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

Optimal result

Integrand size = 49, antiderivative size = 679 \[ \int (a+b \tan (e+f x))^{5/2} \sqrt {c+d \tan (e+f x)} \left (A+B \tan (e+f x)+C \tan ^2(e+f x)\right ) \, dx=-\frac {(a-i b)^{5/2} (i A+B-i C) \sqrt {c-i d} \text {arctanh}\left (\frac {\sqrt {c-i d} \sqrt {a+b \tan (e+f x)}}{\sqrt {a-i b} \sqrt {c+d \tan (e+f x)}}\right )}{f}-\frac {(a+i b)^{5/2} (B-i (A-C)) \sqrt {c+i d} \text {arctanh}\left (\frac {\sqrt {c+i d} \sqrt {a+b \tan (e+f x)}}{\sqrt {a+i b} \sqrt {c+d \tan (e+f x)}}\right )}{f}-\frac {\left (5 a^4 C d^4-20 a^3 b d^3 (c C+2 B d)+30 a^2 b^2 d^2 \left (c^2 C-4 B c d-8 (A-C) d^2\right )-20 a b^3 d \left (c^3 C-2 B c^2 d+8 c (A-C) d^2-16 B d^3\right )+b^4 \left (5 c^4 C-8 B c^3 d+16 c^2 (A-C) d^2+64 B c d^3+128 (A-C) d^4\right )\right ) \text {arctanh}\left (\frac {\sqrt {d} \sqrt {a+b \tan (e+f x)}}{\sqrt {b} \sqrt {c+d \tan (e+f x)}}\right )}{64 b^{3/2} d^{7/2} f}+\frac {\left (64 b \left (a^2 B-b^2 B+2 a b (A-C)\right ) d^3-(b c-a d) \left (16 b (A b+a B-b C) d^2+(b c-a d) (5 b c C-8 b B d-5 a C d)\right )\right ) \sqrt {a+b \tan (e+f x)} \sqrt {c+d \tan (e+f x)}}{64 b d^3 f}+\frac {\left (16 b (A b+a B-b C) d^2+(b c-a d) (5 b c C-8 b B d-5 a C d)\right ) \sqrt {a+b \tan (e+f x)} (c+d \tan (e+f x))^{3/2}}{32 d^3 f}-\frac {(5 b c C-8 b B d-5 a C d) (a+b \tan (e+f x))^{3/2} (c+d \tan (e+f x))^{3/2}}{24 d^2 f}+\frac {C (a+b \tan (e+f x))^{5/2} (c+d \tan (e+f x))^{3/2}}{4 d f} \] Output:

-(a-I*b)^(5/2)*(I*A+B-I*C)*(c-I*d)^(1/2)*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))/f-(a+I*b)^(5/2)*(B-I*(A- 
C))*(c+I*d)^(1/2)*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))/f-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+1/64*(64*b*(B*a^2-B*b^2+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
                                                                                    
                                                                                    
 

Mathematica [A] (verified)

Time = 8.53 (sec) , antiderivative size = 1202, normalized size of antiderivative = 1.77 \[ \int (a+b \tan (e+f x))^{5/2} \sqrt {c+d \tan (e+f x)} \left (A+B \tan (e+f x)+C \tan ^2(e+f x)\right ) \, dx =\text {Too large to display} \] Input:

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]
 

Output:

(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*Tan[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)))*A 
rcTanh[(Sqrt[-c + (Sqrt[-b^2]*d)/b]*Sqrt[a + b*Tan[e + f*x]])/(Sqrt[-a + S 
qrt[-b^2]]*Sqrt[c + d*Tan[e + f*x]])])/(Sqrt[-a + Sqrt[-b^2]]*Sqrt[-c + (S 
qrt[-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]])])/(Sqr 
t[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))]*(5*a^4*C*d^4 - 20*a^3*b*d^3*(c*C + 2*B*...
 

Rubi [A] (verified)

Time = 9.26 (sec) , antiderivative size = 707, normalized size of antiderivative = 1.04, number of steps used = 17, number of rules used = 16, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.327, Rules used = {3042, 4130, 27, 3042, 4130, 27, 3042, 4130, 27, 3042, 4130, 27, 3042, 4138, 2348, 2009}

Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.

