\(\int \frac {x^2 (a+b \arccos (c x))}{(d+e x^2)^3} \, dx\) [649]

Optimal result

Integrand size = 21, antiderivative size = 1092 \[ \int \frac {x^2 (a+b \arccos (c x))}{\left (d+e x^2\right )^3} \, dx =\text {Too large to display} \] Output:

1/16*b*c*(-c^2*x^2+1)^(1/2)/(-d)^(1/2)/e/(c^2*d+e)/((-d)^(1/2)-e^(1/2)*x)+ 
1/16*b*c*(-c^2*x^2+1)^(1/2)/(-d)^(1/2)/e/(c^2*d+e)/((-d)^(1/2)+e^(1/2)*x)- 
1/16*(a+b*arccos(c*x))/(-d)^(1/2)/e^(3/2)/((-d)^(1/2)-e^(1/2)*x)^2-1/16*(a 
+b*arccos(c*x))/d/e^(3/2)/((-d)^(1/2)-e^(1/2)*x)+1/16*(a+b*arccos(c*x))/(- 
d)^(1/2)/e^(3/2)/((-d)^(1/2)+e^(1/2)*x)^2+1/16*(a+b*arccos(c*x))/d/e^(3/2) 
/((-d)^(1/2)+e^(1/2)*x)-1/16*b*c^3*arctanh((e^(1/2)-c^2*(-d)^(1/2)*x)/(c^2 
*d+e)^(1/2)/(-c^2*x^2+1)^(1/2))/e^(3/2)/(c^2*d+e)^(3/2)+1/16*b*c*arctanh(( 
e^(1/2)-c^2*(-d)^(1/2)*x)/(c^2*d+e)^(1/2)/(-c^2*x^2+1)^(1/2))/d/e^(3/2)/(c 
^2*d+e)^(1/2)-1/16*b*c^3*arctanh((e^(1/2)+c^2*(-d)^(1/2)*x)/(c^2*d+e)^(1/2 
)/(-c^2*x^2+1)^(1/2))/e^(3/2)/(c^2*d+e)^(3/2)+1/16*b*c*arctanh((e^(1/2)+c^ 
2*(-d)^(1/2)*x)/(c^2*d+e)^(1/2)/(-c^2*x^2+1)^(1/2))/d/e^(3/2)/(c^2*d+e)^(1 
/2)-1/16*(a+b*arccos(c*x))*ln(1-e^(1/2)*(c*x+I*(-c^2*x^2+1)^(1/2))/(I*c*(- 
d)^(1/2)-(c^2*d+e)^(1/2)))/(-d)^(3/2)/e^(3/2)+1/16*(a+b*arccos(c*x))*ln(1+ 
e^(1/2)*(c*x+I*(-c^2*x^2+1)^(1/2))/(I*c*(-d)^(1/2)-(c^2*d+e)^(1/2)))/(-d)^ 
(3/2)/e^(3/2)-1/16*(a+b*arccos(c*x))*ln(1-e^(1/2)*(c*x+I*(-c^2*x^2+1)^(1/2 
))/(I*c*(-d)^(1/2)+(c^2*d+e)^(1/2)))/(-d)^(3/2)/e^(3/2)+1/16*(a+b*arccos(c 
*x))*ln(1+e^(1/2)*(c*x+I*(-c^2*x^2+1)^(1/2))/(I*c*(-d)^(1/2)+(c^2*d+e)^(1/ 
2)))/(-d)^(3/2)/e^(3/2)+1/16*I*b*polylog(2,e^(1/2)*(c*x+I*(-c^2*x^2+1)^(1/ 
2))/(I*c*(-d)^(1/2)+(c^2*d+e)^(1/2)))/(-d)^(3/2)/e^(3/2)-1/16*I*b*polylog( 
2,-e^(1/2)*(c*x+I*(-c^2*x^2+1)^(1/2))/(I*c*(-d)^(1/2)-(c^2*d+e)^(1/2)))...
 

