\(\int \frac {(d-c^2 d x^2)^{5/2} (a+b \arccos (c x))}{x^6} \, dx\) [91]

Optimal result

Integrand size = 27, antiderivative size = 277 \[ \int \frac {\left (d-c^2 d x^2\right )^{5/2} (a+b \arccos (c x))}{x^6} \, dx=-\frac {b c d^2 \sqrt {d-c^2 d x^2}}{20 x^4 \sqrt {1-c^2 x^2}}+\frac {11 b c^3 d^2 \sqrt {d-c^2 d x^2}}{30 x^2 \sqrt {1-c^2 x^2}}-\frac {c^4 d^2 \sqrt {d-c^2 d x^2} (a+b \arccos (c x))}{x}+\frac {c^2 d \left (d-c^2 d x^2\right )^{3/2} (a+b \arccos (c x))}{3 x^3}-\frac {\left (d-c^2 d x^2\right )^{5/2} (a+b \arccos (c x))}{5 x^5}-\frac {c^5 d^2 \sqrt {d-c^2 d x^2} (a+b \arccos (c x))^2}{2 b \sqrt {1-c^2 x^2}}+\frac {23 b c^5 d^2 \sqrt {d-c^2 d x^2} \log (x)}{15 \sqrt {1-c^2 x^2}} \] Output:

-1/20*b*c*d^2*(-c^2*d*x^2+d)^(1/2)/x^4/(-c^2*x^2+1)^(1/2)+11/30*b*c^3*d^2* 
(-c^2*d*x^2+d)^(1/2)/x^2/(-c^2*x^2+1)^(1/2)-c^4*d^2*(-c^2*d*x^2+d)^(1/2)*( 
a+b*arccos(c*x))/x+1/3*c^2*d*(-c^2*d*x^2+d)^(3/2)*(a+b*arccos(c*x))/x^3-1/ 
5*(-c^2*d*x^2+d)^(5/2)*(a+b*arccos(c*x))/x^5-1/2*c^5*d^2*(-c^2*d*x^2+d)^(1 
/2)*(a+b*arccos(c*x))^2/b/(-c^2*x^2+1)^(1/2)+23/15*b*c^5*d^2*(-c^2*d*x^2+d 
)^(1/2)*ln(x)/(-c^2*x^2+1)^(1/2)
 

Mathematica [A] (verified)

Time = 1.29 (sec) , antiderivative size = 242, normalized size of antiderivative = 0.87 \[ \int \frac {\left (d-c^2 d x^2\right )^{5/2} (a+b \arccos (c x))}{x^6} \, dx=-\frac {b d^2 \sqrt {d-c^2 d x^2} \left (3-11 c^2 x^2+23 c^4 x^4\right ) \arccos (c x)}{15 x^5}+\frac {b c^5 d^2 \sqrt {d-c^2 d x^2} \arccos (c x)^2}{2 \sqrt {1-c^2 x^2}}+a c^5 d^{5/2} \arctan \left (\frac {c x \sqrt {d-c^2 d x^2}}{\sqrt {d} \left (-1+c^2 x^2\right )}\right )-\frac {d^2 \sqrt {d-c^2 d x^2} \left (b c x \left (-3+22 c^2 x^2\right )+4 a \sqrt {1-c^2 x^2} \left (3-11 c^2 x^2+23 c^4 x^4\right )+92 b c^5 x^5 \log (c x)\right )}{60 x^5 \sqrt {1-c^2 x^2}} \] Input:

Integrate[((d - c^2*d*x^2)^(5/2)*(a + b*ArcCos[c*x]))/x^6,x]
 

Output:

-1/15*(b*d^2*Sqrt[d - c^2*d*x^2]*(3 - 11*c^2*x^2 + 23*c^4*x^4)*ArcCos[c*x] 
)/x^5 + (b*c^5*d^2*Sqrt[d - c^2*d*x^2]*ArcCos[c*x]^2)/(2*Sqrt[1 - c^2*x^2] 
) + a*c^5*d^(5/2)*ArcTan[(c*x*Sqrt[d - c^2*d*x^2])/(Sqrt[d]*(-1 + c^2*x^2) 
)] - (d^2*Sqrt[d - c^2*d*x^2]*(b*c*x*(-3 + 22*c^2*x^2) + 4*a*Sqrt[1 - c^2* 
x^2]*(3 - 11*c^2*x^2 + 23*c^4*x^4) + 92*b*c^5*x^5*Log[c*x]))/(60*x^5*Sqrt[ 
1 - c^2*x^2])
 

Rubi [A] (verified)

Time = 1.06 (sec) , antiderivative size = 298, normalized size of antiderivative = 1.08, number of steps used = 11, number of rules used = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.370, Rules used = {5201, 243, 49, 2009, 5201, 244, 2009, 5197, 14, 5153}

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.

