\(\int (a+b \sin (e+f x)) (c+d \sin (e+f x))^{5/2} \, dx\) [730]

Optimal result

Integrand size = 25, antiderivative size = 298 \[ \int (a+b \sin (e+f x)) (c+d \sin (e+f x))^{5/2} \, dx=-\frac {2 \left (15 b c^2+56 a c d+25 b d^2\right ) \cos (e+f x) \sqrt {c+d \sin (e+f x)}}{105 f}-\frac {2 (5 b c+7 a d) \cos (e+f x) (c+d \sin (e+f x))^{3/2}}{35 f}-\frac {2 b \cos (e+f x) (c+d \sin (e+f x))^{5/2}}{7 f}+\frac {2 \left (15 b c^3+161 a c^2 d+145 b c d^2+63 a d^3\right ) E\left (\frac {1}{2} \left (e-\frac {\pi }{2}+f x\right )|\frac {2 d}{c+d}\right ) \sqrt {c+d \sin (e+f x)}}{105 d f \sqrt {\frac {c+d \sin (e+f x)}{c+d}}}-\frac {2 \left (c^2-d^2\right ) \left (15 b c^2+56 a c d+25 b d^2\right ) \operatorname {EllipticF}\left (\frac {1}{2} \left (e-\frac {\pi }{2}+f x\right ),\frac {2 d}{c+d}\right ) \sqrt {\frac {c+d \sin (e+f x)}{c+d}}}{105 d f \sqrt {c+d \sin (e+f x)}} \] Output:

-2/105*(56*a*c*d+15*b*c^2+25*b*d^2)*cos(f*x+e)*(c+d*sin(f*x+e))^(1/2)/f-2/ 
35*(7*a*d+5*b*c)*cos(f*x+e)*(c+d*sin(f*x+e))^(3/2)/f-2/7*b*cos(f*x+e)*(c+d 
*sin(f*x+e))^(5/2)/f-2/105*(161*a*c^2*d+63*a*d^3+15*b*c^3+145*b*c*d^2)*Ell 
ipticE(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/f/((c+d*sin(f*x+e))/(c+d))^(1/2)-2/105*(c^2-d^2)*(56*a*c*d+15*b*c 
^2+25*b*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/f/(c+d*sin(f*x+e))^(1/2)
 

Mathematica [A] (verified)

Time = 2.09 (sec) , antiderivative size = 275, normalized size of antiderivative = 0.92 \[ \int (a+b \sin (e+f x)) (c+d \sin (e+f x))^{5/2} \, dx=\frac {-2 d \left (5 b d \left (27 c^2+5 d^2\right )+7 a \left (15 c^3+17 c d^2\right )\right ) \operatorname {EllipticF}\left (\frac {1}{4} (-2 e+\pi -2 f x),\frac {2 d}{c+d}\right ) \sqrt {\frac {c+d \sin (e+f x)}{c+d}}-2 \left (7 a d \left (23 c^2+9 d^2\right )+5 b \left (3 c^3+29 c d^2\right )\right ) \left ((c+d) E\left (\frac {1}{4} (-2 e+\pi -2 f x)|\frac {2 d}{c+d}\right )-c \operatorname {EllipticF}\left (\frac {1}{4} (-2 e+\pi -2 f x),\frac {2 d}{c+d}\right )\right ) \sqrt {\frac {c+d \sin (e+f x)}{c+d}}-d \cos (e+f x) (c+d \sin (e+f x)) \left (90 b c^2+154 a c d+65 b d^2-15 b d^2 \cos (2 (e+f x))+6 d (15 b c+7 a d) \sin (e+f x)\right )}{105 d f \sqrt {c+d \sin (e+f x)}} \] Input:

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

Output:

(-2*d*(5*b*d*(27*c^2 + 5*d^2) + 7*a*(15*c^3 + 17*c*d^2))*EllipticF[(-2*e + 
 Pi - 2*f*x)/4, (2*d)/(c + d)]*Sqrt[(c + d*Sin[e + f*x])/(c + d)] - 2*(7*a 
*d*(23*c^2 + 9*d^2) + 5*b*(3*c^3 + 29*c*d^2))*((c + d)*EllipticE[(-2*e + P 
i - 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)] - d*Cos[e + f*x]*(c + d*Sin[e + 
 f*x])*(90*b*c^2 + 154*a*c*d + 65*b*d^2 - 15*b*d^2*Cos[2*(e + f*x)] + 6*d* 
(15*b*c + 7*a*d)*Sin[e + f*x]))/(105*d*f*Sqrt[c + d*Sin[e + f*x]])
 

