\(\int \frac {(a+b \tan (e+f x))^{9/2}}{(c+d \tan (e+f x))^{5/2}} \, dx\) [1312]

Optimal result

Integrand size = 29, antiderivative size = 470 \[ \int \frac {(a+b \tan (e+f x))^{9/2}}{(c+d \tan (e+f x))^{5/2}} \, dx=-\frac {i (a-i b)^{9/2} \text {arctanh}\left (\frac {\sqrt {c-i d} \sqrt {a+b \tan (e+f x)}}{\sqrt {a-i b} \sqrt {c+d \tan (e+f x)}}\right )}{(c-i d)^{5/2} f}+\frac {i (a+i b)^{9/2} \text {arctanh}\left (\frac {\sqrt {c+i d} \sqrt {a+b \tan (e+f x)}}{\sqrt {a+i b} \sqrt {c+d \tan (e+f x)}}\right )}{(c+i d)^{5/2} f}-\frac {b^{7/2} (5 b c-9 a d) \text {arctanh}\left (\frac {\sqrt {d} \sqrt {a+b \tan (e+f x)}}{\sqrt {b} \sqrt {c+d \tan (e+f x)}}\right )}{d^{7/2} f}-\frac {2 (b c-a d)^2 (a+b \tan (e+f x))^{5/2}}{3 d \left (c^2+d^2\right ) f (c+d \tan (e+f x))^{3/2}}-\frac {2 (b c-a d)^2 \left (5 b c^2+6 a c d+11 b d^2\right ) (a+b \tan (e+f x))^{3/2}}{3 d^2 \left (c^2+d^2\right )^2 f \sqrt {c+d \tan (e+f x)}}+\frac {b \left (4 a^3 c d^3-4 a^2 b d^2 \left (c^2-2 d^2\right )-4 a b^2 c d \left (c^2+4 d^2\right )+b^3 \left (5 c^4+10 c^2 d^2+d^4\right )\right ) \sqrt {a+b \tan (e+f x)} \sqrt {c+d \tan (e+f x)}}{d^3 \left (c^2+d^2\right )^2 f} \] Output:

-I*(a-I*b)^(9/2)*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))/(c-I*d)^(5/2)/f+I*(a+I*b)^(9/2)*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))/(c+I*d) 
^(5/2)/f-b^(7/2)*(-9*a*d+5*b*c)*arctanh(d^(1/2)*(a+b*tan(f*x+e))^(1/2)/b^( 
1/2)/(c+d*tan(f*x+e))^(1/2))/d^(7/2)/f-2/3*(-a*d+b*c)^2*(a+b*tan(f*x+e))^( 
5/2)/d/(c^2+d^2)/f/(c+d*tan(f*x+e))^(3/2)-2/3*(-a*d+b*c)^2*(6*a*c*d+5*b*c^ 
2+11*b*d^2)*(a+b*tan(f*x+e))^(3/2)/d^2/(c^2+d^2)^2/f/(c+d*tan(f*x+e))^(1/2 
)+b*(4*a^3*c*d^3-4*a^2*b*d^2*(c^2-2*d^2)-4*a*b^2*c*d*(c^2+4*d^2)+b^3*(5*c^ 
4+10*c^2*d^2+d^4))*(a+b*tan(f*x+e))^(1/2)*(c+d*tan(f*x+e))^(1/2)/d^3/(c^2+ 
d^2)^2/f
 

Mathematica [C] (verified)

Result contains higher order function than in optimal. Order 5 vs. order 3 in optimal.

Time = 8.12 (sec) , antiderivative size = 2261, normalized size of antiderivative = 4.81 \[ \int \frac {(a+b \tan (e+f x))^{9/2}}{(c+d \tan (e+f x))^{5/2}} \, dx=\text {Result too large to show} \] Input:

Integrate[(a + b*Tan[e + f*x])^(9/2)/(c + d*Tan[e + f*x])^(5/2),x]
 

Output:

