3.2.28 \(\int (a+b \tan (e+f x))^{5/2} \sqrt {c+d \tan (e+f x)} (A+B \tan (e+f x)+C \tan ^2(e+f x)) \, dx\) [128]

Optimal. Leaf size=679 \[ -\frac {(a-i b)^{5/2} (i A+B-i C) \sqrt {c-i d} \tanh ^{-1}\left (\frac {\sqrt {c-i d} \sqrt {a+b \tan (e+f x)}}{\sqrt {a-i b} \sqrt {c+d \tan (e+f x)}}\right )}{f}-\frac {(a+i b)^{5/2} (B-i (A-C)) \sqrt {c+i d} \tanh ^{-1}\left (\frac {\sqrt {c+i d} \sqrt {a+b \tan (e+f x)}}{\sqrt {a+i b} \sqrt {c+d \tan (e+f x)}}\right )}{f}-\frac {\left (5 a^4 C d^4-20 a^3 b d^3 (c C+2 B d)+30 a^2 b^2 d^2 \left (c^2 C-4 B c d-8 (A-C) d^2\right )-20 a b^3 d \left (c^3 C-2 B c^2 d+8 c (A-C) d^2-16 B d^3\right )+b^4 \left (5 c^4 C-8 B c^3 d+16 c^2 (A-C) d^2+64 B c d^3+128 (A-C) d^4\right )\right ) \tanh ^{-1}\left (\frac {\sqrt {d} \sqrt {a+b \tan (e+f x)}}{\sqrt {b} \sqrt {c+d \tan (e+f x)}}\right )}{64 b^{3/2} d^{7/2} f}+\frac {\left (64 b \left (a^2 B-b^2 B+2 a b (A-C)\right ) d^3-(b c-a d) \left (16 b (A b+a B-b C) d^2+(b c-a d) (5 b c C-8 b B d-5 a C d)\right )\right ) \sqrt {a+b \tan (e+f x)} \sqrt {c+d \tan (e+f x)}}{64 b d^3 f}+\frac {\left (16 b (A b+a B-b C) d^2+(b c-a d) (5 b c C-8 b B d-5 a C d)\right ) \sqrt {a+b \tan (e+f x)} (c+d \tan (e+f x))^{3/2}}{32 d^3 f}-\frac {(5 b c C-8 b B d-5 a C d) (a+b \tan (e+f x))^{3/2} (c+d \tan (e+f x))^{3/2}}{24 d^2 f}+\frac {C (a+b \tan (e+f x))^{5/2} (c+d \tan (e+f x))^{3/2}}{4 d f} \]

[Out]

-1/64*(5*a^4*C*d^4-20*a^3*b*d^3*(2*B*d+C*c)+30*a^2*b^2*d^2*(c^2*C-4*B*c*d-8*(A-C)*d^2)-20*a*b^3*d*(c^3*C-2*B*c
^2*d+8*c*(A-C)*d^2-16*B*d^3)+b^4*(5*c^4*C-8*B*c^3*d+16*c^2*(A-C)*d^2+64*B*c*d^3+128*(A-C)*d^4))*arctanh(d^(1/2
)*(a+b*tan(f*x+e))^(1/2)/b^(1/2)/(c+d*tan(f*x+e))^(1/2))/b^(3/2)/d^(7/2)/f-(a-I*b)^(5/2)*(I*A+B-I*C)*arctanh((
c-I*d)^(1/2)*(a+b*tan(f*x+e))^(1/2)/(a-I*b)^(1/2)/(c+d*tan(f*x+e))^(1/2))*(c-I*d)^(1/2)/f-(a+I*b)^(5/2)*(B-I*(
A-C))*arctanh((c+I*d)^(1/2)*(a+b*tan(f*x+e))^(1/2)/(a+I*b)^(1/2)/(c+d*tan(f*x+e))^(1/2))*(c+I*d)^(1/2)/f+1/64*
(64*b*(a^2*B-b^2*B+2*a*b*(A-C))*d^3-(-a*d+b*c)*(16*b*(A*b+B*a-C*b)*d^2+(-a*d+b*c)*(-8*B*b*d-5*C*a*d+5*C*b*c)))
*(a+b*tan(f*x+e))^(1/2)*(c+d*tan(f*x+e))^(1/2)/b/d^3/f+1/32*(16*b*(A*b+B*a-C*b)*d^2+(-a*d+b*c)*(-8*B*b*d-5*C*a
*d+5*C*b*c))*(a+b*tan(f*x+e))^(1/2)*(c+d*tan(f*x+e))^(3/2)/d^3/f-1/24*(-8*B*b*d-5*C*a*d+5*C*b*c)*(a+b*tan(f*x+
e))^(3/2)*(c+d*tan(f*x+e))^(3/2)/d^2/f+1/4*C*(a+b*tan(f*x+e))^(5/2)*(c+d*tan(f*x+e))^(3/2)/d/f

