3.47 \(\int \frac{1}{(3-2 x)^{11/2} (1+x+2 x^2)^5} \, dx\)

Optimal. Leaf size=407 \[ \frac{x}{28 (3-2 x)^{9/2} \left (2 x^2+x+1\right )^4}+\frac{5 (4377 x+3049)}{153664 (3-2 x)^{9/2} \left (2 x^2+x+1\right )}+\frac{3049 x+1387}{32928 (3-2 x)^{9/2} \left (2 x^2+x+1\right )^2}+\frac{73 x+23}{1176 (3-2 x)^{9/2} \left (2 x^2+x+1\right )^3}-\frac{38225}{240945152 \sqrt{3-2 x}}-\frac{141045}{120472576 (3-2 x)^{3/2}}-\frac{38491}{8605184 (3-2 x)^{5/2}}-\frac{462025}{30118144 (3-2 x)^{7/2}}-\frac{19255}{395136 (3-2 x)^{9/2}}+\frac{5 \sqrt{\frac{1}{2} \left (40815066112 \sqrt{14}-149046503977\right )} \log \left (-2 x-\sqrt{7+2 \sqrt{14}} \sqrt{3-2 x}+\sqrt{14}+3\right )}{6746464256}-\frac{5 \sqrt{\frac{1}{2} \left (40815066112 \sqrt{14}-149046503977\right )} \log \left (-2 x+\sqrt{7+2 \sqrt{14}} \sqrt{3-2 x}+\sqrt{14}+3\right )}{6746464256}+\frac{5 \sqrt{\frac{1}{2} \left (149046503977+40815066112 \sqrt{14}\right )} \tan ^{-1}\left (\frac{\sqrt{7+2 \sqrt{14}}-2 \sqrt{3-2 x}}{\sqrt{2 \sqrt{14}-7}}\right )}{3373232128}-\frac{5 \sqrt{\frac{1}{2} \left (149046503977+40815066112 \sqrt{14}\right )} \tan ^{-1}\left (\frac{2 \sqrt{3-2 x}+\sqrt{7+2 \sqrt{14}}}{\sqrt{2 \sqrt{14}-7}}\right )}{3373232128} \]

[Out]

-19255/(395136*(3 - 2*x)^(9/2)) - 462025/(30118144*(3 - 2*x)^(7/2)) - 38491/(8605184*(3 - 2*x)^(5/2)) - 141045
/(120472576*(3 - 2*x)^(3/2)) - 38225/(240945152*Sqrt[3 - 2*x]) + x/(28*(3 - 2*x)^(9/2)*(1 + x + 2*x^2)^4) + (2
3 + 73*x)/(1176*(3 - 2*x)^(9/2)*(1 + x + 2*x^2)^3) + (1387 + 3049*x)/(32928*(3 - 2*x)^(9/2)*(1 + x + 2*x^2)^2)
+ (5*(3049 + 4377*x))/(153664*(3 - 2*x)^(9/2)*(1 + x + 2*x^2)) + (5*Sqrt[(149046503977 + 40815066112*Sqrt[14]
)/2]*ArcTan[(Sqrt[7 + 2*Sqrt[14]] - 2*Sqrt[3 - 2*x])/Sqrt[-7 + 2*Sqrt[14]]])/3373232128 - (5*Sqrt[(14904650397
7 + 40815066112*Sqrt[14])/2]*ArcTan[(Sqrt[7 + 2*Sqrt[14]] + 2*Sqrt[3 - 2*x])/Sqrt[-7 + 2*Sqrt[14]]])/337323212
8 + (5*Sqrt[(-149046503977 + 40815066112*Sqrt[14])/2]*Log[3 + Sqrt[14] - Sqrt[7 + 2*Sqrt[14]]*Sqrt[3 - 2*x] -
2*x])/6746464256 - (5*Sqrt[(-149046503977 + 40815066112*Sqrt[14])/2]*Log[3 + Sqrt[14] + Sqrt[7 + 2*Sqrt[14]]*S
qrt[3 - 2*x] - 2*x])/6746464256

________________________________________________________________________________________

Rubi [A]  time = 0.676273, antiderivative size = 407, normalized size of antiderivative = 1., number of steps used = 19, number of rules used = 9, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.45, Rules used = {740, 822, 828, 826, 1169, 634, 618, 204, 628} \[ \frac{x}{28 (3-2 x)^{9/2} \left (2 x^2+x+1\right )^4}+\frac{5 (4377 x+3049)}{153664 (3-2 x)^{9/2} \left (2 x^2+x+1\right )}+\frac{3049 x+1387}{32928 (3-2 x)^{9/2} \left (2 x^2+x+1\right )^2}+\frac{73 x+23}{1176 (3-2 x)^{9/2} \left (2 x^2+x+1\right )^3}-\frac{38225}{240945152 \sqrt{3-2 x}}-\frac{141045}{120472576 (3-2 x)^{3/2}}-\frac{38491}{8605184 (3-2 x)^{5/2}}-\frac{462025}{30118144 (3-2 x)^{7/2}}-\frac{19255}{395136 (3-2 x)^{9/2}}+\frac{5 \sqrt{\frac{1}{2} \left (40815066112 \sqrt{14}-149046503977\right )} \log \left (-2 x-\sqrt{7+2 \sqrt{14}} \sqrt{3-2 x}+\sqrt{14}+3\right )}{6746464256}-\frac{5 \sqrt{\frac{1}{2} \left (40815066112 \sqrt{14}-149046503977\right )} \log \left (-2 x+\sqrt{7+2 \sqrt{14}} \sqrt{3-2 x}+\sqrt{14}+3\right )}{6746464256}+\frac{5 \sqrt{\frac{1}{2} \left (149046503977+40815066112 \sqrt{14}\right )} \tan ^{-1}\left (\frac{\sqrt{7+2 \sqrt{14}}-2 \sqrt{3-2 x}}{\sqrt{2 \sqrt{14}-7}}\right )}{3373232128}-\frac{5 \sqrt{\frac{1}{2} \left (149046503977+40815066112 \sqrt{14}\right )} \tan ^{-1}\left (\frac{2 \sqrt{3-2 x}+\sqrt{7+2 \sqrt{14}}}{\sqrt{2 \sqrt{14}-7}}\right )}{3373232128} \]

