\(\int x^2 (a+a \sin (e+f x))^{3/2} \, dx\) [129]

Optimal result

Integrand size = 18, antiderivative size = 271 \[ \int x^2 (a+a \sin (e+f x))^{3/2} \, dx=\frac {32 a x \sqrt {a+a \sin (e+f x)}}{3 f^2}+\frac {224 a \cot \left (\frac {e}{2}+\frac {\pi }{4}+\frac {f x}{2}\right ) \sqrt {a+a \sin (e+f x)}}{9 f^3}-\frac {8 a x^2 \cot \left (\frac {e}{2}+\frac {\pi }{4}+\frac {f x}{2}\right ) \sqrt {a+a \sin (e+f x)}}{3 f}-\frac {32 a \cos ^2\left (\frac {e}{2}+\frac {\pi }{4}+\frac {f x}{2}\right ) \cot \left (\frac {e}{2}+\frac {\pi }{4}+\frac {f x}{2}\right ) \sqrt {a+a \sin (e+f x)}}{27 f^3}-\frac {4 a x^2 \cos \left (\frac {e}{2}+\frac {\pi }{4}+\frac {f x}{2}\right ) \sin \left (\frac {e}{2}+\frac {\pi }{4}+\frac {f x}{2}\right ) \sqrt {a+a \sin (e+f x)}}{3 f}+\frac {16 a x \sin ^2\left (\frac {e}{2}+\frac {\pi }{4}+\frac {f x}{2}\right ) \sqrt {a+a \sin (e+f x)}}{9 f^2} \] Output:

32/3*a*x*(a+a*sin(f*x+e))^(1/2)/f^2+224/9*a*cot(1/2*e+1/4*Pi+1/2*f*x)*(a+a 
*sin(f*x+e))^(1/2)/f^3-8/3*a*x^2*cot(1/2*e+1/4*Pi+1/2*f*x)*(a+a*sin(f*x+e) 
)^(1/2)/f-32/27*a*cos(1/2*e+1/4*Pi+1/2*f*x)^2*cot(1/2*e+1/4*Pi+1/2*f*x)*(a 
+a*sin(f*x+e))^(1/2)/f^3-4/3*a*x^2*cos(1/2*e+1/4*Pi+1/2*f*x)*sin(1/2*e+1/4 
*Pi+1/2*f*x)*(a+a*sin(f*x+e))^(1/2)/f+16/9*a*x*sin(1/2*e+1/4*Pi+1/2*f*x)^2 
*(a+a*sin(f*x+e))^(1/2)/f^2
 

Mathematica [A] (verified)

Time = 0.95 (sec) , antiderivative size = 191, normalized size of antiderivative = 0.70 \[ \int x^2 (a+a \sin (e+f x))^{3/2} \, dx=\frac {2 a \left (-\frac {4 \left (\left (-80-39 f x+9 f^2 x^2\right ) \cos \left (\frac {e}{2}\right )+\left (80-39 f x-9 f^2 x^2\right ) \sin \left (\frac {e}{2}\right )\right )}{\cos \left (\frac {e}{2}\right )+\sin \left (\frac {e}{2}\right )}-\cos (f x) \left (\left (-8+9 f^2 x^2\right ) \cos (e)-12 f x \sin (e)\right )+\left (12 f x \cos (e)+\left (-8+9 f^2 x^2\right ) \sin (e)\right ) \sin (f x)+\frac {8 \left (-80+9 f^2 x^2\right ) \sin \left (\frac {f x}{2}\right )}{\left (\cos \left (\frac {e}{2}\right )+\sin \left (\frac {e}{2}\right )\right ) \left (\cos \left (\frac {1}{2} (e+f x)\right )+\sin \left (\frac {1}{2} (e+f x)\right )\right )}\right ) \sqrt {a (1+\sin (e+f x))}}{27 f^3} \] Input:

Integrate[x^2*(a + a*Sin[e + f*x])^(3/2),x]
 

Output:

(2*a*((-4*((-80 - 39*f*x + 9*f^2*x^2)*Cos[e/2] + (80 - 39*f*x - 9*f^2*x^2) 
*Sin[e/2]))/(Cos[e/2] + Sin[e/2]) - Cos[f*x]*((-8 + 9*f^2*x^2)*Cos[e] - 12 
*f*x*Sin[e]) + (12*f*x*Cos[e] + (-8 + 9*f^2*x^2)*Sin[e])*Sin[f*x] + (8*(-8 
0 + 9*f^2*x^2)*Sin[(f*x)/2])/((Cos[e/2] + Sin[e/2])*(Cos[(e + f*x)/2] + Si 
n[(e + f*x)/2])))*Sqrt[a*(1 + Sin[e + f*x])])/(27*f^3)
 

Rubi [A] (verified)

Time = 0.85 (sec) , antiderivative size = 246, normalized size of antiderivative = 0.91, number of steps used = 14, number of rules used = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.722, Rules used = {3042, 3800, 3042, 3792, 3042, 3113, 2009, 3777, 3042, 3777, 25, 3042, 3118}

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.

\(\displaystyle \int x^2 (a \sin (e+f x)+a)^{3/2} \, dx\)

\(\Big \downarrow \) 3042

\(\displaystyle \int x^2 (a \sin (e+f x)+a)^{3/2}dx\)

\(\Big \downarrow \) 3800

\(\displaystyle 2 a \csc \left (\frac {e}{2}+\frac {f x}{2}+\frac {\pi }{4}\right ) \sqrt {a \sin (e+f x)+a} \int x^2 \sin ^3\left (\frac {e}{2}+\frac {f x}{2}+\frac {\pi }{4}\right )dx\)

\(\Big \downarrow \) 3042

\(\displaystyle 2 a \csc \left (\frac {e}{2}+\frac {f x}{2}+\frac {\pi }{4}\right ) \sqrt {a \sin (e+f x)+a} \int x^2 \sin \left (\frac {e}{2}+\frac {f x}{2}+\frac {\pi }{4}\right )^3dx\)

\(\Big \downarrow \) 3792

\(\displaystyle 2 a \csc \left (\frac {e}{2}+\frac {f x}{2}+\frac {\pi }{4}\right ) \sqrt {a \sin (e+f x)+a} \left (-\frac {8 \int \sin ^3\left (\frac {e}{2}+\frac {f x}{2}+\frac {\pi }{4}\right )dx}{9 f^2}+\frac {2}{3} \int x^2 \sin \left (\frac {e}{2}+\frac {f x}{2}+\frac {\pi }{4}\right )dx+\frac {8 x \sin ^3\left (\frac {e}{2}+\frac {f x}{2}+\frac {\pi }{4}\right )}{9 f^2}-\frac {2 x^2 \sin ^2\left (\frac {e}{2}+\frac {f x}{2}+\frac {\pi }{4}\right ) \cos \left (\frac {e}{2}+\frac {f x}{2}+\frac {\pi }{4}\right )}{3 f}\right )\)

\(\Big \downarrow \) 3042

\(\displaystyle 2 a \csc \left (\frac {e}{2}+\frac {f x}{2}+\frac {\pi }{4}\right ) \sqrt {a \sin (e+f x)+a} \left (-\frac {8 \int \sin \left (\frac {e}{2}+\frac {f x}{2}+\frac {\pi }{4}\right )^3dx}{9 f^2}+\frac {2}{3} \int x^2 \sin \left (\frac {e}{2}+\frac {f x}{2}+\frac {\pi }{4}\right )dx+\frac {8 x \sin ^3\left (\frac {e}{2}+\frac {f x}{2}+\frac {\pi }{4}\right )}{9 f^2}-\frac {2 x^2 \sin ^2\left (\frac {e}{2}+\frac {f x}{2}+\frac {\pi }{4}\right ) \cos \left (\frac {e}{2}+\frac {f x}{2}+\frac {\pi }{4}\right )}{3 f}\right )\)

