\(\int \frac {a+b \arcsin (c x)}{x^3 (d+e x^2)^3} \, dx\) [471]

Optimal result

Integrand size = 21, antiderivative size = 783 \[ \int \frac {a+b \arcsin (c x)}{x^3 \left (d+e x^2\right )^3} \, dx=-\frac {b c \sqrt {1-c^2 x^2}}{2 d^3 x}+\frac {b c e^2 x \sqrt {1-c^2 x^2}}{8 d^3 \left (c^2 d+e\right ) \left (d+e x^2\right )}-\frac {a+b \arcsin (c x)}{2 d^3 x^2}-\frac {e (a+b \arcsin (c x))}{4 d^2 \left (d+e x^2\right )^2}-\frac {e (a+b \arcsin (c x))}{d^3 \left (d+e x^2\right )}+\frac {b c e \arctan \left (\frac {\sqrt {c^2 d+e} x}{\sqrt {d} \sqrt {1-c^2 x^2}}\right )}{d^{7/2} \sqrt {c^2 d+e}}+\frac {b c e \left (2 c^2 d+e\right ) \arctan \left (\frac {\sqrt {c^2 d+e} x}{\sqrt {d} \sqrt {1-c^2 x^2}}\right )}{8 d^{7/2} \left (c^2 d+e\right )^{3/2}}+\frac {3 e (a+b \arcsin (c x)) \log \left (1-\frac {\sqrt {e} e^{i \arcsin (c x)}}{i c \sqrt {-d}-\sqrt {c^2 d+e}}\right )}{2 d^4}+\frac {3 e (a+b \arcsin (c x)) \log \left (1+\frac {\sqrt {e} e^{i \arcsin (c x)}}{i c \sqrt {-d}-\sqrt {c^2 d+e}}\right )}{2 d^4}+\frac {3 e (a+b \arcsin (c x)) \log \left (1-\frac {\sqrt {e} e^{i \arcsin (c x)}}{i c \sqrt {-d}+\sqrt {c^2 d+e}}\right )}{2 d^4}+\frac {3 e (a+b \arcsin (c x)) \log \left (1+\frac {\sqrt {e} e^{i \arcsin (c x)}}{i c \sqrt {-d}+\sqrt {c^2 d+e}}\right )}{2 d^4}-\frac {3 e (a+b \arcsin (c x)) \log \left (1-e^{2 i \arcsin (c x)}\right )}{d^4}-\frac {3 i b e \operatorname {PolyLog}\left (2,-\frac {\sqrt {e} e^{i \arcsin (c x)}}{i c \sqrt {-d}-\sqrt {c^2 d+e}}\right )}{2 d^4}-\frac {3 i b e \operatorname {PolyLog}\left (2,\frac {\sqrt {e} e^{i \arcsin (c x)}}{i c \sqrt {-d}-\sqrt {c^2 d+e}}\right )}{2 d^4}-\frac {3 i b e \operatorname {PolyLog}\left (2,-\frac {\sqrt {e} e^{i \arcsin (c x)}}{i c \sqrt {-d}+\sqrt {c^2 d+e}}\right )}{2 d^4}-\frac {3 i b e \operatorname {PolyLog}\left (2,\frac {\sqrt {e} e^{i \arcsin (c x)}}{i c \sqrt {-d}+\sqrt {c^2 d+e}}\right )}{2 d^4}+\frac {3 i b e \operatorname {PolyLog}\left (2,e^{2 i \arcsin (c x)}\right )}{2 d^4} \] Output:

-1/2*b*c*(-c^2*x^2+1)^(1/2)/d^3/x+1/8*b*c*e^2*x*(-c^2*x^2+1)^(1/2)/d^3/(c^ 
2*d+e)/(e*x^2+d)-1/2*(a+b*arcsin(c*x))/d^3/x^2-1/4*e*(a+b*arcsin(c*x))/d^2 
/(e*x^2+d)^2-e*(a+b*arcsin(c*x))/d^3/(e*x^2+d)+b*c*e*arctan((c^2*d+e)^(1/2 
)*x/d^(1/2)/(-c^2*x^2+1)^(1/2))/d^(7/2)/(c^2*d+e)^(1/2)+1/8*b*c*e*(2*c^2*d 
+e)*arctan((c^2*d+e)^(1/2)*x/d^(1/2)/(-c^2*x^2+1)^(1/2))/d^(7/2)/(c^2*d+e) 
^(3/2)+3/2*e*(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)))/d^4+3/2*e*(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)))/d^4+3/2*e*(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)))/d^4+3/2*e*(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)))/d^4-3*e*(a+b*arcsin(c*x))*ln(1-(I 
*c*x+(-c^2*x^2+1)^(1/2))^2)/d^4+3/2*I*b*e*polylog(2,(I*c*x+(-c^2*x^2+1)^(1 
/2))^2)/d^4-3/2*I*b*e*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)))/d^4-3/2*I*b*e*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)))/d^4-3/2*I*b*e*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)))/d^4-3/2 
*I*b*e*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)))/d^4
 

