\(\int \frac {x (1-c^2 x^2)^{5/2}}{a+b \arcsin (c x)} \, dx\) [335]

Optimal result

Integrand size = 26, antiderivative size = 245 \[ \int \frac {x \left (1-c^2 x^2\right )^{5/2}}{a+b \arcsin (c x)} \, dx=-\frac {5 \operatorname {CosIntegral}\left (\frac {a+b \arcsin (c x)}{b}\right ) \sin \left (\frac {a}{b}\right )}{64 b c^2}-\frac {9 \operatorname {CosIntegral}\left (\frac {3 (a+b \arcsin (c x))}{b}\right ) \sin \left (\frac {3 a}{b}\right )}{64 b c^2}-\frac {5 \operatorname {CosIntegral}\left (\frac {5 (a+b \arcsin (c x))}{b}\right ) \sin \left (\frac {5 a}{b}\right )}{64 b c^2}-\frac {\operatorname {CosIntegral}\left (\frac {7 (a+b \arcsin (c x))}{b}\right ) \sin \left (\frac {7 a}{b}\right )}{64 b c^2}+\frac {5 \cos \left (\frac {a}{b}\right ) \text {Si}\left (\frac {a+b \arcsin (c x)}{b}\right )}{64 b c^2}+\frac {9 \cos \left (\frac {3 a}{b}\right ) \text {Si}\left (\frac {3 (a+b \arcsin (c x))}{b}\right )}{64 b c^2}+\frac {5 \cos \left (\frac {5 a}{b}\right ) \text {Si}\left (\frac {5 (a+b \arcsin (c x))}{b}\right )}{64 b c^2}+\frac {\cos \left (\frac {7 a}{b}\right ) \text {Si}\left (\frac {7 (a+b \arcsin (c x))}{b}\right )}{64 b c^2} \]

[Out]

5/64*cos(a/b)*Si((a+b*arcsin(c*x))/b)/b/c^2+9/64*cos(3*a/b)*Si(3*(a+b*arcsin(c*x))/b)/b/c^2+5/64*cos(5*a/b)*Si
(5*(a+b*arcsin(c*x))/b)/b/c^2+1/64*cos(7*a/b)*Si(7*(a+b*arcsin(c*x))/b)/b/c^2-5/64*Ci((a+b*arcsin(c*x))/b)*sin
(a/b)/b/c^2-9/64*Ci(3*(a+b*arcsin(c*x))/b)*sin(3*a/b)/b/c^2-5/64*Ci(5*(a+b*arcsin(c*x))/b)*sin(5*a/b)/b/c^2-1/
64*Ci(7*(a+b*arcsin(c*x))/b)*sin(7*a/b)/b/c^2

Rubi [A] (verified)

Time = 0.23 (sec) , antiderivative size = 245, normalized size of antiderivative = 1.00, number of steps used = 15, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.192, Rules used = {4809, 4491, 3384, 3380, 3383} \[ \int \frac {x \left (1-c^2 x^2\right )^{5/2}}{a+b \arcsin (c x)} \, dx=-\frac {5 \sin \left (\frac {a}{b}\right ) \operatorname {CosIntegral}\left (\frac {a+b \arcsin (c x)}{b}\right )}{64 b c^2}-\frac {9 \sin \left (\frac {3 a}{b}\right ) \operatorname {CosIntegral}\left (\frac {3 (a+b \arcsin (c x))}{b}\right )}{64 b c^2}-\frac {5 \sin \left (\frac {5 a}{b}\right ) \operatorname {CosIntegral}\left (\frac {5 (a+b \arcsin (c x))}{b}\right )}{64 b c^2}-\frac {\sin \left (\frac {7 a}{b}\right ) \operatorname {CosIntegral}\left (\frac {7 (a+b \arcsin (c x))}{b}\right )}{64 b c^2}+\frac {5 \cos \left (\frac {a}{b}\right ) \text {Si}\left (\frac {a+b \arcsin (c x)}{b}\right )}{64 b c^2}+\frac {9 \cos \left (\frac {3 a}{b}\right ) \text {Si}\left (\frac {3 (a+b \arcsin (c x))}{b}\right )}{64 b c^2}+\frac {5 \cos \left (\frac {5 a}{b}\right ) \text {Si}\left (\frac {5 (a+b \arcsin (c x))}{b}\right )}{64 b c^2}+\frac {\cos \left (\frac {7 a}{b}\right ) \text {Si}\left (\frac {7 (a+b \arcsin (c x))}{b}\right )}{64 b c^2} \]

