\(\int (a+b \sin (e+f x))^{5/2} (c+d \sin (e+f x))^{3/2} \, dx\) [785]

Optimal result

Integrand size = 29, antiderivative size = 1071 \[ \int (a+b \sin (e+f x))^{5/2} (c+d \sin (e+f x))^{3/2} \, dx =\text {Too large to display} \] Output:

1/192*(a+b)^(1/2)*(c-d)*(c+d)^(1/2)*(337*a^2*b*c*d^2+15*a^3*d^3+a*b^2*d*(5 
7*c^2+284*d^2)-b^3*(9*c^3-156*c*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*sin(f*x+e))/b/ 
d^2/(-a*d+b*c)/f+1/64*(c+d)^(1/2)*(60*a^3*b*c*d^3-5*a^4*d^4-20*a*b^3*c*d*( 
c^2-12*d^2)+3*b^4*(c^2+4*d^2)^2+30*a^2*b^2*d^2*(3*c^2+4*d^2))*EllipticPi(( 
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))/b^2/(a+b)^(1/2)/d^3/f-1/192*(337*a^ 
2*b*c*d^2+15*a^3*d^3+a*b^2*d*(57*c^2+284*d^2)-b^3*(9*c^3-156*c*d^2))*cos(f 
*x+e)*(c+d*sin(f*x+e))^(1/2)/d^2/f/(a+b*sin(f*x+e))^(1/2)-1/96*(54*a*b*c*d 
+59*a^2*d^2-9*b^2*(c^2-4*d^2))*cos(f*x+e)*(a+b*sin(f*x+e))^(1/2)*(c+d*sin( 
f*x+e))^(1/2)/d/f-1/192*(a+b)^(3/2)*(15*a^3*d^3-15*a^2*b*d^2*(11*c+2*d)-a* 
b^2*d*(51*c^2+172*c*d+212*d^2)+b^3*(9*c^3-6*c^2*d-156*c*d^2-72*d^3))*Ellip 
ticF((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))/b^2/d^2/(c+d)^(1/2)/f+1/24*b*(-17*a*d+3*b*...
 

Mathematica [A] (warning: unable to verify)

Time = 9.51 (sec) , antiderivative size = 2091, normalized size of antiderivative = 1.95 \[ \int (a+b \sin (e+f x))^{5/2} (c+d \sin (e+f x))^{3/2} \, dx=\text {Result too large to show} \] Input:

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

Output:

((-4*(-(b*c) + a*d)*(-3*b^3*c^3 + 384*a^3*c^2*d + 451*a*b^2*c^2*d + 971*a^ 
2*b*c*d^2 + 228*b^3*c*d^2 + 133*a^3*d^3 + 356*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*S 
qrt[((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)*(-12*a*b^2*c^3 + 664*a^2*b*c^2*d + 228*b^3*c^2*d + 
 532*a^3*c*d^2 + 1160*a*b^2*c*d^2 + 644*a^2*b*d^3 + 144*b^3*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]]) - (Sqrt[((c + d)*Cot[(-e + Pi/2 - f*x)/2]^2)/(-c + d)]*Elli 
pticPi[(-(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))]*Sec[e + f*x]*Sin[(-e + Pi/2 - f*x)/2]^4*Sqrt[((c...
 

Rubi [A] (verified)

Time = 5.99 (sec) , antiderivative size = 1105, normalized size of antiderivative = 1.03, number of steps used = 21, number of rules used = 21, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.724, Rules used = {3042, 3272, 27, 3042, 3528, 27, 3042, 3528, 27, 3042, 3540, 3042, 3532, 25, 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.

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

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 3272

\(\displaystyle \frac {\int \frac {(c+d \sin (e+f x))^{3/2} \left (8 d a^3+5 b^2 d a-b^2 (3 b c-17 a d) \sin ^2(e+f x)+b^3 c-2 b \left (-12 d a^2+b c a-3 b^2 d\right ) \sin (e+f x)\right )}{2 \sqrt {a+b \sin (e+f x)}}dx}{4 d}-\frac {b^2 \cos (e+f x) \sqrt {a+b \sin (e+f x)} (c+d \sin (e+f x))^{5/2}}{4 d f}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {\int \frac {(c+d \sin (e+f x))^{3/2} \left (8 d a^3+5 b^2 d a-b^2 (3 b c-17 a d) \sin ^2(e+f x)+b^3 c-2 b \left (-12 d a^2+b c a-3 b^2 d\right ) \sin (e+f x)\right )}{\sqrt {a+b \sin (e+f x)}}dx}{8 d}-\frac {b^2 \cos (e+f x) \sqrt {a+b \sin (e+f x)} (c+d \sin (e+f x))^{5/2}}{4 d f}\)

\(\Big \downarrow \) 3042

\(\displaystyle \frac {\int \frac {(c+d \sin (e+f x))^{3/2} \left (8 d a^3+5 b^2 d a-b^2 (3 b c-17 a d) \sin (e+f x)^2+b^3 c-2 b \left (-12 d a^2+b c a-3 b^2 d\right ) \sin (e+f x)\right )}{\sqrt {a+b \sin (e+f x)}}dx}{8 d}-\frac {b^2 \cos (e+f x) \sqrt {a+b \sin (e+f x)} (c+d \sin (e+f x))^{5/2}}{4 d f}\)