Mathematica [A] (warning: unable to verify)

Time = 5.80 (sec) , antiderivative size = 1531, normalized size of antiderivative = 1.40 \[ \int \frac {x^2 (a+b \arccos (c x))}{\left (d+e x^2\right )^3} \, dx =\text {Too large to display} \] Input:

Integrate[(x^2*(a + b*ArcCos[c*x]))/(d + e*x^2)^3,x]
 

Output:

((-8*a*Sqrt[e]*x)/(d + e*x^2)^2 + (4*a*Sqrt[e]*x)/(d^2 + d*e*x^2) + (4*a*A 
rcTan[(Sqrt[e]*x)/Sqrt[d]])/d^(3/2) + (2*b*(ArcCos[c*x]/((-I)*Sqrt[d] + Sq 
rt[e]*x) - (c*Log[(2*e*(Sqrt[e] - I*c^2*Sqrt[d]*x + Sqrt[c^2*d + e]*Sqrt[1 
 - c^2*x^2]))/(c*Sqrt[c^2*d + e]*((-I)*Sqrt[d] + Sqrt[e]*x))])/Sqrt[c^2*d 
+ e]))/d + 2*b*(((-I)*c*Sqrt[e]*Sqrt[1 - c^2*x^2])/(Sqrt[d]*(c^2*d + e)*(( 
-I)*Sqrt[d] + Sqrt[e]*x)) - (I*ArcCos[c*x])/(Sqrt[d]*(Sqrt[d] + I*Sqrt[e]* 
x)^2) + (c^3*Log[(-4*e*Sqrt[c^2*d + e]*(Sqrt[e] - I*c^2*Sqrt[d]*x + Sqrt[c 
^2*d + e]*Sqrt[1 - c^2*x^2]))/(c^3*(d + I*Sqrt[d]*Sqrt[e]*x))])/(c^2*d + e 
)^(3/2)) - (2*b*(-(ArcCos[c*x]/(I*Sqrt[d] + Sqrt[e]*x)) + (c*Log[(-2*e*(Sq 
rt[e] + I*c^2*Sqrt[d]*x + Sqrt[c^2*d + e]*Sqrt[1 - c^2*x^2]))/(c*Sqrt[c^2* 
d + e]*(I*Sqrt[d] + Sqrt[e]*x))])/Sqrt[c^2*d + e]))/d + 2*b*((I*c*Sqrt[e]* 
Sqrt[1 - c^2*x^2])/(Sqrt[d]*(c^2*d + e)*(I*Sqrt[d] + Sqrt[e]*x)) + (I*ArcC 
os[c*x])/(Sqrt[d]*(Sqrt[d] - I*Sqrt[e]*x)^2) + (c^3*Log[(-4*e*Sqrt[c^2*d + 
 e]*(Sqrt[e] + I*c^2*Sqrt[d]*x + Sqrt[c^2*d + e]*Sqrt[1 - c^2*x^2]))/(c^3* 
(d - I*Sqrt[d]*Sqrt[e]*x))])/(c^2*d + e)^(3/2)) + (b*(ArcCos[c*x]^2 - 8*Ar 
cSin[Sqrt[1 + (I*c*Sqrt[d])/Sqrt[e]]/Sqrt[2]]*ArcTan[((c*Sqrt[d] + I*Sqrt[ 
e])*Tan[ArcCos[c*x]/2])/Sqrt[c^2*d + e]] + (2*I)*ArcCos[c*x]*Log[1 - (I*(- 
(c*Sqrt[d]) + Sqrt[c^2*d + e])*E^(I*ArcCos[c*x]))/Sqrt[e]] + (4*I)*ArcSin[ 
Sqrt[1 + (I*c*Sqrt[d])/Sqrt[e]]/Sqrt[2]]*Log[1 - (I*(-(c*Sqrt[d]) + Sqrt[c 
^2*d + e])*E^(I*ArcCos[c*x]))/Sqrt[e]] + (2*I)*ArcCos[c*x]*Log[1 + (I*(...
 

