Integrand size = 29, antiderivative size = 731 \[ \int \frac {(3+b \sin (e+f x))^{5/2}}{\sqrt {c+d \sin (e+f x)}} \, dx=-\frac {3 b \sqrt {3+b} (b c-9 d) (c-d) \sqrt {c+d} E\left (\arcsin \left (\frac {\sqrt {3+b} \sqrt {c+d \sin (e+f x)}}{\sqrt {c+d} \sqrt {3+b \sin (e+f x)}}\right )|\frac {(3-b) (c+d)}{(3+b) (c-d)}\right ) \sec (e+f x) \sqrt {-\frac {(b c-3 d) (1-\sin (e+f x))}{(c+d) (3+b \sin (e+f x))}} \sqrt {\frac {(b c-3 d) (1+\sin (e+f x))}{(c-d) (3+b \sin (e+f x))}} (3+b \sin (e+f x))}{4 (b c-3 d) d^2 f}-\frac {\sqrt {c+d} \left (30 b c d-135 d^2-b^2 \left (3 c^2+4 d^2\right )\right ) \operatorname {EllipticPi}\left (\frac {b (c+d)}{(3+b) d},\arcsin \left (\frac {\sqrt {3+b} \sqrt {c+d \sin (e+f x)}}{\sqrt {c+d} \sqrt {3+b \sin (e+f x)}}\right ),\frac {(3-b) (c+d)}{(3+b) (c-d)}\right ) \sec (e+f x) \sqrt {-\frac {(b c-3 d) (1-\sin (e+f x))}{(c+d) (3+b \sin (e+f x))}} \sqrt {\frac {(b c-3 d) (1+\sin (e+f x))}{(c-d) (3+b \sin (e+f x))}} (3+b \sin (e+f x))}{4 \sqrt {3+b} d^3 f}+\frac {3 b^2 (b c-9 d) \cos (e+f x) \sqrt {c+d \sin (e+f x)}}{4 d^2 f \sqrt {3+b \sin (e+f x)}}-\frac {b^2 \cos (e+f x) \sqrt {3+b \sin (e+f x)} \sqrt {c+d \sin (e+f x)}}{2 d f}-\frac {(3+b)^{3/2} (3 b c-21 d-2 b d) \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {c+d} \sqrt {3+b \sin (e+f x)}}{\sqrt {3+b} \sqrt {c+d \sin (e+f x)}}\right ),\frac {(3+b) (c-d)}{(3-b) (c+d)}\right ) \sec (e+f x) \sqrt {\frac {(b c-3 d) (1-\sin (e+f x))}{(3+b) (c+d \sin (e+f x))}} \sqrt {-\frac {(b c-3 d) (1+\sin (e+f x))}{(3-b) (c+d \sin (e+f x))}} (c+d \sin (e+f x))}{4 d^2 \sqrt {c+d} f} \]
-1/4*(10*a*b*c*d-15*a^2*d^2-b^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+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)/d^3/f/(a+b)^(1/2)-3/4*b*(c-d)*(-3*a*d+ b*c)*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-1/4*(a+b)^(3/2)*(-7*a*d+3*b*c-2*b*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)/d^2/f/(c+d)^(1/2)+3/4*b^2*(-3*a*d+b*c)*cos(f*x+e)*(c+d*sin(f*x+e))^(1/2 )/d^2/f/(a+b*sin(f*x+e))^(1/2)-1/2*b^2*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. \(1865\) vs. \(2(731)=1462\).
Time = 21.11 (sec) , antiderivative size = 1865, normalized size of antiderivative = 2.55 \[ \int \frac {(3+b \sin (e+f x))^{5/2}}{\sqrt {c+d \sin (e+f x)}} \, dx =\text {Too large to display} \]
-1/2*(b^2*Cos[e + f*x]*Sqrt[3 + b*Sin[e + f*x]]*Sqrt[c + d*Sin[e + f*x]])/ (d*f) + ((-4*(-(b*c) + 3*d)*(-(b^3*c) + 216*d + 33*b^2*d)*Sqrt[((c + d)*Co t[(-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)*(-12*b^2*c + 216*b*d + 4*b^3*d)*((Sqrt[((c + d)*C ot[(-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 -...
