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

Optimal. Leaf size=697 \[ -\frac{\left (2 a^2 b^2 d^2 \left (8 d^2 (A-C)+20 B c d+15 c^2 C\right )-4 a^3 b d^3 (2 B d+5 c C)+5 a^4 C d^4-4 a b^3 d \left (40 c d^2 (A-C)+30 B c^2 d-16 B d^3+5 c^3 C\right )+b^4 \left (-240 c^2 d^2 (A-C)+128 d^4 (A-C)-40 B c^3 d+320 B c d^3+5 c^4 C\right )\right ) \tanh ^{-1}\left (\frac{\sqrt{d} \sqrt{a+b \tan (e+f x)}}{\sqrt{b} \sqrt{c+d \tan (e+f x)}}\right )}{64 b^{7/2} d^{3/2} f}+\frac{\sqrt{a+b \tan (e+f x)} (c+d \tan (e+f x))^{3/2} \left (48 b d^2 (a B+A b-b C)-5 (b c-a d) (-a C d-8 b B d+b c C)\right )}{96 b^2 d f}+\frac{\sqrt{a+b \tan (e+f x)} \sqrt{c+d \tan (e+f x)} \left (64 b^2 d^2 (a A d+a B c-a C d+A b c-b B d-b c C)+(b c-a d) \left (48 b d^2 (a B+A b-b C)-5 (b c-a d) (-a C d-8 b B d+b c C)\right )\right )}{64 b^3 d f}-\frac{\sqrt{a-i b} (c-i d)^{5/2} (i A+B-i C) \tanh ^{-1}\left (\frac{\sqrt{c-i d} \sqrt{a+b \tan (e+f x)}}{\sqrt{a-i b} \sqrt{c+d \tan (e+f x)}}\right )}{f}+\frac{\sqrt{a+i b} (c+i d)^{5/2} (i A-B-i C) \tanh ^{-1}\left (\frac{\sqrt{c+i d} \sqrt{a+b \tan (e+f x)}}{\sqrt{a+i b} \sqrt{c+d \tan (e+f x)}}\right )}{f}-\frac{(-a C d-8 b B d+b c C) \sqrt{a+b \tan (e+f x)} (c+d \tan (e+f x))^{5/2}}{24 b d f}+\frac{C \sqrt{a+b \tan (e+f x)} (c+d \tan (e+f x))^{7/2}}{4 d f} \]

[Out]

-((Sqrt[a - I*b]*(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]])])/f) + (Sqrt[a + I*b]*(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]])])/f - ((5*a^4*C*d^4 - 4*a^3*b*d^3*(5*c*C + 2*B*
d) + 2*a^2*b^2*d^2*(15*c^2*C + 20*B*c*d + 8*(A - C)*d^2) - 4*a*b^3*d*(5*c^3*C + 30*B*c^2*d + 40*c*(A - C)*d^2
- 16*B*d^3) + b^4*(5*c^4*C - 40*B*c^3*d - 240*c^2*(A - C)*d^2 + 320*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^(7/2)*d^(3/2)*f) + ((64*b^2*d^2*(A*b*c
+ a*B*c - b*c*C + a*A*d - b*B*d - a*C*d) + (b*c - a*d)*(48*b*(A*b + a*B - b*C)*d^2 - 5*(b*c - a*d)*(b*c*C - 8
*b*B*d - a*C*d)))*Sqrt[a + b*Tan[e + f*x]]*Sqrt[c + d*Tan[e + f*x]])/(64*b^3*d*f) + ((48*b*(A*b + a*B - b*C)*d
^2 - 5*(b*c - a*d)*(b*c*C - 8*b*B*d - a*C*d))*Sqrt[a + b*Tan[e + f*x]]*(c + d*Tan[e + f*x])^(3/2))/(96*b^2*d*f
) - ((b*c*C - 8*b*B*d - a*C*d)*Sqrt[a + b*Tan[e + f*x]]*(c + d*Tan[e + f*x])^(5/2))/(24*b*d*f) + (C*Sqrt[a + b
*Tan[e + f*x]]*(c + d*Tan[e + f*x])^(7/2))/(4*d*f)

