3.27 \(\int \frac{2+x}{(2+4 x-3 x^2) (1+3 x-2 x^2)^{5/2}} \, dx\)
Optimal. Leaf size=193 \[ -\frac{2 (14 x+15)}{51 \left (-2 x^2+3 x+1\right )^{3/2}}-\frac{2 (4814 x+291)}{867 \sqrt{-2 x^2+3 x+1}}+\frac{9}{2} \sqrt{\frac{1}{5} \left (17 \sqrt{10}-53\right )} \tan ^{-1}\left (\frac{\left (1+4 \sqrt{10}\right ) x+3 \left (4-\sqrt{10}\right )}{2 \sqrt{1+\sqrt{10}} \sqrt{-2 x^2+3 x+1}}\right )+\frac{9}{2} \sqrt{\frac{1}{5} \left (53+17 \sqrt{10}\right )} \tanh ^{-1}\left (\frac{\left (1-4 \sqrt{10}\right ) x+3 \left (4+\sqrt{10}\right )}{2 \sqrt{\sqrt{10}-1} \sqrt{-2 x^2+3 x+1}}\right ) \]
[Out]
(-2*(15 + 14*x))/(51*(1 + 3*x - 2*x^2)^(3/2)) - (2*(291 + 4814*x))/(867*Sqrt[1 + 3*x - 2*x^2]) + (9*Sqrt[(-53
+ 17*Sqrt[10])/5]*ArcTan[(3*(4 - Sqrt[10]) + (1 + 4*Sqrt[10])*x)/(2*Sqrt[1 + Sqrt[10]]*Sqrt[1 + 3*x - 2*x^2])]
)/2 + (9*Sqrt[(53 + 17*Sqrt[10])/5]*ArcTanh[(3*(4 + Sqrt[10]) + (1 - 4*Sqrt[10])*x)/(2*Sqrt[-1 + Sqrt[10]]*Sqr
t[1 + 3*x - 2*x^2])])/2
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Rubi [A] time = 0.265946, antiderivative size = 193, normalized size of antiderivative = 1.,
number of steps used = 7, number of rules used = 6, integrand size = 30, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.2, Rules used =
{1016, 1060, 1032, 724, 204, 206} \[ -\frac{2 (14 x+15)}{51 \left (-2 x^2+3 x+1\right )^{3/2}}-\frac{2 (4814 x+291)}{867 \sqrt{-2 x^2+3 x+1}}+\frac{9}{2} \sqrt{\frac{1}{5} \left (17 \sqrt{10}-53\right )} \tan ^{-1}\left (\frac{\left (1+4 \sqrt{10}\right ) x+3 \left (4-\sqrt{10}\right )}{2 \sqrt{1+\sqrt{10}} \sqrt{-2 x^2+3 x+1}}\right )+\frac{9}{2} \sqrt{\frac{1}{5} \left (53+17 \sqrt{10}\right )} \tanh ^{-1}\left (\frac{\left (1-4 \sqrt{10}\right ) x+3 \left (4+\sqrt{10}\right )}{2 \sqrt{\sqrt{10}-1} \sqrt{-2 x^2+3 x+1}}\right ) \]
Antiderivative was successfully verified.
