Integrand size = 49, antiderivative size = 679 \[ \int (a+b \tan (e+f x))^{5/2} \sqrt {c+d \tan (e+f x)} \left (A+B \tan (e+f x)+C \tan ^2(e+f x)\right ) \, dx=-\frac {(a-i b)^{5/2} (i A+B-i C) \sqrt {c-i d} \text {arctanh}\left (\frac {\sqrt {c-i d} \sqrt {a+b \tan (e+f x)}}{\sqrt {a-i b} \sqrt {c+d \tan (e+f x)}}\right )}{f}-\frac {(a+i b)^{5/2} (B-i (A-C)) \sqrt {c+i d} \text {arctanh}\left (\frac {\sqrt {c+i d} \sqrt {a+b \tan (e+f x)}}{\sqrt {a+i b} \sqrt {c+d \tan (e+f x)}}\right )}{f}-\frac {\left (5 a^4 C d^4-20 a^3 b d^3 (c C+2 B d)+30 a^2 b^2 d^2 \left (c^2 C-4 B c d-8 (A-C) d^2\right )-20 a b^3 d \left (c^3 C-2 B c^2 d+8 c (A-C) d^2-16 B d^3\right )+b^4 \left (5 c^4 C-8 B c^3 d+16 c^2 (A-C) d^2+64 B c d^3+128 (A-C) d^4\right )\right ) \text {arctanh}\left (\frac {\sqrt {d} \sqrt {a+b \tan (e+f x)}}{\sqrt {b} \sqrt {c+d \tan (e+f x)}}\right )}{64 b^{3/2} d^{7/2} f}+\frac {\left (64 b \left (a^2 B-b^2 B+2 a b (A-C)\right ) d^3-(b c-a d) \left (16 b (A b+a B-b C) d^2+(b c-a d) (5 b c C-8 b B d-5 a C d)\right )\right ) \sqrt {a+b \tan (e+f x)} \sqrt {c+d \tan (e+f x)}}{64 b d^3 f}+\frac {\left (16 b (A b+a B-b C) d^2+(b c-a d) (5 b c C-8 b B d-5 a C d)\right ) \sqrt {a+b \tan (e+f x)} (c+d \tan (e+f x))^{3/2}}{32 d^3 f}-\frac {(5 b c C-8 b B d-5 a C d) (a+b \tan (e+f x))^{3/2} (c+d \tan (e+f x))^{3/2}}{24 d^2 f}+\frac {C (a+b \tan (e+f x))^{5/2} (c+d \tan (e+f x))^{3/2}}{4 d f} \]
[Out]
Time = 11.77 (sec) , antiderivative size = 679, normalized size of antiderivative = 1.00, number of steps used = 16, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.163, Rules used = {3728, 3736, 6857, 65, 223, 212, 95, 214} \[ \int (a+b \tan (e+f x))^{5/2} \sqrt {c+d \tan (e+f x)} \left (A+B \tan (e+f x)+C \tan ^2(e+f x)\right ) \, dx=\frac {\sqrt {a+b \tan (e+f x)} \sqrt {c+d \tan (e+f x)} \left (64 b d^3 \left (a^2 B+2 a b (A-C)-b^2 B\right )-(b c-a d) \left (16 b d^2 (a B+A b-b C)+(b c-a d) (-5 a C d-8 b B d+5 b c C)\right )\right )}{64 b d^3 f}-\frac {\left (5 a^4 C d^4-20 a^3 b d^3 (2 B d+c C)+30 a^2 b^2 d^2 \left (-8 d^2 (A-C)-4 B c d+c^2 C\right )-20 a b^3 d \left (8 c d^2 (A-C)-2 B c^2 d-16 B d^3+c^3 C\right )+b^4 \left (16 c^2 d^2 (A-C)+128 d^4 (A-C)-8 B c^3 d+64 B c d^3+5 c^4 C\right )\right ) \text {arctanh}\left (\frac {\sqrt {d} \sqrt {a+b \tan (e+f x)}}{\sqrt {b} \sqrt {c+d \tan (e+f x)}}\right )}{64 b^{3/2} d^{7/2} f}-\frac {(a-i b)^{5/2} \sqrt {c-i d} (i A+B-i C) \text {arctanh}\left (\frac {\sqrt {c-i d} \sqrt {a+b \tan (e+f x)}}{\sqrt {a-i b} \sqrt {c+d \tan (e+f x)}}\right )}{f}-\frac {(a+i b)^{5/2} \sqrt {c+i d} (B-i (A-C)) \text {arctanh}\left (\frac {\sqrt {c+i d} \sqrt {a+b \tan (e+f x)}}{\sqrt {a+i b} \sqrt {c+d \tan (e+f x)}}\right )}{f}+\frac {\sqrt {a+b \tan (e+f x)} (c+d \tan (e+f x))^{3/2} \left (16 b d^2 (a B+A b-b C)+(b c-a d) (-5 a C d-8 b B d+5 b c C)\right )}{32 d^3 f}-\frac {(-5 a C d-8 b B d+5 b c C) (a+b \tan (e+f x))^{3/2} (c+d \tan (e+f x))^{3/2}}{24 d^2 f}+\frac {C (a+b \tan (e+f x))^{5/2} (c+d \tan (e+f x))^{3/2}}{4 d f} \]
