Integrand size = 49, antiderivative size = 590 \[ \int \frac {(c+d \tan (e+f x))^{5/2} \left (A+B \tan (e+f x)+C \tan ^2(e+f x)\right )}{(a+b \tan (e+f x))^{7/2}} \, dx=-\frac {(i A+B-i C) (c-i d)^{5/2} \text {arctanh}\left (\frac {\sqrt {c-i d} \sqrt {a+b \tan (e+f x)}}{\sqrt {a-i b} \sqrt {c+d \tan (e+f x)}}\right )}{(a-i b)^{7/2} f}-\frac {(B-i (A-C)) (c+i d)^{5/2} \text {arctanh}\left (\frac {\sqrt {c+i d} \sqrt {a+b \tan (e+f x)}}{\sqrt {a+i b} \sqrt {c+d \tan (e+f x)}}\right )}{(a+i b)^{7/2} f}+\frac {2 C d^{5/2} \text {arctanh}\left (\frac {\sqrt {d} \sqrt {a+b \tan (e+f x)}}{\sqrt {b} \sqrt {c+d \tan (e+f x)}}\right )}{b^{7/2} f}-\frac {2 \left (a^6 C d^2+3 a^4 b^2 C d^2-3 a^2 b^4 \left (c^2 C+2 B c d-2 C d^2-A \left (c^2-d^2\right )\right )+b^6 \left (c (c C+2 B d)-A \left (c^2-d^2\right )\right )-a^3 b^3 \left (2 c (A-C) d+B \left (c^2-d^2\right )\right )+3 a b^5 \left (2 c (A-C) d+B \left (c^2-d^2\right )\right )\right ) \sqrt {c+d \tan (e+f x)}}{b^3 \left (a^2+b^2\right )^3 f \sqrt {a+b \tan (e+f x)}}-\frac {2 \left (a^4 C d+b^4 (B c+A d)+2 a b^3 (A c-c C-B d)-a^2 b^2 (B c+(A-3 C) d)\right ) (c+d \tan (e+f x))^{3/2}}{3 b^2 \left (a^2+b^2\right )^2 f (a+b \tan (e+f x))^{3/2}}-\frac {2 \left (A b^2-a (b B-a C)\right ) (c+d \tan (e+f x))^{5/2}}{5 b \left (a^2+b^2\right ) f (a+b \tan (e+f x))^{5/2}} \]
[Out]
Time = 14.55 (sec) , antiderivative size = 590, normalized size of antiderivative = 1.00, number of steps used = 15, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.163, Rules used = {3726, 3736, 6857, 65, 223, 212, 95, 214} \[ \int \frac {(c+d \tan (e+f x))^{5/2} \left (A+B \tan (e+f x)+C \tan ^2(e+f x)\right )}{(a+b \tan (e+f x))^{7/2}} \, dx=-\frac {2 \left (A b^2-a (b B-a C)\right ) (c+d \tan (e+f x))^{5/2}}{5 b f \left (a^2+b^2\right ) (a+b \tan (e+f x))^{5/2}}-\frac {2 (c+d \tan (e+f x))^{3/2} \left (a^4 C d-a^2 b^2 (d (A-3 C)+B c)+2 a b^3 (A c-B d-c C)+b^4 (A d+B c)\right )}{3 b^2 f \left (a^2+b^2\right )^2 (a+b \tan (e+f x))^{3/2}}-\frac {2 \sqrt {c+d \tan (e+f x)} \left (a^6 C d^2+3 a^4 b^2 C d^2-a^3 b^3 \left (2 c d (A-C)+B \left (c^2-d^2\right )\right )-3 a^2 b^4 \left (-A \left (c^2-d^2\right )+2 B c d+c^2 C-2 C d^2\right )+3 a b^5 \left (2 c d (A-C)+B \left (c^2-d^2\right )\right )+b^6 \left (c (2 B d+c C)-A \left (c^2-d^2\right )\right )\right )}{b^3 f \left (a^2+b^2\right )^3 \sqrt {a+b \tan (e+f x)}}-\frac {(c-i d)^{5/2} (i A+B-i C) \text {arctanh}\left (\frac {\sqrt {c-i d} \sqrt {a+b \tan (e+f x)}}{\sqrt {a-i b} \sqrt {c+d \tan (e+f x)}}\right )}{f (a-i b)^{7/2}}-\frac {(c+i d)^{5/2} (B-i (A-C)) \text {arctanh}\left (\frac {\sqrt {c+i d} \sqrt {a+b \tan (e+f x)}}{\sqrt {a+i b} \sqrt {c+d \tan (e+f x)}}\right )}{f (a+i b)^{7/2}}+\frac {2 C d^{5/2} \text {arctanh}\left (\frac {\sqrt {d} \sqrt {a+b \tan (e+f x)}}{\sqrt {b} \sqrt {c+d \tan (e+f x)}}\right )}{b^{7/2} f} \]
[In]
[Out]
Rule 65
Rule 95
Rule 212
Rule 214
Rule 223
Rule 3726
