\(\int x^3 (a+a \cos (x))^{3/2} \, dx\) [164]

Optimal result

Integrand size = 14, antiderivative size = 185 \[ \int x^3 (a+a \cos (x))^{3/2} \, dx=-\frac {1280}{9} a \sqrt {a+a \cos (x)}+16 a x^2 \sqrt {a+a \cos (x)}-\frac {64}{27} a \cos ^2\left (\frac {x}{2}\right ) \sqrt {a+a \cos (x)}+\frac {8}{3} a x^2 \cos ^2\left (\frac {x}{2}\right ) \sqrt {a+a \cos (x)}-\frac {32}{9} a x \cos \left (\frac {x}{2}\right ) \sqrt {a+a \cos (x)} \sin \left (\frac {x}{2}\right )+\frac {4}{3} a x^3 \cos \left (\frac {x}{2}\right ) \sqrt {a+a \cos (x)} \sin \left (\frac {x}{2}\right )-\frac {640}{9} a x \sqrt {a+a \cos (x)} \tan \left (\frac {x}{2}\right )+\frac {8}{3} a x^3 \sqrt {a+a \cos (x)} \tan \left (\frac {x}{2}\right ) \] Output:

-1280/9*a*(a+a*cos(x))^(1/2)+16*a*x^2*(a+a*cos(x))^(1/2)-64/27*a*cos(1/2*x 
)^2*(a+a*cos(x))^(1/2)+8/3*a*x^2*cos(1/2*x)^2*(a+a*cos(x))^(1/2)-32/9*a*x* 
cos(1/2*x)*(a+a*cos(x))^(1/2)*sin(1/2*x)+4/3*a*x^3*cos(1/2*x)*(a+a*cos(x)) 
^(1/2)*sin(1/2*x)-640/9*a*x*(a+a*cos(x))^(1/2)*tan(1/2*x)+8/3*a*x^3*(a+a*c 
os(x))^(1/2)*tan(1/2*x)
 

Mathematica [A] (verified)

Time = 0.41 (sec) , antiderivative size = 67, normalized size of antiderivative = 0.36 \[ \int x^3 (a+a \cos (x))^{3/2} \, dx=\frac {2}{27} a \sqrt {a (1+\cos (x))} \left (-1936+234 x^2+3 x \left (-328+15 x^2\right ) \tan \left (\frac {x}{2}\right )+\cos (x) \left (2 \left (-8+9 x^2\right )+3 x \left (-8+3 x^2\right ) \tan \left (\frac {x}{2}\right )\right )\right ) \] Input:

Integrate[x^3*(a + a*Cos[x])^(3/2),x]
 

Output:

(2*a*Sqrt[a*(1 + Cos[x])]*(-1936 + 234*x^2 + 3*x*(-328 + 15*x^2)*Tan[x/2] 
+ Cos[x]*(2*(-8 + 9*x^2) + 3*x*(-8 + 3*x^2)*Tan[x/2])))/27
 

Rubi [A] (verified)

Time = 1.01 (sec) , antiderivative size = 164, normalized size of antiderivative = 0.89, number of steps used = 20, number of rules used = 20, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 1.429, Rules used = {3042, 3800, 3042, 3792, 3042, 3777, 25, 3042, 3777, 3042, 3777, 25, 3042, 3118, 3791, 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.

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

\(\Big \downarrow \) 3042

\(\displaystyle \int x^3 \left (a \sin \left (x+\frac {\pi }{2}\right )+a\right )^{3/2}dx\)

\(\Big \downarrow \) 3800

\(\displaystyle 2 a \sec \left (\frac {x}{2}\right ) \sqrt {a \cos (x)+a} \int x^3 \cos ^3\left (\frac {x}{2}\right )dx\)

\(\Big \downarrow \) 3042

\(\displaystyle 2 a \sec \left (\frac {x}{2}\right ) \sqrt {a \cos (x)+a} \int x^3 \sin \left (\frac {x}{2}+\frac {\pi }{2}\right )^3dx\)

