3.64 \(\int \frac{\sqrt{3-x+2 x^2}}{(2+3 x+5 x^2)^3} \, dx\)
Optimal. Leaf size=223 \[ \frac{\sqrt{2 x^2-x+3} (10 x+3)}{62 \left (5 x^2+3 x+2\right )^2}+\frac{(13665 x+3464) \sqrt{2 x^2-x+3}}{84568 \left (5 x^2+3 x+2\right )}+\frac{\sqrt{\frac{1}{682} \left (112285869463+79399380740 \sqrt{2}\right )} \tan ^{-1}\left (\frac{\sqrt{\frac{11}{31 \left (112285869463+79399380740 \sqrt{2}\right )}} \left (\left (1235163+872375 \sqrt{2}\right ) x+362788 \sqrt{2}+509587\right )}{\sqrt{2 x^2-x+3}}\right )}{169136}-\frac{\sqrt{\frac{1}{682} \left (79399380740 \sqrt{2}-112285869463\right )} \tanh ^{-1}\left (\frac{\sqrt{\frac{11}{31 \left (79399380740 \sqrt{2}-112285869463\right )}} \left (\left (1235163-872375 \sqrt{2}\right ) x-362788 \sqrt{2}+509587\right )}{\sqrt{2 x^2-x+3}}\right )}{169136} \]
[Out]
((3 + 10*x)*Sqrt[3 - x + 2*x^2])/(62*(2 + 3*x + 5*x^2)^2) + ((3464 + 13665*x)*Sqrt[3 - x + 2*x^2])/(84568*(2 +
3*x + 5*x^2)) + (Sqrt[(112285869463 + 79399380740*Sqrt[2])/682]*ArcTan[(Sqrt[11/(31*(112285869463 + 793993807
40*Sqrt[2]))]*(509587 + 362788*Sqrt[2] + (1235163 + 872375*Sqrt[2])*x))/Sqrt[3 - x + 2*x^2]])/169136 - (Sqrt[(
-112285869463 + 79399380740*Sqrt[2])/682]*ArcTanh[(Sqrt[11/(31*(-112285869463 + 79399380740*Sqrt[2]))]*(509587
- 362788*Sqrt[2] + (1235163 - 872375*Sqrt[2])*x))/Sqrt[3 - x + 2*x^2]])/169136
________________________________________________________________________________________
Rubi [A] time = 0.45864, antiderivative size = 223, normalized size of antiderivative = 1.,
number of steps used = 7, number of rules used = 6, integrand size = 27, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.222, Rules used =
{971, 1060, 1035, 1029, 206, 204} \[ \frac{\sqrt{2 x^2-x+3} (10 x+3)}{62 \left (5 x^2+3 x+2\right )^2}+\frac{(13665 x+3464) \sqrt{2 x^2-x+3}}{84568 \left (5 x^2+3 x+2\right )}+\frac{\sqrt{\frac{1}{682} \left (112285869463+79399380740 \sqrt{2}\right )} \tan ^{-1}\left (\frac{\sqrt{\frac{11}{31 \left (112285869463+79399380740 \sqrt{2}\right )}} \left (\left (1235163+872375 \sqrt{2}\right ) x+362788 \sqrt{2}+509587\right )}{\sqrt{2 x^2-x+3}}\right )}{169136}-\frac{\sqrt{\frac{1}{682} \left (79399380740 \sqrt{2}-112285869463\right )} \tanh ^{-1}\left (\frac{\sqrt{\frac{11}{31 \left (79399380740 \sqrt{2}-112285869463\right )}} \left (\left (1235163-872375 \sqrt{2}\right ) x-362788 \sqrt{2}+509587\right )}{\sqrt{2 x^2-x+3}}\right )}{169136} \]
Antiderivative was successfully verified.
