\(\int \frac {(c+d \tan (e+f x))^{5/2} (A+B \tan (e+f x)+C \tan ^2(e+f x))}{(a+b \tan (e+f x))^{5/2}} \, dx\) [144]

Optimal result

Integrand size = 49, antiderivative size = 545 \[ \int \frac {(c+d \tan (e+f x))^{5/2} \left (A+B \tan (e+f x)+C \tan ^2(e+f x)\right )}{(a+b \tan (e+f x))^{5/2}} \, dx=-\frac {(i A+B-i C) (c-i d)^{5/2} \text {arctanh}\left (\frac {\sqrt {c-i d} \sqrt {a+b \tan (e+f x)}}{\sqrt {a-i b} \sqrt {c+d \tan (e+f x)}}\right )}{(a-i b)^{5/2} f}-\frac {(B-i (A-C)) (c+i d)^{5/2} \text {arctanh}\left (\frac {\sqrt {c+i d} \sqrt {a+b \tan (e+f x)}}{\sqrt {a+i b} \sqrt {c+d \tan (e+f x)}}\right )}{(a+i b)^{5/2} f}+\frac {d^{3/2} (5 b c C+2 b B d-5 a C d) \text {arctanh}\left (\frac {\sqrt {d} \sqrt {a+b \tan (e+f x)}}{\sqrt {b} \sqrt {c+d \tan (e+f x)}}\right )}{b^{7/2} f}-\frac {d \left (2 a^3 b B d-5 a^4 C d-2 a b^3 (2 A c-2 c C-3 B d)+2 a^2 b^2 (B c-5 C d)-b^4 (2 B c+(4 A+C) d)\right ) \sqrt {a+b \tan (e+f x)} \sqrt {c+d \tan (e+f x)}}{b^3 \left (a^2+b^2\right )^2 f}+\frac {2 \left (2 a^3 b B d-5 a^4 C d-b^4 (3 B c+5 A d)-2 a b^3 (3 A c-3 c C-4 B d)+a^2 b^2 (3 B c+(A-11 C) d)\right ) (c+d \tan (e+f x))^{3/2}}{3 b^2 \left (a^2+b^2\right )^2 f \sqrt {a+b \tan (e+f x)}}-\frac {2 \left (A b^2-a (b B-a C)\right ) (c+d \tan (e+f x))^{5/2}}{3 b \left (a^2+b^2\right ) f (a+b \tan (e+f x))^{3/2}} \]

[Out]

-(I*A+B-I*C)*(c-I*d)^(5/2)*arctanh((c-I*d)^(1/2)*(a+b*tan(f*x+e))^(1/2)/(a-I*b)^(1/2)/(c+d*tan(f*x+e))^(1/2))/
(a-I*b)^(5/2)/f-(B-I*(A-C))*(c+I*d)^(5/2)*arctanh((c+I*d)^(1/2)*(a+b*tan(f*x+e))^(1/2)/(a+I*b)^(1/2)/(c+d*tan(
f*x+e))^(1/2))/(a+I*b)^(5/2)/f+d^(3/2)*(2*B*b*d-5*C*a*d+5*C*b*c)*arctanh(d^(1/2)*(a+b*tan(f*x+e))^(1/2)/b^(1/2
)/(c+d*tan(f*x+e))^(1/2))/b^(7/2)/f-d*(2*a^3*b*B*d-5*a^4*C*d-2*a*b^3*(2*A*c-3*B*d-2*C*c)+2*a^2*b^2*(B*c-5*C*d)
-b^4*(2*B*c+(4*A+C)*d))*(a+b*tan(f*x+e))^(1/2)*(c+d*tan(f*x+e))^(1/2)/b^3/(a^2+b^2)^2/f+2/3*(2*a^3*b*B*d-5*a^4
*C*d-b^4*(5*A*d+3*B*c)-2*a*b^3*(3*A*c-4*B*d-3*C*c)+a^2*b^2*(3*B*c+(A-11*C)*d))*(c+d*tan(f*x+e))^(3/2)/b^2/(a^2
+b^2)^2/f/(a+b*tan(f*x+e))^(1/2)-2/3*(A*b^2-a*(B*b-C*a))*(c+d*tan(f*x+e))^(5/2)/b/(a^2+b^2)/f/(a+b*tan(f*x+e))
^(3/2)

Rubi [A] (verified)

Time = 12.75 (sec) , antiderivative size = 545, normalized size of antiderivative = 1.00, number of steps used = 15, number of rules used = 9, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.184, Rules used = {3726, 3728, 3736, 6857, 65, 223, 212, 95, 214} \[ \int \frac {(c+d \tan (e+f x))^{5/2} \left (A+B \tan (e+f x)+C \tan ^2(e+f x)\right )}{(a+b \tan (e+f x))^{5/2}} \, dx=-\frac {2 \left (A b^2-a (b B-a C)\right ) (c+d \tan (e+f x))^{5/2}}{3 b f \left (a^2+b^2\right ) (a+b \tan (e+f x))^{3/2}}+\frac {2 (c+d \tan (e+f x))^{3/2} \left (-5 a^4 C d+2 a^3 b B d+a^2 b^2 (d (A-11 C)+3 B c)-2 a b^3 (3 A c-4 B d-3 c C)-b^4 (5 A d+3 B c)\right )}{3 b^2 f \left (a^2+b^2\right )^2 \sqrt {a+b \tan (e+f x)}}-\frac {d \sqrt {a+b \tan (e+f x)} \sqrt {c+d \tan (e+f x)} \left (-5 a^4 C d+2 a^3 b B d+2 a^2 b^2 (B c-5 C d)-2 a b^3 (2 A c-3 B d-2 c C)-b^4 (d (4 A+C)+2 B c)\right )}{b^3 f \left (a^2+b^2\right )^2}-\frac {(c-i d)^{5/2} (i A+B-i C) \text {arctanh}\left (\frac {\sqrt {c-i d} \sqrt {a+b \tan (e+f x)}}{\sqrt {a-i b} \sqrt {c+d \tan (e+f x)}}\right )}{f (a-i b)^{5/2}}-\frac {(c+i d)^{5/2} (B-i (A-C)) \text {arctanh}\left (\frac {\sqrt {c+i d} \sqrt {a+b \tan (e+f x)}}{\sqrt {a+i b} \sqrt {c+d \tan (e+f x)}}\right )}{f (a+i b)^{5/2}}+\frac {d^{3/2} (-5 a C d+2 b B d+5 b c C) \text {arctanh}\left (\frac {\sqrt {d} \sqrt {a+b \tan (e+f x)}}{\sqrt {b} \sqrt {c+d \tan (e+f x)}}\right )}{b^{7/2} f} \]