Antiderivative was successfully verified.

[In]

Int[1/((3 - 2*x)^(11/2)*(1 + x + 2*x^2)^5),x]

[Out]

-19255/(395136*(3 - 2*x)^(9/2)) - 462025/(30118144*(3 - 2*x)^(7/2)) - 38491/(8605184*(3 - 2*x)^(5/2)) - 141045
/(120472576*(3 - 2*x)^(3/2)) - 38225/(240945152*Sqrt[3 - 2*x]) + x/(28*(3 - 2*x)^(9/2)*(1 + x + 2*x^2)^4) + (2
3 + 73*x)/(1176*(3 - 2*x)^(9/2)*(1 + x + 2*x^2)^3) + (1387 + 3049*x)/(32928*(3 - 2*x)^(9/2)*(1 + x + 2*x^2)^2)
+ (5*(3049 + 4377*x))/(153664*(3 - 2*x)^(9/2)*(1 + x + 2*x^2)) + (5*Sqrt[(149046503977 + 40815066112*Sqrt[14]
)/2]*ArcTan[(Sqrt[7 + 2*Sqrt[14]] - 2*Sqrt[3 - 2*x])/Sqrt[-7 + 2*Sqrt[14]]])/3373232128 - (5*Sqrt[(14904650397
7 + 40815066112*Sqrt[14])/2]*ArcTan[(Sqrt[7 + 2*Sqrt[14]] + 2*Sqrt[3 - 2*x])/Sqrt[-7 + 2*Sqrt[14]]])/337323212
8 + (5*Sqrt[(-149046503977 + 40815066112*Sqrt[14])/2]*Log[3 + Sqrt[14] - Sqrt[7 + 2*Sqrt[14]]*Sqrt[3 - 2*x] -
2*x])/6746464256 - (5*Sqrt[(-149046503977 + 40815066112*Sqrt[14])/2]*Log[3 + Sqrt[14] + Sqrt[7 + 2*Sqrt[14]]*S
qrt[3 - 2*x] - 2*x])/6746464256

Rule 740

Int[((d_.) + (e_.)*(x_))^(m_)*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> Simp[((d + e*x)^(m + 1)*(
b*c*d - b^2*e + 2*a*c*e + c*(2*c*d - b*e)*x)*(a + b*x + c*x^2)^(p + 1))/((p + 1)*(b^2 - 4*a*c)*(c*d^2 - b*d*e
+ a*e^2)), x] + Dist[1/((p + 1)*(b^2 - 4*a*c)*(c*d^2 - b*d*e + a*e^2)), Int[(d + e*x)^m*Simp[b*c*d*e*(2*p - m
+ 2) + b^2*e^2*(m + p + 2) - 2*c^2*d^2*(2*p + 3) - 2*a*c*e^2*(m + 2*p + 3) - c*e*(2*c*d - b*e)*(m + 2*p + 4)*x
, x]*(a + b*x + c*x^2)^(p + 1), x], x] /; FreeQ[{a, b, c, d, e, m}, x] && NeQ[b^2 - 4*a*c, 0] && NeQ[c*d^2 - b
*d*e + a*e^2, 0] && NeQ[2*c*d - b*e, 0] && LtQ[p, -1] && IntQuadraticQ[a, b, c, d, e, m, p, x]

Rule 822

Int[((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> Simp
[((d + e*x)^(m + 1)*(f*(b*c*d - b^2*e + 2*a*c*e) - a*g*(2*c*d - b*e) + c*(f*(2*c*d - b*e) - g*(b*d - 2*a*e))*x
)*(a + b*x + c*x^2)^(p + 1))/((p + 1)*(b^2 - 4*a*c)*(c*d^2 - b*d*e + a*e^2)), x] + Dist[1/((p + 1)*(b^2 - 4*a*
c)*(c*d^2 - b*d*e + a*e^2)), Int[(d + e*x)^m*(a + b*x + c*x^2)^(p + 1)*Simp[f*(b*c*d*e*(2*p - m + 2) + b^2*e^2
*(p + m + 2) - 2*c^2*d^2*(2*p + 3) - 2*a*c*e^2*(m + 2*p + 3)) - g*(a*e*(b*e - 2*c*d*m + b*e*m) - b*d*(3*c*d -
b*e + 2*c*d*p - b*e*p)) + c*e*(g*(b*d - 2*a*e) - f*(2*c*d - b*e))*(m + 2*p + 4)*x, x], x], x] /; FreeQ[{a, b,
c, d, e, f, g, m}, x] && NeQ[b^2 - 4*a*c, 0] && NeQ[c*d^2 - b*d*e + a*e^2, 0] && LtQ[p, -1] && (IntegerQ[m] ||
IntegerQ[p] || IntegersQ[2*m, 2*p])