Rubi [A] (verified)

Time = 3.19 (sec) , antiderivative size = 1092, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.095, Rules used = {5233, 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.

\(\displaystyle \int \frac {x^2 (a+b \arccos (c x))}{\left (d+e x^2\right )^3} \, dx\)

\(\Big \downarrow \) 5233

\(\displaystyle \int \left (\frac {a+b \arccos (c x)}{e \left (d+e x^2\right )^2}-\frac {d (a+b \arccos (c x))}{e \left (d+e x^2\right )^3}\right )dx\)

\(\Big \downarrow \) 2009

\(\displaystyle \frac {b \text {arctanh}\left (\frac {\sqrt {e}-c^2 \sqrt {-d} x}{\sqrt {d c^2+e} \sqrt {1-c^2 x^2}}\right ) c^3}{16 e^{3/2} \left (d c^2+e\right )^{3/2}}+\frac {b \text {arctanh}\left (\frac {\sqrt {-d} x c^2+\sqrt {e}}{\sqrt {d c^2+e} \sqrt {1-c^2 x^2}}\right ) c^3}{16 e^{3/2} \left (d c^2+e\right )^{3/2}}-\frac {b \text {arctanh}\left (\frac {\sqrt {e}-c^2 \sqrt {-d} x}{\sqrt {d c^2+e} \sqrt {1-c^2 x^2}}\right ) c}{16 d e^{3/2} \sqrt {d c^2+e}}-\frac {b \text {arctanh}\left (\frac {\sqrt {-d} x c^2+\sqrt {e}}{\sqrt {d c^2+e} \sqrt {1-c^2 x^2}}\right ) c}{16 d e^{3/2} \sqrt {d c^2+e}}-\frac {b \sqrt {1-c^2 x^2} c}{16 \sqrt {-d} e \left (d c^2+e\right ) \left (\sqrt {-d}-\sqrt {e} x\right )}-\frac {b \sqrt {1-c^2 x^2} c}{16 \sqrt {-d} e \left (d c^2+e\right ) \left (\sqrt {e} x+\sqrt {-d}\right )}-\frac {a+b \arccos (c x)}{16 d e^{3/2} \left (\sqrt {-d}-\sqrt {e} x\right )}+\frac {a+b \arccos (c x)}{16 d e^{3/2} \left (\sqrt {e} x+\sqrt {-d}\right )}-\frac {a+b \arccos (c x)}{16 \sqrt {-d} e^{3/2} \left (\sqrt {-d}-\sqrt {e} x\right )^2}+\frac {a+b \arccos (c x)}{16 \sqrt {-d} e^{3/2} \left (\sqrt {e} x+\sqrt {-d}\right )^2}-\frac {(a+b \arccos (c x)) \log \left (1-\frac {\sqrt {e} e^{i \arccos (c x)}}{c \sqrt {-d}-i \sqrt {d c^2+e}}\right )}{16 (-d)^{3/2} e^{3/2}}+\frac {(a+b \arccos (c x)) \log \left (\frac {e^{i \arccos (c x)} \sqrt {e}}{c \sqrt {-d}-i \sqrt {d c^2+e}}+1\right )}{16 (-d)^{3/2} e^{3/2}}-\frac {(a+b \arccos (c x)) \log \left (1-\frac {\sqrt {e} e^{i \arccos (c x)}}{\sqrt {-d} c+i \sqrt {d c^2+e}}\right )}{16 (-d)^{3/2} e^{3/2}}+\frac {(a+b \arccos (c x)) \log \left (\frac {e^{i \arccos (c x)} \sqrt {e}}{\sqrt {-d} c+i \sqrt {d c^2+e}}+1\right )}{16 (-d)^{3/2} e^{3/2}}-\frac {i b \operatorname {PolyLog}\left (2,-\frac {\sqrt {e} e^{i \arccos (c x)}}{c \sqrt {-d}-i \sqrt {d c^2+e}}\right )}{16 (-d)^{3/2} e^{3/2}}+\frac {i b \operatorname {PolyLog}\left (2,\frac {\sqrt {e} e^{i \arccos (c x)}}{c \sqrt {-d}-i \sqrt {d c^2+e}}\right )}{16 (-d)^{3/2} e^{3/2}}-\frac {i b \operatorname {PolyLog}\left (2,-\frac {\sqrt {e} e^{i \arccos (c x)}}{\sqrt {-d} c+i \sqrt {d c^2+e}}\right )}{16 (-d)^{3/2} e^{3/2}}+\frac {i b \operatorname {PolyLog}\left (2,\frac {\sqrt {e} e^{i \arccos (c x)}}{\sqrt {-d} c+i \sqrt {d c^2+e}}\right )}{16 (-d)^{3/2} e^{3/2}}\)