[In]

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

[Out]

-(((I*A + B - I*C)*(c - I*d)^(5/2)*ArcTanh[(Sqrt[c - I*d]*Sqrt[a + b*Tan[e + f*x]])/(Sqrt[a - I*b]*Sqrt[c + d*
Tan[e + f*x]])])/((a - I*b)^(5/2)*f)) - ((B - I*(A - C))*(c + I*d)^(5/2)*ArcTanh[(Sqrt[c + I*d]*Sqrt[a + b*Tan
[e + f*x]])/(Sqrt[a + I*b]*Sqrt[c + d*Tan[e + f*x]])])/((a + I*b)^(5/2)*f) + (d^(3/2)*(5*b*c*C + 2*b*B*d - 5*a
*C*d)*ArcTanh[(Sqrt[d]*Sqrt[a + b*Tan[e + f*x]])/(Sqrt[b]*Sqrt[c + d*Tan[e + f*x]])])/(b^(7/2)*f) - (d*(2*a^3*
b*B*d - 5*a^4*C*d - 2*a*b^3*(2*A*c - 2*c*C - 3*B*d) + 2*a^2*b^2*(B*c - 5*C*d) - b^4*(2*B*c + (4*A + C)*d))*Sqr
t[a + b*Tan[e + f*x]]*Sqrt[c + d*Tan[e + f*x]])/(b^3*(a^2 + b^2)^2*f) + (2*(2*a^3*b*B*d - 5*a^4*C*d - b^4*(3*B
*c + 5*A*d) - 2*a*b^3*(3*A*c - 3*c*C - 4*B*d) + a^2*b^2*(3*B*c + (A - 11*C)*d))*(c + d*Tan[e + f*x])^(3/2))/(3
*b^2*(a^2 + b^2)^2*f*Sqrt[a + b*Tan[e + f*x]]) - (2*(A*b^2 - a*(b*B - a*C))*(c + d*Tan[e + f*x])^(5/2))/(3*b*(
a^2 + b^2)*f*(a + b*Tan[e + f*x])^(3/2))

Rule 65

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> With[{p = Denominator[m]}, Dist[p/b, Sub
st[Int[x^(p*(m + 1) - 1)*(c - a*(d/b) + d*(x^p/b))^n, x], x, (a + b*x)^(1/p)], x]] /; FreeQ[{a, b, c, d}, x] &
& NeQ[b*c - a*d, 0] && LtQ[-1, m, 0] && LeQ[-1, n, 0] && LeQ[Denominator[n], Denominator[m]] && IntLinearQ[a,
b, c, d, m, n, x]

Rule 95

Int[(((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_))/((e_.) + (f_.)*(x_)), x_Symbol] :> With[{q = Denomin
ator[m]}, Dist[q, Subst[Int[x^(q*(m + 1) - 1)/(b*e - a*f - (d*e - c*f)*x^q), x], x, (a + b*x)^(1/q)/(c + d*x)^
(1/q)], x]] /; FreeQ[{a, b, c, d, e, f}, x] && EqQ[m + n + 1, 0] && RationalQ[n] && LtQ[-1, m, 0] && SimplerQ[
a + b*x, c + d*x]

Rule 212

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1/(Rt[a, 2]*Rt[-b, 2]))*ArcTanh[Rt[-b, 2]*(x/Rt[a, 2])], x]
/; FreeQ[{a, b}, x] && NegQ[a/b] && (GtQ[a, 0] || LtQ[b, 0])

Rule 214

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(Rt[-a/b, 2]/a)*ArcTanh[x/Rt[-a/b, 2]], x] /; FreeQ[{a, b},
x] && NegQ[a/b]

Rule 223

Int[1/Sqrt[(a_) + (b_.)*(x_)^2], x_Symbol] :> Subst[Int[1/(1 - b*x^2), x], x, x/Sqrt[a + b*x^2]] /; FreeQ[{a,
b}, x] &&  !GtQ[a, 0]

Rule 3726

Int[((a_.) + (b_.)*tan[(e_.) + (f_.)*(x_)])^(m_)*((c_.) + (d_.)*tan[(e_.) + (f_.)*(x_)])^(n_)*((A_.) + (B_.)*t
an[(e_.) + (f_.)*(x_)] + (C_.)*tan[(e_.) + (f_.)*(x_)]^2), x_Symbol] :> Simp[(A*d^2 + c*(c*C - B*d))*(a + b*Ta
n[e + f*x])^m*((c + d*Tan[e + f*x])^(n + 1)/(d*f*(n + 1)*(c^2 + d^2))), x] - Dist[1/(d*(n + 1)*(c^2 + d^2)), I
nt[(a + b*Tan[e + f*x])^(m - 1)*(c + d*Tan[e + f*x])^(n + 1)*Simp[A*d*(b*d*m - a*c*(n + 1)) + (c*C - B*d)*(b*c
*m + a*d*(n + 1)) - d*(n + 1)*((A - C)*(b*c - a*d) + B*(a*c + b*d))*Tan[e + f*x] - b*(d*(B*c - A*d)*(m + n + 1
) - C*(c^2*m - d^2*(n + 1)))*Tan[e + f*x]^2, x], x], x] /; FreeQ[{a, b, c, d, e, f, A, B, C}, x] && NeQ[b*c -
a*d, 0] && NeQ[a^2 + b^2, 0] && NeQ[c^2 + d^2, 0] && GtQ[m, 0] && LtQ[n, -1]