Rubi [A] (verified)

Time = 1.60 (sec) , antiderivative size = 309, normalized size of antiderivative = 1.04, number of steps used = 18, number of rules used = 18, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.720, Rules used = {3042, 3232, 27, 3042, 3232, 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.

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

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 3232

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 3232

\(\displaystyle \frac {1}{7} \left (\frac {2}{5} \int \frac {1}{2} \sqrt {c+d \sin (e+f x)} \left (40 b c d+7 a \left (5 c^2+3 d^2\right )+\left (15 b c^2+56 a d c+25 b d^2\right ) \sin (e+f x)\right )dx-\frac {2 (7 a d+5 b c) \cos (e+f x) (c+d \sin (e+f x))^{3/2}}{5 f}\right )-\frac {2 b \cos (e+f x) (c+d \sin (e+f x))^{5/2}}{7 f}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {1}{7} \left (\frac {1}{5} \int \sqrt {c+d \sin (e+f x)} \left (35 a c^2+40 b d c+21 a d^2+\left (15 b c^2+56 a d c+25 b d^2\right ) \sin (e+f x)\right )dx-\frac {2 (7 a d+5 b c) \cos (e+f x) (c+d \sin (e+f x))^{3/2}}{5 f}\right )-\frac {2 b \cos (e+f x) (c+d \sin (e+f x))^{5/2}}{7 f}\)

\(\Big \downarrow \) 3042

\(\displaystyle \frac {1}{7} \left (\frac {1}{5} \int \sqrt {c+d \sin (e+f x)} \left (35 a c^2+40 b d c+21 a d^2+\left (15 b c^2+56 a d c+25 b d^2\right ) \sin (e+f x)\right )dx-\frac {2 (7 a d+5 b c) \cos (e+f x) (c+d \sin (e+f x))^{3/2}}{5 f}\right )-\frac {2 b \cos (e+f x) (c+d \sin (e+f x))^{5/2}}{7 f}\)

\(\Big \downarrow \) 3232

\(\displaystyle \frac {1}{7} \left (\frac {1}{5} \left (\frac {2}{3} \int \frac {105 a c^3+135 b d c^2+119 a d^2 c+25 b d^3+\left (15 b c^3+161 a d c^2+145 b d^2 c+63 a d^3\right ) \sin (e+f x)}{2 \sqrt {c+d \sin (e+f x)}}dx-\frac {2 \left (56 a c d+15 b c^2+25 b d^2\right ) \cos (e+f x) \sqrt {c+d \sin (e+f x)}}{3 f}\right )-\frac {2 (7 a d+5 b c) \cos (e+f x) (c+d \sin (e+f x))^{3/2}}{5 f}\right )-\frac {2 b \cos (e+f x) (c+d \sin (e+f x))^{5/2}}{7 f}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {1}{7} \left (\frac {1}{5} \left (\frac {1}{3} \int \frac {105 a c^3+135 b d c^2+119 a d^2 c+25 b d^3+\left (15 b c^3+161 a d c^2+145 b d^2 c+63 a d^3\right ) \sin (e+f x)}{\sqrt {c+d \sin (e+f x)}}dx-\frac {2 \left (56 a c d+15 b c^2+25 b d^2\right ) \cos (e+f x) \sqrt {c+d \sin (e+f x)}}{3 f}\right )-\frac {2 (7 a d+5 b c) \cos (e+f x) (c+d \sin (e+f x))^{3/2}}{5 f}\right )-\frac {2 b \cos (e+f x) (c+d \sin (e+f x))^{5/2}}{7 f}\)

