\(\int \frac {(a+b \sin (e+f x))^{5/2}}{(c+d \sin (e+f x))^{5/2}} \, dx\) [789]

Optimal result

Integrand size = 29, antiderivative size = 737 \[ \int \frac {(a+b \sin (e+f x))^{5/2}}{(c+d \sin (e+f x))^{5/2}} \, dx=\frac {2 (b c-a d)^2 \cos (e+f x) \sqrt {a+b \sin (e+f x)}}{3 d \left (c^2-d^2\right ) f (c+d \sin (e+f x))^{3/2}}+\frac {2 (a-b) \sqrt {a+b} \left (3 b c^2+4 a c d-7 b d^2\right ) E\left (\arcsin \left (\frac {\sqrt {c+d} \sqrt {a+b \sin (e+f x)}}{\sqrt {a+b} \sqrt {c+d \sin (e+f x)}}\right )|\frac {(a+b) (c-d)}{(a-b) (c+d)}\right ) \sec (e+f x) \sqrt {\frac {(b c-a d) (1-\sin (e+f x))}{(a+b) (c+d \sin (e+f x))}} \sqrt {-\frac {(b c-a d) (1+\sin (e+f x))}{(a-b) (c+d \sin (e+f x))}} (c+d \sin (e+f x))}{3 (c-d)^2 d^2 (c+d)^{3/2} f}-\frac {2 \sqrt {a+b} \left (a^2 d^2 (3 c+d)+a b d \left (3 c^2-4 c d-7 d^2\right )+b^2 \left (3 c^3-6 c^2 d-2 c d^2+9 d^3\right )\right ) \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {c+d} \sqrt {a+b \sin (e+f x)}}{\sqrt {a+b} \sqrt {c+d \sin (e+f x)}}\right ),\frac {(a+b) (c-d)}{(a-b) (c+d)}\right ) \sec (e+f x) \sqrt {\frac {(b c-a d) (1-\sin (e+f x))}{(a+b) (c+d \sin (e+f x))}} \sqrt {-\frac {(b c-a d) (1+\sin (e+f x))}{(a-b) (c+d \sin (e+f x))}} (c+d \sin (e+f x))}{3 (c-d)^2 d^3 (c+d)^{3/2} f}+\frac {2 b^2 \sqrt {a+b} \operatorname {EllipticPi}\left (\frac {(a+b) d}{b (c+d)},\arcsin \left (\frac {\sqrt {c+d} \sqrt {a+b \sin (e+f x)}}{\sqrt {a+b} \sqrt {c+d \sin (e+f x)}}\right ),\frac {(a+b) (c-d)}{(a-b) (c+d)}\right ) \sec (e+f x) \sqrt {\frac {(b c-a d) (1-\sin (e+f x))}{(a+b) (c+d \sin (e+f x))}} \sqrt {-\frac {(b c-a d) (1+\sin (e+f x))}{(a-b) (c+d \sin (e+f x))}} (c+d \sin (e+f x))}{d^3 \sqrt {c+d} f} \] Output:

2/3*(-a*d+b*c)^2*cos(f*x+e)*(a+b*sin(f*x+e))^(1/2)/d/(c^2-d^2)/f/(c+d*sin( 
f*x+e))^(3/2)+2/3*(a-b)*(a+b)^(1/2)*(4*a*c*d+3*b*c^2-7*b*d^2)*EllipticE((c 
+d)^(1/2)*(a+b*sin(f*x+e))^(1/2)/(a+b)^(1/2)/(c+d*sin(f*x+e))^(1/2),((a+b) 
*(c-d)/(a-b)/(c+d))^(1/2))*sec(f*x+e)*((-a*d+b*c)*(1-sin(f*x+e))/(a+b)/(c+ 
d*sin(f*x+e)))^(1/2)*(-(-a*d+b*c)*(1+sin(f*x+e))/(a-b)/(c+d*sin(f*x+e)))^( 
1/2)*(c+d*sin(f*x+e))/(c-d)^2/d^2/(c+d)^(3/2)/f-2/3*(a+b)^(1/2)*(a^2*d^2*( 
3*c+d)+a*b*d*(3*c^2-4*c*d-7*d^2)+b^2*(3*c^3-6*c^2*d-2*c*d^2+9*d^3))*Ellipt 
icF((c+d)^(1/2)*(a+b*sin(f*x+e))^(1/2)/(a+b)^(1/2)/(c+d*sin(f*x+e))^(1/2), 
((a+b)*(c-d)/(a-b)/(c+d))^(1/2))*sec(f*x+e)*((-a*d+b*c)*(1-sin(f*x+e))/(a+ 
b)/(c+d*sin(f*x+e)))^(1/2)*(-(-a*d+b*c)*(1+sin(f*x+e))/(a-b)/(c+d*sin(f*x+ 
e)))^(1/2)*(c+d*sin(f*x+e))/(c-d)^2/d^3/(c+d)^(3/2)/f+2*b^2*(a+b)^(1/2)*El 
lipticPi((c+d)^(1/2)*(a+b*sin(f*x+e))^(1/2)/(a+b)^(1/2)/(c+d*sin(f*x+e))^( 
1/2),(a+b)*d/b/(c+d),((a+b)*(c-d)/(a-b)/(c+d))^(1/2))*sec(f*x+e)*((-a*d+b* 
c)*(1-sin(f*x+e))/(a+b)/(c+d*sin(f*x+e)))^(1/2)*(-(-a*d+b*c)*(1+sin(f*x+e) 
)/(a-b)/(c+d*sin(f*x+e)))^(1/2)*(c+d*sin(f*x+e))/d^3/(c+d)^(1/2)/f
 