\(\Big \downarrow \) 3113

\(\displaystyle 2 a \csc \left (\frac {e}{2}+\frac {f x}{2}+\frac {\pi }{4}\right ) \sqrt {a \sin (e+f x)+a} \left (\frac {16 \int \left (1-\cos ^2\left (\frac {e}{2}+\frac {f x}{2}+\frac {\pi }{4}\right )\right )d\cos \left (\frac {e}{2}+\frac {f x}{2}+\frac {\pi }{4}\right )}{9 f^3}+\frac {2}{3} \int x^2 \sin \left (\frac {e}{2}+\frac {f x}{2}+\frac {\pi }{4}\right )dx+\frac {8 x \sin ^3\left (\frac {e}{2}+\frac {f x}{2}+\frac {\pi }{4}\right )}{9 f^2}-\frac {2 x^2 \sin ^2\left (\frac {e}{2}+\frac {f x}{2}+\frac {\pi }{4}\right ) \cos \left (\frac {e}{2}+\frac {f x}{2}+\frac {\pi }{4}\right )}{3 f}\right )\)

\(\Big \downarrow \) 2009

\(\displaystyle 2 a \csc \left (\frac {e}{2}+\frac {f x}{2}+\frac {\pi }{4}\right ) \sqrt {a \sin (e+f x)+a} \left (\frac {2}{3} \int x^2 \sin \left (\frac {e}{2}+\frac {f x}{2}+\frac {\pi }{4}\right )dx+\frac {16 \left (\cos \left (\frac {e}{2}+\frac {f x}{2}+\frac {\pi }{4}\right )-\frac {1}{3} \cos ^3\left (\frac {e}{2}+\frac {f x}{2}+\frac {\pi }{4}\right )\right )}{9 f^3}+\frac {8 x \sin ^3\left (\frac {e}{2}+\frac {f x}{2}+\frac {\pi }{4}\right )}{9 f^2}-\frac {2 x^2 \sin ^2\left (\frac {e}{2}+\frac {f x}{2}+\frac {\pi }{4}\right ) \cos \left (\frac {e}{2}+\frac {f x}{2}+\frac {\pi }{4}\right )}{3 f}\right )\)

\(\Big \downarrow \) 3777

\(\displaystyle 2 a \csc \left (\frac {e}{2}+\frac {f x}{2}+\frac {\pi }{4}\right ) \sqrt {a \sin (e+f x)+a} \left (\frac {2}{3} \left (\frac {4 \int x \cos \left (\frac {e}{2}+\frac {f x}{2}+\frac {\pi }{4}\right )dx}{f}-\frac {2 x^2 \cos \left (\frac {e}{2}+\frac {f x}{2}+\frac {\pi }{4}\right )}{f}\right )+\frac {16 \left (\cos \left (\frac {e}{2}+\frac {f x}{2}+\frac {\pi }{4}\right )-\frac {1}{3} \cos ^3\left (\frac {e}{2}+\frac {f x}{2}+\frac {\pi }{4}\right )\right )}{9 f^3}+\frac {8 x \sin ^3\left (\frac {e}{2}+\frac {f x}{2}+\frac {\pi }{4}\right )}{9 f^2}-\frac {2 x^2 \sin ^2\left (\frac {e}{2}+\frac {f x}{2}+\frac {\pi }{4}\right ) \cos \left (\frac {e}{2}+\frac {f x}{2}+\frac {\pi }{4}\right )}{3 f}\right )\)

