\(\int \frac {(c+d \tan (e+f x))^{5/2}}{(a+b \tan (e+f x))^{7/2}} \, dx\) [1298]

Optimal result

Integrand size = 29, antiderivative size = 398 \[ \int \frac {(c+d \tan (e+f x))^{5/2}}{(a+b \tan (e+f x))^{7/2}} \, dx=-\frac {i (c-i d)^{5/2} \text {arctanh}\left (\frac {\sqrt {c-i d} \sqrt {a+b \tan (e+f x)}}{\sqrt {a-i b} \sqrt {c+d \tan (e+f x)}}\right )}{(a-i b)^{7/2} f}+\frac {i (c+i d)^{5/2} \text {arctanh}\left (\frac {\sqrt {c+i d} \sqrt {a+b \tan (e+f x)}}{\sqrt {a+i b} \sqrt {c+d \tan (e+f x)}}\right )}{(a+i b)^{7/2} f}-\frac {2 (b c-a d)^2 \sqrt {c+d \tan (e+f x)}}{5 b \left (a^2+b^2\right ) f (a+b \tan (e+f x))^{5/2}}-\frac {2 (b c-a d) \left (10 a b c+a^2 d+11 b^2 d\right ) \sqrt {c+d \tan (e+f x)}}{15 b \left (a^2+b^2\right )^2 f (a+b \tan (e+f x))^{3/2}}+\frac {2 \left (20 a^3 b c d-100 a b^3 c d+2 a^4 d^2+b^4 \left (15 c^2-23 d^2\right )-3 a^2 b^2 \left (15 c^2-13 d^2\right )\right ) \sqrt {c+d \tan (e+f x)}}{15 b \left (a^2+b^2\right )^3 f \sqrt {a+b \tan (e+f x)}} \] Output:

-I*(c-I*d)^(5/2)*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)^(7/2)/f+I*(c+I*d)^(5/2)*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) 
^(7/2)/f-2/5*(-a*d+b*c)^2*(c+d*tan(f*x+e))^(1/2)/b/(a^2+b^2)/f/(a+b*tan(f* 
x+e))^(5/2)-2/15*(-a*d+b*c)*(a^2*d+10*a*b*c+11*b^2*d)*(c+d*tan(f*x+e))^(1/ 
2)/b/(a^2+b^2)^2/f/(a+b*tan(f*x+e))^(3/2)+2/15*(20*a^3*b*c*d-100*a*b^3*c*d 
+2*a^4*d^2+b^4*(15*c^2-23*d^2)-3*a^2*b^2*(15*c^2-13*d^2))*(c+d*tan(f*x+e)) 
^(1/2)/b/(a^2+b^2)^3/f/(a+b*tan(f*x+e))^(1/2)
 

Mathematica [A] (verified)

Time = 4.36 (sec) , antiderivative size = 440, normalized size of antiderivative = 1.11 \[ \int \frac {(c+d \tan (e+f x))^{5/2}}{(a+b \tan (e+f x))^{7/2}} \, dx=\frac {\frac {3 (c+d \tan (e+f x))^{5/2}}{(i a-b) (a+b \tan (e+f x))^{5/2}}-\frac {3 (c+d \tan (e+f x))^{5/2}}{(i a+b) (a+b \tan (e+f x))^{5/2}}-\frac {5 (i c-d) \left (-\frac {3 (c+i d)^{3/2} \text {arctanh}\left (\frac {\sqrt {c+i d} \sqrt {a+b \tan (e+f x)}}{\sqrt {a+i b} \sqrt {c+d \tan (e+f x)}}\right )}{(a+i b)^{5/2}}+\frac {\sqrt {c+d \tan (e+f x)} (4 a c+i b c+3 i a d+(3 b c+a d+4 i b d) \tan (e+f x))}{(a+i b)^2 (a+b \tan (e+f x))^{3/2}}\right )}{a+i b}+\frac {5 (i c+d) \left (\frac {(c+d \tan (e+f x))^{3/2}}{(a+b \tan (e+f x))^{3/2}}+3 (c-i d) \left (\frac {\sqrt {-c+i d} \text {arctanh}\left (\frac {\sqrt {-c+i d} \sqrt {a+b \tan (e+f x)}}{\sqrt {-a+i b} \sqrt {c+d \tan (e+f x)}}\right )}{(-a+i b)^{3/2}}+\frac {\sqrt {c+d \tan (e+f x)}}{(a-i b) \sqrt {a+b \tan (e+f x)}}\right )\right )}{(a-i b)^2}}{15 f} \] Input:

Integrate[(c + d*Tan[e + f*x])^(5/2)/(a + b*Tan[e + f*x])^(7/2),x]
 

Output:

