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

Optimal. Leaf size=679 \[ -\frac{\left (30 a^2 b^2 d^2 \left (-8 d^2 (A-C)-4 B c d+c^2 C\right )-20 a^3 b d^3 (2 B d+c C)+5 a^4 C d^4-20 a b^3 d \left (8 c d^2 (A-C)-2 B c^2 d-16 B d^3+c^3 C\right )+b^4 \left (16 c^2 d^2 (A-C)+128 d^4 (A-C)-8 B c^3 d+64 B c d^3+5 c^4 C\right )\right ) \tanh ^{-1}\left (\frac{\sqrt{d} \sqrt{a+b \tan (e+f x)}}{\sqrt{b} \sqrt{c+d \tan (e+f x)}}\right )}{64 b^{3/2} d^{7/2} f}+\frac{\sqrt{a+b \tan (e+f x)} \sqrt{c+d \tan (e+f x)} \left (64 b d^3 \left (a^2 B+2 a b (A-C)-b^2 B\right )-(b c-a d) \left (16 b d^2 (a B+A b-b C)+(b c-a d) (-5 a C d-8 b B d+5 b c C)\right )\right )}{64 b d^3 f}+\frac{\sqrt{a+b \tan (e+f x)} (c+d \tan (e+f x))^{3/2} \left (16 b d^2 (a B+A b-b C)+(b c-a d) (-5 a C d-8 b B d+5 b c C)\right )}{32 d^3 f}-\frac{(a-i b)^{5/2} \sqrt{c-i d} (i A+B-i C) \tanh ^{-1}\left (\frac{\sqrt{c-i d} \sqrt{a+b \tan (e+f x)}}{\sqrt{a-i b} \sqrt{c+d \tan (e+f x)}}\right )}{f}-\frac{(a+i b)^{5/2} \sqrt{c+i d} (B-i (A-C)) \tanh ^{-1}\left (\frac{\sqrt{c+i d} \sqrt{a+b \tan (e+f x)}}{\sqrt{a+i b} \sqrt{c+d \tan (e+f x)}}\right )}{f}-\frac{(-5 a C d-8 b B d+5 b c C) (a+b \tan (e+f x))^{3/2} (c+d \tan (e+f x))^{3/2}}{24 d^2 f}+\frac{C (a+b \tan (e+f x))^{5/2} (c+d \tan (e+f x))^{3/2}}{4 d f} \]

[Out]

-(((a - I*b)^(5/2)*(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)^(5/2)*(B - I*(A - 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 - ((5*a^4*C*d^4 - 20*a^3*b*d^3*(c*C + 2*B*d
) + 30*a^2*b^2*d^2*(c^2*C - 4*B*c*d - 8*(A - C)*d^2) - 20*a*b^3*d*(c^3*C - 2*B*c^2*d + 8*c*(A - C)*d^2 - 16*B*
d^3) + b^4*(5*c^4*C - 8*B*c^3*d + 16*c^2*(A - C)*d^2 + 64*B*c*d^3 + 128*(A - C)*d^4))*ArcTanh[(Sqrt[d]*Sqrt[a
+ b*Tan[e + f*x]])/(Sqrt[b]*Sqrt[c + d*Tan[e + f*x]])])/(64*b^(3/2)*d^(7/2)*f) + ((64*b*(a^2*B - b^2*B + 2*a*b
*(A - C))*d^3 - (b*c - a*d)*(16*b*(A*b + a*B - b*C)*d^2 + (b*c - a*d)*(5*b*c*C - 8*b*B*d - 5*a*C*d)))*Sqrt[a +
b*Tan[e + f*x]]*Sqrt[c + d*Tan[e + f*x]])/(64*b*d^3*f) + ((16*b*(A*b + a*B - b*C)*d^2 + (b*c - a*d)*(5*b*c*C
- 8*b*B*d - 5*a*C*d))*Sqrt[a + b*Tan[e + f*x]]*(c + d*Tan[e + f*x])^(3/2))/(32*d^3*f) - ((5*b*c*C - 8*b*B*d -
5*a*C*d)*(a + b*Tan[e + f*x])^(3/2)*(c + d*Tan[e + f*x])^(3/2))/(24*d^2*f) + (C*(a + b*Tan[e + f*x])^(5/2)*(c
+ d*Tan[e + f*x])^(3/2))/(4*d*f)

