3.134 \(\int \frac{\sqrt{c+d \tan (e+f x)} (A+B \tan (e+f x)+C \tan ^2(e+f x))}{(a+b \tan (e+f x))^{7/2}} \, dx\)

Optimal. Leaf size=597 \[ \frac{2 \sqrt{c+d \tan (e+f x)} \left (a^3 b^3 \left (80 c d (A-C)+B \left (15 c^2-49 d^2\right )\right )-a^2 b^4 \left (45 A c^2-29 A d^2-90 B c d-45 c^2 C+23 C d^2\right )-a^4 b^2 d (33 A d+25 B c-39 C d)+8 a^5 b B d^2+2 a^6 C d^2-a b^5 \left (40 c d (A-C)+B \left (45 c^2-3 d^2\right )\right )-b^6 \left (5 c (B d+3 c C)-A \left (15 c^2+2 d^2\right )\right )\right )}{15 b f \left (a^2+b^2\right )^3 (b c-a d)^2 \sqrt{a+b \tan (e+f x)}}-\frac{2 \left (A b^2-a (b B-a C)\right ) \sqrt{c+d \tan (e+f x)}}{5 b f \left (a^2+b^2\right ) (a+b \tan (e+f x))^{5/2}}-\frac{2 \sqrt{c+d \tan (e+f x)} \left (-a^2 b^2 (9 A d+5 B c-11 C d)+4 a^3 b B d+a^4 C d+2 a b^3 (5 A c-3 B d-5 c C)+b^4 (A d+5 B c)\right )}{15 b f \left (a^2+b^2\right )^2 (b c-a d) (a+b \tan (e+f x))^{3/2}}-\frac{\sqrt{c-i d} (i A+B-i C) \tanh ^{-1}\left (\frac{\sqrt{c-i d} \sqrt{a+b \tan (e+f x)}}{\sqrt{a-i b} \sqrt{c+d \tan (e+f x)}}\right )}{f (a-i b)^{7/2}}-\frac{\sqrt{c+i d} (B-i (A-C)) \tanh ^{-1}\left (\frac{\sqrt{c+i d} \sqrt{a+b \tan (e+f x)}}{\sqrt{a+i b} \sqrt{c+d \tan (e+f x)}}\right )}{f (a+i b)^{7/2}} \]

[Out]

-(((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*Ta
n[e + f*x]])])/((a - I*b)^(7/2)*f)) - ((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]])])/((a + I*b)^(7/2)*f) - (2*(A*b^2 - a*(b*B - a*C))*Sqrt[c + d
*Tan[e + f*x]])/(5*b*(a^2 + b^2)*f*(a + b*Tan[e + f*x])^(5/2)) - (2*(4*a^3*b*B*d + a^4*C*d + b^4*(5*B*c + A*d)
+ 2*a*b^3*(5*A*c - 5*c*C - 3*B*d) - a^2*b^2*(5*B*c + 9*A*d - 11*C*d))*Sqrt[c + d*Tan[e + f*x]])/(15*b*(a^2 +
b^2)^2*(b*c - a*d)*f*(a + b*Tan[e + f*x])^(3/2)) + (2*(8*a^5*b*B*d^2 + 2*a^6*C*d^2 - a^4*b^2*d*(25*B*c + 33*A*
d - 39*C*d) - a^2*b^4*(45*A*c^2 - 45*c^2*C - 90*B*c*d - 29*A*d^2 + 23*C*d^2) + a^3*b^3*(80*c*(A - C)*d + B*(15
*c^2 - 49*d^2)) - a*b^5*(40*c*(A - C)*d + B*(45*c^2 - 3*d^2)) - b^6*(5*c*(3*c*C + B*d) - A*(15*c^2 + 2*d^2)))*
Sqrt[c + d*Tan[e + f*x]])/(15*b*(a^2 + b^2)^3*(b*c - a*d)^2*f*Sqrt[a + b*Tan[e + f*x]])

