3.2.36 \(\int \frac {a+b \arcsin (c x)}{x (d-c^2 d x^2)^{5/2}} \, dx\) [136]

3.2.36.1 Optimal result

Integrand size = 27, antiderivative size = 291 \[ \int \frac {a+b \arcsin (c x)}{x \left (d-c^2 d x^2\right )^{5/2}} \, dx=-\frac {b c x}{6 d^2 \sqrt {1-c^2 x^2} \sqrt {d-c^2 d x^2}}+\frac {a+b \arcsin (c x)}{3 d \left (d-c^2 d x^2\right )^{3/2}}+\frac {a+b \arcsin (c x)}{d^2 \sqrt {d-c^2 d x^2}}-\frac {2 \sqrt {1-c^2 x^2} (a+b \arcsin (c x)) \text {arctanh}\left (e^{i \arcsin (c x)}\right )}{d^2 \sqrt {d-c^2 d x^2}}-\frac {7 b \sqrt {1-c^2 x^2} \text {arctanh}(c x)}{6 d^2 \sqrt {d-c^2 d x^2}}+\frac {i b \sqrt {1-c^2 x^2} \operatorname {PolyLog}\left (2,-e^{i \arcsin (c x)}\right )}{d^2 \sqrt {d-c^2 d x^2}}-\frac {i b \sqrt {1-c^2 x^2} \operatorname {PolyLog}\left (2,e^{i \arcsin (c x)}\right )}{d^2 \sqrt {d-c^2 d x^2}} \]

output

1/3*(a+b*arcsin(c*x))/d/(-c^2*d*x^2+d)^(3/2)+(a+b*arcsin(c*x))/d^2/(-c^2*d 
*x^2+d)^(1/2)-1/6*b*c*x/d^2/(-c^2*x^2+1)^(1/2)/(-c^2*d*x^2+d)^(1/2)-2*(a+b 
*arcsin(c*x))*arctanh(I*c*x+(-c^2*x^2+1)^(1/2))*(-c^2*x^2+1)^(1/2)/d^2/(-c 
^2*d*x^2+d)^(1/2)-7/6*b*arctanh(c*x)*(-c^2*x^2+1)^(1/2)/d^2/(-c^2*d*x^2+d) 
^(1/2)+I*b*polylog(2,-I*c*x-(-c^2*x^2+1)^(1/2))*(-c^2*x^2+1)^(1/2)/d^2/(-c 
^2*d*x^2+d)^(1/2)-I*b*polylog(2,I*c*x+(-c^2*x^2+1)^(1/2))*(-c^2*x^2+1)^(1/ 
2)/d^2/(-c^2*d*x^2+d)^(1/2)
 

3.2.36.2 Mathematica [A] (verified)

Time = 1.36 (sec) , antiderivative size = 456, normalized size of antiderivative = 1.57 \[ \int \frac {a+b \arcsin (c x)}{x \left (d-c^2 d x^2\right )^{5/2}} \, dx=-\frac {a \left (-4+3 c^2 x^2\right ) \sqrt {d-c^2 d x^2}}{3 d^3 \left (-1+c^2 x^2\right )^2}+\frac {a \log (x)}{d^{5/2}}-\frac {a \log \left (d+\sqrt {d} \sqrt {d-c^2 d x^2}\right )}{d^{5/2}}+\frac {b \left (20 \arcsin (c x)+12 \arcsin (c x) \cos (2 \arcsin (c x))+18 \sqrt {1-c^2 x^2} \arcsin (c x) \log \left (1-e^{i \arcsin (c x)}\right )+6 \arcsin (c x) \cos (3 \arcsin (c x)) \log \left (1-e^{i \arcsin (c x)}\right )-18 \sqrt {1-c^2 x^2} \arcsin (c x) \log \left (1+e^{i \arcsin (c x)}\right )-6 \arcsin (c x) \cos (3 \arcsin (c x)) \log \left (1+e^{i \arcsin (c x)}\right )+21 \sqrt {1-c^2 x^2} \log \left (\cos \left (\frac {1}{2} \arcsin (c x)\right )-\sin \left (\frac {1}{2} \arcsin (c x)\right )\right )+7 \cos (3 \arcsin (c x)) \log \left (\cos \left (\frac {1}{2} \arcsin (c x)\right )-\sin \left (\frac {1}{2} \arcsin (c x)\right )\right )-21 \sqrt {1-c^2 x^2} \log \left (\cos \left (\frac {1}{2} \arcsin (c x)\right )+\sin \left (\frac {1}{2} \arcsin (c x)\right )\right )-7 \cos (3 \arcsin (c x)) \log \left (\cos \left (\frac {1}{2} \arcsin (c x)\right )+\sin \left (\frac {1}{2} \arcsin (c x)\right )\right )+24 i \left (1-c^2 x^2\right )^{3/2} \operatorname {PolyLog}\left (2,-e^{i \arcsin (c x)}\right )-24 i \left (1-c^2 x^2\right )^{3/2} \operatorname {PolyLog}\left (2,e^{i \arcsin (c x)}\right )-2 \sin (2 \arcsin (c x))\right )}{24 d \left (d-c^2 d x^2\right )^{3/2}} \]

