Integrand size = 29, antiderivative size = 867 \[ \int (3+b \sin (e+f x))^{5/2} \sqrt {c+d \sin (e+f x)} \, dx=\frac {\sqrt {3+b} (c-d) \sqrt {c+d} \left (42 b c d+297 d^2-b^2 \left (3 c^2-16 d^2\right )\right ) E\left (\arcsin \left (\frac {\sqrt {3+b} \sqrt {c+d \sin (e+f x)}}{\sqrt {c+d} \sqrt {3+b \sin (e+f x)}}\right )|\frac {(3-b) (c+d)}{(3+b) (c-d)}\right ) \sec (e+f x) \sqrt {-\frac {(b c-3 d) (1-\sin (e+f x))}{(c+d) (3+b \sin (e+f x))}} \sqrt {\frac {(b c-3 d) (1+\sin (e+f x))}{(c-d) (3+b \sin (e+f x))}} (3+b \sin (e+f x))}{24 (b c-3 d) d^2 f}+\frac {\sqrt {c+d} \left (135 b c d^2+135 d^3-15 b^2 d \left (c^2-4 d^2\right )+b^3 \left (c^3+4 c d^2\right )\right ) \operatorname {EllipticPi}\left (\frac {b (c+d)}{(3+b) d},\arcsin \left (\frac {\sqrt {3+b} \sqrt {c+d \sin (e+f x)}}{\sqrt {c+d} \sqrt {3+b \sin (e+f x)}}\right ),\frac {(3-b) (c+d)}{(3+b) (c-d)}\right ) \sec (e+f x) \sqrt {-\frac {(b c-3 d) (1-\sin (e+f x))}{(c+d) (3+b \sin (e+f x))}} \sqrt {\frac {(b c-3 d) (1+\sin (e+f x))}{(c-d) (3+b \sin (e+f x))}} (3+b \sin (e+f x))}{8 b \sqrt {3+b} d^3 f}-\frac {b \left (42 b c d+297 d^2-b^2 \left (3 c^2-16 d^2\right )\right ) \cos (e+f x) \sqrt {c+d \sin (e+f x)}}{24 d^2 f \sqrt {3+b \sin (e+f x)}}+\frac {b (3 b c-39 d) \cos (e+f x) \sqrt {3+b \sin (e+f x)} \sqrt {c+d \sin (e+f x)}}{12 d f}+\frac {(3+b)^{3/2} \left (135 d^2+18 b d (2 c+3 d)-b^2 \left (3 c^2-2 c d-16 d^2\right )\right ) \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {c+d} \sqrt {3+b \sin (e+f x)}}{\sqrt {3+b} \sqrt {c+d \sin (e+f x)}}\right ),\frac {(3+b) (c-d)}{(3-b) (c+d)}\right ) \sec (e+f x) \sqrt {\frac {(b c-3 d) (1-\sin (e+f x))}{(3+b) (c+d \sin (e+f x))}} \sqrt {-\frac {(b c-3 d) (1+\sin (e+f x))}{(3-b) (c+d \sin (e+f x))}} (c+d \sin (e+f x))}{24 b d^2 \sqrt {c+d} f}-\frac {b^2 \cos (e+f x) \sqrt {3+b \sin (e+f x)} (c+d \sin (e+f x))^{3/2}}{3 d f} \]
1/8*(15*a^2*b*c*d^2+5*a^3*d^3-5*a*b^2*d*(c^2-4*d^2)+b^3*(c^3+4*c*d^2))*Ell ipticPi((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+b*sin(f *x+e))*(c+d)^(1/2)*(-(-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)/b/d^3/f/(a+b)^ (1/2)+1/24*(c-d)*(14*a*b*c*d+33*a^2*d^2-b^2*(3*c^2-16*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+b*sin(f*x+e))*(a+b)^(1/2)*(c+d)^(1/ 2)*(-(-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)/d^2/(-a*d+b*c)/f-1/3*b^2*cos(f *x+e)*(c+d*sin(f*x+e))^(3/2)*(a+b*sin(f*x+e))^(1/2)/d/f+1/24*(a+b)^(3/2)*( 15*a^2*d^2+6*a*b*d*(2*c+3*d)-b^2*(3*c^2-2*c*d-16*d^2))*EllipticF((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)*(c+d*sin(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)/b/d^2/f/(c+d)^(1/2)-1/24*b*(14*a*b*c*d+33*a^2*d^2-b^2*(3*c^ 2-16*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)+ 1/12*b*(-13*a*d+3*b*c)*cos(f*x+e)*(a+b*sin(f*x+e))^(1/2)*(c+d*sin(f*x+e))^ (1/2)/d/f
Leaf count is larger than twice the leaf count of optimal. \(1939\) vs. \(2(867)=1734\).