Rule 3728

Int[((a_.) + (b_.)*tan[(e_.) + (f_.)*(x_)])^(m_.)*((c_.) + (d_.)*tan[(e_.) + (f_.)*(x_)])^(n_)*((A_.) + (B_.)*
tan[(e_.) + (f_.)*(x_)] + (C_.)*tan[(e_.) + (f_.)*(x_)]^2), x_Symbol] :> Simp[C*(a + b*Tan[e + f*x])^m*((c + d
*Tan[e + f*x])^(n + 1)/(d*f*(m + n + 1))), x] + Dist[1/(d*(m + n + 1)), Int[(a + b*Tan[e + f*x])^(m - 1)*(c +
d*Tan[e + f*x])^n*Simp[a*A*d*(m + n + 1) - C*(b*c*m + a*d*(n + 1)) + d*(A*b + a*B - b*C)*(m + n + 1)*Tan[e + f
*x] - (C*m*(b*c - a*d) - b*B*d*(m + n + 1))*Tan[e + f*x]^2, x], x], x] /; FreeQ[{a, b, c, d, e, f, A, B, C, n}
, x] && NeQ[b*c - a*d, 0] && NeQ[a^2 + b^2, 0] && NeQ[c^2 + d^2, 0] && GtQ[m, 0] &&  !(IGtQ[n, 0] && ( !Intege
rQ[m] || (EqQ[c, 0] && NeQ[a, 0])))

Rule 3736

Int[((a_.) + (b_.)*tan[(e_.) + (f_.)*(x_)])^(m_)*((c_.) + (d_.)*tan[(e_.) + (f_.)*(x_)])^(n_)*((A_.) + (B_.)*t
an[(e_.) + (f_.)*(x_)] + (C_.)*tan[(e_.) + (f_.)*(x_)]^2), x_Symbol] :> With[{ff = FreeFactors[Tan[e + f*x], x
]}, Dist[ff/f, Subst[Int[(a + b*ff*x)^m*(c + d*ff*x)^n*((A + B*ff*x + C*ff^2*x^2)/(1 + ff^2*x^2)), x], x, Tan[
e + f*x]/ff], x]] /; FreeQ[{a, b, c, d, e, f, A, B, C, m, n}, x] && NeQ[b*c - a*d, 0] && NeQ[a^2 + b^2, 0] &&
NeQ[c^2 + d^2, 0]

Rule 6857

Int[(u_)/((a_) + (b_.)*(x_)^(n_)), x_Symbol] :> With[{v = RationalFunctionExpand[u/(a + b*x^n), x]}, Int[v, x]
/; SumQ[v]] /; FreeQ[{a, b}, x] && IGtQ[n, 0]