Rule 828

Int[(((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_)))/((a_.) + (b_.)*(x_) + (c_.)*(x_)^2), x_Symbol] :> Simp[((
e*f - d*g)*(d + e*x)^(m + 1))/((m + 1)*(c*d^2 - b*d*e + a*e^2)), x] + Dist[1/(c*d^2 - b*d*e + a*e^2), Int[((d
+ e*x)^(m + 1)*Simp[c*d*f - f*b*e + a*e*g - c*(e*f - d*g)*x, x])/(a + b*x + c*x^2), x], x] /; FreeQ[{a, b, c,
d, e, f, g, m}, x] && NeQ[b^2 - 4*a*c, 0] && NeQ[c*d^2 - b*d*e + a*e^2, 0] && FractionQ[m] && LtQ[m, -1]

Rule 826

Int[((f_.) + (g_.)*(x_))/(Sqrt[(d_.) + (e_.)*(x_)]*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)), x_Symbol] :> Dist[2,
Subst[Int[(e*f - d*g + g*x^2)/(c*d^2 - b*d*e + a*e^2 - (2*c*d - b*e)*x^2 + c*x^4), x], x, Sqrt[d + e*x]], x] /
; FreeQ[{a, b, c, d, e, f, g}, x] && NeQ[b^2 - 4*a*c, 0] && NeQ[c*d^2 - b*d*e + a*e^2, 0]

Rule 1169

Int[((d_) + (e_.)*(x_)^2)/((a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4), x_Symbol] :> With[{q = Rt[a/c, 2]}, With[{r =
Rt[2*q - b/c, 2]}, Dist[1/(2*c*q*r), Int[(d*r - (d - e*q)*x)/(q - r*x + x^2), x], x] + Dist[1/(2*c*q*r), Int[(
d*r + (d - e*q)*x)/(q + r*x + x^2), x], x]]] /; FreeQ[{a, b, c, d, e}, x] && NeQ[b^2 - 4*a*c, 0] && NeQ[c*d^2
- b*d*e + a*e^2, 0] && NegQ[b^2 - 4*a*c]

Rule 634

Int[((d_.) + (e_.)*(x_))/((a_) + (b_.)*(x_) + (c_.)*(x_)^2), x_Symbol] :> Dist[(2*c*d - b*e)/(2*c), Int[1/(a +
b*x + c*x^2), x], x] + Dist[e/(2*c), Int[(b + 2*c*x)/(a + b*x + c*x^2), x], x] /; FreeQ[{a, b, c, d, e}, x] &
& NeQ[2*c*d - b*e, 0] && NeQ[b^2 - 4*a*c, 0] &&  !NiceSqrtQ[b^2 - 4*a*c]

Rule 618

Int[((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(-1), x_Symbol] :> Dist[-2, Subst[Int[1/Simp[b^2 - 4*a*c - x^2, x], x]
, x, b + 2*c*x], x] /; FreeQ[{a, b, c}, x] && NeQ[b^2 - 4*a*c, 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 628

Int[((d_) + (e_.)*(x_))/((a_.) + (b_.)*(x_) + (c_.)*(x_)^2), x_Symbol] :> Simp[(d*Log[RemoveContent[a + b*x +
c*x^2, x]])/b, x] /; FreeQ[{a, b, c, d, e}, x] && EqQ[2*c*d - b*e, 0]

