Integrand size = 49, antiderivative size = 535 \[ \int \frac {(c+d \tan (e+f x))^{5/2} \left (A+B \tan (e+f x)+C \tan ^2(e+f x)\right )}{(a+b \tan (e+f x))^{3/2}} \, dx=-\frac {(i A+B-i C) (c-i d)^{5/2} \text {arctanh}\left (\frac {\sqrt {c-i d} \sqrt {a+b \tan (e+f x)}}{\sqrt {a-i b} \sqrt {c+d \tan (e+f x)}}\right )}{(a-i b)^{3/2} f}-\frac {(B-i (A-C)) (c+i d)^{5/2} \text {arctanh}\left (\frac {\sqrt {c+i d} \sqrt {a+b \tan (e+f x)}}{\sqrt {a+i b} \sqrt {c+d \tan (e+f x)}}\right )}{(a+i b)^{3/2} f}+\frac {\sqrt {d} \left (15 a^2 C d^2-6 a b d (5 c C+2 B d)+b^2 \left (15 c^2 C+20 B c d+8 (A-C) d^2\right )\right ) \text {arctanh}\left (\frac {\sqrt {d} \sqrt {a+b \tan (e+f x)}}{\sqrt {b} \sqrt {c+d \tan (e+f x)}}\right )}{4 b^{7/2} f}-\frac {d \left (15 a^3 C d-8 A b^2 (b c-a d)-3 a^2 b (5 c C+4 B d)-b^3 (7 c C+4 B d)+a b^2 (8 B c+7 C d)\right ) \sqrt {a+b \tan (e+f x)} \sqrt {c+d \tan (e+f x)}}{4 b^3 \left (a^2+b^2\right ) f}+\frac {\left (4 A b^2-4 a b B+5 a^2 C+b^2 C\right ) d \sqrt {a+b \tan (e+f x)} (c+d \tan (e+f x))^{3/2}}{2 b^2 \left (a^2+b^2\right ) f}-\frac {2 \left (A b^2-a (b B-a C)\right ) (c+d \tan (e+f x))^{5/2}}{b \left (a^2+b^2\right ) f \sqrt {a+b \tan (e+f x)}} \]
[Out]
Time = 9.89 (sec) , antiderivative size = 535, 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} \[ \int \frac {(c+d \tan (e+f x))^{5/2} \left (A+B \tan (e+f x)+C \tan ^2(e+f x)\right )}{(a+b \tan (e+f x))^{3/2}} \, dx=\frac {\sqrt {d} \left (15 a^2 C d^2-6 a b d (2 B d+5 c C)+b^2 \left (8 d^2 (A-C)+20 B c d+15 c^2 C\right )\right ) \text {arctanh}\left (\frac {\sqrt {d} \sqrt {a+b \tan (e+f x)}}{\sqrt {b} \sqrt {c+d \tan (e+f x)}}\right )}{4 b^{7/2} f}-\frac {2 \left (A b^2-a (b B-a C)\right ) (c+d \tan (e+f x))^{5/2}}{b f \left (a^2+b^2\right ) \sqrt {a+b \tan (e+f x)}}+\frac {d \left (5 a^2 C-4 a b B+4 A b^2+b^2 C\right ) \sqrt {a+b \tan (e+f x)} (c+d \tan (e+f x))^{3/2}}{2 b^2 f \left (a^2+b^2\right )}-\frac {d \sqrt {a+b \tan (e+f x)} \sqrt {c+d \tan (e+f x)} \left (15 a^3 C d-3 a^2 b (4 B d+5 c C)-8 A b^2 (b c-a d)+a b^2 (8 B c+7 C d)-b^3 (4 B d+7 c C)\right )}{4 b^3 f \left (a^2+b^2\right )}-\frac {(c-i d)^{5/2} (i A+B-i C) \text {arctanh}\left (\frac {\sqrt {c-i d} \sqrt {a+b \tan (e+f x)}}{\sqrt {a-i b} \sqrt {c+d \tan (e+f x)}}\right )}{f (a-i b)^{3/2}}-\frac {(c+i d)^{5/2} (B-i (A-C)) \text {arctanh}\left (\frac {\sqrt {c+i d} \sqrt {a+b \tan (e+f x)}}{\sqrt {a+i b} \sqrt {c+d \tan (e+f x)}}\right )}{f (a+i b)^{3/2}} \]
[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}}{b \left (a^2+b^2\right ) f \sqrt {a+b \tan (e+f x)}}+\frac {2 \int \frac {(c+d \tan (e+f x))^{3/2} \left (\frac {1}{2} ((b B-a C) (b c-5 a d)+A b (a c+5 b d))-\frac {1}{2} b ((A-C) (b c-a d)-B (a c+b d)) \tan (e+f x)+\frac {1}{2} \left (4 A b^2-4 a b B+5 a^2 C+b^2 C\right ) d \tan ^2(e+f x)\right )}{\sqrt {a+b \tan (e+f x)}} \, dx}{b \left (a^2+b^2\right )} \\ & = \frac {\left (4 A b^2-4 a b B+5 a^2 