\(\int \frac {a+b \arccos (c x)}{x^4 (d-c^2 d x^2)^{5/2}} \, dx\) [141]

Optimal result

Integrand size = 27, antiderivative size = 310 \[ \int \frac {a+b \arccos (c x)}{x^4 \left (d-c^2 d x^2\right )^{5/2}} \, dx=-\frac {b c^3 \sqrt {d-c^2 d x^2}}{6 d^3 \left (1-c^2 x^2\right )^{3/2}}-\frac {b c \sqrt {d-c^2 d x^2}}{6 d^3 x^2 \sqrt {1-c^2 x^2}}-\frac {a+b \arccos (c x)}{3 d x^3 \left (d-c^2 d x^2\right )^{3/2}}-\frac {2 c^2 (a+b \arccos (c x))}{d x \left (d-c^2 d x^2\right )^{3/2}}+\frac {8 c^4 x (a+b \arccos (c x))}{3 d \left (d-c^2 d x^2\right )^{3/2}}+\frac {16 c^4 x (a+b \arccos (c x))}{3 d^2 \sqrt {d-c^2 d x^2}}+\frac {8 b c^3 \sqrt {d-c^2 d x^2} \log (x)}{3 d^3 \sqrt {1-c^2 x^2}}+\frac {4 b c^3 \sqrt {d-c^2 d x^2} \log \left (1-c^2 x^2\right )}{3 d^3 \sqrt {1-c^2 x^2}} \] Output:

-1/6*b*c^3*(-c^2*d*x^2+d)^(1/2)/d^3/(-c^2*x^2+1)^(3/2)-1/6*b*c*(-c^2*d*x^2 
+d)^(1/2)/d^3/x^2/(-c^2*x^2+1)^(1/2)-1/3*(a+b*arccos(c*x))/d/x^3/(-c^2*d*x 
^2+d)^(3/2)-2*c^2*(a+b*arccos(c*x))/d/x/(-c^2*d*x^2+d)^(3/2)+8/3*c^4*x*(a+ 
b*arccos(c*x))/d/(-c^2*d*x^2+d)^(3/2)+16/3*c^4*x*(a+b*arccos(c*x))/d^2/(-c 
^2*d*x^2+d)^(1/2)+8/3*b*c^3*(-c^2*d*x^2+d)^(1/2)*ln(x)/d^3/(-c^2*x^2+1)^(1 
/2)+4/3*b*c^3*(-c^2*d*x^2+d)^(1/2)*ln(-c^2*x^2+1)/d^3/(-c^2*x^2+1)^(1/2)
 

Mathematica [A] (verified)

Time = 0.35 (sec) , antiderivative size = 277, normalized size of antiderivative = 0.89 \[ \int \frac {a+b \arccos (c x)}{x^4 \left (d-c^2 d x^2\right )^{5/2}} \, dx=-\frac {\sqrt {d-c^2 d x^2} \left (-b c x+b c^3 x^3+2 a \sqrt {1-c^2 x^2}+12 a c^2 x^2 \sqrt {1-c^2 x^2}-48 a c^4 x^4 \sqrt {1-c^2 x^2}+32 a c^6 x^6 \sqrt {1-c^2 x^2}+2 b \sqrt {1-c^2 x^2} \left (1+6 c^2 x^2-24 c^4 x^4+16 c^6 x^6\right ) \arccos (c x)-8 b c^3 x^3 \left (-1+c^2 x^2\right )^2 \log \left (1-\frac {1}{c^2 x^2}\right )+16 b c^3 x^3 \log \left (1-c^2 x^2\right )-32 b c^5 x^5 \log \left (1-c^2 x^2\right )+16 b c^7 x^7 \log \left (1-c^2 x^2\right )\right )}{6 d^3 x^3 \left (1-c^2 x^2\right )^{5/2}} \] Input:

