Optimal. Leaf size=575 \[ \frac{x \left (a^2 \left (\frac{b^4 d}{a^2}-\frac{b^2 (b e+4 c d)}{a}-2 a c f+b^2 f+3 b c e+2 c^2 d\right )+c x^2 \left (2 a^2 c e-a b^2 e-a b (3 c d-a f)+b^3 d\right )\right )}{2 a^3 \left (b^2-4 a c\right ) \left (a+b x^2+c x^4\right )}+\frac{\sqrt{c} \tan ^{-1}\left (\frac{\sqrt{2} \sqrt{c} x}{\sqrt{b-\sqrt{b^2-4 a c}}}\right ) \left (2 a^2 c \left (5 e \sqrt{b^2-4 a c}-6 a f+14 c d\right )-a b^2 \left (3 e \sqrt{b^2-4 a c}-a f+29 c d\right )-a b \left (19 c d \sqrt{b^2-4 a c}-a f \sqrt{b^2-4 a c}-16 a c e\right )+b^3 \left (5 d \sqrt{b^2-4 a c}-3 a e\right )+5 b^4 d\right )}{2 \sqrt{2} a^3 \left (b^2-4 a c\right )^{3/2} \sqrt{b-\sqrt{b^2-4 a c}}}-\frac{\sqrt{c} \tan ^{-1}\left (\frac{\sqrt{2} \sqrt{c} x}{\sqrt{\sqrt{b^2-4 a c}+b}}\right ) \left (2 a^2 c \left (-5 e \sqrt{b^2-4 a c}-6 a f+14 c d\right )-a b^2 \left (-3 e \sqrt{b^2-4 a c}-a f+29 c d\right )+a b \left (19 c d \sqrt{b^2-4 a c}-a f \sqrt{b^2-4 a c}+16 a c e\right )-b^3 \left (5 d \sqrt{b^2-4 a c}+3 a e\right )+5 b^4 d\right )}{2 \sqrt{2} a^3 \left (b^2-4 a c\right )^{3/2} \sqrt{\sqrt{b^2-4 a c}+b}}+\frac{2 b d-a e}{a^3 x}-\frac{d}{3 a^2 x^3} \]
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Rubi [A] time = 9.90565, antiderivative size = 575, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 4, integrand size = 30, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.133, Rules used = {1669, 1664, 1166, 205} \[ \frac{x \left (a^2 \left (\frac{b^4 d}{a^2}-\frac{b^2 (b e+4 c d)}{a}-2 a c f+b^2 f+3 b c e+2 c^2 d\right )+c x^2 \left (2 a^2 c e-a b^2 e-a b (3 c d-a f)+b^3 d\right )\right )}{2 a^3 \left (b^2-4 a c\right ) \left (a+b x^2+c x^4\right )}+\frac{\sqrt{c} \tan ^{-1}\left (\frac{\sqrt{2} \sqrt{c} x}{\sqrt{b-\sqrt{b^2-4 a c}}}\right ) \left (2 a^2 c \left (5 e \sqrt{b^2-4 a c}-6 a f+14 c d\right )-a b^2 \left (3 e \sqrt{b^2-4 a c}-a f+29 c d\right )-a b \left (19 c d \sqrt{b^2-4 a c}-a f \sqrt{b^2-4 a c}-16 a c e\right )+b^3 \left (5 d \sqrt{b^2-4 a c}-3 a e\right )+5 b^4 d\right )}{2 \sqrt{2} a^3 \left (b^2-4 a c\right )^{3/2} \sqrt{b-\sqrt{b^2-4 a c}}}-\frac{\sqrt{c} \tan ^{-1}\left (\frac{\sqrt{2} \sqrt{c} x}{\sqrt{\sqrt{b^2-4 a c}+b}}\right ) \left (2 a^2 c \left (-5 e \sqrt{b^2-4 a c}-6 a f+14 c d\right )-a b^2 \left (-3 e \sqrt{b^2-4 a c}-a f+29 c d\right )+a b \left (19 c d \sqrt{b^2-4 a c}-a f \sqrt{b^2-4 a c}+16 a c e\right )-b^3 \left (5 d \sqrt{b^2-4 a c}+3 a e\right )+5 b^4 d\right )}{2 \sqrt{2} a^3 \left (b^2-4 a c\right )^{3/2} \sqrt{\sqrt{b^2-4 a c}+b}}+\frac{2 b d-a e}{a^3 x}-\frac{d}{3 a^2 x^3} \]
Antiderivative was successfully verified.