[In]
Int[(2 + x)/((2 + 4*x - 3*x^2)*(1 + 3*x - 2*x^2)^(5/2)),x]
[Out]
(-2*(15 + 14*x))/(51*(1 + 3*x - 2*x^2)^(3/2)) - (2*(291 + 4814*x))/(867*Sqrt[1 + 3*x - 2*x^2]) + (9*Sqrt[(-53
+ 17*Sqrt[10])/5]*ArcTan[(3*(4 - Sqrt[10]) + (1 + 4*Sqrt[10])*x)/(2*Sqrt[1 + Sqrt[10]]*Sqrt[1 + 3*x - 2*x^2])]
)/2 + (9*Sqrt[(53 + 17*Sqrt[10])/5]*ArcTanh[(3*(4 + Sqrt[10]) + (1 - 4*Sqrt[10])*x)/(2*Sqrt[-1 + Sqrt[10]]*Sqr
t[1 + 3*x - 2*x^2])])/2
Rule 1016
Int[((g_.) + (h_.)*(x_))*((a_) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_)*((d_) + (e_.)*(x_) + (f_.)*(x_)^2)^(q_), x_Sy
mbol] :> Simp[((a + b*x + c*x^2)^(p + 1)*(d + e*x + f*x^2)^(q + 1)*(g*c*(2*a*c*e - b*(c*d + a*f)) + (g*b - a*h
)*(2*c^2*d + b^2*f - c*(b*e + 2*a*f)) + c*(g*(2*c^2*d + b^2*f - c*(b*e + 2*a*f)) - h*(b*c*d - 2*a*c*e + a*b*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*h - 2*g*c)
*((c*d - a*f)^2 - (b*d - a*e)*(c*e - b*f))*(p + 1) + (b^2*(g*f) - b*(h*c*d + g*c*e + a*h*f) + 2*(g*c*(c*d - a*
f) - a*(-(h*c*e))))*(a*f*(p + 1) - c*d*(p + 2)) - e*((g*c)*(2*a*c*e - b*(c*d + a*f)) + (g*b - a*h)*(2*c^2*d +
b^2*f - c*(b*e + 2*a*f)))*(p + q + 2) - (2*f*((g*c)*(2*a*c*e - b*(c*d + a*f)) + (g*b - a*h)*(2*c^2*d + b^2*f -
c*(b*e + 2*a*f)))*(p + q + 2) - (b^2*g*f - b*(h*c*d + g*c*e + a*h*f) + 2*(g*c*(c*d - a*f) - a*(-(h*c*e))))*(b
*f*(p + 1) - c*e*(2*p + q + 4)))*x - c*f*(b^2*(g*f) - b*(h*c*d + g*c*e + a*h*f) + 2*(g*c*(c*d - a*f) + a*h*c*e
))*(2*p + 2*q + 5)*x^2, x], x], x] /; FreeQ[{a, b, c, d, e, f, g, h, 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
])
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 1032
Int[((g_.) + (h_.)*(x_))/(((a_) + (b_.)*(x_) + (c_.)*(x_)^2)*Sqrt[(d_.) + (e_.)*(x_) + (f_.)*(x_)^2]), x_Symbo
l] :> With[{q = Rt[b^2 - 4*a*c, 2]}, Dist[(2*c*g - h*(b - q))/q, Int[1/((b - q + 2*c*x)*Sqrt[d + e*x + f*x^2])
, x], x] - Dist[(2*c*g - h*(b + q))/q, Int[1/((b + q + 2*c*x)*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] && PosQ[b^2 - 4*a*c]
Rule 724
Int[1/(((d_.) + (e_.)*(x_))*Sqrt[(a_.) + (b_.)*(x_) + (c_.)*(x_)^2]), x_Symbol] :> Dist[-2, Subst[Int[1/(4*c*d
^2 - 4*b*d*e + 4*a*e^2 - x^2), x], x, (2*a*e - b*d - (2*c*d - b*e)*x)/Sqrt[a + b*x + c*x^2]], x] /; FreeQ[{a,
b, c, d, e}, x] && NeQ[b^2 - 4*a*c, 0] && NeQ[2*c*d - b*e, 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])
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])
Rubi steps