Mathematica [B] (warning: unable to verify)

Leaf count is larger than twice the leaf count of optimal. \(2169\) vs. \(2(737)=1474\).

Time = 9.65 (sec) , antiderivative size = 2169, normalized size of antiderivative = 2.94 \[ \int \frac {(a+b \sin (e+f x))^{5/2}}{(c+d \sin (e+f x))^{5/2}} \, dx=\text {Result too large to show} \] Input:

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

Output:

(Sqrt[a + b*Sin[e + f*x]]*Sqrt[c + d*Sin[e + f*x]]*((-2*(b^2*c^2*Cos[e + f 
*x] - 2*a*b*c*d*Cos[e + f*x] + a^2*d^2*Cos[e + f*x]))/(3*d*(-c^2 + d^2)*(c 
 + d*Sin[e + f*x])^2) - (2*(3*b^2*c^3*Cos[e + f*x] + a*b*c^2*d*Cos[e + f*x 
] - 4*a^2*c*d^2*Cos[e + f*x] - 7*b^2*c*d^2*Cos[e + f*x] + 7*a*b*d^3*Cos[e 
+ f*x]))/(3*d*(-c^2 + d^2)^2*(c + d*Sin[e + f*x]))))/f + ((-4*(-(b*c) + a* 
d)*(-(b^3*c^3) + 3*a^3*c^2*d + 2*a*b^2*c^2*d - 8*a^2*b*c*d^2 + b^3*c*d^2 + 
 a^3*d^3 + 2*a*b^2*d^3)*Sqrt[((c + d)*Cot[(-e + Pi/2 - f*x)/2]^2)/(-c + d) 
]*EllipticF[ArcSin[Sqrt[((-a - b)*Csc[(-e + Pi/2 - f*x)/2]^2*(c + d*Sin[e 
+ f*x]))/(-(b*c) + a*d)]/Sqrt[2]], (2*(-(b*c) + a*d))/((a + b)*(-c + d))]* 
Sec[e + f*x]*Sin[(-e + Pi/2 - f*x)/2]^4*Sqrt[((c + d)*Csc[(-e + Pi/2 - f*x 
)/2]^2*(a + b*Sin[e + f*x]))/(-(b*c) + a*d)]*Sqrt[((-a - b)*Csc[(-e + Pi/2 
 - f*x)/2]^2*(c + d*Sin[e + f*x]))/(-(b*c) + a*d)])/((a + b)*(c + d)*Sqrt[ 
a + b*Sin[e + f*x]]*Sqrt[c + d*Sin[e + f*x]]) - 4*(-(b*c) + a*d)*(-4*a*b^2 
*c^3 + 3*a^2*b*c^2*d + b^3*c^2*d + 4*a^3*c*d^2 - 7*a^2*b*d^3 + 3*b^3*d^3)* 
((Sqrt[((c + d)*Cot[(-e + Pi/2 - f*x)/2]^2)/(-c + d)]*EllipticF[ArcSin[Sqr 
t[((-a - b)*Csc[(-e + Pi/2 - f*x)/2]^2*(c + d*Sin[e + f*x]))/(-(b*c) + a*d 
)]/Sqrt[2]], (2*(-(b*c) + a*d))/((a + b)*(-c + d))]*Sec[e + f*x]*Sin[(-e + 
 Pi/2 - f*x)/2]^4*Sqrt[((c + d)*Csc[(-e + Pi/2 - f*x)/2]^2*(a + b*Sin[e + 
f*x]))/(-(b*c) + a*d)]*Sqrt[((-a - b)*Csc[(-e + Pi/2 - f*x)/2]^2*(c + d*Si 
n[e + f*x]))/(-(b*c) + a*d)])/((a + b)*(c + d)*Sqrt[a + b*Sin[e + f*x]]...
 

Rubi [A] (verified)

Time = 3.16 (sec) , antiderivative size = 795, normalized size of antiderivative = 1.08, number of steps used = 12, number of rules used = 12, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.414, Rules used = {3042, 3271, 27, 3042, 3532, 25, 3042, 3290, 3477, 3042, 3297, 3475}

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.

