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

Optimal. Leaf size=682 \[ \frac{\left (6 a^2 b^2 d^2 \left (8 d^2 (A-C)+12 B c d+3 c^2 C\right )-4 a^3 b d^3 (2 B d+3 c C)+3 a^4 C d^4-12 a b^3 d \left (-24 c d^2 (A-C)-6 B c^2 d+16 B d^3+c^3 C\right )+b^4 \left (48 c^2 d^2 (A-C)-128 d^4 (A-C)-8 B c^3 d-192 B c d^3+3 c^4 C\right )\right ) \tanh ^{-1}\left (\frac{\sqrt{d} \sqrt{a+b \tan (e+f x)}}{\sqrt{b} \sqrt{c+d \tan (e+f x)}}\right )}{64 b^{5/2} d^{5/2} f}+\frac{\sqrt{a+b \tan (e+f x)} \sqrt{c+d \tan (e+f x)} \left (64 b d^3 \left (a^2 B+2 a b (A-C)-b^2 B\right )+(b c-a d) \left (48 b d^2 (a B+A b-b C)+(b c-a d) (-3 a C d-8 b B d+3 b c C)\right )\right )}{64 b^2 d^2 f}+\frac{\sqrt{a+b \tan (e+f x)} (c+d \tan (e+f x))^{3/2} \left (48 b d^2 (a B+A b-b C)+(b c-a d) (-3 a C d-8 b B d+3 b c C)\right )}{96 b d^2 f}-\frac{(a-i b)^{3/2} (c-i d)^{3/2} (B+i (A-C)) \tanh ^{-1}\left (\frac{\sqrt{c-i d} \sqrt{a+b \tan (e+f x)}}{\sqrt{a-i b} \sqrt{c+d \tan (e+f x)}}\right )}{f}-\frac{(a+i b)^{3/2} (c+i d)^{3/2} (B-i (A-C)) \tanh ^{-1}\left (\frac{\sqrt{c+i d} \sqrt{a+b \tan (e+f x)}}{\sqrt{a+i b} \sqrt{c+d \tan (e+f x)}}\right )}{f}-\frac{(-3 a C d-8 b B d+3 b c C) \sqrt{a+b \tan (e+f x)} (c+d \tan (e+f x))^{5/2}}{24 d^2 f}+\frac{C (a+b \tan (e+f x))^{3/2} (c+d \tan (e+f x))^{5/2}}{4 d f} \]

[Out]

-(((a - I*b)^(3/2)*(B + I*(A - C))*(c - I*d)^(3/2)*ArcTanh[(Sqrt[c - I*d]*Sqrt[a + b*Tan[e + f*x]])/(Sqrt[a -
I*b]*Sqrt[c + d*Tan[e + f*x]])])/f) - ((a + I*b)^(3/2)*(B - I*(A - C))*(c + I*d)^(3/2)*ArcTanh[(Sqrt[c + I*d]*
Sqrt[a + b*Tan[e + f*x]])/(Sqrt[a + I*b]*Sqrt[c + d*Tan[e + f*x]])])/f + ((3*a^4*C*d^4 - 4*a^3*b*d^3*(3*c*C +
2*B*d) + 6*a^2*b^2*d^2*(3*c^2*C + 12*B*c*d + 8*(A - C)*d^2) - 12*a*b^3*d*(c^3*C - 6*B*c^2*d - 24*c*(A - C)*d^2
 + 16*B*d^3) + b^4*(3*c^4*C - 8*B*c^3*d + 48*c^2*(A - C)*d^2 - 192*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^(5/2)*d^(5/2)*f) + ((64*b*(a^2*B - b^2*
B + 2*a*b*(A - C))*d^3 + (b*c - a*d)*(48*b*(A*b + a*B - b*C)*d^2 + (b*c - a*d)*(3*b*c*C - 8*b*B*d - 3*a*C*d)))
*Sqrt[a + b*Tan[e + f*x]]*Sqrt[c + d*Tan[e + f*x]])/(64*b^2*d^2*f) + ((48*b*(A*b + a*B - b*C)*d^2 + (b*c - a*d
)*(3*b*c*C - 8*b*B*d - 3*a*C*d))*Sqrt[a + b*Tan[e + f*x]]*(c + d*Tan[e + f*x])^(3/2))/(96*b*d^2*f) - ((3*b*c*C
 - 8*b*B*d - 3*a*C*d)*Sqrt[a + b*Tan[e + f*x]]*(c + d*Tan[e + f*x])^(5/2))/(24*d^2*f) + (C*(a + b*Tan[e + f*x]
)^(3/2)*(c + d*Tan[e + f*x])^(5/2))/(4*d*f)

