\(\int \frac {(a+b \arcsin (c x))^2}{(d+e x)^3} \, dx\) [15]

Optimal result

Integrand size = 18, antiderivative size = 401 \[ \int \frac {(a+b \arcsin (c x))^2}{(d+e x)^3} \, dx=\frac {b c \sqrt {1-c^2 x^2} (a+b \arcsin (c x))}{\left (c^2 d^2-e^2\right ) (d+e x)}-\frac {(a+b \arcsin (c x))^2}{2 e (d+e x)^2}-\frac {i b c^3 d (a+b \arcsin (c x)) \log \left (1-\frac {i e e^{i \arcsin (c x)}}{c d-\sqrt {c^2 d^2-e^2}}\right )}{e \left (c^2 d^2-e^2\right )^{3/2}}+\frac {i b c^3 d (a+b \arcsin (c x)) \log \left (1-\frac {i e e^{i \arcsin (c x)}}{c d+\sqrt {c^2 d^2-e^2}}\right )}{e \left (c^2 d^2-e^2\right )^{3/2}}-\frac {b^2 c^2 \log (d+e x)}{e \left (c^2 d^2-e^2\right )}-\frac {b^2 c^3 d \operatorname {PolyLog}\left (2,\frac {i e e^{i \arcsin (c x)}}{c d-\sqrt {c^2 d^2-e^2}}\right )}{e \left (c^2 d^2-e^2\right )^{3/2}}+\frac {b^2 c^3 d \operatorname {PolyLog}\left (2,\frac {i e e^{i \arcsin (c x)}}{c d+\sqrt {c^2 d^2-e^2}}\right )}{e \left (c^2 d^2-e^2\right )^{3/2}} \] Output:

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

Mathematica [A] (verified)

Time = 0.64 (sec) , antiderivative size = 315, normalized size of antiderivative = 0.79 \[ \int \frac {(a+b \arcsin (c x))^2}{(d+e x)^3} \, dx=\frac {\frac {2 b c e \sqrt {1-c^2 x^2} (a+b \arcsin (c x))}{\left (c^2 d^2-e^2\right ) (d+e x)}-\frac {(a+b \arcsin (c x))^2}{(d+e x)^2}-\frac {2 b^2 c^2 \log (d+e x)}{c^2 d^2-e^2}+\frac {2 b c^3 d \left (-i (a+b \arcsin (c x)) \left (\log \left (1+\frac {i e e^{i \arcsin (c x)}}{-c d+\sqrt {c^2 d^2-e^2}}\right )-\log \left (1-\frac {i e e^{i \arcsin (c x)}}{c d+\sqrt {c^2 d^2-e^2}}\right )\right )-b \operatorname {PolyLog}\left (2,\frac {i e e^{i \arcsin (c x)}}{c d-\sqrt {c^2 d^2-e^2}}\right )+b \operatorname {PolyLog}\left (2,\frac {i e e^{i \arcsin (c x)}}{c d+\sqrt {c^2 d^2-e^2}}\right )\right )}{\left (c^2 d^2-e^2\right )^{3/2}}}{2 e} \] Input:

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

Output:

((2*b*c*e*Sqrt[1 - c^2*x^2]*(a + b*ArcSin[c*x]))/((c^2*d^2 - e^2)*(d + e*x 
)) - (a + b*ArcSin[c*x])^2/(d + e*x)^2 - (2*b^2*c^2*Log[d + e*x])/(c^2*d^2 
 - e^2) + (2*b*c^3*d*((-I)*(a + b*ArcSin[c*x])*(Log[1 + (I*e*E^(I*ArcSin[c 
*x]))/(-(c*d) + Sqrt[c^2*d^2 - e^2])] - Log[1 - (I*e*E^(I*ArcSin[c*x]))/(c 
*d + Sqrt[c^2*d^2 - e^2])]) - b*PolyLog[2, (I*e*E^(I*ArcSin[c*x]))/(c*d - 
Sqrt[c^2*d^2 - e^2])] + b*PolyLog[2, (I*e*E^(I*ArcSin[c*x]))/(c*d + Sqrt[c 
^2*d^2 - e^2])]))/(c^2*d^2 - e^2)^(3/2))/(2*e)
 

Rubi [A] (verified)

Time = 1.41 (sec) , antiderivative size = 386, normalized size of antiderivative = 0.96, number of steps used = 14, number of rules used = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.722, Rules used = {5242, 5272, 3042, 3805, 3042, 3147, 16, 3804, 2694, 27, 2620, 2715, 2838}

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.

