\(\int \frac {(a+b \tan (e+f x))^{5/2} (A+B \tan (e+f x)+C \tan ^2(e+f x))}{(c+d \tan (e+f x))^{3/2}} \, dx\) [153]

Optimal result

Integrand size = 49, antiderivative size = 528 \[ \int \frac {(a+b \tan (e+f x))^{5/2} \left (A+B \tan (e+f x)+C \tan ^2(e+f x)\right )}{(c+d \tan (e+f x))^{3/2}} \, dx=-\frac {(a-i b)^{5/2} (i A+B-i C) \text {arctanh}\left (\frac {\sqrt {c-i d} \sqrt {a+b \tan (e+f x)}}{\sqrt {a-i b} \sqrt {c+d \tan (e+f x)}}\right )}{(c-i d)^{3/2} f}-\frac {(a+i b)^{5/2} (B-i (A-C)) \text {arctanh}\left (\frac {\sqrt {c+i d} \sqrt {a+b \tan (e+f x)}}{\sqrt {a+i b} \sqrt {c+d \tan (e+f x)}}\right )}{(c+i d)^{3/2} f}+\frac {\sqrt {b} \left (15 a^2 C d^2-10 a b d (3 c C-2 B d)+b^2 \left (15 c^2 C-12 B c d+8 (A-C) d^2\right )\right ) \text {arctanh}\left (\frac {\sqrt {d} \sqrt {a+b \tan (e+f x)}}{\sqrt {b} \sqrt {c+d \tan (e+f x)}}\right )}{4 d^{7/2} f}-\frac {2 \left (c^2 C-B c d+A d^2\right ) (a+b \tan (e+f x))^{5/2}}{d \left (c^2+d^2\right ) f \sqrt {c+d \tan (e+f x)}}-\frac {b \left (3 (b c-a d) \left (5 c^2 C-4 B c d+(4 A+C) d^2\right )-4 d^2 ((A-C) (b c-a d)+B (a c+b d))\right ) \sqrt {a+b \tan (e+f x)} \sqrt {c+d \tan (e+f x)}}{4 d^3 \left (c^2+d^2\right ) f}+\frac {b \left (5 c^2 C-4 B c d+(4 A+C) d^2\right ) (a+b \tan (e+f x))^{3/2} \sqrt {c+d \tan (e+f x)}}{2 d^2 \left (c^2+d^2\right ) f} \]

[Out]

-(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)^(3/2)/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)^(3/2)/f+1/4*(15*a^2*C*d^2-10*a*b*d*(-2*B*d+3*C*c)+b^2*(15*c^2*C-12*B*c*d+8*(A-C)*d^2))*
arctanh(d^(1/2)*(a+b*tan(f*x+e))^(1/2)/b^(1/2)/(c+d*tan(f*x+e))^(1/2))*b^(1/2)/d^(7/2)/f-1/4*b*(3*(-a*d+b*c)*(
5*c^2*C-4*B*c*d+(4*A+C)*d^2)-4*d^2*((A-C)*(-a*d+b*c)+B*(a*c+b*d)))*(a+b*tan(f*x+e))^(1/2)*(c+d*tan(f*x+e))^(1/
2)/d^3/(c^2+d^2)/f+1/2*b*(5*c^2*C-4*B*c*d+(4*A+C)*d^2)*(c+d*tan(f*x+e))^(1/2)*(a+b*tan(f*x+e))^(3/2)/d^2/(c^2+
d^2)/f-2*(A*d^2-B*c*d+C*c^2)*(a+b*tan(f*x+e))^(5/2)/d/(c^2+d^2)/f/(c+d*tan(f*x+e))^(1/2)

Rubi [A] (verified)