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

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 3271

\(\displaystyle \frac {2 (b c-a d)^2 \cos (e+f x) \sqrt {a+b \sin (e+f x)}}{3 d f \left (c^2-d^2\right ) (c+d \sin (e+f x))^{3/2}}-\frac {2 \int \frac {-3 c d a^3+7 b d^2 a^2-5 b^2 c d a+b^3 c^2-3 b^3 \left (c^2-d^2\right ) \sin ^2(e+f x)-\left (-d^2 a^3+5 b c d a^2+b^2 \left (2 c^2-9 d^2\right ) a+3 b^3 c d\right ) \sin (e+f x)}{2 \sqrt {a+b \sin (e+f x)} (c+d \sin (e+f x))^{3/2}}dx}{3 d \left (c^2-d^2\right )}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {2 (b c-a d)^2 \cos (e+f x) \sqrt {a+b \sin (e+f x)}}{3 d f \left (c^2-d^2\right ) (c+d \sin (e+f x))^{3/2}}-\frac {\int \frac {-3 c d a^3+7 b d^2 a^2-5 b^2 c d a+b^3 c^2-3 b^3 \left (c^2-d^2\right ) \sin ^2(e+f x)-\left (-d^2 a^3+5 b c d a^2+b^2 \left (2 c^2-9 d^2\right ) a+3 b^3 c d\right ) \sin (e+f x)}{\sqrt {a+b \sin (e+f x)} (c+d \sin (e+f x))^{3/2}}dx}{3 d \left (c^2-d^2\right )}\)

\(\Big \downarrow \) 3042

\(\displaystyle \frac {2 (b c-a d)^2 \cos (e+f x) \sqrt {a+b \sin (e+f x)}}{3 d f \left (c^2-d^2\right ) (c+d \sin (e+f x))^{3/2}}-\frac {\int \frac {-3 c d a^3+7 b d^2 a^2-5 b^2 c d a+b^3 c^2-3 b^3 \left (c^2-d^2\right ) \sin (e+f x)^2-\left (-d^2 a^3+5 b c d a^2+b^2 \left (2 c^2-9 d^2\right ) a+3 b^3 c d\right ) \sin (e+f x)}{\sqrt {a+b \sin (e+f x)} (c+d \sin (e+f x))^{3/2}}dx}{3 d \left (c^2-d^2\right )}\)

\(\Big \downarrow \) 3532

\(\displaystyle \frac {2 (b c-a d)^2 \cos (e+f x) \sqrt {a+b \sin (e+f x)}}{3 d f \left (c^2-d^2\right ) (c+d \sin (e+f x))^{3/2}}-\frac {\frac {\int -\frac {-7 a^2 b d^4+3 a^3 c d^3+5 a b^2 c d^3-(b c-a d) \left (6 b^2 c^2+4 a b d c-a^2 d^2-9 b^2 d^2\right ) \sin (e+f x) d-b^3 \left (3 c^4-2 c^2 d^2\right )}{\sqrt {a+b \sin (e+f x)} (c+d \sin (e+f x))^{3/2}}dx}{d^2}-\frac {3 b^3 \left (c^2-d^2\right ) \int \frac {\sqrt {c+d \sin (e+f x)}}{\sqrt {a+b \sin (e+f x)}}dx}{d^2}}{3 d \left (c^2-d^2\right )}\)

\(\Big \downarrow \) 25

\(\displaystyle \frac {2 (b c-a d)^2 \cos (e+f x) \sqrt {a+b \sin (e+f x)}}{3 d f \left (c^2-d^2\right ) (c+d \sin (e+f x))^{3/2}}-\frac {-\frac {\int \frac {-7 a^2 b d^4+3 a^3 c d^3+5 a b^2 c d^3-(b c-a d) \left (\left (6 c^2-9 d^2\right ) b^2+4 a c d b-a^2 d^2\right ) \sin (e+f x) d-b^3 \left (3 c^4-2 c^2 d^2\right )}{\sqrt {a+b \sin (e+f x)} (c+d \sin (e+f x))^{3/2}}dx}{d^2}-\frac {3 b^3 \left (c^2-d^2\right ) \int \frac {\sqrt {c+d \sin (e+f x)}}{\sqrt {a+b \sin (e+f x)}}dx}{d^2}}{3 d \left (c^2-d^2\right )}\)

