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)+1/28*x/(3-2*x)^(9/2)/(2*x^2+x+1)^4+1/1176*(23+73*x)/(3-2*x)^(9/2)/(2*x^2+x+1)^3+1/32928*(1387+3049*x)/(
3-2*x)^(9/2)/(2*x^2+x+1)^2+5/153664*(3049+4377*x)/(3-2*x)^(9/2)/(2*x^2+x+1)-38225/240945152/(3-2*x)^(1/2)+5/13
492928512*ln(3-2*x+14^(1/2)-(3-2*x)^(1/2)*(7+2*14^(1/2))^(1/2))*(-298093007954+81630132224*14^(1/2))^(1/2)-5/1
3492928512*ln(3-2*x+14^(1/2)+(3-2*x)^(1/2)*(7+2*14^(1/2))^(1/2))*(-298093007954+81630132224*14^(1/2))^(1/2)+5/
6746464256*arctan((-2*(3-2*x)^(1/2)+(7+2*14^(1/2))^(1/2))/(-7+2*14^(1/2))^(1/2))*(298093007954+81630132224*14^
(1/2))^(1/2)-5/6746464256*arctan((2*(3-2*x)^(1/2)+(7+2*14^(1/2))^(1/2))/(-7+2*14^(1/2))^(1/2))*(298093007954+8
1630132224*14^(1/2))^(1/2)

________________________________________________________________________________________

Rubi [A]  time = 0.68, antiderivative size = 407, normalized size of antiderivative = 1.00, number of steps used = 19, number of rules used = 9, integrand size = 20, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.450, 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 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 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 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]

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 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 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 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 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]

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*}

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Mathematica [C]  time = 2.13, size = 198, normalized size = 0.49 \[ \frac {45 i \sqrt {14-2 i \sqrt {7}} \left (146319 \sqrt {7}+115739 i\right ) \tanh ^{-1}\left (\frac {\sqrt {6-4 x}}{\sqrt {7-i \sqrt {7}}}\right )-45 i \sqrt {14+2 i \sqrt {7}} \left (146319 \sqrt {7}-115739 i\right ) \tanh ^{-1}\left (\frac {\sqrt {6-4 x}}{\sqrt {7+i \sqrt {7}}}\right )+\frac {56 \left (-88070400 x^{12}[phone] x^{11}-1873554048 x^{10}[phone] x^9-2343370048 x^8+3106712560 x^7-2888625656 x^6+1470758860 x^5-1627773523 x^4+1073855156 x^3-135202154 x^2+429812744 x-40289347\right )}{(3-2 x)^{9/2} \left (2 x^2+x+1\right )^4}}{121436356608} \]

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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fricas [B]  time = 1.52, size = 957, normalized size = 2.35 \[ \text {result too large to display} \]

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)

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giac [B]  time = 2.82, size = 767, normalized size = 1.88 \[ \text {result too large to display} \]

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]

