\(\int \sqrt {\sec (c+d x)} (a+b \sec (c+d x))^{5/2} (A+B \sec (c+d x)+C \sec ^2(c+d x)) \, dx\) [1045]

Optimal result

Integrand size = 45, antiderivative size = 550 \[ \int \sqrt {\sec (c+d x)} (a+b \sec (c+d x))^{5/2} \left (A+B \sec (c+d x)+C \sec ^2(c+d x)\right ) \, dx=\frac {\left (472 a^2 b B+128 b^3 B+4 a b^2 (132 A+89 C)+a^3 (384 A+133 C)\right ) \sqrt {\frac {b+a \cos (c+d x)}{a+b}} \operatorname {EllipticF}\left (\frac {1}{2} (c+d x),\frac {2 a}{a+b}\right ) \sqrt {\sec (c+d x)}}{192 d \sqrt {a+b \sec (c+d x)}}+\frac {\left (40 a^3 b B+160 a b^3 B-5 a^4 C+120 a^2 b^2 (2 A+C)+16 b^4 (4 A+3 C)\right ) \sqrt {\frac {b+a \cos (c+d x)}{a+b}} \operatorname {EllipticPi}\left (2,\frac {1}{2} (c+d x),\frac {2 a}{a+b}\right ) \sqrt {\sec (c+d x)}}{64 b d \sqrt {a+b \sec (c+d x)}}-\frac {\left (264 a^2 b B+128 b^3 B+15 a^3 C+4 a b^2 (108 A+71 C)\right ) E\left (\frac {1}{2} (c+d x)|\frac {2 a}{a+b}\right ) \sqrt {a+b \sec (c+d x)}}{192 b d \sqrt {\frac {b+a \cos (c+d x)}{a+b}} \sqrt {\sec (c+d x)}}+\frac {\left (264 a^2 b B+128 b^3 B+15 a^3 C+4 a b^2 (108 A+71 C)\right ) \sqrt {\sec (c+d x)} \sqrt {a+b \sec (c+d x)} \sin (c+d x)}{192 b d}+\frac {\left (16 A b^2+24 a b B+5 a^2 C+12 b^2 C\right ) \sec ^{\frac {3}{2}}(c+d x) \sqrt {a+b \sec (c+d x)} \sin (c+d x)}{32 d}+\frac {(8 b B+5 a C) \sec ^{\frac {3}{2}}(c+d x) (a+b \sec (c+d x))^{3/2} \sin (c+d x)}{24 d}+\frac {C \sec ^{\frac {3}{2}}(c+d x) (a+b \sec (c+d x))^{5/2} \sin (c+d x)}{4 d} \]

[Out]

1/24*(8*B*b+5*C*a)*sec(d*x+c)^(3/2)*(a+b*sec(d*x+c))^(3/2)*sin(d*x+c)/d+1/4*C*sec(d*x+c)^(3/2)*(a+b*sec(d*x+c)
)^(5/2)*sin(d*x+c)/d+1/192*(472*B*a^2*b+128*B*b^3+4*a*b^2*(132*A+89*C)+a^3*(384*A+133*C))*(cos(1/2*d*x+1/2*c)^
2)^(1/2)/cos(1/2*d*x+1/2*c)*EllipticF(sin(1/2*d*x+1/2*c),2^(1/2)*(a/(a+b))^(1/2))*((b+a*cos(d*x+c))/(a+b))^(1/
2)*sec(d*x+c)^(1/2)/d/(a+b*sec(d*x+c))^(1/2)+1/64*(40*B*a^3*b+160*B*a*b^3-5*a^4*C+120*a^2*b^2*(2*A+C)+16*b^4*(
4*A+3*C))*(cos(1/2*d*x+1/2*c)^2)^(1/2)/cos(1/2*d*x+1/2*c)*EllipticPi(sin(1/2*d*x+1/2*c),2,2^(1/2)*(a/(a+b))^(1
/2))*((b+a*cos(d*x+c))/(a+b))^(1/2)*sec(d*x+c)^(1/2)/b/d/(a+b*sec(d*x+c))^(1/2)+1/32*(16*A*b^2+24*B*a*b+5*C*a^
2+12*C*b^2)*sec(d*x+c)^(3/2)*sin(d*x+c)*(a+b*sec(d*x+c))^(1/2)/d-1/192*(264*B*a^2*b+128*B*b^3+15*a^3*C+4*a*b^2
*(108*A+71*C))*(cos(1/2*d*x+1/2*c)^2)^(1/2)/cos(1/2*d*x+1/2*c)*EllipticE(sin(1/2*d*x+1/2*c),2^(1/2)*(a/(a+b))^
(1/2))*(a+b*sec(d*x+c))^(1/2)/b/d/((b+a*cos(d*x+c))/(a+b))^(1/2)/sec(d*x+c)^(1/2)+1/192*(264*B*a^2*b+128*B*b^3
+15*a^3*C+4*a*b^2*(108*A+71*C))*sin(d*x+c)*sec(d*x+c)^(1/2)*(a+b*sec(d*x+c))^(1/2)/b/d

Rubi [A] (verified)