\begin{align*} \int \frac{2+x}{\left (2+4 x-3 x^2\right ) \left (1+3 x-2 x^2\right )^{5/2}} \, dx &=-\frac{2 (15+14 x)}{51 \left (1+3 x-2 x^2\right )^{3/2}}+\frac{2}{51} \int \frac{-56+\frac{235 x}{2}+84 x^2}{\left (2+4 x-3 x^2\right ) \left (1+3 x-2 x^2\right )^{3/2}} \, dx\\ &=-\frac{2 (15+14 x)}{51 \left (1+3 x-2 x^2\right )^{3/2}}-\frac{2 (291+4814 x)}{867 \sqrt{1+3 x-2 x^2}}+\frac{4}{867} \int \frac{\frac{7803}{2}+\frac{23409 x}{4}}{\left (2+4 x-3 x^2\right ) \sqrt{1+3 x-2 x^2}} \, dx\\ &=-\frac{2 (15+14 x)}{51 \left (1+3 x-2 x^2\right )^{3/2}}-\frac{2 (291+4814 x)}{867 \sqrt{1+3 x-2 x^2}}+\frac{1}{5} \left (27 \left (5-2 \sqrt{10}\right )\right ) \int \frac{1}{\left (4-2 \sqrt{10}-6 x\right ) \sqrt{1+3 x-2 x^2}} \, dx+\frac{1}{5} \left (27 \left (5+2 \sqrt{10}\right )\right ) \int \frac{1}{\left (4+2 \sqrt{10}-6 x\right ) \sqrt{1+3 x-2 x^2}} \, dx\\ &=-\frac{2 (15+14 x)}{51 \left (1+3 x-2 x^2\right )^{3/2}}-\frac{2 (291+4814 x)}{867 \sqrt{1+3 x-2 x^2}}-\frac{1}{5} \left (54 \left (5-2 \sqrt{10}\right )\right ) \operatorname{Subst}\left (\int \frac{1}{144+72 \left (4-2 \sqrt{10}\right )-8 \left (4-2 \sqrt{10}\right )^2-x^2} \, dx,x,\frac{-12-3 \left (4-2 \sqrt{10}\right )-\left (18-4 \left (4-2 \sqrt{10}\right )\right ) x}{\sqrt{1+3 x-2 x^2}}\right )-\frac{1}{5} \left (54 \left (5+2 \sqrt{10}\right )\right ) \operatorname{Subst}\left (\int \frac{1}{144+72 \left (4+2 \sqrt{10}\right )-8 \left (4+2 \sqrt{10}\right )^2-x^2} \, dx,x,\frac{-12-3 \left (4+2 \sqrt{10}\right )-\left (18-4 \left (4+2 \sqrt{10}\right )\right ) x}{\sqrt{1+3 x-2 x^2}}\right )\\ &=-\frac{2 (15+14 x)}{51 \left (1+3 x-2 x^2\right )^{3/2}}-\frac{2 (291+4814 x)}{867 \sqrt{1+3 x-2 x^2}}+\frac{9}{2} \sqrt{\frac{1}{5} \left (-53+17 \sqrt{10}\right )} \tan ^{-1}\left (\frac{3 \left (4-\sqrt{10}\right )+\left (1+4 \sqrt{10}\right ) x}{2 \sqrt{1+\sqrt{10}} \sqrt{1+3 x-2 x^2}}\right )+\frac{9}{2} \sqrt{\frac{1}{5} \left (53+17 \sqrt{10}\right )} \tanh ^{-1}\left (\frac{3 \left (4+\sqrt{10}\right )+\left (1-4 \sqrt{10}\right ) x}{2 \sqrt{-1+\sqrt{10}} \sqrt{1+3 x-2 x^2}}\right )\\ \end{align*}
Mathematica [A] time = 0.608677, size = 185, normalized size = 0.96 \[ -\frac{2 \left (-9628 x^3+13860 x^2+5925 x+546\right )}{867 \left (-2 x^2+3 x+1\right )^{3/2}}+\frac{3}{10} \sqrt{1+\sqrt{10}} \left (7 \sqrt{10}-25\right ) \tan ^{-1}\left (\frac{3 \left (\sqrt{10}-4\right )-\left (1+4 \sqrt{10}\right ) x}{2 \sqrt{1+\sqrt{10}} \sqrt{-2 x^2+3 x+1}}\right )-\frac{3}{10} \sqrt{\sqrt{10}-1} \left (25+7 \sqrt{10}\right ) \tanh ^{-1}\left (\frac{\left (4 \sqrt{10}-1\right ) x-3 \left (4+\sqrt{10}\right )}{2 \sqrt{\sqrt{10}-1} \sqrt{-2 x^2+3 x+1}}\right ) \]
Antiderivative was successfully verified.
