3.30 \(\int \frac{2+x}{(2+4 x-3 x^2) (1+3 x+2 x^2)^{5/2}} \, dx\)

Optimal. Leaf size=197 \[ \frac{2 (22 x+21)}{15 \left (2 x^2+3 x+1\right )^{3/2}}+\frac{2 (230 x+273)}{15 \sqrt{2 x^2+3 x+1}}-\frac{1}{50} \sqrt{\frac{1}{3} \left (4885115+1544809 \sqrt{10}\right )} \tanh ^{-1}\left (\frac{\left (17-4 \sqrt{10}\right ) x+3 \left (4-\sqrt{10}\right )}{2 \sqrt{55-17 \sqrt{10}} \sqrt{2 x^2+3 x+1}}\right )+\frac{1}{50} \sqrt{\frac{1}{3} \left (4885115-1544809 \sqrt{10}\right )} \tanh ^{-1}\left (\frac{\left (17+4 \sqrt{10}\right ) x+3 \left (4+\sqrt{10}\right )}{2 \sqrt{55+17 \sqrt{10}} \sqrt{2 x^2+3 x+1}}\right ) \]

[Out]

(2*(21 + 22*x))/(15*(1 + 3*x + 2*x^2)^(3/2)) + (2*(273 + 230*x))/(15*Sqrt[1 + 3*x + 2*x^2]) - (Sqrt[(4885115 +
1544809*Sqrt[10])/3]*ArcTanh[(3*(4 - Sqrt[10]) + (17 - 4*Sqrt[10])*x)/(2*Sqrt[55 - 17*Sqrt[10]]*Sqrt[1 + 3*x
+ 2*x^2])])/50 + (Sqrt[(4885115 - 1544809*Sqrt[10])/3]*ArcTanh[(3*(4 + Sqrt[10]) + (17 + 4*Sqrt[10])*x)/(2*Sqr
t[55 + 17*Sqrt[10]]*Sqrt[1 + 3*x + 2*x^2])])/50

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Rubi [A]  time = 0.303302, antiderivative size = 197, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 5, integrand size = 30, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.167, Rules used = {1016, 1060, 1032, 724, 206} \[ \frac{2 (22 x+21)}{15 \left (2 x^2+3 x+1\right )^{3/2}}+\frac{2 (230 x+273)}{15 \sqrt{2 x^2+3 x+1}}-\frac{1}{50} \sqrt{\frac{1}{3} \left (4885115+1544809 \sqrt{10}\right )} \tanh ^{-1}\left (\frac{\left (17-4 \sqrt{10}\right ) x+3 \left (4-\sqrt{10}\right )}{2 \sqrt{55-17 \sqrt{10}} \sqrt{2 x^2+3 x+1}}\right )+\frac{1}{50} \sqrt{\frac{1}{3} \left (4885115-1544809 \sqrt{10}\right )} \tanh ^{-1}\left (\frac{\left (17+4 \sqrt{10}\right ) x+3 \left (4+\sqrt{10}\right )}{2 \sqrt{55+17 \sqrt{10}} \sqrt{2 x^2+3 x+1}}\right ) \]

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*(21 + 22*x))/(15*(1 + 3*x + 2*x^2)^(3/2)) + (2*(273 + 230*x))/(15*Sqrt[1 + 3*x + 2*x^2]) - (Sqrt[(4885115 +
1544809*Sqrt[10])/3]*ArcTanh[(3*(4 - Sqrt[10]) + (17 - 4*Sqrt[10])*x)/(2*Sqrt[55 - 17*Sqrt[10]]*Sqrt[1 + 3*x
+ 2*x^2])])/50 + (Sqrt[(4885115 - 1544809*Sqrt[10])/3]*ArcTanh[(3*(4 + Sqrt[10]) + (17 + 4*Sqrt[10])*x)/(2*Sqr
t[55 + 17*Sqrt[10]]*Sqrt[1 + 3*x + 2*x^2])])/50

