Integrand size = 23, antiderivative size = 340 \[ \int \frac {x}{\left (1+2 x+\sqrt {-2+8 x-5 x^2}\right )^3} \, dx=\frac {10 \sqrt {6} \left (3288+1387 \sqrt {6}-\frac {84100 \left (9-4 \sqrt {6}\right ) \sqrt {-2+8 x-5 x^2}}{\left (193-52 \sqrt {6}\right ) \left (4-\sqrt {6}-5 x\right )}\right )}{19343 \left (13+2 \sqrt {6}-\frac {10 \sqrt {6} \sqrt {-2+8 x-5 x^2}}{4-\sqrt {6}-5 x}-\frac {5 \left (13-2 \sqrt {6}\right ) \left (2-8 x+5 x^2\right )}{\left (4-\sqrt {6}-5 x\right )^2}\right )^2}+\frac {2 \sqrt {6} \left (75812+5763 \sqrt {6}-\frac {50460 \sqrt {6} \sqrt {-2+8 x-5 x^2}}{4-\sqrt {6}-5 x}\right )}{444889 \left (13+2 \sqrt {6}-\frac {10 \sqrt {6} \sqrt {-2+8 x-5 x^2}}{4-\sqrt {6}-5 x}-\frac {5 \left (13-2 \sqrt {6}\right ) \left (2-8 x+5 x^2\right )}{\left (4-\sqrt {6}-5 x\right )^2}\right )}+\frac {144 \arctan \left (\frac {6+\frac {\left (12-13 \sqrt {6}\right ) \sqrt {-2+8 x-5 x^2}}{4-\sqrt {6}-5 x}}{\sqrt {138}}\right )}{529 \sqrt {23}} \] Output:
10/19343*6^(1/2)*(3288+1387*6^(1/2)-84100*(9-4*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+2*6^(1/2)*(75812+5763*6^(1/2)-50460*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)+144/ 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 = 13.24 (sec) , antiderivative size = 1147, normalized size of antiderivative = 3.37 \[ \int \frac {x}{\left (1+2 x+\sqrt {-2+8 x-5 x^2}\right )^3} \, dx =\text {Too large to display} \] Input:
Integrate[x/(1 + 2*x + Sqrt[-2 + 8*x - 5*x^2])^3,x]
Output:
(4173 - 21872*x)/(16767*(3 - 4*x + 9*x^2)^2) + (-191021 + 314460*x)/(77128 2*(3 - 4*x + 9*x^2)) + (Sqrt[-2 + 8*x - 5*x^2]*(150 - 208*x + 775*x^2 - 11 74*x^3))/(529*(3 - 4*x + 9*x^2)^2) + (72*ArcTan[(-2 + 9*x)/Sqrt[23]])/(529 *Sqrt[23]) + (36*(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 + 144*(17501 - (2041*I)*Sqrt[23])*x^3 + (81*I)*(15229*I + 1040*Sqrt[2 3])*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] + 65772*Sqrt[23*( 77 - (52*I)*Sqrt[23])]*Sqrt[-2 + 8*x - 5*x^2]) + x*(643448*I + 1971424*Sqr t[23] + 72819*Sqrt[23*(77 - (52*I)*Sqrt[23])]*Sqrt[-2 + 8*x - 5*x^2]))])/( 529*Sqrt[23*(77 - (52*I)*Sqrt[23])]) - (((36*I)/529)*(-13*I + 2*Sqrt[23])* ArcTan[(23*(52606 + (30056*I)*Sqrt[23] + (-450472 - (183872*I)*Sqrt[23])*x + (1707793 + (363428*I)*Sqrt[23])*x^2 + (-2520144 - (293904*I)*Sqrt[23])* x^3 + 81*(15229 + (1040*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])]*S qrt[-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 + 1 752257*Sqrt[23] + 65772*Sqrt[23*(77 + (52*I)*Sqrt[23])]*Sqrt[-2 + 8*x -...
