\(\int (a+b \tan (e+f x))^{3/2} \sqrt {c+d \tan (e+f x)} (A+B \tan (e+f x)+C \tan ^2(e+f x)) \, dx\) [129]

Optimal result

Integrand size = 49, antiderivative size = 505 \[ \int (a+b \tan (e+f x))^{3/2} \sqrt {c+d \tan (e+f x)} \left (A+B \tan (e+f x)+C \tan ^2(e+f x)\right ) \, dx=-\frac {(a-i b)^{3/2} (i A+B-i C) \sqrt {c-i d} \text {arctanh}\left (\frac {\sqrt {c-i d} \sqrt {a+b \tan (e+f x)}}{\sqrt {a-i b} \sqrt {c+d \tan (e+f x)}}\right )}{f}+\frac {(a+i b)^{3/2} (i A-B-i C) \sqrt {c+i d} \text {arctanh}\left (\frac {\sqrt {c+i d} \sqrt {a+b \tan (e+f x)}}{\sqrt {a+i b} \sqrt {c+d \tan (e+f x)}}\right )}{f}-\frac {\left (a^3 C d^3-3 a^2 b d^2 (c C+2 B d)+3 a b^2 d \left (c^2 C-4 B c d-8 (A-C) d^2\right )-b^3 \left (c^3 C-2 B c^2 d+8 c (A-C) d^2-16 B d^3\right )\right ) \text {arctanh}\left (\frac {\sqrt {d} \sqrt {a+b \tan (e+f x)}}{\sqrt {b} \sqrt {c+d \tan (e+f x)}}\right )}{8 b^{3/2} d^{5/2} f}+\frac {\left (8 b (A b+a B-b C) d^2+(b c-a d) (b c C-2 b B d-a C d)\right ) \sqrt {a+b \tan (e+f x)} \sqrt {c+d \tan (e+f x)}}{8 b d^2 f}-\frac {(b c C-2 b B d-a C d) \sqrt {a+b \tan (e+f x)} (c+d \tan (e+f x))^{3/2}}{4 d^2 f}+\frac {C (a+b \tan (e+f x))^{3/2} (c+d \tan (e+f x))^{3/2}}{3 d f} \]

[Out]

-1/8*(a^3*C*d^3-3*a^2*b*d^2*(2*B*d+C*c)+3*a*b^2*d*(c^2*C-4*B*c*d-8*(A-C)*d^2)-b^3*(c^3*C-2*B*c^2*d+8*c*(A-C)*d
^2-16*B*d^3))*arctanh(d^(1/2)*(a+b*tan(f*x+e))^(1/2)/b^(1/2)/(c+d*tan(f*x+e))^(1/2))/b^(3/2)/d^(5/2)/f-(a-I*b)
^(3/2)*(I*A+B-I*C)*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))*(c-I*d)^
(1/2)/f+(a+I*b)^(3/2)*(I*A-B-I*C)*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))*(c+I*d)^(1/2)/f+1/8*(8*b*(A*b+B*a-C*b)*d^2+(-a*d+b*c)*(-2*B*b*d-C*a*d+C*b*c))*(a+b*tan(f*x+e))^(1/2)*(c
+d*tan(f*x+e))^(1/2)/b/d^2/f-1/4*(-2*B*b*d-C*a*d+C*b*c)*(a+b*tan(f*x+e))^(1/2)*(c+d*tan(f*x+e))^(3/2)/d^2/f+1/
3*C*(a+b*tan(f*x+e))^(3/2)*(c+d*tan(f*x+e))^(3/2)/d/f

Rubi [A] (verified)