\(\Big \downarrow \) 3042

\(\displaystyle \frac {2 (b c-a d)^2 \cos (e+f x) \sqrt {a+b \sin (e+f x)}}{3 d f \left (c^2-d^2\right ) (c+d \sin (e+f x))^{3/2}}-\frac {-\frac {\int \frac {-7 a^2 b d^4+3 a^3 c d^3+5 a b^2 c d^3-(b c-a d) \left (\left (6 c^2-9 d^2\right ) b^2+4 a c d b-a^2 d^2\right ) \sin (e+f x) d-b^3 \left (3 c^4-2 c^2 d^2\right )}{\sqrt {a+b \sin (e+f x)} (c+d \sin (e+f x))^{3/2}}dx}{d^2}-\frac {3 b^3 \left (c^2-d^2\right ) \int \frac {\sqrt {c+d \sin (e+f x)}}{\sqrt {a+b \sin (e+f x)}}dx}{d^2}}{3 d \left (c^2-d^2\right )}\)

\(\Big \downarrow \) 3290

\(\displaystyle \frac {2 (b c-a d)^2 \cos (e+f x) \sqrt {a+b \sin (e+f x)}}{3 d f \left (c^2-d^2\right ) (c+d \sin (e+f x))^{3/2}}-\frac {-\frac {\int \frac {-7 a^2 b d^4+3 a^3 c d^3+5 a b^2 c d^3-(b c-a d) \left (\left (6 c^2-9 d^2\right ) b^2+4 a c d b-a^2 d^2\right ) \sin (e+f x) d-b^3 \left (3 c^4-2 c^2 d^2\right )}{\sqrt {a+b \sin (e+f x)} (c+d \sin (e+f x))^{3/2}}dx}{d^2}-\frac {6 b^2 \sqrt {a+b} \left (c^2-d^2\right ) \sec (e+f x) (c+d \sin (e+f x)) \sqrt {\frac {(b c-a d) (1-\sin (e+f x))}{(a+b) (c+d \sin (e+f x))}} \sqrt {-\frac {(b c-a d) (\sin (e+f x)+1)}{(a-b) (c+d \sin (e+f x))}} \operatorname {EllipticPi}\left (\frac {(a+b) d}{b (c+d)},\arcsin \left (\frac {\sqrt {c+d} \sqrt {a+b \sin (e+f x)}}{\sqrt {a+b} \sqrt {c+d \sin (e+f x)}}\right ),\frac {(a+b) (c-d)}{(a-b) (c+d)}\right )}{d^2 f \sqrt {c+d}}}{3 d \left (c^2-d^2\right )}\)

\(\Big \downarrow \) 3477

\(\displaystyle \frac {2 (b c-a d)^2 \cos (e+f x) \sqrt {a+b \sin (e+f x)}}{3 d f \left (c^2-d^2\right ) (c+d \sin (e+f x))^{3/2}}-\frac {-\frac {\frac {\left (a^3 d^3 (3 c+d)-a^2 b d^3 (5 c+7 d)-a b^2 d^2 \left (2 c^2-5 c d-9 d^2\right )+b^3 (-c) \left (3 c^3-6 c^2 d-2 c d^2+9 d^3\right )\right ) \int \frac {1}{\sqrt {a+b \sin (e+f x)} \sqrt {c+d \sin (e+f x)}}dx}{c-d}-\frac {d (b c-a d)^2 \left (4 a c d+3 b c^2-7 b d^2\right ) \int \frac {\sin (e+f x)+1}{\sqrt {a+b \sin (e+f x)} (c+d \sin (e+f x))^{3/2}}dx}{c-d}}{d^2}-\frac {6 b^2 \sqrt {a+b} \left (c^2-d^2\right ) \sec (e+f x) (c+d \sin (e+f x)) \sqrt {\frac {(b c-a d) (1-\sin (e+f x))}{(a+b) (c+d \sin (e+f x))}} \sqrt {-\frac {(b c-a d) (\sin (e+f x)+1)}{(a-b) (c+d \sin (e+f x))}} \operatorname {EllipticPi}\left (\frac {(a+b) d}{b (c+d)},\arcsin \left (\frac {\sqrt {c+d} \sqrt {a+b \sin (e+f x)}}{\sqrt {a+b} \sqrt {c+d \sin (e+f x)}}\right ),\frac {(a+b) (c-d)}{(a-b) (c+d)}\right )}{d^2 f \sqrt {c+d}}}{3 d \left (c^2-d^2\right )}\)