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Rule 1669
Rule 1664
Rule 1166
Rule 205
Rubi steps
\begin{align*} \int \frac{d+e x^2+f x^4}{x^4 \left (a+b x^2+c x^4\right )^2} \, dx &=\frac{x \left (a^2 \left (\frac{b^4 d}{a^2}+2 c^2 d+3 b c e-\frac{b^2 (4 c d+b e)}{a}+b^2 f-2 a c f\right )+c \left (b^3 d-a b^2 e+2 a^2 c e-a b (3 c d-a f)\right ) x^2\right )}{2 a^3 \left (b^2-4 a c\right ) \left (a+b x^2+c x^4\right )}-\frac{\int \frac{-2 \left (b^2-4 a c\right ) d+\frac{2 \left (b^2-4 a c\right ) (b d-a e) x^2}{a}-\left (\frac{b^4 d}{a^2}+6 c^2 d+5 b c e-\frac{b^2 (6 c d+b e)}{a}+b^2 f-6 a c f\right ) x^4-\frac{c \left (b^3 d-a b^2 e+2 a^2 c e-a b (3 c d-a f)\right ) x^6}{a^2}}{x^4 \left (a+b x^2+c x^4\right )} \, dx}{2 a \left (b^2-4 a c\right )}\\ &=\frac{x \left (a^2 \left (\frac{b^4 d}{a^2}+2 c^2 d+3 b c e-\frac{b^2 (4 c d+b e)}{a}+b^2 f-2 a c f\right )+c \left (b^3 d-a b^2 e+2 a^2 c e-a b (3 c d-a f)\right ) x^2\right )}{2 a^3 \left (b^2-4 a c\right ) \left (a+b x^2+c x^4\right )}-\frac{\int \left (\frac{2 \left (-b^2+4 a c\right ) d}{a x^4}+\frac{2 \left (-b^2+4 a c\right ) (-2 b d+a e)}{a^2 x^2}+\frac{-5 b^4 d+3 a b^3 e-13 a^2 b c e-2 a^2 c (7 c d-3 a f)+a b^2 (24 c d-a f)-c \left (5 b^3 d-3 a b^2 e+10 a^2 c e-a b (19 c d-a f)\right ) x^2}{a^2 \left (a+b x^2+c x^4\right )}\right ) \, dx}{2 a \left (b^2-4 a c\right )}\\ &=-\frac{d}{3 a^2 x^3}+\frac{2 b d-a e}{a^3 x}+\frac{x \left (a^2 \left (\frac{b^4 d}{a^2}+2 c^2 d+3 b c e-\frac{b^2 (4 c d+b e)}{a}+b^2 f-2 a c f\right )+c \left (b^3 d-a b^2 e+2 a^2 c e-a b (3 c d-a f)\right ) x^2\right )}{2 a^3 \left (b^2-4 a c\right ) \left (a+b x^2+c x^4\right )}-\frac{\int \frac{-5 b^4 d+3 a b^3 e-13 a^2 b c e-2 a^2 c (7 c d-3 a f)+a b^2 (24 c d-a f)-c \left (5 b^3 d-3 a b^2 e+10 a^2 c e-a b (19 c d-a f)\right ) x^2}{a+b x^2+c x^4} \, dx}{2 a^3 \left (b^2-4 a c\right )}\\ &=-\frac{d}{3 a^2 x^3}+\frac{2 b d-a e}{a^3 x}+\frac{x \left (a^2 \left (\frac{b^4 d}{a^2}+2 c^2 d+3 b c e-\frac{b^2 (4 c d+b e)}{a}+b^2 f-2 a c f\right )+c \left (b^3 d-a b^2 e+2 a^2 c e-a b (3 c d-a f)\right ) x^2\right )}{2 a^3 \left (b^2-4 a c\right ) \left (a+b x^2+c x^4\right )}+\frac{\left (c \left (5 b^4 d+b^3 \left (5 \sqrt{b^2-4 a c} d-3 a e\right )+2 a^2 c \left (14 c d+5 \sqrt{b^2-4 a c} e-6 a f\right )-a b^2 \left (29 c d+3 \sqrt{b^2-4 a c} e-a f\right )-a b \left (19 c \sqrt{b^2-4 a c} d-16 a c e-a \sqrt{b^2-4 a c} f\right )\right )\right ) \int \frac{1}{\frac{b}{2}-\frac{1}{2} \sqrt{b^2-4 a c}+c x^2} \, dx}{4 a^3 \left (b^2-4 a c\right )^{3/2}}-\frac{\left (c \left (5 b^4 d-b^3 \left (5 \sqrt{b^2-4 a c} d+3 a e\right )+2 a^2 c \left (14 c d-5 \sqrt{b^2-4 a c} e-6 a f\right )-a b^2 \left (29 c d-3 \sqrt{b^2-4 a c} e-a f\right )+a b \left (19 c \sqrt{b^2-4 a c} d+16 a c e-a \sqrt{b^2-4 a c} f\right )\right )\right ) \int \frac{1}{\frac{b}{2}+\frac{1}{2} \sqrt{b^2-4 a c}+c x^2} \, dx}{4 a^3 \left (b^2-4 a c\right )^{3/2}}\\ &=-\frac{d}{3 a^2 x^3}+\frac{2 b d-a e}{a^3 