Time = 8.63 (sec) , antiderivative size = 505, normalized size of antiderivative = 1.00, number of steps used = 15, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.163, Rules used = {3728, 3736, 6857, 65, 223, 212, 95, 214} \[ \int (a+b \tan (e+f x))^{3/2} \sqrt {c+d \tan (e+f x)} \left (A+B \tan (e+f x)+C \tan ^2(e+f x)\right ) \, dx=-\frac {\left (a^3 C d^3-3 a^2 b d^2 (2 B d+c C)+3 a b^2 d \left (-8 d^2 (A-C)-4 B c d+c^2 C\right )-\left (b^3 \left (8 c d^2 (A-C)-2 B c^2 d-16 B d^3+c^3 C\right )\right )\right ) \text {arctanh}\left (\frac {\sqrt {d} \sqrt {a+b \tan (e+f x)}}{\sqrt {b} \sqrt {c+d \tan (e+f x)}}\right )}{8 b^{3/2} d^{5/2} f}-\frac {(a-i b)^{3/2} \sqrt {c-i d} (i A+B-i C) \text {arctanh}\left (\frac {\sqrt {c-i d} \sqrt {a+b \tan (e+f x)}}{\sqrt {a-i b} \sqrt {c+d \tan (e+f x)}}\right )}{f}+\frac {(a+i b)^{3/2} \sqrt {c+i d} (i A-B-i C) \text {arctanh}\left (\frac {\sqrt {c+i d} \sqrt {a+b \tan (e+f x)}}{\sqrt {a+i b} \sqrt {c+d \tan (e+f x)}}\right )}{f}+\frac {\sqrt {a+b \tan (e+f x)} \sqrt {c+d \tan (e+f x)} \left (8 b d^2 (a B+A b-b C)+(b c-a d) (-a C d-2 b B d+b c C)\right )}{8 b d^2 f}-\frac {(-a C d-2 b B d+b c C) \sqrt {a+b \tan (e+f x)} (c+d \tan (e+f x))^{3/2}}{4 d^2 f}+\frac {C (a+b \tan (e+f x))^{3/2} (c+d \tan (e+f x))^{3/2}}{3 d f} \]

[In]

Int[(a + b*Tan[e + f*x])^(3/2)*Sqrt[c + d*Tan[e + f*x]]*(A + B*Tan[e + f*x] + C*Tan[e + f*x]^2),x]

[Out]

-(((a - I*b)^(3/2)*(I*A + B - I*C)*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]])])/f) + ((a + I*b)^(3/2)*(I*A - B - I*C)*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]])])/f - ((a^3*C*d^3 - 3*a^2*b*d^2*(c*C + 2*B*d) +
3*a*b^2*d*(c^2*C - 4*B*c*d - 8*(A - C)*d^2) - b^3*(c^3*C - 2*B*c^2*d + 8*c*(A - C)*d^2 - 16*B*d^3))*ArcTanh[(
Sqrt[d]*Sqrt[a + b*Tan[e + f*x]])/(Sqrt[b]*Sqrt[c + d*Tan[e + f*x]])])/(8*b^(3/2)*d^(5/2)*f) + ((8*b*(A*b + a*
B - b*C)*d^2 + (b*c - a*d)*(b*c*C - 2*b*B*d - a*C*d))*Sqrt[a + b*Tan[e + f*x]]*Sqrt[c + d*Tan[e + f*x]])/(8*b*
d^2*f) - ((b*c*C - 2*b*B*d - a*C*d)*Sqrt[a + b*Tan[e + f*x]]*(c + d*Tan[e + f*x])^(3/2))/(4*d^2*f) + (C*(a + b
*Tan[e + f*x])^(3/2)*(c + d*Tan[e + f*x])^(3/2))/(3*d*f)

