3.2.29 \(\int (a+b \tan (e+f x))^{3/2} \sqrt {c+d \tan (e+f x)} (A+B \tan (e+f x)+C \tan ^2(e+f x)) \, dx\) [129]

3.2.29.1 Optimal result

Integrand size = 49, antiderivative size = 505 \[ \int (a+b \tan (e+f x))^{3/2} \sqrt {c+d \tan (e+f x)} \left (A+B \tan (e+f x)+C \tan ^2(e+f x)\right ) \, dx=-\frac {(a-i b)^{3/2} (i A+B-i C) \sqrt {c-i d} \text {arctanh}\left (\frac {\sqrt {c-i d} \sqrt {a+b \tan (e+f x)}}{\sqrt {a-i b} \sqrt {c+d \tan (e+f x)}}\right )}{f}+\frac {(a+i b)^{3/2} (i A-B-i C) \sqrt {c+i d} \text {arctanh}\left (\frac {\sqrt {c+i d} \sqrt {a+b \tan (e+f x)}}{\sqrt {a+i b} \sqrt {c+d \tan (e+f x)}}\right )}{f}-\frac {\left (a^3 C d^3-3 a^2 b d^2 (c C+2 B d)+3 a b^2 d \left (c^2 C-4 B c d-8 (A-C) d^2\right )-b^3 \left (c^3 C-2 B c^2 d+8 c (A-C) d^2-16 B d^3\right )\right ) \text {arctanh}\left (\frac {\sqrt {d} \sqrt {a+b \tan (e+f x)}}{\sqrt {b} \sqrt {c+d \tan (e+f x)}}\right )}{8 b^{3/2} d^{5/2} f}+\frac {\left (8 b (A b+a B-b C) d^2+(b c-a d) (b c C-2 b B d-a C d)\right ) \sqrt {a+b \tan (e+f x)} \sqrt {c+d \tan (e+f x)}}{8 b d^2 f}-\frac {(b c C-2 b B d-a C d) \sqrt {a+b \tan (e+f x)} (c+d \tan (e+f x))^{3/2}}{4 d^2 f}+\frac {C (a+b \tan (e+f x))^{3/2} (c+d \tan (e+f x))^{3/2}}{3 d f} \]

output

-1/8*(a^3*C*d^3-3*a^2*b*d^2*(2*B*d+C*c)+3*a*b^2*d*(c^2*C-4*B*c*d-8*(A-C)*d 
^2)-b^3*(c^3*C-2*B*c^2*d+8*c*(A-C)*d^2-16*B*d^3))*arctanh(d^(1/2)*(a+b*tan 
(f*x+e))^(1/2)/b^(1/2)/(c+d*tan(f*x+e))^(1/2))/b^(3/2)/d^(5/2)/f-(a-I*b)^( 
3/2)*(I*A+B-I*C)*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))*(c-I*d)^(1/2)/f+(a+I*b)^(3/2)*(I*A-B-I*C)*arctan 
h((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 
))*(c+I*d)^(1/2)/f+1/8*(8*b*(A*b+B*a-C*b)*d^2+(-a*d+b*c)*(-2*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/d^2/f-1/4*(-2*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)/d^2/f+1/3*C*(a 
+b*tan(f*x+e))^(3/2)*(c+d*tan(f*x+e))^(3/2)/d/f
 