[In]
Int[Sqrt[3 - x + 2*x^2]/(2 + 3*x + 5*x^2)^3,x]
[Out]
((3 + 10*x)*Sqrt[3 - x + 2*x^2])/(62*(2 + 3*x + 5*x^2)^2) + ((3464 + 13665*x)*Sqrt[3 - x + 2*x^2])/(84568*(2 +
3*x + 5*x^2)) + (Sqrt[(112285869463 + 79399380740*Sqrt[2])/682]*ArcTan[(Sqrt[11/(31*(112285869463 + 793993807
40*Sqrt[2]))]*(509587 + 362788*Sqrt[2] + (1235163 + 872375*Sqrt[2])*x))/Sqrt[3 - x + 2*x^2]])/169136 - (Sqrt[(
-112285869463 + 79399380740*Sqrt[2])/682]*ArcTanh[(Sqrt[11/(31*(-112285869463 + 79399380740*Sqrt[2]))]*(509587
- 362788*Sqrt[2] + (1235163 - 872375*Sqrt[2])*x))/Sqrt[3 - x + 2*x^2]])/169136
Rule 971
Int[((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_)*((d_.) + (e_.)*(x_) + (f_.)*(x_)^2)^(q_), x_Symbol] :> Simp[((b +
2*c*x)*(a + b*x + c*x^2)^(p + 1)*(d + e*x + f*x^2)^q)/((b^2 - 4*a*c)*(p + 1)), x] - Dist[1/((b^2 - 4*a*c)*(p
+ 1)), Int[(a + b*x + c*x^2)^(p + 1)*(d + e*x + f*x^2)^(q - 1)*Simp[2*c*d*(2*p + 3) + b*e*q + (2*b*f*q + 2*c*e
*(2*p + q + 3))*x + 2*c*f*(2*p + 2*q + 3)*x^2, x], x], x] /; FreeQ[{a, b, c, d, e, f}, x] && NeQ[b^2 - 4*a*c,
0] && NeQ[e^2 - 4*d*f, 0] && LtQ[p, -1] && GtQ[q, 0] && !IGtQ[q, 0]
Rule 1060
Int[((a_) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_)*((A_.) + (B_.)*(x_) + (C_.)*(x_)^2)*((d_) + (e_.)*(x_) + (f_.)*(x_
)^2)^(q_), x_Symbol] :> Simp[((a + b*x + c*x^2)^(p + 1)*(d + e*x + f*x^2)^(q + 1)*((A*c - a*C)*(2*a*c*e - b*(c
*d + a*f)) + (A*b - a*B)*(2*c^2*d + b^2*f - c*(b*e + 2*a*f)) + c*(A*(2*c^2*d + b^2*f - c*(b*e + 2*a*f)) - B*(b
*c*d - 2*a*c*e + a*b*f) + C*(b^2*d - a*b*e - 2*a*(c*d - a*f)))*x))/((b^2 - 4*a*c)*((c*d - a*f)^2 - (b*d - a*e)
*(c*e - b*f))*(p + 1)), x] + Dist[1/((b^2 - 4*a*c)*((c*d - a*f)^2 - (b*d - a*e)*(c*e - b*f))*(p + 1)), Int[(a
+ b*x + c*x^2)^(p + 1)*(d + e*x + f*x^2)^q*Simp[(b*B - 2*A*c - 2*a*C)*((c*d - a*f)^2 - (b*d - a*e)*(c*e - b*f)
)*(p + 1) + (b^2*(C*d + A*f) - b*(B*c*d + A*c*e + a*C*e + a*B*f) + 2*(A*c*(c*d - a*f) - a*(c*C*d - B*c*e - a*C
*f)))*(a*f*(p + 1) - c*d*(p + 2)) - e*((A*c - a*C)*(2*a*c*e - b*(c*d + a*f)) + (A*b - a*B)*(2*c^2*d + b^2*f -
c*(b*e + 2*a*f)))*(p + q + 2) - (2*f*((A*c - a*C)*(2*a*c*e - b*(c*d + a*f)) + (A*b - a*B)*(2*c^2*d + b^2*f - c
*(b*e + 2*a*f)))*(p + q + 2) - (b^2*(C*d + A*f) - b*(B*c*d + A*c*e + a*C*e + a*B*f) + 2*(A*c*(c*d - a*f) - a*(
c*C*d - B*c*e - a*C*f)))*(b*f*(p + 1) - c*e*(2*p + q + 4)))*x - c*f*(b^2*(C*d + A*f) - b*(B*c*d + A*c*e + a*C*
e + a*B*f) + 2*(A*c*(c*d - a*f) - a*(c*C*d - B*c*e - a*C*f)))*(2*p + 2*q + 5)*x^2, x], x], x] /; FreeQ[{a, b,
c, d, e, f, A, B, C, q}, x] && NeQ[b^2 - 4*a*c, 0] && NeQ[e^2 - 4*d*f, 0] && LtQ[p, -1] && NeQ[(c*d - a*f)^2 -