[In]
Integrate[(2 + x)/((2 + 4*x - 3*x^2)*(1 + 3*x - 2*x^2)^(5/2)),x]
[Out]
(-2*(546 + 5925*x + 13860*x^2 - 9628*x^3))/(867*(1 + 3*x - 2*x^2)^(3/2)) + (3*Sqrt[1 + Sqrt[10]]*(-25 + 7*Sqrt
[10])*ArcTan[(3*(-4 + Sqrt[10]) - (1 + 4*Sqrt[10])*x)/(2*Sqrt[1 + Sqrt[10]]*Sqrt[1 + 3*x - 2*x^2])])/10 - (3*S
qrt[-1 + Sqrt[10]]*(25 + 7*Sqrt[10])*ArcTanh[(-3*(4 + Sqrt[10]) + (-1 + 4*Sqrt[10])*x)/(2*Sqrt[-1 + Sqrt[10]]*
Sqrt[1 + 3*x - 2*x^2])])/10
________________________________________________________________________________________
Maple [B] time = 0.107, size = 1560, normalized size = 8.1 \begin{align*} \text{result too large to display} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
int((2+x)/(-3*x^2+4*x+2)/(-2*x^2+3*x+1)^(5/2),x)
[Out]
-992/7803/(-1/9-1/9*10^(1/2))/(-2*(x-2/3+1/3*10^(1/2))^2+(1/3+4/3*10^(1/2))*(x-2/3+1/3*10^(1/2))-1/9-1/9*10^(1
/2))^(1/2)*x+2/5*10^(1/2)/(-1/9-1/9*10^(1/2))^2/(1+10^(1/2))^(1/2)*arctan(9/2*(-2/9-2/9*10^(1/2)+(1/3+4/3*10^(
1/2))*(x-2/3+1/3*10^(1/2)))/(1+10^(1/2))^(1/2)/(-18*(x-2/3+1/3*10^(1/2))^2+9*(1/3+4/3*10^(1/2))*(x-2/3+1/3*10^
(1/2))-1-10^(1/2))^(1/2))-128/13005*10^(1/2)/(-1/9-1/9*10^(1/2))/(-2*(x-2/3+1/3*10^(1/2))^2+(1/3+4/3*10^(1/2))
*(x-2/3+1/3*10^(1/2))-1/9-1/9*10^(1/2))^(1/2)+2/5*10^(1/2)/(-1/9+1/9*10^(1/2))^2/(-1+10^(1/2))^(1/2)*arctanh(9
/2*(-2/9+2/9*10^(1/2)+(1/3-4/3*10^(1/2))*(x-2/3-1/3*10^(1/2)))/(-1+10^(1/2))^(1/2)/(-18*(x-2/3-1/3*10^(1/2))^2
+9*(1/3-4/3*10^(1/2))*(x-2/3-1/3*10^(1/2))-1+10^(1/2))^(1/2))+32/765/(-1/9-1/9*10^(1/2))^2/(-2*(x-2/3+1/3*10^(
1/2))^2+(1/3+4/3*10^(1/2))*(x-2/3+1/3*10^(1/2))-1/9-1/9*10^(1/2))^(1/2)*x*10^(1/2)+32/2295/(-1/9-1/9*10^(1/2))
*10^(1/2)/(-2*(x-2/3+1/3*10^(1/2))^2+(1/3+4/3*10^(1/2))*(x-2/3+1/3*10^(1/2))-1/9-1/9*10^(1/2))^(3/2)*x-32/2295
/(-1/9+1/9*10^(1/2))*10^(1/2)/(-2*(x-2/3-1/3*10^(1/2))^2+(1/3-4/3*10^(1/2))*(x-2/3-1/3*10^(1/2))-1/9+1/9*10^(1
/2))^(3/2)*x+248/2601/(-1/9-1/9*10^(1/2))/(-2*(x-2/3+1/3*10^(1/2))^2+(1/3+4/3*10^(1/2))*(x-2/3+1/3*10^(1/2))-1
/9-1/9*10^(1/2))^(1/2)+248/2601/(-1/9+1/9*10^(1/2))/(-2*(x-2/3-1/3*10^(1/2))^2+(1/3-4/3*10^(1/2))*(x-2/3-1/3*1
0^(1/2))-1/9+1/9*10^(1/2))^(1/2)-62/153/(-1/9+1/9*10^(1/2))^2/(-2*(x-2/3-1/3*10^(1/2))^2+(1/3-4/3*10^(1/2))*(x
