Integrand size = 49, antiderivative size = 545 \[ \int \frac {(c+d \tan (e+f x))^{5/2} \left (A+B \tan (e+f x)+C \tan ^2(e+f x)\right )}{(a+b \tan (e+f x))^{5/2}} \, dx=-\frac {(i A+B-i C) (c-i d)^{5/2} \text {arctanh}\left (\frac {\sqrt {c-i d} \sqrt {a+b \tan (e+f x)}}{\sqrt {a-i b} \sqrt {c+d \tan (e+f x)}}\right )}{(a-i b)^{5/2} f}-\frac {(B-i (A-C)) (c+i d)^{5/2} \text {arctanh}\left (\frac {\sqrt {c+i d} \sqrt {a+b \tan (e+f x)}}{\sqrt {a+i b} \sqrt {c+d \tan (e+f x)}}\right )}{(a+i b)^{5/2} f}+\frac {d^{3/2} (5 b c C+2 b B d-5 a C d) \text {arctanh}\left (\frac {\sqrt {d} \sqrt {a+b \tan (e+f x)}}{\sqrt {b} \sqrt {c+d \tan (e+f x)}}\right )}{b^{7/2} f}-\frac {d \left (2 a^3 b B d-5 a^4 C d-2 a b^3 (2 A c-2 c C-3 B d)+2 a^2 b^2 (B c-5 C d)-b^4 (2 B c+(4 A+C) d)\right ) \sqrt {a+b \tan (e+f x)} \sqrt {c+d \tan (e+f x)}}{b^3 \left (a^2+b^2\right )^2 f}+\frac {2 \left (2 a^3 b B d-5 a^4 C d-b^4 (3 B c+5 A d)-2 a b^3 (3 A c-3 c C-4 B d)+a^2 b^2 (3 B c+(A-11 C) d)\right ) (c+d \tan (e+f x))^{3/2}}{3 b^2 \left (a^2+b^2\right )^2 f \sqrt {a+b \tan (e+f x)}}-\frac {2 \left (A b^2-a (b B-a C)\right ) (c+d \tan (e+f x))^{5/2}}{3 b \left (a^2+b^2\right ) f (a+b \tan (e+f x))^{3/2}} \]
[Out]
Time = 12.75 (sec) , antiderivative size = 545, 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} \[ \int \frac {(c+d \tan (e+f x))^{5/2} \left (A+B \tan (e+f x)+C \tan ^2(e+f x)\right )}{(a+b \tan (e+f x))^{5/2}} \, dx=-\frac {2 \left (A b^2-a (b B-a C)\right ) (c+d \tan (e+f x))^{5/2}}{3 b f \left (a^2+b^2\right ) (a+b \tan (e+f x))^{3/2}}+\frac {2 (c+d \tan (e+f x))^{3/2} \left (-5 a^4 C d+2 a^3 b B d+a^2 b^2 (d (A-11 C)+3 B c)-2 a b^3 (3 A c-4 B d-3 c C)-b^4 (5 A d+3 B c)\right )}{3 b^2 f \left (a^2+b^2\right )^2 \sqrt {a+b \tan (e+f x)}}-\frac {d \sqrt {a+b \tan (e+f x)} \sqrt {c+d \tan (e+f x)} \left (-5 a^4 C d+2 a^3 b B d+2 a^2 b^2 (B c-5 C d)-2 a b^3 (2 A c-3 B d-2 c C)-b^4 (d (4 A+C)+2 B c)\right )}{b^3 f \left (a^2+b^2\right )^2}-\frac {(c-i d)^{5/2} (i A+B-i C) \text {arctanh}\left (\frac {\sqrt {c-i d} \sqrt {a+b \tan (e+f x)}}{\sqrt {a-i b} \sqrt {c+d \tan (e+f x)}}\right )}{f (a-i b)^{5/2}}-\frac {(c+i d)^{5/2} (B-i (A-C)) \text {arctanh}\left (\frac {\sqrt {c+i d} \sqrt {a+b \tan (e+f x)}}{\sqrt {a+i b} \sqrt {c+d \tan (e+f x)}}\right )}{f (a+i b)^{5/2}}+\frac {d^{3/2} (-5 a C d+2 b B d+5 b c C) \text {arctanh}\left (\frac {\sqrt {d} \sqrt {a+b \tan (e+f x)}}{\sqrt {b} \sqrt {c+d \tan (e+f x)}}\right )}{b^{7/2} f} \]