Time = 2.44 (sec) , antiderivative size = 550, normalized size of antiderivative = 1.00, number of steps used = 15, number of rules used = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.289, Rules used = {4181, 4187, 4193, 3944, 2886, 2884, 4120, 3941, 2734, 2732, 3943, 2742, 2740} \[ \int \sqrt {\sec (c+d x)} (a+b \sec (c+d x))^{5/2} \left (A+B \sec (c+d x)+C \sec ^2(c+d x)\right ) \, dx=\frac {\sin (c+d x) \sec ^{\frac {3}{2}}(c+d x) \left (5 a^2 C+24 a b B+16 A b^2+12 b^2 C\right ) \sqrt {a+b \sec (c+d x)}}{32 d}+\frac {\sin (c+d x) \sqrt {\sec (c+d x)} \left (15 a^3 C+264 a^2 b B+4 a b^2 (108 A+71 C)+128 b^3 B\right ) \sqrt {a+b \sec (c+d x)}}{192 b d}+\frac {\sqrt {\sec (c+d x)} \left (a^3 (384 A+133 C)+472 a^2 b B+4 a b^2 (132 A+89 C)+128 b^3 B\right ) \sqrt {\frac {a \cos (c+d x)+b}{a+b}} \operatorname {EllipticF}\left (\frac {1}{2} (c+d x),\frac {2 a}{a+b}\right )}{192 d \sqrt {a+b \sec (c+d x)}}-\frac {\left (15 a^3 C+264 a^2 b B+4 a b^2 (108 A+71 C)+128 b^3 B\right ) \sqrt {a+b \sec (c+d x)} E\left (\frac {1}{2} (c+d x)|\frac {2 a}{a+b}\right )}{192 b d \sqrt {\sec (c+d x)} \sqrt {\frac {a \cos (c+d x)+b}{a+b}}}+\frac {\sqrt {\sec (c+d x)} \left (-5 a^4 C+40 a^3 b B+120 a^2 b^2 (2 A+C)+160 a b^3 B+16 b^4 (4 A+3 C)\right ) \sqrt {\frac {a \cos (c+d x)+b}{a+b}} \operatorname {EllipticPi}\left (2,\frac {1}{2} (c+d x),\frac {2 a}{a+b}\right )}{64 b d \sqrt {a+b \sec (c+d x)}}+\frac {(5 a C+8 b B) \sin (c+d x) \sec ^{\frac {3}{2}}(c+d x) (a+b \sec (c+d x))^{3/2}}{24 d}+\frac {C \sin (c+d x) \sec ^{\frac {3}{2}}(c+d x) (a+b \sec (c+d x))^{5/2}}{4 d} \]

[In]

Int[Sqrt[Sec[c + d*x]]*(a + b*Sec[c + d*x])^(5/2)*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2),x]

[Out]

((472*a^2*b*B + 128*b^3*B + 4*a*b^2*(132*A + 89*C) + a^3*(384*A + 133*C))*Sqrt[(b + a*Cos[c + d*x])/(a + b)]*E
llipticF[(c + d*x)/2, (2*a)/(a + b)]*Sqrt[Sec[c + d*x]])/(192*d*Sqrt[a + b*Sec[c + d*x]]) + ((40*a^3*b*B + 160
*a*b^3*B - 5*a^4*C + 120*a^2*b^2*(2*A + C) + 16*b^4*(4*A + 3*C))*Sqrt[(b + a*Cos[c + d*x])/(a + b)]*EllipticPi
[2, (c + d*x)/2, (2*a)/(a + b)]*Sqrt[Sec[c + d*x]])/(64*b*d*Sqrt[a + b*Sec[c + d*x]]) - ((264*a^2*b*B + 128*b^
3*B + 15*a^3*C + 4*a*b^2*(108*A + 71*C))*EllipticE[(c + d*x)/2, (2*a)/(a + b)]*Sqrt[a + b*Sec[c + d*x]])/(192*
b*d*Sqrt[(b + a*Cos[c + d*x])/(a + b)]*Sqrt[Sec[c + d*x]]) + ((264*a^2*b*B + 128*b^3*B + 15*a^3*C + 4*a*b^2*(1
08*A + 71*C))*Sqrt[Sec[c + d*x]]*Sqrt[a + b*Sec[c + d*x]]*Sin[c + d*x])/(192*b*d) + ((16*A*b^2 + 24*a*b*B + 5*
a^2*C + 12*b^2*C)*Sec[c + d*x]^(3/2)*Sqrt[a + b*Sec[c + d*x]]*Sin[c + d*x])/(32*d) + ((8*b*B + 5*a*C)*Sec[c +
d*x]^(3/2)*(a + b*Sec[c + d*x])^(3/2)*Sin[c + d*x])/(24*d) + (C*Sec[c + d*x]^(3/2)*(a + b*Sec[c + d*x])^(5/2)*
Sin[c + d*x])/(4*d)

Rule 2732

Int[Sqrt[(a_) + (b_.)*sin[(c_.) + (d_.)*(x_)]], x_Symbol] :> Simp[2*(Sqrt[a + b]/d)*EllipticE[(1/2)*(c - Pi/2
+ d*x), 2*(b/(a + b))], x] /; FreeQ[{a, b, c, d}, x] && NeQ[a^2 - b^2, 0] && GtQ[a + b, 0]

Rule 2734

Int[Sqrt[(a_) + (b_.)*sin[(c_.) + (d_.)*(x_)]], x_Symbol] :> Dist[Sqrt[a + b*Sin[c + d*x]]/Sqrt[(a + b*Sin[c +
d*x])/(a + b)], Int[Sqrt[a/(a + b) + (b/(a + b))*Sin[c + d*x]], x], x] /; FreeQ[{a, b, c, d}, x] && NeQ[a^2 -
b^2, 0] &&  !GtQ[a + b, 0]

Rule 2740

Int[1/Sqrt[(a_) + (b_.)*sin[(c_.) + (d_.)*(x_)]], x_Symbol] :> Simp[(2/(d*Sqrt[a + b]))*EllipticF[(1/2)*(c - P
i/2 + d*x), 2*(b/(a + b))], x] /; FreeQ[{a, b, c, d}, x] && NeQ[a^2 - b^2, 0] && GtQ[a + b, 0]