\(\Big \downarrow \) 3042

\(\displaystyle \frac {2 (b c-a d)^2 \cos (e+f x) \sqrt {a+b \sin (e+f x)}}{3 d f \left (c^2-d^2\right ) (c+d \sin (e+f x))^{3/2}}-\frac {-\frac {\frac {\left (a^3 d^3 (3 c+d)-a^2 b d^3 (5 c+7 d)-a b^2 d^2 \left (2 c^2-5 c d-9 d^2\right )+b^3 (-c) \left (3 c^3-6 c^2 d-2 c d^2+9 d^3\right )\right ) \int \frac {1}{\sqrt {a+b \sin (e+f x)} \sqrt {c+d \sin (e+f x)}}dx}{c-d}-\frac {d (b c-a d)^2 \left (4 a c d+3 b c^2-7 b d^2\right ) \int \frac {\sin (e+f x)+1}{\sqrt {a+b \sin (e+f x)} (c+d \sin (e+f x))^{3/2}}dx}{c-d}}{d^2}-\frac {6 b^2 \sqrt {a+b} \left (c^2-d^2\right ) \sec (e+f x) (c+d \sin (e+f x)) \sqrt {\frac {(b c-a d) (1-\sin (e+f x))}{(a+b) (c+d \sin (e+f x))}} \sqrt {-\frac {(b c-a d) (\sin (e+f x)+1)}{(a-b) (c+d \sin (e+f x))}} \operatorname {EllipticPi}\left (\frac {(a+b) d}{b (c+d)},\arcsin \left (\frac {\sqrt {c+d} \sqrt {a+b \sin (e+f x)}}{\sqrt {a+b} \sqrt {c+d \sin (e+f x)}}\right ),\frac {(a+b) (c-d)}{(a-b) (c+d)}\right )}{d^2 f \sqrt {c+d}}}{3 d \left (c^2-d^2\right )}\)

\(\Big \downarrow \) 3297

\(\displaystyle \frac {2 (b c-a d)^2 \cos (e+f x) \sqrt {a+b \sin (e+f x)}}{3 d f \left (c^2-d^2\right ) (c+d \sin (e+f x))^{3/2}}-\frac {-\frac {\frac {2 \sqrt {a+b} \left (a^3 d^3 (3 c+d)-a^2 b d^3 (5 c+7 d)-a b^2 d^2 \left (2 c^2-5 c d-9 d^2\right )+b^3 (-c) \left (3 c^3-6 c^2 d-2 c d^2+9 d^3\right )\right ) \sec (e+f x) (c+d \sin (e+f x)) \sqrt {\frac {(b c-a d) (1-\sin (e+f x))}{(a+b) (c+d \sin (e+f x))}} \sqrt {-\frac {(b c-a d) (\sin (e+f x)+1)}{(a-b) (c+d \sin (e+f x))}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {c+d} \sqrt {a+b \sin (e+f x)}}{\sqrt {a+b} \sqrt {c+d \sin (e+f x)}}\right ),\frac {(a+b) (c-d)}{(a-b) (c+d)}\right )}{f (c-d) \sqrt {c+d} (b c-a d)}-\frac {d (b c-a d)^2 \left (4 a c d+3 b c^2-7 b d^2\right ) \int \frac {\sin (e+f x)+1}{\sqrt {a+b \sin (e+f x)} (c+d \sin (e+f x))^{3/2}}dx}{c-d}}{d^2}-\frac {6 b^2 \sqrt {a+b} \left (c^2-d^2\right ) \sec (e+f x) (c+d \sin (e+f x)) \sqrt {\frac {(b c-a d) (1-\sin (e+f x))}{(a+b) (c+d \sin (e+f x))}} \sqrt {-\frac {(b c-a d) (\sin (e+f x)+1)}{(a-b) (c+d \sin (e+f x))}} \operatorname {EllipticPi}\left (\frac {(a+b) d}{b (c+d)},\arcsin \left (\frac {\sqrt {c+d} \sqrt {a+b \sin (e+f x)}}{\sqrt {a+b} \sqrt {c+d \sin (e+f x)}}\right ),\frac {(a+b) (c-d)}{(a-b) (c+d)}\right )}{d^2 f \sqrt {c+d}}}{3 d \left (c^2-d^2\right )}\)