Mathematica [A] (warning: unable to verify)

Time = 5.30 (sec) , antiderivative size = 1065, normalized size of antiderivative = 1.36 \[ \int \frac {a+b \arcsin (c x)}{x^3 \left (d+e x^2\right )^3} \, dx =\text {Too large to display} \] Input:

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

Output:

((-8*a*d)/x^2 - (4*a*d^2*e)/(d + e*x^2)^2 - (16*a*d*e)/(d + e*x^2) - 48*a* 
e*Log[x] + 24*a*e*Log[d + e*x^2] + b*((-8*c*d*Sqrt[1 - c^2*x^2])/x + (c*d* 
e^(3/2)*Sqrt[1 - c^2*x^2])/((c^2*d + e)*((-I)*Sqrt[d] + Sqrt[e]*x)) + (c*d 
*e^(3/2)*Sqrt[1 - c^2*x^2])/((c^2*d + e)*(I*Sqrt[d] + Sqrt[e]*x)) - (8*d*A 
rcSin[c*x])/x^2 - (9*Sqrt[d]*e*ArcSin[c*x])/(Sqrt[d] - I*Sqrt[e]*x) - (d*e 
*ArcSin[c*x])/(Sqrt[d] + I*Sqrt[e]*x)^2 - (9*Sqrt[d]*e*ArcSin[c*x])/(Sqrt[ 
d] + I*Sqrt[e]*x) + (d*e*ArcSin[c*x])/(I*Sqrt[d] + Sqrt[e]*x)^2 + (9*c*Sqr 
t[d]*e*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] - ((9*I)*c*Sqrt[d]*e*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] + 24*e*ArcSin[c 
*x]*Log[1 + (Sqrt[e]*E^(I*ArcSin[c*x]))/(c*Sqrt[d] - Sqrt[c^2*d + e])] + 2 
4*e*ArcSin[c*x]*Log[1 + (Sqrt[e]*E^(I*ArcSin[c*x]))/(-(c*Sqrt[d]) + Sqrt[c 
^2*d + e])] + 24*e*ArcSin[c*x]*Log[1 - (Sqrt[e]*E^(I*ArcSin[c*x]))/(c*Sqrt 
[d] + Sqrt[c^2*d + e])] + 24*e*ArcSin[c*x]*Log[1 + (Sqrt[e]*E^(I*ArcSin[c* 
x]))/(c*Sqrt[d] + Sqrt[c^2*d + e])] - 48*e*ArcSin[c*x]*Log[1 - E^((2*I)*Ar 
cSin[c*x])] + (I*c^3*d^(3/2)*e*Log[(e*Sqrt[c^2*d + e]*(Sqrt[e] - I*c^2*Sqr 
t[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) - (I*c^3*d^(3/2)*e*Log[(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) - (24*I)*e*PolyLog[2, (Sqrt[e]*E^(I*A...
 

Rubi [A] (verified)

Time = 1.81 (sec) , antiderivative size = 783, 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.

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

\(\Big \downarrow \) 5232

\(\displaystyle \int \left (\frac {3 e^2 x (a+b \arcsin (c x))}{d^4 \left (d+e x^2\right )}-\frac {3 e (a+b \arcsin (c x))}{d^4 x}+\frac {2 e^2 x (a+b \arcsin (c x))}{d^3 \left (d+e x^2\right )^2}+\frac {a+b \arcsin (c x)}{d^3 x^3}+\frac {e^2 x (a+b \arcsin (c x))}{d^2 \left (d+e x^2\right )^3}\right )dx\)

