3.92 \(\int \frac{1}{(3-x+2 x^2)^{3/2} (2+3 x+5 x^2)^3} \, dx\)
Optimal. Leaf size=246 \[ -\frac{4353943-6508666 x}{941410976 \sqrt{2 x^2-x+3}}+\frac{5 (17315 x+7318)}{1860496 \sqrt{2 x^2-x+3} \left (5 x^2+3 x+2\right )}+\frac{65 x+4}{1364 \sqrt{2 x^2-x+3} \left (5 x^2+3 x+2\right )^2}+\frac{3 \sqrt{\frac{1}{682} \left (13874275807943+9819738650000 \sqrt{2}\right )} \tan ^{-1}\left (\frac{\sqrt{\frac{11}{31 \left (13874275807943+9819738650000 \sqrt{2}\right )}} \left (\left (13785797+9662095 \sqrt{2}\right ) x+4123702 \sqrt{2}+5538393\right )}{\sqrt{2 x^2-x+3}}\right )}{81861824}-\frac{3 \sqrt{\frac{1}{682} \left (9819738650000 \sqrt{2}-13874275807943\right )} \tanh ^{-1}\left (\frac{\sqrt{\frac{11}{31 \left (9819738650000 \sqrt{2}-13874275807943\right )}} \left (\left (13785797-9662095 \sqrt{2}\right ) x-4123702 \sqrt{2}+5538393\right )}{\sqrt{2 x^2-x+3}}\right )}{81861824} \]
[Out]
-(4353943 - 6508666*x)/(941410976*Sqrt[3 - x + 2*x^2]) + (4 + 65*x)/(1364*Sqrt[3 - x + 2*x^2]*(2 + 3*x + 5*x^2
)^2) + (5*(7318 + 17315*x))/(1860496*Sqrt[3 - x + 2*x^2]*(2 + 3*x + 5*x^2)) + (3*Sqrt[(13874275807943 + 981973
8650000*Sqrt[2])/682]*ArcTan[(Sqrt[11/(31*(13874275807943 + 9819738650000*Sqrt[2]))]*(5538393 + 4123702*Sqrt[2
] + (13785797 + 9662095*Sqrt[2])*x))/Sqrt[3 - x + 2*x^2]])/81861824 - (3*Sqrt[(-13874275807943 + 9819738650000
*Sqrt[2])/682]*ArcTanh[(Sqrt[11/(31*(-13874275807943 + 9819738650000*Sqrt[2]))]*(5538393 - 4123702*Sqrt[2] + (
13785797 - 9662095*Sqrt[2])*x))/Sqrt[3 - x + 2*x^2]])/81861824
________________________________________________________________________________________
Rubi [A] time = 0.525246, antiderivative size = 246, normalized size of antiderivative = 1.,
number of steps used = 8, number of rules used = 6, integrand size = 27, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.222, Rules used =
{974, 1060, 1035, 1029, 206, 204} \[ -\frac{4353943-6508666 x}{941410976 \sqrt{2 x^2-x+3}}+\frac{5 (17315 x+7318)}{1860496 \sqrt{2 x^2-x+3} \left (5 x^2+3 x+2\right )}+\frac{65 x+4}{1364 \sqrt{2 x^2-x+3} \left (5 x^2+3 x+2\right )^2}+\frac{3 \sqrt{\frac{1}{682} \left (13874275807943+9819738650000 \sqrt{2}\right )} \tan ^{-1}\left (\frac{\sqrt{\frac{11}{31 \left (13874275807943+9819738650000 \sqrt{2}\right )}} \left (\left (13785797+9662095 \sqrt{2}\right ) x+4123702 \sqrt{2}+5538393\right )}{\sqrt{2 x^2-x+3}}\right )}{81861824}-\frac{3 \sqrt{\frac{1}{682} \left (9819738650000 \sqrt{2}-13874275807943\right )} \tanh ^{-1}\left (\frac{\sqrt{\frac{11}{31 \left (9819738650000 \sqrt{2}-13874275807943\right )}} \left (\left (13785797-9662095 \sqrt{2}\right ) x-4123702 \sqrt{2}+5538393\right )}{\sqrt{2 x^2-x+3}}\right )}{81861824} \]
Antiderivative was successfully verified.