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 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 {C (a+b \tan (e+f x))^{3/2} (c+d \tan (e+f x))^{3/2}}{3 d f}+\frac {\int \sqrt {a+b \tan (e+f x)} \sqrt {c+d \tan (e+f x)} \left (-\frac {3}{2} (b c C-a (2 A-C) d)+3 (A b+a B-b C) d \tan (e+f x)-\frac {3}{2} (b c C-2 b B d-a C d) \tan ^2(e+f x)\right ) \, dx}{3 d} \\ & = -\frac {(b c C-2 b B d-a C d) \sqrt {a+b \tan (e+f x)} (c+d \tan (e+f x))^{3/2}}{4 d^2 f}+\frac {C (a+b \tan (e+f x))^{3/2} (c+d \tan (e+f x))^{3/2}}{3 d f}+\frac {\int \frac {\sqrt {c+d \tan (e+f x)} \left (\frac {3}{4} \left (a^2 (8 A-7 C) d^2+b^2 c (c C-2 B d)-2 a b d (c C+3 B d)\right )+6 \left (a^2 B-b^2 B+2 a b (A-C)\right ) d^2 \tan (e+f x)+\frac {3}{4} \left (8 b (A b+a B-b C) d^2+(b c-a d) (b c C-2 b B d-a C d)\right ) \tan ^2(e+f x)\right )}{\sqrt {a+b \tan (e+f x)}} \, dx}{6 d^2} \\ & = \frac {\left (8 b (A b+a B-b C) d^2+(b c-a d) (b c C-2 b B d-a C d)\right ) \sqrt {a+b \tan (e+f x)} \sqrt {c+d \tan (e+f x)}}{8 b d^2 f}-\frac {(b c C-2 b B d-a C d) \sqrt {a+b \tan (e+f x)} (c+d \tan (e+f x))^{3/2}}{4 d^2 f}+\frac {C (a+b \tan (e+f x))^{3/2} (c+d \tan (e+f x))^{3/2}}{3 d f}+\frac {\int \frac {-\frac {3}{8} \left (a^3 C d^3-a^2 b d^2 (16 A c-13 c C-10 B d)-b^3 c \left (c^2 C-2 B c d-8 (A-C) d^2\right )+a b^2 d \left (3 c^2 C+20 B c d+8 (A-C) d^2\right )\right )+6 b d^2 \left (2 a b (A c-c C-B d)+a^2 (B c+(A-C) d)-b^2 (B c+(A-C) d)\right ) \tan (e+f x)+\frac {3}{8} \left (16 b \left (a^2 B-b^2 B+2 a b (A-C)\right ) d^3+(b c-a d) \left (8 b (A b+a B-b C) d^2+(b c-a d) (b c C-2 b B d-a C d)\right )\right ) \tan ^2(e+f x)}{\sqrt {a+b \tan (e+f x)} \sqrt {c+d \tan (e+f x)}} \, dx}{6 b d^2} \\ & = \frac {\left (8 b (A b+a B-b C) d^2+(b c-a d) (b c C-2 b B d-a C d)\right ) \sqrt {a+b \tan (e+f x)} \sqrt {c+d \tan (e+f x)}}{8 b d^2 f}-\frac {(b c C-2 b B d-a C d) \sqrt {a+b \tan (e+f x)} (c+d \tan (e+f x))^{3/2}}{4 d^2 f}+\frac {C (a+b \tan (e+f x))^{3/2} (c+d \tan (e+f x))^{3/2}}{3 d f}+\frac {\text {Subst}\left (\int \frac {-\frac {3}{8} \left (a^3 C d^3-a^2 b d^2 (16 A c-13 c C-10 B d)-b^3 c \left (c^2 C-2 B c d-8 (A-C) d^2\right )+a b^2 d \left (3 c^2 C+20 B c d+8 (A-C) d^2\right )\right )+6 b d^2 \left (2 a b (A c-c C-B d)+a^2 (B c+(A-C) d)-b^2 (B c+(A-C) d)\right ) x+\frac {3}{8} \left (16 b \left (a^2 B-b^2 B+2 a b (A-C)\right ) d^3+(b c-a d) \left (8 b (A b+a B-b C) d^2+(b c-a d) (b c C-2 b B d-a C d)\right )\right ) x^2}{\sqrt {a+b x} \sqrt {c+d x} \left (1+x^2\right )} \, dx,x,\tan (e+f x)\right )}{6 b d^2 f} \\ & = \frac {\left (8 b (A b+a B-b C) d^2+(b c-a d) (b c C-2 