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

\(\Big \downarrow \) 5242

\(\displaystyle \frac {b c \int \frac {a+b \arcsin (c x)}{(d+e x)^2 \sqrt {1-c^2 x^2}}dx}{e}-\frac {(a+b \arcsin (c x))^2}{2 e (d+e x)^2}\)

\(\Big \downarrow \) 5272

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

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 3805

\(\displaystyle \frac {b c^2 \left (\frac {c d \int \frac {a+b \arcsin (c x)}{c d+c e x}d\arcsin (c x)}{c^2 d^2-e^2}-\frac {b e \int \frac {\sqrt {1-c^2 x^2}}{c d+c e x}d\arcsin (c x)}{c^2 d^2-e^2}+\frac {e \sqrt {1-c^2 x^2} (a+b \arcsin (c x))}{\left (c^2 d^2-e^2\right ) (c d+c e x)}\right )}{e}-\frac {(a+b \arcsin (c x))^2}{2 e (d+e x)^2}\)

\(\Big \downarrow \) 3042

\(\displaystyle \frac {b c^2 \left (\frac {c d \int \frac {a+b \arcsin (c x)}{c d+e \sin (\arcsin (c x))}d\arcsin (c x)}{c^2 d^2-e^2}-\frac {b e \int \frac {\cos (\arcsin (c x))}{c d+e \sin (\arcsin (c x))}d\arcsin (c x)}{c^2 d^2-e^2}+\frac {e \sqrt {1-c^2 x^2} (a+b \arcsin (c x))}{\left (c^2 d^2-e^2\right ) (c d+c e x)}\right )}{e}-\frac {(a+b \arcsin (c x))^2}{2 e (d+e x)^2}\)

\(\Big \downarrow \) 3147

\(\displaystyle \frac {b c^2 \left (\frac {c d \int \frac {a+b \arcsin (c x)}{c d+e \sin (\arcsin (c x))}d\arcsin (c x)}{c^2 d^2-e^2}-\frac {b \int \frac {1}{c d+c e x}d(c e x)}{c^2 d^2-e^2}+\frac {e \sqrt {1-c^2 x^2} (a+b \arcsin (c x))}{\left (c^2 d^2-e^2\right ) (c d+c e x)}\right )}{e}-\frac {(a+b \arcsin (c x))^2}{2 e (d+e x)^2}\)

\(\Big \downarrow \) 16

\(\displaystyle \frac {b c^2 \left (\frac {c d \int \frac {a+b \arcsin (c x)}{c d+e \sin (\arcsin (c x))}d\arcsin (c x)}{c^2 d^2-e^2}+\frac {e \sqrt {1-c^2 x^2} (a+b \arcsin (c x))}{\left (c^2 d^2-e^2\right ) (c d+c e x)}-\frac {b \log (c d+c e x)}{c^2 d^2-e^2}\right )}{e}-\frac {(a+b \arcsin (c x))^2}{2 e (d+e x)^2}\)

\(\Big \downarrow \) 3804

\(\displaystyle -\frac {(a+b \arcsin (c x))^2}{2 e (d+e x)^2}+\frac {b c^2 \left (\frac {2 c d \int \frac {e^{i \arcsin (c x)} (a+b \arcsin (c x))}{2 c e^{i \arcsin (c x)} d-i e e^{2 i \arcsin (c x)}+i e}d\arcsin (c x)}{c^2 d^2-e^2}+\frac {e \sqrt {1-c^2 x^2} (a+b \arcsin (c x))}{\left (c^2 d^2-e^2\right ) (c d+c e x)}-\frac {b \log (c d+c e x)}{c^2 d^2-e^2}\right )}{e}\)

\(\Big \downarrow \) 2694

\(\displaystyle -\frac {(a+b \arcsin (c x))^2}{2 e (d+e x)^2}+\frac {b c^2 \left (\frac {2 c d \left (\frac {i e \int \frac {e^{i \arcsin (c x)} (a+b \arcsin (c x))}{2 \left (c d-i e e^{i \arcsin (c x)}+\sqrt {c^2 d^2-e^2}\right )}d\arcsin (c x)}{\sqrt {c^2 d^2-e^2}}-\frac {i e \int \frac {e^{i \arcsin (c x)} (a+b \arcsin (c x))}{2 \left (c d-i e e^{i \arcsin (c x)}-\sqrt {c^2 d^2-e^2}\right )}d\arcsin (c x)}{\sqrt {c^2 d^2-e^2}}\right )}{c^2 d^2-e^2}+\frac {e \sqrt {1-c^2 x^2} (a+b \arcsin (c x))}{\left (c^2 d^2-e^2\right ) (c d+c e x)}-\frac {b \log (c d+c e x)}{c^2 d^2-e^2}\right )}{e}\)