\(\Big \downarrow \) 2009

\(\displaystyle \frac {3 e (a+b \arcsin (c x)) \log \left (1-\frac {\sqrt {e} e^{i \arcsin (c x)}}{-\sqrt {c^2 d+e}+i c \sqrt {-d}}\right )}{2 d^4}+\frac {3 e (a+b \arcsin (c x)) \log \left (1+\frac {\sqrt {e} e^{i \arcsin (c x)}}{-\sqrt {c^2 d+e}+i c \sqrt {-d}}\right )}{2 d^4}+\frac {3 e (a+b \arcsin (c x)) \log \left (1-\frac {\sqrt {e} e^{i \arcsin (c x)}}{\sqrt {c^2 d+e}+i c \sqrt {-d}}\right )}{2 d^4}+\frac {3 e (a+b \arcsin (c x)) \log \left (1+\frac {\sqrt {e} e^{i \arcsin (c x)}}{\sqrt {c^2 d+e}+i c \sqrt {-d}}\right )}{2 d^4}-\frac {3 e \log \left (1-e^{2 i \arcsin (c x)}\right ) (a+b \arcsin (c x))}{d^4}-\frac {e (a+b \arcsin (c x))}{d^3 \left (d+e x^2\right )}-\frac {a+b \arcsin (c x)}{2 d^3 x^2}-\frac {e (a+b \arcsin (c x))}{4 d^2 \left (d+e x^2\right )^2}-\frac {3 i b e \operatorname {PolyLog}\left (2,-\frac {\sqrt {e} e^{i \arcsin (c x)}}{i c \sqrt {-d}-\sqrt {d c^2+e}}\right )}{2 d^4}-\frac {3 i b e \operatorname {PolyLog}\left (2,\frac {\sqrt {e} e^{i \arcsin (c x)}}{i c \sqrt {-d}-\sqrt {d c^2+e}}\right )}{2 d^4}-\frac {3 i b e \operatorname {PolyLog}\left (2,-\frac {\sqrt {e} e^{i \arcsin (c x)}}{i \sqrt {-d} c+\sqrt {d c^2+e}}\right )}{2 d^4}-\frac {3 i b e \operatorname {PolyLog}\left (2,\frac {\sqrt {e} e^{i \arcsin (c x)}}{i \sqrt {-d} c+\sqrt {d c^2+e}}\right )}{2 d^4}+\frac {3 i b e \operatorname {PolyLog}\left (2,e^{2 i \arcsin (c x)}\right )}{2 d^4}+\frac {b c e \left (2 c^2 d+e\right ) \arctan \left (\frac {x \sqrt {c^2 d+e}}{\sqrt {d} \sqrt {1-c^2 x^2}}\right )}{8 d^{7/2} \left (c^2 d+e\right )^{3/2}}+\frac {b c e \arctan \left (\frac {x \sqrt {c^2 d+e}}{\sqrt {d} \sqrt {1-c^2 x^2}}\right )}{d^{7/2} \sqrt {c^2 d+e}}+\frac {b c e^2 x \sqrt {1-c^2 x^2}}{8 d^3 \left (c^2 d+e\right ) \left (d+e x^2\right )}-\frac {b c \sqrt {1-c^2 x^2}}{2 d^3 x}\)

Input:

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

Output:

-1/2*(b*c*Sqrt[1 - c^2*x^2])/(d^3*x) + (b*c*e^2*x*Sqrt[1 - c^2*x^2])/(8*d^ 
3*(c^2*d + e)*(d + e*x^2)) - (a + b*ArcSin[c*x])/(2*d^3*x^2) - (e*(a + b*A 
rcSin[c*x]))/(4*d^2*(d + e*x^2)^2) - (e*(a + b*ArcSin[c*x]))/(d^3*(d + e*x 
^2)) + (b*c*e*ArcTan[(Sqrt[c^2*d + e]*x)/(Sqrt[d]*Sqrt[1 - c^2*x^2])])/(d^ 
(7/2)*Sqrt[c^2*d + e]) + (b*c*e*(2*c^2*d + e)*ArcTan[(Sqrt[c^2*d + e]*x)/( 
Sqrt[d]*Sqrt[1 - c^2*x^2])])/(8*d^(7/2)*(c^2*d + e)^(3/2)) + (3*e*(a + b*A 
rcSin[c*x])*Log[1 - (Sqrt[e]*E^(I*ArcSin[c*x]))/(I*c*Sqrt[-d] - Sqrt[c^2*d 
 + e])])/(2*d^4) + (3*e*(a + b*ArcSin[c*x])*Log[1 + (Sqrt[e]*E^(I*ArcSin[c 
*x]))/(I*c*Sqrt[-d] - Sqrt[c^2*d + e])])/(2*d^4) + (3*e*(a + b*ArcSin[c*x] 
)*Log[1 - (Sqrt[e]*E^(I*ArcSin[c*x]))/(I*c*Sqrt[-d] + Sqrt[c^2*d + e])])/( 
2*d^4) + (3*e*(a + b*ArcSin[c*x])*Log[1 + (Sqrt[e]*E^(I*ArcSin[c*x]))/(I*c 
*Sqrt[-d] + Sqrt[c^2*d + e])])/(2*d^4) - (3*e*(a + b*ArcSin[c*x])*Log[1 - 
E^((2*I)*ArcSin[c*x])])/d^4 - (((3*I)/2)*b*e*PolyLog[2, -((Sqrt[e]*E^(I*Ar 
cSin[c*x]))/(I*c*Sqrt[-d] - Sqrt[c^2*d + e]))])/d^4 - (((3*I)/2)*b*e*PolyL 
og[2, (Sqrt[e]*E^(I*ArcSin[c*x]))/(I*c*Sqrt[-d] - Sqrt[c^2*d + e])])/d^4 - 
 (((3*I)/2)*b*e*PolyLog[2, -((Sqrt[e]*E^(I*ArcSin[c*x]))/(I*c*Sqrt[-d] + S 
qrt[c^2*d + e]))])/d^4 - (((3*I)/2)*b*e*PolyLog[2, (Sqrt[e]*E^(I*ArcSin[c* 
x]))/(I*c*Sqrt[-d] + Sqrt[c^2*d + e])])/d^4 + (((3*I)/2)*b*e*PolyLog[2, E^ 
((2*I)*ArcSin[c*x])])/d^4
 

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 = 4.61 (sec) , antiderivative size = 1344, normalized size of antiderivative = 1.72

method	result	size
parts	\(\text {Expression too large to display}\)	\(1344\)
derivativedivides	\(\text {Expression too large to display}\)	\(1395\)
default	\(\text {Expression too large to display}\)	\(1395\)

Input:

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

Output:

-1/2*a/d^3/x^2-3*a/d^4*e*ln(x)+3/2*a/d^4*e*ln(e*x^2+d)-1/4*a/d^2*e/(e*x^2+ 
d)^2-a/d^3*e/(e*x^2+d)+b*c^2*(-1/8*(-8*I*c^8*d^2*e*x^4-6*I*c^6*d*e^2*x^4-4 
*I*c^8*d*e^2*x^6+4*(-c^2*x^2+1)^(1/2)*c^7*d^3*x+8*(-c^2*x^2+1)^(1/2)*c^7*d 
^2*e*x^3+4*(-c^2*x^2+1)^(1/2)*c^7*d*e^2*x^5-3*I*e^3*c^6*x^6-4*I*c^8*d^3*x^ 
2-3*I*c^6*d^2*e*x^2+4*c^6*d^3*arcsin(c*x)+18*arcsin(c*x)*c^6*d^2*e*x^2+12* 
arcsin(c*x)*c^6*d*e^2*x^4+4*(-c^2*x^2+1)^(1/2)*c^5*d^2*e*x+7*(-c^2*x^2+1)^ 
(1/2)*c^5*d*e^2*x^3+3*(-c^2*x^2+1)^(1/2)*e^3*c^5*x^5+4*c^4*d^2*e*arcsin(c* 
x)+18*arcsin(c*x)*c^4*d*e^2*x^2+12*arcsin(c*x)*e^3*c^4*x^4)/c^2/x^2/(c^2*d 
+e)/(c^2*e*x^2+c^2*d)^2/d^3-3/(c^2*d+e)/d^3*e*arcsin(c*x)*ln(1+I*c*x+(-c^2 
*x^2+1)^(1/2))-9/8*I*(c^2*d*(c^2*d+e))^(1/2)/(c^2*d+e)^2/d^4/c^2*arctanh(1 
/4*(2*e*(I*c*x+(-c^2*x^2+1)^(1/2))^2-4*c^2*d-2*e)/(c^4*d^2+c^2*d*e)^(1/2)) 
*e^2-3/(c^2*d+e)*e^2/d^4/c^2*arcsin(c*x)*ln(1+I*c*x+(-c^2*x^2+1)^(1/2))-3/ 
4*I/(c^2*d+e)*e^2/d^4*sum((_R1^2*e-4*c^2*d-e)/(_R1^2*e-2*c^2*d-e)*(I*arcsi 
n(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))/c^2-3/4*I/(c^2*d+ 
e)*e^3/d^4*sum((_R1^2-1)/(_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=Ro 
otOf(e*_Z^4+(-4*c^2*d-2*e)*_Z^2+e))/c^2-3/4*I/(c^2*d+e)*e^2/d^3*sum((_R1^2 
-1)/(_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^...
 