b B d-a C d)\right ) \sqrt {a+b \tan (e+f x)} \sqrt {c+d \tan (e+f x)}}{8 b d^2 f}-\frac {(b c C-2 b B d-a C d) \sqrt {a+b \tan (e+f x)} (c+d \tan (e+f x))^{3/2}}{4 d^2 f}+\frac {C (a+b \tan (e+f x))^{3/2} (c+d \tan (e+f x))^{3/2}}{3 d f}+\frac {\text {Subst}\left (\int \left (-\frac {3 \left (a^3 C d^3-3 a^2 b d^2 (c C+2 B d)+3 a b^2 d \left (c^2 C-4 B c d-8 (A-C) d^2\right )-b^3 \left (c^3 C-2 B c^2 d+8 c (A-C) d^2-16 B d^3\right )\right )}{8 \sqrt {a+b x} \sqrt {c+d x}}+\frac {6 \left (b d^2 \left (a^2 (A c-c C-B d)-b^2 (A c-c C-B d)-2 a b (B c+(A-C) d)\right )+b d^2 \left (2 a b (A c-c C-B d)+a^2 (B c+(A-C) d)-b^2 (B c+(A-C) d)\right ) x\right )}{\sqrt {a+b x} \sqrt {c+d x} \left (1+x^2\right )}\right ) \, dx,x,\tan (e+f x)\right )}{6 b d^2 f} \\ & = \frac {\left (8 b (A b+a B-b C) d^2+(b c-a d) (b c C-2 b B d-a C d)\right ) \sqrt {a+b \tan (e+f x)} \sqrt {c+d \tan (e+f x)}}{8 b d^2 f}-\frac {(b c C-2 b B d-a C d) \sqrt {a+b \tan (e+f x)} (c+d \tan (e+f x))^{3/2}}{4 d^2 f}+\frac {C (a+b \tan (e+f x))^{3/2} (c+d \tan (e+f x))^{3/2}}{3 d f}+\frac {\text {Subst}\left (\int \frac {b d^2 \left (a^2 (A c-c C-B d)-b^2 (A c-c C-B d)-2 a b (B c+(A-C) d)\right )+b d^2 \left (2 a b (A c-c C-B d)+a^2 (B c+(A-C) d)-b^2 (B c+(A-C) d)\right ) x}{\sqrt {a+b x} \sqrt {c+d x} \left (1+x^2\right )} \, dx,x,\tan (e+f x)\right )}{b d^2 f}-\frac {\left (a^3 C d^3-3 a^2 b d^2 (c C+2 B d)+3 a b^2 d \left (c^2 C-4 B c d-8 (A-C) d^2\right )-b^3 \left (c^3 C-2 B c^2 d+8 c (A-C) d^2-16 B d^3\right )\right ) \text {Subst}\left (\int \frac {1}{\sqrt {a+b x} \sqrt {c+d x}} \, dx,x,\tan (e+f x)\right )}{16 b d^2 f} \\ & = \frac {\left (8 b (A b+a B-b C) d^2+(b c-a d) (b c C-2 b B d-a C d)\right ) \sqrt {a+b \tan (e+f x)} \sqrt {c+d \tan (e+f x)}}{8 b d^2 f}-\frac {(b c C-2 b B d-a C d) \sqrt {a+b \tan (e+f x)} (c+d \tan (e+f x))^{3/2}}{4 d^2 f}+\frac {C (a+b \tan (e+f x))^{3/2} (c+d \tan (e+f x))^{3/2}}{3 d f}+\frac {\text {Subst}\left (\int \left (\frac {i b d^2 \left (a^2 (A c-c C-B d)-b^2 (A c-c C-B d)-2 a b (B c+(A-C) d)\right )-b d^2 \left (2 a b (A c-c C-B d)+a^2 (B c+(A-C) d)-b^2 (B c+(A-C) d)\right )}{2 (i-x) \sqrt {a+b x} \sqrt {c+d x}}+\frac {i b d^2 \left (a^2 (A c-c C-B d)-b^2 (A c-c C-B d)-2 a b (B c+(A-C) d)\right )+b d^2 \left (2 a b (A c-c C-B d)+a^2 (B c+(A-C) d)-b^2 (B c+(A-C) d)\right )}{2 (i+x) \sqrt {a+b x} \sqrt {c+d x}}\right ) \, dx,x,\tan (e+f x)\right )}{b d^2 f}-\frac {\left (a^3 C d^3-3 a^2 b d^2 (c C+2 B d)+3 a b^2 d \left (c^2 C-4 B c d-8 (A-C) d^2\right )-b^3 \left (c^3 C-2 B c^2 d+8 c (A-C) d^2-16 B d^3\right )\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 )}{8 b^2 d^2 f} \\ & = \frac {\left (8 b (A b+a B-b C) d^2+(b c-a d) (b c C-2 b B d-a C d)\right ) \sqrt {a+b \tan (e+f x)} \sqrt {c+d \tan (e+f x)}}{8 b d^2 f}-\frac {(b c C-2 b B d-a C d) \sqrt {a+b \tan (e+f x)} (c+d \tan (e+f x))^{3/2}}{4 d^2 f}+\frac {C (a+b \tan (e+f x))^{3/2} (c+d \tan (e+f x))^{3/2}}{3 d f}+\frac {\left ((a-i b)^2 (A-i B-C) (i c+d)\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 f}-\frac {\left (a^3 C d^3-3 a^2 b d^2 (c C+2 B d)+3 a b^2 d \left (c^2 C-4 B c d-8 (A-C) d^2\right )-b^3 \left (c^3 C-2 B c^2 d+8 c (A-C) d^2-16 B d^3\right )\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 )}{8 b^2 d^2 f}+\frac {\left (i b d^2 \left (a^2 (A c-c C-B d)-b^2 (A c-c C-B d)-2 a b (B c+(A-C) d)\right )-b d^2 \left (2 a b (A c-c C-B d)+a^2 (B c+(A-C) d)-b^2 (B c+(A-C) d)\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 d^2 f} \\ & = -\frac {\left (a^3 C d^3-3 a^2 b d^2 (c C+2 B d)+3 a b^2 d \left (c^2 C-4 B c d-8 (A-C) d^2\right )-b^3 \left (c^3 C-2 B c^2 d+8 c (A-C) d^2-16 B d^3\right )\right ) \text {arctanh}\left (\frac {\sqrt {d} \sqrt {a+b \tan (e+f x)}}{\sqrt {b} \sqrt {c+d \tan (e+f x)}}\right )}{8 b^{3/2} d^{5/2} f}+\frac {\left (8 b (A b+a B-b C) d^2+(b c-a d) (b c C-2 b B d-a C d)\right ) \sqrt {a+b \tan (e+f x)} \sqrt {c+d \tan (e+f x)}}{8 b d^2 f}-\frac {(b c C-2 b B d-a C d) \sqrt {a+b \tan (e+f x)} (c+d \tan (e+f x))^{3/2}}{4 d^2 f}+\frac {C (a+b \tan (e+f x))^{3/2} (c+d \tan (e+f x))^{3/2}}{3 d f}+\frac {\left ((a-i b)^2 (A-i B-C) (i c+d)\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 )}{f}+\frac {\left (i b d^2 \left (a^2 (A c-c C-B d)-b^2 (A c-c C-B d)-2 a b (B c+(A-C) d)\right )-b d^2 \left (2 a b (A c-c C-B d)+a^2 (B c+(A-C) d)-b^2 (B c+(A-C) d)\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 d^2 f} \\ & = -\frac {(a-i b)^{3/2} (i A+B-i C) \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 )}{f}-\frac {(a+i b)^{3/2} (B-i (A-C)) \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 )}{f}-\frac {\left (a^3 C d^3-3 a^2 b d^2 (c C+2 B d)+3 a b^2 d \left (c^2 C-4 B c d-8 (A-C) d^2\right )-b^3 \left (c^3 C-2 B c^2 d+8 c (A-C) d^2-16 B d^3\right )\right ) \text {arctanh}\left (\frac {\sqrt {d} \sqrt {a+b \tan (e+f x)}}{\sqrt {b} \sqrt {c+d \tan (e+f x)}}\right )}{8 b^{3/2} d^{5/2} f}+\frac {\left (8 b (A b+a B-b C) d^2+(b c-a d) (b c C-2 b B d-a C d)\right ) \sqrt {a+b \tan (e+f x)} \sqrt {c+d \tan (e+f x)}}{8 b d^2 f}-\frac {(b c C-2 b B d-a C d) \sqrt {a+b \tan (e+f x)} (c+d \tan (e+f x))^{3/2}}{4 d^2 f}+\frac {C (a+b \tan (e+f x))^{3/2} (c+d \tan (e+f x))^{3/2}}{3 d f} \\ \end{align*}