Rule 2742

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

Rule 2884

Int[1/(((a_.) + (b_.)*sin[(e_.) + (f_.)*(x_)])*Sqrt[(c_.) + (d_.)*sin[(e_.) + (f_.)*(x_)]]), x_Symbol] :> Simp
[(2/(f*(a + b)*Sqrt[c + d]))*EllipticPi[2*(b/(a + b)), (1/2)*(e - Pi/2 + f*x), 2*(d/(c + d))], x] /; FreeQ[{a,
b, c, d, e, f}, x] && NeQ[b*c - a*d, 0] && NeQ[a^2 - b^2, 0] && NeQ[c^2 - d^2, 0] && GtQ[c + d, 0]

Rule 2886

Int[1/(((a_.) + (b_.)*sin[(e_.) + (f_.)*(x_)])*Sqrt[(c_.) + (d_.)*sin[(e_.) + (f_.)*(x_)]]), x_Symbol] :> Dist
[Sqrt[(c + d*Sin[e + f*x])/(c + d)]/Sqrt[c + d*Sin[e + f*x]], Int[1/((a + b*Sin[e + f*x])*Sqrt[c/(c + d) + (d/
(c + d))*Sin[e + f*x]]), x], x] /; FreeQ[{a, b, c, d, e, f}, x] && NeQ[b*c - a*d, 0] && NeQ[a^2 - b^2, 0] && N
eQ[c^2 - d^2, 0] &&  !GtQ[c + d, 0]

Rule 3941

Int[Sqrt[csc[(e_.) + (f_.)*(x_)]*(b_.) + (a_)]/Sqrt[csc[(e_.) + (f_.)*(x_)]*(d_.)], x_Symbol] :> Dist[Sqrt[a +
b*Csc[e + f*x]]/(Sqrt[d*Csc[e + f*x]]*Sqrt[b + a*Sin[e + f*x]]), Int[Sqrt[b + a*Sin[e + f*x]], x], x] /; Free
Q[{a, b, d, e, f}, x] && NeQ[a^2 - b^2, 0]

Rule 3943

Int[Sqrt[csc[(e_.) + (f_.)*(x_)]*(d_.)]/Sqrt[csc[(e_.) + (f_.)*(x_)]*(b_.) + (a_)], x_Symbol] :> Dist[Sqrt[d*C
sc[e + f*x]]*(Sqrt[b + a*Sin[e + f*x]]/Sqrt[a + b*Csc[e + f*x]]), Int[1/Sqrt[b + a*Sin[e + f*x]], x], x] /; Fr
eeQ[{a, b, d, e, f}, x] && NeQ[a^2 - b^2, 0]

Rule 3944

Int[(csc[(e_.) + (f_.)*(x_)]*(d_.))^(3/2)/Sqrt[csc[(e_.) + (f_.)*(x_)]*(b_.) + (a_)], x_Symbol] :> Dist[d*Sqrt
[d*Csc[e + f*x]]*(Sqrt[b + a*Sin[e + f*x]]/Sqrt[a + b*Csc[e + f*x]]), Int[1/(Sin[e + f*x]*Sqrt[b + a*Sin[e + f
*x]]), x], x] /; FreeQ[{a, b, d, e, f}, x] && NeQ[a^2 - b^2, 0]

Rule 4120

Int[(csc[(e_.) + (f_.)*(x_)]*(B_.) + (A_))/(Sqrt[csc[(e_.) + (f_.)*(x_)]*(d_.)]*Sqrt[csc[(e_.) + (f_.)*(x_)]*(
b_.) + (a_)]), x_Symbol] :> Dist[A/a, Int[Sqrt[a + b*Csc[e + f*x]]/Sqrt[d*Csc[e + f*x]], x], x] - Dist[(A*b -
a*B)/(a*d), Int[Sqrt[d*Csc[e + f*x]]/Sqrt[a + b*Csc[e + f*x]], x], x] /; FreeQ[{a, b, d, e, f, A, B}, x] && Ne
Q[A*b - a*B, 0] && NeQ[a^2 - b^2, 0]

Rule 4181

Int[((A_.) + csc[(e_.) + (f_.)*(x_)]*(B_.) + csc[(e_.) + (f_.)*(x_)]^2*(C_.))*(csc[(e_.) + (f_.)*(x_)]*(d_.))^
(n_)*(csc[(e_.) + (f_.)*(x_)]*(b_.) + (a_))^(m_), x_Symbol] :> Simp[(-C)*Cot[e + f*x]*(a + b*Csc[e + f*x])^m*(
(d*Csc[e + f*x])^n/(f*(m + n + 1))), x] + Dist[1/(m + n + 1), Int[(a + b*Csc[e + f*x])^(m - 1)*(d*Csc[e + f*x]
)^n*Simp[a*A*(m + n + 1) + a*C*n + ((A*b + a*B)*(m + n + 1) + b*C*(m + n))*Csc[e + f*x] + (b*B*(m + n + 1) + a
*C*m)*Csc[e + f*x]^2, x], x], x] /; FreeQ[{a, b, d, e, f, A, B, C, n}, x] && NeQ[a^2 - b^2, 0] && GtQ[m, 0] &&
!LeQ[n, -1]