Time = 9.81 (sec) , antiderivative size = 528, normalized size of antiderivative = 1.00, number of steps used = 15, number of rules used = 9, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.184, Rules used = {3726, 3728, 3736, 6857, 65, 223, 212, 95, 214} \[ \int \frac {(a+b \tan (e+f x))^{5/2} \left (A+B \tan (e+f x)+C \tan ^2(e+f x)\right )}{(c+d \tan (e+f x))^{3/2}} \, dx=\frac {\sqrt {b} \left (15 a^2 C d^2-10 a b d (3 c C-2 B d)+b^2 \left (8 d^2 (A-C)-12 B c d+15 c^2 C\right )\right ) \text {arctanh}\left (\frac {\sqrt {d} \sqrt {a+b \tan (e+f x)}}{\sqrt {b} \sqrt {c+d \tan (e+f x)}}\right )}{4 d^{7/2} f}-\frac {(a-i b)^{5/2} (i A+B-i C) \text {arctanh}\left (\frac {\sqrt {c-i d} \sqrt {a+b \tan (e+f x)}}{\sqrt {a-i b} \sqrt {c+d \tan (e+f x)}}\right )}{f (c-i d)^{3/2}}-\frac {(a+i b)^{5/2} (B-i (A-C)) \text {arctanh}\left (\frac {\sqrt {c+i d} \sqrt {a+b \tan (e+f x)}}{\sqrt {a+i b} \sqrt {c+d \tan (e+f x)}}\right )}{f (c+i d)^{3/2}}+\frac {b \left (d^2 (4 A+C)-4 B c d+5 c^2 C\right ) (a+b \tan (e+f x))^{3/2} \sqrt {c+d \tan (e+f x)}}{2 d^2 f \left (c^2+d^2\right )}-\frac {2 \left (A d^2-B c d+c^2 C\right ) (a+b \tan (e+f x))^{5/2}}{d f \left (c^2+d^2\right ) \sqrt {c+d \tan (e+f x)}}-\frac {b \sqrt {a+b \tan (e+f x)} \sqrt {c+d \tan (e+f x)} \left (3 (b c-a d) \left (d^2 (4 A+C)-4 B c d+5 c^2 C\right )-4 d^2 ((A-C) (b c-a d)+B (a c+b d))\right )}{4 d^3 f \left (c^2+d^2\right )} \]

[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 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 3726

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*Ta
n[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 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*} \text {integral}& = -\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 {\text {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 {\text {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 {\text {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 ) \text {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 {\text {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 ) \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 )}{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 ) \text {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 ) \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 )}{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 ) \text {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 ) \text {arctanh}\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 ) \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 )}{(c-i 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 )}{\left (c^2+d^2\right ) f} \\ & = -\frac {(a-i b)^{5/2} (i A+B-i C) \text {arctanh}\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)) \text {arctanh}\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 ) \text {arctanh}\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 [B] (verified)

Both result and optimal contain complex but leaf count is larger than twice the leaf count of optimal. \(2245\) vs. \(2(528)=1056\).

Time = 9.61 (sec) , antiderivative size = 2245, normalized size of antiderivative = 4.25 \[ \int \frac {(a+b \tan (e+f x))^{5/2} \left (A+B \tan (e+f x)+C \tan ^2(e+f x)\right )}{(c+d \tan (e+f x))^{3/2}} \, dx=\text {Result too large to show} \]

[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]