________________________________________________________________________________________

Rubi [A]
time = 7.31, antiderivative size = 679, normalized size of antiderivative = 1.00, 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 = {3728, 3736, 6857, 65, 223, 212, 95, 214} \begin {gather*} \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 (16 b d^2 (a B+A b-b C)+(b c-a d) (-5 a C d-8 b B d+5 b c C)\right )\right )}{64 b d^3 f}-\frac {\left (5 a^4 C d^4-20 a^3 b d^3 (2 B d+c C)+30 a^2 b^2 d^2 \left (-8 d^2 (A-C)-4 B c d+c^2 C\right )-20 a b^3 d \left (8 c d^2 (A-C)-2 B c^2 d-16 B d^3+c^3 C\right )+b^4 \left (16 c^2 d^2 (A-C)+128 d^4 (A-C)-8 B c^3 d+64 B c d^3+5 c^4 C\right )\right ) \tanh ^{-1}\left (\frac {\sqrt {d} \sqrt {a+b \tan (e+f x)}}{\sqrt {b} \sqrt {c+d \tan (e+f x)}}\right )}{64 b^{3/2} d^{7/2} f}+\frac {\sqrt {a+b \tan (e+f x)} (c+d \tan (e+f x))^{3/2} \left (16 b d^2 (a B+A b-b C)+(b c-a d) (-5 a C d-8 b B d+5 b c C)\right )}{32 d^3 f}-\frac {(a-i b)^{5/2} \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}-\frac {(a+i b)^{5/2} \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}-\frac {(-5 a C d-8 b B d+5 b c C) (a+b \tan (e+f x))^{3/2} (c+d \tan (e+f x))^{3/2}}{24 d^2 f}+\frac {C (a+b \tan (e+f x))^{5/2} (c+d \tan (e+f x))^{3/2}}{4 d f} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

-(((a - I*b)^(5/2)*(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*Tan[e + f*x]])])/f) - ((a + I*b)^(5/2)*(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]])])/f - ((5*a^4*C*d^4 - 20*a^3*b*d^3*(c*C + 2*B*d
) + 30*a^2*b^2*d^2*(c^2*C - 4*B*c*d - 8*(A - C)*d^2) - 20*a*b^3*d*(c^3*C - 2*B*c^2*d + 8*c*(A - C)*d^2 - 16*B*
d^3) + b^4*(5*c^4*C - 8*B*c^3*d + 16*c^2*(A - C)*d^2 + 64*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^(3/2)*d^(7/2)*f) + ((64*b*(a^2*B - b^2*B + 2*a*b
*(A - C))*d^3 - (b*c - a*d)*(16*b*(A*b + a*B - b*C)*d^2 + (b*c - a*d)*(5*b*c*C - 8*b*B*d - 5*a*C*d)))*Sqrt[a +
b*Tan[e + f*x]]*Sqrt[c + d*Tan[e + f*x]])/(64*b*d^3*f) + ((16*b*(A*b + a*B - b*C)*d^2 + (b*c - a*d)*(5*b*c*C
- 8*b*B*d - 5*a*C*d))*Sqrt[a + b*Tan[e + f*x]]*(c + d*Tan[e + f*x])^(3/2))/(32*d^3*f) - ((5*b*c*C - 8*b*B*d -
5*a*C*d)*(a + b*Tan[e + f*x])^(3/2)*(c + d*Tan[e + f*x])^(3/2))/(24*d^2*f) + (C*(a + b*Tan[e + f*x])^(5/2)*(c
+ d*Tan[e + f*x])^(3/2))/(4*d*f)

