3.153 \(\int \frac{(a+b \tan (e+f x))^{5/2} (A+B \tan (e+f x)+C \tan ^2(e+f x))}{(c+d \tan (e+f x))^{3/2}} \, dx\)

Optimal. Leaf size=528 \[ \frac{\sqrt{b} \left (15 a^2 C d^2-10 a b d (3 c C-2 B d)+b^2 \left (8 d^2 (A-C)-12 B c d+15 c^2 C\right )\right ) \tanh ^{-1}\left (\frac{\sqrt{d} \sqrt{a+b \tan (e+f x)}}{\sqrt{b} \sqrt{c+d \tan (e+f x)}}\right )}{4 d^{7/2} f}+\frac{b \left (d^2 (4 A+C)-4 B c d+5 c^2 C\right ) (a+b \tan (e+f x))^{3/2} \sqrt{c+d \tan (e+f x)}}{2 d^2 f \left (c^2+d^2\right )}-\frac{b \sqrt{a+b \tan (e+f x)} \sqrt{c+d \tan (e+f x)} \left (3 (b c-a d) \left (d^2 (4 A+C)-4 B c d+5 c^2 C\right )-4 d^2 ((A-C) (b c-a d)+B (a c+b d))\right )}{4 d^3 f \left (c^2+d^2\right )}-\frac{2 \left (A d^2-B c d+c^2 C\right ) (a+b \tan (e+f x))^{5/2}}{d f \left (c^2+d^2\right ) \sqrt{c+d \tan (e+f x)}}-\frac{(a-i b)^{5/2} (i A+B-i C) \tanh ^{-1}\left (\frac{\sqrt{c-i d} \sqrt{a+b \tan (e+f x)}}{\sqrt{a-i b} \sqrt{c+d \tan (e+f x)}}\right )}{f (c-i d)^{3/2}}-\frac{(a+i b)^{5/2} (B-i (A-C)) \tanh ^{-1}\left (\frac{\sqrt{c+i d} \sqrt{a+b \tan (e+f x)}}{\sqrt{a+i b} \sqrt{c+d \tan (e+f x)}}\right )}{f (c+i d)^{3/2}} \]

[Out]

-(((a - I*b)^(5/2)*(I*A + B - I*C)*ArcTanh[(Sqrt[c - I*d]*Sqrt[a + b*Tan[e + f*x]])/(Sqrt[a - I*b]*Sqrt[c + d*
Tan[e + f*x]])])/((c - I*d)^(3/2)*f)) - ((a + I*b)^(5/2)*(B - I*(A - C))*ArcTanh[(Sqrt[c + I*d]*Sqrt[a + b*Tan
[e + f*x]])/(Sqrt[a + I*b]*Sqrt[c + d*Tan[e + f*x]])])/((c + I*d)^(3/2)*f) + (Sqrt[b]*(15*a^2*C*d^2 - 10*a*b*d
*(3*c*C - 2*B*d) + b^2*(15*c^2*C - 12*B*c*d + 8*(A - C)*d^2))*ArcTanh[(Sqrt[d]*Sqrt[a + b*Tan[e + f*x]])/(Sqrt
[b]*Sqrt[c + d*Tan[e + f*x]])])/(4*d^(7/2)*f) - (2*(c^2*C - B*c*d + A*d^2)*(a + b*Tan[e + f*x])^(5/2))/(d*(c^2
+ d^2)*f*Sqrt[c + d*Tan[e + f*x]]) - (b*(3*(b*c - a*d)*(5*c^2*C - 4*B*c*d + (4*A + C)*d^2) - 4*d^2*((A - C)*(
b*c - a*d) + B*(a*c + b*d)))*Sqrt[a + b*Tan[e + f*x]]*Sqrt[c + d*Tan[e + f*x]])/(4*d^3*(c^2 + d^2)*f) + (b*(5*
c^2*C - 4*B*c*d + (4*A + C)*d^2)*(a + b*Tan[e + f*x])^(3/2)*Sqrt[c + d*Tan[e + f*x]])/(2*d^2*(c^2 + d^2)*f)

