3.2.13 \(\int \frac {(f+g x) (a+b \arcsin (c x))^2}{(d+e x)^3} \, dx\) [113]

3.2.13.1 Optimal result

Integrand size = 23, antiderivative size = 935 \[ \int \frac {(f+g x) (a+b \arcsin (c x))^2}{(d+e x)^3} \, dx=\frac {a b c (e f-d g) \sqrt {1-c^2 x^2}}{e \left (c^2 d^2-e^2\right ) (d+e x)}+\frac {a b g^2 \arcsin (c x)}{e^2 (e f-d g)}+\frac {b^2 c (e f-d g) \sqrt {1-c^2 x^2} \arcsin (c x)}{e \left (c^2 d^2-e^2\right ) (d+e x)}+\frac {b^2 g^2 \arcsin (c x)^2}{2 e^2 (e f-d g)}-\frac {(f+g x)^2 (a+b \arcsin (c x))^2}{2 (e f-d g) (d+e x)^2}-\frac {a b c \left (2 e^2 g-c^2 d (e f+d g)\right ) \arctan \left (\frac {e+c^2 d x}{\sqrt {c^2 d^2-e^2} \sqrt {1-c^2 x^2}}\right )}{e^2 \left (c^2 d^2-e^2\right )^{3/2}}-\frac {2 i b^2 c g \arcsin (c x) \log \left (1-\frac {i e e^{i \arcsin (c x)}}{c d-\sqrt {c^2 d^2-e^2}}\right )}{e^2 \sqrt {c^2 d^2-e^2}}-\frac {i b^2 c^3 d (e f-d g) \arcsin (c x) \log \left (1-\frac {i e e^{i \arcsin (c x)}}{c d-\sqrt {c^2 d^2-e^2}}\right )}{e^2 \left (c^2 d^2-e^2\right )^{3/2}}+\frac {2 i b^2 c g \arcsin (c x) \log \left (1-\frac {i e e^{i \arcsin (c x)}}{c d+\sqrt {c^2 d^2-e^2}}\right )}{e^2 \sqrt {c^2 d^2-e^2}}+\frac {i b^2 c^3 d (e f-d g) \arcsin (c x) \log \left (1-\frac {i e e^{i \arcsin (c x)}}{c d+\sqrt {c^2 d^2-e^2}}\right )}{e^2 \left (c^2 d^2-e^2\right )^{3/2}}-\frac {b^2 c^2 (e f-d g) \log (d+e x)}{e^2 \left (c^2 d^2-e^2\right )}-\frac {2 b^2 c g \operatorname {PolyLog}\left (2,\frac {i e e^{i \arcsin (c x)}}{c d-\sqrt {c^2 d^2-e^2}}\right )}{e^2 \sqrt {c^2 d^2-e^2}}-\frac {b^2 c^3 d (e f-d g) \operatorname {PolyLog}\left (2,\frac {i e e^{i \arcsin (c x)}}{c d-\sqrt {c^2 d^2-e^2}}\right )}{e^2 \left (c^2 d^2-e^2\right )^{3/2}}+\frac {2 b^2 c g \operatorname {PolyLog}\left (2,\frac {i e e^{i \arcsin (c x)}}{c d+\sqrt {c^2 d^2-e^2}}\right )}{e^2 \sqrt {c^2 d^2-e^2}}+\frac {b^2 c^3 d (e f-d g) \operatorname {PolyLog}\left (2,\frac {i e e^{i \arcsin (c x)}}{c d+\sqrt {c^2 d^2-e^2}}\right )}{e^2 \left (c^2 d^2-e^2\right )^{3/2}} \]

output

a*b*g^2*arcsin(c*x)/e^2/(-d*g+e*f)+1/2*b^2*g^2*arcsin(c*x)^2/e^2/(-d*g+e*f 
)-1/2*(g*x+f)^2*(a+b*arcsin(c*x))^2/(-d*g+e*f)/(e*x+d)^2-a*b*c*(2*e^2*g-c^ 
2*d*(d*g+e*f))*arctan((c^2*d*x+e)/(c^2*d^2-e^2)^(1/2)/(-c^2*x^2+1)^(1/2))/ 
e^2/(c^2*d^2-e^2)^(3/2)-b^2*c^2*(-d*g+e*f)*ln(e*x+d)/e^2/(c^2*d^2-e^2)-I*b 
^2*c^3*d*(-d*g+e*f)*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^2/(c^2*d^2-e^2)^(3/2)+I*b^2*c^3*d*(-d*g+e*f)*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^2/( 
c^2*d^2-e^2)^(3/2)-b^2*c^3*d*(-d*g+e*f)*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^2/(c^2*d^2-e^2)^(3/2)+b^2*c^3*d*(-d*g+ 
e*f)*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 
^2/(c^2*d^2-e^2)^(3/2)-2*I*b^2*c*g*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^2/(c^2*d^2-e^2)^(1/2)+2*I*b^2*c*g*ar 
csin(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 
^2/(c^2*d^2-e^2)^(1/2)-2*b^2*c*g*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^2/(c^2*d^2-e^2)^(1/2)+2*b^2*c*g*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^2/(c^2*d^2-e^2)^( 
1/2)+a*b*c*(-d*g+e*f)*(-c^2*x^2+1)^(1/2)/e/(c^2*d^2-e^2)/(e*x+d)+b^2*c*(-d 
*g+e*f)*arcsin(c*x)*(-c^2*x^2+1)^(1/2)/e/(c^2*d^2-e^2)/(e*x+d)
 