\(\Big \downarrow \) 3792

\(\displaystyle 2 a \sec \left (\frac {x}{2}\right ) \sqrt {a \cos (x)+a} \left (\frac {2}{3} \int x^3 \cos \left (\frac {x}{2}\right )dx-\frac {8}{3} \int x \cos ^3\left (\frac {x}{2}\right )dx+\frac {2}{3} x^3 \sin \left (\frac {x}{2}\right ) \cos ^2\left (\frac {x}{2}\right )+\frac {4}{3} x^2 \cos ^3\left (\frac {x}{2}\right )\right )\)

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 3777

\(\displaystyle 2 a \sec \left (\frac {x}{2}\right ) \sqrt {a \cos (x)+a} \left (\frac {2}{3} \left (6 \int -x^2 \sin \left (\frac {x}{2}\right )dx+2 x^3 \sin \left (\frac {x}{2}\right )\right )-\frac {8}{3} \int x \sin \left (\frac {x}{2}+\frac {\pi }{2}\right )^3dx+\frac {2}{3} x^3 \sin \left (\frac {x}{2}\right ) \cos ^2\left (\frac {x}{2}\right )+\frac {4}{3} x^2 \cos ^3\left (\frac {x}{2}\right )\right )\)

\(\Big \downarrow \) 25

\(\displaystyle 2 a \sec \left (\frac {x}{2}\right ) \sqrt {a \cos (x)+a} \left (\frac {2}{3} \left (2 x^3 \sin \left (\frac {x}{2}\right )-6 \int x^2 \sin \left (\frac {x}{2}\right )dx\right )-\frac {8}{3} \int x \sin \left (\frac {x}{2}+\frac {\pi }{2}\right )^3dx+\frac {2}{3} x^3 \sin \left (\frac {x}{2}\right ) \cos ^2\left (\frac {x}{2}\right )+\frac {4}{3} x^2 \cos ^3\left (\frac {x}{2}\right )\right )\)

\(\Big \downarrow \) 3042

\(\displaystyle 2 a \sec \left (\frac {x}{2}\right ) \sqrt {a \cos (x)+a} \left (\frac {2}{3} \left (2 x^3 \sin \left (\frac {x}{2}\right )-6 \int x^2 \sin \left (\frac {x}{2}\right )dx\right )-\frac {8}{3} \int x \sin \left (\frac {x}{2}+\frac {\pi }{2}\right )^3dx+\frac {2}{3} x^3 \sin \left (\frac {x}{2}\right ) \cos ^2\left (\frac {x}{2}\right )+\frac {4}{3} x^2 \cos ^3\left (\frac {x}{2}\right )\right )\)

\(\Big \downarrow \) 3777

\(\displaystyle 2 a \sec \left (\frac {x}{2}\right ) \sqrt {a \cos (x)+a} \left (\frac {2}{3} \left (2 x^3 \sin \left (\frac {x}{2}\right )-6 \left (4 \int x \cos \left (\frac {x}{2}\right )dx-2 x^2 \cos \left (\frac {x}{2}\right )\right )\right )-\frac {8}{3} \int x \sin \left (\frac {x}{2}+\frac {\pi }{2}\right )^3dx+\frac {2}{3} x^3 \sin \left (\frac {x}{2}\right ) \cos ^2\left (\frac {x}{2}\right )+\frac {4}{3} x^2 \cos ^3\left (\frac {x}{2}\right )\right )\)

\(\Big \downarrow \) 3042

\(\displaystyle 2 a \sec \left (\frac {x}{2}\right ) \sqrt {a \cos (x)+a} \left (\frac {2}{3} \left (2 x^3 \sin \left (\frac {x}{2}\right )-6 \left (4 \int x \sin \left (\frac {x}{2}+\frac {\pi }{2}\right )dx-2 x^2 \cos \left (\frac {x}{2}\right )\right )\right )-\frac {8}{3} \int x \sin \left (\frac {x}{2}+\frac {\pi }{2}\right )^3dx+\frac {2}{3} x^3 \sin \left (\frac {x}{2}\right ) \cos ^2\left (\frac {x}{2}\right )+\frac {4}{3} x^2 \cos ^3\left (\frac {x}{2}\right )\right )\)