C+b^2 C\right ) d \sqrt {a+b \tan (e+f x)} (c+d \tan (e+f x))^{3/2}}{2 b^2 \left (a^2+b^2\right ) f}-\frac {2 \left (A b^2-a (b B-a C)\right ) (c+d \tan (e+f x))^{5/2}}{b \left (a^2+b^2\right ) f \sqrt {a+b \tan (e+f x)}}+\frac {\int \frac {\sqrt {c+d \tan (e+f x)} \left (\frac {1}{4} \left (-\left (\left (4 A b^2-4 a b B+5 a^2 C+b^2 C\right ) d (b c+3 a d)\right )+4 b c ((b B-a C) (b c-5 a d)+A b (a c+5 b d))\right )+b^2 \left (2 a A c d-2 a c C d-A b \left (c^2-d^2\right )+a B \left (c^2-d^2\right )+b \left (c^2 C+2 B c d-C d^2\right )\right ) \tan (e+f x)+\frac {1}{4} d \left (3 \left (4 A b^2-4 a b B+5 a^2 C+b^2 C\right ) (b c-a d)-4 b^2 ((A-C) (b c-a d)-B (a c+b d))\right ) \tan ^2(e+f x)\right )}{\sqrt {a+b \tan (e+f x)}} \, dx}{b^2 \left (a^2+b^2\right )} \\ & = -\frac {d \left (15 a^3 C d-8 A b^2 (b c-a d)-3 a^2 b (5 c C+4 B d)-b^3 (7 c C+4 B d)+a b^2 (8 B c+7 C d)\right ) \sqrt {a+b \tan (e+f x)} \sqrt {c+d \tan (e+f x)}}{4 b^3 \left (a^2+b^2\right ) f}+\frac {\left (4 A b^2-4 a b B+5 a^2 C+b^2 C\right ) d \sqrt {a+b \tan (e+f x)} (c+d \tan (e+f x))^{3/2}}{2 b^2 \left (a^2+b^2\right ) f}-\frac {2 \left (A b^2-a (b B-a C)\right ) (c+d \tan (e+f x))^{5/2}}{b \left (a^2+b^2\right ) f \sqrt {a+b \tan (e+f x)}}+\frac {\int \frac {\frac {1}{8} \left (15 a^4 C d^3-6 a^3 b d^2 (5 c C+2 B d)+b^4 c \left (8 B c^2+24 A c d-9 c C d-4 B d^2\right )+a^2 b^2 d \left (15 c^2 C+20 B c d+(8 A+7 C) d^2\right )-2 a b^3 \left (4 c^3 C+12 B c^2 d+3 c C d^2+2 B d^3-4 A \left (c^3-3 c d^2\right )\right )\right )+b^3 \left (a A d \left (3 c^2-d^2\right )-A b \left (c^3-3 c d^2\right )+b \left (c^3 C+3 B c^2 d-3 c C d^2-B d^3\right )-a \left (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 {1}{8} \left (a^2+b^2\right ) d \left (15 a^2 C d^2-6 a b d (5 c C+2 B d)+b^2 \left (15 c^2 C+20 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}{b^3 \left (a^2+b^2\right )} \\ & = -\frac {d \left (15 a^3 C d-8 A b^2 (b c-a d)-3 a^2 b (5 c C+4 B d)-b^3 (7 c C+4 B d)+a b^2 (8 B c+7 C d)\right ) \sqrt {a+b \tan (e+f x)} \sqrt {c+d \tan (e+f x)}}{4 b^3 \left (a^2+b^2\right ) f}+\frac {\left (4 A b^2-4 a b B+5 a^2 C+b^2 C\right ) d \sqrt {a+b \tan (e+f x)} (c+d \tan (e+f x))^{3/2}}{2 b^2 \left (a^2+b^2\right ) f}-\frac {2 \left (A b^2-a (b B-a C)\right ) (c+d \tan (e+f x))^{5/2}}{b \left (a^2+b^2\right ) f \sqrt {a+b \tan (e+f x)}}+\frac {\text {Subst}\left (\int \frac {\frac {1}{8} \left (15 a^4 C d^3-6 a^3 b d^2 (5 c C+2 B d)+b^4 c \left (8 B c^2+24 A c d-9 c C d-4 B d^2\right )+a^2 b^2 d \left (15 c^2 C+20 B c d+(8 A+7 C) d^2\right )-2 a b^3 \left (4 c^3 C+12 B c^2 d+3 c C d^2+2 B d^3-4 A \left (c^3-3 c d^2\right )\right )\right )+b^3 \left (a A d \left (3 c^2-d^2\right )-A b \left (c^3-3 c d^2\right )+b \left (c^3 C+3 B c^2 d-3 c C d^2-B d^3\right )-a \left (C d \left (3 c^2-d^2\right )-B \left (c^3-3 c d^2\right )\right )\right ) x+\frac {1}{8} \left (a^2+b^2\right ) d \left (15 a^2 C d^2-6 