3.2.29.2 Mathematica [A] (verified)

Time = 9.10 (sec) , antiderivative size = 835, normalized size of antiderivative = 1.65 \[ \int (a+b \tan (e+f x))^{3/2} \sqrt {c+d \tan (e+f x)} \left (A+B \tan (e+f x)+C \tan ^2(e+f x)\right ) \, dx=\frac {C (a+b \tan (e+f x))^{3/2} (c+d \tan (e+f x))^{3/2}}{3 d f}+\frac {-\frac {3 (b c C-2 b B d-a C d) \sqrt {a+b \tan (e+f x)} (c+d \tan (e+f x))^{3/2}}{4 d f}+\frac {\frac {3 \left (8 b (A b+a B-b C) d^2+(b c-a d) (b c C-2 b B d-a C d)\right ) \sqrt {a+b \tan (e+f x)} \sqrt {c+d \tan (e+f x)}}{4 b f}+\frac {\frac {6 b d^2 \left (\sqrt {-b^2} \left (a^2 (A c-c C-B d)-b^2 (A c-c C-B d)-2 a b (B c+(A-C) d)\right )+b \left (2 a b (A c-c C-B d)+a^2 (B c+(A-C) d)-b^2 (B c+(A-C) d)\right )\right ) \text {arctanh}\left (\frac {\sqrt {-c+\frac {\sqrt {-b^2} d}{b}} \sqrt {a+b \tan (e+f x)}}{\sqrt {-a+\sqrt {-b^2}} \sqrt {c+d \tan (e+f x)}}\right )}{\sqrt {-a+\sqrt {-b^2}} \sqrt {-c+\frac {\sqrt {-b^2} d}{b}}}+\frac {6 b d^2 \left (\sqrt {-b^2} \left (a^2 (A c-c C-B d)-b^2 (A c-c C-B d)-2 a b (B c+(A-C) d)\right )-b \left (2 a b (A c-c C-B d)+a^2 (B c+(A-C) d)-b^2 (B c+(A-C) d)\right )\right ) \text {arctanh}\left (\frac {\sqrt {c+\frac {\sqrt {-b^2} d}{b}} \sqrt {a+b \tan (e+f x)}}{\sqrt {a+\sqrt {-b^2}} \sqrt {c+d \tan (e+f x)}}\right )}{\sqrt {a+\sqrt {-b^2}} \sqrt {c+\frac {\sqrt {-b^2} d}{b}}}-\frac {3 \sqrt {b} \sqrt {c-\frac {a d}{b}} \left (a^3 C d^3-3 a^2 b d^2 (c C+2 B d)+3 a b^2 d \left (c^2 C-4 B c d-8 (A-C) d^2\right )-b^3 \left (c^3 C-2 B c^2 d+8 c (A-C) d^2-16 B d^3\right )\right ) \text {arcsinh}\left (\frac {\sqrt {d} \sqrt {a+b \tan (e+f x)}}{\sqrt {b} \sqrt {c-\frac {a d}{b}}}\right ) \sqrt {\frac {b c+b d \tan (e+f x)}{b c-a d}}}{4 \sqrt {d} \sqrt {c+d \tan (e+f x)}}}{b^2 f}}{2 d}}{3 d} \]

input

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

output

(C*(a + b*Tan[e + f*x])^(3/2)*(c + d*Tan[e + f*x])^(3/2))/(3*d*f) + ((-3*( 
b*c*C - 2*b*B*d - a*C*d)*Sqrt[a + b*Tan[e + f*x]]*(c + d*Tan[e + f*x])^(3/ 
2))/(4*d*f) + ((3*(8*b*(A*b + a*B - b*C)*d^2 + (b*c - a*d)*(b*c*C - 2*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*d^2*(Sqrt[-b^2]*(a^2*(A*c - c*C - B*d) - b^2*(A*c - c*C - B*d) - 2*a* 
b*(B*c + (A - C)*d)) + b*(2*a*b*(A*c - c*C - B*d) + a^2*(B*c + (A - C)*d) 
- b^2*(B*c + (A - C)*d)))*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*d^2*(Sqrt[-b^2]*(a^2*(A 
*c - c*C - B*d) - b^2*(A*c - c*C - B*d) - 2*a*b*(B*c + (A - C)*d)) - b*(2* 
a*b*(A*c - c*C - B*d) + a^2*(B*c + (A - C)*d) - b^2*(B*c + (A - C)*d)))*Ar 
cTanh[(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]*Sqrt[c - (a*d)/b]*(a^3*C*d^3 - 3*a^2*b*d^2*(c*C + 
 2*B*d) + 3*a*b^2*d*(c^2*C - 4*B*c*d - 8*(A - C)*d^2) - b^3*(c^3*C - 2*B*c 
^2*d + 8*c*(A - C)*d^2 - 16*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*d))/(3*d)
 

3.2.29.3 Rubi [A] (verified)

Time = 3.68 (sec) , antiderivative size = 523, normalized size of antiderivative = 1.04, 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.