\(\Big \downarrow \) 3042

\(\displaystyle \frac {1}{7} \left (\frac {1}{5} \left (\frac {1}{3} \int \frac {105 a c^3+135 b d c^2+119 a d^2 c+25 b d^3+\left (15 b c^3+161 a d c^2+145 b d^2 c+63 a d^3\right ) \sin (e+f x)}{\sqrt {c+d \sin (e+f x)}}dx-\frac {2 \left (56 a c d+15 b c^2+25 b d^2\right ) \cos (e+f x) \sqrt {c+d \sin (e+f x)}}{3 f}\right )-\frac {2 (7 a d+5 b c) \cos (e+f x) (c+d \sin (e+f x))^{3/2}}{5 f}\right )-\frac {2 b \cos (e+f x) (c+d \sin (e+f x))^{5/2}}{7 f}\)

\(\Big \downarrow \) 3231

\(\displaystyle \frac {1}{7} \left (\frac {1}{5} \left (\frac {1}{3} \left (\frac {\left (161 a c^2 d+63 a d^3+15 b c^3+145 b c d^2\right ) \int \sqrt {c+d \sin (e+f x)}dx}{d}-\frac {\left (c^2-d^2\right ) \left (56 a c d+15 b c^2+25 b d^2\right ) \int \frac {1}{\sqrt {c+d \sin (e+f x)}}dx}{d}\right )-\frac {2 \left (56 a c d+15 b c^2+25 b d^2\right ) \cos (e+f x) \sqrt {c+d \sin (e+f x)}}{3 f}\right )-\frac {2 (7 a d+5 b c) \cos (e+f x) (c+d \sin (e+f x))^{3/2}}{5 f}\right )-\frac {2 b \cos (e+f x) (c+d \sin (e+f x))^{5/2}}{7 f}\)

\(\Big \downarrow \) 3042

\(\displaystyle \frac {1}{7} \left (\frac {1}{5} \left (\frac {1}{3} \left (\frac {\left (161 a c^2 d+63 a d^3+15 b c^3+145 b c d^2\right ) \int \sqrt {c+d \sin (e+f x)}dx}{d}-\frac {\left (c^2-d^2\right ) \left (56 a c d+15 b c^2+25 b d^2\right ) \int \frac {1}{\sqrt {c+d \sin (e+f x)}}dx}{d}\right )-\frac {2 \left (56 a c d+15 b c^2+25 b d^2\right ) \cos (e+f x) \sqrt {c+d \sin (e+f x)}}{3 f}\right )-\frac {2 (7 a d+5 b c) \cos (e+f x) (c+d \sin (e+f x))^{3/2}}{5 f}\right )-\frac {2 b \cos (e+f x) (c+d \sin (e+f x))^{5/2}}{7 f}\)

\(\Big \downarrow \) 3134

\(\displaystyle \frac {1}{7} \left (\frac {1}{5} \left (\frac {1}{3} \left (\frac {\left (161 a c^2 d+63 a d^3+15 b c^3+145 b c d^2\right ) \sqrt {c+d \sin (e+f x)} \int \sqrt {\frac {c}{c+d}+\frac {d \sin (e+f x)}{c+d}}dx}{d \sqrt {\frac {c+d \sin (e+f x)}{c+d}}}-\frac {\left (c^2-d^2\right ) \left (56 a c d+15 b c^2+25 b d^2\right ) \int \frac {1}{\sqrt {c+d \sin (e+f x)}}dx}{d}\right )-\frac {2 \left (56 a c d+15 b c^2+25 b d^2\right ) \cos (e+f x) \sqrt {c+d \sin (e+f x)}}{3 f}\right )-\frac {2 (7 a d+5 b c) \cos (e+f x) (c+d \sin (e+f x))^{3/2}}{5 f}\right )-\frac {2 b \cos (e+f x) (c+d \sin (e+f x))^{5/2}}{7 f}\)

\(\Big \downarrow \) 3042

\(\displaystyle \frac {1}{7} \left (\frac {1}{5} \left (\frac {1}{3} \left (\frac {\left (161 a c^2 d+63 a d^3+15 b c^3+145 b c d^2\right ) \sqrt {c+d \sin (e+f x)} \int \sqrt {\frac {c}{c+d}+\frac {d \sin (e+f x)}{c+d}}dx}{d \sqrt {\frac {c+d \sin (e+f x)}{c+d}}}-\frac {\left (c^2-d^2\right ) \left (56 a c d+15 b c^2+25 b d^2\right ) \int \frac {1}{\sqrt {c+d \sin (e+f x)}}dx}{d}\right )-\frac {2 \left (56 a c d+15 b c^2+25 b d^2\right ) \cos (e+f x) \sqrt {c+d \sin (e+f x)}}{3 f}\right )-\frac {2 (7 a d+5 b c) \cos (e+f x) (c+d \sin (e+f x))^{3/2}}{5 f}\right )-\frac {2 b \cos (e+f x) (c+d \sin (e+f x))^{5/2}}{7 f}\)