[In]

Int[(x*(1 - c^2*x^2)^(5/2))/(a + b*ArcSin[c*x]),x]

[Out]

(-5*CosIntegral[(a + b*ArcSin[c*x])/b]*Sin[a/b])/(64*b*c^2) - (9*CosIntegral[(3*(a + b*ArcSin[c*x]))/b]*Sin[(3
*a)/b])/(64*b*c^2) - (5*CosIntegral[(5*(a + b*ArcSin[c*x]))/b]*Sin[(5*a)/b])/(64*b*c^2) - (CosIntegral[(7*(a +
b*ArcSin[c*x]))/b]*Sin[(7*a)/b])/(64*b*c^2) + (5*Cos[a/b]*SinIntegral[(a + b*ArcSin[c*x])/b])/(64*b*c^2) + (9
*Cos[(3*a)/b]*SinIntegral[(3*(a + b*ArcSin[c*x]))/b])/(64*b*c^2) + (5*Cos[(5*a)/b]*SinIntegral[(5*(a + b*ArcSi
n[c*x]))/b])/(64*b*c^2) + (Cos[(7*a)/b]*SinIntegral[(7*(a + b*ArcSin[c*x]))/b])/(64*b*c^2)

Rule 3380

Int[sin[(e_.) + (f_.)*(x_)]/((c_.) + (d_.)*(x_)), x_Symbol] :> Simp[SinIntegral[e + f*x]/d, x] /; FreeQ[{c, d,
e, f}, x] && EqQ[d*e - c*f, 0]

Rule 3383

Int[sin[(e_.) + (f_.)*(x_)]/((c_.) + (d_.)*(x_)), x_Symbol] :> Simp[CosIntegral[e - Pi/2 + f*x]/d, x] /; FreeQ
[{c, d, e, f}, x] && EqQ[d*(e - Pi/2) - c*f, 0]

Rule 3384

Int[sin[(e_.) + (f_.)*(x_)]/((c_.) + (d_.)*(x_)), x_Symbol] :> Dist[Cos[(d*e - c*f)/d], Int[Sin[c*(f/d) + f*x]
/(c + d*x), x], x] + Dist[Sin[(d*e - c*f)/d], Int[Cos[c*(f/d) + f*x]/(c + d*x), x], x] /; FreeQ[{c, d, e, f},
x] && NeQ[d*e - c*f, 0]

Rule 4491

Int[Cos[(a_.) + (b_.)*(x_)]^(p_.)*((c_.) + (d_.)*(x_))^(m_.)*Sin[(a_.) + (b_.)*(x_)]^(n_.), x_Symbol] :> Int[E
xpandTrigReduce[(c + d*x)^m, Sin[a + b*x]^n*Cos[a + b*x]^p, x], x] /; FreeQ[{a, b, c, d, m}, x] && IGtQ[n, 0]
&& IGtQ[p, 0]

Rule 4809

Int[((a_.) + ArcSin[(c_.)*(x_)]*(b_.))^(n_.)*(x_)^(m_.)*((d_) + (e_.)*(x_)^2)^(p_.), x_Symbol] :> Dist[(1/(b*c
^(m + 1)))*Simp[(d + e*x^2)^p/(1 - c^2*x^2)^p], Subst[Int[x^n*Sin[-a/b + x/b]^m*Cos[-a/b + x/b]^(2*p + 1), x],
x, a + b*ArcSin[c*x]], x] /; FreeQ[{a, b, c, d, e, n}, x] && EqQ[c^2*d + e, 0] && IGtQ[2*p + 2, 0] && IGtQ[m,
0]

