3.145 \(\int \frac{(c+d \tan (e+f x))^{5/2} (A+B \tan (e+f x)+C \tan ^2(e+f x))}{(a+b \tan (e+f x))^{7/2}} \, dx\)

Optimal. Leaf size=590 \[ -\frac{2 \sqrt{c+d \tan (e+f x)} \left (-a^3 b^3 \left (2 c d (A-C)+B \left (c^2-d^2\right )\right )-3 a^2 b^4 \left (-A \left (c^2-d^2\right )+2 B c d+c^2 C-2 C d^2\right )+3 a^4 b^2 C d^2+a^6 C d^2+3 a b^5 \left (2 c d (A-C)+B \left (c^2-d^2\right )\right )+b^6 \left (c (2 B d+c C)-A \left (c^2-d^2\right )\right )\right )}{b^3 f \left (a^2+b^2\right )^3 \sqrt{a+b \tan (e+f x)}}-\frac{2 \left (A b^2-a (b B-a C)\right ) (c+d \tan (e+f x))^{5/2}}{5 b f \left (a^2+b^2\right ) (a+b \tan (e+f x))^{5/2}}-\frac{2 (c+d \tan (e+f x))^{3/2} \left (-a^2 b^2 (d (A-3 C)+B c)+a^4 C d+2 a b^3 (A c-B d-c C)+b^4 (A d+B c)\right )}{3 b^2 f \left (a^2+b^2\right )^2 (a+b \tan (e+f x))^{3/2}}-\frac{(c-i d)^{5/2} (i A+B-i C) \tanh ^{-1}\left (\frac{\sqrt{c-i d} \sqrt{a+b \tan (e+f x)}}{\sqrt{a-i b} \sqrt{c+d \tan (e+f x)}}\right )}{f (a-i b)^{7/2}}-\frac{(c+i d)^{5/2} (B-i (A-C)) \tanh ^{-1}\left (\frac{\sqrt{c+i d} \sqrt{a+b \tan (e+f x)}}{\sqrt{a+i b} \sqrt{c+d \tan (e+f x)}}\right )}{f (a+i b)^{7/2}}+\frac{2 C d^{5/2} \tanh ^{-1}\left (\frac{\sqrt{d} \sqrt{a+b \tan (e+f x)}}{\sqrt{b} \sqrt{c+d \tan (e+f x)}}\right )}{b^{7/2} f} \]

[Out]

-(((I*A + B - I*C)*(c - I*d)^(5/2)*ArcTanh[(Sqrt[c - I*d]*Sqrt[a + b*Tan[e + f*x]])/(Sqrt[a - I*b]*Sqrt[c + d*
Tan[e + f*x]])])/((a - I*b)^(7/2)*f)) - ((B - I*(A - C))*(c + I*d)^(5/2)*ArcTanh[(Sqrt[c + I*d]*Sqrt[a + b*Tan
[e + f*x]])/(Sqrt[a + I*b]*Sqrt[c + d*Tan[e + f*x]])])/((a + I*b)^(7/2)*f) + (2*C*d^(5/2)*ArcTanh[(Sqrt[d]*Sqr
t[a + b*Tan[e + f*x]])/(Sqrt[b]*Sqrt[c + d*Tan[e + f*x]])])/(b^(7/2)*f) - (2*(a^6*C*d^2 + 3*a^4*b^2*C*d^2 - 3*
a^2*b^4*(c^2*C + 2*B*c*d - 2*C*d^2 - A*(c^2 - d^2)) + b^6*(c*(c*C + 2*B*d) - A*(c^2 - d^2)) - a^3*b^3*(2*c*(A
- C)*d + B*(c^2 - d^2)) + 3*a*b^5*(2*c*(A - C)*d + B*(c^2 - d^2)))*Sqrt[c + d*Tan[e + f*x]])/(b^3*(a^2 + b^2)^
3*f*Sqrt[a + b*Tan[e + f*x]]) - (2*(a^4*C*d + b^4*(B*c + A*d) + 2*a*b^3*(A*c - c*C - B*d) - a^2*b^2*(B*c + (A
- 3*C)*d))*(c + d*Tan[e + f*x])^(3/2))/(3*b^2*(a^2 + b^2)^2*f*(a + b*Tan[e + f*x])^(3/2)) - (2*(A*b^2 - a*(b*B
- a*C))*(c + d*Tan[e + f*x])^(5/2))/(5*b*(a^2 + b^2)*f*(a + b*Tan[e + f*x])^(5/2))