Rubi steps \begin{align*} \text {integral}& = -\frac {2 \left (A b^2-a (b B-a C)\right ) (c+d \tan (e+f x))^{5/2}}{3 b \left (a^2+b^2\right ) f (a+b \tan (e+f x))^{3/2}}+\frac {2 \int \frac {(c+d \tan (e+f x))^{3/2} \left (\frac {1}{2} ((b B-a C) (3 b c-5 a d)+A b (3 a c+5 b d))-\frac {3}{2} b ((A-C) (b c-a d)-B (a c+b d)) \tan (e+f x)+\frac {1}{2} \left (2 A b^2-2 a b B+5 a^2 C+3 b^2 C\right ) d \tan ^2(e+f x)\right )}{(a+b \tan (e+f x))^{3/2}} \, dx}{3 b \left (a^2+b^2\right )} \\ & = \frac {2 \left (2 a^3 b B d-5 a^4 C d-b^4 (3 B c+5 A d)-2 a b^3 (3 A c-3 c C-4 B d)+a^2 b^2 (3 B c+(A-11 C) d)\right ) (c+d \tan (e+f x))^{3/2}}{3 b^2 \left (a^2+b^2\right )^2 f \sqrt {a+b \tan (e+f x)}}-\frac {2 \left (A b^2-a (b B-a C)\right ) (c+d \tan (e+f x))^{5/2}}{3 b \left (a^2+b^2\right ) f (a+b \tan (e+f x))^{3/2}}+\frac {4 \int \frac {\sqrt {c+d \tan (e+f x)} \left (\frac {1}{4} \left (b (a c+3 b d) ((b B-a C) (3 b c-5 a d)+A b (3 a c+5 b d))+(b c-3 a d) \left (2 a^2 b B d-5 a^3 C d-A b^2 (3 b c-a d)+3 b^3 (c C+B d)+3 a b^2 (B c-2 C d)\right )\right )+\frac {3}{4} b^2 \left (2 a b \left (c^2 C+2 B c d-C d^2-A \left (c^2-d^2\right )\right )+a^2 \left (2 c (A-C) d+B \left (c^2-d^2\right )\right )-b^2 \left (2 c (A-C) d+B \left (c^2-d^2\right )\right )\right ) \tan (e+f x)-\frac {3}{4} d \left (2 a^3 b B d-5 a^4 C d-2 a b^3 (2 A c-2 c C-3 B d)+2 a^2 b^2 (B c-5 C d)-b^4 (2 B c+(4 A+C) d)\right ) \tan ^2(e+f x)\right )}{\sqrt {a+b \tan (e+f x)}} \, dx}{3 b^2 \left (a^2+b^2\right )^2} \\ & = -\frac {d \left (2 a^3 b B d-5 a^4 C d-2 a b^3 (2 A c-2 c C-3 B d)+2 a^2 b^2 (B c-5 C d)-b^4 (2 B c+(4 A+C) d)\right ) \sqrt {a+b \tan (e+f x)} \sqrt {c+d \tan (e+f x)}}{b^3 \left (a^2+b^2\right )^2 f}+\frac {2 \left (2 a^3 b B d-5 a^4 C d-b^4 (3 B c+5 A d)-2 a b^3 (3 A c-3 c C-4 B d)+a^2 b^2 (3 B c+(A-11 C) d)\right ) (c+d \tan (e+f x))^{3/2}}{3 b^2 \left (a^2+b^2\right )^2 f \sqrt {a+b \tan (e+f x)}}-\frac {2 \left (A b^2-a (b B-a C)\right ) (c+d \tan (e+f x))^{5/2}}{3 b \left (a^2+b^2\right ) f (a+b \tan (e+f x))^{3/2}}+\frac {4 \int \frac {-\frac {3}{8} \left (5 a^5 C d^3+10 a^3 b^2 C d^3-a^4 b d^2 (5 c C+2 B d)-2 a^2 b^3 \left (A c^3-c^3 C-3 B c^2 d-3 A c d^2+8 c C d^2+3 B d^3\right )-b^5 c \left (2 c^2 C+6 B c d-C d^2-2 A \left (c^2-3 d^2\right )\right )-a b^4 \left (4 A d \left (3 c^2-d^2\right )-C d \left (12 c^2+d^2\right )+4 B \left (c^3-3 c d^2\right )\right )\right )+\frac {3}{4} b^3 \left (2 a b \left (c^3 C+3 B c^2 d-3 c C d^2-B d^3-A \left (c^3-3 c d^2\right )\right )+a^2 \left ((A-C) d \left (3 c^2-d^2\right )+B \left (c^3-3 c d^2\right )\right )-b^2 \left ((A-C) d \left (3 c^2-d^2\right )+B \left (c^3-3 c d^2\right )\right )\right ) \tan (e+f x)+\frac {3}{8} \left (a^2+b^2\right )^2 d^2 (5 b c C+2 b B d-5 a C d) \tan ^2(e+f x)}{\sqrt {a+b \tan (e+f x)} \sqrt {c+d \tan (e+f x)}} \, dx}{3 b^3 \left (a^2+b^2\right )^2} \\ & = -\frac {d \left (2 a^3 b B d-5 a^4 C d-2 a b^3 (2 A c-2 c C-3 B d)+2 a^2 b^2 (B c-5 C d)-b^4 (2 B c+(4 A+C) d)\right ) \sqrt {a+b \tan (e+f x)} \sqrt {c+d \tan (e+f x)}}{b^3 \left (a^2+b^2\right )^2 f}+\frac {2 \left (2 a^3 b B d-5 a^4 C d-b^4 (3 B c+5 A d)-2 a b^3 (3 A c-3 c C-4 B d)+a^2 b^2 (3 B c+(A-11 C) d)\right ) (c+d \tan (e+f x))^{3/2}}{3 b^2 \left (a^2+b^2\right )^2 f \sqrt {a+b \tan (e+f x)}}-\frac {2 \left (A b^2-a (b B-a C)\right ) (c+d \tan (e+f x))^{5/2}}{3 b \left (a^2+b^2\right ) f (a+b \tan (e+f x))^{3/2}}+\frac {4 \text {Subst}\left (\int \frac {-\frac {3}{8} \left (5 a^5 C d^3+10 a^3 b^2 C d^3-a^4 b d^2 (5 c C+2 B d)-2 a^2 b^3 \left (A c^3-c^3 C-3 B c^2 d-3 A c d^2+8 c C d^2+3 B d^3\right )-b^5 c \left (2 c^2 C+6 B c d-C d^2-2 A \left (c^2-3 d^2\right )\right )-a b^4 \left (4 A d \left (3 c^2-d^2\right )-C d \left (12 c^2+d^2\right )+4 B \left (c^3-3 c d^2\right )\right )\right )+\frac {3}{4} b^3 \left (2 a b \left (c^3 C+3 B c^2 d-3 c C d^2-B d^3-A \left (c^3-3 c d^2\right )\right )+a^2 \left ((A-C) d \left (3 c^2-d^2\right )+B \left (c^3-3 c d^2\right )\right )-b^2 \left ((A-C) d \left (3 c^2-d^2\right )+B \left (c^3-3 c d^2\right )\right )\right ) x+\frac {3}{8} \left (a^2+b^2\right )^2 d^2 (5 b c C+2 b B d-5 a C d) x^2}{\sqrt {a+b x} \sqrt {c+d x} \left (1+x^2\right )} \, dx,x,\tan (e+f x)\right )}{3 b^3 \left (a^2+b^2\right )^2 f} \\ & = -\frac {d \left (2 a^3 b B d-5 a^4 C d-2 a b^3 (2 A c-2 c C-3 B d)+2 a^2 b^2 (B