\(\displaystyle \int \frac {\left (d-c^2 d x^2\right )^{5/2} (a+b \arccos (c x))}{x^6} \, dx\)

\(\Big \downarrow \) 5201

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

\(\Big \downarrow \) 243

\(\displaystyle -c^2 d \int \frac {\left (d-c^2 d x^2\right )^{3/2} (a+b \arccos (c x))}{x^4}dx-\frac {b c d^2 \sqrt {d-c^2 d x^2} \int \frac {\left (1-c^2 x^2\right )^2}{x^6}dx^2}{10 \sqrt {1-c^2 x^2}}-\frac {\left (d-c^2 d x^2\right )^{5/2} (a+b \arccos (c x))}{5 x^5}\)

\(\Big \downarrow \) 49

\(\displaystyle -c^2 d \int \frac {\left (d-c^2 d x^2\right )^{3/2} (a+b \arccos (c x))}{x^4}dx-\frac {b c d^2 \sqrt {d-c^2 d x^2} \int \left (\frac {c^4}{x^2}-\frac {2 c^2}{x^4}+\frac {1}{x^6}\right )dx^2}{10 \sqrt {1-c^2 x^2}}-\frac {\left (d-c^2 d x^2\right )^{5/2} (a+b \arccos (c x))}{5 x^5}\)

\(\Big \downarrow \) 2009

\(\displaystyle -c^2 d \int \frac {\left (d-c^2 d x^2\right )^{3/2} (a+b \arccos (c x))}{x^4}dx-\frac {\left (d-c^2 d x^2\right )^{5/2} (a+b \arccos (c x))}{5 x^5}-\frac {b c d^2 \sqrt {d-c^2 d x^2} \left (c^4 \log \left (x^2\right )+\frac {2 c^2}{x^2}-\frac {1}{2 x^4}\right )}{10 \sqrt {1-c^2 x^2}}\)

\(\Big \downarrow \) 5201

\(\displaystyle -c^2 d \left (c^2 (-d) \int \frac {\sqrt {d-c^2 d x^2} (a+b \arccos (c x))}{x^2}dx-\frac {b c d \sqrt {d-c^2 d x^2} \int \frac {1-c^2 x^2}{x^3}dx}{3 \sqrt {1-c^2 x^2}}-\frac {\left (d-c^2 d x^2\right )^{3/2} (a+b \arccos (c x))}{3 x^3}\right )-\frac {\left (d-c^2 d x^2\right )^{5/2} (a+b \arccos (c x))}{5 x^5}-\frac {b c d^2 \sqrt {d-c^2 d x^2} \left (c^4 \log \left (x^2\right )+\frac {2 c^2}{x^2}-\frac {1}{2 x^4}\right )}{10 \sqrt {1-c^2 x^2}}\)

\(\Big \downarrow \) 244

\(\displaystyle -c^2 d \left (c^2 (-d) \int \frac {\sqrt {d-c^2 d x^2} (a+b \arccos (c x))}{x^2}dx-\frac {b c d \sqrt {d-c^2 d x^2} \int \left (\frac {1}{x^3}-\frac {c^2}{x}\right )dx}{3 \sqrt {1-c^2 x^2}}-\frac {\left (d-c^2 d x^2\right )^{3/2} (a+b \arccos (c x))}{3 x^3}\right )-\frac {\left (d-c^2 d x^2\right )^{5/2} (a+b \arccos (c x))}{5 x^5}-\frac {b c d^2 \sqrt {d-c^2 d x^2} \left (c^4 \log \left (x^2\right )+\frac {2 c^2}{x^2}-\frac {1}{2 x^4}\right )}{10 \sqrt {1-c^2 x^2}}\)

\(\Big \downarrow \) 2009

\(\displaystyle -c^2 d \left (c^2 (-d) \int \frac {\sqrt {d-c^2 d x^2} (a+b \arccos (c x))}{x^2}dx-\frac {\left (d-c^2 d x^2\right )^{3/2} (a+b \arccos (c x))}{3 x^3}-\frac {b c d \sqrt {d-c^2 d x^2} \left (c^2 (-\log (x))-\frac {1}{2 x^2}\right )}{3 \sqrt {1-c^2 x^2}}\right )-\frac {\left (d-c^2 d x^2\right )^{5/2} (a+b \arccos (c x))}{5 x^5}-\frac {b c d^2 \sqrt {d-c^2 d x^2} \left (c^4 \log \left (x^2\right )+\frac {2 c^2}{x^2}-\frac {1}{2 x^4}\right )}{10 \sqrt {1-c^2 x^2}}\)