________________________________________________________________________________________

Rubi [A]  time = 14.0201, antiderivative size = 590, normalized size of antiderivative = 1., number of steps used = 15, number of rules used = 8, integrand size = 49, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.163, Rules used = {3645, 3655, 6725, 63, 217, 206, 93, 208} \[ -\frac{2 \sqrt{c+d \tan (e+f x)} \left (-a^3 b^3 \left (2 c d (A-C)+B \left (c^2-d^2\right )\right )-3 a^2 b^4 \left (-A \left (c^2-d^2\right )+2 B c d+c^2 C-2 C d^2\right )+3 a^4 b^2 C d^2+a^6 C d^2+3 a b^5 \left (2 c d (A-C)+B \left (c^2-d^2\right )\right )+b^6 \left (c (2 B d+c C)-A \left (c^2-d^2\right )\right )\right )}{b^3 f \left (a^2+b^2\right )^3 \sqrt{a+b \tan (e+f x)}}-\frac{2 \left (A b^2-a (b B-a C)\right ) (c+d \tan (e+f x))^{5/2}}{5 b f \left (a^2+b^2\right ) (a+b \tan (e+f x))^{5/2}}-\frac{2 (c+d \tan (e+f x))^{3/2} \left (-a^2 b^2 (d (A-3 C)+B c)+a^4 C d+2 a b^3 (A c-B d-c C)+b^4 (A d+B c)\right )}{3 b^2 f \left (a^2+b^2\right )^2 (a+b \tan (e+f x))^{3/2}}-\frac{(c-i d)^{5/2} (i A+B-i C) \tanh ^{-1}\left (\frac{\sqrt{c-i d} \sqrt{a+b \tan (e+f x)}}{\sqrt{a-i b} \sqrt{c+d \tan (e+f x)}}\right )}{f (a-i b)^{7/2}}-\frac{(c+i d)^{5/2} (B-i (A-C)) \tanh ^{-1}\left (\frac{\sqrt{c+i d} \sqrt{a+b \tan (e+f x)}}{\sqrt{a+i b} \sqrt{c+d \tan (e+f x)}}\right )}{f (a+i b)^{7/2}}+\frac{2 C d^{5/2} \tanh ^{-1}\left (\frac{\sqrt{d} \sqrt{a+b \tan (e+f x)}}{\sqrt{b} \sqrt{c+d \tan (e+f x)}}\right )}{b^{7/2} f} \]

Antiderivative was successfully verified.

[In]

Int[((c + d*Tan[e + f*x])^(5/2)*(A + B*Tan[e + f*x] + C*Tan[e + f*x]^2))/(a + b*Tan[e + f*x])^(7/2),x]

[Out]