(b*d - a*e)*(c*e - b*f), 0] && !( !IntegerQ[p] && ILtQ[q, -1]) && !IGtQ[q, 0]
Rule 1035
Int[((g_.) + (h_.)*(x_))/(((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)*Sqrt[(d_.) + (e_.)*(x_) + (f_.)*(x_)^2]), x_Symb
ol] :> With[{q = Rt[(c*d - a*f)^2 - (b*d - a*e)*(c*e - b*f), 2]}, Dist[1/(2*q), Int[Simp[h*(b*d - a*e) - g*(c*
d - a*f - q) - (g*(c*e - b*f) - h*(c*d - a*f + q))*x, x]/((a + b*x + c*x^2)*Sqrt[d + e*x + f*x^2]), x], x] - D
ist[1/(2*q), Int[Simp[h*(b*d - a*e) - g*(c*d - a*f + q) - (g*(c*e - b*f) - h*(c*d - a*f - q))*x, x]/((a + b*x
+ c*x^2)*Sqrt[d + e*x + f*x^2]), x], x]] /; FreeQ[{a, b, c, d, e, f, g, h}, x] && NeQ[b^2 - 4*a*c, 0] && NeQ[e
^2 - 4*d*f, 0] && NeQ[b*d - a*e, 0] && NegQ[b^2 - 4*a*c]
Rule 1029
Int[((g_.) + (h_.)*(x_))/(((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)*Sqrt[(d_.) + (e_.)*(x_) + (f_.)*(x_)^2]), x_Symb
ol] :> Dist[-2*g*(g*b - 2*a*h), Subst[Int[1/Simp[g*(g*b - 2*a*h)*(b^2 - 4*a*c) - (b*d - a*e)*x^2, x], x], x, S
imp[g*b - 2*a*h - (b*h - 2*g*c)*x, x]/Sqrt[d + e*x + f*x^2]], x] /; FreeQ[{a, b, c, d, e, f, g, h}, x] && NeQ[
b^2 - 4*a*c, 0] && NeQ[e^2 - 4*d*f, 0] && NeQ[b*d - a*e, 0] && EqQ[h^2*(b*d - a*e) - 2*g*h*(c*d - a*f) + g^2*(
c*e - b*f), 0]
Rule 206
Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1*ArcTanh[(Rt[-b, 2]*x)/Rt[a, 2]])/(Rt[a, 2]*Rt[-b, 2]), x]
/; FreeQ[{a, b}, x] && NegQ[a/b] && (GtQ[a, 0] || LtQ[b, 0])
Rule 204
Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> -Simp[ArcTan[(Rt[-b, 2]*x)/Rt[-a, 2]]/(Rt[-a, 2]*Rt[-b, 2]), x] /
; FreeQ[{a, b}, x] && PosQ[a/b] && (LtQ[a, 0] || LtQ[b, 0])
Rubi steps
\begin{align*} \int \frac{\sqrt{3-x+2 x^2}}{\left (2+3 x+5 x^2\right )^3} \, dx &=\frac{(3+10 x) \sqrt{3-x+2 x^2}}{62 \left (2+3 x+5 x^2\right )^2}-\frac{1}{62} \int \frac{-\frac{183}{2}+31 x-40 x^2}{\sqrt{3-x+2 x^2} \left (2+3 x+5 x^2\right )^2} \, dx\\ &=\frac{(3+10 x) \sqrt{3-x+2 x^2}}{62 \left (2+3 x+5 x^2\right )^2}+\frac{(3464+13665 x) \sqrt{3-x+2 x^2}}{84568 \left (2+3 x+5 x^2\right )}-\frac{\int \frac{-213004+\frac{358655 x}{4}}{\sqrt{3-x+2 x^2} \left (2+3 x+5 x^2\right )} \, dx}{465124}\\ &=\frac{(3+10 x) \sqrt{3-x+2 x^2}}{62 \left (2+3 x+5 x^2\right )^2}+\frac{(3464+13665 x) \sqrt{3-x+2 x^2}}{84568 \left (2+3 x+5 x^2\right )}-\frac{\int \frac{\frac{121}{4} \left (110061-77456 \sqrt{2}\right )-\frac{121}{4} \left (44851-32605 \sqrt{2}\right ) x}{\sqrt{3-x+2 x^2} \left (2+3 x+5 x^2\right )} \, dx}{10232728 \sqrt{2}}+\frac{\int \frac{\frac{121}{4} \left (110061+77456 \sqrt{2}\right )-\frac{121}{4} \left (44851+32605 \sqrt{2}\right ) x}{\sqrt{3-x+2 x^2} \left (2+3 x+5 x^2\right )} \, dx}{10232728 \sqrt{2}}\\ &=\frac{(3+10 x) \sqrt{3-x+2 x^2}}{62 \left (2+3 x+5 x^2\right )^2}+\frac{(3464+13665 x) \sqrt{3-x+2 x^2}}{84568 \left (2+3 x+5 x^2\right )}-\frac{\left (11 \left (158798761480-112285869463 \sqrt{2}\right )\right ) \operatorname{Subst}\left (\int \frac{1}{-\frac{453871}{16} \left (112285869463-79399380740 \sqrt{2}\right )-11 x^2} \, dx,x,\frac{\frac{121}{4} \left (509587-362788 \sqrt{2}\right )+\frac{121}{4} \left (1235163-872375 \sqrt{2}\right ) x}{\sqrt{3-x+2 x^2}}\right )}{123008}-\frac{\left (11 \left (158798761480+112285869463 \sqrt{2}\right )\right ) \operatorname{Subst}\left (\int \frac{1}{-\frac{453871}{16} \left (112285869463+79399380740 \sqrt{2}\right )-11 x^2} \, dx,x,\frac{\frac{121}{4} \left (509587+362788 \sqrt{2}\right )+\frac{121}{4} \left (1235163+872375 \sqrt{2}\right ) x}{\sqrt{3-x+2 x^2}}\right )}{123008}\\ &=\frac{(3+10 x) \sqrt{3-x+2 x^2}}{62 \left (2+3 x+5 x^2\right )^2}+\frac{(3464+13665 x) \sqrt{3-x+2 x^2}}{84568 \left (2+3 x+5 x^2\right )}+\frac{\sqrt{\frac{1}{682} \left (112285869463+79399380740 \sqrt{2}\right )} \tan ^{-1}\left (\frac{\sqrt{\frac{11}{31 \left (112285869463+79399380740 \sqrt{2}\right )}} \left (509587+362788 \sqrt{2}+\left (1235163+872375 \sqrt{2}\right ) x\right )}{\sqrt{3-x+2 x^2}}\right )}{169136}-\frac{\sqrt{\frac{1}{682} \left (-112285869463+79399380740 \sqrt{2}\right )} \tanh ^{-1}\left (\frac{\sqrt{\frac{11}{31 \left (-112285869463+79399380740 \sqrt{2}\right )}} \left (509587-362788 \sqrt{2}+\left (1235163-872375 \sqrt{2}\right ) x\right )}{\sqrt{3-x+2 x^2}}\right )}{169136}\\ \end{align*}
Mathematica [C] time = 2.06866, size = 299, normalized size = 1.34 \[ \frac{5 \left (\frac{i \sqrt{286+22 i \sqrt{31}} \left (258253 \sqrt{31}+1004586 i\right ) \tanh ^{-1}\left (\frac{\left (-22-4 i \sqrt{31}\right ) x+i \sqrt{31}+63}{2 \sqrt{286+22 i \sqrt{31}} \sqrt{2 x^2-x+3}}\right )}{\left (\sqrt{31}-13 i\right )^2}+\frac{2000 \left (1364 \left (\sqrt{31}+13 i\right ) \sqrt{2 x^2-x+3} \left (68325 x^3+58315 x^2+51362 x+11020\right )-5 \sqrt{286-22 i \sqrt{31}} \left (174475 \sqrt{31}-202151 i\right ) \left (5 x^2+3 x+2\right )^2 \tanh ^{-1}\left (\frac{\left (-22+4 i \sqrt{31}\right ) x-i \sqrt{31}+63}{2 \sqrt{286-22 i \sqrt{31}} \sqrt{2 x^2-x+3}}\right )\right )}{\left (\sqrt{31}-13 i\right ) \left (\sqrt{31}+13 i\right )^2 \left (-10 i x+\sqrt{31}-3 i\right )^2 \left (10 i x+\sqrt{31}+3 i\right )^2}\right )}{14418844} \]
Antiderivative was successfully verified.
