\(\int (a+b \sin (e+f x))^3 \sqrt {c+d \sin (e+f x)} \, dx\) [746]

Optimal result

Integrand size = 27, antiderivative size = 375 \[ \int (a+b \sin (e+f x))^3 \sqrt {c+d \sin (e+f x)} \, dx=\frac {2 b \left (42 a b c d-105 a^2 d^2-b^2 \left (8 c^2+25 d^2\right )\right ) \cos (e+f x) \sqrt {c+d \sin (e+f x)}}{105 d^2 f}+\frac {8 b^2 (b c-4 a d) \cos (e+f x) (c+d \sin (e+f x))^{3/2}}{35 d^2 f}-\frac {2 b^2 \cos (e+f x) (a+b \sin (e+f x)) (c+d \sin (e+f x))^{3/2}}{7 d f}+\frac {2 \left (105 a^2 b c d^2+105 a^3 d^3-21 a b^2 d \left (2 c^2-9 d^2\right )+b^3 \left (8 c^3+19 c d^2\right )\right ) E\left (\frac {1}{2} \left (e-\frac {\pi }{2}+f x\right )|\frac {2 d}{c+d}\right ) \sqrt {c+d \sin (e+f x)}}{105 d^3 f \sqrt {\frac {c+d \sin (e+f x)}{c+d}}}+\frac {2 b \left (c^2-d^2\right ) \left (42 a b c d-105 a^2 d^2-b^2 \left (8 c^2+25 d^2\right )\right ) \operatorname {EllipticF}\left (\frac {1}{2} \left (e-\frac {\pi }{2}+f x\right ),\frac {2 d}{c+d}\right ) \sqrt {\frac {c+d \sin (e+f x)}{c+d}}}{105 d^3 f \sqrt {c+d \sin (e+f x)}} \] Output:

2/105*b*(42*a*b*c*d-105*a^2*d^2-b^2*(8*c^2+25*d^2))*cos(f*x+e)*(c+d*sin(f* 
x+e))^(1/2)/d^2/f+8/35*b^2*(-4*a*d+b*c)*cos(f*x+e)*(c+d*sin(f*x+e))^(3/2)/ 
d^2/f-2/7*b^2*cos(f*x+e)*(a+b*sin(f*x+e))*(c+d*sin(f*x+e))^(3/2)/d/f-2/105 
*(105*a^2*b*c*d^2+105*a^3*d^3-21*a*b^2*d*(2*c^2-9*d^2)+b^3*(8*c^3+19*c*d^2 
))*EllipticE(cos(1/2*e+1/4*Pi+1/2*f*x),2^(1/2)*(d/(c+d))^(1/2))*(c+d*sin(f 
*x+e))^(1/2)/d^3/f/((c+d*sin(f*x+e))/(c+d))^(1/2)+2/105*b*(c^2-d^2)*(42*a* 
b*c*d-105*a^2*d^2-b^2*(8*c^2+25*d^2))*InverseJacobiAM(1/2*e-1/4*Pi+1/2*f*x 
,2^(1/2)*(d/(c+d))^(1/2))*((c+d*sin(f*x+e))/(c+d))^(1/2)/d^3/f/(c+d*sin(f* 
x+e))^(1/2)
 

Mathematica [A] (verified)

Time = 4.88 (sec) , antiderivative size = 306, normalized size of antiderivative = 0.82 \[ \int (a+b \sin (e+f x))^3 \sqrt {c+d \sin (e+f x)} \, dx=\frac {-4 \left (d^2 \left (105 a^3 c d+147 a b^2 c d+105 a^2 b d^2+b^3 \left (2 c^2+25 d^2\right )\right ) \operatorname {EllipticF}\left (\frac {1}{4} (-2 e+\pi -2 f x),\frac {2 d}{c+d}\right )+\left (105 a^2 b c d^2+105 a^3 d^3+21 a b^2 d \left (-2 c^2+9 d^2\right )+b^3 \left (8 c^3+19 c d^2\right )\right ) \left ((c+d) E\left (\frac {1}{4} (-2 e+\pi -2 f x)|\frac {2 d}{c+d}\right )-c \operatorname {EllipticF}\left (\frac {1}{4} (-2 e+\pi -2 f x),\frac {2 d}{c+d}\right )\right )\right ) \sqrt {\frac {c+d \sin (e+f x)}{c+d}}+b d (c+d \sin (e+f x)) \left (\left (-84 a b c d-420 a^2 d^2+b^2 \left (16 c^2-115 d^2\right )\right ) \cos (e+f x)+3 b d (5 b d \cos (3 (e+f x))-2 (b c+21 a d) \sin (2 (e+f x)))\right )}{210 d^3 f \sqrt {c+d \sin (e+f x)}} \] Input:

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

Output:

(-4*(d^2*(105*a^3*c*d + 147*a*b^2*c*d + 105*a^2*b*d^2 + b^3*(2*c^2 + 25*d^ 
2))*EllipticF[(-2*e + Pi - 2*f*x)/4, (2*d)/(c + d)] + (105*a^2*b*c*d^2 + 1 
05*a^3*d^3 + 21*a*b^2*d*(-2*c^2 + 9*d^2) + b^3*(8*c^3 + 19*c*d^2))*((c + d 
)*EllipticE[(-2*e + Pi - 2*f*x)/4, (2*d)/(c + d)] - c*EllipticF[(-2*e + Pi 
 - 2*f*x)/4, (2*d)/(c + d)]))*Sqrt[(c + d*Sin[e + f*x])/(c + d)] + b*d*(c 
+ d*Sin[e + f*x])*((-84*a*b*c*d - 420*a^2*d^2 + b^2*(16*c^2 - 115*d^2))*Co 
s[e + f*x] + 3*b*d*(5*b*d*Cos[3*(e + f*x)] - 2*(b*c + 21*a*d)*Sin[2*(e + f 
*x)])))/(210*d^3*f*Sqrt[c + d*Sin[e + f*x]])
 

Rubi [A] (verified)

Time = 1.96 (sec) , antiderivative size = 389, normalized size of antiderivative = 1.04, number of steps used = 18, number of rules used = 18, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.667, Rules used = {3042, 3272, 27, 3042, 3502, 27, 3042, 3232, 27, 3042, 3231, 3042, 3134, 3042, 3132, 3142, 3042, 3140}

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.

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

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 3272

\(\displaystyle \frac {2 \int \frac {1}{2} \sqrt {c+d \sin (e+f x)} \left (7 d a^3+3 b^2 d a-4 b^2 (b c-4 a d) \sin ^2(e+f x)+2 b^3 c-b \left (-21 d a^2+2 b c a-5 b^2 d\right ) \sin (e+f x)\right )dx}{7 d}-\frac {2 b^2 \cos (e+f x) (a+b \sin (e+f x)) (c+d \sin (e+f x))^{3/2}}{7 d f}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {\int \sqrt {c+d \sin (e+f x)} \left (7 d a^3+3 b^2 d a-4 b^2 (b c-4 a d) \sin ^2(e+f x)+2 b^3 c-b \left (-21 d a^2+2 b c a-5 b^2 d\right ) \sin (e+f x)\right )dx}{7 d}-\frac {2 b^2 \cos (e+f x) (a+b \sin (e+f x)) (c+d \sin (e+f x))^{3/2}}{7 d f}\)

\(\Big \downarrow \) 3042

\(\displaystyle \frac {\int \sqrt {c+d \sin (e+f x)} \left (7 d a^3+3 b^2 d a-4 b^2 (b c-4 a d) \sin (e+f x)^2+2 b^3 c-b \left (-21 d a^2+2 b c a-5 b^2 d\right ) \sin (e+f x)\right )dx}{7 d}-\frac {2 b^2 \cos (e+f x) (a+b \sin (e+f x)) (c+d \sin (e+f x))^{3/2}}{7 d f}\)

\(\Big \downarrow \) 3502

\(\displaystyle \frac {\frac {2 \int -\frac {1}{2} \sqrt {c+d \sin (e+f x)} \left (d \left (-35 d a^3-63 b^2 d a+2 b^3 c\right )+b \left (-\left (\left (8 c^2+25 d^2\right ) b^2\right )+42 a c d b-105 a^2 d^2\right ) \sin (e+f x)\right )dx}{5 d}+\frac {8 b^2 (b c-4 a d) \cos (e+f x) (c+d \sin (e+f x))^{3/2}}{5 d f}}{7 d}-\frac {2 b^2 \cos (e+f x) (a+b \sin (e+f x)) (c+d \sin (e+f x))^{3/2}}{7 d f}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {\frac {8 b^2 (b c-4 a d) \cos (e+f x) (c+d \sin (e+f x))^{3/2}}{5 d f}-\frac {\int \sqrt {c+d \sin (e+f x)} \left (d \left (-35 d a^3-63 b^2 d a+2 b^3 c\right )+b \left (-\left (\left (8 c^2+25 d^2\right ) b^2\right )+42 a c d b-105 a^2 d^2\right ) \sin (e+f x)\right )dx}{5 d}}{7 d}-\frac {2 b^2 \cos (e+f x) (a+b \sin (e+f x)) (c+d \sin (e+f x))^{3/2}}{7 d f}\)