-(((I*A + B - I*C)*(c - I*d)^(5/2)*ArcTanh[(Sqrt[c - I*d]*Sqrt[a + b*Tan[e + f*x]])/(Sqrt[a - I*b]*Sqrt[c + d*
Tan[e + f*x]])])/((a - I*b)^(7/2)*f)) - ((B - I*(A - C))*(c + I*d)^(5/2)*ArcTanh[(Sqrt[c + I*d]*Sqrt[a + b*Tan
[e + f*x]])/(Sqrt[a + I*b]*Sqrt[c + d*Tan[e + f*x]])])/((a + I*b)^(7/2)*f) + (2*C*d^(5/2)*ArcTanh[(Sqrt[d]*Sqr
t[a + b*Tan[e + f*x]])/(Sqrt[b]*Sqrt[c + d*Tan[e + f*x]])])/(b^(7/2)*f) - (2*(a^6*C*d^2 + 3*a^4*b^2*C*d^2 - 3*
a^2*b^4*(c^2*C + 2*B*c*d - 2*C*d^2 - A*(c^2 - d^2)) + b^6*(c*(c*C + 2*B*d) - A*(c^2 - d^2)) - a^3*b^3*(2*c*(A
- C)*d + B*(c^2 - d^2)) + 3*a*b^5*(2*c*(A - C)*d + B*(c^2 - d^2)))*Sqrt[c + d*Tan[e + f*x]])/(b^3*(a^2 + b^2)^
3*f*Sqrt[a + b*Tan[e + f*x]]) - (2*(a^4*C*d + b^4*(B*c + A*d) + 2*a*b^3*(A*c - c*C - B*d) - a^2*b^2*(B*c + (A
- 3*C)*d))*(c + d*Tan[e + f*x])^(3/2))/(3*b^2*(a^2 + b^2)^2*f*(a + b*Tan[e + f*x])^(3/2)) - (2*(A*b^2 - a*(b*B
- a*C))*(c + d*Tan[e + f*x])^(5/2))/(5*b*(a^2 + b^2)*f*(a + b*Tan[e + f*x])^(5/2))