[In]
Integrate[Sqrt[3 - x + 2*x^2]/(2 + 3*x + 5*x^2)^3,x]
[Out]
(5*((I*Sqrt[286 + (22*I)*Sqrt[31]]*(1004586*I + 258253*Sqrt[31])*ArcTanh[(63 + I*Sqrt[31] + (-22 - (4*I)*Sqrt[
31])*x)/(2*Sqrt[286 + (22*I)*Sqrt[31]]*Sqrt[3 - x + 2*x^2])])/(-13*I + Sqrt[31])^2 + (2000*(1364*(13*I + Sqrt[
31])*Sqrt[3 - x + 2*x^2]*(11020 + 51362*x + 58315*x^2 + 68325*x^3) - 5*Sqrt[286 - (22*I)*Sqrt[31]]*(-202151*I
+ 174475*Sqrt[31])*(2 + 3*x + 5*x^2)^2*ArcTanh[(63 - I*Sqrt[31] + (-22 + (4*I)*Sqrt[31])*x)/(2*Sqrt[286 - (22*
I)*Sqrt[31]]*Sqrt[3 - x + 2*x^2])]))/((-13*I + Sqrt[31])*(13*I + Sqrt[31])^2*(-3*I + Sqrt[31] - (10*I)*x)^2*(3
*I + Sqrt[31] + (10*I)*x)^2)))/14418844
________________________________________________________________________________________
Maple [B] time = 0.428, size = 44343, normalized size = 198.9 \begin{align*} \text{output too large to display} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
int((2*x^2-x+3)^(1/2)/(5*x^2+3*x+2)^3,x)
[Out]
result too large to display
________________________________________________________________________________________
Maxima [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\sqrt{2 \, x^{2} - x + 3}}{{\left (5 \, x^{2} + 3 \, x + 2\right )}^{3}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate((2*x^2-x+3)^(1/2)/(5*x^2+3*x+2)^3,x, algorithm="maxima")
[Out]
integrate(sqrt(2*x^2 - x + 3)/(5*x^2 + 3*x + 2)^3, x)
________________________________________________________________________________________
Fricas [B] time = 6.15615, size = 10298, normalized size = 46.18 \begin{align*} \text{result too large to display} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate((2*x^2-x+3)^(1/2)/(5*x^2+3*x+2)^3,x, algorithm="fricas")
[Out]
1/65052151896952926425996714240*(14205421276*788032707736935368450^(1/4)*sqrt(39699690370)*sqrt(341)*sqrt(2)*(
25*x^4 + 30*x^3 + 29*x^2 + 12*x + 4)*sqrt(112285869463*sqrt(2) + 158798761480)*arctan(1/8610476622139712875910
57659551879544939625*(2461380802940*sqrt(39699690370)*(22*788032707736935368450^(3/4)*sqrt(341)*(667937076*x^7
- 2573871186*x^6 + 5404850058*x^5 - 8671430212*x^4 + 4348809776*x^3 - 2064441888*x^2 - sqrt(2)*(473555282*x^7
- 1821195871*x^6 + 3826055542*x^5 - 6128133137*x^4 + 3070797960*x^3 - 1452037320*x^2 - 3352976640*x + 2366869
248) - 4733738496*x + 3352976640) + 615345200735*788032707736935368450^(1/4)*sqrt(341)*(50730703*x^7 - 7788334
17*x^6 + 4116367112*x^5 - 9392273180*x^4 + 12133646496*x^3 - 7660912032*x^2 - sqrt(2)*(35938543*x^7 - 55154677