((3*(c + d*Tan[e + f*x])^(5/2))/((I*a - b)*(a + b*Tan[e + f*x])^(5/2)) - ( 
3*(c + d*Tan[e + f*x])^(5/2))/((I*a + b)*(a + b*Tan[e + f*x])^(5/2)) - (5* 
(I*c - d)*((-3*(c + I*d)^(3/2)*ArcTanh[(Sqrt[c + I*d]*Sqrt[a + b*Tan[e + f 
*x]])/(Sqrt[a + I*b]*Sqrt[c + d*Tan[e + f*x]])])/(a + I*b)^(5/2) + (Sqrt[c 
 + d*Tan[e + f*x]]*(4*a*c + I*b*c + (3*I)*a*d + (3*b*c + a*d + (4*I)*b*d)* 
Tan[e + f*x]))/((a + I*b)^2*(a + b*Tan[e + f*x])^(3/2))))/(a + I*b) + (5*( 
I*c + d)*((c + d*Tan[e + f*x])^(3/2)/(a + b*Tan[e + f*x])^(3/2) + 3*(c - I 
*d)*((Sqrt[-c + I*d]*ArcTanh[(Sqrt[-c + I*d]*Sqrt[a + b*Tan[e + f*x]])/(Sq 
rt[-a + I*b]*Sqrt[c + d*Tan[e + f*x]])])/(-a + I*b)^(3/2) + Sqrt[c + d*Tan 
[e + f*x]]/((a - I*b)*Sqrt[a + b*Tan[e + f*x]]))))/(a - I*b)^2)/(15*f)
 

Rubi [A] (verified)

Time = 3.29 (sec) , antiderivative size = 501, normalized size of antiderivative = 1.26, number of steps used = 16, number of rules used = 15, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.517, Rules used = {3042, 4048, 27, 3042, 4132, 27, 3042, 4132, 27, 3042, 4099, 3042, 4098, 104, 221}

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.

\(\displaystyle \int \frac {(c+d \tan (e+f x))^{5/2}}{(a+b \tan (e+f x))^{7/2}} \, dx\)

\(\Big \downarrow \) 3042

\(\displaystyle \int \frac {(c+d \tan (e+f x))^{5/2}}{(a+b \tan (e+f x))^{7/2}}dx\)

\(\Big \downarrow \) 4048

\(\displaystyle \frac {2 \int \frac {a^2 d^3+11 b^2 c^2 d+\left (\left (a^2+5 b^2\right ) d^2-4 b c (b c-2 a d)\right ) \tan ^2(e+f x) d+a b c \left (5 c^2-7 d^2\right )+5 b \left (a d \left (3 c^2-d^2\right )-b \left (c^3-3 c d^2\right )\right ) \tan (e+f x)}{2 (a+b \tan (e+f x))^{5/2} \sqrt {c+d \tan (e+f x)}}dx}{5 b \left (a^2+b^2\right )}-\frac {2 (b c-a d)^2 \sqrt {c+d \tan (e+f x)}}{5 b f \left (a^2+b^2\right ) (a+b \tan (e+f x))^{5/2}}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {\int \frac {a^2 d^3+11 b^2 c^2 d+\left (\left (a^2+5 b^2\right ) d^2-4 b c (b c-2 a d)\right ) \tan ^2(e+f x) d+a b c \left (5 c^2-7 d^2\right )-5 b \left (b c^3-3 a d c^2-3 b d^2 c+a d^3\right ) \tan (e+f x)}{(a+b \tan (e+f x))^{5/2} \sqrt {c+d \tan (e+f x)}}dx}{5 b \left (a^2+b^2\right )}-\frac {2 (b c-a d)^2 \sqrt {c+d \tan (e+f x)}}{5 b f \left (a^2+b^2\right ) (a+b \tan (e+f x))^{5/2}}\)

\(\Big \downarrow \) 3042

\(\displaystyle \frac {\int \frac {a^2 d^3+11 b^2 c^2 d+\left (\left (a^2+5 b^2\right ) d^2-4 b c (b c-2 a d)\right ) \tan (e+f x)^2 d+a b c \left (5 c^2-7 d^2\right )-5 b \left (b c^3-3 a d c^2-3 b d^2 c+a d^3\right ) \tan (e+f x)}{(a+b \tan (e+f x))^{5/2} \sqrt {c+d \tan (e+f x)}}dx}{5 b \left (a^2+b^2\right )}-\frac {2 (b c-a d)^2 \sqrt {c+d \tan (e+f x)}}{5 b f \left (a^2+b^2\right ) (a+b \tan (e+f x))^{5/2}}\)