x}+\frac{x \left (a^2 \left (\frac{b^4 d}{a^2}+2 c^2 d+3 b c e-\frac{b^2 (4 c d+b e)}{a}+b^2 f-2 a c f\right )+c \left (b^3 d-a b^2 e+2 a^2 c e-a b (3 c d-a f)\right ) x^2\right )}{2 a^3 \left (b^2-4 a c\right ) \left (a+b x^2+c x^4\right )}+\frac{\sqrt{c} \left (5 b^4 d+b^3 \left (5 \sqrt{b^2-4 a c} d-3 a e\right )+2 a^2 c \left (14 c d+5 \sqrt{b^2-4 a c} e-6 a f\right )-a b^2 \left (29 c d+3 \sqrt{b^2-4 a c} e-a f\right )-a b \left (19 c \sqrt{b^2-4 a c} d-16 a c e-a \sqrt{b^2-4 a c} f\right )\right ) \tan ^{-1}\left (\frac{\sqrt{2} \sqrt{c} x}{\sqrt{b-\sqrt{b^2-4 a c}}}\right )}{2 \sqrt{2} a^3 \left (b^2-4 a c\right )^{3/2} \sqrt{b-\sqrt{b^2-4 a c}}}-\frac{\sqrt{c} \left (5 b^4 d-b^3 \left (5 \sqrt{b^2-4 a c} d+3 a e\right )+2 a^2 c \left (14 c d-5 \sqrt{b^2-4 a c} e-6 a f\right )-a b^2 \left (29 c d-3 \sqrt{b^2-4 a c} e-a f\right )+a b \left (19 c \sqrt{b^2-4 a c} d+16 a c e-a \sqrt{b^2-4 a c} f\right )\right ) \tan ^{-1}\left (\frac{\sqrt{2} \sqrt{c} x}{\sqrt{b+\sqrt{b^2-4 a c}}}\right )}{2 \sqrt{2} a^3 \left (b^2-4 a c\right )^{3/2} \sqrt{b+\sqrt{b^2-4 a c}}}\\ \end{align*}
Mathematica [A] time = 1.96036, size = 548, normalized size = 0.95 \[ \frac{\frac{6 x \left (2 a^2 c \left (c \left (d+e x^2\right )-a f\right )+a b^2 \left (a f-c \left (4 d+e x^2\right )\right )+b^3 \left (c d x^2-a e\right )+a b c \left (3 a e+a f x^2-3 c d x^2\right )+b^4 d\right )}{\left (b^2-4 a c\right ) \left (a+b x^2+c x^4\right )}+\frac{3 \sqrt{2} \sqrt{c} \tan ^{-1}\left (\frac{\sqrt{2} \sqrt{c} x}{\sqrt{b-\sqrt{b^2-4 a c}}}\right ) \left (2 a^2 c \left (5 e \sqrt{b^2-4 a c}-6 a f+14 c d\right )+a b^2 \left (-3 e \sqrt{b^2-4 a c}+a f-29 c d\right )+a b \left (-19 c d \sqrt{b^2-4 a c}+a f \sqrt{b^2-4 a c}+16 a c e\right )+b^3 \left (5 d \sqrt{b^2-4 a c}-3 a e\right )+5 b^4 d\right )}{\left (b^2-4 a c\right )^{3/2} \sqrt{b-\sqrt{b^2-4 a c}}}+\frac{3 \sqrt{2} \sqrt{c} \tan ^{-1}\left (\frac{\sqrt{2} \sqrt{c} x}{\sqrt{\sqrt{b^2-4 a c}+b}}\right ) \left (2 a^2 c \left (5 e \sqrt{b^2-4 a c}+6 a f-14 c d\right )-a b^2 \left (3 e \sqrt{b^2-4 a c}+a f-29 c d\right )+a b \left (-19 c d \sqrt{b^2-4 a c}+a f \sqrt{b^2-4 a c}-16 a c e\right )+b^3 \left (5 d \sqrt{b^2-4 a c}+3 a e\right )-5 b^4 d\right )}{\left (b^2-4 a c\right )^{3/2} \sqrt{\sqrt{b^2-4 a c}+b}}+\frac{24 b d-12 a e}{x}-\frac{4 a d}{x^3}}{12 a^3} \]
Antiderivative was successfully verified.
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Maple [B] time = 0.048, size = 2180, normalized size = 3.8 \begin{align*} \text{result too large to display} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F(-1)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [F(-1)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F(-1)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F(-2)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: NotImplementedError} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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