________________________________________________________________________________________

Rubi [A]  time = 9.92627, antiderivative size = 679, normalized size of antiderivative = 1., number of steps used = 16, number of rules used = 8, integrand size = 49, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.163, Rules used = {3647, 3655, 6725, 63, 217, 206, 93, 208} \[ -\frac{\left (30 a^2 b^2 d^2 \left (-8 d^2 (A-C)-4 B c d+c^2 C\right )-20 a^3 b d^3 (2 B d+c C)+5 a^4 C d^4-20 a b^3 d \left (8 c d^2 (A-C)-2 B c^2 d-16 B d^3+c^3 C\right )+b^4 \left (16 c^2 d^2 (A-C)+128 d^4 (A-C)-8 B c^3 d+64 B c d^3+5 c^4 C\right )\right ) \tanh ^{-1}\left (\frac{\sqrt{d} \sqrt{a+b \tan (e+f x)}}{\sqrt{b} \sqrt{c+d \tan (e+f x)}}\right )}{64 b^{3/2} d^{7/2} f}+\frac{\sqrt{a+b \tan (e+f x)} \sqrt{c+d \tan (e+f x)} \left (64 b d^3 \left (a^2 B+2 a b (A-C)-b^2 B\right )-(b c-a d) \left (16 b d^2 (a B+A b-b C)+(b c-a d) (-5 a C d-8 b B d+5 b c C)\right )\right )}{64 b d^3 f}+\frac{\sqrt{a+b \tan (e+f x)} (c+d \tan (e+f x))^{3/2} \left (16 b d^2 (a B+A b-b C)+(b c-a d) (-5 a C d-8 b B d+5 b c C)\right )}{32 d^3 f}-\frac{(a-i b)^{5/2} \sqrt{c-i d} (i A+B-i C) \tanh ^{-1}\left (\frac{\sqrt{c-i d} \sqrt{a+b \tan (e+f x)}}{\sqrt{a-i b} \sqrt{c+d \tan (e+f x)}}\right )}{f}-\frac{(a+i b)^{5/2} \sqrt{c+i d} (B-i (A-C)) \tanh ^{-1}\left (\frac{\sqrt{c+i d} \sqrt{a+b \tan (e+f x)}}{\sqrt{a+i b} \sqrt{c+d \tan (e+f x)}}\right )}{f}-\frac{(-5 a C d-8 b B d+5 b c C) (a+b \tan (e+f x))^{3/2} (c+d \tan (e+f x))^{3/2}}{24 d^2 f}+\frac{C (a+b \tan (e+f x))^{5/2} (c+d \tan (e+f x))^{3/2}}{4 d f} \]

Antiderivative was successfully verified.

[In]

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

[Out]

-(((a - I*b)^(5/2)*(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)^(5/2)*(B - I*(A - 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 - ((5*a^4*C*d^4 - 20*a^3*b*d^3*(c*C + 2*B*d
) + 30*a^2*b^2*d^2*(c^2*C - 4*B*c*d - 8*(A - C)*d^2) - 20*a*b^3*d*(c^3*C - 2*B*c^2*d + 8*c*(A - C)*d^2 - 16*B*
d^3) + b^4*(5*c^4*C - 8*B*c^3*d + 16*c^2*(A - C)*d^2 + 64*B*c*d^3 + 128*(A - C)*d^4))*ArcTanh[(Sqrt[d]*Sqrt[a
+ b*Tan[e + f*x]])/(Sqrt[b]*Sqrt[c + d*Tan[e + f*x]])])/(64*b^(3/2)*d^(7/2)*f) + ((64*b*(a^2*B - b^2*B + 2*a*b
*(A - C))*d^3 - (b*c - a*d)*(16*b*(A*b + a*B - b*C)*d^2 + (b*c - a*d)*(5*b*c*C - 8*b*B*d - 5*a*C*d)))*Sqrt[a +
b*Tan[e + f*x]]*Sqrt[c + d*Tan[e + f*x]])/(64*b*d^3*f) + ((16*b*(A*b + a*B - b*C)*d^2 + (b*c - a*d)*(5*b*c*C
- 8*b*B*d - 5*a*C*d))*Sqrt[a + b*Tan[e + f*x]]*(c + d*Tan[e + f*x])^(3/2))/(32*d^3*f) - ((5*b*c*C - 8*b*B*d -
5*a*C*d)*(a + b*Tan[e + f*x])^(3/2)*(c + d*Tan[e + f*x])^(3/2))/(24*d^2*f) + (C*(a + b*Tan[e + f*x])^(5/2)*(c
+ d*Tan[e + f*x])^(3/2))/(4*d*f)

Rule 3647

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 3655

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 6725

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]