________________________________________________________________________________________

Rubi [A]  time = 11.8965, antiderivative size = 682, 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} \[ \frac{\left (6 a^2 b^2 d^2 \left (8 d^2 (A-C)+12 B c d+3 c^2 C\right )-4 a^3 b d^3 (2 B d+3 c C)+3 a^4 C d^4-12 a b^3 d \left (-24 c d^2 (A-C)-6 B c^2 d+16 B d^3+c^3 C\right )+b^4 \left (48 c^2 d^2 (A-C)-128 d^4 (A-C)-8 B c^3 d-192 B c d^3+3 c^4 C\right )\right ) \tanh ^{-1}\left (\frac{\sqrt{d} \sqrt{a+b \tan (e+f x)}}{\sqrt{b} \sqrt{c+d \tan (e+f x)}}\right )}{64 b^{5/2} d^{5/2} f}+\frac{\sqrt{a+b \tan (e+f x)} \sqrt{c+d \tan (e+f x)} \left (64 b d^3 \left (a^2 B+2 a b (A-C)-b^2 B\right )+(b c-a d) \left (48 b d^2 (a B+A b-b C)+(b c-a d) (-3 a C d-8 b B d+3 b c C)\right )\right )}{64 b^2 d^2 f}+\frac{\sqrt{a+b \tan (e+f x)} (c+d \tan (e+f x))^{3/2} \left (48 b d^2 (a B+A b-b C)+(b c-a d) (-3 a C d-8 b B d+3 b c C)\right )}{96 b d^2 f}-\frac{(a-i b)^{3/2} (c-i d)^{3/2} (B+i (A-C)) \tanh ^{-1}\left (\frac{\sqrt{c-i d} \sqrt{a+b \tan (e+f x)}}{\sqrt{a-i b} \sqrt{c+d \tan (e+f x)}}\right )}{f}-\frac{(a+i b)^{3/2} (c+i d)^{3/2} (B-i (A-C)) \tanh ^{-1}\left (\frac{\sqrt{c+i d} \sqrt{a+b \tan (e+f x)}}{\sqrt{a+i b} \sqrt{c+d \tan (e+f x)}}\right )}{f}-\frac{(-3 a C d-8 b B d+3 b c C) \sqrt{a+b \tan (e+f x)} (c+d \tan (e+f x))^{5/2}}{24 d^2 f}+\frac{C (a+b \tan (e+f x))^{3/2} (c+d \tan (e+f x))^{5/2}}{4 d f} \]

Antiderivative was successfully verified.

[In]

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

[Out]