________________________________________________________________________________________

Rubi [A]  time = 10.4159, antiderivative size = 697, 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} \[ -\frac{\left (2 a^2 b^2 d^2 \left (8 d^2 (A-C)+20 B c d+15 c^2 C\right )-4 a^3 b d^3 (2 B d+5 c C)+5 a^4 C d^4-4 a b^3 d \left (40 c d^2 (A-C)+30 B c^2 d-16 B d^3+5 c^3 C\right )+b^4 \left (-240 c^2 d^2 (A-C)+128 d^4 (A-C)-40 B c^3 d+320 B c d^3+5 c^4 C\right )\right ) \tanh ^{-1}\left (\frac{\sqrt{d} \sqrt{a+b \tan (e+f x)}}{\sqrt{b} \sqrt{c+d \tan (e+f x)}}\right )}{64 b^{7/2} d^{3/2} f}+\frac{\sqrt{a+b \tan (e+f x)} (c+d \tan (e+f x))^{3/2} \left (48 b d^2 (a B+A b-b C)-5 (b c-a d) (-a C d-8 b B d+b c C)\right )}{96 b^2 d f}+\frac{\sqrt{a+b \tan (e+f x)} \sqrt{c+d \tan (e+f x)} \left (64 b^2 d^2 (a A d+a B c-a C d+A b c-b B d-b c C)+(b c-a d) \left (48 b d^2 (a B+A b-b C)-5 (b c-a d) (-a C d-8 b B d+b c C)\right )\right )}{64 b^3 d f}-\frac{\sqrt{a-i b} (c-i d)^{5/2} (i A+B-i C) \tanh ^{-1}\left (\frac{\sqrt{c-i d} \sqrt{a+b \tan (e+f x)}}{\sqrt{a-i b} \sqrt{c+d \tan (e+f x)}}\right )}{f}+\frac{\sqrt{a+i b} (c+i d)^{5/2} (i A-B-i C) \tanh ^{-1}\left (\frac{\sqrt{c+i d} \sqrt{a+b \tan (e+f x)}}{\sqrt{a+i b} \sqrt{c+d \tan (e+f x)}}\right )}{f}-\frac{(-a C d-8 b B d+b c C) \sqrt{a+b \tan (e+f x)} (c+d \tan (e+f x))^{5/2}}{24 b d f}+\frac{C \sqrt{a+b \tan (e+f x)} (c+d \tan (e+f x))^{7/2}}{4 d f} \]

Antiderivative was successfully verified.

[In]

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

[Out]