c-5 C d)-b^4 (2 B c+(4 A+C) d)\right ) \sqrt {a+b \tan (e+f x)} \sqrt {c+d \tan (e+f x)}}{b^3 \left (a^2+b^2\right )^2 f}+\frac {2 \left (2 a^3 b B d-5 a^4 C d-b^4 (3 B c+5 A d)-2 a b^3 (3 A c-3 c C-4 B d)+a^2 b^2 (3 B c+(A-11 C) d)\right ) (c+d \tan (e+f x))^{3/2}}{3 b^2 \left (a^2+b^2\right )^2 f \sqrt {a+b \tan (e+f x)}}-\frac {2 \left (A b^2-a (b B-a C)\right ) (c+d \tan (e+f x))^{5/2}}{3 b \left (a^2+b^2\right ) f (a+b \tan (e+f x))^{3/2}}+\frac {4 \text {Subst}\left (\int \left (\frac {3 \left (a^2+b^2\right )^2 d^2 (5 b c C+2 b B d-5 a C d)}{8 \sqrt {a+b x} \sqrt {c+d x}}+\frac {3 \left (-b^3 \left (b^2 \left (A c^3-c^3 C-3 B c^2 d-3 A c d^2+3 c C d^2+B d^3\right )+a^2 \left (c^3 C+3 B c^2 d-3 c C d^2-B d^3-A \left (c^3-3 c d^2\right )\right )-2 a b \left ((A-C) d \left (3 c^2-d^2\right )+B \left (c^3-3 c d^2\right )\right )\right )-b^3 \left (2 a b \left (A c^3-c^3 C-3 B c^2 d-3 A c d^2+3 c C d^2+B d^3\right )-a^2 \left ((A-C) d \left (3 c^2-d^2\right )+B \left (c^3-3 c d^2\right )\right )+b^2 \left ((A-C) d \left (3 c^2-d^2\right )+B \left (c^3-3 c d^2\right )\right )\right ) x\right )}{4 \sqrt {a+b x} \sqrt {c+d x} \left (1+x^2\right )}\right ) \, dx,x,\tan (e+f x)\right )}{3 b^3 \left (a^2+b^2\right )^2 f} \\ & = -\frac {d \left (2 a^3 b B d-5 a^4 C d-2 a b^3 (2 A c-2 c C-3 B d)+2 a^2 b^2 (B c-5 C d)-b^4 (2 B c+(4 A+C) d)\right ) \sqrt {a+b \tan (e+f x)} \sqrt {c+d \tan (e+f x)}}{b^3 \left (a^2+b^2\right )^2 f}+\frac {2 \left (2 a^3 b B d-5 a^4 C d-b^4 (3 B c+5 A d)-2 a b^3 (3 A c-3 c C-4 B d)+a^2 b^2 (3 B c+(A-11 C) d)\right ) (c+d \tan (e+f x))^{3/2}}{3 b^2 \left (a^2+b^2\right )^2 f \sqrt {a+b \tan (e+f x)}}-\frac {2 \left (A b^2-a (b B-a C)\right ) (c+d \tan (e+f x))^{5/2}}{3 b \left (a^2+b^2\right ) f (a+b \tan (e+f x))^{3/2}}+\frac {\text {Subst}\left (\int \frac {-b^3 \left (b^2 \left (A c^3-c^3 C-3 B c^2 d-3 A c d^2+3 c C d^2+B d^3\right )+a^2 \left (c^3 C+3 B c^2 d-3 c C d^2-B d^3-A \left (c^3-3 c d^2\right )\right )-2 a b \left ((A-C) d \left (3 c^2-d^2\right )+B \left (c^3-3 c d^2\right )\right )\right )-b^3 \left (2 a b \left (A c^3-c^3 C-3 B c^2 d-3 A c d^2+3 c C d^2+B d^3\right )-a^2 \left ((A-C) d \left (3 c^2-d^2\right )+B \left (c^3-3 c d^2\right )\right )+b^2 \left ((A-C) d \left (3 c^2-d^2\right )+B \left (c^3-3 c d^2\right )\right )\right ) x}{\sqrt {a+b x} \sqrt {c+d x} \left (1+x^2\right )} \, dx,x,\tan (e+f x)\right )}{b^3 \left (a^2+b^2\right )^2 f}+\frac {\left (d^2 (5 b c C+2 b B d-5 a C d)\right ) \text {Subst}\left (\int \frac {1}{\sqrt {a+b x} \sqrt {c+d x}} \, dx,x,\tan (e+f x)\right )}{2 b^3 f} \\ & = -\frac {d \left (2 a^3 b B d-5 a^4 C d-2 a b^3 (2 A c-2 c C-3 B d)+2 a^2 b^2 (B c-5 C d)-b^4 (2 B c+(4 A+C) d)\right ) \sqrt {a+b \tan (e+f x)} \sqrt {c+d \tan (e+f x)}}{b^3 \left (a^2+b^2\right )^2 f}+\frac {2 \left (2 a^3 b B d-5 a^4 C d-b^4 (3 B c+5 A d)-2 a b^3 (3 A c-3 c C-4 B d)+a^2 b^2 (3 B c+(A-11 C) d)\right ) (c+d \tan (e+f x))^{3/2}}{3 b^2 \left (a^2+b^2\right )^2 f \sqrt {a+b \tan (e+f x)}}-\frac {2 \left (A b^2-a (b B-a C)\right ) (c+d \tan (e+f x))^{5/2}}{3 b \left (a^2+b^2\right ) f (a+b \tan (e+f x))^{3/2}}+\frac {\text {Subst}\left (\int \left (\frac {-i b^3 \left (b^2 \left (A c^3-c^3 C-3 B c^2 d-3 A c d^2+3 c C d^2+B d^3\right )+a^2 \left (c^3 C+3 B c^2 d-3 c C d^2-B d^3-A \left (c^3-3 c d^2\right )\right )-2 a b \left ((A-C) d \left (3 c^2-d^2\right )+B \left (c^3-3 c d^2\right )\right )\right )+b^3 \left (2 a b \left (A c^3-c^3 C-3 B c^2 d-3 A c d^2+3 c C d^2+B d^3\right )-a^2 \left ((A-C) d \left (3 c^2-d^2\right )+B \left (c^3-3 c d^2\right )\right )+b^2 \left ((A-C) d \left (3 c^2-d^2\right )+B \left (c^3-3 c d^2\right )\right )\right )}{2 (i-x) \sqrt {a+b x} \sqrt {c+d x}}+\frac {-i b^3 \left (b^2 \left (A c^3-c^3 C-3 B c^2 d-3 A c d^2+3 c C d^2+B d^3\right )+a^2 \left (c^3 C+3 B c^2 d-3 c C d^2-B d^3-A \left (c^3-3 c d^2\right )\right )-2 a b \left ((A-C) d \left (3 c^2-d^2\right )+B \left (c^3-3 c d^2\right )\right )\right )-b^3 \left (2 a b \left (A c^3-c^3 C-3 B c^2 d-3 A c d^2+3 c C d^2+B d^3\right )-a^2 \left ((A-C) d \left (3 c^2-d^2\right )+B \left (c^3-3 c d^2\right )\right )+b^2 \left ((A-C) d \left (3 c^2-d^2\right )+B \left (c^3-3 c d^2\right )\right )\right )}{2 (i+x) \sqrt {a+b x} \sqrt {c+d x}}\right ) \, dx,x,\tan (e+f x)\right )}{b^3 \left (a^2+b^2\right )^2 f}+\frac {\left (d^2 (5 b c C+2 b B d-5 a C d)\right ) \text {Subst}\left (\int \frac {1}{\sqrt {c-\frac {a d}{b}+\frac {d x^2}{b}}} \, dx,x,\sqrt {a+b \tan (e+f x)}\right )}{b^4 f} \\ & = -\frac {d \left (2 a^3 b B d-5 a^4 C d-2 a b^3 (2 A c-2 c C-3 B d)+2 a^2 b^2 (B c-5 C d)-b^4 (2 B c+(4 A+C) d)\right ) \sqrt {a+b \tan (e+f x)} \sqrt {c+d \tan (e+f x)}}{b^3 \left (a^2+b^2\right )^2 f}+\frac {2 \left (2 a^3 b B d-5 a^4 C d-b^4 (3 B c+5 A d)-2 a b^3 (3 A c-3 c C-4 B d)+a^2 b^2 (3 B c+(A-11 C) d)\right ) (c+d \tan (e+f x))^{3/2}}{3 b^2 \left (a^2+b^2\right )^2 f \sqrt {a+b \tan (e+f x)}}-\frac {2 \left (A b^2-a (b B-a C)\right ) (c+d \tan (e+f x))^{5/2}}{3 b \left (a^2+b^2\right ) f (a+b \tan (e+f x))^{3/2}}+\frac {\left ((i A+B-i C) (c-i d)^3\right ) \text {Subst}\left (\int \frac {1}{(i+x) \sqrt {a+b x} \sqrt {c+d x}} \, dx,x,\tan (e+f x)\right )}{2 (a-i b)^2 f}+\frac {\left (d^2 (5 b c C+2 b B d-5 a C d)\right ) \text {Subst}\left (\int \frac {1}{1-\frac {d x^2}{b}} \, dx,x,\frac {\sqrt {a+b \tan (e+f x)}}{\sqrt {c+d \tan (e+f x)}}\right )}{b^4 f}+\frac {\left (-i b^3 \left (b^2 \left (A c^3-c^3 C-3 B c^2 d-3 A c d^2+3 c C d^2+B d^3\right )+a^2 \left (c^3 C+3 B c^2 d-3 c C d^2-B d^3-A \left (c^3-3 c d^2\right )\right )-2 a b \left ((A-C) d \left (3 c^2-d^2\right )+B \left (c^3-3 c d^2\right )\right )\right )+b^3 \left (2 a b \left (A c^3-c^3 C-3 B c^2 d-3 A c d^2+3 c C d^2+B d^3\right )-a^2 \left ((A-C) d \left (3 c^2-d^2\right )+B \left (c^3-3 c d^2\right )\right )+b^2 \left ((A-C) d \left (3 c^2-d^2\right )+B \left (c^3-3 c d^2\right )\right )\right )\right ) \text {Subst}\left (\int \frac {1}{(i-x) \sqrt {a+b x} \sqrt {c+d x}} \, dx,x,\tan (e+f x)\right )}{2 b^3 \left (a^2+b^2\right )^2 f} \\ & = \frac {d^{3/2} (5 b c C+2 b B d-5 a C d) \text {arctanh}\left (\frac {\sqrt {d} \sqrt {a+b \tan (e+f x)}}{\sqrt {b} \sqrt {c+d \tan (e+f x)}}\right )}{b^{7/2} f}-\frac {d \left (2 a^3 b B d-5 a^4 C d-2 a b^3 (2 A c-2 c C-3 B d)+2 a^2 b^2 (B c-5 C d)-b^4 (2 B c+(4 A+C) d)\right ) \sqrt {a+b \tan (e+f x)} \sqrt {c+d \tan (e+f x)}}{b^3 \left (a^2+b^2\right )^2 f}+\frac {2 \left (2 a^3 b B d-5 a^4 C d-b^4 (3 B c+5 A d)-2 a b^3 (3 A c-3 c C-4 B d)+a^2 b^2 (3 B c+(A-11 C) d)\right ) (c+d \tan (e+f x))^{3/2}}{3 b^2 \left (a^2+b^2\right )^2 f \sqrt {a+b \tan (e+f x)}}-\frac {2 \left (A b^2-a (b B-a C)\right ) (c+d \tan (e+f x))^{5/2}}{3 b \left (a^2+b^2\right ) f (a+b \tan (e+f x))^{3/2}}+\frac {\left ((i A+B-i C) (c-i d)^3\right ) \text {Subst}\left (\int \frac {1}{-a+i b-(-c+i d) x^2} \, dx,x,\frac {\sqrt {a+b \tan (e+f x)}}{\sqrt {c+d \tan (e+f x)}}\right )}{(a-i b)^2 f}+\frac {\left (-i b^3 \left (b^2 \left (A c^3-c^3 C-3 B c^2 d-3 A c d^2+3 c C d^2+B d^3\right )+a^2 \left (c^3 C+3 B c^2 d-3 c C d^2-B d^3-A \left (c^3-3 c d^2\right )\right )-2 a b \left ((A-C) d \left (3 c^2-d^2\right )+B \left (c^3-3 c d^2\right )\right )\right )+b^3 \left (2 a b \left (A c^3-c^3 C-3 B c^2 d-3 A c d^2+3 c C d^2+B d^3\right )-a^2 \left ((A-C) d \left (3 c^2-d^2\right )+B \left (c^3-3 c d^2\right )\right )+b^2 \left ((A-C) d \left (3 c^2-d^2\right )+B \left (c^3-3 c d^2\right )\right )\right )\right ) \text {Subst}\left (\int \frac {1}{a+i b-(c+i d) x^2} \, dx,x,\frac {\sqrt {a+b \tan (e+f x)}}{\sqrt {c+d \tan (e+f x)}}\right )}{b^3 \left (a^2+b^2\right )^2 f} \\ & = -\frac {(i A+B-i C) (c-i d)^{5/2} \text {arctanh}\left (\frac {\sqrt {c-i d} \sqrt {a+b \tan (e+f x)}}{\sqrt {a-i b} \sqrt {c+d \tan (e+f x)}}\right )}{(a-i b)^{5/2} f}-\frac {(B-i (A-C)) (c+i d)^{5/2} \text {arctanh}\left (\frac {\sqrt {c+i d} \sqrt {a+b \tan (e+f x)}}{\sqrt {a+i b} \sqrt {c+d \tan (e+f x)}}\right )}{(a+i b)^{5/2} f}+\frac {d^{3/2} (5 b c C+2 b B d-5 a C d) \text {arctanh}\left (\frac {\sqrt {d} \sqrt {a+b \tan (e+f x)}}{\sqrt {b} \sqrt {c+d \tan (e+f x)}}\right )}{b^{7/2} f}-\frac {d \left (2 a^3 b B d-5 a^4 C d-2 a b^3 (2 A c-2 c C-3 B d)+2 a^2 b^2 (B c-5 C d)-b^4 (2 B c+(4 A+C) d)\right ) \sqrt {a+b \tan (e+f x)} \sqrt {c+d \tan (e+f x)}}{b^3 \left (a^2+b^2\right )^2 f}+\frac {2 \left (2 a^3 b B d-5 a^4 C d-b^4 (3 B c+5 A d)-2 a b^3 (3 A c-3 c C-4 B d)+a^2 b^2 (3 B c+(A-11 C) d)\right ) (c+d \tan (e+f x))^{3/2}}{3 b^2 \left (a^2+b^2\right )^2 f \sqrt {a+b \tan (e+f x)}}-\frac {2 \left (A b^2-a (b B-a C)\right ) (c+d \tan (e+f x))^{5/2}}{3 b \left (a^2+b^2\right ) f (a+b \tan (e+f x))^{3/2}} \\ \end{align*}