Mathematica [A] (verified)

Time = 9.10 (sec) , antiderivative size = 835, normalized size of antiderivative = 1.65 \[ \int (a+b \tan (e+f x))^{3/2} \sqrt {c+d \tan (e+f x)} \left (A+B \tan (e+f x)+C \tan ^2(e+f x)\right ) \, dx=\frac {C (a+b \tan (e+f x))^{3/2} (c+d \tan (e+f x))^{3/2}}{3 d f}+\frac {-\frac {3 (b c C-2 b B d-a C d) \sqrt {a+b \tan (e+f x)} (c+d \tan (e+f x))^{3/2}}{4 d f}+\frac {\frac {3 \left (8 b (A b+a B-b C) d^2+(b c-a d) (b c C-2 b B d-a C d)\right ) \sqrt {a+b \tan (e+f x)} \sqrt {c+d \tan (e+f x)}}{4 b f}+\frac {\frac {6 b d^2 \left (\sqrt {-b^2} \left (a^2 (A c-c C-B d)-b^2 (A c-c C-B d)-2 a b (B c+(A-C) d)\right )+b \left (2 a b (A c-c C-B d)+a^2 (B c+(A-C) d)-b^2 (B c+(A-C) d)\right )\right ) \text {arctanh}\left (\frac {\sqrt {-c+\frac {\sqrt {-b^2} d}{b}} \sqrt {a+b \tan (e+f x)}}{\sqrt {-a+\sqrt {-b^2}} \sqrt {c+d \tan (e+f x)}}\right )}{\sqrt {-a+\sqrt {-b^2}} \sqrt {-c+\frac {\sqrt {-b^2} d}{b}}}+\frac {6 b d^2 \left (\sqrt {-b^2} \left (a^2 (A c-c C-B d)-b^2 (A c-c C-B d)-2 a b (B c+(A-C) d)\right )-b \left (2 a b (A c-c C-B d)+a^2 (B c+(A-C) d)-b^2 (B c+(A-C) d)\right )\right ) \text {arctanh}\left (\frac {\sqrt {c+\frac {\sqrt {-b^2} d}{b}} \sqrt {a+b \tan (e+f x)}}{\sqrt {a+\sqrt {-b^2}} \sqrt {c+d \tan (e+f x)}}\right )}{\sqrt {a+\sqrt {-b^2}} \sqrt {c+\frac {\sqrt {-b^2} d}{b}}}-\frac {3 \sqrt {b} \sqrt {c-\frac {a d}{b}} \left (a^3 C d^3-3 a^2 b d^2 (c C+2 B d)+3 a b^2 d \left (c^2 C-4 B c d-8 (A-C) d^2\right )-b^3 \left (c^3 C-2 B c^2 d+8 c (A-C) d^2-16 B d^3\right )\right ) \text {arcsinh}\left (\frac {\sqrt {d} \sqrt {a+b \tan (e+f x)}}{\sqrt {b} \sqrt {c-\frac {a d}{b}}}\right ) \sqrt {\frac {b c+b d \tan (e+f x)}{b c-a d}}}{4 \sqrt {d} \sqrt {c+d \tan (e+f x)}}}{b^2 f}}{2 d}}{3 d} \]

