3.2.36 \(\int \sqrt {a+b \tan (e+f x)} (c+d \tan (e+f x))^{3/2} (A+B \tan (e+f x)+C \tan ^2(e+f x)) \, dx\) [136]

3.2.36.1 Optimal result

Integrand size = 49, antiderivative size = 508 \[ \int \sqrt {a+b \tan (e+f x)} (c+d \tan (e+f x))^{3/2} \left (A+B \tan (e+f x)+C \tan ^2(e+f x)\right ) \, dx=-\frac {\sqrt {a-i b} (i A+B-i C) (c-i d)^{3/2} \text {arctanh}\left (\frac {\sqrt {c-i d} \sqrt {a+b \tan (e+f x)}}{\sqrt {a-i b} \sqrt {c+d \tan (e+f x)}}\right )}{f}-\frac {\sqrt {a+i b} (B-i (A-C)) (c+i d)^{3/2} \text {arctanh}\left (\frac {\sqrt {c+i d} \sqrt {a+b \tan (e+f x)}}{\sqrt {a+i b} \sqrt {c+d \tan (e+f x)}}\right )}{f}+\frac {\left (a^3 C d^3-a^2 b d^2 (3 c C+2 B d)+a b^2 d \left (3 c^2 C+12 B c d+8 (A-C) d^2\right )-b^3 \left (c^3 C-6 B c^2 d-24 c (A-C) d^2+16 B d^3\right )\right ) \text {arctanh}\left (\frac {\sqrt {d} \sqrt {a+b \tan (e+f x)}}{\sqrt {b} \sqrt {c+d \tan (e+f x)}}\right )}{8 b^{5/2} d^{3/2} f}+\frac {\left (8 b (A b+a B-b C) d^2-(b c-a d) (b c C-6 b B d-a C d)\right ) \sqrt {a+b \tan (e+f x)} \sqrt {c+d \tan (e+f x)}}{8 b^2 d f}-\frac {(b c C-6 b B d-a C d) \sqrt {a+b \tan (e+f x)} (c+d \tan (e+f x))^{3/2}}{12 b d f}+\frac {C \sqrt {a+b \tan (e+f x)} (c+d \tan (e+f x))^{5/2}}{3 d f} \]

output

1/8*(a^3*C*d^3-a^2*b*d^2*(2*B*d+3*C*c)+a*b^2*d*(3*c^2*C+12*B*c*d+8*(A-C)*d 
^2)-b^3*(c^3*C-6*B*c^2*d-24*c*(A-C)*d^2+16*B*d^3))*arctanh(d^(1/2)*(a+b*ta 
n(f*x+e))^(1/2)/b^(1/2)/(c+d*tan(f*x+e))^(1/2))/b^(5/2)/d^(3/2)/f-(I*A+B-I 
*C)*(c-I*d)^(3/2)*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))*(a-I*b)^(1/2)/f-(B-I*(A-C))*(c+I*d)^(3/2)*arcta 
nh((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))*(a+I*b)^(1/2)/f+1/8*(8*b*(A*b+B*a-C*b)*d^2-(-a*d+b*c)*(-6*B*b*d-C*a*d+ 
C*b*c))*(a+b*tan(f*x+e))^(1/2)*(c+d*tan(f*x+e))^(1/2)/b^2/d/f-1/12*(-6*B*b 
*d-C*a*d+C*b*c)*(a+b*tan(f*x+e))^(1/2)*(c+d*tan(f*x+e))^(3/2)/b/d/f+1/3*C* 
(a+b*tan(f*x+e))^(1/2)*(c+d*tan(f*x+e))^(5/2)/d/f
 

