Integrand size = 21, antiderivative size = 346 \[ \int \frac {1}{\left (1+2 x+\sqrt {-2+8 x-5 x^2}\right )^3} \, dx=\frac {2 \sqrt {6} \left (2 \left (351-2411 \sqrt {6}\right )+\frac {21025 \left (156+61 \sqrt {6}\right ) \sqrt {-2+8 x-5 x^2}}{\left (193-52 \sqrt {6}\right ) \left (4-\sqrt {6}-5 x\right )}\right )}{19343 \left (13+2 \sqrt {6}-\frac {10 \sqrt {6} \sqrt {-2+8 x-5 x^2}}{4-\sqrt {6}-5 x}-\frac {5 \left (13-2 \sqrt {6}\right ) \left (2-8 x+5 x^2\right )}{\left (4-\sqrt {6}-5 x\right )^2}\right )^2}+\frac {\sqrt {6} \left (4 \left (73103+17147 \sqrt {6}\right )-\frac {163995 \sqrt {6} \sqrt {-2+8 x-5 x^2}}{4-\sqrt {6}-5 x}\right )}{444889 \left (13+2 \sqrt {6}-\frac {10 \sqrt {6} \sqrt {-2+8 x-5 x^2}}{4-\sqrt {6}-5 x}-\frac {5 \left (13-2 \sqrt {6}\right ) \left (2-8 x+5 x^2\right )}{\left (4-\sqrt {6}-5 x\right )^2}\right )}+\frac {234 \arctan \left (\frac {6+\frac {\left (12-13 \sqrt {6}\right ) \sqrt {-2+8 x-5 x^2}}{4-\sqrt {6}-5 x}}{\sqrt {138}}\right )}{529 \sqrt {23}} \] Output:
2/19343*6^(1/2)*(702-4822*6^(1/2)+21025*(156+61*6^(1/2))*(-5*x^2+8*x-2)^(1 /2)/(193-52*6^(1/2))/(4-6^(1/2)-5*x))/(13+2*6^(1/2)-10*6^(1/2)*(-5*x^2+8*x -2)^(1/2)/(4-6^(1/2)-5*x)-5*(13-2*6^(1/2))*(5*x^2-8*x+2)/(4-6^(1/2)-5*x)^2 )^2+6^(1/2)*(292412+68588*6^(1/2)-163995*6^(1/2)*(-5*x^2+8*x-2)^(1/2)/(4-6 ^(1/2)-5*x))/(5783557+889778*6^(1/2)-4448890*6^(1/2)*(-5*x^2+8*x-2)^(1/2)/ (4-6^(1/2)-5*x)-2224445*(13-2*6^(1/2))*(5*x^2-8*x+2)/(4-6^(1/2)-5*x)^2)+23 4/12167*arctan(1/138*(6+(12-13*6^(1/2))*(-5*x^2+8*x-2)^(1/2)/(4-6^(1/2)-5* x))*138^(1/2))*23^(1/2)
Result contains complex when optimal does not.
Time = 13.67 (sec) , antiderivative size = 1131, normalized size of antiderivative = 3.27 \[ \int \frac {1}{\left (1+2 x+\sqrt {-2+8 x-5 x^2}\right )^3} \, dx =\text {Too large to display} \] Input:
Integrate[(1 + 2*x + Sqrt[-2 + 8*x - 5*x^2])^(-3),x]
Output:
((-2116*(1812 + 1391*x))/(3 - 4*x + 9*x^2)^2 + (92*(3713 + 9477*x))/(3 - 4 *x + 9*x^2) - (3726*Sqrt[-2 + 8*x - 5*x^2]*(-752 + 1205*x - 1064*x^2 + 249 3*x^3))/(3 - 4*x + 9*x^2)^2 + 37908*Sqrt[23]*ArcTan[(-2 + 9*x)/Sqrt[23]] + (18954*(13 - (2*I)*Sqrt[23])*ArcTan[(23*(-52606 + (30056*I)*Sqrt[23] + 8* (56309 - (22984*I)*Sqrt[23])*x + (-1707793 + (363428*I)*Sqrt[23])*x^2 + 14 4*(17501 - (2041*I)*Sqrt[23])*x^3 + (81*I)*(15229*I + 1040*Sqrt[23])*x^4)) /(27378*(-460*I + 67*Sqrt[23])*x^4 - 4*(260*(-782*I + 203*Sqrt[23]) + 7047 *Sqrt[23*(77 - (52*I)*Sqrt[23])]*Sqrt[-2 + 8*x - 5*x^2]) + 9*x^3*(52*(6387 1*I + 494*Sqrt[23]) + 11745*Sqrt[23*(77 - (52*I)*Sqrt[23])]*Sqrt[-2 + 8*x - 5*x^2]) - 2*x^2*(9295312*I + 1752257*Sqrt[23] + 65772*Sqrt[23*(77 - (52* I)*Sqrt[23])]*Sqrt[-2 + 8*x - 5*x^2]) + x*(643448*I + 1971424*Sqrt[23] + 7 2819*Sqrt[23*(77 - (52*I)*Sqrt[23])]*Sqrt[-2 + 8*x - 5*x^2]))])/Sqrt[77/23 - (52*I)/Sqrt[23]] - ((18954*I)*(-13*I + 2*Sqrt[23])*ArcTan[(23*(52606 + (30056*I)*Sqrt[23] + (-450472 - (183872*I)*Sqrt[23])*x + (1707793 + (36342 8*I)*Sqrt[23])*x^2 + (-2520144 - (293904*I)*Sqrt[23])*x^3 + 81*(15229 + (1 040*I)*Sqrt[23])*x^4))/(27378*(460*I + 67*Sqrt[23])*x^4 - 4*(260*(782*I + 203*Sqrt[23]) + 7047*Sqrt[23*(77 + (52*I)*Sqrt[23])]*Sqrt[-2 + 8*x - 5*x^2 ]) + 9*x^3*(52*(-63871*I + 494*Sqrt[23]) + 11745*Sqrt[23*(77 + (52*I)*Sqrt [23])]*Sqrt[-2 + 8*x - 5*x^2]) - 2*x^2*(-9295312*I + 1752257*Sqrt[23] + 65 772*Sqrt[23*(77 + (52*I)*Sqrt[23])]*Sqrt[-2 + 8*x - 5*x^2]) + x*(-64344...
Time = 0.82 (sec) , antiderivative size = 295, normalized size of antiderivative = 0.85, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.095, Rules used = {7293, 2009}
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.
| \(\displaystyle \int \frac {1}{\left (\sqrt {-5 x^2+8 x-2}+2 x+1\right )^3} \, dx\) |
| \(\Big \downarrow \) 7293 |
| \(\displaystyle \int \left (\frac {317-198 x}{81 \left (9 x^2-4 x+3\right )^2}-\frac {7 \sqrt {-5 x^2+8 x-2}}{9 \left (9 x^2-4 x+3\right )^2}+\frac {4 (830 x-339)}{81 \left (9 x^2-4 x+3\right )^3}-\frac {208 x \sqrt {-5 x^2+8 x-2}}{9 \left (9 x^2-4 x+3\right )^3}+\frac {4 \sqrt {-5 x^2+8 x-2}}{3 \left (9 x^2-4 x+3\right )^3}\right )dx\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {117 \arctan \left (\frac {8-13 x}{\sqrt {23} \sqrt {-5 x^2+8 x-2}}\right )}{529 \sqrt {23}}-\frac {117 \arctan \left (\frac {2-9 x}{\sqrt {23}}\right )}{529 \sqrt {23}}-\frac {\sqrt {-5 x^2+8 x-2} (23351-82206 x)}{1334667 \left (9 x^2-4 x+3\right )}-\frac {40-2457 x}{3726 \left (9 x^2-4 x+3\right )}+\frac {1391 (2-9 x)}{28566 \left (9 x^2-4 x+3\right )}+\frac {7 (2-9 x) \sqrt {-5 x^2+8 x-2}}{414 \left (9 x^2-4 x+3\right )}-\frac {130 (1758 x+215) \sqrt {-5 x^2+8 x-2}}{1334667 \left (9 x^2-4 x+3\right )}-\frac {1391 x+1812}{1863 \left (9 x^2-4 x+3\right )^2}-\frac {(2-9 x) \sqrt {-5 x^2+8 x-2}}{69 \left (9 x^2-4 x+3\right )^2}+\frac {52 (3-2 x) \sqrt {-5 x^2+8 x-2}}{207 \left (9 x^2-4 x+3\right )^2}\) |
Input:
Int[(1 + 2*x + Sqrt[-2 + 8*x - 5*x^2])^(-3),x]
Output:
-1/1863*(1812 + 1391*x)/(3 - 4*x + 9*x^2)^2 - ((2 - 9*x)*Sqrt[-2 + 8*x - 5 *x^2])/(69*(3 - 4*x + 9*x^2)^2) + (52*(3 - 2*x)*Sqrt[-2 + 8*x - 5*x^2])/(2 07*(3 - 4*x + 9*x^2)^2) - (40 - 2457*x)/(3726*(3 - 4*x + 9*x^2)) + (1391*( 2 - 9*x))/(28566*(3 - 4*x + 9*x^2)) - ((23351 - 82206*x)*Sqrt[-2 + 8*x - 5 *x^2])/(1334667*(3 - 4*x + 9*x^2)) + (7*(2 - 9*x)*Sqrt[-2 + 8*x - 5*x^2])/ (414*(3 - 4*x + 9*x^2)) - (130*(215 + 1758*x)*Sqrt[-2 + 8*x - 5*x^2])/(133 4667*(3 - 4*x + 9*x^2)) - (117*ArcTan[(2 - 9*x)/Sqrt[23]])/(529*Sqrt[23]) + (117*ArcTan[(8 - 13*x)/(Sqrt[23]*Sqrt[-2 + 8*x - 5*x^2])])/(529*Sqrt[23] )
Result contains higher order function than in optimal. Order 9 vs. order 3.
Time = 0.47 (sec) , antiderivative size = 130, normalized size of antiderivative = 0.38
| method | result | size |
| trager | \(\frac {\left (30537 x^{3}-17667 x^{2}+25891 x -11094\right ) x}{4761 \left (9 x^{2}-4 x +3\right )^{2}}-\frac {\left (2493 x^{3}-1064 x^{2}+1205 x -752\right ) \sqrt {-5 x^{2}+8 x -2}}{1058 \left (9 x^{2}-4 x +3\right )^{2}}+\frac {117 \operatorname {RootOf}\left (\textit {\_Z}^{2}+23\right ) \ln \left (\frac {13 \operatorname {RootOf}\left (\textit {\_Z}^{2}+23\right ) x -8 \operatorname {RootOf}\left (\textit {\_Z}^{2}+23\right )+23 \sqrt {-5 x^{2}+8 x -2}}{\operatorname {RootOf}\left (\textit {\_Z}^{2}+23\right ) x -2 x +3}\right )}{12167}\) | \(130\) |
| default | \(\text {Expression too large to display}\) | \(19721\) |
Input:
int(1/(1+2*x+(-5*x^2+8*x-2)^(1/2))^3,x,method=_RETURNVERBOSE)
Output:
1/4761*(30537*x^3-17667*x^2+25891*x-11094)*x/(9*x^2-4*x+3)^2-1/1058*(2493* x^3-1064*x^2+1205*x-752)/(9*x^2-4*x+3)^2*(-5*x^2+8*x-2)^(1/2)+117/12167*Ro otOf(_Z^2+23)*ln((13*RootOf(_Z^2+23)*x-8*RootOf(_Z^2+23)+23*(-5*x^2+8*x-2) ^(1/2))/(RootOf(_Z^2+23)*x-2*x+3))
Time = 0.08 (sec) , antiderivative size = 174, normalized size of antiderivative = 0.50 \[ \int \frac {1}{\left (1+2 x+\sqrt {-2+8 x-5 x^2}\right )^3} \, dx=\frac {435942 \, x^{3} + 2106 \, \sqrt {23} {\left (81 \, x^{4} - 72 \, x^{3} + 70 \, x^{2} - 24 \, x + 9\right )} \arctan \left (\frac {1}{23} \, \sqrt {23} {\left (9 \, x - 2\right )}\right ) + 1053 \, \sqrt {23} {\left (81 \, x^{4} - 72 \, x^{3} + 70 \, x^{2} - 24 \, x + 9\right )} \arctan \left (\frac {\sqrt {23} {\left (142 \, x^{2} - 196 \, x + 55\right )} \sqrt {-5 \, x^{2} + 8 \, x - 2}}{23 \, {\left (65 \, x^{3} - 144 \, x^{2} + 90 \, x - 16\right )}}\right ) - 22954 \, x^{2} - 207 \, {\left (2493 \, x^{3} - 1064 \, x^{2} + 1205 \, x - 752\right )} \sqrt {-5 \, x^{2} + 8 \, x - 2} - 94116 \, x - 156078}{219006 \, {\left (81 \, x^{4} - 72 \, x^{3} + 70 \, x^{2} - 24 \, x + 9\right )}} \] Input:
integrate(1/(1+2*x+(-5*x^2+8*x-2)^(1/2))^3,x, algorithm="fricas")
Output:
1/219006*(435942*x^3 + 2106*sqrt(23)*(81*x^4 - 72*x^3 + 70*x^2 - 24*x + 9) *arctan(1/23*sqrt(23)*(9*x - 2)) + 1053*sqrt(23)*(81*x^4 - 72*x^3 + 70*x^2 - 24*x + 9)*arctan(1/23*sqrt(23)*(142*x^2 - 196*x + 55)*sqrt(-5*x^2 + 8*x - 2)/(65*x^3 - 144*x^2 + 90*x - 16)) - 22954*x^2 - 207*(2493*x^3 - 1064*x ^2 + 1205*x - 752)*sqrt(-5*x^2 + 8*x - 2) - 94116*x - 156078)/(81*x^4 - 72 *x^3 + 70*x^2 - 24*x + 9)
\[ \int \frac {1}{\left (1+2 x+\sqrt {-2+8 x-5 x^2}\right )^3} \, dx=\int \frac {1}{\left (2 x + \sqrt {- 5 x^{2} + 8 x - 2} + 1\right )^{3}}\, dx \] Input:
integrate(1/(1+2*x+(-5*x**2+8*x-2)**(1/2))**3,x)
Output:
Integral((2*x + sqrt(-5*x**2 + 8*x - 2) + 1)**(-3), x)
\[ \int \frac {1}{\left (1+2 x+\sqrt {-2+8 x-5 x^2}\right )^3} \, dx=\int { \frac {1}{{\left (2 \, x + \sqrt {-5 \, x^{2} + 8 \, x - 2} + 1\right )}^{3}} \,d x } \] Input:
integrate(1/(1+2*x+(-5*x^2+8*x-2)^(1/2))^3,x, algorithm="maxima")
Output:
integrate((2*x + sqrt(-5*x^2 + 8*x - 2) + 1)^(-3), x)
Leaf count of result is larger than twice the leaf count of optimal. 611 vs. \(2 (265) = 530\).
Time = 0.17 (sec) , antiderivative size = 611, normalized size of antiderivative = 1.77 \[ \int \frac {1}{\left (1+2 x+\sqrt {-2+8 x-5 x^2}\right )^3} \, dx =\text {Too large to display} \] Input:
integrate(1/(1+2*x+(-5*x^2+8*x-2)^(1/2))^3,x, algorithm="giac")
Output:
117/12167*sqrt(23)*arctan(1/23*sqrt(23)*(9*x - 2)) + 117/529*(5*sqrt(6) + 13*sqrt(5))*arctan(-(26*sqrt(6) + 12*sqrt(5) - 139*(sqrt(5)*sqrt(-5*x^2 + 8*x - 2) - sqrt(6))/(5*x - 4))/(5*sqrt(138) + 13*sqrt(115)))/(5*sqrt(138) + 13*sqrt(115)) + 117/529*(5*sqrt(6) - 13*sqrt(5))*arctan((26*sqrt(6) - 12 *sqrt(5) - 139*(sqrt(5)*sqrt(-5*x^2 + 8*x - 2) - sqrt(6))/(5*x - 4))/(5*sq rt(138) - 13*sqrt(115)))/(5*sqrt(138) - 13*sqrt(115)) + 1/4761*(9477*x^3 - 499*x^2 - 2046*x - 3393)/(9*x^2 - 4*x + 3)^2 - 6/10220809*(325017862*sqrt (30) + 612698517*sqrt(5)*(sqrt(5)*sqrt(-5*x^2 + 8*x - 2) - sqrt(6))^7/(5*x - 4)^7 + 425483214*sqrt(30)*(sqrt(5)*sqrt(-5*x^2 + 8*x - 2) - sqrt(6))^6/ (5*x - 4)^6 - 3416769843*sqrt(5)*(sqrt(5)*sqrt(-5*x^2 + 8*x - 2) - sqrt(6) )^5/(5*x - 4)^5 + 3469285170*sqrt(30)*(sqrt(5)*sqrt(-5*x^2 + 8*x - 2) - sq