input

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

output

-1/3*(a*(-4 + 3*c^2*x^2)*Sqrt[d - c^2*d*x^2])/(d^3*(-1 + c^2*x^2)^2) + (a* 
Log[x])/d^(5/2) - (a*Log[d + Sqrt[d]*Sqrt[d - c^2*d*x^2]])/d^(5/2) + (b*(2 
0*ArcSin[c*x] + 12*ArcSin[c*x]*Cos[2*ArcSin[c*x]] + 18*Sqrt[1 - c^2*x^2]*A 
rcSin[c*x]*Log[1 - E^(I*ArcSin[c*x])] + 6*ArcSin[c*x]*Cos[3*ArcSin[c*x]]*L 
og[1 - E^(I*ArcSin[c*x])] - 18*Sqrt[1 - c^2*x^2]*ArcSin[c*x]*Log[1 + E^(I* 
ArcSin[c*x])] - 6*ArcSin[c*x]*Cos[3*ArcSin[c*x]]*Log[1 + E^(I*ArcSin[c*x]) 
] + 21*Sqrt[1 - c^2*x^2]*Log[Cos[ArcSin[c*x]/2] - Sin[ArcSin[c*x]/2]] + 7* 
Cos[3*ArcSin[c*x]]*Log[Cos[ArcSin[c*x]/2] - Sin[ArcSin[c*x]/2]] - 21*Sqrt[ 
1 - c^2*x^2]*Log[Cos[ArcSin[c*x]/2] + Sin[ArcSin[c*x]/2]] - 7*Cos[3*ArcSin 
[c*x]]*Log[Cos[ArcSin[c*x]/2] + Sin[ArcSin[c*x]/2]] + (24*I)*(1 - c^2*x^2) 
^(3/2)*PolyLog[2, -E^(I*ArcSin[c*x])] - (24*I)*(1 - c^2*x^2)^(3/2)*PolyLog 
[2, E^(I*ArcSin[c*x])] - 2*Sin[2*ArcSin[c*x]]))/(24*d*(d - c^2*d*x^2)^(3/2 
))
 

3.2.36.3 Rubi [A] (verified)

Time = 0.99 (sec) , antiderivative size = 260, normalized size of antiderivative = 0.89, number of steps used = 11, number of rules used = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.370, Rules used = {5208, 215, 219, 5208, 219, 5218, 3042, 4671, 2715, 2838}

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.

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

\(\Big \downarrow \) 5208

\(\displaystyle \frac {\int \frac {a+b \arcsin (c x)}{x \left (d-c^2 d x^2\right )^{3/2}}dx}{d}-\frac {b c \sqrt {1-c^2 x^2} \int \frac {1}{\left (1-c^2 x^2\right )^2}dx}{3 d^2 \sqrt {d-c^2 d x^2}}+\frac {a+b \arcsin (c x)}{3 d \left (d-c^2 d x^2\right )^{3/2}}\)