\(\Big \downarrow \) 3132

\(\displaystyle \frac {1}{7} \left (\frac {1}{5} \left (\frac {1}{3} \left (\frac {2 \left (161 a c^2 d+63 a d^3+15 b c^3+145 b c d^2\right ) \sqrt {c+d \sin (e+f x)} E\left (\frac {1}{2} \left (e+f x-\frac {\pi }{2}\right )|\frac {2 d}{c+d}\right )}{d f \sqrt {\frac {c+d \sin (e+f x)}{c+d}}}-\frac {\left (c^2-d^2\right ) \left (56 a c d+15 b c^2+25 b d^2\right ) \int \frac {1}{\sqrt {c+d \sin (e+f x)}}dx}{d}\right )-\frac {2 \left (56 a c d+15 b c^2+25 b d^2\right ) \cos (e+f x) \sqrt {c+d \sin (e+f x)}}{3 f}\right )-\frac {2 (7 a d+5 b c) \cos (e+f x) (c+d \sin (e+f x))^{3/2}}{5 f}\right )-\frac {2 b \cos (e+f x) (c+d \sin (e+f x))^{5/2}}{7 f}\)

\(\Big \downarrow \) 3142

\(\displaystyle \frac {1}{7} \left (\frac {1}{5} \left (\frac {1}{3} \left (\frac {2 \left (161 a c^2 d+63 a d^3+15 b c^3+145 b c d^2\right ) \sqrt {c+d \sin (e+f x)} E\left (\frac {1}{2} \left (e+f x-\frac {\pi }{2}\right )|\frac {2 d}{c+d}\right )}{d f \sqrt {\frac {c+d \sin (e+f x)}{c+d}}}-\frac {\left (c^2-d^2\right ) \left (56 a c d+15 b c^2+25 b d^2\right ) \sqrt {\frac {c+d \sin (e+f x)}{c+d}} \int \frac {1}{\sqrt {\frac {c}{c+d}+\frac {d \sin (e+f x)}{c+d}}}dx}{d \sqrt {c+d \sin (e+f x)}}\right )-\frac {2 \left (56 a c d+15 b c^2+25 b d^2\right ) \cos (e+f x) \sqrt {c+d \sin (e+f x)}}{3 f}\right )-\frac {2 (7 a d+5 b c) \cos (e+f x) (c+d \sin (e+f x))^{3/2}}{5 f}\right )-\frac {2 b \cos (e+f x) (c+d \sin (e+f x))^{5/2}}{7 f}\)

\(\Big \downarrow \) 3042

\(\displaystyle \frac {1}{7} \left (\frac {1}{5} \left (\frac {1}{3} \left (\frac {2 \left (161 a c^2 d+63 a d^3+15 b c^3+145 b c d^2\right ) \sqrt {c+d \sin (e+f x)} E\left (\frac {1}{2} \left (e+f x-\frac {\pi }{2}\right )|\frac {2 d}{c+d}\right )}{d f \sqrt {\frac {c+d \sin (e+f x)}{c+d}}}-\frac {\left (c^2-d^2\right ) \left (56 a c d+15 b c^2+25 b d^2\right ) \sqrt {\frac {c+d \sin (e+f x)}{c+d}} \int \frac {1}{\sqrt {\frac {c}{c+d}+\frac {d \sin (e+f x)}{c+d}}}dx}{d \sqrt {c+d \sin (e+f x)}}\right )-\frac {2 \left (56 a c d+15 b c^2+25 b d^2\right ) \cos (e+f x) \sqrt {c+d \sin (e+f x)}}{3 f}\right )-\frac {2 (7 a d+5 b c) \cos (e+f x) (c+d \sin (e+f x))^{3/2}}{5 f}\right )-\frac {2 b \cos (e+f x) (c+d \sin (e+f x))^{5/2}}{7 f}\)

