\(\int \frac {(d-c^2 d x^2)^{5/2} (a+b \arcsin (c x))}{x^2} \, dx\) [87]

Optimal result

Integrand size = 27, antiderivative size = 306 \[ \int \frac {\left (d-c^2 d x^2\right )^{5/2} (a+b \arcsin (c x))}{x^2} \, dx=-\frac {b c^3 d^2 x^2 \sqrt {d-c^2 d x^2}}{16 \sqrt {1-c^2 x^2}}+\frac {b c^5 d^2 x^4 \sqrt {d-c^2 d x^2}}{4 \sqrt {1-c^2 x^2}}-\frac {5}{16} b c d^2 \left (1-c^2 x^2\right )^{3/2} \sqrt {d-c^2 d x^2}-\frac {15}{8} c^2 d^2 x \sqrt {d-c^2 d x^2} (a+b \arcsin (c x))-\frac {5}{4} c^2 d x \left (d-c^2 d x^2\right )^{3/2} (a+b \arcsin (c x))-\frac {\left (d-c^2 d x^2\right )^{5/2} (a+b \arcsin (c x))}{x}-\frac {15 c d^2 \sqrt {d-c^2 d x^2} (a+b \arcsin (c x))^2}{16 b \sqrt {1-c^2 x^2}}+\frac {b c d^2 \sqrt {d-c^2 d x^2} \log (x)}{\sqrt {1-c^2 x^2}} \] Output:

-1/16*b*c^3*d^2*x^2*(-c^2*d*x^2+d)^(1/2)/(-c^2*x^2+1)^(1/2)+1/4*b*c^5*d^2* 
x^4*(-c^2*d*x^2+d)^(1/2)/(-c^2*x^2+1)^(1/2)-5/16*b*c*d^2*(-c^2*x^2+1)^(3/2 
)*(-c^2*d*x^2+d)^(1/2)-15/8*c^2*d^2*x*(-c^2*d*x^2+d)^(1/2)*(a+b*arcsin(c*x 
))-5/4*c^2*d*x*(-c^2*d*x^2+d)^(3/2)*(a+b*arcsin(c*x))-(-c^2*d*x^2+d)^(5/2) 
*(a+b*arcsin(c*x))/x-15/16*c*d^2*(-c^2*d*x^2+d)^(1/2)*(a+b*arcsin(c*x))^2/ 
b/(-c^2*x^2+1)^(1/2)+b*c*d^2*(-c^2*d*x^2+d)^(1/2)*ln(x)/(-c^2*x^2+1)^(1/2)
 

Mathematica [A] (verified)

Time = 0.67 (sec) , antiderivative size = 257, normalized size of antiderivative = 0.84 \[ \int \frac {\left (d-c^2 d x^2\right )^{5/2} (a+b \arcsin (c x))}{x^2} \, dx=\frac {d^2 \left (-120 b c x \sqrt {d-c^2 d x^2} \arcsin (c x)^2+240 a c \sqrt {d} x \sqrt {1-c^2 x^2} \arctan \left (\frac {c x \sqrt {d-c^2 d x^2}}{\sqrt {d} \left (-1+c^2 x^2\right )}\right )+\sqrt {d-c^2 d x^2} \left (-32 b c x \cos (2 \arcsin (c x))-b c x \cos (4 \arcsin (c x))+16 \left (a \sqrt {1-c^2 x^2} \left (-8-9 c^2 x^2+2 c^4 x^4\right )+8 b c x \log (c x)\right )\right )-4 b \sqrt {d-c^2 d x^2} \arcsin (c x) \left (32 \sqrt {1-c^2 x^2}+16 c x \sin (2 \arcsin (c x))+c x \sin (4 \arcsin (c x))\right )\right )}{128 x \sqrt {1-c^2 x^2}} \] Input:

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

Output:

