Integrand size = 29, antiderivative size = 726 \[ \int (3+b \sin (e+f x))^{3/2} \sqrt {c+d \sin (e+f x)} \, dx=\frac {\sqrt {3+b} (c-d) \sqrt {c+d} (b c+15 d) E\left (\arcsin \left (\frac {\sqrt {3+b} \sqrt {c+d \sin (e+f x)}}{\sqrt {c+d} \sqrt {3+b \sin (e+f x)}}\right )|\frac {(3-b) (c+d)}{(3+b) (c-d)}\right ) \sec (e+f x) \sqrt {-\frac {(b c-3 d) (1-\sin (e+f x))}{(c+d) (3+b \sin (e+f x))}} \sqrt {\frac {(b c-3 d) (1+\sin (e+f x))}{(c-d) (3+b \sin (e+f x))}} (3+b \sin (e+f x))}{4 (b c-3 d) d f}+\frac {\sqrt {c+d} \left (18 b c d+27 d^2-b^2 \left (c^2-4 d^2\right )\right ) \operatorname {EllipticPi}\left (\frac {b (c+d)}{(3+b) d},\arcsin \left (\frac {\sqrt {3+b} \sqrt {c+d \sin (e+f x)}}{\sqrt {c+d} \sqrt {3+b \sin (e+f x)}}\right ),\frac {(3-b) (c+d)}{(3+b) (c-d)}\right ) \sec (e+f x) \sqrt {-\frac {(b c-3 d) (1-\sin (e+f x))}{(c+d) (3+b \sin (e+f x))}} \sqrt {\frac {(b c-3 d) (1+\sin (e+f x))}{(c-d) (3+b \sin (e+f x))}} (3+b \sin (e+f x))}{4 b \sqrt {3+b} d^2 f}-\frac {b (b c+15 d) \cos (e+f x) \sqrt {c+d \sin (e+f x)}}{4 d f \sqrt {3+b \sin (e+f x)}}-\frac {b \cos (e+f x) \sqrt {3+b \sin (e+f x)} \sqrt {c+d \sin (e+f x)}}{2 f}+\frac {(3+b)^{3/2} (9 d+b (c+2 d)) \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {c+d} \sqrt {3+b \sin (e+f x)}}{\sqrt {3+b} \sqrt {c+d \sin (e+f x)}}\right ),\frac {(3+b) (c-d)}{(3-b) (c+d)}\right ) \sec (e+f x) \sqrt {\frac {(b c-3 d) (1-\sin (e+f x))}{(3+b) (c+d \sin (e+f x))}} \sqrt {-\frac {(b c-3 d) (1+\sin (e+f x))}{(3-b) (c+d \sin (e+f x))}} (c+d \sin (e+f x))}{4 b d \sqrt {c+d} f} \]
1/4*(6*a*b*c*d+3*a^2*d^2-b^2*(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+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^2/f/(a+b)^(1/2)+1/4*(c-d)*(5*a*d+b*c)*E llipticE((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/(-a*d+b*c)/f+1/ 4*(a+b)^(3/2)*(3*a*d+b*(c+2*d))*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/f/(c +d)^(1/2)-1/4*b*(5*a*d+b*c)*cos(f*x+e)*(c+d*sin(f*x+e))^(1/2)/d/f/(a+b*sin (f*x+e))^(1/2)-1/2*b*cos(f*x+e)*(a+b*sin(f*x+e))^(1/2)*(c+d*sin(f*x+e))^(1 /2)/f
Leaf count is larger than twice the leaf count of optimal. \(1849\) vs. \(2(726)=1452\).
Time = 13.73 (sec) , antiderivative size = 1849, normalized size of antiderivative = 2.55 \[ \int (3+b \sin (e+f x))^{3/2} \sqrt {c+d \sin (e+f x)} \, dx =\text {Too large to display} \]
-1/2*(b*Cos[e + f*x]*Sqrt[3 + b*Sin[e + f*x]]*Sqrt[c + d*Sin[e + f*x]])/f + ((4*(-(b*c) + 3*d)*(-72*c - 3*b^2*c - 21*b*d)*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)*Csc[(-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]]) + 4* (-(b*c) + 3*d)*(-36*b*c - 72*d - 4*b^2*d)*((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)* Csc[(-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))]* Sec[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 + P...