Time = 9.09 (sec) , antiderivative size = 1939, normalized size of antiderivative = 2.24 \[ \int (3+b \sin (e+f x))^{5/2} \sqrt {c+d \sin (e+f x)} \, dx =\text {Too large to display} \]
((-4*(-(b*c) + 3*d)*(-(b^3*c^2) + 1296*c*d + 174*b^2*c*d + 531*b*d^2 + 16* b^3*d^2)*Sqrt[((c + d)*Cot[(-e + Pi/2 - f*x)/2]^2)/(-c + d)]*EllipticF[Arc Sin[Sqrt[((-3 - b)*Csc[(-e + Pi/2 - f*x)/2]^2*(c + d*Sin[e + f*x]))/(-(b*c ) + 3*d)]/Sqrt[2]], (2*(-(b*c) + 3*d))/((3 + b)*(-c + d))]*Sec[e + f*x]*Si n[(-e + Pi/2 - f*x)/2]^4*Sqrt[((c + d)*Csc[(-e + Pi/2 - f*x)/2]^2*(3 + b*S in[e + f*x]))/(-(b*c) + 3*d)]*Sqrt[((-3 - b)*Csc[(-e + Pi/2 - f*x)/2]^2*(c + d*Sin[e + f*x]))/(-(b*c) + 3*d)])/((3 + b)*(c + d)*Sqrt[3 + b*Sin[e + f *x]]*Sqrt[c + d*Sin[e + f*x]]) - 4*(-(b*c) + 3*d)*(-12*b^2*c^2 + 828*b*c*d + 28*b^3*c*d + 1296*d^2 + 228*b^2*d^2)*((Sqrt[((c + d)*Cot[(-e + Pi/2 - f *x)/2]^2)/(-c + d)]*EllipticF[ArcSin[Sqrt[((-3 - b)*Csc[(-e + Pi/2 - f*x)/ 2]^2*(c + d*Sin[e + f*x]))/(-(b*c) + 3*d)]/Sqrt[2]], (2*(-(b*c) + 3*d))/(( 3 + b)*(-c + d))]*Sec[e + f*x]*Sin[(-e + Pi/2 - f*x)/2]^4*Sqrt[((c + d)*Cs c[(-e + Pi/2 - f*x)/2]^2*(3 + b*Sin[e + f*x]))/(-(b*c) + 3*d)]*Sqrt[((-3 - b)*Csc[(-e + Pi/2 - f*x)/2]^2*(c + d*Sin[e + f*x]))/(-(b*c) + 3*d)])/((3 + b)*(c + d)*Sqrt[3 + 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) + 3*d)/((3 + b)*d), ArcSin[Sqrt[((-3 - b)*Csc[(-e + Pi/2 - f*x)/2]^2*(c + d*Sin[e + f*x]))/(-(b*c) + 3*d)]/Sqrt[2]], (2*(-(b*c) + 3*d))/((3 + b)*(-c + d))]*Se c[e + f*x]*Sin[(-e + Pi/2 - f*x)/2]^4*Sqrt[((c + d)*Csc[(-e + Pi/2 - f*x)/ 2]^2*(3 + b*Sin[e + f*x]))/(-(b*c) + 3*d)]*Sqrt[((-3 - b)*Csc[(-e + Pi/...
Time = 4.31 (sec) , antiderivative size = 908, normalized size of antiderivative = 1.05, number of steps used = 18, number of rules used = 18, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.621, Rules used = {3042, 3272, 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.