Rubi steps \begin{align*} \text {integral}& = -\frac {\text {Subst}\left (\int \frac {\cos ^6\left (\frac {a}{b}-\frac {x}{b}\right ) \sin \left (\frac {a}{b}-\frac {x}{b}\right )}{x} \, dx,x,a+b \arcsin (c x)\right )}{b c^2} \\ & = -\frac {\text {Subst}\left (\int \left (\frac {\sin \left (\frac {7 a}{b}-\frac {7 x}{b}\right )}{64 x}+\frac {5 \sin \left (\frac {5 a}{b}-\frac {5 x}{b}\right )}{64 x}+\frac {9 \sin \left (\frac {3 a}{b}-\frac {3 x}{b}\right )}{64 x}+\frac {5 \sin \left (\frac {a}{b}-\frac {x}{b}\right )}{64 x}\right ) \, dx,x,a+b \arcsin (c x)\right )}{b c^2} \\ & = -\frac {\text {Subst}\left (\int \frac {\sin \left (\frac {7 a}{b}-\frac {7 x}{b}\right )}{x} \, dx,x,a+b \arcsin (c x)\right )}{64 b c^2}-\frac {5 \text {Subst}\left (\int \frac {\sin \left (\frac {5 a}{b}-\frac {5 x}{b}\right )}{x} \, dx,x,a+b \arcsin (c x)\right )}{64 b c^2}-\frac {5 \text {Subst}\left (\int \frac {\sin \left (\frac {a}{b}-\frac {x}{b}\right )}{x} \, dx,x,a+b \arcsin (c x)\right )}{64 b c^2}-\frac {9 \text {Subst}\left (\int \frac {\sin \left (\frac {3 a}{b}-\frac {3 x}{b}\right )}{x} \, dx,x,a+b \arcsin (c x)\right )}{64 b c^2} \\ & = \frac {\left (5 \cos \left (\frac {a}{b}\right )\right ) \text {Subst}\left (\int \frac {\sin \left (\frac {x}{b}\right )}{x} \, dx,x,a+b \arcsin (c x)\right )}{64 b c^2}+\frac {\left (9 \cos \left (\frac {3 a}{b}\right )\right ) \text {Subst}\left (\int \frac {\sin \left (\frac {3 x}{b}\right )}{x} \, dx,x,a+b \arcsin (c x)\right )}{64 b c^2}+\frac {\left (5 \cos \left (\frac {5 a}{b}\right )\right ) \text {Subst}\left (\int \frac {\sin \left (\frac {5 x}{b}\right )}{x} \, dx,x,a+b \arcsin (c x)\right )}{64 b c^2}+\frac {\cos \left (\frac {7 a}{b}\right ) \text {Subst}\left (\int \frac {\sin \left (\frac {7 x}{b}\right )}{x} \, dx,x,a+b \arcsin (c x)\right )}{64 b c^2}-\frac {\left (5 \sin \left (\frac {a}{b}\right )\right ) \text {Subst}\left (\int \frac {\cos \left (\frac {x}{b}\right )}{x} \, dx,x,a+b \arcsin (c x)\right )}{64 b c^2}-\frac {\left (9 \sin \left (\frac {3 a}{b}\right )\right ) \text {Subst}\left (\int \frac {\cos \left (\frac {3 x}{b}\right )}{x} \, dx,x,a+b \arcsin (c x)\right )}{64 b c^2}-\frac {\left (5 \sin \left (\frac {5 a}{b}\right )\right ) \text {Subst}\left (\int \frac {\cos \left (\frac {5 