3.2.36.2 Mathematica [A] (verified)

Time = 9.23 (sec) , antiderivative size = 867, normalized size of antiderivative = 1.71 \[ \int \sqrt {a+b \tan (e+f x)} (c+d \tan (e+f x))^{3/2} \left (A+B \tan (e+f x)+C \tan ^2(e+f x)\right ) \, dx=\frac {C \sqrt {a+b \tan (e+f x)} (c+d \tan (e+f x))^{5/2}}{3 d f}+\frac {\frac {(-b c C+6 b B d+a C d) \sqrt {a+b \tan (e+f x)} (c+d \tan (e+f x))^{3/2}}{4 b f}+\frac {\frac {3 \left (8 b (A b+a B-b C) d^2-(b c-a d) (b c C-6 b B d-a C d)\right ) \sqrt {a+b \tan (e+f x)} \sqrt {c+d \tan (e+f x)}}{4 b f}+\frac {\frac {6 b^2 d \left (b \left (2 a A c d-2 a c C d+A b \left (c^2-d^2\right )+a B \left (c^2-d^2\right )-b \left (c^2 C+2 B c d-C d^2\right )\right )-\sqrt {-b^2} \left (a \left (c^2 C+2 B c d-C d^2-A \left (c^2-d^2\right )\right )+b \left (2 c (A-C) d+B \left (c^2-d^2\right )\right )\right )\right ) \text {arctanh}\left (\frac {\sqrt {-c+\frac {\sqrt {-b^2} d}{b}} \sqrt {a+b \tan (e+f x)}}{\sqrt {-a+\sqrt {-b^2}} \sqrt {c+d \tan (e+f x)}}\right )}{\sqrt {-a+\sqrt {-b^2}} \sqrt {-c+\frac {\sqrt {-b^2} d}{b}}}-\frac {6 b^2 d \left (b \left (2 a A c d-2 a c C d+A b \left (c^2-d^2\right )+a B \left (c^2-d^2\right )-b \left (c^2 C+2 B c d-C d^2\right )\right )+\sqrt {-b^2} \left (a \left (c^2 C+2 B c d-C d^2-A \left (c^2-d^2\right )\right )+b \left (2 c (A-C) d+B \left (c^2-d^2\right )\right )\right )\right ) \text {arctanh}\left (\frac {\sqrt {c+\frac {\sqrt {-b^2} d}{b}} \sqrt {a+b \tan (e+f x)}}{\sqrt {a+\sqrt {-b^2}} \sqrt {c+d \tan (e+f x)}}\right )}{\sqrt {a+\sqrt {-b^2}} \sqrt {c+\frac {\sqrt {-b^2} d}{b}}}+\frac {3 \sqrt {b} \sqrt {c-\frac {a d}{b}} \left (a^3 C d^3-a^2 b d^2 (3 c C+2 B d)+a b^2 d \left (3 c^2 C+12 B c d+8 (A-C) d^2\right )-b^3 \left (c^3 C-6 B c^2 d-24 c (A-C) d^2+16 B d^3\right )\right ) \text {arcsinh}\left (\frac {\sqrt {d} \sqrt {a+b \tan (e+f x)}}{\sqrt {b} \sqrt {c-\frac {a d}{b}}}\right ) \sqrt {\frac {b c+b d \tan (e+f x)}{b c-a d}}}{4 \sqrt {d} \sqrt {c+d \tan (e+f x)}}}{b^2 f}}{2 b}}{3 d} \]

input

Integrate[Sqrt[a + b*Tan[e + f*x]]*(c + d*Tan[e + f*x])^(3/2)*(A + B*Tan[e 
 + f*x] + C*Tan[e + f*x]^2),x]
 