Rule 65

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 95

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 212

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1/(Rt[a, 2]*Rt[-b, 2]))*ArcTanh[Rt[-b, 2]*(x/Rt[a, 2])], x]
/; FreeQ[{a, b}, x] && NegQ[a/b] && (GtQ[a, 0] || LtQ[b, 0])

Rule 214

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(Rt[-a/b, 2]/a)*ArcTanh[x/Rt[-a/b, 2]], x] /; FreeQ[{a, b},
x] && NegQ[a/b]

Rule 223

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 3728

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 3736

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 6857

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]

Rubi steps

\begin {align*} \int (a+b \tan (e+f x))^{5/2} \sqrt {c+d \tan (e+f x)} \left (A+B \tan (e+f x)+C \tan ^2(e+f x)\right ) \, dx &=\frac {C (a+b \tan (e+f x))^{5/2} (c+d \tan (e+f x))^{3/2}}{4 d f}+\frac {\int (a+b \tan (e+f x))^{3/2} \sqrt {c+d \tan (e+f x)} \left (\frac {1}{2} (-5 b c C+a (8 A-3 C) d)+4 (A b+a B-b C) d \tan (e+f x)-\frac {1}{2} (5 b c C-8 b B d-5 a C d) \tan ^2(e+f x)\right ) \, dx}{4 d}\\ &=-\frac {(5 b c C-8 b B d-5 a C d) (a+b \tan (e+f x))^{3/2} (c+d \tan (e+f x))^{3/2}}{24 d^2 f}+\frac {C (a+b \tan (e+f x))^{5/2} (c+d \tan (e+f x))^{3/2}}{4 d f}+\frac {\int \sqrt {a+b \tan (e+f x)} \sqrt {c+d \tan (e+f x)} \left (\frac {3}{4} \left (a^2 (16 A-11 C) d^2+b^2 c (5 c C-8 B d)-2 a b d (5 c C+4 B d)\right )+12 \left (a^2 B-b^2 B+2 a b (A-C)\right ) d^2 \tan (e+f x)+\frac {3}{4} \left (16 b (A b+a B-b C) d^2+(b c-a d) (5 b c C-8 b B d-5 a C d)\right ) \tan ^2(e+f x)\right ) \, dx}{12 d^2}\\ &=\frac {\left (16 b (A b+a B-b C) d^2+(b c-a d) (5 b c C-8 b B d-5 a C d)\right ) \sqrt {a+b \tan (e+f x)} (c+d \tan (e+f x))^{3/2}}{32 d^3 f}-\frac {(5 b c C-8 b B d-5 a C d) (a+b \tan (e+f x))^{3/2} (c+d \tan (e+f x))^{3/2}}{24 d^2 f}+\frac {C (a+b \tan (e+f x))^{5/2} (c+d \tan (e+f x))^{3/2}}{4 d f}+\frac {\int \frac {\sqrt {c+d \tan (e+f x)} \left (\frac {3}{8} \left (a^3 (64 A-59 C) d^3-a^2 b d^2 (15 c C+104 B d)+a b^2 d \left (15 c^2 C-32 B c d-48 (A-C) d^2\right )-b^3 c \left (5 c^2 C-8 B c d+16 (A-C) d^2\right )\right )+24 \left (a^3 B-3 a b^2 B+3 a^2 b (A-C)-b^3 (A-C)\right ) d^3 \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 (16 b (A b+a B-b C) d^2+(b c-a d) (5 b c C-8 b B d-5 a C d)\right )\right ) \tan ^2(e+f x)\right )}{\sqrt {a+b \tan (e+f x)}} \, dx}{24 d^3}\\ &=\frac {\left (64 b \left (a^2 B-b^2 B+2 a b (A-C)\right ) d^3-(b c-a d) \left (16 b (A b+a B-b C) d^2+(b c-a d) (5 b c C-8 b B d-5 a C d)\right )\right ) \sqrt {a+b \tan (e+f x)} \sqrt {c+d \tan (e+f x)}}{64 b d^3 f}+\frac {\left (16 b (A b+a B-b C) d^2+(b c-a d) (5 b c C-8 b B d-5 a C d)\right ) \sqrt {a+b \tan (e+f x)} (c+d \tan (e+f x))^{3/2}}{32 d^3 f}-\frac {(5 b c C-8 b B d-5 a C d) (a+b \tan (e+f x))^{3/2} (c+d \tan (e+f x))^{3/2}}{24 d^2 f}+\frac {C (a+b \tan (e+f x))^{5/2} (c+d \tan (e+f x))^{3/2}}{4 d f}+\frac {\int \frac {-\frac {3}{16} \left (5 a^4 C d^4-4 a^3 b d^3 (32 A c-27 c C-22 B d)+6 a^2 b^2 d^2 \left (5 c^2 C+44 B c d+24 (A-C) d^2\right )+b^4 c \left (5 c^3 C-8 B c^2 d+16 c (A-C) d^2-64 B d^3\right )-4 a b^3 d \left (5 c^3 C-10 B c^2 d-56 c (A-C) d^2+16 B d^3\right )\right )+24 b 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 {3}{16} \left (128 b \left (a^3 B-3 a b^2 B+3 a^2 b (A-C)-b^3 (A-C)\right ) d^4+(b c-a d) \left (64 b \left (a^2 B-b^2 B+2 a b (A-C)\right ) d^3-(b c-a d) \left (16 b (A b+a B-b C) d^2+(b c-a d) (5 b c C-8 b B d-5 a C d)\right )\right )\right ) \tan ^2(e+f x)}{\sqrt {a+b \tan (e+f x)} \sqrt {c+d \tan (e+f x)}} \, dx}{24 b d^3}\\ &=\frac {\left (64 b \left (a^2 B-b^2 B+2 a b (A-C)\right ) d^3-(b c-a d) \left (16 b (A b+a B-b C) d^2+(b c-a d) (5 b c C-8 b B d-5 a C d)\right )\right ) \sqrt {a+b \tan (e+f x)} \sqrt {c+d \tan (e+f x)}}{64 b d^3 f}+\frac {\left (16 b (A b+a B-b C) d^2+(b c-a d) (5 b c C-8 b B d-5 a C d)\right ) \sqrt {a+b \tan (e+f x)} (c+d \tan (e+f x))^{3/2}}{32 d^3 f}-\frac {(5 b c C-8 b B d-5 a C d) (a+b \tan (e+f x))^{3/2} (c+d \tan (e+f x))^{3/2}}{24 d^2 f}+\frac {C (a+b \tan (e+f x))^{5/2} (c+d \tan (e+f x))^{3/2}}{4 d f}+\frac {\text {Subst}\left (\int \frac {-\frac {3}{16} \left (5 a^4 C d^4-4 a^3 b d^3 (32 A c-27 c C-22 B d)+6 a^2 b^2 d^2 \left (5 c^2 C+44 B c d+24 (A-C) d^2\right )+b^4 c \left (5 c^3 C-8 B c^2 d+16 c (A-C) d^2-64 B d^3\right )-4 a b^3 d \left (5 c^3 C-10 B c^2 d-56 c (A-C) d^2+16 B d^3\right )\right )+24 b 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 {3}{16} \left (128 b \left (a^3 B-3 a b^2 B+3 a^2 b (A-C)-b^3 (A-C)\right ) d^4+(b c-a d) \left (64 b \left (a^2 B-b^2 B+2 a b (A-C)\right ) d^3-(b c-a d) \left (16 b (A b+a B-b C) d^2+(b c-a d) (5 b c C-8 b B d-5 a C d)\right )\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 d^3 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 (16 b (A b+a B-b C) d^2+(b c-a d) (5 b c C-8 b B d-5 a C d)\right )\right ) \sqrt {a+b \tan (e+f x)} \sqrt {c+d \tan (e+f x)}}{64 b d^3 f}+\frac {\left (16 b (A b+a B-b C) d^2+(b c-a d) (5 b c C-8 b B d-5 a C d)\right ) \sqrt {a+b \tan (e+f x)} (c+d \tan (e+f x))^{3/2}}{32 d^3 f}-\frac {(5 b c C-8 b B d-5 a C d) (a+b \tan (e+f x))^{3/2} (c+d \tan (e+f x))^{3/2}}{24 d^2 f}+\frac {C (a+b \tan (e+f x))^{5/2} (c+d \tan (e+f x))^{3/2}}{4 d f}+\frac {\text {Subst}\left (\int \left (-\frac {3 \left (5 a^4 C d^4-20 a^3 b d^3 (c C+2 B d)+30 a^2 b^2 d^2 \left (c^2 C-4 B c d-8 (A-C) d^2\right )-20 a b^3 d \left (c^3 C-2 B c^2 d+8 c (A-C) d^2-16 B d^3\right )+b^4 \left (5 c^4 C-8 B c^3 d+16 c^2 (A-C) d^2+64 B c d^3+128 (A-C) d^4\right )\right )}{16 \sqrt {a+b x} \sqrt {c+d x}}+\frac {24 \left (b 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 )+b 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\right )}{\sqrt {a+b x} \sqrt {c+d x} \left (1+x^2\right )}\right ) \, dx,x,\tan (e+f x)\right )}{24 b d^3 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 (16 b (A b+a B-b C) d^2+(b c-a d) (5 b c C-8 b B d-5 a C d)\right )\right ) \sqrt {a+b \tan (e+f x)} \sqrt {c+d \tan (e+f x)}}{64 b d^3 f}+\frac {\left (16 b (A b+a B-b C) d^2+(b c-a d) (5 b c C-8 b B d-5 a C d)\right ) \sqrt {a+b \tan (e+f x)} (c+d \tan (e+f x))^{3/2}}{32 d^3 f}-\frac {(5 b c C-8 b B d-5 a C d) (a+b \tan (e+f x))^{3/2} (c+d \tan (e+f x))^{3/2}}{24 d^2 f}+\frac {C (a+b \tan (e+f x))^{5/2} (c+d \tan (e+f x))^{3/2}}{4 d f}+\frac {\text {Subst}\left (\int \frac {b 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 )+b 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 )}{b d^3 f}-\frac {\left (5 a^4 C d^4-20 a^3 b d^3 (c C+2 B d)+30 a^2 b^2 d^2 \left (c^2 C-4 B c d-8 (A-C) d^2\right )-20 a b^3 d \left (c^3 C-2 B c^2 d+8 c (A-C) d^2-16 B d^3\right )+b^4 \left (5 c^4 C-8 B c^3 d+16 c^2 (A-C) d^2+64 B c d^3+128 (A-C) d^4\right )\right ) \text {Subst}\left (\int \frac {1}{\sqrt {a+b x} \sqrt {c+d x}} \, dx,x,\tan (e+f x)\right )}{128 b d^3 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 (16 b (A b+a B-b C) d^2+(b c-a d) (5 b c C-8 b B d-5 a C d)\right )\right ) \sqrt {a+b \tan (e+f x)} \sqrt {c+d \tan (e+f x)}}{64 b d^3 f}+\frac {\left (16 b (A b+a B-b C) d^2+(b c-a d) (5 b c C-8 b B d-5 a C d)\right ) \sqrt {a+b \tan (e+f x)} (c+d \tan (e+f x))^{3/2}}{32 d^3 f}-\frac {(5 b c C-8 b B d-5 a C d) (a+b \tan (e+f x))^{3/2} (c+d \tan (e+f x))^{3/2}}{24 d^2 f}+\frac {C (a+b \tan (e+f x))^{5/2} (c+d \tan (e+f x))^{3/2}}{4 d f}+\frac {\text {Subst}\left (\int \left (\frac {-b 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 b 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 {b 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 b 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 )}{b d^3 f}-\frac {\left (5 a^4 C d^4-20 a^3 b d^3 (c C+2 B d)+30 a^2 b^2 d^2 \left (c^2 C-4 B c d-8 (A-C) d^2\right )-20 a b^3 d \left (c^3 C-2 B c^2 d+8 c (A-C) d^2-16 B d^3\right )+b^4 \left (5 c^4 C-8 B c^3 d+16 c^2 (A-C) d^2+64 B c d^3+128 (A-C) d^4\right )\right ) \text {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^2 d^3 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 (16 b (A b+a B-b C) d^2+(b c-a d) (5 b c C-8 b B d-5 a C d)\right )\right ) \sqrt {a+b \tan (e+f x)} \sqrt {c+d \tan (e+f x)}}{64 b d^3 f}+\frac {\left (16 b (A b+a B-b C) d^2+(b c-a d) (5 b c C-8 b B d-5 a C d)\right ) \sqrt {a+b \tan (e+f x)} (c+d \tan (e+f x))^{3/2}}{32 d^3 f}-\frac {(5 b c C-8 b B d-5 a C d) (a+b \tan (e+f x))^{3/2} (c+d \tan (e+f x))^{3/2}}{24 d^2 f}+\frac {C (a+b \tan (e+f x))^{5/2} (c+d \tan (e+f x))^{3/2}}{4 d f}+\frac {\left ((a-i b)^3 (A-i B-C) (i c+d)\right ) \text {Subst}\left (\int \frac {1}{(i+x) \sqrt {a+b x} \sqrt {c+d x}} \, dx,x,\tan (e+f x)\right )}{2 f}-\frac {\left (5 a^4 C d^4-20 a^3 b d^3 (c C+2 B d)+30 a^2 b^2 d^2 \left (c^2 C-4 B c d-8 (A-C) d^2\right )-20 a b^3 d \left (c^3 C-2 B c^2 d+8 c (A-C) d^2-16 B d^3\right )+b^4 \left (5 c^4 C-8 B c^3 d+16 c^2 (A-C) d^2+64 B c d^3+128 (A-C) d^4\right )\right ) \text {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^2 d^3 f}+\frac {\left (-b 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 b 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 ) \text {Subst}\left (\int \frac {1}{(i-x) \sqrt {a+b x} \sqrt {c+d x}} \, dx,x,\tan (e+f x)\right )}{2 b d^3 f}\\ &=-\frac {\left (5 a^4 C d^4-20 a^3 b d^3 (c C+2 B d)+30 a^2 b^2 d^2 \left (c^2 C-4 B c d-8 (A-C) d^2\right )-20 a b^3 d \left (c^3 C-2 B c^2 d+8 c (A-C) d^2-16 B d^3\right )+b^4 \left (5 c^4 C-8 B c^3 d+16 c^2 (A-C) d^2+64 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^{3/2} d^{7/2} 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 (16 b (A b+a B-b C) d^2+(b c-a d) (5 b c C-8 b B d-5 a C d)\right )\right ) \sqrt {a+b \tan (e+f x)} \sqrt {c+d \tan (e+f x)}}{64 b d^3 f}+\frac {\left (16 b (A b+a B-b C) d^2+(b c-a d) (5 b c C-8 b B d-5 a C d)\right ) \sqrt {a+b \tan (e+f x)} (c+d \tan (e+f x))^{3/2}}{32 d^3 f}-\frac {(5 b c C-8 b B d-5 a C d) (a+b \tan (e+f x))^{3/2} (c+d \tan (e+f x))^{3/2}}{24 d^2 f}+\frac {C (a+b \tan (e+f x))^{5/2} (c+d \tan (e+f x))^{3/2}}{4 d f}+\frac {\left ((a-i b)^3 (A-i B-C) (i c+d)\right ) \text {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 (-b 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 b 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 ) \text {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 d^3 f}\\ &=-\frac {(a-i b)^{5/2} (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 )}{f}-\frac {(a+i b)^{5/2} (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 )}{f}-\frac {\left (5 a^4 C d^4-20 a^3 b d^3 (c C+2 B d)+30 a^2 b^2 d^2 \left (c^2 C-4 B c d-8 (A-C) d^2\right )-20 a b^3 d \left (c^3 C-2 B c^2 d+8 c (A-C) d^2-16 B d^3\right )+b^4 \left (5 c^4 C-8 B c^3 d+16 c^2 (A-C) d^2+64 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^{3/2} d^{7/2} 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 (16 b (A b+a B-b C) d^2+(b c-a d) (5 b c C-8 b B d-5 a C d)\right )\right ) \sqrt {a+b \tan (e+f x)} \sqrt {c+d \tan (e+f x)}}{64 b d^3 f}+\frac {\left (16 b (A b+a B-b C) d^2+(b c-a d) (5 b c C-8 b B d-5 a C d)\right ) \sqrt {a+b \tan (e+f x)} (c+d \tan (e+f x))^{3/2}}{32 d^3 f}-\frac {(5 b c C-8 b B d-5 a C d) (a+b \tan (e+f x))^{3/2} (c+d \tan (e+f x))^{3/2}}{24 d^2 f}+\frac {C (a+b \tan (e+f x))^{5/2} (c+d \tan (e+f x))^{3/2}}{4 d f}\\ \end {align*}

