\sqrt{de}}}} \right ){\frac{1}{\sqrt{de}}}}+{\frac{e{x}^{5}ab}{8\, \left ( e{x}^{2}+d \right ) ^{3}{d}^{2}}}-{\frac{5\,{x}^{3}bcd}{3\,{e}^{2} \left ( e{x}^{2}+d \right ) ^{3}}}-{\frac{acdx}{8\,{e}^{2} \left ( e{x}^{2}+d \right ) ^{3}}}-{\frac{5\,bc{d}^{2}x}{8\,{e}^{3} \left ( e{x}^{2}+d \right ) ^{3}}}+{\frac{ab}{8\,{d}^{2}e}\arctan \left ({ex{\frac{1}{\sqrt{de}}}} \right ){\frac{1}{\sqrt{de}}}}+{\frac{ac}{8\,d{e}^{2}}\arctan \left ({ex{\frac{1}{\sqrt{de}}}} \right ){\frac{1}{\sqrt{de}}}}+{\frac{{x}^{5}{b}^{2}}{16\, \left ( e{x}^{2}+d \right ) ^{3}d}}+{\frac{11\,{a}^{2}x}{16\, \left ( e{x}^{2}+d \right ) ^{3}d}}-{\frac{{x}^{3}{b}^{2}}{6\,e \left ( e{x}^{2}+d \right ) ^{3}}}+{\frac{5\,{a}^{2}}{16\,{d}^{3}}\arctan \left ({ex{\frac{1}{\sqrt{de}}}} \right ){\frac{1}{\sqrt{de}}}}+{\frac{{c}^{2}x}{{e}^{4}}} \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 = 1.68979, size = 2130, normalized size = 8.52 \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 = 110.179, size = 457, normalized size = 1.83 \begin{align*} \frac{c^{2} x}{e^{4}} - \frac{\sqrt{- \frac{1}{d^{7} e^{9}}} \left (5 a^{2} e^{4} + 2 a b d e^{3} + 2 a c d^{2} e^{2} + b^{2} d^{2} e^{2} + 10 b c d^{3} e - 35 c^{2} d^{4}\right ) \log{\left (- d^{4} e^{4} \sqrt{- \frac{1}{d^{7} e^{9}}} + x \right )}}{32} + \frac{\sqrt{- \frac{1}{d^{7} e^{9}}} \left (5 a^{2} e^{4} + 2 a b d e^{3} + 2 a c d^{2} e^{2} + b^{2} d^{2} e^{2} + 10 b c d^{3} e - 35 c^{2} d^{4}\right ) \log{\left (d^{4} e^{4} \sqrt{- \frac{1}{d^{7} e^{9}}} + x \right )}}{32} + \frac{x^{5} \left (15 a^{2} e^{6} + 6 a b d e^{5} + 6 a c d^{2} e^{4} + 3 b^{2} d^{2} e^{4} - 66 b c d^{3} e^{3} + 87 c^{2} d^{4} e^{2}\right ) + x^{3} \left (40 a^{2} d e^{5} + 16 a b d^{2} e^{4} - 16 a c d^{3} e^{3} - 8 b^{2} d^{3} e^{3} - 80 b c d^{4} e^{2} + 136 c^{2} d^{5} e\right ) + x \left (33 a^{2} d^{2} e^{4} - 6 a b d^{3} e^{3} - 6 a c d^{4} e^{2} - 3 b^{2} d^{4} e^{2} - 30 b c d^{5} e + 57 c^{2} d^{6}\right )}{48 d^{6} e^{4} + 144 d^{5} e^{5} x^{2} + 144 d^{4} e^{6} x^{4} + 48 d^{3} e^{7} x^{6}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.11388, size = 400, normalized size = 1.6 \begin{align*} c^{2} x e^{\left (-4\right )} - \frac{{\left (35 \, c^{2} d^{4} - 10 \, b c d^{3} e - b^{2} d^{2} e^{2} - 2 \, a c d^{2} e^{2} - 2 \, a b d e^{3} - 5 \, a^{2} e^{4}\right )} \arctan \left (\frac{x e^{\frac{1}{2}}}{\sqrt{d}}\right ) e^{\left (-\frac{9}{2}\right )}}{16 \, d^{\frac{7}{2}}} + \frac{{\left (87 \, c^{2} d^{4} x^{5} e^{2} - 66 \, b c d^{3} x^{5} e^{3} + 136 \, c^{2} d^{5} x^{3} e + 3 \, b^{2} d^{2} x^{5} e^{4} + 6 \, a c d^{2} x^{5} e^{4} - 80 \, b c d^{4} x^{3} e^{2} + 57 \, c^{2} d^{6} x + 6 \, a b d x^{5} e^{5} - 8 \, b^{2} d^{3} x^{3} e^{3} - 16 \, a c d^{3} x^{3} e^{3} - 30 \, b c d^{5} x e + 15 \, a^{2} x^{5} e^{6} + 16 \, a b d^{2} x^{3} e^{4} - 3 \, b^{2} d^{4} x e^{2} - 6 \, a c d^{4} x e^{2} + 40 \, a^{2} d x^{3} e^{5} - 6 \, a b d^{3} x e^{3} + 33 \, a^{2} d^{2} x e^{4}\right )} e^{\left (-4\right )}}{48 \,{\left (x^{2} e + d\right )}^{3} d^{3}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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