\(\displaystyle \int (a+b \tan (e+f x))^{5/2} \sqrt {c+d \tan (e+f x)} \left (A+B \tan (e+f x)+C \tan ^2(e+f x)\right ) \, dx\)

\(\Big \downarrow \) 3042

\(\displaystyle \int (a+b \tan (e+f x))^{5/2} \sqrt {c+d \tan (e+f x)} \left (A+B \tan (e+f x)+C \tan (e+f x)^2\right )dx\)

\(\Big \downarrow \) 4130

\(\displaystyle \frac {\int -\frac {1}{2} (a+b \tan (e+f x))^{3/2} \sqrt {c+d \tan (e+f x)} \left ((5 b c C-5 a d C-8 b B d) \tan ^2(e+f x)-8 (A b-C b+a B) d \tan (e+f x)+5 b c C-a (8 A-3 C) d\right )dx}{4 d}+\frac {C (a+b \tan (e+f x))^{5/2} (c+d \tan (e+f x))^{3/2}}{4 d f}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {C (a+b \tan (e+f x))^{5/2} (c+d \tan (e+f x))^{3/2}}{4 d f}-\frac {\int (a+b \tan (e+f x))^{3/2} \sqrt {c+d \tan (e+f x)} \left ((5 b c C-5 a d C-8 b B d) \tan ^2(e+f x)-8 (A b-C b+a B) d \tan (e+f x)+5 b c C-a (8 A-3 C) d\right )dx}{8 d}\)

\(\Big \downarrow \) 3042

\(\displaystyle \frac {C (a+b \tan (e+f x))^{5/2} (c+d \tan (e+f x))^{3/2}}{4 d f}-\frac {\int (a+b \tan (e+f x))^{3/2} \sqrt {c+d \tan (e+f x)} \left ((5 b c C-5 a d C-8 b B d) \tan (e+f x)^2-8 (A b-C b+a B) d \tan (e+f x)+5 b c C-a (8 A-3 C) d\right )dx}{8 d}\)

\(\Big \downarrow \) 4130

\(\displaystyle \frac {C (a+b \tan (e+f x))^{5/2} (c+d \tan (e+f x))^{3/2}}{4 d f}-\frac {\frac {\int -\frac {3}{2} \sqrt {a+b \tan (e+f x)} \sqrt {c+d \tan (e+f x)} \left (c (5 c C-8 B d) b^2-2 a d (5 c C+4 B d) b+a^2 (16 A-11 C) d^2+\left (16 b (A b-C b+a B) d^2+(b c-a d) (5 b c C-5 a d C-8 b B d)\right ) \tan ^2(e+f x)+16 \left (B a^2+2 b (A-C) a-b^2 B\right ) d^2 \tan (e+f x)\right )dx}{3 d}+\frac {(-5 a C d-8 b B d+5 b c C) (a+b \tan (e+f x))^{3/2} (c+d \tan (e+f x))^{3/2}}{3 d f}}{8 d}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {C (a+b \tan (e+f x))^{5/2} (c+d \tan (e+f x))^{3/2}}{4 d f}-\frac {\frac {(-5 a C d-8 b B d+5 b c C) (a+b \tan (e+f x))^{3/2} (c+d \tan (e+f x))^{3/2}}{3 d f}-\frac {\int \sqrt {a+b \tan (e+f x)} \sqrt {c+d \tan (e+f x)} \left (c (5 c C-8 B d) b^2-2 a d (5 c C+4 B d) b+a^2 (16 A-11 C) d^2+\left (16 b (A b-C b+a B) d^2+(b c-a d) (5 b c C-5 a d C-8 b B d)\right ) \tan ^2(e+f x)+16 \left (B a^2+2 b (A-C) a-b^2 B\right ) d^2 \tan (e+f x)\right )dx}{2 d}}{8 d}\)

\(\Big \downarrow \) 3042

\(\displaystyle \frac {C (a+b \tan (e+f x))^{5/2} (c+d \tan (e+f x))^{3/2}}{4 d f}-\frac {\frac {(-5 a C d-8 b B d+5 b c C) (a+b \tan (e+f x))^{3/2} (c+d \tan (e+f x))^{3/2}}{3 d f}-\frac {\int \sqrt {a+b \tan (e+f x)} \sqrt {c+d \tan (e+f x)} \left (c (5 c C-8 B d) b^2-2 a d (5 c C+4 B d) b+a^2 (16 A-11 C) d^2+\left (16 b (A b-C b+a B) d^2+(b c-a d) (5 b c C-5 a d C-8 b B d)\right ) \tan (e+f x)^2+16 \left (B a^2+2 b (A-C) a-b^2 B\right ) d^2 \tan (e+f x)\right )dx}{2 d}}{8 d}\)