\(\Big \downarrow \) 3528

\(\displaystyle \frac {\frac {\int \frac {\sqrt {c+d \sin (e+f x)} \left (b^2 \left (-9 \left (c^2-4 d^2\right ) b^2+54 a c d b+59 a^2 d^2\right ) \sin ^2(e+f x)+2 b \left (24 d^2 a^3+55 b c d a^2-b^2 \left (3 c^2-49 d^2\right ) a+15 b^3 c d\right ) \sin (e+f x)+b \left (48 c d a^3+51 b d^2 a^2+38 b^2 c d a+3 b^3 c^2\right )\right )}{2 \sqrt {a+b \sin (e+f x)}}dx}{3 b}+\frac {b (3 b c-17 a d) \cos (e+f x) \sqrt {a+b \sin (e+f x)} (c+d \sin (e+f x))^{3/2}}{3 f}}{8 d}-\frac {b^2 \cos (e+f x) \sqrt {a+b \sin (e+f x)} (c+d \sin (e+f x))^{5/2}}{4 d f}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {\frac {\int \frac {\sqrt {c+d \sin (e+f x)} \left (b^2 \left (-9 \left (c^2-4 d^2\right ) b^2+54 a c d b+59 a^2 d^2\right ) \sin ^2(e+f x)+2 b \left (24 d^2 a^3+55 b c d a^2-b^2 \left (3 c^2-49 d^2\right ) a+15 b^3 c d\right ) \sin (e+f x)+b \left (48 c d a^3+51 b d^2 a^2+38 b^2 c d a+3 b^3 c^2\right )\right )}{\sqrt {a+b \sin (e+f x)}}dx}{6 b}+\frac {b (3 b c-17 a d) \cos (e+f x) \sqrt {a+b \sin (e+f x)} (c+d \sin (e+f x))^{3/2}}{3 f}}{8 d}-\frac {b^2 \cos (e+f x) \sqrt {a+b \sin (e+f x)} (c+d \sin (e+f x))^{5/2}}{4 d f}\)

\(\Big \downarrow \) 3042

\(\displaystyle \frac {\frac {\int \frac {\sqrt {c+d \sin (e+f x)} \left (b^2 \left (-9 \left (c^2-4 d^2\right ) b^2+54 a c d b+59 a^2 d^2\right ) \sin (e+f x)^2+2 b \left (24 d^2 a^3+55 b c d a^2-b^2 \left (3 c^2-49 d^2\right ) a+15 b^3 c d\right ) \sin (e+f x)+b \left (48 c d a^3+51 b d^2 a^2+38 b^2 c d a+3 b^3 c^2\right )\right )}{\sqrt {a+b \sin (e+f x)}}dx}{6 b}+\frac {b (3 b c-17 a d) \cos (e+f x) \sqrt {a+b \sin (e+f x)} (c+d \sin (e+f x))^{3/2}}{3 f}}{8 d}-\frac {b^2 \cos (e+f x) \sqrt {a+b \sin (e+f x)} (c+d \sin (e+f x))^{5/2}}{4 d f}\)

\(\Big \downarrow \) 3528

\(\displaystyle \frac {\frac {\frac {\int \frac {\left (-\left (\left (9 c^3-156 c d^2\right ) b^3\right )+a d \left (57 c^2+284 d^2\right ) b^2+337 a^2 c d^2 b+15 a^3 d^3\right ) \sin ^2(e+f x) b^2+\left (d \left (192 c^2+59 d^2\right ) a^3+317 b c d^2 a^2+b^2 d \left (197 c^2+36 d^2\right ) a+3 b^3 \left (c^3+12 d^2 c\right )\right ) b^2+2 \left (133 c d^2 a^3+b d \left (166 c^2+161 d^2\right ) a^2-b^2 c \left (3 c^2-290 d^2\right ) a+3 b^3 d \left (19 c^2+12 d^2\right )\right ) \sin (e+f x) b^2}{2 \sqrt {a+b \sin (e+f x)} \sqrt {c+d \sin (e+f x)}}dx}{2 b}-\frac {b \left (59 a^2 d^2+54 a b c d-9 b^2 \left (c^2-4 d^2\right )\right ) \cos (e+f x) \sqrt {a+b \sin (e+f x)} \sqrt {c+d \sin (e+f x)}}{2 f}}{6 b}+\frac {b (3 b c-17 a d) \cos (e+f x) \sqrt {a+b \sin (e+f x)} (c+d \sin (e+f x))^{3/2}}{3 f}}{8 d}-\frac {b^2 \cos (e+f x) \sqrt {a+b \sin (e+f x)} (c+d \sin (e+f x))^{5/2}}{4 d f}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {\frac {\frac {\int \frac {\left (-\left (\left (9 c^3-156 c d^2\right ) b^3\right )+a d \left (57 c^2+284 d^2\right ) b^2+337 a^2 c d^2 b+15 a^3 d^3\right ) \sin ^2(e+f x) b^2+\left (d \left (192 c^2+59 d^2\right ) a^3+317 b c d^2 a^2+b^2 d \left (197 c^2+36 d^2\right ) a+3 b^3 \left (c^3+12 d^2 c\right )\right ) b^2+2 \left (133 c d^2 a^3+b d \left (166 c^2+161 d^2\right ) a^2-b^2 c \left (3 c^2-290 d^2\right ) a+3 b^3 d \left (19 c^2+12 d^2\right )\right ) \sin (e+f x) b^2}{\sqrt {a+b \sin (e+f x)} \sqrt {c+d \sin (e+f x)}}dx}{4 b}-\frac {b \left (59 a^2 d^2+54 a b c d-9 b^2 \left (c^2-4 d^2\right )\right ) \cos (e+f x) \sqrt {a+b \sin (e+f x)} \sqrt {c+d \sin (e+f x)}}{2 f}}{6 b}+\frac {b (3 b c-17 a d) \cos (e+f x) \sqrt {a+b \sin (e+f x)} (c+d \sin (e+f x))^{3/2}}{3 f}}{8 d}-\frac {b^2 \cos (e+f x) \sqrt {a+b \sin (e+f x)} (c+d \sin (e+f x))^{5/2}}{4 d f}\)