Rule 63

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 217

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 206

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

Rule 93

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 208

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

Rubi steps

\begin{align*} \int (a+b \tan (e+f x))^{5/2} \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))^{5/2} (c+d \tan (e+f x))^{3/2}}{4 d f}+\frac{\int (a+b \tan (e+f x))^{3/2} \sqrt{c+d \tan (e+f x)} \left (\frac{1}{2} (-5 b c C+a (8 A-3 C) d)+4 (A b+a B-b C) d \tan (e+f x)-\frac{1}{2} (5 b c C-8 b B d-5 a C d) \tan ^2(e+f x)\right ) \, dx}{4 d}\\ &=-\frac{(5 b c C-8 b B d-5 a C d) (a+b \tan (e+f x))^{3/2} (c+d \tan (e+f x))^{3/2}}{24 d^2 f}+\frac{C (a+b \tan (e+f x))^{5/2} (c+d \tan (e+f x))^{3/2}}{4 d f}+\frac{\int \sqrt{a+b \tan (e+f x)} \sqrt{c+d \tan (e+f x)} \left (\frac{3}{4} \left (a^2 (16 A-11 C) d^2+b^2 c (5 c C-8 B d)-2 a b d (5 c C+4 B d)\right )+12 \left (a^2 B-b^2 B+2 a b (A-C)\right ) d^2 \tan (e+f x)+\frac{3}{4} \left (16 b (A b+a B-b C) d^2+(b c-a d) (5 b c C-8 b B d-5 a C d)\right ) \tan ^2(e+f x)\right ) \, dx}{12 d^2}\\ &=\frac{\left (16 b (A b+a B-b C) d^2+(b c-a d) (5 b c C-8 b B d-5 a C d)\right ) \sqrt{a+b \tan (e+f x)} (c+d \tan (e+f x))^{3/2}}{32 d^3 f}-\frac{(5 b c C-8 b B d-5 a C d) (a+b \tan (e+f x))^{3/2} (c+d \tan (e+f x))^{3/2}}{24 d^2 f}+\frac{C (a+b \tan (e+f x))^{5/2} (c+d \tan (e+f x))^{3/2}}{4 d f}+\frac{\int \frac{\sqrt{c+d \tan (e+f x)} \left (\frac{3}{8} \left (a^3 (64 A-59 C) d^3-a^2 b d^2 (15 c C+104 B d)+a b^2 d \left (15 c^2 C-32 B c d-48 (A-C) d^2\right )-b^3 c \left (5 c^2 C-8 B c d+16 (A-C) d^2\right )\right )+24 \left (a^3 B-3 a b^2 B+3 a^2 b (A-C)-b^3 (A-C)\right ) d^3 \tan (e+f x)+\frac{3}{8} \left (64 b \left (a^2 B-b^2 B+2 a b (A-C)\right ) d^3-(b c-a d) \left (16 b (A b+a B-b C) d^2+(b c-a d) (5 b c C-8 b B d-5 a C d)\right )\right ) \tan ^2(e+f x)\right )}{\sqrt{a+b \tan (e+f x)}} \, dx}{24 d^3}\\ &=\frac{\left (64 b \left (a^2 B-b^2 B+2 a b (A-C)\right ) d^3-(b c-a d) \left (16 b (A b+a B-b C) d^2+(b c-a d) (5 b c C-8 b B d-5 a C d)\right )\right ) \sqrt{a+b \tan (e+f x)} \sqrt{c+d \tan (e+f x)}}{64 b d^3 f}+\frac{\left (16 b (A b+a B-b C) d^2+(b c-a d) (5 b c C-8 b B d-5 a C d)\right ) \sqrt{a+b \tan (e+f x)} (c+d \tan (e+f x))^{3/2}}{32 d^3 f}-\frac{(5 b c C-8 b B d-5 a C d) (a+b \tan (e+f x))^{3/2} (c+d \tan (e+f x))^{3/2}}{24 d^2 f}+\frac{C (a+b \tan (e+f x))^{5/2} (c+d \tan (e+f x))^{3/2}}{4 d f}+\frac{\int \frac{-\frac{3}{16} \left (5 a^4 C d^4-4 a^3 b d^3 (32 A c-27 c C-22 B d)+6 a^2 b^2 d^2 \left (5 c^2 C+44 B c d+24 (A-C) d^2\right )+b^4 c \left (5 c^3 C-8 B c^2 d+16 c (A-C) d^2-64 B d^3\right )-4 a b^3 d \left (5 c^3 C-10 B c^2 d-56 c (A-C) d^2+16 B d^3\right )\right )+24 b d^3 \left (3 a^2 b (A c-c C-B d)-b^3 (A c-c C-B d)+a^3 (B c+(A-C) d)-3 a b^2 (B c+(A-C) d)\right ) \tan (e+f x)+\frac{3}{16} \left (128 b \left (a^3 B-3 a b^2 B+3 a^2 b (A-C)-b^3 (A-C)\right ) d^4+(b c-a d) \left (64 b \left (a^2 B-b^2 B+2 a b (A-C)\right ) d^3-(b c-a d) \left (16 b (A b+a B-b C) d^2+(b c-a d) (5 b c C-8 b B d-5 a C d)\right )\right )\right ) \tan ^2(e+f x)}{\sqrt{a+b \tan (e+f x)} \sqrt{c+d \tan (e+f x)}} \, dx}{24 b d^3}\\ &=\frac{\left (64 b \left (a^2 B-b^2 B+2 a b (A-C)\right ) d^3-(b c-a d) \left (16 b (A b+a B-b C) d^2+(b c-a d) (5 b c C-8 b B d-5 a C d)\right )\right ) \sqrt{a+b \tan (e+f x)} \sqrt{c+d \tan (e+f x)}}{64 b d^3 f}+\frac{\left (16 b (A b+a B-b C) d^2+(b c-a d) (5 b c C-8 b B d-5 a C d)\right ) \sqrt{a+b \tan (e+f x)} (c+d \tan (e+f x))^{3/2}}{32 d^3 f}-\frac{(5 b c C-8 b B d-5 a C d) (a+b \tan (e+f x))^{3/2} (c+d \tan (e+f x))^{3/2}}{24 d^2 f}+\frac{C (a+b \tan (e+f x))^{5/2} (c+d \tan (e+f x))^{3/2}}{4 d f}+\frac{\operatorname{Subst}\left (\int \frac{-\frac{3}{16} \left (5 a^4 C d^4-4 a^3 b d^3 (32 A c-27 c C-22 B d)+6 a^2 b^2 d^2 \left (5 c^2 C+44 B c d+24 (A-C) d^2\right )+b^4 c \left (5 c^3 C-8 B c^2 d+16 c (A-C) d^2-64 B d^3\right )-4 a b^3 d \left (5 c^3 C-10 B c^2 d-56 c (A-C) d^2+16 B d^3\right )\right )+24 b d^3 \left (3 a^2 b (A c-c C-B d)-b^3 (A c-c C-B d)+a^3 (B c+(A-C) d)-3 a b^2 (B c+(A-C) d)\right ) x+\frac{3}{16} \left (128 b \left (a^3 B-3 a b^2 B+3 a^2 b (A-C)-b^3 (A-C)\right ) d^4+(b c-a d) \left (64 b \left (a^2 B-b^2 B+2 a b (A-C)\right ) d^3-(b c-a d) \left (16 b (A b+a B-b C) d^2+(b c-a d) (5 b c C-8 b B d-5 a C d)\right )\right )\right ) x^2}{\sqrt{a+b x} \sqrt{c+d x} \left (1+x^2\right )} \, dx,x,\tan (e+f x)\right )}{24 b d^3 f}\\ &=\frac{\left (64 b \left (a^2 B-b^2 B+2 a b (A-C)\right ) d^3-(b c-a d) \left (16 b (A b+a B-b C) d^2+(b c-a d) (5 b c C-8 b B d-5 a C d)\right )\right ) \sqrt{a+b \tan (e+f x)} \sqrt{c+d \tan (e+f x)}}{64 b d^3 f}+\frac{\left (16 b (A b+a B-b C) d^2+(b c-a d) (5 b c C-8 b B d-5 a C d)\right ) \sqrt{a+b \tan (e+f x)} (c+d \tan (e+f x))^{3/2}}{32 d^3 f}-\frac{(5 b c C-8 b B d-5 a C d) (a+b \tan (e+f x))^{3/2} (c+d \tan (e+f x))^{3/2}}{24 d^2 f}+\frac{C (a+b \tan (e+f x))^{5/2} (c+d \tan (e+f x))^{3/2}}{4 d f}+\frac{\operatorname{Subst}\left (\int \left (-\frac{3 \left (5 a^4 C d^4-20 a^3 b d^3 (c C+2 B d)+30 a^2 b^2 d^2 \left (c^2 C-4 B c d-8 (A-C) d^2\right )-20 a b^3 d \left (c^3 C-2 B c^2 d+8 c (A-C) d^2-16 B d^3\right )+b^4 \left (5 c^4 C-8 B c^3 d+16 c^2 (A-C) d^2+64 B c d^3+128 (A-C) d^4\right )\right )}{16 \sqrt{a+b x} \sqrt{c+d x}}+\frac{24 \left (b d^3 \left (a^3 (A c-c C-B d)-3 a b^2 (A c-c C-B d)-3 a^2 b (B c+(A-C) d)+b^3 (B c+(A-C) d)\right )+b d^3 \left (3 a^2 b (A c-c C-B d)-b^3 (A c-c C-B d)+a^3 (B c+(A-C) d)-3 a 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 )}{24 b d^3 f}\\ &=\frac{\left (64 b \left (a^2 B-b^2 B+2 a b (A-C)\right ) d^3-(b c-a d) \left (16 b (A b+a B-b C) d^2+(b c-a d) (5 b c C-8 b B d-5 a C d)\right )\right ) \sqrt{a+b \tan (e+f x)} \sqrt{c+d \tan (e+f x)}}{64 b d^3 f}+\frac{\left (16 b (A b+a B-b C) d^2+(b c-a d) (5 b c C-8 b B d-5 a C d)\right ) \sqrt{a+b \tan (e+f x)} (c+d \tan (e+f x))^{3/2}}{32 d^3 f}-\frac{(5 b c C-8 b B d-5 a C d) (a+b \tan (e+f x))^{3/2} (c+d \tan (e+f x))^{3/2}}{24 d^2 f}+\frac{C (a+b \tan (e+f x))^{5/2} (c+d \tan (e+f x))^{3/2}}{4 d f}+\frac{\operatorname{Subst}\left (\int \frac{b d^3 \left (a^3 (A c-c C-B d)-3 a b^2 (A c-c C-B d)-3 a^2 b (B c+(A-C) d)+b^3 (B c+(A-C) d)\right )+b d^3 \left (3 a^2 b (A c-c C-B d)-b^3 (A c-c C-B d)+a^3 (B c+(A-C) d)-3 a 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^3 f}-\frac{\left (5 a^4 C d^4-20 a^3 b d^3 (c C+2 B d)+30 a^2 b^2 d^2 \left (c^2 C-4 B c d-8 (A-C) d^2\right )-20 a b^3 d \left (c^3 C-2 B c^2 d+8 c (A-C) d^2-16 B d^3\right )+b^4 \left (5 c^4 C-8 B c^3 d+16 c^2 (A-C) d^2+64 B c d^3+128 (A-C) d^4\right )\right ) \operatorname{Subst}\left (\int \frac{1}{\sqrt{a+b x} \sqrt{c+d x}} \, dx,x,\tan (e+f x)\right )}{128 b d^3 f}\\ &=\frac{\left (64 b \left (a^2 B-b^2 B+2 a b (A-C)\right ) d^3-(b c-a d) \left (16 b (A b+a B-b C) d^2+(b c-a d) (5 b c C-8 b B d-5 a C d)\right )\right ) \sqrt{a+b \tan (e+f x)} \sqrt{c+d \tan (e+f x)}}{64 b d^3 f}+\frac{\left (16 b (A b+a B-b C) d^2+(b c-a d) (5 b c C-8 b B d-5 a C d)\right ) \sqrt{a+b \tan (e+f x)} (c+d \tan (e+f x))^{3/2}}{32 d^3 f}-\frac{(5 b c C-8 b B d-5 a C d) (a+b \tan (e+f x))^{3/2} (c+d \tan (e+f x))^{3/2}}{24 d^2 f}+\frac{C (a+b \tan (e+f x))^{5/2} (c+d \tan (e+f x))^{3/2}}{4 d f}+\frac{\operatorname{Subst}\left (\int \left (\frac{-b d^3 \left (3 a^2 b (A c-c C-B d)-b^3 (A c-c C-B d)+a^3 (B c+(A-C) d)-3 a b^2 (B c+(A-C) d)\right )+i b d^3 \left (a^3 (A c-c C-B d)-3 a b^2 (A c-c C-B d)-3 a^2 b (B c+(A-C) d)+b^3 (B c+(A-C) d)\right )}{2 (i-x) \sqrt{a+b x} \sqrt{c+d x}}+\frac{b d^3 \left (3 a^2 b (A c-c C-B d)-b^3 (A c-c C-B d)+a^3 (B c+(A-C) d)-3 a b^2 (B c+(A-C) d)\right )+i b d^3 \left (a^3 (A c-c C-B d)-3 a b^2 (A c-c C-B d)-3 a^2 b (B c+(A-C) d)+b^3 (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^3 f}-\frac{\left (5 a^4 C d^4-20 a^3 b d^3 (c C+2 B d)+30 a^2 b^2 d^2 \left (c^2 C-4 B c d-8 (A-C) d^2\right )-20 a b^3 d \left (c^3 C-2 B c^2 d+8 c (A-C) d^2-16 B d^3\right )+b^4 \left (5 c^4 C-8 B c^3 d+16 c^2 (A-C) d^2+64 B c d^3+128 (A-C) d^4\right )\right ) \operatorname{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 )}{64 b^2 d^3 f}\\ &=\frac{\left (64 b \left (a^2 B-b^2 B+2 a b (A-C)\right ) d^3-(b c-a d) \left (16 b (A b+a B-b C) d^2+(b c-a d) (5 b c C-8 b B d-5 a C d)\right )\right ) \sqrt{a+b \tan (e+f x)} \sqrt{c+d \tan (e+f x)}}{64 b d^3 f}+\frac{\left (16 b (A b+a B-b C) d^2+(b c-a d) (5 b c C-8 b B d-5 a C d)\right ) \sqrt{a+b \tan (e+f x)} (c+d \tan (e+f x))^{3/2}}{32 d^3 f}-\frac{(5 b c C-8 b B d-5 a C d) (a+b \tan (e+f x))^{3/2} (c+d \tan (e+f x))^{3/2}}{24 d^2 f}+\frac{C (a+b \tan (e+f x))^{5/2} (c+d \tan (e+f x))^{3/2}}{4 d f}+\frac{\left ((a-i b)^3 (A-i B-C) (i c+d)\right ) \operatorname{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 (5 a^4 C d^4-20 a^3 b d^3 (c C+2 B d)+30 a^2 b^2 d^2 \left (c^2 C-4 B c d-8 (A-C) d^2\right )-20 a b^3 d \left (c^3 C-2 B c^2 d+8 c (A-C) d^2-16 B d^3\right )+b^4 \left (5 c^4 C-8 B c^3 d+16 c^2 (A-C) d^2+64 B c d^3+128 (A-C) d^4\right )\right ) \operatorname{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 )}{64 b^2 d^3 f}+\frac{\left (-b d^3 \left (3 a^2 b (A c-c C-B d)-b^3 (A c-c C-B d)+a^3 (B c+(A-C) d)-3 a b^2 (B c+(A-C) d)\right )+i b d^3 \left (a^3 (A c-c C-B d)-3 a b^2 (A c-c C-B d)-3 a^2 b (B c+(A-C) d)+b^3 (B c+(A-C) d)\right )\right ) \operatorname{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^3 f}\\ &=-\frac{\left (5 a^4 C d^4-20 a^3 b d^3 (c C+2 B d)+30 a^2 b^2 d^2 \left (c^2 C-4 B c d-8 (A-C) d^2\right )-20 a b^3 d \left (c^3 C-2 B c^2 d+8 c (A-C) d^2-16 B d^3\right )+b^4 \left (5 c^4 C-8 B c^3 d+16 c^2 (A-C) d^2+64 B c d^3+128 (A-C) d^4\right )\right ) \tanh ^{-1}\left (\frac{\sqrt{d} \sqrt{a+b \tan (e+f x)}}{\sqrt{b} \sqrt{c+d \tan (e+f x)}}\right )}{64 b^{3/2} d^{7/2} f}+\frac{\left (64 b \left (a^2 B-b^2 B+2 a b (A-C)\right ) d^3-(b c-a d) \left (16 b (A b+a B-b C) d^2+(b c-a d) (5 b c C-8 b B d-5 a C d)\right )\right ) \sqrt{a+b \tan (e+f x)} \sqrt{c+d \tan (e+f x)}}{64 b d^3 f}+\frac{\left (16 b (A b+a B-b C) d^2+(b c-a d) (5 b c C-8 b B d-5 a C d)\right ) \sqrt{a+b \tan (e+f x)} (c+d \tan (e+f x))^{3/2}}{32 d^3 f}-\frac{(5 b c C-8 b B d-5 a C d) (a+b \tan (e+f x))^{3/2} (c+d \tan (e+f x))^{3/2}}{24 d^2 f}+\frac{C (a+b \tan (e+f x))^{5/2} (c+d \tan (e+f x))^{3/2}}{4 d f}+\frac{\left ((a-i b)^3 (A-i B-C) (i c+d)\right ) \operatorname{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 (-b d^3 \left (3 a^2 b (A c-c C-B d)-b^3 (A c-c C-B d)+a^3 (B c+(A-C) d)-3 a b^2 (B c+(A-C) d)\right )+i b d^3 \left (a^3 (A c-c C-B d)-3 a b^2 (A c-c C-B d)-3 a^2 b (B c+(A-C) d)+b^3 (B c+(A-C) d)\right )\right ) \operatorname{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^3 f}\\ &=-\frac{(a-i b)^{5/2} (i A+B-i C) \sqrt{c-i d} \tanh ^{-1}\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)^{5/2} (B-i (A-C)) \sqrt{c+i d} \tanh ^{-1}\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 (5 a^4 C d^4-20 a^3 b d^3 (c C+2 B d)+30 a^2 b^2 d^2 \left (c^2 C-4 B c d-8 (A-C) d^2\right )-20 a b^3 d \left (c^3 C-2 B c^2 d+8 c (A-C) d^2-16 B d^3\right )+b^4 \left (5 c^4 C-8 B c^3 d+16 c^2 (A-C) d^2+64 B c d^3+128 (A-C) d^4\right )\right ) \tanh ^{-1}\left (\frac{\sqrt{d} \sqrt{a+b \tan (e+f x)}}{\sqrt{b} \sqrt{c+d \tan (e+f x)}}\right )}{64 b^{3/2} d^{7/2} f}+\frac{\left (64 b \left (a^2 B-b^2 B+2 a b (A-C)\right ) d^3-(b c-a d) \left (16 b (A b+a B-b C) d^2+(b c-a d) (5 b c C-8 b B d-5 a C d)\right )\right ) \sqrt{a+b \tan (e+f x)} \sqrt{c+d \tan (e+f x)}}{64 b d^3 f}+\frac{\left (16 b (A b+a B-b C) d^2+(b c-a d) (5 b c C-8 b B d-5 a C d)\right ) \sqrt{a+b \tan (e+f x)} (c+d \tan (e+f x))^{3/2}}{32 d^3 f}-\frac{(5 b c C-8 b B d-5 a C d) (a+b \tan (e+f x))^{3/2} (c+d \tan (e+f x))^{3/2}}{24 d^2 f}+\frac{C (a+b \tan (e+f x))^{5/2} (c+d \tan (e+f x))^{3/2}}{4 d f}\\ \end{align*}