Rubi steps

\begin{align*} \int \frac{1}{(3-2 x)^{11/2} \left (1+x+2 x^2\right )^5} \, dx &=\frac{x}{28 (3-2 x)^{9/2} \left (1+x+2 x^2\right )^4}+\frac{1}{784} \int \frac{700-644 x}{(3-2 x)^{11/2} \left (1+x+2 x^2\right )^4} \, dx\\ &=\frac{x}{28 (3-2 x)^{9/2} \left (1+x+2 x^2\right )^4}+\frac{23+73 x}{1176 (3-2 x)^{9/2} \left (1+x+2 x^2\right )^3}+\frac{\int \frac{325752-543704 x}{(3-2 x)^{11/2} \left (1+x+2 x^2\right )^3} \, dx}{460992}\\ &=\frac{x}{28 (3-2 x)^{9/2} \left (1+x+2 x^2\right )^4}+\frac{23+73 x}{1176 (3-2 x)^{9/2} \left (1+x+2 x^2\right )^3}+\frac{1387+3049 x}{32928 (3-2 x)^{9/2} \left (1+x+2 x^2\right )^2}+\frac{\int \frac{54660480-250993680 x}{(3-2 x)^{11/2} \left (1+x+2 x^2\right )^2} \, dx}{180708864}\\ &=\frac{x}{28 (3-2 x)^{9/2} \left (1+x+2 x^2\right )^4}+\frac{23+73 x}{1176 (3-2 x)^{9/2} \left (1+x+2 x^2\right )^3}+\frac{1387+3049 x}{32928 (3-2 x)^{9/2} \left (1+x+2 x^2\right )^2}+\frac{5 (3049+4377 x)}{153664 (3-2 x)^{9/2} \left (1+x+2 x^2\right )}+\frac{\int \frac{-25503229920-55488454560 x}{(3-2 x)^{11/2} \left (1+x+2 x^2\right )} \, dx}{35418937344}\\ &=-\frac{19255}{395136 (3-2 x)^{9/2}}+\frac{x}{28 (3-2 x)^{9/2} \left (1+x+2 x^2\right )^4}+\frac{23+73 x}{1176 (3-2 x)^{9/2} \left (1+x+2 x^2\right )^3}+\frac{1387+3049 x}{32928 (3-2 x)^{9/2} \left (1+x+2 x^2\right )^2}+\frac{5 (3049+4377 x)}{153664 (3-2 x)^{9/2} \left (1+x+2 x^2\right )}+\frac{\int \frac{-93048930240-434943647040 x}{(3-2 x)^{9/2} \left (1+x+2 x^2\right )} \, dx}{991730245632}\\ &=-\frac{19255}{395136 (3-2 x)^{9/2}}-\frac{462025}{30118144 (3-2 x)^{7/2}}+\frac{x}{28 (3-2 x)^{9/2} \left (1+x+2 x^2\right )^4}+\frac{23+73 x}{1176 (3-2 x)^{9/2} \left (1+x+2 x^2\right )^3}+\frac{1387+3049 x}{32928 (3-2 x)^{9/2} \left (1+x+2 x^2\right )^2}+\frac{5 (3049+4377 x)}{153664 (3-2 x)^{9/2} \left (1+x+2 x^2\right )}+\frac{\int \frac{125495852160-2981857603200 x}{(3-2 x)^{7/2} \left (1+x+2 x^2\right )} \, dx}{27768446877696}\\ &=-\frac{19255}{395136 (3-2 x)^{9/2}}-\frac{462025}{30118144 (3-2 x)^{7/2}}-\frac{38491}{8605184 (3-2 x)^{5/2}}+\frac{x}{28 (3-2 x)^{9/2} \left (1+x+2 x^2\right )^4}+\frac{23+73 x}{1176 (3-2 x)^{9/2} \left (1+x+2 x^2\right )^3}+\frac{1387+3049 x}{32928 (3-2 x)^{9/2} \left (1+x+2 x^2\right )^2}+\frac{5 (3049+4377 x)}{153664 (3-2 x)^{9/2} \left (1+x+2 x^2\right )}+\frac{\int \frac{6967682023680-17389162210560 x}{(3-2 x)^{5/2} \left (1+x+2 x^2\right )} \, dx}{777516512575488}\\ &=-\frac{19255}{395136 (3-2 x)^{9/2}}-\frac{462025}{30118144 (3-2 x)^{7/2}}-\frac{38491}{8605184 (3-2 x)^{5/2}}-\frac{141045}{120472576 (3-2 x)^{3/2}}+\frac{x}{28 (3-2 x)^{9/2} \left (1+x+2 x^2\right )^4}+\frac{23+73 x}{1176 (3-2 x)^{9/2} \left (1+x+2 x^2\right )^3}+\frac{1387+3049 x}{32928 (3-2 x)^{9/2} \left (1+x+2 x^2\right )^2}+\frac{5 (3049+4377 x)}{153664 (3-2 x)^{9/2} \left (1+x+2 x^2\right )}+\frac{\int \frac{90519780610560-76464245168640 x}{(3-2 x)^{3/2} \left (1+x+2 x^2\right )} \, dx}{21770462352113664}\\ &=-\frac{19255}{395136 (3-2 x)^{9/2}}-\frac{462025}{30118144 (3-2 x)^{7/2}}-\frac{38491}{8605184 (3-2 x)^{5/2}}-\frac{141045}{120472576 (3-2 x)^{3/2}}-\frac{38225}{240945152 \sqrt{3-2 x}}+\frac{x}{28 (3-2 x)^{9/2} \left (1+x+2 x^2\right )^4}+\frac{23+73 x}{1176 (3-2 x)^{9/2} \left (1+x+2 x^2\right )^3}+\frac{1387+3049 x}{32928 (3-2 x)^{9/2} \left (1+x+2 x^2\right )^2}+\frac{5 (3049+4377 x)}{153664 (3-2 x)^{9/2} \left (1+x+2 x^2\right )}+\frac{\int \frac{877086735221760-96706348569600 x}{\sqrt{3-2 x} \left (1+x+2 x^2\right )} \, dx}{609572945859182592}\\ &=-\frac{19255}{395136 (3-2 x)^{9/2}}-\frac{462025}{30118144 (3-2 x)^{7/2}}-\frac{38491}{8605184 (3-2 x)^{5/2}}-\frac{141045}{120472576 (3-2 x)^{3/2}}-\frac{38225}{240945152 \sqrt{3-2 x}}+\frac{x}{28 (3-2 x)^{9/2} \left (1+x+2 x^2\right )^4}+\frac{23+73 x}{1176 (3-2 x)^{9/2} \left (1+x+2 x^2\right )^3}+\frac{1387+3049 x}{32928 (3-2 x)^{9/2} \left (1+x+2 x^2\right )^2}+\frac{5 (3049+4377 x)}{153664 (3-2 x)^{9/2} \left (1+x+2 x^2\right )}+\frac{\operatorname{Subst}\left (\int \frac{-1464054424734720-96706348569600 x^2}{28-14 x^2+2 x^4} \, dx,x,\sqrt{3-2 x}\right )}{304786472929591296}\\ &=-\frac{19255}{395136 (3-2 x)^{9/2}}-\frac{462025}{30118144 (3-2 x)^{7/2}}-\frac{38491}{8605184 (3-2 x)^{5/2}}-\frac{141045}{120472576 (3-2 x)^{3/2}}-\frac{38225}{240945152 \sqrt{3-2 x}}+\frac{x}{28 (3-2 