3.2.13.2 Mathematica [A] (verified)

Time = 1.81 (sec) , antiderivative size = 574, normalized size of antiderivative = 0.61 \[ \int \frac {(f+g x) (a+b \arcsin (c x))^2}{(d+e x)^3} \, dx=\frac {-\frac {(e f-d g) (a+b \arcsin (c x))^2}{(d+e x)^2}-\frac {2 g (a+b \arcsin (c x))^2}{d+e x}+\frac {4 b c g \left (-i (a+b \arcsin (c x)) \left (\log \left (1+\frac {i e e^{i \arcsin (c x)}}{-c d+\sqrt {c^2 d^2-e^2}}\right )-\log \left (1-\frac {i e e^{i \arcsin (c x)}}{c d+\sqrt {c^2 d^2-e^2}}\right )\right )-b \operatorname {PolyLog}\left (2,\frac {i e e^{i \arcsin (c x)}}{c d-\sqrt {c^2 d^2-e^2}}\right )+b \operatorname {PolyLog}\left (2,\frac {i e e^{i \arcsin (c x)}}{c d+\sqrt {c^2 d^2-e^2}}\right )\right )}{\sqrt {c^2 d^2-e^2}}+\frac {2 b c (e f-d g) \left (e \sqrt {c^2 d^2-e^2} \sqrt {1-c^2 x^2} (a+b \arcsin (c x))-b c \sqrt {c^2 d^2-e^2} (d+e x) \log (d+e x)-i c^2 d (d+e x) \left ((a+b \arcsin (c x)) \left (\log \left (1+\frac {i e e^{i \arcsin (c x)}}{-c d+\sqrt {c^2 d^2-e^2}}\right )-\log \left (1-\frac {i e e^{i \arcsin (c x)}}{c d+\sqrt {c^2 d^2-e^2}}\right )\right )-i b \operatorname {PolyLog}\left (2,\frac {i e e^{i \arcsin (c x)}}{c d-\sqrt {c^2 d^2-e^2}}\right )+i b \operatorname {PolyLog}\left (2,\frac {i e e^{i \arcsin (c x)}}{c d+\sqrt {c^2 d^2-e^2}}\right )\right )\right )}{\left (c^2 d^2-e^2\right )^{3/2} (d+e x)}}{2 e^2} \]

input

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

output

(-(((e*f - d*g)*(a + b*ArcSin[c*x])^2)/(d + e*x)^2) - (2*g*(a + b*ArcSin[c 
*x])^2)/(d + e*x) + (4*b*c*g*((-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])]))/Sqrt[c^2*d^2 - e^2] + (2*b*c*(e*f - d*g)*(e*Sqrt 
[c^2*d^2 - e^2]*Sqrt[1 - c^2*x^2]*(a + b*ArcSin[c*x]) - b*c*Sqrt[c^2*d^2 - 
 e^2]*(d + e*x)*Log[d + e*x] - I*c^2*d*(d + e*x)*((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])]) - I*b*PolyLog[2, (I*e*E 
^(I*ArcSin[c*x]))/(c*d - Sqrt[c^2*d^2 - e^2])] + I*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)*(d + 
e*x)))/(2*e^2)
 

3.2.13.3 Rubi [A] (verified)

Time = 2.76 (sec) , antiderivative size = 955, normalized size of antiderivative = 1.02, number of steps used = 4, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.174, Rules used = {5254, 27, 5298, 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.