\(\Big \downarrow \) 3042

\(\displaystyle 2 a \csc \left (\frac {e}{2}+\frac {f x}{2}+\frac {\pi }{4}\right ) \sqrt {a \sin (e+f x)+a} \left (\frac {2}{3} \left (\frac {4 \int x \sin \left (\frac {e}{2}+\frac {f x}{2}+\frac {3 \pi }{4}\right )dx}{f}-\frac {2 x^2 \cos \left (\frac {e}{2}+\frac {f x}{2}+\frac {\pi }{4}\right )}{f}\right )+\frac {16 \left (\cos \left (\frac {e}{2}+\frac {f x}{2}+\frac {\pi }{4}\right )-\frac {1}{3} \cos ^3\left (\frac {e}{2}+\frac {f x}{2}+\frac {\pi }{4}\right )\right )}{9 f^3}+\frac {8 x \sin ^3\left (\frac {e}{2}+\frac {f x}{2}+\frac {\pi }{4}\right )}{9 f^2}-\frac {2 x^2 \sin ^2\left (\frac {e}{2}+\frac {f x}{2}+\frac {\pi }{4}\right ) \cos \left (\frac {e}{2}+\frac {f x}{2}+\frac {\pi }{4}\right )}{3 f}\right )\)

\(\Big \downarrow \) 3777

\(\displaystyle 2 a \csc \left (\frac {e}{2}+\frac {f x}{2}+\frac {\pi }{4}\right ) \sqrt {a \sin (e+f x)+a} \left (\frac {2}{3} \left (\frac {4 \left (\frac {2 \int -\sin \left (\frac {e}{2}+\frac {f x}{2}+\frac {\pi }{4}\right )dx}{f}+\frac {2 x \sin \left (\frac {e}{2}+\frac {f x}{2}+\frac {\pi }{4}\right )}{f}\right )}{f}-\frac {2 x^2 \cos \left (\frac {e}{2}+\frac {f x}{2}+\frac {\pi }{4}\right )}{f}\right )+\frac {16 \left (\cos \left (\frac {e}{2}+\frac {f x}{2}+\frac {\pi }{4}\right )-\frac {1}{3} \cos ^3\left (\frac {e}{2}+\frac {f x}{2}+\frac {\pi }{4}\right )\right )}{9 f^3}+\frac {8 x \sin ^3\left (\frac {e}{2}+\frac {f x}{2}+\frac {\pi }{4}\right )}{9 f^2}-\frac {2 x^2 \sin ^2\left (\frac {e}{2}+\frac {f x}{2}+\frac {\pi }{4}\right ) \cos \left (\frac {e}{2}+\frac {f x}{2}+\frac {\pi }{4}\right )}{3 f}\right )\)

\(\Big \downarrow \) 25

\(\displaystyle 2 a \csc \left (\frac {e}{2}+\frac {f x}{2}+\frac {\pi }{4}\right ) \sqrt {a \sin (e+f x)+a} \left (\frac {2}{3} \left (\frac {4 \left (\frac {2 x \sin \left (\frac {e}{2}+\frac {f x}{2}+\frac {\pi }{4}\right )}{f}-\frac {2 \int \sin \left (\frac {e}{2}+\frac {f x}{2}+\frac {\pi }{4}\right )dx}{f}\right )}{f}-\frac {2 x^2 \cos \left (\frac {e}{2}+\frac {f x}{2}+\frac {\pi }{4}\right )}{f}\right )+\frac {16 \left (\cos \left (\frac {e}{2}+\frac {f x}{2}+\frac {\pi }{4}\right )-\frac {1}{3} \cos ^3\left (\frac {e}{2}+\frac {f x}{2}+\frac {\pi }{4}\right )\right )}{9 f^3}+\frac {8 x \sin ^3\left (\frac {e}{2}+\frac {f x}{2}+\frac {\pi }{4}\right )}{9 f^2}-\frac {2 x^2 \sin ^2\left (\frac {e}{2}+\frac {f x}{2}+\frac {\pi }{4}\right ) \cos \left (\frac {e}{2}+\frac {f x}{2}+\frac {\pi }{4}\right )}{3 f}\right )\)