x)^{9/2} \left (1+x+2 x^2\right )^4}+\frac{23+73 x}{1176 (3-2 x)^{9/2} \left (1+x+2 x^2\right )^3}+\frac{1387+3049 x}{32928 (3-2 x)^{9/2} \left (1+x+2 x^2\right )^2}+\frac{5 (3049+4377 x)}{153664 (3-2 x)^{9/2} \left (1+x+2 x^2\right )}+\frac{\operatorname{Subst}\left (\int \frac{-1464054424734720 \sqrt{7+2 \sqrt{14}}-\left (-1464054424734720+96706348569600 \sqrt{14}\right ) x}{\sqrt{14}-\sqrt{7+2 \sqrt{14}} x+x^2} \, dx,x,\sqrt{3-2 x}\right )}{1219145891718365184 \sqrt{14 \left (7+2 \sqrt{14}\right )}}+\frac{\operatorname{Subst}\left (\int \frac{-1464054424734720 \sqrt{7+2 \sqrt{14}}+\left (-1464054424734720+96706348569600 \sqrt{14}\right ) x}{\sqrt{14}+\sqrt{7+2 \sqrt{14}} x+x^2} \, dx,x,\sqrt{3-2 x}\right )}{1219145891718365184 \sqrt{14 \left (7+2 \sqrt{14}\right )}}\\ &=-\frac{19255}{395136 (3-2 x)^{9/2}}-\frac{462025}{30118144 (3-2 x)^{7/2}}-\frac{38491}{8605184 (3-2 x)^{5/2}}-\frac{141045}{120472576 (3-2 x)^{3/2}}-\frac{38225}{240945152 \sqrt{3-2 x}}+\frac{x}{28 (3-2 x)^{9/2} \left (1+x+2 x^2\right )^4}+\frac{23+73 x}{1176 (3-2 x)^{9/2} \left (1+x+2 x^2\right )^3}+\frac{1387+3049 x}{32928 (3-2 x)^{9/2} \left (1+x+2 x^2\right )^2}+\frac{5 (3049+4377 x)}{153664 (3-2 x)^{9/2} \left (1+x+2 x^2\right )}-\frac{\left (5 \left (107030+115739 \sqrt{14}\right )\right ) \operatorname{Subst}\left (\int \frac{1}{\sqrt{14}-\sqrt{7+2 \sqrt{14}} x+x^2} \, dx,x,\sqrt{3-2 x}\right )}{13492928512}-\frac{\left (5 \left (107030+115739 \sqrt{14}\right )\right ) \operatorname{Subst}\left (\int \frac{1}{\sqrt{14}+\sqrt{7+2 \sqrt{14}} x+x^2} \, dx,x,\sqrt{3-2 x}\right )}{13492928512}+\frac{\left (5 \sqrt{\frac{1}{2} \left (-149046503977+40815066112 \sqrt{14}\right )}\right ) \operatorname{Subst}\left (\int \frac{-\sqrt{7+2 \sqrt{14}}+2 x}{\sqrt{14}-\sqrt{7+2 \sqrt{14}} x+x^2} \, dx,x,\sqrt{3-2 x}\right )}{6746464256}-\frac{\left (5 \sqrt{\frac{1}{2} \left (-149046503977+40815066112 \sqrt{14}\right )}\right ) \operatorname{Subst}\left (\int \frac{\sqrt{7+2 \sqrt{14}}+2 x}{\sqrt{14}+\sqrt{7+2 \sqrt{14}} x+x^2} \, dx,x,\sqrt{3-2 x}\right )}{6746464256}\\ &=-\frac{19255}{395136 (3-2 x)^{9/2}}-\frac{462025}{30118144 (3-2 x)^{7/2}}-\frac{38491}{8605184 (3-2 x)^{5/2}}-\frac{141045}{120472576 (3-2 x)^{3/2}}-\frac{38225}{240945152 \sqrt{3-2 x}}+\frac{x}{28 (3-2 x)^{9/2} \left (1+x+2 x^2\right )^4}+\frac{23+73 x}{1176 (3-2 x)^{9/2} \left (1+x+2 x^2\right )^3}+\frac{1387+3049 x}{32928 (3-2 x)^{9/2} \left (1+x+2 x^2\right )^2}+\frac{5 (3049+4377 x)}{153664 (3-2 x)^{9/2} \left (1+x+2 x^2\right )}+\frac{5 \sqrt{\frac{1}{2} \left (-149046503977+40815066112 \sqrt{14}\right )} \log \left (3+\sqrt{14}-\sqrt{7+2 \sqrt{14}} \sqrt{3-2 x}-2 x\right )}{6746464256}-\frac{5 \sqrt{\frac{1}{2} \left (-149046503977+40815066112 \sqrt{14}\right )} \log \left (3+\sqrt{14}+\sqrt{7+2 \sqrt{14}} \sqrt{3-2 x}-2 x\right )}{6746464256}+\frac{\left (5 \left (107030+115739 \sqrt{14}\right )\right ) \operatorname{Subst}\left (\int \frac{1}{7-2 \sqrt{14}-x^2} \, dx,x,-\sqrt{7+2 \sqrt{14}}+2 \sqrt{3-2 x}\right )}{6746464256}+\frac{\left (5 \left (107030+115739 \sqrt{14}\right )\right ) \operatorname{Subst}\left (\int \frac{1}{7-2 \sqrt{14}-x^2} \, dx,x,\sqrt{7+2 \sqrt{14}}+2 \sqrt{3-2 x}\right )}{6746464256}\\ &=-\frac{19255}{395136 (3-2 x)^{9/2}}-\frac{462025}{30118144 (3-2 x)^{7/2}}-\frac{38491}{8605184 (3-2 x)^{5/2}}-\frac{141045}{120472576 (3-2 x)^{3/2}}-\frac{38225}{240945152 \sqrt{3-2 x}}+\frac{x}{28 (3-2 x)^{9/2} \left (1+x+2 x^2\right )^4}+\frac{23+73 x}{1176 (3-2 x)^{9/2} \left (1+x+2 x^2\right )^3}+\frac{1387+3049 x}{32928 (3-2 x)^{9/2} \left (1+x+2 x^2\right )^2}+\frac{5 (3049+4377 x)}{153664 (3-2 x)^{9/2} \left (1+x+2 x^2\right )}+\frac{5 \sqrt{298093007954+81630132224 \sqrt{14}} \tan ^{-1}\left (\frac{\sqrt{7+2 \sqrt{14}}-2 \sqrt{3-2 x}}{\sqrt{-7+2 \sqrt{14}}}\right )}{6746464256}-\frac{5 \sqrt{298093007954+81630132224 \sqrt{14}} \tan ^{-1}\left (\frac{\sqrt{7+2 \sqrt{14}}+2 \sqrt{3-2 x}}{\sqrt{-7+2 \sqrt{14}}}\right )}{6746464256}+\frac{5 \sqrt{\frac{1}{2} \left (-149046503977+40815066112 \sqrt{14}\right )} \log \left (3+\sqrt{14}-\sqrt{7+2 \sqrt{14}} \sqrt{3-2 x}-2 x\right )}{6746464256}-\frac{5 \sqrt{\frac{1}{2} \left (-149046503977+40815066112 \sqrt{14}\right )} \log \left (3+\sqrt{14}+\sqrt{7+2 \sqrt{14}} \sqrt{3-2 x}-2 x\right )}{6746464256}\\ \end{align*}