-2/3-1/3*10^(1/2))-1/9+1/9*10^(1/2))^(1/2)*x-26/255/(-1/9+1/9*10^(1/2))^2/(-2*(x-2/3-1/3*10^(1/2))^2+(1/3-4/3*
10^(1/2))*(x-2/3-1/3*10^(1/2))-1/9+1/9*10^(1/2))^(1/2)*10^(1/2)+1/2/(-1/9+1/9*10^(1/2))^2/(-1+10^(1/2))^(1/2)*
arctanh(9/2*(-2/9+2/9*10^(1/2)+(1/3-4/3*10^(1/2))*(x-2/3-1/3*10^(1/2)))/(-1+10^(1/2))^(1/2)/(-18*(x-2/3-1/3*10
^(1/2))^2+9*(1/3-4/3*10^(1/2))*(x-2/3-1/3*10^(1/2))-1+10^(1/2))^(1/2))-62/459/(-1/9+1/9*10^(1/2))/(-2*(x-2/3-1
/3*10^(1/2))^2+(1/3-4/3*10^(1/2))*(x-2/3-1/3*10^(1/2))-1/9+1/9*10^(1/2))^(3/2)*x-26/765/(-1/9+1/9*10^(1/2))*10
^(1/2)/(-2*(x-2/3-1/3*10^(1/2))^2+(1/3-4/3*10^(1/2))*(x-2/3-1/3*10^(1/2))-1/9+1/9*10^(1/2))^(3/2)+26/255/(-1/9
-1/9*10^(1/2))^2/(-2*(x-2/3+1/3*10^(1/2))^2+(1/3+4/3*10^(1/2))*(x-2/3+1/3*10^(1/2))-1/9-1/9*10^(1/2))^(1/2)*10
^(1/2)-1/2/(-1/9-1/9*10^(1/2))^2/(1+10^(1/2))^(1/2)*arctan(9/2*(-2/9-2/9*10^(1/2)+(1/3+4/3*10^(1/2))*(x-2/3+1/
3*10^(1/2)))/(1+10^(1/2))^(1/2)/(-18*(x-2/3+1/3*10^(1/2))^2+9*(1/3+4/3*10^(1/2))*(x-2/3+1/3*10^(1/2))-1-10^(1/
2))^(1/2))+128/13005*10^(1/2)/(-1/9+1/9*10^(1/2))/(-2*(x-2/3-1/3*10^(1/2))^2+(1/3-4/3*10^(1/2))*(x-2/3-1/3*10^
(1/2))-1/9+1/9*10^(1/2))^(1/2)-992/7803/(-1/9+1/9*10^(1/2))/(-2*(x-2/3-1/3*10^(1/2))^2+(1/3-4/3*10^(1/2))*(x-2
/3-1/3*10^(1/2))-1/9+1/9*10^(1/2))^(1/2)*x-62/153/(-1/9-1/9*10^(1/2))^2/(-2*(x-2/3+1/3*10^(1/2))^2+(1/3+4/3*10
^(1/2))*(x-2/3+1/3*10^(1/2))-1/9-1/9*10^(1/2))^(1/2)*x+26/765/(-1/9-1/9*10^(1/2))*10^(1/2)/(-2*(x-2/3+1/3*10^(
1/2))^2+(1/3+4/3*10^(1/2))*(x-2/3+1/3*10^(1/2))-1/9-1/9*10^(1/2))^(3/2)-62/459/(-1/9-1/9*10^(1/2))/(-2*(x-2/3+
1/3*10^(1/2))^2+(1/3+4/3*10^(1/2))*(x-2/3+1/3*10^(1/2))-1/9-1/9*10^(1/2))^(3/2)*x-32/765/(-1/9+1/9*10^(1/2))^2
/(-2*(x-2/3-1/3*10^(1/2))^2+(1/3-4/3*10^(1/2))*(x-2/3-1/3*10^(1/2))-1/9+1/9*10^(1/2))^(1/2)*x*10^(1/2)+7/51/(-
1/9+1/9*10^(1/2))^2/(-2*(x-2/3-1/3*10^(1/2))^2+(1/3-4/3*10^(1/2))*(x-2/3-1/3*10^(1/2))-1/9+1/9*10^(1/2))^(1/2)
+7/153/(-1/9-1/9*10^(1/2))/(-2*(x-2/3+1/3*10^(1/2))^2+(1/3+4/3*10^(1/2))*(x-2/3+1/3*10^(1/2))-1/9-1/9*10^(1/2)
)^(3/2)+7/153/(-1/9+1/9*10^(1/2))/(-2*(x-2/3-1/3*10^(1/2))^2+(1/3-4/3*10^(1/2))*(x-2/3-1/3*10^(1/2))-1/9+1/9*1
0^(1/2))^(3/2)+7/51/(-1/9-1/9*10^(1/2))^2/(-2*(x-2/3+1/3*10^(1/2))^2+(1/3+4/3*10^(1/2))*(x-2/3+1/3*10^(1/2))-1