\(\Big \downarrow \) 4132

\(\displaystyle \frac {-\frac {2 \int -\frac {-2 d (b c-a d)^2 \left (d a^2+10 b c a+11 b^2 d\right ) \tan ^2(e+f x)-15 b (b c-a d) \left (-\left (\left (3 c^2 d-d^3\right ) a^2\right )+2 b c \left (c^2-3 d^2\right ) a+b^2 d \left (3 c^2-d^2\right )\right ) \tan (e+f x)+(b c-a d) \left (-c \left (15 c^2-23 d^2\right ) b^3+2 a d \left (35 c^2-4 d^2\right ) b^2+3 a^2 c \left (5 c^2-9 d^2\right ) b+2 a^3 d^3\right )}{2 (a+b \tan (e+f x))^{3/2} \sqrt {c+d \tan (e+f x)}}dx}{3 \left (a^2+b^2\right ) (b c-a d)}-\frac {2 (b c-a d) \left (a^2 d+10 a b c+11 b^2 d\right ) \sqrt {c+d \tan (e+f x)}}{3 f \left (a^2+b^2\right ) (a+b \tan (e+f x))^{3/2}}}{5 b \left (a^2+b^2\right )}-\frac {2 (b c-a d)^2 \sqrt {c+d \tan (e+f x)}}{5 b f \left (a^2+b^2\right ) (a+b \tan (e+f x))^{5/2}}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {\frac {\int \frac {-2 d (b c-a d)^2 \left (d a^2+10 b c a+11 b^2 d\right ) \tan ^2(e+f x)-15 b (b c-a d) \left (-\left (\left (3 c^2 d-d^3\right ) a^2\right )+2 b c \left (c^2-3 d^2\right ) a+b^2 d \left (3 c^2-d^2\right )\right ) \tan (e+f x)+(b c-a d) \left (-c \left (15 c^2-23 d^2\right ) b^3+2 a d \left (35 c^2-4 d^2\right ) b^2+3 a^2 c \left (5 c^2-9 d^2\right ) b+2 a^3 d^3\right )}{(a+b \tan (e+f x))^{3/2} \sqrt {c+d \tan (e+f x)}}dx}{3 \left (a^2+b^2\right ) (b c-a d)}-\frac {2 (b c-a d) \left (a^2 d+10 a b c+11 b^2 d\right ) \sqrt {c+d \tan (e+f x)}}{3 f \left (a^2+b^2\right ) (a+b \tan (e+f x))^{3/2}}}{5 b \left (a^2+b^2\right )}-\frac {2 (b c-a d)^2 \sqrt {c+d \tan (e+f x)}}{5 b f \left (a^2+b^2\right ) (a+b \tan (e+f x))^{5/2}}\)

\(\Big \downarrow \) 3042

\(\displaystyle \frac {\frac {\int \frac {-2 d (b c-a d)^2 \left (d a^2+10 b c a+11 b^2 d\right ) \tan (e+f x)^2-15 b (b c-a d) \left (-\left (\left (3 c^2 d-d^3\right ) a^2\right )+2 b c \left (c^2-3 d^2\right ) a+b^2 d \left (3 c^2-d^2\right )\right ) \tan (e+f x)+(b c-a d) \left (-c \left (15 c^2-23 d^2\right ) b^3+2 a d \left (35 c^2-4 d^2\right ) b^2+3 a^2 c \left (5 c^2-9 d^2\right ) b+2 a^3 d^3\right )}{(a+b \tan (e+f x))^{3/2} \sqrt {c+d \tan (e+f x)}}dx}{3 \left (a^2+b^2\right ) (b c-a d)}-\frac {2 (b c-a d) \left (a^2 d+10 a b c+11 b^2 d\right ) \sqrt {c+d \tan (e+f x)}}{3 f \left (a^2+b^2\right ) (a+b \tan (e+f x))^{3/2}}}{5 b \left (a^2+b^2\right )}-\frac {2 (b c-a d)^2 \sqrt {c+d \tan (e+f x)}}{5 b f \left (a^2+b^2\right ) (a+b \tan (e+f x))^{5/2}}\)

\(\Big \downarrow \) 4132

\(\displaystyle \frac {\frac {\frac {2 (b c-a d) \left (2 a^4 d^2+20 a^3 b c d-3 a^2 b^2 \left (15 c^2-13 d^2\right )-100 a b^3 c d+b^4 \left (15 c^2-23 d^2\right )\right ) \sqrt {c+d \tan (e+f x)}}{f \left (a^2+b^2\right ) \sqrt {a+b \tan (e+f x)}}-\frac {2 \int -\frac {15 \left (b (b c-a d)^2 (a c+b d) \left (\left (c^2-3 d^2\right ) a^2+8 b c d a-b^2 \left (3 c^2-d^2\right )\right )-b (b c-a d)^3 \left (\left (3 c^2-d^2\right ) a^2+8 b c d a-b^2 \left (c^2-3 d^2\right )\right ) \tan (e+f x)\right )}{2 \sqrt {a+b \tan (e+f x)} \sqrt {c+d \tan (e+f x)}}dx}{\left (a^2+b^2\right ) (b c-a d)}}{3 \left (a^2+b^2\right ) (b c-a d)}-\frac {2 (b c-a d) \left (a^2 d+10 a b c+11 b^2 d\right ) \sqrt {c+d \tan (e+f x)}}{3 f \left (a^2+b^2\right ) (a+b \tan (e+f x))^{3/2}}}{5 b \left (a^2+b^2\right )}-\frac {2 (b c-a d)^2 \sqrt {c+d \tan (e+f x)}}{5 b f \left (a^2+b^2\right ) (a+b \tan (e+f x))^{5/2}}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {\frac {\frac {15 \int \frac {b (b c-a d)^2 (a c+b d) \left (\left (c^2-3 d^2\right ) a^2+8 b c d a-b^2 \left (3 c^2-d^2\right )\right )-b (b c-a d)^3 \left (\left (3 c^2-d^2\right ) a^2+8 b c d a-b^2 \left (c^2-3 d^2\right )\right ) \tan (e+f x)}{\sqrt {a+b \tan (e+f x)} \sqrt {c+d \tan (e+f x)}}dx}{\left (a^2+b^2\right ) (b c-a d)}+\frac {2 (b c-a d) \left (2 a^4 d^2+20 a^3 b c d-3 a^2 b^2 \left (15 c^2-13 d^2\right )-100 a b^3 c d+b^4 \left (15 c^2-23 d^2\right )\right ) \sqrt {c+d \tan (e+f x)}}{f \left (a^2+b^2\right ) \sqrt {a+b \tan (e+f x)}}}{3 \left (a^2+b^2\right ) (b c-a d)}-\frac {2 (b c-a d) \left (a^2 d+10 a b c+11 b^2 d\right ) \sqrt {c+d \tan (e+f x)}}{3 f \left (a^2+b^2\right ) (a+b \tan (e+f x))^{3/2}}}{5 b \left (a^2+b^2\right )}-\frac {2 (b c-a d)^2 \sqrt {c+d \tan (e+f x)}}{5 b f \left (a^2+b^2\right ) (a+b \tan (e+f x))^{5/2}}\)