-(((a - I*b)^(3/2)*(B + I*(A - C))*(c - I*d)^(3/2)*ArcTanh[(Sqrt[c - I*d]*Sqrt[a + b*Tan[e + f*x]])/(Sqrt[a -
I*b]*Sqrt[c + d*Tan[e + f*x]])])/f) - ((a + I*b)^(3/2)*(B - I*(A - C))*(c + I*d)^(3/2)*ArcTanh[(Sqrt[c + I*d]*
Sqrt[a + b*Tan[e + f*x]])/(Sqrt[a + I*b]*Sqrt[c + d*Tan[e + f*x]])])/f + ((3*a^4*C*d^4 - 4*a^3*b*d^3*(3*c*C +
2*B*d) + 6*a^2*b^2*d^2*(3*c^2*C + 12*B*c*d + 8*(A - C)*d^2) - 12*a*b^3*d*(c^3*C - 6*B*c^2*d - 24*c*(A - C)*d^2
 + 16*B*d^3) + b^4*(3*c^4*C - 8*B*c^3*d + 48*c^2*(A - C)*d^2 - 192*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^(5/2)*d^(5/2)*f) + ((64*b*(a^2*B - b^2*
B + 2*a*b*(A - C))*d^3 + (b*c - a*d)*(48*b*(A*b + a*B - b*C)*d^2 + (b*c - a*d)*(3*b*c*C - 8*b*B*d - 3*a*C*d)))
*Sqrt[a + b*Tan[e + f*x]]*Sqrt[c + d*Tan[e + f*x]])/(64*b^2*d^2*f) + ((48*b*(A*b + a*B - b*C)*d^2 + (b*c - a*d
)*(3*b*c*C - 8*b*B*d - 3*a*C*d))*Sqrt[a + b*Tan[e + f*x]]*(c + d*Tan[e + f*x])^(3/2))/(96*b*d^2*f) - ((3*b*c*C
 - 8*b*B*d - 3*a*C*d)*Sqrt[a + b*Tan[e + f*x]]*(c + d*Tan[e + f*x])^(5/2))/(24*d^2*f) + (C*(a + b*Tan[e + f*x]
)^(3/2)*(c + d*Tan[e + f*x])^(5/2))/(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))^{3/2} (c+d \tan (e+f x))^{3/2} \left (A+B \tan (e+f x)+C \tan ^2(e+f x)\right ) \, dx &=\frac{C (a+b \tan (e+f x))^{3/2} (c+d \tan (e+f x))^{5/2}}{4 d f}+\frac{\int \sqrt{a+b \tan (e+f x)} (c+d \tan (e+f x))^{3/2} \left (\frac{1}{2} (-3 b c C+a (8 A-5 C) d)+4 (A b+a B-b C) d \tan (e+f x)-\frac{1}{2} (3 b c C-8 b B d-3 a C d) \tan ^2(e+f x)\right ) \, dx}{4 d}\\ &=-\frac{(3 b c C-8 b B d-3 a C d) \sqrt{a+b \tan (e+f x)} (c+d \tan (e+f x))^{5/2}}{24 d^2 f}+\frac{C (a+b \tan (e+f x))^{3/2} (c+d \tan (e+f x))^{5/2}}{4 d f}+\frac{\int \frac{(c+d \tan (e+f x))^{3/2} \left (\frac{1}{4} \left (3 a^2 (16 A-15 C) d^2+b^2 c (3 c C-8 B d)-2 a b d (3 c C+20 B d)\right )+12 \left (a^2 B-b^2 B+2 a b (A-C)\right ) d^2 \tan (e+f x)+\frac{1}{4} \left (48 b (A b+a B-b C) d^2+(b c-a d) (3 b c C-8 b B d-3 a C d)\right ) \tan ^2(e+f x)\right )}{\sqrt{a+b \tan (e+f x)}} \, dx}{12 d^2}\\ &=\frac{\left (48 b (A b+a B-b C) d^2+(b c-a d) (3 b c C-8 b B d-3 a C d)\right ) \sqrt{a+b \tan (e+f x)} (c+d \tan (e+f x))^{3/2}}{96 b d^2 f}-\frac{(3 b c C-8 b B d-3 a C d) \sqrt{a+b \tan (e+f x)} (c+d \tan (e+f x))^{5/2}}{24 d^2 f}+\frac{C (a+b \tan (e+f x))^{3/2} (c+d \tan (e+f x))^{5/2}}{4 d f}+\frac{\int \frac{\sqrt{c+d \tan (e+f x)} \left (-\frac{3}{8} \left (3 a^3 C d^3-a^2 b d^2 (64 A c-55 c C-56 B d)-b^3 c \left (3 c^2 C-8 B c d-16 (A-C) d^2\right )+a b^2 d \left (9 c^2 C+64 B c d+48 (A-C) d^2\right )\right )+24 b 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{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 (48 b (A b+a B-b C) d^2+(b c-a d) (3 b c C-8 b B d-3 a C d)\right )\right ) \tan ^2(e+f x)\right )}{\sqrt{a+b \tan (e+f x)}} \, dx}{24 b d^2}\\ &=\frac{\left (64 b \left (a^2 B-b^2 B+2 a b (A-C)\right ) d^3+(b c-a d) \left (48 b (A b+a B-b C) d^2+(b c-a d) (3 b c C-8 b B d-3 a C d)\right )\right ) \sqrt{a+b \tan (e+f x)} \sqrt{c+d \tan (e+f x)}}{64 b^2 d^2 f}+\frac{\left (48 b (A b+a B-b C) d^2+(b c-a d) (3 b c C-8 b B d-3 a C d)\right ) \sqrt{a+b \tan (e+f x)} (c+d \tan (e+f x))^{3/2}}{96 b d^2 f}-\frac{(3 b c C-8 b B d-3 a C d) \sqrt{a+b \tan (e+f x)} (c+d \tan (e+f x))^{5/2}}{24 d^2 f}+\frac{C (a+b \tan (e+f x))^{3/2} (c+d \tan (e+f x))^{5/2}}{4 d f}+\frac{\int \frac{\frac{3}{16} \left (3 a^4 C d^4-4 