Time = 3.24 (sec) , antiderivative size = 758, normalized size of antiderivative = 1.04, number of steps used = 14, number of rules used = 14, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.483, Rules used = {3042, 3272, 27, 3042, 3540, 3042, 3532, 25, 3042, 3290, 3477, 3042, 3297, 3475}
Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.
| \(\displaystyle \int \frac {(a+b \sin (e+f x))^{5/2}}{\sqrt {c+d \sin (e+f x)}} \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \frac {(a+b \sin (e+f x))^{5/2}}{\sqrt {c+d \sin (e+f x)}}dx\) |
| \(\Big \downarrow \) 3272 |
| \(\displaystyle \frac {\int \frac {4 d a^3+b^2 d a-3 b^2 (b c-3 a d) \sin ^2(e+f x)+b^3 c-2 b \left (-6 d a^2+b c a-b^2 d\right ) \sin (e+f x)}{2 \sqrt {a+b \sin (e+f x)} \sqrt {c+d \sin (e+f x)}}dx}{2 d}-\frac {b^2 \cos (e+f x) \sqrt {a+b \sin (e+f x)} \sqrt {c+d \sin (e+f x)}}{2 d f}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\int \frac {4 d a^3+b^2 d a-3 b^2 (b c-3 a d) \sin ^2(e+f x)+b^3 c-2 b \left (-6 d a^2+b c a-b^2 d\right ) \sin (e+f x)}{\sqrt {a+b \sin (e+f x)} \sqrt {c+d \sin (e+f x)}}dx}{4 d}-\frac {b^2 \cos (e+f x) \sqrt {a+b \sin (e+f x)} \sqrt {c+d \sin (e+f x)}}{2 d f}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\int \frac {4 d a^3+b^2 d a-3 b^2 (b c-3 a d) \sin (e+f x)^2+b^3 c-2 b \left (-6 d a^2+b c a-b^2 d\right ) \sin (e+f x)}{\sqrt {a+b \sin (e+f x)} \sqrt {c+d \sin (e+f x)}}dx}{4 d}-\frac {b^2 \cos (e+f x) \sqrt {a+b \sin (e+f x)} \sqrt {c+d \sin (e+f x)}}{2 d f}\) |
| \(\Big \downarrow \) 3540 |
| \(\displaystyle \frac {\frac {\int \frac {8 d^2 a^4+11 b^2 d^2 a^2-10 b^3 c d a+3 b^4 c^2-b^2 \left (-\left (\left (3 c^2+4 d^2\right ) b^2\right )+10 a c d b-15 a^2 d^2\right ) \sin ^2(e+f x)-2 b \left (-16 d^2 a^3+11 b c d a^2-3 b^2 \left (c^2+d^2\right ) a-b^3 c d\right ) \sin (e+f x)}{(a+b \sin (e+f x))^{3/2} \sqrt {c+d \sin (e+f x)}}dx}{2 d}+\frac {3 b^2 (b c-3 a d) \cos (e+f x) \sqrt {c+d \sin (e+f x)}}{d f \sqrt {a+b \sin (e+f x)}}}{4 d}-\frac {b^2 \cos (e+f x) \sqrt {a+b \sin (e+f x)} \sqrt {c+d \sin (e+f x)}}{2 d f}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\frac {\int \frac {8 d^2 a^4+11 b^2 d^2 a^2-10 b^3 c d a+3 b^4 c^2-b^2 \left (-\left (\left (3 c^2+4 d^2\right ) b^2\right )+10 a c d b-15 a^2 d^2\right ) \sin (e+f x)^2-2 b \left (-16 d^2 a^3+11 b c d a^2-3 b^2 \left (c^2+d^2\right ) a-b^3 c d\right ) \sin (e+f x)}{(a+b \sin (e+f x))^{3/2} \sqrt {c+d \sin (e+f x)}}dx}{2 d}+\frac {3 b^2 (b c-3 a d) \cos (e+f x) \sqrt {c+d \sin (e+f x)}}{d f \sqrt {a+b \sin (e+f x)}}}{4 d}-\frac {b^2 \cos (e+f x) \sqrt {a+b \sin (e+f x)} \sqrt {c+d \sin (e+f x)}}{2 d f}\) |