Rule 4187

Int[((A_.) + csc[(e_.) + (f_.)*(x_)]*(B_.) + csc[(e_.) + (f_.)*(x_)]^2*(C_.))*(csc[(e_.) + (f_.)*(x_)]*(d_.))^
(n_)*(csc[(e_.) + (f_.)*(x_)]*(b_.) + (a_))^(m_), x_Symbol] :> Simp[(-C)*d*Cot[e + f*x]*(a + b*Csc[e + f*x])^(
m + 1)*((d*Csc[e + f*x])^(n - 1)/(b*f*(m + n + 1))), x] + Dist[d/(b*(m + n + 1)), Int[(a + b*Csc[e + f*x])^m*(
d*Csc[e + f*x])^(n - 1)*Simp[a*C*(n - 1) + (A*b*(m + n + 1) + b*C*(m + n))*Csc[e + f*x] + (b*B*(m + n + 1) - a
*C*n)*Csc[e + f*x]^2, x], x], x] /; FreeQ[{a, b, d, e, f, A, B, C, m}, x] && NeQ[a^2 - b^2, 0] && GtQ[n, 0]

Rule 4193

Int[((A_.) + csc[(e_.) + (f_.)*(x_)]*(B_.) + csc[(e_.) + (f_.)*(x_)]^2*(C_.))/(Sqrt[csc[(e_.) + (f_.)*(x_)]*(d
_.)]*Sqrt[csc[(e_.) + (f_.)*(x_)]*(b_.) + (a_)]), x_Symbol] :> Dist[C/d^2, Int[(d*Csc[e + f*x])^(3/2)/Sqrt[a +
b*Csc[e + f*x]], x], x] + Int[(A + B*Csc[e + f*x])/(Sqrt[d*Csc[e + f*x]]*Sqrt[a + b*Csc[e + f*x]]), x] /; Fre
eQ[{a, b, d, e, f, A, B, C}, x] && NeQ[a^2 - b^2, 0]