output

(C*Sqrt[a + b*Tan[e + f*x]]*(c + d*Tan[e + f*x])^(5/2))/(3*d*f) + (((-(b*c 
*C) + 6*b*B*d + a*C*d)*Sqrt[a + b*Tan[e + f*x]]*(c + d*Tan[e + f*x])^(3/2) 
)/(4*b*f) + ((3*(8*b*(A*b + a*B - b*C)*d^2 - (b*c - a*d)*(b*c*C - 6*b*B*d 
- a*C*d))*Sqrt[a + b*Tan[e + f*x]]*Sqrt[c + d*Tan[e + f*x]])/(4*b*f) + ((6 
*b^2*d*(b*(2*a*A*c*d - 2*a*c*C*d + A*b*(c^2 - d^2) + a*B*(c^2 - d^2) - b*( 
c^2*C + 2*B*c*d - C*d^2)) - Sqrt[-b^2]*(a*(c^2*C + 2*B*c*d - C*d^2 - A*(c^ 
2 - d^2)) + b*(2*c*(A - C)*d + B*(c^2 - d^2))))*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]])])/(Sqrt[-a + Sqrt[-b^2]]*Sqrt[-c + (Sqrt[-b^2]*d)/b]) - (6*b^2 
*d*(b*(2*a*A*c*d - 2*a*c*C*d + A*b*(c^2 - d^2) + a*B*(c^2 - d^2) - b*(c^2* 
C + 2*B*c*d - C*d^2)) + Sqrt[-b^2]*(a*(c^2*C + 2*B*c*d - C*d^2 - A*(c^2 - 
d^2)) + b*(2*c*(A - C)*d + B*(c^2 - d^2))))*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]])])/(Sqrt[a + Sqrt[-b^2]]*Sqrt[c + (Sqrt[-b^2]*d)/b]) + (3*Sqrt[b]*Sqr 
t[c - (a*d)/b]*(a^3*C*d^3 - a^2*b*d^2*(3*c*C + 2*B*d) + a*b^2*d*(3*c^2*C + 
 12*B*c*d + 8*(A - C)*d^2) - b^3*(c^3*C - 6*B*c^2*d - 24*c*(A - C)*d^2 + 1 
6*B*d^3))*ArcSinh[(Sqrt[d]*Sqrt[a + b*Tan[e + f*x]])/(Sqrt[b]*Sqrt[c - (a* 
d)/b])]*Sqrt[(b*c + b*d*Tan[e + f*x])/(b*c - a*d)])/(4*Sqrt[d]*Sqrt[c + d* 
Tan[e + f*x]]))/(b^2*f))/(2*b))/(3*d)
 

3.2.36.3 Rubi [A] (verified)

Time = 3.66 (sec) , antiderivative size = 524, normalized size of antiderivative = 1.03, number of steps used = 14, number of rules used = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.265, Rules used = {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.

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

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 4130

\(\displaystyle \frac {\int -\frac {(c+d \tan (e+f x))^{3/2} \left ((b c C-a d C-6 b B d) \tan ^2(e+f x)-6 (A b-C b+a B) d \tan (e+f x)+b c C-a (6 A-5 C) d\right )}{2 \sqrt {a+b \tan (e+f x)}}dx}{3 d}+\frac {C \sqrt {a+b \tan (e+f x)} (c+d \tan (e+f x))^{5/2}}{3 d f}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {C \sqrt {a+b \tan (e+f x)} (c+d \tan (e+f x))^{5/2}}{3 d f}-\frac {\int \frac {(c+d \tan (e+f x))^{3/2} \left ((b c C-a d C-6 b B d) \tan ^2(e+f x)-6 (A b-C b+a B) d \tan (e+f x)+b c C-a (6 A-5 C) d\right )}{\sqrt {a+b \tan (e+f x)}}dx}{6 d}\)

\(\Big \downarrow \) 3042

\(\displaystyle \frac {C \sqrt {a+b \tan (e+f x)} (c+d \tan (e+f x))^{5/2}}{3 d f}-\frac {\int \frac {(c+d \tan (e+f x))^{3/2} \left ((b c C-a d C-6 b B d) \tan (e+f x)^2-6 (A b-C b+a B) d \tan (e+f x)+b c C-a (6 A-5 C) d\right )}{\sqrt {a+b \tan (e+f x)}}dx}{6 d}\)