[In]
Int[1/((3 - x + 2*x^2)^(3/2)*(2 + 3*x + 5*x^2)^3),x]
[Out]
-(4353943 - 6508666*x)/(941410976*Sqrt[3 - x + 2*x^2]) + (4 + 65*x)/(1364*Sqrt[3 - x + 2*x^2]*(2 + 3*x + 5*x^2
)^2) + (5*(7318 + 17315*x))/(1860496*Sqrt[3 - x + 2*x^2]*(2 + 3*x + 5*x^2)) + (3*Sqrt[(13874275807943 + 981973
8650000*Sqrt[2])/682]*ArcTan[(Sqrt[11/(31*(13874275807943 + 9819738650000*Sqrt[2]))]*(5538393 + 4123702*Sqrt[2
] + (13785797 + 9662095*Sqrt[2])*x))/Sqrt[3 - x + 2*x^2]])/81861824 - (3*Sqrt[(-13874275807943 + 9819738650000
*Sqrt[2])/682]*ArcTanh[(Sqrt[11/(31*(-13874275807943 + 9819738650000*Sqrt[2]))]*(5538393 - 4123702*Sqrt[2] + (
13785797 - 9662095*Sqrt[2])*x))/Sqrt[3 - x + 2*x^2]])/81861824
Rule 974
Int[((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_)*((d_.) + (e_.)*(x_) + (f_.)*(x_)^2)^(q_), x_Symbol] :> Simp[((2*a
*c^2*e - b^2*c*e + b^3*f + b*c*(c*d - 3*a*f) + c*(2*c^2*d + b^2*f - c*(b*e + 2*a*f))*x)*(a + b*x + c*x^2)^(p +
1)*(d + e*x + f*x^2)^(q + 1))/((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[2*c*((c*d - a*f)^2 - (b*d - a*e)*(c*e - b*f))*(p + 1) - (2*c^2*d + b^2*f - c*(b*e + 2*a*f))*(a*f*(
p + 1) - c*d*(p + 2)) - e*(b^2*c*e - 2*a*c^2*e - b^3*f - b*c*(c*d - 3*a*f))*(p + q + 2) + (2*f*(2*a*c^2*e - b^
2*c*e + b^3*f + b*c*(c*d - 3*a*f))*(p + q + 2) - (2*c^2*d + b^2*f - c*(b*e + 2*a*f))*(b*f*(p + 1) - c*e*(2*p +
q + 4)))*x + c*f*(2*c^2*d + b^2*f - c*(b*e + 2*a*f))*(2*p + 2*q + 5)*x^2, x], x], x] /; FreeQ[{a, b, c, d, e,
f, 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 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{1}{\left (3-x+2 x^2\right )^{3/2} \left (2+3 x+5 x^2\right )^3} \, dx &=\frac{4+65 x}{1364 \sqrt{3-x+2 x^2} \left (2+3 x+5 x^2\right )^2}-\frac{\int \frac{-5731+\frac{7557 x}{2}-5720 x^2}{\left (3-x+2 x^2\right )^{3/2} \left (2+3 x+5 x^2\right )^2} \, dx}{15004}\\ &=\frac{4+65 x}{1364 \sqrt{3-x+2 x^2} \left (2+3 x+5 x^2\right )^2}+\frac{5 (7318+17315 x)}{1860496 \sqrt{3-x+2 x^2} \left (2+3 x+5 x^2\right )}-\frac{\int \frac{-\frac{29276797}{2}+\frac{3439425 x}{4}-20951150 x^2}{\left (3-x+2 x^2\right )^{3/2} \left (2+3 x+5 x^2\right )} \, dx}{112560008}\\ &=-\frac{4353943-6508666 x}{941410976 \sqrt{3-x+2 x^2}}+\frac{4+65 x}{1364 \sqrt{3-x+2 x^2} \left (2+3 x+5 x^2\right )^2}+\frac{5 (7318+17315 x)}{1860496 \sqrt{3-x+2 x^2} \left (2+3 x+5 x^2\right )}-\frac{\int \frac{-\frac{38923847853}{4}+\frac{36293395215 x}{8}}{\sqrt{3-x+2 x^2} \left (2+3 x+5 x^2\right )} \, dx}{313254502264}\\ &=-\frac{4353943-6508666 x}{941410976 \sqrt{3-x+2 x^2}}+\frac{4+65 x}{1364 \sqrt{3-x+2 x^2} \left (2+3 x+5 x^2\right )^2}+\frac{5 (7318+17315 x)}{1860496 \sqrt{3-x+2 x^2} \left (2+3 x+5 x^2\right )}-\frac{\int \frac{\frac{1010229}{8} \left (1242839-847654 \sqrt{2}\right )-\frac{1010229}{8} \left (452469-395185 \sqrt{2}\right ) x}{\sqrt{3-x+2 x^2} \left (2+3 x+5 x^2\right )} \, dx}{6891599049808 \sqrt{2}}+\frac{\int \frac{\frac{1010229}{8} \left (1242839+847654 \sqrt{2}\right )-\frac{1010229}{8} \left (452469+395185 \sqrt{2}\right ) x}{\sqrt{3-x+2 x^2} \left (2+3 x+5 x^2\right )} \, dx}{6891599049808 \sqrt{2}}\\ &=-\frac{4353943-6508666 x}{941410976 \sqrt{3-x+2 x^2}}+\frac{4+65 x}{1364 \sqrt{3-x+2 x^2} \left (2+3 x+5 x^2\right )^2}+\frac{5 (7318+17315 x)}{1860496 \sqrt{3-x+2 x^2} \left (2+3 x+5 x^2\right )}-\frac{\left (2277 \left (19639477300000-13874275807943 \sqrt{2}\right )\right ) \operatorname{Subst}\left (\int \frac{1}{-\frac{31637441605671}{64} \left (13874275807943-9819738650000 \sqrt{2}\right )-11 x^2} \, dx,x,\frac{\frac{1010229}{8} \left (5538393-4123702 \sqrt{2}\right )+\frac{1010229}{8} \left (13785797-9662095 \sqrt{2}\right ) x}{\sqrt{3-x+2 x^2}}\right )}{984064}-\frac{\left (2277 \left (19639477300000+13874275807943 \sqrt{2}\right )\right ) \operatorname{Subst}\left (\int \frac{1}{-\frac{31637441605671}{64} \left (13874275807943+9819738650000 \sqrt{2}\right )-11 x^2} \, dx,x,\frac{\frac{1010229}{8} \left (5538393+4123702 \sqrt{2}\right )+\frac{1010229}{8} \left (13785797+9662095 \sqrt{2}\right ) x}{\sqrt{3-x+2 x^2}}\right )}{984064}\\ &=-\frac{4353943-6508666 x}{941410976 \sqrt{3-x+2 x^2}}+\frac{4+65 x}{1364 \sqrt{3-x+2 x^2} \left (2+3 x+5 x^2\right )^2}+\frac{5 (7318+17315 x)}{1860496 \sqrt{3-x+2 x^2} \left (2+3 x+5 x^2\right )}+\frac{3 \sqrt{\frac{1}{682} \left (13874275807943+9819738650000 \sqrt{2}\right )} \tan ^{-1}\left (\frac{\sqrt{\frac{11}{31 \left (13874275807943+9819738650000 \sqrt{2}\right )}} \left (5538393+4123702 \sqrt{2}+\left (13785797+9662095 \sqrt{2}\right ) x\right )}{\sqrt{3-x+2 x^2}}\right )}{81861824}-\frac{3 \sqrt{\frac{1}{682} \left (-13874275807943+9819738650000 \sqrt{2}\right )} \tanh ^{-1}\left (\frac{\sqrt{\frac{11}{31 \left (-13874275807943+9819738650000 \sqrt{2}\right )}} \left (5538393-4123702 \sqrt{2}+\left (13785797-9662095 \sqrt{2}\right ) x\right )}{\sqrt{3-x+2 x^2}}\right )}{81861824}\\ \end{align*}
Mathematica [C] time = 2.24957, size = 231, normalized size = 0.94 \[ \frac{\frac{27280 \left (162716650 x^5+86411405 x^4+277167774 x^3+175833195 x^2+161806828 x+22374044\right )}{\sqrt{2 x^2-x+3} \left (5 x^2+3 x+2\right )^2}+69 \sqrt{286-22 i \sqrt{31}} \left (13785797 \sqrt{31}+14026539 i\right ) \tan ^{-1}\left (\frac{-2 \left (2 \sqrt{31}+11 i\right ) x+\sqrt{31}+63 i}{2 \sqrt{286-22 i \sqrt{31}} \sqrt{2 x^2-x+3}}\right )-69 i \sqrt{286+22 i \sqrt{31}} \left (13785797 \sqrt{31}-14026539 i\right ) \tanh ^{-1}\left (\frac{\left (-22-4 i \sqrt{31}\right ) x+i \sqrt{31}+63}{2 \sqrt{286+22 i \sqrt{31}} \sqrt{2 x^2-x+3}}\right )}{25681691425280} \]
Antiderivative was successfully verified.