Rubi steps \begin{align*} \text {integral}& = \frac {C \sec ^{\frac {3}{2}}(c+d x) (a+b \sec (c+d x))^{5/2} \sin (c+d x)}{4 d}+\frac {1}{4} \int \sqrt {\sec (c+d x)} (a+b \sec (c+d x))^{3/2} \left (\frac {1}{2} a (8 A+C)+(4 A b+4 a B+3 b C) \sec (c+d x)+\frac {1}{2} (8 b B+5 a C) \sec ^2(c+d x)\right ) \, dx \\ & = \frac {(8 b B+5 a C) \sec ^{\frac {3}{2}}(c+d x) (a+b \sec (c+d x))^{3/2} \sin (c+d x)}{24 d}+\frac {C \sec ^{\frac {3}{2}}(c+d x) (a+b \sec (c+d x))^{5/2} \sin (c+d x)}{4 d}+\frac {1}{12} \int \sqrt {\sec (c+d x)} \sqrt {a+b \sec (c+d x)} \left (\frac {1}{4} a (48 a A+8 b B+11 a C)+\frac {1}{2} \left (24 a^2 B+16 b^2 B+a b (48 A+31 C)\right ) \sec (c+d x)+\frac {3}{4} \left (16 A b^2+24 a b B+5 a^2 C+12 b^2 C\right ) \sec ^2(c+d x)\right ) \, dx \\ & = \frac {\left (16 A b^2+24 a b B+5 a^2 C+12 b^2 C\right ) \sec ^{\frac {3}{2}}(c+d x) \sqrt {a+b \sec (c+d x)} \sin (c+d x)}{32 d}+\frac {(8 b B+5 a C) \sec ^{\frac {3}{2}}(c+d x) (a+b \sec (c+d x))^{3/2} \sin (c+d x)}{24 d}+\frac {C \sec ^{\frac {3}{2}}(c+d x) (a+b \sec (c+d x))^{5/2} \sin (c+d x)}{4 d}+\frac {1}{24} \int \frac {\sqrt {\sec (c+d x)} \left (\frac {1}{8} a \left (192 a^2 A+48 A b^2+104 a b B+59 a^2 C+36 b^2 C\right )+\frac {1}{4} \left (96 a^3 B+152 a b^2 B+12 b^3 (4 A+3 C)+a^2 b (288 A+161 C)\right ) \sec (c+d x)+\frac {1}{8} \left (264 a^2 b B+128 b^3 B+15 a^3 C+4 a b^2 (108 A+71 C)\right ) \sec ^2(c+d x)\right )}{\sqrt {a+b \sec (c+d x)}} \, dx \\ & = \frac {\left (264 a^2 b B+128 b^3 B+15 a^3 C+4 a b^2 (108 A+71 C)\right ) \sqrt {\sec (c+d x)} \sqrt {a+b \sec (c+d x)} \sin (c+d x)}{192 b d}+\frac {\left (16 A b^2+24 a b B+5 a^2 C+12 b^2 C\right ) \sec ^{\frac {3}{2}}(c+d x) \sqrt {a+b \sec (c+d x)} \sin (c+d x)}{32 d}+\frac {(8 b B+5 a C) \sec ^{\frac {3}{2}}(c+d x) (a+b \sec (c+d x))^{3/2} \sin (c+d x)}{24 d}+\frac {C \sec ^{\frac {3}{2}}(c+d x) (a+b \sec (c+d x))^{5/2} \sin (c+d x)}{4 d}+\frac {\int \frac {-\frac {1}{16} a \left (264 a^2 b B+128 b^3 B+15 a^3 C+4 a b^2 (108 A+71 C)\right )+\frac {1}{8} a b \left (104 a b B+12 b^2 (4 A+3 C)+a^2 (192 A+59 C)\right ) \sec (c+d x)+\frac {3}{16} \left (40 a^3 b B+160 a b^3 B-5 a^4 C+120 a^2 b^2 (2 A+C)+16 b^4 (4 A+3 C)\right ) \sec ^2(c+d x)}{\sqrt {\sec (c+d x)} \sqrt {a+b \sec (c+d x)}} \, dx}{24 b} \\ & = \frac {\left (264 a^2 b B+128 b^3 B+15 a^3 C+4 a b^2 (108 A+71 C)\right ) \sqrt {\sec (c+d x)} \sqrt {a+b \sec (c+d x)} \sin (c+d x)}{192 b d}+\frac {\left (16 A b^2+24 a b B+5 a^2 C+12 b^2 C\right ) \sec ^{\frac {3}{2}}(c+d x) \sqrt {a+b \sec (c+d x)} \sin (c+d x)}{32 d}+\frac {(8 b B+5 a C) \sec ^{\frac {3}{2}}(c+d x) (a+b \sec (c+d x))^{3/2} \sin (c+d x)}{24 d}+\frac {C \sec ^{\frac {3}{2}}(c+d x) (a+b \sec (c+d x))^{5/2} \sin (c+d x)}{4 d}+\frac {\int \frac {-\frac {1}{16} a \left (264 a^2 b B+128 b^3 B+15 a^3 C+4 a b^2 (108 A+71 C)\right )+\frac {1}{8} a b \left (104 a b B+12 b^2 (4 A+3 C)+a^2 (192 A+59 C)\right ) \sec (c+d x)}{\sqrt {\sec (c+d x)} \sqrt {a+b \sec (c+d x)}} \, dx}{24 b}+\frac {\left (40 a^3 b B+160 a b^3 B-5 a^4 C+120 a^2 b^2 (2 A+C)+16 b^4 (4 A+3 C)\right ) \int \frac {\sec ^{\frac {3}{2}}(c+d x)}{\sqrt {a+b \sec (c+d x)}} \, dx}{128 b} \\ & = \frac {\left (264 a^2 b B+128 b^3 B+15 a^3 C+4 a b^2 (108 A+71 C)\right ) \sqrt {\sec (c+d x)} \sqrt {a+b \sec (c+d x)} \sin (c+d x)}{192 b d}+\frac {\left (16 A b^2+24 a b B+5 a^2 C+12 b^2 C\right ) \sec ^{\frac {3}{2}}(c+d x) \sqrt {a+b \sec (c+d x)} \sin (c+d x)}{32 d}+\frac {(8 b B+5 a C) \sec ^{\frac {3}{2}}(c+d x) (a+b \sec (c+d x))^{3/2} \sin (c+d x)}{24 d}+\frac {C \sec ^{\frac {3}{2}}(c+d x) (a+b \sec (c+d x))^{5/2} \sin (c+d x)}{4 d}-\frac {\left (264 a^2 b B+128 b^3 B+15 a^3 C+4 a b^2 (108 A+71 C)\right ) \int \frac {\sqrt {a+b \sec (c+d x)}}{\sqrt {\sec (c+d x)}} \, dx}{384 b}+\frac {1}{384} \left (472 a^2 b B+128 b^3 B+4 a b^2 (132 A+89 C)+a^3 (384 A+133 C)\right ) \int \frac {\sqrt {\sec (c+d