\(\Big \downarrow \) 3042

\(\displaystyle \frac {\frac {\frac {15 \int \frac {b (b c-a d)^2 (a c+b d) \left (\left (c^2-3 d^2\right ) a^2+8 b c d a-b^2 \left (3 c^2-d^2\right )\right )-b (b c-a d)^3 \left (\left (3 c^2-d^2\right ) a^2+8 b c d a-b^2 \left (c^2-3 d^2\right )\right ) \tan (e+f x)}{\sqrt {a+b \tan (e+f x)} \sqrt {c+d \tan (e+f x)}}dx}{\left (a^2+b^2\right ) (b c-a d)}+\frac {2 (b c-a d) \left (2 a^4 d^2+20 a^3 b c d-3 a^2 b^2 \left (15 c^2-13 d^2\right )-100 a b^3 c d+b^4 \left (15 c^2-23 d^2\right )\right ) \sqrt {c+d \tan (e+f x)}}{f \left (a^2+b^2\right ) \sqrt {a+b \tan (e+f x)}}}{3 \left (a^2+b^2\right ) (b c-a d)}-\frac {2 (b c-a d) \left (a^2 d+10 a b c+11 b^2 d\right ) \sqrt {c+d \tan (e+f x)}}{3 f \left (a^2+b^2\right ) (a+b \tan (e+f x))^{3/2}}}{5 b \left (a^2+b^2\right )}-\frac {2 (b c-a d)^2 \sqrt {c+d \tan (e+f x)}}{5 b f \left (a^2+b^2\right ) (a+b \tan (e+f x))^{5/2}}\)

\(\Big \downarrow \) 4099

\(\displaystyle -\frac {2 (b c-a d)^2 \sqrt {c+d \tan (e+f x)}}{5 b f \left (a^2+b^2\right ) (a+b \tan (e+f x))^{5/2}}+\frac {-\frac {2 (b c-a d) \left (a^2 d+10 a b c+11 b^2 d\right ) \sqrt {c+d \tan (e+f x)}}{3 f \left (a^2+b^2\right ) (a+b \tan (e+f x))^{3/2}}+\frac {\frac {2 (b c-a d) \left (2 a^4 d^2+20 a^3 b c d-3 a^2 b^2 \left (15 c^2-13 d^2\right )-100 a b^3 c d+b^4 \left (15 c^2-23 d^2\right )\right ) \sqrt {c+d \tan (e+f x)}}{f \left (a^2+b^2\right ) \sqrt {a+b \tan (e+f x)}}+\frac {15 \left (\frac {1}{2} b (a+i b)^3 (c-i d)^3 (b c-a d)^2 \int \frac {i \tan (e+f x)+1}{\sqrt {a+b \tan (e+f x)} \sqrt {c+d \tan (e+f x)}}dx+\frac {1}{2} b (a-i b)^3 (c+i d)^3 (b c-a d)^2 \int \frac {1-i \tan (e+f x)}{\sqrt {a+b \tan (e+f x)} \sqrt {c+d \tan (e+f x)}}dx\right )}{\left (a^2+b^2\right ) (b c-a d)}}{3 \left (a^2+b^2\right ) (b c-a d)}}{5 b \left (a^2+b^2\right )}\)