\(\Big \downarrow \) 3140

\(\displaystyle \frac {1}{7} \left (\frac {1}{5} \left (\frac {1}{3} \left (\frac {2 \left (161 a c^2 d+63 a d^3+15 b c^3+145 b c d^2\right ) \sqrt {c+d \sin (e+f x)} E\left (\frac {1}{2} \left (e+f x-\frac {\pi }{2}\right )|\frac {2 d}{c+d}\right )}{d f \sqrt {\frac {c+d \sin (e+f x)}{c+d}}}-\frac {2 \left (c^2-d^2\right ) \left (56 a c d+15 b c^2+25 b d^2\right ) \sqrt {\frac {c+d \sin (e+f x)}{c+d}} \operatorname {EllipticF}\left (\frac {1}{2} \left (e+f x-\frac {\pi }{2}\right ),\frac {2 d}{c+d}\right )}{d f \sqrt {c+d \sin (e+f x)}}\right )-\frac {2 \left (56 a c d+15 b c^2+25 b d^2\right ) \cos (e+f x) \sqrt {c+d \sin (e+f x)}}{3 f}\right )-\frac {2 (7 a d+5 b c) \cos (e+f x) (c+d \sin (e+f x))^{3/2}}{5 f}\right )-\frac {2 b \cos (e+f x) (c+d \sin (e+f x))^{5/2}}{7 f}\)

Input:

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

Output:

(-2*b*Cos[e + f*x]*(c + d*Sin[e + f*x])^(5/2))/(7*f) + ((-2*(5*b*c + 7*a*d 
)*Cos[e + f*x]*(c + d*Sin[e + f*x])^(3/2))/(5*f) + ((-2*(15*b*c^2 + 56*a*c 
*d + 25*b*d^2)*Cos[e + f*x]*Sqrt[c + d*Sin[e + f*x]])/(3*f) + ((2*(15*b*c^ 
3 + 161*a*c^2*d + 145*b*c*d^2 + 63*a*d^3)*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*(c^2 - d^2)*(15*b*c^2 + 56*a*c*d + 25*b*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)/7
 

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]
 

Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(1834\) vs. \(2(279)=558\).

Time = 6.67 (sec) , antiderivative size = 1835, normalized size of antiderivative = 6.16

method	result	size
parts	\(\text {Expression too large to display}\)	\(1835\)
default	\(\text {Expression too large to display}\)	\(1839\)

Input:

int((a+b*sin(f*x+e))*(c+d*sin(f*x+e))^(5/2),x,method=_RETURNVERBOSE)
 

Output:

2/15*a*(15*c^4*((c+d*sin(f*x+e))/(c-d))^(1/2)*(-(sin(f*x+e)-1)*d/(c+d))^(1 
/2)*(-d*(1+sin(f*x+e))/(c-d))^(1/2)*EllipticF(((c+d*sin(f*x+e))/(c-d))^(1/ 
2),((c-d)/(c+d))^(1/2))+8*((c+d*sin(f*x+e))/(c-d))^(1/2)*(-(sin(f*x+e)-1)* 
d/(c+d))^(1/2)*(-d*(1+sin(f*x+e))/(c-d))^(1/2)*EllipticF(((c+d*sin(f*x+e)) 
/(c-d))^(1/2),((c-d)/(c+d))^(1/2))*c^3*d-6*c^2*((c+d*sin(f*x+e))/(c-d))^(1 
/2)*(-(sin(f*x+e)-1)*d/(c+d))^(1/2)*(-d*(1+sin(f*x+e))/(c-d))^(1/2)*Ellipt 
icF(((c+d*sin(f*x+e))/(c-d))^(1/2),((c-d)/(c+d))^(1/2))*d^2-8*c*((c+d*sin( 
f*x+e))/(c-d))^(1/2)*(-(sin(f*x+e)-1)*d/(c+d))^(1/2)*(-d*(1+sin(f*x+e))/(c 
-d))^(1/2)*EllipticF(((c+d*sin(f*x+e))/(c-d))^(1/2),((c-d)/(c+d))^(1/2))*d 
^3-9*((c+d*sin(f*x+e))/(c-d))^(1/2)*(-(sin(f*x+e)-1)*d/(c+d))^(1/2)*(-d*(1 
+sin(f*x+e))/(c-d))^(1/2)*EllipticF(((c+d*sin(f*x+e))/(c-d))^(1/2),((c-d)/ 
(c+d))^(1/2))*d^4-23*((c+d*sin(f*x+e))/(c-d))^(1/2)*(-(sin(f*x+e)-1)*d/(c+ 
d))^(1/2)*(-d*(1+sin(f*x+e))/(c-d))^(1/2)*EllipticE(((c+d*sin(f*x+e))/(c-d 
))^(1/2),((c-d)/(c+d))^(1/2))*c^4+14*((c+d*sin(f*x+e))/(c-d))^(1/2)*(-(sin 
(f*x+e)-1)*d/(c+d))^(1/2)*(-d*(1+sin(f*x+e))/(c-d))^(1/2)*EllipticE(((c+d* 
sin(f*x+e))/(c-d))^(1/2),((c-d)/(c+d))^(1/2))*c^2*d^2+9*((c+d*sin(f*x+e))/ 
(c-d))^(1/2)*(-(sin(f*x+e)-1)*d/(c+d))^(1/2)*(-d*(1+sin(f*x+e))/(c-d))^(1/ 
2)*EllipticE(((c+d*sin(f*x+e))/(c-d))^(1/2),((c-d)/(c+d))^(1/2))*d^4+3*sin 
(f*x+e)^4*d^4+14*sin(f*x+e)^3*c*d^3+11*sin(f*x+e)^2*c^2*d^2-3*d^4*sin(f*x+ 
e)^2-14*sin(f*x+e)*c*d^3-11*d^2*c^2)/d/cos(f*x+e)/(c+d*sin(f*x+e))^(1/2...
 

Fricas [C] (verification not implemented)

Result contains complex when optimal does not.

Time = 0.13 (sec) , antiderivative size = 567, normalized size of antiderivative = 1.90 \[ \int (a+b \sin (e+f x)) (c+d \sin (e+f x))^{5/2} \, dx =\text {Too large to display} \] Input:

integrate((a+b*sin(f*x+e))*(c+d*sin(f*x+e))^(5/2),x, algorithm="fricas")
 

Output:

-2/315*((30*b*c^4 + 7*a*c^3*d - 115*b*c^2*d^2 - 231*a*c*d^3 - 75*b*d^4)*sq 
rt(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) + 
(30*b*c^4 + 7*a*c^3*d - 115*b*c^2*d^2 - 231*a*c*d^3 - 75*b*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*(15* 
I*b*c^3*d + 161*I*a*c^2*d^2 + 145*I*b*c*d^3 + 63*I*a*d^4)*sqrt(1/2*I*d)*we 
ierstrassZeta(-4/3*(4*c^2 - 3*d^2)/d^2, -8/27*(8*I*c^3 - 9*I*c*d^2)/d^3, w 
eierstrassPInverse(-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*I*b*c 
^3*d - 161*I*a*c^2*d^2 - 145*I*b*c*d^3 - 63*I*a*d^4)*sqrt(-1/2*I*d)*weiers 
trassZeta(-4/3*(4*c^2 - 3*d^2)/d^2, -8/27*(-8*I*c^3 + 9*I*c*d^2)/d^3, weie 
rstrassPInverse(-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*d^4*co 
s(f*x + e)^3 - 3*(15*b*c*d^3 + 7*a*d^4)*cos(f*x + e)*sin(f*x + e) - (45*b* 
c^2*d^2 + 77*a*c*d^3 + 40*b*d^4)*cos(f*x + e))*sqrt(d*sin(f*x + e) + c))/( 
d^2*f)
 

Sympy [F]

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

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

Output:

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

Maxima [F]

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

integrate((a+b*sin(f*x+e))*(c+d*sin(f*x+e))^(5/2),x, algorithm="maxima")
 

Output:

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

Giac [F]

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

integrate((a+b*sin(f*x+e))*(c+d*sin(f*x+e))^(5/2),x, algorithm="giac")
 

Output:

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

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

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

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

Output:

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