\(\Big \downarrow \) 3777

\(\displaystyle 2 a \sec \left (\frac {x}{2}\right ) \sqrt {a \cos (x)+a} \left (\frac {2}{3} \left (2 x^3 \sin \left (\frac {x}{2}\right )-6 \left (4 \left (2 \int -\sin \left (\frac {x}{2}\right )dx+2 x \sin \left (\frac {x}{2}\right )\right )-2 x^2 \cos \left (\frac {x}{2}\right )\right )\right )-\frac {8}{3} \int x \sin \left (\frac {x}{2}+\frac {\pi }{2}\right )^3dx+\frac {2}{3} x^3 \sin \left (\frac {x}{2}\right ) \cos ^2\left (\frac {x}{2}\right )+\frac {4}{3} x^2 \cos ^3\left (\frac {x}{2}\right )\right )\)

\(\Big \downarrow \) 25

\(\displaystyle 2 a \sec \left (\frac {x}{2}\right ) \sqrt {a \cos (x)+a} \left (\frac {2}{3} \left (2 x^3 \sin \left (\frac {x}{2}\right )-6 \left (4 \left (2 x \sin \left (\frac {x}{2}\right )-2 \int \sin \left (\frac {x}{2}\right )dx\right )-2 x^2 \cos \left (\frac {x}{2}\right )\right )\right )-\frac {8}{3} \int x \sin \left (\frac {x}{2}+\frac {\pi }{2}\right )^3dx+\frac {2}{3} x^3 \sin \left (\frac {x}{2}\right ) \cos ^2\left (\frac {x}{2}\right )+\frac {4}{3} x^2 \cos ^3\left (\frac {x}{2}\right )\right )\)

\(\Big \downarrow \) 3042

\(\displaystyle 2 a \sec \left (\frac {x}{2}\right ) \sqrt {a \cos (x)+a} \left (\frac {2}{3} \left (2 x^3 \sin \left (\frac {x}{2}\right )-6 \left (4 \left (2 x \sin \left (\frac {x}{2}\right )-2 \int \sin \left (\frac {x}{2}\right )dx\right )-2 x^2 \cos \left (\frac {x}{2}\right )\right )\right )-\frac {8}{3} \int x \sin \left (\frac {x}{2}+\frac {\pi }{2}\right )^3dx+\frac {2}{3} x^3 \sin \left (\frac {x}{2}\right ) \cos ^2\left (\frac {x}{2}\right )+\frac {4}{3} x^2 \cos ^3\left (\frac {x}{2}\right )\right )\)

\(\Big \downarrow \) 3118

\(\displaystyle 2 a \sec \left (\frac {x}{2}\right ) \sqrt {a \cos (x)+a} \left (-\frac {8}{3} \int x \sin \left (\frac {x}{2}+\frac {\pi }{2}\right )^3dx+\frac {2}{3} x^3 \sin \left (\frac {x}{2}\right ) \cos ^2\left (\frac {x}{2}\right )+\frac {4}{3} x^2 \cos ^3\left (\frac {x}{2}\right )+\frac {2}{3} \left (2 x^3 \sin \left (\frac {x}{2}\right )-6 \left (4 \left (2 x \sin \left (\frac {x}{2}\right )+4 \cos \left (\frac {x}{2}\right )\right )-2 x^2 \cos \left (\frac {x}{2}\right )\right )\right )\right )\)