\(\Big \downarrow \) 4130

\(\displaystyle \frac {C \sqrt {a+b \tan (e+f x)} (c+d \tan (e+f x))^{5/2}}{3 d f}-\frac {\frac {\int \frac {3 \sqrt {c+d \tan (e+f x)} \left (c (c C+2 B d) b^2-2 a d (4 A c-3 C c-3 B d) b-8 d (A b c+a B c-b C c+a A d-b B d-a C d) \tan (e+f x) b+a^2 C d^2-\left (8 b (A b-C b+a B) d^2-(b c-a d) (b c C-a d C-6 b B d)\right ) \tan ^2(e+f x)\right )}{2 \sqrt {a+b \tan (e+f x)}}dx}{2 b}+\frac {(-a C d-6 b B d+b c C) \sqrt {a+b \tan (e+f x)} (c+d \tan (e+f x))^{3/2}}{2 b f}}{6 d}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {C \sqrt {a+b \tan (e+f x)} (c+d \tan (e+f x))^{5/2}}{3 d f}-\frac {\frac {3 \int \frac {\sqrt {c+d \tan (e+f x)} \left (c (c C+2 B d) b^2-2 a d (4 A c-3 C c-3 B d) b-8 d (A b c+a B c-b C c+a A d-b B d-a C d) \tan (e+f x) b+a^2 C d^2-\left (8 b (A b-C b+a B) d^2-(b c-a d) (b c C-a d C-6 b B d)\right ) \tan ^2(e+f x)\right )}{\sqrt {a+b \tan (e+f x)}}dx}{4 b}+\frac {(-a C d-6 b B d+b c C) \sqrt {a+b \tan (e+f x)} (c+d \tan (e+f x))^{3/2}}{2 b f}}{6 d}\)

\(\Big \downarrow \) 3042

\(\displaystyle \frac {C \sqrt {a+b \tan (e+f x)} (c+d \tan (e+f x))^{5/2}}{3 d f}-\frac {\frac {3 \int \frac {\sqrt {c+d \tan (e+f x)} \left (c (c C+2 B d) b^2-2 a d (4 A c-3 C c-3 B d) b-8 d (A b c+a B c-b C c+a A d-b B d-a C d) \tan (e+f x) b+a^2 C d^2-\left (8 b (A b-C b+a B) d^2-(b c-a d) (b c C-a d C-6 b B d)\right ) \tan (e+f x)^2\right )}{\sqrt {a+b \tan (e+f x)}}dx}{4 b}+\frac {(-a C d-6 b B d+b c C) \sqrt {a+b \tan (e+f x)} (c+d \tan (e+f x))^{3/2}}{2 b f}}{6 d}\)