Time = 3.14 (sec) , antiderivative size = 745, normalized size of antiderivative = 1.03, number of steps used = 15, number of rules used = 15, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.517, Rules used = {3042, 3300, 27, 3042, 3540, 25, 3042, 3532, 25, 3042, 3290, 3477, 3042, 3297, 3475}
Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.
| \(\displaystyle \int (a+b \sin (e+f x))^{3/2} \sqrt {c+d \sin (e+f x)} \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int (a+b \sin (e+f x))^{3/2} \sqrt {c+d \sin (e+f x)}dx\) |
| \(\Big \downarrow \) 3300 |
| \(\displaystyle \frac {\int \frac {b d (b c+5 a d) \sin ^2(e+f x)+2 d \left (2 d a^2+3 b c a+b^2 d\right ) \sin (e+f x)+d \left (4 c a^2+b d a+b^2 c\right )}{2 \sqrt {a+b \sin (e+f x)} \sqrt {c+d \sin (e+f x)}}dx}{2 d}-\frac {b \cos (e+f x) \sqrt {a+b \sin (e+f x)} \sqrt {c+d \sin (e+f x)}}{2 f}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\int \frac {b d (b c+5 a d) \sin ^2(e+f x)+2 d \left (2 d a^2+3 b c a+b^2 d\right ) \sin (e+f x)+d \left (4 c a^2+b d a+b^2 c\right )}{\sqrt {a+b \sin (e+f x)} \sqrt {c+d \sin (e+f x)}}dx}{4 d}-\frac {b \cos (e+f x) \sqrt {a+b \sin (e+f x)} \sqrt {c+d \sin (e+f x)}}{2 f}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\int \frac {b d (b c+5 a d) \sin (e+f x)^2+2 d \left (2 d a^2+3 b c a+b^2 d\right ) \sin (e+f x)+d \left (4 c a^2+b d a+b^2 c\right )}{\sqrt {a+b \sin (e+f x)} \sqrt {c+d \sin (e+f x)}}dx}{4 d}-\frac {b \cos (e+f x) \sqrt {a+b \sin (e+f x)} \sqrt {c+d \sin (e+f x)}}{2 f}\) |
| \(\Big \downarrow \) 3540 |
| \(\displaystyle \frac {\frac {\int -\frac {-b d \left (-\left (\left (c^2-4 d^2\right ) b^2\right )+6 a c d b+3 a^2 d^2\right ) \sin ^2(e+f x)-2 d \left (4 d^2 a^3+5 b c d a^2-b^2 \left (c^2-3 d^2\right ) a+b^3 c d\right ) \sin (e+f x)+d \left (-8 c d a^3-7 b d^2 a^2+2 b^2 c d a+b^3 c^2\right )}{(a+b \sin (e+f x))^{3/2} \sqrt {c+d \sin (e+f x)}}dx}{2 d}-\frac {b (5 a d+b c) \cos (e+f x) \sqrt {c+d \sin (e+f x)}}{f \sqrt {a+b \sin (e+f x)}}}{4 d}-\frac {b \cos (e+f x) \sqrt {a+b \sin (e+f x)} \sqrt {c+d \sin (e+f x)}}{2 f}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {-\frac {\int \frac {-b d \left (-\left (\left (c^2-4 d^2\right ) b^2\right )+6 a c d b+3 a^2 d^2\right ) \sin ^2(e+f x)-2 d \left (4 d^2 a^3+5 b c d a^2-b^2 \left (c^2-3 d^2\right ) a+b^3 c d\right ) \sin (e+f x)+d \left (-8 c d a^3-7 b d^2 a^2+2 b^2 c d a+b^3 c^2\right )}{(a+b \sin (e+f x))^{3/2} \sqrt {c+d \sin (e+f x)}}dx}{2 d}-\frac {b (5 a d+b c) \cos (e+f x) \sqrt {c+d \sin (e+f x)}}{f \sqrt {a+b \sin (e+f x)}}}{4 d}-\frac {b \cos (e+f x) \sqrt {a+b \sin (e+f x)} \sqrt {c+d \sin (e+f x)}}{2 f}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {-\frac {\int \frac {-b d \left (-\left (\left (c^2-4 d^2\right ) b^2\right )+6 a c d b+3 a^2 d^2\right ) \sin (e+f x)^2-2 d \left (4 d^2 a^3+5 b c d a^2-b^2 \left (c^2-3 d^2\right ) a+b^3 c d\right ) \sin (e+f x)+d \left (-8 c d a^3-7 b d^2 a^2+2 b^2 c d a+b^3 c^2\right )}{(a+b \sin (e+f x))^{3/2} \sqrt {c+d \sin (e+f x)}}dx}{2 d}-\frac {b (5 a d+b c) \cos (e+f x) \sqrt {c+d \sin (e+f x)}}{f \sqrt {a+b \sin (e+f x)}}}{4 d}-\frac {b \cos (e+f x) \sqrt {a+b \sin (e+f x)} \sqrt {c+d \sin (e+f x)}}{2 f}\) |
| \(\Big \downarrow \) 3532 |
| \(\displaystyle \frac {-\frac {\frac {\int -\frac {b \left (a^2-b^2\right ) d (b c-a d) (b c+3 a d)-2 b^2 \left (a^2-b^2\right ) d^2 (b c-a d) \sin (e+f x)}{(a+b \sin (e+f x))^{3/2} \sqrt {c+d \sin (e+f x)}}dx}{b^2}-\frac {d \left (3 a^2 d^2+6 a b c d-\left (b^2 \left (c^2-4 d^2\right )\right )\right ) \int \frac {\sqrt {a+b \sin (e+f x)}}{\sqrt {c+d \sin (e+f x)}}dx}{b}}{2 d}-\frac {b (5 a d+b c) \cos (e+f x) \sqrt {c+d \sin (e+f x)}}{f \sqrt {a+b \sin (e+f x)}}}{4 d}-\frac {b \cos (e+f x) \sqrt {a+b \sin (e+f x)} \sqrt {c+d \sin (e+f x)}}{2 f}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {-\frac {-\frac {d \left (3 a^2 d^2+6 a b c d-\left (b^2 \left (c^2-4 d^2\right )\right )\right ) \int \frac {\sqrt {a+b \sin (e+f x)}}{\sqrt {c+d \sin (e+f x)}}dx}{b}-\frac {\int \frac {b \left (a^2-b^2\right ) d (b c-a d) (b c+3 a d)-2 b^2 \left (a^2-b^2\right ) d^2 (b c-a d) \sin (e+f x)}{(a+b \sin (e+f x))^{3/2} \sqrt {c+d \sin (e+f x)}}dx}{b^2}}{2 d}-\frac {b (5 a d+b c) \cos (e+f x) \sqrt {c+d \sin (e+f x)}}{f \sqrt {a+b \sin (e+f x)}}}{4 d}-\frac {b \cos (e+f x) \sqrt {a+b \sin (e+f x)} \sqrt {c+d \sin (e+f x)}}{2 f}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {-\frac {-\frac {d \left (3 a^2 d^2+6 a b c d-\left (b^2 \left (c^2-4 d^2\right )\right )\right ) \int \frac {\sqrt {a+b \sin (e+f x)}}{\sqrt {c+d \sin (e+f x)}}dx}{b}-\frac {\int \frac {b \left (a^2-b^2\right ) d (b c-a d) (b c+3 a d)-2 b^2 \left (a^2-b^2\right ) d^2 (b c-a d) \sin (e+f x)}{(a+b \sin (e+f x))^{3/2} \sqrt {c+d \sin (e+f x)}}dx}{b^2}}{2 d}-\frac {b (5 a d+b c) \cos (e+f x) \sqrt {c+d \sin (e+f x)}}{f \sqrt {a+b \sin (e+f x)}}}{4 d}-\frac {b \cos (e+f x) \sqrt {a+b \sin (e+f x)} \sqrt {c+d \sin (e+f x)}}{2 f}\) |
| \(\Big \downarrow \) 3290 |