x}{b}\right )}{x} \, dx,x,a+b \arcsin (c x)\right )}{64 b c^2}-\frac {\sin \left (\frac {7 a}{b}\right ) \text {Subst}\left (\int \frac {\cos \left (\frac {7 x}{b}\right )}{x} \, dx,x,a+b \arcsin (c x)\right )}{64 b c^2} \\ & = -\frac {5 \operatorname {CosIntegral}\left (\frac {a+b \arcsin (c x)}{b}\right ) \sin \left (\frac {a}{b}\right )}{64 b c^2}-\frac {9 \operatorname {CosIntegral}\left (\frac {3 (a+b \arcsin (c x))}{b}\right ) \sin \left (\frac {3 a}{b}\right )}{64 b c^2}-\frac {5 \operatorname {CosIntegral}\left (\frac {5 (a+b \arcsin (c x))}{b}\right ) \sin \left (\frac {5 a}{b}\right )}{64 b c^2}-\frac {\operatorname {CosIntegral}\left (\frac {7 (a+b \arcsin (c x))}{b}\right ) \sin \left (\frac {7 a}{b}\right )}{64 b c^2}+\frac {5 \cos \left (\frac {a}{b}\right ) \text {Si}\left (\frac {a+b \arcsin (c x)}{b}\right )}{64 b c^2}+\frac {9 \cos \left (\frac {3 a}{b}\right ) \text {Si}\left (\frac {3 (a+b \arcsin (c x))}{b}\right )}{64 b c^2}+\frac {5 \cos \left (\frac {5 a}{b}\right ) \text {Si}\left (\frac {5 (a+b \arcsin (c x))}{b}\right )}{64 b c^2}+\frac {\cos \left (\frac {7 a}{b}\right ) \text {Si}\left (\frac {7 (a+b \arcsin (c x))}{b}\right )}{64 b c^2} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.79 (sec) , antiderivative size = 180, normalized size of antiderivative = 0.73 \[ \int \frac {x \left (1-c^2 x^2\right )^{5/2}}{a+b \arcsin (c x)} \, dx=\frac {-5 \operatorname {CosIntegral}\left (\frac {a}{b}+\arcsin (c x)\right ) \sin \left (\frac {a}{b}\right )-9 \operatorname {CosIntegral}\left (3 \left (\frac {a}{b}+\arcsin (c x)\right )\right ) \sin \left (\frac {3 a}{b}\right )-5 \operatorname {CosIntegral}\left (5 \left (\frac {a}{b}+\arcsin (c x)\right )\right ) \sin \left (\frac {5 a}{b}\right )-\operatorname {CosIntegral}\left (7 \left (\frac {a}{b}+\arcsin (c x)\right )\right ) \sin \left (\frac {7 a}{b}\right )+5 \cos \left (\frac {a}{b}\right ) \text {Si}\left (\frac {a}{b}+\arcsin (c x)\right )+9 \cos \left (\frac {3 a}{b}\right ) \text {Si}\left (3 \left (\frac {a}{b}+\arcsin (c x)\right )\right )+5 \cos \left (\frac {5 a}{b}\right ) \text {Si}\left (5 \left (\frac {a}{b}+\arcsin (c x)\right )\right )+\cos \left (\frac {7 a}{b}\right ) \text {Si}\left (7 \left (\frac {a}{b}+\arcsin (c x)\right )\right )}{64 b c^2} \]