\(\Big \downarrow \) 4130

\(\displaystyle \frac {C (a+b \tan (e+f x))^{5/2} (c+d \tan (e+f x))^{3/2}}{4 d f}-\frac {\frac {(-5 a C d-8 b B d+5 b c C) (a+b \tan (e+f x))^{3/2} (c+d \tan (e+f x))^{3/2}}{3 d f}-\frac {\frac {\int \frac {\sqrt {c+d \tan (e+f x)} \left (-c \left (5 C c^2-8 B d c+16 (A-C) d^2\right ) b^3+a d \left (15 C c^2-32 B d c-48 (A-C) d^2\right ) b^2-a^2 d^2 (15 c C+104 B d) b+a^3 (64 A-59 C) d^3+\left (64 b \left (B a^2+2 b (A-C) a-b^2 B\right ) d^3-(b c-a d) \left (16 b (A b-C b+a B) d^2+(b c-a d) (5 b c C-5 a d C-8 b B d)\right )\right ) \tan ^2(e+f x)+64 \left (B a^3+3 b (A-C) a^2-3 b^2 B a-b^3 (A-C)\right ) d^3 \tan (e+f x)\right )}{2 \sqrt {a+b \tan (e+f x)}}dx}{2 d}+\frac {\sqrt {a+b \tan (e+f x)} (c+d \tan (e+f x))^{3/2} \left (16 b d^2 (a B+A b-b C)+(b c-a d) (-5 a C d-8 b B d+5 b c C)\right )}{2 d f}}{2 d}}{8 d}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {C (a+b \tan (e+f x))^{5/2} (c+d \tan (e+f x))^{3/2}}{4 d f}-\frac {\frac {(-5 a C d-8 b B d+5 b c C) (a+b \tan (e+f x))^{3/2} (c+d \tan (e+f x))^{3/2}}{3 d f}-\frac {\frac {\int \frac {\sqrt {c+d \tan (e+f x)} \left (-c \left (5 C c^2-8 B d c+16 (A-C) d^2\right ) b^3+a d \left (15 C c^2-32 B d c-48 (A-C) d^2\right ) b^2-a^2 d^2 (15 c C+104 B d) b+a^3 (64 A-59 C) d^3+\left (64 b \left (B a^2+2 b (A-C) a-b^2 B\right ) d^3-(b c-a d) \left (16 b (A b-C b+a B) d^2+(b c-a d) (5 b c C-5 a d C-8 b B d)\right )\right ) \tan ^2(e+f x)+64 \left (B a^3+3 b (A-C) a^2-3 b^2 B a-b^3 (A-C)\right ) d^3 \tan (e+f x)\right )}{\sqrt {a+b \tan (e+f x)}}dx}{4 d}+\frac {\sqrt {a+b \tan (e+f x)} (c+d \tan (e+f x))^{3/2} \left (16 b d^2 (a B+A b-b C)+(b c-a d) (-5 a C d-8 b B d+5 b c C)\right )}{2 d f}}{2 d}}{8 d}\)

\(\Big \downarrow \) 3042

\(\displaystyle \frac {C (a+b \tan (e+f x))^{5/2} (c+d \tan (e+f x))^{3/2}}{4 d f}-\frac {\frac {(-5 a C d-8 b B d+5 b c C) (a+b \tan (e+f x))^{3/2} (c+d \tan (e+f x))^{3/2}}{3 d f}-\frac {\frac {\int \frac {\sqrt {c+d \tan (e+f x)} \left (-c \left (5 C c^2-8 B d c+16 (A-C) d^2\right ) b^3+a d \left (15 C c^2-32 B d c-48 (A-C) d^2\right ) b^2-a^2 d^2 (15 c C+104 B d) b+a^3 (64 A-59 C) d^3+\left (64 b \left (B a^2+2 b (A-C) a-b^2 B\right ) d^3-(b c-a d) \left (16 b (A b-C b+a B) d^2+(b c-a d) (5 b c C-5 a d C-8 b B d)\right )\right ) \tan (e+f x)^2+64 \left (B a^3+3 b (A-C) a^2-3 b^2 B a-b^3 (A-C)\right ) d^3 \tan (e+f x)\right )}{\sqrt {a+b \tan (e+f x)}}dx}{4 d}+\frac {\sqrt {a+b \tan (e+f x)} (c+d \tan (e+f x))^{3/2} \left (16 b d^2 (a B+A b-b C)+(b c-a d) (-5 a C d-8 b B d+5 b c C)\right )}{2 d f}}{2 d}}{8 d}\)