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

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 4130

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 4130

\(\displaystyle \frac {C (a+b \tan (e+f x))^{3/2} (c+d \tan (e+f x))^{3/2}}{3 d f}-\frac {\frac {\int -\frac {\sqrt {c+d \tan (e+f x)} \left (c (c C-2 B d) b^2-2 a d (c C+3 B d) b+a^2 (8 A-7 C) d^2+\left (8 b (A b-C b+a B) d^2+(b c-a d) (b c C-a d C-2 b B d)\right ) \tan ^2(e+f x)+8 \left (B a^2+2 b (A-C) a-b^2 B\right ) d^2 \tan (e+f x)\right )}{2 \sqrt {a+b \tan (e+f x)}}dx}{2 d}+\frac {(-a C d-2 b B d+b c C) \sqrt {a+b \tan (e+f x)} (c+d \tan (e+f x))^{3/2}}{2 d f}}{2 d}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {C (a+b \tan (e+f x))^{3/2} (c+d \tan (e+f x))^{3/2}}{3 d f}-\frac {\frac {(-a C d-2 b B d+b c C) \sqrt {a+b \tan (e+f x)} (c+d \tan (e+f x))^{3/2}}{2 d f}-\frac {\int \frac {\sqrt {c+d \tan (e+f x)} \left (c (c C-2 B d) b^2-2 a d (c C+3 B d) b+a^2 (8 A-7 C) d^2+\left (8 b (A b-C b+a B) d^2+(b c-a d) (b c C-a d C-2 b B d)\right ) \tan ^2(e+f x)+8 \left (B a^2+2 b (A-C) a-b^2 B\right ) d^2 \tan (e+f x)\right )}{\sqrt {a+b \tan (e+f x)}}dx}{4 d}}{2 d}\)

\(\Big \downarrow \) 3042

\(\displaystyle \frac {C (a+b \tan (e+f x))^{3/2} (c+d \tan (e+f x))^{3/2}}{3 d f}-\frac {\frac {(-a C d-2 b B d+b c C) \sqrt {a+b \tan (e+f x)} (c+d \tan (e+f x))^{3/2}}{2 d f}-\frac {\int \frac {\sqrt {c+d \tan (e+f x)} \left (c (c C-2 B d) b^2-2 a d (c C+3 B d) b+a^2 (8 A-7 C) d^2+\left (8 b (A b-C b+a B) d^2+(b c-a d) (b c C-a d C-2 b B d)\right ) \tan (e+f x)^2+8 \left (B a^2+2 b (A-C) a-b^2 B\right ) d^2 \tan (e+f x)\right )}{\sqrt {a+b \tan (e+f x)}}dx}{4 d}}{2 d}\)

\(\Big \downarrow \) 4130

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 4138

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

\(\Big \downarrow \) 2348

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

\(\Big \downarrow \) 2009

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

input

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

output

(C*(a + b*Tan[e + f*x])^(3/2)*(c + d*Tan[e + f*x])^(3/2))/(3*d*f) - (((b*c 
*C - 2*b*B*d - a*C*d)*Sqrt[a + b*Tan[e + f*x]]*(c + d*Tan[e + f*x])^(3/2)) 
/(2*d*f) - (-1/2*(16*(a - I*b)^(3/2)*b*(B + I*(A - C))*Sqrt[c - I*d]*d^2*A 
rcTanh[(Sqrt[c - I*d]*Sqrt[a + b*Tan[e + f*x]])/(Sqrt[a - I*b]*Sqrt[c + d* 
Tan[e + f*x]])] - 16*(a + I*b)^(3/2)*b*(I*A - B - I*C)*Sqrt[c + I*d]*d^2*A 
rcTanh[(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 - 3*a^2*b*d^2*(c*C + 2*B*d) + 3*a*b^2*d*(c 
^2*C - 4*B*c*d - 8*(A - C)*d^2) - b^3*(c^3*C - 2*B*c^2*d + 8*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 - 2*b*B*d - a*C*d))*Sqrt[a + b*Tan[e + f*x]]*Sqrt[c 
+ d*Tan[e + f*x]])/(b*f))/(4*d))/(2*d)
 

3.2.29.3.1 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]
 

3.2.29.4 Maple [F(-1)]

Timed out.

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

input

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

output

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

3.2.29.5 Fricas [F(-1)]

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

input

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

output

Timed out
 

3.2.29.6 Sympy [F]

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

input

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

output

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

3.2.29.7 Maxima [F]

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

input

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

3.2.29.8 Giac [F(-1)]

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

input

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

output

Timed out
 

3.2.29.9 Mupad [F(-1)]

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

input

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

output

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