Input:

Int[(x^2*(a + b*ArcCos[c*x]))/(d + e*x^2)^3,x]
 

Output:

-1/16*(b*c*Sqrt[1 - c^2*x^2])/(Sqrt[-d]*e*(c^2*d + e)*(Sqrt[-d] - Sqrt[e]* 
x)) - (b*c*Sqrt[1 - c^2*x^2])/(16*Sqrt[-d]*e*(c^2*d + e)*(Sqrt[-d] + Sqrt[ 
e]*x)) - (a + b*ArcCos[c*x])/(16*Sqrt[-d]*e^(3/2)*(Sqrt[-d] - Sqrt[e]*x)^2 
) - (a + b*ArcCos[c*x])/(16*d*e^(3/2)*(Sqrt[-d] - Sqrt[e]*x)) + (a + b*Arc 
Cos[c*x])/(16*Sqrt[-d]*e^(3/2)*(Sqrt[-d] + Sqrt[e]*x)^2) + (a + b*ArcCos[c 
*x])/(16*d*e^(3/2)*(Sqrt[-d] + Sqrt[e]*x)) + (b*c^3*ArcTanh[(Sqrt[e] - c^2 
*Sqrt[-d]*x)/(Sqrt[c^2*d + e]*Sqrt[1 - c^2*x^2])])/(16*e^(3/2)*(c^2*d + e) 
^(3/2)) - (b*c*ArcTanh[(Sqrt[e] - c^2*Sqrt[-d]*x)/(Sqrt[c^2*d + e]*Sqrt[1 
- c^2*x^2])])/(16*d*e^(3/2)*Sqrt[c^2*d + e]) + (b*c^3*ArcTanh[(Sqrt[e] + c 
^2*Sqrt[-d]*x)/(Sqrt[c^2*d + e]*Sqrt[1 - c^2*x^2])])/(16*e^(3/2)*(c^2*d + 
e)^(3/2)) - (b*c*ArcTanh[(Sqrt[e] + c^2*Sqrt[-d]*x)/(Sqrt[c^2*d + e]*Sqrt[ 
1 - c^2*x^2])])/(16*d*e^(3/2)*Sqrt[c^2*d + e]) - ((a + b*ArcCos[c*x])*Log[ 
1 - (Sqrt[e]*E^(I*ArcCos[c*x]))/(c*Sqrt[-d] - I*Sqrt[c^2*d + e])])/(16*(-d 
)^(3/2)*e^(3/2)) + ((a + b*ArcCos[c*x])*Log[1 + (Sqrt[e]*E^(I*ArcCos[c*x]) 
)/(c*Sqrt[-d] - I*Sqrt[c^2*d + e])])/(16*(-d)^(3/2)*e^(3/2)) - ((a + b*Arc 
Cos[c*x])*Log[1 - (Sqrt[e]*E^(I*ArcCos[c*x]))/(c*Sqrt[-d] + I*Sqrt[c^2*d + 
 e])])/(16*(-d)^(3/2)*e^(3/2)) + ((a + b*ArcCos[c*x])*Log[1 + (Sqrt[e]*E^( 
I*ArcCos[c*x]))/(c*Sqrt[-d] + I*Sqrt[c^2*d + e])])/(16*(-d)^(3/2)*e^(3/2)) 
 - ((I/16)*b*PolyLog[2, -((Sqrt[e]*E^(I*ArcCos[c*x]))/(c*Sqrt[-d] - I*Sqrt 
[c^2*d + e]))])/((-d)^(3/2)*e^(3/2)) + ((I/16)*b*PolyLog[2, (Sqrt[e]*E^...
 