\(\Big \downarrow \) 215

\(\displaystyle \frac {\int \frac {a+b \arcsin (c x)}{x \left (d-c^2 d x^2\right )^{3/2}}dx}{d}-\frac {b c \sqrt {1-c^2 x^2} \left (\frac {1}{2} \int \frac {1}{1-c^2 x^2}dx+\frac {x}{2 \left (1-c^2 x^2\right )}\right )}{3 d^2 \sqrt {d-c^2 d x^2}}+\frac {a+b \arcsin (c x)}{3 d \left (d-c^2 d x^2\right )^{3/2}}\)

\(\Big \downarrow \) 219

\(\displaystyle \frac {\int \frac {a+b \arcsin (c x)}{x \left (d-c^2 d x^2\right )^{3/2}}dx}{d}+\frac {a+b \arcsin (c x)}{3 d \left (d-c^2 d x^2\right )^{3/2}}-\frac {b c \sqrt {1-c^2 x^2} \left (\frac {\text {arctanh}(c x)}{2 c}+\frac {x}{2 \left (1-c^2 x^2\right )}\right )}{3 d^2 \sqrt {d-c^2 d x^2}}\)

\(\Big \downarrow \) 5208

\(\displaystyle \frac {\frac {\int \frac {a+b \arcsin (c x)}{x \sqrt {d-c^2 d x^2}}dx}{d}-\frac {b c \sqrt {1-c^2 x^2} \int \frac {1}{1-c^2 x^2}dx}{d \sqrt {d-c^2 d x^2}}+\frac {a+b \arcsin (c x)}{d \sqrt {d-c^2 d x^2}}}{d}+\frac {a+b \arcsin (c x)}{3 d \left (d-c^2 d x^2\right )^{3/2}}-\frac {b c \sqrt {1-c^2 x^2} \left (\frac {\text {arctanh}(c x)}{2 c}+\frac {x}{2 \left (1-c^2 x^2\right )}\right )}{3 d^2 \sqrt {d-c^2 d x^2}}\)

\(\Big \downarrow \) 219

\(\displaystyle \frac {\frac {\int \frac {a+b \arcsin (c x)}{x \sqrt {d-c^2 d x^2}}dx}{d}+\frac {a+b \arcsin (c x)}{d \sqrt {d-c^2 d x^2}}-\frac {b \sqrt {1-c^2 x^2} \text {arctanh}(c x)}{d \sqrt {d-c^2 d x^2}}}{d}+\frac {a+b \arcsin (c x)}{3 d \left (d-c^2 d x^2\right )^{3/2}}-\frac {b c \sqrt {1-c^2 x^2} \left (\frac {\text {arctanh}(c x)}{2 c}+\frac {x}{2 \left (1-c^2 x^2\right )}\right )}{3 d^2 \sqrt {d-c^2 d x^2}}\)

\(\Big \downarrow \) 5218

\(\displaystyle \frac {\frac {\sqrt {1-c^2 x^2} \int \frac {a+b \arcsin (c x)}{c x}d\arcsin (c x)}{d \sqrt {d-c^2 d x^2}}+\frac {a+b \arcsin (c x)}{d \sqrt {d-c^2 d x^2}}-\frac {b \sqrt {1-c^2 x^2} \text {arctanh}(c x)}{d \sqrt {d-c^2 d x^2}}}{d}+\frac {a+b \arcsin (c x)}{3 d \left (d-c^2 d x^2\right )^{3/2}}-\frac {b c \sqrt {1-c^2 x^2} \left (\frac {\text {arctanh}(c x)}{2 c}+\frac {x}{2 \left (1-c^2 x^2\right )}\right )}{3 d^2 \sqrt {d-c^2 d x^2}}\)

\(\Big \downarrow \) 3042

\(\displaystyle \frac {\frac {\sqrt {1-c^2 x^2} \int (a+b \arcsin (c x)) \csc (\arcsin (c x))d\arcsin (c x)}{d \sqrt {d-c^2 d x^2}}+\frac {a+b \arcsin (c x)}{d \sqrt {d-c^2 d x^2}}-\frac {b \sqrt {1-c^2 x^2} \text {arctanh}(c x)}{d \sqrt {d-c^2 d x^2}}}{d}+\frac {a+b \arcsin (c x)}{3 d \left (d-c^2 d x^2\right )^{3/2}}-\frac {b c \sqrt {1-c^2 x^2} \left (\frac {\text {arctanh}(c x)}{2 c}+\frac {x}{2 \left (1-c^2 x^2\right )}\right )}{3 d^2 \sqrt {d-c^2 d x^2}}\)