\(\Big \downarrow \) 3042

\(\displaystyle \frac {\frac {\frac {\int \frac {\left (-\left (\left (9 c^3-156 c d^2\right ) b^3\right )+a d \left (57 c^2+284 d^2\right ) b^2+337 a^2 c d^2 b+15 a^3 d^3\right ) \sin (e+f x)^2 b^2+\left (d \left (192 c^2+59 d^2\right ) a^3+317 b c d^2 a^2+b^2 d \left (197 c^2+36 d^2\right ) a+3 b^3 \left (c^3+12 d^2 c\right )\right ) b^2+2 \left (133 c d^2 a^3+b d \left (166 c^2+161 d^2\right ) a^2-b^2 c \left (3 c^2-290 d^2\right ) a+3 b^3 d \left (19 c^2+12 d^2\right )\right ) \sin (e+f x) b^2}{\sqrt {a+b \sin (e+f x)} \sqrt {c+d \sin (e+f x)}}dx}{4 b}-\frac {b \left (59 a^2 d^2+54 a b c d-9 b^2 \left (c^2-4 d^2\right )\right ) \cos (e+f x) \sqrt {a+b \sin (e+f x)} \sqrt {c+d \sin (e+f x)}}{2 f}}{6 b}+\frac {b (3 b c-17 a d) \cos (e+f x) \sqrt {a+b \sin (e+f x)} (c+d \sin (e+f x))^{3/2}}{3 f}}{8 d}-\frac {b^2 \cos (e+f x) \sqrt {a+b \sin (e+f x)} (c+d \sin (e+f x))^{5/2}}{4 d f}\)

\(\Big \downarrow \) 3540

\(\displaystyle \frac {\frac {\frac {\frac {\int \frac {3 \left (3 \left (c^2+4 d^2\right )^2 b^4-20 a c d \left (c^2-12 d^2\right ) b^3+30 a^2 d^2 \left (3 c^2+4 d^2\right ) b^2+60 a^3 c d^3 b-5 a^4 d^4\right ) \sin ^2(e+f x) b^2+\left (d^2 \left (384 c^2+133 d^2\right ) a^4+956 b c d^3 a^3+2 b^2 d^2 \left (57 c^2+178 d^2\right ) a^2-4 b^3 c d \left (15 c^2+14 d^2\right ) a+3 b^4 \left (3 c^4-52 c^2 d^2\right )\right ) b^2+2 \left (251 c d^3 a^4+b d^2 \left (187 c^2+381 d^2\right ) a^3-b^2 c d \left (63 c^2-613 d^2\right ) a^2+b^3 \left (9 c^4+155 d^2 c^2+108 d^4\right ) a+3 b^4 c d \left (c^2+12 d^2\right )\right ) \sin (e+f x) b^2}{(a+b \sin (e+f x))^{3/2} \sqrt {c+d \sin (e+f x)}}dx}{2 d}-\frac {b^2 \left (15 a^3 d^3+337 a^2 b c d^2+a b^2 d \left (57 c^2+284 d^2\right )-\left (b^3 \left (9 c^3-156 c d^2\right )\right )\right ) \cos (e+f x) \sqrt {c+d \sin (e+f x)}}{d f \sqrt {a+b \sin (e+f x)}}}{4 b}-\frac {b \left (59 a^2 d^2+54 a b c d-9 b^2 \left (c^2-4 d^2\right )\right ) \cos (e+f x) \sqrt {a+b \sin (e+f x)} \sqrt {c+d \sin (e+f x)}}{2 f}}{6 b}+\frac {b (3 b c-17 a d) \cos (e+f x) \sqrt {a+b \sin (e+f x)} (c+d \sin (e+f x))^{3/2}}{3 f}}{8 d}-\frac {b^2 \cos (e+f x) \sqrt {a+b \sin (e+f x)} (c+d \sin (e+f x))^{5/2}}{4 d f}\)

\(\Big \downarrow \) 3042

\(\displaystyle \frac {\frac {\frac {\frac {\int \frac {3 \left (3 \left (c^2+4 d^2\right )^2 b^4-20 a c d \left (c^2-12 d^2\right ) b^3+30 a^2 d^2 \left (3 c^2+4 d^2\right ) b^2+60 a^3 c d^3 b-5 a^4 d^4\right ) \sin (e+f x)^2 b^2+\left (d^2 \left (384 c^2+133 d^2\right ) a^4+956 b c d^3 a^3+2 b^2 d^2 \left (57 c^2+178 d^2\right ) a^2-4 b^3 c d \left (15 c^2+14 d^2\right ) a+3 b^4 \left (3 c^4-52 c^2 d^2\right )\right ) b^2+2 \left (251 c d^3 a^4+b d^2 \left (187 c^2+381 d^2\right ) a^3-b^2 c d \left (63 c^2-613 d^2\right ) a^2+b^3 \left (9 c^4+155 d^2 c^2+108 d^4\right ) a+3 b^4 c d \left (c^2+12 d^2\right )\right ) \sin (e+f x) b^2}{(a+b \sin (e+f x))^{3/2} \sqrt {c+d \sin (e+f x)}}dx}{2 d}-\frac {b^2 \left (15 a^3 d^3+337 a^2 b c d^2+a b^2 d \left (57 c^2+284 d^2\right )-\left (b^3 \left (9 c^3-156 c d^2\right )\right )\right ) \cos (e+f x) \sqrt {c+d \sin (e+f x)}}{d f \sqrt {a+b \sin (e+f x)}}}{4 b}-\frac {b \left (59 a^2 d^2+54 a b c d-9 b^2 \left (c^2-4 d^2\right )\right ) \cos (e+f x) \sqrt {a+b \sin (e+f x)} \sqrt {c+d \sin (e+f x)}}{2 f}}{6 b}+\frac {b (3 b c-17 a d) \cos (e+f x) \sqrt {a+b \sin (e+f x)} (c+d \sin (e+f x))^{3/2}}{3 f}}{8 d}-\frac {b^2 \cos (e+f x) \sqrt {a+b \sin (e+f x)} (c+d \sin (e+f x))^{5/2}}{4 d f}\)