\(\Big \downarrow \) 3042

\(\displaystyle -\frac {2 (b c-a d)^2 \sqrt {c+d \tan (e+f x)}}{5 b f \left (a^2+b^2\right ) (a+b \tan (e+f x))^{5/2}}+\frac {-\frac {2 (b c-a d) \left (a^2 d+10 a b c+11 b^2 d\right ) \sqrt {c+d \tan (e+f x)}}{3 f \left (a^2+b^2\right ) (a+b \tan (e+f x))^{3/2}}+\frac {\frac {2 (b c-a d) \left (2 a^4 d^2+20 a^3 b c d-3 a^2 b^2 \left (15 c^2-13 d^2\right )-100 a b^3 c d+b^4 \left (15 c^2-23 d^2\right )\right ) \sqrt {c+d \tan (e+f x)}}{f \left (a^2+b^2\right ) \sqrt {a+b \tan (e+f x)}}+\frac {15 \left (\frac {1}{2} b (a+i b)^3 (c-i d)^3 (b c-a d)^2 \int \frac {i \tan (e+f x)+1}{\sqrt {a+b \tan (e+f x)} \sqrt {c+d \tan (e+f x)}}dx+\frac {1}{2} b (a-i b)^3 (c+i d)^3 (b c-a d)^2 \int \frac {1-i \tan (e+f x)}{\sqrt {a+b \tan (e+f x)} \sqrt {c+d \tan (e+f x)}}dx\right )}{\left (a^2+b^2\right ) (b c-a d)}}{3 \left (a^2+b^2\right ) (b c-a d)}}{5 b \left (a^2+b^2\right )}\)

\(\Big \downarrow \) 4098

\(\displaystyle -\frac {2 (b c-a d)^2 \sqrt {c+d \tan (e+f x)}}{5 b f \left (a^2+b^2\right ) (a+b \tan (e+f x))^{5/2}}+\frac {-\frac {2 (b c-a d) \left (a^2 d+10 a b c+11 b^2 d\right ) \sqrt {c+d \tan (e+f x)}}{3 f \left (a^2+b^2\right ) (a+b \tan (e+f x))^{3/2}}+\frac {\frac {2 (b c-a d) \left (2 a^4 d^2+20 a^3 b c d-3 a^2 b^2 \left (15 c^2-13 d^2\right )-100 a b^3 c d+b^4 \left (15 c^2-23 d^2\right )\right ) \sqrt {c+d \tan (e+f x)}}{f \left (a^2+b^2\right ) \sqrt {a+b \tan (e+f x)}}+\frac {15 \left (\frac {b (a+i b)^3 (c-i d)^3 (b c-a d)^2 \int \frac {1}{(1-i \tan (e+f x)) \sqrt {a+b \tan (e+f x)} \sqrt {c+d \tan (e+f x)}}d\tan (e+f x)}{2 f}+\frac {b (a-i b)^3 (c+i d)^3 (b c-a d)^2 \int \frac {1}{(i \tan (e+f x)+1) \sqrt {a+b \tan (e+f x)} \sqrt {c+d \tan (e+f x)}}d\tan (e+f x)}{2 f}\right )}{\left (a^2+b^2\right ) (b c-a d)}}{3 \left (a^2+b^2\right ) (b c-a d)}}{5 b \left (a^2+b^2\right )}\)

\(\Big \downarrow \) 104

\(\displaystyle -\frac {2 (b c-a d)^2 \sqrt {c+d \tan (e+f x)}}{5 b f \left (a^2+b^2\right ) (a+b \tan (e+f x))^{5/2}}+\frac {-\frac {2 (b c-a d) \left (a^2 d+10 a b c+11 b^2 d\right ) \sqrt {c+d \tan (e+f x)}}{3 f \left (a^2+b^2\right ) (a+b \tan (e+f x))^{3/2}}+\frac {\frac {2 (b c-a d) \left (2 a^4 d^2+20 a^3 b c d-3 a^2 b^2 \left (15 c^2-13 d^2\right )-100 a b^3 c d+b^4 \left (15 c^2-23 d^2\right )\right ) \sqrt {c+d \tan (e+f x)}}{f \left (a^2+b^2\right ) \sqrt {a+b \tan (e+f x)}}+\frac {15 \left (\frac {b (a+i b)^3 (c-i d)^3 (b c-a d)^2 \int \frac {1}{i a+b-\frac {(i c+d) (a+b \tan (e+f x))}{c+d \tan (e+f x)}}d\frac {\sqrt {a+b \tan (e+f x)}}{\sqrt {c+d \tan (e+f x)}}}{f}+\frac {b (a-i b)^3 (c+i d)^3 (b c-a d)^2 \int \frac {1}{-i a+b+\frac {(i c-d) (a+b \tan (e+f x))}{c+d \tan (e+f x)}}d\frac {\sqrt {a+b \tan (e+f x)}}{\sqrt {c+d \tan (e+f x)}}}{f}\right )}{\left (a^2+b^2\right ) (b c-a d)}}{3 \left (a^2+b^2\right ) (b c-a d)}}{5 b \left (a^2+b^2\right )}\)