x)}}{\sqrt {a+b \sec (c+d x)}} \, dx+\frac {\left (\left (40 a^3 b B+160 a b^3 B-5 a^4 C+120 a^2 b^2 (2 A+C)+16 b^4 (4 A+3 C)\right ) \sqrt {b+a \cos (c+d x)} \sqrt {\sec (c+d x)}\right ) \int \frac {\sec (c+d x)}{\sqrt {b+a \cos (c+d x)}} \, dx}{128 b \sqrt {a+b \sec (c+d x)}} \\ & = \frac {\left (264 a^2 b B+128 b^3 B+15 a^3 C+4 a b^2 (108 A+71 C)\right ) \sqrt {\sec (c+d x)} \sqrt {a+b \sec (c+d x)} \sin (c+d x)}{192 b d}+\frac {\left (16 A b^2+24 a b B+5 a^2 C+12 b^2 C\right ) \sec ^{\frac {3}{2}}(c+d x) \sqrt {a+b \sec (c+d x)} \sin (c+d x)}{32 d}+\frac {(8 b B+5 a C) \sec ^{\frac {3}{2}}(c+d x) (a+b \sec (c+d x))^{3/2} \sin (c+d x)}{24 d}+\frac {C \sec ^{\frac {3}{2}}(c+d x) (a+b \sec (c+d x))^{5/2} \sin (c+d x)}{4 d}+\frac {\left (\left (472 a^2 b B+128 b^3 B+4 a b^2 (132 A+89 C)+a^3 (384 A+133 C)\right ) \sqrt {b+a \cos (c+d x)} \sqrt {\sec (c+d x)}\right ) \int \frac {1}{\sqrt {b+a \cos (c+d x)}} \, dx}{384 \sqrt {a+b \sec (c+d x)}}+\frac {\left (\left (40 a^3 b B+160 a b^3 B-5 a^4 C+120 a^2 b^2 (2 A+C)+16 b^4 (4 A+3 C)\right ) \sqrt {\frac {b+a \cos (c+d x)}{a+b}} \sqrt {\sec (c+d x)}\right ) \int \frac {\sec (c+d x)}{\sqrt {\frac {b}{a+b}+\frac {a \cos (c+d x)}{a+b}}} \, dx}{128 b \sqrt {a+b \sec (c+d x)}}-\frac {\left (\left (264 a^2 b B+128 b^3 B+15 a^3 C+4 a b^2 (108 A+71 C)\right ) \sqrt {a+b \sec (c+d x)}\right ) \int \sqrt {b+a \cos (c+d x)} \, dx}{384 b \sqrt {b+a \cos (c+d x)} \sqrt {\sec (c+d x)}} \\ & = \frac {\left (40 a^3 b B+160 a b^3 B-5 a^4 C+120 a^2 b^2 (2 A+C)+16 b^4 (4 A+3 C)\right ) \sqrt {\frac {b+a \cos (c+d x)}{a+b}} \operatorname {EllipticPi}\left (2,\frac {1}{2} (c+d x),\frac {2 a}{a+b}\right ) \sqrt {\sec (c+d x)}}{64 b d \sqrt {a+b \sec (c+d x)}}+\frac {\left (264 a^2 b B+128 b^3 B+15 a^3 C+4 a b^2 (108 A+71 C)\right ) \sqrt {\sec (c+d x)} \sqrt {a+b \sec (c+d x)} \sin (c+d x)}{192 b d}+\frac {\left (16 A b^2+24 a b B+5 a^2 C+12 b^2 C\right ) \sec ^{\frac {3}{2}}(c+d x) \sqrt {a+b \sec (c+d x)} \sin (c+d x)}{32 d}+\frac {(8 b B+5 a C) \sec ^{\frac {3}{2}}(c+d x) (a+b \sec (c+d x))^{3/2} \sin (c+d x)}{24 d}+\frac {C \sec ^{\frac {3}{2}}(c+d x) (a+b \sec (c+d x))^{5/2} \sin (c+d x)}{4 d}+\frac {\left (\left (472 a^2 b B+128 b^3 B+4 a b^2 (132 A+89 C)+a^3 (384 A+133 C)\right ) \sqrt {\frac {b+a \cos (c+d x)}{a+b}} \sqrt {\sec (c+d x)}\right ) \int \frac {1}{\sqrt {\frac {b}{a+b}+\frac {a \cos (c+d x)}{a+b}}} \, dx}{384 \sqrt {a+b \sec (c+d x)}}-\frac {\left (\left (264 a^2 b B+128 b^3 B+15 a^3 C+4 a b^2 (108 A+71 C)\right ) \sqrt {a+b \sec (c+d x)}\right ) \int \sqrt {\frac {b}{a+b}+\frac {a \cos (c+d x)}{a+b}} \, dx}{384 b \sqrt {\frac {b+a \cos (c+d x)}{a+b}} \sqrt {\sec (c+d x)}} \\ & = \frac {\left (472 a^2 b B+128 b^3 B+4 a b^2 (132 A+89 C)+a^3 (384 A+133 C)\right ) \sqrt {\frac {b+a \cos (c+d x)}{a+b}} \operatorname {EllipticF}\left (\frac {1}{2} (c+d x),\frac {2 a}{a+b}\right ) \sqrt {\sec (c+d x)}}{192 d \sqrt {a+b \sec (c+d x)}}+\frac {\left (40 a^3 b B+160 a b^3 B-5 a^4 C+120 a^2 b^2 (2 A+C)+16 b^4 (4 A+3 C)\right ) \sqrt {\frac {b+a \cos (c+d x)}{a+b}} \operatorname {EllipticPi}\left (2,\frac {1}{2} (c+d x),\frac {2 a}{a+b}\right ) \sqrt {\sec (c+d x)}}{64 b d \sqrt {a+b \sec (c+d x)}}-\frac {\left (264 a^2 b B+128 b^3 B+15 a^3 C+4 a b^2 (108 A+71 C)\right ) E\left (\frac {1}{2} (c+d x)|\frac {2 a}{a+b}\right ) \sqrt {a+b \sec (c+d x)}}{192 b d \sqrt {\frac {b+a \cos (c+d x)}{a+b}} \sqrt {\sec (c+d x)}}+\frac {\left (264 a^2 b B+128 b^3 B+15 a^3 C+4 a b^2 (108 A+71 C)\right ) \sqrt {\sec (c+d x)} \sqrt {a+b \sec (c+d x)} \sin (c+d x)}{192 b d}+\frac {\left (16 A b^2+24 a b B+5 a^2 C+12 b^2 C\right ) \sec ^{\frac {3}{2}}(c+d x) \sqrt {a+b \sec (c+d x)} \sin (c+d x)}{32 d}+\frac {(8 b B+5 a C) \sec ^{\frac {3}{2}}(c+d x) (a+b \sec (c+d x))^{3/2} \sin (c+d x)}{24 d}+\frac {C \sec ^{\frac {3}{2}}(c+d x) (a+b \sec (c+d x))^{5/2} \sin (c+d x)}{4 d} \\ \end{align*}

