3.71 \(\int \frac{(3-x+2 x^2)^{3/2}}{(2+3 x+5 x^2)^3} \, dx\)
Optimal. Leaf size=223 \[ \frac{(10 x+3) \left (2 x^2-x+3\right )^{3/2}}{62 \left (5 x^2+3 x+2\right )^2}+\frac{3 (696 x+277) \sqrt{2 x^2-x+3}}{3844 \left (5 x^2+3 x+2\right )}+\frac{3 \sqrt{\frac{1}{682} \left (366990269+259509026 \sqrt{2}\right )} \tan ^{-1}\left (\frac{\sqrt{\frac{11}{31 \left (366990269+259509026 \sqrt{2}\right )}} \left (\left (70517+49942 \sqrt{2}\right ) x+20575 \sqrt{2}+29367\right )}{\sqrt{2 x^2-x+3}}\right )}{7688}-\frac{3 \sqrt{\frac{1}{682} \left (259509026 \sqrt{2}-366990269\right )} \tanh ^{-1}\left (\frac{\sqrt{\frac{11}{31 \left (259509026 \sqrt{2}-366990269\right )}} \left (\left (70517-49942 \sqrt{2}\right ) x-20575 \sqrt{2}+29367\right )}{\sqrt{2 x^2-x+3}}\right )}{7688} \]
[Out]
((3 + 10*x)*(3 - x + 2*x^2)^(3/2))/(62*(2 + 3*x + 5*x^2)^2) + (3*(277 + 696*x)*Sqrt[3 - x + 2*x^2])/(3844*(2 +
3*x + 5*x^2)) + (3*Sqrt[(366990269 + 259509026*Sqrt[2])/682]*ArcTan[(Sqrt[11/(31*(366990269 + 259509026*Sqrt[
2]))]*(29367 + 20575*Sqrt[2] + (70517 + 49942*Sqrt[2])*x))/Sqrt[3 - x + 2*x^2]])/7688 - (3*Sqrt[(-366990269 +
259509026*Sqrt[2])/682]*ArcTanh[(Sqrt[11/(31*(-366990269 + 259509026*Sqrt[2]))]*(29367 - 20575*Sqrt[2] + (7051
7 - 49942*Sqrt[2])*x))/Sqrt[3 - x + 2*x^2]])/7688
________________________________________________________________________________________
Rubi [A] time = 0.432945, 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, 1013, 1035, 1029, 206, 204} \[ \frac{(10 x+3) \left (2 x^2-x+3\right )^{3/2}}{62 \left (5 x^2+3 x+2\right )^2}+\frac{3 (696 x+277) \sqrt{2 x^2-x+3}}{3844 \left (5 x^2+3 x+2\right )}+\frac{3 \sqrt{\frac{1}{682} \left (366990269+259509026 \sqrt{2}\right )} \tan ^{-1}\left (\frac{\sqrt{\frac{11}{31 \left (366990269+259509026 \sqrt{2}\right )}} \left (\left (70517+49942 \sqrt{2}\right ) x+20575 \sqrt{2}+29367\right )}{\sqrt{2 x^2-x+3}}\right )}{7688}-\frac{3 \sqrt{\frac{1}{682} \left (259509026 \sqrt{2}-366990269\right )} \tanh ^{-1}\left (\frac{\sqrt{\frac{11}{31 \left (259509026 \sqrt{2}-366990269\right )}} \left (\left (70517-49942 \sqrt{2}\right ) x-20575 \sqrt{2}+29367\right )}{\sqrt{2 x^2-x+3}}\right )}{7688} \]
Antiderivative was successfully verified.