\(\Big \downarrow \) 27

\(\displaystyle -\frac {(a+b \arcsin (c x))^2}{2 e (d+e x)^2}+\frac {b c^2 \left (\frac {2 c d \left (\frac {i e \int \frac {e^{i \arcsin (c x)} (a+b \arcsin (c x))}{c d-i e e^{i \arcsin (c x)}+\sqrt {c^2 d^2-e^2}}d\arcsin (c x)}{2 \sqrt {c^2 d^2-e^2}}-\frac {i e \int \frac {e^{i \arcsin (c x)} (a+b \arcsin (c x))}{c d-i e e^{i \arcsin (c x)}-\sqrt {c^2 d^2-e^2}}d\arcsin (c x)}{2 \sqrt {c^2 d^2-e^2}}\right )}{c^2 d^2-e^2}+\frac {e \sqrt {1-c^2 x^2} (a+b \arcsin (c x))}{\left (c^2 d^2-e^2\right ) (c d+c e x)}-\frac {b \log (c d+c e x)}{c^2 d^2-e^2}\right )}{e}\)

\(\Big \downarrow \) 2620

\(\displaystyle -\frac {(a+b \arcsin (c x))^2}{2 e (d+e x)^2}+\frac {b c^2 \left (\frac {2 c d \left (\frac {i e \left (\frac {(a+b \arcsin (c x)) \log \left (1-\frac {i e e^{i \arcsin (c x)}}{\sqrt {c^2 d^2-e^2}+c d}\right )}{e}-\frac {b \int \log \left (1-\frac {i e e^{i \arcsin (c x)}}{c d+\sqrt {c^2 d^2-e^2}}\right )d\arcsin (c x)}{e}\right )}{2 \sqrt {c^2 d^2-e^2}}-\frac {i e \left (\frac {(a+b \arcsin (c x)) \log \left (1-\frac {i e e^{i \arcsin (c x)}}{c d-\sqrt {c^2 d^2-e^2}}\right )}{e}-\frac {b \int \log \left (1-\frac {i e e^{i \arcsin (c x)}}{c d-\sqrt {c^2 d^2-e^2}}\right )d\arcsin (c x)}{e}\right )}{2 \sqrt {c^2 d^2-e^2}}\right )}{c^2 d^2-e^2}+\frac {e \sqrt {1-c^2 x^2} (a+b \arcsin (c x))}{\left (c^2 d^2-e^2\right ) (c d+c e x)}-\frac {b \log (c d+c e x)}{c^2 d^2-e^2}\right )}{e}\)

\(\Big \downarrow \) 2715

\(\displaystyle -\frac {(a+b \arcsin (c x))^2}{2 e (d+e x)^2}+\frac {b c^2 \left (\frac {2 c d \left (\frac {i e \left (\frac {i b \int e^{-i \arcsin (c x)} \log \left (1-\frac {i e e^{i \arcsin (c x)}}{c d+\sqrt {c^2 d^2-e^2}}\right )de^{i \arcsin (c x)}}{e}+\frac {(a+b \arcsin (c x)) \log \left (1-\frac {i e e^{i \arcsin (c x)}}{\sqrt {c^2 d^2-e^2}+c d}\right )}{e}\right )}{2 \sqrt {c^2 d^2-e^2}}-\frac {i e \left (\frac {i b \int e^{-i \arcsin (c x)} \log \left (1-\frac {i e e^{i \arcsin (c x)}}{c d-\sqrt {c^2 d^2-e^2}}\right )de^{i \arcsin (c x)}}{e}+\frac {(a+b \arcsin (c x)) \log \left (1-\frac {i e e^{i \arcsin (c x)}}{c d-\sqrt {c^2 d^2-e^2}}\right )}{e}\right )}{2 \sqrt {c^2 d^2-e^2}}\right )}{c^2 d^2-e^2}+\frac {e \sqrt {1-c^2 x^2} (a+b \arcsin (c x))}{\left (c^2 d^2-e^2\right ) (c d+c e x)}-\frac {b \log (c d+c e x)}{c^2 d^2-e^2}\right )}{e}\)

\(\Big \downarrow \) 2838

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

Input:

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

Output:

-1/2*(a + b*ArcSin[c*x])^2/(e*(d + e*x)^2) + (b*c^2*((e*Sqrt[1 - c^2*x^2]* 
(a + b*ArcSin[c*x]))/((c^2*d^2 - e^2)*(c*d + c*e*x)) - (b*Log[c*d + c*e*x] 
)/(c^2*d^2 - e^2) + (2*c*d*(((-1/2*I)*e*(((a + b*ArcSin[c*x])*Log[1 - (I*e 
*E^(I*ArcSin[c*x]))/(c*d - Sqrt[c^2*d^2 - e^2])])/e - (I*b*PolyLog[2, (I*e 
*E^(I*ArcSin[c*x]))/(c*d - Sqrt[c^2*d^2 - e^2])])/e))/Sqrt[c^2*d^2 - e^2] 
+ ((I/2)*e*(((a + b*ArcSin[c*x])*Log[1 - (I*e*E^(I*ArcSin[c*x]))/(c*d + Sq 
rt[c^2*d^2 - e^2])])/e - (I*b*PolyLog[2, (I*e*E^(I*ArcSin[c*x]))/(c*d + Sq 
rt[c^2*d^2 - e^2])])/e))/Sqrt[c^2*d^2 - e^2]))/(c^2*d^2 - e^2)))/e
 