| \(\displaystyle \int (a+b \sin (e+f x))^{5/2} \sqrt {c+d \sin (e+f x)} \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int (a+b \sin (e+f x))^{5/2} \sqrt {c+d \sin (e+f x)}dx\) |
| \(\Big \downarrow \) 3272 |
| \(\displaystyle \frac {\int \frac {\sqrt {c+d \sin (e+f x)} \left (6 d a^3+3 b^2 d a-b^2 (3 b c-13 a d) \sin ^2(e+f x)+b^3 c-2 b \left (-9 d a^2+b c a-2 b^2 d\right ) \sin (e+f x)\right )}{2 \sqrt {a+b \sin (e+f x)}}dx}{3 d}-\frac {b^2 \cos (e+f x) \sqrt {a+b \sin (e+f x)} (c+d \sin (e+f x))^{3/2}}{3 d f}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\int \frac {\sqrt {c+d \sin (e+f x)} \left (6 d a^3+3 b^2 d a-b^2 (3 b c-13 a d) \sin ^2(e+f x)+b^3 c-2 b \left (-9 d a^2+b c a-2 b^2 d\right ) \sin (e+f x)\right )}{\sqrt {a+b \sin (e+f x)}}dx}{6 d}-\frac {b^2 \cos (e+f x) \sqrt {a+b \sin (e+f x)} (c+d \sin (e+f x))^{3/2}}{3 d f}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\int \frac {\sqrt {c+d \sin (e+f x)} \left (6 d a^3+3 b^2 d a-b^2 (3 b c-13 a d) \sin (e+f x)^2+b^3 c-2 b \left (-9 d a^2+b c a-2 b^2 d\right ) \sin (e+f x)\right )}{\sqrt {a+b \sin (e+f x)}}dx}{6 d}-\frac {b^2 \cos (e+f x) \sqrt {a+b \sin (e+f x)} (c+d \sin (e+f x))^{3/2}}{3 d f}\) |
| \(\Big \downarrow \) 3528 |
| \(\displaystyle \frac {\frac {\int \frac {b^2 \left (-\left (\left (3 c^2-16 d^2\right ) b^2\right )+14 a c d b+33 a^2 d^2\right ) \sin ^2(e+f x)+2 b \left (12 d^2 a^3+23 b c d a^2-b^2 \left (c^2-19 d^2\right ) a+7 b^3 c d\right ) \sin (e+f x)+b \left (24 c d a^3+13 b d^2 a^2+22 b^2 c d a+b^3 c^2\right )}{2 \sqrt {a+b \sin (e+f x)} \sqrt {c+d \sin (e+f x)}}dx}{2 b}+\frac {b (3 b c-13 a d) \cos (e+f x) \sqrt {a+b \sin (e+f x)} \sqrt {c+d \sin (e+f x)}}{2 f}}{6 d}-\frac {b^2 \cos (e+f x) \sqrt {a+b \sin (e+f x)} (c+d \sin (e+f x))^{3/2}}{3 d f}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\frac {\int \frac {b^2 \left (-\left (\left (3 c^2-16 d^2\right ) b^2\right )+14 a c d b+33 a^2 d^2\right ) \sin ^2(e+f x)+2 b \left (12 d^2 a^3+23 b c d a^2-b^2 \left (c^2-19 d^2\right ) a+7 b^3 c d\right ) \sin (e+f x)+b \left (24 c d a^3+13 b d^2 a^2+22 b^2 c d a+b^3 c^2\right )}{\sqrt {a+b \sin (e+f x)} \sqrt {c+d \sin (e+f x)}}dx}{4 b}+\frac {b (3 b c-13 a d) \cos (e+f x) \sqrt {a+b \sin (e+f x)} \sqrt {c+d \sin (e+f x)}}{2 f}}{6 d}-\frac {b^2 \cos (e+f x) \sqrt {a+b \sin (e+f x)} (c+d \sin (e+f x))^{3/2}}{3 d f}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\frac {\int \frac {b^2 \left (-\left (\left (3 c^2-16 d^2\right ) b^2\right )+14 a c d b+33 a^2 d^2\right ) \sin (e+f x)^2+2 b \left (12 d^2 a^3+23 b c d a^2-b^2 \left (c^2-19 d^2\right ) a+7 b^3 c d\right ) \sin (e+f x)+b \left (24 c d a^3+13 b d^2 a^2+22 b^2 c d a+b^3 c^2\right )}{\sqrt {a+b \sin (e+f x)} \sqrt {c+d \sin (e+f x)}}dx}{4 b}+\frac {b (3 b c-13 a d) \cos (e+f x) \sqrt {a+b \sin (e+f x)} \sqrt {c+d \sin (e+f x)}}{2 f}}{6 d}-\frac {b^2 \cos (e+f x) \sqrt {a+b \sin (e+f x)} (c+d \sin (e+f x))^{3/2}}{3 d f}\) |
| \(\Big \downarrow \) 3540 |