Time = 0.99 (sec) , antiderivative size = 330, normalized size of antiderivative = 0.97, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.087, 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.
| \(\displaystyle \int \frac {x}{\left (\sqrt {-5 x^2+8 x-2}+2 x+1\right )^3} \, dx\) |
| \(\Big \downarrow \) 7293 |
| \(\displaystyle \int \left (-\frac {7 \sqrt {-5 x^2+8 x-2} x}{9 \left (9 x^2-4 x+3\right )^2}-\frac {724 \sqrt {-5 x^2+8 x-2} x}{81 \left (9 x^2-4 x+3\right )^3}-\frac {22}{81 \left (9 x^2-4 x+3\right )}+\frac {2061 x+3914}{729 \left (9 x^2-4 x+3\right )^2}-\frac {208 \sqrt {-5 x^2+8 x-2}}{81 \left (9 x^2-4 x+3\right )^2}+\frac {4 (269 x-2490)}{729 \left (9 x^2-4 x+3\right )^3}+\frac {208 \sqrt {-5 x^2+8 x-2}}{27 \left (9 x^2-4 x+3\right )^3}\right )dx\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {72 \arctan \left (\frac {8-13 x}{\sqrt {23} \sqrt {-5 x^2+8 x-2}}\right )}{529 \sqrt {23}}-\frac {72 \arctan \left (\frac {2-9 x}{\sqrt {23}}\right )}{529 \sqrt {23}}-\frac {52 \sqrt {-5 x^2+8 x-2} (23351-82206 x)}{12012003 \left (9 x^2-4 x+3\right )}-\frac {14011-39348 x}{33534 \left (9 x^2-4 x+3\right )}+\frac {10936 (2-9 x)}{128547 \left (9 x^2-4 x+3\right )}+\frac {104 (2-9 x) \sqrt {-5 x^2+8 x-2}}{1863 \left (9 x^2-4 x+3\right )}+\frac {7 (3-2 x) \sqrt {-5 x^2+8 x-2}}{414 \left (9 x^2-4 x+3\right )}-\frac {905 (1758 x+215) \sqrt {-5 x^2+8 x-2}}{24024006 \left (9 x^2-4 x+3\right )}+\frac {4173-21872 x}{16767 \left (9 x^2-4 x+3\right )^2}-\frac {52 (2-9 x) \sqrt {-5 x^2+8 x-2}}{621 \left (9 x^2-4 x+3\right )^2}+\frac {181 (3-2 x) \sqrt {-5 x^2+8 x-2}}{1863 \left (9 x^2-4 x+3\right )^2}\) |
Input:
Int[x/(1 + 2*x + Sqrt[-2 + 8*x - 5*x^2])^3,x]
Output:
(4173 - 21872*x)/(16767*(3 - 4*x + 9*x^2)^2) - (52*(2 - 9*x)*Sqrt[-2 + 8*x - 5*x^2])/(621*(3 - 4*x + 9*x^2)^2) + (181*(3 - 2*x)*Sqrt[-2 + 8*x - 5*x^ 2])/(1863*(3 - 4*x + 9*x^2)^2) - (14011 - 39348*x)/(33534*(3 - 4*x + 9*x^2 )) + (10936*(2 - 9*x))/(128547*(3 - 4*x + 9*x^2)) - (52*(23351 - 82206*x)* Sqrt[-2 + 8*x - 5*x^2])/(12012003*(3 - 4*x + 9*x^2)) + (104*(2 - 9*x)*Sqrt [-2 + 8*x - 5*x^2])/(1863*(3 - 4*x + 9*x^2)) + (7*(3 - 2*x)*Sqrt[-2 + 8*x - 5*x^2])/(414*(3 - 4*x + 9*x^2)) - (905*(215 + 1758*x)*Sqrt[-2 + 8*x - 5* x^2])/(24024006*(3 - 4*x + 9*x^2)) - (72*ArcTan[(2 - 9*x)/Sqrt[23]])/(529* Sqrt[23]) + (72*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 = 0.51 (sec) , antiderivative size = 130, normalized size of antiderivative = 0.38
| method | result | size |
| trager | \(\frac {\left (14115 x^{3}-900 x^{2}-53 x -1296\right ) x}{3174 \left (9 x^{2}-4 x +3\right )^{2}}-\frac {\left (1174 x^{3}-775 x^{2}+208 x -150\right ) \sqrt {-5 x^{2}+8 x -2}}{529 \left (9 x^{2}-4 x +3\right )^{2}}+\frac {72 \operatorname {RootOf}\left (\textit {\_Z}^{2}+23\right ) \ln \left (\frac {13 \operatorname {RootOf}\left (\textit {\_Z}^{2}+23\right ) x -8 \operatorname {RootOf}\left (\textit {\_Z}^{2}+23\right )+23 \sqrt {-5 x^{2}+8 x -2}}{\operatorname {RootOf}\left (\textit {\_Z}^{2}+23\right ) x -2 x +3}\right )}{12167}\) | \(130\) |