[In]
Integrate[1/((3 - x + 2*x^2)^(3/2)*(2 + 3*x + 5*x^2)^3),x]
[Out]
((27280*(22374044 + 161806828*x + 175833195*x^2 + 277167774*x^3 + 86411405*x^4 + 162716650*x^5))/(Sqrt[3 - x +
2*x^2]*(2 + 3*x + 5*x^2)^2) + 69*Sqrt[286 - (22*I)*Sqrt[31]]*(14026539*I + 13785797*Sqrt[31])*ArcTan[(63*I +
Sqrt[31] - 2*(11*I + 2*Sqrt[31])*x)/(2*Sqrt[286 - (22*I)*Sqrt[31]]*Sqrt[3 - x + 2*x^2])] - (69*I)*Sqrt[286 + (
22*I)*Sqrt[31]]*(-14026539*I + 13785797*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])])/25681691425280
________________________________________________________________________________________
Maple [B] time = 0.266, size = 18981, normalized size = 77.2 \begin{align*} \text{output too large to display} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
int(1/(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{1}{{\left (5 \, x^{2} + 3 \, x + 2\right )}^{3}{\left (2 \, x^{2} - x + 3\right )}^{\frac{3}{2}}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate(1/(2*x^2-x+3)^(3/2)/(5*x^2+3*x+2)^3,x, algorithm="maxima")
[Out]
integrate(1/((5*x^2 + 3*x + 2)^3*(2*x^2 - x + 3)^(3/2)), x)
________________________________________________________________________________________
Fricas [B] time = 5.4732, size = 11105, normalized size = 45.14 \begin{align*} \text{result too large to display} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate(1/(2*x^2-x+3)^(3/2)/(5*x^2+3*x+2)^3,x, algorithm="fricas")
[Out]
1/33652296632397026886019646994897920*(920746859815884*1928545343086076450^(1/4)*sqrt(1963947730)*sqrt(341)*sq
rt(2)*(50*x^6 + 35*x^5 + 103*x^4 + 85*x^3 + 83*x^2 + 32*x + 12)*sqrt(13874275807943*sqrt(2) + 19639477300000)*
arctan(1/2252270155289097943751876925347228391692375*(2800589462980*sqrt(1963947730)*(22*1928545343086076450^(
3/4)*sqrt(341)*(7361410004*x^7 - 28555361914*x^6 + 59872788262*x^5 - 96593638888*x^4 + 48573560944*x^3 - 23355
012672*x^2 - sqrt(2)*(5311119598*x^7 - 20292577289*x^6 + 42695479118*x^5 - 68006818683*x^4 + 33985514680*x^3 -
15860251800*x^2 - 37489478400*x + 26167456512) - 52334913024*x + 37489478400) + 30441189815*19285453430860764
50^(1/4)*sqrt(341)*(560592897*x^7 - 8616399363*x^6 + 45618625128*x^5 - 104316505460*x^4 + 134890825824*x^3 - 8
5859939808*x^2 - sqrt(2)*(402019087*x^7 - 6162703212*x^6 + 32499503540*x^5 - 73942829952*x^4 + 95407993440*x^3
- 59600016000*x^2 - 68177562624*x + 47773380096) - 95546760192*x + 68177562624))*sqrt(2*x^2 - x + 3)*sqrt(138
74275807943*sqrt(2) + 19639477300000) + 6393541085981955453231134497759874144159000*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(1963947730/3471424919)*(sqrt(1963947730)*(22*1928545343086076450^(3/4)*sqrt(341)*(26184810824*x^7
- 37618468196*x^6 + 121297463436*x^5 - 48741866816*x^4 + 58784153184*x^3 + 51583129344*x^2 - sqrt(2)*(19194187
986*x^7 - 27528525721*x^6 + 88457613411*x^5 - 33685377680*x^4 + 38926767960*x^3 + 41764674816*x^2 - 4176467481
6*x) - 51583129344*x) + 30441189815*1928545343086076450^(1/4)*sqrt(341)*(1998926311*x^7 - 25858659004*x^6 + 99