Mathematica [C] (warning: unable to verify)

Result contains complex when optimal does not.

Time = 15.83 (sec) , antiderivative size = 925, normalized size of antiderivative = 1.68 \[ \int \sqrt {\sec (c+d x)} (a+b \sec (c+d x))^{5/2} \left (A+B \sec (c+d x)+C \sec ^2(c+d x)\right ) \, dx=-\frac {(a+b \sec (c+d x))^{5/2} \left (A+B \sec (c+d x)+C \sec ^2(c+d x)\right ) \left (\frac {2 \left (-768 a^3 A b-192 a A b^3-416 a^2 b^2 B-236 a^3 b C-144 a b^3 C\right ) \sqrt {\frac {b+a \cos (c+d x)}{a+b}} \operatorname {EllipticF}\left (\frac {1}{2} (c+d x),\frac {2 a}{a+b}\right )}{\sqrt {b+a \cos (c+d x)}}+\frac {2 \left (-1008 a^2 A b^2-384 A b^4+24 a^3 b B-832 a b^3 B+45 a^4 C-436 a^2 b^2 C-288 b^4 C\right ) \sqrt {\frac {b+a \cos (c+d x)}{a+b}} \operatorname {EllipticPi}\left (2,\frac {1}{2} (c+d x),\frac {2 a}{a+b}\right )}{\sqrt {b+a \cos (c+d x)}}+\frac {2 i \left (432 a^2 A b^2+264 a^3 b B+128 a b^3 B+15 a^4 C+284 a^2 b^2 C\right ) \sqrt {\frac {a-a \cos (c+d x)}{a+b}} \sqrt {\frac {a+a \cos (c+d x)}{a-b}} \cos (2 (c+d x)) \left (-2 b (a+b) E\left (i \text {arcsinh}\left (\sqrt {\frac {1}{a-b}} \sqrt {b+a \cos (c+d x)}\right )|\frac {-a+b}{a+b}\right )+a \left (2 b \operatorname {EllipticF}\left (i \text {arcsinh}\left (\sqrt {\frac {1}{a-b}} \sqrt {b+a \cos (c+d x)}\right ),\frac {-a+b}{a+b}\right )+a \operatorname {EllipticPi}\left (1-\frac {a}{b},i \text {arcsinh}\left (\sqrt {\frac {1}{a-b}} \sqrt {b+a \cos (c+d x)}\right ),\frac {-a+b}{a+b}\right )\right )\right ) \sin (c+d x)}{\sqrt {\frac {1}{a-b}} b \sqrt {1-\cos ^2(c+d x)} \sqrt {\frac {a^2-a^2 \cos ^2(c+d x)}{a^2}} \left (-a^2+2 b^2-4 b (b+a \cos (c+d x))+2 (b+a \cos (c+d x))^2\right )}\right )}{384 b d (b+a \cos (c+d x))^{5/2} (A+2 C+2 B \cos (c+d x)+A \cos (2 c+2 d x)) \sec ^{\frac {9}{2}}(c+d x)}+\frac {(a+b \sec (c+d x))^{5/2} \left (A+B \sec (c+d x)+C \sec ^2(c+d x)\right ) \left (\frac {1}{12} \sec ^3(c+d x) \left (8 b^2 B \sin (c+d x)+17 a b C \sin (c+d x)\right )+\frac {1}{48} \sec ^2(c+d x) \left (48 A b^2 \sin (c+d x)+104 a b B \sin (c+d x)+59 a^2 C \sin (c+d x)+36 b^2 C \sin (c+d x)\right )+\frac {\sec (c+d x) \left (432 a A b^2 \sin (c+d x)+264 a^2 b B \sin (c+d x)+128 b^3 B \sin (c+d x)+15 a^3 C \sin (c+d x)+284 a b^2 C \sin (c+d x)\right )}{96 b}+\frac {1}{2} b^2 C \sec ^3(c+d x) \tan (c+d x)\right )}{d (b+a \cos (c+d x))^2 (A+2 C+2 B \cos (c+d x)+A \cos (2 c+2 d x)) \sec ^{\frac {9}{2}}(c+d x)} \]

[In]

Integrate[Sqrt[Sec[c + d*x]]*(a + b*Sec[c + d*x])^(5/2)*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2),x]

[Out]

-1/384*((a + b*Sec[c + d*x])^(5/2)*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2)*((2*(-768*a^3*A*b - 192*a*A*b^3 - 4
16*a^2*b^2*B - 236*a^3*b*C - 144*a*b^3*C)*Sqrt[(b + a*Cos[c + d*x])/(a + b)]*EllipticF[(c + d*x)/2, (2*a)/(a +
b)])/Sqrt[b + a*Cos[c + d*x]] + (2*(-1008*a^2*A*b^2 - 384*A*b^4 + 24*a^3*b*B - 832*a*b^3*B + 45*a^4*C - 436*a
^2*b^2*C - 288*b^4*C)*Sqrt[(b + a*Cos[c + d*x])/(a + b)]*EllipticPi[2, (c + d*x)/2, (2*a)/(a + b)])/Sqrt[b + a
*Cos[c + d*x]] + ((2*I)*(432*a^2*A*b^2 + 264*a^3*b*B + 128*a*b^3*B + 15*a^4*C + 284*a^2*b^2*C)*Sqrt[(a - a*Cos
[c + d*x])/(a + b)]*Sqrt[(a + a*Cos[c + d*x])/(a - b)]*Cos[2*(c + d*x)]*(-2*b*(a + b)*EllipticE[I*ArcSinh[Sqrt
[(a - b)^(-1)]*Sqrt[b + a*Cos[c + d*x]]], (-a + b)/(a + b)] + a*(2*b*EllipticF[I*ArcSinh[Sqrt[(a - b)^(-1)]*Sq
rt[b + a*Cos[c + d*x]]], (-a + b)/(a + b)] + a*EllipticPi[1 - a/b, I*ArcSinh[Sqrt[(a - b)^(-1)]*Sqrt[b + a*Cos
[c + d*x]]], (-a + b)/(a + b)]))*Sin[c + d*x])/(Sqrt[(a - b)^(-1)]*b*Sqrt[1 - Cos[c + d*x]^2]*Sqrt[(a^2 - a^2*
Cos[c + d*x]^2)/a^2]*(-a^2 + 2*b^2 - 4*b*(b + a*Cos[c + d*x]) + 2*(b + a*Cos[c + d*x])^2))))/(b*d*(b + a*Cos[c
+ d*x])^(5/2)*(A + 2*C + 2*B*Cos[c + d*x] + A*Cos[2*c + 2*d*x])*Sec[c + d*x]^(9/2)) + ((a + b*Sec[c + d*x])^(
5/2)*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2)*((Sec[c + d*x]^3*(8*b^2*B*Sin[c + d*x] + 17*a*b*C*Sin[c + d*x]))/
12 + (Sec[c + d*x]^2*(48*A*b^2*Sin[c + d*x] + 104*a*b*B*Sin[c + d*x] + 59*a^2*C*Sin[c + d*x] + 36*b^2*C*Sin[c
+ d*x]))/48 + (Sec[c + d*x]*(432*a*A*b^2*Sin[c + d*x] + 264*a^2*b*B*Sin[c + d*x] + 128*b^3*B*Sin[c + d*x] + 15
*a^3*C*Sin[c + d*x] + 284*a*b^2*C*Sin[c + d*x]))/(96*b) + (b^2*C*Sec[c + d*x]^3*Tan[c + d*x])/2))/(d*(b + a*Co
s[c + d*x])^2*(A + 2*C + 2*B*Cos[c + d*x] + A*Cos[2*c + 2*d*x])*Sec[c + d*x]^(9/2))