(C*(a + b*Tan[e + f*x])^(5/2))/(2*d*f*Sqrt[c + d*Tan[e + f*x]]) + (((-5*b*c*C + 4*b*B*d + 5*a*C*d)*(a + b*Tan[
e + f*x])^(3/2))/(2*d*f*Sqrt[c + d*Tan[e + f*x]]) + ((8*(-a + I*b)^(5/2)*(I*A + B - I*C)*d^2*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) - (8*(a + I*
b)^(5/2)*(B - I*(A - C))*d^2*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) + (8*(a - I*b)^2*(I*A + B - I*C)*d^2*Sqrt[a + b*Tan[e + f*x]])/((c - I*d)*f*Sqr
t[c + d*Tan[e + f*x]]) + (8*(a + I*b)^2*(B - I*(A - C))*d^2*Sqrt[a + b*Tan[e + f*x]])/((c + I*d)*f*Sqrt[c + d*
Tan[e + f*x]]) + (30*a^2*C*(b*c - a*d)*(b/((b^2*c)/(b*c - a*d) - (a*b*d)/(b*c - a*d)))^(3/2)*((b^2*c)/(b*c - a
*d) - (a*b*d)/(b*c - a*d))^2*Sqrt[(b*(c + d*Tan[e + f*x]))/(b*c - a*d)]*(-1 - (b*d*(a + b*Tan[e + f*x]))/((b*c
- a*d)*((b^2*c)/(b*c - a*d) - (a*b*d)/(b*c - a*d))))*(-((b*d*(a + b*Tan[e + f*x]))/((b*c - a*d)*((b^2*c)/(b*c
- a*d) - (a*b*d)/(b*c - a*d))*(-1 - (b*d*(a + b*Tan[e + f*x]))/((b*c - a*d)*((b^2*c)/(b*c - a*d) - (a*b*d)/(b
*c - a*d)))))) - (Sqrt[b]*Sqrt[d]*ArcSinh[(Sqrt[b]*Sqrt[d]*Sqrt[a + b*Tan[e + f*x]])/(Sqrt[b*c - a*d]*Sqrt[(b^
2*c)/(b*c - a*d) - (a*b*d)/(b*c - a*d)])]*Sqrt[a + b*Tan[e + f*x]])/(Sqrt[b*c - a*d]*Sqrt[(b^2*c)/(b*c - a*d)
- (a*b*d)/(b*c - a*d)]*Sqrt[1 + (b*d*(a + b*Tan[e + f*x]))/((b*c - a*d)*((b^2*c)/(b*c - a*d) - (a*b*d)/(b*c -
a*d)))])))/(b^2*f*Sqrt[a + b*Tan[e + f*x]]*Sqrt[c + d*Tan[e + f*x]]*Sqrt[1 + (b*d*(a + b*Tan[e + f*x]))/((b*c
- a*d)*((b^2*c)/(b*c - a*d) - (a*b*d)/(b*c - a*d)))]) - (20*a*(b*c - a*d)*(3*c*C - 2*B*d)*(b/((b^2*c)/(b*c - a
*d) - (a*b*d)/(b*c - a*d)))^(3/2)*((b^2*c)/(b*c - a*d) - (a*b*d)/(b*c - a*d))^2*Sqrt[(b*(c + d*Tan[e + f*x]))/
(b*c - a*d)]*(-1 - (b*d*(a + b*Tan[e + f*x]))/((b*c - a*d)*((b^2*c)/(b*c - a*d) - (a*b*d)/(b*c - a*d))))*(-((b
*d*(a + b*Tan[e + f*x]))/((b*c - a*d)*((b^2*c)/(b*c - a*d) - (a*b*d)/(b*c - a*d))*(-1 - (b*d*(a + b*Tan[e + f*
x]))/((b*c - a*d)*((b^2*c)/(b*c - a*d) - (a*b*d)/(b*c - a*d)))))) - (Sqrt[b]*Sqrt[d]*ArcSinh[(Sqrt[b]*Sqrt[d]*
Sqrt[a + b*Tan[e + f*x]])/(Sqrt[b*c - a*d]*Sqrt[(b^2*c)/(b*c - a*d) - (a*b*d)/(b*c - a*d)])]*Sqrt[a + b*Tan[e
+ f*x]])/(Sqrt[b*c - a*d]*Sqrt[(b^2*c)/(b*c - a*d) - (a*b*d)/(b*c - a*d)]*Sqrt[1 + (b*d*(a + b*Tan[e + f*x]))/
((b*c - a*d)*((b^2*c)/(b*c - a*d) - (a*b*d)/(b*c - a*d)))])))/(b*d*f*Sqrt[a + b*Tan[e + f*x]]*Sqrt[c + d*Tan[e
+ f*x]]*Sqrt[1 + (b*d*(a + b*Tan[e + f*x]))/((b*c - a*d)*((b^2*c)/(b*c - a*d) - (a*b*d)/(b*c - a*d)))]) + (2*
(b*c - a*d)*(15*c^2*C - 12*B*c*d + 8*(A - C)*d^2)*(b/((b^2*c)/(b*c - a*d) - (a*b*d)/(b*c - a*d)))^(3/2)*((b^2*
c)/(b*c - a*d) - (a*b*d)/(b*c - a*d))^2*Sqrt[(b*(c + d*Tan[e + f*x]))/(b*c - a*d)]*(-1 - (b*d*(a + b*Tan[e + f
*x]))/((b*c - a*d)*((b^2*c)/(b*c - a*d) - (a*b*d)/(b*c - a*d))))*(-((b*d*(a + b*Tan[e + f*x]))/((b*c - a*d)*((
b^2*c)/(b*c - a*d) - (a*b*d)/(b*c - a*d))*(-1 - (b*d*(a + b*Tan[e + f*x]))/((b*c - a*d)*((b^2*c)/(b*c - a*d) -
(a*b*d)/(b*c - a*d)))))) - (Sqrt[b]*Sqrt[d]*ArcSinh[(Sqrt[b]*Sqrt[d]*Sqrt[a + b*Tan[e + f*x]])/(Sqrt[b*c - a*
d]*Sqrt[(b^2*c)/(b*c - a*d) - (a*b*d)/(b*c - a*d)])]*Sqrt[a + b*Tan[e + f*x]])/(Sqrt[b*c - a*d]*Sqrt[(b^2*c)/(
b*c - a*d) - (a*b*d)/(b*c - a*d)]*Sqrt[1 + (b*d*(a + b*Tan[e + f*x]))/((b*c - a*d)*((b^2*c)/(b*c - a*d) - (a*b
*d)/(b*c - a*d)))])))/(d^2*f*Sqrt[a + b*Tan[e + f*x]]*Sqrt[c + d*Tan[e + f*x]]*Sqrt[1 + (b*d*(a + b*Tan[e + f*
x]))/((b*c - a*d)*((b^2*c)/(b*c - a*d) - (a*b*d)/(b*c - a*d)))]))/(4*d))/(2*d)