[In]
[Out]
Rule 65
Rule 95
Rule 212
Rule 214
Rule 223
Rule 3726
Rule 3728
Rule 3736
Rule 6857
Rubi steps \begin{align*} \text {integral}& = -\frac {2 \left (A b^2-a (b B-a C)\right ) (c+d \tan (e+f x))^{5/2}}{3 b \left (a^2+b^2\right ) f (a+b \tan (e+f x))^{3/2}}+\frac {2 \int \frac {(c+d \tan (e+f x))^{3/2} \left (\frac {1}{2} ((b B-a C) (3 b c-5 a d)+A b (3 a c+5 b d))-\frac {3}{2} b ((A-C) (b c-a d)-B (a c+b d)) \tan (e+f x)+\frac {1}{2} \left (2 A b^2-2 a b B+5 a^2 C+3 b^2 C\right ) d \tan ^2(e+f x)\right )}{(a+b \tan (e+f x))^{3/2}} \, dx}{3 b \left (a^2+b^2\right )} \\ & = \frac {2 \left (2 a^3 b B d-5 a^4 C d-b^4 (3 B c+5 A d)-2 a b^3 (3 A c-3 c C-4 B d)+a^2 b^2 (3 B c+(A-11 C) d)\right ) (c+d \tan (e+f x))^{3/2}}{3 b^2 \left (a^2+b^2\right )^2 f \sqrt {a+b \tan (e+f x)}}-\frac {2 \left (A b^2-a (b B-a C)\right ) (c+d \tan (e+f x))^{5/2}}{3 b \left (a^2+b^2\right ) f (a+b \tan (e+f x))^{3/2}}+\frac {4 \int \frac {\sqrt {c+d \tan (e+f x)} \left (\frac {1}{4} \left (b (a c+3 b d) ((b B-a C) (3 b c-5 a d)+A b (3 a c+5 b d))+(b c-3 a d) \left (2 a^2 b B d-5 a^3 C d-A b^2 (3 b c-a d)+3 b^3 (c C+B d)+3 a b^2 (B c-2 C d)\right )\right )+\frac {3}{4} b^2 \left (2 a b \left (c^2 C+2 B c d-C d^2-A \left (c^2-d^2\right )\right )+a^2 \left (2 c (A-C) d+B \left (c^2-d^2\right )\right )-b^2 \left (2 c (A-C) d+B \left (c^2-d^2\right )\right )\right ) \tan (e+f x)-\frac {3}{4} d \left (2 a^3 b B d-5 a^4 C d-2 a b^3 (2 A c-2 c C-3 B d)+2 a^2 b^2 (B c-5 C d)-b^4 (2 B c+(4 A+C) d)\right ) \tan ^2(e+f x)\right )}{\sqrt {a+b \tan (e+f x)}} \, dx}{3 b^2 \left (a^2+b^2\right )^2} \\ & = -\frac {d \left (2 a^3 b B d-5 a^4 C d-2 a b^3 (2 A c-2 c C-3 B d)+2 a^2 b^2 (B c-5 C d)-b^4 (2 B c+(4 A+C) d)\right ) \sqrt {a+b \tan (e+f x)} \sqrt {c+d \tan (e+f x)}}{b^3 \left (a^2+b^2\right )^2 f}+\frac {2 \left (2 a^3 b B d-5 a^4 C d-b^4 (3 B c+5 A d)-2 a b^3 (3 A c-3 c C-4 B d)+a^2 b^2 (3 B c+(A-11 C) d)\right ) (c+d \tan (e+f x))^{3/2}}{3 b^2 \left (a^2+b^2\right )^2 f \sqrt {a+b \tan (e+f x)}}-\frac {2 \left (A b^2-a (b B-a C)\right ) (c+d \tan (e+f x))^{5/2}}{3 b \left (a^2+b^2\right ) f (a+b \tan (e+f x))^{3/2}}+\frac {4 \int \frac {-\frac {3}{8} \left (5 a^5 C d^3+10 a^3 b^2 C d^3-a^4 b d^2 (5 c C+2 B d)-2 a^2 b^3 \left (A c^3-c^3 C-3 B c^2 d-3 A c d^2+8 c C d^2+3 B d^3\right )-b^5 c \left (2 c^2 C+6 B c d-C d^2-2 A \left (c^2-3 d^2\right )\right )-a b^4 \left (4 A d \left (3 c^2-d^2\right )-C d \left (12 c^2+d^2\right )+4 B \left (c^3-3 c d^2\right )\right )\right )+\frac {3}{4} b^3 \left (2 a b \left (c^3 C+3 B c^2 d-3 c C d^2-B d^3-A \left (c^3-3 c d^2\right )\right )+a^2 \left ((A-C) d \left (3 c^2-d^2\right )+B \left (c^3-3 c d^2\right )\right )-b^2 \left ((A-C) d \left (3 c^2-d^2\right )+B \left (c^3-3 c d^2\right )\right )\right ) \tan (e+f x)+\frac {3}{8} \left (a^2+b^2\right )^2 d^2 (5 b c C+2 b B d-5 a C d) \tan ^2(e+f x)}{\sqrt {a+b \tan (e+f x)} \sqrt {c+d \tan (e+f x)}} \, dx}{3 b^3 \left (a^2+b^2\right )^2} \\ & = -\frac {d \left (2 a^3 b B d-5 a^4 C d-2 a b^3 (2 A c-2 c C-3 B d)+2 a^2 b^2 (B c-5 C d)-b^4 (2 B c+(4 A+C) d)\right ) \sqrt {a+b \tan (e+f x)} \sqrt {c+d \tan (e+f x)}}{b^3 \left (a^2+b^2\right )^2 f}+\frac {2 \left (2 a^3 b B d-5 a^4 C d-b^4 (3 B c+5 A d)-2 a b^3 (3 A c-3 c C-4 B d)+a^2 b^2 (3 B c+(A-11 C) d)\right ) (c+d \tan (e+f x))^{3/2}}{3 b^2 \left (a^2+b^2\right )^2 f \sqrt {a+b \tan (e+f x)}}-\frac {2 \left (A b^2-a (b B-a C)\right ) (c+d \tan (e+f x))^{5/2}}{3 b \left (a^2+b^2\right ) f (a+b \tan (e+f x))^{3/2}}+\frac {4 \text {Subst}\left (\int \frac {-\frac {3}{8} \left (5 a^5 C d^3+10 a^3 b^2 C d^3-a^4 b d^2 (5 c C+2 B d)-2 a^2 b^3 \left (A c^3-c^3 C-3 B c^2 d-3 A c d^2+8 c C d^2+3 B d^3\right )-b^5 c \left (2 c^2 C+6 B c d-C d^2-2 A \left (c^2-3 d^2\right )\right )-a b^4 \left (4 A d \left (3 c^2-d^2\right )-C d \left (12 c^2+d^2\right )+4 B \left (c^3-3 c d^2\right )\right )\right )+\frac {3}{4} b^3 \left (2 a b \left (c^3 C+3 B c^2 d-3 c C d^2-B d^3-A \left (c^3-3 c d^2\right )\right )+a^2 \left ((A-C) d \left (3 c^2-d^2\right )+B \left (c^3-3 c d^2\right )\right )-b^2 \left ((A-C) d \left (3 c^2-d^2\right )+B \left (c^3-3 c d^2\right )\right )\right ) x+\frac {3}{8} \left (a^2+b^2\right )^2 d^2 (5 b c C+2 b B d-5 a C d) x^2}{\sqrt {a+b x} \sqrt {c+d x} \left (1+x^2\right )} \, dx,x,\tan (e+f x)\right )}{3 b^3 \left (a^2+b^2\right )^2 f} \\ & = -\frac {d \left (2 a^3 b B d-5 a^4 C d-2 a b^3 (2 A c-2 c C-3 B d)+2 a^2 b^2 (B c-5 C d)-b^4 (2 B c+(4 A+C) d)\right ) \sqrt {a+b \tan (e+f x)} \sqrt {c+d \tan (e+f x)}}{b^3 \left (a^2+b^2\right )^2 f}+\frac {2 \left (2 a^3 b B d-5 a^4 C d-b^4 (3 B c+5 A d)-2 a b^3 (3 A c-3 c C-4 B d)+a^2 b^2 (3 B c+(A-11 C) d)\right ) (c+d \tan (e+f x))^{3/2}}{3 b^2 \left (a^2+b^2\right )^2 f \sqrt {a+b \tan (e+f x)}}-\frac {2 \left (A b^2-a (b B-a C)\right ) (c+d \tan (e+f x))^{5/2}}{3 b \left (a^2+b^2\right ) f (a+b \tan (e+f x))^{3/2}}+\frac {4 \text {Subst}\left (\int \left (\frac {3 \left (a^2+b^2\right )^2 d^2 (5 b c C+2 b B d-5 a C d)}{8 \sqrt {a+b x} \sqrt {c+d x}}+\frac {3 \left (-b^3 \left (b^2 \left (A c^3-c^3 C-3 B c^2 d-3 A c d^2+3 c C d^2+B d^3\right )+a^2 \left (c^3 C+3 B c^2 d-3 c C d^2-B d^3-A \left (c^3-3 c d^2\right )\right )-2 a b \left ((A-C) d \left (3 c^2-d^2\right )+B \left (c^3-3 c d^2\right )\right )\right )-b^3 \left (2 a b \left (A c^3-c^3 C-3 B c^2 d-3 A c d^2+3 c C d^2+B d^3\right )-a^2 \left ((A-C) d \left (3 c^2-d^2\right )+B \left (c^3-3 c d^2\right )\right )+b^2 \left ((A-C) d \left (3 c^2-d^2\right )+B \left (c^3-3 c d^2\right )\right )\right ) x\right )}{4 \sqrt {a+b x} \sqrt {c+d x} \left (1+x^2\right )}\right ) \, dx,x,\tan (e+f x)\right )}{3 b^3 \left (a^2+b^2\right )^2 f} \\ & = -\frac {d \left (2 a^3 b B d-5 a^4 C d-2 a b^3 (2 A c-2 c C-3 B d)+2 a^2 b^2 (B c-5 C d)-b^4 (2 B c+(4 A+C) d)\right ) \sqrt {a+b \tan (e+f x)} \sqrt {c+d \tan (e+f x)}}{b^3 \left (a^2+b^2\right )^2 f}+\frac {2 \left (2 a^3 b B d-5 a^4 C d-b^4 (3 B c+5 A d)-2 a b^3 (3 A c-3 c C-4 B d)+a^2 b^2 (3 B c+(A-11 C) d)\right ) (c+d \tan (e+f x))^{3/2}}{3 b^2 \left (a^2+b^2\right )^2 f \sqrt {a+b \tan (e+f x)}}-\frac {2 \left (A b^2-a (b B-a C)\right ) (c+d \tan (e+f x))^{5/2}}{3 b \left (a^2+b^2\right ) f (a+b \tan (e+f x))^{3/2}}+\frac {\text {Subst}\left (\int \frac {-b^3 \left (b^2 \left (A c^3-c^3 C-3 B c^2 d-3 A c d^2+3 c C d^2+B d^3\right )+a^2 \left (c^3 C+3 B c^2 d-3 c C d^2-B d^3-A \left (c^3-3 c d^2\right )\right )-2 a b \left ((A-C) d \left (3 c^2-d^2\right )+B \left (c^3-3 c d^2\right )\right )\right )-b^3 \left (2 a b \left (A c^3-c^3 C-3 B c^2 d-3 A c d^2+3 c C d^2+B d^3\right )-a^2 \left ((A-C) d \left (3 c^2-d^2\right )+B \left (c^3-3 c d^2\right )\right )+b^2 \left ((A-C) d \left (3 c^2-d^2\right )+B \left (c^3-3 c d^2\right )\right )\right ) x}{\sqrt {a+b x} \sqrt {c+d x} \left (1+x^2\right )} \, dx,x,\tan (e+f x)\right )}{b^3 \left (a^2+b^2\right )^2 f}+\frac {\left (d^2 (5 b c C+2 b B d-5 a C d)\right ) \text {Subst}\left (\int \frac {1}{\sqrt {a+b x} \sqrt {c+d x}} \, dx,x,\tan (e+f x)\right )}{2 b^3 f} \\ & = -\frac {d \left (2 a^3 b B d-5 a^4 C d-2 a b^3 (2 A c-2 c C-3 B d)+2 a^2 b^2 (B c-5 C d)-b^4 (2 B c+(4 A+C) d)\right ) \sqrt {a+b \tan (e+f x)} \sqrt {c+d \tan (e+f x)}}{b^3 \left (a^2+b^2\right )^2 f}+\frac {2 \left (2 a^3 b B d-5 a^4 C d-b^4 (3 B c+5 A d)-2 a b^3 (3 A c-3 c C-4 B d)+a^2 b^2 (3 B c+(A-11 C) d)\right ) (c+d \tan (e+f x))^{3/2}}{3 b^2 \left (a^2+b^2\right )^2 f \sqrt {a+b \tan (e+f x)}}-\frac {2 \left (A b^2-a (b B-a C)\right ) (c+d \tan (e+f x))^{5/2}}{3 b \left (a^2+b^2\right ) f (a+b \tan (e+f x))^{3/2}}+\frac {\text {Subst}\left (\int \left (\frac {-i b^3 \left (b^2 \left (A c^3-c^3 C-3 B c^2 d-3 A c d^2+3 c C d^2+B d^3\right )+a^2 \left (c^3 C+3 B c^2 d-3 c C d^2-B d^3-A \left (c^3-3 c d^2\right )\right )-2 a b \left ((A-C) d \left (3 c^2-d^2\right )+B \left (c^3-3 c d^2\right )\right )\right )+b^3 \left (2 a b \left (A c^3-c^3 C-3 B c^2 d-3 A c d^2+3 c C d^2+B d^3\right )-a^2 \left ((A-C) d \left (3 c^2-d^2\right )+B \left (c^3-3 c d^2\right )\right )+b^2 \left ((A-C) d \left (3 c^2-d^2\right )+B \left (c^3-3 c d^2\right )\right )\right )}{2 (i-x) \sqrt {a+b x} \sqrt {c+d x}}+\frac {-i b^3 \left (b^2 \left (A c^3-c^3 C-3 B c^2 d-3 A c d^2+3 c C d^2+B d^3\right )+a^2 \left (c^3 C+3 B c^2 d-3 c C d^2-B d^3-A \left (c^3-3 c d^2\right )\right )-2 a b \left ((A-C) d \left (3 c^2-d^2\right )+B \left (c^3-3 c d^2\right )\right )\right )-b^3 \left (2 a b \left (A c^3-c^3 C-3 B c^2 d-3 A c d^2+3 c C d^2+B d^3\right )-a^2 \left ((A-C) d \left (3 c^2-d^2\right )+B \left (c^3-3 c d^2\right )\right )+b^2 \left ((A-C) d \left (3 c^2-d^2\right )+B \left (c^3-3 c d^2\right )\right )\right )}{2 (i+x) \sqrt {a+b x} \sqrt {c+d x}}\right ) \, dx,x,\tan (e+f x)\right )}{b^3 \left (a^2+b^2\right )^2 f}+\frac {\left (d^2 (5 b c C+2 b B d-5 a C d)\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 )}{b^4 f} \\ & = -\frac {d \left (2 a^3 b B d-5 a^4 C d-2 a b^3 (2 A c-2 c C-3 B d)+2 a^2 b^2 (B c-5 C d)-b^4 (2 B c+(4 A+C) d)\right ) \sqrt {a+b \tan (e+f x)} \sqrt {c+d \tan (e+f x)}}{b^3 \left (a^2+b^2\right )^2 f}+\frac {2 \left (2 a^3 