[In]

Integrate[(x*(1 - c^2*x^2)^(5/2))/(a + b*ArcSin[c*x]),x]

[Out]

(-5*CosIntegral[a/b + ArcSin[c*x]]*Sin[a/b] - 9*CosIntegral[3*(a/b + ArcSin[c*x])]*Sin[(3*a)/b] - 5*CosIntegra
l[5*(a/b + ArcSin[c*x])]*Sin[(5*a)/b] - CosIntegral[7*(a/b + ArcSin[c*x])]*Sin[(7*a)/b] + 5*Cos[a/b]*SinIntegr
al[a/b + ArcSin[c*x]] + 9*Cos[(3*a)/b]*SinIntegral[3*(a/b + ArcSin[c*x])] + 5*Cos[(5*a)/b]*SinIntegral[5*(a/b
+ ArcSin[c*x])] + Cos[(7*a)/b]*SinIntegral[7*(a/b + ArcSin[c*x])])/(64*b*c^2)

Maple [A] (verified)

Time = 0.11 (sec) , antiderivative size = 185, normalized size of antiderivative = 0.76

method	result	size
default	\(\frac {\operatorname {Si}\left (7 \arcsin \left (c x \right )+\frac {7 a}{b}\right ) \cos \left (\frac {7 a}{b}\right )-\operatorname {Ci}\left (7 \arcsin \left (c x \right )+\frac {7 a}{b}\right ) \sin \left (\frac {7 a}{b}\right )+5 \,\operatorname {Si}\left (5 \arcsin \left (c x \right )+\frac {5 a}{b}\right ) \cos \left (\frac {5 a}{b}\right )-5 \,\operatorname {Ci}\left (5 \arcsin \left (c x \right )+\frac {5 a}{b}\right ) \sin \left (\frac {5 a}{b}\right )+9 \,\operatorname {Si}\left (3 \arcsin \left (c x \right )+\frac {3 a}{b}\right ) \cos \left (\frac {3 a}{b}\right )-9 \,\operatorname {Ci}\left (3 \arcsin \left (c x \right )+\frac {3 a}{b}\right ) \sin \left (\frac {3 a}{b}\right )+5 \,\operatorname {Si}\left (\arcsin \left (c x \right )+\frac {a}{b}\right ) \cos \left (\frac {a}{b}\right )-5 \,\operatorname {Ci}\left (\arcsin \left (c x \right )+\frac {a}{b}\right ) \sin \left (\frac {a}{b}\right )}{64 c^{2} b}\)	\(185\)

[In]

int(x*(-c^2*x^2+1)^(5/2)/(a+b*arcsin(c*x)),x,method=_RETURNVERBOSE)

[Out]

1/64/c^2*(Si(7*arcsin(c*x)+7*a/b)*cos(7*a/b)-Ci(7*arcsin(c*x)+7*a/b)*sin(7*a/b)+5*Si(5*arcsin(c*x)+5*a/b)*cos(
5*a/b)-5*Ci(5*arcsin(c*x)+5*a/b)*sin(5*a/b)+9*Si(3*arcsin(c*x)+3*a/b)*cos(3*a/b)-9*Ci(3*arcsin(c*x)+3*a/b)*sin
(3*a/b)+5*Si(arcsin(c*x)+a/b)*cos(a/b)-5*Ci(arcsin(c*x)+a/b)*sin(a/b))/b

Fricas [F]

\[ \int \frac {x \left (1-c^2 x^2\right )^{5/2}}{a+b \arcsin (c x)} \, dx=\int { \frac {{\left (-c^{2} x^{2} + 1\right )}^{\frac {5}{2}} x}{b \arcsin \left (c x\right ) + a} \,d x } \]

[In]

integrate(x*(-c^2*x^2+1)^(5/2)/(a+b*arcsin(c*x)),x, algorithm="fricas")

[Out]

integral((c^4*x^5 - 2*c^2*x^3 + x)*sqrt(-c^2*x^2 + 1)/(b*arcsin(c*x) + a), x)

Sympy [F]

\[ \int \frac {x \left (1-c^2 x^2\right )^{5/2}}{a+b \arcsin (c x)} \, dx=\int \frac {x \left (- \left (c x - 1\right ) \left (c x + 1\right )\right )^{\frac {5}{2}}}{a + b \operatorname {asin}{\left (c x \right )}}\, dx \]

[In]

integrate(x*(-c**2*x**2+1)**(5/2)/(a+b*asin(c*x)),x)

[Out]

Integral(x*(-(c*x - 1)*(c*x + 1))**(5/2)/(a + b*asin(c*x)), x)

Maxima [F]

\[ \int \frac {x \left (1-c^2 x^2\right )^{5/2}}{a+b \arcsin (c x)} \, dx=\int { \frac {{\left (-c^{2} x^{2} + 1\right )}^{\frac {5}{2}} x}{b \arcsin \left (c x\right ) + a} \,d x } \]

[In]

integrate(x*(-c^2*x^2+1)^(5/2)/(a+b*arcsin(c*x)),x, algorithm="maxima")

[Out]

integrate((-c^2*x^2 + 1)^(5/2)*x/(b*arcsin(c*x) + a), x)

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 614 vs. \(2 (229) = 458\).