________________________________________________________________________________________

Rubi [A]  time = 8.18832, antiderivative size = 528, normalized size of antiderivative = 1., number of steps used = 15, number of rules used = 9, integrand size = 49, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.184, Rules used = {3645, 3647, 3655, 6725, 63, 217, 206, 93, 208} \[ \frac{\sqrt{b} \left (15 a^2 C d^2-10 a b d (3 c C-2 B d)+b^2 \left (8 d^2 (A-C)-12 B c d+15 c^2 C\right )\right ) \tanh ^{-1}\left (\frac{\sqrt{d} \sqrt{a+b \tan (e+f x)}}{\sqrt{b} \sqrt{c+d \tan (e+f x)}}\right )}{4 d^{7/2} f}+\frac{b \left (d^2 (4 A+C)-4 B c d+5 c^2 C\right ) (a+b \tan (e+f x))^{3/2} \sqrt{c+d \tan (e+f x)}}{2 d^2 f \left (c^2+d^2\right )}-\frac{b \sqrt{a+b \tan (e+f x)} \sqrt{c+d \tan (e+f x)} \left (3 (b c-a d) \left (d^2 (4 A+C)-4 B c d+5 c^2 C\right )-4 d^2 ((A-C) (b c-a d)+B (a c+b d))\right )}{4 d^3 f \left (c^2+d^2\right )}-\frac{2 \left (A d^2-B c d+c^2 C\right ) (a+b \tan (e+f x))^{5/2}}{d f \left (c^2+d^2\right ) \sqrt{c+d \tan (e+f x)}}-\frac{(a-i b)^{5/2} (i A+B-i C) \tanh ^{-1}\left (\frac{\sqrt{c-i d} \sqrt{a+b \tan (e+f x)}}{\sqrt{a-i b} \sqrt{c+d \tan (e+f x)}}\right )}{f (c-i d)^{3/2}}-\frac{(a+i b)^{5/2} (B-i (A-C)) \tanh ^{-1}\left (\frac{\sqrt{c+i d} \sqrt{a+b \tan (e+f x)}}{\sqrt{a+i b} \sqrt{c+d \tan (e+f x)}}\right )}{f (c+i d)^{3/2}} \]

Antiderivative was successfully verified.

[In]

Int[((a + b*Tan[e + f*x])^(5/2)*(A + B*Tan[e + f*x] + C*Tan[e + f*x]^2))/(c + d*Tan[e + f*x])^(3/2),x]

[Out]

-(((a - I*b)^(5/2)*(I*A + B - I*C)*ArcTanh[(Sqrt[c - I*d]*Sqrt[a + b*Tan[e + f*x]])/(Sqrt[a - I*b]*Sqrt[c + d*
Tan[e + f*x]])])/((c - I*d)^(3/2)*f)) - ((a + I*b)^(5/2)*(B - I*(A - C))*ArcTanh[(Sqrt[c + I*d]*Sqrt[a + b*Tan
[e + f*x]])/(Sqrt[a + I*b]*Sqrt[c + d*Tan[e + f*x]])])/((c + I*d)^(3/2)*f) + (Sqrt[b]*(15*a^2*C*d^2 - 10*a*b*d
*(3*c*C - 2*B*d) + b^2*(15*c^2*C - 12*B*c*d + 8*(A - C)*d^2))*ArcTanh[(Sqrt[d]*Sqrt[a + b*Tan[e + f*x]])/(Sqrt
[b]*Sqrt[c + d*Tan[e + f*x]])])/(4*d^(7/2)*f) - (2*(c^2*C - B*c*d + A*d^2)*(a + b*Tan[e + f*x])^(5/2))/(d*(c^2
+ d^2)*f*Sqrt[c + d*Tan[e + f*x]]) - (b*(3*(b*c - a*d)*(5*c^2*C - 4*B*c*d + (4*A + C)*d^2) - 4*d^2*((A - C)*(
b*c - a*d) + B*(a*c + b*d)))*Sqrt[a + b*Tan[e + f*x]]*Sqrt[c + d*Tan[e + f*x]])/(4*d^3*(c^2 + d^2)*f) + (b*(5*
c^2*C - 4*B*c*d + (4*A + C)*d^2)*(a + b*Tan[e + f*x])^(3/2)*Sqrt[c + d*Tan[e + f*x]])/(2*d^2*(c^2 + d^2)*f)