((-1/2*I)*(-a - I*b)*((-2*b^2*(b/((b^2*c)/(b*c - a*d) - (a*b*d)/(b*c - a*d 
)))^(5/2)*Hypergeometric2F1[5/2, 7/2, 9/2, -((b*d*(a + b*Tan[e + f*x]))/(( 
b*c - a*d)*((b^2*c)/(b*c - a*d) - (a*b*d)/(b*c - a*d))))]*(a + b*Tan[e + f 
*x])^(7/2)*Sqrt[(b*(c + d*Tan[e + f*x]))/(b*c - a*d)])/(7*(b*c - a*d)^2*Sq 
rt[c + d*Tan[e + f*x]]) - (-a - I*b)*((2*(b*c - a*d)*(b/((b^2*c)/(b*c - a* 
d) - (a*b*d)/(b*c - a*d)))^(5/2)*((b^2*c)/(b*c - a*d) - (a*b*d)/(b*c - a*d 
))^3*Sqrt[(b*(c + d*Tan[e + f*x]))/(b*c - a*d)]*(-1 - (b*d*(a + b*Tan[e + 
f*x]))/((b*c - a*d)*((b^2*c)/(b*c - a*d) - (a*b*d)/(b*c - a*d))))^2*((b^2* 
d^2*(a + b*Tan[e + f*x])^2)/(3*(b*c - a*d)^2*((b^2*c)/(b*c - a*d) - (a*b*d 
)/(b*c - a*d))^2*(-1 - (b*d*(a + b*Tan[e + f*x]))/((b*c - a*d)*((b^2*c)/(b 
*c - a*d) - (a*b*d)/(b*c - a*d))))^2) - (b*d*(a + b*Tan[e + f*x]))/((b*c - 
 a*d)*((b^2*c)/(b*c - a*d) - (a*b*d)/(b*c - a*d))*(-1 - (b*d*(a + b*Tan[e 
+ f*x]))/((b*c - a*d)*((b^2*c)/(b*c - a*d) - (a*b*d)/(b*c - a*d))))) - (Sq 
rt[b]*Sqrt[d]*ArcSinh[(Sqrt[b]*Sqrt[d]*Sqrt[a + b*Tan[e + f*x]])/(Sqrt[b*c 
 - a*d]*Sqrt[(b^2*c)/(b*c - a*d) - (a*b*d)/(b*c - a*d)])]*Sqrt[a + b*Tan[e 
 + f*x]])/(Sqrt[b*c - a*d]*Sqrt[(b^2*c)/(b*c - a*d) - (a*b*d)/(b*c - a*d)] 
*Sqrt[1 + (b*d*(a + b*Tan[e + f*x]))/((b*c - a*d)*((b^2*c)/(b*c - a*d) - ( 
a*b*d)/(b*c - a*d)))])))/(b*d^3*Sqrt[a + b*Tan[e + f*x]]*Sqrt[c + d*Tan[e 
+ f*x]]*(1 + (b*d*(a + b*Tan[e + f*x]))/((b*c - a*d)*((b^2*c)/(b*c - a*d) 
- (a*b*d)/(b*c - a*d))))^(3/2)) - (-a - I*b)*((-2*(a + b*Tan[e + f*x])^...
 

Rubi [A] (verified)

Time = 4.17 (sec) , antiderivative size = 526, normalized size of antiderivative = 1.12, number of steps used = 14, number of rules used = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.448, Rules used = {3042, 4048, 27, 3042, 4128, 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.

\(\displaystyle \int \frac {(a+b \tan (e+f x))^{9/2}}{(c+d \tan (e+f x))^{5/2}} \, dx\)

\(\Big \downarrow \) 3042

\(\displaystyle \int \frac {(a+b \tan (e+f x))^{9/2}}{(c+d \tan (e+f x))^{5/2}}dx\)

\(\Big \downarrow \) 4048

\(\displaystyle \frac {2 \int \frac {(a+b \tan (e+f x))^{3/2} \left (3 c d a^3+11 b d^2 a^2-13 b^2 c d a+5 b^3 c^2-b \left (2 a d (2 b c-a d)-b^2 \left (5 c^2+3 d^2\right )\right ) \tan ^2(e+f x)+3 d \left (-d a^3+3 b c a^2+3 b^2 d a-b^3 c\right ) \tan (e+f x)\right )}{2 (c+d \tan (e+f x))^{3/2}}dx}{3 d \left (c^2+d^2\right )}-\frac {2 (b c-a d)^2 (a+b \tan (e+f x))^{5/2}}{3 d f \left (c^2+d^2\right ) (c+d \tan (e+f x))^{3/2}}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {\int \frac {(a+b \tan (e+f x))^{3/2} \left (3 c d a^3+11 b d^2 a^2-13 b^2 c d a+5 b^3 c^2-b \left (2 a d (2 b c-a d)-b^2 \left (5 c^2+3 d^2\right )\right ) \tan ^2(e+f x)+3 d \left (-d a^3+3 b c a^2+3 b^2 d a-b^3 c\right ) \tan (e+f x)\right )}{(c+d \tan (e+f x))^{3/2}}dx}{3 d \left (c^2+d^2\right )}-\frac {2 (b c-a d)^2 (a+b \tan (e+f x))^{5/2}}{3 d f \left (c^2+d^2\right ) (c+d \tan (e+f x))^{3/2}}\)