\(\Big \downarrow \) 3532

\(\displaystyle \frac {\frac {\frac {\frac {\frac {\int -\frac {2 \left (a^2-b^2\right ) d (b c-a d) \left (3 b^2 c^2+86 a b d c+15 a^2 d^2+36 b^2 d^2\right ) \sin (e+f x) b^3+\left (a^2-b^2\right ) (b c-a d) \left (9 c^3 b^3-156 c d^2 b^3-212 a d^3 b^2-51 a c^2 d b^2-165 a^2 c d^2 b+15 a^3 d^3\right ) b^2}{(a+b \sin (e+f x))^{3/2} \sqrt {c+d \sin (e+f x)}}dx}{b^2}+3 \left (-5 a^4 d^4+60 a^3 b c d^3+30 a^2 b^2 d^2 \left (3 c^2+4 d^2\right )-20 a b^3 c d \left (c^2-12 d^2\right )+3 b^4 \left (c^2+4 d^2\right )^2\right ) \int \frac {\sqrt {a+b \sin (e+f x)}}{\sqrt {c+d \sin (e+f x)}}dx}{2 d}-\frac {b^2 \left (15 a^3 d^3+337 a^2 b c d^2+a b^2 d \left (57 c^2+284 d^2\right )-\left (b^3 \left (9 c^3-156 c d^2\right )\right )\right ) \cos (e+f x) \sqrt {c+d \sin (e+f x)}}{d f \sqrt {a+b \sin (e+f x)}}}{4 b}-\frac {b \left (59 a^2 d^2+54 a b c d-9 b^2 \left (c^2-4 d^2\right )\right ) \cos (e+f x) \sqrt {a+b \sin (e+f x)} \sqrt {c+d \sin (e+f x)}}{2 f}}{6 b}+\frac {b (3 b c-17 a d) \cos (e+f x) \sqrt {a+b \sin (e+f x)} (c+d \sin (e+f x))^{3/2}}{3 f}}{8 d}-\frac {b^2 \cos (e+f x) \sqrt {a+b \sin (e+f x)} (c+d \sin (e+f x))^{5/2}}{4 d f}\)

\(\Big \downarrow \) 25

\(\displaystyle \frac {\frac {\frac {\frac {3 \left (-5 a^4 d^4+60 a^3 b c d^3+30 a^2 b^2 d^2 \left (3 c^2+4 d^2\right )-20 a b^3 c d \left (c^2-12 d^2\right )+3 b^4 \left (c^2+4 d^2\right )^2\right ) \int \frac {\sqrt {a+b \sin (e+f x)}}{\sqrt {c+d \sin (e+f x)}}dx-\frac {\int -\frac {b^2 \left (a^2-b^2\right ) (b c-a d) \left (-3 \left (3 c^3-52 c d^2\right ) b^3+a d \left (51 c^2+212 d^2\right ) b^2+165 a^2 c d^2 b-15 a^3 d^3\right )-2 b^3 \left (a^2-b^2\right ) d (b c-a d) \left (3 \left (c^2+12 d^2\right ) b^2+86 a c d b+15 a^2 d^2\right ) \sin (e+f x)}{(a+b \sin (e+f x))^{3/2} \sqrt {c+d \sin (e+f x)}}dx}{b^2}}{2 d}-\frac {b^2 \left (15 a^3 d^3+337 a^2 b c d^2+a b^2 d \left (57 c^2+284 d^2\right )-\left (b^3 \left (9 c^3-156 c d^2\right )\right )\right ) \cos (e+f x) \sqrt {c+d \sin (e+f x)}}{d f \sqrt {a+b \sin (e+f x)}}}{4 b}-\frac {b \left (59 a^2 d^2+54 a b c d-9 b^2 \left (c^2-4 d^2\right )\right ) \cos (e+f x) \sqrt {a+b \sin (e+f x)} \sqrt {c+d \sin (e+f x)}}{2 f}}{6 b}+\frac {b (3 b c-17 a d) \cos (e+f x) \sqrt {a+b \sin (e+f x)} (c+d \sin (e+f x))^{3/2}}{3 f}}{8 d}-\frac {b^2 \cos (e+f x) \sqrt {a+b \sin (e+f x)} (c+d \sin (e+f x))^{5/2}}{4 d f}\)