Fricas [F]

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

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

Output:

integral((b*arcsin(c*x) + a)/(e^3*x^9 + 3*d*e^2*x^7 + 3*d^2*e*x^5 + d^3*x^ 
3), x)
 

Sympy [F(-1)]

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

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

Output:

Timed out
 

Maxima [F]

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

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

Output:

-1/4*a*((6*e^2*x^4 + 9*d*e*x^2 + 2*d^2)/(d^3*e^2*x^6 + 2*d^4*e*x^4 + d^5*x 
^2) - 6*e*log(e*x^2 + d)/d^4 + 12*e*log(x)/d^4) + b*integrate(arctan2(c*x, 
 sqrt(c*x + 1)*sqrt(-c*x + 1))/(e^3*x^9 + 3*d*e^2*x^7 + 3*d^2*e*x^5 + d^3* 
x^3), x)
 

Giac [F(-1)]

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

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

Output:

Timed out
                                                                                    
                                                                                    
 

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

\[ \int \frac {a+b \arcsin (c x)}{x^3 \left (d+e x^2\right )^3} \, dx=\frac {4 \left (\int \frac {\mathit {asin} \left (c x \right )}{e^{3} x^{9}+3 d \,e^{2} x^{7}+3 d^{2} e \,x^{5}+d^{3} x^{3}}d x \right ) b \,d^{6} x^{2}+8 \left (\int \frac {\mathit {asin} \left (c x \right )}{e^{3} x^{9}+3 d \,e^{2} x^{7}+3 d^{2} e \,x^{5}+d^{3} x^{3}}d x \right ) b \,d^{5} e \,x^{4}+4 \left (\int \frac {\mathit {asin} \left (c x \right )}{e^{3} x^{9}+3 d \,e^{2} x^{7}+3 d^{2} e \,x^{5}+d^{3} x^{3}}d x \right ) b \,d^{4} e^{2} x^{6}+6 \,\mathrm {log}\left (e \,x^{2}+d \right ) a \,d^{2} e \,x^{2}+12 \,\mathrm {log}\left (e \,x^{2}+d \right ) a d \,e^{2} x^{4}+6 \,\mathrm {log}\left (e \,x^{2}+d \right ) a \,e^{3} x^{6}-12 \,\mathrm {log}\left (x \right ) a \,d^{2} e \,x^{2}-24 \,\mathrm {log}\left (x \right ) a d \,e^{2} x^{4}-12 \,\mathrm {log}\left (x \right ) a \,e^{3} x^{6}-2 a \,d^{3}-6 a \,d^{2} e \,x^{2}+3 a \,e^{3} x^{6}}{4 d^{4} x^{2} \left (e^{2} x^{4}+2 d e \,x^{2}+d^{2}\right )} \] Input:

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

Output:

(4*int(asin(c*x)/(d**3*x**3 + 3*d**2*e*x**5 + 3*d*e**2*x**7 + e**3*x**9),x 
)*b*d**6*x**2 + 8*int(asin(c*x)/(d**3*x**3 + 3*d**2*e*x**5 + 3*d*e**2*x**7 
 + e**3*x**9),x)*b*d**5*e*x**4 + 4*int(asin(c*x)/(d**3*x**3 + 3*d**2*e*x** 
5 + 3*d*e**2*x**7 + e**3*x**9),x)*b*d**4*e**2*x**6 + 6*log(d + e*x**2)*a*d 
**2*e*x**2 + 12*log(d + e*x**2)*a*d*e**2*x**4 + 6*log(d + e*x**2)*a*e**3*x 
**6 - 12*log(x)*a*d**2*e*x**2 - 24*log(x)*a*d*e**2*x**4 - 12*log(x)*a*e**3 
*x**6 - 2*a*d**3 - 6*a*d**2*e*x**2 + 3*a*e**3*x**6)/(4*d**4*x**2*(d**2 + 2 
*d*e*x**2 + e**2*x**4))