-((Sqrt[a - I*b]*(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]])])/f) + (Sqrt[a + I*b]*(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]])])/f - ((5*a^4*C*d^4 - 4*a^3*b*d^3*(5*c*C + 2*B*
d) + 2*a^2*b^2*d^2*(15*c^2*C + 20*B*c*d + 8*(A - C)*d^2) - 4*a*b^3*d*(5*c^3*C + 30*B*c^2*d + 40*c*(A - C)*d^2
- 16*B*d^3) + b^4*(5*c^4*C - 40*B*c^3*d - 240*c^2*(A - C)*d^2 + 320*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^(7/2)*d^(3/2)*f) + ((64*b^2*d^2*(A*b*c
+ a*B*c - b*c*C + a*A*d - b*B*d - a*C*d) + (b*c - a*d)*(48*b*(A*b + a*B - b*C)*d^2 - 5*(b*c - a*d)*(b*c*C - 8
*b*B*d - a*C*d)))*Sqrt[a + b*Tan[e + f*x]]*Sqrt[c + d*Tan[e + f*x]])/(64*b^3*d*f) + ((48*b*(A*b + a*B - b*C)*d
^2 - 5*(b*c - a*d)*(b*c*C - 8*b*B*d - a*C*d))*Sqrt[a + b*Tan[e + f*x]]*(c + d*Tan[e + f*x])^(3/2))/(96*b^2*d*f
) - ((b*c*C - 8*b*B*d - a*C*d)*Sqrt[a + b*Tan[e + f*x]]*(c + d*Tan[e + f*x])^(5/2))/(24*b*d*f) + (C*Sqrt[a + b
*Tan[e + f*x]]*(c + d*Tan[e + f*x])^(7/2))/(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 \sqrt{a+b \tan (e+f x)} (c+d \tan (e+f x))^{5/2} \left (A+B \tan (e+f x)+C \tan ^2(e+f x)\right ) \, dx &=\frac{C \sqrt{a+b \tan (e+f x)} (c+d \tan (e+f x))^{7/2}}{4 d f}+\frac{\int \frac{(c+d \tan (e+f x))^{5/2} \left (\frac{1}{2} (-b c C+a (8 A-7 C) d)+4 (A b+a B-b C) d \tan (e+f x)-\frac{1}{2} (b c C-8 b B d-a C d) \tan ^2(e+f x)\right )}{\sqrt{a+b \tan (e+f x)}} \, dx}{4 d}\\ &=-\frac{(b c C-8 b B d-a C d) \sqrt{a+b \tan (e+f x)} (c+d \tan (e+f x))^{5/2}}{24 b d f}+\frac{C \sqrt{a+b \tan (e+f x)} (c+d \tan (e+f x))^{7/2}}{4 d f}+\frac{\int \frac{(c+d \tan (e+f x))^{3/2} \left (\frac{1}{4} (-6 b c (b c C-a (8 A-7 C) d)+(b c+5 a d) (b c C-8 b B d-a C d))+12 b d (A b c+a B c-b c C+a A d-b B d-a C d) \tan (e+f x)+\frac{1}{4} \left (48 b (A b+a B-b C) d^2-5 (b c-a d) (b c C-8 b B d-a C d)\right ) \tan ^2(e+f x)\right )}{\sqrt{a+b \tan (e+f x)}} \, dx}{12 b d}\\ &=\frac{\left (48 b (A b+a B-b C) d^2-5 (b c-a d) (b c C-8 b B d-a C d)\right ) \sqrt{a+b \tan (e+f x)} (c+d \tan (e+f x))^{3/2}}{96 b^2 d f}-\frac{(b c C-8 b B d-a C d) \sqrt{a+b \tan (e+f x)} (c+d \tan (e+f x))^{5/2}}{24 b d f}+\frac{C \sqrt{a+b \tan (e+f x)} (c+d \tan (e+f x))^{7/2}}{4 d f}+\frac{\int \frac{\sqrt{c+d \tan (e+f x)} \left (\frac{3}{8} \left (5 a^3 C d^3-a^2 b d^2 (15 c C+8 B d)-b^3 c \left (5 c^2 C+24 B c d+16 (A-C) d^2\right )+a b^2 d \left (64 A c^2-49 c^2 C-96 B c d-48 A d^2+48 C d^2\right )\right )+24 b^2 d \left (2 a A c d-2 a c C d+A b \left (c^2-d^2\right )+a B \left (c^2-d^2\right )-b \left (c^2 C+2 B c d-C d^2\right )\right ) \tan (e+f x)+\frac{3}{8} \left (64 b^2 d^2 (A b c+a B c-b c C+a A d-b B d-a C d)+(b c-a d) \left (48 b (A b+a B-b C) d^2-5 (b c-a d) (b c C-8 b B d-a C d)\right )\right ) \tan ^2(e+f x)\right )}{\sqrt{a+b \tan (e+f x)}} \, dx}{24 b^2 d}\\ &=\frac{\left (64 b^2 d^2 (A b c+a B c-b c C+a A d-b B d-a C d)+(b c-a d) \left (48 b (A b+a B-b C) d^2-5 (b c-a d) (b c C-8 b B d-a C d)\right )\right ) \sqrt{a+b \tan (e+f x)} \sqrt{c+d \tan (e+f x)}}{64 b^3 d f}+\frac{\left (48 b (A b+a B-b C) d^2-5 (b c-a d) (b c