Defintions of rubi rules used

rule 16

Int[(c_.)/((a_.) + (b_.)*(x_)), x_Symbol] :> Simp[c*(Log[RemoveContent[a + 
b*x, x]]/b), x] /; FreeQ[{a, b, c}, x]
 

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 2620

Int[(((F_)^((g_.)*((e_.) + (f_.)*(x_))))^(n_.)*((c_.) + (d_.)*(x_))^(m_.))/ 
((a_) + (b_.)*((F_)^((g_.)*((e_.) + (f_.)*(x_))))^(n_.)), x_Symbol] :> Simp 
[((c + d*x)^m/(b*f*g*n*Log[F]))*Log[1 + b*((F^(g*(e + f*x)))^n/a)], x] - Si 
mp[d*(m/(b*f*g*n*Log[F]))   Int[(c + d*x)^(m - 1)*Log[1 + b*((F^(g*(e + f*x 
)))^n/a)], x], x] /; FreeQ[{F, a, b, c, d, e, f, g, n}, x] && IGtQ[m, 0]
 

rule 2694

Int[((F_)^(u_)*((f_.) + (g_.)*(x_))^(m_.))/((a_.) + (b_.)*(F_)^(u_) + (c_.) 
*(F_)^(v_)), x_Symbol] :> With[{q = Rt[b^2 - 4*a*c, 2]}, Simp[2*(c/q)   Int 
[(f + g*x)^m*(F^u/(b - q + 2*c*F^u)), x], x] - Simp[2*(c/q)   Int[(f + g*x) 
^m*(F^u/(b + q + 2*c*F^u)), x], x]] /; FreeQ[{F, a, b, c, f, g}, x] && EqQ[ 
v, 2*u] && LinearQ[u, x] && NeQ[b^2 - 4*a*c, 0] && IGtQ[m, 0]
 

rule 2715

Int[Log[(a_) + (b_.)*((F_)^((e_.)*((c_.) + (d_.)*(x_))))^(n_.)], x_Symbol] 
:> Simp[1/(d*e*n*Log[F])   Subst[Int[Log[a + b*x]/x, x], x, (F^(e*(c + d*x) 
))^n], x] /; FreeQ[{F, a, b, c, d, e, n}, x] && GtQ[a, 0]
 

rule 2838

Int[Log[(c_.)*((d_) + (e_.)*(x_)^(n_.))]/(x_), x_Symbol] :> Simp[-PolyLog[2 
, (-c)*e*x^n]/n, x] /; FreeQ[{c, d, e, n}, x] && EqQ[c*d, 1]
 

rule 3042

Int[u_, x_Symbol] :> Int[DeactivateTrig[u, x], x] /; FunctionOfTrigOfLinear 
Q[u, x]
 

rule 3147

Int[cos[(e_.) + (f_.)*(x_)]^(p_.)*((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m 
_.), x_Symbol] :> Simp[1/(b^p*f)   Subst[Int[(a + x)^m*(b^2 - x^2)^((p - 1) 
/2), x], x, b*Sin[e + f*x]], x] /; FreeQ[{a, b, e, f, m}, x] && IntegerQ[(p 
 - 1)/2] && NeQ[a^2 - b^2, 0]
 

rule 3804

Int[((c_.) + (d_.)*(x_))^(m_.)/((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)]), x_Sy 
mbol] :> Simp[2   Int[(c + d*x)^m*(E^(I*(e + f*x))/(I*b + 2*a*E^(I*(e + f*x 
)) - I*b*E^(2*I*(e + f*x)))), x], x] /; FreeQ[{a, b, c, d, e, f}, x] && NeQ 
[a^2 - b^2, 0] && IGtQ[m, 0]
 

rule 3805

Int[((c_.) + (d_.)*(x_))^(m_.)/((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)])^2, x_ 
Symbol] :> Simp[b*(c + d*x)^m*(Cos[e + f*x]/(f*(a^2 - b^2)*(a + b*Sin[e + f 
*x]))), x] + (Simp[a/(a^2 - b^2)   Int[(c + d*x)^m/(a + b*Sin[e + f*x]), x] 
, x] - Simp[b*d*(m/(f*(a^2 - b^2)))   Int[(c + d*x)^(m - 1)*(Cos[e + f*x]/( 
a + b*Sin[e + f*x])), x], x]) /; FreeQ[{a, b, c, d, e, f}, x] && NeQ[a^2 - 
b^2, 0] && IGtQ[m, 0]
 

