\(\int \frac {x^4 (a+b \arcsin (c x))}{(d+e x^2)^2} \, dx\) [463]

Optimal result

Integrand size = 21, antiderivative size = 787 \[ \int \frac {x^4 (a+b \arcsin (c x))}{\left (d+e x^2\right )^2} \, dx=\frac {a x}{e^2}+\frac {b \sqrt {1-c^2 x^2}}{c e^2}+\frac {b x \arcsin (c x)}{e^2}-\frac {d (a+b \arcsin (c x))}{4 e^{5/2} \left (\sqrt {-d}-\sqrt {e} x\right )}+\frac {d (a+b \arcsin (c x))}{4 e^{5/2} \left (\sqrt {-d}+\sqrt {e} x\right )}+\frac {b c d \text {arctanh}\left (\frac {\sqrt {e}-c^2 \sqrt {-d} x}{\sqrt {c^2 d+e} \sqrt {1-c^2 x^2}}\right )}{4 e^{5/2} \sqrt {c^2 d+e}}+\frac {b c d \text {arctanh}\left (\frac {\sqrt {e}+c^2 \sqrt {-d} x}{\sqrt {c^2 d+e} \sqrt {1-c^2 x^2}}\right )}{4 e^{5/2} \sqrt {c^2 d+e}}+\frac {3 \sqrt {-d} (a+b \arcsin (c x)) \log \left (1-\frac {\sqrt {e} e^{i \arcsin (c x)}}{i c \sqrt {-d}-\sqrt {c^2 d+e}}\right )}{4 e^{5/2}}-\frac {3 \sqrt {-d} (a+b \arcsin (c x)) \log \left (1+\frac {\sqrt {e} e^{i \arcsin (c x)}}{i c \sqrt {-d}-\sqrt {c^2 d+e}}\right )}{4 e^{5/2}}+\frac {3 \sqrt {-d} (a+b \arcsin (c x)) \log \left (1-\frac {\sqrt {e} e^{i \arcsin (c x)}}{i c \sqrt {-d}+\sqrt {c^2 d+e}}\right )}{4 e^{5/2}}-\frac {3 \sqrt {-d} (a+b \arcsin (c x)) \log \left (1+\frac {\sqrt {e} e^{i \arcsin (c x)}}{i c \sqrt {-d}+\sqrt {c^2 d+e}}\right )}{4 e^{5/2}}+\frac {3 i b \sqrt {-d} \operatorname {PolyLog}\left (2,-\frac {\sqrt {e} e^{i \arcsin (c x)}}{i c \sqrt {-d}-\sqrt {c^2 d+e}}\right )}{4 e^{5/2}}-\frac {3 i b \sqrt {-d} \operatorname {PolyLog}\left (2,\frac {\sqrt {e} e^{i \arcsin (c x)}}{i c \sqrt {-d}-\sqrt {c^2 d+e}}\right )}{4 e^{5/2}}+\frac {3 i b \sqrt {-d} \operatorname {PolyLog}\left (2,-\frac {\sqrt {e} e^{i \arcsin (c x)}}{i c \sqrt {-d}+\sqrt {c^2 d+e}}\right )}{4 e^{5/2}}-\frac {3 i b \sqrt {-d} \operatorname {PolyLog}\left (2,\frac {\sqrt {e} e^{i \arcsin (c x)}}{i c \sqrt {-d}+\sqrt {c^2 d+e}}\right )}{4 e^{5/2}} \] Output:

a*x/e^2+b*(-c^2*x^2+1)^(1/2)/c/e^2+b*x*arcsin(c*x)/e^2-1/4*d*(a+b*arcsin(c 
*x))/e^(5/2)/((-d)^(1/2)-e^(1/2)*x)+1/4*d*(a+b*arcsin(c*x))/e^(5/2)/((-d)^ 
(1/2)+e^(1/2)*x)+1/4*b*c*d*arctanh((e^(1/2)-c^2*(-d)^(1/2)*x)/(c^2*d+e)^(1 
/2)/(-c^2*x^2+1)^(1/2))/e^(5/2)/(c^2*d+e)^(1/2)+1/4*b*c*d*arctanh((e^(1/2) 
+c^2*(-d)^(1/2)*x)/(c^2*d+e)^(1/2)/(-c^2*x^2+1)^(1/2))/e^(5/2)/(c^2*d+e)^( 
1/2)+3/4*(-d)^(1/2)*(a+b*arcsin(c*x))*ln(1-e^(1/2)*(I*c*x+(-c^2*x^2+1)^(1/ 
2))/(I*c*(-d)^(1/2)-(c^2*d+e)^(1/2)))/e^(5/2)-3/4*(-d)^(1/2)*(a+b*arcsin(c 
*x))*ln(1+e^(1/2)*(I*c*x+(-c^2*x^2+1)^(1/2))/(I*c*(-d)^(1/2)-(c^2*d+e)^(1/ 
2)))/e^(5/2)+3/4*(-d)^(1/2)*(a+b*arcsin(c*x))*ln(1-e^(1/2)*(I*c*x+(-c^2*x^ 
2+1)^(1/2))/(I*c*(-d)^(1/2)+(c^2*d+e)^(1/2)))/e^(5/2)-3/4*(-d)^(1/2)*(a+b* 
arcsin(c*x))*ln(1+e^(1/2)*(I*c*x+(-c^2*x^2+1)^(1/2))/(I*c*(-d)^(1/2)+(c^2* 
d+e)^(1/2)))/e^(5/2)+3/4*I*b*(-d)^(1/2)*polylog(2,-e^(1/2)*(I*c*x+(-c^2*x^ 
2+1)^(1/2))/(I*c*(-d)^(1/2)-(c^2*d+e)^(1/2)))/e^(5/2)-3/4*I*b*(-d)^(1/2)*p 
olylog(2,e^(1/2)*(I*c*x+(-c^2*x^2+1)^(1/2))/(I*c*(-d)^(1/2)-(c^2*d+e)^(1/2 
)))/e^(5/2)+3/4*I*b*(-d)^(1/2)*polylog(2,-e^(1/2)*(I*c*x+(-c^2*x^2+1)^(1/2 
))/(I*c*(-d)^(1/2)+(c^2*d+e)^(1/2)))/e^(5/2)-3/4*I*b*(-d)^(1/2)*polylog(2, 
e^(1/2)*(I*c*x+(-c^2*x^2+1)^(1/2))/(I*c*(-d)^(1/2)+(c^2*d+e)^(1/2)))/e^(5/ 
2)
 

Mathematica [A] (verified)

Time = 1.05 (sec) , antiderivative size = 649, normalized size of antiderivative = 0.82 \[ \int \frac {x^4 (a+b \arcsin (c x))}{\left (d+e x^2\right )^2} \, dx=\frac {8 a \sqrt {e} x+\frac {4 a d \sqrt {e} x}{d+e x^2}-12 a \sqrt {d} \arctan \left (\frac {\sqrt {e} x}{\sqrt {d}}\right )+b \left (\frac {8 \sqrt {e} \left (\sqrt {1-c^2 x^2}+c x \arcsin (c x)\right )}{c}+2 i d \left (\frac {\arcsin (c x)}{\sqrt {d}+i \sqrt {e} x}-\frac {c \arctan \left (\frac {i \sqrt {e}+c^2 \sqrt {d} x}{\sqrt {c^2 d+e} \sqrt {1-c^2 x^2}}\right )}{\sqrt {c^2 d+e}}\right )+2 d \left (\frac {\arcsin (c x)}{i \sqrt {d}+\sqrt {e} x}+\frac {c \text {arctanh}\left (\frac {\sqrt {e}+i c^2 \sqrt {d} x}{\sqrt {c^2 d+e} \sqrt {1-c^2 x^2}}\right )}{\sqrt {c^2 d+e}}\right )+3 \sqrt {d} \left (\arcsin (c x) \left (\arcsin (c x)+2 i \left (\log \left (1+\frac {\sqrt {e} e^{i \arcsin (c x)}}{c \sqrt {d}-\sqrt {c^2 d+e}}\right )+\log \left (1+\frac {\sqrt {e} e^{i \arcsin (c x)}}{c \sqrt {d}+\sqrt {c^2 d+e}}\right )\right )\right )+2 \operatorname {PolyLog}\left (2,\frac {\sqrt {e} e^{i \arcsin (c x)}}{-c \sqrt {d}+\sqrt {c^2 d+e}}\right )+2 \operatorname {PolyLog}\left (2,-\frac {\sqrt {e} e^{i \arcsin (c x)}}{c \sqrt {d}+\sqrt {c^2 d+e}}\right )\right )-3 \sqrt {d} \left (\arcsin (c x) \left (\arcsin (c x)+2 i \left (\log \left (1+\frac {\sqrt {e} e^{i \arcsin (c x)}}{-c \sqrt {d}+\sqrt {c^2 d+e}}\right )+\log \left (1-\frac {\sqrt {e} e^{i \arcsin (c x)}}{c \sqrt {d}+\sqrt {c^2 d+e}}\right )\right )\right )+2 \operatorname {PolyLog}\left (2,\frac {\sqrt {e} e^{i \arcsin (c x)}}{c \sqrt {d}-\sqrt {c^2 d+e}}\right )+2 \operatorname {PolyLog}\left (2,\frac {\sqrt {e} e^{i \arcsin (c x)}}{c \sqrt {d}+\sqrt {c^2 d+e}}\right )\right )\right )}{8 e^{5/2}} \] Input:

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

Output:

(8*a*Sqrt[e]*x + (4*a*d*Sqrt[e]*x)/(d + e*x^2) - 12*a*Sqrt[d]*ArcTan[(Sqrt 
[e]*x)/Sqrt[d]] + b*((8*Sqrt[e]*(Sqrt[1 - c^2*x^2] + c*x*ArcSin[c*x]))/c + 
 (2*I)*d*(ArcSin[c*x]/(Sqrt[d] + I*Sqrt[e]*x) - (c*ArcTan[(I*Sqrt[e] + c^2 
*Sqrt[d]*x)/(Sqrt[c^2*d + e]*Sqrt[1 - c^2*x^2])])/Sqrt[c^2*d + e]) + 2*d*( 
ArcSin[c*x]/(I*Sqrt[d] + Sqrt[e]*x) + (c*ArcTanh[(Sqrt[e] + I*c^2*Sqrt[d]* 
x)/(Sqrt[c^2*d + e]*Sqrt[1 - c^2*x^2])])/Sqrt[c^2*d + e]) + 3*Sqrt[d]*(Arc 
Sin[c*x]*(ArcSin[c*x] + (2*I)*(Log[1 + (Sqrt[e]*E^(I*ArcSin[c*x]))/(c*Sqrt 
[d] - Sqrt[c^2*d + e])] + Log[1 + (Sqrt[e]*E^(I*ArcSin[c*x]))/(c*Sqrt[d] + 
 Sqrt[c^2*d + e])])) + 2*PolyLog[2, (Sqrt[e]*E^(I*ArcSin[c*x]))/(-(c*Sqrt[ 
d]) + Sqrt[c^2*d + e])] + 2*PolyLog[2, -((Sqrt[e]*E^(I*ArcSin[c*x]))/(c*Sq 
rt[d] + Sqrt[c^2*d + e]))]) - 3*Sqrt[d]*(ArcSin[c*x]*(ArcSin[c*x] + (2*I)* 
(Log[1 + (Sqrt[e]*E^(I*ArcSin[c*x]))/(-(c*Sqrt[d]) + Sqrt[c^2*d + e])] + L 
og[1 - (Sqrt[e]*E^(I*ArcSin[c*x]))/(c*Sqrt[d] + Sqrt[c^2*d + e])])) + 2*Po 
lyLog[2, (Sqrt[e]*E^(I*ArcSin[c*x]))/(c*Sqrt[d] - Sqrt[c^2*d + e])] + 2*Po 
lyLog[2, (Sqrt[e]*E^(I*ArcSin[c*x]))/(c*Sqrt[d] + Sqrt[c^2*d + e])])))/(8* 
e^(5/2))
 

Rubi [A] (verified)

Time = 2.82 (sec) , antiderivative size = 787, 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 = {5232, 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.