________________________________________________________________________________________

Mathematica [A]
time = 8.58, size = 1202, normalized size = 1.77 \begin {gather*} \frac {C (a+b \tan (e+f x))^{5/2} (c+d \tan (e+f x))^{3/2}}{4 d f}+\frac {\frac {(-5 b c C+8 b B d+5 a C d) (a+b \tan (e+f x))^{3/2} (c+d \tan (e+f x))^{3/2}}{6 d f}+\frac {\frac {3 \left (16 b (A b+a B-b C) d^2+(b c-a d) (5 b c C-8 b B d-5 a C d)\right ) \sqrt {a+b \tan (e+f x)} (c+d \tan (e+f x))^{3/2}}{8 d f}+\frac {\frac {\left (24 b \left (a^2 B-b^2 B+2 a b (A-C)\right ) d^3-\frac {3}{8} (b c-a d) \left (16 b (A b+a B-b C) d^2+(b c-a d) (5 b c C-8 b B d-5 a C d)\right )\right ) \sqrt {a+b \tan (e+f x)} \sqrt {c+d \tan (e+f x)}}{b f}+\frac {\frac {24 b d^3 \left (b \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 )+\sqrt {-b^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 )\right ) \tanh ^{-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 )}{\sqrt {-a+\sqrt {-b^2}} \sqrt {-c+\frac {\sqrt {-b^2} d}{b}}}-\frac {24 b d^3 \left (b \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 )-\sqrt {-b^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 )\right ) \tanh ^{-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 )}{\sqrt {a+\sqrt {-b^2}} \sqrt {c+\frac {\sqrt {-b^2} d}{b}}}-\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 (5 a^4 C d^4-20 a^3 b d^3 (c C+2 B d)+30 a^2 b^2 d^2 \left (c^2 C-4 B c d-8 (A-C) d^2\right )-20 a b^3 d \left (c^3 C-2 B c^2 d+8 c (A-C) d^2-16 B d^3\right )+b^4 \left (5 c^4 C-8 B c^3 d+16 c^2 (A-C) d^2+64 B c d^3+128 (A-C) d^4\right )\right ) \sinh ^{-1}\left (\frac {\sqrt {d} \sqrt {a+b \tan (e+f x)}}{\sqrt {b} \sqrt {c-\frac {a d}{b}} \sqrt {\frac {c}{c-\frac {a d}{b}}-\frac {a d}{b \left (c-\frac {a d}{b}\right )}}}\right ) \sqrt {\frac {c+d \tan (e+f x)}{c-\frac {a d}{b}}}}{8 \sqrt {d} \sqrt {c+d \tan (e+f x)}}}{b^2 f}}{2 d}}{3 d}}{4 d} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(C*(a + b*Tan[e + f*x])^(5/2)*(c + d*Tan[e + f*x])^(3/2))/(4*d*f) + (((-5*b*c*C + 8*b*B*d + 5*a*C*d)*(a + b*Ta
n[e + f*x])^(3/2)*(c + d*Tan[e + f*x])^(3/2))/(6*d*f) + ((3*(16*b*(A*b + a*B - b*C)*d^2 + (b*c - a*d)*(5*b*c*C
- 8*b*B*d - 5*a*C*d))*Sqrt[a + b*Tan[e + f*x]]*(c + d*Tan[e + f*x])^(3/2))/(8*d*f) + (((24*b*(a^2*B - b^2*B +
2*a*b*(A - C))*d^3 - (3*(b*c - a*d)*(16*b*(A*b + a*B - b*C)*d^2 + (b*c - a*d)*(5*b*c*C - 8*b*B*d - 5*a*C*d)))
/8)*Sqrt[a + b*Tan[e + f*x]]*Sqrt[c + d*Tan[e + f*x]])/(b*f) + ((24*b*d^3*(b*(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)) + Sqrt[-b^2]*(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)))*ArcTanh[(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]])])/(Sqrt[-a + Sqrt[-b^2]]*Sqrt
[-c + (Sqrt[-b^2]*d)/b]) - (24*b*d^3*(b*(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)) - Sqrt[-b^2]*(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)))*ArcTanh[(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]])])/(Sqrt[a + Sqrt[-b^2]]*Sqrt[c + (Sqrt[-b^2]*d)/b]) - (3*Sqrt[b]*Sqr
t[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))]*(5*a^4*C*d^4 - 20*a^3*b*d^3*(c*C + 2*B*d) + 30*a^2*b^2*d^2*(c^2*C - 4*B*c*d - 8*(A - C)*d^2) - 20*a*b^
3*d*(c^3*C - 2*B*c^2*d + 8*c*(A - C)*d^2 - 16*B*d^3) + b^4*(5*c^4*C - 8*B*c^3*d + 16*c^2*(A - C)*d^2 + 64*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*Sqrt[d]*Sqrt[c + d*Tan[e + f*x]
]))/(b^2*f))/(2*d))/(3*d))/(4*d)