Rule 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}}{5 b \left (a^2+b^2\right ) f (a+b \tan (e+f x))^{5/2}}+\frac {2 \int \frac {(c+d \tan (e+f x))^{3/2} \left (\frac {5}{2} ((b B-a C) (b c-a d)+A b (a c+b d))-\frac {5}{2} b ((A-C) (b c-a d)-B (a c+b d)) \tan (e+f x)+\frac {5}{2} \left (a^2+b^2\right ) C d \tan ^2(e+f x)\right )}{(a+b \tan (e+f x))^{5/2}} \, dx}{5 b \left (a^2+b^2\right )} \\ & = -\frac {2 \left (a^4 C d+b^4 (B c+A d)+2 a b^3 (A c-c C-B d)-a^2 b^2 (B c+(A-3 C) d)\right ) (c+d \tan (e+f x))^{3/2}}{3 b^2 \left (a^2+b^2\right )^2 f (a+b \tan (e+f x))^{3/2}}-\frac {2 \left (A b^2-a (b B-a C)\right ) (c+d \tan (e+f x))^{5/2}}{5 b \left (a^2+b^2\right ) f (a+b \tan (e+f x))^{5/2}}+\frac {4 \int \frac {\sqrt {c+d \tan (e+f x)} \left (\frac {15}{4} \left (b (a c+b d) ((b B-a C) (b c-a d)+A b (a c+b d))-(b c-a d) \left (a^3 C d+A b^2 (b c-a d)-b^3 (c C+B d)-a b^2 (B c-2 C d)\right )\right )-\frac {15}{4} b^2 ((a c+b d) ((A-C) (b c-a d)-B (a c+b d))+(b c-a d) (b B c+b (A-C) d+a (A c-c C-B d))) \tan (e+f x)+\frac {15}{4} \left (a^2+b^2\right )^2 C d^2 \tan ^2(e+f x)\right )}{(a+b \tan (e+f x))^{3/2}} \, dx}{15 b^2 \left (a^2+b^2\right )^2} \\ & = -\frac {2 \left (a^6 C d^2+3 a^4 b^2 C d^2-3 a^2 b^4 \left (c^2 C+2 B c d-2 C d^2-A \left (c^2-d^2\right )\right )+b^6 \left (c (c C+2 B d)-A \left (c^2-d^2\right )\right )-a^3 b^3 \left (2 c (A-C) d+B \left (c^2-d^2\right )\right )+3 a b^5 \left (2 c (A-C) d+B \left (c^2-d^2\right )\right )\right ) \sqrt {c+d \tan (e+f x)}}{b^3 \left (a^2+b^2\right )^3 f \sqrt {a+b \tan (e+f x)}}-\frac {2 \left (a^4 C d+b^4 (B c+A d)+2 a b^3 (A c-c C-B d)-a^2 b^2 (B c+(A-3 C) d)\right ) (c+d \tan (e+f x))^{3/2}}{3 b^2 \left (a^2+b^2\right )^2 f (a+b \tan (e+f x))^{3/2}}-\frac {2 \left (A b^2-a (b B-a C)\right ) (c+d \tan (e+f x))^{5/2}}{5 b \left (a^2+b^2\right ) f (a+b \tan (e+f x))^{5/2}}+\frac {8 \int \frac {\frac {15}{8} \left (a^6 C d^3+3 a^4 b^2 C d^3-b^6 \left (B c^3+3 A c^2 d-3 c^2 C d-3 B c d^2-A d^3\right )-3 a b^5 \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 )+3 a^2 b^4 \left (B c^3+3 A c^2 d-3 c^2 C d-3 B c d^2-A d^3+2 C d^3\right )-a^3 b^3 \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 )\right )-\frac {15}{8} b^3 \left (3 a^2 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 )+b^3 \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^3 \left ((A-C) d \left (3 c^2-d^2\right )+B \left (c^3-3 c d^2\right )\right )+3 a 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 {15}{8} \left (a^2+b^2\right )^3 C d^3 \tan ^2(e+f x)}{\sqrt {a+b \tan (e+f x)} \sqrt {c+d \tan (e+f x)}} \, dx}{15 b^3 \left (a^2+b^2\right )^3} \\ & = -\frac {2 \left (a^6 C d^2+3 a^4 b^2 C d^2-3 a^2 b^4 \left (c^2 C+2 B c d-2 C d^2-A \left (c^2-d^2\right )\right )+b^6 \left (c (c C+2 B d)-A \left (c^2-d^2\right )\right )-a^3 b^3 \left (2 c (A-C) d+B \left (c^2-d^2\right )\right )+3 a b^5 \left (2 c (A-C) d+B \left (c^2-d^2\right )\right )\right ) \sqrt {c+d \tan (e+f x)}}{b^3 \left (a^2+b^2\right )^3 f \sqrt {a+b \tan (e+f x)}}-\frac {2 \left (a^4 C d+b^4 (B c+A d)+2 a b^3 (A c-c C-B d)-a^2 b^2 (B c+(A-3 C) d)\right ) (c+d \tan (e+f x))^{3/2}}{3 b^2 \left (a^2+b^2\right )^2 f (a+b \tan (e+f x))^{3/2}}-\frac {2 \left (A b^2-a (b B-a C)\right ) (c+d \tan (e+f x))^{5/2}}{5 b \left (a^2+b^2\right ) f (a+b \tan (e+f x))^{5/2}}+\frac {8 \text {Subst}\left (\int \frac {\frac {15}{8} \left (a^6 C d^3+3 a^4 b^2 C d^3-b^6 \left (B c^3+3 A c^2 d-3 c^2 C d-3 B c d^2-A d^3\right )-3 a b^5 \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 )+3 a^2 b^4 \left (B c^3+3 A c^2 d-3 c^2 C d-3 B c d^2-A d^3+2 C d^3\right )-a^3 b^3 \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 )\right )-\frac {15}{8} b^3 \left (3 a^2 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 )+b^3 \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^3 \left ((A-C) d \left (3 c^2-d^2\right )+B \left (c^3-3 c d^2\right )\right )+3 a 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 {15}{8} \left (a^2+b^2\right )^3 C d^3 x^2}{\sqrt {a+b x} \sqrt {c+d x} \left (1+x^2\right )} \, dx,x,\tan (e+f x)\right )}{15 b^3 \left (a^2+b^2\right )^3 f} \\ & = -\frac {2 \left (a^6 C d^2+3 a^4 b^2 C d^2-3 a^2 b^4 \left (c^2 C+2 B c d-2 C d^2-A \left (c^2-d^2\right )\right )+b^6 \left (c (c C+2 B d)-A \left (c^2-d^2\right )\right )-a^3 b^3 \left (2 c (A-C) d+B \left (c^2-d^2\right )\right )+3 a b^5 \left (2 c (A-C) d+B \left (c^2-d^2\right )\right )\right ) \sqrt {c+d \tan (e+f x)}}{b^3 \left (a^2+b^2\right )^3 f \sqrt {a+b \tan (e+f x)}}-\frac {2 \left (a^4 C d+b^4 (B c+A d)+2 a b^3 (A c-c C-B d)-a^2 b^2 (B c+(A-3 C) d)\right ) (c+d \tan (e+f x))^{3/2}}{3 b^2 \left (a^2+b^2\right )^2 f (a+b \tan (e+f x))^{3/2}}-\frac {2 \left (A b^2-a (b B-a C)\right ) (c+d \tan (e+f x))^{5/2}}{5 b \left (a^2+b^2\right ) f (a+b \tan (e+f x))^{5/2}}+\frac {8 \text {Subst}\left (\int \left (\frac {15 \left (a^2+b^2\right )^3 C d^3}{8 \sqrt {a+b x} \sqrt {c+d x}}+\frac {15 \left (-b^3 \left (3 a 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^3 \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 )-3 a^2 b \left ((A-C) d \left (3 c^2-d^2\right )+B \left (c^3-3 c d^2\right )\right )+b^3 \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 (3 a^2 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 )+b^3 \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^3 \left ((A-C) d \left (3 c^2-d^2\right )+B \left (c^3-3 c d^2\right )\right )+3 a 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 )}{8 \sqrt {a+b x} \sqrt {c+d x} \left (1+x^2\right )}\right ) \, dx,x,\tan (e+f x)\right )}{15 b^3 \left (a^2+b^2\right )^3 f} \\ & = -\frac {2 \left (a^6 C d^2+3 a^4 b^2 C d^2-3 a^2 b^4 \left (c^2 C+2 B c d-2 C d^2-A \left (c^2-d^2\right )\right )+b^6 \left (c (c C+2 B d)-A \left (c^2-d^2\right )\right )-a^3 b^3 \left (2 c (A-C) d+B \left (c^2-d^2\right )\right )+3 a b^5 \left (2 c (A-C) d+B \left (c^2-d^2\right )\right )\right ) \sqrt {c+d \tan (e+f x)}}{b^3 \left (a^2+b^2\right )^3 f \sqrt {a+b \tan (e+f x)}}-\frac {2 \left (a^4 C d+b^4 (B c+A d)+2 a b^3 (A c-c C-B d)-a^2 b^2 (B c+(A-3 C) d)\right ) (c+d \tan (e+f x))^{3/2}}{3 b^2 \left (a^2+b^2\right )^2 f (a+b \tan (e+f x))^{3/2}}-\frac {2 \left (A b^2-a (b B-a C)\right ) (c+d \tan (e+f x))^{5/2}}{5 b \left (a^2+b^2\right ) f (a+b \tan (e+f x))^{5/2}}+\frac {\text {Subst}\left (\int \frac {-b^3 \left (3 a 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^3 \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 )-3 a^2 b \left ((A-C) d \left (3 c^2-d^2\right )+B \left (c^3-3 c d^2\right )\right )+b^3 \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 (3 a^2 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 )+b^3 \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^3 \left ((A-C) d \left (3 c^2-d^2\right )+B \left (c^3-3 c d^2\right )\right )+3 a 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 )^3 f}+\frac {\left (C d^3\right ) \text {Subst}\left (\int \frac {1}{\sqrt {a+b x} \sqrt {c+d x}} \, dx,x,\tan (e+f x)\right )}{b^3 f} \\ & = -\frac {2 \left (a^6 C d^2+3 a^4 b^2 C d^2-3 a^2 b^4 \left (c^2 C+2 B c d-2 C d^2-A \left (c^2-d^2\right )\right )+b^6 \left (c (c C+2 B d)-A \left (c^2-d^2\right )\right )-a^3 b^3 \left (2 c (A-C) d+B \left (c^2-d^2\right )\right )+3 a b^5 \left (2 c (A-C) d+B \left (c^2-d^2\right )\right )\right ) \sqrt {c+d \tan (e+f x)}}{b^3 \left (a^2+b^2\right )^3 f \sqrt {a+b \tan (e+f x)}}-\frac {2 \left (a^4 C d+b^4 (B c+A d)+2 a b^3 (A c-c C-B d)-a^2 b^2 (B c+(A-3 C) d)\right ) (c+d \tan (e+f x))^{3/2}}{3 b^2 \left (a^2+b^2\right )^2 f (a+b \tan (e+f x))^{3/2}}-\frac {2 \left (A b^2-a (b B-a C)\right ) (c+d \tan (e+f x))^{5/2}}{5 b \left (a^2+b^2\right ) f (a+b \tan (e+f x))^{5/2}}+\frac {\text {Subst}\left (\int \left (\frac {b^3 \left (3 a^2 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 )+b^3 \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^3 \left ((A-C) d \left (3 c^2-d^2\right )+B \left (c^3-3 c d^2\right )\right )+3 a b^2 \left ((A-C) d \left (3 c^2-d^2\right )+B \left (c^3-3 c d^2\right )\right )\right )-i b^3 \left (3 a 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^3 \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 )-3 a^2 b \left ((A-C) d \left (3 c^2-d^2\right )+B \left (c^3-3 c d^2\right )\right )+b^3 \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 {-b^3 \left (3 a^2 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 )+b^3 \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^3 \left ((A-C) d \left (3 c^2-d^2\right )+B \left (c^3-3 c d^2\right )\right )+3 a b^2 \left ((A-C) d \left (3 c^2-d^2\right )+B \left (c^3-3 c d^2\right )\right )\right )-i b^3 \left (3 a 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^3 \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 )-3 a^2 b \left ((A-C) d \left (3 c^2-d^2\right )+B \left (c^3-3 c d^2\right )\right )+b^3 \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 )^3 f}+\frac {\left (2 C d^3\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 {2 \left (a^6 C d^2+3 a^4 b^2 C d^2-3 a^2 b^4 \left (c^2 C+2 B c d-2 C d^2-A \left (c^2-d^2\right )\right )+b^6 \left (c (c C+2 B d)-A \left (c^2-d^2\right )\right )-a^3 b^3 \left (2 c (A-C) d+B \left (c^2-d^2\right )\right )+3 a b^5 \left (2 c (A-C) d+B \left (c^2-d^2\right )\right )\right ) \sqrt {c+d \tan (e+f x)}}{b^3 \left (a^2+b^2\right )^3 f \sqrt {a+b \tan (e+f x)}}-\frac {2 \left (a^4 C d+b^4 (B c+A d)+2 a b^3 (A c-c C-B d)-a^2 b^2 (B c+(A-3 C) d)\right ) (c+d \tan (e+f x))^{3/2}}{3 b^2 \left (a^2+b^2\right )^2 f (a+b \tan (e+f x))^{3/2}}-\frac {2 \left (A b^2-a (b B-a C)\right ) (c+d \tan (e+f x))^{5/2}}{5 b \left (a^2+b^2\right ) f (a+b \tan (e+f x))^{5/2}}+\frac {\left ((A-i B-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 (i a+b)^3 f}+\frac {\left ((A+i B-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 (i a-b)^3 f}+\frac {\left (2 C d^3\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 {2 C d^{5/2} \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 {2 \left (a^6 C d^2+3 a^4 b^2 C d^2-3 a^2 b^4 \left (c^2 C+2 B c d-2 C d^2-A \left (c^2-d^2\right )\right )+b^6 \left (c (c C+2 B d)-A \left (c^2-d^2\right )\right )-a^3 b^3 \left (2 c (A-C) d+B \left (c^2-d^2\right )\right )+3 a b^5 \left (2 c (A-C) d+B \left (c^2-d^2\right )\right )\right ) \sqrt {c+d \tan (e+f x)}}{b^3 \left (a^2+b^2\right )^3 f \sqrt {a+b \tan (e+f x)}}-\frac {2 \left (a^4 C d+b^4 (B c+A d)+2 a b^3 (A c-c C-B d)-a^2 b^2 (B c+(A-3 C) d)\right ) (c+d \tan (e+f x))^{3/2}}{3 b^2 \left (a^2+b^2\right )^2 f (a+b \tan (e+f x))^{3/2}}-\frac {2 \left (A b^2-a (b B-a C)\right ) (c+d \tan (e+f x))^{5/2}}{5 b \left (a^2+b^2\right ) f (a+b \tan (e+f x))^{5/2}}+\frac {\left ((A-i B-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 )}{(i a+b)^3 f}+\frac {\left ((A+i B-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 )}{(i a-b)^3 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)^{7/2} 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)^{7/2} f}+\frac {2 C d^{5/2} \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 {2 \left (a^6 C d^2+3 a^4 b^2 C d^2-3 a^2 b^4 \left (c^2 C+2 B c d-2 C d^2-A \left (c^2-d^2\right )\right )+b^6 \left (c (c C+2 B d)-A \left (c^2-d^2\right )\right )-a^3 b^3 \left (2 c (A-C) d+B \left (c^2-d^2\right )\right )+3 a b^5 \left (2 c (A-C) d+B \left (c^2-d^2\right )\right )\right ) \sqrt {c+d \tan (e+f x)}}{b^3 \left (a^2+b^2\right )^3 f \sqrt {a+b \tan (e+f x)}}-\frac {2 \left (a^4 C d+b^4 (B c+A d)+2 a b^3 (A c-c C-B d)-a^2 b^2 (B c+(A-3 C) d)\right ) (c+d \tan (e+f x))^{3/2}}{3 b^2 \left (a^2+b^2\right )^2 f (a+b \tan (e+f x))^{3/2}}-\frac {2 \left (A b^2-a (b B-a C)\right ) (c+d \tan (e+f x))^{5/2}}{5 b \left (a^2+b^2\right ) f (a+b \tan (e+f x))^{5/2}} \\ \end{align*}
Result contains higher order function than in optimal. Order 5 vs. order 3 in optimal.
