Optimal. Leaf size=143 \[ -\frac {d x \left (59-35 x^2\right )}{3456 \left (x^4-5 x^2+4\right )}+\frac {d x \left (17-5 x^2\right )}{144 \left (x^4-5 x^2+4\right )^2}-\frac {313 d \tanh ^{-1}\left (\frac {x}{2}\right )}{20736}+\frac {13}{648} d \tanh ^{-1}(x)-\frac {1}{81} e \log \left (1-x^2\right )+\frac {1}{81} e \log \left (4-x^2\right )-\frac {e \left (5-2 x^2\right )}{54 \left (x^4-5 x^2+4\right )}+\frac {e \left (5-2 x^2\right )}{36 \left (x^4-5 x^2+4\right )^2} \]
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Rubi [A] time = 0.08, antiderivative size = 143, normalized size of antiderivative = 1.00, 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} \begin {gather*} -\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 ) \end {gather*}
Antiderivative was successfully verified.
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Rule 12
Rule 31
Rule 207
Rule 614
Rule 616
Rule 1092
Rule 1107
Rule 1166
Rule 1178
Rule 1673
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*}
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Mathematica [A] time = 0.10, size = 128, normalized size = 0.90 \begin {gather*} \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} \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [F] time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {d+e x}{\left (4-5 x^2+x^4\right )^3} \, dx \end {gather*}
Verification is not applicable to the result.
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fricas [B] time = 1.35, size = 307, normalized size = 2.15 \begin {gather*} \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 {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.33, size = 123, normalized size = 0.86 \begin {gather*} -\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 {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.02, size = 186, normalized size = 1.30 \begin {gather*} -\frac {313 d \ln \left (x +2\right )}{41472}+\frac {313 d \ln \left (x -2\right )}{41472}-\frac {13 d \ln \left (x -1\right )}{1296}+\frac {13 d \ln \left (x +1\right )}{1296}+\frac {e \ln \left (x +2\right )}{81}+\frac {e \ln \left (x -2\right )}{81}-\frac {e \ln \left (x -1\right )}{81}-\frac {e \ln \left (x +1\right )}{81}+\frac {19 d}{6912 \left (x -2\right )}-\frac {d}{3456 \left (x -2\right )^{2}}+\frac {d}{432 x +432}-\frac {d}{432 \left (x +1\right )^{2}}+\frac {d}{432 x -432}+\frac {d}{432 \left (x -1\right )^{2}}+\frac {19 d}{6912 \left (x +2\right )}+\frac {d}{3456 \left (x +2\right )^{2}}+\frac {17 e}{3456 \left (x -2\right )}-\frac {e}{1728 \left (x -2\right )^{2}}-\frac {e}{144 \left (x +1\right )}+\frac {e}{432 \left (x +1\right )^{2}}+\frac {e}{144 x -144}+\frac {e}{432 \left (x -1\right )^{2}}-\frac {17 e}{3456 \left (x +2\right )}-\frac {e}{1728 \left (x +2\right )^{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 1.06, size = 121, normalized size = 0.85 \begin {gather*} -\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 {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.09, size = 118, normalized size = 0.83 \begin {gather*} \ln \left (x+1\right )\,\left (\frac {13\,d}{1296}-\frac {e}{81}\right )-\ln \left (x-1\right )\,\left (\frac {13\,d}{1296}+\frac {e}{81}\right )+\ln \left (x-2\right )\,\left (\frac {313\,d}{41472}+\frac {e}{81}\right )-\ln \left (x+2\right )\,\left (\frac {313\,d}{41472}-\frac {e}{81}\right )+\frac {\frac {35\,d\,x^7}{3456}+\frac {e\,x^6}{27}-\frac {13\,d\,x^5}{192}-\frac {5\,e\,x^4}{18}+\frac {35\,d\,x^3}{384}+\frac {5\,e\,x^2}{9}+\frac {43\,d\,x}{864}-\frac {25\,e}{108}}{x^8-10\,x^6+33\,x^4-40\,x^2+16} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [B] time = 3.69, size = 668, normalized size = 4.67 \begin {gather*} \frac {\left (13 d - 16 e\right ) \log {\left (x + \frac {- 1106258459719280 d^{4} e - 13113710954343 d^{4} \left (13 d - 16 e\right ) - 817263343042560 d^{2} e^{3} + 153628968222720 d^{2} e^{2} \left (13 d - 16 e\right ) + 9530197557248 d^{2} e \left (13 d - 16 e\right )^{2} + 88038005760 d^{2} \left (13 d - 16 e\right )^{3} + 5035763255214080 e^{5} + 142661633703936 e^{4} \left (13 d - 16 e\right ) - 19670950215680 e^{3} \left (13 d - 16 e\right )^{2} - 557272006656 e^{2} \left (13 d - 16 e\right )^{3}}{22941256248261 d^{5} - 2312740746035200 d^{3} e^{2} + 4473912813420544 d e^{4}} \right )}}{1296} - \frac {\left (13 d + 16 e\right ) \log {\left (x + \frac {- 1106258459719280 d^{4} e + 13113710954343 d^{4} \left (13 d + 16 e\right ) - 817263343042560 d^{2} e^{3} - 153628968222720 d^{2} e^{2} \left (13 d + 16 e\right ) + 9530197557248 d^{2} e \left (13 d + 16 e\right )^{2} - 88038005760 d^{2} \left (13 d + 16 e\right )^{3} + 5035763255214080 e^{5} - 142661633703936 e^{4} \left (13 d + 16 e\right ) - 19670950215680 e^{3} \left (13 d + 16 e\right )^{2} + 557272006656 e^{2} \left (13 d + 16 e\right )^{3}}{22941256248261 d^{5} - 2312740746035200 d^{3} e^{2} + 4473912813420544 d e^{4}} \right )}}{1296} - \frac {\left (313 d - 512 e\right ) \log {\left (x + \frac {- 1106258459719280 d^{4} e + \frac {13113710954343 d^{4} \left (313 d - 512 e\right )}{32} - 817263343042560 d^{2} e^{3} - 4800905256960 d^{2} e^{2} \left (313 d - 512 e\right ) + 9306833552 d^{2} e \left (313 d - 512 e\right )^{2} - \frac {85974615 d^{2} \left (313 d - 512 e\right )^{3}}{32} + 5035763255214080 e^{5} - 4458176053248 e^{4} \left (313 d - 512 e\right ) - 19209912320 e^{3} \left (313 d - 512 e\right )^{2} + 17006592 e^{2} \left (313 d - 512 e\right )^{3}}{22941256248261 d^{5} - 2312740746035200 d^{3} e^{2} + 4473912813420544 d e^{4}} \right )}}{41472} + \frac {\left (313 d + 512 e\right ) \log {\left (x + \frac {- 1106258459719280 d^{4} e - \frac {13113710954343 d^{4} \left (313 d + 512 e\right )}{32} - 817263343042560 d^{2} e^{3} + 4800905256960 d^{2} e^{2} \left (313 d + 512 e\right ) + 9306833552 d^{2} e \left (313 d + 512 e\right )^{2} + \frac {85974615 d^{2} \left (313 d + 512 e\right )^{3}}{32} + 5035763255214080 e^{5} + 4458176053248 e^{4} \left (313 d + 512 e\right ) - 19209912320 e^{3} \left (313 d + 512 e\right )^{2} - 17006592 e^{2} \left (313 d + 512 e\right )^{3}}{22941256248261 d^{5} - 2312740746035200 d^{3} e^{2} + 4473912813420544 d e^{4}} \right )}}{41472} + \frac {35 d x^{7} - 234 d x^{5} + 315 d x^{3} + 172 d x + 128 e x^{6} - 960 e x^{4} + 1920 e x^{2} - 800 e}{3456 x^{8} - 34560 x^{6} + 114048 x^{4} - 138240 x^{2} + 55296} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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