a^3 b d^3 (3 c C+2 B d)-4 a b^3 d \left (3 c^3 C+46 B c^2 d+56 c (A-C) d^2-16 B d^3\right )+b^4 c \left (3 c^3 C-8 B c^2 d-80 c (A-C) d^2+64 B d^3\right )-2 a^2 b^2 d^2 \left (55 c^2 C+92 B c d-40 C d^2-8 A \left (8 c^2-5 d^2\right )\right )\right )-24 b^2 d^2 \left (2 a b \left (c^2 C+2 B c d-C d^2-A \left (c^2-d^2\right )\right )-a^2 \left (2 c (A-C) d+B \left (c^2-d^2\right )\right )+b^2 \left (2 c (A-C) d+B \left (c^2-d^2\right )\right )\right ) \tan (e+f x)+\frac{3}{16} \left (3 a^4 C d^4-4 a^3 b d^3 (3 c C+2 B d)+6 a^2 b^2 d^2 \left (3 c^2 C+12 B c d+8 (A-C) d^2\right )-12 a b^3 d \left (c^3 C-6 B c^2 d-24 c (A-C) d^2+16 B d^3\right )+b^4 \left (3 c^4 C-8 B c^3 d+48 c^2 (A-C) d^2-192 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^2 d^2}\\ &=\frac{\left (64 b \left (a^2 B-b^2 B+2 a b (A-C)\right ) d^3+(b c-a d) \left (48 b (A b+a B-b C) d^2+(b c-a d) (3 b c C-8 b B d-3 a C d)\right )\right ) \sqrt{a+b \tan (e+f x)} \sqrt{c+d \tan (e+f x)}}{64 b^2 d^2 f}+\frac{\left (48 b (A b+a B-b C) d^2+(b c-a d) (3 b c C-8 b B d-3 a C d)\right ) \sqrt{a+b \tan (e+f x)} (c+d \tan (e+f x))^{3/2}}{96 b d^2 f}-\frac{(3 b c C-8 b B d-3 a C d) \sqrt{a+b \tan (e+f x)} (c+d \tan (e+f x))^{5/2}}{24 d^2 f}+\frac{C (a+b \tan (e+f x))^{3/2} (c+d \tan (e+f x))^{5/2}}{4 d f}+\frac{\operatorname{Subst}\left (\int \frac{\frac{3}{16} \left (3 a^4 C d^4-4 a^3 b d^3 (3 c C+2 B d)-4 a b^3 d \left (3 c^3 C+46 B c^2 d+56 c (A-C) d^2-16 B d^3\right )+b^4 c \left (3 c^3 C-8 B c^2 d-80 c (A-C) d^2+64 B d^3\right )-2 a^2 b^2 d^2 \left (55 c^2 C+92 B c d-40 C d^2-8 A \left (8 c^2-5 d^2\right )\right )\right )-24 b^2 d^2 \left (2 a b \left (c^2 C+2 B c d-C d^2-A \left (c^2-d^2\right )\right )-a^2 \left (2 c (A-C) d+B \left (c^2-d^2\right )\right )+b^2 \left (2 c (A-C) d+B \left (c^2-d^2\right )\right )\right ) x+\frac{3}{16} \left (3 a^4 C d^4-4 a^3 b d^3 (3 c C+2 B d)+6 a^2 b^2 d^2 \left (3 c^2 C+12 B c d+8 (A-C) d^2\right )-12 a b^3 d \left (c^3 C-6 B c^2 d-24 c (A-C) d^2+16 B d^3\right )+b^4 \left (3 c^4 C-8 B c^3 d+48 c^2 (A-C) d^2-192 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^2 d^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 (48 b (A b+a B-b C) d^2+(b c-a d) (3 b c C-8 b B d-3 a C d)\right )\right ) \sqrt{a+b \tan (e+f x)} \sqrt{c+d \tan (e+f x)}}{64 b^2 d^2 f}+\frac{\left (48 b (A b+a B-b C) d^2+(b c-a d) (3 b c C-8 b B d-3 a C d)\right ) \sqrt{a+b \tan (e+f x)} (c+d \tan (e+f x))^{3/2}}{96 b d^2 f}-\frac{(3 b c C-8 b B d-3 a C d) \sqrt{a+b \tan (e+f x)} (c+d \tan (e+f x))^{5/2}}{24 d^2 f}+\frac{C (a+b \tan (e+f x))^{3/2} (c+d \tan (e+f x))^{5/2}}{4 d f}+\frac{\operatorname{Subst}\left (\int \left (\frac{3 \left (3 a^4 C d^4-4 a^3 b d^3 (3 c C+2 B d)+6 a^2 b^2 d^2 \left (3 c^2 C+12 B c d+8 (A-C) d^2\right )-12 a b^3 d \left (c^3 C-6 B c^2 d-24 c (A-C) d^2+16 B d^3\right )+b^4 \left (3 c^4 C-8 B c^3 d+48 c^2 (A-C) d^2-192 B c d^3-128 (A-C) d^4\right )\right )}{16 \sqrt{a+b x} \sqrt{c+d x}}+\frac{24 \left (-b^2 d^2 \left (a^2 \left (c^2 C+2 B c d-C d^2-A \left (c^2-d^2\right )\right )-b^2 \left (c^2 C+2 B c d-C d^2-A \left (c^2-d^2\right )\right )+2 a b \left (2 c (A-C) d+B \left (c^2-d^2\right )\right )\right )-b^2 d^2 \left (2 a b \left (c^2 C+2 B c d-C d^2-A \left (c^2-d^2\right )\right )-a^2 \left (2 c (A-C) d+B \left (c^2-d^2\right )\right )+b^2 \left (2 c (A-C) d+B \left (c^2-d^2\right )\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^2 d^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 (48 b (A b+a B-b C) d^2+(b c-a d) (3 b c C-8 b B d-3 a C d)\right )\right ) \sqrt{a+b \tan (e+f x)} \sqrt{c+d \tan (e+f x)}}{64 b^2 d^2 f}+\frac{\left (48 b (A b+a B-b C) d^2+(b c-a d) (3 b c C-8 b B d-3 a C d)\right ) \sqrt{a+b \tan (e+f x)} (c+d \tan (e+f x))^{3/2}}{96 b d^2 f}-\frac{(3 b c C-8 b B d-3 a C d) \sqrt{a+b \tan (e+f x)} (c+d \tan (e+f x))^{5/2}}{24 d^2 f}+\frac{C (a+b \tan (e+f x))^{3/2} (c+d \tan (e+f x))^{5/2}}{4 d f}+\frac{\operatorname{Subst}\left (\int \frac{-b^2 d^2 \left (a^2 \left (c^2 C+2 B c d-C d^2-A \left (c^2-d^2\right )\right )-b^2 \left (c^2 C+2 B c d-C d^2-A \left (c^2-d^2\right )\right )+2 a b \left (2 c (A-C) d+B \left (c^2-d^2\right )\right )\right )-b^2 d^2 \left (2 a b \left (c^2 C+2 B c d-C d^2-A \left (c^2-d^2\right )\right )-a^2 \left (2 c (A-C) d+B \left (c^2-d^2\right )\right )+b^2 \left (2 c (A-C) d+B \left (c^2-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^2 d^2 f}+\frac{\left (3 a^4 C d^4-4 a^3 b d^3 (3 c C+2 B d)+6 a^2 b^2 d^2 \left (3 c^2 C+12 B c d+8 (A-C) d^2\right )-12 a b^3 d \left (c^3 C-6 B c^2 d-24 c (A-C) d^2+16 B d^3\right )+b^4 \left (3 c^4 C-8 B c^3 d+48 c^2 (A-C) d^2-192 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^2 d^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 (48 b (A b+a B-b C) d^2+(b c-a d) (3 b c C-8 b B d-3 a C d)\right )\right ) \sqrt{a+b \tan (e+f x)} \sqrt{c+d \tan (e+f x)}}{64 b^2 d^2 f}+\frac{\left (48 b (A b+a B-b C) d^2+(b c-a d) (3 b c C-8 b B d-3 a C d)\right ) \sqrt{a+b \tan (e+f x)} (c+d \tan (e+f x))^{3/2}}{96 b d^2 f}-\frac{(3 b c C-8 b B d-3 a C d) \sqrt{a+b \tan (e+f x)} (c+d \tan (e+f x))^{5/2}}{24 d^2 f}+\frac{C (a+b \tan (e+f x))^{3/2} (c+d \tan (e+f x))^{5/2}}{4 d f}+\frac{\operatorname{Subst}\left (\int \left (\frac{-i b^2 d^2 \left (a^2 \left (c^2 C+2 B c d-C d^2-A \left (c^2-d^2\right )\right )-b^2 \left (c^2 C+2 B c d-C d^2-A \left (c^2-d^2\right )\right )+2 a b \left (2 c (A-C) d+B \left (c^2-d^2\right )\right )\right )+b^2 d^2 \left (2 a b \left (c^2 C+2 B c d-C d^2-A \left (c^2-d^2\right )\right )-a^2 \left (2 c (A-C) d+B \left (c^2-d^2\right )\right )+b^2 \left (2 c (A-C) d+B \left (c^2-d^2\right )\right )\right )}{2 (i-x) \sqrt{a+b x} \sqrt{c+d x}}+\frac{-i b^2 d^2 \left (a^2 \left (c^2 C+2 B c d-C d^2-A \left (c^2-d^2\right )\right )-b^2 \left (c^2 C+2 B c d-C d^2-A \left (c^2-d^2\right )\right )+2 a b \left (2 c (A-C) d+B \left (c^2-d^2\right )\right )\right )-b^2 d^2 \left (2 a b \left (c^2 C+2 B c d-C d^2-A \left (c^2-d^2\right )\right )-a^2 \left (2 c (A-C) d+B \left (c^2-d^2\right )\right )+b^2 \left (2 c (A-C) d+B \left (c^2-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^2 d^2 f}+\frac{\left (3 a^4 C d^4-4 a^3 b d^3 (3 c C+2 B d)+6 a^2 b^2 d^2 \left (3 c^2 C+12 B c d+8 (A-C) d^2\right )-12 a b^3 d \left (c^3 C-6 B c^2 d-24 c (A-C) d^2+16 B d^3\right )+b^4 \left (3 c^4 C-8 B c^3 d+48 c^2 (A-C) d^2-192 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^3 d^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 (48 b (A b+a B-b C) d^2+(b c-a d) (3 b c C-8 b B d-3 a C d)\right )\right ) \sqrt{a+b \tan (e+f x)} \sqrt{c+d \tan (e+f x)}}{64 b^2 d^2 f}+\frac{\left (48 b (A b+a B-b C) d^2+(b c-a d) (3 b c C-8 b B d-3 a C d)\right ) \sqrt{a+b \tan (e+f x)} (c+d \tan (e+f x))^{3/2}}{96 b