\(\Big \downarrow \) 4671

\(\displaystyle \frac {\frac {\sqrt {1-c^2 x^2} \left (-b \int \log \left (1-e^{i \arcsin (c x)}\right )d\arcsin (c x)+b \int \log \left (1+e^{i \arcsin (c x)}\right )d\arcsin (c x)-2 \text {arctanh}\left (e^{i \arcsin (c x)}\right ) (a+b \arcsin (c x))\right )}{d \sqrt {d-c^2 d x^2}}+\frac {a+b \arcsin (c x)}{d \sqrt {d-c^2 d x^2}}-\frac {b \sqrt {1-c^2 x^2} \text {arctanh}(c x)}{d \sqrt {d-c^2 d x^2}}}{d}+\frac {a+b \arcsin (c x)}{3 d \left (d-c^2 d x^2\right )^{3/2}}-\frac {b c \sqrt {1-c^2 x^2} \left (\frac {\text {arctanh}(c x)}{2 c}+\frac {x}{2 \left (1-c^2 x^2\right )}\right )}{3 d^2 \sqrt {d-c^2 d x^2}}\)

\(\Big \downarrow \) 2715

\(\displaystyle \frac {\frac {\sqrt {1-c^2 x^2} \left (i b \int e^{-i \arcsin (c x)} \log \left (1-e^{i \arcsin (c x)}\right )de^{i \arcsin (c x)}-i b \int e^{-i \arcsin (c x)} \log \left (1+e^{i \arcsin (c x)}\right )de^{i \arcsin (c x)}-2 \text {arctanh}\left (e^{i \arcsin (c x)}\right ) (a+b \arcsin (c x))\right )}{d \sqrt {d-c^2 d x^2}}+\frac {a+b \arcsin (c x)}{d \sqrt {d-c^2 d x^2}}-\frac {b \sqrt {1-c^2 x^2} \text {arctanh}(c x)}{d \sqrt {d-c^2 d x^2}}}{d}+\frac {a+b \arcsin (c x)}{3 d \left (d-c^2 d x^2\right )^{3/2}}-\frac {b c \sqrt {1-c^2 x^2} \left (\frac {\text {arctanh}(c x)}{2 c}+\frac {x}{2 \left (1-c^2 x^2\right )}\right )}{3 d^2 \sqrt {d-c^2 d x^2}}\)

\(\Big \downarrow \) 2838

\(\displaystyle \frac {\frac {\sqrt {1-c^2 x^2} \left (-2 \text {arctanh}\left (e^{i \arcsin (c x)}\right ) (a+b \arcsin (c x))+i b \operatorname {PolyLog}\left (2,-e^{i \arcsin (c x)}\right )-i b \operatorname {PolyLog}\left (2,e^{i \arcsin (c x)}\right )\right )}{d \sqrt {d-c^2 d x^2}}+\frac {a+b \arcsin (c x)}{d \sqrt {d-c^2 d x^2}}-\frac {b \sqrt {1-c^2 x^2} \text {arctanh}(c x)}{d \sqrt {d-c^2 d x^2}}}{d}+\frac {a+b \arcsin (c x)}{3 d \left (d-c^2 d x^2\right )^{3/2}}-\frac {b c \sqrt {1-c^2 x^2} \left (\frac {\text {arctanh}(c x)}{2 c}+\frac {x}{2 \left (1-c^2 x^2\right )}\right )}{3 d^2 \sqrt {d-c^2 d x^2}}\)

input

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

output

(a + b*ArcSin[c*x])/(3*d*(d - c^2*d*x^2)^(3/2)) - (b*c*Sqrt[1 - c^2*x^2]*( 
x/(2*(1 - c^2*x^2)) + ArcTanh[c*x]/(2*c)))/(3*d^2*Sqrt[d - c^2*d*x^2]) + ( 
(a + b*ArcSin[c*x])/(d*Sqrt[d - c^2*d*x^2]) - (b*Sqrt[1 - c^2*x^2]*ArcTanh 
[c*x])/(d*Sqrt[d - c^2*d*x^2]) + (Sqrt[1 - c^2*x^2]*(-2*(a + b*ArcSin[c*x] 
)*ArcTanh[E^(I*ArcSin[c*x])] + I*b*PolyLog[2, -E^(I*ArcSin[c*x])] - I*b*Po 
lyLog[2, E^(I*ArcSin[c*x])]))/(d*Sqrt[d - c^2*d*x^2]))/d
 