\(\Big \downarrow \) 3042

\(\displaystyle \frac {\frac {8 b^2 (b c-4 a d) \cos (e+f x) (c+d \sin (e+f x))^{3/2}}{5 d f}-\frac {\int \sqrt {c+d \sin (e+f x)} \left (d \left (-35 d a^3-63 b^2 d a+2 b^3 c\right )+b \left (-\left (\left (8 c^2+25 d^2\right ) b^2\right )+42 a c d b-105 a^2 d^2\right ) \sin (e+f x)\right )dx}{5 d}}{7 d}-\frac {2 b^2 \cos (e+f x) (a+b \sin (e+f x)) (c+d \sin (e+f x))^{3/2}}{7 d f}\)

\(\Big \downarrow \) 3232

\(\displaystyle \frac {\frac {8 b^2 (b c-4 a d) \cos (e+f x) (c+d \sin (e+f x))^{3/2}}{5 d f}-\frac {\frac {2}{3} \int -\frac {d \left (105 c d a^3+105 b d^2 a^2+147 b^2 c d a+b^3 \left (2 c^2+25 d^2\right )\right )+\left (\left (8 c^3+19 d^2 c\right ) b^3-21 a d \left (2 c^2-9 d^2\right ) b^2+105 a^2 c d^2 b+105 a^3 d^3\right ) \sin (e+f x)}{2 \sqrt {c+d \sin (e+f x)}}dx-\frac {2 b \left (-105 a^2 d^2+42 a b c d-\left (b^2 \left (8 c^2+25 d^2\right )\right )\right ) \cos (e+f x) \sqrt {c+d \sin (e+f x)}}{3 f}}{5 d}}{7 d}-\frac {2 b^2 \cos (e+f x) (a+b \sin (e+f x)) (c+d \sin (e+f x))^{3/2}}{7 d f}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {\frac {8 b^2 (b c-4 a d) \cos (e+f x) (c+d \sin (e+f x))^{3/2}}{5 d f}-\frac {-\frac {1}{3} \int \frac {d \left (105 c d a^3+105 b d^2 a^2+147 b^2 c d a+b^3 \left (2 c^2+25 d^2\right )\right )+\left (\left (8 c^3+19 d^2 c\right ) b^3-21 a d \left (2 c^2-9 d^2\right ) b^2+105 a^2 c d^2 b+105 a^3 d^3\right ) \sin (e+f x)}{\sqrt {c+d \sin (e+f x)}}dx-\frac {2 b \left (-105 a^2 d^2+42 a b c d-\left (b^2 \left (8 c^2+25 d^2\right )\right )\right ) \cos (e+f x) \sqrt {c+d \sin (e+f x)}}{3 f}}{5 d}}{7 d}-\frac {2 b^2 \cos (e+f x) (a+b \sin (e+f x)) (c+d \sin (e+f x))^{3/2}}{7 d f}\)

\(\Big \downarrow \) 3042

\(\displaystyle \frac {\frac {8 b^2 (b c-4 a d) \cos (e+f x) (c+d \sin (e+f x))^{3/2}}{5 d f}-\frac {-\frac {1}{3} \int \frac {d \left (105 c d a^3+105 b d^2 a^2+147 b^2 c d a+b^3 \left (2 c^2+25 d^2\right )\right )+\left (\left (8 c^3+19 d^2 c\right ) b^3-21 a d \left (2 c^2-9 d^2\right ) b^2+105 a^2 c d^2 b+105 a^3 d^3\right ) \sin (e+f x)}{\sqrt {c+d \sin (e+f x)}}dx-\frac {2 b \left (-105 a^2 d^2+42 a b c d-\left (b^2 \left (8 c^2+25 d^2\right )\right )\right ) \cos (e+f x) \sqrt {c+d \sin (e+f x)}}{3 f}}{5 d}}{7 d}-\frac {2 b^2 \cos (e+f x) (a+b \sin (e+f x)) (c+d \sin (e+f x))^{3/2}}{7 d f}\)

\(\Big \downarrow \) 3231

\(\displaystyle \frac {\frac {8 b^2 (b c-4 a d) \cos (e+f x) (c+d \sin (e+f x))^{3/2}}{5 d f}-\frac {\frac {1}{3} \left (-\frac {b \left (c^2-d^2\right ) \left (-105 a^2 d^2+42 a b c d-\left (b^2 \left (8 c^2+25 d^2\right )\right )\right ) \int \frac {1}{\sqrt {c+d \sin (e+f x)}}dx}{d}-\frac {\left (105 a^3 d^3+105 a^2 b c d^2-21 a b^2 d \left (2 c^2-9 d^2\right )+b^3 \left (8 c^3+19 c d^2\right )\right ) \int \sqrt {c+d \sin (e+f x)}dx}{d}\right )-\frac {2 b \left (-105 a^2 d^2+42 a b c d-\left (b^2 \left (8 c^2+25 d^2\right )\right )\right ) \cos (e+f x) \sqrt {c+d \sin (e+f x)}}{3 f}}{5 d}}{7 d}-\frac {2 b^2 \cos (e+f x) (a+b \sin (e+f x)) (c+d \sin (e+f x))^{3/2}}{7 d f}\)