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

\(\Big \downarrow \) 5232

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

\(\Big \downarrow \) 2009

\(\displaystyle \frac {3 \sqrt {-d} (a+b \arcsin (c x)) \log \left (1-\frac {\sqrt {e} e^{i \arcsin (c x)}}{-\sqrt {c^2 d+e}+i c \sqrt {-d}}\right )}{4 e^{5/2}}-\frac {3 \sqrt {-d} (a+b \arcsin (c x)) \log \left (1+\frac {\sqrt {e} e^{i \arcsin (c x)}}{-\sqrt {c^2 d+e}+i c \sqrt {-d}}\right )}{4 e^{5/2}}+\frac {3 \sqrt {-d} (a+b \arcsin (c x)) \log \left (1-\frac {\sqrt {e} e^{i \arcsin (c x)}}{\sqrt {c^2 d+e}+i c \sqrt {-d}}\right )}{4 e^{5/2}}-\frac {3 \sqrt {-d} (a+b \arcsin (c x)) \log \left (1+\frac {\sqrt {e} e^{i \arcsin (c x)}}{\sqrt {c^2 d+e}+i c \sqrt {-d}}\right )}{4 e^{5/2}}-\frac {d (a+b \arcsin (c x))}{4 e^{5/2} \left (\sqrt {-d}-\sqrt {e} x\right )}+\frac {d (a+b \arcsin (c x))}{4 e^{5/2} \left (\sqrt {-d}+\sqrt {e} x\right )}+\frac {a x}{e^2}+\frac {3 i b \sqrt {-d} \operatorname {PolyLog}\left (2,-\frac {\sqrt {e} e^{i \arcsin (c x)}}{i c \sqrt {-d}-\sqrt {d c^2+e}}\right )}{4 e^{5/2}}-\frac {3 i b \sqrt {-d} \operatorname {PolyLog}\left (2,\frac {\sqrt {e} e^{i \arcsin (c x)}}{i c \sqrt {-d}-\sqrt {d c^2+e}}\right )}{4 e^{5/2}}+\frac {3 i b \sqrt {-d} \operatorname {PolyLog}\left (2,-\frac {\sqrt {e} e^{i \arcsin (c x)}}{i \sqrt {-d} c+\sqrt {d c^2+e}}\right )}{4 e^{5/2}}-\frac {3 i b \sqrt {-d} \operatorname {PolyLog}\left (2,\frac {\sqrt {e} e^{i \arcsin (c x)}}{i \sqrt {-d} c+\sqrt {d c^2+e}}\right )}{4 e^{5/2}}+\frac {b x \arcsin (c x)}{e^2}+\frac {b c d \text {arctanh}\left (\frac {\sqrt {e}-c^2 \sqrt {-d} x}{\sqrt {1-c^2 x^2} \sqrt {c^2 d+e}}\right )}{4 e^{5/2} \sqrt {c^2 d+e}}+\frac {b c d \text {arctanh}\left (\frac {c^2 \sqrt {-d} x+\sqrt {e}}{\sqrt {1-c^2 x^2} \sqrt {c^2 d+e}}\right )}{4 e^{5/2} \sqrt {c^2 d+e}}+\frac {b \sqrt {1-c^2 x^2}}{c e^2}\)

Input:

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

Output:

(a*x)/e^2 + (b*Sqrt[1 - c^2*x^2])/(c*e^2) + (b*x*ArcSin[c*x])/e^2 - (d*(a 
+ b*ArcSin[c*x]))/(4*e^(5/2)*(Sqrt[-d] - Sqrt[e]*x)) + (d*(a + b*ArcSin[c* 
x]))/(4*e^(5/2)*(Sqrt[-d] + Sqrt[e]*x)) + (b*c*d*ArcTanh[(Sqrt[e] - c^2*Sq 
rt[-d]*x)/(Sqrt[c^2*d + e]*Sqrt[1 - c^2*x^2])])/(4*e^(5/2)*Sqrt[c^2*d + e] 
) + (b*c*d*ArcTanh[(Sqrt[e] + c^2*Sqrt[-d]*x)/(Sqrt[c^2*d + e]*Sqrt[1 - c^ 
2*x^2])])/(4*e^(5/2)*Sqrt[c^2*d + e]) + (3*Sqrt[-d]*(a + b*ArcSin[c*x])*Lo 
g[1 - (Sqrt[e]*E^(I*ArcSin[c*x]))/(I*c*Sqrt[-d] - Sqrt[c^2*d + e])])/(4*e^ 
(5/2)) - (3*Sqrt[-d]*(a + b*ArcSin[c*x])*Log[1 + (Sqrt[e]*E^(I*ArcSin[c*x] 
))/(I*c*Sqrt[-d] - Sqrt[c^2*d + e])])/(4*e^(5/2)) + (3*Sqrt[-d]*(a + b*Arc 
Sin[c*x])*Log[1 - (Sqrt[e]*E^(I*ArcSin[c*x]))/(I*c*Sqrt[-d] + Sqrt[c^2*d + 
 e])])/(4*e^(5/2)) - (3*Sqrt[-d]*(a + b*ArcSin[c*x])*Log[1 + (Sqrt[e]*E^(I 
*ArcSin[c*x]))/(I*c*Sqrt[-d] + Sqrt[c^2*d + e])])/(4*e^(5/2)) + (((3*I)/4) 
*b*Sqrt[-d]*PolyLog[2, -((Sqrt[e]*E^(I*ArcSin[c*x]))/(I*c*Sqrt[-d] - Sqrt[ 
c^2*d + e]))])/e^(5/2) - (((3*I)/4)*b*Sqrt[-d]*PolyLog[2, (Sqrt[e]*E^(I*Ar 
cSin[c*x]))/(I*c*Sqrt[-d] - Sqrt[c^2*d + e])])/e^(5/2) + (((3*I)/4)*b*Sqrt 
[-d]*PolyLog[2, -((Sqrt[e]*E^(I*ArcSin[c*x]))/(I*c*Sqrt[-d] + Sqrt[c^2*d + 
 e]))])/e^(5/2) - (((3*I)/4)*b*Sqrt[-d]*PolyLog[2, (Sqrt[e]*E^(I*ArcSin[c* 
x]))/(I*c*Sqrt[-d] + Sqrt[c^2*d + e])])/e^(5/2)
 

Defintions of rubi rules used

rule 2009

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

rule 5232

Int[((a_.) + ArcSin[(c_.)*(x_)]*(b_.))^(n_.)*((f_.)*(x_))^(m_.)*((d_) + (e_ 
.)*(x_)^2)^(p_.), x_Symbol] :> Int[ExpandIntegrand[(a + b*ArcSin[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 = 5.41 (sec) , antiderivative size = 914, normalized size of antiderivative = 1.16