[In]
Int[(3 - x + 2*x^2)^(3/2)/(2 + 3*x + 5*x^2)^3,x]
[Out]
((3 + 10*x)*(3 - x + 2*x^2)^(3/2))/(62*(2 + 3*x + 5*x^2)^2) + (3*(277 + 696*x)*Sqrt[3 - x + 2*x^2])/(3844*(2 +
3*x + 5*x^2)) + (3*Sqrt[(366990269 + 259509026*Sqrt[2])/682]*ArcTan[(Sqrt[11/(31*(366990269 + 259509026*Sqrt[
2]))]*(29367 + 20575*Sqrt[2] + (70517 + 49942*Sqrt[2])*x))/Sqrt[3 - x + 2*x^2]])/7688 - (3*Sqrt[(-366990269 +
259509026*Sqrt[2])/682]*ArcTanh[(Sqrt[11/(31*(-366990269 + 259509026*Sqrt[2]))]*(29367 - 20575*Sqrt[2] + (7051
7 - 49942*Sqrt[2])*x))/Sqrt[3 - x + 2*x^2]])/7688
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 1013
Int[((g_.) + (h_.)*(x_))*((a_) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_)*((d_) + (e_.)*(x_) + (f_.)*(x_)^2)^(q_), x_Sy
mbol] :> Simp[((g*b - 2*a*h - (b*h - 2*g*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[e*q
*(g*b - 2*a*h) - d*(b*h - 2*g*c)*(2*p + 3) + (2*f*q*(g*b - 2*a*h) - e*(b*h - 2*g*c)*(2*p + q + 3))*x - f*(b*h
- 2*g*c)*(2*p + 2*q + 3)*x^2, x], 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] && LtQ[p, -1] && GtQ[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{\left (3-x+2 x^2\right )^{3/2}}{\left (2+3 x+5 x^2\right )^3} \, dx &=\frac{(3+10 x) \left (3-x+2 x^2\right )^{3/2}}{62 \left (2+3 x+5 x^2\right )^2}-\frac{1}{62} \int \frac{\left (-\frac{189}{2}+33 x\right ) \sqrt{3-x+2 x^2}}{\left (2+3 x+5 x^2\right )^2} \, dx\\ &=\frac{(3+10 x) \left (3-x+2 x^2\right )^{3/2}}{62 \left (2+3 x+5 x^2\right )^2}+\frac{3 (277+696 x) \sqrt{3-x+2 x^2}}{3844 \left (2+3 x+5 x^2\right )}+\frac{\int \frac{\frac{13359}{4}-1353 x}{\sqrt{3-x+2 x^2} \left (2+3 x+5 x^2\right )} \, dx}{1922}\\ &=\frac{(3+10 x) \left (3-x+2 x^2\right )^{3/2}}{62 \left (2+3 x+5 x^2\right )^2}+\frac{3 (277+696 x) \sqrt{3-x+2 x^2}}{3844 \left (2+3 x+5 x^2\right )}+\frac{\int \frac{-\frac{33}{4} \left (6257-4453 \sqrt{2}\right )+\frac{33}{4} \left (2649-1804 \sqrt{2}\right ) x}{\sqrt{3-x+2 x^2} \left (2+3 x+5 x^2\right )} \, dx}{42284 \sqrt{2}}-\frac{\int \frac{-\frac{33}{4} \left (6257+4453 \sqrt{2}\right )+\frac{33}{4} \left (2649+1804 \sqrt{2}\right ) x}{\sqrt{3-x+2 x^2} \left (2+3 x+5 x^2\right )} \, dx}{42284 \sqrt{2}}\\ &=\frac{(3+10 x) \left (3-x+2 x^2\right )^{3/2}}{62 \left (2+3 x+5 x^2\right )^2}+\frac{3 (277+696 x) \sqrt{3-x+2 x^2}}{3844 \left (2+3 x+5 x^2\right )}+\frac{\left (99 \left (519018052-366990269 \sqrt{2}\right )\right ) \operatorname{Subst}\left (\int \frac{1}{-\frac{33759}{16} \left (366990269-259509026 \sqrt{2}\right )-11 x^2} \, dx,x,\frac{-\frac{33}{4} \left (29367-20575 \sqrt{2}\right )-\frac{33}{4} \left (70517-49942 \sqrt{2}\right ) x}{\sqrt{3-x+2 x^2}}\right )}{61504}+\frac{\left (99 \left (519018052+366990269 \sqrt{2}\right )\right ) \operatorname{Subst}\left (\int \frac{1}{-\frac{33759}{16} \left (366990269+259509026 \sqrt{2}\right )-11 x^2} \, dx,x,\frac{-\frac{33}{4} \left (29367+20575 \sqrt{2}\right )-\frac{33}{4} \left (70517+49942 \sqrt{2}\right ) x}{\sqrt{3-x+2 x^2}}\right )}{61504}\\ &=\frac{(3+10 x) \left (3-x+2 x^2\right )^{3/2}}{62 \left (2+3 x+5 x^2\right )^2}+\frac{3 (277+696 x) \sqrt{3-x+2 x^2}}{3844 \left (2+3 x+5 x^2\right )}+\frac{3 \sqrt{\frac{1}{682} \left (366990269+259509026 \sqrt{2}\right )} \tan ^{-1}\left (\frac{\sqrt{\frac{11}{31 \left (366990269+259509026 \sqrt{2}\right )}} \left (29367+20575 \sqrt{2}+\left (70517+49942 \sqrt{2}\right ) x\right )}{\sqrt{3-x+2 x^2}}\right )}{7688}-\frac{3 \sqrt{\frac{1}{682} \left (-366990269+259509026 \sqrt{2}\right )} \tanh ^{-1}\left (\frac{\sqrt{\frac{11}{31 \left (-366990269+259509026 \sqrt{2}\right )}} \left (29367-20575 \sqrt{2}+\left (70517-49942 \sqrt{2}\right ) x\right )}{\sqrt{3-x+2 x^2}}\right )}{7688}\\ \end{align*}
Mathematica [C] time = 5.38845, size = 1262, normalized size = 5.66 \[ \text{result too large to display} \]
Antiderivative was successfully verified.
[In]
Integrate[(3 - x + 2*x^2)^(3/2)/(2 + 3*x + 5*x^2)^3,x]
[Out]
(((248000*I)*Sqrt[31]*(3 - x + 2*x^2)^(3/2))/(3*I + Sqrt[31] + (10*I)*x)^2 + (744000*(3 - x + 2*x^2)^(3/2))/(3
- I*Sqrt[31] + 10*x) + ((248000*I)*Sqrt[31]*(3 - x + 2*x^2)^(3/2))/(3 + I*Sqrt[31] + 10*x)^2 + (744000*(3 - x
+ 2*x^2)^(3/2))/(3 + I*Sqrt[31] + 10*x) + (3*I)*Sqrt[31]*(20*(1199 + (98*I)*Sqrt[31] - 20*(11 + (2*I)*Sqrt[31
])*x)*Sqrt[3 - x + 2*x^2] + Sqrt[2]*(13453 + (4406*I)*Sqrt[31])*ArcSinh[(1 - 4*x)/Sqrt[23]] - (352*Sqrt[286 +
(22*I)*Sqrt[31]]*(-69*I + 13*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])) + 558*(20*(27 + (4*I)*Sqrt[31] - 20*x)*Sqrt[3 - x + 2*
x^2] + Sqrt[2]*(569 + (88*I)*Sqrt[31])*ArcSinh[(1 - 4*x)/Sqrt[23]] - (4*Sqrt[286 + (22*I)*Sqrt[31]]*(-81*I + 3
7*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])) + (744*Sqrt[31]*(220*(-439 + (497*I)*Sqrt[31] + 20*(69 + (13*I)*Sqrt[31])*x)*Sqrt
[3 - x + 2*x^2] + 88*Sqrt[2]*(4426 - (398*I)*Sqrt[31] + 5*(47 - (281*I)*Sqrt[31])*x)*ArcSinh[(-1 + 4*x)/Sqrt[2
3]] + Sqrt[286 + (22*I)*Sqrt[31]]*(19548 - (4904*I)*Sqrt[31] + (-23345 - (8565*I)*Sqrt[31])*x)*ArcTanh[(63 + I
*Sqrt[31] + (-22 - (4*I)*Sqrt[31])*x)/(2*Sqrt[286 + (22*I)*Sqrt[31]]*Sqrt[3 - x + 2*x^2])]))/(11*(-13*I + Sqrt
[31])^2*(-3*I + Sqrt[31] - (10*I)*x)) + (744*Sqrt[31]*(220*(-439 - (497*I)*Sqrt[31] + 20*(69 - (13*I)*Sqrt[31]
)*x)*Sqrt[3 - x + 2*x^2] + 88*Sqrt[2]*(4426 + (398*I)*Sqrt[31] + 5*(47 + (281*I)*Sqrt[31])*x)*ArcSinh[(-1 + 4*
x)/Sqrt[23]] + Sqrt[286 - (22*I)*Sqrt[31]]*(-19548 - (4904*I)*Sqrt[31] + 5*(4669 - (1713*I)*Sqrt[31])*x)*ArcTa