\(\Big \downarrow \) 3791

\(\displaystyle 2 a \sec \left (\frac {x}{2}\right ) \sqrt {a \cos (x)+a} \left (-\frac {8}{3} \left (\frac {2}{3} \int x \cos \left (\frac {x}{2}\right )dx+\frac {4}{9} \cos ^3\left (\frac {x}{2}\right )+\frac {2}{3} x \sin \left (\frac {x}{2}\right ) \cos ^2\left (\frac {x}{2}\right )\right )+\frac {2}{3} x^3 \sin \left (\frac {x}{2}\right ) \cos ^2\left (\frac {x}{2}\right )+\frac {4}{3} x^2 \cos ^3\left (\frac {x}{2}\right )+\frac {2}{3} \left (2 x^3 \sin \left (\frac {x}{2}\right )-6 \left (4 \left (2 x \sin \left (\frac {x}{2}\right )+4 \cos \left (\frac {x}{2}\right )\right )-2 x^2 \cos \left (\frac {x}{2}\right )\right )\right )\right )\)

\(\Big \downarrow \) 3042

\(\displaystyle 2 a \sec \left (\frac {x}{2}\right ) \sqrt {a \cos (x)+a} \left (-\frac {8}{3} \left (\frac {2}{3} \int x \sin \left (\frac {x}{2}+\frac {\pi }{2}\right )dx+\frac {4}{9} \cos ^3\left (\frac {x}{2}\right )+\frac {2}{3} x \sin \left (\frac {x}{2}\right ) \cos ^2\left (\frac {x}{2}\right )\right )+\frac {2}{3} x^3 \sin \left (\frac {x}{2}\right ) \cos ^2\left (\frac {x}{2}\right )+\frac {4}{3} x^2 \cos ^3\left (\frac {x}{2}\right )+\frac {2}{3} \left (2 x^3 \sin \left (\frac {x}{2}\right )-6 \left (4 \left (2 x \sin \left (\frac {x}{2}\right )+4 \cos \left (\frac {x}{2}\right )\right )-2 x^2 \cos \left (\frac {x}{2}\right )\right )\right )\right )\)

\(\Big \downarrow \) 3777

\(\displaystyle 2 a \sec \left (\frac {x}{2}\right ) \sqrt {a \cos (x)+a} \left (-\frac {8}{3} \left (\frac {2}{3} \left (2 \int -\sin \left (\frac {x}{2}\right )dx+2 x \sin \left (\frac {x}{2}\right )\right )+\frac {4}{9} \cos ^3\left (\frac {x}{2}\right )+\frac {2}{3} x \sin \left (\frac {x}{2}\right ) \cos ^2\left (\frac {x}{2}\right )\right )+\frac {2}{3} x^3 \sin \left (\frac {x}{2}\right ) \cos ^2\left (\frac {x}{2}\right )+\frac {4}{3} x^2 \cos ^3\left (\frac {x}{2}\right )+\frac {2}{3} \left (2 x^3 \sin \left (\frac {x}{2}\right )-6 \left (4 \left (2 x \sin \left (\frac {x}{2}\right )+4 \cos \left (\frac {x}{2}\right )\right )-2 x^2 \cos \left (\frac {x}{2}\right )\right )\right )\right )\)

\(\Big \downarrow \) 25

\(\displaystyle 2 a \sec \left (\frac {x}{2}\right ) \sqrt {a \cos (x)+a} \left (-\frac {8}{3} \left (\frac {2}{3} \left (2 x \sin \left (\frac {x}{2}\right )-2 \int \sin \left (\frac {x}{2}\right )dx\right )+\frac {4}{9} \cos ^3\left (\frac {x}{2}\right )+\frac {2}{3} x \sin \left (\frac {x}{2}\right ) \cos ^2\left (\frac {x}{2}\right )\right )+\frac {2}{3} x^3 \sin \left (\frac {x}{2}\right ) \cos ^2\left (\frac {x}{2}\right )+\frac {4}{3} x^2 \cos ^3\left (\frac {x}{2}\right )+\frac {2}{3} \left (2 x^3 \sin \left (\frac {x}{2}\right )-6 \left (4 \left (2 x \sin \left (\frac {x}{2}\right )+4 \cos \left (\frac {x}{2}\right )\right )-2 x^2 \cos \left (\frac {x}{2}\right )\right )\right )\right )\)