________________________________________________________________________________________

Rubi [A]  time = 3.58853, antiderivative size = 597, normalized size of antiderivative = 1., number of steps used = 10, number of rules used = 6, integrand size = 49, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.122, Rules used = {3645, 3649, 3616, 3615, 93, 208} \[ \frac{2 \sqrt{c+d \tan (e+f x)} \left (a^3 b^3 \left (80 c d (A-C)+B \left (15 c^2-49 d^2\right )\right )-a^2 b^4 \left (45 A c^2-29 A d^2-90 B c d-45 c^2 C+23 C d^2\right )-a^4 b^2 d (33 A d+25 B c-39 C d)+8 a^5 b B d^2+2 a^6 C d^2-a b^5 \left (40 c d (A-C)+B \left (45 c^2-3 d^2\right )\right )-b^6 \left (5 c (B d+3 c C)-A \left (15 c^2+2 d^2\right )\right )\right )}{15 b f \left (a^2+b^2\right )^3 (b c-a d)^2 \sqrt{a+b \tan (e+f x)}}-\frac{2 \left (A b^2-a (b B-a C)\right ) \sqrt{c+d \tan (e+f x)}}{5 b f \left (a^2+b^2\right ) (a+b \tan (e+f x))^{5/2}}-\frac{2 \sqrt{c+d \tan (e+f x)} \left (-a^2 b^2 (9 A d+5 B c-11 C d)+4 a^3 b B d+a^4 C d+2 a b^3 (5 A c-3 B d-5 c C)+b^4 (A d+5 B c)\right )}{15 b f \left (a^2+b^2\right )^2 (b c-a d) (a+b \tan (e+f x))^{3/2}}-\frac{\sqrt{c-i d} (i A+B-i C) \tanh ^{-1}\left (\frac{\sqrt{c-i d} \sqrt{a+b \tan (e+f x)}}{\sqrt{a-i b} \sqrt{c+d \tan (e+f x)}}\right )}{f (a-i b)^{7/2}}-\frac{\sqrt{c+i d} (B-i (A-C)) \tanh ^{-1}\left (\frac{\sqrt{c+i d} \sqrt{a+b \tan (e+f x)}}{\sqrt{a+i b} \sqrt{c+d \tan (e+f x)}}\right )}{f (a+i b)^{7/2}} \]

Antiderivative was successfully verified.

[In]

Int[(Sqrt[c + d*Tan[e + f*x]]*(A + B*Tan[e + f*x] + C*Tan[e + f*x]^2))/(a + b*Tan[e + f*x])^(7/2),x]

[Out]

-(((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*Ta
n[e + f*x]])])/((a - I*b)^(7/2)*f)) - ((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]])])/((a + I*b)^(7/2)*f) - (2*(A*b^2 - a*(b*B - a*C))*Sqrt[c + d
*Tan[e + f*x]])/(5*b*(a^2 + b^2)*f*(a + b*Tan[e + f*x])^(5/2)) - (2*(4*a^3*b*B*d + a^4*C*d + b^4*(5*B*c + A*d)
+ 2*a*b^3*(5*A*c - 5*c*C - 3*B*d) - a^2*b^2*(5*B*c + 9*A*d - 11*C*d))*Sqrt[c + d*Tan[e + f*x]])/(15*b*(a^2 +
b^2)^2*(b*c - a*d)*f*(a + b*Tan[e + f*x])^(3/2)) + (2*(8*a^5*b*B*d^2 + 2*a^6*C*d^2 - a^4*b^2*d*(25*B*c + 33*A*
d - 39*C*d) - a^2*b^4*(45*A*c^2 - 45*c^2*C - 90*B*c*d - 29*A*d^2 + 23*C*d^2) + a^3*b^3*(80*c*(A - C)*d + B*(15
*c^2 - 49*d^2)) - a*b^5*(40*c*(A - C)*d + B*(45*c^2 - 3*d^2)) - b^6*(5*c*(3*c*C + B*d) - A*(15*c^2 + 2*d^2)))*
Sqrt[c + d*Tan[e + f*x]])/(15*b*(a^2 + b^2)^3*(b*c - a*d)^2*f*Sqrt[a + b*Tan[e + f*x]])

Rule 3645

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*T
an[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 3649