\(\Big \downarrow \) 3042

\(\displaystyle \frac {\int \frac {(a+b \tan (e+f x))^{3/2} \left (3 c d a^3+11 b d^2 a^2-13 b^2 c d a+5 b^3 c^2-b \left (2 a d (2 b c-a d)-b^2 \left (5 c^2+3 d^2\right )\right ) \tan (e+f x)^2+3 d \left (-d a^3+3 b c a^2+3 b^2 d a-b^3 c\right ) \tan (e+f x)\right )}{(c+d \tan (e+f x))^{3/2}}dx}{3 d \left (c^2+d^2\right )}-\frac {2 (b c-a d)^2 (a+b \tan (e+f x))^{5/2}}{3 d f \left (c^2+d^2\right ) (c+d \tan (e+f x))^{3/2}}\)

\(\Big \downarrow \) 4128

\(\displaystyle \frac {\frac {2 \int \frac {3 \sqrt {a+b \tan (e+f x)} \left (d^2 \left (c^2-d^2\right ) a^4+12 b c d^3 a^3-2 b^2 d^2 \left (5 c^2-7 d^2\right ) a^2-4 b^3 c d \left (c^2+6 d^2\right ) a+b \left (\left (5 c^4+10 d^2 c^2+d^4\right ) b^3-4 a c d \left (c^2+4 d^2\right ) b^2-4 a^2 d^2 \left (c^2-2 d^2\right ) b+4 a^3 c d^3\right ) \tan ^2(e+f x)+b^4 \left (5 c^4+11 d^2 c^2\right )+2 d^2 \left (c a^2+2 b d a-b^2 c\right ) \left (-d a^2+2 b c a+b^2 d\right ) \tan (e+f x)\right )}{2 \sqrt {c+d \tan (e+f x)}}dx}{d \left (c^2+d^2\right )}-\frac {2 (b c-a d)^2 \left (6 a c d+5 b c^2+11 b d^2\right ) (a+b \tan (e+f x))^{3/2}}{d f \left (c^2+d^2\right ) \sqrt {c+d \tan (e+f x)}}}{3 d \left (c^2+d^2\right )}-\frac {2 (b c-a d)^2 (a+b \tan (e+f x))^{5/2}}{3 d f \left (c^2+d^2\right ) (c+d \tan (e+f x))^{3/2}}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {\frac {3 \int \frac {\sqrt {a+b \tan (e+f x)} \left (d^2 \left (c^2-d^2\right ) a^4+12 b c d^3 a^3-2 b^2 d^2 \left (5 c^2-7 d^2\right ) a^2-4 b^3 c d \left (c^2+6 d^2\right ) a+b \left (\left (5 c^4+10 d^2 c^2+d^4\right ) b^3-4 a c d \left (c^2+4 d^2\right ) b^2-4 a^2 d^2 \left (c^2-2 d^2\right ) b+4 a^3 c d^3\right ) \tan ^2(e+f x)+b^4 \left (5 c^4+11 d^2 c^2\right )+2 d^2 \left (c a^2+2 b d a-b^2 c\right ) \left (-d a^2+2 b c a+b^2 d\right ) \tan (e+f x)\right )}{\sqrt {c+d \tan (e+f x)}}dx}{d \left (c^2+d^2\right )}-\frac {2 (b c-a d)^2 \left (6 a c d+5 b c^2+11 b d^2\right ) (a+b \tan (e+f x))^{3/2}}{d f \left (c^2+d^2\right ) \sqrt {c+d \tan (e+f x)}}}{3 d \left (c^2+d^2\right )}-\frac {2 (b c-a d)^2 (a+b \tan (e+f x))^{5/2}}{3 d f \left (c^2+d^2\right ) (c+d \tan (e+f x))^{3/2}}\)

