\(\int \frac {(d-c^2 d x^2)^{3/2} (a+b \arcsin (c x))}{x^4} \, dx\) [72]

Optimal result

Integrand size = 27, antiderivative size = 191 \[ \int \frac {\left (d-c^2 d x^2\right )^{3/2} (a+b \arcsin (c x))}{x^4} \, dx=-\frac {b c d \sqrt {d-c^2 d x^2}}{6 x^2 \sqrt {1-c^2 x^2}}+\frac {c^2 d \sqrt {d-c^2 d x^2} (a+b \arcsin (c x))}{x}-\frac {\left (d-c^2 d x^2\right )^{3/2} (a+b \arcsin (c x))}{3 x^3}+\frac {c^3 d \sqrt {d-c^2 d x^2} (a+b \arcsin (c x))^2}{2 b \sqrt {1-c^2 x^2}}-\frac {4 b c^3 d \sqrt {d-c^2 d x^2} \log (x)}{3 \sqrt {1-c^2 x^2}} \] Output:

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

Mathematica [A] (verified)

Time = 0.56 (sec) , antiderivative size = 211, normalized size of antiderivative = 1.10 \[ \int \frac {\left (d-c^2 d x^2\right )^{3/2} (a+b \arcsin (c x))}{x^4} \, dx=\frac {b d \left (-1+4 c^2 x^2\right ) \sqrt {d-c^2 d x^2} \arcsin (c x)}{3 x^3}+\frac {b c^3 d \sqrt {d-c^2 d x^2} \arcsin (c x)^2}{2 \sqrt {1-c^2 x^2}}-a c^3 d^{3/2} \arctan \left (\frac {c x \sqrt {d-c^2 d x^2}}{\sqrt {d} \left (-1+c^2 x^2\right )}\right )-\frac {d \sqrt {d-c^2 d x^2} \left (b c x+2 a \left (1-4 c^2 x^2\right ) \sqrt {1-c^2 x^2}+8 b c^3 x^3 \log (c x)\right )}{6 x^3 \sqrt {1-c^2 x^2}} \] Input:

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

Output:

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

Rubi [A] (verified)

Time = 0.79 (sec) , antiderivative size = 198, normalized size of antiderivative = 1.04, number of steps used = 6, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.222, Rules used = {5200, 244, 2009, 5196, 14, 5152}

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.

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

\(\Big \downarrow \) 5200

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

\(\Big \downarrow \) 244

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

\(\Big \downarrow \) 2009

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

\(\Big \downarrow \) 5196

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

\(\Big \downarrow \) 14

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

\(\Big \downarrow \) 5152

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

Input:

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

Output:

-1/3*((d - c^2*d*x^2)^(3/2)*(a + b*ArcSin[c*x]))/x^3 + (b*c*d*Sqrt[d - c^2 
*d*x^2]*(-1/2*1/x^2 - c^2*Log[x]))/(3*Sqrt[1 - c^2*x^2]) - c^2*d*(-((Sqrt[ 
d - c^2*d*x^2]*(a + b*ArcSin[c*x]))/x) - (c*Sqrt[d - c^2*d*x^2]*(a + b*Arc 
Sin[c*x])^2)/(2*b*Sqrt[1 - c^2*x^2]) + (b*c*Sqrt[d - c^2*d*x^2]*Log[x])/Sq 
rt[1 - c^2*x^2])
 

Defintions of rubi rules used

rule 14

Int[(a_.)/(x_), x_Symbol] :> Simp[a*Log[x], x] /; FreeQ[a, x]
 

rule 244

Int[((c_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^2)^(p_.), x_Symbol] :> Int[Expand 
Integrand[(c*x)^m*(a + b*x^2)^p, x], x] /; FreeQ[{a, b, c, m}, x] && IGtQ[p 
, 0]
 

rule 2009

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

rule 5152

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

rule 5196

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

rule 5200

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*((a + b*ArcS 
in[c*x])^n/(f*(m + 1))), x] + (-Simp[2*e*(p/(f^2*(m + 1)))   Int[(f*x)^(m + 
 2)*(d + e*x^2)^(p - 1)*(a + b*ArcSin[c*x])^n, x], x] - Simp[b*c*(n/(f*(m + 
 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} 
, x] && EqQ[c^2*d + e, 0] && GtQ[n, 0] && GtQ[p, 0] && LtQ[m, -1]
 

Maple [C] (verified)

Result contains complex when optimal does not.

