3.85 \(\int \frac{1}{\sqrt{3-x+2 x^2} (2+3 x+5 x^2)^3} \, dx\)
Optimal. Leaf size=223 \[ \frac{\sqrt{2 x^2-x+3} (65 x+4)}{1364 \left (5 x^2+3 x+2\right )^2}+\frac{(86265 x+26794) \sqrt{2 x^2-x+3}}{1860496 \left (5 x^2+3 x+2\right )}+\frac{25 \sqrt{\frac{1}{682} \left (6414867847+4536374600 \sqrt{2}\right )} \tan ^{-1}\left (\frac{\sqrt{\frac{11}{31 \left (6414867847+4536374600 \sqrt{2}\right )}} \left (\left (294669+208915 \sqrt{2}\right ) x+85754 \sqrt{2}+123161\right )}{\sqrt{2 x^2-x+3}}\right )}{3720992}-\frac{25 \sqrt{\frac{1}{682} \left (4536374600 \sqrt{2}-6414867847\right )} \tanh ^{-1}\left (\frac{\sqrt{\frac{11}{31 \left (4536374600 \sqrt{2}-6414867847\right )}} \left (\left (294669-208915 \sqrt{2}\right ) x-85754 \sqrt{2}+123161\right )}{\sqrt{2 x^2-x+3}}\right )}{3720992} \]
[Out]
((4 + 65*x)*Sqrt[3 - x + 2*x^2])/(1364*(2 + 3*x + 5*x^2)^2) + ((26794 + 86265*x)*Sqrt[3 - x + 2*x^2])/(1860496
*(2 + 3*x + 5*x^2)) + (25*Sqrt[(6414867847 + 4536374600*Sqrt[2])/682]*ArcTan[(Sqrt[11/(31*(6414867847 + 453637
4600*Sqrt[2]))]*(123161 + 85754*Sqrt[2] + (294669 + 208915*Sqrt[2])*x))/Sqrt[3 - x + 2*x^2]])/3720992 - (25*Sq
rt[(-6414867847 + 4536374600*Sqrt[2])/682]*ArcTanh[(Sqrt[11/(31*(-6414867847 + 4536374600*Sqrt[2]))]*(123161 -
85754*Sqrt[2] + (294669 - 208915*Sqrt[2])*x))/Sqrt[3 - x + 2*x^2]])/3720992
________________________________________________________________________________________
Rubi [A] time = 0.469453, 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 =
{974, 1060, 1035, 1029, 206, 204} \[ \frac{\sqrt{2 x^2-x+3} (65 x+4)}{1364 \left (5 x^2+3 x+2\right )^2}+\frac{(86265 x+26794) \sqrt{2 x^2-x+3}}{1860496 \left (5 x^2+3 x+2\right )}+\frac{25 \sqrt{\frac{1}{682} \left (6414867847+4536374600 \sqrt{2}\right )} \tan ^{-1}\left (\frac{\sqrt{\frac{11}{31 \left (6414867847+4536374600 \sqrt{2}\right )}} \left (\left (294669+208915 \sqrt{2}\right ) x+85754 \sqrt{2}+123161\right )}{\sqrt{2 x^2-x+3}}\right )}{3720992}-\frac{25 \sqrt{\frac{1}{682} \left (4536374600 \sqrt{2}-6414867847\right )} \tanh ^{-1}\left (\frac{\sqrt{\frac{11}{31 \left (4536374600 \sqrt{2}-6414867847\right )}} \left (\left (294669-208915 \sqrt{2}\right ) x-85754 \sqrt{2}+123161\right )}{\sqrt{2 x^2-x+3}}\right )}{3720992} \]
Antiderivative was successfully verified.