C-8 b B d-a C d)\right ) \sqrt{a+b \tan (e+f x)} (c+d \tan (e+f x))^{3/2}}{96 b^2 d f}-\frac{(b c C-8 b B d-a C d) \sqrt{a+b \tan (e+f x)} (c+d \tan (e+f x))^{5/2}}{24 b d f}+\frac{C \sqrt{a+b \tan (e+f x)} (c+d \tan (e+f x))^{7/2}}{4 d f}+\frac{\int \frac{-\frac{3}{16} \left (5 a^4 C d^4-4 a^3 b d^3 (5 c C+2 B d)+2 a^2 b^2 d^2 \left (15 c^2 C+20 B c d+8 (A-C) d^2\right )+b^4 c \left (5 c^3 C+88 B c^2 d+144 c (A-C) d^2-64 B d^3\right )+4 a b^3 d \left (27 c^3 C+66 B c^2 d-56 c C d^2-16 B d^3-8 A \left (4 c^3-7 c d^2\right )\right )\right )+24 b^3 d \left (A \left (b c^3+3 a c^2 d-3 b c d^2-a d^3\right )-b \left (c^3 C+3 B c^2 d-3 c C d^2-B d^3\right )+a \left (B c^3-3 c^2 C d-3 B c d^2+C d^3\right )\right ) \tan (e+f x)-\frac{3}{16} \left (5 a^4 C d^4-4 a^3 b d^3 (5 c C+2 B d)+2 a^2 b^2 d^2 \left (15 c^2 C+20 B c d+8 (A-C) d^2\right )-4 a b^3 d \left (5 c^3 C+30 B c^2 d+40 c (A-C) d^2-16 B d^3\right )+b^4 \left (5 c^4 C-40 B c^3 d-240 c^2 (A-C) d^2+320 B c d^3+128 (A-C) d^4\right )\right ) \tan ^2(e+f x)}{\sqrt{a+b \tan (e+f x)} \sqrt{c+d \tan (e+f x)}} \, dx}{24 b^3 d}\\ &=\frac{\left (64 b^2 d^2 (A b c+a B c-b c C+a A d-b B d-a C d)+(b c-a d) \left (48 b (A b+a B-b C) d^2-5 (b c-a d) (b c C-8 b B d-a C d)\right )\right ) \sqrt{a+b \tan (e+f x)} \sqrt{c+d \tan (e+f x)}}{64 b^3 d f}+\frac{\left (48 b (A b+a B-b C) d^2-5 (b c-a d) (b c C-8 b B d-a C d)\right ) \sqrt{a+b \tan (e+f x)} (c+d \tan (e+f x))^{3/2}}{96 b^2 d f}-\frac{(b c C-8 b B d-a C d) \sqrt{a+b \tan (e+f x)} (c+d \tan (e+f x))^{5/2}}{24 b d f}+\frac{C \sqrt{a+b \tan (e+f x)} (c+d \tan (e+f x))^{7/2}}{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 (5 c C+2 B d)+2 a^2 b^2 d^2 \left (15 c^2 C+20 B c d+8 (A-C) d^2\right )+b^4 c \left (5 c^3 C+88 B c^2 d+144 c (A-C) d^2-64 B d^3\right )+4 a b^3 d \left (27 c^3 C+66 B c^2 d-56 c C d^2-16 B d^3-8 A \left (4 c^3-7 c d^2\right )\right )\right )+24 b^3 d \left (A \left (b c^3+3 a c^2 d-3 b c d^2-a d^3\right )-b \left (c^3 C+3 B c^2 d-3 c C d^2-B d^3\right )+a \left (B c^3-3 c^2 C d-3 B c d^2+C d^3\right )\right ) x-\frac{3}{16} \left (5 a^4 C d^4-4 a^3 b d^3 (5 c C+2 B d)+2 a^2 b^2 d^2 \left (15 c^2 C+20 B c d+8 (A-C) d^2\right )-4 a b^3 d \left (5 c^3 C+30 B c^2 d+40 c (A-C) d^2-16 B d^3\right )+b^4 \left (5 c^4 C-40 B c^3 d-240 c^2 (A-C) d^2+320 B c d^3+128 (A-C) d^4\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^3 d f}\\ &=\frac{\left (64 b^2 d^2 (A b c+a B c-b c C+a A d-b B d-a C d)+(b c-a d) \left (48 b (A b+a B-b C) d^2-5 (b c-a d) (b c C-8 b B d-a C d)\right )\right ) \sqrt{a+b \tan (e+f x)} \sqrt{c+d \tan (e+f x)}}{64 b^3 d f}+\frac{\left (48 b (A b+a B-b C) d^2-5 (b c-a d) (b c C-8 b B d-a C d)\right ) \sqrt{a+b \tan (e+f x)} (c+d \tan (e+f x))^{3/2}}{96 b^2 d f}-\frac{(b c C-8 b B d-a C d) \sqrt{a+b \tan (e+f x)} (c+d \tan (e+f x))^{5/2}}{24 b d f}+\frac{C \sqrt{a+b \tan (e+f x)} (c+d \tan (e+f x))^{7/2}}{4 d f}+\frac{\operatorname{Subst}\left (\int \left (-\frac{3 \left (5 a^4 C d^4-4 a^3 b d^3 (5 c C+2 B d)+2 a^2 b^2 d^2 \left (15 c^2 C+20 B c d+8 (A-C) d^2\right )-4 a b^3 