| \(\Big \downarrow \) 3532 |
| \(\displaystyle \frac {\frac {\frac {\int -\frac {2 \left (a^2-b^2\right ) d (b c-a d) \sin (e+f x) b^3+\left (a^2-b^2\right ) (3 b c-7 a d) (b c-a d) b^2}{(a+b \sin (e+f x))^{3/2} \sqrt {c+d \sin (e+f x)}}dx}{b^2}-\left (-15 a^2 d^2+10 a b c d-\left (b^2 \left (3 c^2+4 d^2\right )\right )\right ) \int \frac {\sqrt {a+b \sin (e+f x)}}{\sqrt {c+d \sin (e+f x)}}dx}{2 d}+\frac {3 b^2 (b c-3 a d) \cos (e+f x) \sqrt {c+d \sin (e+f x)}}{d f \sqrt {a+b \sin (e+f x)}}}{4 d}-\frac {b^2 \cos (e+f x) \sqrt {a+b \sin (e+f x)} \sqrt {c+d \sin (e+f x)}}{2 d f}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {\frac {-\left (\left (-15 a^2 d^2+10 a b c d-\left (b^2 \left (3 c^2+4 d^2\right )\right )\right ) \int \frac {\sqrt {a+b \sin (e+f x)}}{\sqrt {c+d \sin (e+f x)}}dx\right )-\frac {\int \frac {2 \left (a^2-b^2\right ) d (b c-a d) \sin (e+f x) b^3+\left (a^2-b^2\right ) (3 b c-7 a d) (b c-a d) b^2}{(a+b \sin (e+f x))^{3/2} \sqrt {c+d \sin (e+f x)}}dx}{b^2}}{2 d}+\frac {3 b^2 (b c-3 a d) \cos (e+f x) \sqrt {c+d \sin (e+f x)}}{d f \sqrt {a+b \sin (e+f x)}}}{4 d}-\frac {b^2 \cos (e+f x) \sqrt {a+b \sin (e+f x)} \sqrt {c+d \sin (e+f x)}}{2 d f}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\frac {-\left (\left (-15 a^2 d^2+10 a b c d-\left (b^2 \left (3 c^2+4 d^2\right )\right )\right ) \int \frac {\sqrt {a+b \sin (e+f x)}}{\sqrt {c+d \sin (e+f x)}}dx\right )-\frac {\int \frac {2 \left (a^2-b^2\right ) d (b c-a d) \sin (e+f x) b^3+\left (a^2-b^2\right ) (3 b c-7 a d) (b c-a d) b^2}{(a+b \sin (e+f x))^{3/2} \sqrt {c+d \sin (e+f x)}}dx}{b^2}}{2 d}+\frac {3 b^2 (b c-3 a d) \cos (e+f x) \sqrt {c+d \sin (e+f x)}}{d f \sqrt {a+b \sin (e+f x)}}}{4 d}-\frac {b^2 \cos (e+f x) \sqrt {a+b \sin (e+f x)} \sqrt {c+d \sin (e+f x)}}{2 d f}\) |
| \(\Big \downarrow \) 3290 |
| \(\displaystyle \frac {\frac {-\frac {\int \frac {2 \left (a^2-b^2\right ) d (b c-a d) \sin (e+f x) b^3+\left (a^2-b^2\right ) (3 b c-7 a d) (b c-a d) b^2}{(a+b \sin (e+f x))^{3/2} \sqrt {c+d \sin (e+f x)}}dx}{b^2}-\frac {2 \sqrt {c+d} \left (-15 a^2 d^2+10 a b c d-\left (b^2 \left (3 c^2+4 d^2\right )\right )\right ) \sec (e+f x) (a+b \sin (e+f x)) \sqrt {-\frac {(b c-a d) (1-\sin (e+f