\(\Big \downarrow \) 4130

\(\displaystyle \frac {C (a+b \tan (e+f x))^{5/2} (c+d \tan (e+f x))^{3/2}}{4 d f}-\frac {\frac {(-5 a C d-8 b B d+5 b c C) (a+b \tan (e+f x))^{3/2} (c+d \tan (e+f x))^{3/2}}{3 d f}-\frac {\frac {\frac {\int -\frac {c \left (5 C c^3-8 B d c^2+16 (A-C) d^2 c-64 B d^3\right ) b^4-4 a d \left (5 C c^3-10 B d c^2-56 (A-C) d^2 c+16 B d^3\right ) b^3+6 a^2 d^2 \left (5 C c^2+44 B d c+24 (A-C) d^2\right ) b^2-4 a^3 d^3 (32 A c-27 C c-22 B d) b-128 d^3 \left ((B c+(A-C) d) a^3+3 b (A c-C c-B d) a^2-3 b^2 (B c+(A-C) d) a-b^3 (A c-C c-B d)\right ) \tan (e+f x) b+5 a^4 C d^4-\left (128 b \left (B a^3+3 b (A-C) a^2-3 b^2 B a-b^3 (A-C)\right ) d^4+(b c-a d) \left (64 b \left (B a^2+2 b (A-C) a-b^2 B\right ) d^3-(b c-a d) \left (16 b (A b-C b+a B) d^2+(b c-a d) (5 b c C-5 a d C-8 b B d)\right )\right )\right ) \tan ^2(e+f x)}{2 \sqrt {a+b \tan (e+f x)} \sqrt {c+d \tan (e+f x)}}dx}{b}+\frac {\sqrt {a+b \tan (e+f x)} \sqrt {c+d \tan (e+f x)} \left (64 b d^3 \left (a^2 B+2 a b (A-C)-b^2 B\right )-(b c-a d) \left (16 b d^2 (a B+A b-b C)+(b c-a d) (-5 a C d-8 b B d+5 b c C)\right )\right )}{b f}}{4 d}+\frac {\sqrt {a+b \tan (e+f x)} (c+d \tan (e+f x))^{3/2} \left (16 b d^2 (a B+A b-b C)+(b c-a d) (-5 a C d-8 b B d+5 b c C)\right )}{2 d f}}{2 d}}{8 d}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {C (a+b \tan (e+f x))^{5/2} (c+d \tan (e+f x))^{3/2}}{4 d f}-\frac {\frac {(-5 a C d-8 b B d+5 b c C) (a+b \tan (e+f x))^{3/2} (c+d \tan (e+f x))^{3/2}}{3 d f}-\frac {\frac {\frac {\sqrt {a+b \tan (e+f x)} \sqrt {c+d \tan (e+f x)} \left (64 b d^3 \left (a^2 B+2 a b (A-C)-b^2 B\right )-(b c-a d) \left (16 b d^2 (a B+A b-b C)+(b c-a d) (-5 a C d-8 b B d+5 b c C)\right )\right )}{b f}-\frac {\int \frac {c \left (5 C c^3-8 B d c^2+16 (A-C) d^2 c-64 B d^3\right ) b^4-4 a d \left (5 C c^3-10 B d c^2-56 (A-C) d^2 c+16 B d^3\right ) b^3+6 a^2 d^2 \left (5 C c^2+44 B d c+24 (A-C) d^2\right ) b^2-4 a^3 d^3 (32 A c-27 C c-22 B d) b-128 d^3 \left ((B c+(A-C) d) a^3+3 b (A c-C c-B d) a^2-3 b^2 (B c+(A-C) d) a-b^3 (A c-C c-B d)\right ) \tan (e+f x) b+5 a^4 C d^4-\left (128 b \left (B a^3+3 b (A-C) a^2-3 b^2 B a-b^3 (A-C)\right ) d^4+(b c-a d) \left (64 b \left (B a^2+2 b (A-C) a-b^2 B\right ) d^3-(b c-a d) \left (16 b (A b-C b+a B) d^2+(b c-a d) (5 b c C-5 a d C-8 b B d)\right )\right )\right ) \tan ^2(e+f x)}{\sqrt {a+b \tan (e+f x)} \sqrt {c+d \tan (e+f x)}}dx}{2 b}}{4 d}+\frac {\sqrt {a+b \tan (e+f x)} (c+d \tan (e+f x))^{3/2} \left (16 b d^2 (a B+A b-b C)+(b c-a d) (-5 a C d-8 b B d+5 b c C)\right )}{2 d f}}{2 d}}{8 d}\)