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*b^2 - a*(b*B - a*C))*(a + b*T
an[e + f*x])^(m + 1)*(c + d*Tan[e + f*x])^(n + 1))/(f*(m + 1)*(b*c - a*d)*(a^2 + b^2)), x] + Dist[1/((m + 1)*(
b*c - a*d)*(a^2 + b^2)), Int[(a + b*Tan[e + f*x])^(m + 1)*(c + d*Tan[e + f*x])^n*Simp[A*(a*(b*c - a*d)*(m + 1)
- b^2*d*(m + n + 2)) + (b*B - a*C)*(b*c*(m + 1) + a*d*(n + 1)) - (m + 1)*(b*c - a*d)*(A*b - a*B - b*C)*Tan[e
+ f*x] - d*(A*b^2 - a*(b*B - a*C))*(m + n + 2)*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] && LtQ[m, -1] &&  !(ILtQ[n, -1] && ( !I
ntegerQ[m] || (EqQ[c, 0] && NeQ[a, 0])))

Rule 3616

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

Rule 3615

Int[((a_.) + (b_.)*tan[(e_.) + (f_.)*(x_)])^(m_)*((A_.) + (B_.)*tan[(e_.) + (f_.)*(x_)])*((c_.) + (d_.)*tan[(e
_.) + (f_.)*(x_)])^(n_), x_Symbol] :> Dist[A^2/f, Subst[Int[((a + b*x)^m*(c + d*x)^n)/(A - B*x), x], x, Tan[e
+ f*x]], x] /; FreeQ[{a, b, c, d, e, f, A, B, m, n}, x] && NeQ[b*c - a*d, 0] && NeQ[a^2 + b^2, 0] && EqQ[A^2 +
B^2, 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 \frac{\sqrt{c+d \tan (e+f x)} \left (A+B \tan (e+f x)+C \tan ^2(e+f x)\right )}{(a+b \tan (e+f x))^{7/2}} \, dx &=-\frac{2 \left (A b^2-a (b B-a C)\right ) \sqrt{c+d \tan (e+f x)}}{5 b \left (a^2+b^2\right ) f (a+b \tan (e+f x))^{5/2}}+\frac{2 \int \frac{\frac{1}{2} ((b B-a C) (5 b c-a d)+A b (5 a c+b d))-\frac{5}{2} b ((A-C) (b c-a d)-B (a c+b d)) \tan (e+f x)-\frac{1}{2} \left (4 A b^2-4 a b B-a^2 C-5 b^2 C\right ) d \tan ^2(e+f x)}{(a+b \tan (e+f x))^{5/2} \sqrt{c+d \tan (e+f x)}} \, dx}{5 b \left (a^2+b^2\right )}\\ &=-\frac{2 \left (A b^2-a (b B-a C)\right ) \sqrt{c+d \tan (e+f x)}}{5 b \left (a^2+b^2\right ) f (a+b \tan (e+f x))^{5/2}}-\frac{2 \left (4 a^3 b B d+a^4 C d+b^4 (5 B c+A d)+2 a b^3 (5 A c-5 c C-3 B d)-a^2 b^2 (5 B c+9 A d-11 C d)\right ) \sqrt{c+d \tan (e+f x)}}{15 b \left (a^2+b^2\right )^2 (b c-a d) f (a+b \tan (e+f x))^{3/2}}-\frac{4 \int \frac{\frac{1}{4} \left (2 \left (b^2 d-\frac{3}{2} a (b c-a d)\right ) ((b B-a C) (5 b c-a d)+A b (5 a c+b d))+(3 b c-a d) \left (4 a^2 b B d+a^3 C d+A b^2 (5 b c-9 a d)-5 b^3 (c C+B d)-5 a b^2 (B c-2 C d)\right )\right )+\frac{15}{4} b (b c-a d) \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{1}{2} d \left (4 a^3 b B d+a^4 C d+b^4 (5 B c+A d)+2 a b^3 (5 A c-5 c C-3 B d)-a^2 b^2 (5 B c+9 A d-11 C d)\right ) \tan ^2(e+f x)}{(a+b \tan (e+f x))^{3/2} \sqrt{c+d \tan (e+f x)}} \, dx}{15 b \left (a^2+b^2\right )^2 (b c-a d)}\\ &=-\frac{2 \left (A b^2-a (b B-a C)\right ) \sqrt{c+d \tan (e+f x)}}{5 b \left (a^2+b^2\right ) f (a+b \tan (e+f x))^{5/2}}-\frac{2 \left (4 a^3 b B d+a^4 C d+b^4 (5 B c+A d)+2 a b^3 (5 A c-5 c C-3 B d)-a^2 b^2 (5 B c+9 A d-11 C d)\right ) \sqrt{c+d \tan (e+f x)}}{15 b \left (a^2+b^2\right )^2 (b c-a d) f (a+b \tan (e+f x))^{3/2}}+\frac{2 \left (8 a^5 b B d^2+2 a^6 C d^2-a^4 b^2 d (25 B c+33 A d-39 C d)-a^2 b^4 \left (45 A c^2-45 c^2 C-90 B c d-29 A d^2+23 C d^2\right )+a^3 b^3 \left (80 c (A-C) d+B \left (15 c^2-49 d^2\right )\right )-a b^5 \left (40 c (A-C) d+B \left (45 c^2-3 d^2\right )\right )-b^6 \left (5 c (3 c C+B d)-A \left (15 c^2+2 d^2\right )\right )\right ) \sqrt{c+d \tan (e+f x)}}{15 b \left (a^2+b^2\right )^3 (b c-a d)^2 f \sqrt{a+b \tan (e+f x)}}+\frac{8 \int \frac{\frac{15}{8} b (b c-a d)^2 \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 )-\frac{15}{8} b (b c-a d)^2 \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)}{\sqrt{a+b \tan (e+f x)} \sqrt{c+d \tan (e+f x)}} \, dx}{15 b \left (a^2+b^2\right )^3 (b c-a d)^2}\\ &=-\frac{2 \left (A b^2-a (b B-a C)\right ) \sqrt{c+d \tan (e+f x)}}{5 b \left (a^2+b^2\right ) f (a+b \tan (e+f x))^{5/2}}-\frac{2 \left (4 a^3 b B d+a^4 C d+b^4 (5 B c+A d)+2 a b^3 (5 A c-5 c C-3 B d)-a^2 b^2 (5 B c+9 A d-11 C d)\right ) \sqrt{c+d \tan (e+f x)}}{15 b \left (a^2+b^2\right )^2 (b c-a d) f (a+b \tan (e+f x))^{3/2}}+\frac{2 \left (8 a^5 b B d^2+2 a^6 C d^2-a^4 b^2 d (25 B c+33 A d-39 C d)-a^2 b^4 \left (45 A c^2-45 c^2 C-90 B c d-29 A d^2+23 C d^2\right )+a^3 b^3 \left (80 c (A-C) d+B \left (15 c^2-49 d^2\right )\right )-a b^5 \left (40 c (A-C) d+B \left (45 c^2-3 d^2\right )\right )-b^6 \left (5 c (3 c C+B d)-A \left (15 c^2+2 d^2\right )\right )\right ) \sqrt{c+d \tan (e+f x)}}{15 b \left (a^2+b^2\right )^3 (b c-a d)^2 f \sqrt{a+b \tan (e+f x)}}+\frac{((A-i B-C) (c-i d)) \int \frac{1+i \tan (e+f x)}{\sqrt{a+b \tan (e+f x)} \sqrt{c+d \tan (e+f x)}} \, dx}{2 (a-i b)^3}+\frac{((A+i B-C) (c+i d)) \int \frac{1-i \tan (e+f x)}{\sqrt{a+b \tan (e+f x)} \sqrt{c+d \tan (e+f x)}} \, dx}{2 (a+i b)^3}\\ &=-\frac{2 \left (A b^2-a (b B-a C)\right ) \sqrt{c+d \tan (e+f x)}}{5 b \left (a^2+b^2\right ) f (a+b \tan (e+f x))^{5/2}}-\frac{2 \left (4 a^3 b B d+a^4 C d+b^4 (5 B c+A d)+2 a b^3 (5 A c-5 c C-3 B d)-a^2 b^2 (5 B c+9 A d-11 C d)\right ) \sqrt{c+d \tan (e+f x)}}{15 b \left (a^2+b^2\right )^2 (b c-a d) f (a+b \tan (e+f x))^{3/2}}+\frac{2 \left (8 a^5 b B d^2+2 a^6 C d^2-a^4 b^2 d (25 B c+33 A d-39 C d)-a^2 b^4 \left (45 A c^2-45 c^2 C-90 B c d-29 A d^2+23 C d^2\right )+a^3 b^3 \left (80 c (A-C) d+B \left (15 c^2-49 d^2\right )\right )-a b^5 \left (40 c (A-C) d+B \left (45 c^2-3 d^2\right )\right )-b^6 \left (5 c (3 c C+B d)-A \left (15 c^2+2 d^2\right )\right )\right ) \sqrt{c+d \tan (e+f x)}}{15 b \left (a^2+b^2\right )^3 (b c-a d)^2 f \sqrt{a+b \tan (e+f x)}}+\frac{((A-i B-C) (c-i d)) \operatorname{Subst}\left (\int \frac{1}{(1-i x) \sqrt{a+b x} \sqrt{c+d x}} \, dx,x,\tan (e+f x)\right )}{2 (a-i b)^3 f}+\frac{((A+i B-C) (c+i d)) \operatorname{Subst}\left (\int \frac{1}{(1+i x) \sqrt{a+b x} \sqrt{c+d x}} \, dx,x,\tan (e+f x)\right )}{2 (a+i b)^3 f}\\ &=-\frac{2 \left (A b^2-a (b B-a C)\right ) \sqrt{c+d \tan (e+f x)}}{5 b \left (a^2+b^2\right ) f (a+b \tan (e+f x))^{5/2}}-\frac{2 \left (4 a^3 b B d+a^4 C d+b^4 (5 B c+A d)+2 a b^3 (5 A c-5 c C-3 B d)-a^2 b^2 (5 B c+9 A d-11 C d)\right ) \sqrt{c+d \tan (e+f x)}}{15 b \left (a^2+b^2\right )^2 (b c-a d) f (a+b \tan (e+f x))^{3/2}}+\frac{2 \left (8 a^5 b B d^2+2 a^6 C d^2-a^4 b^2 d (25 B c+33 A d-39 C d)-a^2 b^4 \left (45 A c^2-45 c^2 C-90 B c d-29 A d^2+23 C d^2\right )+a^3 b^3 \left (80 c (A-C) d+B \left (15 c^2-49 d^2\right )\right )-a b^5 \left (40 c (A-C) d+B \left (45 c^2-3 d^2\right )\right )-b^6 \left (5 c (3 c C+B d)-A \left (15 c^2+2 d^2\right )\right )\right ) \sqrt{c+d \tan (e+f x)}}{15 b \left (a^2+b^2\right )^3 (b c-a d)^2 f \sqrt{a+b \tan (e+f x)}}+\frac{((A-i B-C) (c-i d)) \operatorname{Subst}\left (\int \frac{1}{i a+b-(i c+d) x^2} \, dx,x,\frac{\sqrt{a+b \tan (e+f x)}}{\sqrt{c+d \tan (e+f x)}}\right )}{(a-i b)^3 f}+\frac{((A+i B-C) (c+i d)) \operatorname{Subst}\left (\int \frac{1}{-i a+b-(-i c+d) x^2} \, dx,x,\frac{\sqrt{a+b \tan (e+f x)}}{\sqrt{c+d \tan (e+f x)}}\right )}{(a+i b)^3 f}\\ &=-\frac{(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 )}{(a-i b)^{7/2} f}-\frac{(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 )}{(a+i b)^{7/2} f}-\frac{2 \left (A b^2-a (b B-a C)\right ) \sqrt{c+d \tan (e+f x)}}{5 b \left (a^2+b^2\right ) f (a+b \tan (e+f x))^{5/2}}-\frac{2 \left (4 a^3 b B d+a^4 C d+b^4 (5 B c+A d)+2 a b^3 (5 A c-5 c C-3 B d)-a^2 b^2 (5 B c+9 A d-11 C d)\right ) \sqrt{c+d \tan (e+f x)}}{15 b \left (a^2+b^2\right )^2 (b c-a d) f (a+b \tan (e+f x))^{3/2}}+\frac{2 \left (8 a^5 b B d^2+2 a^6 C d^2-a^4 b^2 d (25 B c+33 A d-39 C d)-a^2 b^4 \left (45 A c^2-45 c^2 C-90 B c d-29 A d^2+23 C d^2\right )+a^3 b^3 \left (80 c (A-C) d+B \left (15 c^2-49 d^2\right )\right )-a b^5 \left (40 c (A-C) d+B \left (45 c^2-3 d^2\right )\right )-b^6 \left (5 c (3 c C+B d)-A \left (15 c^2+2 d^2\right )\right )\right ) \sqrt{c+d \tan (e+f x)}}{15 b \left (a^2+b^2\right )^3 (b c-a d)^2 f \sqrt{a+b \tan (e+f x)}}\\ \end{align*}