Defintions of rubi rules used

rule 2009

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

rule 5233

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

Maple [C] (warning: unable to verify)

Result contains higher order function than in optimal. Order 9 vs. order 4.

Time = 10.10 (sec) , antiderivative size = 1226, normalized size of antiderivative = 1.12

method	result	size
parts	\(\text {Expression too large to display}\)	\(1226\)
derivativedivides	\(\text {Expression too large to display}\)	\(1241\)
default	\(\text {Expression too large to display}\)	\(1241\)

Input:

int(x^2*(a+b*arccos(c*x))/(e*x^2+d)^3,x,method=_RETURNVERBOSE)
 

Output:

a*((1/8/d*x^3-1/8/e*x)/(e*x^2+d)^2+1/8/d/e/(d*e)^(1/2)*arctan(e*x/(d*e)^(1 
/2)))+b/c^3*(1/8*c^4*(arccos(c*x)*c^5*d*e*x^3-arccos(c*x)*d^2*c^5*x+c^4*d* 
e*x^2*(-c^2*x^2+1)^(1/2)+d^2*c^4*(-c^2*x^2+1)^(1/2)+arccos(c*x)*e^2*c^3*x^ 
3-arccos(c*x)*c^3*d*e*x)/e/d/(c^2*d+e)/(c^2*e*x^2+c^2*d)^2-1/8*I*((2*c^2*d 
+2*(c^2*d*(c^2*d+e))^(1/2)+e)*e)^(1/2)*(-2*d*c^2*(c^2*d*(c^2*d+e))^(1/2)+2 
*c^4*d^2+2*c^2*d*e-(c^2*d*(c^2*d+e))^(1/2)*e)*c^4*arctan(e*(c*x+I*(-c^2*x^ 
2+1)^(1/2))/((2*c^2*d+2*(c^2*d*(c^2*d+e))^(1/2)+e)*e)^(1/2))/(c^2*d+e)^2/e 
^3/d+1/8*I*((2*c^2*d+2*(c^2*d*(c^2*d+e))^(1/2)+e)*e)^(1/2)*(2*c^2*d-2*(c^2 
*d*(c^2*d+e))^(1/2)+e)*arctan(e*(c*x+I*(-c^2*x^2+1)^(1/2))/((2*c^2*d+2*(c^ 
2*d*(c^2*d+e))^(1/2)+e)*e)^(1/2))*c^4/e^3/d/(c^2*d+e)-1/8*I*(-(2*c^2*d-2*( 
c^2*d*(c^2*d+e))^(1/2)+e)*e)^(1/2)*(2*c^4*d^2+2*d*c^2*(c^2*d*(c^2*d+e))^(1 
/2)+2*c^2*d*e+(c^2*d*(c^2*d+e))^(1/2)*e)*c^4*arctanh(e*(c*x+I*(-c^2*x^2+1) 
^(1/2))/((-2*c^2*d+2*(c^2*d*(c^2*d+e))^(1/2)-e)*e)^(1/2))/(c^2*d+e)^2/e^3/ 
d+1/8*I*(-(2*c^2*d-2*(c^2*d*(c^2*d+e))^(1/2)+e)*e)^(1/2)*(2*c^2*d+2*(c^2*d 
*(c^2*d+e))^(1/2)+e)*arctanh(e*(c*x+I*(-c^2*x^2+1)^(1/2))/((-2*c^2*d+2*(c^ 
2*d*(c^2*d+e))^(1/2)-e)*e)^(1/2))*c^4/e^3/d/(c^2*d+e)-1/16*I/(c^2*d+e)/d*c 
^4*sum(_R1/(_R1^2*e+2*c^2*d+e)*(I*arccos(c*x)*ln((_R1-c*x-I*(-c^2*x^2+1)^( 
1/2))/_R1)+dilog((_R1-c*x-I*(-c^2*x^2+1)^(1/2))/_R1)),_R1=RootOf(e*_Z^4+(4 
*c^2*d+2*e)*_Z^2+e))-1/16*I/(c^2*d+e)*c^6/e*sum(_R1/(_R1^2*e+2*c^2*d+e)*(I 
*arccos(c*x)*ln((_R1-c*x-I*(-c^2*x^2+1)^(1/2))/_R1)+dilog((_R1-c*x-I*(-...
 