d^2 f}-\frac{(3 b c C-8 b B d-3 a C d) \sqrt{a+b \tan (e+f x)} (c+d \tan (e+f x))^{5/2}}{24 d^2 f}+\frac{C (a+b \tan (e+f x))^{3/2} (c+d \tan (e+f x))^{5/2}}{4 d f}+\frac{\left ((a-i b)^2 (B+i (A-C)) (c-i d)^2\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 (3 a^4 C d^4-4 a^3 b d^3 (3 c C+2 B d)+6 a^2 b^2 d^2 \left (3 c^2 C+12 B c d+8 (A-C) d^2\right )-12 a b^3 d \left (c^3 C-6 B c^2 d-24 c (A-C) d^2+16 B d^3\right )+b^4 \left (3 c^4 C-8 B c^3 d+48 c^2 (A-C) d^2-192 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^3 d^2 f}+\frac{\left (-i b^2 d^2 \left (a^2 \left (c^2 C+2 B c d-C d^2-A \left (c^2-d^2\right )\right )-b^2 \left (c^2 C+2 B c d-C d^2-A \left (c^2-d^2\right )\right )+2 a b \left (2 c (A-C) d+B \left (c^2-d^2\right )\right )\right )+b^2 d^2 \left (2 a b \left (c^2 C+2 B c d-C d^2-A \left (c^2-d^2\right )\right )-a^2 \left (2 c (A-C) d+B \left (c^2-d^2\right )\right )+b^2 \left (2 c (A-C) d+B \left (c^2-d^2\right )\right )\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^2 d^2 f}\\ &=\frac{\left (3 a^4 C d^4-4 a^3 b d^3 (3 c C+2 B d)+6 a^2 b^2 d^2 \left (3 c^2 C+12 B c d+8 (A-C) d^2\right )-12 a b^3 d \left (c^3 C-6 B c^2 d-24 c (A-C) d^2+16 B d^3\right )+b^4 \left (3 c^4 C-8 B c^3 d+48 c^2 (A-C) d^2-192 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^{5/2} d^{5/2} 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 (48 b (A b+a B-b C) d^2+(b c-a d) (3 b c C-8 b B d-3 a C d)\right )\right ) \sqrt{a+b \tan (e+f x)} \sqrt{c+d \tan (e+f x)}}{64 b^2 d^2 f}+\frac{\left (48 b (A b+a B-b C) d^2+(b c-a d) (3 b c C-8 b B d-3 a C d)\right ) \sqrt{a+b \tan (e+f x)} (c+d \tan (e+f x))^{3/2}}{96 b d^2 f}-\frac{(3 b c C-8 b B d-3 a C d) \sqrt{a+b \tan (e+f x)} (c+d \tan (e+f x))^{5/2}}{24 d^2 f}+\frac{C (a+b \tan (e+f x))^{3/2} (c+d \tan (e+f x))^{5/2}}{4 d f}+\frac{\left ((a-i b)^2 (B+i (A-C)) (c-i d)^2\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 b^2 d^2 \left (a^2 \left (c^2 C+2 B c d-C d^2-A \left (c^2-d^2\right )\right )-b^2 \left (c^2 C+2 B c d-C d^2-A \left (c^2-d^2\right )\right )+2 a b \left (2 c (A-C) d+B \left (c^2-d^2\right )\right )\right )+b^2 d^2 \left (2 a b \left (c^2 C+2 B c d-C d^2-A \left (c^2-d^2\right )\right )-a^2 \left (2 c (A-C) d+B \left (c^2-d^2\right )\right )+b^2 \left (2 c (A-C) d+B \left (c^2-d^2\right )\right )\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^2 d^2 f}\\ &=-\frac{(a-i b)^{3/2} (B+i (A-C)) (c-i d)^{3/2} \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{(a+i b)^{3/2} (B-i (A-C)) (c+i d)^{3/2} \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 (3 a^4 C d^4-4 a^3 b d^3 (3 c C+2 B d)+6 a^2 b^2 d^2 \left (3 c^2 C+12 B c d+8 (A-C) d^2\right )-12 a b^3 d \left (c^3 C-6 B c^2 d-24 c (A-C) d^2+16 B d^3\right )+b^4 \left (3 c^4 C-8 B c^3 d+48 c^2 (A-C) d^2-192 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^{5/2} d^{5/2} 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 (48 b (A b+a B-b C) d^2+(b c-a d) (3 b c C-8 b B d-3 a C d)\right )\right ) \sqrt{a+b \tan (e+f x)} \sqrt{c+d \tan (e+f x)}}{64 b^2 d^2 f}+\frac{\left (48 b (A b+a B-b C) d^2+(b c-a d) (3 b c C-8 b B d-3 a C d)\right ) \sqrt{a+b \tan (e+f x)} (c+d \tan (e+f x))^{3/2}}{96 b d^2 f}-\frac{(3 b c C-8 b B d-3 a C d) \sqrt{a+b \tan (e+f x)} (c+d \tan (e+f x))^{5/2}}{24 d^2 f}+\frac{C (a+b \tan (e+f x))^{3/2} (c+d \tan (e+f x))^{5/2}}{4 d f}\\ \end{align*}