\(\Big \downarrow \) 3475

\(\displaystyle \frac {2 (b c-a d)^2 \cos (e+f x) \sqrt {a+b \sin (e+f x)}}{3 d \left (c^2-d^2\right ) f (c+d \sin (e+f x))^{3/2}}-\frac {-\frac {6 \sqrt {a+b} \left (c^2-d^2\right ) \operatorname {EllipticPi}\left (\frac {(a+b) d}{b (c+d)},\arcsin \left (\frac {\sqrt {c+d} \sqrt {a+b \sin (e+f x)}}{\sqrt {a+b} \sqrt {c+d \sin (e+f x)}}\right ),\frac {(a+b) (c-d)}{(a-b) (c+d)}\right ) \sec (e+f x) \sqrt {\frac {(b c-a d) (1-\sin (e+f x))}{(a+b) (c+d \sin (e+f x))}} \sqrt {-\frac {(b c-a d) (\sin (e+f x)+1)}{(a-b) (c+d \sin (e+f x))}} (c+d \sin (e+f x)) b^2}{d^2 \sqrt {c+d} f}-\frac {\frac {2 (a-b) \sqrt {a+b} d \left (3 b c^2+4 a d c-7 b d^2\right ) E\left (\arcsin \left (\frac {\sqrt {c+d} \sqrt {a+b \sin (e+f x)}}{\sqrt {a+b} \sqrt {c+d \sin (e+f x)}}\right )|\frac {(a+b) (c-d)}{(a-b) (c+d)}\right ) \sec (e+f x) \sqrt {\frac {(b c-a d) (1-\sin (e+f x))}{(a+b) (c+d \sin (e+f x))}} \sqrt {-\frac {(b c-a d) (\sin (e+f x)+1)}{(a-b) (c+d \sin (e+f x))}} (c+d \sin (e+f x))}{(c-d) \sqrt {c+d} f}+\frac {2 \sqrt {a+b} \left (-c \left (3 c^3-6 d c^2-2 d^2 c+9 d^3\right ) b^3-a d^2 \left (2 c^2-5 d c-9 d^2\right ) b^2-a^2 d^3 (5 c+7 d) b+a^3 d^3 (3 c+d)\right ) \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {c+d} \sqrt {a+b \sin (e+f x)}}{\sqrt {a+b} \sqrt {c+d \sin (e+f x)}}\right ),\frac {(a+b) (c-d)}{(a-b) (c+d)}\right ) \sec (e+f x) \sqrt {\frac {(b c-a d) (1-\sin (e+f x))}{(a+b) (c+d \sin (e+f x))}} \sqrt {-\frac {(b c-a d) (\sin (e+f x)+1)}{(a-b) (c+d \sin (e+f x))}} (c+d \sin (e+f x))}{(c-d) \sqrt {c+d} (b c-a d) f}}{d^2}}{3 d \left (c^2-d^2\right )}\)

Input:

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

Output:

(2*(b*c - a*d)^2*Cos[e + f*x]*Sqrt[a + b*Sin[e + f*x]])/(3*d*(c^2 - d^2)*f 
*(c + d*Sin[e + f*x])^(3/2)) - ((-6*b^2*Sqrt[a + b]*(c^2 - d^2)*EllipticPi 
[((a + b)*d)/(b*(c + d)), ArcSin[(Sqrt[c + d]*Sqrt[a + b*Sin[e + f*x]])/(S 
qrt[a + b]*Sqrt[c + d*Sin[e + f*x]])], ((a + b)*(c - d))/((a - b)*(c + d)) 
]*Sec[e + f*x]*Sqrt[((b*c - a*d)*(1 - Sin[e + f*x]))/((a + b)*(c + d*Sin[e 
 + f*x]))]*Sqrt[-(((b*c - a*d)*(1 + Sin[e + f*x]))/((a - b)*(c + d*Sin[e + 
 f*x])))]*(c + d*Sin[e + f*x]))/(d^2*Sqrt[c + d]*f) - ((2*(a - b)*Sqrt[a + 
 b]*d*(3*b*c^2 + 4*a*c*d - 7*b*d^2)*EllipticE[ArcSin[(Sqrt[c + d]*Sqrt[a + 
 b*Sin[e + f*x]])/(Sqrt[a + b]*Sqrt[c + d*Sin[e + f*x]])], ((a + b)*(c - d 
))/((a - b)*(c + d))]*Sec[e + f*x]*Sqrt[((b*c - a*d)*(1 - Sin[e + f*x]))/( 
(a + b)*(c + d*Sin[e + f*x]))]*Sqrt[-(((b*c - a*d)*(1 + Sin[e + f*x]))/((a 
 - b)*(c + d*Sin[e + f*x])))]*(c + d*Sin[e + f*x]))/((c - d)*Sqrt[c + d]*f 
) + (2*Sqrt[a + b]*(a^3*d^3*(3*c + d) - a^2*b*d^3*(5*c + 7*d) - a*b^2*d^2* 
(2*c^2 - 5*c*d - 9*d^2) - b^3*c*(3*c^3 - 6*c^2*d - 2*c*d^2 + 9*d^3))*Ellip 
ticF[ArcSin[(Sqrt[c + d]*Sqrt[a + b*Sin[e + f*x]])/(Sqrt[a + b]*Sqrt[c + d 
*Sin[e + f*x]])], ((a + b)*(c - d))/((a - b)*(c + d))]*Sec[e + f*x]*Sqrt[( 
(b*c - a*d)*(1 - Sin[e + f*x]))/((a + b)*(c + d*Sin[e + f*x]))]*Sqrt[-(((b 
*c - a*d)*(1 + Sin[e + f*x]))/((a - b)*(c + d*Sin[e + f*x])))]*(c + d*Sin[ 
e + f*x]))/((c - d)*Sqrt[c + d]*(b*c - a*d)*f))/d^2)/(3*d*(c^2 - d^2))
 

Defintions of rubi rules used

rule 25

Int[-(Fx_), x_Symbol] :> Simp[Identity[-1]   Int[Fx, x], x]
 