-5/24179327893504*sqrt(7)*(856240*14^(3/4)*sqrt(2)*(sqrt(14) + 4)^(3/2) + 2568720*14^(3/4)*sqrt(2)*sqrt(sqrt(1
4) + 4)*(sqrt(14) - 4) + 183480*14^(3/4)*sqrt(7)*(sqrt(14) + 4)*sqrt(-8*sqrt(14) + 32) - 7645*14^(3/4)*sqrt(7)
*(-8*sqrt(14) + 32)^(3/2) + 103702144*14^(1/4)*sqrt(2)*sqrt(sqrt(14) + 4) + 7407296*14^(1/4)*sqrt(7)*sqrt(-8*s
qrt(14) + 32))*arctan(1/28*14^(3/4)*(14^(1/4)*sqrt(1/2)*sqrt(sqrt(14) + 4) + 2*sqrt(-2*x + 3))/sqrt(-1/8*sqrt(
14) + 1/2)) - 5/24179327893504*sqrt(7)*(856240*14^(3/4)*sqrt(2)*(sqrt(14) + 4)^(3/2) + 2568720*14^(3/4)*sqrt(2
)*sqrt(sqrt(14) + 4)*(sqrt(14) - 4) + 183480*14^(3/4)*sqrt(7)*(sqrt(14) + 4)*sqrt(-8*sqrt(14) + 32) - 7645*14^
(3/4)*sqrt(7)*(-8*sqrt(14) + 32)^(3/2) + 103702144*14^(1/4)*sqrt(2)*sqrt(sqrt(14) + 4) + 7407296*14^(1/4)*sqrt
(7)*sqrt(-8*sqrt(14) + 32))*arctan(-1/28*14^(3/4)*(14^(1/4)*sqrt(1/2)*sqrt(sqrt(14) + 4) - 2*sqrt(-2*x + 3))/s
qrt(-1/8*sqrt(14) + 1/2)) - 5/48358655787008*sqrt(7)*(122320*14^(3/4)*sqrt(7)*sqrt(2)*(sqrt(14) + 4)^(3/2) + 3
66960*14^(3/4)*sqrt(7)*sqrt(2)*sqrt(sqrt(14) + 4)*(sqrt(14) - 4) - 1284360*14^(3/4)*(sqrt(14) + 4)*sqrt(-8*sqr
t(14) + 32) + 53515*14^(3/4)*(-8*sqrt(14) + 32)^(3/2) + 14814592*14^(1/4)*sqrt(7)*sqrt(2)*sqrt(sqrt(14) + 4) -
51851072*14^(1/4)*sqrt(-8*sqrt(14) + 32))*log(14^(1/4)*sqrt(1/2)*sqrt(-2*x + 3)*sqrt(sqrt(14) + 4) - 2*x + sq
rt(14) + 3) + 5/48358655787008*sqrt(7)*(122320*14^(3/4)*sqrt(7)*sqrt(2)*(sqrt(14) + 4)^(3/2) + 366960*14^(3/4)
*sqrt(7)*sqrt(2)*sqrt(sqrt(14) + 4)*(sqrt(14) - 4) - 1284360*14^(3/4)*(sqrt(14) + 4)*sqrt(-8*sqrt(14) + 32) +
53515*14^(3/4)*(-8*sqrt(14) + 32)^(3/2) + 14814592*14^(1/4)*sqrt(7)*sqrt(2)*sqrt(sqrt(14) + 4) - 51851072*14^(
1/4)*sqrt(-8*sqrt(14) + 32))*log(-14^(1/4)*sqrt(1/2)*sqrt(-2*x + 3)*sqrt(sqrt(14) + 4) - 2*x + sqrt(14) + 3) +
1/5059848192*(1578405*(2*x - 3)^7*sqrt(-2*x + 3) + 37939930*(2*x - 3)^6*sqrt(-2*x + 3) + 400127266*(2*x - 3)^
5*sqrt(-2*x + 3) + 2394090608*(2*x - 3)^4*sqrt(-2*x + 3) + 8763772549*(2*x - 3)^3*sqrt(-2*x + 3) + 19602865030
*(2*x - 3)^2*sqrt(-2*x + 3) - 24778425644*(-2*x + 3)^(3/2) + 13623638952*sqrt(-2*x + 3))/((2*x - 3)^2 + 14*x -
7)^4 + 1/59295096*(9090*(2*x - 3)^4 - 3885*(2*x - 3)^3 + 2394*(2*x - 3)^2 - 2520*x + 4172)/((2*x - 3)^4*sqrt(
-2*x + 3))