a b d (5 c C+2 B d)+b^2 \left (15 c^2 C+20 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 )}{b^3 \left (a^2+b^2\right ) f} \\ & = -\frac {d \left (15 a^3 C d-8 A b^2 (b c-a d)-3 a^2 b (5 c C+4 B d)-b^3 (7 c C+4 B d)+a b^2 (8 B c+7 C d)\right ) \sqrt {a+b \tan (e+f x)} \sqrt {c+d \tan (e+f x)}}{4 b^3 \left (a^2+b^2\right ) f}+\frac {\left (4 A b^2-4 a b B+5 a^2 C+b^2 C\right ) d \sqrt {a+b \tan (e+f x)} (c+d \tan (e+f x))^{3/2}}{2 b^2 \left (a^2+b^2\right ) f}-\frac {2 \left (A b^2-a (b B-a C)\right ) (c+d \tan (e+f x))^{5/2}}{b \left (a^2+b^2\right ) f \sqrt {a+b \tan (e+f x)}}+\frac {\text {Subst}\left (\int \left (\frac {\left (a^2+b^2\right ) d \left (15 a^2 C d^2-6 a b d (5 c C+2 B d)+b^2 \left (15 c^2 C+20 B c d+8 (A-C) d^2\right )\right )}{8 \sqrt {a+b x} \sqrt {c+d x}}+\frac {-b^3 \left (a \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 )-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 (b \left (c^3 C+3 B c^2 d-3 c C d^2-B d^3\right )+a \left (B c^3-3 c^2 C d-3 B c d^2+C d^3\right )+A \left (a 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 )}\right ) \, dx,x,\tan (e+f x)\right )}{b^3 \left (a^2+b^2\right ) f} \\ & = -\frac {d \left (15 a^3 C d-8 A b^2 (b c-a d)-3 a^2 b (5 c C+4 B d)-b^3 (7 c C+4 B d)+a b^2 (8 B c+7 C d)\right ) \sqrt {a+b \tan (e+f x)} \sqrt {c+d \tan (e+f x)}}{4 b^3 \left (a^2+b^2\right ) f}+\frac {\left (4 A b^2-4 a b B+5 a^2 C+b^2 C\right ) d \sqrt {a+b \tan (e+f x)} (c+d \tan (e+f x))^{3/2}}{2 b^2 \left (a^2+b^2\right ) f}-\frac {2 \left (A b^2-a (b B-a C)\right ) (c+d \tan (e+f x))^{5/2}}{b \left (a^2+b^2\right ) f \sqrt {a+b \tan (e+f x)}}+\frac {\text {Subst}\left (\int \frac {-b^3 \left (a \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 )-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 (b \left (c^3 C+3 B c^2 d-3 c C d^2-B d^3\right )+a \left (B c^3-3 c^2 C d-3 B c d^2+C d^3\right )+A \left (a 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 ) f}+\frac {\left (d \left (15 a^2 C d^2-6 a b d (5 c C+2 B d)+b^2 \left (15 c^2 C+20 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 b^3 f} \\ & = -\frac {d \left (15 a^3 C d-8 A b^2 (b c-a d)-3 a^2 b (5 c C+4 B d)-b^3 (7 c C+4 B d)+a b^2 (8 B c+7 C d)\right ) \sqrt {a+b \tan (e+f x)} \sqrt {c+d \tan (e+f x)}}{4 b^3 \left (a^2+b^2\right ) f}+\frac {\left (4 A b^2-4 a b B+5 a^2 C+b^2 C\right ) d \sqrt {a+b \tan (e+f x)} (c+d \tan (e+f x))^{3/2}}{2 b^2 \left (a^2+b^2\right ) f}-\frac {2 \left (A b^2-a (b B-a C)\right ) (c+d \tan (e+f x))^{5/2}}{b \left (a^2+b^2\right ) f \sqrt {a+b \tan (e+f x)}}+\frac {\text {Subst}\left (\int \left (\frac {-b^3 \left (b \left (c^3 C+3 B c^2 d-3 c C d^2-B d^3\right )+a \left (B c^3-3 c^2 C d-3 B c d^2+C d^3\right )+A \left (a d \left (3 c^2-d^2\right )-b \left (c^3-3 c d^2\right )\right )\right )-i b^3 \left (a \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 )-b \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 (b \left (c^3 C+3 B c^2 d-3 c C d^2-B d^3\right )+a \left (B c^3-3 c^2 C d-3 B c d^2+C d^3\right )+A \left (a d \left (3 c^2-d^2\right )-b \left (c^3-3 c d^2\right )\right )\right )-i b^3 \left (a \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 )-b \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 ) f}+\frac {\left (d \left (15 a^2 C d^2-6 a b d (5 c C+2 B d)+b^2 \left (15 c^2 C+20 B c d+8 (A-C) d^2\right )\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 b^4 f} \\ & = -\frac {d \left (15 a^3 C d-8 A b^2 (b c-a d)-3 a^2 b (5 c C+4 B d)-b^3 (7 c C+4 B d)+a b^2 (8 B c+7 C d)\right ) \sqrt {a+b \tan (e+f x)} \sqrt {c+d \tan (e+f x)}}{4 b^3 \left (a^2+b^2\right ) f}+\frac {\left (4 A b^2-4 a b B+5 a^2 C+b^2 C\right ) d \sqrt {a+b \tan (e+f x)} (c+d \tan (e+f x))^{3/2}}{2 b^2 \left (a^2+b^2\right ) f}-\frac {2 \left (A b^2-a (b B-a C)\right ) (c+d \tan (e+f x))^{5/2}}{b \left (a^2+b^2\right ) f \sqrt {a+b \tan (e+f x)}}+\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) f}+\frac {\left ((i a+b) (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 \left (a^2+b^2\right ) f}+\frac {\left (d \left (15 a^2 C d^2-6 a b d (5 c C+2 B d)+b^2 \left (15 c^2 C+20 B c d+8 (A-C) d^2\right )\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 b^4 f} \\ & = \frac {\sqrt {d} \left (15 a^2 C d^2-6 a b d (5 c C+2 B d)+b^2 \left (15 c^2 C+20 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 b^{7/2} f}-\frac {d \left (15 a^3 C d-8 A b^2 (b c-a d)-3 a^2 b (5 c C+4 B d)-b^3 (7 c C+4 B d)+a b^2 (8 B c+7 C d)\right ) \sqrt {a+b \tan (e+f x)} \sqrt {c+d \tan (e+f x)}}{4 b^3 \left (a^2+b^2\right ) f}+\frac {\left (4 A b^2-4 a b B+5 a^2 C+b^2 C\right ) d \sqrt {a+b \tan (e+f x)} (c+d \tan (e+f x))^{3/2}}{2 b^2 \left (a^2+b^2\right ) f}-\frac {2 \left (A b^2-a (b B-a C)\right ) (c+d \tan (e+f x))^{5/2}}{b \left (a^2+b^2\right ) f \sqrt {a+b \tan (e+f x)}}+\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) f}+\frac {\left ((i a+b) (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 )}{\left (a^2+b^2\right ) 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)^{3/2} 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)^{3/2} f}+\frac {\sqrt {d} \left (15 a^2 C d^2-6 a b d (5 c C+2 B d)+b^2 \left (15 c^2 C+20 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 b^{7/2} f}-\frac {d \left (15 a^3 C d-8 A b^2 (b c-a d)-3 a^2 b (5 c C+4 B d)-b^3 (7 c C+4 B d)+a b^2 (8 B c+7 C d)\right ) \sqrt {a+b \tan (e+f x)} \sqrt {c+d \tan (e+f x)}}{4 b^3 \left (a^2+b^2\right ) f}+\frac {\left (4 A b^2-4 a b B+5 a^2 C+b^2 C\right ) d \sqrt {a+b \tan (e+f x)} (c+d \tan (e+f x))^{3/2}}{2 b^2 \left (a^2+b^2\right ) f}-\frac {2 \left (A b^2-a (b B-a C)\right ) (c+d \tan (e+f x))^{5/2}}{b \left (a^2+b^2\right ) f \sqrt {a+b \tan (e+f x)}} \\ \end{align*}
Both result and optimal contain complex but leaf count is larger than twice the leaf count of optimal. \(1774\) vs. \(2(535)=1070\).
