\(\int \frac {a+b \arcsin (c x)}{x^4 (d-c^2 d x^2)^{3/2}} \, dx\) [126]

Optimal result

Integrand size = 27, antiderivative size = 234 \[ \int \frac {a+b \arcsin (c x)}{x^4 \left (d-c^2 d x^2\right )^{3/2}} \, dx=-\frac {b c \sqrt {1-c^2 x^2}}{6 d x^2 \sqrt {d-c^2 d x^2}}+\frac {a+b \arcsin (c x)}{d x^3 \sqrt {d-c^2 d x^2}}-\frac {4 \sqrt {d-c^2 d x^2} (a+b \arcsin (c x))}{3 d^2 x^3}-\frac {8 c^2 \sqrt {d-c^2 d x^2} (a+b \arcsin (c x))}{3 d^2 x}+\frac {5 b c^3 \sqrt {1-c^2 x^2} \log (x)}{3 d \sqrt {d-c^2 d x^2}}+\frac {b c^3 \sqrt {1-c^2 x^2} \log \left (1-c^2 x^2\right )}{2 d \sqrt {d-c^2 d x^2}} \] Output:

-1/6*b*c*(-c^2*x^2+1)^(1/2)/d/x^2/(-c^2*d*x^2+d)^(1/2)+(a+b*arcsin(c*x))/d 
/x^3/(-c^2*d*x^2+d)^(1/2)-4/3*(-c^2*d*x^2+d)^(1/2)*(a+b*arcsin(c*x))/d^2/x 
^3-8/3*c^2*(-c^2*d*x^2+d)^(1/2)*(a+b*arcsin(c*x))/d^2/x+5/3*b*c^3*(-c^2*x^ 
2+1)^(1/2)*ln(x)/d/(-c^2*d*x^2+d)^(1/2)+1/2*b*c^3*(-c^2*x^2+1)^(1/2)*ln(-c 
^2*x^2+1)/d/(-c^2*d*x^2+d)^(1/2)
 

Mathematica [A] (verified)

Time = 0.44 (sec) , antiderivative size = 224, normalized size of antiderivative = 0.96 \[ \int \frac {a+b \arcsin (c x)}{x^4 \left (d-c^2 d x^2\right )^{3/2}} \, dx=-\frac {\sqrt {d-c^2 d x^2} \left (b c x-b c^3 x^3+2 a \sqrt {1-c^2 x^2}+8 a c^2 x^2 \sqrt {1-c^2 x^2}-16 a c^4 x^4 \sqrt {1-c^2 x^2}-2 b \sqrt {1-c^2 x^2} \left (-1-4 c^2 x^2+8 c^4 x^4\right ) \arcsin (c x)-5 b c^3 x^3 \left (-1+c^2 x^2\right ) \log \left (1-\frac {1}{c^2 x^2}\right )-8 b c^3 x^3 \log \left (1-c^2 x^2\right )+8 b c^5 x^5 \log \left (1-c^2 x^2\right )\right )}{6 d^2 x^3 \left (1-c^2 x^2\right )^{3/2}} \] Input:

Integrate[(a + b*ArcSin[c*x])/(x^4*(d - c^2*d*x^2)^(3/2)),x]
 

Output:

-1/6*(Sqrt[d - c^2*d*x^2]*(b*c*x - b*c^3*x^3 + 2*a*Sqrt[1 - c^2*x^2] + 8*a 
*c^2*x^2*Sqrt[1 - c^2*x^2] - 16*a*c^4*x^4*Sqrt[1 - c^2*x^2] - 2*b*Sqrt[1 - 
 c^2*x^2]*(-1 - 4*c^2*x^2 + 8*c^4*x^4)*ArcSin[c*x] - 5*b*c^3*x^3*(-1 + c^2 
*x^2)*Log[1 - 1/(c^2*x^2)] - 8*b*c^3*x^3*Log[1 - c^2*x^2] + 8*b*c^5*x^5*Lo 
g[1 - c^2*x^2]))/(d^2*x^3*(1 - c^2*x^2)^(3/2))
 

Rubi [A] (verified)

Time = 0.52 (sec) , antiderivative size = 173, normalized size of antiderivative = 0.74, number of steps used = 6, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.185, Rules used = {5194, 27, 1578, 1195, 2009}

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.