nh[(-63 + I*Sqrt[31] + (22 - (4*I)*Sqrt[31])*x)/(2*Sqrt[286 - (22*I)*Sqrt[31]]*Sqrt[3 - x + 2*x^2])]))/(11*(13
*I + Sqrt[31])^2*(3*I + Sqrt[31] + (10*I)*x)) + (3*Sqrt[31]*(20*(12549 - (2473*I)*Sqrt[31] + (20*I)*(81*I + 37
*Sqrt[31])*x)*Sqrt[3 - x + 2*x^2] + Sqrt[2]*(38303 - (70731*I)*Sqrt[31])*ArcSinh[(1 - 4*x)/Sqrt[23]] + (352*I)
*Sqrt[286 - (22*I)*Sqrt[31]]*(69*I + 13*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]) + 558*(-20*(-27 + (4*I)*Sqrt[31] + 20*x)*Sqr
t[3 - x + 2*x^2] + Sqrt[2]*(569 - (88*I)*Sqrt[31])*ArcSinh[(1 - 4*x)/Sqrt[23]] - (4*Sqrt[286 - (22*I)*Sqrt[31]
]*(81*I + 37*Sqrt[31])*ArcTanh[(63 - I*Sqrt[31] + (-22 + (4*I)*Sqrt[31])*x)/(2*Sqrt[286 - (22*I)*Sqrt[31]]*Sqr
t[3 - x + 2*x^2])])/(13*I + Sqrt[31])))/4766560
________________________________________________________________________________________
Maple [B] time = 0.607, size = 81552, normalized size = 365.7 \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)^(3/2)/(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{{\left (2 \, x^{2} - x + 3\right )}^{\frac{3}{2}}}{{\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)^(3/2)/(5*x^2+3*x+2)^3,x, algorithm="maxima")
[Out]
integrate((2*x^2 - x + 3)^(3/2)/(5*x^2 + 3*x + 2)^3, x)
________________________________________________________________________________________
Fricas [B] time = 4.94277, size = 9586, normalized size = 42.99 \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)^(3/2)/(5*x^2+3*x+2)^3,x, algorithm="fricas")
[Out]
-1/85773071417697924109696*(189113268*134689869150937352^(1/4)*sqrt(129754513)*sqrt(341)*sqrt(2)*(25*x^4 + 30*
x^3 + 29*x^2 + 12*x + 4)*sqrt(366990269*sqrt(2) + 519018052)*arctan(1/1067259092343193675559267622545473*(1608
9559612*sqrt(129754513)*(11*134689869150937352^(3/4)*sqrt(341)*(38305160*x^7 - 147261352*x^6 + 309398878*x^5 -
495410374*x^4 + 248212864*x^3 - 117285552*x^2 - sqrt(2)*(26988622*x^7 - 104036813*x^6 + 218448200*x^5 - 35057
9241*x^4 + 175844824*x^3 - 83534472*x^2 - 191303424*x + 135585792) - 271171584*x + 191303424) + 4022389903*134
689869150937352^(1/4)*sqrt(341)*(2906601*x^7 - 44604657*x^6 + 235604928*x^5 - 537156764*x^4 + 693706464*x^3 -
436717728*x^2 - sqrt(2)*(2050114*x^7 - 31475955*x^6 + 166375268*x^5 - 379661892*x^4 + 490500864*x^3 - 30982780
8*x^2 - 348696576*x + 246965760) - 493931520*x + 348696576))*sqrt(2*x^2 - x + 3)*sqrt(366990269*sqrt(2) + 5190
18052) + 3029638713748420756426308089806504*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 - 456192) - 2*sqrt(259509026/713)*(sqrt(1297545
13)*(11*134689869150937352^(3/4)*sqrt(341)*(5980372*x^7 - 8582986*x^6 + 27618126*x^5 - 10751392*x^4 + 12649968