\(\Big \downarrow \) 3042

\(\displaystyle \frac {\frac {3 \int \frac {\sqrt {a+b \tan (e+f x)} \left (d^2 \left (c^2-d^2\right ) a^4+12 b c d^3 a^3-2 b^2 d^2 \left (5 c^2-7 d^2\right ) a^2-4 b^3 c d \left (c^2+6 d^2\right ) a+b \left (\left (5 c^4+10 d^2 c^2+d^4\right ) b^3-4 a c d \left (c^2+4 d^2\right ) b^2-4 a^2 d^2 \left (c^2-2 d^2\right ) b+4 a^3 c d^3\right ) \tan (e+f x)^2+b^4 \left (5 c^4+11 d^2 c^2\right )+2 d^2 \left (c a^2+2 b d a-b^2 c\right ) \left (-d a^2+2 b c a+b^2 d\right ) \tan (e+f x)\right )}{\sqrt {c+d \tan (e+f x)}}dx}{d \left (c^2+d^2\right )}-\frac {2 (b c-a d)^2 \left (6 a c d+5 b c^2+11 b d^2\right ) (a+b \tan (e+f x))^{3/2}}{d f \left (c^2+d^2\right ) \sqrt {c+d \tan (e+f x)}}}{3 d \left (c^2+d^2\right )}-\frac {2 (b c-a d)^2 (a+b \tan (e+f x))^{5/2}}{3 d f \left (c^2+d^2\right ) (c+d \tan (e+f x))^{3/2}}\)

\(\Big \downarrow \) 4130

\(\displaystyle \frac {\frac {3 \left (\frac {\int \frac {2 d^3 \left (c^2-d^2\right ) a^5+20 b c d^4 a^4-20 b^2 d^3 \left (c^2-d^2\right ) a^3-40 b^3 c d^4 a^2+b^4 d \left (9 c^4+28 d^2 c^2-d^4\right ) a-b^4 (5 b c-9 a d) \left (c^2+d^2\right )^2 \tan ^2(e+f x)-b^5 c \left (5 c^4+10 d^2 c^2+d^4\right )-2 d^3 \left (2 c d a^5-5 b \left (c^2-d^2\right ) a^4-20 b^2 c d a^3+10 b^3 \left (c^2-d^2\right ) a^2+10 b^4 c d a-b^5 \left (c^2-d^2\right )\right ) \tan (e+f x)}{2 \sqrt {a+b \tan (e+f x)} \sqrt {c+d \tan (e+f x)}}dx}{d}+\frac {b \left (4 a^3 c d^3-4 a^2 b d^2 \left (c^2-2 d^2\right )-4 a b^2 c d \left (c^2+4 d^2\right )+b^3 \left (5 c^4+10 c^2 d^2+d^4\right )\right ) \sqrt {a+b \tan (e+f x)} \sqrt {c+d \tan (e+f x)}}{d f}\right )}{d \left (c^2+d^2\right )}-\frac {2 (b c-a d)^2 \left (6 a c d+5 b c^2+11 b d^2\right ) (a+b \tan (e+f x))^{3/2}}{d f \left (c^2+d^2\right ) \sqrt {c+d \tan (e+f x)}}}{3 d \left (c^2+d^2\right )}-\frac {2 (b c-a d)^2 (a+b \tan (e+f x))^{5/2}}{3 d f \left (c^2+d^2\right ) (c+d \tan (e+f x))^{3/2}}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {\frac {3 \left (\frac {\int \frac {2 d^3 \left (c^2-d^2\right ) a^5+20 b c d^4 a^4-20 b^2 d^3 \left (c^2-d^2\right ) a^3-40 b^3 c d^4 a^2+b^4 d \left (9 c^4+28 d^2 c^2-d^4\right ) a-b^4 (5 b c-9 a d) \left (c^2+d^2\right )^2 \tan ^2(e+f x)-b^5 c \left (5 c^4+10 d^2 c^2+d^4\right )-2 d^3 \left (2 c d a^5-5 b \left (c^2-d^2\right ) a^4-20 b^2 c d a^3+10 b^3 \left (c^2-d^2\right ) a^2+10 b^4 c d a-b^5 \left (c^2-d^2\right )\right ) \tan (e+f x)}{\sqrt {a+b \tan (e+f x)} \sqrt {c+d \tan (e+f x)}}dx}{2 d}+\frac {b \left (4 a^3 c d^3-4 a^2 b d^2 \left (c^2-2 d^2\right )-4 a b^2 c d \left (c^2+4 d^2\right )+b^3 \left (5 c^4+10 c^2 d^2+d^4\right )\right ) \sqrt {a+b \tan (e+f x)} \sqrt {c+d \tan (e+f x)}}{d f}\right )}{d \left (c^2+d^2\right )}-\frac {2 (b c-a d)^2 \left (6 a c d+5 b c^2+11 b d^2\right ) (a+b \tan (e+f x))^{3/2}}{d f \left (c^2+d^2\right ) \sqrt {c+d \tan (e+f x)}}}{3 d \left (c^2+d^2\right )}-\frac {2 (b c-a d)^2 (a+b \tan (e+f x))^{5/2}}{3 d f \left (c^2+d^2\right ) (c+d \tan (e+f x))^{3/2}}\)