| default | \(\text {Expression too large to display}\) | \(23877\) |
Input:
int(x/(1+2*x+(-5*x^2+8*x-2)^(1/2))^3,x,method=_RETURNVERBOSE)
Output:
1/3174*(14115*x^3-900*x^2-53*x-1296)*x/(9*x^2-4*x+3)^2-1/529*(1174*x^3-775 *x^2+208*x-150)/(9*x^2-4*x+3)^2*(-5*x^2+8*x-2)^(1/2)+72/12167*RootOf(_Z^2+ 23)*ln((13*RootOf(_Z^2+23)*x-8*RootOf(_Z^2+23)+23*(-5*x^2+8*x-2)^(1/2))/(R ootOf(_Z^2+23)*x-2*x+3))
Time = 0.12 (sec) , antiderivative size = 174, normalized size of antiderivative = 0.51 \[ \int \frac {x}{\left (1+2 x+\sqrt {-2+8 x-5 x^2}\right )^3} \, dx=\frac {7232580 \, x^{3} + 11664 \, \sqrt {23} {\left (81 \, x^{4} - 72 \, x^{3} + 70 \, x^{2} - 24 \, x + 9\right )} \arctan \left (\frac {1}{23} \, \sqrt {23} {\left (9 \, x - 2\right )}\right ) + 5832 \, \sqrt {23} {\left (81 \, x^{4} - 72 \, x^{3} + 70 \, x^{2} - 24 \, x + 9\right )} \arctan \left (\frac {\sqrt {23} {\left (142 \, x^{2} - 196 \, x + 55\right )} \sqrt {-5 \, x^{2} + 8 \, x - 2}}{23 \, {\left (65 \, x^{3} - 144 \, x^{2} + 90 \, x - 16\right )}}\right ) - 7607963 \, x^{2} - 3726 \, {\left (1174 \, x^{3} - 775 \, x^{2} + 208 \, x - 150\right )} \sqrt {-5 \, x^{2} + 8 \, x - 2} + 1792344 \, x - 973935}{1971054 \, {\left (81 \, x^{4} - 72 \, x^{3} + 70 \, x^{2} - 24 \, x + 9\right )}} \] Input:
integrate(x/(1+2*x+(-5*x^2+8*x-2)^(1/2))^3,x, algorithm="fricas")
Output:
1/1971054*(7232580*x^3 + 11664*sqrt(23)*(81*x^4 - 72*x^3 + 70*x^2 - 24*x + 9)*arctan(1/23*sqrt(23)*(9*x - 2)) + 5832*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)) - 7607963*x^2 - 3726*(1174*x^3 - 775*x^2 + 208*x - 150)*sqrt(-5*x^2 + 8*x - 2) + 1792344*x - 973935)/(81*x^ 4 - 72*x^3 + 70*x^2 - 24*x + 9)
\[ \int \frac {x}{\left (1+2 x+\sqrt {-2+8 x-5 x^2}\right )^3} \, dx=\int \frac {x}{\left (2 x + \sqrt {- 5 x^{2} + 8 x - 2} + 1\right )^{3}}\, dx \] Input:
integrate(x/(1+2*x+(-5*x**2+8*x-2)**(1/2))**3,x)
Output:
Integral(x/(2*x + sqrt(-5*x**2 + 8*x - 2) + 1)**3, x)
\[ \int \frac {x}{\left (1+2 x+\sqrt {-2+8 x-5 x^2}\right )^3} \, dx=\int { \frac {x}{{\left (2 \, x + \sqrt {-5 \, x^{2} + 8 \, x - 2} + 1\right )}^{3}} \,d x } \] Input:
integrate(x/(1+2*x+(-5*x^2+8*x-2)^(1/2))^3,x, algorithm="maxima")
Output:
integrate(x/(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 = 0.21 (sec) , antiderivative size = 611, normalized size of antiderivative = 1.80 \[ \int \frac {x}{\left (1+2 x+\sqrt {-2+8 x-5 x^2}\right )^3} \, dx =\text {Too large to display} \] Input:
integrate(x/(1+2*x+(-5*x^2+8*x-2)^(1/2))^3,x, algorithm="giac")
Output:
72/12167*sqrt(23)*arctan(1/23*sqrt(23)*(9*x - 2)) + 72/529*(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)) + 72/529*(5*sqrt(6) - 13*sqrt(5))*arctan((26*sqrt(6) - 12*sq rt(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)) + 1/85698*(314460*x^3 - 330781*x^2 + 77928*x - 42345)/(9*x^2 - 4*x + 3)^2 - 12/10220809*(48901451 *sqrt(30) + 370766376*sqrt(5)*(sqrt(5)*sqrt(-5*x^2 + 8*x - 2) - sqrt(6))^7 /(5*x - 4)^7 - 187632663*sqrt(30)*(sqrt(5)*sqrt(-5*x^2 + 8*x - 2) - sqrt(6 ))^6/(5*x - 4)^6 + 134958936*sqrt(5)*(sqrt(5)*sqrt(-5*x^2 + 8*x - 2) - sqr t(6))^5/(5*x - 4)^5 + 521980785*sqrt(30)*(sqrt(5)*sqrt(-5*x^2 + 8*x - 2) - sqrt(6))^4/(5*x - 4)^4 - 2134408440*sqrt(5)*(sqrt(5)*sqrt(-5*x^2 + 8*x - 2) - sqrt(6))^3/(5*x - 4)^3 + 699471731*sqrt(30)*(sqrt(5)*sqrt(-5*x^2 + 8* x - 2) - sqrt(6))^2/(5*x - 4)^2 - 809824008*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. \[ \int \frac {x}{\left (1+2 x+\sqrt {-2+8 x-5 x^2}\right )^3} \, dx=\int \frac {x}{{\left (2\,x+\sqrt {-5\,x^2+8\,x-2}+1\right )}^3} \,d x \] Input:
int(x/(2*x + (8*x - 5*x^2 - 2)^(1/2) + 1)^3,x)
Output:
int(x/(2*x + (8*x - 5*x^2 - 2)^(1/2) + 1)^3, x)
\[ \int \frac {x}{\left (1+2 x+\sqrt {-2+8 x-5 x^2}\right )^3} \, dx=\text {too large to display} \] Input:
int(x/(1+2*x+(-5*x^2+8*x-2)^(1/2))^3,x)
Output:
(3880064717747954795119000528147445517011040829409477705730943362699581805 1393325478885664613138250000000*sqrt(5)*asin((5*x - 4)/sqrt(6))*tan(asin(( 5*x - 4)/sqrt(6))/2)**16*x**8 - 248324141935869106887616033801436513088706 613082206573166780375212773235528917283064868253524084800000000*sqrt(5)*as in((5*x - 4)/sqrt(6))*tan(asin((5*x - 4)/sqrt(6))/2)**16*x**7 + 6875666287 98358567249630691121199678678637970975455298728144057417836017262320045177 448488384124920000000*sqrt(5)*asin((5*x - 4)/sqrt(6))*tan(asin((5*x - 4)/s qrt(6))/2)**16*x**6 - 1075335506486733925085187654520887300382651007465614 538320739313699164692520037360590533192667999808000000*sqrt(5)*asin((5*x - 4)/sqrt(6))*tan(asin((5*x - 4)/sqrt(6))/2)**16*x**5 + 1068181435945276056 65351003658382026371914579622573051711206303320043651422102461078577830774 5161487200000*sqrt(5)*asin((5*x - 4)/sqrt(6))*tan(asin((5*x - 4)/sqrt(6))/ 2)**16*x**4 - 728735713370882376309558636753715749809794518813856122146894 479285129946511150373337789646745733414400000*sqrt(5)*asin((5*x - 4)/sqrt( 6))*tan(asin((5*x - 4)/sqrt(6))/2)**16*x**3 + 3494930475560289062116616396 95518328711865785957632995822158840239730527817077358212860559167829816000 000*sqrt(5)*asin((5*x - 4)/sqrt(6))*tan(asin((5*x - 4)/sqrt(6))/2)**16*x** 2 - 1065956643147043578387008832210874197510622124868179575840823420241197 12541684185643819751926100876800000*sqrt(5)*asin((5*x - 4)/sqrt(6))*tan(as in((5*x - 4)/sqrt(6))/2)**16*x + 20530258045967510342103347409082344311...