x))}{(c+d) (a+b \sin (e+f x))}} \sqrt {\frac {(b c-a d) (\sin (e+f x)+1)}{(c-d) (a+b \sin (e+f x))}} \operatorname {EllipticPi}\left (\frac {b (c+d)}{(a+b) d},\arcsin \left (\frac {\sqrt {a+b} \sqrt {c+d \sin (e+f x)}}{\sqrt {c+d} \sqrt {a+b \sin (e+f x)}}\right ),\frac {(a-b) (c+d)}{(a+b) (c-d)}\right )}{d f \sqrt {a+b}}}{2 d}+\frac {3 b^2 (b c-3 a d) \cos (e+f x) \sqrt {c+d \sin (e+f x)}}{d f \sqrt {a+b \sin (e+f x)}}}{4 d}-\frac {b^2 \cos (e+f x) \sqrt {a+b \sin (e+f x)} \sqrt {c+d \sin (e+f x)}}{2 d f}\) |
| \(\Big \downarrow \) 3477 |
| \(\displaystyle \frac {\frac {-\frac {b^2 (a+b) (b c-a d) (-7 a d+3 b c-2 b d) \int \frac {1}{\sqrt {a+b \sin (e+f x)} \sqrt {c+d \sin (e+f x)}}dx-3 b^3 (a+b) (b c-3 a d) (b c-a d) \int \frac {\sin (e+f x)+1}{(a+b \sin (e+f x))^{3/2} \sqrt {c+d \sin (e+f x)}}dx}{b^2}-\frac {2 \sqrt {c+d} \left (-15 a^2 d^2+10 a b c d-\left (b^2 \left (3 c^2+4 d^2\right )\right )\right ) \sec (e+f x) (a+b \sin (e+f x)) \sqrt {-\frac {(b c-a d) (1-\sin (e+f x))}{(c+d) (a+b \sin (e+f x))}} \sqrt {\frac {(b c-a d) (\sin (e+f x)+1)}{(c-d) (a+b \sin (e+f x))}} \operatorname {EllipticPi}\left (\frac {b (c+d)}{(a+b) d},\arcsin \left (\frac {\sqrt {a+b} \sqrt {c+d \sin (e+f x)}}{\sqrt {c+d} \sqrt {a+b \sin (e+f x)}}\right ),\frac {(a-b) (c+d)}{(a+b) (c-d)}\right )}{d f \sqrt {a+b}}}{2 d}+\frac {3 b^2 (b c-3 a d) \cos (e+f x) \sqrt {c+d \sin (e+f x)}}{d f \sqrt {a+b \sin (e+f x)}}}{4 d}-\frac {b^2 \cos (e+f x) \sqrt {a+b \sin (e+f x)} \sqrt {c+d \sin (e+f x)}}{2 d f}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\frac {-\frac {b^2 (a+b) (b c-a d) (-7 a d+3 b c-2 b d) \int \frac {1}{\sqrt {a+b \sin (e+f x)} \sqrt {c+d \sin (e+f x)}}dx-3 b^3 (a+b) (b c-3 a d) (b c-a d) \int \frac {\sin (e+f x)+1}{(a+b \sin (e+f x))^{3/2} \sqrt {c+d \sin (e+f x)}}dx}{b^2}-\frac {2 \sqrt {c+d} \left (-15 a^2 d^2+10 a b c d-\left (b^2 \left (3 c^2+4 d^2\right )\right )\right ) \sec (e+f x) (a+b \sin (e+f x)) \sqrt {-\frac {(b c-a d) (1-\sin (e+f x))}{(c+d) (a+b \sin (e+f x))}} \sqrt {\frac {(b c-a d) (\sin (e+f x)+1)}{(c-d) (a+b \sin (e+f x))}} \operatorname {EllipticPi}\left (\frac {b (c+d)}{(a+b) d},\arcsin \left (\frac {\sqrt {a+b} \sqrt {c+d \sin (e+f x)}}{\sqrt {c+d} \sqrt {a+b \sin (e+f x)}}\right ),\frac {(a-b) (c+d)}{(a+b) (c-d)}\right )}{d f \sqrt {a+b}}}{2 d}+\frac {3 b^2 (b c-3 a d) \cos (e+f x) \sqrt {c+d \sin (e+f x)}}{d f \sqrt {a+b \sin (e+f x)}}}{4 d}-\frac {b^2 \cos (e+f x) \sqrt {a+b \sin (e+f x)} \sqrt {c+d \sin (e+f x)}}{2 d f}\) |