Integrate[(a + b*ArcCos[c*x])/(x^4*(d - c^2*d*x^2)^(5/2)),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] + 
12*a*c^2*x^2*Sqrt[1 - c^2*x^2] - 48*a*c^4*x^4*Sqrt[1 - c^2*x^2] + 32*a*c^6 
*x^6*Sqrt[1 - c^2*x^2] + 2*b*Sqrt[1 - c^2*x^2]*(1 + 6*c^2*x^2 - 24*c^4*x^4 
 + 16*c^6*x^6)*ArcCos[c*x] - 8*b*c^3*x^3*(-1 + c^2*x^2)^2*Log[1 - 1/(c^2*x 
^2)] + 16*b*c^3*x^3*Log[1 - c^2*x^2] - 32*b*c^5*x^5*Log[1 - c^2*x^2] + 16* 
b*c^7*x^7*Log[1 - c^2*x^2]))/(d^3*x^3*(1 - c^2*x^2)^(5/2))
 

Rubi [A] (verified)

Time = 0.66 (sec) , antiderivative size = 222, normalized size of antiderivative = 0.72, number of steps used = 6, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.185, Rules used = {5195, 27, 2331, 2123, 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.

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

\(\Big \downarrow \) 5195

\(\displaystyle \frac {b c \sqrt {d-c^2 d x^2} \int -\frac {16 c^6 x^6-24 c^4 x^4+6 c^2 x^2+1}{3 d^3 x^3 \left (1-c^2 x^2\right )^2}dx}{\sqrt {1-c^2 x^2}}-\frac {2 c^2 (a+b \arccos (c x))}{d x \left (d-c^2 d x^2\right )^{3/2}}-\frac {a+b \arccos (c x)}{3 d x^3 \left (d-c^2 d x^2\right )^{3/2}}+\frac {16 c^4 x (a+b \arccos (c x))}{3 d^2 \sqrt {d-c^2 d x^2}}+\frac {8 c^4 x (a+b \arccos (c x))}{3 d \left (d-c^2 d x^2\right )^{3/2}}\)

\(\Big \downarrow \) 27

\(\displaystyle -\frac {b c \sqrt {d-c^2 d x^2} \int \frac {16 c^6 x^6-24 c^4 x^4+6 c^2 x^2+1}{x^3 \left (1-c^2 x^2\right )^2}dx}{3 d^3 \sqrt {1-c^2 x^2}}-\frac {2 c^2 (a+b \arccos (c x))}{d x \left (d-c^2 d x^2\right )^{3/2}}-\frac {a+b \arccos (c x)}{3 d x^3 \left (d-c^2 d x^2\right )^{3/2}}+\frac {16 c^4 x (a+b \arccos (c x))}{3 d^2 \sqrt {d-c^2 d x^2}}+\frac {8 c^4 x (a+b \arccos (c x))}{3 d \left (d-c^2 d x^2\right )^{3/2}}\)

\(\Big \downarrow \) 2331

\(\displaystyle -\frac {b c \sqrt {d-c^2 d x^2} \int \frac {16 c^6 x^6-24 c^4 x^4+6 c^2 x^2+1}{x^4 \left (1-c^2 x^2\right )^2}dx^2}{6 d^3 \sqrt {1-c^2 x^2}}-\frac {2 c^2 (a+b \arccos (c x))}{d x \left (d-c^2 d x^2\right )^{3/2}}-\frac {a+b \arccos (c x)}{3 d x^3 \left (d-c^2 d x^2\right )^{3/2}}+\frac {16 c^4 x (a+b \arccos (c x))}{3 d^2 \sqrt {d-c^2 d x^2}}+\frac {8 c^4 x (a+b \arccos (c x))}{3 d \left (d-c^2 d x^2\right )^{3/2}}\)

\(\Big \downarrow \) 2123

\(\displaystyle -\frac {b c \sqrt {d-c^2 d x^2} \int \left (\frac {8 c^4}{c^2 x^2-1}-\frac {c^4}{\left (c^2 x^2-1\right )^2}+\frac {8 c^2}{x^2}+\frac {1}{x^4}\right )dx^2}{6 d^3 \sqrt {1-c^2 x^2}}-\frac {2 c^2 (a+b \arccos (c x))}{d x \left (d-c^2 d x^2\right )^{3/2}}-\frac {a+b \arccos (c x)}{3 d x^3 \left (d-c^2 d x^2\right )^{3/2}}+\frac {16 c^4 x (a+b \arccos (c x))}{3 d^2 \sqrt {d-c^2 d x^2}}+\frac {8 c^4 x (a+b \arccos (c x))}{3 d \left (d-c^2 d x^2\right )^{3/2}}\)