\(\Big \downarrow \) 3042

\(\displaystyle \frac {\frac {8 b^2 (b c-4 a d) \cos (e+f x) (c+d \sin (e+f x))^{3/2}}{5 d f}-\frac {\frac {1}{3} \left (-\frac {b \left (c^2-d^2\right ) \left (-105 a^2 d^2+42 a b c d-\left (b^2 \left (8 c^2+25 d^2\right )\right )\right ) \int \frac {1}{\sqrt {c+d \sin (e+f x)}}dx}{d}-\frac {\left (105 a^3 d^3+105 a^2 b c d^2-21 a b^2 d \left (2 c^2-9 d^2\right )+b^3 \left (8 c^3+19 c d^2\right )\right ) \int \sqrt {c+d \sin (e+f x)}dx}{d}\right )-\frac {2 b \left (-105 a^2 d^2+42 a b c d-\left (b^2 \left (8 c^2+25 d^2\right )\right )\right ) \cos (e+f x) \sqrt {c+d \sin (e+f x)}}{3 f}}{5 d}}{7 d}-\frac {2 b^2 \cos (e+f x) (a+b \sin (e+f x)) (c+d \sin (e+f x))^{3/2}}{7 d f}\)

\(\Big \downarrow \) 3134

\(\displaystyle \frac {\frac {8 b^2 (b c-4 a d) \cos (e+f x) (c+d \sin (e+f x))^{3/2}}{5 d f}-\frac {\frac {1}{3} \left (-\frac {b \left (c^2-d^2\right ) \left (-105 a^2 d^2+42 a b c d-\left (b^2 \left (8 c^2+25 d^2\right )\right )\right ) \int \frac {1}{\sqrt {c+d \sin (e+f x)}}dx}{d}-\frac {\left (105 a^3 d^3+105 a^2 b c d^2-21 a b^2 d \left (2 c^2-9 d^2\right )+b^3 \left (8 c^3+19 c d^2\right )\right ) \sqrt {c+d \sin (e+f x)} \int \sqrt {\frac {c}{c+d}+\frac {d \sin (e+f x)}{c+d}}dx}{d \sqrt {\frac {c+d \sin (e+f x)}{c+d}}}\right )-\frac {2 b \left (-105 a^2 d^2+42 a b c d-\left (b^2 \left (8 c^2+25 d^2\right )\right )\right ) \cos (e+f x) \sqrt {c+d \sin (e+f x)}}{3 f}}{5 d}}{7 d}-\frac {2 b^2 \cos (e+f x) (a+b \sin (e+f x)) (c+d \sin (e+f x))^{3/2}}{7 d f}\)

\(\Big \downarrow \) 3042

\(\displaystyle \frac {\frac {8 b^2 (b c-4 a d) \cos (e+f x) (c+d \sin (e+f x))^{3/2}}{5 d f}-\frac {\frac {1}{3} \left (-\frac {b \left (c^2-d^2\right ) \left (-105 a^2 d^2+42 a b c d-\left (b^2 \left (8 c^2+25 d^2\right )\right )\right ) \int \frac {1}{\sqrt {c+d \sin (e+f x)}}dx}{d}-\frac {\left (105 a^3 d^3+105 a^2 b c d^2-21 a b^2 d \left (2 c^2-9 d^2\right )+b^3 \left (8 c^3+19 c d^2\right )\right ) \sqrt {c+d \sin (e+f x)} \int \sqrt {\frac {c}{c+d}+\frac {d \sin (e+f x)}{c+d}}dx}{d \sqrt {\frac {c+d \sin (e+f x)}{c+d}}}\right )-\frac {2 b \left (-105 a^2 d^2+42 a b c d-\left (b^2 \left (8 c^2+25 d^2\right )\right )\right ) \cos (e+f x) \sqrt {c+d \sin (e+f x)}}{3 f}}{5 d}}{7 d}-\frac {2 b^2 \cos (e+f x) (a+b \sin (e+f x)) (c+d \sin (e+f x))^{3/2}}{7 d f}\)