| \(\displaystyle \frac {\frac {\frac {\int \frac {3 b^2 \left (\left (c^3+4 d^2 c\right ) b^3-5 a d \left (c^2-4 d^2\right ) b^2+15 a^2 c d^2 b+5 a^3 d^3\right ) \sin ^2(e+f x)+2 b \left (24 d^3 a^4+37 b c d^2 a^3-b^2 d \left (16 c^2-51 d^2\right ) a^2+b^3 c \left (3 c^2+20 d^2\right ) a+b^4 c^2 d\right ) \sin (e+f x)+b \left (48 c d^2 a^4+59 b d^3 a^3+25 b^2 c d^2 a^2-b^3 \left (15 c^2 d-16 d^3\right ) a+b^4 \left (3 c^3-16 c d^2\right )\right )}{(a+b \sin (e+f x))^{3/2} \sqrt {c+d \sin (e+f x)}}dx}{2 d}-\frac {b^2 \left (33 a^2 d^2+14 a b c d-\left (b^2 \left (3 c^2-16 d^2\right )\right )\right ) \cos (e+f x) \sqrt {c+d \sin (e+f x)}}{d f \sqrt {a+b \sin (e+f x)}}}{4 b}+\frac {b (3 b c-13 a d) \cos (e+f x) \sqrt {a+b \sin (e+f x)} \sqrt {c+d \sin (e+f x)}}{2 f}}{6 d}-\frac {b^2 \cos (e+f x) \sqrt {a+b \sin (e+f x)} (c+d \sin (e+f x))^{3/2}}{3 d f}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\frac {\frac {\int \frac {3 b^2 \left (\left (c^3+4 d^2 c\right ) b^3-5 a d \left (c^2-4 d^2\right ) b^2+15 a^2 c d^2 b+5 a^3 d^3\right ) \sin (e+f x)^2+2 b \left (24 d^3 a^4+37 b c d^2 a^3-b^2 d \left (16 c^2-51 d^2\right ) a^2+b^3 c \left (3 c^2+20 d^2\right ) a+b^4 c^2 d\right ) \sin (e+f x)+b \left (48 c d^2 a^4+59 b d^3 a^3+25 b^2 c d^2 a^2-b^3 \left (15 c^2 d-16 d^3\right ) a+b^4 \left (3 c^3-16 c d^2\right )\right )}{(a+b \sin (e+f x))^{3/2} \sqrt {c+d \sin (e+f x)}}dx}{2 d}-\frac {b^2 \left (33 a^2 d^2+14 a b c d-\left (b^2 \left (3 c^2-16 d^2\right )\right )\right ) \cos (e+f x) \sqrt {c+d \sin (e+f x)}}{d f \sqrt {a+b \sin (e+f x)}}}{4 b}+\frac {b (3 b c-13 a d) \cos (e+f x) \sqrt {a+b \sin (e+f x)} \sqrt {c+d \sin (e+f x)}}{2 f}}{6 d}-\frac {b^2 \cos (e+f x) \sqrt {a+b \sin (e+f x)} (c+d \sin (e+f x))^{3/2}}{3 d f}\) |
| \(\Big \downarrow \) 3532 |
| \(\displaystyle \frac {\frac {\frac {\frac {\int -\frac {2 \left (a^2-b^2\right ) d (b c-a d) (b c+9 a d) \sin (e+f x) b^3+\left (a^2-b^2\right ) (b c-a d) \left (3 b^2 c^2-12 a b d c-15 a^2 d^2-16 b^2 d^2\right ) b^2}{(a+b \sin (e+f x))^{3/2} \sqrt {c+d \sin (e+f x)}}dx}{b^2}+3 \left (5 a^3 d^3+15 a^2 b c d^2-5 a b^2 d \left (c^2-4 d^2\right )+b^3 \left (c^3+4 c d^2\right )\right ) \int \frac {\sqrt {a+b \sin (e+f x)}}{\sqrt {c+d \sin (e+f x)}}dx}{2 d}-\frac {b^2 \left (33 a^2 d^2+14 a b c d-\left (b^2 \left (3 c^2-16 d^2\right )\right )\right ) \cos (e+f x) \sqrt {c+d \sin (e+f x)}}{d f \sqrt {a+b \sin (e+f x)}}}{4 b}+\frac {b (3 b c-13 a d) \cos (e+f x) \sqrt {a+b \sin (e+f x)} \sqrt {c+d \sin (e+f x)}}{2 f}}{6 d}-\frac {b^2 \cos (e+f x) \sqrt {a+b \sin (e+f x)} (c+d \sin (e+f x))^{3/2}}{3 d f}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {\frac {\frac {3 \left (5 a^3 d^3+15 a^2 b c d^2-5 a b^2 d \left (c^2-4 d^2\right )+b^3 \left (c^3+4 c d^2\right )\right ) \int \frac {\sqrt {a+b \sin (e+f x)}}{\sqrt {c+d \sin (e+f x)}}dx-\frac {\int -\frac {b^2 \left (a^2-b^2\right ) (b c-a d) \left (-\left (\left (3 c^2-16 d^2\right ) b^2\right )+12 a c d b+15 a^2 d^2\right )-2 b^3 \left (a^2-b^2\right ) d (b c-a d) (b c+9 a d) \sin (e+f x)}{(a+b \sin (e+f x))^{3/2} \sqrt {c+d \sin (e+f x)}}dx}{b^2}}{2 d}-\frac {b^2 \left (33 a^2 d^2+14 a b c d-\left (b^2 \left (3 c^2-16 d^2\right )\right )\right ) \cos (e+f x) \sqrt {c+d \sin (e+f x)}}{d f \sqrt {a+b \sin (e+f x)}}}{4 b}+\frac {b (3 b c-13 a d) \cos (e+f x) \sqrt {a+b \sin (e+f x)} \sqrt {c+d \sin (e+f x)}}{2 f}}{6 d}-\frac {b^2 \cos (e+f x) \sqrt {a+b \sin (e+f x)} (c+d \sin (e+f x))^{3/2}}{3 d f}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {\frac {\frac {\frac {\int \frac {b^2 \left (a^2-b^2\right ) (b c-a d) \left (-\left (\left (3 c^2-16 d^2\right ) b^2\right )+12 a c d b+15 a^2 d^2\right )-2 b^3 \left (a^2-b^2\right ) d (b c-a d) (b c+9 a d) \sin (e+f x)}{(a+b \sin (e+f x))^{3/2} \sqrt {c+d \sin (e+f x)}}dx}{b^2}+3 \left (5 a^3 d^3+15 a^2 b c d^2-5 a b^2 d \left (c^2-4 d^2\right )+b^3 \left (c^3+4 c d^2\right )\right ) \int \frac {\sqrt {a+b \sin (e+f x)}}{\sqrt {c+d \sin (e+f x)}}dx}{2 d}-\frac {b^2 \left (33 a^2 d^2+14 a b c d-\left (b^2 \left (3 c^2-16 d^2\right )\right )\right ) \cos (e+f x) \sqrt {c+d \sin (e+f x)}}{d f \sqrt {a+b \sin (e+f x)}}}{4 b}+\frac {b (3 b c-13 a d) \cos (e+f x) \sqrt {a+b \sin (e+f x)} \sqrt {c+d \sin (e+f x)}}{2 f}}{6 d}-\frac {b^2 \cos (e+f x) \sqrt {a+b \sin (e+f x)} (c+d \sin (e+f x))^{3/2}}{3 d f}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\frac {\frac {\frac {\int \frac {b^2 \left (a^2-b^2\right ) (b c-a d) \left (-\left (\left (3 c^2-16 d^2\right ) b^2\right )+12 a c d b+15 a^2 d^2\right )-2 b^3 \left (a^2-b^2\right ) d (b c-a d) (b c+9 a d) \sin (e+f x)}{(a+b \sin (e+f x))^{3/2} \sqrt {c+d \sin (e+f x)}}dx}{b^2}+3 \left (5 a^3 d^3+15 a^2 b c d^2-5 a b^2 d \left (c^2-4 d^2\right )+b^3 \left (c^3+4 c d^2\right )\right ) \int \frac {\sqrt {a+b \sin (e+f x)}}{\sqrt {c+d \sin (e+f x)}}dx}{2 d}-\frac {b^2 \left (33 a^2 d^2+14 a b c d-\left (b^2 \left (3 c^2-16 d^2\right )\right )\right ) \cos (e+f x) \sqrt {c+d \sin (e+f x)}}{d f \sqrt {a+b \sin (e+f x)}}}{4 b}+\frac {b (3 b c-13 a d) \cos (e+f x) \sqrt {a+b \sin (e+f x)} \sqrt {c+d \sin (e+f x)}}{2 f}}{6 d}-\frac {b^2 \cos (e+f x) \sqrt {a+b \sin (e+f x)} (c+d \sin (e+f x))^{3/2}}{3 d f}\) |
| \(\Big \downarrow \) 3290 |
| \(\displaystyle \frac {\frac {\frac {\frac {\int \frac {b^2 \left (a^2-b^2\right ) (b c-a d) \left (-\left (\left (3 c^2-16 d^2\right ) b^2\right )+12 a c d b+15 a^2 d^2\right )-2 b^3 \left (a^2-b^2\right ) d (b c-a d) (b c+9 a d) \sin (e+f x)}{(a+b \sin (e+f x))^{3/2} \sqrt {c+d \sin (e+f x)}}dx}{b^2}+\frac {6 \sqrt {c+d} \left (5 a^3 d^3+15 a^2 b c d^2-5 a b^2 d \left (c^2-4 d^2\right )+b^3 \left (c^3+4 c d^2\right )\right ) \sec (e+f x) (a+b \sin (e+f x)) \sqrt {-\frac {(b c-a d) (1-\sin (e+f x))}{(c+d) (a+b \sin (e+f x))}} \sqrt {\frac {(b c-a d) (\sin (e+f x)+1)}{(c-d) (a+b \sin (e+f x))}} \operatorname {EllipticPi}\left (\frac {b (c+d)}{(a+b) d},\arcsin \left (\frac {\sqrt {a+b} \sqrt {c+d \sin (e+f x)}}{\sqrt {c+d} \sqrt {a+b \sin (e+f x)}}\right ),\frac {(a-b) (c+d)}{(a+b) (c-d)}\right )}{d f \sqrt {a+b}}}{2 d}-\frac {b^2 \left (33 a^2 d^2+14 a b c d-\left (b^2 \left (3 c^2-16 d^2\right )\right )\right ) \cos (e+f x) \sqrt {c+d \sin (e+f x)}}{d f \sqrt {a+b \sin (e+f x)}}}{4 b}+\frac {b (3 b c-13 a d) \cos (e+f x) \sqrt {a+b \sin (e+f x)} \sqrt {c+d \sin (e+f x)}}{2 f}}{6 d}-\frac {b^2 \cos (e+f x) \sqrt {a+b \sin (e+f x)} (c+d \sin (e+f x))^{3/2}}{3 d f}\) |