rule 5242

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

rule 5272

Int[(((a_.) + ArcSin[(c_.)*(x_)]*(b_.))^(n_.)*((f_) + (g_.)*(x_))^(m_.))/Sq 
rt[(d_) + (e_.)*(x_)^2], x_Symbol] :> Simp[1/(c^(m + 1)*Sqrt[d])   Subst[In 
t[(a + b*x)^n*(c*f + g*Sin[x])^m, x], x, ArcSin[c*x]], x] /; FreeQ[{a, b, c 
, d, e, f, g, n}, x] && EqQ[c^2*d + e, 0] && IntegerQ[m] && GtQ[d, 0] && (G 
tQ[m, 0] || IGtQ[n, 0])
 

Maple [B] (verified)

Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 965 vs. \(2 (407 ) = 814\).

Time = 1.90 (sec) , antiderivative size = 966, normalized size of antiderivative = 2.41

method	result	size
derivativedivides	\(\frac {-\frac {a^{2} c^{3}}{2 \left (c x e +c d \right )^{2} e}+b^{2} c^{3} \left (-\frac {\arcsin \left (c x \right ) \left (2 i c^{2} d^{2}+4 i c^{2} d e x +2 i e^{2} c^{2} x^{2}+c^{2} d^{2} \arcsin \left (c x \right )-2 \sqrt {-c^{2} x^{2}+1}\, c d e -2 \sqrt {-c^{2} x^{2}+1}\, c \,e^{2} x -e^{2} \arcsin \left (c x \right )\right )}{2 \left (c x e +c d \right )^{2} \left (c^{2} d^{2}-e^{2}\right ) e}-\frac {\ln \left (i \left (i c x +\sqrt {-c^{2} x^{2}+1}\right )^{2} e -2 d c \left (i c x +\sqrt {-c^{2} x^{2}+1}\right )-i e \right )}{e \left (c^{2} d^{2}-e^{2}\right )}+\frac {2 \ln \left (i c x +\sqrt {-c^{2} x^{2}+1}\right )}{e \left (c^{2} d^{2}-e^{2}\right )}-\frac {\sqrt {-c^{2} d^{2}+e^{2}}\, \arcsin \left (c x \right ) \ln \left (\frac {i d c +\left (i c x +\sqrt {-c^{2} x^{2}+1}\right ) e -\sqrt {-c^{2} d^{2}+e^{2}}}{i d c -\sqrt {-c^{2} d^{2}+e^{2}}}\right ) c d}{e \left (c^{2} d^{2}-e^{2}\right )^{2}}+\frac {\sqrt {-c^{2} d^{2}+e^{2}}\, \arcsin \left (c x \right ) \ln \left (\frac {i d c +\left (i c x +\sqrt {-c^{2} x^{2}+1}\right ) e +\sqrt {-c^{2} d^{2}+e^{2}}}{i d c +\sqrt {-c^{2} d^{2}+e^{2}}}\right ) c d}{e \left (c^{2} d^{2}-e^{2}\right )^{2}}+\frac {i \sqrt {-c^{2} d^{2}+e^{2}}\, \operatorname {dilog}\left (\frac {i d c +\left (i c x +\sqrt {-c^{2} x^{2}+1}\right ) e -\sqrt {-c^{2} d^{2}+e^{2}}}{i d c -\sqrt {-c^{2} d^{2}+e^{2}}}\right ) c d}{e \left (c^{2} d^{2}-e^{2}\right )^{2}}-\frac {i \sqrt {-c^{2} d^{2}+e^{2}}\, \operatorname {dilog}\left (\frac {i d c +\left (i c x +\sqrt {-c^{2} x^{2}+1}\right ) e +\sqrt {-c^{2} d^{2}+e^{2}}}{i d c +\sqrt {-c^{2} d^{2}+e^{2}}}\right ) c d}{e \left (c^{2} d^{2}-e^{2}\right )^{2}}\right )-\frac {a b \,c^{3} \arcsin \left (c x \right )}{\left (c x e +c d \right )^{2} e}+\frac {a b \,c^{3} \sqrt {-\left (c x +\frac {d c}{e}\right )^{2}+\frac {2 d c \left (c x +\frac {d c}{e}\right )}{e}-\frac {c^{2} d^{2}-e^{2}}{e^{2}}}}{e \left (c^{2} d^{2}-e^{2}\right ) \left (c x +\frac {d c}{e}\right )}-\frac {a b \,c^{4} d \ln \left (\frac {-\frac {2 \left (c^{2} d^{2}-e^{2}\right )}{e^{2}}+\frac {2 d c \left (c x +\frac {d c}{e}\right )}{e}+2 \sqrt {-\frac {c^{2} d^{2}-e^{2}}{e^{2}}}\, \sqrt {-\left (c x +\frac {d c}{e}\right )^{2}+\frac {2 d c \left (c x +\frac {d c}{e}\right )}{e}-\frac {c^{2} d^{2}-e^{2}}{e^{2}}}}{c x +\frac {d c}{e}}\right )}{e^{2} \left (c^{2} d^{2}-e^{2}\right ) \sqrt {-\frac {c^{2} d^{2}-e^{2}}{e^{2}}}}}{c}\)	\(966\)
default	\(\frac {-\frac {a^{2} c^{3}}{2 \left (c x e +c d \right )^{2} e}+b^{2} c^{3} \left (-\frac {\arcsin \left (c x \right ) \left (2 i c^{2} d^{2}+4 i c^{2} d e x +2 i e^{2} c^{2} x^{2}+c^{2} d^{2} \arcsin \left (c x \right )-2 \sqrt {-c^{2} x^{2}+1}\, c d e -2 \sqrt {-c^{2} x^{2}+1}\, c \,e^{2} x -e^{2} \arcsin \left (c x \right )\right )}{2 \left (c x e +c d \right )^{2} \left (c^{2} d^{2}-e^{2}\right ) e}-\frac {\ln \left (i \left (i c x +\sqrt {-c^{2} x^{2}+1}\right )^{2} e -2 d c \left (i c x +\sqrt {-c^{2} x^{2}+1}\right )-i e \right )}{e \left (c^{2} d^{2}-e^{2}\right )}+\frac {2 \ln \left (i c x +\sqrt {-c^{2} x^{2}+1}\right )}{e \left (c^{2} d^{2}-e^{2}\right )}-\frac {\sqrt {-c^{2} d^{2}+e^{2}}\, \arcsin \left (c x \right ) \ln \left (\frac {i d c +\left (i c x +\sqrt {-c^{2} x^{2}+1}\right ) e -\sqrt {-c^{2} d^{2}+e^{2}}}{i d c -\sqrt {-c^{2} d^{2}+e^{2}}}\right ) c d}{e \left (c^{2} d^{2}-e^{2}\right )^{2}}+\frac {\sqrt {-c^{2} d^{2}+e^{2}}\, \arcsin \left (c x \right ) \ln \left (\frac {i d c +\left (i c x +\sqrt {-c^{2} x^{2}+1}\right ) e +\sqrt {-c^{2} d^{2}+e^{2}}}{i d c +\sqrt {-c^{2} d^{2}+e^{2}}}\right ) c d}{e \left (c^{2} d^{2}-e^{2}\right )^{2}}+\frac {i \sqrt {-c^{2} d^{2}+e^{2}}\, \operatorname {dilog}\left (\frac {i d c +\left (i c x +\sqrt {-c^{2} x^{2}+1}\right ) e -\sqrt {-c^{2} d^{2}+e^{2}}}{i d c -\sqrt {-c^{2} d^{2}+e^{2}}}\right ) c d}{e \left (c^{2} d^{2}-e^{2}\right )^{2}}-\frac {i \sqrt {-c^{2} d^{2}+e^{2}}\, \operatorname {dilog}\left (\frac {i d c +\left (i c x +\sqrt {-c^{2} x^{2}+1}\right ) e +\sqrt {-c^{2} d^{2}+e^{2}}}{i d c +\sqrt {-c^{2} d^{2}+e^{2}}}\right ) c d}{e \left (c^{2} d^{2}-e^{2}\right )^{2}}\right )-\frac {a b \,c^{3} \arcsin \left (c x \right )}{\left (c x e +c d \right )^{2} e}+\frac {a b \,c^{3} \sqrt {-\left (c x +\frac {d c}{e}\right )^{2}+\frac {2 d c \left (c x +\frac {d c}{e}\right )}{e}-\frac {c^{2} d^{2}-e^{2}}{e^{2}}}}{e \left (c^{2} d^{2}-e^{2}\right ) \left (c x +\frac {d c}{e}\right )}-\frac {a b \,c^{4} d \ln \left (\frac {-\frac {2 \left (c^{2} d^{2}-e^{2}\right )}{e^{2}}+\frac {2 d c \left (c x +\frac {d c}{e}\right )}{e}+2 \sqrt {-\frac {c^{2} d^{2}-e^{2}}{e^{2}}}\, \sqrt {-\left (c x +\frac {d c}{e}\right )^{2}+\frac {2 d c \left (c x +\frac {d c}{e}\right )}{e}-\frac {c^{2} d^{2}-e^{2}}{e^{2}}}}{c x +\frac {d c}{e}}\right )}{e^{2} \left (c^{2} d^{2}-e^{2}\right ) \sqrt {-\frac {c^{2} d^{2}-e^{2}}{e^{2}}}}}{c}\)	\(966\)