d \left (5 c^3 C+30 B c^2 d+40 c (A-C) d^2-16 B d^3\right )+b^4 \left (5 c^4 C-40 B c^3 d-240 c^2 (A-C) d^2+320 B c d^3+128 (A-C) d^4\right )\right )}{16 \sqrt{a+b x} \sqrt{c+d x}}+\frac{24 \left (-b^3 d \left (a \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 )+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 d \left (A \left (b c^3+3 a c^2 d-3 b c d^2-a d^3\right )-b \left (c^3 C+3 B c^2 d-3 c C d^2-B d^3\right )+a \left (B c^3-3 c^2 C d-3 B c d^2+C d^3\right )\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^3 d f}\\ &=\frac{\left (64 b^2 d^2 (A b c+a B c-b c C+a A d-b B d-a C d)+(b c-a d) \left (48 b (A b+a B-b C) d^2-5 (b c-a d) (b c C-8 b B d-a C d)\right )\right ) \sqrt{a+b \tan (e+f x)} \sqrt{c+d \tan (e+f x)}}{64 b^3 d f}+\frac{\left (48 b (A b+a B-b C) d^2-5 (b c-a d) (b c C-8 b B d-a C d)\right ) \sqrt{a+b \tan (e+f x)} (c+d \tan (e+f x))^{3/2}}{96 b^2 d f}-\frac{(b c C-8 b B d-a C d) \sqrt{a+b \tan (e+f x)} (c+d \tan (e+f x))^{5/2}}{24 b d f}+\frac{C \sqrt{a+b \tan (e+f x)} (c+d \tan (e+f x))^{7/2}}{4 d f}+\frac{\operatorname{Subst}\left (\int \frac{-b^3 d \left (a \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 )+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 d \left (A \left (b c^3+3 a c^2 d-3 b c d^2-a d^3\right )-b \left (c^3 C+3 B c^2 d-3 c C d^2-B d^3\right )+a \left (B c^3-3 c^2 C d-3 B c d^2+C d^3\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 d f}-\frac{\left (5 a^4 C d^4-4 a^3 b d^3 (5 c C+2 B d)+2 a^2 b^2 d^2 \left (15 c^2 C+20 B c d+8 (A-C) d^2\right )-4 a b^3 d \left (5 c^3 C+30 B c^2 d+40 c (A-C) d^2-16 B d^3\right )+b^4 \left (5 c^4 C-40 B c^3 d-240 c^2 (A-C) d^2+320 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^3 d f}\\ &=\frac{\left (64 b^2 d^2 (A b c+a B c-b c C+a A d-b B d-a C d)+(b c-a d) \left (48 b (A b+a B-b C) d^2-5 (b c-a d) (b c C-8 b B d-a C d)\right )\right ) \sqrt{a+b \tan (e+f x)} \sqrt{c+d \tan (e+f x)}}{64 b^3 d f}+\frac{\left (48 b (A b+a B-b C) d^2-5 (b c-a d) (b c C-8 b B d-a C d)\right ) \sqrt{a+b \tan (e+f x)} (c+d \tan (e+f x))^{3/2}}{96 b^2 d f}-\frac{(b c C-8 b B d-a C d) \sqrt{a+b \tan (e+f x)} (c+d \tan (e+f x))^{5/2}}{24 b d f}+\frac{C \sqrt{a+b \tan (e+f x)} (c+d \tan (e+f x))^{7/2}}{4 d f}+\frac{\operatorname{Subst}\left (\int \left (\frac{-b^3 d \left (A \left (b c^3+3 a c^2 d-3 b c d^2-a d^3\right )-b \left (c^3 C+3 B c^2 d-3 c C d^2-B d^3\right )+a \left (B c^3-3 c^2 C d-3 B c d^2+C d^3\right )\right )-i b^3 d \left (a \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 )+b \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{b^3 d \left (A \left (b c^3+3 a c^2 d-3 b c d^2-a d^3\right )-b \left (c^3 C+3 B c^2 d-3 c C d^2-B d^3\right )+a \left (B c^3-3 c^2 C d-3 B c d^2+C d^3\right )\right )-i b^3 d \left (a \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 )+b \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 d f}-\frac{\left (5 a^4 C d^4-4 a^3 b d^3 (5 c C+2 B d)+2 a^2 b^2 d^2 \left (15 c^2 C+20 B c d+8 (A-C) d^2\right )-4 a b^3 d \left (5 c^3 C+30 B c^2 d+40 c (A-C) d^2-16 B