738083860*x^5 - 129415692096*x^4 + 167446420704*x^3 + 96037622784*x^2 - 22*sqrt(2)*(65886479*x^7 - 852213084*x
^6 + 3285070260*x^5 - 4244909760*x^4 + 5424792480*x^3 + 3393259776*x^2 - 3393259776*x) - 96037622784*x))*sqrt(
2*x^2 - x + 3)*sqrt(13874275807943*sqrt(2) + 19639477300000) + 2282926923240949861309948624550*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)*(1555
0*x^8 - 118051*x^7 + 244047*x^6 - 707374*x^5 + 1053960*x^4 - 1667952*x^3 + 1209600*x^2 - 1036800*x) + 3276288*
x) + 103769405601861357332270392025*sqrt(31)*(254591*x^8 - 4815126*x^7 + 32303580*x^6 - 90866808*x^5 + 1087819
20*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(-(1928545343086076450^(1/4)*sqrt(1963947730)*sqrt(341)*sqrt(31)*s
qrt(2*x^2 - x + 3)*(sqrt(2)*(2995431*x + 1523456) - 4518887*x - 1471975)*sqrt(13874275807943*sqrt(2) + 1963947
7300000) - 160519269124568199977215*x^2 - 144139751866959199979540*sqrt(2)*(2*x^2 - x + 3) + 49466142117979179
9929785*x - 655180690304359999907000)/x^2) + 72653875977067675604899255656362206183625*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)*(134
8*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 - 948
87936))/(2585191*x^8 - 4661200*x^7 + 14191920*x^6 + 490880*x^5 - 13562944*x^4 + 44249088*x^3 - 34615296*x^2 -
24772608*x + 18579456)) + 920746859815884*1928545343086076450^(1/4)*sqrt(1963947730)*sqrt(341)*sqrt(2)*(50*x^6
+ 35*x^5 + 103*x^4 + 85*x^3 + 83*x^2 + 32*x + 12)*sqrt(13874275807943*sqrt(2) + 19639477300000)*arctan(1/2252
270155289097943751876925347228391692375*(2800589462980*sqrt(1963947730)*(22*1928545343086076450^(3/4)*sqrt(341
)*(7361410004*x^7 - 28555361914*x^6 + 59872788262*x^5 - 96593638888*x^4 + 48573560944*x^3 - 23355012672*x^2 -
sqrt(2)*(5311119598*x^7 - 20292577289*x^6 + 42695479118*x^5 - 68006818683*x^4 + 33985514680*x^3 - 15860251800*
x^2 - 37489478400*x + 26167456512) - 52334913024*x + 37489478400) + 30441189815*1928545343086076450^(1/4)*sqrt
(341)*(560592897*x^7 - 8616399363*x^6 + 45618625128*x^5 - 104316505460*x^4 + 134890825824*x^3 - 85859939808*x^
2 - sqrt(2)*(402019087*x^7 - 6162703212*x^6 + 32499503540*x^5 - 73942829952*x^4 + 95407993440*x^3 - 5960001600
0*x^2 - 68177562624*x + 47773380096) - 95546760192*x + 68177562624))*sqrt(2*x^2 - x + 3)*sqrt(13874275807943*s
qrt(2) + 19639477300000) - 6393541085981955453231134497759874144159000*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(1
963947730/3471424919)*(sqrt(1963947730)*(22*1928545343086076450^(3/4)*sqrt(341)*(26184810824*x^7 - 37618468196
*x^6 + 121297463436*x^5 - 48741866816*x^4 + 58784153184*x^3 + 51583129344*x^2 - sqrt(2)*(19194187986*x^7 - 275
28525721*x^6 + 88457613411*x^5 - 33685377680*x^4 + 38926767960*x^3 + 41764674816*x^2 - 41764674816*x) - 515831
29344*x) + 30441189815*1928545343086076450^(1/4)*sqrt(341)*(1998926311*x^7 - 25858659004*x^6 + 99738083860*x^5
- 129415692096*x^4 + 167446420704*x^3 + 96037622784*x^2 - 22*sqrt(2)*(65886479*x^7 - 852213084*x^6 + 32850702
60*x^5 - 4244909760*x^4 + 5424792480*x^3 + 3393259776*x^2 - 3393259776*x) - 96037622784*x))*sqrt(2*x^2 - x + 3