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maple [A]  time = 0.12, size = 584, normalized size = 1.43 \[ -\frac {731595 \left (7+2 \sqrt {14}\right ) \sqrt {14}\, \arctan \left (\frac {2 \sqrt {-2 x +3}-\sqrt {7+2 \sqrt {14}}}{\sqrt {-7+2 \sqrt {14}}}\right )}{6746464256 \sqrt {-7+2 \sqrt {14}}}+\frac {1424965 \left (7+2 \sqrt {14}\right ) \arctan \left (\frac {2 \sqrt {-2 x +3}-\sqrt {7+2 \sqrt {14}}}{\sqrt {-7+2 \sqrt {14}}}\right )}{3373232128 \sqrt {-7+2 \sqrt {14}}}-\frac {578695 \sqrt {14}\, \arctan \left (\frac {2 \sqrt {-2 x +3}-\sqrt {7+2 \sqrt {14}}}{\sqrt {-7+2 \sqrt {14}}}\right )}{3373232128 \sqrt {-7+2 \sqrt {14}}}-\frac {731595 \left (7+2 \sqrt {14}\right ) \sqrt {14}\, \arctan \left (\frac {2 \sqrt {-2 x +3}+\sqrt {7+2 \sqrt {14}}}{\sqrt {-7+2 \sqrt {14}}}\right )}{6746464256 \sqrt {-7+2 \sqrt {14}}}+\frac {1424965 \left (7+2 \sqrt {14}\right ) \arctan \left (\frac {2 \sqrt {-2 x +3}+\sqrt {7+2 \sqrt {14}}}{\sqrt {-7+2 \sqrt {14}}}\right )}{3373232128 \sqrt {-7+2 \sqrt {14}}}-\frac {578695 \sqrt {14}\, \arctan \left (\frac {2 \sqrt {-2 x +3}+\sqrt {7+2 \sqrt {14}}}{\sqrt {-7+2 \sqrt {14}}}\right )}{3373232128 \sqrt {-7+2 \sqrt {14}}}-\frac {731595 \sqrt {7+2 \sqrt {14}}\, \sqrt {14}\, \ln \left (-2 x +3+\sqrt {14}-\sqrt {-2 x +3}\, \sqrt {7+2 \sqrt {14}}\right )}{13492928512}+\frac {1424965 \sqrt {7+2 \sqrt {14}}\, \ln \left (-2 x +3+\sqrt {14}-\sqrt {-2 x +3}\, \sqrt {7+2 \sqrt {14}}\right )}{6746464256}+\frac {731595 \sqrt {7+2 \sqrt {14}}\, \sqrt {14}\, \ln \left (-2 x +3+\sqrt {14}+\sqrt {-2 x +3}\, \sqrt {7+2 \sqrt {14}}\right )}{13492928512}-\frac {1424965 \sqrt {7+2 \sqrt {14}}\, \ln \left (-2 x +3+\sqrt {14}+\sqrt {-2 x +3}\, \sqrt {7+2 \sqrt {14}}\right )}{6746464256}+\frac {\frac {567651623 \sqrt {-2 x +3}}{32}-\frac {6194606411 \left (-2 x +3\right )^{\frac {3}{2}}}{192}+\frac {9801432515 \left (-2 x +3\right )^{\frac {5}{2}}}{384}-\frac {8763772549 \left (-2 x +3\right )^{\frac {7}{2}}}{768}+\frac {149630663 \left (-2 x +3\right )^{\frac {9}{2}}}{48}-\frac {200063633 \left (-2 x +3\right )^{\frac {11}{2}}}{384}+\frac {18969965 \left (-2 x +3\right )^{\frac {13}{2}}}{384}-\frac {526135 \left (-2 x +3\right )^{\frac {15}{2}}}{256}}{6588344 \left (14 x +\left (-2 x +3\right )^{2}-7\right )^{4}}+\frac {1}{151263 \left (-2 x +3\right )^{\frac {9}{2}}}+\frac {5}{235298 \left (-2 x +3\right )^{\frac {7}{2}}}+\frac {19}{470596 \left (-2 x +3\right )^{\frac {5}{2}}}+\frac {185}{2823576 \left (-2 x +3\right )^{\frac {3}{2}}}+\frac {505}{3294172 \sqrt {-2 x +3}} \]

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]

1/6588344*(567651623/32*(3-2*x)^(1/2)-6194606411/192*(3-2*x)^(3/2)[phone]/384*(3-2*x)^(5/2)-8763772549/768
*(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*1
4^(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)+1/151263/(3-2*x)^(9/2)+5/23
5298/(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)

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maxima [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {1}{{\left (2 \, x^{2} + x + 1\right )}^{5} {\left (-2 \, x + 3\right )}^{\frac {11}{2}}}\,{d x} \]