\(\Big \downarrow \) 25

\(\displaystyle \frac {\frac {\frac {\frac {\frac {\int \frac {b^2 \left (a^2-b^2\right ) (b c-a d) \left (-\left (\left (9 c^3-156 c d^2\right ) b^3\right )+a d \left (51 c^2+212 d^2\right ) b^2+165 a^2 c d^2 b-15 a^3 d^3\right )-2 b^3 \left (a^2-b^2\right ) d (b c-a d) \left (3 \left (c^2+12 d^2\right ) b^2+86 a c d b+15 a^2 d^2\right ) \sin (e+f x)}{(a+b \sin (e+f x))^{3/2} \sqrt {c+d \sin (e+f x)}}dx}{b^2}+3 \left (-5 a^4 d^4+60 a^3 b c d^3+30 a^2 b^2 d^2 \left (3 c^2+4 d^2\right )-20 a b^3 c d \left (c^2-12 d^2\right )+3 b^4 \left (c^2+4 d^2\right )^2\right ) \int \frac {\sqrt {a+b \sin (e+f x)}}{\sqrt {c+d \sin (e+f x)}}dx}{2 d}-\frac {b^2 \left (15 a^3 d^3+337 a^2 b c d^2+a b^2 d \left (57 c^2+284 d^2\right )-\left (b^3 \left (9 c^3-156 c d^2\right )\right )\right ) \cos (e+f x) \sqrt {c+d \sin (e+f x)}}{d f \sqrt {a+b \sin (e+f x)}}}{4 b}-\frac {b \left (59 a^2 d^2+54 a b c d-9 b^2 \left (c^2-4 d^2\right )\right ) \cos (e+f x) \sqrt {a+b \sin (e+f x)} \sqrt {c+d \sin (e+f x)}}{2 f}}{6 b}+\frac {b (3 b c-17 a d) \cos (e+f x) \sqrt {a+b \sin (e+f x)} (c+d \sin (e+f x))^{3/2}}{3 f}}{8 d}-\frac {b^2 \cos (e+f x) \sqrt {a+b \sin (e+f x)} (c+d \sin (e+f x))^{5/2}}{4 d f}\)

\(\Big \downarrow \) 3042

\(\displaystyle \frac {\frac {\frac {\frac {\frac {\int \frac {b^2 \left (a^2-b^2\right ) (b c-a d) \left (-\left (\left (9 c^3-156 c d^2\right ) b^3\right )+a d \left (51 c^2+212 d^2\right ) b^2+165 a^2 c d^2 b-15 a^3 d^3\right )-2 b^3 \left (a^2-b^2\right ) d (b c-a d) \left (3 \left (c^2+12 d^2\right ) b^2+86 a c d b+15 a^2 d^2\right ) \sin (e+f x)}{(a+b \sin (e+f x))^{3/2} \sqrt {c+d \sin (e+f x)}}dx}{b^2}+3 \left (-5 a^4 d^4+60 a^3 b c d^3+30 a^2 b^2 d^2 \left (3 c^2+4 d^2\right )-20 a b^3 c d \left (c^2-12 d^2\right )+3 b^4 \left (c^2+4 d^2\right )^2\right ) \int \frac {\sqrt {a+b \sin (e+f x)}}{\sqrt {c+d \sin (e+f x)}}dx}{2 d}-\frac {b^2 \left (15 a^3 d^3+337 a^2 b c d^2+a b^2 d \left (57 c^2+284 d^2\right )-\left (b^3 \left (9 c^3-156 c d^2\right )\right )\right ) \cos (e+f x) \sqrt {c+d \sin (e+f x)}}{d f \sqrt {a+b \sin (e+f x)}}}{4 b}-\frac {b \left (59 a^2 d^2+54 a b c d-9 b^2 \left (c^2-4 d^2\right )\right ) \cos (e+f x) \sqrt {a+b \sin (e+f x)} \sqrt {c+d \sin (e+f x)}}{2 f}}{6 b}+\frac {b (3 b c-17 a d) \cos (e+f x) \sqrt {a+b \sin (e+f x)} (c+d \sin (e+f x))^{3/2}}{3 f}}{8 d}-\frac {b^2 \cos (e+f x) \sqrt {a+b \sin (e+f x)} (c+d \sin (e+f x))^{5/2}}{4 d f}\)

\(\Big \downarrow \) 3290

\(\displaystyle \frac {\frac {\frac {\frac {\frac {\int \frac {b^2 \left (a^2-b^2\right ) (b c-a d) \left (-\left (\left (9 c^3-156 c d^2\right ) b^3\right )+a d \left (51 c^2+212 d^2\right ) b^2+165 a^2 c d^2 b-15 a^3 d^3\right )-2 b^3 \left (a^2-b^2\right ) d (b c-a d) \left (3 \left (c^2+12 d^2\right ) b^2+86 a c d b+15 a^2 d^2\right ) \sin (e+f x)}{(a+b \sin (e+f x))^{3/2} \sqrt {c+d \sin (e+f x)}}dx}{b^2}+\frac {6 \sqrt {c+d} \left (-5 a^4 d^4+60 a^3 b c d^3+30 a^2 b^2 d^2 \left (3 c^2+4 d^2\right )-20 a b^3 c d \left (c^2-12 d^2\right )+3 b^4 \left (c^2+4 d^2\right )^2\right ) \sec (e+f x) (a+b \sin (e+f x)) \sqrt {-\frac {(b c-a d) (1-\sin (e+f x))}{(c+d) (a+b \sin (e+f x))}} \sqrt {\frac {(b c-a d) (\sin (e+f x)+1)}{(c-d) (a+b \sin (e+f x))}} \operatorname {EllipticPi}\left (\frac {b (c+d)}{(a+b) d},\arcsin \left (\frac {\sqrt {a+b} \sqrt {c+d \sin (e+f x)}}{\sqrt {c+d} \sqrt {a+b \sin (e+f x)}}\right ),\frac {(a-b) (c+d)}{(a+b) (c-d)}\right )}{d f \sqrt {a+b}}}{2 d}-\frac {b^2 \left (15 a^3 d^3+337 a^2 b c d^2+a b^2 d \left (57 c^2+284 d^2\right )-\left (b^3 \left (9 c^3-156 c d^2\right )\right )\right ) \cos (e+f x) \sqrt {c+d \sin (e+f x)}}{d f \sqrt {a+b \sin (e+f x)}}}{4 b}-\frac {b \left (59 a^2 d^2+54 a b c d-9 b^2 \left (c^2-4 d^2\right )\right ) \cos (e+f x) \sqrt {a+b \sin (e+f x)} \sqrt {c+d \sin (e+f x)}}{2 f}}{6 b}+\frac {b (3 b c-17 a d) \cos (e+f x) \sqrt {a+b \sin (e+f x)} (c+d \sin (e+f x))^{3/2}}{3 f}}{8 d}-\frac {b^2 \cos (e+f x) \sqrt {a+b \sin (e+f x)} (c+d \sin (e+f x))^{5/2}}{4 d f}\)