\(\Big \downarrow \) 221

\(\displaystyle -\frac {2 (b c-a d)^2 \sqrt {c+d \tan (e+f x)}}{5 b f \left (a^2+b^2\right ) (a+b \tan (e+f x))^{5/2}}+\frac {-\frac {2 (b c-a d) \left (a^2 d+10 a b c+11 b^2 d\right ) \sqrt {c+d \tan (e+f x)}}{3 f \left (a^2+b^2\right ) (a+b \tan (e+f x))^{3/2}}+\frac {\frac {2 (b c-a d) \left (2 a^4 d^2+20 a^3 b c d-3 a^2 b^2 \left (15 c^2-13 d^2\right )-100 a b^3 c d+b^4 \left (15 c^2-23 d^2\right )\right ) \sqrt {c+d \tan (e+f x)}}{f \left (a^2+b^2\right ) \sqrt {a+b \tan (e+f x)}}+\frac {15 \left (\frac {i b (a-i b)^3 (c+i d)^{5/2} (b c-a d)^2 \text {arctanh}\left (\frac {\sqrt {c+i d} \sqrt {a+b \tan (e+f x)}}{\sqrt {a+i b} \sqrt {c+d \tan (e+f x)}}\right )}{f \sqrt {a+i b}}-\frac {i b (a+i b)^3 (c-i d)^{5/2} (b c-a d)^2 \text {arctanh}\left (\frac {\sqrt {c-i d} \sqrt {a+b \tan (e+f x)}}{\sqrt {a-i b} \sqrt {c+d \tan (e+f x)}}\right )}{f \sqrt {a-i b}}\right )}{\left (a^2+b^2\right ) (b c-a d)}}{3 \left (a^2+b^2\right ) (b c-a d)}}{5 b \left (a^2+b^2\right )}\)

Input:

Int[(c + d*Tan[e + f*x])^(5/2)/(a + b*Tan[e + f*x])^(7/2),x]
 

Output:

(-2*(b*c - a*d)^2*Sqrt[c + d*Tan[e + f*x]])/(5*b*(a^2 + b^2)*f*(a + b*Tan[ 
e + f*x])^(5/2)) + ((-2*(b*c - a*d)*(10*a*b*c + a^2*d + 11*b^2*d)*Sqrt[c + 
 d*Tan[e + f*x]])/(3*(a^2 + b^2)*f*(a + b*Tan[e + f*x])^(3/2)) + ((15*(((- 
I)*(a + I*b)^3*b*(c - I*d)^(5/2)*(b*c - a*d)^2*ArcTanh[(Sqrt[c - I*d]*Sqrt 
[a + b*Tan[e + f*x]])/(Sqrt[a - I*b]*Sqrt[c + d*Tan[e + f*x]])])/(Sqrt[a - 
 I*b]*f) + (I*(a - I*b)^3*b*(c + I*d)^(5/2)*(b*c - a*d)^2*ArcTanh[(Sqrt[c 
+ I*d]*Sqrt[a + b*Tan[e + f*x]])/(Sqrt[a + I*b]*Sqrt[c + d*Tan[e + f*x]])] 
)/(Sqrt[a + I*b]*f)))/((a^2 + b^2)*(b*c - a*d)) + (2*(b*c - a*d)*(20*a^3*b 
*c*d - 100*a*b^3*c*d + 2*a^4*d^2 + b^4*(15*c^2 - 23*d^2) - 3*a^2*b^2*(15*c 
^2 - 13*d^2))*Sqrt[c + d*Tan[e + f*x]])/((a^2 + b^2)*f*Sqrt[a + b*Tan[e + 
f*x]]))/(3*(a^2 + b^2)*(b*c - a*d)))/(5*b*(a^2 + b^2))
 

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 104

Int[(((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_))/((e_.) + (f_.)*(x 
_)), x_] :> With[{q = Denominator[m]}, Simp[q   Subst[Int[x^(q*(m + 1) - 1) 
/(b*e - a*f - (d*e - c*f)*x^q), x], x, (a + b*x)^(1/q)/(c + d*x)^(1/q)], x] 
] /; FreeQ[{a, b, c, d, e, f}, x] && EqQ[m + n + 1, 0] && RationalQ[n] && L 
tQ[-1, m, 0] && SimplerQ[a + b*x, c + d*x]
 

rule 221

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(Rt[-a/b, 2]/a)*ArcTanh[x 
/Rt[-a/b, 2]], x] /; FreeQ[{a, b}, x] && NegQ[a/b]
 

rule 3042

Int[u_, x_Symbol] :> Int[DeactivateTrig[u, x], x] /; FunctionOfTrigOfLinear 
Q[u, x]
 

rule 4048

Int[((a_.) + (b_.)*tan[(e_.) + (f_.)*(x_)])^(m_)*((c_.) + (d_.)*tan[(e_.) + 
 (f_.)*(x_)])^(n_), x_Symbol] :> Simp[(b*c - a*d)^2*(a + b*Tan[e + f*x])^(m 
 - 2)*((c + d*Tan[e + f*x])^(n + 1)/(d*f*(n + 1)*(c^2 + d^2))), x] - Simp[1 
/(d*(n + 1)*(c^2 + d^2))   Int[(a + b*Tan[e + f*x])^(m - 3)*(c + d*Tan[e + 
f*x])^(n + 1)*Simp[a^2*d*(b*d*(m - 2) - a*c*(n + 1)) + b*(b*c - 2*a*d)*(b*c 
*(m - 2) + a*d*(n + 1)) - d*(n + 1)*(3*a^2*b*c - b^3*c - a^3*d + 3*a*b^2*d) 
*Tan[e + f*x] - b*(a*d*(2*b*c - a*d)*(m + n - 1) - b^2*(c^2*(m - 2) - d^2*( 
n + 1)))*Tan[e + f*x]^2, x], x], x] /; FreeQ[{a, b, c, d, e, f}, x] && NeQ[ 
b*c - a*d, 0] && NeQ[a^2 + b^2, 0] && NeQ[c^2 + d^2, 0] && GtQ[m, 2] && LtQ 
[n, -1] && IntegerQ[2*m]
 