\(\Big \downarrow \) 3132

\(\displaystyle \frac {\frac {8 b^2 (b c-4 a d) \cos (e+f x) (c+d \sin (e+f x))^{3/2}}{5 d f}-\frac {\frac {1}{3} \left (-\frac {b \left (c^2-d^2\right ) \left (-105 a^2 d^2+42 a b c d-\left (b^2 \left (8 c^2+25 d^2\right )\right )\right ) \int \frac {1}{\sqrt {c+d \sin (e+f x)}}dx}{d}-\frac {2 \left (105 a^3 d^3+105 a^2 b c d^2-21 a b^2 d \left (2 c^2-9 d^2\right )+b^3 \left (8 c^3+19 c d^2\right )\right ) \sqrt {c+d \sin (e+f x)} E\left (\frac {1}{2} \left (e+f x-\frac {\pi }{2}\right )|\frac {2 d}{c+d}\right )}{d f \sqrt {\frac {c+d \sin (e+f x)}{c+d}}}\right )-\frac {2 b \left (-105 a^2 d^2+42 a b c d-\left (b^2 \left (8 c^2+25 d^2\right )\right )\right ) \cos (e+f x) \sqrt {c+d \sin (e+f x)}}{3 f}}{5 d}}{7 d}-\frac {2 b^2 \cos (e+f x) (a+b \sin (e+f x)) (c+d \sin (e+f x))^{3/2}}{7 d f}\)

\(\Big \downarrow \) 3142

\(\displaystyle \frac {\frac {8 b^2 (b c-4 a d) \cos (e+f x) (c+d \sin (e+f x))^{3/2}}{5 d f}-\frac {\frac {1}{3} \left (-\frac {b \left (c^2-d^2\right ) \left (-105 a^2 d^2+42 a b c d-\left (b^2 \left (8 c^2+25 d^2\right )\right )\right ) \sqrt {\frac {c+d \sin (e+f x)}{c+d}} \int \frac {1}{\sqrt {\frac {c}{c+d}+\frac {d \sin (e+f x)}{c+d}}}dx}{d \sqrt {c+d \sin (e+f x)}}-\frac {2 \left (105 a^3 d^3+105 a^2 b c d^2-21 a b^2 d \left (2 c^2-9 d^2\right )+b^3 \left (8 c^3+19 c d^2\right )\right ) \sqrt {c+d \sin (e+f x)} E\left (\frac {1}{2} \left (e+f x-\frac {\pi }{2}\right )|\frac {2 d}{c+d}\right )}{d f \sqrt {\frac {c+d \sin (e+f x)}{c+d}}}\right )-\frac {2 b \left (-105 a^2 d^2+42 a b c d-\left (b^2 \left (8 c^2+25 d^2\right )\right )\right ) \cos (e+f x) \sqrt {c+d \sin (e+f x)}}{3 f}}{5 d}}{7 d}-\frac {2 b^2 \cos (e+f x) (a+b \sin (e+f x)) (c+d \sin (e+f x))^{3/2}}{7 d f}\)

\(\Big \downarrow \) 3042

\(\displaystyle \frac {\frac {8 b^2 (b c-4 a d) \cos (e+f x) (c+d \sin (e+f x))^{3/2}}{5 d f}-\frac {\frac {1}{3} \left (-\frac {b \left (c^2-d^2\right ) \left (-105 a^2 d^2+42 a b c d-\left (b^2 \left (8 c^2+25 d^2\right )\right )\right ) \sqrt {\frac {c+d \sin (e+f x)}{c+d}} \int \frac {1}{\sqrt {\frac {c}{c+d}+\frac {d \sin (e+f x)}{c+d}}}dx}{d \sqrt {c+d \sin (e+f x)}}-\frac {2 \left (105 a^3 d^3+105 a^2 b c d^2-21 a b^2 d \left (2 c^2-9 d^2\right )+b^3 \left (8 c^3+19 c d^2\right )\right ) \sqrt {c+d \sin (e+f x)} E\left (\frac {1}{2} \left (e+f x-\frac {\pi }{2}\right )|\frac {2 d}{c+d}\right )}{d f \sqrt {\frac {c+d \sin (e+f x)}{c+d}}}\right )-\frac {2 b \left (-105 a^2 d^2+42 a b c d-\left (b^2 \left (8 c^2+25 d^2\right )\right )\right ) \cos (e+f x) \sqrt {c+d \sin (e+f x)}}{3 f}}{5 d}}{7 d}-\frac {2 b^2 \cos (e+f x) (a+b \sin (e+f x)) (c+d \sin (e+f x))^{3/2}}{7 d f}\)