3.2.36.3.1 Defintions of rubi rules used

rule 215

Int[((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> Simp[(-x)*((a + b*x^2)^(p + 1) 
/(2*a*(p + 1))), x] + Simp[(2*p + 3)/(2*a*(p + 1))   Int[(a + b*x^2)^(p + 1 
), x], x] /; FreeQ[{a, b}, x] && LtQ[p, -1] && (IntegerQ[4*p] || IntegerQ[6 
*p])
 

rule 219

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1/(Rt[a, 2]*Rt[-b, 2]))* 
ArcTanh[Rt[-b, 2]*(x/Rt[a, 2])], x] /; FreeQ[{a, b}, x] && NegQ[a/b] && (Gt 
Q[a, 0] || LtQ[b, 0])
 

rule 2715

Int[Log[(a_) + (b_.)*((F_)^((e_.)*((c_.) + (d_.)*(x_))))^(n_.)], x_Symbol] 
:> Simp[1/(d*e*n*Log[F])   Subst[Int[Log[a + b*x]/x, x], x, (F^(e*(c + d*x) 
))^n], x] /; FreeQ[{F, a, b, c, d, e, n}, x] && GtQ[a, 0]
 

rule 2838

Int[Log[(c_.)*((d_) + (e_.)*(x_)^(n_.))]/(x_), x_Symbol] :> Simp[-PolyLog[2 
, (-c)*e*x^n]/n, x] /; FreeQ[{c, d, e, n}, x] && EqQ[c*d, 1]
 

rule 3042

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

rule 4671

Int[csc[(e_.) + (f_.)*(x_)]*((c_.) + (d_.)*(x_))^(m_.), x_Symbol] :> Simp[- 
2*(c + d*x)^m*(ArcTanh[E^(I*(e + f*x))]/f), x] + (-Simp[d*(m/f)   Int[(c + 
d*x)^(m - 1)*Log[1 - E^(I*(e + f*x))], x], x] + Simp[d*(m/f)   Int[(c + d*x 
)^(m - 1)*Log[1 + E^(I*(e + f*x))], x], x]) /; FreeQ[{c, d, e, f}, x] && IG 
tQ[m, 0]
 

rule 5208

Int[((a_.) + ArcSin[(c_.)*(x_)]*(b_.))^(n_.)*((f_.)*(x_))^(m_)*((d_) + (e_. 
)*(x_)^2)^(p_), x_Symbol] :> Simp[(-(f*x)^(m + 1))*(d + e*x^2)^(p + 1)*((a 
+ b*ArcSin[c*x])^n/(2*d*f*(p + 1))), x] + (Simp[(m + 2*p + 3)/(2*d*(p + 1)) 
   Int[(f*x)^m*(d + e*x^2)^(p + 1)*(a + b*ArcSin[c*x])^n, x], x] + Simp[b*c 
*(n/(2*f*(p + 1)))*Simp[(d + e*x^2)^p/(1 - c^2*x^2)^p]   Int[(f*x)^(m + 1)* 
(1 - c^2*x^2)^(p + 1/2)*(a + b*ArcSin[c*x])^(n - 1), x], x]) /; FreeQ[{a, b 
, c, d, e, f, m}, x] && EqQ[c^2*d + e, 0] && GtQ[n, 0] && LtQ[p, -1] &&  !G 
tQ[m, 1] && (IntegerQ[m] || IntegerQ[p] || EqQ[n, 1])
 

rule 5218

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

3.2.36.4 Maple [A] (verified)

Time = 0.22 (sec) , antiderivative size = 512, normalized size of antiderivative = 1.76