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 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 (21+22 x)}{15 \left (1+3 x+2 x^2\right )^{3/2}}-\frac{2}{45} \int \frac{-480-\frac{813 x}{2}+396 x^2}{\left (2+4 x-3 x^2\right ) \left (1+3 x+2 x^2\right )^{3/2}} \, dx\\ &=\frac{2 (21+22 x)}{15 \left (1+3 x+2 x^2\right )^{3/2}}+\frac{2 (273+230 x)}{15 \sqrt{1+3 x+2 x^2}}+\frac{4}{675} \int \frac{\frac{23355}{2}-\frac{27135 x}{4}}{\left (2+4 x-3 x^2\right ) \sqrt{1+3 x+2 x^2}} \, dx\\ &=\frac{2 (21+22 x)}{15 \left (1+3 x+2 x^2\right )^{3/2}}+\frac{2 (273+230 x)}{15 \sqrt{1+3 x+2 x^2}}-\frac{1}{25} \left (3 \left (335-106 \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}{25} \left (3 \left (335+106 \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 (21+22 x)}{15 \left (1+3 x+2 x^2\right )^{3/2}}+\frac{2 (273+230 x)}{15 \sqrt{1+3 x+2 x^2}}+\frac{1}{25} \left (6 \left (335-106 \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}{25} \left (6 \left (335+106 \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 (21+22 x)}{15 \left (1+3 x+2 x^2\right )^{3/2}}+\frac{2 (273+230 x)}{15 \sqrt{1+3 x+2 x^2}}-\frac{1}{50} \sqrt{\frac{1}{3} \left (4885115+1544809 \sqrt{10}\right )} \tanh ^{-1}\left (\frac{3 \left (4-\sqrt{10}\right )+\left (17-4 \sqrt{10}\right ) x}{2 \sqrt{55-17 \sqrt{10}} \sqrt{1+3 x+2 x^2}}\right )+\frac{1}{50} \sqrt{\frac{1}{3} \left (4885115-1544809 \sqrt{10}\right )} \tanh ^{-1}\left (\frac{3 \left (4+\sqrt{10}\right )+\left (17+4 \sqrt{10}\right ) x}{2 \sqrt{55+17 \sqrt{10}} \sqrt{1+3 x+2 x^2}}\right )\\ \end{align*}

Mathematica [A]  time = 0.72194, size = 190, normalized size = 0.96 \[ \frac{1}{450} \left (\frac{60 \left (460 x^3+1236 x^2+1071 x+294\right )}{\left (2 x^2+3 x+1\right )^{3/2}}+\sqrt{55-17 \sqrt{10}} \left (7289+2305 \sqrt{10}\right ) \tanh ^{-1}\left (\frac{\left (4 \sqrt{10}-17\right ) x+3 \left (\sqrt{10}-4\right )}{2 \sqrt{55-17 \sqrt{10}} \sqrt{2 x^2+3 x+1}}\right )-\sqrt{55+17 \sqrt{10}} \left (2305 \sqrt{10}-7289\right ) \tanh ^{-1}\left (\frac{-\left (17+4 \sqrt{10}\right ) x-3 \left (4+\sqrt{10}\right )}{2 \sqrt{55+17 \sqrt{10}} \sqrt{2 x^2+3 x+1}}\right )\right ) \]

Antiderivative was successfully verified.

[In]

Integrate[(2 + x)/((2 + 4*x - 3*x^2)*(1 + 3*x + 2*x^2)^(5/2)),x]

[Out]

((60*(294 + 1071*x + 1236*x^2 + 460*x^3))/(1 + 3*x + 2*x^2)^(3/2) + Sqrt[55 - 17*Sqrt[10]]*(7289 + 2305*Sqrt[1
0])*ArcTanh[(3*(-4 + Sqrt[10]) + (-17 + 4*Sqrt[10])*x)/(2*Sqrt[55 - 17*Sqrt[10]]*Sqrt[1 + 3*x + 2*x^2])] - Sqr
t[55 + 17*Sqrt[10]]*(-7289 + 2305*Sqrt[10])*ArcTanh[(-3*(4 + Sqrt[10]) - (17 + 4*Sqrt[10])*x)/(2*Sqrt[55 + 17*
Sqrt[10]]*Sqrt[1 + 3*x + 2*x^2])])/450