| \(\Big \downarrow \) 3477 |
| \(\displaystyle \frac {\frac {\frac {\frac {b^2 (a+b) (b c-a d) \left (15 a^2 d^2+6 a b d (2 c+3 d)-\left (b^2 \left (3 c^2-2 c d-16 d^2\right )\right )\right ) \int \frac {1}{\sqrt {a+b \sin (e+f x)} \sqrt {c+d \sin (e+f x)}}dx-b^3 (a+b) (b c-a d) \left (33 a^2 d^2+14 a b c d-\left (b^2 \left (3 c^2-16 d^2\right )\right )\right ) \int \frac {\sin (e+f x)+1}{(a+b \sin (e+f x))^{3/2} \sqrt {c+d \sin (e+f x)}}dx}{b^2}+\frac {6 \sqrt {c+d} \left (5 a^3 d^3+15 a^2 b c d^2-5 a b^2 d \left (c^2-4 d^2\right )+b^3 \left (c^3+4 c d^2\right )\right ) \sec (e+f x) (a+b \sin (e+f x)) \sqrt {-\frac {(b c-a d) (1-\sin (e+f x))}{(c+d) (a+b \sin (e+f x))}} \sqrt {\frac {(b c-a d) (\sin (e+f x)+1)}{(c-d) (a+b \sin (e+f x))}} \operatorname {EllipticPi}\left (\frac {b (c+d)}{(a+b) d},\arcsin \left (\frac {\sqrt {a+b} \sqrt {c+d \sin (e+f x)}}{\sqrt {c+d} \sqrt {a+b \sin (e+f x)}}\right ),\frac {(a-b) (c+d)}{(a+b) (c-d)}\right )}{d f \sqrt {a+b}}}{2 d}-\frac {b^2 \left (33 a^2 d^2+14 a b c d-\left (b^2 \left (3 c^2-16 d^2\right )\right )\right ) \cos (e+f x) \sqrt {c+d \sin (e+f x)}}{d f \sqrt {a+b \sin (e+f x)}}}{4 b}+\frac {b (3 b c-13 a d) \cos (e+f x) \sqrt {a+b \sin (e+f x)} \sqrt {c+d \sin (e+f x)}}{2 f}}{6 d}-\frac {b^2 \cos (e+f x) \sqrt {a+b \sin (e+f x)} (c+d \sin (e+f x))^{3/2}}{3 d f}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\frac {\frac {\frac {b^2 (a+b) (b c-a d) \left (15 a^2 d^2+6 a b d (2 c+3 d)-\left (b^2 \left (3 c^2-2 c d-16 d^2\right )\right )\right ) \int \frac {1}{\sqrt {a+b \sin (e+f x)} \sqrt {c+d \sin (e+f x)}}dx-b^3 (a+b) (b c-a d) \left (33 a^2 d^2+14 a b c d-\left (b^2 \left (3 c^2-16 d^2\right )\right )\right ) \int \frac {\sin (e+f x)+1}{(a+b \sin (e+f x))^{3/2} \sqrt {c+d \sin (e+f x)}}dx}{b^2}+\frac {6 \sqrt {c+d} \left (5 a^3 d^3+15 a^2 b c d^2-5 a b^2 d \left (c^2-4 d^2\right )+b^3 \left (c^3+4 c d^2\right )\right ) \sec (e+f x) (a+b \sin (e+f x)) \sqrt {-\frac {(b c-a d) (1-\sin (e+f x))}{(c+d) (a+b \sin (e+f x))}} \sqrt {\frac {(b c-a d) (\sin (e+f x)+1)}{(c-d) (a+b \sin (e+f x))}} \operatorname {EllipticPi}\left (\frac {b (c+d)}{(a+b) d},\arcsin \left (\frac {\sqrt {a+b} \sqrt {c+d \sin (e+f x)}}{\sqrt {c+d} \sqrt {a+b \sin (e+f x)}}\right ),\frac {(a-b) (c+d)}{(a+b) (c-d)}\right )}{d f \sqrt {a+b}}}{2 d}-\frac {b^2 \left (33 a^2 d^2+14 a b c d-\left (b^2 \left (3 c^2-16 d^2\right )\right )\right ) \cos (e+f x) \sqrt {c+d \sin (e+f x)}}{d f \sqrt {a+b \sin (e+f x)}}}{4 b}+\frac {b (3 b c-13 a d) \cos (e+f x) \sqrt {a+b \sin (e+f x)} \sqrt {c+d \sin (e+f x)}}{2 f}}{6 d}-\frac {b^2 \cos (e+f x) \sqrt {a+b \sin (e+f x)} (c+d \sin (e+f x))^{3/2}}{3 d f}\) |