[In]
Int[1/(Sqrt[3 - x + 2*x^2]*(2 + 3*x + 5*x^2)^3),x]
[Out]
((4 + 65*x)*Sqrt[3 - x + 2*x^2])/(1364*(2 + 3*x + 5*x^2)^2) + ((26794 + 86265*x)*Sqrt[3 - x + 2*x^2])/(1860496
*(2 + 3*x + 5*x^2)) + (25*Sqrt[(6414867847 + 4536374600*Sqrt[2])/682]*ArcTan[(Sqrt[11/(31*(6414867847 + 453637
4600*Sqrt[2]))]*(123161 + 85754*Sqrt[2] + (294669 + 208915*Sqrt[2])*x))/Sqrt[3 - x + 2*x^2]])/3720992 - (25*Sq
rt[(-6414867847 + 4536374600*Sqrt[2])/682]*ArcTanh[(Sqrt[11/(31*(-6414867847 + 4536374600*Sqrt[2]))]*(123161 -
85754*Sqrt[2] + (294669 - 208915*Sqrt[2])*x))/Sqrt[3 - x + 2*x^2]])/3720992
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}{\sqrt{3-x+2 x^2} \left (2+3 x+5 x^2\right )^3} \, dx &=\frac{(4+65 x) \sqrt{3-x+2 x^2}}{1364 \left (2+3 x+5 x^2\right )^2}-\frac{\int \frac{-5775+\frac{6479 x}{2}-2860 x^2}{\sqrt{3-x+2 x^2} \left (2+3 x+5 x^2\right )^2} \, dx}{15004}\\ &=\frac{(4+65 x) \sqrt{3-x+2 x^2}}{1364 \left (2+3 x+5 x^2\right )^2}+\frac{(26794+86265 x) \sqrt{3-x+2 x^2}}{1860496 \left (2+3 x+5 x^2\right )}-\frac{\int \frac{-\frac{28220225}{2}+\frac{22521125 x}{4}}{\sqrt{3-x+2 x^2} \left (2+3 x+5 x^2\right )} \, dx}{112560008}\\ &=\frac{(4+65 x) \sqrt{3-x+2 x^2}}{1364 \left (2+3 x+5 x^2\right )^2}+\frac{(26794+86265 x) \sqrt{3-x+2 x^2}}{1860496 \left (2+3 x+5 x^2\right )}-\frac{\int \frac{\frac{33275}{4} \left (26103-18658 \sqrt{2}\right )-\frac{33275}{4} \left (11213-7445 \sqrt{2}\right ) x}{\sqrt{3-x+2 x^2} \left (2+3 x+5 x^2\right )} \, dx}{2476320176 \sqrt{2}}+\frac{\int \frac{\frac{33275}{4} \left (26103+18658 \sqrt{2}\right )-\frac{33275}{4} \left (11213+7445 \sqrt{2}\right ) x}{\sqrt{3-x+2 x^2} \left (2+3 x+5 x^2\right )} \, dx}{2476320176 \sqrt{2}}\\ &=\frac{(4+65 x) \sqrt{3-x+2 x^2}}{1364 \left (2+3 x+5 x^2\right )^2}+\frac{(26794+86265 x) \sqrt{3-x+2 x^2}}{1860496 \left (2+3 x+5 x^2\right )}-\frac{\left (6875 \left (9072749200-6414867847 \sqrt{2}\right )\right ) \operatorname{Subst}\left (\int \frac{1}{-\frac{34323994375}{16} \left (6414867847-4536374600 \sqrt{2}\right )-11 x^2} \, dx,x,\frac{\frac{33275}{4} \left (123161-85754 \sqrt{2}\right )+\frac{33275}{4} \left (294669-208915 \sqrt{2}\right ) x}{\sqrt{3-x+2 x^2}}\right )}{246016}-\frac{\left (6875 \left (9072749200+6414867847 \sqrt{2}\right )\right ) \operatorname{Subst}\left (\int \frac{1}{-\frac{34323994375}{16} \left (6414867847+4536374600 \sqrt{2}\right )-11 x^2} \, dx,x,\frac{\frac{33275}{4} \left (123161+85754 \sqrt{2}\right )+\frac{33275}{4} \left (294669+208915 \sqrt{2}\right ) x}{\sqrt{3-x+2 x^2}}\right )}{246016}\\ &=\frac{(4+65 x) \sqrt{3-x+2 x^2}}{1364 \left (2+3 x+5 x^2\right )^2}+\frac{(26794+86265 x) \sqrt{3-x+2 x^2}}{1860496 \left (2+3 x+5 x^2\right )}+\frac{25 \sqrt{\frac{1}{682} \left (6414867847+4536374600 \sqrt{2}\right )} \tan ^{-1}\left (\frac{\sqrt{\frac{11}{31 \left (6414867847+4536374600 \sqrt{2}\right )}} \left (123161+85754 \sqrt{2}+\left (294669+208915 \sqrt{2}\right ) x\right )}{\sqrt{3-x+2 x^2}}\right )}{3720992}-\frac{25 \sqrt{\frac{1}{682} \left (-6414867847+4536374600 \sqrt{2}\right )} \tanh ^{-1}\left (\frac{\sqrt{\frac{11}{31 \left (-6414867847+4536374600 \sqrt{2}\right )}} \left (123161-85754 \sqrt{2}+\left (294669-208915 \sqrt{2}\right ) x\right )}{\sqrt{3-x+2 x^2}}\right )}{3720992}\\ \end{align*}
Mathematica [C] time = 6.23773, size = 1279, normalized size = 5.74 \[ \text{result too large to display} \]
Antiderivative was successfully verified.