method	result	size
default	\(\frac {a}{3 d \left (-c^{2} d \,x^{2}+d \right )^{\frac {3}{2}}}+\frac {a}{d^{2} \sqrt {-c^{2} d \,x^{2}+d}}-\frac {a \ln \left (\frac {2 d +2 \sqrt {d}\, \sqrt {-c^{2} d \,x^{2}+d}}{x}\right )}{d^{\frac {5}{2}}}-\frac {i b \sqrt {-d \left (c^{2} x^{2}-1\right )}\, \sqrt {-c^{2} x^{2}+1}\, \left (6 i \sqrt {-c^{2} x^{2}+1}\, \arcsin \left (c x \right ) c^{2} x^{2}+6 \operatorname {dilog}\left (1+i c x +\sqrt {-c^{2} x^{2}+1}\right ) c^{4} x^{4}+6 \operatorname {dilog}\left (i c x +\sqrt {-c^{2} x^{2}+1}\right ) c^{4} x^{4}+14 \arctan \left (i c x +\sqrt {-c^{2} x^{2}+1}\right ) c^{4} x^{4}-i x^{3} c^{3}-8 i \sqrt {-c^{2} x^{2}+1}\, \arcsin \left (c x \right )+i c x -12 \operatorname {dilog}\left (1+i c x +\sqrt {-c^{2} x^{2}+1}\right ) c^{2} x^{2}-12 \operatorname {dilog}\left (i c x +\sqrt {-c^{2} x^{2}+1}\right ) c^{2} x^{2}-28 \arctan \left (i c x +\sqrt {-c^{2} x^{2}+1}\right ) c^{2} x^{2}+6 i \arcsin \left (c x \right ) \ln \left (1+i c x +\sqrt {-c^{2} x^{2}+1}\right ) x^{4} c^{4}+6 i \arcsin \left (c x \right ) \ln \left (1+i c x +\sqrt {-c^{2} x^{2}+1}\right )-12 i \arcsin \left (c x \right ) \ln \left (1+i c x +\sqrt {-c^{2} x^{2}+1}\right ) x^{2} c^{2}+6 \operatorname {dilog}\left (1+i c x +\sqrt {-c^{2} x^{2}+1}\right )+6 \operatorname {dilog}\left (i c x +\sqrt {-c^{2} x^{2}+1}\right )+14 \arctan \left (i c x +\sqrt {-c^{2} x^{2}+1}\right )\right )}{6 \left (c^{6} x^{6}-3 c^{4} x^{4}+3 c^{2} x^{2}-1\right ) d^{3}}\)	\(512\)
parts	\(\frac {a}{3 d \left (-c^{2} d \,x^{2}+d \right )^{\frac {3}{2}}}+\frac {a}{d^{2} \sqrt {-c^{2} d \,x^{2}+d}}-\frac {a \ln \left (\frac {2 d +2 \sqrt {d}\, \sqrt {-c^{2} d \,x^{2}+d}}{x}\right )}{d^{\frac {5}{2}}}-\frac {i b \sqrt {-d \left (c^{2} x^{2}-1\right )}\, \sqrt {-c^{2} x^{2}+1}\, \left (6 i \sqrt {-c^{2} x^{2}+1}\, \arcsin \left (c x \right ) c^{2} x^{2}+6 \operatorname {dilog}\left (1+i c x +\sqrt {-c^{2} x^{2}+1}\right ) c^{4} x^{4}+6 \operatorname {dilog}\left (i c x +\sqrt {-c^{2} x^{2}+1}\right ) c^{4} x^{4}+14 \arctan \left (i c x +\sqrt {-c^{2} x^{2}+1}\right ) c^{4} x^{4}-i x^{3} c^{3}-8 i \sqrt {-c^{2} x^{2}+1}\, \arcsin \left (c x \right )+i c x -12 \operatorname {dilog}\left (1+i c x +\sqrt {-c^{2} x^{2}+1}\right ) c^{2} x^{2}-12 \operatorname {dilog}\left (i c x +\sqrt {-c^{2} x^{2}+1}\right ) c^{2} x^{2}-28 \arctan \left (i c x +\sqrt {-c^{2} x^{2}+1}\right ) c^{2} x^{2}+6 i \arcsin \left (c x \right ) \ln \left (1+i c x +\sqrt {-c^{2} x^{2}+1}\right ) x^{4} c^{4}+6 i \arcsin \left (c x \right ) \ln \left (1+i c x +\sqrt {-c^{2} x^{2}+1}\right )-12 i \arcsin \left (c x \right ) \ln \left (1+i c x +\sqrt {-c^{2} x^{2}+1}\right ) x^{2} c^{2}+6 \operatorname {dilog}\left (1+i c x +\sqrt {-c^{2} x^{2}+1}\right )+6 \operatorname {dilog}\left (i c x +\sqrt {-c^{2} x^{2}+1}\right )+14 \arctan \left (i c x +\sqrt {-c^{2} x^{2}+1}\right )\right )}{6 \left (c^{6} x^{6}-3 c^{4} x^{4}+3 c^{2} x^{2}-1\right ) d^{3}}\)	\(512\)