\(\Big \downarrow \) 3042

\(\displaystyle \frac {\frac {3 \left (\frac {\int \frac {2 d^3 \left (c^2-d^2\right ) a^5+20 b c d^4 a^4-20 b^2 d^3 \left (c^2-d^2\right ) a^3-40 b^3 c d^4 a^2+b^4 d \left (9 c^4+28 d^2 c^2-d^4\right ) a-b^4 (5 b c-9 a d) \left (c^2+d^2\right )^2 \tan (e+f x)^2-b^5 c \left (5 c^4+10 d^2 c^2+d^4\right )-2 d^3 \left (2 c d a^5-5 b \left (c^2-d^2\right ) a^4-20 b^2 c d a^3+10 b^3 \left (c^2-d^2\right ) a^2+10 b^4 c d a-b^5 \left (c^2-d^2\right )\right ) \tan (e+f x)}{\sqrt {a+b \tan (e+f x)} \sqrt {c+d \tan (e+f x)}}dx}{2 d}+\frac {b \left (4 a^3 c d^3-4 a^2 b d^2 \left (c^2-2 d^2\right )-4 a b^2 c d \left (c^2+4 d^2\right )+b^3 \left (5 c^4+10 c^2 d^2+d^4\right )\right ) \sqrt {a+b \tan (e+f x)} \sqrt {c+d \tan (e+f x)}}{d f}\right )}{d \left (c^2+d^2\right )}-\frac {2 (b c-a d)^2 \left (6 a c d+5 b c^2+11 b d^2\right ) (a+b \tan (e+f x))^{3/2}}{d f \left (c^2+d^2\right ) \sqrt {c+d \tan (e+f x)}}}{3 d \left (c^2+d^2\right )}-\frac {2 (b c-a d)^2 (a+b \tan (e+f x))^{5/2}}{3 d f \left (c^2+d^2\right ) (c+d \tan (e+f x))^{3/2}}\)

\(\Big \downarrow \) 4138

\(\displaystyle \frac {\frac {3 \left (\frac {\int \frac {2 d^3 \left (c^2-d^2\right ) a^5+20 b c d^4 a^4-20 b^2 d^3 \left (c^2-d^2\right ) a^3-40 b^3 c d^4 a^2+b^4 d \left (9 c^4+28 d^2 c^2-d^4\right ) a-b^4 (5 b c-9 a d) \left (c^2+d^2\right )^2 \tan ^2(e+f x)-b^5 c \left (5 c^4+10 d^2 c^2+d^4\right )-2 d^3 \left (2 c d a^5-5 b \left (c^2-d^2\right ) a^4-20 b^2 c d a^3+10 b^3 \left (c^2-d^2\right ) a^2+10 b^4 c d a-b^5 \left (c^2-d^2\right )\right ) \tan (e+f x)}{\sqrt {a+b \tan (e+f x)} \sqrt {c+d \tan (e+f x)} \left (\tan ^2(e+f x)+1\right )}d\tan (e+f x)}{2 d f}+\frac {b \left (4 a^3 c d^3-4 a^2 b d^2 \left (c^2-2 d^2\right )-4 a b^2 c d \left (c^2+4 d^2\right )+b^3 \left (5 c^4+10 c^2 d^2+d^4\right )\right ) \sqrt {a+b \tan (e+f x)} \sqrt {c+d \tan (e+f x)}}{d f}\right )}{d \left (c^2+d^2\right )}-\frac {2 (b c-a d)^2 \left (6 a c d+5 b c^2+11 b d^2\right ) (a+b \tan (e+f x))^{3/2}}{d f \left (c^2+d^2\right ) \sqrt {c+d \tan (e+f x)}}}{3 d \left (c^2+d^2\right )}-\frac {2 (b c-a d)^2 (a+b \tan (e+f x))^{5/2}}{3 d f \left (c^2+d^2\right ) (c+d \tan (e+f x))^{3/2}}\)