\(\Big \downarrow \) 3042

\(\displaystyle 2 a \sec \left (\frac {x}{2}\right ) \sqrt {a \cos (x)+a} \left (-\frac {8}{3} \left (\frac {2}{3} \left (2 x \sin \left (\frac {x}{2}\right )-2 \int \sin \left (\frac {x}{2}\right )dx\right )+\frac {4}{9} \cos ^3\left (\frac {x}{2}\right )+\frac {2}{3} x \sin \left (\frac {x}{2}\right ) \cos ^2\left (\frac {x}{2}\right )\right )+\frac {2}{3} x^3 \sin \left (\frac {x}{2}\right ) \cos ^2\left (\frac {x}{2}\right )+\frac {4}{3} x^2 \cos ^3\left (\frac {x}{2}\right )+\frac {2}{3} \left (2 x^3 \sin \left (\frac {x}{2}\right )-6 \left (4 \left (2 x \sin \left (\frac {x}{2}\right )+4 \cos \left (\frac {x}{2}\right )\right )-2 x^2 \cos \left (\frac {x}{2}\right )\right )\right )\right )\)

\(\Big \downarrow \) 3118

\(\displaystyle 2 a \sec \left (\frac {x}{2}\right ) \sqrt {a \cos (x)+a} \left (\frac {2}{3} x^3 \sin \left (\frac {x}{2}\right ) \cos ^2\left (\frac {x}{2}\right )+\frac {4}{3} x^2 \cos ^3\left (\frac {x}{2}\right )+\frac {2}{3} \left (2 x^3 \sin \left (\frac {x}{2}\right )-6 \left (4 \left (2 x \sin \left (\frac {x}{2}\right )+4 \cos \left (\frac {x}{2}\right )\right )-2 x^2 \cos \left (\frac {x}{2}\right )\right )\right )-\frac {8}{3} \left (\frac {4}{9} \cos ^3\left (\frac {x}{2}\right )+\frac {2}{3} x \sin \left (\frac {x}{2}\right ) \cos ^2\left (\frac {x}{2}\right )+\frac {2}{3} \left (2 x \sin \left (\frac {x}{2}\right )+4 \cos \left (\frac {x}{2}\right )\right )\right )\right )\)

Input:

Int[x^3*(a + a*Cos[x])^(3/2),x]
 

Output:

2*a*Sqrt[a + a*Cos[x]]*Sec[x/2]*((4*x^2*Cos[x/2]^3)/3 + (2*x^3*Cos[x/2]^2* 
Sin[x/2])/3 - (8*((4*Cos[x/2]^3)/9 + (2*x*Cos[x/2]^2*Sin[x/2])/3 + (2*(4*C 
os[x/2] + 2*x*Sin[x/2]))/3))/3 + (2*(2*x^3*Sin[x/2] - 6*(-2*x^2*Cos[x/2] + 
 4*(4*Cos[x/2] + 2*x*Sin[x/2]))))/3)
 

Defintions of rubi rules used

rule 25

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

rule 3042

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

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 3791

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

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]

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

Input:

int(x^3*(a+a*cos(x))^(3/2),x)
 

Output:

int(x^3*(a+a*cos(x))^(3/2),x)
 

Fricas [F(-2)]

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

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

Output:

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

Sympy [F]

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

integrate(x**3*(a+a*cos(x))**(3/2),x)
 

Output:

Integral(x**3*(a*(cos(x) + 1))**(3/2), x)
 

Maxima [A] (verification not implemented)