\(\displaystyle \int \frac {a+b \arcsin (c x)}{x^4 \left (d-c^2 d x^2\right )^{3/2}} \, dx\)

\(\Big \downarrow \) 5194

\(\displaystyle -\frac {b c \sqrt {d-c^2 d x^2} \int -\frac {-8 c^4 x^4+4 c^2 x^2+1}{3 d^2 x^3 \left (1-c^2 x^2\right )}dx}{\sqrt {1-c^2 x^2}}-\frac {4 c^2 (a+b \arcsin (c x))}{3 d x \sqrt {d-c^2 d x^2}}-\frac {a+b \arcsin (c x)}{3 d x^3 \sqrt {d-c^2 d x^2}}+\frac {8 c^4 x (a+b \arcsin (c x))}{3 d \sqrt {d-c^2 d x^2}}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {b c \sqrt {d-c^2 d x^2} \int \frac {-8 c^4 x^4+4 c^2 x^2+1}{x^3 \left (1-c^2 x^2\right )}dx}{3 d^2 \sqrt {1-c^2 x^2}}-\frac {4 c^2 (a+b \arcsin (c x))}{3 d x \sqrt {d-c^2 d x^2}}-\frac {a+b \arcsin (c x)}{3 d x^3 \sqrt {d-c^2 d x^2}}+\frac {8 c^4 x (a+b \arcsin (c x))}{3 d \sqrt {d-c^2 d x^2}}\)

\(\Big \downarrow \) 1578

\(\displaystyle \frac {b c \sqrt {d-c^2 d x^2} \int \frac {-8 c^4 x^4+4 c^2 x^2+1}{x^4 \left (1-c^2 x^2\right )}dx^2}{6 d^2 \sqrt {1-c^2 x^2}}-\frac {4 c^2 (a+b \arcsin (c x))}{3 d x \sqrt {d-c^2 d x^2}}-\frac {a+b \arcsin (c x)}{3 d x^3 \sqrt {d-c^2 d x^2}}+\frac {8 c^4 x (a+b \arcsin (c x))}{3 d \sqrt {d-c^2 d x^2}}\)

\(\Big \downarrow \) 1195

\(\displaystyle \frac {b c \sqrt {d-c^2 d x^2} \int \left (\frac {3 c^4}{c^2 x^2-1}+\frac {5 c^2}{x^2}+\frac {1}{x^4}\right )dx^2}{6 d^2 \sqrt {1-c^2 x^2}}-\frac {4 c^2 (a+b \arcsin (c x))}{3 d x \sqrt {d-c^2 d x^2}}-\frac {a+b \arcsin (c x)}{3 d x^3 \sqrt {d-c^2 d x^2}}+\frac {8 c^4 x (a+b \arcsin (c x))}{3 d \sqrt {d-c^2 d x^2}}\)

\(\Big \downarrow \) 2009

\(\displaystyle -\frac {4 c^2 (a+b \arcsin (c x))}{3 d x \sqrt {d-c^2 d x^2}}-\frac {a+b \arcsin (c x)}{3 d x^3 \sqrt {d-c^2 d x^2}}+\frac {8 c^4 x (a+b \arcsin (c x))}{3 d \sqrt {d-c^2 d x^2}}+\frac {b c \sqrt {d-c^2 d x^2} \left (5 c^2 \log \left (x^2\right )+3 c^2 \log \left (1-c^2 x^2\right )-\frac {1}{x^2}\right )}{6 d^2 \sqrt {1-c^2 x^2}}\)

Input:

Int[(a + b*ArcSin[c*x])/(x^4*(d - c^2*d*x^2)^(3/2)),x]
 

Output:

-1/3*(a + b*ArcSin[c*x])/(d*x^3*Sqrt[d - c^2*d*x^2]) - (4*c^2*(a + b*ArcSi 
n[c*x]))/(3*d*x*Sqrt[d - c^2*d*x^2]) + (8*c^4*x*(a + b*ArcSin[c*x]))/(3*d* 
Sqrt[d - c^2*d*x^2]) + (b*c*Sqrt[d - c^2*d*x^2]*(-x^(-2) + 5*c^2*Log[x^2] 
+ 3*c^2*Log[1 - c^2*x^2]))/(6*d^2*Sqrt[1 - c^2*x^2])
 

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 1195

Int[((d_.) + (e_.)*(x_))^(m_.)*((f_.) + (g_.)*(x_))^(n_.)*((a_.) + (b_.)*(x 
_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> Int[ExpandIntegrand[(d + e*x)^m*(f + 
 g*x)^n*(a + b*x + c*x^2)^p, x], x] /; FreeQ[{a, b, c, d, e, f, g, m, n}, x 
] && IGtQ[p, 0]
 

rule 1578

Int[(x_)^(m_.)*((d_) + (e_.)*(x_)^2)^(q_.)*((a_) + (b_.)*(x_)^2 + (c_.)*(x_ 
)^4)^(p_.), x_Symbol] :> Simp[1/2   Subst[Int[x^((m - 1)/2)*(d + e*x)^q*(a 
+ b*x + c*x^2)^p, x], x, x^2], x] /; FreeQ[{a, b, c, d, e, p, q}, x] && Int 
egerQ[(m - 1)/2]
 

rule 2009

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

rule 5194

Int[((a_.) + ArcSin[(c_.)*(x_)]*(b_.))*(x_)^(m_)*((d_) + (e_.)*(x_)^2)^(p_) 
, x_Symbol] :> With[{u = IntHide[x^m*(d + e*x^2)^p, x]}, Simp[(a + b*ArcSin 
[c*x])   u, x] - Simp[b*c*Simp[Sqrt[d + e*x^2]/Sqrt[1 - c^2*x^2]]   Int[Sim 
plifyIntegrand[u/Sqrt[d + e*x^2], x], x], x]] /; FreeQ[{a, b, c, d, e}, x] 
&& EqQ[c^2*d + e, 0] && IntegerQ[p - 1/2] && NeQ[p, -2^(-1)] && (IGtQ[(m + 
1)/2, 0] || ILtQ[(m + 2*p + 3)/2, 0])
 

Maple [C] (verified)

Result contains complex when optimal does not.

Time = 0.57 (sec) , antiderivative size = 1048, normalized size of antiderivative = 4.48

method	result	size
default	\(\text {Expression too large to display}\)	\(1048\)
parts	\(\text {Expression too large to display}\)	\(1048\)

Input:

int((a+b*arcsin(c*x))/x^4/(-c^2*d*x^2+d)^(3/2),x,method=_RETURNVERBOSE)
 

Output:

a*(-1/3/d/x^3/(-c^2*d*x^2+d)^(1/2)+4/3*c^2*(-1/d/x/(-c^2*d*x^2+d)^(1/2)+2* 
c^2/d*x/(-c^2*d*x^2+d)^(1/2)))+4/3*I*b*(-d*(c^2*x^2-1))^(1/2)/(8*c^4*x^4-7 
*c^2*x^2-1)/d^2*x*c^4-16/3*I*b*(-d*(c^2*x^2-1))^(1/2)/(8*c^4*x^4-7*c^2*x^2 
-1)/d^2*x^3*(-c^2*x^2+1)*c^6-8/3*I*b*(-d*(c^2*x^2-1))^(1/2)/(8*c^4*x^4-7*c 
^2*x^2-1)/d^2*(-c^2*x^2+1)^(1/2)*arcsin(c*x)*c^3+32/3*I*b*(-d*(c^2*x^2-1)) 
^(1/2)/(8*c^4*x^4-7*c^2*x^2-1)/d^2*x^5*(-c^2*x^2+1)*c^8+4*I*b*(-d*(c^2*x^2 
-1))^(1/2)/(8*c^4*x^4-7*c^2*x^2-1)/d^2*x^3*c^6+32/3*I*b*(-d*(c^2*x^2-1))^( 
1/2)/(8*c^4*x^4-7*c^2*x^2-1)/d^2*x^7*c^10+16/3*I*b*(-d*(c^2*x^2-1))^(1/2)* 
(-c^2*x^2+1)^(1/2)/d^2/(c^2*x^2-1)*arcsin(c*x)*c^3-64/3*b*(-d*(c^2*x^2-1)) 
^(1/2)/(8*c^4*x^4-7*c^2*x^2-1)/d^2*x^3*arcsin(c*x)*c^6-4/3*I*b*(-d*(c^2*x^ 
2-1))^(1/2)/(8*c^4*x^4-7*c^2*x^2-1)/d^2*x*(-c^2*x^2+1)*c^4-16*I*b*(-d*(c^2 
*x^2-1))^(1/2)/(8*c^4*x^4-7*c^2*x^2-1)/d^2*x^5*c^8-64/3*I*b*(-d*(c^2*x^2-1 
))^(1/2)/(8*c^4*x^4-7*c^2*x^2-1)/d^2*x^2*(-c^2*x^2+1)^(1/2)*arcsin(c*x)*c^ 
5+8*b*(-d*(c^2*x^2-1))^(1/2)/(8*c^4*x^4-7*c^2*x^2-1)/d^2*x*arcsin(c*x)*c^4 
+4/3*b*(-d*(c^2*x^2-1))^(1/2)/(8*c^4*x^4-7*c^2*x^2-1)/d^2*c^3*(-c^2*x^2+1) 
^(1/2)+4*b*(-d*(c^2*x^2-1))^(1/2)/(8*c^4*x^4-7*c^2*x^2-1)/d^2/x*arcsin(c*x 
)*c^2+1/6*b*(-d*(c^2*x^2-1))^(1/2)/(8*c^4*x^4-7*c^2*x^2-1)/d^2/x^2*(-c^2*x 
^2+1)^(1/2)*c+1/3*b*(-d*(c^2*x^2-1))^(1/2)/(8*c^4*x^4-7*c^2*x^2-1)/d^2/x^3 
*arcsin(c*x)-b*(-d*(c^2*x^2-1))^(1/2)*(-c^2*x^2+1)^(1/2)/d^2/(c^2*x^2-1)*l 
n(1+(I*c*x+(-c^2*x^2+1)^(1/2))^2)*c^3-5/3*b*(-d*(c^2*x^2-1))^(1/2)*(-c^...
 

Fricas [F]

\[ \int \frac {a+b \arcsin (c x)}{x^4 \left (d-c^2 d x^2\right )^{3/2}} \, dx=\int { \frac {b \arcsin \left (c x\right ) + a}{{\left (-c^{2} d x^{2} + d\right )}^{\frac {3}{2}} x^{4}} \,d x } \] Input:

integrate((a+b*arcsin(c*x))/x^4/(-c^2*d*x^2+d)^(3/2),x, algorithm="fricas" 
)
 

Output:

integral(sqrt(-c^2*d*x^2 + d)*(b*arcsin(c*x) + a)/(c^4*d^2*x^8 - 2*c^2*d^2 
*x^6 + d^2*x^4), x)
                                                                                    
                                                                                    
 