\(\Big \downarrow \) 4130

\(\displaystyle \frac {C \sqrt {a+b \tan (e+f x)} (c+d \tan (e+f x))^{5/2}}{3 d f}-\frac {\frac {3 \left (\frac {\int -\frac {-c \left (C c^2+10 B d c+8 (A-C) d^2\right ) b^3-a d \left (13 C c^2+20 B d c-8 C d^2-8 A \left (2 c^2-d^2\right )\right ) b^2+16 d \left (2 a A c d-2 a c C d+A b \left (c^2-d^2\right )+a B \left (c^2-d^2\right )-b \left (C c^2+2 B d c-C d^2\right )\right ) \tan (e+f x) b^2-a^2 d^2 (3 c C+2 B d) b+a^3 C d^3+\left (-\left (\left (C c^3-6 B d c^2-24 (A-C) d^2 c+16 B d^3\right ) b^3\right )+a d \left (3 C c^2+12 B d c+8 (A-C) d^2\right ) b^2-a^2 d^2 (3 c C+2 B d) b+a^3 C d^3\right ) \tan ^2(e+f x)}{2 \sqrt {a+b \tan (e+f x)} \sqrt {c+d \tan (e+f x)}}dx}{b}-\frac {\sqrt {a+b \tan (e+f x)} \sqrt {c+d \tan (e+f x)} \left (8 b d^2 (a B+A b-b C)-(b c-a d) (-a C d-6 b B d+b c C)\right )}{b f}\right )}{4 b}+\frac {(-a C d-6 b B d+b c C) \sqrt {a+b \tan (e+f x)} (c+d \tan (e+f x))^{3/2}}{2 b f}}{6 d}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {C \sqrt {a+b \tan (e+f x)} (c+d \tan (e+f x))^{5/2}}{3 d f}-\frac {\frac {3 \left (-\frac {\int \frac {-c \left (C c^2+10 B d c+8 (A-C) d^2\right ) b^3+a d \left (16 A c^2-13 C c^2-20 B d c-8 A d^2+8 C d^2\right ) b^2+16 d \left (2 a A c d-2 a c C d+A b \left (c^2-d^2\right )+a B \left (c^2-d^2\right )-b \left (C c^2+2 B d c-C d^2\right )\right ) \tan (e+f x) b^2-a^2 d^2 (3 c C+2 B d) b+a^3 C d^3+\left (-\left (\left (C c^3-6 B d c^2-24 (A-C) d^2 c+16 B d^3\right ) b^3\right )+a d \left (3 C c^2+12 B d c+8 (A-C) d^2\right ) b^2-a^2 d^2 (3 c C+2 B d) b+a^3 C d^3\right ) \tan ^2(e+f x)}{\sqrt {a+b \tan (e+f x)} \sqrt {c+d \tan (e+f x)}}dx}{2 b}-\frac {\sqrt {a+b \tan (e+f x)} \sqrt {c+d \tan (e+f x)} \left (8 b d^2 (a B+A b-b C)-(b c-a d) (-a C d-6 b B d+b c C)\right )}{b f}\right )}{4 b}+\frac {(-a C d-6 b B d+b c C) \sqrt {a+b \tan (e+f x)} (c+d \tan (e+f x))^{3/2}}{2 b f}}{6 d}\)

\(\Big \downarrow \) 3042

\(\displaystyle \frac {C \sqrt {a+b \tan (e+f x)} (c+d \tan (e+f x))^{5/2}}{3 d f}-\frac {\frac {3 \left (-\frac {\int \frac {-c \left (C c^2+10 B d c+8 (A-C) d^2\right ) b^3+a d \left (16 A c^2-13 C c^2-20 B d c-8 A d^2+8 C d^2\right ) b^2+16 d \left (2 a A c d-2 a c C d+A b \left (c^2-d^2\right )+a B \left (c^2-d^2\right )-b \left (C c^2+2 B d c-C d^2\right )\right ) \tan (e+f x) b^2-a^2 d^2 (3 c C+2 B d) b+a^3 C d^3+\left (-\left (\left (C c^3-6 B d c^2-24 (A-C) d^2 c+16 B d^3\right ) b^3\right )+a d \left (3 C c^2+12 B d c+8 (A-C) d^2\right ) b^2-a^2 d^2 (3 c C+2 B d) b+a^3 C d^3\right ) \tan (e+f x)^2}{\sqrt {a+b \tan (e+f x)} \sqrt {c+d \tan (e+f x)}}dx}{2 b}-\frac {\sqrt {a+b \tan (e+f x)} \sqrt {c+d \tan (e+f x)} \left (8 b d^2 (a B+A b-b C)-(b c-a d) (-a C d-6 b B d+b c C)\right )}{b f}\right )}{4 b}+\frac {(-a C d-6 b B d+b c C) \sqrt {a+b \tan (e+f x)} (c+d \tan (e+f x))^{3/2}}{2 b f}}{6 d}\)