Time = 0.53 (sec) , antiderivative size = 276, normalized size of antiderivative = 1.45

method	result	size
default	\(-\frac {a \left (-c^{2} d \,x^{2}+d \right )^{\frac {5}{2}}}{3 d \,x^{3}}+\frac {2 a \,c^{2} \left (-c^{2} d \,x^{2}+d \right )^{\frac {5}{2}}}{3 d x}+\frac {2 a \,c^{4} x \left (-c^{2} d \,x^{2}+d \right )^{\frac {3}{2}}}{3}+a \,c^{4} d x \sqrt {-c^{2} d \,x^{2}+d}+\frac {a \,c^{4} d^{2} \arctan \left (\frac {\sqrt {c^{2} d}\, x}{\sqrt {-c^{2} d \,x^{2}+d}}\right )}{\sqrt {c^{2} d}}-\frac {b \sqrt {-d \left (c^{2} x^{2}-1\right )}\, \sqrt {-c^{2} x^{2}+1}\, \left (3 \arcsin \left (c x \right )^{2} c^{3} x^{3}+8 i \arcsin \left (c x \right ) x^{3} c^{3}-8 \ln \left (\left (i c x +\sqrt {-c^{2} x^{2}+1}\right )^{2}-1\right ) x^{3} c^{3}+8 \arcsin \left (c x \right ) \sqrt {-c^{2} x^{2}+1}\, c^{2} x^{2}-2 \arcsin \left (c x \right ) \sqrt {-c^{2} x^{2}+1}-c x \right ) d}{6 \left (c^{2} x^{2}-1\right ) x^{3}}\)	\(276\)
parts	\(-\frac {a \left (-c^{2} d \,x^{2}+d \right )^{\frac {5}{2}}}{3 d \,x^{3}}+\frac {2 a \,c^{2} \left (-c^{2} d \,x^{2}+d \right )^{\frac {5}{2}}}{3 d x}+\frac {2 a \,c^{4} x \left (-c^{2} d \,x^{2}+d \right )^{\frac {3}{2}}}{3}+a \,c^{4} d x \sqrt {-c^{2} d \,x^{2}+d}+\frac {a \,c^{4} d^{2} \arctan \left (\frac {\sqrt {c^{2} d}\, x}{\sqrt {-c^{2} d \,x^{2}+d}}\right )}{\sqrt {c^{2} d}}-\frac {b \sqrt {-d \left (c^{2} x^{2}-1\right )}\, \sqrt {-c^{2} x^{2}+1}\, \left (3 \arcsin \left (c x \right )^{2} c^{3} x^{3}+8 i \arcsin \left (c x \right ) x^{3} c^{3}-8 \ln \left (\left (i c x +\sqrt {-c^{2} x^{2}+1}\right )^{2}-1\right ) x^{3} c^{3}+8 \arcsin \left (c x \right ) \sqrt {-c^{2} x^{2}+1}\, c^{2} x^{2}-2 \arcsin \left (c x \right ) \sqrt {-c^{2} x^{2}+1}-c x \right ) d}{6 \left (c^{2} x^{2}-1\right ) x^{3}}\)	\(276\)

Input:

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

Output:

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

Fricas [F]

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

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

Output:

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

Sympy [F]

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

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

Output:

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

Maxima [F]

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

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

Output:

-b*sqrt(d)*integrate((c^2*d*x^2 - d)*sqrt(c*x + 1)*sqrt(-c*x + 1)*arctan2( 
c*x, sqrt(c*x + 1)*sqrt(-c*x + 1))/x^4, x) + 1/3*(3*sqrt(-c^2*d*x^2 + d)*c 
^4*d*x + 3*c^3*d^(3/2)*arcsin(c*x) + 2*(-c^2*d*x^2 + d)^(3/2)*c^2/x - (-c^ 
2*d*x^2 + d)^(5/2)/(d*x^3))*a
 

Giac [F(-2)]

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

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

Output:

Exception raised: TypeError >> an error occurred running a Giac command:IN 
PUT:sage2:=int(sage0,sageVARx):;OUTPUT:sym2poly/r2sym(const gen & e,const 
index_m & i,const vecteur & l) Error: Bad Argument Value
 

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

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

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

Output:

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