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Maple [B]  time = 0.101, size = 878, normalized size = 4.5 \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]

-1/20*(8+10^(1/2))*10^(1/2)*(1/9/(55/9+17/9*10^(1/2))/(2*(x-2/3-1/3*10^(1/2))^2+(17/3+4/3*10^(1/2))*(x-2/3-1/3
*10^(1/2))+55/9+17/9*10^(1/2))^(3/2)-1/6*(17/3+4/3*10^(1/2))/(55/9+17/9*10^(1/2))*(2/3*(4*x+3)/(440/9+136/9*10
^(1/2)-(17/3+4/3*10^(1/2))^2)/(2*(x-2/3-1/3*10^(1/2))^2+(17/3+4/3*10^(1/2))*(x-2/3-1/3*10^(1/2))+55/9+17/9*10^
(1/2))^(3/2)+32/3/(440/9+136/9*10^(1/2)-(17/3+4/3*10^(1/2))^2)^2*(4*x+3)/(2*(x-2/3-1/3*10^(1/2))^2+(17/3+4/3*1
0^(1/2))*(x-2/3-1/3*10^(1/2))+55/9+17/9*10^(1/2))^(1/2))+1/3/(55/9+17/9*10^(1/2))*(1/(55/9+17/9*10^(1/2))/(2*(
x-2/3-1/3*10^(1/2))^2+(17/3+4/3*10^(1/2))*(x-2/3-1/3*10^(1/2))+55/9+17/9*10^(1/2))^(1/2)-(17/3+4/3*10^(1/2))/(
55/9+17/9*10^(1/2))*(4*x+3)/(440/9+136/9*10^(1/2)-(17/3+4/3*10^(1/2))^2)/(2*(x-2/3-1/3*10^(1/2))^2+(17/3+4/3*1
0^(1/2))*(x-2/3-1/3*10^(1/2))+55/9+17/9*10^(1/2))^(1/2)-3/(55/9+17/9*10^(1/2))/(55+17*10^(1/2))^(1/2)*arctanh(
9/2*(110/9+34/9*10^(1/2)+(17/3+4/3*10^(1/2))*(x-2/3-1/3*10^(1/2)))/(55+17*10^(1/2))^(1/2)/(18*(x-2/3-1/3*10^(1
/2))^2+9*(17/3+4/3*10^(1/2))*(x-2/3-1/3*10^(1/2))+55+17*10^(1/2))^(1/2))))-1/20*(-8+10^(1/2))*10^(1/2)*(1/9/(5
5/9-17/9*10^(1/2))/(2*(x-2/3+1/3*10^(1/2))^2+(17/3-4/3*10^(1/2))*(x-2/3+1/3*10^(1/2))+55/9-17/9*10^(1/2))^(3/2
)-1/6*(17/3-4/3*10^(1/2))/(55/9-17/9*10^(1/2))*(2/3*(4*x+3)/(440/9-136/9*10^(1/2)-(17/3-4/3*10^(1/2))^2)/(2*(x
-2/3+1/3*10^(1/2))^2+(17/3-4/3*10^(1/2))*(x-2/3+1/3*10^(1/2))+55/9-17/9*10^(1/2))^(3/2)+32/3/(440/9-136/9*10^(
1/2)-(17/3-4/3*10^(1/2))^2)^2*(4*x+3)/(2*(x-2/3+1/3*10^(1/2))^2+(17/3-4/3*10^(1/2))*(x-2/3+1/3*10^(1/2))+55/9-
17/9*10^(1/2))^(1/2))+1/3/(55/9-17/9*10^(1/2))*(1/(55/9-17/9*10^(1/2))/(2*(x-2/3+1/3*10^(1/2))^2+(17/3-4/3*10^
(1/2))*(x-2/3+1/3*10^(1/2))+55/9-17/9*10^(1/2))^(1/2)-(17/3-4/3*10^(1/2))/(55/9-17/9*10^(1/2))*(4*x+3)/(440/9-
136/9*10^(1/2)-(17/3-4/3*10^(1/2))^2)/(2*(x-2/3+1/3*10^(1/2))^2+(17/3-4/3*10^(1/2))*(x-2/3+1/3*10^(1/2))+55/9-
17/9*10^(1/2))^(1/2)-3/(55/9-17/9*10^(1/2))/(55-17*10^(1/2))^(1/2)*arctanh(9/2*(110/9-34/9*10^(1/2)+(17/3-4/3*
10^(1/2))*(x-2/3+1/3*10^(1/2)))/(55-17*10^(1/2))^(1/2)/(18*(x-2/3+1/3*10^(1/2))^2+9*(17/3-4/3*10^(1/2))*(x-2/3
+1/3*10^(1/2))+55-17*10^(1/2))^(1/2))))