\(\Big \downarrow \) 4138

\(\displaystyle \frac {C \sqrt {a+b \tan (e+f x)} (c+d \tan (e+f x))^{5/2}}{3 d f}-\frac {\frac {3 \left (-\frac {\int \frac {-c \left (C c^2+10 B d c+8 (A-C) d^2\right ) b^3+a d \left (16 A c^2-13 C c^2-20 B d c-8 A d^2+8 C d^2\right ) b^2+16 d \left (2 a A c d-2 a c C d+A b \left (c^2-d^2\right )+a B \left (c^2-d^2\right )-b \left (C c^2+2 B d c-C d^2\right )\right ) \tan (e+f x) b^2-a^2 d^2 (3 c C+2 B d) b+a^3 C d^3+\left (-\left (\left (C c^3-6 B d c^2-24 (A-C) d^2 c+16 B d^3\right ) b^3\right )+a d \left (3 C c^2+12 B d c+8 (A-C) d^2\right ) b^2-a^2 d^2 (3 c C+2 B d) b+a^3 C d^3\right ) \tan ^2(e+f x)}{\sqrt {a+b \tan (e+f x)} \sqrt {c+d \tan (e+f x)} \left (\tan ^2(e+f x)+1\right )}d\tan (e+f x)}{2 b f}-\frac {\sqrt {a+b \tan (e+f x)} \sqrt {c+d \tan (e+f x)} \left (8 b d^2 (a B+A b-b C)-(b c-a d) (-a C d-6 b B d+b c C)\right )}{b f}\right )}{4 b}+\frac {(-a C d-6 b B d+b c C) \sqrt {a+b \tan (e+f x)} (c+d \tan (e+f x))^{3/2}}{2 b f}}{6 d}\)

\(\Big \downarrow \) 2348

\(\displaystyle \frac {C \sqrt {a+b \tan (e+f x)} (c+d \tan (e+f x))^{5/2}}{3 d f}-\frac {\frac {(b c C-a d C-6 b B d) \sqrt {a+b \tan (e+f x)} (c+d \tan (e+f x))^{3/2}}{2 b f}+\frac {3 \left (-\frac {\sqrt {a+b \tan (e+f x)} \sqrt {c+d \tan (e+f x)} \left (8 b (A b-C b+a B) d^2-(b c-a d) (b c C-a d C-6 b B d)\right )}{b f}-\frac {\int \left (\frac {-16 B d^3 b^3+24 A c d^2 b^3-24 c C d^2 b^3-c^3 C b^3+6 B c^2 d b^3+8 a A d^3 b^2-8 a C d^3 b^2+12 a B c d^2 b^2+3 a c^2 C d b^2-2 a^2 B d^3 b-3 a^2 c C d^2 b+a^3 C d^3}{\sqrt {a+b \tan (e+f x)} \sqrt {c+d \tan (e+f x)}}+\frac {16 A d^3 b^3-16 C d^3 b^3+32 B c d^2 b^3-16 A c^2 d b^3+16 c^2 C d b^3+16 a B d^3 b^2-32 a A c d^2 b^2+32 a c C d^2 b^2-16 a B c^2 d b^2+i \left (16 B d^3 b^3-32 A c d^2 b^3+32 c C d^2 b^3-16 B c^2 d b^3-16 a A d^3 b^2+16 a C d^3 b^2-32 a B c d^2 b^2+16 a A c^2 d b^2-16 a c^2 C d b^2\right )}{2 (i-\tan (e+f x)) \sqrt {a+b \tan (e+f x)} \sqrt {c+d \tan (e+f x)}}+\frac {-16 A d^3 b^3+16 C d^3 b^3-32 B c d^2 b^3+16 A c^2 d b^3-16 c^2 C d b^3-16 a B d^3 b^2+32 a A c d^2 b^2-32 a c C d^2 b^2+16 a B c^2 d b^2+i \left (16 B d^3 b^3-32 A c d^2 b^3+32 c C d^2 b^3-16 B c^2 d b^3-16 a A d^3 b^2+16 a C d^3 b^2-32 a B c d^2 b^2+16 a A c^2 d b^2-16 a c^2 C d b^2\right )}{2 (\tan (e+f x)+i) \sqrt {a+b \tan (e+f x)} \sqrt {c+d \tan (e+f x)}}\right )d\tan (e+f x)}{2 b f}\right )}{4 b}}{6 d}\)