Maple [F(-1)]

Timed out.

\[\int \frac {\left (a +b \tan \left (f x +e \right )\right )^{\frac {5}{2}} \left (A +B \tan \left (f x +e \right )+C \tan \left (f x +e \right )^{2}\right )}{\left (c +d \tan \left (f x +e \right )\right )^{\frac {3}{2}}}d x\]

[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)

Fricas [F(-1)]

Timed out. \[ \int \frac {(a+b \tan (e+f x))^{5/2} \left (A+B \tan (e+f x)+C \tan ^2(e+f x)\right )}{(c+d \tan (e+f x))^{3/2}} \, dx=\text {Timed out} \]

[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

Sympy [F]

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

[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]

Integral((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)

Maxima [F(-1)]

Timed out. \[ \int \frac {(a+b \tan (e+f x))^{5/2} \left (A+B \tan (e+f x)+C \tan ^2(e+f x)\right )}{(c+d \tan (e+f x))^{3/2}} \, dx=\text {Timed out} \]

[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

Giac [F(-1)]

Timed out. \[ \int \frac {(a+b \tan (e+f x))^{5/2} \left (A+B \tan (e+f x)+C \tan ^2(e+f x)\right )}{(c+d \tan (e+f x))^{3/2}} \, dx=\text {Timed out} \]

[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]

Timed out

Mupad [F(-1)]

Timed out. \[ \int \frac {(a+b \tan (e+f x))^{5/2} \left (A+B \tan (e+f x)+C \tan ^2(e+f x)\right )}{(c+d \tan (e+f x))^{3/2}} \, dx=\int \frac {{\left (a+b\,\mathrm {tan}\left (e+f\,x\right )\right )}^{5/2}\,\left (C\,{\mathrm {tan}\left (e+f\,x\right )}^2+B\,\mathrm {tan}\left (e+f\,x\right )+A\right )}{{\left (c+d\,\mathrm {tan}\left (e+f\,x\right )\right )}^{3/2}} \,d x \]

[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]

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)