\(\Big \downarrow \) 3042

\(\displaystyle \frac {C (a+b \tan (e+f x))^{5/2} (c+d \tan (e+f x))^{3/2}}{4 d f}-\frac {\frac {(-5 a C d-8 b B d+5 b c C) (a+b \tan (e+f x))^{3/2} (c+d \tan (e+f x))^{3/2}}{3 d f}-\frac {\frac {\frac {\sqrt {a+b \tan (e+f x)} \sqrt {c+d \tan (e+f x)} \left (64 b d^3 \left (a^2 B+2 a b (A-C)-b^2 B\right )-(b c-a d) \left (16 b d^2 (a B+A b-b C)+(b c-a d) (-5 a C d-8 b B d+5 b c C)\right )\right )}{b f}-\frac {\int \frac {c \left (5 C c^3-8 B d c^2+16 (A-C) d^2 c-64 B d^3\right ) b^4-4 a d \left (5 C c^3-10 B d c^2-56 (A-C) d^2 c+16 B d^3\right ) b^3+6 a^2 d^2 \left (5 C c^2+44 B d c+24 (A-C) d^2\right ) b^2-4 a^3 d^3 (32 A c-27 C c-22 B d) b-128 d^3 \left ((B c+(A-C) d) a^3+3 b (A c-C c-B d) a^2-3 b^2 (B c+(A-C) d) a-b^3 (A c-C c-B d)\right ) \tan (e+f x) b+5 a^4 C d^4-\left (128 b \left (B a^3+3 b (A-C) a^2-3 b^2 B a-b^3 (A-C)\right ) d^4+(b c-a d) \left (64 b \left (B a^2+2 b (A-C) a-b^2 B\right ) d^3-(b c-a d) \left (16 b (A b-C b+a B) d^2+(b c-a d) (5 b c C-5 a d C-8 b B d)\right )\right )\right ) \tan (e+f x)^2}{\sqrt {a+b \tan (e+f x)} \sqrt {c+d \tan (e+f x)}}dx}{2 b}}{4 d}+\frac {\sqrt {a+b \tan (e+f x)} (c+d \tan (e+f x))^{3/2} \left (16 b d^2 (a B+A b-b C)+(b c-a d) (-5 a C d-8 b B d+5 b c C)\right )}{2 d f}}{2 d}}{8 d}\)

\(\Big \downarrow \) 4138

\(\displaystyle \frac {C (a+b \tan (e+f x))^{5/2} (c+d \tan (e+f x))^{3/2}}{4 d f}-\frac {\frac {(-5 a C d-8 b B d+5 b c C) (a+b \tan (e+f x))^{3/2} (c+d \tan (e+f x))^{3/2}}{3 d f}-\frac {\frac {\frac {\sqrt {a+b \tan (e+f x)} \sqrt {c+d \tan (e+f x)} \left (64 b d^3 \left (a^2 B+2 a b (A-C)-b^2 B\right )-(b c-a d) \left (16 b d^2 (a B+A b-b C)+(b c-a d) (-5 a C d-8 b B d+5 b c C)\right )\right )}{b f}-\frac {\int \frac {c \left (5 C c^3-8 B d c^2+16 (A-C) d^2 c-64 B d^3\right ) b^4-4 a d \left (5 C c^3-10 B d c^2-56 (A-C) d^2 c+16 B d^3\right ) b^3+6 a^2 d^2 \left (5 C c^2+44 B d c+24 (A-C) d^2\right ) b^2-4 a^3 d^3 (32 A c-27 C c-22 B d) b-128 d^3 \left ((B c+(A-C) d) a^3+3 b (A c-C c-B d) a^2-3 b^2 (B c+(A-C) d) a-b^3 (A c-C c-B d)\right ) \tan (e+f x) b+5 a^4 C d^4-\left (128 b \left (B a^3+3 b (A-C) a^2-3 b^2 B a-b^3 (A-C)\right ) d^4+(b c-a d) \left (64 b \left (B a^2+2 b (A-C) a-b^2 B\right ) d^3-(b c-a d) \left (16 b (A b-C b+a B) d^2+(b c-a d) (5 b c C-5 a d C-8 b B d)\right )\right )\right ) \tan ^2(e+f x)}{\sqrt {a+b \tan (e+f x)} \sqrt {c+d \tan (e+f x)} \left (\tan ^2(e+f x)+1\right )}d\tan (e+f x)}{2 b f}}{4 d}+\frac {\sqrt {a+b \tan (e+f x)} (c+d \tan (e+f x))^{3/2} \left (16 b d^2 (a B+A b-b C)+(b c-a d) (-5 a C d-8 b B d+5 b c C)\right )}{2 d f}}{2 d}}{8 d}\)