parts	\(-\frac {a^{2}}{2 \left (e x +d \right )^{2} e}+\frac {b^{2} \left (-\frac {c^{3} \arcsin \left (c x \right ) \left (2 i c^{2} d^{2}+4 i c^{2} d e x +2 i e^{2} c^{2} x^{2}+c^{2} d^{2} \arcsin \left (c x \right )-2 \sqrt {-c^{2} x^{2}+1}\, c d e -2 \sqrt {-c^{2} x^{2}+1}\, c \,e^{2} x -e^{2} \arcsin \left (c x \right )\right )}{2 \left (c x e +c d \right )^{2} \left (c^{2} d^{2}-e^{2}\right ) e}-\frac {c^{3} \ln \left (e \left (i c x +\sqrt {-c^{2} x^{2}+1}\right )^{2}+2 i \left (i c x +\sqrt {-c^{2} x^{2}+1}\right ) c d -e \right )}{e \left (c^{2} d^{2}-e^{2}\right )}+\frac {2 c^{3} \ln \left (i c x +\sqrt {-c^{2} x^{2}+1}\right )}{e \left (c^{2} d^{2}-e^{2}\right )}-\frac {\sqrt {-c^{2} d^{2}+e^{2}}\, c^{4} \arcsin \left (c x \right ) \ln \left (\frac {i d c +\left (i c x +\sqrt {-c^{2} x^{2}+1}\right ) e -\sqrt {-c^{2} d^{2}+e^{2}}}{i d c -\sqrt {-c^{2} d^{2}+e^{2}}}\right ) d}{e \left (c^{2} d^{2}-e^{2}\right )^{2}}+\frac {\sqrt {-c^{2} d^{2}+e^{2}}\, c^{4} \arcsin \left (c x \right ) \ln \left (\frac {i d c +\left (i c x +\sqrt {-c^{2} x^{2}+1}\right ) e +\sqrt {-c^{2} d^{2}+e^{2}}}{i d c +\sqrt {-c^{2} d^{2}+e^{2}}}\right ) d}{e \left (c^{2} d^{2}-e^{2}\right )^{2}}+\frac {i \sqrt {-c^{2} d^{2}+e^{2}}\, c^{4} \operatorname {dilog}\left (\frac {i d c +\left (i c x +\sqrt {-c^{2} x^{2}+1}\right ) e -\sqrt {-c^{2} d^{2}+e^{2}}}{i d c -\sqrt {-c^{2} d^{2}+e^{2}}}\right ) d}{e \left (c^{2} d^{2}-e^{2}\right )^{2}}-\frac {i \sqrt {-c^{2} d^{2}+e^{2}}\, c^{4} \operatorname {dilog}\left (\frac {i d c +\left (i c x +\sqrt {-c^{2} x^{2}+1}\right ) e +\sqrt {-c^{2} d^{2}+e^{2}}}{i d c +\sqrt {-c^{2} d^{2}+e^{2}}}\right ) d}{e \left (c^{2} d^{2}-e^{2}\right )^{2}}\right )}{c}-\frac {a b \,c^{2} \arcsin \left (c x \right )}{\left (c x e +c d \right )^{2} e}+\frac {a b \,c^{2} \sqrt {-\left (c x +\frac {d c}{e}\right )^{2}+\frac {2 d c \left (c x +\frac {d c}{e}\right )}{e}-\frac {c^{2} d^{2}-e^{2}}{e^{2}}}}{e \left (c^{2} d^{2}-e^{2}\right ) \left (c x +\frac {d c}{e}\right )}-\frac {a b \,c^{3} d \ln \left (\frac {-\frac {2 \left (c^{2} d^{2}-e^{2}\right )}{e^{2}}+\frac {2 d c \left (c x +\frac {d c}{e}\right )}{e}+2 \sqrt {-\frac {c^{2} d^{2}-e^{2}}{e^{2}}}\, \sqrt {-\left (c x +\frac {d c}{e}\right )^{2}+\frac {2 d c \left (c x +\frac {d c}{e}\right )}{e}-\frac {c^{2} d^{2}-e^{2}}{e^{2}}}}{c x +\frac {d c}{e}}\right )}{e^{2} \left (c^{2} d^{2}-e^{2}\right ) \sqrt {-\frac {c^{2} d^{2}-e^{2}}{e^{2}}}}\)	\(971\)