[In]

Integrate[(a + b*Tan[e + f*x])^(3/2)*Sqrt[c + d*Tan[e + f*x]]*(A + B*Tan[e + f*x] + C*Tan[e + f*x]^2),x]

[Out]

(C*(a + b*Tan[e + f*x])^(3/2)*(c + d*Tan[e + f*x])^(3/2))/(3*d*f) + ((-3*(b*c*C - 2*b*B*d - a*C*d)*Sqrt[a + b*
Tan[e + f*x]]*(c + d*Tan[e + f*x])^(3/2))/(4*d*f) + ((3*(8*b*(A*b + a*B - b*C)*d^2 + (b*c - a*d)*(b*c*C - 2*b*
B*d - a*C*d))*Sqrt[a + b*Tan[e + f*x]]*Sqrt[c + d*Tan[e + f*x]])/(4*b*f) + ((6*b*d^2*(Sqrt[-b^2]*(a^2*(A*c - c
*C - B*d) - b^2*(A*c - c*C - B*d) - 2*a*b*(B*c + (A - C)*d)) + b*(2*a*b*(A*c - c*C - B*d) + a^2*(B*c + (A - C)
*d) - b^2*(B*c + (A - C)*d)))*ArcTanh[(Sqrt[-c + (Sqrt[-b^2]*d)/b]*Sqrt[a + b*Tan[e + f*x]])/(Sqrt[-a + Sqrt[-
b^2]]*Sqrt[c + d*Tan[e + f*x]])])/(Sqrt[-a + Sqrt[-b^2]]*Sqrt[-c + (Sqrt[-b^2]*d)/b]) + (6*b*d^2*(Sqrt[-b^2]*(
a^2*(A*c - c*C - B*d) - b^2*(A*c - c*C - B*d) - 2*a*b*(B*c + (A - C)*d)) - b*(2*a*b*(A*c - c*C - B*d) + a^2*(B
*c + (A - C)*d) - b^2*(B*c + (A - C)*d)))*ArcTanh[(Sqrt[c + (Sqrt[-b^2]*d)/b]*Sqrt[a + b*Tan[e + f*x]])/(Sqrt[
a + Sqrt[-b^2]]*Sqrt[c + d*Tan[e + f*x]])])/(Sqrt[a + Sqrt[-b^2]]*Sqrt[c + (Sqrt[-b^2]*d)/b]) - (3*Sqrt[b]*Sqr
t[c - (a*d)/b]*(a^3*C*d^3 - 3*a^2*b*d^2*(c*C + 2*B*d) + 3*a*b^2*d*(c^2*C - 4*B*c*d - 8*(A - C)*d^2) - b^3*(c^3
*C - 2*B*c^2*d + 8*c*(A - C)*d^2 - 16*B*d^3))*ArcSinh[(Sqrt[d]*Sqrt[a + b*Tan[e + f*x]])/(Sqrt[b]*Sqrt[c - (a*
d)/b])]*Sqrt[(b*c + b*d*Tan[e + f*x])/(b*c - a*d)])/(4*Sqrt[d]*Sqrt[c + d*Tan[e + f*x]]))/(b^2*f))/(2*d))/(3*d
)

Maple [F(-1)]

Timed out.

\[\int \sqrt {c +d \tan \left (f x +e \right )}\, \left (a +b \tan \left (f x +e \right )\right )^{\frac {3}{2}} \left (A +B \tan \left (f x +e \right )+C \tan \left (f x +e \right )^{2}\right )d x\]

[In]

int((c+d*tan(f*x+e))^(1/2)*(a+b*tan(f*x+e))^(3/2)*(A+B*tan(f*x+e)+C*tan(f*x+e)^2),x)

[Out]

int((c+d*tan(f*x+e))^(1/2)*(a+b*tan(f*x+e))^(3/2)*(A+B*tan(f*x+e)+C*tan(f*x+e)^2),x)