\[\frac {a \,c^{4} \left (\frac {c x}{e^{2}}-\frac {c^{2} d \left (-\frac {c x}{2 \left (c^{2} e \,x^{2}+c^{2} d \right )}+\frac {3 \arctan \left (\frac {e x}{\sqrt {d e}}\right )}{2 c \sqrt {d e}}\right )}{e^{2}}\right )+b \,c^{4} \left (\frac {\left (-i \sqrt {-c^{2} x^{2}+1}+c x \right ) \left (\arcsin \left (c x \right )+i\right )}{2 e^{2}}+\frac {\left (i \sqrt {-c^{2} x^{2}+1}+c x \right ) \left (\arcsin \left (c x \right )-i\right )}{2 e^{2}}+\frac {\arcsin \left (c x \right ) c^{3} d x}{2 e^{2} \left (c^{2} e \,x^{2}+c^{2} d \right )}-\frac {\sqrt {\left (2 c^{2} d +2 \sqrt {c^{2} d \left (c^{2} d +e \right )}+e \right ) e}\, \left (-2 \sqrt {c^{2} d \left (c^{2} d +e \right )}\, c^{2} d +2 c^{4} d^{2}+2 c^{2} d e -\sqrt {c^{2} d \left (c^{2} d +e \right )}\, e \right ) c^{2} d \,\operatorname {arctanh}\left (\frac {e \left (i c x +\sqrt {-c^{2} x^{2}+1}\right )}{\sqrt {\left (2 c^{2} d +2 \sqrt {c^{2} d \left (c^{2} d +e \right )}+e \right ) e}}\right )}{2 e^{5} \left (c^{2} d +e \right )}+\frac {\sqrt {\left (2 c^{2} d +2 \sqrt {c^{2} d \left (c^{2} d +e \right )}+e \right ) e}\, \left (2 c^{2} d -2 \sqrt {c^{2} d \left (c^{2} d +e \right )}+e \right ) \operatorname {arctanh}\left (\frac {e \left (i c x +\sqrt {-c^{2} x^{2}+1}\right )}{\sqrt {\left (2 c^{2} d +2 \sqrt {c^{2} d \left (c^{2} d +e \right )}+e \right ) e}}\right ) c^{2} d}{2 e^{5}}-\frac {\sqrt {-e \left (2 c^{2} d -2 \sqrt {c^{2} d \left (c^{2} d +e \right )}+e \right )}\, \left (2 c^{4} d^{2}+2 \sqrt {c^{2} d \left (c^{2} d +e \right )}\, c^{2} d +2 c^{2} d e +\sqrt {c^{2} d \left (c^{2} d +e \right )}\, e \right ) c^{2} d \arctan \left (\frac {e \left (i c x +\sqrt {-c^{2} x^{2}+1}\right )}{\sqrt {\left (-2 c^{2} d +2 \sqrt {c^{2} d \left (c^{2} d +e \right )}-e \right ) e}}\right )}{2 e^{5} \left (c^{2} d +e \right )}+\frac {\sqrt {-e \left (2 c^{2} d -2 \sqrt {c^{2} d \left (c^{2} d +e \right )}+e \right )}\, \left (2 c^{2} d +2 \sqrt {c^{2} d \left (c^{2} d +e \right )}+e \right ) \arctan \left (\frac {e \left (i c x +\sqrt {-c^{2} x^{2}+1}\right )}{\sqrt {\left (-2 c^{2} d +2 \sqrt {c^{2} d \left (c^{2} d +e \right )}-e \right ) e}}\right ) c^{2} d}{2 e^{5}}-\frac {3 d \,c^{2} \left (\munderset {\textit {\_R1} =\operatorname {RootOf}\left (e \,\textit {\_Z}^{4}+\left (-4 c^{2} d -2 e \right ) \textit {\_Z}^{2}+e \right )}{\sum }\frac {i \arcsin \left (c x \right ) \ln \left (\frac {\textit {\_R1} -i c x -\sqrt {-c^{2} x^{2}+1}}{\textit {\_R1}}\right )+\operatorname {dilog}\left (\frac {\textit {\_R1} -i c x -\sqrt {-c^{2} x^{2}+1}}{\textit {\_R1}}\right )}{\textit {\_R1} \left (\textit {\_R1}^{2} e -2 c^{2} d -e \right )}\right )}{4 e^{2}}-\frac {3 d \,c^{2} \left (\munderset {\textit {\_R1} =\operatorname {RootOf}\left (e \,\textit {\_Z}^{4}+\left (-4 c^{2} d -2 e \right ) \textit {\_Z}^{2}+e \right )}{\sum }\frac {\textit {\_R1} \left (i \arcsin \left (c x \right ) \ln \left (\frac {\textit {\_R1} -i c x -\sqrt {-c^{2} x^{2}+1}}{\textit {\_R1}}\right )+\operatorname {dilog}\left (\frac {\textit {\_R1} -i c x -\sqrt {-c^{2} x^{2}+1}}{\textit {\_R1}}\right )\right )}{\textit {\_R1}^{2} e -2 c^{2} d -e}\right )}{4 e^{2}}\right )}{c^{5}}\]

Input:

int(x^4*(a+b*arcsin(c*x))/(e*x^2+d)^2,x)
 

Output:

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

Fricas [F]

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

integrate(x^4*(a+b*arcsin(c*x))/(e*x^2+d)^2,x, algorithm="fricas")
 

Output:

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

Sympy [F]

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

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

Output:

Integral(x**4*(a + b*asin(c*x))/(d + e*x**2)**2, x)
 

Maxima [F(-2)]

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

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

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

integrate(x^4*(a+b*arcsin(c*x))/(e*x^2+d)^2,x, algorithm="giac")
 

Output:

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

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

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

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

Output:

( - 3*sqrt(e)*sqrt(d)*atan((e*x)/(sqrt(e)*sqrt(d)))*a*d - 3*sqrt(e)*sqrt(d 
)*atan((e*x)/(sqrt(e)*sqrt(d)))*a*e*x**2 + 2*int((asin(c*x)*x**4)/(d**2 + 
2*d*e*x**2 + e**2*x**4),x)*b*d*e**3 + 2*int((asin(c*x)*x**4)/(d**2 + 2*d*e 
*x**2 + e**2*x**4),x)*b*e**4*x**2 + 3*a*d*e*x + 2*a*e**2*x**3)/(2*e**3*(d 
+ e*x**2))