\(\Big \downarrow \) 3477

\(\displaystyle \frac {\frac {b (3 b c-17 a d) \cos (e+f x) \sqrt {a+b \sin (e+f x)} (c+d \sin (e+f x))^{3/2}}{3 f}+\frac {\frac {\frac {\frac {6 \sqrt {c+d} \left (3 \left (c^2+4 d^2\right )^2 b^4-20 a c d \left (c^2-12 d^2\right ) b^3+30 a^2 d^2 \left (3 c^2+4 d^2\right ) b^2+60 a^3 c d^3 b-5 a^4 d^4\right ) \operatorname {EllipticPi}\left (\frac {b (c+d)}{(a+b) d},\arcsin \left (\frac {\sqrt {a+b} \sqrt {c+d \sin (e+f x)}}{\sqrt {c+d} \sqrt {a+b \sin (e+f x)}}\right ),\frac {(a-b) (c+d)}{(a+b) (c-d)}\right ) \sec (e+f x) \sqrt {-\frac {(b c-a d) (1-\sin (e+f x))}{(c+d) (a+b \sin (e+f x))}} \sqrt {\frac {(b c-a d) (\sin (e+f x)+1)}{(c-d) (a+b \sin (e+f x))}} (a+b \sin (e+f x))}{\sqrt {a+b} d f}+\frac {-\left ((a+b) (b c-a d) \left (-\left (\left (9 c^3-156 c d^2\right ) b^3\right )+a d \left (57 c^2+284 d^2\right ) b^2+337 a^2 c d^2 b+15 a^3 d^3\right ) \int \frac {\sin (e+f x)+1}{(a+b \sin (e+f x))^{3/2} \sqrt {c+d \sin (e+f x)}}dx b^3\right )-\frac {\left (a^2-b^2\right ) (b c-a d) \left (\left (9 c^3-6 d c^2-156 d^2 c-72 d^3\right ) b^3-a d \left (51 c^2+172 d c+212 d^2\right ) b^2-15 a^2 d^2 (11 c+2 d) b+15 a^3 d^3\right ) \int \frac {1}{\sqrt {a+b \sin (e+f x)} \sqrt {c+d \sin (e+f x)}}dx b^2}{a-b}}{b^2}}{2 d}-\frac {b^2 \left (-\left (\left (9 c^3-156 c d^2\right ) b^3\right )+a d \left (57 c^2+284 d^2\right ) b^2+337 a^2 c d^2 b+15 a^3 d^3\right ) \cos (e+f x) \sqrt {c+d \sin (e+f x)}}{d f \sqrt {a+b \sin (e+f x)}}}{4 b}-\frac {b \left (-9 \left (c^2-4 d^2\right ) b^2+54 a c d b+59 a^2 d^2\right ) \cos (e+f x) \sqrt {a+b \sin (e+f x)} \sqrt {c+d \sin (e+f x)}}{2 f}}{6 b}}{8 d}-\frac {b^2 \cos (e+f x) \sqrt {a+b \sin (e+f x)} (c+d \sin (e+f x))^{5/2}}{4 d f}\)

\(\Big \downarrow \) 3042

\(\displaystyle \frac {\frac {b (3 b c-17 a d) \cos (e+f x) \sqrt {a+b \sin (e+f x)} (c+d \sin (e+f x))^{3/2}}{3 f}+\frac {\frac {\frac {\frac {6 \sqrt {c+d} \left (3 \left (c^2+4 d^2\right )^2 b^4-20 a c d \left (c^2-12 d^2\right ) b^3+30 a^2 d^2 \left (3 c^2+4 d^2\right ) b^2+60 a^3 c d^3 b-5 a^4 d^4\right ) \operatorname {EllipticPi}\left (\frac {b (c+d)}{(a+b) d},\arcsin \left (\frac {\sqrt {a+b} \sqrt {c+d \sin (e+f x)}}{\sqrt {c+d} \sqrt {a+b \sin (e+f x)}}\right ),\frac {(a-b) (c+d)}{(a+b) (c-d)}\right ) \sec (e+f x) \sqrt {-\frac {(b c-a d) (1-\sin (e+f x))}{(c+d) (a+b \sin (e+f x))}} \sqrt {\frac {(b c-a d) (\sin (e+f x)+1)}{(c-d) (a+b \sin (e+f x))}} (a+b \sin (e+f x))}{\sqrt {a+b} d f}+\frac {-\left ((a+b) (b c-a d) \left (-\left (\left (9 c^3-156 c d^2\right ) b^3\right )+a d \left (57 c^2+284 d^2\right ) b^2+337 a^2 c d^2 b+15 a^3 d^3\right ) \int \frac {\sin (e+f x)+1}{(a+b \sin (e+f x))^{3/2} \sqrt {c+d \sin (e+f x)}}dx b^3\right )-\frac {\left (a^2-b^2\right ) (b c-a d) \left (\left (9 c^3-6 d c^2-156 d^2 c-72 d^3\right ) b^3-a d \left (51 c^2+172 d c+212 d^2\right ) b^2-15 a^2 d^2 (11 c+2 d) b+15 a^3 d^3\right ) \int \frac {1}{\sqrt {a+b \sin (e+f x)} \sqrt {c+d \sin (e+f x)}}dx b^2}{a-b}}{b^2}}{2 d}-\frac {b^2 \left (-\left (\left (9 c^3-156 c d^2\right ) b^3\right )+a d \left (57 c^2+284 d^2\right ) b^2+337 a^2 c d^2 b+15 a^3 d^3\right ) \cos (e+f x) \sqrt {c+d \sin (e+f x)}}{d f \sqrt {a+b \sin (e+f x)}}}{4 b}-\frac {b \left (-9 \left (c^2-4 d^2\right ) b^2+54 a c d b+59 a^2 d^2\right ) \cos (e+f x) \sqrt {a+b \sin (e+f x)} \sqrt {c+d \sin (e+f x)}}{2 f}}{6 b}}{8 d}-\frac {b^2 \cos (e+f x) \sqrt {a+b \sin (e+f x)} (c+d \sin (e+f x))^{5/2}}{4 d f}\)