[In]
Integrate[1/(Sqrt[3 - x + 2*x^2]*(2 + 3*x + 5*x^2)^3),x]
[Out]
(2500*Sqrt[3 - x + 2*x^2])/(341*Sqrt[31]*(13*I + Sqrt[31])*(3 - I*Sqrt[31] + 10*x)^2) + (7500*Sqrt[3 - x + 2*x
^2])/(10571*(13 - I*Sqrt[31])*(3 - I*Sqrt[31] + 10*x)) - (2500*Sqrt[3 - x + 2*x^2])/(341*Sqrt[31]*(13*I - Sqrt
[31])*(3 + I*Sqrt[31] + 10*x)^2) + (7500*Sqrt[3 - x + 2*x^2])/(10571*(13 + I*Sqrt[31])*(3 + I*Sqrt[31] + 10*x)
) - (375*Sqrt[(2*(13 - I*Sqrt[31]))/11]*(11 - (2*I)*Sqrt[31])*ArcTanh[(63 - I*Sqrt[31] - 2*(11 - (2*I)*Sqrt[31
])*x)/(2*Sqrt[22*(13 - I*Sqrt[31])]*Sqrt[3 - x + 2*x^2])])/(10571*(13*I + Sqrt[31])^2) - (750*Sqrt[(2*(13 - I*
Sqrt[31]))/341]*ArcTanh[(63 - I*Sqrt[31] - 2*(11 - (2*I)*Sqrt[31])*x)/(2*Sqrt[22*(13 - I*Sqrt[31])]*Sqrt[3 - x
+ 2*x^2])])/(961*(13*I + Sqrt[31])) + (750*Sqrt[(2*(13 + I*Sqrt[31]))/341]*ArcTanh[(63 + I*Sqrt[31] - 2*(11 +
(2*I)*Sqrt[31])*x)/(2*Sqrt[22*(13 + I*Sqrt[31])]*Sqrt[3 - x + 2*x^2])])/(961*(13*I - Sqrt[31])) - (375*Sqrt[(
2*(13 + I*Sqrt[31]))/11]*(11 + (2*I)*Sqrt[31])*ArcTanh[(63 + I*Sqrt[31] - 2*(11 + (2*I)*Sqrt[31])*x)/(2*Sqrt[2
2*(13 + I*Sqrt[31])]*Sqrt[3 - x + 2*x^2])])/(10571*(13*I - Sqrt[31])^2) + (((500*I)/31)*(-(((20*(-3 + I*Sqrt[3
1]) - 10*(27 - (4*I)*Sqrt[31]))*Sqrt[3 - x + 2*x^2])/((300 - 10*(-3 + I*Sqrt[31]) + 2*(-3 + I*Sqrt[31])^2)*(-3
+ I*Sqrt[31] - 10*x))) + (2*Sqrt[22*(13 - I*Sqrt[31])]*(20*(-3 + I*Sqrt[31]) + 10*(27 - (4*I)*Sqrt[31]) - 2*(
600 + 2*(-3 + I*Sqrt[31])*(27 - (4*I)*Sqrt[31])))*ArcTanh[(-63 + I*Sqrt[31] - (-10 + 4*(-3 + I*Sqrt[31]))*x)/(
2*Sqrt[22*(13 - I*Sqrt[31])]*Sqrt[3 - x + 2*x^2])])/((300 - 10*(-3 + I*Sqrt[31]) + 2*(-3 + I*Sqrt[31])^2)*(120
0 - 40*(-3 + I*Sqrt[31]) + 8*(-3 + I*Sqrt[31])^2))))/(Sqrt[31]*(300 - 10*(-3 + I*Sqrt[31]) + 2*(-3 + I*Sqrt[31
])^2)) + (((500*I)/31)*(-(((-20*(3 + I*Sqrt[31]) + 10*(-27 - (4*I)*Sqrt[31]))*Sqrt[3 - x + 2*x^2])/((300 + 10*
(3 + I*Sqrt[31]) + 2*(3 + I*Sqrt[31])^2)*(3 + I*Sqrt[31] + 10*x))) + (2*Sqrt[22*(13 + I*Sqrt[31])]*(-20*(3 + I
*Sqrt[31]) - 10*(-27 - (4*I)*Sqrt[31]) - 2*(600 + 2*(3 + I*Sqrt[31])*(-27 - (4*I)*Sqrt[31])))*ArcTanh[(63 + I*
Sqrt[31] - (10 + 4*(3 + I*Sqrt[31]))*x)/(2*Sqrt[22*(13 + I*Sqrt[31])]*Sqrt[3 - x + 2*x^2])])/((300 + 10*(3 + I
*Sqrt[31]) + 2*(3 + I*Sqrt[31])^2)*(1200 + 40*(3 + I*Sqrt[31]) + 8*(3 + I*Sqrt[31])^2))))/(Sqrt[31]*(300 + 10*
(3 + I*Sqrt[31]) + 2*(3 + I*Sqrt[31])^2))
________________________________________________________________________________________
Maple [B] time = 0.191, size = 13040, normalized size = 58.5 \begin{align*} \text{output too large to display} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