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 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 \frac{(a+b \tan (e+f x))^{5/2} \left (A+B \tan (e+f x)+C \tan ^2(e+f x)\right )}{(c+d \tan (e+f x))^{3/2}} \, dx &=-\frac{2 \left (c^2 C-B c d+A d^2\right ) (a+b \tan (e+f x))^{5/2}}{d \left (c^2+d^2\right ) f \sqrt{c+d \tan (e+f x)}}+\frac{2 \int \frac{(a+b \tan (e+f x))^{3/2} \left (\frac{1}{2} (A d (a c+5 b d)+(5 b c-a d) (c C-B d))+\frac{1}{2} d ((A-C) (b c-a d)+B (a c+b d)) \tan (e+f x)+\frac{1}{2} b \left (5 c^2 C-4 B c d+(4 A+C) d^2\right ) \tan ^2(e+f x)\right )}{\sqrt{c+d \tan (e+f x)}} \, dx}{d \left (c^2+d^2\right )}\\ &=-\frac{2 \left (c^2 C-B c d+A d^2\right ) (a+b \tan (e+f x))^{5/2}}{d \left (c^2+d^2\right ) f \sqrt{c+d \tan (e+f x)}}+\frac{b \left (5 c^2 C-4 B c d+(4 A+C) d^2\right ) (a+b \tan (e+f x))^{3/2} \sqrt{c+d \tan (e+f x)}}{2 d^2 \left (c^2+d^2\right ) f}+\frac{\int \frac{\sqrt{a+b \tan (e+f x)} \left (\frac{1}{4} \left (-b (3 b c+a d) \left (5 c^2 C-4 B c d+(4 A+C) d^2\right )+4 a d (A d (a c+5 b d)+(5 b c-a d) (c C-B d))\right )+d^2 \left (2 a b (A c-c C+B d)+a^2 (B c-(A-C) d)-b^2 (B c-(A-C) d)\right ) \tan (e+f x)-\frac{1}{4} b \left (3 (b c-a d) \left (5 c^2 C-4 B c d+(4 A+C) d^2\right )-4 d^2 ((A-C) (b c-a d)+B (a c+b d))\right ) \tan ^2(e+f x)\right )}{\sqrt{c+d \tan (e+f x)}} \, dx}{d^2 \left (c^2+d^2\right )}\\ &=-\frac{2 \left (c^2 C-B c d+A d^2\right ) (a+b \tan (e+f x))^{5/2}}{d \left (c^2+d^2\right ) f \sqrt{c+d \tan (e+f x)}}-\frac{b \left (3 (b c-a d) \left (5 c^2 C-4 B c d+(4 A+C) d^2\right )-4 d^2 ((A-C) (b c-a d)+B (a c+b d))\right ) \sqrt{a+b \tan (e+f x)} \sqrt{c+d \tan (e+f x)}}{4 d^3 \left (c^2+d^2\right ) f}+\frac{b \left (5 c^2 C-4 B c d+(4 A+C) d^2\right ) (a+b \tan (e+f x))^{3/2} \sqrt{c+d \tan (e+f x)}}{2 d^2 \left (c^2+d^2\right ) f}+\frac{\int \frac{\frac{1}{8} \left (8 a^3 d^3 (A c-c C+B d)+3 a^2 b d^2 \left (5 c^2 C-8 B c d+(8 A-3 C) d^2\right )+b^3 c \left (15 c^3 C-12 B c^2 d+c (8 A+7 C) d^2-4 B d^3\right )-2 a b^2 d \left (15 c^3 C-10 B c^2 d+3 c (4 A+C) d^2+2 B d^3\right )\right )+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{1}{8} b \left (c^2+d^2\right ) \left (15 a^2 C d^2-10 a b d (3 c C-2 B d)+b^2 \left (15 c^2 C-12 B c d+8 (A-C) d^2\right )\right ) \tan ^2(e+f x)}{\sqrt{a+b \tan (e+f x)} \sqrt{c+d \tan (e+f x)}} \, dx}{d^3 \left (c^2+d^2\right )}\\ &=-\frac{2 \left (c^2 C-B c d+A d^2\right ) (a+b \tan (e+f x))^{5/2}}{d \left (c^2+d^2\right ) f \sqrt{c+d \tan (e+f x)}}-\frac{b \left (3 (b c-a d) \left (5 c^2 C-4 B c d+(4 A+C) d^2\right )-4 d^2 ((A-C) (b c-a d)+B (a c+b d))\right ) \sqrt{a+b \tan (e+f x)} \sqrt{c+d \tan (e+f x)}}{4 d^3 \left (c^2+d^2\right ) f}+\frac{b \left (5 c^2 C-4 B c d+(4 A+C) d^2\right ) (a+b \tan (e+f x))^{3/2} \sqrt{c+d \tan (e+f x)}}{2 d^2 \left (c^2+d^2\right ) f}+\frac{\operatorname{Subst}\left (\int \frac{\frac{1}{8} \left (8 a^3 d^3 (A c-c C+B d)+3 a^2 b d^2 \left (5 c^2 C-8 B c d+(8 A-3 C) d^2\right )+b^3 c \left (15 c^3 C-12 B c^2 d+c (8 A+7 C) d^2-4 B d^3\right )-2 a b^2 d \left (15 c^3 C-10 B c^2 d+3 c (4 A+C) d^2+2 B d^3\right )\right )+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{1}{8} b \left (c^2+d^2\right ) \left (15 a^2 C d^2-10 a b d (3 c C-2 B d)+b^2 \left (15 c^2 C-12 B c d+8 (A-C) d^2\right )\right ) x^2}{\sqrt{a+b x} \sqrt{c+d x} \left (1+x^2\right )} \, dx,x,\tan (e+f x)\right )}{d^3 \left (c^2+d^2\right ) f}\\ &=-\frac{2 \left (c^2 C-B c d+A d^2\right ) (a+b \tan (e+f x))^{5/2}}{d \left (c^2+d^2\right ) f \sqrt{c+d \tan (e+f x)}}-\frac{b \left (3 (b c-a d) \left (5 c^2 C-4 B c d+(4 A+C) d^2\right )-4 d^2 ((A-C) (b c-a d)+B (a c+b d))\right ) \sqrt{a+b \tan (e+f x)} \sqrt{c+d \tan (e+f x)}}{4 d^3 \left (c^2+d^2\right ) f}+\frac{b \left (5 c^2 C-4 B c d+(4 A+C) d^2\right ) (a+b \tan (e+f x))^{3/2} \sqrt{c+d \tan (e+f x)}}{2 d^2 \left (c^2+d^2\right ) f}+\frac{\operatorname{Subst}\left (\int \left (\frac{b \left (c^2+d^2\right ) \left (15 a^2 C d^2-10 a b d (3 c C-2 B d)+b^2 \left (15 c^2 C-12 B c d+8 (A-C) d^2\right )\right )}{8 \sqrt{a+b x} \sqrt{c+d x}}+\frac{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 )+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 )}\right ) \, dx,x,\tan (e+f x)\right )}{d^3 \left (c^2+d^2\right ) f}\\ &=-\frac{2 \left (c^2 C-B c d+A d^2\right ) (a+b \tan (e+f x))^{5/2}}{d \left (c^2+d^2\right ) f \sqrt{c+d \tan (e+f x)}}-\frac{b \left (3 (b c-a d) \left (5 c^2 C-4 B c d+(4 A+C) d^2\right )-4 d^2 ((A-C) (b c-a d)+B (a c+b d))\right ) \sqrt{a+b \tan (e+f x)} \sqrt{c+d \tan (e+f x)}}{4 d^3 \left (c^2+d^2\right ) f}+\frac{b \left (5 c^2 C-4 B c d+(4 A+C) d^2\right ) (a+b \tan (e+f x))^{3/2} \sqrt{c+d \tan (e+f x)}}{2 d^2 \left (c^2+d^2\right ) f}+\frac{\operatorname{Subst}\left (\int \frac{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 )+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 )}{d^3 \left (c^2+d^2\right ) f}+\frac{\left (b \left (15 a^2 C d^2-10 a b d (3 c C-2 B d)+b^2 \left (15 c^2 C-12 B c d+8 (A-C) d^2\right )\right )\right ) \operatorname{Subst}\left (\int \frac{1}{\sqrt{a+b x} \sqrt{c+d x}} \, dx,x,\tan (e+f x)\right )}{8 d^3 f}\\ &=-\frac{2 \left (c^2 C-B c d+A d^2\right ) (a+b \tan (e+f x))^{5/2}}{d \left (c^2+d^2\right ) f \sqrt{c+d \tan (e+f x)}}-\frac{b \left (3 (b c-a d) \left (5 c^2 C-4 B c d+(4 A+C) d^2\right )-4 d^2 ((A-C) (b c-a d)+B (a c+b d))\right ) \sqrt{a+b \tan (e+f x)} \sqrt{c+d \tan (e+f x)}}{4 d^3 \left (c^2+d^2\right ) f}+\frac{b \left (5 c^2 C-4 B c d+(4 A+C) d^2\right ) (a+b \tan (e+f x))^{3/2} \sqrt{c+d \tan (e+f x)}}{2 d^2 \left (c^2+d^2\right ) f}+\frac{\operatorname{Subst}\left (\int \left (\frac{-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 