\(\Big \downarrow \) 3297

\(\displaystyle \frac {\frac {b (3 b c-17 a d) \cos (e+f x) \sqrt {a+b \sin (e+f x)} (c+d \sin (e+f x))^{3/2}}{3 f}+\frac {\frac {\frac {\frac {6 \sqrt {c+d} \left (3 \left (c^2+4 d^2\right )^2 b^4-20 a c d \left (c^2-12 d^2\right ) b^3+30 a^2 d^2 \left (3 c^2+4 d^2\right ) b^2+60 a^3 c d^3 b-5 a^4 d^4\right ) \operatorname {EllipticPi}\left (\frac {b (c+d)}{(a+b) d},\arcsin \left (\frac {\sqrt {a+b} \sqrt {c+d \sin (e+f x)}}{\sqrt {c+d} \sqrt {a+b \sin (e+f x)}}\right ),\frac {(a-b) (c+d)}{(a+b) (c-d)}\right ) \sec (e+f x) \sqrt {-\frac {(b c-a d) (1-\sin (e+f x))}{(c+d) (a+b \sin (e+f x))}} \sqrt {\frac {(b c-a d) (\sin (e+f x)+1)}{(c-d) (a+b \sin (e+f x))}} (a+b \sin (e+f x))}{\sqrt {a+b} d f}+\frac {-\left ((a+b) (b c-a d) \left (-\left (\left (9 c^3-156 c d^2\right ) b^3\right )+a d \left (57 c^2+284 d^2\right ) b^2+337 a^2 c d^2 b+15 a^3 d^3\right ) \int \frac {\sin (e+f x)+1}{(a+b \sin (e+f x))^{3/2} \sqrt {c+d \sin (e+f x)}}dx b^3\right )-\frac {2 \sqrt {a+b} \left (a^2-b^2\right ) \left (\left (9 c^3-6 d c^2-156 d^2 c-72 d^3\right ) b^3-a d \left (51 c^2+172 d c+212 d^2\right ) b^2-15 a^2 d^2 (11 c+2 d) b+15 a^3 d^3\right ) \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {c+d} \sqrt {a+b \sin (e+f x)}}{\sqrt {a+b} \sqrt {c+d \sin (e+f x)}}\right ),\frac {(a+b) (c-d)}{(a-b) (c+d)}\right ) \sec (e+f x) \sqrt {\frac {(b c-a d) (1-\sin (e+f x))}{(a+b) (c+d \sin (e+f x))}} \sqrt {-\frac {(b c-a d) (\sin (e+f x)+1)}{(a-b) (c+d \sin (e+f x))}} (c+d \sin (e+f x)) b^2}{(a-b) \sqrt {c+d} f}}{b^2}}{2 d}-\frac {b^2 \left (-\left (\left (9 c^3-156 c d^2\right ) b^3\right )+a d \left (57 c^2+284 d^2\right ) b^2+337 a^2 c d^2 b+15 a^3 d^3\right ) \cos (e+f x) \sqrt {c+d \sin (e+f x)}}{d f \sqrt {a+b \sin (e+f x)}}}{4 b}-\frac {b \left (-9 \left (c^2-4 d^2\right ) b^2+54 a c d b+59 a^2 d^2\right ) \cos (e+f x) \sqrt {a+b \sin (e+f x)} \sqrt {c+d \sin (e+f x)}}{2 f}}{6 b}}{8 d}-\frac {b^2 \cos (e+f x) \sqrt {a+b \sin (e+f x)} (c+d \sin (e+f x))^{5/2}}{4 d f}\)

\(\Big \downarrow \) 3475

\(\displaystyle \frac {\frac {b (3 b c-17 a d) \cos (e+f x) \sqrt {a+b \sin (e+f x)} (c+d \sin (e+f x))^{3/2}}{3 f}+\frac {\frac {\frac {\frac {6 \sqrt {c+d} \left (3 \left (c^2+4 d^2\right )^2 b^4-20 a c d \left (c^2-12 d^2\right ) b^3+30 a^2 d^2 \left (3 c^2+4 d^2\right ) b^2+60 a^3 c d^3 b-5 a^4 d^4\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}+\frac {\frac {2 b^3 \sqrt {a+b} (c-d) \sqrt {c+d} \left (-\left (\left (9 c^3-156 c d^2\right ) b^3\right )+a d \left (57 c^2+284 d^2\right ) b^2+337 a^2 c d^2 b+15 a^3 d^3\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 c-a d) f}-\frac {2 b^2 \sqrt {a+b} \left (a^2-b^2\right ) \left (\left (9 c^3-6 d c^2-156 d^2 c-72 d^3\right ) b^3-a d \left (51 c^2+172 d c+212 d^2\right ) b^2-15 a^2 d^2 (11 c+2 d) b+15 a^3 d^3\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))}{(a-b) \sqrt {c+d} f}}{b^2}}{2 d}-\frac {b^2 \left (-\left (\left (9 c^3-156 c d^2\right ) b^3\right )+a d \left (57 c^2+284 d^2\right ) b^2+337 a^2 c d^2 b+15 a^3 d^3\right ) \cos (e+f x) \sqrt {c+d \sin (e+f x)}}{d f \sqrt {a+b \sin (e+f x)}}}{4 b}-\frac {b \left (-9 \left (c^2-4 d^2\right ) b^2+54 a c d b+59 a^2 d^2\right ) \cos (e+f x) \sqrt {a+b \sin (e+f x)} \sqrt {c+d \sin (e+f x)}}{2 f}}{6 b}}{8 d}-\frac {b^2 \cos (e+f x) \sqrt {a+b \sin (e+f x)} (c+d \sin (e+f x))^{5/2}}{4 d f}\)