8*x^6 + 2913578540*x^5 - 6643469608*x^4 + 8580088800*x^3 - 5403919680*x^2 - 6107913216*x + 4313793024) - 86275
86048*x + 6107913216))*sqrt(2*x^2 - x + 3)*sqrt(112285869463*sqrt(2) + 158798761480) + 24442643314461120421939
70130340819353377000*sqrt(31)*sqrt(2)*(28180*x^8 - 254666*x^7 + 704270*x^6 - 1385256*x^5 + 1549144*x^4 - 64204
8*x^3 - 98496*x^2 - sqrt(2)*(8746*x^8 - 102335*x^7 + 396104*x^6 - 783113*x^5 + 1320710*x^4 - 752088*x^3 + 3961
44*x^2 + 546048*x - 539136) + 1154304*x - 456192) - 2*sqrt(39699690370/160673)*(sqrt(39699690370)*(22*78803270
7736935368450^(3/4)*sqrt(341)*(104024992*x^7 - 149335248*x^6 + 480784368*x^5 - 188730368*x^4 + 223535232*x^3 +
214417152*x^2 - sqrt(2)*(73906058*x^7 - 106073653*x^6 + 341348823*x^5 - 133050960*x^4 + 156704760*x^3 + 15433
8048*x^2 - 154338048*x) - 214417152*x) + 615345200735*788032707736935368450^(1/4)*sqrt(341)*(7903323*x^7 - 102
233612*x^6 + 394216580*x^5 - 510585408*x^4 + 657060192*x^3 + 391744512*x^2 - 4*sqrt(2)*(1401761*x^7 - 18132196
*x^6 + 69912940*x^5 - 90501120*x^4 + 116274240*x^3 + 70118784*x^2 - 70118784*x) - 391744512*x))*sqrt(2*x^2 - x
+ 3)*sqrt(112285869463*sqrt(2) + 158798761480) + 43175912524323866211143695850*sqrt(31)*sqrt(2)*(123408*x^8 -
914152*x^7 + 1578888*x^6 - 3293072*x^5 + 396480*x^4 + 798336*x^3 - 3822336*x^2 - sqrt(2)*(15550*x^8 - 118051*
x^7 + 244047*x^6 - 707374*x^5 + 1053960*x^4 - 1667952*x^3 + 1209600*x^2 - 1036800*x) + 3276288*x) + 1962541478
378357555051986175*sqrt(31)*(254591*x^8 - 4815126*x^7 + 32303580*x^6 - 90866808*x^5 + 108781920*x^4 - 74219328
*x^3 - 168956928*x^2 - 15488*sqrt(2)*(4*x^8 - 76*x^7 + 517*x^6 - 1536*x^5 + 2385*x^4 - 3618*x^3 + 2268*x^2 - 1
944*x) + 144820224*x))*sqrt(-(788032707736935368450^(1/4)*sqrt(39699690370)*sqrt(341)*sqrt(31)*sqrt(2*x^2 - x
+ 3)*(sqrt(2)*(12053*x + 5138) - 17191*x - 6915)*sqrt(112285869463*sqrt(2) + 158798761480) - 15018255698585818
0945*x^2 - 134857806273015509420*sqrt(2)*(2*x^2 - x + 3) + 462807471527848680055*x - 612990028513706861000)/x^
2) + 27775731039160364115840569662963856288375*sqrt(31)*(2828123*x^8 - 9696916*x^7 + 53385560*x^6 - 142835344*
x^5 + 254146592*x^4 - 249300096*x^3 + 37981440*x^2 - 7744*sqrt(2)*(1348*x^8 - 2692*x^7 + 9789*x^6 - 10070*x^5
+ 15569*x^4 - 5568*x^3 + 1080*x^2 + 4320*x - 5184) + 223064064*x - 94887936))/(2585191*x^8 - 4661200*x^7 + 141
91920*x^6 + 490880*x^5 - 13562944*x^4 + 44249088*x^3 - 34615296*x^2 - 24772608*x + 18579456)) + 14205421276*78
8032707736935368450^(1/4)*sqrt(39699690370)*sqrt(341)*sqrt(2)*(25*x^4 + 30*x^3 + 29*x^2 + 12*x + 4)*sqrt(11228