Time = 7.14 (sec) , antiderivative size = 641, normalized size of antiderivative = 1.09 \[ \int \frac {(c+d \tan (e+f x))^{5/2} \left (A+B \tan (e+f x)+C \tan ^2(e+f x)\right )}{(a+b \tan (e+f x))^{7/2}} \, dx=\frac {(B+i (A-C)) (c+d \tan (e+f x))^{5/2}}{5 (a-i b) f (a+b \tan (e+f x))^{5/2}}-\frac {(i A-B-i C) (c+d \tan (e+f x))^{5/2}}{5 (a+i b) f (a+b \tan (e+f x))^{5/2}}-\frac {2 C (b c-a d)^2 \operatorname {Hypergeometric2F1}\left (-\frac {5}{2},-\frac {5}{2},-\frac {3}{2},-\frac {d (a+b \tan (e+f x))}{b c-a d}\right ) \sqrt {c+d \tan (e+f x)}}{5 b^3 f (a+b \tan (e+f x))^{5/2} \sqrt {\frac {b (c+d \tan (e+f x))}{b c-a d}}}+\frac {(A-i B-C) (i c+d) \left (\frac {(c+d \tan (e+f x))^{3/2}}{(a-i b) (a+b \tan (e+f x))^{3/2}}+\frac {3 (c-i d) \left (\frac {\sqrt {-c+i d} \text {arctanh}\left (\frac {\sqrt {-c+i d} \sqrt {a+b \tan (e+f x)}}{\sqrt {-a+i b} \sqrt {c+d \tan (e+f x)}}\right )}{(-a+i b)^{3/2}}+\frac {\sqrt {c+d \tan (e+f x)}}{(a-i b) \sqrt {a+b \tan (e+f x)}}\right )}{a-i b}\right )}{3 (a-i b) f}-\frac {(A+i B-C) (i c-d) \left (\frac {(c+d \tan (e+f x))^{3/2}}{(a+i b) (a+b \tan (e+f x))^{3/2}}-\frac {3 (c+i d) \left (\frac {\sqrt {c+i d} \text {arctanh}\left (\frac {\sqrt {c+i d} \sqrt {a+b \tan (e+f x)}}{\sqrt {a+i b} \sqrt {c+d \tan (e+f x)}}\right )}{(a+i b)^{3/2}}-\frac {\sqrt {c+d \tan (e+f x)}}{(a+i b) \sqrt {a+b \tan (e+f x)}}\right )}{a+i b}\right )}{3 (a+i b) f} \]
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Timed out.
\[\int \frac {\left (c +d \tan \left (f x +e \right )\right )^{\frac {5}{2}} \left (A +B \tan \left (f x +e \right )+C \tan \left (f x +e \right )^{2}\right )}{\left (a +b \tan \left (f x +e \right )\right )^{\frac {7}{2}}}d x\]
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Timed out. \[ \int \frac {(c+d \tan (e+f x))^{5/2} \left (A+B \tan (e+f x)+C \tan ^2(e+f x)\right )}{(a+b \tan (e+f x))^{7/2}} \, dx=\text {Timed out} \]
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\[ \int \frac {(c+d \tan (e+f x))^{5/2} \left (A+B \tan (e+f x)+C \tan ^2(e+f x)\right )}{(a+b \tan (e+f x))^{7/2}} \, dx=\int \frac {\left (c + d \tan {\left (e + f x \right )}\right )^{\frac {5}{2}} \left (A + B \tan {\left (e + f x \right )} + C \tan ^{2}{\left (e + f x \right )}\right )}{\left (a + b \tan {\left (e + f x \right )}\right )^{\frac {7}{2}}}\, dx \]
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Timed out. \[ \int \frac {(c+d \tan (e+f x))^{5/2} \left (A+B \tan (e+f x)+C \tan ^2(e+f x)\right )}{(a+b \tan (e+f x))^{7/2}} \, dx=\text {Timed out} \]
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Timed out. \[ \int \frac {(c+d \tan (e+f x))^{5/2} \left (A+B \tan (e+f x)+C \tan ^2(e+f x)\right )}{(a+b \tan (e+f x))^{7/2}} \, dx=\text {Timed out} \]
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Timed out. \[ \int \frac {(c+d \tan (e+f x))^{5/2} \left (A+B \tan (e+f x)+C \tan ^2(e+f x)\right )}{(a+b \tan (e+f x))^{7/2}} \, dx=\text {Hanged} \]
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