Mathematica [C]  time = 2.07546, size = 198, normalized size = 0.49 \[ \frac{\frac{56 \left (-88070400 x^{12}[phone] x^{11}-1873554048 x^{10}[phone] x^9-2343370048 x^8+3106712560 x^7-2888625656 x^6+1470758860 x^5-1627773523 x^4+1073855156 x^3-135202154 x^2+429812744 x-40289347\right )}{(3-2 x)^{9/2} \left (2 x^2+x+1\right )^4}+45 i \sqrt{14-2 i \sqrt{7}} \left (146319 \sqrt{7}+115739 i\right ) \tanh ^{-1}\left (\frac{\sqrt{6-4 x}}{\sqrt{7-i \sqrt{7}}}\right )-45 i \sqrt{14+2 i \sqrt{7}} \left (146319 \sqrt{7}-115739 i\right ) \tanh ^{-1}\left (\frac{\sqrt{6-4 x}}{\sqrt{7+i \sqrt{7}}}\right )}{121436356608} \]

Antiderivative was successfully verified.

[In]

Integrate[1/((3 - 2*x)^(11/2)*(1 + x + 2*x^2)^5),x]

[Out]

((56*(-40289347 + 429812744*x - 135202154*x^2 + 1073855156*x^3 - 1627773523*x^4 + 1470758860*x^5 - 2888625656*
x^6 + 3106712560*x^7 - 2343370048*x^8 + 2443779648*x^9 - 1873554048*x^10 + 677249280*x^11 - 88070400*x^12))/((
3 - 2*x)^(9/2)*(1 + x + 2*x^2)^4) + (45*I)*Sqrt[14 - (2*I)*Sqrt[7]]*(115739*I + 146319*Sqrt[7])*ArcTanh[Sqrt[6
- 4*x]/Sqrt[7 - I*Sqrt[7]]] - (45*I)*Sqrt[14 + (2*I)*Sqrt[7]]*(-115739*I + 146319*Sqrt[7])*ArcTanh[Sqrt[6 - 4
*x]/Sqrt[7 + I*Sqrt[7]]])/121436356608

