Optimal. Leaf size=143 \[ -\frac{d x \left (59-35 x^2\right )}{3456 \left (x^4-5 x^2+4\right )}+\frac{d x \left (17-5 x^2\right )}{144 \left (x^4-5 x^2+4\right )^2}-\frac{313 d \tanh ^{-1}\left (\frac{x}{2}\right )}{20736}+\frac{13}{648} d \tanh ^{-1}(x)-\frac{e \left (5-2 x^2\right )}{54 \left (x^4-5 x^2+4\right )}+\frac{e \left (5-2 x^2\right )}{36 \left (x^4-5 x^2+4\right )^2}-\frac{1}{81} e \log \left (1-x^2\right )+\frac{1}{81} e \log \left (4-x^2\right ) \]
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Rubi [A] time = 0.0758338, antiderivative size = 143, normalized size of antiderivative = 1., number of steps used = 14, number of rules used = 10, integrand size = 18, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.556, Rules used = {1673, 12, 1092, 1178, 1166, 207, 1107, 614, 616, 31} \[ -\frac{d x \left (59-35 x^2\right )}{3456 \left (x^4-5 x^2+4\right )}+\frac{d x \left (17-5 x^2\right )}{144 \left (x^4-5 x^2+4\right )^2}-\frac{313 d \tanh ^{-1}\left (\frac{x}{2}\right )}{20736}+\frac{13}{648} d \tanh ^{-1}(x)-\frac{e \left (5-2 x^2\right )}{54 \left (x^4-5 x^2+4\right )}+\frac{e \left (5-2 x^2\right )}{36 \left (x^4-5 x^2+4\right )^2}-\frac{1}{81} e \log \left (1-x^2\right )+\frac{1}{81} e \log \left (4-x^2\right ) \]
Antiderivative was successfully verified.
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Rule 1673
Rule 12
Rule 1092
Rule 1178
Rule 1166
Rule 207
Rule 1107
Rule 614
Rule 616
Rule 31
Rubi steps
\begin{align*} \int \frac{d+e x}{\left (4-5 x^2+x^4\right )^3} \, dx &=\int \frac{d}{\left (4-5 x^2+x^4\right )^3} \, dx+\int \frac{e x}{\left (4-5 x^2+x^4\right )^3} \, dx\\ &=d \int \frac{1}{\left (4-5 x^2+x^4\right )^3} \, dx+e \int \frac{x}{\left (4-5 x^2+x^4\right )^3} \, dx\\ &=\frac{d x \left (17-5 x^2\right )}{144 \left (4-5 x^2+x^4\right )^2}-\frac{1}{144} d \int \frac{-19+25 x^2}{\left (4-5 x^2+x^4\right )^2} \, dx+\frac{1}{2} e \operatorname{Subst}\left (\int \frac{1}{\left (4-5 x+x^2\right )^3} \, dx,x,x^2\right )\\ &=\frac{d x \left (17-5 x^2\right )}{144 \left (4-5 x^2+x^4\right )^2}+\frac{e \left (5-2 x^2\right )}{36 \left (4-5 x^2+x^4\right )^2}-\frac{d x \left (59-35 x^2\right )}{3456 \left (4-5 x^2+x^4\right )}+\frac{d \int \frac{519+105 x^2}{4-5 x^2+x^4} \, dx}{10368}-\frac{1}{6} e \operatorname{Subst}\left (\int \frac{1}{\left (4-5 x+x^2\right )^2} \, dx,x,x^2\right )\\ &=\frac{d x \left (17-5 x^2\right )}{144 \left (4-5 x^2+x^4\right )^2}+\frac{e \left (5-2 x^2\right )}{36 \left (4-5 x^2+x^4\right )^2}-\frac{d x \left (59-35 x^2\right )}{3456 \left (4-5 x^2+x^4\right )}-\frac{e \left (5-2 x^2\right )}{54 \left (4-5 x^2+x^4\right )}-\frac{1}{648} (13 d) \int \frac{1}{-1+x^2} \, dx+\frac{(313 d) \int \frac{1}{-4+x^2} \, dx}{10368}+\frac{1}{27} e \operatorname{Subst}\left (\int \frac{1}{4-5 x+x^2} \, dx,x,x^2\right )\\ &=\frac{d x \left (17-5 x^2\right )}{144 \left (4-5 x^2+x^4\right )^2}+\frac{e \left (5-2 x^2\right )}{36 \left (4-5 x^2+x^4\right )^2}-\frac{d x \left (59-35 x^2\right )}{3456 \left (4-5 x^2+x^4\right )}-\frac{e \left (5-2 x^2\right )}{54 \left (4-5 x^2+x^4\right )}-\frac{313 d \tanh ^{-1}\left (\frac{x}{2}\right )}{20736}+\frac{13}{648} d \tanh ^{-1}(x)+\frac{1}{81} e \operatorname{Subst}\left (\int \frac{1}{-4+x} \, dx,x,x^2\right )-\frac{1}{81} e \operatorname{Subst}\left (\int \frac{1}{-1+x} \, dx,x,x^2\right )\\ &=\frac{d x \left (17-5 x^2\right )}{144 \left (4-5 x^2+x^4\right )^2}+\frac{e \left (5-2 x^2\right )}{36 \left (4-5 x^2+x^4\right )^2}-\frac{d x \left (59-35 x^2\right )}{3456 \left (4-5 x^2+x^4\right )}-\frac{e \left (5-2 x^2\right )}{54 \left (4-5 x^2+x^4\right )}-\frac{313 d \tanh ^{-1}\left (\frac{x}{2}\right )}{20736}+\frac{13}{648} d \tanh ^{-1}(x)-\frac{1}{81} e \log \left (1-x^2\right )+\frac{1}{81} e \log \left (4-x^2\right )\\ \end{align*}
Mathematica [A] time = 0.098161, size = 128, normalized size = 0.9 \[ \frac{\frac{288 \left (d x \left (17-5 x^2\right )+e \left (20-8 x^2\right )\right )}{\left (x^4-5 x^2+4\right )^2}+\frac{12 \left (d x \left (35 x^2-59\right )+64 e \left (2 x^2-5\right )\right )}{x^4-5 x^2+4}-32 (13 d+16 e) \log (1-x)+(313 d+512 e) \log (2-x)+32 (13 d-16 e) \log (x+1)+(512 e-313 d) \log (x+2)}{41472} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.019, size = 186, normalized size = 1.3 \begin{align*} -{\frac{313\,\ln \left ( 2+x \right ) d}{41472}}+{\frac{\ln \left ( 2+x \right ) e}{81}}+{\frac{19\,d}{13824+6912\,x}}-{\frac{17\,e}{6912+3456\,x}}+{\frac{d}{3456\, \left ( 2+x \right ) ^{2}}}-{\frac{e}{1728\, \left ( 2+x \right ) ^{2}}}+{\frac{d}{432+432\,x}}-{\frac{e}{144+144\,x}}-{\frac{d}{432\, \left ( 1+x \right ) ^{2}}}+{\frac{e}{432\, \left ( 1+x \right ) ^{2}}}+{\frac{13\,\ln \left ( 1+x \right ) d}{1296}}-{\frac{\ln \left ( 1+x \right ) e}{81}}+{\frac{19\,d}{6912\,x-13824}}+{\frac{17\,e}{3456\,x-6912}}-{\frac{d}{3456\, \left ( x-2 \right ) ^{2}}}-{\frac{e}{1728\, \left ( x-2 \right ) ^{2}}}+{\frac{313\,\ln \left ( x-2 \right ) d}{41472}}+{\frac{\ln \left ( x-2 \right ) e}{81}}-{\frac{13\,\ln \left ( x-1 \right ) d}{1296}}-{\frac{\ln \left ( x-1 \right ) e}{81}}+{\frac{d}{432\,x-432}}+{\frac{e}{144\,x-144}}+{\frac{d}{432\, \left ( x-1 \right ) ^{2}}}+{\frac{e}{432\, \left ( x-1 \right ) ^{2}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 0.940788, size = 163, normalized size = 1.14 \begin{align*} -\frac{1}{41472} \,{\left (313 \, d - 512 \, e\right )} \log \left (x + 2\right ) + \frac{1}{1296} \,{\left (13 \, d - 16 \, e\right )} \log \left (x + 1\right ) - \frac{1}{1296} \,{\left (13 \, d + 16 \, e\right )} \log \left (x - 1\right ) + \frac{1}{41472} \,{\left (313 \, d + 512 \, e\right )} \log \left (x - 2\right ) + \frac{35 \, d x^{7} + 128 \, e x^{6} - 234 \, d x^{5} - 960 \, e x^{4} + 315 \, d x^{3} + 1920 \, e x^{2} + 172 \, d x - 800 \, e}{3456 \,{\left (x^{8} - 10 \, x^{6} + 33 \, x^{4} - 40 \, x^{2} + 16\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] time = 2.07694, size = 869, normalized size = 6.08 \begin{align*} \frac{420 \, d x^{7} + 1536 \, e x^{6} - 2808 \, d x^{5} - 11520 \, e x^{4} + 3780 \, d x^{3} + 23040 \, e x^{2} + 2064 \, d x -{\left ({\left (313 \, d - 512 \, e\right )} x^{8} - 10 \,{\left (313 \, d - 512 \, e\right )} x^{6} + 33 \,{\left (313 \, d - 512 \, e\right )} x^{4} - 40 \,{\left (313 \, d - 512 \, e\right )} x^{2} + 5008 \, d - 8192 \, e\right )} \log \left (x + 2\right ) + 32 \,{\left ({\left (13 \, d - 16 \, e\right )} x^{8} - 10 \,{\left (13 \, d - 16 \, e\right )} x^{6} + 33 \,{\left (13 \, d - 16 \, e\right )} x^{4} - 40 \,{\left (13 \, d - 16 \, e\right )} x^{2} + 208 \, d - 256 \, e\right )} \log \left (x + 1\right ) - 32 \,{\left ({\left (13 \, d + 16 \, e\right )} x^{8} - 10 \,{\left (13 \, d + 16 \, e\right )} x^{6} + 33 \,{\left (13 \, d + 16 \, e\right )} x^{4} - 40 \,{\left (13 \, d + 16 \, e\right )} x^{2} + 208 \, d + 256 \, e\right )} \log \left (x - 1\right ) +{\left ({\left (313 \, d + 512 \, e\right )} x^{8} - 10 \,{\left (313 \, d + 512 \, e\right )} x^{6} + 33 \,{\left (313 \, d + 512 \, e\right )} x^{4} - 40 \,{\left (313 \, d + 512 \, e\right )} x^{2} + 5008 \, d + 8192 \, e\right )} \log \left (x - 2\right ) - 9600 \, e}{41472 \,{\left (x^{8} - 10 \, x^{6} + 33 \, x^{4} - 40 \, x^{2} + 16\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [B] time = 2.79382, size = 668, normalized size = 4.67 \begin{align*} \text{result too large to display} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.13233, size = 166, normalized size = 1.16 \begin{align*} -\frac{1}{41472} \,{\left (313 \, d - 512 \, e\right )} \log \left ({\left | x + 2 \right |}\right ) + \frac{1}{1296} \,{\left (13 \, d - 16 \, e\right )} \log \left ({\left | x + 1 \right |}\right ) - \frac{1}{1296} \,{\left (13 \, d + 16 \, e\right )} \log \left ({\left | x - 1 \right |}\right ) + \frac{1}{41472} \,{\left (313 \, d + 512 \, e\right )} \log \left ({\left | x - 2 \right |}\right ) + \frac{35 \, d x^{7} + 128 \, x^{6} e - 234 \, d x^{5} - 960 \, x^{4} e + 315 \, d x^{3} + 1920 \, x^{2} e + 172 \, d x - 800 \, e}{3456 \,{\left (x^{4} - 5 \, x^{2} + 4\right )}^{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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