Mathematica [C] (verified)

Result contains higher order function than in optimal. Order 5 vs. order 3 in optimal.

Time = 7.63 (sec) , antiderivative size = 802, normalized size of antiderivative = 1.47 \[ \int \frac {(c+d \tan (e+f x))^{5/2} \left (A+B \tan (e+f x)+C \tan ^2(e+f x)\right )}{(a+b \tan (e+f x))^{5/2}} \, dx=\frac {C (c+d \tan (e+f x))^{5/2}}{b f (a+b \tan (e+f x))^{3/2}}+\frac {-\frac {2 b (A-i B-C) (c-i d) (c+d \tan (e+f x))^{3/2}}{3 (i a+b) f (a+b \tan (e+f x))^{3/2}}+\frac {2 b (A+i B-C) (c+i d) (c+d \tan (e+f x))^{3/2}}{3 (i a-b) f (a+b \tan (e+f x))^{3/2}}-\frac {10 c C (b c-a d) \operatorname {Hypergeometric2F1}\left (-\frac {3}{2},-\frac {3}{2},-\frac {1}{2},-\frac {d (a+b \tan (e+f x))}{b c-a d}\right ) \sqrt {c+d \tan (e+f x)}}{3 b f (a+b \tan (e+f x))^{3/2} \sqrt {\frac {b (c+d \tan (e+f x))}{b c-a d}}}-\frac {4 B d (b c-a d) \operatorname {Hypergeometric2F1}\left (-\frac {3}{2},-\frac {3}{2},-\frac {1}{2},-\frac {d (a+b \tan (e+f x))}{b c-a d}\right ) \sqrt {c+d \tan (e+f x)}}{3 b f (a+b \tan (e+f x))^{3/2} \sqrt {\frac {b (c+d \tan (e+f x))}{b c-a d}}}+\frac {10 a C d (b c-a d) \operatorname {Hypergeometric2F1}\left (-\frac {3}{2},-\frac {3}{2},-\frac {1}{2},-\frac {d (a+b \tan (e+f x))}{b c-a d}\right ) \sqrt {c+d \tan (e+f x)}}{3 b^2 f (a+b \tan (e+f x))^{3/2} \sqrt {\frac {b (c+d \tan (e+f x))}{b c-a d}}}+\frac {2 b (i A+B-i C) (c-i d)^2 \left (\frac {\sqrt {-c+i d} \text {arctanh}\left (\frac {\sqrt {-c+i d} \sqrt {a+b \tan (e+f x)}}{\sqrt {-a+i b} \sqrt {c+d \tan (e+f x)}}\right )}{(-a+i b)^{3/2}}+\frac {\sqrt {c+d \tan (e+f x)}}{(a-i b) \sqrt {a+b \tan (e+f x)}}\right )}{(a-i b) f}-\frac {2 b (A+i B-C) (c+i d)^2 \left (\frac {\sqrt {c+i d} \text {arctanh}\left (\frac {\sqrt {c+i d} \sqrt {a+b \tan (e+f x)}}{\sqrt {a+i b} \sqrt {c+d \tan (e+f x)}}\right )}{(a+i b)^{3/2}}-\frac {\sqrt {c+d \tan (e+f x)}}{(a+i b) \sqrt {a+b \tan (e+f x)}}\right )}{(i a-b) f}}{2 b} \]