rule 4098

Int[((a_.) + (b_.)*tan[(e_.) + (f_.)*(x_)])^(m_)*((A_.) + (B_.)*tan[(e_.) + 
 (f_.)*(x_)])*((c_.) + (d_.)*tan[(e_.) + (f_.)*(x_)])^(n_), x_Symbol] :> Si 
mp[A^2/f   Subst[Int[(a + b*x)^m*((c + d*x)^n/(A - B*x)), x], x, Tan[e + f* 
x]], x] /; FreeQ[{a, b, c, d, e, f, A, B, m, n}, x] && NeQ[b*c - a*d, 0] && 
 NeQ[a^2 + b^2, 0] && EqQ[A^2 + B^2, 0]
 

rule 4099

Int[((a_.) + (b_.)*tan[(e_.) + (f_.)*(x_)])^(m_)*((A_.) + (B_.)*tan[(e_.) + 
 (f_.)*(x_)])*((c_.) + (d_.)*tan[(e_.) + (f_.)*(x_)])^(n_), x_Symbol] :> Si 
mp[(A + I*B)/2   Int[(a + b*Tan[e + f*x])^m*(c + d*Tan[e + f*x])^n*(1 - I*T 
an[e + f*x]), x], x] + Simp[(A - I*B)/2   Int[(a + b*Tan[e + f*x])^m*(c + d 
*Tan[e + f*x])^n*(1 + I*Tan[e + f*x]), x], x] /; FreeQ[{a, b, c, d, e, f, A 
, B, m, n}, x] && NeQ[b*c - a*d, 0] && NeQ[a^2 + b^2, 0] && NeQ[A^2 + B^2, 
0]
 

rule 4132

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[(A*b^2 - a*(b*B - a*C))*(a + b*Tan[e + 
 f*x])^(m + 1)*((c + d*Tan[e + f*x])^(n + 1)/(f*(m + 1)*(b*c - a*d)*(a^2 + 
b^2))), x] + Simp[1/((m + 1)*(b*c - a*d)*(a^2 + b^2))   Int[(a + b*Tan[e + 
f*x])^(m + 1)*(c + d*Tan[e + f*x])^n*Simp[A*(a*(b*c - a*d)*(m + 1) - b^2*d* 
(m + n + 2)) + (b*B - a*C)*(b*c*(m + 1) + a*d*(n + 1)) - (m + 1)*(b*c - a*d 
)*(A*b - a*B - b*C)*Tan[e + f*x] - d*(A*b^2 - a*(b*B - a*C))*(m + n + 2)*Ta 
n[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] && LtQ[m, -1] && 
!(ILtQ[n, -1] && ( !IntegerQ[m] || (EqQ[c, 0] && NeQ[a, 0])))
 

Maple [F(-1)]

Timed out.

\[\int \frac {\left (c +d \tan \left (f x +e \right )\right )^{\frac {5}{2}}}{\left (a +b \tan \left (f x +e \right )\right )^{\frac {7}{2}}}d x\]

Input:

int((c+d*tan(f*x+e))^(5/2)/(a+b*tan(f*x+e))^(7/2),x)
 

Output:

int((c+d*tan(f*x+e))^(5/2)/(a+b*tan(f*x+e))^(7/2),x)
 

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 38721 vs. \(2 (336) = 672\).

Time = 60.96 (sec) , antiderivative size = 38721, normalized size of antiderivative = 97.29 \[ \int \frac {(c+d \tan (e+f x))^{5/2}}{(a+b \tan (e+f x))^{7/2}} \, dx=\text {Too large to display} \] Input:

integrate((c+d*tan(f*x+e))^(5/2)/(a+b*tan(f*x+e))^(7/2),x, algorithm="fric 
as")
 

Output:

Too large to include
 

Sympy [F]

\[ \int \frac {(c+d \tan (e+f x))^{5/2}}{(a+b \tan (e+f x))^{7/2}} \, dx=\int \frac {\left (c + d \tan {\left (e + f x \right )}\right )^{\frac {5}{2}}}{\left (a + b \tan {\left (e + f x \right )}\right )^{\frac {7}{2}}}\, dx \] Input:

integrate((c+d*tan(f*x+e))**(5/2)/(a+b*tan(f*x+e))**(7/2),x)
 

Output:

Integral((c + d*tan(e + f*x))**(5/2)/(a + b*tan(e + f*x))**(7/2), x)
 

Maxima [F(-2)]

Exception generated. \[ \int \frac {(c+d \tan (e+f x))^{5/2}}{(a+b \tan (e+f x))^{7/2}} \, dx=\text {Exception raised: ValueError} \] Input:

integrate((c+d*tan(f*x+e))^(5/2)/(a+b*tan(f*x+e))^(7/2),x, algorithm="maxi 
ma")
 