int(1/(5*x^2+3*x+2)^3/(2*x^2-x+3)^(1/2),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} \sqrt{2 \, x^{2} - x + 3}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate(1/(5*x^2+3*x+2)^3/(2*x^2-x+3)^(1/2),x, algorithm="maxima")
[Out]
integrate(1/((5*x^2 + 3*x + 2)^3*sqrt(2*x^2 - x + 3)), x)
________________________________________________________________________________________
Fricas [B] time = 5.17811, size = 9927, normalized size = 44.52 \begin{align*} \text{result too large to display} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate(1/(5*x^2+3*x+2)^3/(2*x^2-x+3)^(1/2),x, algorithm="fricas")
[Out]
-1/212344000027477426346822144*(46113488900*4115738902305032^(1/4)*sqrt(22681873)*sqrt(341)*sqrt(2)*(25*x^4 +
30*x^3 + 29*x^2 + 12*x + 4)*sqrt(6414867847*sqrt(2) + 9072749200)*arctan(1/3836668309294009530058322373948769*
(64688701796*sqrt(22681873)*(11*4115738902305032^(3/4)*sqrt(341)*(160344708*x^7 - 615873378*x^6 + 1294230774*x
^5 - 2070733376*x^4 + 1037098288*x^3 - 489164544*x^2 - sqrt(2)*(112700446*x^7 - 434839553*x^6 + 912850886*x^5
- 1466127691*x^4 + 735661560*x^3 - 350098200*x^2 - 799200000*x + 567316224) - 1134632448*x + 799200000) + 7031
38063*4115738902305032^(1/4)*sqrt(341)*(12162569*x^7 - 186616851*x^6 + 985490056*x^5 - 2246141620*x^4 + 290038
2048*x^3 - 1823848416*x^2 - sqrt(2)*(8564099*x^7 - 131508024*x^6 + 695288980*x^5 - 1587105104*x^4 + 2050714080
*x^3 - 1296806400*x^2 - 1457077248*x + 1033108992) - 2066217984*x + 1457077248))*sqrt(2*x^2 - x + 3)*sqrt(6414
867847*sqrt(2) + 9072749200) + 10891187458641059311133302222822312*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(45363
746/479849)*(sqrt(22681873)*(11*4115738902305032^(3/4)*sqrt(341)*(576322648*x^7 - 827050092*x^6 + 2660713572*x
^5 - 1032439232*x^4 + 1211604768*x^3 + 1213394688*x^2 - sqrt(2)*(403157522*x^7 - 578844217*x^6 + 1864129347*x^
5 - 735062160*x^4 + 873708120*x^3 + 823986432*x^2 - 823986432*x) - 1213394688*x) + 703138063*4115738902305032^
(1/4)*sqrt(341)*(43684647*x^7 - 565067708*x^6 + 2178643220*x^5 - 2819241792*x^4 + 3618371808*x^3 + 2197767168*
x^2 - 2*sqrt(2)*(15328963*x^7 - 198290348*x^6 + 764653220*x^5 - 990717120*x^4 + 1276256160*x^3 + 755350272*x^2
- 755350272*x) - 2197767168*x))*sqrt(2*x^2 - x + 3)*sqrt(6414867847*sqrt(2) + 9072749200) + 16836305500436726
2339322*sqrt(31)*sqrt(2)*(123408*x^8 - 914152*x^7 + 1578888*x^6 - 3293072*x^5 + 396480*x^4 + 798336*x^3 - 3822