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Maple [A]  time = 0.089, size = 584, normalized size = 1.4 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(3-2*x)^(11/2)/(2*x^2+x+1)^5,x)

[Out]

1/151263/(3-2*x)^(9/2)+5/235298/(3-2*x)^(7/2)+19/470596/(3-2*x)^(5/2)+185/2823576/(3-2*x)^(3/2)+505/3294172/(3
-2*x)^(1/2)+1/6588344*(567651623/32*(3-2*x)^(1/2)-6194606411/192*(3-2*x)^(3/2)[phone]/384*(3-2*x)^(5/2)-87
63772549/768*(3-2*x)^(7/2)[phone]/48*(3-2*x)^(9/2)-200063633/384*(3-2*x)^(11/2)[phone]/384*(3-2*x)^(13/2)
-526135/256*(3-2*x)^(15/2))/((3-2*x)^2-7+14*x)^4+731595/13492928512*ln(3-2*x+14^(1/2)+(3-2*x)^(1/2)*(7+2*14^(1
/2))^(1/2))*(7+2*14^(1/2))^(1/2)*14^(1/2)-1424965/6746464256*ln(3-2*x+14^(1/2)+(3-2*x)^(1/2)*(7+2*14^(1/2))^(1
/2))*(7+2*14^(1/2))^(1/2)-731595/6746464256/(-7+2*14^(1/2))^(1/2)*arctan((2*(3-2*x)^(1/2)+(7+2*14^(1/2))^(1/2)
)/(-7+2*14^(1/2))^(1/2))*(7+2*14^(1/2))*14^(1/2)+1424965/3373232128/(-7+2*14^(1/2))^(1/2)*arctan((2*(3-2*x)^(1
/2)+(7+2*14^(1/2))^(1/2))/(-7+2*14^(1/2))^(1/2))*(7+2*14^(1/2))-578695/3373232128/(-7+2*14^(1/2))^(1/2)*arctan
((2*(3-2*x)^(1/2)+(7+2*14^(1/2))^(1/2))/(-7+2*14^(1/2))^(1/2))*14^(1/2)-731595/13492928512*ln(3-2*x+14^(1/2)-(
3-2*x)^(1/2)*(7+2*14^(1/2))^(1/2))*(7+2*14^(1/2))^(1/2)*14^(1/2)+1424965/6746464256*ln(3-2*x+14^(1/2)-(3-2*x)^
(1/2)*(7+2*14^(1/2))^(1/2))*(7+2*14^(1/2))^(1/2)-731595/6746464256/(-7+2*14^(1/2))^(1/2)*arctan((2*(3-2*x)^(1/
2)-(7+2*14^(1/2))^(1/2))/(-7+2*14^(1/2))^(1/2))*(7+2*14^(1/2))*14^(1/2)+1424965/3373232128/(-7+2*14^(1/2))^(1/
2)*arctan((2*(3-2*x)^(1/2)-(7+2*14^(1/2))^(1/2))/(-7+2*14^(1/2))^(1/2))*(7+2*14^(1/2))-578695/3373232128/(-7+2
*14^(1/2))^(1/2)*arctan((2*(3-2*x)^(1/2)-(7+2*14^(1/2))^(1/2))/(-7+2*14^(1/2))^(1/2))*14^(1/2)

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Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{1}{{\left (2 \, x^{2} + x + 1\right )}^{5}{\left (-2 \, x + 3\right )}^{\frac{11}{2}}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(3-2*x)^(11/2)/(2*x^2+x+1)^5,x, algorithm="maxima")

[Out]

integrate(1/((2*x^2 + x + 1)^5*(-2*x + 3)^(11/2)), x)

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Fricas [B]  time = 3.1791, size = 5156, normalized size = 12.67 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(3-2*x)^(11/2)/(2*x^2+x+1)^5,x, algorithm="fricas")

[Out]