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

\(\Big \downarrow \) 5254

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 5298

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

\(\Big \downarrow \) 2009

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

input

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

output

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

3.2.13.3.1 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 5254

Int[((a_.) + ArcSin[(c_.)*(x_)]*(b_.))^(n_)*((d_) + (e_.)*(x_))^(m_)*((f_.) 
 + (g_.)*(x_))^(p_.), x_Symbol] :> With[{u = IntHide[(f + g*x)^p*(d + e*x)^ 
m, x]}, Simp[(a + b*ArcSin[c*x])^n   u, x] - Simp[b*c*n   Int[SimplifyInteg 
rand[u*((a + b*ArcSin[c*x])^(n - 1)/Sqrt[1 - c^2*x^2]), x], x], x]] /; Free 
Q[{a, b, c, d, e, f, g}, x] && IGtQ[n, 0] && IGtQ[p, 0] && ILtQ[m, 0] && Lt 
Q[m + p + 1, 0]
 

rule 5298

Int[(ArcSin[(c_.)*(x_)]*(b_.) + (a_))^(n_.)*(RFx_)*((d_) + (e_.)*(x_)^2)^(p 
_), x_Symbol] :> Int[ExpandIntegrand[(d + e*x^2)^p, RFx*(a + b*ArcSin[c*x]) 
^n, x], x] /; FreeQ[{a, b, c, d, e}, x] && RationalFunctionQ[RFx, x] && IGt 
Q[n, 0] && EqQ[c^2*d + e, 0] && IntegerQ[p - 1/2]
 

3.2.13.4 Maple [B] (verified)

Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 2320 vs. \(2 (941 ) = 1882\).

Time = 4.98 (sec) , antiderivative size = 2321, normalized size of antiderivative = 2.48

method	result	size
derivativedivides	\(\text {Expression too large to display}\)	\(2321\)
default	\(\text {Expression too large to display}\)	\(2321\)
parts	\(\text {Expression too large to display}\)	\(2336\)

input

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

output

1/c*(a^2*c^2*(1/2*c*(d*g-e*f)/e^2/(c*e*x+c*d)^2-g/e^2/(c*e*x+c*d))+b^2*c^2 
*(-1/2*arcsin(c*x)*(2*(-c^2*x^2+1)^(1/2)*c^2*d^2*e*g-2*(-c^2*x^2+1)^(1/2)* 
c^2*d*e^2*f-2*(-c^2*x^2+1)^(1/2)*c^2*e^3*f*x+2*I*c^3*d^2*e*f+4*I*c^3*d*e^2 
*f*x+2*arcsin(c*x)*c^3*d^2*e*g*x-2*I*c^3*d*e^2*g*x^2+2*I*c^3*e^3*f*x^2-2*a 
rcsin(c*x)*e^3*g*c*x-4*I*c^3*d^2*e*g*x-e^2*c*d*g*arcsin(c*x)-2*I*c^3*d^3*g 
+2*(-c^2*x^2+1)^(1/2)*c^2*d*e^2*g*x+c^3*d^3*g*arcsin(c*x)+e*c^3*d^2*f*arcs 
in(c*x)-e^3*c*f*arcsin(c*x))/(c^2*d^2-e^2)/(c*e*x+c*d)^2/e^2+2/e/(c^2*d^2- 
e^2)*c*f*ln(I*c*x+(-c^2*x^2+1)^(1/2))-1/e/(c^2*d^2-e^2)*c*f*ln(I*e*(I*c*x+ 
(-c^2*x^2+1)^(1/2))^2-2*d*c*(I*c*x+(-c^2*x^2+1)^(1/2))-I*e)-2/e^2/(c^2*d^2 
-e^2)*c*d*g*ln(I*c*x+(-c^2*x^2+1)^(1/2))+1/e^2/(c^2*d^2-e^2)*c*d*g*ln(I*e* 
(I*c*x+(-c^2*x^2+1)^(1/2))^2-2*d*c*(I*c*x+(-c^2*x^2+1)^(1/2))-I*e)+2*(-c^2 
*d^2+e^2)^(1/2)/(c^2*d^2-e^2)^2*g*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)))-2*(-c^2*d^2 
+e^2)^(1/2)/(c^2*d^2-e^2)^2*g*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)))-I/e*(-c^2*d^2+e 
^2)^(1/2)/(c^2*d^2-e^2)^2*c^2*d*f*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)))+2*I*(-c^2*d^2+e^2)^( 
1/2)/(c^2*d^2-e^2)^2*g*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)))-1/e^2*(-c^2*d^2+e^2)^(1/2)/(c^2 
*d^2-e^2)^2*c^2*d^2*g*arcsin(c*x)*ln((I*d*c+(I*c*x+(-c^2*x^2+1)^(1/2))*...
 

3.2.13.5 Fricas [F]

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

input

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

output

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

3.2.13.6 Sympy [F]

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

input

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

output

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

3.2.13.7 Maxima [F(-2)]

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

input

integrate((g*x+f)*(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
 

3.2.13.8 Giac [F]

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

input

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

output

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

3.2.13.9 Mupad [F(-1)]

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

input

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

output

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