Mathematica [A]  time = 8.17153, size = 1316, normalized size = 1.93 \[ \frac{C (a+b \tan (e+f x))^{3/2} (c+d \tan (e+f x))^{5/2}}{4 d f}+\frac{\frac{(-3 b c C+3 a d C+8 b B d) \sqrt{a+b \tan (e+f x)} (c+d \tan (e+f x))^{5/2}}{6 d f}+\frac{\frac{\left (48 b (A b-C b+a B) d^2+(b c-a d) (3 b c C-3 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 \left (B a^2+2 b (A-C) a-b^2 B\right ) d^3-\frac{3}{8} (a d-b c) \left (48 b (A b-C b+a B) d^2+(b c-a d) (3 b c C-3 a d C-8 b B d)\right )\right )}{b f}+\frac{\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 (3 C c^4-8 B d c^3+48 (A-C) d^2 c^2-192 B d^3 c-128 (A-C) d^4\right ) b^4-12 a d \left (C c^3-6 B d c^2-24 (A-C) d^2 c+16 B d^3\right ) b^3+6 a^2 d^2 \left (3 C c^2+12 B d c+8 (A-C) d^2\right ) b^2-4 a^3 d^3 (3 c C+2 B d) b+3 a^4 C d^4\right ) \sqrt{\frac{c+d \tan (e+f x)}{c-\frac{a d}{b}}} \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 )}{8 \sqrt{d} \sqrt{c+d \tan (e+f x)}}+\frac{24 \left (b^5 d^2 \left (-\left (2 c (A-C) d+B \left (c^2-d^2\right )\right ) a^2+2 b \left (C c^2+2 B d c-C d^2-A \left (c^2-d^2\right )\right ) a+b^2 \left (2 c (A-C) d+B \left (c^2-d^2\right )\right )\right )-b^4 \sqrt{-b^2} d^2 \left (\left (C c^2+2 B d c-C d^2-A \left (c^2-d^2\right )\right ) a^2+2 b \left (2 c (A-C) d+B \left (c^2-d^2\right )\right ) a-b^2 \left (C c^2+2 B d c-C d^2-A \left (c^2-d^2\right )\right )\right )\right ) \tan ^{-1}\left (\frac{\sqrt{-c-\frac{\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^2 \sqrt{a+\sqrt{-b^2}} \sqrt{-c-\frac{\sqrt{-b^2} d}{b}}}+\frac{24 \left (-d^2 \left (-\left (2 c (A-C) d+B \left (c^2-d^2\right )\right ) a^2+2 b \left (C c^2+2 B d c-C d^2-A \left (c^2-d^2\right )\right ) a+b^2 \left (2 c (A-C) d+B \left (c^2-d^2\right )\right )\right ) b^5-\sqrt{-b^2} d^2 \left (\left (C c^2+2 B d c-C d^2-A \left (c^2-d^2\right )\right ) a^2+2 b \left (2 c (A-C) d+B \left (c^2-d^2\right )\right ) a-b^2 \left (C c^2+2 B d c-C d^2-A \left (c^2-d^2\right )\right )\right ) b^4\right ) \tan ^{-1}\left (\frac{\sqrt{c-\frac{\sqrt{-b^2} d}{b}} \sqrt{a+b \tan (e+f x)}}{\sqrt{\sqrt{-b^2}-a} \sqrt{c+d \tan (e+f x)}}\right )}{b^2 \sqrt{\sqrt{-b^2}-a} \sqrt{c-\frac{\sqrt{-b^2} d}{b}}}}{b^2 f}}{2 b}}{3 d}}{4 d} \]