| \(\Big \downarrow \) 3297 |
| \(\displaystyle \frac {\frac {-\frac {\frac {2 b^2 (a+b)^{3/2} (-7 a d+3 b c-2 b d) \sec (e+f x) (c+d \sin (e+f x)) \sqrt {\frac {(b c-a d) (1-\sin (e+f x))}{(a+b) (c+d \sin (e+f x))}} \sqrt {-\frac {(b c-a d) (\sin (e+f x)+1)}{(a-b) (c+d \sin (e+f x))}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {c+d} \sqrt {a+b \sin (e+f x)}}{\sqrt {a+b} \sqrt {c+d \sin (e+f x)}}\right ),\frac {(a+b) (c-d)}{(a-b) (c+d)}\right )}{f \sqrt {c+d}}-3 b^3 (a+b) (b c-3 a d) (b c-a d) \int \frac {\sin (e+f x)+1}{(a+b \sin (e+f x))^{3/2} \sqrt {c+d \sin (e+f x)}}dx}{b^2}-\frac {2 \sqrt {c+d} \left (-15 a^2 d^2+10 a b c d-\left (b^2 \left (3 c^2+4 d^2\right )\right )\right ) \sec (e+f x) (a+b \sin (e+f x)) \sqrt {-\frac {(b c-a d) (1-\sin (e+f x))}{(c+d) (a+b \sin (e+f x))}} \sqrt {\frac {(b c-a d) (\sin (e+f x)+1)}{(c-d) (a+b \sin (e+f x))}} \operatorname {EllipticPi}\left (\frac {b (c+d)}{(a+b) d},\arcsin \left (\frac {\sqrt {a+b} \sqrt {c+d \sin (e+f x)}}{\sqrt {c+d} \sqrt {a+b \sin (e+f x)}}\right ),\frac {(a-b) (c+d)}{(a+b) (c-d)}\right )}{d f \sqrt {a+b}}}{2 d}+\frac {3 b^2 (b c-3 a d) \cos (e+f x) \sqrt {c+d \sin (e+f x)}}{d f \sqrt {a+b \sin (e+f x)}}}{4 d}-\frac {b^2 \cos (e+f x) \sqrt {a+b \sin (e+f x)} \sqrt {c+d \sin (e+f x)}}{2 d f}\) |
| \(\Big \downarrow \) 3475 |
| \(\displaystyle \frac {\frac {-\frac {2 \sqrt {c+d} \left (-15 a^2 d^2+10 a b c d-\left (b^2 \left (3 c^2+4 d^2\right )\right )\right ) \sec (e+f x) (a+b \sin (e+f x)) \sqrt {-\frac {(b c-a d) (1-\sin (e+f x))}{(c+d) (a+b \sin (e+f x))}} \sqrt {\frac {(b c-a d) (\sin (e+f x)+1)}{(c-d) (a+b \sin (e+f x))}} \operatorname {EllipticPi}\left (\frac {b (c+d)}{(a+b) d},\arcsin \left (\frac {\sqrt {a+b} \sqrt {c+d \sin (e+f x)}}{\sqrt {c+d} \sqrt {a+b \sin (e+f x)}}\right ),\frac {(a-b) (c+d)}{(a+b) (c-d)}\right )}{d f \sqrt {a+b}}-\frac {\frac {6 b^3 \sqrt {a+b} (c-d) \sqrt {c+d} (b c-3 a d) \sec (e+f x) (a+b \sin (e+f x)) \sqrt {-\frac {(b c-a d) (1-\sin (e+f x))}{(c+d) (a+b \sin (e+f x))}} \sqrt {\frac {(b c-a d) (\sin (e+f x)+1)}{(c-d) (a+b \sin (e+f x))}} E\left (\arcsin \left (\frac {\sqrt {a+b} \sqrt {c+d \sin (e+f x)}}{\sqrt {c+d} \sqrt {a+b \sin (e+f x)}}\right )|\frac {(a-b) (c+d)}{(a+b) (c-d)}\right )}{f (b c-a d)}+\frac {2 b^2 (a+b)^{3/2} (-7 a d+3 b c-2 b d) \sec (e+f x) (c+d \sin (e+f x)) \sqrt {\frac {(b