d^3\right )+b^4 \left (5 c^4 C-40 B c^3 d-240 c^2 (A-C) d^2+320 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^4 d f}\\ &=\frac{\left (64 b^2 d^2 (A b c+a B c-b c C+a A d-b B d-a C d)+(b c-a d) \left (48 b (A b+a B-b C) d^2-5 (b c-a d) (b c C-8 b B d-a C d)\right )\right ) \sqrt{a+b \tan (e+f x)} \sqrt{c+d \tan (e+f x)}}{64 b^3 d f}+\frac{\left (48 b (A b+a B-b C) d^2-5 (b c-a d) (b c C-8 b B d-a C d)\right ) \sqrt{a+b \tan (e+f x)} (c+d \tan (e+f x))^{3/2}}{96 b^2 d f}-\frac{(b c C-8 b B d-a C d) \sqrt{a+b \tan (e+f x)} (c+d \tan (e+f x))^{5/2}}{24 b d f}+\frac{C \sqrt{a+b \tan (e+f x)} (c+d \tan (e+f x))^{7/2}}{4 d f}+\frac{\left ((i a+b) (A-i B-C) (c-i d)^3\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 ((i a-b) (A+i B-C) (c+i d)^3\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^4-4 a^3 b d^3 (5 c C+2 B d)+2 a^2 b^2 d^2 \left (15 c^2 C+20 B c d+8 (A-C) d^2\right )-4 a b^3 d \left (5 c^3 C+30 B c^2 d+40 c (A-C) d^2-16 B d^3\right )+b^4 \left (5 c^4 C-40 B c^3 d-240 c^2 (A-C) d^2+320 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^4 d f}\\ &=-\frac{\left (5 a^4 C d^4-4 a^3 b d^3 (5 c C+2 B d)+2 a^2 b^2 d^2 \left (15 c^2 C+20 B c d+8 (A-C) d^2\right )-4 a b^3 d \left (5 c^3 C+30 B c^2 d+40 c (A-C) d^2-16 B d^3\right )+b^4 \left (5 c^4 C-40 B c^3 d-240 c^2 (A-C) d^2+320 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^{7/2} d^{3/2} f}+\frac{\left (64 b^2 d^2 (A b c+a B c-b c C+a A d-b B d-a C d)+(b c-a d) \left (48 b (A b+a B-b C) d^2-5 (b c-a d) (b c C-8 b B d-a C d)\right )\right ) \sqrt{a+b \tan (e+f x)} \sqrt{c+d \tan (e+f x)}}{64 b^3 d f}+\frac{\left (48 b (A b+a B-b C) d^2-5 (b c-a d) (b c C-8 b B d-a C d)\right ) \sqrt{a+b \tan (e+f x)} (c+d \tan (e+f x))^{3/2}}{96 b^2 d f}-\frac{(b c C-8 b B d-a C d) \sqrt{a+b \tan (e+f x)} (c+d \tan (e+f x))^{5/2}}{24 b d f}+\frac{C \sqrt{a+b \tan (e+f x)} (c+d \tan (e+f x))^{7/2}}{4 d f}+\frac{\left ((i a+b) (A-i B-C) (c-i d)^3\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 ((i a-b) (A+i B-C) (c+i d)^3\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{\sqrt{a-i b} (i A+B-i C) (c-i d)^{5/2} \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{\sqrt{a+i b} (B-i (A-C)) (c+i d)^{5/2} \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^4-4 a^3 b d^3 (5 c C+2 B d)+2 a^2 b^2 d^2 \left (15 c^2 C+20 B c d+8 (A-C) d^2\right )-4 a b^3 d \left (5 c^3 C+30 B c^2 d+40 c (A-C) d^2-16 B d^3\right )+b^4 \left (5 c^4 C-40 B c^3 d-240 c^2 (A-C) d^2+320 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^{7/2} d^{3/2} f}+\frac{\left (64 b^2 d^2 (A b c+a B c-b c C+a A d-b B d-a C d)+(b c-a d) \left (48 b (A b+a B-b C) d^2-5 (b c-a d) (b c C-8 b B d-a C d)\right )\right ) \sqrt{a+b \tan (e+f x)} \sqrt{c+d \tan (e+f x)}}{64 b^3 d f}+\frac{\left (48 b (A b+a B-b C) d^2-5 (b c-a d) (b c C-8 b B d-a C d)\right ) \sqrt{a+b \tan (e+f x)} (c+d \tan (e+f x))^{3/2}}{96 b^2 d f}-\frac{(b c C-8 b B d-a C d) \sqrt{a+b \tan (e+f x)} (c+d \tan (e+f x))^{5/2}}{24 b d f}+\frac{C \sqrt{a+b \tan (e+f x)} (c+d \tan (e+f x))^{7/2}}{4 d f}\\ \end{align*}