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Maxima [B]  time = 1.76204, size = 1723, normalized size = 8.75 \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]

-1/300*sqrt(10)*(980*sqrt(10)*x/(17*sqrt(10)*(2*x^2 + 3*x + 1)^(3/2) + 55*(2*x^2 + 3*x + 1)^(3/2)) - 980*sqrt(
10)*x/(17*sqrt(10)*(2*x^2 + 3*x + 1)^(3/2) - 55*(2*x^2 + 3*x + 1)^(3/2)) + 5292*sqrt(10)*x/(374*sqrt(10)*sqrt(
2*x^2 + 3*x + 1) + 1183*sqrt(2*x^2 + 3*x + 1)) - 5292*sqrt(10)*x/(374*sqrt(10)*sqrt(2*x^2 + 3*x + 1) - 1183*sq
rt(2*x^2 + 3*x + 1)) - 15680*sqrt(10)*x/(17*sqrt(10)*sqrt(2*x^2 + 3*x + 1) + 55*sqrt(2*x^2 + 3*x + 1)) + 15680
*sqrt(10)*x/(17*sqrt(10)*sqrt(2*x^2 + 3*x + 1) - 55*sqrt(2*x^2 + 3*x + 1)) + 3520*x/(17*sqrt(10)*(2*x^2 + 3*x
+ 1)^(3/2) + 55*(2*x^2 + 3*x + 1)^(3/2)) + 3520*x/(17*sqrt(10)*(2*x^2 + 3*x + 1)^(3/2) - 55*(2*x^2 + 3*x + 1)^
(3/2)) + 19008*x/(374*sqrt(10)*sqrt(2*x^2 + 3*x + 1) + 1183*sqrt(2*x^2 + 3*x + 1)) + 19008*x/(374*sqrt(10)*sqr
t(2*x^2 + 3*x + 1) - 1183*sqrt(2*x^2 + 3*x + 1)) - 56320*x/(17*sqrt(10)*sqrt(2*x^2 + 3*x + 1) + 55*sqrt(2*x^2
+ 3*x + 1)) - 56320*x/(17*sqrt(10)*sqrt(2*x^2 + 3*x + 1) - 55*sqrt(2*x^2 + 3*x + 1)) + 750*sqrt(10)/(17*sqrt(1
0)*(2*x^2 + 3*x + 1)^(3/2) + 55*(2*x^2 + 3*x + 1)^(3/2)) - 750*sqrt(10)/(17*sqrt(10)*(2*x^2 + 3*x + 1)^(3/2) -
55*(2*x^2 + 3*x + 1)^(3/2)) + 4050*sqrt(10)/(374*sqrt(10)*sqrt(2*x^2 + 3*x + 1) + 1183*sqrt(2*x^2 + 3*x + 1))
- 4050*sqrt(10)/(374*sqrt(10)*sqrt(2*x^2 + 3*x + 1) - 1183*sqrt(2*x^2 + 3*x + 1)) - 11760*sqrt(10)/(17*sqrt(1