c-a d) (1-\sin (e+f x))}{(a+b) (c+d \sin (e+f x))}} \sqrt {-\frac {(b c-a d) (\sin (e+f x)+1)}{(a-b) (c+d \sin (e+f x))}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {c+d} \sqrt {a+b \sin (e+f x)}}{\sqrt {a+b} \sqrt {c+d \sin (e+f x)}}\right ),\frac {(a+b) (c-d)}{(a-b) (c+d)}\right )}{f \sqrt {c+d}}}{b^2}}{2 d}+\frac {3 b^2 (b c-3 a d) \cos (e+f x) \sqrt {c+d \sin (e+f x)}}{d f \sqrt {a+b \sin (e+f x)}}}{4 d}-\frac {b^2 \cos (e+f x) \sqrt {a+b \sin (e+f x)} \sqrt {c+d \sin (e+f x)}}{2 d f}\) |
-1/2*(b^2*Cos[e + f*x]*Sqrt[a + b*Sin[e + f*x]]*Sqrt[c + d*Sin[e + f*x]])/ (d*f) + ((3*b^2*(b*c - 3*a*d)*Cos[e + f*x]*Sqrt[c + d*Sin[e + f*x]])/(d*f* Sqrt[a + b*Sin[e + f*x]]) + ((-2*Sqrt[c + d]*(10*a*b*c*d - 15*a^2*d^2 - b^ 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) - ((6*b^3*Sqrt[a + b]*(c - d)*Sqrt[c + d]*(b*c - 3*a*d)*EllipticE[Arc Sin[(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)*(3*b*c - 7*a*d - 2*b*d)*Elliptic F[ArcSin[(Sqrt[c + d]*Sqrt[a + b*Sin[e + f*x]])/(Sqrt[a + b]*Sqrt[c + d*Si n[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))/(4*d)
3.8.80.3.1 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_)] + (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 has leaf size over 500,000. Avoiding possible recursion issues.
Time = 21.70 (sec) , antiderivative size = 656608, normalized size of antiderivative = 898.23
\[ \int \frac {(3+b \sin (e+f x))^{5/2}}{\sqrt {c+d \sin (e+f x)}} \, dx=\int { \frac {{\left (b \sin \left (f x + e\right ) + a\right )}^{\frac {5}{2}}}{\sqrt {d \sin \left (f x + e\right ) + c}} \,d x } \]
integral(-(b^2*cos(f*x + e)^2 - 2*a*b*sin(f*x + e) - a^2 - b^2)*sqrt(b*sin (f*x + e) + a)/sqrt(d*sin(f*x + e) + c), x)
Timed out. \[ \int \frac {(3+b \sin (e+f x))^{5/2}}{\sqrt {c+d \sin (e+f x)}} \, dx=\text {Timed out} \]
\[ \int \frac {(3+b \sin (e+f x))^{5/2}}{\sqrt {c+d \sin (e+f x)}} \, dx=\int { \frac {{\left (b \sin \left (f x + e\right ) + a\right )}^{\frac {5}{2}}}{\sqrt {d \sin \left (f x + e\right ) + c}} \,d x } \]
\[ \int \frac {(3+b \sin (e+f x))^{5/2}}{\sqrt {c+d \sin (e+f x)}} \, dx=\int { \frac {{\left (b \sin \left (f x + e\right ) + a\right )}^{\frac {5}{2}}}{\sqrt {d \sin \left (f x + e\right ) + c}} \,d x } \]
Timed out. \[ \int \frac {(3+b \sin (e+f x))^{5/2}}{\sqrt {c+d \sin (e+f x)}} \, dx=\int \frac {{\left (a+b\,\sin \left (e+f\,x\right )\right )}^{5/2}}{\sqrt {c+d\,\sin \left (e+f\,x\right )}} \,d x \]