*x^3 + 12517632*x^2 - sqrt(2)*(4201650*x^7 - 6032009*x^6 + 19421619*x^5 - 7633552*x^4 + 9050328*x^3 + 8640000*
x^2 - 8640000*x) - 12517632*x) + 4022389903*134689869150937352^(1/4)*sqrt(341)*(453599*x^7 - 5867420*x^6 + 226
22900*x^5 - 29282112*x^4 + 37610208*x^3 + 22726656*x^2 - sqrt(2)*(319303*x^7 - 4130364*x^6 + 15927060*x^5 - 20
630592*x^4 + 26556768*x^3 + 15800832*x^2 - 15800832*x) - 22726656*x))*sqrt(2*x^2 - x + 3)*sqrt(366990269*sqrt(
2) + 519018052) + 8186887989068712800954*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) + 372131272230396036407*sqrt(31)*(254591*x^8 - 4815
126*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 - 1944*x) + 144820224*x))*sqrt(-(1346898691509373
52^(1/4)*sqrt(129754513)*sqrt(341)*sqrt(31)*sqrt(2*x^2 - x + 3)*(sqrt(2)*(696*x + 277) - 973*x - 419)*sqrt(366
990269*sqrt(2) + 519018052) - 4356437317274441*x^2 - 3911902897144396*sqrt(2)*(2*x^2 - x + 3) + 13424939487927
359*x - 17781376805201800)/x^2) + 34427712656232054050298955565983*sqrt(31)*(2828123*x^8 - 9696916*x^7 + 53385
560*x^6 - 142835344*x^5 + 254146592*x^4 - 249300096*x^3 + 37981440*x^2 - 7744*sqrt(2)*(1348*x^8 - 2692*x^7 + 9
789*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 + 14191920*x^6 + 490880*x^5 - 13562944*x^4 + 44249088*x^3 - 34615296*x^2 - 24772608*x + 1857945
6)) + 189113268*134689869150937352^(1/4)*sqrt(129754513)*sqrt(341)*sqrt(2)*(25*x^4 + 30*x^3 + 29*x^2 + 12*x +
4)*sqrt(366990269*sqrt(2) + 519018052)*arctan(1/1067259092343193675559267622545473*(16089559612*sqrt(129754513
)*(11*134689869150937352^(3/4)*sqrt(341)*(38305160*x^7 - 147261352*x^6 + 309398878*x^5 - 495410374*x^4 + 24821
2864*x^3 - 117285552*x^2 - sqrt(2)*(26988622*x^7 - 104036813*x^6 + 218448200*x^5 - 350579241*x^4 + 175844824*x
^3 - 83534472*x^2 - 191303424*x + 135585792) - 271171584*x + 191303424) + 4022389903*134689869150937352^(1/4)*
sqrt(341)*(2906601*x^7 - 44604657*x^6 + 235604928*x^5 - 537156764*x^4 + 693706464*x^3 - 436717728*x^2 - sqrt(2
)*(2050114*x^7 - 31475955*x^6 + 166375268*x^5 - 379661892*x^4 + 490500864*x^3 - 309827808*x^2 - 348696576*x +
246965760) - 493931520*x + 348696576))*sqrt(2*x^2 - x + 3)*sqrt(366990269*sqrt(2) + 519018052) - 3029638713748
420756426308089806504*sqrt(31)*sqrt(2)*(28180*x^8 - 254666*x^7 + 704270*x^6 - 1385256*x^5 + 1549144*x^4 - 6420
48*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 + 396
144*x^2 + 546048*x - 539136) + 1154304*x - 456192) - 2*sqrt(259509026/713)*(sqrt(129754513)*(11*13468986915093
7352^(3/4)*sqrt(341)*(5980372*x^7 - 8582986*x^6 + 27618126*x^5 - 10751392*x^4 + 12649968*x^3 + 12517632*x^2 -
sqrt(2)*(4201650*x^7 - 6032009*x^6 + 19421619*x^5 - 7633552*x^4 + 9050328*x^3 + 8640000*x^2 - 8640000*x) - 125