Input:

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

Output:

-1/4*(b^2*Cos[e + f*x]*Sqrt[a + b*Sin[e + f*x]]*(c + d*Sin[e + f*x])^(5/2) 
)/(d*f) + ((b*(3*b*c - 17*a*d)*Cos[e + f*x]*Sqrt[a + b*Sin[e + f*x]]*(c + 
d*Sin[e + f*x])^(3/2))/(3*f) + (-1/2*(b*(54*a*b*c*d + 59*a^2*d^2 - 9*b^2*( 
c^2 - 4*d^2))*Cos[e + f*x]*Sqrt[a + b*Sin[e + f*x]]*Sqrt[c + d*Sin[e + f*x 
]])/f + (-((b^2*(337*a^2*b*c*d^2 + 15*a^3*d^3 + a*b^2*d*(57*c^2 + 284*d^2) 
 - b^3*(9*c^3 - 156*c*d^2))*Cos[e + f*x]*Sqrt[c + d*Sin[e + f*x]])/(d*f*Sq 
rt[a + b*Sin[e + f*x]])) + ((6*Sqrt[c + d]*(60*a^3*b*c*d^3 - 5*a^4*d^4 - 2 
0*a*b^3*c*d*(c^2 - 12*d^2) + 3*b^4*(c^2 + 4*d^2)^2 + 30*a^2*b^2*d^2*(3*c^2 
 + 4*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])))]*Sqrt[((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^3*Sqrt[a + b]*(c - d)*Sqrt[c + d]*(337*a^2*b*c*d^2 + 15*a^3*d^3 + a*b 
^2*d*(57*c^2 + 284*d^2) - b^3*(9*c^3 - 156*c*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 - 
 Sin[e + f*x]))/((c + d)*(a + b*Sin[e + f*x])))]*Sqrt[((b*c - a*d)*(1 + Si 
n[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*Sqrt[a + b]*(a^2 - b^2)*(15*a^3*d^3 - 15*a^2*b*d^2*(1...
 

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 3272

Int[((a_.) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_)*((c_.) + (d_.)*sin[(e_.) + 
 (f_.)*(x_)])^(n_), x_Symbol] :> Simp[(-b^2)*Cos[e + f*x]*(a + b*Sin[e + f* 
x])^(m - 2)*((c + d*Sin[e + f*x])^(n + 1)/(d*f*(m + n))), x] + Simp[1/(d*(m 
 + n))   Int[(a + b*Sin[e + f*x])^(m - 3)*(c + d*Sin[e + f*x])^n*Simp[a^3*d 
*(m + n) + b^2*(b*c*(m - 2) + a*d*(n + 1)) - b*(a*b*c - b^2*d*(m + n - 1) - 
 3*a^2*d*(m + n))*Sin[e + f*x] - b^2*(b*c*(m - 1) - a*d*(3*m + 2*n - 2))*Si 
n[e + f*x]^2, x], x], x] /; FreeQ[{a, b, c, d, e, f, n}, x] && NeQ[b*c - a* 
d, 0] && NeQ[a^2 - b^2, 0] && NeQ[c^2 - d^2, 0] && GtQ[m, 2] && (IntegerQ[m 
] || IntegersQ[2*m, 2*n]) &&  !(IGtQ[n, 2] && ( !IntegerQ[m] || (EqQ[a, 0] 
&& NeQ[c, 0])))
 

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 3528

Int[((a_.) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_.)*((c_.) + (d_.)*sin[(e_.) 
+ (f_.)*(x_)])^(n_.)*((A_.) + (B_.)*sin[(e_.) + (f_.)*(x_)] + (C_.)*sin[(e_ 
.) + (f_.)*(x_)]^2), x_Symbol] :> Simp[(-C)*Cos[e + f*x]*(a + b*Sin[e + f*x 
])^m*((c + d*Sin[e + f*x])^(n + 1)/(d*f*(m + n + 2))), x] + Simp[1/(d*(m + 
n + 2))   Int[(a + b*Sin[e + f*x])^(m - 1)*(c + d*Sin[e + f*x])^n*Simp[a*A* 
d*(m + n + 2) + C*(b*c*m + a*d*(n + 1)) + (d*(A*b + a*B)*(m + n + 2) - C*(a 
*c - b*d*(m + n + 1)))*Sin[e + f*x] + (C*(a*d*m - b*c*(m + 1)) + b*B*d*(m + 
 n + 2))*Sin[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[a, 0] && NeQ[c, 0])))
 

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 [C] (warning: unable to verify)

Result contains complex when optimal does not.

Time = 40.54 (sec) , antiderivative size = 366209, normalized size of antiderivative = 341.93

method	result	size
default	\(\text {Expression too large to display}\)	\(366209\)

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

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

Output:

Timed out
 

Maxima [F]

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

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

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