Time = 8.66 (sec) , antiderivative size = 1774, normalized size of antiderivative = 3.32 \[ \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))^{3/2}} \, dx=\frac {C (c+d \tan (e+f x))^{5/2}}{2 b f \sqrt {a+b \tan (e+f x)}}+\frac {\frac {(5 b c C+4 b B d-5 a C d) (c+d \tan (e+f x))^{3/2}}{2 b f \sqrt {a+b \tan (e+f x)}}+\frac {\frac {8 b^2 (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)^{3/2} f}-\frac {8 b^2 (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)^{3/2} f}+\frac {8 b^2 (i A+B-i C) (c-i d)^2 \sqrt {c+d \tan (e+f x)}}{(a-i b) f \sqrt {a+b \tan (e+f x)}}+\frac {8 b^2 (A+i B-C) (c+i d)^2 \sqrt {c+d \tan (e+f x)}}{(i a-b) f \sqrt {a+b \tan (e+f x)}}+\frac {30 a^2 C d^2 \sqrt {c+d \tan (e+f x)} \left (1+\frac {b d (a+b \tan (e+f x))}{(b c-a d) \left (\frac {b^2 c}{b c-a d}-\frac {a b d}{b c-a d}\right )}\right )^{3/2} \left (1-\frac {\sqrt {b} \sqrt {d} \text {arcsinh}\left (\frac {\sqrt {b} \sqrt {d} \sqrt {a+b \tan (e+f x)}}{\sqrt {b c-a d} \sqrt {\frac {b^2 c}{b c-a d}-\frac {a b d}{b c-a d}}}\right ) \sqrt {a+b \tan (e+f x)}}{\sqrt {b c-a d} \sqrt {\frac {b^2 c}{b c-a d}-\frac {a b d}{b c-a d}} \sqrt {1+\frac {b d (a+b \tan (e+f x))}{(b c-a d) \left (\frac {b^2 c}{b c-a d}-\frac {a b d}{b c-a d}\right )}}}\right )}{b \sqrt {\frac {b}{\frac {b^2 c}{b c-a d}-\frac {a b d}{b c-a d}}} f \sqrt {a+b \tan (e+f x)} \sqrt {\frac {b (c+d \tan (e+f x))}{b c-a d}} \left (-1-\frac {b d (a+b \tan (e+f x))}{(b c-a d) \left (\frac {b^2 c}{b c-a d}-\frac {a b d}{b c-a d}\right )}\right )}-\frac {12 a d (5 c C+2 B d) \sqrt {c+d \tan (e+f x)} \left (1+\frac {b d (a+b \tan (e+f x))}{(b c-a d) \left (\frac {b^2 c}{b c-a d}-\frac {a b d}{b c-a d}\right )}\right )^{3/2} \left (1-\frac {\sqrt {b} \sqrt {d} \text {arcsinh}\left (\frac {\sqrt {b} \sqrt {d} \sqrt {a+b \tan (e+f x)}}{\sqrt {b c-a d} \sqrt {\frac {b^2 c}{b c-a d}-\frac {a b d}{b c-a d}}}\right ) \sqrt {a+b \tan (e+f x)}}{\sqrt {b c-a d} \sqrt {\frac {b^2 c}{b c-a d}-\frac {a b d}{b c-a d}} \sqrt {1+\frac {b d (a+b \tan (e+f x))}{(b c-a d) \left (\frac {b^2 c}{b c-a d}-\frac {a b d}{b c-a d}\right )}}}\right )}{\sqrt {\frac {b}{\frac {b^2 c}{b c-a d}-\frac {a b d}{b c-a d}}} f \sqrt {a+b \tan (e+f x)} \sqrt {\frac {b (c+d \tan (e+f x))}{b c-a d}} \left (-1-\frac {b d (a+b \tan (e+f x))}{(b c-a d) \left (\frac {b^2 c}{b c-a d}-\frac {a b d}{b c-a d}\right )}\right )}+\frac {2 b \left (15 c^2 C+20 B c d+8 (A-C) d^2\right ) \sqrt {c+d \tan (e+f x)} \left (1+\frac {b d (a+b \tan (e+f x))}{(b c-a d) \left (\frac {b^2 c}{b c-a d}-\frac {a b d}{b c-a d}\right )}\right )^{3/2} \left (1-\frac {\sqrt {b} \sqrt {d} \text {arcsinh}\left (\frac {\sqrt {b} \sqrt {d} \sqrt {a+b \tan (e+f x)}}{\sqrt {b c-a d} \sqrt {\frac {b^2 c}{b c-a d}-\frac {a b d}{b c-a d}}}\right ) \sqrt {a+b \tan (e+f x)}}{\sqrt {b c-a d} \sqrt {\frac {b^2 c}{b c-a d}-\frac {a b d}{b c-a d}} \sqrt {1+\frac {b d (a+b \tan (e+f x))}{(b c-a d) \left (\frac {b^2 c}{b c-a d}-\frac {a b d}{b c-a d}\right )}}}\right )}{\sqrt {\frac {b}{\frac {b^2 c}{b c-a d}-\frac {a b d}{b c-a d}}} f \sqrt {a+b \tan (e+f x)} \sqrt {\frac {b (c+d \tan (e+f x))}{b c-a d}} \left (-1-\frac {b d (a+b \tan (e+f x))}{(b c-a d) \left (\frac {b^2 c}{b c-a d}-\frac {a b d}{b c-a d}\right )}\right )}}{4 b}}{2 b} \]
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Timed out.
\[\int \frac {\left (c +d \tan \left (f x +e \right )\right )^{\frac {5}{2}} \left (A +B \tan \left (f x +e \right )+C \tan \left (f x +e \right )^{2}\right )}{\left (a +b \tan \left (f x +e \right )\right )^{\frac {3}{2}}}d x\]
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Timed out. \[ \int \frac {(c+d \tan (e+f x))^{5/2} \left (A+B \tan (e+f x)+C \tan ^2(e+f x)\right )}{(a+b \tan (e+f x))^{3/2}} \, dx=\text {Timed out} \]
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\[ \int \frac {(c+d \tan (e+f x))^{5/2} \left (A+B \tan (e+f x)+C \tan ^2(e+f x)\right )}{(a+b \tan (e+f x))^{3/2}} \, dx=\int \frac {\left (c + d \tan {\left (e + f x \right )}\right )^{\frac {5}{2}} \left (A + B \tan {\left (e + f x \right )} + C \tan ^{2}{\left (e + f x \right )}\right )}{\left (a + b \tan {\left (e + f x \right )}\right )^{\frac {3}{2}}}\, dx \]
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Timed out. \[ \int \frac {(c+d \tan (e+f x))^{5/2} \left (A+B \tan (e+f x)+C \tan ^2(e+f x)\right )}{(a+b \tan (e+f x))^{3/2}} \, dx=\text {Timed out} \]
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Timed out. \[ \int \frac {(c+d \tan (e+f x))^{5/2} \left (A+B \tan (e+f x)+C \tan ^2(e+f x)\right )}{(a+b \tan (e+f x))^{3/2}} \, dx=\text {Timed out} \]
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Timed out. \[ \int \frac {(c+d \tan (e+f x))^{5/2} \left (A+B \tan (e+f x)+C \tan ^2(e+f x)\right )}{(a+b \tan (e+f x))^{3/2}} \, dx=\text {Hanged} \]
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