Sympy [F]

\[ \int \frac {a+b \arcsin (c x)}{x^4 \left (d-c^2 d x^2\right )^{3/2}} \, dx=\int \frac {a + b \operatorname {asin}{\left (c x \right )}}{x^{4} \left (- d \left (c x - 1\right ) \left (c x + 1\right )\right )^{\frac {3}{2}}}\, dx \] Input:

integrate((a+b*asin(c*x))/x**4/(-c**2*d*x**2+d)**(3/2),x)
 

Output:

Integral((a + b*asin(c*x))/(x**4*(-d*(c*x - 1)*(c*x + 1))**(3/2)), x)
 

Maxima [F]

\[ \int \frac {a+b \arcsin (c x)}{x^4 \left (d-c^2 d x^2\right )^{3/2}} \, dx=\int { \frac {b \arcsin \left (c x\right ) + a}{{\left (-c^{2} d x^{2} + d\right )}^{\frac {3}{2}} x^{4}} \,d x } \] Input:

integrate((a+b*arcsin(c*x))/x^4/(-c^2*d*x^2+d)^(3/2),x, algorithm="maxima" 
)
 

Output:

1/3*(8*c^4*x/(sqrt(-c^2*d*x^2 + d)*d) - 4*c^2/(sqrt(-c^2*d*x^2 + d)*d*x) - 
 1/(sqrt(-c^2*d*x^2 + d)*d*x^3))*a - b*integrate(arctan2(c*x, sqrt(c*x + 1 
)*sqrt(-c*x + 1))/((c^2*d*x^6 - d*x^4)*sqrt(c*x + 1)*sqrt(-c*x + 1)), x)/s 
qrt(d)
 

Giac [F(-2)]

Exception generated. \[ \int \frac {a+b \arcsin (c x)}{x^4 \left (d-c^2 d x^2\right )^{3/2}} \, dx=\text {Exception raised: RuntimeError} \] Input:

integrate((a+b*arcsin(c*x))/x^4/(-c^2*d*x^2+d)^(3/2),x, algorithm="giac")
 

Output:

Exception raised: RuntimeError >> an error occurred running a Giac command 
:INPUT:sage2OUTPUT:sym2poly/r2sym(const gen & e,const index_m & i,const ve 
cteur & l) Error: Bad Argument Value
 

Mupad [F(-1)]

Timed out. \[ \int \frac {a+b \arcsin (c x)}{x^4 \left (d-c^2 d x^2\right )^{3/2}} \, dx=\int \frac {a+b\,\mathrm {asin}\left (c\,x\right )}{x^4\,{\left (d-c^2\,d\,x^2\right )}^{3/2}} \,d x \] Input:

int((a + b*asin(c*x))/(x^4*(d - c^2*d*x^2)^(3/2)),x)
 

Output:

int((a + b*asin(c*x))/(x^4*(d - c^2*d*x^2)^(3/2)), x)
 

Reduce [F]

\[ \int \frac {a+b \arcsin (c x)}{x^4 \left (d-c^2 d x^2\right )^{3/2}} \, dx=\frac {-3 \sqrt {-c^{2} x^{2}+1}\, \left (\int \frac {\mathit {asin} \left (c x \right )}{\sqrt {-c^{2} x^{2}+1}\, c^{2} x^{6}-\sqrt {-c^{2} x^{2}+1}\, x^{4}}d x \right ) b \,x^{3}+8 a \,c^{4} x^{4}-4 a \,c^{2} x^{2}-a}{3 \sqrt {d}\, \sqrt {-c^{2} x^{2}+1}\, d \,x^{3}} \] Input:

int((a+b*asin(c*x))/x^4/(-c^2*d*x^2+d)^(3/2),x)
 

Output:

( - 3*sqrt( - c**2*x**2 + 1)*int(asin(c*x)/(sqrt( - c**2*x**2 + 1)*c**2*x* 
*6 - sqrt( - c**2*x**2 + 1)*x**4),x)*b*x**3 + 8*a*c**4*x**4 - 4*a*c**2*x** 
2 - a)/(3*sqrt(d)*sqrt( - c**2*x**2 + 1)*d*x**3)