336*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) + 7652866136562148288151*sqrt(31)*(254591*x^8 - 4815126*x^7 + 32303580*x^6 - 9086680
8*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 + 23
85*x^4 - 3618*x^3 + 2268*x^2 - 1944*x) + 144820224*x))*sqrt(-(4115738902305032^(1/4)*sqrt(22681873)*sqrt(341)*
sqrt(31)*sqrt(2*x^2 - x + 3)*(sqrt(2)*(67187*x + 26012) - 93199*x - 41175)*sqrt(6414867847*sqrt(2) + 907274920
0) - 512510746420187753*x^2 - 460213731479352268*sqrt(2)*(2*x^2 - x + 3) + 1579369851213231647*x - 20918805976
33419400)/x^2) + 123763493848193855808332979804799*sqrt(31)*(2828123*x^8 - 9696916*x^7 + 53385560*x^6 - 142835
344*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 +
14191920*x^6 + 490880*x^5 - 13562944*x^4 + 44249088*x^3 - 34615296*x^2 - 24772608*x + 18579456)) + 4611348890
0*4115738902305032^(1/4)*sqrt(22681873)*sqrt(341)*sqrt(2)*(25*x^4 + 30*x^3 + 29*x^2 + 12*x + 4)*sqrt(641486784
7*sqrt(2) + 9072749200)*arctan(1/3836668309294009530058322373948769*(64688701796*sqrt(22681873)*(11*4115738902
305032^(3/4)*sqrt(341)*(160344708*x^7 - 615873378*x^6 + 1294230774*x^5 - 2070733376*x^4 + 1037098288*x^3 - 489
164544*x^2 - sqrt(2)*(112700446*x^7 - 434839553*x^6 + 912850886*x^5 - 1466127691*x^4 + 735661560*x^3 - 3500982
00*x^2 - 799200000*x + 567316224) - 1134632448*x + 799200000) + 703138063*4115738902305032^(1/4)*sqrt(341)*(12
162569*x^7 - 186616851*x^6 + 985490056*x^5 - 2246141620*x^4 + 2900382048*x^3 - 1823848416*x^2 - sqrt(2)*(85640
99*x^7 - 131508024*x^6 + 695288980*x^5 - 1587105104*x^4 + 2050714080*x^3 - 1296806400*x^2 - 1457077248*x + 103
3108992) - 2066217984*x + 1457077248))*sqrt(2*x^2 - x + 3)*sqrt(6414867847*sqrt(2) + 9072749200) - 10891187458
641059311133302222822312*sqrt(31)*sqrt(2)*(28180*x^8 - 254666*x^7 + 704270*x^6 - 1385256*x^5 + 1549144*x^4 - 6
42048*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(45363746/479849)*(sqrt(22681873)*(11*4115738902
305032^(3/4)*sqrt(341)*(576322648*x^7 - 827050092*x^6 + 2660713572*x^5 - 1032439232*x^4 + 1211604768*x^3 + 121
3394688*x^2 - sqrt(2)*(403157522*x^7 - 578844217*x^6 + 1864129347*x^5 - 735062160*x^4 + 873708120*x^3 + 823986
432*x^2 - 823986432*x) - 1213394688*x) + 703138063*4115738902305032^(1/4)*sqrt(341)*(43684647*x^7 - 565067708*
x^6 + 2178643220*x^5 - 2819241792*x^4 + 3618371808*x^3 + 2197767168*x^2 - 2*sqrt(2)*(15328963*x^7 - 198290348*
x^6 + 764653220*x^5 - 990717120*x^4 + 1276256160*x^3 + 755350272*x^2 - 755350272*x) - 2197767168*x))*sqrt(2*x^