| \(\displaystyle \frac {-\frac {-\frac {\int \frac {b \left (a^2-b^2\right ) d (b c-a d) (b c+3 a d)-2 b^2 \left (a^2-b^2\right ) d^2 (b c-a d) \sin (e+f x)}{(a+b \sin (e+f x))^{3/2} \sqrt {c+d \sin (e+f x)}}dx}{b^2}-\frac {2 \sqrt {c+d} \left (3 a^2 d^2+6 a b c d-\left (b^2 \left (c^2-4 d^2\right )\right )\right ) \sec (e+f x) (a+b \sin (e+f x)) \sqrt {-\frac {(b c-a d) (1-\sin (e+f x))}{(c+d) (a+b \sin (e+f x))}} \sqrt {\frac {(b c-a d) (\sin (e+f x)+1)}{(c-d) (a+b \sin (e+f x))}} \operatorname {EllipticPi}\left (\frac {b (c+d)}{(a+b) d},\arcsin \left (\frac {\sqrt {a+b} \sqrt {c+d \sin (e+f x)}}{\sqrt {c+d} \sqrt {a+b \sin (e+f x)}}\right ),\frac {(a-b) (c+d)}{(a+b) (c-d)}\right )}{b f \sqrt {a+b}}}{2 d}-\frac {b (5 a d+b c) \cos (e+f x) \sqrt {c+d \sin (e+f x)}}{f \sqrt {a+b \sin (e+f x)}}}{4 d}-\frac {b \cos (e+f x) \sqrt {a+b \sin (e+f x)} \sqrt {c+d \sin (e+f x)}}{2 f}\) |
| \(\Big \downarrow \) 3477 |
| \(\displaystyle \frac {-\frac {-\frac {b d (a+b) (b c-a d) (3 a d+b (c+2 d)) \int \frac {1}{\sqrt {a+b \sin (e+f x)} \sqrt {c+d \sin (e+f x)}}dx-b^2 d (a+b) (b c-a d) (5 a d+b c) \int \frac {\sin (e+f x)+1}{(a+b \sin (e+f x))^{3/2} \sqrt {c+d \sin (e+f x)}}dx}{b^2}-\frac {2 \sqrt {c+d} \left (3 a^2 d^2+6 a b c d-\left (b^2 \left (c^2-4 d^2\right )\right )\right ) \sec (e+f x) (a+b \sin (e+f x)) \sqrt {-\frac {(b c-a d) (1-\sin (e+f x))}{(c+d) (a+b \sin (e+f x))}} \sqrt {\frac {(b c-a d) (\sin (e+f x)+1)}{(c-d) (a+b \sin (e+f x))}} \operatorname {EllipticPi}\left (\frac {b (c+d)}{(a+b) d},\arcsin \left (\frac {\sqrt {a+b} \sqrt {c+d \sin (e+f x)}}{\sqrt {c+d} \sqrt {a+b \sin (e+f x)}}\right ),\frac {(a-b) (c+d)}{(a+b) (c-d)}\right )}{b f \sqrt {a+b}}}{2 d}-\frac {b (5 a d+b c) \cos (e+f x) \sqrt {c+d \sin (e+f x)}}{f \sqrt {a+b \sin (e+f x)}}}{4 d}-\frac {b \cos (e+f x) \sqrt {a+b \sin (e+f x)} \sqrt {c+d \sin (e+f x)}}{2 f}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {-\frac {-\frac {b d (a+b) (b c-a d) (3 a d+b (c+2 d)) \int \frac {1}{\sqrt {a+b \sin (e+f x)} \sqrt {c+d \sin (e+f x)}}dx-b^2 d (a+b) (b c-a d) (5 a d+b c) \int \frac {\sin (e+f x)+1}{(a+b \sin (e+f x))^{3/2} \sqrt {c+d \sin (e+f x)}}dx}{b^2}-\frac {2 \sqrt {c+d} \left (3 a^2 d^2+6 a b c d-\left (b^2 \left (c^2-4 d^2\right )\right )\right ) \sec (e+f x) (a+b \sin (e+f x)) \sqrt {-\frac {(b c-a d) (1-\sin (e+f x))}{(c+d) (a+b \sin (e+f x))}} \sqrt {\frac {(b c-a d) (\sin (e+f x)+1)}{(c-d) (a+b \sin (e+f x))}} \operatorname {EllipticPi}\left (\frac {b (c+d)}{(a+b) d},\arcsin \left (\frac {\sqrt {a+b} \sqrt {c+d \sin (e+f x)}}{\sqrt {c+d} \sqrt {a+b \sin (e+f x)}}\right ),\frac {(a-b) (c+d)}{(a+b) (c-d)}\right )}{b f \sqrt {a+b}}}{2 d}-\frac {b (5 a d+b c) \cos (e+f x) \sqrt {c+d \sin (e+f x)}}{f \sqrt {a+b \sin (e+f x)}}}{4 d}-\frac {b \cos (e+f x) \sqrt {a+b \sin (e+f x)} \sqrt {c+d \sin (e+f x)}}{2 f}\) |
| \(\Big \downarrow \) 3297 |