\(\Big \downarrow \) 5197

\(\displaystyle -c^2 d \left (c^2 (-d) \left (-\frac {c^2 \sqrt {d-c^2 d x^2} \int \frac {a+b \arccos (c x)}{\sqrt {1-c^2 x^2}}dx}{\sqrt {1-c^2 x^2}}-\frac {b c \sqrt {d-c^2 d x^2} \int \frac {1}{x}dx}{\sqrt {1-c^2 x^2}}-\frac {\sqrt {d-c^2 d x^2} (a+b \arccos (c x))}{x}\right )-\frac {\left (d-c^2 d x^2\right )^{3/2} (a+b \arccos (c x))}{3 x^3}-\frac {b c d \sqrt {d-c^2 d x^2} \left (c^2 (-\log (x))-\frac {1}{2 x^2}\right )}{3 \sqrt {1-c^2 x^2}}\right )-\frac {\left (d-c^2 d x^2\right )^{5/2} (a+b \arccos (c x))}{5 x^5}-\frac {b c d^2 \sqrt {d-c^2 d x^2} \left (c^4 \log \left (x^2\right )+\frac {2 c^2}{x^2}-\frac {1}{2 x^4}\right )}{10 \sqrt {1-c^2 x^2}}\)

\(\Big \downarrow \) 14

\(\displaystyle -c^2 d \left (c^2 (-d) \left (-\frac {c^2 \sqrt {d-c^2 d x^2} \int \frac {a+b \arccos (c x)}{\sqrt {1-c^2 x^2}}dx}{\sqrt {1-c^2 x^2}}-\frac {\sqrt {d-c^2 d x^2} (a+b \arccos (c x))}{x}-\frac {b c \log (x) \sqrt {d-c^2 d x^2}}{\sqrt {1-c^2 x^2}}\right )-\frac {\left (d-c^2 d x^2\right )^{3/2} (a+b \arccos (c x))}{3 x^3}-\frac {b c d \sqrt {d-c^2 d x^2} \left (c^2 (-\log (x))-\frac {1}{2 x^2}\right )}{3 \sqrt {1-c^2 x^2}}\right )-\frac {\left (d-c^2 d x^2\right )^{5/2} (a+b \arccos (c x))}{5 x^5}-\frac {b c d^2 \sqrt {d-c^2 d x^2} \left (c^4 \log \left (x^2\right )+\frac {2 c^2}{x^2}-\frac {1}{2 x^4}\right )}{10 \sqrt {1-c^2 x^2}}\)

\(\Big \downarrow \) 5153

\(\displaystyle -\frac {\left (d-c^2 d x^2\right )^{5/2} (a+b \arccos (c x))}{5 x^5}-c^2 d \left (c^2 (-d) \left (\frac {c \sqrt {d-c^2 d x^2} (a+b \arccos (c x))^2}{2 b \sqrt {1-c^2 x^2}}-\frac {\sqrt {d-c^2 d x^2} (a+b \arccos (c x))}{x}-\frac {b c \log (x) \sqrt {d-c^2 d x^2}}{\sqrt {1-c^2 x^2}}\right )-\frac {\left (d-c^2 d x^2\right )^{3/2} (a+b \arccos (c x))}{3 x^3}-\frac {b c d \sqrt {d-c^2 d x^2} \left (c^2 (-\log (x))-\frac {1}{2 x^2}\right )}{3 \sqrt {1-c^2 x^2}}\right )-\frac {b c d^2 \sqrt {d-c^2 d x^2} \left (c^4 \log \left (x^2\right )+\frac {2 c^2}{x^2}-\frac {1}{2 x^4}\right )}{10 \sqrt {1-c^2 x^2}}\)

Input:

Int[((d - c^2*d*x^2)^(5/2)*(a + b*ArcCos[c*x]))/x^6,x]
 

Output:

-1/5*((d - c^2*d*x^2)^(5/2)*(a + b*ArcCos[c*x]))/x^5 - c^2*d*(-1/3*((d - c 
^2*d*x^2)^(3/2)*(a + b*ArcCos[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*ArcCos[c*x]))/x) + (c*Sqrt[d - c^2*d*x^2]*(a + b*ArcCos[c*x])^2) 
/(2*b*Sqrt[1 - c^2*x^2]) - (b*c*Sqrt[d - c^2*d*x^2]*Log[x])/Sqrt[1 - c^2*x 
^2])) - (b*c*d^2*Sqrt[d - c^2*d*x^2]*(-1/2*1/x^4 + (2*c^2)/x^2 + c^4*Log[x 
^2]))/(10*Sqrt[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 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 5153