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 3042

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

rule 3271

Int[((a_.) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_)*((c_.) + (d_.)*sin[(e_.) + 
 (f_.)*(x_)])^(n_), x_Symbol] :> Simp[(-(b^2*c^2 - 2*a*b*c*d + a^2*d^2))*Co 
s[e + f*x]*(a + b*Sin[e + f*x])^(m - 2)*((c + d*Sin[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*Sin 
[e + f*x])^(m - 3)*(c + d*Sin[e + f*x])^(n + 1)*Simp[b*(m - 2)*(b*c - a*d)^ 
2 + a*d*(n + 1)*(c*(a^2 + b^2) - 2*a*b*d) + (b*(n + 1)*(a*b*c^2 + c*d*(a^2 
+ b^2) - 3*a*b*d^2) - a*(n + 2)*(b*c - a*d)^2)*Sin[e + f*x] + b*(b^2*(c^2 - 
 d^2) - m*(b*c - a*d)^2 + d*n*(2*a*b*c - d*(a^2 + b^2)))*Sin[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[m] || 
IntegersQ[2*m, 2*n])
 

rule 3290

Int[Sqrt[(a_) + (b_.)*sin[(e_.) + (f_.)*(x_)]]/Sqrt[(c_.) + (d_.)*sin[(e_.) 
 + (f_.)*(x_)]], x_Symbol] :> Simp[2*((a + b*Sin[e + f*x])/(d*f*Rt[(a + b)/ 
(c + d), 2]*Cos[e + f*x]))*Sqrt[(b*c - a*d)*((1 + Sin[e + f*x])/((c - d)*(a 
 + b*Sin[e + f*x])))]*Sqrt[(-(b*c - a*d))*((1 - Sin[e + f*x])/((c + d)*(a + 
 b*Sin[e + f*x])))]*EllipticPi[b*((c + d)/(d*(a + b))), ArcSin[Rt[(a + b)/( 
c + d), 2]*(Sqrt[c + d*Sin[e + f*x]]/Sqrt[a + b*Sin[e + f*x]])], (a - b)*(( 
c + d)/((a + b)*(c - d)))], 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] && PosQ[(a + b)/(c + d)]
 

rule 3297

Int[1/(Sqrt[(a_) + (b_.)*sin[(e_.) + (f_.)*(x_)]]*Sqrt[(c_) + (d_.)*sin[(e_ 
.) + (f_.)*(x_)]]), x_Symbol] :> Simp[2*((c + d*Sin[e + f*x])/(f*(b*c - a*d 
)*Rt[(c + d)/(a + b), 2]*Cos[e + f*x]))*Sqrt[(b*c - a*d)*((1 - Sin[e + f*x] 
)/((a + b)*(c + d*Sin[e + f*x])))]*Sqrt[(-(b*c - a*d))*((1 + Sin[e + f*x])/ 
((a - b)*(c + d*Sin[e + f*x])))]*EllipticF[ArcSin[Rt[(c + d)/(a + b), 2]*(S 
qrt[a + b*Sin[e + f*x]]/Sqrt[c + d*Sin[e + f*x]])], (a + b)*((c - d)/((a - 
b)*(c + d)))], x] /; FreeQ[{a, b, c, d, e, f}, x] && NeQ[b*c - a*d, 0] && N 
eQ[a^2 - b^2, 0] && NeQ[c^2 - d^2, 0] && PosQ[(c + d)/(a + b)]
 

rule 3475

Int[((A_) + (B_.)*sin[(e_.) + (f_.)*(x_)])/(((a_) + (b_.)*sin[(e_.) + (f_.) 
*(x_)])^(3/2)*Sqrt[(c_) + (d_.)*sin[(e_.) + (f_.)*(x_)]]), x_Symbol] :> Sim 
p[-2*A*(c - d)*((a + b*Sin[e + f*x])/(f*(b*c - a*d)^2*Rt[(a + b)/(c + d), 2 
]*Cos[e + f*x]))*Sqrt[(b*c - a*d)*((1 + Sin[e + f*x])/((c - d)*(a + b*Sin[e 
 + f*x])))]*Sqrt[(-(b*c - a*d))*((1 - Sin[e + f*x])/((c + d)*(a + b*Sin[e + 
 f*x])))]*EllipticE[ArcSin[Rt[(a + b)/(c + d), 2]*(Sqrt[c + d*Sin[e + f*x]] 
/Sqrt[a + b*Sin[e + f*x]])], (a - b)*((c + d)/((a + b)*(c - d)))], x] /; Fr 
eeQ[{a, b, c, d, e, f, A, B}, x] && NeQ[b*c - a*d, 0] && NeQ[a^2 - b^2, 0] 
&& NeQ[c^2 - d^2, 0] && EqQ[A, B] && PosQ[(a + b)/(c + d)]
 