1/852282865707923134247251378176*(2263908918780*22241759018113166^(1/4)*sqrt(79716926)*sqrt(14)*sqrt(7)*(512*x
^13 - 2816*x^12 + 5632*x^11 - 5888*x^10 + 6848*x^9 - 8992*x^8 + 6112*x^7 - 4240*x^6 + 4994*x^5 - 1707*x^4 + 93
6*x^3 - 1242*x^2 - 162*x - 243)*sqrt(21292357711*sqrt(14) + 81630132224)*arctan(1/1005218715695186946952690868
5753437228729401815040*22241759018113166^(3/4)*sqrt(12577271771)*sqrt(79716926)*sqrt(-2089731384934400*2224175
9018113166^(1/4)*sqrt(79716926)*sqrt(-2*x + 3)*sqrt(21292357711*sqrt(14) + 81630132224)*(7645*sqrt(14) - 11573
9) - 4190418993502514995568679111884800*x + 2095209496751257497784339555942400*sqrt(14) + 62856284902537724933
53018667827200)*(115739*sqrt(14)*sqrt(7) - 107030*sqrt(7))*sqrt(21292357711*sqrt(14) + 81630132224) - 1/195818
4534851295802906658902*22241759018113166^(3/4)*sqrt(79716926)*(115739*sqrt(14)*sqrt(7) - 107030*sqrt(7))*sqrt(
-2*x + 3)*sqrt(21292357711*sqrt(14) + 81630132224) - 2/7*sqrt(14)*sqrt(7) - sqrt(7)) + 2263908918780*222417590
18113166^(1/4)*sqrt(79716926)*sqrt(14)*sqrt(7)*(512*x^13 - 2816*x^12 + 5632*x^11 - 5888*x^10 + 6848*x^9 - 8992
*x^8 + 6112*x^7 - 4240*x^6 + 4994*x^5 - 1707*x^4 + 936*x^3 - 1242*x^2 - 162*x - 243)*sqrt(21292357711*sqrt(14)
+ 81630132224)*arctan(1/24628619072593968384668700756050455442*22241759018113166^(3/4)*sqrt(12577271771)*sqrt
(22241759018113166^(1/4)*sqrt(79716926)*sqrt(-2*x + 3)*sqrt(21292357711*sqrt(14) + 81630132224)*(7645*sqrt(14)
- 115739) - 2005242886101391892*x + 1002621443050695946*sqrt(14) + 3007864329152087838)*(115739*sqrt(14)*sqrt
(7) - 107030*sqrt(7))*sqrt(21292357711*sqrt(14) + 81630132224) - 1/1958184534851295802906658902*22241759018113
166^(3/4)*sqrt(79716926)*(115739*sqrt(14)*sqrt(7) - 107030*sqrt(7))*sqrt(-2*x + 3)*sqrt(21292357711*sqrt(14) +
81630132224) + 2/7*sqrt(14)*sqrt(7) + sqrt(7)) + 315*22241759018113166^(1/4)*sqrt(79716926)*(41794627698688*x
^13 - 229870452342784*x^12 + 459740904685568*x^11 - 480638218534912*x^10 + 559003145469952*x^9 - 7340181489582
08*x^8 + 498923368153088*x^7 - 346111760629760*x^6 + 407660880326656*x^5 - 139342635706368*x^4 + 7640580376166
4*x^3 - 101384624222208*x^2 - 21292357711*sqrt(14)*(512*x^13 - 2816*x^12 + 5632*x^11 - 5888*x^10 + 6848*x^9 -
8992*x^8 + 6112*x^7 - 4240*x^6 + 4994*x^5 - 1707*x^4 + 936*x^3 - 1242*x^2 - 162*x - 243) - 13224081420288*x -
19836122130432)*sqrt(21292357711*sqrt(14) + 81630132224)*log(2089731384934400/12577271771*22241759018113166^(1
/4)*sqrt(79716926)*sqrt(-2*x + 3)*sqrt(21292357711*sqrt(14) + 81630132224)*(7645*sqrt(14) - 115739) - 33317392
4345386159308800*x + 166586962172693079654400*sqrt(14) + 499760886518079238963200) - 315*22241759018113166^(1/
4)*sqrt(79716926)*(41794627698688*x^13 - 229870452342784*x^12 + 459740904685568*x^11 - 480638218534912*x^10 +
559003145469952*x^9 - 734018148958208*x^8 + 498923368153088*x^7 - 346111760629760*x^6 + 407660880326656*x^5 -
139342635706368*x^4 + 76405803761664*x^3 - 101384624222208*x^2 - 21292357711*sqrt(14)*(512*x^13 - 2816*x^12 +
5632*x^11 - 5888*x^10 + 6848*x^9 - 8992*x^8 + 6112*x^7 - 4240*x^6 + 4994*x^5 - 1707*x^4 + 936*x^3 - 1242*x^2 -
162*x - 243) - 13224081420288*x - 19836122130432)*sqrt(21292357711*sqrt(14) + 81630132224)*log(-2089731384934
400/12577271771*22241759018113166^(1/4)*sqrt(79716926)*sqrt(-2*x + 3)*sqrt(21292357711*sqrt(14) + 81630132224)
*(7645*sqrt(14) - 115739) - 333173924345386159308800*x + 166586962172693079654400*sqrt(14) + 49976088651807923
8963200) + 393027605675872810832*(88070400*x^12 - 677249280*x^11 + 1873554048*x^10 - 2443779648*x^9 + 23433700
48*x^8 - 3106712560*x^7 + 2888625656*x^6 - 1470758860*x^5 + 1627773523*x^4 - 1073855156*x^3 + 135202154*x^2 -
429812744*x + 40289347)*sqrt(-2*x + 3))/(512*x^13 - 2816*x^12 + 5632*x^11 - 5888*x^10 + 6848*x^9 - 8992*x^8 +
6112*x^7 - 4240*x^6 + 4994*x^5 - 1707*x^4 + 936*x^3 - 1242*x^2 - 162*x - 243)

________________________________________________________________________________________

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(1/(3-2*x)**(11/2)/(2*x**2+x+1)**5,x)

[Out]

Timed out

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{1}{{\left (2 \, x^{2} + x + 1\right )}^{5}{\left (-2 \, x + 3\right )}^{\frac{11}{2}}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(3-2*x)^(11/2)/(2*x^2+x+1)^5,x, algorithm="giac")

[Out]

integrate(1/((2*x^2 + x + 1)^5*(-2*x + 3)^(11/2)), x)