input

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

output

1/3*a/d/(-c^2*d*x^2+d)^(3/2)+a/d^2/(-c^2*d*x^2+d)^(1/2)-a/d^(5/2)*ln((2*d+ 
2*d^(1/2)*(-c^2*d*x^2+d)^(1/2))/x)-1/6*I*b*(-d*(c^2*x^2-1))^(1/2)*(-c^2*x^ 
2+1)^(1/2)*(6*I*arcsin(c*x)*(-c^2*x^2+1)^(1/2)*x^2*c^2+6*dilog(1+I*c*x+(-c 
^2*x^2+1)^(1/2))*c^4*x^4+6*dilog(I*c*x+(-c^2*x^2+1)^(1/2))*c^4*x^4+14*arct 
an(I*c*x+(-c^2*x^2+1)^(1/2))*c^4*x^4-I*x^3*c^3-8*I*(-c^2*x^2+1)^(1/2)*arcs 
in(c*x)+I*c*x-12*dilog(1+I*c*x+(-c^2*x^2+1)^(1/2))*c^2*x^2-12*dilog(I*c*x+ 
(-c^2*x^2+1)^(1/2))*c^2*x^2-28*arctan(I*c*x+(-c^2*x^2+1)^(1/2))*c^2*x^2+6* 
I*arcsin(c*x)*ln(1+I*c*x+(-c^2*x^2+1)^(1/2))*x^4*c^4+6*I*arcsin(c*x)*ln(1+ 
I*c*x+(-c^2*x^2+1)^(1/2))-12*I*arcsin(c*x)*ln(1+I*c*x+(-c^2*x^2+1)^(1/2))* 
x^2*c^2+6*dilog(1+I*c*x+(-c^2*x^2+1)^(1/2))+6*dilog(I*c*x+(-c^2*x^2+1)^(1/ 
2))+14*arctan(I*c*x+(-c^2*x^2+1)^(1/2)))/(c^6*x^6-3*c^4*x^4+3*c^2*x^2-1)/d 
^3
 

3.2.36.5 Fricas [F]

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

input

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

output

integral(-sqrt(-c^2*d*x^2 + d)*(b*arcsin(c*x) + a)/(c^6*d^3*x^7 - 3*c^4*d^ 
3*x^5 + 3*c^2*d^3*x^3 - d^3*x), x)
 

3.2.36.6 Sympy [F]

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

input

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

output

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

3.2.36.7 Maxima [F]

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

input

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

output

-1/3*a*(3*log(2*sqrt(-c^2*d*x^2 + d)*sqrt(d)/abs(x) + 2*d/abs(x))/d^(5/2) 
- 3/(sqrt(-c^2*d*x^2 + d)*d^2) - 1/((-c^2*d*x^2 + d)^(3/2)*d)) + b*integra 
te(arctan2(c*x, sqrt(c*x + 1)*sqrt(-c*x + 1))/((c^4*d^2*x^5 - 2*c^2*d^2*x^ 
3 + d^2*x)*sqrt(c*x + 1)*sqrt(-c*x + 1)), x)/sqrt(d)
 

3.2.36.8 Giac [F]

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

input

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

output

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

3.2.36.9 Mupad [F(-1)]

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

input

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

output

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