rule 3477

Int[((A_.) + (B_.)*sin[(e_.) + (f_.)*(x_)])/(((a_.) + (b_.)*sin[(e_.) + (f_ 
.)*(x_)])^(3/2)*Sqrt[(c_) + (d_.)*sin[(e_.) + (f_.)*(x_)]]), x_Symbol] :> S 
imp[(A - B)/(a - b)   Int[1/(Sqrt[a + b*Sin[e + f*x]]*Sqrt[c + d*Sin[e + f* 
x]]), x], x] - Simp[(A*b - a*B)/(a - b)   Int[(1 + Sin[e + f*x])/((a + b*Si 
n[e + f*x])^(3/2)*Sqrt[c + d*Sin[e + f*x]]), x], x] /; FreeQ[{a, b, c, d, e 
, f, A, B}, x] && NeQ[b*c - a*d, 0] && NeQ[a^2 - b^2, 0] && NeQ[c^2 - d^2, 
0] && NeQ[A, B]
 

rule 3532

Int[((A_.) + (B_.)*sin[(e_.) + (f_.)*(x_)] + (C_.)*sin[(e_.) + (f_.)*(x_)]^ 
2)/(((a_.) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(3/2)*Sqrt[(c_.) + (d_.)*sin[(e 
_.) + (f_.)*(x_)]]), x_Symbol] :> Simp[C/b^2   Int[Sqrt[a + b*Sin[e + f*x]] 
/Sqrt[c + d*Sin[e + f*x]], x], x] + Simp[1/b^2   Int[(A*b^2 - a^2*C + b*(b* 
B - 2*a*C)*Sin[e + f*x])/((a + b*Sin[e + f*x])^(3/2)*Sqrt[c + d*Sin[e + f*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]
 

Maple [B] (warning: unable to verify)

result has leaf size over 500,000. Avoiding possible recursion issues.

Time = 56.63 (sec) , antiderivative size = 709455, normalized size of antiderivative = 962.63

method	result	size
default	\(\text {Expression too large to display}\)	\(709455\)

Input:

int((a+b*sin(f*x+e))^(5/2)/(c+d*sin(f*x+e))^(5/2),x,method=_RETURNVERBOSE)
 

Output:

result too large to display
 

Fricas [F(-1)]

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

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

Output:

Timed out
 

Sympy [F(-1)]

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

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

Output:

Timed out
 

Maxima [F]

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

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

Output:

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

Giac [F]

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

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

Output:

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

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

\[ \int \frac {(a+b \sin (e+f x))^{5/2}}{(c+d \sin (e+f x))^{5/2}} \, dx=\left (\int \frac {\sqrt {\sin \left (f x +e \right ) d +c}\, \sqrt {\sin \left (f x +e \right ) b +a}\, \sin \left (f x +e \right )^{2}}{\sin \left (f x +e \right )^{3} d^{3}+3 \sin \left (f x +e \right )^{2} c \,d^{2}+3 \sin \left (f x +e \right ) c^{2} d +c^{3}}d x \right ) b^{2}+2 \left (\int \frac {\sqrt {\sin \left (f x +e \right ) d +c}\, \sqrt {\sin \left (f x +e \right ) b +a}\, \sin \left (f x +e \right )}{\sin \left (f x +e \right )^{3} d^{3}+3 \sin \left (f x +e \right )^{2} c \,d^{2}+3 \sin \left (f x +e \right ) c^{2} d +c^{3}}d x \right ) a b +\left (\int \frac {\sqrt {\sin \left (f x +e \right ) d +c}\, \sqrt {\sin \left (f x +e \right ) b +a}}{\sin \left (f x +e \right )^{3} d^{3}+3 \sin \left (f x +e \right )^{2} c \,d^{2}+3 \sin \left (f x +e \right ) c^{2} d +c^{3}}d x \right ) a^{2} \] Input:

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

Output:

int((sqrt(sin(e + f*x)*d + c)*sqrt(sin(e + f*x)*b + a)*sin(e + f*x)**2)/(s 
in(e + f*x)**3*d**3 + 3*sin(e + f*x)**2*c*d**2 + 3*sin(e + f*x)*c**2*d + c 
**3),x)*b**2 + 2*int((sqrt(sin(e + f*x)*d + c)*sqrt(sin(e + f*x)*b + a)*si 
n(e + f*x))/(sin(e + f*x)**3*d**3 + 3*sin(e + f*x)**2*c*d**2 + 3*sin(e + f 
*x)*c**2*d + c**3),x)*a*b + int((sqrt(sin(e + f*x)*d + c)*sqrt(sin(e + f*x 
)*b + a))/(sin(e + f*x)**3*d**3 + 3*sin(e + f*x)**2*c*d**2 + 3*sin(e + f*x 
)*c**2*d + c**3),x)*a**2