2 - x + 3)*sqrt(6414867847*sqrt(2) + 9072749200) - 168363055004367262339322*sqrt(31)*sqrt(2)*(123408*x^8 - 914
152*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) - 76528661365621
48288151*sqrt(31)*(254591*x^8 - 4815126*x^7 + 32303580*x^6 - 90866808*x^5 + 108781920*x^4 - 74219328*x^3 - 168
956928*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) + 1
44820224*x))*sqrt((4115738902305032^(1/4)*sqrt(22681873)*sqrt(341)*sqrt(31)*sqrt(2*x^2 - x + 3)*(sqrt(2)*(6718
7*x + 26012) - 93199*x - 41175)*sqrt(6414867847*sqrt(2) + 9072749200) + 512510746420187753*x^2 + 4602137314793
52268*sqrt(2)*(2*x^2 - x + 3) - 1579369851213231647*x + 2091880597633419400)/x^2) - 12376349384819385580833297
9804799*sqrt(31)*(2828123*x^8 - 9696916*x^7 + 53385560*x^6 - 142835344*x^5 + 254146592*x^4 - 249300096*x^3 + 3
7981440*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 + 432
0*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 + 18579456)) - 25*4115738902305032^(1/4)*sqrt(22681873)*sqrt(341)*sq
rt(31)*(226818730000*x^4 + 272182476000*x^3 + 263109726800*x^2 - 6414867847*sqrt(2)*(25*x^4 + 30*x^3 + 29*x^2
+ 12*x + 4) + 108872990400*x + 36290996800)*sqrt(6414867847*sqrt(2) + 9072749200)*log(1134093650000000/479849*
(4115738902305032^(1/4)*sqrt(22681873)*sqrt(341)*sqrt(31)*sqrt(2*x^2 - x + 3)*(sqrt(2)*(67187*x + 26012) - 931
99*x - 41175)*sqrt(6414867847*sqrt(2) + 9072749200) + 512510746420187753*x^2 + 460213731479352268*sqrt(2)*(2*x
^2 - x + 3) - 1579369851213231647*x + 2091880597633419400)/x^2) + 25*4115738902305032^(1/4)*sqrt(22681873)*sqr
t(341)*sqrt(31)*(226818730000*x^4 + 272182476000*x^3 + 263109726800*x^2 - 6414867847*sqrt(2)*(25*x^4 + 30*x^3
+ 29*x^2 + 12*x + 4) + 108872990400*x + 36290996800)*sqrt(6414867847*sqrt(2) + 9072749200)*log(-11340936500000
00/479849*(4115738902305032^(1/4)*sqrt(22681873)*sqrt(341)*sqrt(31)*sqrt(2*x^2 - x + 3)*(sqrt(2)*(67187*x + 26
012) - 93199*x - 41175)*sqrt(6414867847*sqrt(2) + 9072749200) - 512510746420187753*x^2 - 460213731479352268*sq
rt(2)*(2*x^2 - x + 3) + 1579369851213231647*x - 2091880597633419400)/x^2) - 114133005406879362464*(431325*x^3
+ 392765*x^2 + 341572*x + 59044)*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{1}{\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(1/(5*x**2+3*x+2)**3/(2*x**2-x+3)**(1/2),x)
[Out]
Integral(1/(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(1/(5*x^2+3*x+2)^3/(2*x^2-x+3)^(1/2),x, algorithm="giac")
[Out]
Exception raised: TypeError