\(\Big \downarrow \) 2348

\(\displaystyle \frac {C (a+b \tan (e+f x))^{5/2} (c+d \tan (e+f x))^{3/2}}{4 d f}-\frac {\frac {(5 b c C-5 a d C-8 b B d) (a+b \tan (e+f x))^{3/2} (c+d \tan (e+f x))^{3/2}}{3 d f}-\frac {\frac {\left (16 b (A b-C b+a B) d^2+(b c-a d) (5 b c C-5 a d C-8 b B d)\right ) \sqrt {a+b \tan (e+f x)} (c+d \tan (e+f x))^{3/2}}{2 d f}+\frac {\frac {\left (64 b \left (B a^2+2 b (A-C) a-b^2 B\right ) d^3-(b c-a d) \left (16 b (A b-C b+a B) d^2+(b c-a d) (5 b c C-5 a d C-8 b B d)\right )\right ) \sqrt {a+b \tan (e+f x)} \sqrt {c+d \tan (e+f x)}}{b f}-\frac {\int \left (\frac {128 A d^4 b^4-128 C d^4 b^4+64 B c d^3 b^4+16 A c^2 d^2 b^4-16 c^2 C d^2 b^4+5 c^4 C b^4-8 B c^3 d b^4+320 a B d^4 b^3-160 a A c d^3 b^3+160 a c C d^3 b^3+40 a B c^2 d^2 b^3-20 a c^3 C d b^3-240 a^2 A d^4 b^2+240 a^2 C d^4 b^2-120 a^2 B c d^3 b^2+30 a^2 c^2 C d^2 b^2-40 a^3 B d^4 b-20 a^3 c C d^3 b+5 a^4 C d^4}{\sqrt {a+b \tan (e+f x)} \sqrt {c+d \tan (e+f x)}}+\frac {128 B d^4 b^4-128 A c d^3 b^4+128 c C d^3 b^4-384 a A d^4 b^3+384 a C d^4 b^3-384 a B c d^3 b^3-384 a^2 B d^4 b^2+384 a^2 A c d^3 b^2-384 a^2 c C d^3 b^2+128 a^3 A d^4 b-128 a^3 C d^4 b+128 a^3 B c d^3 b+i \left (-128 A d^4 b^4+128 C d^4 b^4-128 B c d^3 b^4-384 a B d^4 b^3+384 a A c d^3 b^3-384 a c C d^3 b^3+384 a^2 A d^4 b^2-384 a^2 C d^4 b^2+384 a^2 B c d^3 b^2+128 a^3 B d^4 b-128 a^3 A c d^3 b+128 a^3 c C d^3 b\right )}{2 (i-\tan (e+f x)) \sqrt {a+b \tan (e+f x)} \sqrt {c+d \tan (e+f x)}}+\frac {-128 B d^4 b^4+128 A c d^3 b^4-128 c C d^3 b^4+384 a A d^4 b^3-384 a C d^4 b^3+384 a B c d^3 b^3+384 a^2 B d^4 b^2-384 a^2 A c d^3 b^2+384 a^2 c C d^3 b^2-128 a^3 A d^4 b+128 a^3 C d^4 b-128 a^3 B c d^3 b+i \left (-128 A d^4 b^4+128 C d^4 b^4-128 B c d^3 b^4-384 a B d^4 b^3+384 a A c d^3 b^3-384 a c C d^3 b^3+384 a^2 A d^4 b^2-384 a^2 C d^4 b^2+384 a^2 B c d^3 b^2+128 a^3 B d^4 b-128 a^3 A c d^3 b+128 a^3 c C d^3 b\right )}{2 (\tan (e+f x)+i) \sqrt {a+b \tan (e+f x)} \sqrt {c+d \tan (e+f x)}}\right )d\tan (e+f x)}{2 b f}}{4 d}}{2 d}}{8 d}\)

