Integrand size = 53, antiderivative size = 281 \[ \int \frac {-7-4 x-5 x^2}{20+70 x-84 x^2-193 x^3+36 x^4+103 x^5+89 x^6-120 x^7+30 x^8} \, dx=\frac {27331243886191 \sqrt {\frac {33}{590 \left (838674112505756+48951578419025 \sqrt {295}\right )}} \arctan \left (\sqrt {-\frac {551}{231}+\frac {40 \sqrt {295}}{231}}+2 \sqrt {\frac {10}{231} \left (-8+\sqrt {295}\right )} x\right )-\sqrt {\frac {7}{590} \left (838674112505756+48951578419025 \sqrt {295}\right )} \text {arctanh}\left (\sqrt {\frac {551}{231}+\frac {40 \sqrt {295}}{231}}-2 \sqrt {\frac {10}{231} \left (8+\sqrt {295}\right )} x\right )-\frac {\left (168408125+14036453 \sqrt {295}\right ) \log \left (10+\left (-5-\sqrt {295}\right ) x+10 x^2\right )}{1180}-\frac {\left (168408125-14036453 \sqrt {295}\right ) \log \left (10+\left (-5+\sqrt {295}\right ) x+10 x^2\right )}{1180}}{22703142}+\frac {\text {RootSum}\left [2+9 \text {$\#$1}+2 \text {$\#$1}^2-9 \text {$\#$1}^3+3 \text {$\#$1}^4\&,\frac {-13555697 \log (x-\text {$\#$1})-6761246 \log (x-\text {$\#$1}) \text {$\#$1}+3434127 \log (x-\text {$\#$1}) \text {$\#$1}^2+1712625 \log (x-\text {$\#$1}) \text {$\#$1}^3}{9+4 \text {$\#$1}-27 \text {$\#$1}^2+12 \text {$\#$1}^3}\&\right ]}{22703142} \] Output:
79454293/66*33^(1/2)/(494817726378396040+28881431267224750*295^(1/2))^(1/2 )*arctan(1/231*(-127281+9240*295^(1/2))^(1/2)+2/231*(-18480+2310*295^(1/2) )^(1/2)*x)+1/13394853780*(3463724084648772280+202170018870573250*295^(1/2) )^(1/2)*arctanh(-1/231*(127281+9240*295^(1/2))^(1/2)+2/231*(18480+2310*295 ^(1/2))^(1/2)*x)-1/26789707560*(168408125+14036453*295^(1/2))*ln(10+(-5-29 5^(1/2))*x+10*x^2)-1/26789707560*(168408125-14036453*295^(1/2))*ln(10+(-5+ 295^(1/2))*x+10*x^2)+1/22703142*RootSum(_Z1 -> 3*_Z1^4-9*_Z1^3+2*_Z1^2+9*_ Z1+2,_Z1 -> (-13555697*ln(x-_Z1)-6761246*ln(x-_Z1)*_Z1+3434127*ln(x-_Z1)*_ Z1^2+1712625*ln(x-_Z1)*_Z1^3)/(12*_Z1^3-27*_Z1^2+4*_Z1+9))
Time = 0.04 (sec) , antiderivative size = 199, normalized size of antiderivative = 0.71 \[ \int \frac {-7-4 x-5 x^2}{20+70 x-84 x^2-193 x^3+36 x^4+103 x^5+89 x^6-120 x^7+30 x^8} \, dx=\frac {\text {RootSum}\left [2+9 \text {$\#$1}+2 \text {$\#$1}^2-9 \text {$\#$1}^3+3 \text {$\#$1}^4\&,\frac {-13555697 \log (x-\text {$\#$1})-6761246 \log (x-\text {$\#$1}) \text {$\#$1}+3434127 \log (x-\text {$\#$1}) \text {$\#$1}^2+1712625 \log (x-\text {$\#$1}) \text {$\#$1}^3}{9+4 \text {$\#$1}-27 \text {$\#$1}^2+12 \text {$\#$1}^3}\&\right ]}{22703142}-\frac {\text {RootSum}\left [10-10 \text {$\#$1}-7 \text {$\#$1}^2-10 \text {$\#$1}^3+10 \text {$\#$1}^4\&,\frac {11682512 \log (x-\text {$\#$1})+26807235 \log (x-\text {$\#$1}) \text {$\#$1}+22864590 \log (x-\text {$\#$1}) \text {$\#$1}^2+5708750 \log (x-\text {$\#$1}) \text {$\#$1}^3}{-5-7 \text {$\#$1}-15 \text {$\#$1}^2+20 \text {$\#$1}^3}\&\right ]}{45406284} \] Input:
Integrate[(-7 - 4*x - 5*x^2)/(20 + 70*x - 84*x^2 - 193*x^3 + 36*x^4 + 103* x^5 + 89*x^6 - 120*x^7 + 30*x^8),x]
Output:
RootSum[2 + 9*#1 + 2*#1^2 - 9*#1^3 + 3*#1^4 & , (-13555697*Log[x - #1] - 6 761246*Log[x - #1]*#1 + 3434127*Log[x - #1]*#1^2 + 1712625*Log[x - #1]*#1^ 3)/(9 + 4*#1 - 27*#1^2 + 12*#1^3) & ]/22703142 - RootSum[10 - 10*#1 - 7*#1 ^2 - 10*#1^3 + 10*#1^4 & , (11682512*Log[x - #1] + 26807235*Log[x - #1]*#1 + 22864590*Log[x - #1]*#1^2 + 5708750*Log[x - #1]*#1^3)/(-5 - 7*#1 - 15*# 1^2 + 20*#1^3) & ]/45406284
Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.
| \(\displaystyle \int \frac {-5 x^2-4 x-7}{30 x^8-120 x^7+89 x^6+103 x^5+36 x^4-193 x^3-84 x^2+70 x+20} \, dx\) |
| \(\Big \downarrow \) 2462 |
| \(\displaystyle \int \left (\frac {-5708750 x^3-22864590 x^2-26807235 x-11682512}{22703142 \left (10 x^4-10 x^3-7 x^2-10 x+10\right )}+\frac {1712625 x^3+3434127 x^2-6761246 x-13555697}{22703142 \left (3 x^4-9 x^3+2 x^2+9 x+2\right )}\right )dx\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle -\frac {59360663 \int \frac {1}{3 x^4-9 x^3+2 x^2+9 x+2}dx}{90812568}-\frac {7332121 \int \frac {x}{3 x^4-9 x^3+2 x^2+9 x+2}dx}{22703142}+\frac {9716711 \int \frac {x^2}{3 x^4-9 x^3+2 x^2+9 x+2}dx}{30270856}-\frac {\left (11105503 \sqrt {5}-2875418 \sqrt {59}\right ) \arctan \left (\frac {-20 x-\sqrt {295}+5}{\sqrt {10 \left (8+\sqrt {295}\right )}}\right )}{3243306 \sqrt {590 \left (8+\sqrt {295}\right )}}-\frac {\left (11105503 \sqrt {5}+2875418 \sqrt {59}\right ) \text {arctanh}\left (\frac {-20 x+\sqrt {295}+5}{\sqrt {10 \left (\sqrt {295}-8\right )}}\right )}{3243306 \sqrt {590 \left (\sqrt {295}-8\right )}}-\frac {\left (168408125-14036453 \sqrt {295}\right ) \log \left (10 x^2-\left (5-\sqrt {295}\right ) x+10\right )}{26789707560}-\frac {\left (168408125+14036453 \sqrt {295}\right ) \log \left (10 x^2-\left (5+\sqrt {295}\right ) x+10\right )}{26789707560}+\frac {570875 \log \left (3 x^4-9 x^3+2 x^2+9 x+2\right )}{90812568}\) |
Input:
Int[(-7 - 4*x - 5*x^2)/(20 + 70*x - 84*x^2 - 193*x^3 + 36*x^4 + 103*x^5 + 89*x^6 - 120*x^7 + 30*x^8),x]
Output:
$Aborted
Time = 0.11 (sec) , antiderivative size = 92, normalized size of antiderivative = 0.33
| method | result | size |
| risch | \(\left (\munderset {\textit {\_R} =\operatorname {RootOf}\left (2892539212794 \textit {\_Z}^{4}-72733471125 \textit {\_Z}^{3}-15447799570 \textit {\_Z}^{2}+845714817 \textit {\_Z} -11926899\right )}{\sum }\textit {\_R} \ln \left (404748573250166694515167681788 \textit {\_R}^{3}+24237872922602901082566583860 \textit {\_R}^{2}-1155408599958166938987224783 \textit {\_R} +9967012827048091416020589 x -22384516712413089943282635\right )\right )+\left (\munderset {\textit {\_R} =\operatorname {RootOf}\left (1095350773005720 \textit {\_Z}^{4}+27542812027500 \textit {\_Z}^{3}-10490512063449 \textit {\_Z}^{2}+56698109100 \textit {\_Z} -200672500\right )}{\sum }\textit {\_R} \ln \left (-148622067869253730215610357494360 \textit {\_R}^{3}-5187777152535315110375002669500 \textit {\_R}^{2}+1411414920506128393395409151337 \textit {\_R} +6050214123593112330770975000 x -181735037503505656661621650\right )\right )\) | \(92\) |
| default | \(\frac {\left (14036453 \sqrt {295}-168408125\right ) \ln \left (\sqrt {295}\, x +10 x^{2}-5 x +10\right )}{26789707560}+\frac {\left (26939741 \sqrt {295}-344634104-\frac {\left (14036453 \sqrt {295}-168408125\right ) \left (-5+\sqrt {295}\right )}{20}\right ) \arctan \left (\frac {\sqrt {295}+20 x -5}{\sqrt {80+10 \sqrt {295}}}\right )}{669742689 \sqrt {80+10 \sqrt {295}}}-\frac {\left (168408125+14036453 \sqrt {295}\right ) \ln \left (10 x^{2}-5 x -\sqrt {295}\, x +10\right )}{26789707560}+\frac {\left (26939741 \sqrt {295}+344634104-\frac {\left (168408125+14036453 \sqrt {295}\right ) \left (-5-\sqrt {295}\right )}{20}\right ) \operatorname {arctanh}\left (\frac {20 x -5-\sqrt {295}}{\sqrt {-80+10 \sqrt {295}}}\right )}{669742689 \sqrt {-80+10 \sqrt {295}}}+\frac {\left (\munderset {\textit {\_R} =\operatorname {RootOf}\left (3 \textit {\_Z}^{4}-9 \textit {\_Z}^{3}+2 \textit {\_Z}^{2}+9 \textit {\_Z} +2\right )}{\sum }\frac {\left (1712625 \textit {\_R}^{3}+3434127 \textit {\_R}^{2}-6761246 \textit {\_R} -13555697\right ) \ln \left (x -\textit {\_R} \right )}{12 \textit {\_R}^{3}-27 \textit {\_R}^{2}+4 \textit {\_R} +9}\right )}{22703142}\) | \(224\) |
Input:
int((-5*x^2-4*x-7)/(30*x^8-120*x^7+89*x^6+103*x^5+36*x^4-193*x^3-84*x^2+70 *x+20),x,method=_RETURNVERBOSE)
Output:
sum(_R*ln(404748573250166694515167681788*_R^3+2423787292260290108256658386 0*_R^2-1155408599958166938987224783*_R+9967012827048091416020589*x-2238451 6712413089943282635),_R=RootOf(2892539212794*_Z^4-72733471125*_Z^3-1544779 9570*_Z^2+845714817*_Z-11926899))+sum(_R*ln(-14862206786925373021561035749 4360*_R^3-5187777152535315110375002669500*_R^2+141141492050612839339540915 1337*_R+6050214123593112330770975000*x-181735037503505656661621650),_R=Roo tOf(1095350773005720*_Z^4+27542812027500*_Z^3-10490512063449*_Z^2+56698109 100*_Z-200672500))
Default grade assigned because unable to parse optimal
Time = 0.91 (sec) , antiderivative size = 7957, normalized size of antiderivative = 28.32 \[ \int \frac {-7-4 x-5 x^2}{20+70 x-84 x^2-193 x^3+36 x^4+103 x^5+89 x^6-120 x^7+30 x^8} \, dx=\text {Too large to display} \] Input:
integrate((-5*x^2-4*x-7)/(30*x^8-120*x^7+89*x^6+103*x^5+36*x^4-193*x^3-84* x^2+70*x+20),x, algorithm="fricas")
Output:
Too large to include
Default grade assigned because unable to parse optimal