Time = 0.33 (sec) , antiderivative size = 614, normalized size of antiderivative = 2.51 \[ \int \frac {x \left (1-c^2 x^2\right )^{5/2}}{a+b \arcsin (c x)} \, dx=-\frac {\cos \left (\frac {a}{b}\right )^{6} \operatorname {Ci}\left (\frac {7 \, a}{b} + 7 \, \arcsin \left (c x\right )\right ) \sin \left (\frac {a}{b}\right )}{b c^{2}} + \frac {\cos \left (\frac {a}{b}\right )^{7} \operatorname {Si}\left (\frac {7 \, a}{b} + 7 \, \arcsin \left (c x\right )\right )}{b c^{2}} + \frac {5 \, \cos \left (\frac {a}{b}\right )^{4} \operatorname {Ci}\left (\frac {7 \, a}{b} + 7 \, \arcsin \left (c x\right )\right ) \sin \left (\frac {a}{b}\right )}{4 \, b c^{2}} - \frac {5 \, \cos \left (\frac {a}{b}\right )^{4} \operatorname {Ci}\left (\frac {5 \, a}{b} + 5 \, \arcsin \left (c x\right )\right ) \sin \left (\frac {a}{b}\right )}{4 \, b c^{2}} - \frac {7 \, \cos \left (\frac {a}{b}\right )^{5} \operatorname {Si}\left (\frac {7 \, a}{b} + 7 \, \arcsin \left (c x\right )\right )}{4 \, b c^{2}} + \frac {5 \, \cos \left (\frac {a}{b}\right )^{5} \operatorname {Si}\left (\frac {5 \, a}{b} + 5 \, \arcsin \left (c x\right )\right )}{4 \, b c^{2}} - \frac {3 \, \cos \left (\frac {a}{b}\right )^{2} \operatorname {Ci}\left (\frac {7 \, a}{b} + 7 \, \arcsin \left (c x\right )\right ) \sin \left (\frac {a}{b}\right )}{8 \, b c^{2}} + \frac {15 \, \cos \left (\frac {a}{b}\right )^{2} \operatorname {Ci}\left (\frac {5 \, a}{b} + 5 \, \arcsin \left (c x\right )\right ) \sin \left (\frac {a}{b}\right )}{16 \, b c^{2}} - \frac {9 \, \cos \left (\frac {a}{b}\right )^{2} \operatorname {Ci}\left (\frac {3 \, a}{b} + 3 \, \arcsin \left (c x\right )\right ) \sin \left (\frac {a}{b}\right )}{16 \, b c^{2}} + \frac {7 \, \cos \left (\frac {a}{b}\right )^{3} \operatorname {Si}\left (\frac {7 \, a}{b} + 7 \, \arcsin \left (c x\right )\right )}{8 \, b c^{2}} - \frac {25 \, \cos \left (\frac {a}{b}\right )^{3} \operatorname {Si}\left (\frac {5 \, a}{b} + 5 \, \arcsin \left (c x\right )\right )}{16 \, b c^{2}} + \frac {9 \, \cos \left (\frac {a}{b}\right )^{3} \operatorname {Si}\left (\frac {3 \, a}{b} + 3 \, \arcsin \left (c x\right )\right )}{16 \, b c^{2}} + \frac {\operatorname {Ci}\left (\frac {7 \, a}{b} + 7 \, \arcsin \left (c x\right )\right ) \sin \left (\frac {a}{b}\right )}{64 \, b c^{2}} - \frac {5 \, \operatorname {Ci}\left (\frac {5 \, a}{b} + 5 \, \arcsin \left (c x\right )\right ) \sin \left (\frac {a}{b}\right )}{64 \, b c^{2}} + \frac {9 \, \operatorname {Ci}\left (\frac {3 \, a}{b} + 3 \, \arcsin \left (c x\right )\right ) \sin \left (\frac {a}{b}\right )}{64 \, b c^{2}} - \frac {5 \, \operatorname {Ci}\left (\frac {a}{b} + \arcsin \left (c x\right )\right ) \sin \left (\frac {a}{b}\right )}{64 \, b c^{2}} - \frac {7 \, \cos \left (\frac {a}{b}\right ) \operatorname {Si}\left (\frac {7 \, a}{b} + 7 \, \arcsin \left (c x\right )\right )}{64 \, b c^{2}} + \frac {25 \, \cos \left (\frac {a}{b}\right ) \operatorname {Si}\left (\frac {5 \, a}{b} + 5 \, \arcsin \left (c x\right )\right )}{64 \, b c^{2}} - \frac {27 \, \cos \left (\frac {a}{b}\right ) \operatorname {Si}\left (\frac {3 \, a}{b} + 3 \, \arcsin \left (c x\right )\right )}{64 \, b c^{2}} + \frac {5 \, \cos \left (\frac {a}{b}\right ) \operatorname {Si}\left (\frac {a}{b} + \arcsin \left (c x\right )\right )}{64 \, b c^{2}} \]

