\(\int \frac {(d-c^2 d x^2)^{5/2} (a+b \arcsin (c x))}{x^4} \, dx\) [88]

Optimal result

Integrand size = 27, antiderivative size = 277 \[ \int \frac {\left (d-c^2 d x^2\right )^{5/2} (a+b \arcsin (c x))}{x^4} \, dx=-\frac {b c d^2 \sqrt {d-c^2 d x^2}}{6 x^2 \sqrt {1-c^2 x^2}}-\frac {b c^5 d^2 x^2 \sqrt {d-c^2 d x^2}}{4 \sqrt {1-c^2 x^2}}+\frac {5}{2} c^4 d^2 x \sqrt {d-c^2 d x^2} (a+b \arcsin (c x))+\frac {5 c^2 d \left (d-c^2 d x^2\right )^{3/2} (a+b \arcsin (c x))}{3 x}-\frac {\left (d-c^2 d x^2\right )^{5/2} (a+b \arcsin (c x))}{3 x^3}+\frac {5 c^3 d^2 \sqrt {d-c^2 d x^2} (a+b \arcsin (c x))^2}{4 b \sqrt {1-c^2 x^2}}-\frac {7 b c^3 d^2 \sqrt {d-c^2 d x^2} \log (x)}{3 \sqrt {1-c^2 x^2}} \] Output:

-1/6*b*c*d^2*(-c^2*d*x^2+d)^(1/2)/x^2/(-c^2*x^2+1)^(1/2)-1/4*b*c^5*d^2*x^2 
*(-c^2*d*x^2+d)^(1/2)/(-c^2*x^2+1)^(1/2)+5/2*c^4*d^2*x*(-c^2*d*x^2+d)^(1/2 
)*(a+b*arcsin(c*x))+5/3*c^2*d*(-c^2*d*x^2+d)^(3/2)*(a+b*arcsin(c*x))/x-1/3 
*(-c^2*d*x^2+d)^(5/2)*(a+b*arcsin(c*x))/x^3+5/4*c^3*d^2*(-c^2*d*x^2+d)^(1/ 
2)*(a+b*arcsin(c*x))^2/b/(-c^2*x^2+1)^(1/2)-7/3*b*c^3*d^2*(-c^2*d*x^2+d)^( 
1/2)*ln(x)/(-c^2*x^2+1)^(1/2)
 

Mathematica [A] (verified)

Time = 0.97 (sec) , antiderivative size = 243, normalized size of antiderivative = 0.88 \[ \int \frac {\left (d-c^2 d x^2\right )^{5/2} (a+b \arcsin (c x))}{x^4} \, dx=\frac {1}{24} d^2 \left (\frac {4 b \sqrt {d-c^2 d x^2} \left (-2+14 c^2 x^2+3 c^4 x^4\right ) \arcsin (c x)}{x^3}+\frac {30 b c^3 \sqrt {d-c^2 d x^2} \arcsin (c x)^2}{\sqrt {1-c^2 x^2}}-60 a c^3 \sqrt {d} \arctan \left (\frac {c x \sqrt {d-c^2 d x^2}}{\sqrt {d} \left (-1+c^2 x^2\right )}\right )+\frac {\sqrt {d-c^2 d x^2} \left (4 a \sqrt {1-c^2 x^2} \left (-2+14 c^2 x^2+3 c^4 x^4\right )+b \left (-4 c x+3 c^3 x^3-6 c^5 x^5\right )-56 b c^3 x^3 \log (c x)\right )}{x^3 \sqrt {1-c^2 x^2}}\right ) \] Input:

Integrate[((d - c^2*d*x^2)^(5/2)*(a + b*ArcSin[c*x]))/x^4,x]
 

Output:

(d^2*((4*b*Sqrt[d - c^2*d*x^2]*(-2 + 14*c^2*x^2 + 3*c^4*x^4)*ArcSin[c*x])/ 
x^3 + (30*b*c^3*Sqrt[d - c^2*d*x^2]*ArcSin[c*x]^2)/Sqrt[1 - c^2*x^2] - 60* 
a*c^3*Sqrt[d]*ArcTan[(c*x*Sqrt[d - c^2*d*x^2])/(Sqrt[d]*(-1 + c^2*x^2))] + 
 (Sqrt[d - c^2*d*x^2]*(4*a*Sqrt[1 - c^2*x^2]*(-2 + 14*c^2*x^2 + 3*c^4*x^4) 
 + b*(-4*c*x + 3*c^3*x^3 - 6*c^5*x^5) - 56*b*c^3*x^3*Log[c*x]))/(x^3*Sqrt[ 
1 - c^2*x^2])))/24
 

Rubi [A] (verified)

Time = 0.98 (sec) , antiderivative size = 296, normalized size of antiderivative = 1.07, 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, 5200, 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.

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

\(\Big \downarrow \) 5200

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

\(\Big \downarrow \) 243

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

\(\Big \downarrow \) 49

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

\(\Big \downarrow \) 2009

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

\(\Big \downarrow \) 5200

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

\(\Big \downarrow \) 244

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

\(\Big \downarrow \) 2009

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

\(\Big \downarrow \) 5156

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

\(\Big \downarrow \) 15

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

\(\Big \downarrow \) 5152

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

Input:

Int[((d - c^2*d*x^2)^(5/2)*(a + b*ArcSin[c*x]))/x^4,x]
 

Output:

-1/3*((d - c^2*d*x^2)^(5/2)*(a + b*ArcSin[c*x]))/x^3 - (5*c^2*d*(-(((d - c 
^2*d*x^2)^(3/2)*(a + b*ArcSin[c*x]))/x) - 3*c^2*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 + (Sqrt[d - c^2*d*x^2]*(a + b*ArcSin[c*x])^2)/(4*b*c*Sqrt[1 - c^2*x^2] 
)) + (b*c*d*Sqrt[d - c^2*d*x^2]*(-1/2*(c^2*x^2) + Log[x]))/Sqrt[1 - c^2*x^ 
2]))/3 + (b*c*d^2*Sqrt[d - c^2*d*x^2]*(-x^(-2) + c^4*x^2 - 2*c^2*Log[x^2]) 
)/(6*Sqrt[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 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 = 0.57 (sec) , antiderivative size = 343, normalized size of antiderivative = 1.24