________________________________________________________________________________________

Maple [F]
time = 180.00, size = 0, normalized size = 0.00 \[\int \sqrt {c +d \tan \left (f x +e \right )}\, \left (a +b \tan \left (f x +e \right )\right )^{\frac {5}{2}} \left (A +B \tan \left (f x +e \right )+C \left (\tan ^{2}\left (f x +e \right )\right )\right )\, dx\]

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))^(5/2)*(A+B*tan(f*x+e)+C*tan(f*x+e)^2),x)

[Out]

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

________________________________________________________________________________________

Maxima [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \end {gather*}

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))^(5/2)*(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)^(5/2)*sqrt(d*tan(f*x + e) + c), x)

________________________________________________________________________________________

Fricas [F(-1)] Timed out
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}

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))^(5/2)*(A+B*tan(f*x+e)+C*tan(f*x+e)^2),x, algorithm="fricas")

[Out]

Timed out

________________________________________________________________________________________

Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \left (a + b \tan {\left (e + f x \right )}\right )^{\frac {5}{2}} \sqrt {c + d \tan {\left (e + f x \right )}} \left (A + B \tan {\left (e + f x \right )} + C \tan ^{2}{\left (e + f x \right )}\right )\, dx \end {gather*}

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))**(5/2)*(A+B*tan(f*x+e)+C*tan(f*x+e)**2),x)

[Out]

Integral((a + b*tan(e + f*x))**(5/2)*sqrt(c + d*tan(e + f*x))*(A + B*tan(e + f*x) + C*tan(e + f*x)**2), x)

________________________________________________________________________________________

Giac [F(-2)]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: TypeError} \end {gather*}

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))^(5/2)*(A+B*tan(f*x+e)+C*tan(f*x+e)^2),x, algorithm="giac")

[Out]

Exception raised: TypeError >> An error occurred running a Giac command:INPUT:sage2:=int(sage0,sageVARx):;OUTP
UT:Warning, need to choose a branch for the root of a polynomial with parameters. This might be wrong.The choi
ce was done

________________________________________________________________________________________

Mupad [F]
time = 0.00, size = -1, normalized size = -0.00 \begin {gather*} \int {\left (a+b\,\mathrm {tan}\left (e+f\,x\right )\right )}^{5/2}\,\sqrt {c+d\,\mathrm {tan}\left (e+f\,x\right )}\,\left (C\,{\mathrm {tan}\left (e+f\,x\right )}^2+B\,\mathrm {tan}\left (e+f\,x\right )+A\right ) \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________