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

rule 5197

Int[((a_.) + ArcCos[(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*ArcC 
os[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*ArcCos[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*ArcCos[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 5201

Int[((a_.) + ArcCos[(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*ArcC 
os[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*ArcCos[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*ArcCos[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 = 0.94 (sec) , antiderivative size = 2615, normalized size of antiderivative = 9.44

method	result	size
default	\(\text {Expression too large to display}\)	\(2615\)
parts	\(\text {Expression too large to display}\)	\(2615\)

Input:

int((-c^2*d*x^2+d)^(5/2)*(a+b*arccos(c*x))/x^6,x,method=_RETURNVERBOSE)
 

Output:

9/5*b*(-d*(c^2*x^2-1))^(1/2)*d^2/(1035*c^8*x^8-765*c^6*x^6+325*c^4*x^4-75* 
c^2*x^2+9)/x^5/(c^2*x^2-1)*arccos(c*x)-175/4*b*(-d*(c^2*x^2-1))^(1/2)*d^2/ 
(1035*c^8*x^8-765*c^6*x^6+325*c^4*x^4-75*c^2*x^2+9)/(c^2*x^2-1)*c^5*(-c^2* 
x^2+1)^(1/2)+23/15*b*(-d*(c^2*x^2-1))^(1/2)*(-c^2*x^2+1)^(1/2)/(c^2*x^2-1) 
*ln(1+(c*x+I*(-c^2*x^2+1)^(1/2))^2)*d^2*c^5-1/2*b*(-d*(c^2*x^2-1))^(1/2)*( 
-c^2*x^2+1)^(1/2)/(c^2*x^2-1)*arccos(c*x)^2*d^2*c^5+69/5*I*b*(-d*(c^2*x^2- 
1))^(1/2)*d^2/(1035*c^8*x^8-765*c^6*x^6+325*c^4*x^4-75*c^2*x^2+9)/(c^2*x^2 
-1)*(-c^2*x^2+1)^(1/2)*arccos(c*x)*c^5+759/20*I*b*(-d*(c^2*x^2-1))^(1/2)*d 
^2/(1035*c^8*x^8-765*c^6*x^6+325*c^4*x^4-75*c^2*x^2+9)*x^3/(c^2*x^2-1)*(-c 
^2*x^2+1)*c^8-69/20*I*b*(-d*(c^2*x^2-1))^(1/2)*d^2/(1035*c^8*x^8-765*c^6*x 
^6+325*c^4*x^4-75*c^2*x^2+9)*x/(c^2*x^2-1)*(-c^2*x^2+1)*c^6+5819/30*I*b*(- 
d*(c^2*x^2-1))^(1/2)*d^2/(1035*c^8*x^8-765*c^6*x^6+325*c^4*x^4-75*c^2*x^2+ 
9)*x^7/(c^2*x^2-1)*(-c^2*x^2+1)*c^12-7153/60*I*b*(-d*(c^2*x^2-1))^(1/2)*d^ 
2/(1035*c^8*x^8-765*c^6*x^6+325*c^4*x^4-75*c^2*x^2+9)*x^5/(c^2*x^2-1)*(-c^ 
2*x^2+1)*c^10-1/5*a/d/x^5*(-c^2*d*x^2+d)^(7/2)-8/15*a*c^6*x*(-c^2*d*x^2+d) 
^(5/2)+69/20*I*b*(-d*(c^2*x^2-1))^(1/2)*d^2/(1035*c^8*x^8-765*c^6*x^6+325* 
c^4*x^4-75*c^2*x^2+9)*x/(c^2*x^2-1)*c^6-207/5*I*b*(-d*(c^2*x^2-1))^(1/2)*d 
^2/(1035*c^8*x^8-765*c^6*x^6+325*c^4*x^4-75*c^2*x^2+9)*x^3/(c^2*x^2-1)*c^8 
-46/15*I*b*(-d*(c^2*x^2-1))^(1/2)*(-c^2*x^2+1)^(1/2)/(c^2*x^2-1)*arccos(c* 
x)*d^2*c^5+5819/30*I*b*(-d*(c^2*x^2-1))^(1/2)*d^2/(1035*c^8*x^8-765*c^6...
 