Output:

Exception raised: ValueError >> Computation failed since Maxima requested 
additional constraints; using the 'assume' command before evaluation *may* 
 help (example of legal syntax is 'assume(((2*b*d+2*a*c)^2>0)', see `assum 
e?` for mo
 

Giac [F]

\[ \int \frac {(c+d \tan (e+f x))^{5/2}}{(a+b \tan (e+f x))^{7/2}} \, dx=\int { \frac {{\left (d \tan \left (f x + e\right ) + c\right )}^{\frac {5}{2}}}{{\left (b \tan \left (f x + e\right ) + a\right )}^{\frac {7}{2}}} \,d x } \] Input:

integrate((c+d*tan(f*x+e))^(5/2)/(a+b*tan(f*x+e))^(7/2),x, algorithm="giac 
")
 

Output:

integrate((d*tan(f*x + e) + c)^(5/2)/(b*tan(f*x + e) + a)^(7/2), x)
 

Mupad [F(-1)]

Timed out. \[ \int \frac {(c+d \tan (e+f x))^{5/2}}{(a+b \tan (e+f x))^{7/2}} \, dx=\int \frac {{\left (c+d\,\mathrm {tan}\left (e+f\,x\right )\right )}^{5/2}}{{\left (a+b\,\mathrm {tan}\left (e+f\,x\right )\right )}^{7/2}} \,d x \] Input:

int((c + d*tan(e + f*x))^(5/2)/(a + b*tan(e + f*x))^(7/2),x)
 

Output:

int((c + d*tan(e + f*x))^(5/2)/(a + b*tan(e + f*x))^(7/2), x)
 

Reduce [F]

\[ \int \frac {(c+d \tan (e+f x))^{5/2}}{(a+b \tan (e+f x))^{7/2}} \, dx=\text {too large to display} \] Input:

int((c+d*tan(f*x+e))^(5/2)/(a+b*tan(f*x+e))^(7/2),x)
 

Output:

(4*sqrt(tan(e + f*x)*d + c)*sqrt(tan(e + f*x)*b + a)*tan(e + f*x)**2*a*b*d 
**5 + 28*sqrt(tan(e + f*x)*d + c)*sqrt(tan(e + f*x)*b + a)*tan(e + f*x)**2 
*b**2*c*d**4 + 10*sqrt(tan(e + f*x)*d + c)*sqrt(tan(e + f*x)*b + a)*tan(e 
+ f*x)*a**2*d**5 + 68*sqrt(tan(e + f*x)*d + c)*sqrt(tan(e + f*x)*b + a)*ta 
n(e + f*x)*a*b*c*d**4 - 14*sqrt(tan(e + f*x)*d + c)*sqrt(tan(e + f*x)*b + 
a)*tan(e + f*x)*b**2*c**2*d**3 + 70*sqrt(tan(e + f*x)*d + c)*sqrt(tan(e + 
f*x)*b + a)*a**2*c*d**4 - 56*sqrt(tan(e + f*x)*d + c)*sqrt(tan(e + f*x)*b 
+ a)*a*b*c**2*d**3 + 18*sqrt(tan(e + f*x)*d + c)*sqrt(tan(e + f*x)*b + a)* 
b**2*c**3*d**2 + 45*int((sqrt(tan(e + f*x)*d + c)*sqrt(tan(e + f*x)*b + a) 
*tan(e + f*x))/(tan(e + f*x)**5*b**4*d + 4*tan(e + f*x)**4*a*b**3*d + tan( 
e + f*x)**4*b**4*c + 6*tan(e + f*x)**3*a**2*b**2*d + 4*tan(e + f*x)**3*a*b 
**3*c + 4*tan(e + f*x)**2*a**3*b*d + 6*tan(e + f*x)**2*a**2*b**2*c + tan(e 
 + f*x)*a**4*d + 4*tan(e + f*x)*a**3*b*c + a**4*c),x)*tan(e + f*x)**3*a**3 
*b**3*c**2*d**4*f - 15*int((sqrt(tan(e + f*x)*d + c)*sqrt(tan(e + f*x)*b + 
 a)*tan(e + f*x))/(tan(e + f*x)**5*b**4*d + 4*tan(e + f*x)**4*a*b**3*d + t 
an(e + f*x)**4*b**4*c + 6*tan(e + f*x)**3*a**2*b**2*d + 4*tan(e + f*x)**3* 
a*b**3*c + 4*tan(e + f*x)**2*a**3*b*d + 6*tan(e + f*x)**2*a**2*b**2*c + ta 
n(e + f*x)*a**4*d + 4*tan(e + f*x)*a**3*b*c + a**4*c),x)*tan(e + f*x)**3*a 
**3*b**3*d**6*f - 135*int((sqrt(tan(e + f*x)*d + c)*sqrt(tan(e + f*x)*b + 
a)*tan(e + f*x))/(tan(e + f*x)**5*b**4*d + 4*tan(e + f*x)**4*a*b**3*d +...