\(\Big \downarrow \) 2009

\(\displaystyle -\frac {2 c^2 (a+b \arccos (c x))}{d x \left (d-c^2 d x^2\right )^{3/2}}-\frac {a+b \arccos (c x)}{3 d x^3 \left (d-c^2 d x^2\right )^{3/2}}+\frac {16 c^4 x (a+b \arccos (c x))}{3 d^2 \sqrt {d-c^2 d x^2}}+\frac {8 c^4 x (a+b \arccos (c x))}{3 d \left (d-c^2 d x^2\right )^{3/2}}-\frac {b c \sqrt {d-c^2 d x^2} \left (-\frac {c^2}{1-c^2 x^2}+8 c^2 \log \left (x^2\right )+8 c^2 \log \left (1-c^2 x^2\right )-\frac {1}{x^2}\right )}{6 d^3 \sqrt {1-c^2 x^2}}\)

Input:

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

Output:

-1/3*(a + b*ArcCos[c*x])/(d*x^3*(d - c^2*d*x^2)^(3/2)) - (2*c^2*(a + b*Arc 
Cos[c*x]))/(d*x*(d - c^2*d*x^2)^(3/2)) + (8*c^4*x*(a + b*ArcCos[c*x]))/(3* 
d*(d - c^2*d*x^2)^(3/2)) + (16*c^4*x*(a + b*ArcCos[c*x]))/(3*d^2*Sqrt[d - 
c^2*d*x^2]) - (b*c*Sqrt[d - c^2*d*x^2]*(-x^(-2) - c^2/(1 - c^2*x^2) + 8*c^ 
2*Log[x^2] + 8*c^2*Log[1 - c^2*x^2]))/(6*d^3*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 2009

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

rule 2123

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

rule 2331

Int[(Pq_)*(x_)^(m_.)*((a_) + (b_.)*(x_)^2)^(p_.), x_Symbol] :> Simp[1/2   S 
ubst[Int[x^((m - 1)/2)*SubstFor[x^2, Pq, x]*(a + b*x)^p, x], x, x^2], x] /; 
 FreeQ[{a, b, p}, x] && PolyQ[Pq, x^2] && IntegerQ[(m - 1)/2]
 

rule 5195

Int[((a_.) + ArcCos[(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*ArcCos 
[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 = 0.89 (sec) , antiderivative size = 1878, normalized size of antiderivative = 6.06

method	result	size
default	\(\text {Expression too large to display}\)	\(1878\)
parts	\(\text {Expression too large to display}\)	\(1878\)

Input:

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

Output:

2*b*(-d*(c^2*x^2-1))^(1/2)/(12*c^8*x^8-36*c^6*x^6+35*c^4*x^4-10*c^2*x^2-1) 
/d^3*x^2*c^5*(-c^2*x^2+1)^(1/2)+12*b*(-d*(c^2*x^2-1))^(1/2)/(12*c^8*x^8-36 
*c^6*x^6+35*c^4*x^4-10*c^2*x^2-1)/d^3*x*arccos(c*x)*c^4+6*b*(-d*(c^2*x^2-1 
))^(1/2)/(12*c^8*x^8-36*c^6*x^6+35*c^4*x^4-10*c^2*x^2-1)/d^3/x*arccos(c*x) 
*c^2-1/6*b*(-d*(c^2*x^2-1))^(1/2)/(12*c^8*x^8-36*c^6*x^6+35*c^4*x^4-10*c^2 
*x^2-1)/d^3/x^2*(-c^2*x^2+1)^(1/2)*c+128/3*I*b*(-d*(c^2*x^2-1))^(1/2)/(12* 
c^8*x^8-36*c^6*x^6+35*c^4*x^4-10*c^2*x^2-1)/d^3*x^11*c^14-448/3*I*b*(-d*(c 
^2*x^2-1))^(1/2)/(12*c^8*x^8-36*c^6*x^6+35*c^4*x^4-10*c^2*x^2-1)/d^3*x^9*c 
^12+560/3*I*b*(-d*(c^2*x^2-1))^(1/2)/(12*c^8*x^8-36*c^6*x^6+35*c^4*x^4-10* 
c^2*x^2-1)/d^3*x^7*c^10+1/3*b*(-d*(c^2*x^2-1))^(1/2)/(12*c^8*x^8-36*c^6*x^ 
6+35*c^4*x^4-10*c^2*x^2-1)/d^3/x^3*arccos(c*x)-2*b*(-d*(c^2*x^2-1))^(1/2)/ 
(12*c^8*x^8-36*c^6*x^6+35*c^4*x^4-10*c^2*x^2-1)/d^3*(-c^2*x^2+1)^(1/2)*c^3 
+64*I*b*(-d*(c^2*x^2-1))^(1/2)/(12*c^8*x^8-36*c^6*x^6+35*c^4*x^4-10*c^2*x^ 
2-1)/d^3*x^6*(-c^2*x^2+1)^(1/2)*arccos(c*x)*c^9-128*I*b*(-d*(c^2*x^2-1))^( 
1/2)/(12*c^8*x^8-36*c^6*x^6+35*c^4*x^4-10*c^2*x^2-1)/d^3*x^4*(-c^2*x^2+1)^ 
(1/2)*arccos(c*x)*c^7+176/3*I*b*(-d*(c^2*x^2-1))^(1/2)/(12*c^8*x^8-36*c^6* 
x^6+35*c^4*x^4-10*c^2*x^2-1)/d^3*x^2*(-c^2*x^2+1)^(1/2)*arccos(c*x)*c^5-40 
/3*I*b*(-d*(c^2*x^2-1))^(1/2)/(12*c^8*x^8-36*c^6*x^6+35*c^4*x^4-10*c^2*x^2 
-1)/d^3*x^3*(-c^2*x^2+1)*c^6+128/3*I*b*(-d*(c^2*x^2-1))^(1/2)/(12*c^8*x^8- 
36*c^6*x^6+35*c^4*x^4-10*c^2*x^2-1)/d^3*x^9*(-c^2*x^2+1)*c^12-320/3*I*b...
 

Fricas [F]

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

integrate((a+b*arccos(c*x))/x^4/(-c^2*d*x^2+d)^(5/2),x, algorithm="fricas" 
)
 

Output:

integral(-sqrt(-c^2*d*x^2 + d)*(b*arccos(c*x) + a)/(c^6*d^3*x^10 - 3*c^4*d 
^3*x^8 + 3*c^2*d^3*x^6 - d^3*x^4), x)
                                                                                    
                                                                                    
 

Sympy [F(-1)]

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

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

Output:

Timed out
 

Maxima [A] (verification not implemented)

Time = 0.13 (sec) , antiderivative size = 255, normalized size of antiderivative = 0.82 \[ \int \frac {a+b \arccos (c x)}{x^4 \left (d-c^2 d x^2\right )^{5/2}} \, dx=-\frac {1}{6} \, b c {\left (\frac {8 \, c^{2} \log \left (c x + 1\right )}{d^{\frac {5}{2}}} + \frac {8 \, c^{2} \log \left (c x - 1\right )}{d^{\frac {5}{2}}} + \frac {16 \, c^{2} \log \left (x\right )}{d^{\frac {5}{2}}} + \frac {1}{c^{2} d^{\frac {5}{2}} x^{4} - d^{\frac {5}{2}} x^{2}}\right )} + \frac {1}{3} \, {\left (\frac {16 \, c^{4} x}{\sqrt {-c^{2} d x^{2} + d} d^{2}} + \frac {8 \, c^{4} x}{{\left (-c^{2} d x^{2} + d\right )}^{\frac {3}{2}} d} - \frac {6 \, c^{2}}{{\left (-c^{2} d x^{2} + d\right )}^{\frac {3}{2}} d x} - \frac {1}{{\left (-c^{2} d x^{2} + d\right )}^{\frac {3}{2}} d x^{3}}\right )} b \arccos \left (c x\right ) + \frac {1}{3} \, {\left (\frac {16 \, c^{4} x}{\sqrt {-c^{2} d x^{2} + d} d^{2}} + \frac {8 \, c^{4} x}{{\left (-c^{2} d x^{2} + d\right )}^{\frac {3}{2}} d} - \frac {6 \, c^{2}}{{\left (-c^{2} d x^{2} + d\right )}^{\frac {3}{2}} d x} - \frac {1}{{\left (-c^{2} d x^{2} + d\right )}^{\frac {3}{2}} d x^{3}}\right )} a \] Input:

integrate((a+b*arccos(c*x))/x^4/(-c^2*d*x^2+d)^(5/2),x, algorithm="maxima" 
)
 

Output:

-1/6*b*c*(8*c^2*log(c*x + 1)/d^(5/2) + 8*c^2*log(c*x - 1)/d^(5/2) + 16*c^2 
*log(x)/d^(5/2) + 1/(c^2*d^(5/2)*x^4 - d^(5/2)*x^2)) + 1/3*(16*c^4*x/(sqrt 
(-c^2*d*x^2 + d)*d^2) + 8*c^4*x/((-c^2*d*x^2 + d)^(3/2)*d) - 6*c^2/((-c^2* 
d*x^2 + d)^(3/2)*d*x) - 1/((-c^2*d*x^2 + d)^(3/2)*d*x^3))*b*arccos(c*x) + 
1/3*(16*c^4*x/(sqrt(-c^2*d*x^2 + d)*d^2) + 8*c^4*x/((-c^2*d*x^2 + d)^(3/2) 
*d) - 6*c^2/((-c^2*d*x^2 + d)^(3/2)*d*x) - 1/((-c^2*d*x^2 + d)^(3/2)*d*x^3 
))*a
 

Giac [F(-2)]

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

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

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

Output:

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

Reduce [F]

\[ \int \frac {a+b \arccos (c x)}{x^4 \left (d-c^2 d x^2\right )^{5/2}} \, dx=\frac {3 \sqrt {-c^{2} x^{2}+1}\, \left (\int \frac {\mathit {acos} \left (c x \right )}{\sqrt {-c^{2} x^{2}+1}\, c^{4} x^{8}-2 \sqrt {-c^{2} x^{2}+1}\, c^{2} x^{6}+\sqrt {-c^{2} x^{2}+1}\, x^{4}}d x \right ) b \,c^{2} x^{5}-3 \sqrt {-c^{2} x^{2}+1}\, \left (\int \frac {\mathit {acos} \left (c x \right )}{\sqrt {-c^{2} x^{2}+1}\, c^{4} x^{8}-2 \sqrt {-c^{2} x^{2}+1}\, c^{2} x^{6}+\sqrt {-c^{2} x^{2}+1}\, x^{4}}d x \right ) b \,x^{3}+16 a \,c^{6} x^{6}-24 a \,c^{4} x^{4}+6 a \,c^{2} x^{2}+a}{3 \sqrt {d}\, \sqrt {-c^{2} x^{2}+1}\, d^{2} x^{3} \left (c^{2} x^{2}-1\right )} \] Input:

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

Output:

(3*sqrt( - c**2*x**2 + 1)*int(acos(c*x)/(sqrt( - c**2*x**2 + 1)*c**4*x**8 
- 2*sqrt( - c**2*x**2 + 1)*c**2*x**6 + sqrt( - c**2*x**2 + 1)*x**4),x)*b*c 
**2*x**5 - 3*sqrt( - c**2*x**2 + 1)*int(acos(c*x)/(sqrt( - c**2*x**2 + 1)* 
c**4*x**8 - 2*sqrt( - c**2*x**2 + 1)*c**2*x**6 + sqrt( - c**2*x**2 + 1)*x* 
*4),x)*b*x**3 + 16*a*c**6*x**6 - 24*a*c**4*x**4 + 6*a*c**2*x**2 + a)/(3*sq 
rt(d)*sqrt( - c**2*x**2 + 1)*d**2*x**3*(c**2*x**2 - 1))