b B d-5 a^4 C d-b^4 (3 B c+5 A d)-2 a b^3 (3 A c-3 c C-4 B d)+a^2 b^2 (3 B c+(A-11 C) d)\right ) (c+d \tan (e+f x))^{3/2}}{3 b^2 \left (a^2+b^2\right )^2 f \sqrt {a+b \tan (e+f x)}}-\frac {2 \left (A b^2-a (b B-a C)\right ) (c+d \tan (e+f x))^{5/2}}{3 b \left (a^2+b^2\right ) f (a+b \tan (e+f x))^{3/2}}+\frac {\left ((i A+B-i C) (c-i d)^3\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 (a-i b)^2 f}+\frac {\left (d^2 (5 b c C+2 b B d-5 a C d)\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 )}{b^4 f}+\frac {\left (-i b^3 \left (b^2 \left (A c^3-c^3 C-3 B c^2 d-3 A c d^2+3 c C d^2+B d^3\right )+a^2 \left (c^3 C+3 B c^2 d-3 c C d^2-B d^3-A \left (c^3-3 c d^2\right )\right )-2 a b \left ((A-C) d \left (3 c^2-d^2\right )+B \left (c^3-3 c d^2\right )\right )\right )+b^3 \left (2 a b \left (A c^3-c^3 C-3 B c^2 d-3 A c d^2+3 c C d^2+B d^3\right )-a^2 \left ((A-C) d \left (3 c^2-d^2\right )+B \left (c^3-3 c d^2\right )\right )+b^2 \left ((A-C) d \left (3 c^2-d^2\right )+B \left (c^3-3 c d^2\right )\right )\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^3 \left (a^2+b^2\right )^2 f} \\ & = \frac {d^{3/2} (5 b c C+2 b B d-5 a C d) \text {arctanh}\left (\frac {\sqrt {d} \sqrt {a+b \tan (e+f x)}}{\sqrt {b} \sqrt {c+d \tan (e+f x)}}\right )}{b^{7/2} f}-\frac {d \left (2 a^3 b B d-5 a^4 C d-2 a b^3 (2 A c-2 c C-3 B d)+2 a^2 b^2 (B c-5 C d)-b^4 (2 B c+(4 A+C) d)\right ) \sqrt {a+b \tan (e+f x)} \sqrt {c+d \tan (e+f x)}}{b^3 \left (a^2+b^2\right )^2 f}+\frac {2 \left (2 a^3 b B d-5 a^4 C d-b^4 (3 B c+5 A d)-2 a b^3 (3 A c-3 c C-4 B d)+a^2 b^2 (3 B c+(A-11 C) d)\right ) (c+d \tan (e+f x))^{3/2}}{3 b^2 \left (a^2+b^2\right )^2 f \sqrt {a+b \tan (e+f x)}}-\frac {2 \left (A b^2-a (b B-a C)\right ) (c+d \tan (e+f x))^{5/2}}{3 b \left (a^2+b^2\right ) f (a+b \tan (e+f x))^{3/2}}+\frac {\left ((i A+B-i C) (c-i d)^3\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 )}{(a-i b)^2 f}+\frac {\left (-i b^3 \left (b^2 \left (A c^3-c^3 C-3 B c^2 d-3 A c d^2+3 c C d^2+B d^3\right )+a^2 \left (c^3 C+3 B c^2 d-3 c C d^2-B d^3-A \left (c^3-3 c d^2\right )\right )-2 a b \left ((A-C) d \left (3 c^2-d^2\right )+B \left (c^3-3 c d^2\right )\right )\right )+b^3 \left (2 a b \left (A c^3-c^3 C-3 B c^2 d-3 A c d^2+3 c C d^2+B d^3\right )-a^2 \left ((A-C) d \left (3 c^2-d^2\right )+B \left (c^3-3 c d^2\right )\right )+b^2 \left ((A-C) d \left (3 c^2-d^2\right )+B \left (c^3-3 c d^2\right )\right )\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^3 \left (a^2+b^2\right )^2 f} \\ & = -\frac {(i A+B-i C) (c-i d)^{5/2} \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 )}{(a-i b)^{5/2} f}-\frac {(B-i (A-C)) (c+i d)^{5/2} \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 )}{(a+i b)^{5/2} f}+\frac {d^{3/2} (5 b c C+2 b B d-5 a C d) \text {arctanh}\left (\frac {\sqrt {d} \sqrt {a+b \tan (e+f x)}}{\sqrt {b} \sqrt {c+d \tan (e+f x)}}\right )}{b^{7/2} f}-\frac {d \left (2 a^3 b B d-5 a^4 C d-2 a b^3 (2 A c-2 c C-3 B d)+2 a^2 b^2 (B c-5 C d)-b^4 (2 B c+(4 A+C) d)\right ) \sqrt {a+b \tan (e+f x)} \sqrt {c+d \tan (e+f x)}}{b^3 \left (a^2+b^2\right )^2 f}+\frac {2 \left (2 a^3 b B d-5 a^4 C d-b^4 (3 B c+5 A d)-2 a b^3 (3 A c-3 c C-4 B d)+a^2 b^2 (3 B c+(A-11 C) d)\right ) (c+d \tan (e+f x))^{3/2}}{3 b^2 \left (a^2+b^2\right )^2 f \sqrt {a+b \tan (e+f x)}}-\frac {2 \left (A b^2-a (b B-a C)\right ) (c+d \tan (e+f x))^{5/2}}{3 b \left (a^2+b^2\right ) f (a+b \tan (e+f x))^{3/2}} \\ \end{align*}
Result contains higher order function than in optimal. Order 5 vs. order 3 in optimal.
Time = 7.63 (sec) , antiderivative size = 802, normalized size of antiderivative = 1.47 \[ \int \frac {(c+d \tan (e+f x))^{5/2} \left (A+B \tan (e+f x)+C \tan ^2(e+f x)\right )}{(a+b \tan (e+f x))^{5/2}} \, dx=\frac {C (c+d \tan (e+f x))^{5/2}}{b f (a+b \tan (e+f x))^{3/2}}+\frac {-\frac {2 b (A-i B-C) (c-i d) (c+d \tan (e+f x))^{3/2}}{3 (i a+b) f (a+b \tan (e+f x))^{3/2}}+\frac {2 b (A+i B-C) (c+i d) (c+d \tan (e+f x))^{3/2}}{3 (i a-b) f (a+b \tan (e+f x))^{3/2}}-\frac {10 c C (b c-a d) \operatorname {Hypergeometric2F1}\left (-\frac {3}{2},-\frac {3}{2},-\frac {1}{2},-\frac {d (a+b \tan (e+f x))}{b c-a d}\right ) \sqrt {c+d \tan (e+f x)}}{3 b f (a+b \tan (e+f x))^{3/2} \sqrt {\frac {b (c+d \tan (e+f x))}{b c-a d}}}-\frac {4 B d (b c-a d) \operatorname {Hypergeometric2F1}\left (-\frac {3}{2},-\frac {3}{2},-\frac {1}{2},-\frac {d (a+b \tan (e+f x))}{b c-a d}\right ) \sqrt {c+d \tan (e+f x)}}{3 b f (a+b \tan (e+f x))^{3/2} \sqrt {\frac {b (c+d \tan (e+f x))}{b c-a d}}}+\frac {10 a C d (b c-a d) \operatorname {Hypergeometric2F1}\left (-\frac {3}{2},-\frac {3}{2},-\frac {1}{2},-\frac {d (a+b \tan (e+f x))}{b c-a d}\right ) \sqrt {c+d \tan (e+f x)}}{3 b^2 f (a+b \tan (e+f x))^{3/2} \sqrt {\frac {b (c+d \tan (e+f x))}{b c-a d}}}+\frac {2 b (i A+B-i C) (c-i d)^2 \left (\frac {\sqrt {-c+i d} \text {arctanh}\left (\frac {\sqrt {-c+i d} \sqrt {a+b \tan (e+f x)}}{\sqrt {-a+i b} \sqrt {c+d \tan (e+f x)}}\right )}{(-a+i b)^{3/2}}+\frac {\sqrt {c+d \tan (e+f x)}}{(a-i b) \sqrt {a+b \tan (e+f x)}}\right )}{(a-i b) f}-\frac {2 b (A+i B-C) (c+i d)^2 \left (\frac {\sqrt {c+i d} \text {arctanh}\left (\frac {\sqrt {c+i d} \sqrt {a+b \tan (e+f x)}}{\sqrt {a+i b} \sqrt {c+d \tan (e+f x)}}\right )}{(a+i b)^{3/2}}-\frac {\sqrt {c+d \tan (e+f x)}}{(a+i b) \sqrt {a+b \tan (e+f x)}}\right )}{(i a-b) f}}{2 b} \]
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Timed out.
\[\int \frac {\left (c +d \tan \left (f x +e \right )\right )^{\frac {5}{2}} \left (A +B \tan \left (f x +e \right )+C \tan \left (f x +e \right )^{2}\right )}{\left (a +b \tan \left (f x +e \right )\right )^{\frac {5}{2}}}d x\]
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Timed out. \[ \int \frac {(c+d \tan (e+f x))^{5/2} \left (A+B \tan (e+f x)+C \tan ^2(e+f x)\right )}{(a+b \tan (e+f x))^{5/2}} \, dx=\text {Timed out} \]
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\[ \int \frac {(c+d \tan (e+f x))^{5/2} \left (A+B \tan (e+f x)+C \tan ^2(e+f x)\right )}{(a+b \tan (e+f x))^{5/2}} \, dx=\int \frac {\left (c + d \tan {\left (e + f x \right )}\right )^{\frac {5}{2}} \left (A + B \tan {\left (e + f x \right )} + C \tan ^{2}{\left (e + f x \right )}\right )}{\left (a + b \tan {\left (e + f x \right )}\right )^{\frac {5}{2}}}\, dx \]
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Timed out. \[ \int \frac {(c+d \tan (e+f x))^{5/2} \left (A+B \tan (e+f x)+C \tan ^2(e+f x)\right )}{(a+b \tan (e+f x))^{5/2}} \, dx=\text {Timed out} \]
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Timed out. \[ \int \frac {(c+d \tan (e+f x))^{5/2} \left (A+B \tan (e+f x)+C \tan ^2(e+f x)\right )}{(a+b \tan (e+f x))^{5/2}} \, dx=\text {Timed out} \]
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Timed out. \[ \int \frac {(c+d \tan (e+f x))^{5/2} \left (A+B \tan (e+f x)+C \tan ^2(e+f x)\right )}{(a+b \tan (e+f x))^{5/2}} \, dx=\text {Hanged} \]
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