\(\int \frac {(a+b \sin (e+f x))^{5/2}}{(c+d \sin (e+f x))^{3/2}} \, dx\) [788]

Optimal result

Integrand size = 29, antiderivative size = 780 \[ \int \frac {(a+b \sin (e+f x))^{5/2}}{(c+d \sin (e+f x))^{3/2}} \, dx=-\frac {\sqrt {a+b} \left (4 a b c d-2 a^2 d^2-b^2 \left (3 c^2-d^2\right )\right ) E\left (\arcsin \left (\frac {\sqrt {a+b} \sqrt {c+d \sin (e+f x)}}{\sqrt {c+d} \sqrt {a+b \sin (e+f x)}}\right )|\frac {(a-b) (c+d)}{(a+b) (c-d)}\right ) \sec (e+f x) \sqrt {-\frac {(b c-a d) (1-\sin (e+f x))}{(c+d) (a+b \sin (e+f x))}} \sqrt {\frac {(b c-a d) (1+\sin (e+f x))}{(c-d) (a+b \sin (e+f x))}} (a+b \sin (e+f x))}{d^2 \sqrt {c+d} (b c-a d) f}-\frac {b \sqrt {c+d} (3 b c-5 a d) \operatorname {EllipticPi}\left (\frac {b (c+d)}{(a+b) d},\arcsin \left (\frac {\sqrt {a+b} \sqrt {c+d \sin (e+f x)}}{\sqrt {c+d} \sqrt {a+b \sin (e+f x)}}\right ),\frac {(a-b) (c+d)}{(a+b) (c-d)}\right ) \sec (e+f x) \sqrt {-\frac {(b c-a d) (1-\sin (e+f x))}{(c+d) (a+b \sin (e+f x))}} \sqrt {\frac {(b c-a d) (1+\sin (e+f x))}{(c-d) (a+b \sin (e+f x))}} (a+b \sin (e+f x))}{\sqrt {a+b} d^3 f}+\frac {2 (b c-a d)^2 \cos (e+f x) \sqrt {a+b \sin (e+f x)}}{d \left (c^2-d^2\right ) f \sqrt {c+d \sin (e+f x)}}+\frac {b \left (4 a b c d-2 a^2 d^2-b^2 \left (3 c^2-d^2\right )\right ) \cos (e+f x) \sqrt {c+d \sin (e+f x)}}{d^2 \left (c^2-d^2\right ) f \sqrt {a+b \sin (e+f x)}}-\frac {(a+b)^{3/2} (2 a d-b (3 c+d)) \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {c+d} \sqrt {a+b \sin (e+f x)}}{\sqrt {a+b} \sqrt {c+d \sin (e+f x)}}\right ),\frac {(a+b) (c-d)}{(a-b) (c+d)}\right ) \sec (e+f x) \sqrt {\frac {(b c-a d) (1-\sin (e+f x))}{(a+b) (c+d \sin (e+f x))}} \sqrt {-\frac {(b c-a d) (1+\sin (e+f x))}{(a-b) (c+d \sin (e+f x))}} (c+d \sin (e+f x))}{d^2 (c+d)^{3/2} f} \] Output:

-(a+b)^(1/2)*(4*a*b*c*d-2*a^2*d^2-b^2*(3*c^2-d^2))*EllipticE((a+b)^(1/2)*( 
c+d*sin(f*x+e))^(1/2)/(c+d)^(1/2)/(a+b*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))/(c+d)/(a+b*sin(f*x+ 
e)))^(1/2)*((-a*d+b*c)*(1+sin(f*x+e))/(c-d)/(a+b*sin(f*x+e)))^(1/2)*(a+b*s 
in(f*x+e))/d^2/(c+d)^(1/2)/(-a*d+b*c)/f-b*(c+d)^(1/2)*(-5*a*d+3*b*c)*Ellip 
ticPi((a+b)^(1/2)*(c+d*sin(f*x+e))^(1/2)/(c+d)^(1/2)/(a+b*sin(f*x+e))^(1/2 
),b*(c+d)/(a+b)/d,((a-b)*(c+d)/(a+b)/(c-d))^(1/2))*sec(f*x+e)*(-(-a*d+b*c) 
*(1-sin(f*x+e))/(c+d)/(a+b*sin(f*x+e)))^(1/2)*((-a*d+b*c)*(1+sin(f*x+e))/( 
c-d)/(a+b*sin(f*x+e)))^(1/2)*(a+b*sin(f*x+e))/(a+b)^(1/2)/d^3/f+2*(-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))^(1/2 
)+b*(4*a*b*c*d-2*a^2*d^2-b^2*(3*c^2-d^2))*cos(f*x+e)*(c+d*sin(f*x+e))^(1/2 
)/d^2/(c^2-d^2)/f/(a+b*sin(f*x+e))^(1/2)-(a+b)^(3/2)*(2*a*d-b*(3*c+d))*Ell 
ipticF((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))/d^2/(c+d)^(3/2)/f
 

Mathematica [B] (warning: unable to verify)

Leaf count is larger than twice the leaf count of optimal. \(2006\) vs. \(2(780)=1560\).

Time = 9.37 (sec) , antiderivative size = 2006, normalized size of antiderivative = 2.57 \[ \int \frac {(a+b \sin (e+f x))^{5/2}}{(c+d \sin (e+f x))^{3/2}} \, dx=\text {Result too large to show} \] Input:

Integrate[(a + b*Sin[e + f*x])^(5/2)/(c + d*Sin[e + f*x])^(3/2),x]
 

Output:

(-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]) 
*Sqrt[a + b*Sin[e + f*x]])/(d*(-c^2 + d^2)*f*Sqrt[c + d*Sin[e + f*x]]) - ( 
(-4*(-(b*c) + a*d)*(-(b^3*c^2) - 2*a^3*c*d - 2*a*b^2*c*d + 4*a^2*b*d^2 + b 
^3*d^2)*Sqrt[((c + d)*Cot[(-e + Pi/2 - f*x)/2]^2)/(-c + d)]*EllipticF[ArcS 
in[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*Si 
n[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^2 + 2*a^2*b*c 
*d - 2*b^3*c*d - 2*a^3*d^2 + 6*a*b^2*d^2)*((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]]) - (Sqrt[ 
((c + d)*Cot[(-e + Pi/2 - f*x)/2]^2)/(-c + d)]*EllipticPi[(-(b*c) + a*d)/( 
(a + b)*d), 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)...
 

Rubi [A] (verified)

Time = 3.98 (sec) , antiderivative size = 819, normalized size of antiderivative = 1.05, number of steps used = 14, number of rules used = 14, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.483, Rules used = {3042, 3271, 27, 3042, 3540, 25, 3042, 3532, 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.

\(\displaystyle \int \frac {(a+b \sin (e+f x))^{5/2}}{(c+d \sin (e+f x))^{3/2}} \, dx\)

\(\Big \downarrow \) 3042

\(\displaystyle \int \frac {(a+b \sin (e+f x))^{5/2}}{(c+d \sin (e+f x))^{3/2}}dx\)

\(\Big \downarrow \) 3271

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 3540

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

\(\Big \downarrow \) 25

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

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 3532

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

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 3290

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

\(\Big \downarrow \) 3477

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

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 3297

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

\(\Big \downarrow \) 3475

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

Input:

Int[(a + b*Sin[e + f*x])^(5/2)/(c + d*Sin[e + f*x])^(3/2),x]
 

Output:

(2*(b*c - a*d)^2*Cos[e + f*x]*Sqrt[a + b*Sin[e + f*x]])/(d*(c^2 - d^2)*f*S 
qrt[c + d*Sin[e + f*x]]) - (-((b*(4*a*b*c*d - 2*a^2*d^2 - b^2*(3*c^2 - d^2 
))*Cos[e + f*x]*Sqrt[c + d*Sin[e + f*x]])/(d*f*Sqrt[a + b*Sin[e + f*x]])) 
- ((-2*b*Sqrt[c + d]*(3*b*c - 5*a*d)*(c^2 - d^2)*EllipticPi[(b*(c + d))/(( 
a + b)*d), ArcSin[(Sqrt[a + b]*Sqrt[c + d*Sin[e + f*x]])/(Sqrt[c + d]*Sqrt 
[a + b*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]))/((c + d)*(a + b*Sin[e + f*x])))]*S 
qrt[((b*c - a*d)*(1 + Sin[e + f*x]))/((c - d)*(a + b*Sin[e + f*x]))]*(a + 
b*Sin[e + f*x]))/(Sqrt[a + b]*d*f) + ((-2*b^2*Sqrt[a + b]*(c - d)*Sqrt[c + 
 d]*(4*a*b*c*d - 2*a^2*d^2 - b^2*(3*c^2 - d^2))*EllipticE[ArcSin[(Sqrt[a + 
 b]*Sqrt[c + d*Sin[e + f*x]])/(Sqrt[c + d]*Sqrt[a + b*Sin[e + f*x]])], ((a 
 - b)*(c + d))/((a + b)*(c - d))]*Sec[e + f*x]*Sqrt[-(((b*c - a*d)*(1 - Si 
n[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]))]*(a + b*Sin[e + f*x]))/((b*c - a* 
d)*f) - (2*b^2*(a + b)^(3/2)*(c - d)*(2*a*d - b*(3*c + d))*EllipticF[ArcSi 
n[(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])) 
/(Sqrt[c + d]*f))/b^2)/(2*d))/(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]
 

rule 3540

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

Maple [B] (warning: unable to verify)

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

Time = 36.77 (sec) , antiderivative size = 562817, normalized size of antiderivative = 721.56

method	result	size
default	\(\text {Expression too large to display}\)	\(562817\)

Input:

int((a+b*sin(f*x+e))^(5/2)/(c+d*sin(f*x+e))^(3/2),x,method=_RETURNVERBOSE)
                                                                                    
                                                                                    
 

Output:

result too large to display
 

Fricas [F(-1)]

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

integrate((a+b*sin(f*x+e))^(5/2)/(c+d*sin(f*x+e))^(3/2),x, algorithm="fric 
as")
 

Output:

Timed out
 

Sympy [F(-1)]

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

integrate((a+b*sin(f*x+e))**(5/2)/(c+d*sin(f*x+e))**(3/2),x)
 

Output:

Timed out
 

Maxima [F]

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

integrate((a+b*sin(f*x+e))^(5/2)/(c+d*sin(f*x+e))^(3/2),x, algorithm="maxi 
ma")
 

Output:

integrate((b*sin(f*x + e) + a)^(5/2)/(d*sin(f*x + e) + c)^(3/2), x)
 

Giac [F]

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

integrate((a+b*sin(f*x+e))^(5/2)/(c+d*sin(f*x+e))^(3/2),x, algorithm="giac 
")
 

Output:

integrate((b*sin(f*x + e) + a)^(5/2)/(d*sin(f*x + e) + c)^(3/2), x)
 

Mupad [F(-1)]

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

int((a + b*sin(e + f*x))^(5/2)/(c + d*sin(e + f*x))^(3/2),x)
 

Output:

int((a + b*sin(e + f*x))^(5/2)/(c + d*sin(e + f*x))^(3/2), x)
 

Reduce [F]

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

int((a+b*sin(f*x+e))^(5/2)/(c+d*sin(f*x+e))^(3/2),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)**2*d**2 + 2*sin(e + f*x)*c*d + c**2),x)*b**2 + 2*int((sqrt(sin 
(e + f*x)*d + c)*sqrt(sin(e + f*x)*b + a)*sin(e + f*x))/(sin(e + f*x)**2*d 
**2 + 2*sin(e + f*x)*c*d + c**2),x)*a*b + int((sqrt(sin(e + f*x)*d + c)*sq 
rt(sin(e + f*x)*b + a))/(sin(e + f*x)**2*d**2 + 2*sin(e + f*x)*c*d + c**2) 
,x)*a**2