Fricas [F]

\[ \int \frac {\left (d-c^2 d x^2\right )^{5/2} (a+b \arccos (c x))}{x^6} \, dx=\int { \frac {{\left (-c^{2} d x^{2} + d\right )}^{\frac {5}{2}} {\left (b \arccos \left (c x\right ) + a\right )}}{x^{6}} \,d x } \] Input:

integrate((-c^2*d*x^2+d)^(5/2)*(a+b*arccos(c*x))/x^6,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)*arccos(c*x))*sqrt(-c^2*d*x^2 + d)/x^6, x)
 

Sympy [F]

\[ \int \frac {\left (d-c^2 d x^2\right )^{5/2} (a+b \arccos (c x))}{x^6} \, dx=\int \frac {\left (- d \left (c x - 1\right ) \left (c x + 1\right )\right )^{\frac {5}{2}} \left (a + b \operatorname {acos}{\left (c x \right )}\right )}{x^{6}}\, dx \] Input:

integrate((-c**2*d*x**2+d)**(5/2)*(a+b*acos(c*x))/x**6,x)
 

Output:

Integral((-d*(c*x - 1)*(c*x + 1))**(5/2)*(a + b*acos(c*x))/x**6, x)
 

Maxima [F]

\[ \int \frac {\left (d-c^2 d x^2\right )^{5/2} (a+b \arccos (c x))}{x^6} \, dx=\int { \frac {{\left (-c^{2} d x^{2} + d\right )}^{\frac {5}{2}} {\left (b \arccos \left (c x\right ) + a\right )}}{x^{6}} \,d x } \] Input:

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

Giac [F(-2)]

Exception generated. \[ \int \frac {\left (d-c^2 d x^2\right )^{5/2} (a+b \arccos (c x))}{x^6} \, dx=\text {Exception raised: TypeError} \] Input:

integrate((-c^2*d*x^2+d)^(5/2)*(a+b*arccos(c*x))/x^6,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. \[ \int \frac {\left (d-c^2 d x^2\right )^{5/2} (a+b \arccos (c x))}{x^6} \, dx=\int \frac {\left (a+b\,\mathrm {acos}\left (c\,x\right )\right )\,{\left (d-c^2\,d\,x^2\right )}^{5/2}}{x^6} \,d x \] Input:

int(((a + b*acos(c*x))*(d - c^2*d*x^2)^(5/2))/x^6,x)
 

Output:

int(((a + b*acos(c*x))*(d - c^2*d*x^2)^(5/2))/x^6, x)
 

Reduce [F]

\[ \int \frac {\left (d-c^2 d x^2\right )^{5/2} (a+b \arccos (c x))}{x^6} \, dx=\frac {\sqrt {d}\, d^{2} \left (15 \mathit {acos} \left (c x \right )^{2} b \,c^{5} x^{5}-30 \mathit {asin} \left (c x \right ) a \,c^{5} x^{5}-46 \sqrt {-c^{2} x^{2}+1}\, a \,c^{4} x^{4}+22 \sqrt {-c^{2} x^{2}+1}\, a \,c^{2} x^{2}-6 \sqrt {-c^{2} x^{2}+1}\, a +30 \left (\int \frac {\mathit {acos} \left (c x \right )}{\sqrt {-c^{2} x^{2}+1}\, x^{2}}d x \right ) b \,c^{4} x^{5}+30 \left (\int \frac {\sqrt {-c^{2} x^{2}+1}\, \mathit {acos} \left (c x \right )}{x^{6}}d x \right ) b \,x^{5}-60 \left (\int \frac {\sqrt {-c^{2} x^{2}+1}\, \mathit {acos} \left (c x \right )}{x^{4}}d x \right ) b \,c^{2} x^{5}\right )}{30 x^{5}} \] Input:

int((-c^2*d*x^2+d)^(5/2)*(a+b*acos(c*x))/x^6,x)
 

Output:

(sqrt(d)*d**2*(15*acos(c*x)**2*b*c**5*x**5 - 30*asin(c*x)*a*c**5*x**5 - 46 
*sqrt( - c**2*x**2 + 1)*a*c**4*x**4 + 22*sqrt( - c**2*x**2 + 1)*a*c**2*x** 
2 - 6*sqrt( - c**2*x**2 + 1)*a + 30*int(acos(c*x)/(sqrt( - c**2*x**2 + 1)* 
x**2),x)*b*c**4*x**5 + 30*int((sqrt( - c**2*x**2 + 1)*acos(c*x))/x**6,x)*b 
*x**5 - 60*int((sqrt( - c**2*x**2 + 1)*acos(c*x))/x**4,x)*b*c**2*x**5))/(3 
0*x**5)