\(\Big \downarrow \) 2009

\(\displaystyle \frac {C (a+b \tan (e+f x))^{5/2} (c+d \tan (e+f x))^{3/2}}{4 d f}-\frac {\frac {(-5 a C d-8 b B d+5 b c C) (a+b \tan (e+f x))^{3/2} (c+d \tan (e+f x))^{3/2}}{3 d f}-\frac {\frac {\sqrt {a+b \tan (e+f x)} (c+d \tan (e+f x))^{3/2} \left (16 b d^2 (a B+A b-b C)+(b c-a d) (-5 a C d-8 b B d+5 b c C)\right )}{2 d f}+\frac {\frac {\sqrt {a+b \tan (e+f x)} \sqrt {c+d \tan (e+f x)} \left (64 b d^3 \left (a^2 B+2 a b (A-C)-b^2 B\right )-(b c-a d) \left (16 b d^2 (a B+A b-b C)+(b c-a d) (-5 a C d-8 b B d+5 b c C)\right )\right )}{b f}-\frac {\frac {2 \left (5 a^4 C d^4-20 a^3 b d^3 (2 B d+c C)+30 a^2 b^2 d^2 \left (-8 d^2 (A-C)-4 B c d+c^2 C\right )-20 a b^3 d \left (8 c d^2 (A-C)-2 B c^2 d-16 B d^3+c^3 C\right )+b^4 \left (16 c^2 d^2 (A-C)+128 d^4 (A-C)-8 B c^3 d+64 B c d^3+5 c^4 C\right )\right ) \text {arctanh}\left (\frac {\sqrt {d} \sqrt {a+b \tan (e+f x)}}{\sqrt {b} \sqrt {c+d \tan (e+f x)}}\right )}{\sqrt {b} \sqrt {d}}+128 b d^3 (a-i b)^{5/2} \sqrt {c-i d} (B+i (A-C)) \text {arctanh}\left (\frac {\sqrt {c-i d} \sqrt {a+b \tan (e+f x)}}{\sqrt {a-i b} \sqrt {c+d \tan (e+f x)}}\right )-128 b d^3 (a+i b)^{5/2} \sqrt {c+i d} (i A-B-i C) \text {arctanh}\left (\frac {\sqrt {c+i d} \sqrt {a+b \tan (e+f x)}}{\sqrt {a+i b} \sqrt {c+d \tan (e+f x)}}\right )}{2 b f}}{4 d}}{2 d}}{8 d}\)

Input:

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]
 

Output:

(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*Tan[e + f*x])^(3/2)*(c + d*Tan[e + f*x])^ 
(3/2))/(3*d*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))/(2* 
d*f) + (-1/2*(128*(a - I*b)^(5/2)*b*(B + I*(A - C))*Sqrt[c - I*d]*d^3*ArcT 
anh[(Sqrt[c - I*d]*Sqrt[a + b*Tan[e + f*x]])/(Sqrt[a - I*b]*Sqrt[c + d*Tan 
[e + f*x]])] - 128*(a + I*b)^(5/2)*b*(I*A - B - I*C)*Sqrt[c + I*d]*d^3*Arc 
Tanh[(Sqrt[c + I*d]*Sqrt[a + b*Tan[e + f*x]])/(Sqrt[a + I*b]*Sqrt[c + d*Ta 
n[e + f*x]])] + (2*(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]])])/(Sqrt[b]*Sqrt[d]))/(b*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]])/(b*f))/(4*d))/(2*d))/(8*d)
 

Defintions of rubi rules used

rule 27

Int[(a_)*(Fx_), x_Symbol] :> Simp[a   Int[Fx, x], x] /; FreeQ[a, x] &&  !Ma 
tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
 

rule 2009

Int[u_, x_Symbol] :> Simp[IntSum[u, x], x] /; SumQ[u]
 

rule 2348

Int[(Px_)*((c_) + (d_.)*(x_))^(m_.)*((e_) + (f_.)*(x_))^(n_.)*((a_.) + (b_. 
)*(x_)^2)^(p_.), x_Symbol] :> Int[ExpandIntegrand[Px*(c + d*x)^m*(e + f*x)^ 
n*(a + b*x^2)^p, x], x] /; FreeQ[{a, b, c, d, e, f, m, n, p}, x] && PolyQ[P 
x, x] && (IntegerQ[p] || (IntegerQ[2*p] && IntegerQ[m] && ILtQ[n, 0])) && 
!(IGtQ[m, 0] && IGtQ[n, 0])
 

rule 3042

Int[u_, x_Symbol] :> Int[DeactivateTrig[u, x], x] /; FunctionOfTrigOfLinear 
Q[u, x]
 

rule 4130

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] + Simp[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] && ( !IntegerQ[m] || (EqQ[ 
c, 0] && NeQ[a, 0])))
 

rule 4138

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] :> With[{ff = FreeFactors[Tan[e + f*x], x]}, S 
imp[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]
 

Maple [F(-1)]

Timed out.