Fricas [F(-1)]

Timed out. \[ \int (a+b \tan (e+f x))^{3/2} \sqrt {c+d \tan (e+f x)} \left (A+B \tan (e+f x)+C \tan ^2(e+f x)\right ) \, dx=\text {Timed out} \]

[In]

integrate((c+d*tan(f*x+e))^(1/2)*(a+b*tan(f*x+e))^(3/2)*(A+B*tan(f*x+e)+C*tan(f*x+e)^2),x, algorithm="fricas")

[Out]

Timed out

Sympy [F]

\[ \int (a+b \tan (e+f x))^{3/2} \sqrt {c+d \tan (e+f x)} \left (A+B \tan (e+f x)+C \tan ^2(e+f x)\right ) \, dx=\int \left (a + b \tan {\left (e + f x \right )}\right )^{\frac {3}{2}} \sqrt {c + d \tan {\left (e + f x \right )}} \left (A + B \tan {\left (e + f x \right )} + C \tan ^{2}{\left (e + f x \right )}\right )\, dx \]

[In]

integrate((c+d*tan(f*x+e))**(1/2)*(a+b*tan(f*x+e))**(3/2)*(A+B*tan(f*x+e)+C*tan(f*x+e)**2),x)

[Out]

Integral((a + b*tan(e + f*x))**(3/2)*sqrt(c + d*tan(e + f*x))*(A + B*tan(e + f*x) + C*tan(e + f*x)**2), x)

Maxima [F]

\[ \int (a+b \tan (e+f x))^{3/2} \sqrt {c+d \tan (e+f x)} \left (A+B \tan (e+f x)+C \tan ^2(e+f x)\right ) \, dx=\int { {\left (C \tan \left (f x + e\right )^{2} + B \tan \left (f x + e\right ) + A\right )} {\left (b \tan \left (f x + e\right ) + a\right )}^{\frac {3}{2}} \sqrt {d \tan \left (f x + e\right ) + c} \,d x } \]

[In]

integrate((c+d*tan(f*x+e))^(1/2)*(a+b*tan(f*x+e))^(3/2)*(A+B*tan(f*x+e)+C*tan(f*x+e)^2),x, algorithm="maxima")

[Out]

integrate((C*tan(f*x + e)^2 + B*tan(f*x + e) + A)*(b*tan(f*x + e) + a)^(3/2)*sqrt(d*tan(f*x + e) + c), x)

Giac [F(-1)]

Timed out. \[ \int (a+b \tan (e+f x))^{3/2} \sqrt {c+d \tan (e+f x)} \left (A+B \tan (e+f x)+C \tan ^2(e+f x)\right ) \, dx=\text {Timed out} \]

[In]

integrate((c+d*tan(f*x+e))^(1/2)*(a+b*tan(f*x+e))^(3/2)*(A+B*tan(f*x+e)+C*tan(f*x+e)^2),x, algorithm="giac")

[Out]

Timed out

Mupad [F(-1)]

Timed out. \[ \int (a+b \tan (e+f x))^{3/2} \sqrt {c+d \tan (e+f x)} \left (A+B \tan (e+f x)+C \tan ^2(e+f x)\right ) \, dx=\int {\left (a+b\,\mathrm {tan}\left (e+f\,x\right )\right )}^{3/2}\,\sqrt {c+d\,\mathrm {tan}\left (e+f\,x\right )}\,\left (C\,{\mathrm {tan}\left (e+f\,x\right )}^2+B\,\mathrm {tan}\left (e+f\,x\right )+A\right ) \,d x \]

[In]

int((a + b*tan(e + f*x))^(3/2)*(c + d*tan(e + f*x))^(1/2)*(A + B*tan(e + f*x) + C*tan(e + f*x)^2),x)

[Out]

int((a + b*tan(e + f*x))^(3/2)*(c + d*tan(e + f*x))^(1/2)*(A + B*tan(e + f*x) + C*tan(e + f*x)^2), x)