Mathematica [A]  time = 7.15435, size = 1108, normalized size = 1.86 \[ -\frac{\sqrt{c+d \tan (e+f x)} C}{2 b f (a+b \tan (e+f x))^{5/2}}-\frac{-\frac{2 \sqrt{c+d \tan (e+f x)} \left (\frac{1}{2} b^2 (-4 A b c+5 b C c-a C d)-a \left (-2 (B c+(A-C) d) b^2-\frac{1}{2} a (b c C-a d C-4 b B d)\right )\right )}{5 \left (a^2+b^2\right ) (b c-a d) f (a+b \tan (e+f x))^{5/2}}-\frac{2 \left (-\frac{2 \sqrt{c+d \tan (e+f x)} \left (b^2 (b c-a d) \left (C d a^2+b (5 A c-5 C c-B d) a+b^2 (5 B c+A d)\right )-a \left (a \left (-C a^2-4 b B a+4 A b^2-5 b^2 C\right ) d (b c-a d)-5 b^2 (b c-a d) (A b c-a B c-b C c-a A d-b B d+a C d)\right )\right )}{3 \left (a^2+b^2\right ) (b c-a d) f (a+b \tan (e+f x))^{3/2}}-\frac{2 \left (-\frac{15 b \left (\frac{(i A+B-i C) \sqrt{c-i d} \tan ^{-1}\left (\frac{\sqrt{c-i d} \sqrt{a+b \tan (e+f x)}}{\sqrt{i b-a} \sqrt{c+d \tan (e+f x)}}\right ) (a+i b)^3}{\sqrt{i b-a}}+\frac{(i a+b)^3 (A+i B-C) \sqrt{-c-i d} \tan ^{-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 )}{\sqrt{a+i b}}\right ) (b c-a d)^2}{2 \left (a^2+b^2\right ) f}-\frac{2 \left (b^2 \left ((b c-a d) \left (b^2 d-\frac{3}{2} a (b c-a d)\right ) \left (C d a^2+b (5 A c-5 C c-B d) a+b^2 (5 B c+A d)\right )+\left (\frac{a d}{2}-\frac{3 b c}{2}\right ) \left (a \left (-C a^2-4 b B a+4 A b^2-5 b^2 C\right ) d (b c-a d)-5 b^2 (b c-a d) (A b c-a B c-b C c-a A d-b B d+a C d)\right )\right )-a \left (\frac{3}{2} b (b c-a d) \left (b \left (-C a^2-4 b B a+4 A b^2-5 b^2 C\right ) d (b c-a d)+5 a b (A b c-a B c-b C c-a A d-b B d+a C d) (b c-a d)+b \left (C d a^2+b (5 A c-5 C c-B d) a+b^2 (5 B c+A d)\right ) (b c-a d)\right )-a d \left (b^2 (b c-a d) \left (C d a^2+b (5 A c-5 C c-B d) a+b^2 (5 B c+A d)\right )-a \left (a \left (-C a^2-4 b B a+4 A b^2-5 b^2 C\right ) d (b c-a d)-5 b^2 (b c-a d) (A b c-a B c-b C c-a A d-b B d+a C d)\right )\right )\right )\right ) \sqrt{c+d \tan (e+f x)}}{\left (a^2+b^2\right ) f \sqrt{a+b \tan (e+f x)} (b c-a d)}\right )}{3 \left (a^2+b^2\right ) (b c-a d)}\right )}{5 \left (a^2+b^2\right ) (b c-a d)}}{2 b} \]