\(\Big \downarrow \) 3042

\(\displaystyle 2 a \csc \left (\frac {e}{2}+\frac {f x}{2}+\frac {\pi }{4}\right ) \sqrt {a \sin (e+f x)+a} \left (\frac {2}{3} \left (\frac {4 \left (\frac {2 x \sin \left (\frac {e}{2}+\frac {f x}{2}+\frac {\pi }{4}\right )}{f}-\frac {2 \int \sin \left (\frac {e}{2}+\frac {f x}{2}+\frac {\pi }{4}\right )dx}{f}\right )}{f}-\frac {2 x^2 \cos \left (\frac {e}{2}+\frac {f x}{2}+\frac {\pi }{4}\right )}{f}\right )+\frac {16 \left (\cos \left (\frac {e}{2}+\frac {f x}{2}+\frac {\pi }{4}\right )-\frac {1}{3} \cos ^3\left (\frac {e}{2}+\frac {f x}{2}+\frac {\pi }{4}\right )\right )}{9 f^3}+\frac {8 x \sin ^3\left (\frac {e}{2}+\frac {f x}{2}+\frac {\pi }{4}\right )}{9 f^2}-\frac {2 x^2 \sin ^2\left (\frac {e}{2}+\frac {f x}{2}+\frac {\pi }{4}\right ) \cos \left (\frac {e}{2}+\frac {f x}{2}+\frac {\pi }{4}\right )}{3 f}\right )\)

\(\Big \downarrow \) 3118

\(\displaystyle 2 a \csc \left (\frac {e}{2}+\frac {f x}{2}+\frac {\pi }{4}\right ) \sqrt {a \sin (e+f x)+a} \left (\frac {16 \left (\cos \left (\frac {e}{2}+\frac {f x}{2}+\frac {\pi }{4}\right )-\frac {1}{3} \cos ^3\left (\frac {e}{2}+\frac {f x}{2}+\frac {\pi }{4}\right )\right )}{9 f^3}+\frac {2}{3} \left (\frac {4 \left (\frac {4 \cos \left (\frac {e}{2}+\frac {f x}{2}+\frac {\pi }{4}\right )}{f^2}+\frac {2 x \sin \left (\frac {e}{2}+\frac {f x}{2}+\frac {\pi }{4}\right )}{f}\right )}{f}-\frac {2 x^2 \cos \left (\frac {e}{2}+\frac {f x}{2}+\frac {\pi }{4}\right )}{f}\right )+\frac {8 x \sin ^3\left (\frac {e}{2}+\frac {f x}{2}+\frac {\pi }{4}\right )}{9 f^2}-\frac {2 x^2 \sin ^2\left (\frac {e}{2}+\frac {f x}{2}+\frac {\pi }{4}\right ) \cos \left (\frac {e}{2}+\frac {f x}{2}+\frac {\pi }{4}\right )}{3 f}\right )\)

Input:

Int[x^2*(a + a*Sin[e + f*x])^(3/2),x]
 

Output:

2*a*Csc[e/2 + Pi/4 + (f*x)/2]*((16*(Cos[e/2 + Pi/4 + (f*x)/2] - Cos[e/2 + 
Pi/4 + (f*x)/2]^3/3))/(9*f^3) - (2*x^2*Cos[e/2 + Pi/4 + (f*x)/2]*Sin[e/2 + 
 Pi/4 + (f*x)/2]^2)/(3*f) + (8*x*Sin[e/2 + Pi/4 + (f*x)/2]^3)/(9*f^2) + (2 
*((-2*x^2*Cos[e/2 + Pi/4 + (f*x)/2])/f + (4*((4*Cos[e/2 + Pi/4 + (f*x)/2]) 
/f^2 + (2*x*Sin[e/2 + Pi/4 + (f*x)/2])/f))/f))/3)*Sqrt[a + a*Sin[e + f*x]]
 