[In]
[Out]
Rule 65
Rule 95
Rule 212
Rule 214
Rule 223
Rule 3728
Rule 3736
Rule 6857
Rubi steps \begin{align*} \text {integral}& = \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 ) \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}+\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} \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} \\ \end{align*}
Time = 10.25 (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=\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 ) \text {arctanh}\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 ) \text {arctanh}\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 ) \text {arcsinh}\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} \]
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\[\int \sqrt {c +d \tan \left (f x +e \right )}\, \left (a +b \tan \left (f x +e \right )\right )^{\frac {5}{2}} \left (A +B \tan \left (f x +e \right )+C \tan \left (f x +e \right )^{2}\right )d x\]
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Timed out. \[ \int (a+b \tan (e+f x))^{5/2} \sqrt {c+d \tan (e+f x)} \left (A+B \tan (e+f x)+C \tan ^2(e+f x)\right ) \, dx=\text {Timed out} \]
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\[ \int (a+b \tan (e+f x))^{5/2} \sqrt {c+d \tan (e+f x)} \left (A+B \tan (e+f x)+C \tan ^2(e+f x)\right ) \, dx=\int \left (a + b \tan {\left (e + f x \right )}\right )^{\frac {5}{2}} \sqrt {c + d \tan {\left (e + f x \right )}} \left (A + B \tan {\left (e + f x \right )} + C \tan ^{2}{\left (e + f x \right )}\right )\, dx \]
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\[ \int (a+b \tan (e+f x))^{5/2} \sqrt {c+d \tan (e+f x)} \left (A+B \tan (e+f x)+C \tan ^2(e+f x)\right ) \, dx=\int { {\left (C \tan \left (f x + e\right )^{2} + B \tan \left (f x + e\right ) + A\right )} {\left (b \tan \left (f x + e\right ) + a\right )}^{\frac {5}{2}} \sqrt {d \tan \left (f x + e\right ) + c} \,d x } \]
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Timed out. \[ \int (a+b \tan (e+f x))^{5/2} \sqrt {c+d \tan (e+f x)} \left (A+B \tan (e+f x)+C \tan ^2(e+f x)\right ) \, dx=\text {Timed out} \]
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Timed out. \[ \int (a+b \tan (e+f x))^{5/2} \sqrt {c+d \tan (e+f x)} \left (A+B \tan (e+f x)+C \tan ^2(e+f x)\right ) \, dx=\int {\left (a+b\,\mathrm {tan}\left (e+f\,x\right )\right )}^{5/2}\,\sqrt {c+d\,\mathrm {tan}\left (e+f\,x\right )}\,\left (C\,{\mathrm {tan}\left (e+f\,x\right )}^2+B\,\mathrm {tan}\left (e+f\,x\right )+A\right ) \,d x \]
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