Mathematica [A]  time = 9.72995, size = 1202, normalized size = 1.77 \[ \frac{C (c+d \tan (e+f x))^{3/2} (a+b \tan (e+f x))^{5/2}}{4 d f}+\frac{\frac{(-5 b c C+5 a d C+8 b B d) (a+b \tan (e+f x))^{3/2} (c+d \tan (e+f x))^{3/2}}{6 d f}+\frac{\frac{3 \left (16 b (A b-C b+a B) d^2+(b c-a d) (5 b c C-5 a d C-8 b B d)\right ) \sqrt{a+b \tan (e+f x)} (c+d \tan (e+f x))^{3/2}}{8 d f}+\frac{\frac{\sqrt{a+b \tan (e+f x)} \sqrt{c+d \tan (e+f x)} \left (24 b \left (B a^2+2 b (A-C) a-b^2 B\right ) d^3-\frac{3}{8} (b c-a d) \left (16 b (A b-C b+a B) d^2+(b c-a d) (5 b c C-5 a d C-8 b B d)\right )\right )}{b f}+\frac{\frac{24 b \left (b \left ((B c+(A-C) d) a^3+3 b (A c-C c-B d) a^2-3 b^2 (B c+(A-C) d) a-b^3 (A c-C c-B d)\right )+\sqrt{-b^2} \left ((A c-C c-B d) a^3-3 b (B c+(A-C) d) a^2-3 b^2 (A c-C c-B d) a+b^3 (B c+(A-C) d)\right )\right ) \tan ^{-1}\left (\frac{\sqrt{c+\frac{b d}{\sqrt{-b^2}}} \sqrt{a+b \tan (e+f x)}}{\sqrt{\sqrt{-b^2}-a} \sqrt{c+d \tan (e+f x)}}\right ) d^3}{\sqrt{\sqrt{-b^2}-a} \sqrt{c+\frac{b d}{\sqrt{-b^2}}}}-\frac{24 b \left (b \left ((B c+(A-C) d) a^3+3 b (A c-C c-B d) a^2-3 b^2 (B c+(A-C) d) a-b^3 (A c-C c-B d)\right )-\sqrt{-b^2} \left ((A c-C c-B d) a^3-3 b (B c+(A-C) d) a^2-3 b^2 (A c-C c-B d) a+b^3 (B c+(A-C) d)\right )\right ) \tan ^{-1}\left (\frac{\sqrt{-\frac{b 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)}}\right ) d^3}{\sqrt{a+\sqrt{-b^2}} \sqrt{-\frac{b c+\sqrt{-b^2} d}{b}}}-\frac{3 \sqrt{b} \sqrt{c-\frac{a d}{b}} \sqrt{\frac{1}{\frac{c}{c-\frac{a d}{b}}-\frac{a d}{b \left (c-\frac{a d}{b}\right )}}} \sqrt{\frac{c}{c-\frac{a d}{b}}-\frac{a d}{b \left (c-\frac{a d}{b}\right )}} \left (\left (5 C c^4-8 B d c^3+16 (A-C) d^2 c^2+64 B d^3 c+128 (A-C) d^4\right ) b^4-20 a d \left (C c^3-2 B d c^2+8 (A-C) d^2 c-16 B d^3\right ) b^3+30 a^2 d^2 \left (C c^2-4 B d c-8 (A-C) d^2\right ) b^2-20 a^3 d^3 (c C+2 B d) b+5 a^4 C d^4\right ) \sinh ^{-1}\left (\frac{\sqrt{d} \sqrt{a+b \tan (e+f x)}}{\sqrt{b} \sqrt{c-\frac{a d}{b}} \sqrt{\frac{c}{c-\frac{a d}{b}}-\frac{a d}{b \left (c-\frac{a d}{b}\right )}}}\right ) \sqrt{\frac{c+d \tan (e+f x)}{c-\frac{a d}{b}}}}{8 \sqrt{c+d \tan (e+f x)} \sqrt{d}}}{b^2 f}}{2 d}}{3 d}}{4 d} \]