rt(6))^4/(5*x - 4)^4 - 9872341005*sqrt(5)*(sqrt(5)*sqrt(-5*x^2 + 8*x - 2) - sqrt(6))^3/(5*x - 4)^3 + 2976396202*sqrt(30)*(sqrt(5)*sqrt(-5*x^2 + 8*x - 2) - sqrt(6))^2/(5*x - 4)^2 - 3530844501*sqrt(5)*(sqrt(5)*sqrt(-5*x^2 + 8*x - 2) - sqrt(6))/(5*x - 4))/(104*sqrt(6)*(sqrt(5)*sqrt(-5*x^2 + 8*x - 2 ) - sqrt(6))^3/(5*x - 4)^3 + 104*sqrt(6)*(sqrt(5)*sqrt(-5*x^2 + 8*x - 2) - sqrt(6))/(5*x - 4) - 139*(sqrt(5)*sqrt(-5*x^2 + 8*x - 2) - sqrt(6))^4/(5* x - 4)^4 - 494*(sqrt(5)*sqrt(-5*x^2 + 8*x - 2) - sqrt(6))^2/(5*x - 4)^2 - 139)^2
Timed out. \[ \int \frac {1}{\left (1+2 x+\sqrt {-2+8 x-5 x^2}\right )^3} \, dx=\int \frac {1}{{\left (2\,x+\sqrt {-5\,x^2+8\,x-2}+1\right )}^3} \,d x \] Input:
int(1/(2*x + (8*x - 5*x^2 - 2)^(1/2) + 1)^3,x)
Output:
int(1/(2*x + (8*x - 5*x^2 - 2)^(1/2) + 1)^3, x)
\[ \int \frac {1}{\left (1+2 x+\sqrt {-2+8 x-5 x^2}\right )^3} \, dx=\text {too large to display} \] Input:
int(1/(1+2*x+(-5*x^2+8*x-2)^(1/2))^3,x)
Output:
(6305105166340426542068375858239598965142941347790401271812782964386820433 3514153903189204996349656250000*sqrt(5)*asin((5*x - 4)/sqrt(6))*tan(asin(( 5*x - 4)/sqrt(6))/2)**16*x**8 - 403526730645787298692376054927334333769148 246258585681396018109720756507734490584980410911976637800000000*sqrt(5)*as in((5*x - 4)/sqrt(6))*tan(asin((5*x - 4)/sqrt(6))/2)**16*x**7 + 1117295771 79733267178064987307194947785278670283511486043323409330398352805127007341 3353793624202995000000*sqrt(5)*asin((5*x - 4)/sqrt(6))*tan(asin((5*x - 4)/ sqrt(6))/2)**16*x**6 - 174742019804094262826342993859644186312180788713162 3624771201384761142625345060710959616438085499688000000*sqrt(5)*asin((5*x - 4)/sqrt(6))*tan(asin((5*x - 4)/sqrt(6))/2)**16*x**5 + 173579483341107359 20619538094487079285436119188668120903071024289507093356091649925268897500 85887416700000*sqrt(5)*asin((5*x - 4)/sqrt(6))*tan(asin((5*x - 4)/sqrt(6)) /2)**16*x**4 - 11841955342276838615030327847247880934409160930725161984887 03528838336163080619356673908175961816798400000*sqrt(5)*asin((5*x - 4)/sqr t(6))*tan(asin((5*x - 4)/sqrt(6))/2)**16*x**3 + 56792620227854697259395016 45052172841567819021811536182110081153895621077027507070958984086477234510 00000*sqrt(5)*asin((5*x - 4)/sqrt(6))*tan(asin((5*x - 4)/sqrt(6))/2)**16*x **2 - 17321795451139458148788893523426705709547609529107918107413380578919 4532880236801671207096879913924800000*sqrt(5)*asin((5*x - 4)/sqrt(6))*tan( asin((5*x - 4)/sqrt(6))/2)**16*x + 333616693246972043059179395397588095...