\(\Big \downarrow \) 2009

\(\displaystyle \frac {C \sqrt {a+b \tan (e+f x)} (c+d \tan (e+f x))^{5/2}}{3 d f}-\frac {\frac {(-a C d-6 b B d+b c C) \sqrt {a+b \tan (e+f x)} (c+d \tan (e+f x))^{3/2}}{2 b f}+\frac {3 \left (-\frac {\sqrt {a+b \tan (e+f x)} \sqrt {c+d \tan (e+f x)} \left (8 b d^2 (a B+A b-b C)-(b c-a d) (-a C d-6 b B d+b c C)\right )}{b f}-\frac {\frac {2 \left (a^3 C d^3-a^2 b d^2 (2 B d+3 c C)+a b^2 d \left (8 d^2 (A-C)+12 B c d+3 c^2 C\right )-\left (b^3 \left (-24 c d^2 (A-C)-6 B c^2 d+16 B d^3+c^3 C\right )\right )\right ) \text {arctanh}\left (\frac {\sqrt {d} \sqrt {a+b \tan (e+f x)}}{\sqrt {b} \sqrt {c+d \tan (e+f x)}}\right )}{\sqrt {b} \sqrt {d}}-16 b^2 d \sqrt {a-i b} (c-i d)^{3/2} (i A+B-i C) \text {arctanh}\left (\frac {\sqrt {c-i d} \sqrt {a+b \tan (e+f x)}}{\sqrt {a-i b} \sqrt {c+d \tan (e+f x)}}\right )-16 b^2 d \sqrt {a+i b} (c+i d)^{3/2} (B-i (A-C)) \text {arctanh}\left (\frac {\sqrt {c+i d} \sqrt {a+b \tan (e+f x)}}{\sqrt {a+i b} \sqrt {c+d \tan (e+f x)}}\right )}{2 b f}\right )}{4 b}}{6 d}\)

input

Int[Sqrt[a + b*Tan[e + f*x]]*(c + d*Tan[e + f*x])^(3/2)*(A + B*Tan[e + f*x 
] + C*Tan[e + f*x]^2),x]
 

output

(C*Sqrt[a + b*Tan[e + f*x]]*(c + d*Tan[e + f*x])^(5/2))/(3*d*f) - (((b*c*C 
 - 6*b*B*d - a*C*d)*Sqrt[a + b*Tan[e + f*x]]*(c + d*Tan[e + f*x])^(3/2))/( 
2*b*f) + (3*(-1/2*(-16*Sqrt[a - I*b]*b^2*(I*A + B - I*C)*(c - I*d)^(3/2)*d 
*ArcTanh[(Sqrt[c - I*d]*Sqrt[a + b*Tan[e + f*x]])/(Sqrt[a - I*b]*Sqrt[c + 
d*Tan[e + f*x]])] - 16*Sqrt[a + I*b]*b^2*(B - I*(A - C))*(c + I*d)^(3/2)*d 
*ArcTanh[(Sqrt[c + I*d]*Sqrt[a + b*Tan[e + f*x]])/(Sqrt[a + I*b]*Sqrt[c + 
d*Tan[e + f*x]])] + (2*(a^3*C*d^3 - a^2*b*d^2*(3*c*C + 2*B*d) + a*b^2*d*(3 
*c^2*C + 12*B*c*d + 8*(A - C)*d^2) - b^3*(c^3*C - 6*B*c^2*d - 24*c*(A - C) 
*d^2 + 16*B*d^3))*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) - ((8*b*(A*b + a*B - b*C) 
*d^2 - (b*c - a*d)*(b*c*C - 6*b*B*d - a*C*d))*Sqrt[a + b*Tan[e + f*x]]*Sqr 
t[c + d*Tan[e + f*x]])/(b*f)))/(4*b))/(6*d)
 