| \(\displaystyle \frac {-\frac {-\frac {\frac {2 b d (a+b)^{3/2} (3 a d+b (c+2 d)) \sec (e+f x) (c+d \sin (e+f x)) \sqrt {\frac {(b c-a d) (1-\sin (e+f x))}{(a+b) (c+d \sin (e+f x))}} \sqrt {-\frac {(b c-a d) (\sin (e+f x)+1)}{(a-b) (c+d \sin (e+f x))}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {c+d} \sqrt {a+b \sin (e+f x)}}{\sqrt {a+b} \sqrt {c+d \sin (e+f x)}}\right ),\frac {(a+b) (c-d)}{(a-b) (c+d)}\right )}{f \sqrt {c+d}}-b^2 d (a+b) (b c-a d) (5 a d+b c) \int \frac {\sin (e+f x)+1}{(a+b \sin (e+f x))^{3/2} \sqrt {c+d \sin (e+f x)}}dx}{b^2}-\frac {2 \sqrt {c+d} \left (3 a^2 d^2+6 a b c d-\left (b^2 \left (c^2-4 d^2\right )\right )\right ) \sec (e+f x) (a+b \sin (e+f x)) \sqrt {-\frac {(b c-a d) (1-\sin (e+f x))}{(c+d) (a+b \sin (e+f x))}} \sqrt {\frac {(b c-a d) (\sin (e+f x)+1)}{(c-d) (a+b \sin (e+f x))}} \operatorname {EllipticPi}\left (\frac {b (c+d)}{(a+b) d},\arcsin \left (\frac {\sqrt {a+b} \sqrt {c+d \sin (e+f x)}}{\sqrt {c+d} \sqrt {a+b \sin (e+f x)}}\right ),\frac {(a-b) (c+d)}{(a+b) (c-d)}\right )}{b f \sqrt {a+b}}}{2 d}-\frac {b (5 a d+b c) \cos (e+f x) \sqrt {c+d \sin (e+f x)}}{f \sqrt {a+b \sin (e+f x)}}}{4 d}-\frac {b \cos (e+f x) \sqrt {a+b \sin (e+f x)} \sqrt {c+d \sin (e+f x)}}{2 f}\) |
| \(\Big \downarrow \) 3475 |
| \(\displaystyle \frac {-\frac {-\frac {2 \sqrt {c+d} \left (3 a^2 d^2+6 a b c d-\left (b^2 \left (c^2-4 d^2\right )\right )\right ) \sec (e+f x) (a+b \sin (e+f x)) \sqrt {-\frac {(b c-a d) (1-\sin (e+f x))}{(c+d) (a+b \sin (e+f x))}} \sqrt {\frac {(b c-a d) (\sin (e+f x)+1)}{(c-d) (a+b \sin (e+f x))}} \operatorname {EllipticPi}\left (\frac {b (c+d)}{(a+b) d},\arcsin \left (\frac {\sqrt {a+b} \sqrt {c+d \sin (e+f x)}}{\sqrt {c+d} \sqrt {a+b \sin (e+f x)}}\right ),\frac {(a-b) (c+d)}{(a+b) (c-d)}\right )}{b f \sqrt {a+b}}-\frac {\frac {2 b^2 d \sqrt {a+b} (c-d) \sqrt {c+d} (5 a d+b c) \sec (e+f x) (a+b \sin (e+f x)) \sqrt {-\frac {(b c-a d) (1-\sin (e+f x))}{(c+d) (a+b \sin (e+f x))}} \sqrt {\frac {(b c-a d) (\sin (e+f x)+1)}{(c-d) (a+b \sin (e+f x))}} E\left (\arcsin \left (\frac {\sqrt {a+b} \sqrt {c+d \sin (e+f x)}}{\sqrt {c+d} \sqrt {a+b \sin (e+f x)}}\right )|\frac {(a-b) (c+d)}{(a+b) (c-d)}\right )}{f (b c-a d)}+\frac {2 b d (a+b)^{3/2} (3 a d+b (c+2 d)) \sec (e+f x) (c+d \sin (e+f x)) \sqrt {\frac {(b c-a d) (1-\sin (e+f x))}{(a+b) (c+d \sin (e+f x))}} \sqrt {-\frac {(b c-a d) (\sin (e+f x)+1)}{(a-b) (c+d \sin (e+f x))}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {c+d} \sqrt {a+b \sin (e+f x)}}{\sqrt {a+b} \sqrt {c+d \sin (e+f x)}}\right ),\frac {(a+b) (c-d)}{(a-b) (c+d)}\right )}{f \sqrt {c+d}}}{b^2}}{2 d}-\frac {b (5 a d+b c) \cos (e+f x) \sqrt {c+d \sin (e+f x)}}{f \sqrt {a+b \sin (e+f x)}}}{4 d}-\frac {b \cos (e+f x) \sqrt {a+b \sin (e+f x)} \sqrt {c+d \sin (e+f x)}}{2 f}\) |