(d^2*(-120*b*c*x*Sqrt[d - c^2*d*x^2]*ArcSin[c*x]^2 + 240*a*c*Sqrt[d]*x*Sqr 
t[1 - c^2*x^2]*ArcTan[(c*x*Sqrt[d - c^2*d*x^2])/(Sqrt[d]*(-1 + c^2*x^2))] 
+ Sqrt[d - c^2*d*x^2]*(-32*b*c*x*Cos[2*ArcSin[c*x]] - b*c*x*Cos[4*ArcSin[c 
*x]] + 16*(a*Sqrt[1 - c^2*x^2]*(-8 - 9*c^2*x^2 + 2*c^4*x^4) + 8*b*c*x*Log[ 
c*x])) - 4*b*Sqrt[d - c^2*d*x^2]*ArcSin[c*x]*(32*Sqrt[1 - c^2*x^2] + 16*c* 
x*Sin[2*ArcSin[c*x]] + c*x*Sin[4*ArcSin[c*x]])))/(128*x*Sqrt[1 - c^2*x^2])
 

Rubi [A] (verified)

Time = 0.95 (sec) , antiderivative size = 300, normalized size of antiderivative = 0.98, number of steps used = 11, number of rules used = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.370, Rules used = {5200, 243, 49, 2009, 5158, 244, 2009, 5156, 15, 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.

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

\(\Big \downarrow \) 5200

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

\(\Big \downarrow \) 243

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

\(\Big \downarrow \) 49

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

\(\Big \downarrow \) 2009

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

\(\Big \downarrow \) 5158

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

\(\Big \downarrow \) 244

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

\(\Big \downarrow \) 2009

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

\(\Big \downarrow \) 5156

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

\(\Big \downarrow \) 15

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

\(\Big \downarrow \) 5152

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

Input:

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

Output:

-(((d - c^2*d*x^2)^(5/2)*(a + b*ArcSin[c*x]))/x) - 5*c^2*d*(-1/4*(b*c*d*Sq 
rt[d - c^2*d*x^2]*(x^2/2 - (c^2*x^4)/4))/Sqrt[1 - c^2*x^2] + (x*(d - c^2*d 
*x^2)^(3/2)*(a + b*ArcSin[c*x]))/4 + (3*d*(-1/4*(b*c*x^2*Sqrt[d - c^2*d*x^ 
2])/Sqrt[1 - c^2*x^2] + (x*Sqrt[d - c^2*d*x^2]*(a + b*ArcSin[c*x]))/2 + (S 
qrt[d - c^2*d*x^2]*(a + b*ArcSin[c*x])^2)/(4*b*c*Sqrt[1 - c^2*x^2])))/4) + 
 (b*c*d^2*Sqrt[d - c^2*d*x^2]*(-2*c^2*x^2 + (c^4*x^4)/2 + Log[x^2]))/(2*Sq 
rt[1 - c^2*x^2])
 

Defintions of rubi rules used

rule 15

Int[(a_.)*(x_)^(m_.), x_Symbol] :> Simp[a*(x^(m + 1)/(m + 1)), x] /; FreeQ[ 
{a, m}, x] && NeQ[m, -1]
 

rule 49

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

rule 243

Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> Simp[1/2   Subst[In 
t[x^((m - 1)/2)*(a + b*x)^p, x], x, x^2], x] /; FreeQ[{a, b, m, p}, x] && I 
ntegerQ[(m - 1)/2]
 

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 5156

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

rule 5158

Int[((a_.) + ArcSin[(c_.)*(x_)]*(b_.))^(n_.)*((d_) + (e_.)*(x_)^2)^(p_.), x 
_Symbol] :> Simp[x*(d + e*x^2)^p*((a + b*ArcSin[c*x])^n/(2*p + 1)), x] + (S 
imp[2*d*(p/(2*p + 1))   Int[(d + e*x^2)^(p - 1)*(a + b*ArcSin[c*x])^n, x], 
x] - Simp[b*c*(n/(2*p + 1))*Simp[(d + e*x^2)^p/(1 - c^2*x^2)^p]   Int[x*(1 
- c^2*x^2)^(p - 1/2)*(a + b*ArcSin[c*x])^(n - 1), x], x]) /; FreeQ[{a, b, c 
, d, e}, x] && EqQ[c^2*d + e, 0] && GtQ[n, 0] && GtQ[p, 0]
 