| \(\Big \downarrow \) 3297 |
| \(\displaystyle \frac {\frac {\frac {\frac {\frac {2 b^2 (a+b)^{3/2} \left (15 a^2 d^2+6 a b d (2 c+3 d)-\left (b^2 \left (3 c^2-2 c d-16 d^2\right )\right )\right ) \sec (e+f x) (c+d \sin (e+f x)) \sqrt {\frac {(b c-a d) (1-\sin (e+f x))}{(a+b) (c+d \sin (e+f x))}} \sqrt {-\frac {(b c-a d) (\sin (e+f x)+1)}{(a-b) (c+d \sin (e+f x))}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {c+d} \sqrt {a+b \sin (e+f x)}}{\sqrt {a+b} \sqrt {c+d \sin (e+f x)}}\right ),\frac {(a+b) (c-d)}{(a-b) (c+d)}\right )}{f \sqrt {c+d}}-b^3 (a+b) (b c-a d) \left (33 a^2 d^2+14 a b c d-\left (b^2 \left (3 c^2-16 d^2\right )\right )\right ) \int \frac {\sin (e+f x)+1}{(a+b \sin (e+f x))^{3/2} \sqrt {c+d \sin (e+f x)}}dx}{b^2}+\frac {6 \sqrt {c+d} \left (5 a^3 d^3+15 a^2 b c d^2-5 a b^2 d \left (c^2-4 d^2\right )+b^3 \left (c^3+4 c d^2\right )\right ) \sec (e+f x) (a+b \sin (e+f x)) \sqrt {-\frac {(b c-a d) (1-\sin (e+f x))}{(c+d) (a+b \sin (e+f x))}} \sqrt {\frac {(b c-a d) (\sin (e+f x)+1)}{(c-d) (a+b \sin (e+f x))}} \operatorname {EllipticPi}\left (\frac {b (c+d)}{(a+b) d},\arcsin \left (\frac {\sqrt {a+b} \sqrt {c+d \sin (e+f x)}}{\sqrt {c+d} \sqrt {a+b \sin (e+f x)}}\right ),\frac {(a-b) (c+d)}{(a+b) (c-d)}\right )}{d f \sqrt {a+b}}}{2 d}-\frac {b^2 \left (33 a^2 d^2+14 a b c d-\left (b^2 \left (3 c^2-16 d^2\right )\right )\right ) \cos (e+f x) \sqrt {c+d \sin (e+f x)}}{d f \sqrt {a+b \sin (e+f x)}}}{4 b}+\frac {b (3 b c-13 a d) \cos (e+f x) \sqrt {a+b \sin (e+f x)} \sqrt {c+d \sin (e+f x)}}{2 f}}{6 d}-\frac {b^2 \cos (e+f x) \sqrt {a+b \sin (e+f x)} (c+d \sin (e+f x))^{3/2}}{3 d f}\) |
| \(\Big \downarrow \) 3475 |
| \(\displaystyle \frac {\frac {b (3 b c-13 a d) \sqrt {a+b \sin (e+f x)} \sqrt {c+d \sin (e+f x)} \cos (e+f x)}{2 f}+\frac {\frac {\frac {6 \sqrt {c+d} \left (\left (c^3+4 d^2 c\right ) b^3-5 a d \left (c^2-4 d^2\right ) b^2+15 a^2 c d^2 b+5 a^3 d^3\right ) \operatorname {EllipticPi}\left (\frac {b (c+d)}{(a+b) d},\arcsin \left (\frac {\sqrt {a+b} \sqrt {c+d \sin (e+f x)}}{\sqrt {c+d} \sqrt {a+b \sin (e+f x)}}\right ),\frac {(a-b) (c+d)}{(a+b) (c-d)}\right ) \sec (e+f x) \sqrt {-\frac {(b c-a d) (1-\sin (e+f x))}{(c+d) (a+b \sin (e+f x))}} \sqrt {\frac {(b c-a d) (\sin (e+f x)+1)}{(c-d) (a+b \sin (e+f x))}} (a+b \sin (e+f x))}{\sqrt {a+b} d f}+\frac {\frac {2 \sqrt {a+b} (c-d) \sqrt {c+d} \left (-\left (\left (3 c^2-16 d^2\right ) b^2\right )+14 a c d b+33 a^2 d^2\right ) E\left (\arcsin \left (\frac {\sqrt {a+b} \sqrt {c+d \sin (e+f x)}}{\sqrt {c+d} \sqrt {a+b \sin (e+f x)}}\right )|\frac {(a-b) (c+d)}{(a+b) (c-d)}\right ) \sec (e+f x) \sqrt {-\frac {(b c-a d) (1-\sin (e+f x))}{(c+d) (a+b \sin (e+f x))}} \sqrt {\frac {(b c-a d) (\sin (e+f x)+1)}{(c-d) (a+b \sin (e+f x))}} (a+b \sin (e+f x)) b^3}{(b c-a d) f}+\frac {2 (a+b)^{3/2} \left (-\left (\left (3 c^2-2 d c-16 d^2\right ) b^2\right )+6 a d (2 