-1/2*(b*Cos[e + f*x]*Sqrt[a + b*Sin[e + f*x]]*Sqrt[c + d*Sin[e + f*x]])/f + (-((b*(b*c + 5*a*d)*Cos[e + f*x]*Sqrt[c + d*Sin[e + f*x]])/(f*Sqrt[a + b *Sin[e + f*x]])) - ((-2*Sqrt[c + d]*(6*a*b*c*d + 3*a^2*d^2 - b^2*(c^2 - 4* d^2))*EllipticPi[(b*(c + d))/((a + b)*d), ArcSin[(Sqrt[a + b]*Sqrt[c + d*S in[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*Sqrt[a + b]*f) - ((2*b^ 2*Sqrt[a + b]*(c - d)*d*Sqrt[c + d]*(b*c + 5*a*d)*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*(a + b)^(3/2)*d*(3*a*d + b*(c + 2*d))*EllipticF[ArcSin[(Sqr t[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 + S in[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))/(4*d)
3.8.73.3.1 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[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_)])^(m_)*((c_.) + (d_.)*sin[(e_.) + (f_.)*(x_)])^(n_), x_Symbol] :> Simp[(-b)*Cos[e + f*x]*(a + b*Sin[e + f*x] )^(m - 1)*((c + d*Sin[e + f*x])^n/(f*(m + n))), x] + Simp[1/(d*(m + n)) I nt[(a + b*Sin[e + f*x])^(m - 2)*(c + d*Sin[e + f*x])^(n - 1)*Simp[a^2*c*d*( m + n) + b*d*(b*c*(m - 1) + a*d*n) + (a*d*(2*b*c + a*d)*(m + n) - b*d*(a*c - b*d*(m + n - 1)))*Sin[e + f*x] + b*d*(b*c*n + a*d*(2*m + n - 1))*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] && LtQ[0, m, 2] && LtQ[-1, n, 2] && NeQ[m + n, 0] && (IntegerQ[m] || IntegersQ[2*m, 2*n])
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_)] + (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 = 12.45 (sec) , antiderivative size = 250087, normalized size of antiderivative = 344.47
\[ \int (3+b \sin (e+f x))^{3/2} \sqrt {c+d \sin (e+f x)} \, dx=\int { {\left (b \sin \left (f x + e\right ) + a\right )}^{\frac {3}{2}} \sqrt {d \sin \left (f x + e\right ) + c} \,d x } \]
\[ \int (3+b \sin (e+f x))^{3/2} \sqrt {c+d \sin (e+f x)} \, dx=\int \left (a + b \sin {\left (e + f x \right )}\right )^{\frac {3}{2}} \sqrt {c + d \sin {\left (e + f x \right )}}\, dx \]
\[ \int (3+b \sin (e+f x))^{3/2} \sqrt {c+d \sin (e+f x)} \, dx=\int { {\left (b \sin \left (f x + e\right ) + a\right )}^{\frac {3}{2}} \sqrt {d \sin \left (f x + e\right ) + c} \,d x } \]
\[ \int (3+b \sin (e+f x))^{3/2} \sqrt {c+d \sin (e+f x)} \, dx=\int { {\left (b \sin \left (f x + e\right ) + a\right )}^{\frac {3}{2}} \sqrt {d \sin \left (f x + e\right ) + c} \,d x } \]
Timed out. \[ \int (3+b \sin (e+f x))^{3/2} \sqrt {c+d \sin (e+f x)} \, dx=\int {\left (a+b\,\sin \left (e+f\,x\right )\right )}^{3/2}\,\sqrt {c+d\,\sin \left (e+f\,x\right )} \,d x \]