3.2.36.3.1 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]
 

3.2.36.4 Maple [F(-1)]

Timed out.

\[\int \sqrt {a +b \tan \left (f x +e \right )}\, \left (c +d \tan \left (f x +e \right )\right )^{\frac {3}{2}} \left (A +B \tan \left (f x +e \right )+C \tan \left (f x +e \right )^{2}\right )d x\]

input

int((a+b*tan(f*x+e))^(1/2)*(c+d*tan(f*x+e))^(3/2)*(A+B*tan(f*x+e)+C*tan(f* 
x+e)^2),x)
 

output

int((a+b*tan(f*x+e))^(1/2)*(c+d*tan(f*x+e))^(3/2)*(A+B*tan(f*x+e)+C*tan(f* 
x+e)^2),x)
 

3.2.36.5 Fricas [F(-1)]

Timed out. \[ \int \sqrt {a+b \tan (e+f x)} (c+d \tan (e+f x))^{3/2} \left (A+B \tan (e+f x)+C \tan ^2(e+f x)\right ) \, dx=\text {Timed out} \]

input

integrate((a+b*tan(f*x+e))^(1/2)*(c+d*tan(f*x+e))^(3/2)*(A+B*tan(f*x+e)+C* 
tan(f*x+e)^2),x, algorithm="fricas")
 

output

Timed out
 

3.2.36.6 Sympy [F]

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

input

integrate((a+b*tan(f*x+e))**(1/2)*(c+d*tan(f*x+e))**(3/2)*(A+B*tan(f*x+e)+ 
C*tan(f*x+e)**2),x)
 

output

Integral(sqrt(a + b*tan(e + f*x))*(c + d*tan(e + f*x))**(3/2)*(A + B*tan(e 
 + f*x) + C*tan(e + f*x)**2), x)
 

3.2.36.7 Maxima [F]

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

input

integrate((a+b*tan(f*x+e))^(1/2)*(c+d*tan(f*x+e))^(3/2)*(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)*sqrt(b*tan(f*x + e) + a) 
*(d*tan(f*x + e) + c)^(3/2), x)
 

3.2.36.8 Giac [F(-1)]

Timed out. \[ \int \sqrt {a+b \tan (e+f x)} (c+d \tan (e+f x))^{3/2} \left (A+B \tan (e+f x)+C \tan ^2(e+f x)\right ) \, dx=\text {Timed out} \]

input

integrate((a+b*tan(f*x+e))^(1/2)*(c+d*tan(f*x+e))^(3/2)*(A+B*tan(f*x+e)+C* 
tan(f*x+e)^2),x, algorithm="giac")
 

output

Timed out
 

3.2.36.9 Mupad [F(-1)]

Timed out. \[ \int \sqrt {a+b \tan (e+f x)} (c+d \tan (e+f x))^{3/2} \left (A+B \tan (e+f x)+C \tan ^2(e+f x)\right ) \, dx=\int \sqrt {a+b\,\mathrm {tan}\left (e+f\,x\right )}\,{\left (c+d\,\mathrm {tan}\left (e+f\,x\right )\right )}^{3/2}\,\left (C\,{\mathrm {tan}\left (e+f\,x\right )}^2+B\,\mathrm {tan}\left (e+f\,x\right )+A\right ) \,d x \]

input

int((a + b*tan(e + f*x))^(1/2)*(c + d*tan(e + f*x))^(3/2)*(A + B*tan(e + f 
*x) + C*tan(e + f*x)^2),x)
 

output

int((a + b*tan(e + f*x))^(1/2)*(c + d*tan(e + f*x))^(3/2)*(A + B*tan(e + f 
*x) + C*tan(e + f*x)^2), x)