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

Input:

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

Output:

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

Fricas [F(-1)]

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

integrate((a+b*tan(f*x+e))^(5/2)*(c+d*tan(f*x+e))^(1/2)*(A+B*tan(f*x+e)+C* 
tan(f*x+e)^2),x, algorithm="fricas")
 

Output:

Timed out
 

Sympy [F]

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

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

Output:

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)
 

Maxima [F]

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

integrate((a+b*tan(f*x+e))^(5/2)*(c+d*tan(f*x+e))^(1/2)*(A+B*tan(f*x+e)+C* 
tan(f*x+e)^2),x, algorithm="maxima")
 

Output:

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)
 

Giac [F]

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

integrate((a+b*tan(f*x+e))^(5/2)*(c+d*tan(f*x+e))^(1/2)*(A+B*tan(f*x+e)+C* 
tan(f*x+e)^2),x, algorithm="giac")
 

Output:

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)
 

Mupad [F(-1)]

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

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)
 

Output:

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)
 

Reduce [F]

\[ \int (a+b \tan (e+f x))^{5/2} \sqrt {c+d \tan (e+f x)} \left (A+B \tan (e+f x)+C \tan ^2(e+f x)\right ) \, dx=\left (\int \sqrt {d \tan \left (f x +e \right )+c}\, \sqrt {\tan \left (f x +e \right ) b +a}\, \tan \left (f x +e \right )^{4}d x \right ) b^{2} c +2 \left (\int \sqrt {d \tan \left (f x +e \right )+c}\, \sqrt {\tan \left (f x +e \right ) b +a}\, \tan \left (f x +e \right )^{3}d x \right ) a b c +\left (\int \sqrt {d \tan \left (f x +e \right )+c}\, \sqrt {\tan \left (f x +e \right ) b +a}\, \tan \left (f x +e \right )^{3}d x \right ) b^{3}+\left (\int \sqrt {d \tan \left (f x +e \right )+c}\, \sqrt {\tan \left (f x +e \right ) b +a}\, \tan \left (f x +e \right )^{2}d x \right ) a^{2} c +3 \left (\int \sqrt {d \tan \left (f x +e \right )+c}\, \sqrt {\tan \left (f x +e \right ) b +a}\, \tan \left (f x +e \right )^{2}d x \right ) a \,b^{2}+3 \left (\int \sqrt {d \tan \left (f x +e \right )+c}\, \sqrt {\tan \left (f x +e \right ) b +a}\, \tan \left (f x +e \right )d x \right ) a^{2} b +\left (\int \sqrt {d \tan \left (f x +e \right )+c}\, \sqrt {\tan \left (f x +e \right ) b +a}d x \right ) a^{3} \] Input:

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

Output:

int(sqrt(tan(e + f*x)*d + c)*sqrt(tan(e + f*x)*b + a)*tan(e + f*x)**4,x)*b 
**2*c + 2*int(sqrt(tan(e + f*x)*d + c)*sqrt(tan(e + f*x)*b + a)*tan(e + f* 
x)**3,x)*a*b*c + int(sqrt(tan(e + f*x)*d + c)*sqrt(tan(e + f*x)*b + a)*tan 
(e + f*x)**3,x)*b**3 + int(sqrt(tan(e + f*x)*d + c)*sqrt(tan(e + f*x)*b + 
a)*tan(e + f*x)**2,x)*a**2*c + 3*int(sqrt(tan(e + f*x)*d + c)*sqrt(tan(e + 
 f*x)*b + a)*tan(e + f*x)**2,x)*a*b**2 + 3*int(sqrt(tan(e + f*x)*d + c)*sq 
rt(tan(e + f*x)*b + a)*tan(e + f*x),x)*a**2*b + int(sqrt(tan(e + f*x)*d + 
c)*sqrt(tan(e + f*x)*b + a),x)*a**3