)*sqrt(13874275807943*sqrt(2) + 19639477300000) - 2282926923240949861309948624550*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 - 11805
1*x^7 + 244047*x^6 - 707374*x^5 + 1053960*x^4 - 1667952*x^3 + 1209600*x^2 - 1036800*x) + 3276288*x) - 10376940
5601861357332270392025*sqrt(31)*(254591*x^8 - 4815126*x^7 + 32303580*x^6 - 90866808*x^5 + 108781920*x^4 - 7421
9328*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((1928545343086076450^(1/4)*sqrt(1963947730)*sqrt(341)*sqrt(31)*sqrt(2*x^2 - x
+ 3)*(sqrt(2)*(2995431*x + 1523456) - 4518887*x - 1471975)*sqrt(13874275807943*sqrt(2) + 19639477300000) + 160
519269124568199977215*x^2 + 144139751866959199979540*sqrt(2)*(2*x^2 - x + 3) - 494661421179791799929785*x + 65
5180690304359999907000)/x^2) - 72653875977067675604899255656362206183625*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))/(25851
91*x^8 - 4661200*x^7 + 14191920*x^6 + 490880*x^5 - 13562944*x^4 + 44249088*x^3 - 34615296*x^2 - 24772608*x + 1
8579456)) + 69*1928545343086076450^(1/4)*sqrt(1963947730)*sqrt(341)*sqrt(31)*(981973865000000*x^6 + 6873817055
00000*x^5 + 2022866161900000*x^4 + 1669355570500000*x^3 + 1630076615900000*x^2 - 13874275807943*sqrt(2)*(50*x^
6 + 35*x^5 + 103*x^4 + 85*x^3 + 83*x^2 + 32*x + 12) + 628463273600000*x + 235673727600000)*sqrt(13874275807943
*sqrt(2) + 19639477300000)*log(1767552957000000000/3471424919*(1928545343086076450^(1/4)*sqrt(1963947730)*sqrt
(341)*sqrt(31)*sqrt(2*x^2 - x + 3)*(sqrt(2)*(2995431*x + 1523456) - 4518887*x - 1471975)*sqrt(13874275807943*s
qrt(2) + 19639477300000) + 160519269124568199977215*x^2 + 144139751866959199979540*sqrt(2)*(2*x^2 - x + 3) - 4
94661421179791799929785*x + 655180690304359999907000)/x^2) - 69*1928545343086076450^(1/4)*sqrt(1963947730)*sqr
t(341)*sqrt(31)*(981973865000000*x^6 + 687381705500000*x^5 + 2022866161900000*x^4 + 1669355570500000*x^3 + 163
0076615900000*x^2 - 13874275807943*sqrt(2)*(50*x^6 + 35*x^5 + 103*x^4 + 85*x^3 + 83*x^2 + 32*x + 12) + 6284632
73600000*x + 235673727600000)*sqrt(13874275807943*sqrt(2) + 19639477300000)*log(-1767552957000000000/347142491
9*(1928545343086076450^(1/4)*sqrt(1963947730)*sqrt(341)*sqrt(31)*sqrt(2*x^2 - x + 3)*(sqrt(2)*(2995431*x + 152
3456) - 4518887*x - 1471975)*sqrt(13874275807943*sqrt(2) + 19639477300000) - 160519269124568199977215*x^2 - 14
4139751866959199979540*sqrt(2)*(2*x^2 - x + 3) + 494661421179791799929785*x - 655180690304359999907000)/x^2) +
35746658463005881594925920*(162716650*x^5 + 86411405*x^4 + 277167774*x^3 + 175833195*x^2 + 161806828*x + 2237
4044)*sqrt(2*x^2 - x + 3))/(50*x^6 + 35*x^5 + 103*x^4 + 85*x^3 + 83*x^2 + 32*x + 12)
________________________________________________________________________________________
Sympy [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{1}{\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(1/(2*x**2-x+3)**(3/2)/(5*x**2+3*x+2)**3,x)
[Out]
Integral(1/((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(1/(2*x^2-x+3)^(3/2)/(5*x^2+3*x+2)^3,x, algorithm="giac")
[Out]
Exception raised: TypeError