17632*x) + 4022389903*134689869150937352^(1/4)*sqrt(341)*(453599*x^7 - 5867420*x^6 + 22622900*x^5 - 29282112*x
^4 + 37610208*x^3 + 22726656*x^2 - sqrt(2)*(319303*x^7 - 4130364*x^6 + 15927060*x^5 - 20630592*x^4 + 26556768*
x^3 + 15800832*x^2 - 15800832*x) - 22726656*x))*sqrt(2*x^2 - x + 3)*sqrt(366990269*sqrt(2) + 519018052) - 8186
887989068712800954*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 + 1
209600*x^2 - 1036800*x) + 3276288*x) - 372131272230396036407*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 - 153
6*x^5 + 2385*x^4 - 3618*x^3 + 2268*x^2 - 1944*x) + 144820224*x))*sqrt((134689869150937352^(1/4)*sqrt(129754513
)*sqrt(341)*sqrt(31)*sqrt(2*x^2 - x + 3)*(sqrt(2)*(696*x + 277) - 973*x - 419)*sqrt(366990269*sqrt(2) + 519018
052) + 4356437317274441*x^2 + 3911902897144396*sqrt(2)*(2*x^2 - x + 3) - 13424939487927359*x + 177813768052018
00)/x^2) - 34427712656232054050298955565983*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 + 1
5569*x^4 - 5568*x^3 + 1080*x^2 + 4320*x - 5184) + 223064064*x - 94887936))/(2585191*x^8 - 4661200*x^7 + 141919
20*x^6 + 490880*x^5 - 13562944*x^4 + 44249088*x^3 - 34615296*x^2 - 24772608*x + 18579456)) - 3*134689869150937
352^(1/4)*sqrt(129754513)*sqrt(341)*sqrt(31)*(12975451300*x^4 + 15570541560*x^3 + 15051523508*x^2 - 366990269*
sqrt(2)*(25*x^4 + 30*x^3 + 29*x^2 + 12*x + 4) + 6228216624*x + 2076072208)*sqrt(366990269*sqrt(2) + 519018052)
*log(9342324936/713*(134689869150937352^(1/4)*sqrt(129754513)*sqrt(341)*sqrt(31)*sqrt(2*x^2 - x + 3)*(sqrt(2)*
(696*x + 277) - 973*x - 419)*sqrt(366990269*sqrt(2) + 519018052) + 4356437317274441*x^2 + 3911902897144396*sqr
t(2)*(2*x^2 - x + 3) - 13424939487927359*x + 17781376805201800)/x^2) + 3*134689869150937352^(1/4)*sqrt(1297545
13)*sqrt(341)*sqrt(31)*(12975451300*x^4 + 15570541560*x^3 + 15051523508*x^2 - 366990269*sqrt(2)*(25*x^4 + 30*x
^3 + 29*x^2 + 12*x + 4) + 6228216624*x + 2076072208)*sqrt(366990269*sqrt(2) + 519018052)*log(-9342324936/713*(
134689869150937352^(1/4)*sqrt(129754513)*sqrt(341)*sqrt(31)*sqrt(2*x^2 - x + 3)*(sqrt(2)*(696*x + 277) - 973*x
- 419)*sqrt(366990269*sqrt(2) + 519018052) - 4356437317274441*x^2 - 3911902897144396*sqrt(2)*(2*x^2 - x + 3)
+ 13424939487927359*x - 17781376805201800)/x^2) - 22313494125311634784*(11680*x^3 + 10171*x^2 + 8343*x + 2220)
*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{\left (2 x^{2} - x + 3\right )^{\frac{3}{2}}}{\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)**(3/2)/(5*x**2+3*x+2)**3,x)
[Out]
Integral((2*x**2 - x + 3)**(3/2)/(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)^(3/2)/(5*x^2+3*x+2)^3,x, algorithm="giac")
[Out]
Exception raised: TypeError