method	result	size
default	\(-\frac {a \left (-c^{2} d \,x^{2}+d \right )^{\frac {7}{2}}}{3 d \,x^{3}}+\frac {4 a \,c^{2} \left (-c^{2} d \,x^{2}+d \right )^{\frac {7}{2}}}{3 d x}+\frac {4 a \,c^{4} x \left (-c^{2} d \,x^{2}+d \right )^{\frac {5}{2}}}{3}+\frac {5 a \,c^{4} d x \left (-c^{2} d \,x^{2}+d \right )^{\frac {3}{2}}}{3}+\frac {5 a \,c^{4} d^{2} x \sqrt {-c^{2} d \,x^{2}+d}}{2}+\frac {5 a \,c^{4} d^{3} \arctan \left (\frac {\sqrt {c^{2} d}\, x}{\sqrt {-c^{2} d \,x^{2}+d}}\right )}{2 \sqrt {c^{2} d}}-\frac {b \sqrt {-d \left (c^{2} x^{2}-1\right )}\, \sqrt {-c^{2} x^{2}+1}\, \left (12 \arcsin \left (c x \right ) \sqrt {-c^{2} x^{2}+1}\, c^{4} x^{4}-6 c^{5} x^{5}+56 i \arcsin \left (c x \right ) x^{3} c^{3}+30 \arcsin \left (c x \right )^{2} c^{3} x^{3}-56 \ln \left (\left (i c x +\sqrt {-c^{2} x^{2}+1}\right )^{2}-1\right ) x^{3} c^{3}+56 \arcsin \left (c x \right ) \sqrt {-c^{2} x^{2}+1}\, c^{2} x^{2}+3 c^{3} x^{3}-8 \arcsin \left (c x \right ) \sqrt {-c^{2} x^{2}+1}-4 c x \right ) d^{2}}{24 \left (c^{2} x^{2}-1\right ) x^{3}}\)	\(343\)
parts	\(-\frac {a \left (-c^{2} d \,x^{2}+d \right )^{\frac {7}{2}}}{3 d \,x^{3}}+\frac {4 a \,c^{2} \left (-c^{2} d \,x^{2}+d \right )^{\frac {7}{2}}}{3 d x}+\frac {4 a \,c^{4} x \left (-c^{2} d \,x^{2}+d \right )^{\frac {5}{2}}}{3}+\frac {5 a \,c^{4} d x \left (-c^{2} d \,x^{2}+d \right )^{\frac {3}{2}}}{3}+\frac {5 a \,c^{4} d^{2} x \sqrt {-c^{2} d \,x^{2}+d}}{2}+\frac {5 a \,c^{4} d^{3} \arctan \left (\frac {\sqrt {c^{2} d}\, x}{\sqrt {-c^{2} d \,x^{2}+d}}\right )}{2 \sqrt {c^{2} d}}-\frac {b \sqrt {-d \left (c^{2} x^{2}-1\right )}\, \sqrt {-c^{2} x^{2}+1}\, \left (12 \arcsin \left (c x \right ) \sqrt {-c^{2} x^{2}+1}\, c^{4} x^{4}-6 c^{5} x^{5}+56 i \arcsin \left (c x \right ) x^{3} c^{3}+30 \arcsin \left (c x \right )^{2} c^{3} x^{3}-56 \ln \left (\left (i c x +\sqrt {-c^{2} x^{2}+1}\right )^{2}-1\right ) x^{3} c^{3}+56 \arcsin \left (c x \right ) \sqrt {-c^{2} x^{2}+1}\, c^{2} x^{2}+3 c^{3} x^{3}-8 \arcsin \left (c x \right ) \sqrt {-c^{2} x^{2}+1}-4 c x \right ) d^{2}}{24 \left (c^{2} x^{2}-1\right ) x^{3}}\)	\(343\)

Input:

int((-c^2*d*x^2+d)^(5/2)*(a+b*arcsin(c*x))/x^4,x,method=_RETURNVERBOSE)
 

Output:

-1/3*a/d/x^3*(-c^2*d*x^2+d)^(7/2)+4/3*a*c^2/d/x*(-c^2*d*x^2+d)^(7/2)+4/3*a 
*c^4*x*(-c^2*d*x^2+d)^(5/2)+5/3*a*c^4*d*x*(-c^2*d*x^2+d)^(3/2)+5/2*a*c^4*d 
^2*x*(-c^2*d*x^2+d)^(1/2)+5/2*a*c^4*d^3/(c^2*d)^(1/2)*arctan((c^2*d)^(1/2) 
*x/(-c^2*d*x^2+d)^(1/2))-1/24*b*(-d*(c^2*x^2-1))^(1/2)*(-c^2*x^2+1)^(1/2)/ 
(c^2*x^2-1)/x^3*(12*arcsin(c*x)*(-c^2*x^2+1)^(1/2)*c^4*x^4-6*c^5*x^5+56*I* 
arcsin(c*x)*x^3*c^3+30*arcsin(c*x)^2*c^3*x^3-56*ln((I*c*x+(-c^2*x^2+1)^(1/ 
2))^2-1)*x^3*c^3+56*arcsin(c*x)*(-c^2*x^2+1)^(1/2)*c^2*x^2+3*c^3*x^3-8*arc 
sin(c*x)*(-c^2*x^2+1)^(1/2)-4*c*x)*d^2
 

Fricas [F]

\[ \int \frac {\left (d-c^2 d x^2\right )^{5/2} (a+b \arcsin (c x))}{x^4} \, dx=\int { \frac {{\left (-c^{2} d x^{2} + d\right )}^{\frac {5}{2}} {\left (b \arcsin \left (c x\right ) + a\right )}}{x^{4}} \,d x } \] Input:

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

Sympy [F(-1)]

Timed out. \[ \int \frac {\left (d-c^2 d x^2\right )^{5/2} (a+b \arcsin (c x))}{x^4} \, dx=\text {Timed out} \] Input:

integrate((-c**2*d*x**2+d)**(5/2)*(a+b*asin(c*x))/x**4,x)
 

Output:

Timed out
 

Maxima [F]

\[ \int \frac {\left (d-c^2 d x^2\right )^{5/2} (a+b \arcsin (c x))}{x^4} \, dx=\int { \frac {{\left (-c^{2} d x^{2} + d\right )}^{\frac {5}{2}} {\left (b \arcsin \left (c x\right ) + a\right )}}{x^{4}} \,d x } \] Input:

integrate((-c^2*d*x^2+d)^(5/2)*(a+b*arcsin(c*x))/x^4,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^4, x) + 1/6*(10*(- 
c^2*d*x^2 + d)^(3/2)*c^4*d*x + 15*sqrt(-c^2*d*x^2 + d)*c^4*d^2*x + 15*c^3* 
d^(5/2)*arcsin(c*x) + 8*(-c^2*d*x^2 + d)^(5/2)*c^2/x - 2*(-c^2*d*x^2 + d)^ 
(7/2)/(d*x^3))*a
 

Giac [F(-2)]

Exception generated. \[ \int \frac {\left (d-c^2 d x^2\right )^{5/2} (a+b \arcsin (c x))}{x^4} \, dx=\text {Exception raised: TypeError} \] Input:

integrate((-c^2*d*x^2+d)^(5/2)*(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. \[ \int \frac {\left (d-c^2 d x^2\right )^{5/2} (a+b \arcsin (c x))}{x^4} \, dx=\int \frac {\left (a+b\,\mathrm {asin}\left (c\,x\right )\right )\,{\left (d-c^2\,d\,x^2\right )}^{5/2}}{x^4} \,d x \] Input:

int(((a + b*asin(c*x))*(d - c^2*d*x^2)^(5/2))/x^4,x)
 

Output:

int(((a + b*asin(c*x))*(d - c^2*d*x^2)^(5/2))/x^4, x)
 

Reduce [F]

\[ \int \frac {\left (d-c^2 d x^2\right )^{5/2} (a+b \arcsin (c x))}{x^4} \, dx=\frac {\sqrt {d}\, d^{2} \left (6 \mathit {asin} \left (c x \right )^{2} b \,c^{3} x^{3}+15 \mathit {asin} \left (c x \right ) a \,c^{3} x^{3}+3 \sqrt {-c^{2} x^{2}+1}\, a \,c^{4} x^{4}+14 \sqrt {-c^{2} x^{2}+1}\, a \,c^{2} x^{2}-2 \sqrt {-c^{2} x^{2}+1}\, a -12 \left (\int \frac {\mathit {asin} \left (c x \right )}{\sqrt {-c^{2} x^{2}+1}\, x^{2}}d x \right ) b \,c^{2} x^{3}+6 \left (\int \frac {\sqrt {-c^{2} x^{2}+1}\, \mathit {asin} \left (c x \right )}{x^{4}}d x \right ) b \,x^{3}+6 \left (\int \sqrt {-c^{2} x^{2}+1}\, \mathit {asin} \left (c x \right )d x \right ) b \,c^{4} x^{3}\right )}{6 x^{3}} \] Input:

int((-c^2*d*x^2+d)^(5/2)*(a+b*asin(c*x))/x^4,x)
 

Output:

(sqrt(d)*d**2*(6*asin(c*x)**2*b*c**3*x**3 + 15*asin(c*x)*a*c**3*x**3 + 3*s 
qrt( - c**2*x**2 + 1)*a*c**4*x**4 + 14*sqrt( - c**2*x**2 + 1)*a*c**2*x**2 
- 2*sqrt( - c**2*x**2 + 1)*a - 12*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*int(sqrt( - c**2*x**2 + 1)*asin(c*x),x)*b*c**4*x**3))/(6*x**3)