Time = 0.13 (sec) , antiderivative size = 98, normalized size of antiderivative = 0.53 \[ \int x^3 (a+a \cos (x))^{3/2} \, dx=\frac {1}{27} \, {\left (81 \, \sqrt {2} a x^{3} \sin \left (\frac {1}{2} \, x\right ) + 486 \, \sqrt {2} a x^{2} \cos \left (\frac {1}{2} \, x\right ) - 1944 \, \sqrt {2} a x \sin \left (\frac {1}{2} \, x\right ) - 3888 \, \sqrt {2} a \cos \left (\frac {1}{2} \, x\right ) + 2 \, {\left (9 \, \sqrt {2} a x^{2} - 8 \, \sqrt {2} a\right )} \cos \left (\frac {3}{2} \, x\right ) + 3 \, {\left (3 \, \sqrt {2} a x^{3} - 8 \, \sqrt {2} a x\right )} \sin \left (\frac {3}{2} \, x\right )\right )} \sqrt {a} \] Input:

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

Output:

1/27*(81*sqrt(2)*a*x^3*sin(1/2*x) + 486*sqrt(2)*a*x^2*cos(1/2*x) - 1944*sq 
rt(2)*a*x*sin(1/2*x) - 3888*sqrt(2)*a*cos(1/2*x) + 2*(9*sqrt(2)*a*x^2 - 8* 
sqrt(2)*a)*cos(3/2*x) + 3*(3*sqrt(2)*a*x^3 - 8*sqrt(2)*a*x)*sin(3/2*x))*sq 
rt(a)
 

Giac [A] (verification not implemented)

Time = 0.37 (sec) , antiderivative size = 113, normalized size of antiderivative = 0.61 \[ \int x^3 (a+a \cos (x))^{3/2} \, dx=\frac {1}{27} \, \sqrt {2} {\left (2 \, {\left (9 \, a x^{2} \mathrm {sgn}\left (\cos \left (\frac {1}{2} \, x\right )\right ) - 8 \, a \mathrm {sgn}\left (\cos \left (\frac {1}{2} \, x\right )\right )\right )} \cos \left (\frac {3}{2} \, x\right ) + 486 \, {\left (a x^{2} \mathrm {sgn}\left (\cos \left (\frac {1}{2} \, x\right )\right ) - 8 \, a \mathrm {sgn}\left (\cos \left (\frac {1}{2} \, x\right )\right )\right )} \cos \left (\frac {1}{2} \, x\right ) + 3 \, {\left (3 \, a x^{3} \mathrm {sgn}\left (\cos \left (\frac {1}{2} \, x\right )\right ) - 8 \, a x \mathrm {sgn}\left (\cos \left (\frac {1}{2} \, x\right )\right )\right )} \sin \left (\frac {3}{2} \, x\right ) + 81 \, {\left (a x^{3} \mathrm {sgn}\left (\cos \left (\frac {1}{2} \, x\right )\right ) - 24 \, a x \mathrm {sgn}\left (\cos \left (\frac {1}{2} \, x\right )\right )\right )} \sin \left (\frac {1}{2} \, x\right )\right )} \sqrt {a} \] Input:

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

Output:

1/27*sqrt(2)*(2*(9*a*x^2*sgn(cos(1/2*x)) - 8*a*sgn(cos(1/2*x)))*cos(3/2*x) 
 + 486*(a*x^2*sgn(cos(1/2*x)) - 8*a*sgn(cos(1/2*x)))*cos(1/2*x) + 3*(3*a*x 
^3*sgn(cos(1/2*x)) - 8*a*x*sgn(cos(1/2*x)))*sin(3/2*x) + 81*(a*x^3*sgn(cos 
(1/2*x)) - 24*a*x*sgn(cos(1/2*x)))*sin(1/2*x))*sqrt(a)
 

Mupad [F(-1)]

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

int(x^3*(a + a*cos(x))^(3/2),x)
 

Output:

int(x^3*(a + a*cos(x))^(3/2), x)
 

Reduce [F]

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

int(x^3*(a+a*cos(x))^(3/2),x)
 

Output:

sqrt(a)*a*(int(sqrt(cos(x) + 1)*cos(x)*x**3,x) + int(sqrt(cos(x) + 1)*x**3 
,x))