Warning: Unable to verify antiderivative.

[In]

Integrate[(a + b*Tan[e + f*x])^(5/2)*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])^(5/2)*(c + d*Tan[e + f*x])^(3/2))/(4*d*f) + (((-5*b*c*C + 8*b*B*d + 5*a*C*d)*(a + b*Ta
n[e + f*x])^(3/2)*(c + d*Tan[e + f*x])^(3/2))/(6*d*f) + ((3*(16*b*(A*b + a*B - b*C)*d^2 + (b*c - a*d)*(5*b*c*C
- 8*b*B*d - 5*a*C*d))*Sqrt[a + b*Tan[e + f*x]]*(c + d*Tan[e + f*x])^(3/2))/(8*d*f) + (((24*b*(a^2*B - b^2*B +
2*a*b*(A - C))*d^3 - (3*(b*c - a*d)*(16*b*(A*b + a*B - b*C)*d^2 + (b*c - a*d)*(5*b*c*C - 8*b*B*d - 5*a*C*d)))
/8)*Sqrt[a + b*Tan[e + f*x]]*Sqrt[c + d*Tan[e + f*x]])/(b*f) + ((24*b*d^3*(b*(3*a^2*b*(A*c - c*C - B*d) - b^3*
(A*c - c*C - B*d) + a^3*(B*c + (A - C)*d) - 3*a*b^2*(B*c + (A - C)*d)) + Sqrt[-b^2]*(a^3*(A*c - c*C - B*d) - 3
*a*b^2*(A*c - c*C - B*d) - 3*a^2*b*(B*c + (A - C)*d) + b^3*(B*c + (A - C)*d)))*ArcTan[(Sqrt[c + (b*d)/Sqrt[-b^
2]]*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
+ (b*d)/Sqrt[-b^2]]) - (24*b*d^3*(b*(3*a^2*b*(A*c - c*C - B*d) - b^3*(A*c - c*C - B*d) + a^3*(B*c + (A - C)*d
) - 3*a*b^2*(B*c + (A - C)*d)) - Sqrt[-b^2]*(a^3*(A*c - c*C - B*d) - 3*a*b^2*(A*c - c*C - B*d) - 3*a^2*b*(B*c
+ (A - C)*d) + b^3*(B*c + (A - C)*d)))*ArcTan[(Sqrt[-((b*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[-((b*c + Sqrt[-b^2]*d)/b)]) - (3*Sqrt[
b]*Sqrt[c - (a*d)/b]*Sqrt[(c/(c - (a*d)/b) - (a*d)/(b*(c - (a*d)/b)))^(-1)]*Sqrt[c/(c - (a*d)/b) - (a*d)/(b*(c
- (a*d)/b))]*(5*a^4*C*d^4 - 20*a^3*b*d^3*(c*C + 2*B*d) + 30*a^2*b^2*d^2*(c^2*C - 4*B*c*d - 8*(A - C)*d^2) - 2
0*a*b^3*d*(c^3*C - 2*B*c^2*d + 8*c*(A - C)*d^2 - 16*B*d^3) + b^4*(5*c^4*C - 8*B*c^3*d + 16*c^2*(A - C)*d^2 + 6
4*B*c*d^3 + 128*(A - C)*d^4))*ArcSinh[(Sqrt[d]*Sqrt[a + b*Tan[e + f*x]])/(Sqrt[b]*Sqrt[c - (a*d)/b]*Sqrt[c/(c
- (a*d)/b) - (a*d)/(b*(c - (a*d)/b))])]*Sqrt[(c + d*Tan[e + f*x])/(c - (a*d)/b)])/(8*Sqrt[d]*Sqrt[c + d*Tan[e
+ f*x]]))/(b^2*f))/(2*d))/(3*d))/(4*d)

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Maple [F]  time = 180., size = 0, normalized size = 0. \begin{align*} \int \sqrt{c+d\tan \left ( fx+e \right ) } \left ( a+b\tan \left ( fx+e \right ) \right ) ^{{\frac{5}{2}}} \left ( A+B\tan \left ( fx+e \right ) +C \left ( \tan \left ( fx+e \right ) \right ) ^{2} \right ) \, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

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Maxima [F(-1)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Timed out

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Fricas [F(-1)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Timed out

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Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Timed out

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Giac [F(-1)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Timed out