Mathematica [A]  time = 9.26618, size = 1261, normalized size = 1.81 \[ \frac{C \sqrt{a+b \tan (e+f x)} (c+d \tan (e+f x))^{7/2}}{4 d f}+\frac{\frac{(-b c C+a d C+8 b B d) \sqrt{a+b \tan (e+f x)} (c+d \tan (e+f x))^{5/2}}{6 b f}+\frac{\frac{\left (48 b (A b-C b+a B) d^2-5 (b c-a d) (b c C-a d C-8 b B d)\right ) \sqrt{a+b \tan (e+f x)} (c+d \tan (e+f x))^{3/2}}{8 b f}+\frac{\frac{\sqrt{a+b \tan (e+f x)} \sqrt{c+d \tan (e+f x)} \left (24 b^2 d^2 (A b c+a B c-b C c+a A d-b B d-a C d)-\frac{3}{8} (a d-b c) \left (48 b (A b-C b+a B) d^2-5 (b c-a d) (b c C-a d C-8 b B d)\right )\right )}{b f}+\frac{-\frac{24 d \left (\sqrt{-b^2} \left (b (A-C) d \left (3 c^2-d^2\right )+b B \left (c^3-3 c d^2\right )-a \left (A c^3-C c^3-3 B d c^2-3 A d^2 c+3 C d^2 c+B d^3\right )\right )-b \left (A \left (b c^3+3 a d c^2-3 b d^2 c-a d^3\right )-b \left (C c^3+3 B d c^2-3 C d^2 c-B d^3\right )+a \left (B c^3-3 C d c^2-3 B d^2 c+C d^3\right )\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 ) b^3}{\sqrt{\sqrt{-b^2}-a} \sqrt{c+\frac{b d}{\sqrt{-b^2}}}}-\frac{24 d \left (\sqrt{-b^2} \left (b (A-C) d \left (3 c^2-d^2\right )+b B \left (c^3-3 c d^2\right )-a \left (A c^3-C c^3-3 B d c^2-3 A d^2 c+3 C d^2 c+B d^3\right )\right )+b \left (A \left (b c^3+3 a d c^2-3 b d^2 c-a d^3\right )-b \left (C c^3+3 B d c^2-3 C d^2 c-B d^3\right )+a \left (B c^3-3 C d c^2-3 B d^2 c+C d^3\right )\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 ) b^3}{\sqrt{a+\sqrt{-b^2}} \sqrt{-\frac{b c+\sqrt{-b^2} d}{b}}}-\frac{3 \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^4-40 B d c^3-240 (A-C) d^2 c^2+320 B d^3 c+128 (A-C) d^4\right ) b^4-4 a d \left (5 C c^3+30 B d c^2+40 (A-C) d^2 c-16 B d^3\right ) b^3+2 a^2 d^2 \left (15 C c^2+20 B d c+8 (A-C) d^2\right ) b^2-4 a^3 d^3 (5 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}}} \sqrt{b}}{8 \sqrt{d} \sqrt{c+d \tan (e+f x)}}}{b^2 f}}{2 b}}{3 b}}{4 d} \]