5869463*sqrt(2) + 158798761480)*arctan(1/861047662213971287591057659551879544939625*(2461380802940*sqrt(396996
90370)*(22*788032707736935368450^(3/4)*sqrt(341)*(667937076*x^7 - 2573871186*x^6 + 5404850058*x^5 - 8671430212
*x^4 + 4348809776*x^3 - 2064441888*x^2 - sqrt(2)*(473555282*x^7 - 1821195871*x^6 + 3826055542*x^5 - 6128133137
*x^4 + 3070797960*x^3 - 1452037320*x^2 - 3352976640*x + 2366869248) - 4733738496*x + 3352976640) + 61534520073
5*788032707736935368450^(1/4)*sqrt(341)*(50730703*x^7 - 778833417*x^6 + 4116367112*x^5 - 9392273180*x^4 + 1213
3646496*x^3 - 7660912032*x^2 - sqrt(2)*(35938543*x^7 - 551546778*x^6 + 2913578540*x^5 - 6643469608*x^4 + 85800
88800*x^3 - 5403919680*x^2 - 6107913216*x + 4313793024) - 8627586048*x + 6107913216))*sqrt(2*x^2 - x + 3)*sqrt
(112285869463*sqrt(2) + 158798761480) - 2444264331446112042193970130340819353377000*sqrt(31)*sqrt(2)*(28180*x^
8 - 254666*x^7 + 704270*x^6 - 1385256*x^5 + 1549144*x^4 - 642048*x^3 - 98496*x^2 - sqrt(2)*(8746*x^8 - 102335*
x^7 + 396104*x^6 - 783113*x^5 + 1320710*x^4 - 752088*x^3 + 396144*x^2 + 546048*x - 539136) + 1154304*x - 45619
2) - 2*sqrt(39699690370/160673)*(sqrt(39699690370)*(22*788032707736935368450^(3/4)*sqrt(341)*(104024992*x^7 -
149335248*x^6 + 480784368*x^5 - 188730368*x^4 + 223535232*x^3 + 214417152*x^2 - sqrt(2)*(73906058*x^7 - 106073
653*x^6 + 341348823*x^5 - 133050960*x^4 + 156704760*x^3 + 154338048*x^2 - 154338048*x) - 214417152*x) + 615345
200735*788032707736935368450^(1/4)*sqrt(341)*(7903323*x^7 - 102233612*x^6 + 394216580*x^5 - 510585408*x^4 + 65
7060192*x^3 + 391744512*x^2 - 4*sqrt(2)*(1401761*x^7 - 18132196*x^6 + 69912940*x^5 - 90501120*x^4 + 116274240*
x^3 + 70118784*x^2 - 70118784*x) - 391744512*x))*sqrt(2*x^2 - x + 3)*sqrt(112285869463*sqrt(2) + 158798761480)
- 43175912524323866211143695850*sqrt(31)*sqrt(2)*(123408*x^8 - 914152*x^7 + 1578888*x^6 - 3293072*x^5 + 39648
0*x^4 + 798336*x^3 - 3822336*x^2 - sqrt(2)*(15550*x^8 - 118051*x^7 + 244047*x^6 - 707374*x^5 + 1053960*x^4 - 1
667952*x^3 + 1209600*x^2 - 1036800*x) + 3276288*x) - 1962541478378357555051986175*sqrt(31)*(254591*x^8 - 48151
26*x^7 + 32303580*x^6 - 90866808*x^5 + 108781920*x^4 - 74219328*x^3 - 168956928*x^2 - 15488*sqrt(2)*(4*x^8 - 7
6*x^7 + 517*x^6 - 1536*x^5 + 2385*x^4 - 3618*x^3 + 2268*x^2 - 1944*x) + 144820224*x))*sqrt((788032707736935368
450^(1/4)*sqrt(39699690370)*sqrt(341)*sqrt(31)*sqrt(2*x^2 - x + 3)*(sqrt(2)*(12053*x + 5138) - 17191*x - 6915)
*sqrt(112285869463*sqrt(2) + 158798761480) + 150182556985858180945*x^2 + 134857806273015509420*sqrt(2)*(2*x^2