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{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 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 )}{d^3 \left (c^2+d^2\right ) f}+\frac{\left (15 a^2 C d^2-10 a b d (3 c C-2 B d)+b^2 \left (15 c^2 C-12 B c d+8 (A-C) d^2\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 )}{4 d^3 f}\\ &=-\frac{2 \left (c^2 C-B c d+A d^2\right ) (a+b \tan (e+f x))^{5/2}}{d \left (c^2+d^2\right ) f \sqrt{c+d \tan (e+f x)}}-\frac{b \left (3 (b c-a d) \left (5 c^2 C-4 B c d+(4 A+C) d^2\right )-4 d^2 ((A-C) (b c-a d)+B (a c+b d))\right ) \sqrt{a+b \tan (e+f x)} \sqrt{c+d \tan (e+f x)}}{4 d^3 \left (c^2+d^2\right ) f}+\frac{b \left (5 c^2 C-4 B c d+(4 A+C) d^2\right ) (a+b \tan (e+f x))^{3/2} \sqrt{c+d \tan (e+f x)}}{2 d^2 \left (c^2+d^2\right ) 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 \left (c^2+d^2\right ) f}+\frac{\left (15 a^2 C d^2-10 a b d (3 c C-2 B d)+b^2 \left (15 c^2 C-12 B c d+8 (A-C) d^2\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 )}{4 d^3 f}+\frac{\left (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 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 d^3 \left (c^2+d^2\right ) f}\\ &=\frac{\sqrt{b} \left (15 a^2 C d^2-10 a b d (3 c C-2 B d)+b^2 \left (15 c^2 C-12 B c d+8 (A-C) d^2\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 )}{4 d^{7/2} f}-\frac{2 \left (c^2 C-B c d+A d^2\right ) (a+b \tan (e+f x))^{5/2}}{d \left (c^2+d^2\right ) f \sqrt{c+d \tan (e+f x)}}-\frac{b \left (3 (b c-a d) \left (5 c^2 C-4 B c d+(4 A+C) d^2\right )-4 d^2 ((A-C) (b c-a d)+B (a c+b d))\right ) \sqrt{a+b \tan (e+f x)} \sqrt{c+d \tan (e+f x)}}{4 d^3 \left (c^2+d^2\right ) f}+\frac{b \left (5 c^2 C-4 B c d+(4 A+C) d^2\right ) (a+b \tan (e+f x))^{3/2} \sqrt{c+d \tan (e+f x)}}{2 d^2 \left (c^2+d^2\right ) f}+\frac{\left ((a-i b)^3 (i A+B-i C)\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 )}{(c-i 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 )}{\left (c^2+d^2\right ) f}\\ &=-\frac{(a-i b)^{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 )}{(c-i d)^{3/2} f}-\frac{(a+i b)^{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 )}{(c+i d)^{3/2} f}+\frac{\sqrt{b} \left (15 a^2 C d^2-10 a b d (3 c C-2 B d)+b^2 \left (15 c^2 C-12 B c d+8 (A-C) d^2\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 )}{4 d^{7/2} f}-\frac{2 \left (c^2 C-B c d+A d^2\right ) (a+b \tan (e+f x))^{5/2}}{d \left (c^2+d^2\right ) f \sqrt{c+d \tan (e+f x)}}-\frac{b \left (3 (b c-a d) \left (5 c^2 C-4 B c d+(4 A+C) d^2\right )-4 d^2 ((A-C) (b c-a d)+B (a c+b d))\right ) \sqrt{a+b \tan (e+f x)} \sqrt{c+d \tan (e+f x)}}{4 d^3 \left (c^2+d^2\right ) f}+\frac{b \left (5 c^2 C-4 B c d+(4 A+C) d^2\right ) (a+b \tan (e+f x))^{3/2} \sqrt{c+d \tan (e+f x)}}{2 d^2 \left (c^2+d^2\right ) f}\\ \end{align*}