Maple [C] (verified)

Result contains complex when optimal does not.

Time = 32.07 (sec) , antiderivative size = 9727, normalized size of antiderivative = 17.69

method	result	size
default	\(\text {Expression too large to display}\)	\(9727\)
parts	\(\text {Expression too large to display}\)	\(9787\)

[In]

int((a+b*sec(d*x+c))^(5/2)*(A+B*sec(d*x+c)+C*sec(d*x+c)^2)*sec(d*x+c)^(1/2),x,method=_RETURNVERBOSE)

[Out]

result too large to display

Fricas [F(-1)]

Timed out. \[ \int \sqrt {\sec (c+d x)} (a+b \sec (c+d x))^{5/2} \left (A+B \sec (c+d x)+C \sec ^2(c+d x)\right ) \, dx=\text {Timed out} \]

[In]

integrate((a+b*sec(d*x+c))^(5/2)*(A+B*sec(d*x+c)+C*sec(d*x+c)^2)*sec(d*x+c)^(1/2),x, algorithm="fricas")

[Out]

Timed out

Sympy [F(-1)]

Timed out. \[ \int \sqrt {\sec (c+d x)} (a+b \sec (c+d x))^{5/2} \left (A+B \sec (c+d x)+C \sec ^2(c+d x)\right ) \, dx=\text {Timed out} \]

[In]

integrate((a+b*sec(d*x+c))**(5/2)*(A+B*sec(d*x+c)+C*sec(d*x+c)**2)*sec(d*x+c)**(1/2),x)

[Out]

Timed out

Maxima [F]

\[ \int \sqrt {\sec (c+d x)} (a+b \sec (c+d x))^{5/2} \left (A+B \sec (c+d x)+C \sec ^2(c+d x)\right ) \, dx=\int { {\left (C \sec \left (d x + c\right )^{2} + B \sec \left (d x + c\right ) + A\right )} {\left (b \sec \left (d x + c\right ) + a\right )}^{\frac {5}{2}} \sqrt {\sec \left (d x + c\right )} \,d x } \]

[In]

integrate((a+b*sec(d*x+c))^(5/2)*(A+B*sec(d*x+c)+C*sec(d*x+c)^2)*sec(d*x+c)^(1/2),x, algorithm="maxima")

[Out]

integrate((C*sec(d*x + c)^2 + B*sec(d*x + c) + A)*(b*sec(d*x + c) + a)^(5/2)*sqrt(sec(d*x + c)), x)

Giac [F]

\[ \int \sqrt {\sec (c+d x)} (a+b \sec (c+d x))^{5/2} \left (A+B \sec (c+d x)+C \sec ^2(c+d x)\right ) \, dx=\int { {\left (C \sec \left (d x + c\right )^{2} + B \sec \left (d x + c\right ) + A\right )} {\left (b \sec \left (d x + c\right ) + a\right )}^{\frac {5}{2}} \sqrt {\sec \left (d x + c\right )} \,d x } \]

[In]

integrate((a+b*sec(d*x+c))^(5/2)*(A+B*sec(d*x+c)+C*sec(d*x+c)^2)*sec(d*x+c)^(1/2),x, algorithm="giac")

[Out]

integrate((C*sec(d*x + c)^2 + B*sec(d*x + c) + A)*(b*sec(d*x + c) + a)^(5/2)*sqrt(sec(d*x + c)), x)

Mupad [F(-1)]

Timed out. \[ \int \sqrt {\sec (c+d x)} (a+b \sec (c+d x))^{5/2} \left (A+B \sec (c+d x)+C \sec ^2(c+d x)\right ) \, dx=\int {\left (a+\frac {b}{\cos \left (c+d\,x\right )}\right )}^{5/2}\,\sqrt {\frac {1}{\cos \left (c+d\,x\right )}}\,\left (A+\frac {B}{\cos \left (c+d\,x\right )}+\frac {C}{{\cos \left (c+d\,x\right )}^2}\right ) \,d x \]

[In]

int((a + b/cos(c + d*x))^(5/2)*(1/cos(c + d*x))^(1/2)*(A + B/cos(c + d*x) + C/cos(c + d*x)^2),x)

[Out]

int((a + b/cos(c + d*x))^(5/2)*(1/cos(c + d*x))^(1/2)*(A + B/cos(c + d*x) + C/cos(c + d*x)^2), x)