[In]

integrate(x*(-c^2*x^2+1)^(5/2)/(a+b*arcsin(c*x)),x, algorithm="giac")

[Out]

-cos(a/b)^6*cos_integral(7*a/b + 7*arcsin(c*x))*sin(a/b)/(b*c^2) + cos(a/b)^7*sin_integral(7*a/b + 7*arcsin(c*
x))/(b*c^2) + 5/4*cos(a/b)^4*cos_integral(7*a/b + 7*arcsin(c*x))*sin(a/b)/(b*c^2) - 5/4*cos(a/b)^4*cos_integra
l(5*a/b + 5*arcsin(c*x))*sin(a/b)/(b*c^2) - 7/4*cos(a/b)^5*sin_integral(7*a/b + 7*arcsin(c*x))/(b*c^2) + 5/4*c
os(a/b)^5*sin_integral(5*a/b + 5*arcsin(c*x))/(b*c^2) - 3/8*cos(a/b)^2*cos_integral(7*a/b + 7*arcsin(c*x))*sin
(a/b)/(b*c^2) + 15/16*cos(a/b)^2*cos_integral(5*a/b + 5*arcsin(c*x))*sin(a/b)/(b*c^2) - 9/16*cos(a/b)^2*cos_in
tegral(3*a/b + 3*arcsin(c*x))*sin(a/b)/(b*c^2) + 7/8*cos(a/b)^3*sin_integral(7*a/b + 7*arcsin(c*x))/(b*c^2) -
25/16*cos(a/b)^3*sin_integral(5*a/b + 5*arcsin(c*x))/(b*c^2) + 9/16*cos(a/b)^3*sin_integral(3*a/b + 3*arcsin(c
*x))/(b*c^2) + 1/64*cos_integral(7*a/b + 7*arcsin(c*x))*sin(a/b)/(b*c^2) - 5/64*cos_integral(5*a/b + 5*arcsin(
c*x))*sin(a/b)/(b*c^2) + 9/64*cos_integral(3*a/b + 3*arcsin(c*x))*sin(a/b)/(b*c^2) - 5/64*cos_integral(a/b + a
rcsin(c*x))*sin(a/b)/(b*c^2) - 7/64*cos(a/b)*sin_integral(7*a/b + 7*arcsin(c*x))/(b*c^2) + 25/64*cos(a/b)*sin_
integral(5*a/b + 5*arcsin(c*x))/(b*c^2) - 27/64*cos(a/b)*sin_integral(3*a/b + 3*arcsin(c*x))/(b*c^2) + 5/64*co
s(a/b)*sin_integral(a/b + arcsin(c*x))/(b*c^2)

Mupad [F(-1)]

Timed out. \[ \int \frac {x \left (1-c^2 x^2\right )^{5/2}}{a+b \arcsin (c x)} \, dx=\int \frac {x\,{\left (1-c^2\,x^2\right )}^{5/2}}{a+b\,\mathrm {asin}\left (c\,x\right )} \,d x \]

[In]

int((x*(1 - c^2*x^2)^(5/2))/(a + b*asin(c*x)),x)

[Out]

int((x*(1 - c^2*x^2)^(5/2))/(a + b*asin(c*x)), x)