0)*sqrt(2*x^2 + 3*x + 1) + 55*sqrt(2*x^2 + 3*x + 1)) + 11760*sqrt(10)/(17*sqrt(10)*sqrt(2*x^2 + 3*x + 1) - 55*
sqrt(2*x^2 + 3*x + 1)) + 2760/(17*sqrt(10)*(2*x^2 + 3*x + 1)^(3/2) + 55*(2*x^2 + 3*x + 1)^(3/2)) + 2760/(17*sq
rt(10)*(2*x^2 + 3*x + 1)^(3/2) - 55*(2*x^2 + 3*x + 1)^(3/2)) + 14904/(374*sqrt(10)*sqrt(2*x^2 + 3*x + 1) + 118
3*sqrt(2*x^2 + 3*x + 1)) + 14904/(374*sqrt(10)*sqrt(2*x^2 + 3*x + 1) - 1183*sqrt(2*x^2 + 3*x + 1)) - 42240/(17
*sqrt(10)*sqrt(2*x^2 + 3*x + 1) + 55*sqrt(2*x^2 + 3*x + 1)) - 42240/(17*sqrt(10)*sqrt(2*x^2 + 3*x + 1) - 55*sq
rt(2*x^2 + 3*x + 1)) - 1215*sqrt(10)*log(2/9*sqrt(10) + 2/3*sqrt(2*x^2 + 3*x + 1)*sqrt(17*sqrt(10) + 55)/abs(6
*x - 2*sqrt(10) - 4) + 34/9*sqrt(10)/abs(6*x - 2*sqrt(10) - 4) + 110/9/abs(6*x - 2*sqrt(10) - 4) + 17/18)/(17*
sqrt(10) + 55)^(5/2) - 5*sqrt(10)*log(-2/9*sqrt(10) + 2*sqrt(2*x^2 + 3*x + 1)*sqrt(-17/9*sqrt(10) + 55/9)/abs(
6*x + 2*sqrt(10) - 4) - 34/9*sqrt(10)/abs(6*x + 2*sqrt(10) - 4) + 110/9/abs(6*x + 2*sqrt(10) - 4) + 17/18)/(-1
7/9*sqrt(10) + 55/9)^(5/2) - 9720*log(2/9*sqrt(10) + 2/3*sqrt(2*x^2 + 3*x + 1)*sqrt(17*sqrt(10) + 55)/abs(6*x
- 2*sqrt(10) - 4) + 34/9*sqrt(10)/abs(6*x - 2*sqrt(10) - 4) + 110/9/abs(6*x - 2*sqrt(10) - 4) + 17/18)/(17*sqr
t(10) + 55)^(5/2) + 40*log(-2/9*sqrt(10) + 2*sqrt(2*x^2 + 3*x + 1)*sqrt(-17/9*sqrt(10) + 55/9)/abs(6*x + 2*sqr
t(10) - 4) - 34/9*sqrt(10)/abs(6*x + 2*sqrt(10) - 4) + 110/9/abs(6*x + 2*sqrt(10) - 4) + 17/18)/(-17/9*sqrt(10
) + 55/9)^(5/2))