c+3 d) b+15 a^2 d^2\right ) \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {c+d} \sqrt {a+b \sin (e+f x)}}{\sqrt {a+b} \sqrt {c+d \sin (e+f x)}}\right ),\frac {(a+b) (c-d)}{(a-b) (c+d)}\right ) \sec (e+f x) \sqrt {\frac {(b c-a d) (1-\sin (e+f x))}{(a+b) (c+d \sin (e+f x))}} \sqrt {-\frac {(b c-a d) (\sin (e+f x)+1)}{(a-b) (c+d \sin (e+f x))}} (c+d \sin (e+f x)) b^2}{\sqrt {c+d} f}}{b^2}}{2 d}-\frac {b^2 \left (-\left (\left (3 c^2-16 d^2\right ) b^2\right )+14 a c d b+33 a^2 d^2\right ) \cos (e+f x) \sqrt {c+d \sin (e+f x)}}{d f \sqrt {a+b \sin (e+f x)}}}{4 b}}{6 d}-\frac {b^2 \cos (e+f x) \sqrt {a+b \sin (e+f x)} (c+d \sin (e+f x))^{3/2}}{3 d f}\) |
-1/3*(b^2*Cos[e + f*x]*Sqrt[a + b*Sin[e + f*x]]*(c + d*Sin[e + f*x])^(3/2) )/(d*f) + ((b*(3*b*c - 13*a*d)*Cos[e + f*x]*Sqrt[a + b*Sin[e + f*x]]*Sqrt[ c + d*Sin[e + f*x]])/(2*f) + (-((b^2*(14*a*b*c*d + 33*a^2*d^2 - b^2*(3*c^2 - 16*d^2))*Cos[e + f*x]*Sqrt[c + d*Sin[e + f*x]])/(d*f*Sqrt[a + b*Sin[e + f*x]])) + ((6*Sqrt[c + d]*(15*a^2*b*c*d^2 + 5*a^3*d^3 - 5*a*b^2*d*(c^2 - 4*d^2) + b^3*(c^3 + 4*c*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]))/(S qrt[a + b]*d*f) + ((2*b^3*Sqrt[a + b]*(c - d)*Sqrt[c + d]*(14*a*b*c*d + 33 *a^2*d^2 - b^2*(3*c^2 - 16*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 + 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)*(15*a^2*d^2 + 6*a*b*d*(2*c + 3*d) - b^2*(3*c^2 - 2*c*d - 16* d^2))*EllipticF[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]...
3.8.79.3.1 Defintions of rubi rules used
Int[(a_)*(Fx_), x_Symbol] :> Simp[a Int[Fx, x], x] /; FreeQ[a, x] && !Ma tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
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])))
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)]
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)]
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)]
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]
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])))
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]
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]
Result contains complex when optimal does not.
Time = 17.20 (sec) , antiderivative size = 361707, normalized size of antiderivative = 417.19
Timed out. \[ \int (3+b \sin (e+f x))^{5/2} \sqrt {c+d \sin (e+f x)} \, dx=\text {Timed out} \]
Timed out. \[ \int (3+b \sin (e+f x))^{5/2} \sqrt {c+d \sin (e+f x)} \, dx=\text {Timed out} \]
\[ \int (3+b \sin (e+f x))^{5/2} \sqrt {c+d \sin (e+f x)} \, dx=\int { {\left (b \sin \left (f x + e\right ) + a\right )}^{\frac {5}{2}} \sqrt {d \sin \left (f x + e\right ) + c} \,d x } \]
\[ \int (3+b \sin (e+f x))^{5/2} \sqrt {c+d \sin (e+f x)} \, dx=\int { {\left (b \sin \left (f x + e\right ) + a\right )}^{\frac {5}{2}} \sqrt {d \sin \left (f x + e\right ) + c} \,d x } \]
Timed out. \[ \int (3+b \sin (e+f x))^{5/2} \sqrt {c+d \sin (e+f x)} \, dx=\int {\left (a+b\,\sin \left (e+f\,x\right )\right )}^{5/2}\,\sqrt {c+d\,\sin \left (e+f\,x\right )} \,d x \]