/9-1/9*10^(1/2))^(1/2)-512/39015/(-1/9+1/9*10^(1/2))/(-2*(x-2/3-1/3*10^(1/2))^2+(1/3-4/3*10^(1/2))*(x-2/3-1/3*
10^(1/2))-1/9+1/9*10^(1/2))^(1/2)*x*10^(1/2)+512/39015/(-1/9-1/9*10^(1/2))/(-2*(x-2/3+1/3*10^(1/2))^2+(1/3+4/3
*10^(1/2))*(x-2/3+1/3*10^(1/2))-1/9-1/9*10^(1/2))^(1/2)*x*10^(1/2)
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Maxima [B] time = 1.73312, size = 1723, normalized size = 8.93 \begin{align*} \text{result too large to display} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate((2+x)/(-3*x^2+4*x+2)/(-2*x^2+3*x+1)^(5/2),x, algorithm="maxima")
[Out]
1/17340*sqrt(10)*(2108*sqrt(10)*x/(sqrt(10)*(-2*x^2 + 3*x + 1)^(3/2) + (-2*x^2 + 3*x + 1)^(3/2)) - 2108*sqrt(1
0)*x/(sqrt(10)*(-2*x^2 + 3*x + 1)^(3/2) - (-2*x^2 + 3*x + 1)^(3/2)) - 56916*sqrt(10)*x/(2*sqrt(10)*sqrt(-2*x^2
+ 3*x + 1) + 11*sqrt(-2*x^2 + 3*x + 1)) + 56916*sqrt(10)*x/(2*sqrt(10)*sqrt(-2*x^2 + 3*x + 1) - 11*sqrt(-2*x^
2 + 3*x + 1)) + 1984*sqrt(10)*x/(sqrt(10)*sqrt(-2*x^2 + 3*x + 1) + sqrt(-2*x^2 + 3*x + 1)) - 1984*sqrt(10)*x/(
sqrt(10)*sqrt(-2*x^2 + 3*x + 1) - sqrt(-2*x^2 + 3*x + 1)) - 70227*sqrt(10)*arcsin(8/17*sqrt(17)*sqrt(10)*x/abs
(6*x + 2*sqrt(10) - 4) + 2/17*sqrt(17)*x/abs(6*x + 2*sqrt(10) - 4) - 6/17*sqrt(17)*sqrt(10)/abs(6*x + 2*sqrt(1
0) - 4) + 24/17*sqrt(17)/abs(6*x + 2*sqrt(10) - 4))/(2*sqrt(10)*sqrt(sqrt(10) + 1) + 11*sqrt(sqrt(10) + 1)) -
2176*x/(sqrt(10)*(-2*x^2 + 3*x + 1)^(3/2) + (-2*x^2 + 3*x + 1)^(3/2)) - 2176*x/(sqrt(10)*(-2*x^2 + 3*x + 1)^(3
/2) - (-2*x^2 + 3*x + 1)^(3/2)) + 58752*x/(2*sqrt(10)*sqrt(-2*x^2 + 3*x + 1) + 11*sqrt(-2*x^2 + 3*x + 1)) + 58
752*x/(2*sqrt(10)*sqrt(-2*x^2 + 3*x + 1) - 11*sqrt(-2*x^2 + 3*x + 1)) - 2048*x/(sqrt(10)*sqrt(-2*x^2 + 3*x + 1
) + sqrt(-2*x^2 + 3*x + 1)) - 2048*x/(sqrt(10)*sqrt(-2*x^2 + 3*x + 1) - sqrt(-2*x^2 + 3*x + 1)) + 561816*arcsi
n(8/17*sqrt(17)*sqrt(10)*x/abs(6*x + 2*sqrt(10) - 4) + 2/17*sqrt(17)*x/abs(6*x + 2*sqrt(10) - 4) - 6/17*sqrt(1
7)*sqrt(10)/abs(6*x + 2*sqrt(10) - 4) + 24/17*sqrt(17)/abs(6*x + 2*sqrt(10) - 4))/(2*sqrt(10)*sqrt(sqrt(10) +
1) + 11*sqrt(sqrt(10) + 1)) - 714*sqrt(10)/(sqrt(10)*(-2*x^2 + 3*x + 1)^(3/2) + (-2*x^2 + 3*x + 1)^(3/2)) + 71