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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mupad [B]  time = 0.47, size = 343, normalized size = 0.84 \[ -\frac {\frac {272\,x}{441}-\frac {164\,{\left (2\,x-3\right )}^2}{441}+\frac {1966\,{\left (2\,x-3\right )}^3}{3087}-\frac {9091\,{\left (2\,x-3\right )}^4}{3087}-\frac {32070727\,{\left (2\,x-3\right )}^5}{5531904}-\frac {41014777\,{\left (2\,x-3\right )}^6}{11063808}-\frac {141921511\,{\left (2\,x-3\right )}^7}{154893312}+\frac {23262655\,{\left (2\,x-3\right )}^8}{309786624}+\frac {1571659\,{\left (2\,x-3\right )}^9}{15059072}+\frac {468427\,{\left (2\,x-3\right )}^{10}}{17210368}+\frac {394105\,{\left (2\,x-3\right )}^{11}}{120472576}+\frac {38225\,{\left (2\,x-3\right )}^{12}}{240945152}-\frac {520}{441}}{38416\,{\left (3-2\,x\right )}^{9/2}-76832\,{\left (3-2\,x\right )}^{11/2}+68600\,{\left (3-2\,x\right )}^{13/2}-35672\,{\left (3-2\,x\right )}^{15/2}+11809\,{\left (3-2\,x\right )}^{17/2}-2548\,{\left (3-2\,x\right )}^{19/2}+350\,{\left (3-2\,x\right )}^{21/2}-28\,{\left (3-2\,x\right )}^{23/2}+{\left (3-2\,x\right )}^{25/2}}+\frac {\mathrm {atan}\left (\frac {\sqrt {3-2\,x}\,\sqrt {-149046503977+\sqrt {7}\,12577271771{}\mathrm {i}}\,1572158971375{}\mathrm {i}}{391663056253676053933850624\,\left (-\frac {230036728532618625}{27975932589548289566703616}+\frac {\sqrt {7}\,181960107187971125{}\mathrm {i}}{195831528126838026966925312}\right )}-\frac {1572158971375\,\sqrt {7}\,\sqrt {3-2\,x}\,\sqrt {-149046503977+\sqrt {7}\,12577271771{}\mathrm {i}}}{391663056253676053933850624\,\left (-\frac {230036728532618625}{27975932589548289566703616}+\frac {\sqrt {7}\,181960107187971125{}\mathrm {i}}{195831528126838026966925312}\right )}\right )\,\sqrt {-149046503977+\sqrt {7}\,12577271771{}\mathrm {i}}\,5{}\mathrm {i}}{3373232128}-\frac {\mathrm {atan}\left (\frac {\sqrt {3-2\,x}\,\sqrt {-149046503977-\sqrt {7}\,12577271771{}\mathrm {i}}\,1572158971375{}\mathrm {i}}{391663056253676053933850624\,\left (\frac {230036728532618625}{27975932589548289566703616}+\frac {\sqrt {7}\,181960107187971125{}\mathrm {i}}{195831528126838026966925312}\right )}+\frac {1572158971375\,\sqrt {7}\,\sqrt {3-2\,x}\,\sqrt {-149046503977-\sqrt {7}\,12577271771{}\mathrm {i}}}{391663056253676053933850624\,\left (\frac {230036728532618625}{27975932589548289566703616}+\frac {\sqrt {7}\,181960107187971125{}\mathrm {i}}{195831528126838026966925312}\right )}\right )\,\sqrt {-149046503977-\sqrt {7}\,12577271771{}\mathrm {i}}\,5{}\mathrm {i}}{3373232128} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

(atan(((3 - 2*x)^(1/2)*(7^(1/2)*12577271771i - 149046503977)^(1/2)*1572158971375i)/(39166305625367605393385062
4*((7^(1/2)*181960107187971125i)/195831528126838026966925312 - 230036728532618625/27975932589548289566703616))
- (1572158971375*7^(1/2)*(3 - 2*x)^(1/2)*(7^(1/2)*12577271771i - 149046503977)^(1/2))/(3916630562536760539338
50624*((7^(1/2)*181960107187971125i)/195831528126838026966925312 - 230036728532618625/279759325895482895667036
16)))*(7^(1/2)*12577271771i - 149046503977)^(1/2)*5i)/3373232128 - ((272*x)/441 - (164*(2*x - 3)^2)/441 + (196
6*(2*x - 3)^3)/3087 - (9091*(2*x - 3)^4)/3087 - (32070727*(2*x - 3)^5)/5531904 - (41014777*(2*x - 3)^6)/110638
08 - (141921511*(2*x - 3)^7)/154893312 + (23262655*(2*x - 3)^8)/309786624 + (1571659*(2*x - 3)^9)/15059072 + (
468427*(2*x - 3)^10)/17210368 + (394105*(2*x - 3)^11)/120472576 + (38225*(2*x - 3)^12)/240945152 - 520/441)/(3
8416*(3 - 2*x)^(9/2) - 76832*(3 - 2*x)^(11/2) + 68600*(3 - 2*x)^(13/2) - 35672*(3 - 2*x)^(15/2) + 11809*(3 - 2
*x)^(17/2) - 2548*(3 - 2*x)^(19/2) + 350*(3 - 2*x)^(21/2) - 28*(3 - 2*x)^(23/2) + (3 - 2*x)^(25/2)) - (atan(((
3 - 2*x)^(1/2)*(- 7^(1/2)*12577271771i - 149046503977)^(1/2)*1572158971375i)/(391663056253676053933850624*((7^
(1/2)*181960107187971125i)/195831528126838026966925312 + 230036728532618625/27975932589548289566703616)) + (15
72158971375*7^(1/2)*(3 - 2*x)^(1/2)*(- 7^(1/2)*12577271771i - 149046503977)^(1/2))/(39166305625367605393385062
4*((7^(1/2)*181960107187971125i)/195831528126838026966925312 + 230036728532618625/27975932589548289566703616))
)*(- 7^(1/2)*12577271771i - 149046503977)^(1/2)*5i)/3373232128

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sympy [F(-1)]  time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]

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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