\(\Big \downarrow \) 2348

\(\displaystyle \frac {\frac {3 \left (\frac {b \sqrt {a+b \tan (e+f x)} \sqrt {c+d \tan (e+f x)} \left (\left (5 c^4+10 d^2 c^2+d^4\right ) b^3-4 a c d \left (c^2+4 d^2\right ) b^2-4 a^2 d^2 \left (c^2-2 d^2\right ) b+4 a^3 c d^3\right )}{d f}+\frac {\int \left (-\frac {(5 b c-9 a d) \left (c^2+d^2\right )^2 b^4}{\sqrt {a+b \tan (e+f x)} \sqrt {c+d \tan (e+f x)}}+\frac {4 c d^4 a^5+10 b d^5 a^4-10 b c^2 d^3 a^4-40 b^2 c d^4 a^3-20 b^3 d^5 a^2+20 b^3 c^2 d^3 a^2+20 b^4 c d^4 a+2 b^5 d^5-2 b^5 c^2 d^3+i \left (-2 d^5 a^5+2 c^2 d^3 a^5+20 b c d^4 a^4+20 b^2 d^5 a^3-20 b^2 c^2 d^3 a^3-40 b^3 c d^4 a^2-10 b^4 d^5 a+10 b^4 c^2 d^3 a+4 b^5 c d^4\right )}{2 (i-\tan (e+f x)) \sqrt {a+b \tan (e+f x)} \sqrt {c+d \tan (e+f x)}}+\frac {-4 c d^4 a^5-10 b d^5 a^4+10 b c^2 d^3 a^4+40 b^2 c d^4 a^3+20 b^3 d^5 a^2-20 b^3 c^2 d^3 a^2-20 b^4 c d^4 a-2 b^5 d^5+2 b^5 c^2 d^3+i \left (-2 d^5 a^5+2 c^2 d^3 a^5+20 b c d^4 a^4+20 b^2 d^5 a^3-20 b^2 c^2 d^3 a^3-40 b^3 c d^4 a^2-10 b^4 d^5 a+10 b^4 c^2 d^3 a+4 b^5 c d^4\right )}{2 (\tan (e+f x)+i) \sqrt {a+b \tan (e+f x)} \sqrt {c+d \tan (e+f x)}}\right )d\tan (e+f x)}{2 d f}\right )}{d \left (c^2+d^2\right )}-\frac {2 (b c-a d)^2 \left (5 b c^2+6 a d c+11 b d^2\right ) (a+b \tan (e+f x))^{3/2}}{d \left (c^2+d^2\right ) f \sqrt {c+d \tan (e+f x)}}}{3 d \left (c^2+d^2\right )}-\frac {2 (b c-a d)^2 (a+b \tan (e+f x))^{5/2}}{3 d \left (c^2+d^2\right ) f (c+d \tan (e+f x))^{3/2}}\)

\(\Big \downarrow \) 2009

\(\displaystyle -\frac {2 (b c-a d)^2 (a+b \tan (e+f x))^{5/2}}{3 d f \left (c^2+d^2\right ) (c+d \tan (e+f x))^{3/2}}+\frac {-\frac {2 (b c-a d)^2 \left (6 a c d+5 b c^2+11 b d^2\right ) (a+b \tan (e+f x))^{3/2}}{d f \left (c^2+d^2\right ) \sqrt {c+d \tan (e+f x)}}+\frac {3 \left (\frac {b \left (4 a^3 c d^3-4 a^2 b d^2 \left (c^2-2 d^2\right )-4 a b^2 c d \left (c^2+4 d^2\right )+b^3 \left (5 c^4+10 c^2 d^2+d^4\right )\right ) \sqrt {a+b \tan (e+f x)} \sqrt {c+d \tan (e+f x)}}{d f}+\frac {-\frac {2 b^{7/2} \left (c^2+d^2\right )^2 (5 b c-9 a d) \text {arctanh}\left (\frac {\sqrt {d} \sqrt {a+b \tan (e+f x)}}{\sqrt {b} \sqrt {c+d \tan (e+f x)}}\right )}{\sqrt {d}}-\frac {2 i d^3 (a-i b)^{9/2} (c+i d)^2 \text {arctanh}\left (\frac {\sqrt {c-i d} \sqrt {a+b \tan (e+f x)}}{\sqrt {a-i b} \sqrt {c+d \tan (e+f x)}}\right )}{\sqrt {c-i d}}+\frac {2 i d^3 (a+i b)^{9/2} (c-i d)^2 \text {arctanh}\left (\frac {\sqrt {c+i d} \sqrt {a+b \tan (e+f x)}}{\sqrt {a+i b} \sqrt {c+d \tan (e+f x)}}\right )}{\sqrt {c+i d}}}{2 d f}\right )}{d \left (c^2+d^2\right )}}{3 d \left (c^2+d^2\right )}\)

Input:

Int[(a + b*Tan[e + f*x])^(9/2)/(c + d*Tan[e + f*x])^(5/2),x]
 

Output:

(-2*(b*c - a*d)^2*(a + b*Tan[e + f*x])^(5/2))/(3*d*(c^2 + d^2)*f*(c + d*Ta 
n[e + f*x])^(3/2)) + ((-2*(b*c - a*d)^2*(5*b*c^2 + 6*a*c*d + 11*b*d^2)*(a 
+ b*Tan[e + f*x])^(3/2))/(d*(c^2 + d^2)*f*Sqrt[c + d*Tan[e + f*x]]) + (3*( 
(((-2*I)*(a - I*b)^(9/2)*(c + I*d)^2*d^3*ArcTanh[(Sqrt[c - I*d]*Sqrt[a + b 
*Tan[e + f*x]])/(Sqrt[a - I*b]*Sqrt[c + d*Tan[e + f*x]])])/Sqrt[c - I*d] + 
 ((2*I)*(a + I*b)^(9/2)*(c - I*d)^2*d^3*ArcTanh[(Sqrt[c + I*d]*Sqrt[a + b* 
Tan[e + f*x]])/(Sqrt[a + I*b]*Sqrt[c + d*Tan[e + f*x]])])/Sqrt[c + I*d] - 
(2*b^(7/2)*(5*b*c - 9*a*d)*(c^2 + d^2)^2*ArcTanh[(Sqrt[d]*Sqrt[a + b*Tan[e 
 + f*x]])/(Sqrt[b]*Sqrt[c + d*Tan[e + f*x]])])/Sqrt[d])/(2*d*f) + (b*(4*a^ 
3*c*d^3 - 4*a^2*b*d^2*(c^2 - 2*d^2) - 4*a*b^2*c*d*(c^2 + 4*d^2) + b^3*(5*c 
^4 + 10*c^2*d^2 + d^4))*Sqrt[a + b*Tan[e + f*x]]*Sqrt[c + d*Tan[e + f*x]]) 
/(d*f)))/(d*(c^2 + d^2)))/(3*d*(c^2 + d^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 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 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 4128

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*d^2 + c*(c*C - B*d))*(a + b*Tan[e + 
 f*x])^m*((c + d*Tan[e + f*x])^(n + 1)/(d*f*(n + 1)*(c^2 + d^2))), x] - Sim 
p[1/(d*(n + 1)*(c^2 + d^2))   Int[(a + b*Tan[e + f*x])^(m - 1)*(c + d*Tan[e 
 + f*x])^(n + 1)*Simp[A*d*(b*d*m - a*c*(n + 1)) + (c*C - B*d)*(b*c*m + a*d* 
(n + 1)) - d*(n + 1)*((A - C)*(b*c - a*d) + B*(a*c + b*d))*Tan[e + f*x] - b 
*(d*(B*c - A*d)*(m + n + 1) - C*(c^2*m - d^2*(n + 1)))*Tan[e + f*x]^2, x], 
x], x] /; FreeQ[{a, b, c, d, e, f, A, B, C}, x] && NeQ[b*c - a*d, 0] && NeQ 
[a^2 + b^2, 0] && NeQ[c^2 + d^2, 0] && GtQ[m, 0] && LtQ[n, -1]
 

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]
 

Maple [F(-1)]

Timed out.

\[\int \frac {\left (a +b \tan \left (f x +e \right )\right )^{\frac {9}{2}}}{\left (c +d \tan \left (f x +e \right )\right )^{\frac {5}{2}}}d x\]

Input:

int((a+b*tan(f*x+e))^(9/2)/(c+d*tan(f*x+e))^(5/2),x)
 

Output:

int((a+b*tan(f*x+e))^(9/2)/(c+d*tan(f*x+e))^(5/2),x)
 

Fricas [F(-1)]

Timed out. \[ \int \frac {(a+b \tan (e+f x))^{9/2}}{(c+d \tan (e+f x))^{5/2}} \, dx=\text {Timed out} \] Input:

integrate((a+b*tan(f*x+e))^(9/2)/(c+d*tan(f*x+e))^(5/2),x, algorithm="fric 
as")
 

Output:

Timed out
 

Sympy [F(-1)]

Timed out. \[ \int \frac {(a+b \tan (e+f x))^{9/2}}{(c+d \tan (e+f x))^{5/2}} \, dx=\text {Timed out} \] Input:

integrate((a+b*tan(f*x+e))**(9/2)/(c+d*tan(f*x+e))**(5/2),x)
 

Output:

Timed out
 

Maxima [F(-1)]

Timed out. \[ \int \frac {(a+b \tan (e+f x))^{9/2}}{(c+d \tan (e+f x))^{5/2}} \, dx=\text {Timed out} \] Input:

integrate((a+b*tan(f*x+e))^(9/2)/(c+d*tan(f*x+e))^(5/2),x, algorithm="maxi 
ma")
 

Output:

Timed out
 

Giac [F]

\[ \int \frac {(a+b \tan (e+f x))^{9/2}}{(c+d \tan (e+f x))^{5/2}} \, dx=\int { \frac {{\left (b \tan \left (f x + e\right ) + a\right )}^{\frac {9}{2}}}{{\left (d \tan \left (f x + e\right ) + c\right )}^{\frac {5}{2}}} \,d x } \] Input:

integrate((a+b*tan(f*x+e))^(9/2)/(c+d*tan(f*x+e))^(5/2),x, algorithm="giac 
")
 

Output:

integrate((b*tan(f*x + e) + a)^(9/2)/(d*tan(f*x + e) + c)^(5/2), x)
 

Mupad [F(-1)]

Timed out. \[ \int \frac {(a+b \tan (e+f x))^{9/2}}{(c+d \tan (e+f x))^{5/2}} \, dx=\int \frac {{\left (a+b\,\mathrm {tan}\left (e+f\,x\right )\right )}^{9/2}}{{\left (c+d\,\mathrm {tan}\left (e+f\,x\right )\right )}^{5/2}} \,d x \] Input:

int((a + b*tan(e + f*x))^(9/2)/(c + d*tan(e + f*x))^(5/2),x)
 

Output:

int((a + b*tan(e + f*x))^(9/2)/(c + d*tan(e + f*x))^(5/2), x)
                                                                                    
                                                                                    
 

Reduce [F]

\[ \int \frac {(a+b \tan (e+f x))^{9/2}}{(c+d \tan (e+f x))^{5/2}} \, dx=\text {too large to display} \] Input:

int((a+b*tan(f*x+e))^(9/2)/(c+d*tan(f*x+e))^(5/2),x)
 

Output:

(4*sqrt(tan(e + f*x)*d + c)*sqrt(tan(e + f*x)*b + a)*tan(e + f*x)*a**3*b** 
3*d + 12*sqrt(tan(e + f*x)*d + c)*sqrt(tan(e + f*x)*b + a)*tan(e + f*x)*a* 
*2*b**4*c - 20*sqrt(tan(e + f*x)*d + c)*sqrt(tan(e + f*x)*b + a)*a**4*b**2 
*d + 36*sqrt(tan(e + f*x)*d + c)*sqrt(tan(e + f*x)*b + a)*a**3*b**3*c + 3* 
int((sqrt(tan(e + f*x)*d + c)*sqrt(tan(e + f*x)*b + a)*tan(e + f*x)**4)/(t 
an(e + f*x)**3*d**3 + 3*tan(e + f*x)**2*c*d**2 + 3*tan(e + f*x)*c**2*d + c 
**3),x)*tan(e + f*x)**2*a**2*b**4*d**4*f - 6*int((sqrt(tan(e + f*x)*d + c) 
*sqrt(tan(e + f*x)*b + a)*tan(e + f*x)**4)/(tan(e + f*x)**3*d**3 + 3*tan(e 
 + f*x)**2*c*d**2 + 3*tan(e + f*x)*c**2*d + c**3),x)*tan(e + f*x)**2*a*b** 
5*c*d**3*f + 3*int((sqrt(tan(e + f*x)*d + c)*sqrt(tan(e + f*x)*b + a)*tan( 
e + f*x)**4)/(tan(e + f*x)**3*d**3 + 3*tan(e + f*x)**2*c*d**2 + 3*tan(e + 
f*x)*c**2*d + c**3),x)*tan(e + f*x)**2*b**6*c**2*d**2*f + 6*int((sqrt(tan( 
e + f*x)*d + c)*sqrt(tan(e + f*x)*b + a)*tan(e + f*x)**4)/(tan(e + f*x)**3 
*d**3 + 3*tan(e + f*x)**2*c*d**2 + 3*tan(e + f*x)*c**2*d + c**3),x)*tan(e 
+ f*x)*a**2*b**4*c*d**3*f - 12*int((sqrt(tan(e + f*x)*d + c)*sqrt(tan(e + 
f*x)*b + a)*tan(e + f*x)**4)/(tan(e + f*x)**3*d**3 + 3*tan(e + f*x)**2*c*d 
**2 + 3*tan(e + f*x)*c**2*d + c**3),x)*tan(e + f*x)*a*b**5*c**2*d**2*f + 6 
*int((sqrt(tan(e + f*x)*d + c)*sqrt(tan(e + f*x)*b + a)*tan(e + f*x)**4)/( 
tan(e + f*x)**3*d**3 + 3*tan(e + f*x)**2*c*d**2 + 3*tan(e + f*x)*c**2*d + 
c**3),x)*tan(e + f*x)*b**6*c**3*d*f + 3*int((sqrt(tan(e + f*x)*d + c)*s...