4*sqrt(10)/(sqrt(10)*(-2*x^2 + 3*x + 1)^(3/2) - (-2*x^2 + 3*x + 1)^(3/2)) + 19278*sqrt(10)/(2*sqrt(10)*sqrt(-2
*x^2 + 3*x + 1) + 11*sqrt(-2*x^2 + 3*x + 1)) - 19278*sqrt(10)/(2*sqrt(10)*sqrt(-2*x^2 + 3*x + 1) - 11*sqrt(-2*
x^2 + 3*x + 1)) - 1488*sqrt(10)/(sqrt(10)*sqrt(-2*x^2 + 3*x + 1) + sqrt(-2*x^2 + 3*x + 1)) + 1488*sqrt(10)/(sq
rt(10)*sqrt(-2*x^2 + 3*x + 1) - sqrt(-2*x^2 + 3*x + 1)) - 5304/(sqrt(10)*(-2*x^2 + 3*x + 1)^(3/2) + (-2*x^2 +
3*x + 1)^(3/2)) - 5304/(sqrt(10)*(-2*x^2 + 3*x + 1)^(3/2) - (-2*x^2 + 3*x + 1)^(3/2)) + 143208/(2*sqrt(10)*sqr
t(-2*x^2 + 3*x + 1) + 11*sqrt(-2*x^2 + 3*x + 1)) + 143208/(2*sqrt(10)*sqrt(-2*x^2 + 3*x + 1) - 11*sqrt(-2*x^2
+ 3*x + 1)) + 1536/(sqrt(10)*sqrt(-2*x^2 + 3*x + 1) + sqrt(-2*x^2 + 3*x + 1)) + 1536/(sqrt(10)*sqrt(-2*x^2 + 3
*x + 1) - sqrt(-2*x^2 + 3*x + 1)) + 70227*sqrt(10)*log(-2/9*sqrt(10) + 2/3*sqrt(-2*x^2 + 3*x + 1)*sqrt(sqrt(10
) - 1)/abs(6*x - 2*sqrt(10) - 4) + 2/9*sqrt(10)/abs(6*x - 2*sqrt(10) - 4) - 2/9/abs(6*x - 2*sqrt(10) - 4) + 1/
18)/(sqrt(10) - 1)^(5/2) + 561816*log(-2/9*sqrt(10) + 2/3*sqrt(-2*x^2 + 3*x + 1)*sqrt(sqrt(10) - 1)/abs(6*x -
2*sqrt(10) - 4) + 2/9*sqrt(10)/abs(6*x - 2*sqrt(10) - 4) - 2/9/abs(6*x - 2*sqrt(10) - 4) + 1/18)/(sqrt(10) - 1
)^(5/2))
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Fricas [B] time = 1.49064, size = 1304, normalized size = 6.76 \begin{align*} -\frac{43680 \, x^{4} - 131040 \, x^{3} - 31212 \, \sqrt{5}{\left (4 \, x^{4} - 12 \, x^{3} + 5 \, x^{2} + 6 \, x + 1\right )} \sqrt{17 \, \sqrt{10} - 53} \arctan \left (\frac{\sqrt{2}{\left (\sqrt{10} \sqrt{5} x + 10 \, \sqrt{5} x\right )} \sqrt{17 \, \sqrt{10} - 53} \sqrt{\frac{6 \, x^{2} + \sqrt{10}{\left (3 \, x^{2} + 2 \, x\right )} - 2 \, \sqrt{-2 \, x^{2} + 3 \, x + 1}{\left (\sqrt{10} x + 2 \, x + 2\right )} + 10 \, x + 4}{x^{2}}} + 2 \,{\left (\sqrt{10} \sqrt{5}{\left (6 \, x + 1\right )} - \sqrt{-2 \, x^{2} + 3 \, x + 1}{\left (\sqrt{10} \sqrt{5} + 10 \, \sqrt{5}\right )} + 5 \, \sqrt{5}{\left (3 \, x + 2\right )}\right )} \sqrt{17 \, \sqrt{10} - 53}}{90 \, x}\right ) - 7803 \, \sqrt{5}{\left (4 \, x^{4} - 12 \, x^{3} + 5 \, x^{2} + 6 \, x + 1\right )} \sqrt{17 \, \sqrt{10} + 53} \log \left (\frac{9 \,{\left (45 \, \sqrt{10} x +{\left (13 \, \sqrt{10} \sqrt{5} x - 40 \, \sqrt{5} x\right )} \sqrt{17 \, \sqrt{10} + 53} - 90 \, x + 90 \, \sqrt{-2 \, x^{2} + 3 \, x + 1} - 90\right )}}{x}\right ) + 7803 \, \sqrt{5}{\left (4 \, x^{4} - 12 \, x^{3} + 5 \, x^{2} + 6 \, x + 1\right )} \sqrt{17 \, \sqrt{10} + 53} \log \left (\frac{9 \,{\left (45 \, \sqrt{10} x -{\left (13 \, \sqrt{10} \sqrt{5} x - 40 \, \sqrt{5} x\right )} \sqrt{17 \, \sqrt{10} + 53} - 90 \, x + 90 \, \sqrt{-2 \, x^{2} + 3 \, x + 1} - 90\right )}}{x}\right ) + 54600 \, x^{2} - 20 \,{\left (9628 \, x^{3} - 13860 \, x^{2} - 5925 \, x - 546\right )} \sqrt{-2 \, x^{2} + 3 \, x + 1} + 65520 \, x + 10920}{8670 \,{\left (4 \, x^{4} - 12 \, x^{3} + 5 \, x^{2} + 6 \, x + 1\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate((2+x)/(-3*x^2+4*x+2)/(-2*x^2+3*x+1)^(5/2),x, algorithm="fricas")
[Out]
-1/8670*(43680*x^4 - 131040*x^3 - 31212*sqrt(5)*(4*x^4 - 12*x^3 + 5*x^2 + 6*x + 1)*sqrt(17*sqrt(10) - 53)*arct
an(1/90*(sqrt(2)*(sqrt(10)*sqrt(5)*x + 10*sqrt(5)*x)*sqrt(17*sqrt(10) - 53)*sqrt((6*x^2 + sqrt(10)*(3*x^2 + 2*
x) - 2*sqrt(-2*x^2 + 3*x + 1)*(sqrt(10)*x + 2*x + 2) + 10*x + 4)/x^2) + 2*(sqrt(10)*sqrt(5)*(6*x + 1) - sqrt(-
2*x^2 + 3*x + 1)*(sqrt(10)*sqrt(5) + 10*sqrt(5)) + 5*sqrt(5)*(3*x + 2))*sqrt(17*sqrt(10) - 53))/x) - 7803*sqrt
(5)*(4*x^4 - 12*x^3 + 5*x^2 + 6*x + 1)*sqrt(17*sqrt(10) + 53)*log(9*(45*sqrt(10)*x + (13*sqrt(10)*sqrt(5)*x -
40*sqrt(5)*x)*sqrt(17*sqrt(10) + 53) - 90*x + 90*sqrt(-2*x^2 + 3*x + 1) - 90)/x) + 7803*sqrt(5)*(4*x^4 - 12*x^
3 + 5*x^2 + 6*x + 1)*sqrt(17*sqrt(10) + 53)*log(9*(45*sqrt(10)*x - (13*sqrt(10)*sqrt(5)*x - 40*sqrt(5)*x)*sqrt
(17*sqrt(10) + 53) - 90*x + 90*sqrt(-2*x^2 + 3*x + 1) - 90)/x) + 54600*x^2 - 20*(9628*x^3 - 13860*x^2 - 5925*x
- 546)*sqrt(-2*x^2 + 3*x + 1) + 65520*x + 10920)/(4*x^4 - 12*x^3 + 5*x^2 + 6*x + 1)
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Sympy [F(-1)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate((2+x)/(-3*x**2+4*x+2)/(-2*x**2+3*x+1)**(5/2),x)
[Out]
Timed out
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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)/(-3*x^2+4*x+2)/(-2*x^2+3*x+1)^(5/2),x, algorithm="giac")
[Out]
Exception raised: TypeError