[In]

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

[Out]

(C*(c + d*Tan[e + f*x])^(5/2))/(b*f*(a + b*Tan[e + f*x])^(3/2)) + ((-2*b*(A - I*B - C)*(c - I*d)*(c + d*Tan[e
+ f*x])^(3/2))/(3*(I*a + b)*f*(a + b*Tan[e + f*x])^(3/2)) + (2*b*(A + I*B - C)*(c + I*d)*(c + d*Tan[e + f*x])^
(3/2))/(3*(I*a - b)*f*(a + b*Tan[e + f*x])^(3/2)) - (10*c*C*(b*c - a*d)*Hypergeometric2F1[-3/2, -3/2, -1/2, -(
(d*(a + b*Tan[e + f*x]))/(b*c - a*d))]*Sqrt[c + d*Tan[e + f*x]])/(3*b*f*(a + b*Tan[e + f*x])^(3/2)*Sqrt[(b*(c
+ d*Tan[e + f*x]))/(b*c - a*d)]) - (4*B*d*(b*c - a*d)*Hypergeometric2F1[-3/2, -3/2, -1/2, -((d*(a + b*Tan[e +
f*x]))/(b*c - a*d))]*Sqrt[c + d*Tan[e + f*x]])/(3*b*f*(a + b*Tan[e + f*x])^(3/2)*Sqrt[(b*(c + d*Tan[e + f*x]))
/(b*c - a*d)]) + (10*a*C*d*(b*c - a*d)*Hypergeometric2F1[-3/2, -3/2, -1/2, -((d*(a + b*Tan[e + f*x]))/(b*c - a
*d))]*Sqrt[c + d*Tan[e + f*x]])/(3*b^2*f*(a + b*Tan[e + f*x])^(3/2)*Sqrt[(b*(c + d*Tan[e + f*x]))/(b*c - a*d)]
) + (2*b*(I*A + B - I*C)*(c - I*d)^2*((Sqrt[-c + I*d]*ArcTanh[(Sqrt[-c + I*d]*Sqrt[a + b*Tan[e + f*x]])/(Sqrt[
-a + I*b]*Sqrt[c + d*Tan[e + f*x]])])/(-a + I*b)^(3/2) + Sqrt[c + d*Tan[e + f*x]]/((a - I*b)*Sqrt[a + b*Tan[e
+ f*x]])))/((a - I*b)*f) - (2*b*(A + I*B - C)*(c + I*d)^2*((Sqrt[c + I*d]*ArcTanh[(Sqrt[c + I*d]*Sqrt[a + b*Ta
n[e + f*x]])/(Sqrt[a + I*b]*Sqrt[c + d*Tan[e + f*x]])])/(a + I*b)^(3/2) - Sqrt[c + d*Tan[e + f*x]]/((a + I*b)*
Sqrt[a + b*Tan[e + f*x]])))/((I*a - b)*f))/(2*b)

Maple [F(-1)]

Timed out.

\[\int \frac {\left (c +d \tan \left (f x +e \right )\right )^{\frac {5}{2}} \left (A +B \tan \left (f x +e \right )+C \tan \left (f x +e \right )^{2}\right )}{\left (a +b \tan \left (f x +e \right )\right )^{\frac {5}{2}}}d x\]

[In]

int((c+d*tan(f*x+e))^(5/2)*(A+B*tan(f*x+e)+C*tan(f*x+e)^2)/(a+b*tan(f*x+e))^(5/2),x)

[Out]

int((c+d*tan(f*x+e))^(5/2)*(A+B*tan(f*x+e)+C*tan(f*x+e)^2)/(a+b*tan(f*x+e))^(5/2),x)

Fricas [F(-1)]

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

[In]

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

[Out]

Timed out

Sympy [F]

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

[In]

integrate((c+d*tan(f*x+e))**(5/2)*(A+B*tan(f*x+e)+C*tan(f*x+e)**2)/(a+b*tan(f*x+e))**(5/2),x)

[Out]

Integral((c + d*tan(e + f*x))**(5/2)*(A + B*tan(e + f*x) + C*tan(e + f*x)**2)/(a + b*tan(e + f*x))**(5/2), x)

Maxima [F(-1)]

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

[In]

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

[Out]

Timed out

Giac [F(-1)]

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

[In]

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

[Out]

Timed out

Mupad [F(-1)]

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

[In]

int(((c + d*tan(e + f*x))^(5/2)*(A + B*tan(e + f*x) + C*tan(e + f*x)^2))/(a + b*tan(e + f*x))^(5/2),x)

[Out]

\text{Hanged}