Time = 2.76 (sec) , antiderivative size = 150, normalized size of antiderivative = 0.53 \[ \int \frac {-7-4 x-5 x^2}{20+70 x-84 x^2-193 x^3+36 x^4+103 x^5+89 x^6-120 x^7+30 x^8} \, dx=\text {Too large to display} \] Input:
integrate((-5*x**2-4*x-7)/(30*x**8-120*x**7+89*x**6+103*x**5+36*x**4-193*x **3-84*x**2+70*x+20),x)
Output:
-RootSum(2892539212794*_t**4 + 72733471125*_t**3 - 15447799570*_t**2 - 845 714817*_t - 11926899, Lambda(_t, _t*log(-125321034572133108556905574656481 30376764004458547980354328503286333368883028804271998285360510110808520862 8132942226839246843430340765774*_t**7/761387234033768129319773871117548342 21231842929823943546301087592150460424717449154725674297626622822292107379 21905738369661875 + 809315046154666812991652724725301121212071603089692100 41686801073063917906649687075863206584045141473191257211034476523361475490 57532*_t**6/30455489361350725172790954844701933688492737171929577418520435 0368601841698869796618902697190506491289168429516876229534786475 + 4047988 77530009335272429088791616466981271868409535633781796907803640739748459997 39313782685930947558169610366342536537085760206962937991*_t**5/15227744680 67536258639547742235096684424636858596478870926021751843009208494348983094 51348595253245644584214758438114767393237500 + 384090448206105367031357014 22344894886037857818575423307719396406438057749510748614539407769863295174 1590197284034701457516906557863961*_t**4/609109787227014503455819096894038 67376985474343859154837040870073720368339773959323780539438101298257833685 903375245906957295000 - 91494097986588684421891279403635119694971092419708 38495163208768750167466870253697286070190988800815165861948482456984690284 981302499*_t**3/9136646808405217551837286453410580106547821151578873225556 130511058055250966093898567080915715194738675052885506286886043594250 -...
\[ \int \frac {-7-4 x-5 x^2}{20+70 x-84 x^2-193 x^3+36 x^4+103 x^5+89 x^6-120 x^7+30 x^8} \, dx=\int { -\frac {5 \, x^{2} + 4 \, x + 7}{30 \, x^{8} - 120 \, x^{7} + 89 \, x^{6} + 103 \, x^{5} + 36 \, x^{4} - 193 \, x^{3} - 84 \, x^{2} + 70 \, x + 20} \,d x } \] Input:
integrate((-5*x^2-4*x-7)/(30*x^8-120*x^7+89*x^6+103*x^5+36*x^4-193*x^3-84* x^2+70*x+20),x, algorithm="maxima")
Output:
-integrate((5*x^2 + 4*x + 7)/(30*x^8 - 120*x^7 + 89*x^6 + 103*x^5 + 36*x^4 - 193*x^3 - 84*x^2 + 70*x + 20), x)
\[ \int \frac {-7-4 x-5 x^2}{20+70 x-84 x^2-193 x^3+36 x^4+103 x^5+89 x^6-120 x^7+30 x^8} \, dx=\int { -\frac {5 \, x^{2} + 4 \, x + 7}{30 \, x^{8} - 120 \, x^{7} + 89 \, x^{6} + 103 \, x^{5} + 36 \, x^{4} - 193 \, x^{3} - 84 \, x^{2} + 70 \, x + 20} \,d x } \] Input:
integrate((-5*x^2-4*x-7)/(30*x^8-120*x^7+89*x^6+103*x^5+36*x^4-193*x^3-84* x^2+70*x+20),x, algorithm="giac")
Output:
integrate(-(5*x^2 + 4*x + 7)/(30*x^8 - 120*x^7 + 89*x^6 + 103*x^5 + 36*x^4 - 193*x^3 - 84*x^2 + 70*x + 20), x)
Time = 9.90 (sec) , antiderivative size = 771, normalized size of antiderivative = 2.74 \[ \int \frac {-7-4 x-5 x^2}{20+70 x-84 x^2-193 x^3+36 x^4+103 x^5+89 x^6-120 x^7+30 x^8} \, dx=\text {Too large to display} \] Input:
int(-(4*x + 5*x^2 + 7)/(70*x - 84*x^2 - 193*x^3 + 36*x^4 + 103*x^5 + 89*x^ 6 - 120*x^7 + 30*x^8 + 20),x)
Output:
symsum(log((34396962819377653*root(z^4 + (570875*z^3)/22703142 - (53880390 67*z^2)/562583858760 + (5338805*z)/103140374106 - 1433375/7823934092898, z , k)^2*x)/2214337500000000 - (398341*x)/32805000000 - (808108441*root(z^4 + (570875*z^3)/22703142 - (5388039067*z^2)/562583858760 + (5338805*z)/1031 40374106 - 1433375/7823934092898, z, k)*x)/22781250000 - (44761017427*root (z^4 + (570875*z^3)/22703142 - (5388039067*z^2)/562583858760 + (5338805*z) /103140374106 - 1433375/7823934092898, z, k))/3280500000000 - (39954030731 81761061*root(z^4 + (570875*z^3)/22703142 - (5388039067*z^2)/562583858760 + (5338805*z)/103140374106 - 1433375/7823934092898, z, k)^3*x)/22143375000 00000 + (54179246638818556019*root(z^4 + (570875*z^3)/22703142 - (53880390 67*z^2)/562583858760 + (5338805*z)/103140374106 - 1433375/7823934092898, z , k)^4*x)/738112500000000 - (470116923498806117933*root(z^4 + (570875*z^3) /22703142 - (5388039067*z^2)/562583858760 + (5338805*z)/103140374106 - 143 3375/7823934092898, z, k)^5*x)/492075000000000 - (1325395687143820022269*r oot(z^4 + (570875*z^3)/22703142 - (5388039067*z^2)/562583858760 + (5338805 *z)/103140374106 - 1433375/7823934092898, z, k)^6*x)/328050000000000 + (88 14932846054050405921*root(z^4 + (570875*z^3)/22703142 - (5388039067*z^2)/5 62583858760 + (5338805*z)/103140374106 - 1433375/7823934092898, z, k)^7*x) /82012500000000 + (32150255721172381*root(z^4 + (570875*z^3)/22703142 - (5 388039067*z^2)/562583858760 + (5338805*z)/103140374106 - 1433375/782393...
\[ \int \frac {-7-4 x-5 x^2}{20+70 x-84 x^2-193 x^3+36 x^4+103 x^5+89 x^6-120 x^7+30 x^8} \, dx=-5 \left (\int \frac {x^{2}}{30 x^{8}-120 x^{7}+89 x^{6}+103 x^{5}+36 x^{4}-193 x^{3}-84 x^{2}+70 x +20}d x \right )-4 \left (\int \frac {x}{30 x^{8}-120 x^{7}+89 x^{6}+103 x^{5}+36 x^{4}-193 x^{3}-84 x^{2}+70 x +20}d x \right )-7 \left (\int \frac {1}{30 x^{8}-120 x^{7}+89 x^{6}+103 x^{5}+36 x^{4}-193 x^{3}-84 x^{2}+70 x +20}d x \right ) \] Input:
int((-5*x^2-4*x-7)/(30*x^8-120*x^7+89*x^6+103*x^5+36*x^4-193*x^3-84*x^2+70 *x+20),x)
Output:
- 5*int(x**2/(30*x**8 - 120*x**7 + 89*x**6 + 103*x**5 + 36*x**4 - 193*x** 3 - 84*x**2 + 70*x + 20),x) - 4*int(x/(30*x**8 - 120*x**7 + 89*x**6 + 103* x**5 + 36*x**4 - 193*x**3 - 84*x**2 + 70*x + 20),x) - 7*int(1/(30*x**8 - 1 20*x**7 + 89*x**6 + 103*x**5 + 36*x**4 - 193*x**3 - 84*x**2 + 70*x + 20),x )