Fricas [F]

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

integrate(x^2*(a+b*arccos(c*x))/(e*x^2+d)^3,x, algorithm="fricas")
 

Output:

integral((b*x^2*arccos(c*x) + a*x^2)/(e^3*x^6 + 3*d*e^2*x^4 + 3*d^2*e*x^2 
+ d^3), x)
 

Sympy [F(-1)]

Timed out. \[ \int \frac {x^2 (a+b \arccos (c x))}{\left (d+e x^2\right )^3} \, dx=\text {Timed out} \] Input:

integrate(x**2*(a+b*acos(c*x))/(e*x**2+d)**3,x)
 

Output:

Timed out
 

Maxima [F(-2)]

Exception generated. \[ \int \frac {x^2 (a+b \arccos (c x))}{\left (d+e x^2\right )^3} \, dx=\text {Exception raised: ValueError} \] Input:

integrate(x^2*(a+b*arccos(c*x))/(e*x^2+d)^3,x, algorithm="maxima")
 

Output:

Exception raised: ValueError >> Computation failed since Maxima requested 
additional constraints; using the 'assume' command before evaluation *may* 
 help (example of legal syntax is 'assume(e>0)', see `assume?` for more de 
tails)Is e
 

Giac [F]

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

integrate(x^2*(a+b*arccos(c*x))/(e*x^2+d)^3,x, algorithm="giac")
 

Output:

integrate((b*arccos(c*x) + a)*x^2/(e*x^2 + d)^3, x)
                                                                                    
                                                                                    
 

Mupad [F(-1)]

Timed out. \[ \int \frac {x^2 (a+b \arccos (c x))}{\left (d+e x^2\right )^3} \, dx=\int \frac {x^2\,\left (a+b\,\mathrm {acos}\left (c\,x\right )\right )}{{\left (e\,x^2+d\right )}^3} \,d x \] Input:

int((x^2*(a + b*acos(c*x)))/(d + e*x^2)^3,x)
 

Output:

int((x^2*(a + b*acos(c*x)))/(d + e*x^2)^3, x)
 

Reduce [F]

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

int(x^2*(a+b*acos(c*x))/(e*x^2+d)^3,x)
 

Output:

(sqrt(e)*sqrt(d)*atan((e*x)/(sqrt(e)*sqrt(d)))*a*d**2 + 2*sqrt(e)*sqrt(d)* 
atan((e*x)/(sqrt(e)*sqrt(d)))*a*d*e*x**2 + sqrt(e)*sqrt(d)*atan((e*x)/(sqr 
t(e)*sqrt(d)))*a*e**2*x**4 + 8*int((acos(c*x)*x**2)/(d**3 + 3*d**2*e*x**2 
+ 3*d*e**2*x**4 + e**3*x**6),x)*b*d**4*e**2 + 16*int((acos(c*x)*x**2)/(d** 
3 + 3*d**2*e*x**2 + 3*d*e**2*x**4 + e**3*x**6),x)*b*d**3*e**3*x**2 + 8*int 
((acos(c*x)*x**2)/(d**3 + 3*d**2*e*x**2 + 3*d*e**2*x**4 + e**3*x**6),x)*b* 
d**2*e**4*x**4 - a*d**2*e*x + a*d*e**2*x**3)/(8*d**2*e**2*(d**2 + 2*d*e*x* 
*2 + e**2*x**4))