- x + 3) - 462807471527848680055*x + 612990028513706861000)/x^2) - 27775731039160364115840569662963856288375*s
qrt(31)*(2828123*x^8 - 9696916*x^7 + 53385560*x^6 - 142835344*x^5 + 254146592*x^4 - 249300096*x^3 + 37981440*x
^2 - 7744*sqrt(2)*(1348*x^8 - 2692*x^7 + 9789*x^6 - 10070*x^5 + 15569*x^4 - 5568*x^3 + 1080*x^2 + 4320*x - 518
4) + 223064064*x - 94887936))/(2585191*x^8 - 4661200*x^7 + 14191920*x^6 + 490880*x^5 - 13562944*x^4 + 44249088
*x^3 - 34615296*x^2 - 24772608*x + 18579456)) + 788032707736935368450^(1/4)*sqrt(39699690370)*sqrt(341)*sqrt(3
1)*(3969969037000*x^4 + 4763962844400*x^3 + 4605164082920*x^2 - 112285869463*sqrt(2)*(25*x^4 + 30*x^3 + 29*x^2
+ 12*x + 4) + 1905585137760*x + 635195045920)*sqrt(112285869463*sqrt(2) + 158798761480)*log(635195045920/1606
73*(788032707736935368450^(1/4)*sqrt(39699690370)*sqrt(341)*sqrt(31)*sqrt(2*x^2 - x + 3)*(sqrt(2)*(12053*x + 5
138) - 17191*x - 6915)*sqrt(112285869463*sqrt(2) + 158798761480) + 150182556985858180945*x^2 + 134857806273015
509420*sqrt(2)*(2*x^2 - x + 3) - 462807471527848680055*x + 612990028513706861000)/x^2) - 788032707736935368450
^(1/4)*sqrt(39699690370)*sqrt(341)*sqrt(31)*(3969969037000*x^4 + 4763962844400*x^3 + 4605164082920*x^2 - 11228
5869463*sqrt(2)*(25*x^4 + 30*x^3 + 29*x^2 + 12*x + 4) + 1905585137760*x + 635195045920)*sqrt(112285869463*sqrt
(2) + 158798761480)*log(-635195045920/160673*(788032707736935368450^(1/4)*sqrt(39699690370)*sqrt(341)*sqrt(31)
*sqrt(2*x^2 - x + 3)*(sqrt(2)*(12053*x + 5138) - 17191*x - 6915)*sqrt(112285869463*sqrt(2) + 158798761480) - 1
50182556985858180945*x^2 - 134857806273015509420*sqrt(2)*(2*x^2 - x + 3) + 462807471527848680055*x - 612990028
513706861000)/x^2) + 769228926981280465731680*(68325*x^3 + 58315*x^2 + 51362*x + 11020)*sqrt(2*x^2 - x + 3))/(
25*x^4 + 30*x^3 + 29*x^2 + 12*x + 4)
________________________________________________________________________________________
Sympy [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\sqrt{2 x^{2} - x + 3}}{\left (5 x^{2} + 3 x + 2\right )^{3}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate((2*x**2-x+3)**(1/2)/(5*x**2+3*x+2)**3,x)
[Out]
Integral(sqrt(2*x**2 - x + 3)/(5*x**2 + 3*x + 2)**3, x)
________________________________________________________________________________________
Giac [F(-2)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: TypeError} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate((2*x^2-x+3)^(1/2)/(5*x^2+3*x+2)^3,x, algorithm="giac")
[Out]
Exception raised: TypeError