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Fricas [B]  time = 1.45697, size = 1422, normalized size = 7.22 \begin{align*} \frac{23520 \, x^{4} + 70560 \, x^{3} + \sqrt{3}{\left (4 \, x^{4} + 12 \, x^{3} + 13 \, x^{2} + 6 \, x + 1\right )} \sqrt{1544809 \, \sqrt{10} + 4885115} \log \left (-\frac{243 \, \sqrt{10} x +{\left (893 \, \sqrt{10} \sqrt{3} x - 2824 \, \sqrt{3} x\right )} \sqrt{1544809 \, \sqrt{10} + 4885115} + 486 \, x - 486 \, \sqrt{2 \, x^{2} + 3 \, x + 1} + 486}{x}\right ) - \sqrt{3}{\left (4 \, x^{4} + 12 \, x^{3} + 13 \, x^{2} + 6 \, x + 1\right )} \sqrt{1544809 \, \sqrt{10} + 4885115} \log \left (-\frac{243 \, \sqrt{10} x -{\left (893 \, \sqrt{10} \sqrt{3} x - 2824 \, \sqrt{3} x\right )} \sqrt{1544809 \, \sqrt{10} + 4885115} + 486 \, x - 486 \, \sqrt{2 \, x^{2} + 3 \, x + 1} + 486}{x}\right ) + \sqrt{3}{\left (4 \, x^{4} + 12 \, x^{3} + 13 \, x^{2} + 6 \, x + 1\right )} \sqrt{-1544809 \, \sqrt{10} + 4885115} \log \left (\frac{243 \, \sqrt{10} x +{\left (893 \, \sqrt{10} \sqrt{3} x + 2824 \, \sqrt{3} x\right )} \sqrt{-1544809 \, \sqrt{10} + 4885115} - 486 \, x + 486 \, \sqrt{2 \, x^{2} + 3 \, x + 1} - 486}{x}\right ) - \sqrt{3}{\left (4 \, x^{4} + 12 \, x^{3} + 13 \, x^{2} + 6 \, x + 1\right )} \sqrt{-1544809 \, \sqrt{10} + 4885115} \log \left (\frac{243 \, \sqrt{10} x -{\left (893 \, \sqrt{10} \sqrt{3} x + 2824 \, \sqrt{3} x\right )} \sqrt{-1544809 \, \sqrt{10} + 4885115} - 486 \, x + 486 \, \sqrt{2 \, x^{2} + 3 \, x + 1} - 486}{x}\right ) + 76440 \, x^{2} + 20 \,{\left (460 \, x^{3} + 1236 \, x^{2} + 1071 \, x + 294\right )} \sqrt{2 \, x^{2} + 3 \, x + 1} + 35280 \, x + 5880}{150 \,{\left (4 \, x^{4} + 12 \, x^{3} + 13 \, 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]

1/150*(23520*x^4 + 70560*x^3 + sqrt(3)*(4*x^4 + 12*x^3 + 13*x^2 + 6*x + 1)*sqrt(1544809*sqrt(10) + 4885115)*lo
g(-(243*sqrt(10)*x + (893*sqrt(10)*sqrt(3)*x - 2824*sqrt(3)*x)*sqrt(1544809*sqrt(10) + 4885115) + 486*x - 486*
sqrt(2*x^2 + 3*x + 1) + 486)/x) - sqrt(3)*(4*x^4 + 12*x^3 + 13*x^2 + 6*x + 1)*sqrt(1544809*sqrt(10) + 4885115)
*log(-(243*sqrt(10)*x - (893*sqrt(10)*sqrt(3)*x - 2824*sqrt(3)*x)*sqrt(1544809*sqrt(10) + 4885115) + 486*x - 4
86*sqrt(2*x^2 + 3*x + 1) + 486)/x) + sqrt(3)*(4*x^4 + 12*x^3 + 13*x^2 + 6*x + 1)*sqrt(-1544809*sqrt(10) + 4885
115)*log((243*sqrt(10)*x + (893*sqrt(10)*sqrt(3)*x + 2824*sqrt(3)*x)*sqrt(-1544809*sqrt(10) + 4885115) - 486*x
+ 486*sqrt(2*x^2 + 3*x + 1) - 486)/x) - sqrt(3)*(4*x^4 + 12*x^3 + 13*x^2 + 6*x + 1)*sqrt(-1544809*sqrt(10) +
4885115)*log((243*sqrt(10)*x - (893*sqrt(10)*sqrt(3)*x + 2824*sqrt(3)*x)*sqrt(-1544809*sqrt(10) + 4885115) - 4
86*x + 486*sqrt(2*x^2 + 3*x + 1) - 486)/x) + 76440*x^2 + 20*(460*x^3 + 1236*x^2 + 1071*x + 294)*sqrt(2*x^2 + 3
*x + 1) + 35280*x + 5880)/(4*x^4 + 12*x^3 + 13*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