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 = 0.60 (sec) , antiderivative size = 306, normalized size of antiderivative = 1.00

method	result	size
default	\(-\frac {a \left (-c^{2} d \,x^{2}+d \right )^{\frac {7}{2}}}{d x}-a \,c^{2} x \left (-c^{2} d \,x^{2}+d \right )^{\frac {5}{2}}-\frac {5 a \,c^{2} d x \left (-c^{2} d \,x^{2}+d \right )^{\frac {3}{2}}}{4}-\frac {15 a \,c^{2} d^{2} x \sqrt {-c^{2} d \,x^{2}+d}}{8}-\frac {15 a \,c^{2} d^{3} \arctan \left (\frac {\sqrt {c^{2} d}\, x}{\sqrt {-c^{2} d \,x^{2}+d}}\right )}{8 \sqrt {c^{2} d}}-\frac {b \sqrt {-d \left (c^{2} x^{2}-1\right )}\, \sqrt {-c^{2} x^{2}+1}\, \left (32 \arcsin \left (c x \right ) \sqrt {-c^{2} x^{2}+1}\, c^{4} x^{4}-8 c^{5} x^{5}-144 \arcsin \left (c x \right ) \sqrt {-c^{2} x^{2}+1}\, c^{2} x^{2}+72 c^{3} x^{3}-120 \arcsin \left (c x \right )^{2} c x -128 i \arcsin \left (c x \right ) x c +128 \ln \left (\left (i c x +\sqrt {-c^{2} x^{2}+1}\right )^{2}-1\right ) x c -128 \arcsin \left (c x \right ) \sqrt {-c^{2} x^{2}+1}-33 c x \right ) d^{2}}{128 x \left (c^{2} x^{2}-1\right )}\)	\(306\)
parts	\(-\frac {a \left (-c^{2} d \,x^{2}+d \right )^{\frac {7}{2}}}{d x}-a \,c^{2} x \left (-c^{2} d \,x^{2}+d \right )^{\frac {5}{2}}-\frac {5 a \,c^{2} d x \left (-c^{2} d \,x^{2}+d \right )^{\frac {3}{2}}}{4}-\frac {15 a \,c^{2} d^{2} x \sqrt {-c^{2} d \,x^{2}+d}}{8}-\frac {15 a \,c^{2} d^{3} \arctan \left (\frac {\sqrt {c^{2} d}\, x}{\sqrt {-c^{2} d \,x^{2}+d}}\right )}{8 \sqrt {c^{2} d}}-\frac {b \sqrt {-d \left (c^{2} x^{2}-1\right )}\, \sqrt {-c^{2} x^{2}+1}\, \left (32 \arcsin \left (c x \right ) \sqrt {-c^{2} x^{2}+1}\, c^{4} x^{4}-8 c^{5} x^{5}-144 \arcsin \left (c x \right ) \sqrt {-c^{2} x^{2}+1}\, c^{2} x^{2}+72 c^{3} x^{3}-120 \arcsin \left (c x \right )^{2} c x -128 i \arcsin \left (c x \right ) x c +128 \ln \left (\left (i c x +\sqrt {-c^{2} x^{2}+1}\right )^{2}-1\right ) x c -128 \arcsin \left (c x \right ) \sqrt {-c^{2} x^{2}+1}-33 c x \right ) d^{2}}{128 x \left (c^{2} x^{2}-1\right )}\)	\(306\)

Input:

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

Output:

-a/d/x*(-c^2*d*x^2+d)^(7/2)-a*c^2*x*(-c^2*d*x^2+d)^(5/2)-5/4*a*c^2*d*x*(-c 
^2*d*x^2+d)^(3/2)-15/8*a*c^2*d^2*x*(-c^2*d*x^2+d)^(1/2)-15/8*a*c^2*d^3/(c^ 
2*d)^(1/2)*arctan((c^2*d)^(1/2)*x/(-c^2*d*x^2+d)^(1/2))-1/128*b*(-d*(c^2*x 
^2-1))^(1/2)*(-c^2*x^2+1)^(1/2)/x/(c^2*x^2-1)*(32*arcsin(c*x)*(-c^2*x^2+1) 
^(1/2)*c^4*x^4-8*c^5*x^5-144*arcsin(c*x)*(-c^2*x^2+1)^(1/2)*c^2*x^2+72*c^3 
*x^3-120*arcsin(c*x)^2*c*x-128*I*arcsin(c*x)*x*c+128*ln((I*c*x+(-c^2*x^2+1 
)^(1/2))^2-1)*x*c-128*arcsin(c*x)*(-c^2*x^2+1)^(1/2)-33*c*x)*d^2
 

Fricas [F]

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

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

Output:

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

Sympy [F(-1)]

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

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

Output:

Timed out
 

Maxima [F]

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

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

Output:

b*sqrt(d)*integrate((c^4*d^2*x^4 - 2*c^2*d^2*x^2 + d^2)*sqrt(c*x + 1)*sqrt 
(-c*x + 1)*arctan2(c*x, sqrt(c*x + 1)*sqrt(-c*x + 1))/x^2, x) - 1/8*(10*(- 
c^2*d*x^2 + d)^(3/2)*c^2*d*x + 15*sqrt(-c^2*d*x^2 + d)*c^2*d^2*x + 15*c*d^ 
(5/2)*arcsin(c*x) + 8*(-c^2*d*x^2 + d)^(5/2)/x)*a
 

Giac [F(-2)]

Exception generated. \[ \int \frac {\left (d-c^2 d x^2\right )^{5/2} (a+b \arcsin (c x))}{x^2} \, dx=\text {Exception raised: TypeError} \] Input:

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

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

Output:

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

Reduce [F]

\[ \int \frac {\left (d-c^2 d x^2\right )^{5/2} (a+b \arcsin (c x))}{x^2} \, dx=\frac {\sqrt {d}\, d^{2} \left (-4 \mathit {asin} \left (c x \right )^{2} b c x -15 \mathit {asin} \left (c x \right ) a c x +2 \sqrt {-c^{2} x^{2}+1}\, a \,c^{4} x^{4}-9 \sqrt {-c^{2} x^{2}+1}\, a \,c^{2} x^{2}-8 \sqrt {-c^{2} x^{2}+1}\, a +8 \left (\int \frac {\mathit {asin} \left (c x \right )}{\sqrt {-c^{2} x^{2}+1}\, x^{2}}d x \right ) b x +8 \left (\int \sqrt {-c^{2} x^{2}+1}\, \mathit {asin} \left (c x \right ) x^{2}d x \right ) b \,c^{4} x -16 \left (\int \sqrt {-c^{2} x^{2}+1}\, \mathit {asin} \left (c x \right )d x \right ) b \,c^{2} x \right )}{8 x} \] Input:

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

Output:

(sqrt(d)*d**2*( - 4*asin(c*x)**2*b*c*x - 15*asin(c*x)*a*c*x + 2*sqrt( - c* 
*2*x**2 + 1)*a*c**4*x**4 - 9*sqrt( - c**2*x**2 + 1)*a*c**2*x**2 - 8*sqrt( 
- c**2*x**2 + 1)*a + 8*int(asin(c*x)/(sqrt( - c**2*x**2 + 1)*x**2),x)*b*x 
+ 8*int(sqrt( - c**2*x**2 + 1)*asin(c*x)*x**2,x)*b*c**4*x - 16*int(sqrt( - 
 c**2*x**2 + 1)*asin(c*x),x)*b*c**2*x))/(8*x)