Defintions of rubi rules used

rule 25

Int[-(Fx_), x_Symbol] :> Simp[Identity[-1]   Int[Fx, x], x]
 

rule 2009

Int[u_, x_Symbol] :> Simp[IntSum[u, x], x] /; SumQ[u]
 

rule 3042

Int[u_, x_Symbol] :> Int[DeactivateTrig[u, x], x] /; FunctionOfTrigOfLinear 
Q[u, x]
 

rule 3113

Int[sin[(c_.) + (d_.)*(x_)]^(n_), x_Symbol] :> Simp[-d^(-1)   Subst[Int[Exp 
and[(1 - x^2)^((n - 1)/2), x], x], x, Cos[c + d*x]], x] /; FreeQ[{c, d}, x] 
 && IGtQ[(n - 1)/2, 0]
 

rule 3118

Int[sin[(c_.) + (d_.)*(x_)], x_Symbol] :> Simp[-Cos[c + d*x]/d, x] /; FreeQ 
[{c, d}, x]
 

rule 3777

Int[((c_.) + (d_.)*(x_))^(m_.)*sin[(e_.) + (f_.)*(x_)], x_Symbol] :> Simp[( 
-(c + d*x)^m)*(Cos[e + f*x]/f), x] + Simp[d*(m/f)   Int[(c + d*x)^(m - 1)*C 
os[e + f*x], x], x] /; FreeQ[{c, d, e, f}, x] && GtQ[m, 0]
 

rule 3792

Int[((c_.) + (d_.)*(x_))^(m_)*((b_.)*sin[(e_.) + (f_.)*(x_)])^(n_), x_Symbo 
l] :> Simp[d*m*(c + d*x)^(m - 1)*((b*Sin[e + f*x])^n/(f^2*n^2)), x] + (-Sim 
p[b*(c + d*x)^m*Cos[e + f*x]*((b*Sin[e + f*x])^(n - 1)/(f*n)), x] + Simp[b^ 
2*((n - 1)/n)   Int[(c + d*x)^m*(b*Sin[e + f*x])^(n - 2), x], x] - Simp[d^2 
*m*((m - 1)/(f^2*n^2))   Int[(c + d*x)^(m - 2)*(b*Sin[e + f*x])^n, x], x]) 
/; FreeQ[{b, c, d, e, f}, x] && GtQ[n, 1] && GtQ[m, 1]
 

rule 3800

Int[((c_.) + (d_.)*(x_))^(m_.)*((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(n_), 
 x_Symbol] :> Simp[(2*a)^IntPart[n]*((a + b*Sin[e + f*x])^FracPart[n]/Sin[e 
/2 + a*(Pi/(4*b)) + f*(x/2)]^(2*FracPart[n]))   Int[(c + d*x)^m*Sin[e/2 + a 
*(Pi/(4*b)) + f*(x/2)]^(2*n), x], x] /; FreeQ[{a, b, c, d, e, f, m}, x] && 
EqQ[a^2 - b^2, 0] && IntegerQ[n + 1/2] && (GtQ[n, 0] || IGtQ[m, 0])
 

Maple [F]

\[\int x^{2} \left (a +a \sin \left (f x +e \right )\right )^{\frac {3}{2}}d x\]

Input:

int(x^2*(a+a*sin(f*x+e))^(3/2),x)
 

Output:

int(x^2*(a+a*sin(f*x+e))^(3/2),x)
 

Fricas [F(-2)]

Exception generated. \[ \int x^2 (a+a \sin (e+f x))^{3/2} \, dx=\text {Exception raised: TypeError} \] Input:

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

Output:

Exception raised: TypeError >>  Error detected within library code:   inte 
grate: implementation incomplete (has polynomial part)
 

Sympy [F]

\[ \int x^2 (a+a \sin (e+f x))^{3/2} \, dx=\int x^{2} \left (a \left (\sin {\left (e + f x \right )} + 1\right )\right )^{\frac {3}{2}}\, dx \] Input:

integrate(x**2*(a+a*sin(f*x+e))**(3/2),x)
 

Output:

Integral(x**2*(a*(sin(e + f*x) + 1))**(3/2), x)
 

Maxima [F]

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

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

Output:

integrate((a*sin(f*x + e) + a)^(3/2)*x^2, x)
                                                                                    
                                                                                    
 

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 499 vs. \(2 (205) = 410\).