Input:

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

Output:

1/c*(-1/2*a^2*c^3/(c*e*x+c*d)^2/e+b^2*c^3*(-1/2*arcsin(c*x)*(2*I*c^2*d^2+4 
*I*c^2*d*e*x+2*I*e^2*c^2*x^2+c^2*d^2*arcsin(c*x)-2*(-c^2*x^2+1)^(1/2)*c*d* 
e-2*(-c^2*x^2+1)^(1/2)*c*e^2*x-e^2*arcsin(c*x))/(c*e*x+c*d)^2/(c^2*d^2-e^2 
)/e-1/e/(c^2*d^2-e^2)*ln(I*(I*c*x+(-c^2*x^2+1)^(1/2))^2*e-2*d*c*(I*c*x+(-c 
^2*x^2+1)^(1/2))-I*e)+2/e/(c^2*d^2-e^2)*ln(I*c*x+(-c^2*x^2+1)^(1/2))-1/e*( 
-c^2*d^2+e^2)^(1/2)/(c^2*d^2-e^2)^2*arcsin(c*x)*ln((I*d*c+(I*c*x+(-c^2*x^2 
+1)^(1/2))*e-(-c^2*d^2+e^2)^(1/2))/(I*d*c-(-c^2*d^2+e^2)^(1/2)))*c*d+1/e*( 
-c^2*d^2+e^2)^(1/2)/(c^2*d^2-e^2)^2*arcsin(c*x)*ln((I*d*c+(I*c*x+(-c^2*x^2 
+1)^(1/2))*e+(-c^2*d^2+e^2)^(1/2))/(I*d*c+(-c^2*d^2+e^2)^(1/2)))*c*d+I/e*( 
-c^2*d^2+e^2)^(1/2)/(c^2*d^2-e^2)^2*dilog((I*d*c+(I*c*x+(-c^2*x^2+1)^(1/2) 
)*e-(-c^2*d^2+e^2)^(1/2))/(I*d*c-(-c^2*d^2+e^2)^(1/2)))*c*d-I/e*(-c^2*d^2+ 
e^2)^(1/2)/(c^2*d^2-e^2)^2*dilog((I*d*c+(I*c*x+(-c^2*x^2+1)^(1/2))*e+(-c^2 
*d^2+e^2)^(1/2))/(I*d*c+(-c^2*d^2+e^2)^(1/2)))*c*d)-a*b*c^3/(c*e*x+c*d)^2/ 
e*arcsin(c*x)+a*b*c^3/e/(c^2*d^2-e^2)/(c*x+d*c/e)*(-(c*x+d*c/e)^2+2*d*c/e* 
(c*x+d*c/e)-(c^2*d^2-e^2)/e^2)^(1/2)-a*b*c^4/e^2*d/(c^2*d^2-e^2)/(-(c^2*d^ 
2-e^2)/e^2)^(1/2)*ln((-2*(c^2*d^2-e^2)/e^2+2*d*c/e*(c*x+d*c/e)+2*(-(c^2*d^ 
2-e^2)/e^2)^(1/2)*(-(c*x+d*c/e)^2+2*d*c/e*(c*x+d*c/e)-(c^2*d^2-e^2)/e^2)^( 
1/2))/(c*x+d*c/e)))
 

Fricas [F]

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

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

Output:

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

Sympy [F]

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

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

Output:

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

Maxima [F(-2)]

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

integrate((a+b*arcsin(c*x))^2/(e*x+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-c*d)*(e+c*d)>0)', see `assume 
?` for mor
 

Giac [F(-2)]

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

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

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

Output:

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

Reduce [F]

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

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

Output:

(4*int(asin(c*x)/(d**3 + 3*d**2*e*x + 3*d*e**2*x**2 + e**3*x**3),x)*a*b*d* 
*2*e + 8*int(asin(c*x)/(d**3 + 3*d**2*e*x + 3*d*e**2*x**2 + e**3*x**3),x)* 
a*b*d*e**2*x + 4*int(asin(c*x)/(d**3 + 3*d**2*e*x + 3*d*e**2*x**2 + e**3*x 
**3),x)*a*b*e**3*x**2 + 2*int(asin(c*x)**2/(d**3 + 3*d**2*e*x + 3*d*e**2*x 
**2 + e**3*x**3),x)*b**2*d**2*e + 4*int(asin(c*x)**2/(d**3 + 3*d**2*e*x + 
3*d*e**2*x**2 + e**3*x**3),x)*b**2*d*e**2*x + 2*int(asin(c*x)**2/(d**3 + 3 
*d**2*e*x + 3*d*e**2*x**2 + e**3*x**3),x)*b**2*e**3*x**2 - a**2)/(2*e*(d** 
2 + 2*d*e*x + e**2*x**2))