Mathematica [C]  time = 44.2267, size = 1653959, normalized size = 3132.5 \[ \text{Result too large to show} \]

Warning: Unable to verify antiderivative.

[In]

Integrate[((a + b*Tan[e + f*x])^(5/2)*(A + B*Tan[e + f*x] + C*Tan[e + f*x]^2))/(c + d*Tan[e + f*x])^(3/2),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 ( a+b\tan \left ( fx+e \right ) \right ) ^{{\frac{5}{2}}} \left ( c+d\tan \left ( fx+e \right ) \right ) ^{-{\frac{3}{2}}}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a+b*tan(f*x+e))^(5/2)*(A+B*tan(f*x+e)+C*tan(f*x+e)^2)/(c+d*tan(f*x+e))^(3/2),x)

[Out]

int((a+b*tan(f*x+e))^(5/2)*(A+B*tan(f*x+e)+C*tan(f*x+e)^2)/(c+d*tan(f*x+e))^(3/2),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))^(5/2)*(A+B*tan(f*x+e)+C*tan(f*x+e)^2)/(c+d*tan(f*x+e))^(3/2),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))^(5/2)*(A+B*tan(f*x+e)+C*tan(f*x+e)^2)/(c+d*tan(f*x+e))^(3/2),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))**(5/2)*(A+B*tan(f*x+e)+C*tan(f*x+e)**2)/(c+d*tan(f*x+e))**(3/2),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((a+b*tan(f*x+e))^(5/2)*(A+B*tan(f*x+e)+C*tan(f*x+e)^2)/(c+d*tan(f*x+e))^(3/2),x, algorithm="giac")

[Out]

Exception raised: TypeError