Antiderivative was successfully verified.

[In]

Integrate[(Sqrt[c + d*Tan[e + f*x]]*(A + B*Tan[e + f*x] + C*Tan[e + f*x]^2))/(a + b*Tan[e + f*x])^(7/2),x]

[Out]

-(C*Sqrt[c + d*Tan[e + f*x]])/(2*b*f*(a + b*Tan[e + f*x])^(5/2)) - ((-2*((b^2*(-4*A*b*c + 5*b*c*C - a*C*d))/2
- a*(-2*b^2*(B*c + (A - C)*d) - (a*(b*c*C - 4*b*B*d - a*C*d))/2))*Sqrt[c + d*Tan[e + f*x]])/(5*(a^2 + b^2)*(b*
c - a*d)*f*(a + b*Tan[e + f*x])^(5/2)) - (2*((-2*(b^2*(b*c - a*d)*(a^2*C*d + b^2*(5*B*c + A*d) + a*b*(5*A*c -
5*c*C - B*d)) - a*(a*(4*A*b^2 - 4*a*b*B - a^2*C - 5*b^2*C)*d*(b*c - a*d) - 5*b^2*(b*c - a*d)*(A*b*c - a*B*c -
b*c*C - a*A*d - b*B*d + a*C*d)))*Sqrt[c + d*Tan[e + f*x]])/(3*(a^2 + b^2)*(b*c - a*d)*f*(a + b*Tan[e + f*x])^(
3/2)) - (2*((-15*b*(b*c - a*d)^2*(((I*a + b)^3*(A + I*B - C)*Sqrt[-c - I*d]*ArcTan[(Sqrt[-c - I*d]*Sqrt[a + b*
Tan[e + f*x]])/(Sqrt[a + I*b]*Sqrt[c + d*Tan[e + f*x]])])/Sqrt[a + I*b] + ((a + I*b)^3*(I*A + B - I*C)*Sqrt[c
- I*d]*ArcTan[(Sqrt[c - I*d]*Sqrt[a + b*Tan[e + f*x]])/(Sqrt[-a + I*b]*Sqrt[c + d*Tan[e + f*x]])])/Sqrt[-a + I
*b]))/(2*(a^2 + b^2)*f) - (2*(b^2*((b*c - a*d)*(b^2*d - (3*a*(b*c - a*d))/2)*(a^2*C*d + b^2*(5*B*c + A*d) + a*
b*(5*A*c - 5*c*C - B*d)) + ((-3*b*c)/2 + (a*d)/2)*(a*(4*A*b^2 - 4*a*b*B - a^2*C - 5*b^2*C)*d*(b*c - a*d) - 5*b
^2*(b*c - a*d)*(A*b*c - a*B*c - b*c*C - a*A*d - b*B*d + a*C*d))) - a*((3*b*(b*c - a*d)*(b*(4*A*b^2 - 4*a*b*B -
a^2*C - 5*b^2*C)*d*(b*c - a*d) + 5*a*b*(b*c - a*d)*(A*b*c - a*B*c - b*c*C - a*A*d - b*B*d + a*C*d) + b*(b*c -
a*d)*(a^2*C*d + b^2*(5*B*c + A*d) + a*b*(5*A*c - 5*c*C - B*d))))/2 - a*d*(b^2*(b*c - a*d)*(a^2*C*d + b^2*(5*B
*c + A*d) + a*b*(5*A*c - 5*c*C - B*d)) - a*(a*(4*A*b^2 - 4*a*b*B - a^2*C - 5*b^2*C)*d*(b*c - a*d) - 5*b^2*(b*c
- a*d)*(A*b*c - a*B*c - b*c*C - a*A*d - b*B*d + a*C*d)))))*Sqrt[c + d*Tan[e + f*x]])/((a^2 + b^2)*(b*c - a*d)
*f*Sqrt[a + b*Tan[e + f*x]])))/(3*(a^2 + b^2)*(b*c - a*d))))/(5*(a^2 + b^2)*(b*c - a*d)))/(2*b)

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Maple [F]  time = 180., size = 0, normalized size = 0. \begin{align*} \int{(A+B\tan \left ( fx+e \right ) +C \left ( \tan \left ( fx+e \right ) \right ) ^{2})\sqrt{c+d\tan \left ( fx+e \right ) } \left ( a+b\tan \left ( fx+e \right ) \right ) ^{-{\frac{7}{2}}}}\, 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)+C*tan(f*x+e)^2)/(a+b*tan(f*x+e))^(7/2),x)

[Out]

int((c+d*tan(f*x+e))^(1/2)*(A+B*tan(f*x+e)+C*tan(f*x+e)^2)/(a+b*tan(f*x+e))^(7/2),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)+C*tan(f*x+e)^2)/(a+b*tan(f*x+e))^(7/2),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)+C*tan(f*x+e)^2)/(a+b*tan(f*x+e))^(7/2),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)+C*tan(f*x+e)**2)/(a+b*tan(f*x+e))**(7/2),x)

[Out]

Timed out

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Giac [F(-2)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: TypeError} \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)+C*tan(f*x+e)^2)/(a+b*tan(f*x+e))^(7/2),x, algorithm="giac")

[Out]

Exception raised: TypeError