Time = 0.40 (sec) , antiderivative size = 499, normalized size of antiderivative = 1.84 \[ \int x^2 (a+a \sin (e+f x))^{3/2} \, dx=\text {Too large to display} \] Input:

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

Output:

1/108*sqrt(2)*sqrt(a)*(648*(pi*a*sgn(cos(-1/4*pi + 1/2*f*x + 1/2*e)) - (pi 
 - 2*f*x - 2*e)*a*sgn(cos(-1/4*pi + 1/2*f*x + 1/2*e)) - 2*a*e*sgn(cos(-1/4 
*pi + 1/2*f*x + 1/2*e)))*cos(-1/4*pi + 1/2*f*x + 1/2*e)/f^2 + 24*(pi*a*sgn 
(cos(-1/4*pi + 1/2*f*x + 1/2*e)) - (pi - 2*f*x - 2*e)*a*sgn(cos(-1/4*pi + 
1/2*f*x + 1/2*e)) - 2*a*e*sgn(cos(-1/4*pi + 1/2*f*x + 1/2*e)))*cos(-3/4*pi 
 + 3/2*f*x + 3/2*e)/f^2 + 81*(pi^2*a*sgn(cos(-1/4*pi + 1/2*f*x + 1/2*e)) - 
 2*pi*(pi - 2*f*x - 2*e)*a*sgn(cos(-1/4*pi + 1/2*f*x + 1/2*e)) + (pi - 2*f 
*x - 2*e)^2*a*sgn(cos(-1/4*pi + 1/2*f*x + 1/2*e)) - 4*pi*a*e*sgn(cos(-1/4* 
pi + 1/2*f*x + 1/2*e)) + 4*(pi - 2*f*x - 2*e)*a*e*sgn(cos(-1/4*pi + 1/2*f* 
x + 1/2*e)) + 4*a*e^2*sgn(cos(-1/4*pi + 1/2*f*x + 1/2*e)) - 32*a*sgn(cos(- 
1/4*pi + 1/2*f*x + 1/2*e)))*sin(-1/4*pi + 1/2*f*x + 1/2*e)/f^2 + (9*pi^2*a 
*sgn(cos(-1/4*pi + 1/2*f*x + 1/2*e)) - 18*pi*(pi - 2*f*x - 2*e)*a*sgn(cos( 
-1/4*pi + 1/2*f*x + 1/2*e)) + 9*(pi - 2*f*x - 2*e)^2*a*sgn(cos(-1/4*pi + 1 
/2*f*x + 1/2*e)) - 36*pi*a*e*sgn(cos(-1/4*pi + 1/2*f*x + 1/2*e)) + 36*(pi 
- 2*f*x - 2*e)*a*e*sgn(cos(-1/4*pi + 1/2*f*x + 1/2*e)) + 36*a*e^2*sgn(cos( 
-1/4*pi + 1/2*f*x + 1/2*e)) - 32*a*sgn(cos(-1/4*pi + 1/2*f*x + 1/2*e)))*si 
n(-3/4*pi + 3/2*f*x + 3/2*e)/f^2)/f
 

Mupad [F(-1)]

Timed out. \[ \int x^2 (a+a \sin (e+f x))^{3/2} \, dx=\int x^2\,{\left (a+a\,\sin \left (e+f\,x\right )\right )}^{3/2} \,d x \] Input:

int(x^2*(a + a*sin(e + f*x))^(3/2),x)
 

Output:

int(x^2*(a + a*sin(e + f*x))^(3/2), x)
 

Reduce [F]

\[ \int x^2 (a+a \sin (e+f x))^{3/2} \, dx=\sqrt {a}\, a \left (\int \sqrt {\sin \left (f x +e \right )+1}\, x^{2}d x +\int \sqrt {\sin \left (f x +e \right )+1}\, \sin \left (f x +e \right ) x^{2}d x \right ) \] Input:

int(x^2*(a+a*sin(f*x+e))^(3/2),x)
 

Output:

sqrt(a)*a*(int(sqrt(sin(e + f*x) + 1)*x**2,x) + int(sqrt(sin(e + f*x) + 1) 
*sin(e + f*x)*x**2,x))