\(\Big \downarrow \) 3140

\(\displaystyle \frac {\frac {8 b^2 (b c-4 a d) \cos (e+f x) (c+d \sin (e+f x))^{3/2}}{5 d f}-\frac {\frac {1}{3} \left (-\frac {2 b \left (c^2-d^2\right ) \left (-105 a^2 d^2+42 a b c d-\left (b^2 \left (8 c^2+25 d^2\right )\right )\right ) \sqrt {\frac {c+d \sin (e+f x)}{c+d}} \operatorname {EllipticF}\left (\frac {1}{2} \left (e+f x-\frac {\pi }{2}\right ),\frac {2 d}{c+d}\right )}{d f \sqrt {c+d \sin (e+f x)}}-\frac {2 \left (105 a^3 d^3+105 a^2 b c d^2-21 a b^2 d \left (2 c^2-9 d^2\right )+b^3 \left (8 c^3+19 c d^2\right )\right ) \sqrt {c+d \sin (e+f x)} E\left (\frac {1}{2} \left (e+f x-\frac {\pi }{2}\right )|\frac {2 d}{c+d}\right )}{d f \sqrt {\frac {c+d \sin (e+f x)}{c+d}}}\right )-\frac {2 b \left (-105 a^2 d^2+42 a b c d-\left (b^2 \left (8 c^2+25 d^2\right )\right )\right ) \cos (e+f x) \sqrt {c+d \sin (e+f x)}}{3 f}}{5 d}}{7 d}-\frac {2 b^2 \cos (e+f x) (a+b \sin (e+f x)) (c+d \sin (e+f x))^{3/2}}{7 d f}\)

Input:

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

Output:

(-2*b^2*Cos[e + f*x]*(a + b*Sin[e + f*x])*(c + d*Sin[e + f*x])^(3/2))/(7*d 
*f) + ((8*b^2*(b*c - 4*a*d)*Cos[e + f*x]*(c + d*Sin[e + f*x])^(3/2))/(5*d* 
f) - ((-2*b*(42*a*b*c*d - 105*a^2*d^2 - b^2*(8*c^2 + 25*d^2))*Cos[e + f*x] 
*Sqrt[c + d*Sin[e + f*x]])/(3*f) + ((-2*(105*a^2*b*c*d^2 + 105*a^3*d^3 - 2 
1*a*b^2*d*(2*c^2 - 9*d^2) + b^3*(8*c^3 + 19*c*d^2))*EllipticE[(e - Pi/2 + 
f*x)/2, (2*d)/(c + d)]*Sqrt[c + d*Sin[e + f*x]])/(d*f*Sqrt[(c + d*Sin[e + 
f*x])/(c + d)]) - (2*b*(c^2 - d^2)*(42*a*b*c*d - 105*a^2*d^2 - b^2*(8*c^2 
+ 25*d^2))*EllipticF[(e - Pi/2 + f*x)/2, (2*d)/(c + d)]*Sqrt[(c + d*Sin[e 
+ f*x])/(c + d)])/(d*f*Sqrt[c + d*Sin[e + f*x]]))/3)/(5*d))/(7*d)
 

Defintions of rubi rules used

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 3132

Int[Sqrt[(a_) + (b_.)*sin[(c_.) + (d_.)*(x_)]], x_Symbol] :> Simp[2*(Sqrt[a 
 + b]/d)*EllipticE[(1/2)*(c - Pi/2 + d*x), 2*(b/(a + b))], x] /; FreeQ[{a, 
b, c, d}, x] && NeQ[a^2 - b^2, 0] && GtQ[a + b, 0]
 

rule 3134

Int[Sqrt[(a_) + (b_.)*sin[(c_.) + (d_.)*(x_)]], x_Symbol] :> Simp[Sqrt[a + 
b*Sin[c + d*x]]/Sqrt[(a + b*Sin[c + d*x])/(a + b)]   Int[Sqrt[a/(a + b) + ( 
b/(a + b))*Sin[c + d*x]], x], x] /; FreeQ[{a, b, c, d}, x] && NeQ[a^2 - b^2 
, 0] &&  !GtQ[a + b, 0]
 

rule 3140

Int[1/Sqrt[(a_) + (b_.)*sin[(c_.) + (d_.)*(x_)]], x_Symbol] :> Simp[(2/(d*S 
qrt[a + b]))*EllipticF[(1/2)*(c - Pi/2 + d*x), 2*(b/(a + b))], x] /; FreeQ[ 
{a, b, c, d}, x] && NeQ[a^2 - b^2, 0] && GtQ[a + b, 0]
 

rule 3142

Int[1/Sqrt[(a_) + (b_.)*sin[(c_.) + (d_.)*(x_)]], x_Symbol] :> Simp[Sqrt[(a 
 + b*Sin[c + d*x])/(a + b)]/Sqrt[a + b*Sin[c + d*x]]   Int[1/Sqrt[a/(a + b) 
 + (b/(a + b))*Sin[c + d*x]], x], x] /; FreeQ[{a, b, c, d}, x] && NeQ[a^2 - 
 b^2, 0] &&  !GtQ[a + b, 0]
 

rule 3231

Int[((c_.) + (d_.)*sin[(e_.) + (f_.)*(x_)])/Sqrt[(a_) + (b_.)*sin[(e_.) + ( 
f_.)*(x_)]], x_Symbol] :> Simp[(b*c - a*d)/b   Int[1/Sqrt[a + b*Sin[e + f*x 
]], x], x] + Simp[d/b   Int[Sqrt[a + b*Sin[e + f*x]], x], x] /; FreeQ[{a, b 
, c, d, e, f}, x] && NeQ[b*c - a*d, 0] && NeQ[a^2 - b^2, 0]
 