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 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 \frac{(c+d \tan (e+f x))^{5/2} \left (A+B \tan (e+f x)+C \tan ^2(e+f x)\right )}{(a+b \tan (e+f x))^{7/2}} \, dx &=-\frac{2 \left (A b^2-a (b B-a C)\right ) (c+d \tan (e+f x))^{5/2}}{5 b \left (a^2+b^2\right ) f (a+b \tan (e+f x))^{5/2}}+\frac{2 \int \frac{(c+d \tan (e+f x))^{3/2} \left (\frac{5}{2} ((b B-a C) (b c-a d)+A b (a c+b d))-\frac{5}{2} b ((A-C) (b c-a d)-B (a c+b d)) \tan (e+f x)+\frac{5}{2} \left (a^2+b^2\right ) C d \tan ^2(e+f x)\right )}{(a+b \tan (e+f x))^{5/2}} \, dx}{5 b \left (a^2+b^2\right )}\\ &=-\frac{2 \left (a^4 C d+b^4 (B c+A d)+2 a b^3 (A c-c C-B d)-a^2 b^2 (B c+(A-3 C) d)\right ) (c+d \tan (e+f x))^{3/2}}{3 b^2 \left (a^2+b^2\right )^2 f (a+b \tan (e+f x))^{3/2}}-\frac{2 \left (A b^2-a (b B-a C)\right ) (c+d \tan (e+f x))^{5/2}}{5 b \left (a^2+b^2\right ) f (a+b \tan (e+f x))^{5/2}}+\frac{4 \int \frac{\sqrt{c+d \tan (e+f x)} \left (\frac{15}{4} \left (b (a c+b d) ((b B-a C) (b c-a d)+A b (a c+b d))-(b c-a d) \left (a^3 C d+A b^2 (b c-a d)-b^3 (c C+B d)-a b^2 (B c-2 C d)\right )\right )-\frac{15}{4} b^2 ((a c+b d) ((A-C) (b c-a d)-B (a c+b d))+(b c-a d) (b B c+b (A-C) d+a (A c-c C-B d))) \tan (e+f x)+\frac{15}{4} \left (a^2+b^2\right )^2 C d^2 \tan ^2(e+f x)\right )}{(a+b \tan (e+f x))^{3/2}} \, dx}{15 b^2 \left (a^2+b^2\right )^2}\\ &=-\frac{2 \left (a^6 C d^2+3 a^4 b^2 C d^2-3 a^2 b^4 \left (c^2 C+2 B c d-2 C d^2-A \left (c^2-d^2\right )\right )+b^6 \left (c (c C+2 B d)-A \left (c^2-d^2\right )\right )-a^3 b^3 \left (2 c (A-C) d+B \left (c^2-d^2\right )\right )+3 a b^5 \left (2 c (A-C) d+B \left (c^2-d^2\right )\right )\right ) \sqrt{c+d \tan (e+f x)}}{b^3 \left (a^2+b^2\right )^3 f \sqrt{a+b \tan (e+f x)}}-\frac{2 \left (a^4 C d+b^4 (B c+A d)+2 a b^3 (A c-c C-B d)-a^2 b^2 (B c+(A-3 C) d)\right ) (c+d \tan (e+f x))^{3/2}}{3 b^2 \left (a^2+b^2\right )^2 f (a+b \tan (e+f x))^{3/2}}-\frac{2 \left (A b^2-a (b B-a C)\right ) (c+d \tan (e+f x))^{5/2}}{5 b \left (a^2+b^2\right ) f (a+b \tan (e+f x))^{5/2}}+\frac{8 \int \frac{\frac{15}{8} \left (a^6 C d^3+3 a^4 b^2 C d^3-b^6 \left (B c^3+3 A c^2 d-3 c^2 C d-3 B c d^2-A d^3\right )-3 a b^5 \left (A c^3-c^3 C-3 B c^2 d-3 A c d^2+3 c C d^2+B d^3\right )+3 a^2 b^4 \left (B c^3+3 A c^2 d-3 c^2 C d-3 B c d^2-A d^3+2 C d^3\right )-a^3 b^3 \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 )\right )-\frac{15}{8} b^3 \left (3 a^2 b \left (A c^3-c^3 C-3 B c^2 d-3 A c d^2+3 c C d^2+B d^3\right )+b^3 \left (c^3 C+3 B c^2 d-3 c C d^2-B d^3-A \left (c^3-3 c d^2\right )\right )-a^3 \left ((A-C) d \left (3 c^2-d^2\right )+B \left (c^3-3 c d^2\right )\right )+3 a b^2 \left ((A-C) d \left (3 c^2-d^2\right )+B \left (c^3-3 c d^2\right )\right )\right ) \tan (e+f x)+\frac{15}{8} \left (a^2+b^2\right )^3 C d^3 \tan ^2(e+f x)}{\sqrt{a+b \tan (e+f x)} \sqrt{c+d \tan (e+f x)}} \, dx}{15 b^3 \left (a^2+b^2\right )^3}\\ &=-\frac{2 \left (a^6 C d^2+3 a^4 b^2 C d^2-3 a^2 b^4 \left (c^2 C+2 B c d-2 C d^2-A \left (c^2-d^2\right )\right )+b^6 \left (c (c C+2 B d)-A \left (c^2-d^2\right )\right )-a^3 b^3 \left (2 c (A-C) d+B \left (c^2-d^2\right )\right )+3 a b^5 \left (2 c (A-C) d+B \left (c^2-d^2\right )\right )\right ) \sqrt{c+d \tan (e+f x)}}{b^3 \left (a^2+b^2\right )^3 f \sqrt{a+b \tan (e+f x)}}-\frac{2 \left (a^4 C d+b^4 (B c+A d)+2 a b^3 (A c-c C-B d)-a^2 b^2 (B c+(A-3 C) d)\right ) (c+d \tan (e+f x))^{3/2}}{3 b^2 \left (a^2+b^2\right )^2 f (a+b \tan (e+f x))^{3/2}}-\frac{2 \left (A b^2-a (b B-a C)\right ) (c+d \tan (e+f x))^{5/2}}{5 b \left (a^2+b^2\right ) f (a+b \tan (e+f x))^{5/2}}+\frac{8 \operatorname{Subst}\left (\int \frac{\frac{15}{8} \left (a^6 C d^3+3 a^4 b^2 C d^3-b^6 \left (B c^3+3 A c^2 d-3 c^2 C d-3 B c d^2-A d^3\right )-3 a b^5 \left (A c^3-c^3 C-3 B c^2 d-3 A c d^2+3 c C d^2+B d^3\right )+3 a^2 b^4 \left (B c^3+3 A c^2 d-3 c^2 C d-3 B c d^2-A d^3+2 C d^3\right )-a^3 b^3 \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 )\right )-\frac{15}{8} b^3 \left (3 a^2 b \left (A c^3-c^3 C-3 B c^2 d-3 A c d^2+3 c C d^2+B d^3\right )+b^3 \left (c^3 C+3 B c^2 d-3 c C d^2-B d^3-A \left (c^3-3 c d^2\right )\right )-a^3 \left ((A-C) d \left (3 c^2-d^2\right )+B \left (c^3-3 c d^2\right )\right )+3 a b^2 \left ((A-C) d \left (3 c^2-d^2\right )+B \left (c^3-3 c d^2\right )\right )\right ) x+\frac{15}{8} \left (a^2+b^2\right )^3 C d^3 x^2}{\sqrt{a+b x} \sqrt{c+d x} \left (1+x^2\right )} \, dx,x,\tan (e+f x)\right )}{15 b^3 \left (a^2+b^2\right )^3 f}\\ &=-\frac{2 \left (a^6 C d^2+3 a^4 b^2 C d^2-3 a^2 b^4 \left (c^2 C+2 B c d-2 C d^2-A \left (c^2-d^2\right )\right )+b^6 \left (c (c C+2 B d)-A \left (c^2-d^2\right )\right )-a^3 b^3 \left (2 c (A-C) d+B \left (c^2-d^2\right )\right )+3 a b^5 \left (2 c (A-C) d+B \left (c^2-d^2\right )\right )\right ) \sqrt{c+d \tan (e+f x)}}{b^3 \left (a^2+b^2\right )^3 f \sqrt{a+b \tan (e+f x)}}-\frac{2 \left (a^4 C d+b^4 (B c+A d)+2 a b^3 (A c-c C-B d)-a^2 b^2 (B c+(A-3 C) d)\right ) (c+d \tan (e+f x))^{3/2}}{3 b^2 \left (a^2+b^2\right )^2 f (a+b \tan (e+f x))^{3/2}}-\frac{2 \left (A b^2-a (b B-a C)\right ) (c+d \tan (e+f x))^{5/2}}{5 b \left (a^2+b^2\right ) f (a+b \tan (e+f x))^{5/2}}+\frac{8 \operatorname{Subst}\left (\int \left (\frac{15 \left (a^2+b^2\right )^3 C d^3}{8 \sqrt{a+b x} \sqrt{c+d x}}+\frac{15 \left (-b^3 \left (3 a b^2 \left (A c^3-c^3 C-3 B c^2 d-3 A c d^2+3 c C d^2+B d^3\right )+a^3 \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 )-3 a^2 b \left ((A-C) d \left (3 c^2-d^2\right )+B \left (c^3-3 c d^2\right )\right )+b^3 \left ((A-C) d \left (3 c^2-d^2\right )+B \left (c^3-3 c d^2\right )\right )\right )-b^3 \left (3 a^2 b \left (A c^3-c^3 C-3 B c^2 d-3 A c d^2+3 c C d^2+B d^3\right )+b^3 \left (c^3 C+3 B c^2 d-3 c C d^2-B d^3-A \left (c^3-3 c d^2\right )\right )-a^3 \left ((A-C) d \left (3 c^2-d^2\right )+B \left (c^3-3 c d^2\right )\right )+3 a b^2 \left ((A-C) d \left (3 c^2-d^2\right )+B \left (c^3-3 c d^2\right )\right )\right ) x\right )}{8 \sqrt{a+b x} \sqrt{c+d x} \left (1+x^2\right )}\right ) \, dx,x,\tan (e+f x)\right )}{15 b^3 \left (a^2+b^2\right )^3 f}\\ &=-\frac{2 \left (a^6 C d^2+3 a^4 b^2 C d^2-3 a^2 b^4 \left (c^2 C+2 B c d-2 C d^2-A \left (c^2-d^2\right )\right )+b^6 \left (c (c C+2 B d)-A \left (c^2-d^2\right )\right )-a^3 b^3 \left (2 c (A-C) d+B \left (c^2-d^2\right )\right )+3 a b^5 \left (2 c (A-C) d+B \left (c^2-d^2\right )\right )\right ) \sqrt{c+d \tan (e+f x)}}{b^3 \left (a^2+b^2\right )^3 f \sqrt{a+b \tan (e+f x)}}-\frac{2 \left (a^4 C d+b^4 (B c+A d)+2 a b^3 (A c-c C-B d)-a^2 b^2 (B c+(A-3 C) d)\right ) (c+d \tan (e+f x))^{3/2}}{3 b^2 \left (a^2+b^2\right )^2 f (a+b \tan (e+f x))^{3/2}}-\frac{2 \left (A b^2-a (b B-a C)\right ) (c+d \tan (e+f x))^{5/2}}{5 b \left (a^2+b^2\right ) f (a+b \tan (e+f x))^{5/2}}+\frac{\operatorname{Subst}\left (\int \frac{-b^3 \left (3 a b^2 \left (A c^3-c^3 C-3 B c^2 d-3 A c d^2+3 c C d^2+B d^3\right )+a^3 \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 )-3 a^2 b \left ((A-C) d \left (3 c^2-d^2\right )+B \left (c^3-3 c d^2\right )\right )+b^3 \left ((A-C) d \left (3 c^2-d^2\right )+B \left (c^3-3 c d^2\right )\right )\right )-b^3 \left (3 a^2 b \left (A c^3-c^3 C-3 B c^2 d-3 A c d^2+3 c C d^2+B d^3\right )+b^3 \left (c^3 C+3 B c^2 d-3 c C d^2-B d^3-A \left (c^3-3 c d^2\right )\right )-a^3 \left ((A-C) d \left (3 c^2-d^2\right )+B \left (c^3-3 c d^2\right )\right )+3 a b^2 \left ((A-C) d \left (3 c^2-d^2\right )+B \left (c^3-3 c d^2\right )\right )\right ) x}{\sqrt{a+b x} \sqrt{c+d x} \left (1+x^2\right )} \, dx,x,\tan (e+f x)\right )}{b^3 \left (a^2+b^2\right )^3 f}+\frac{\left (C d^3\right ) \operatorname{Subst}\left (\int \frac{1}{\sqrt{a+b x} \sqrt{c+d x}} \, dx,x,\tan (e+f x)\right )}{b^3 f}\\ &=-\frac{2 \left (a^6 C d^2+3 a^4 b^2 C d^2-3 a^2 b^4 \left (c^2 C+2 B c d-2 C d^2-A \left (c^2-d^2\right )\right )+b^6 \left (c (c C+2 B d)-A \left (c^2-d^2\right )\right )-a^3 b^3 \left (2 c (A-C) d+B \left (c^2-d^2\right )\right )+3 a b^5 \left (2 c (A-C) d+B \left (c^2-d^2\right )\right )\right ) \sqrt{c+d \tan (e+f x)}}{b^3 \left (a^2+b^2\right )^3 f \sqrt{a+b \tan (e+f x)}}-\frac{2 \left (a^4 C d+b^4 (B c+A d)+2 a b^3 (A c-c C-B d)-a^2 b^2 (B c+(A-3 C) d)\right ) (c+d \tan (e+f x))^{3/2}}{3 b^2 \left (a^2+b^2\right )^2 f (a+b \tan (e+f x))^{3/2}}-\frac{2 \left (A b^2-a (b B-a C)\right ) (c+d \tan (e+f x))^{5/2}}{5 b \left (a^2+b^2\right ) f (a+b \tan (e+f x))^{5/2}}+\frac{\operatorname{Subst}\left (\int \left (\frac{b^3 \left (3 a^2 b \left (A c^3-c^3 C-3 B c^2 d-3 A c d^2+3 c C d^2+B d^3\right )+b^3 \left (c^3 C+3 B c^2 d-3 c C d^2-B d^3-A \left (c^3-3 c d^2\right )\right )-a^3 \left ((A-C) d \left (3 c^2-d^2\right )+B \left (c^3-3 c d^2\right )\right )+3 a b^2 \left ((A-C) d \left (3 c^2-d^2\right )+B \left (c^3-3 c d^2\right )\right )\right )-i b^3 \left (3 a b^2 \left (A c^3-c^3 C-3 B c^2 d-3 A c d^2+3 c C d^2+B d^3\right )+a^3 \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 )-3 a^2 b \left ((A-C) d \left (3 c^2-d^2\right )+B \left (c^3-3 c d^2\right )\right )+b^3 \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 \left (3 a^2 b \left (A c^3-c^3 C-3 B c^2 d-3 A c d^2+3 c C d^2+B d^3\right )+b^3 \left (c^3 C+3 B c^2 d-3 c C d^2-B d^3-A \left (c^3-3 c d^2\right )\right )-a^3 \left ((A-C) d \left (3 c^2-d^2\right )+B \left (c^3-3 c d^2\right )\right )+3 a b^2 \left ((A-C) d \left (3 c^2-d^2\right )+B \left (c^3-3 c d^2\right )\right )\right )-i b^3 \left (3 a b^2 \left (A c^3-c^3 C-3 B c^2 d-3 A c d^2+3 c C d^2+B d^3\right )+a^3 \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 )-3 a^2 b \left ((A-C) d \left (3 c^2-d^2\right )+B \left (c^3-3 c d^2\right )\right )+b^3 \left ((A-C) d \left (3 c^2-d^2\right )+B \left (c^3-3 c d^2\right )\right )\right )}{2 (i+x) \sqrt{a+b x} \sqrt{c+d x}}\right ) \, dx,x,\tan (e+f x)\right )}{b^3 \left (a^2+b^2\right )^3 f}+\frac{\left (2 C d^3\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 )}{b^4 f}\\ &=-\frac{2 \left (a^6 C d^2+3 a^4 b^2 C d^2-3 a^2 b^4 \left (c^2 C+2 B c d-2 C d^2-A \left (c^2-d^2\right )\right )+b^6 \left (c (c C+2 B d)-A \left (c^2-d^2\right )\right )-a^3 b^3 \left (2 c (A-C) d+B \left (c^2-d^2\right )\right )+3 a b^5 \left (2 c (A-C) d+B \left (c^2-d^2\right )\right )\right ) \sqrt{c+d \tan (e+f x)}}{b^3 \left (a^2+b^2\right )^3 f \sqrt{a+b \tan (e+f x)}}-\frac{2 \left (a^4 C d+b^4 (B c+A d)+2 a b^3 (A c-c C-B d)-a^2 b^2 (B c+(A-3 C) d)\right ) (c+d \tan (e+f x))^{3/2}}{3 b^2 \left (a^2+b^2\right )^2 f (a+b \tan (e+f x))^{3/2}}-\frac{2 \left (A b^2-a (b B-a C)\right ) (c+d \tan (e+f x))^{5/2}}{5 b \left (a^2+b^2\right ) f (a+b \tan (e+f x))^{5/2}}+\frac{\left ((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 (i a+b)^3 f}+\frac{\left ((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 (i a-b)^3 f}+\frac{\left (2 C d^3\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 )}{b^4 f}\\ &=\frac{2 C d^{5/2} \tanh ^{-1}\left (\frac{\sqrt{d} \sqrt{a+b \tan (e+f x)}}{\sqrt{b} \sqrt{c+d \tan (e+f x)}}\right )}{b^{7/2} f}-\frac{2 \left (a^6 C d^2+3 a^4 b^2 C d^2-3 a^2 b^4 \left (c^2 C+2 B c d-2 C d^2-A \left (c^2-d^2\right )\right )+b^6 \left (c (c C+2 B d)-A \left (c^2-d^2\right )\right )-a^3 b^3 \left (2 c (A-C) d+B \left (c^2-d^2\right )\right )+3 a b^5 \left (2 c (A-C) d+B \left (c^2-d^2\right )\right )\right ) \sqrt{c+d \tan (e+f x)}}{b^3 \left (a^2+b^2\right )^3 f \sqrt{a+b \tan (e+f x)}}-\frac{2 \left (a^4 C d+b^4 (B c+A d)+2 a b^3 (A c-c C-B d)-a^2 b^2 (B c+(A-3 C) d)\right ) (c+d \tan (e+f x))^{3/2}}{3 b^2 \left (a^2+b^2\right )^2 f (a+b \tan (e+f x))^{3/2}}-\frac{2 \left (A b^2-a (b B-a C)\right ) (c+d \tan (e+f x))^{5/2}}{5 b \left (a^2+b^2\right ) f (a+b \tan (e+f x))^{5/2}}+\frac{\left ((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 )}{(i a+b)^3 f}+\frac{\left ((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 )}{(i a-b)^3 f}\\ &=-\frac{(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 )}{(a-i b)^{7/2} f}-\frac{(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 )}{(a+i b)^{7/2} f}+\frac{2 C d^{5/2} \tanh ^{-1}\left (\frac{\sqrt{d} \sqrt{a+b \tan (e+f x)}}{\sqrt{b} \sqrt{c+d \tan (e+f x)}}\right )}{b^{7/2} f}-\frac{2 \left (a^6 C d^2+3 a^4 b^2 C d^2-3 a^2 b^4 \left (c^2 C+2 B c d-2 C d^2-A \left (c^2-d^2\right )\right )+b^6 \left (c (c C+2 B d)-A \left (c^2-d^2\right )\right )-a^3 b^3 \left (2 c (A-C) d+B \left (c^2-d^2\right )\right )+3 a b^5 \left (2 c (A-C) d+B \left (c^2-d^2\right )\right )\right ) \sqrt{c+d \tan (e+f x)}}{b^3 \left (a^2+b^2\right )^3 f \sqrt{a+b \tan (e+f x)}}-\frac{2 \left (a^4 C d+b^4 (B c+A d)+2 a b^3 (A c-c C-B d)-a^2 b^2 (B c+(A-3 C) d)\right ) (c+d \tan (e+f x))^{3/2}}{3 b^2 \left (a^2+b^2\right )^2 f (a+b \tan (e+f x))^{3/2}}-\frac{2 \left (A b^2-a (b B-a C)\right ) (c+d \tan (e+f x))^{5/2}}{5 b \left (a^2+b^2\right ) f (a+b \tan (e+f x))^{5/2}}\\ \end{align*}

Mathematica [C]  time = 48.3229, size = 2345519, normalized size = 3975.46 \[ \text{Result too large to show} \]

Warning: Unable to verify antiderivative.

[In]

Integrate[((c + d*Tan[e + f*x])^(5/2)*(A + B*Tan[e + f*x] + C*Tan[e + f*x]^2))/(a + b*Tan[e + f*x])^(7/2),x]

[Out]

Result too large to show

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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}) \left ( c+d\tan \left ( fx+e \right ) \right ) ^{{\frac{5}{2}}} \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))^(5/2)*(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))^(5/2)*(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))^(5/2)*(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))^(5/2)*(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))**(5/2)*(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))^(5/2)*(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