Warning: Unable to verify antiderivative.

[In]

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

[Out]

(C*Sqrt[a + b*Tan[e + f*x]]*(c + d*Tan[e + f*x])^(7/2))/(4*d*f) + (((-(b*c*C) + 8*b*B*d + a*C*d)*Sqrt[a + b*Ta
n[e + f*x]]*(c + d*Tan[e + f*x])^(5/2))/(6*b*f) + (((48*b*(A*b + a*B - b*C)*d^2 - 5*(b*c - a*d)*(b*c*C - 8*b*B
*d - a*C*d))*Sqrt[a + b*Tan[e + f*x]]*(c + d*Tan[e + f*x])^(3/2))/(8*b*f) + (((24*b^2*d^2*(A*b*c + a*B*c - b*c
*C + a*A*d - b*B*d - a*C*d) - (3*(-(b*c) + a*d)*(48*b*(A*b + a*B - b*C)*d^2 - 5*(b*c - a*d)*(b*c*C - 8*b*B*d -
a*C*d)))/8)*Sqrt[a + b*Tan[e + f*x]]*Sqrt[c + d*Tan[e + f*x]])/(b*f) + ((-24*b^3*d*(Sqrt[-b^2]*(b*(A - C)*d*(
3*c^2 - d^2) + b*B*(c^3 - 3*c*d^2) - a*(A*c^3 - c^3*C - 3*B*c^2*d - 3*A*c*d^2 + 3*c*C*d^2 + B*d^3)) - b*(A*(b*
c^3 + 3*a*c^2*d - 3*b*c*d^2 - a*d^3) - b*(c^3*C + 3*B*c^2*d - 3*c*C*d^2 - B*d^3) + a*(B*c^3 - 3*c^2*C*d - 3*B*
c*d^2 + C*d^3)))*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^3*d*(Sqrt[-b^2]*(b*(A - C)*d*(3*
c^2 - d^2) + b*B*(c^3 - 3*c*d^2) - a*(A*c^3 - c^3*C - 3*B*c^2*d - 3*A*c*d^2 + 3*c*C*d^2 + B*d^3)) + b*(A*(b*c^
3 + 3*a*c^2*d - 3*b*c*d^2 - a*d^3) - b*(c^3*C + 3*B*c^2*d - 3*c*C*d^2 - B*d^3) + a*(B*c^3 - 3*c^2*C*d - 3*B*c*
d^2 + C*d^3)))*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]*Sqr
t[(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 - 4*a^3*b*d^3*(5*c*C + 2*B*d) + 2*a^2*b^2*d^2*(15*c^2*C + 20*B*c*d + 8*(A - C)*d^2) - 4*a*b^3*d*(5*c^3*C +
30*B*c^2*d + 40*c*(A - C)*d^2 - 16*B*d^3) + b^4*(5*c^4*C - 40*B*c^3*d - 240*c^2*(A - C)*d^2 + 320*B*c*d^3 + 12
8*(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*b))/(3*b))/(4*d)

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Maple [F]  time = 180., size = 0, normalized size = 0. \begin{align*} \int \sqrt{a+b\tan \left ( fx+e \right ) } \left ( c+d\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((a+b*tan(f*x+e))^(1/2)*(c+d*tan(f*x+e))^(5/2)*(A+B*tan(f*x+e)+C*tan(f*x+e)^2),x)

[Out]

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

[Out]

Timed out