rule 3232

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

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 3502

Int[((a_.) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_.)*((A_.) + (B_.)*sin[(e_.) 
+ (f_.)*(x_)] + (C_.)*sin[(e_.) + (f_.)*(x_)]^2), x_Symbol] :> Simp[(-C)*Co 
s[e + f*x]*((a + b*Sin[e + f*x])^(m + 1)/(b*f*(m + 2))), x] + Simp[1/(b*(m 
+ 2))   Int[(a + b*Sin[e + f*x])^m*Simp[A*b*(m + 2) + b*C*(m + 1) + (b*B*(m 
 + 2) - a*C)*Sin[e + f*x], x], x], x] /; FreeQ[{a, b, e, f, A, B, C, m}, x] 
 &&  !LtQ[m, -1]
 

Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(1543\) vs. \(2(356)=712\).

Time = 8.08 (sec) , antiderivative size = 1544, normalized size of antiderivative = 4.12

method	result	size
default	\(\text {Expression too large to display}\)	\(1544\)
parts	\(\text {Expression too large to display}\)	\(2445\)

Input:

int((a+b*sin(f*x+e))^3*(c+d*sin(f*x+e))^(1/2),x,method=_RETURNVERBOSE)
 

Output:

(-(-c-d*sin(f*x+e))*cos(f*x+e)^2)^(1/2)*(2*a^3*c*(c/d-1)*((c+d*sin(f*x+e)) 
/(c-d))^(1/2)*(d*(1-sin(f*x+e))/(c+d))^(1/2)*(-d*(1+sin(f*x+e))/(c-d))^(1/ 
2)/(-(-c-d*sin(f*x+e))*cos(f*x+e)^2)^(1/2)*EllipticF(((c+d*sin(f*x+e))/(c- 
d))^(1/2),((c-d)/(c+d))^(1/2))+b^2*(3*a*d+b*c)*(-2/5/d*sin(f*x+e)*(-(-c-d* 
sin(f*x+e))*cos(f*x+e)^2)^(1/2)+8/15*c/d^2*(-(-c-d*sin(f*x+e))*cos(f*x+e)^ 
2)^(1/2)+4/15*c/d*(c/d-1)*((c+d*sin(f*x+e))/(c-d))^(1/2)*(d*(1-sin(f*x+e)) 
/(c+d))^(1/2)*(-d*(1+sin(f*x+e))/(c-d))^(1/2)/(-(-c-d*sin(f*x+e))*cos(f*x+ 
e)^2)^(1/2)*EllipticF(((c+d*sin(f*x+e))/(c-d))^(1/2),((c-d)/(c+d))^(1/2))+ 
2*(3/5+8/15*c^2/d^2)*(c/d-1)*((c+d*sin(f*x+e))/(c-d))^(1/2)*(d*(1-sin(f*x+ 
e))/(c+d))^(1/2)*(-d*(1+sin(f*x+e))/(c-d))^(1/2)/(-(-c-d*sin(f*x+e))*cos(f 
*x+e)^2)^(1/2)*((-c/d-1)*EllipticE(((c+d*sin(f*x+e))/(c-d))^(1/2),((c-d)/( 
c+d))^(1/2))+EllipticF(((c+d*sin(f*x+e))/(c-d))^(1/2),((c-d)/(c+d))^(1/2)) 
))+3*a*b*(a*d+b*c)*(-2/3/d*(-(-c-d*sin(f*x+e))*cos(f*x+e)^2)^(1/2)+2/3*(c/ 
d-1)*((c+d*sin(f*x+e))/(c-d))^(1/2)*(d*(1-sin(f*x+e))/(c+d))^(1/2)*(-d*(1+ 
sin(f*x+e))/(c-d))^(1/2)/(-(-c-d*sin(f*x+e))*cos(f*x+e)^2)^(1/2)*EllipticF 
(((c+d*sin(f*x+e))/(c-d))^(1/2),((c-d)/(c+d))^(1/2))-4/3*c/d*(c/d-1)*((c+d 
*sin(f*x+e))/(c-d))^(1/2)*(d*(1-sin(f*x+e))/(c+d))^(1/2)*(-d*(1+sin(f*x+e) 
)/(c-d))^(1/2)/(-(-c-d*sin(f*x+e))*cos(f*x+e)^2)^(1/2)*((-c/d-1)*EllipticE 
(((c+d*sin(f*x+e))/(c-d))^(1/2),((c-d)/(c+d))^(1/2))+EllipticF(((c+d*sin(f 
*x+e))/(c-d))^(1/2),((c-d)/(c+d))^(1/2))))+2*a^2*(a*d+3*b*c)*(c/d-1)*((...
 