Warning: Unable to verify antiderivative.

[In]

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

[Out]

(C*(a + b*Tan[e + f*x])^(3/2)*(c + d*Tan[e + f*x])^(5/2))/(4*d*f) + (((-3*b*c*C + 8*b*B*d + 3*a*C*d)*Sqrt[a +
b*Tan[e + f*x]]*(c + d*Tan[e + f*x])^(5/2))/(6*d*f) + (((48*b*(A*b + a*B - b*C)*d^2 + (b*c - a*d)*(3*b*c*C - 8
*b*B*d - 3*a*C*d))*Sqrt[a + b*Tan[e + f*x]]*(c + d*Tan[e + f*x])^(3/2))/(8*b*f) + (((24*b*(a^2*B - b^2*B + 2*a
*b*(A - C))*d^3 - (3*(-(b*c) + a*d)*(48*b*(A*b + a*B - b*C)*d^2 + (b*c - a*d)*(3*b*c*C - 8*b*B*d - 3*a*C*d)))/
8)*Sqrt[a + b*Tan[e + f*x]]*Sqrt[c + d*Tan[e + f*x]])/(b*f) + ((24*(-(b^4*Sqrt[-b^2]*d^2*(a^2*(c^2*C + 2*B*c*d
 - C*d^2 - A*(c^2 - d^2)) - b^2*(c^2*C + 2*B*c*d - C*d^2 - A*(c^2 - d^2)) + 2*a*b*(2*c*(A - C)*d + B*(c^2 - d^
2)))) + b^5*d^2*(2*a*b*(c^2*C + 2*B*c*d - C*d^2 - A*(c^2 - d^2)) - a^2*(2*c*(A - C)*d + B*(c^2 - d^2)) + b^2*(
2*c*(A - C)*d + B*(c^2 - d^2))))*ArcTan[(Sqrt[-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]])])/(b^2*Sqrt[a + Sqrt[-b^2]]*Sqrt[-c - (Sqrt[-b^2]*d)/b]) + (24*(-(b^4*Sqrt[-b
^2]*d^2*(a^2*(c^2*C + 2*B*c*d - C*d^2 - A*(c^2 - d^2)) - b^2*(c^2*C + 2*B*c*d - C*d^2 - A*(c^2 - d^2)) + 2*a*b
*(2*c*(A - C)*d + B*(c^2 - d^2)))) - b^5*d^2*(2*a*b*(c^2*C + 2*B*c*d - C*d^2 - A*(c^2 - d^2)) - a^2*(2*c*(A -
C)*d + B*(c^2 - d^2)) + b^2*(2*c*(A - C)*d + B*(c^2 - d^2))))*ArcTan[(Sqrt[c - (Sqrt[-b^2]*d)/b]*Sqrt[a + b*Ta
n[e + f*x]])/(Sqrt[-a + Sqrt[-b^2]]*Sqrt[c + d*Tan[e + f*x]])])/(b^2*Sqrt[-a + Sqrt[-b^2]]*Sqrt[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))]*(3*a^4*C*d^4 - 4*a^3*b*d^3*(3*c*C + 2*B*d) + 6*a^2*b^2*d^2*(3*c^2*C + 12*B*c*
d + 8*(A - C)*d^2) - 12*a*b^3*d*(c^3*C - 6*B*c^2*d - 24*c*(A - C)*d^2 + 16*B*d^3) + b^4*(3*c^4*C - 8*B*c^3*d +
 48*c^2*(A - C)*d^2 - 192*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*S
qrt[d]*Sqrt[c + d*Tan[e + f*x]]))/(b^2*f))/(2*b))/(3*d))/(4*d)

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Maple [F]  time = 180., size = 0, normalized size = 0. \begin{align*} \int \left ( a+b\tan \left ( fx+e \right ) \right ) ^{{\frac{3}{2}}} \left ( c+d\tan \left ( fx+e \right ) \right ) ^{{\frac{3}{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))^(3/2)*(c+d*tan(f*x+e))^(3/2)*(A+B*tan(f*x+e)+C*tan(f*x+e)^2),x)

[Out]

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

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Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int{\left (C \tan \left (f x + e\right )^{2} + B \tan \left (f x + e\right ) + A\right )}{\left (b \tan \left (f x + e\right ) + a\right )}^{\frac{3}{2}}{\left (d \tan \left (f x + e\right ) + c\right )}^{\frac{3}{2}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

[Out]

Timed out