Fricas [C] (verification not implemented)

Result contains complex when optimal does not.

Time = 0.12 (sec) , antiderivative size = 718, normalized size of antiderivative = 1.91 \[ \int (a+b \sin (e+f x))^3 \sqrt {c+d \sin (e+f x)} \, dx =\text {Too large to display} \] Input:

integrate((a+b*sin(f*x+e))^3*(c+d*sin(f*x+e))^(1/2),x, algorithm="fricas")
 

Output:

-2/315*((16*b^3*c^4 - 84*a*b^2*c^3*d + 2*(105*a^2*b + 16*b^3)*c^2*d^2 - 21 
*(5*a^3 + 3*a*b^2)*c*d^3 - 15*(21*a^2*b + 5*b^3)*d^4)*sqrt(1/2*I*d)*weiers 
trassPInverse(-4/3*(4*c^2 - 3*d^2)/d^2, -8/27*(8*I*c^3 - 9*I*c*d^2)/d^3, 1 
/3*(3*d*cos(f*x + e) - 3*I*d*sin(f*x + e) - 2*I*c)/d) + (16*b^3*c^4 - 84*a 
*b^2*c^3*d + 2*(105*a^2*b + 16*b^3)*c^2*d^2 - 21*(5*a^3 + 3*a*b^2)*c*d^3 - 
 15*(21*a^2*b + 5*b^3)*d^4)*sqrt(-1/2*I*d)*weierstrassPInverse(-4/3*(4*c^2 
 - 3*d^2)/d^2, -8/27*(-8*I*c^3 + 9*I*c*d^2)/d^3, 1/3*(3*d*cos(f*x + e) + 3 
*I*d*sin(f*x + e) + 2*I*c)/d) + 3*(8*I*b^3*c^3*d - 42*I*a*b^2*c^2*d^2 + I* 
(105*a^2*b + 19*b^3)*c*d^3 + 21*I*(5*a^3 + 9*a*b^2)*d^4)*sqrt(1/2*I*d)*wei 
erstrassZeta(-4/3*(4*c^2 - 3*d^2)/d^2, -8/27*(8*I*c^3 - 9*I*c*d^2)/d^3, we 
ierstrassPInverse(-4/3*(4*c^2 - 3*d^2)/d^2, -8/27*(8*I*c^3 - 9*I*c*d^2)/d^ 
3, 1/3*(3*d*cos(f*x + e) - 3*I*d*sin(f*x + e) - 2*I*c)/d)) + 3*(-8*I*b^3*c 
^3*d + 42*I*a*b^2*c^2*d^2 - I*(105*a^2*b + 19*b^3)*c*d^3 - 21*I*(5*a^3 + 9 
*a*b^2)*d^4)*sqrt(-1/2*I*d)*weierstrassZeta(-4/3*(4*c^2 - 3*d^2)/d^2, -8/2 
7*(-8*I*c^3 + 9*I*c*d^2)/d^3, weierstrassPInverse(-4/3*(4*c^2 - 3*d^2)/d^2 
, -8/27*(-8*I*c^3 + 9*I*c*d^2)/d^3, 1/3*(3*d*cos(f*x + e) + 3*I*d*sin(f*x 
+ e) + 2*I*c)/d)) - 3*(15*b^3*d^4*cos(f*x + e)^3 - 3*(b^3*c*d^3 + 21*a*b^2 
*d^4)*cos(f*x + e)*sin(f*x + e) + (4*b^3*c^2*d^2 - 21*a*b^2*c*d^3 - 5*(21* 
a^2*b + 8*b^3)*d^4)*cos(f*x + e))*sqrt(d*sin(f*x + e) + c))/(d^4*f)
 

Sympy [F]

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

integrate((a+b*sin(f*x+e))**3*(c+d*sin(f*x+e))**(1/2),x)
 

Output:

Integral((a + b*sin(e + f*x))**3*sqrt(c + d*sin(e + f*x)), x)
 

Maxima [F]

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

integrate((a+b*sin(f*x+e))^3*(c+d*sin(f*x+e))^(1/2),x, algorithm="maxima")
 

Output:

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

Giac [F]

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

integrate((a+b*sin(f*x+e))^3*(c+d*sin(f*x+e))^(1/2),x, algorithm="giac")
 

Output:

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

Mupad [F(-1)]

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

int((a + b*sin(e + f*x))^3*(c + d*sin(e + f*x))^(1/2),x)
 

Output:

int((a + b*sin(e + f*x))^3*(c + d*sin(e + f*x))^(1/2), x)
 

Reduce [F]

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

int((a+b*sin(f*x+e))^3*(c+d*sin(f*x+e))^(1/2),x)
 

Output:

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