\(\int \frac {(f+g x) (a+b \arcsin (c x))}{(d+e x)^3} \, dx\) [36]

Optimal result

Integrand size = 21, antiderivative size = 202 \[ \int \frac {(f+g x) (a+b \arcsin (c x))}{(d+e x)^3} \, dx=\frac {b c (e f-d g) \sqrt {1-c^2 x^2}}{2 e \left (c^2 d^2-e^2\right ) (d+e x)}+\frac {b g^2 \arcsin (c x)}{2 e^2 (e f-d g)}-\frac {(f+g x)^2 (a+b \arcsin (c x))}{2 (e f-d g) (d+e x)^2}-\frac {b c \left (2 e^2 g-c^2 d (e f+d g)\right ) \arctan \left (\frac {e+c^2 d x}{\sqrt {c^2 d^2-e^2} \sqrt {1-c^2 x^2}}\right )}{2 e^2 \left (c^2 d^2-e^2\right )^{3/2}} \] Output:

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

Mathematica [A] (verified)

Time = 0.42 (sec) , antiderivative size = 263, normalized size of antiderivative = 1.30 \[ \int \frac {(f+g x) (a+b \arcsin (c x))}{(d+e x)^3} \, dx=\frac {\frac {a (-e f+d g)}{(d+e x)^2}-\frac {2 a g}{d+e x}-\frac {b c e (e f-d g) \sqrt {1-c^2 x^2}}{\left (-c^2 d^2+e^2\right ) (d+e x)}-\frac {b (d g+e (f+2 g x)) \arcsin (c x)}{(d+e x)^2}+\frac {b c \left (-2 e^2 g+c^2 d (e f+d g)\right ) \log (d+e x)}{(c d-e) (c d+e) \sqrt {-c^2 d^2+e^2}}+\frac {b c \left (-2 e^2 g+c^2 d (e f+d g)\right ) \log \left (e+c^2 d x+\sqrt {-c^2 d^2+e^2} \sqrt {1-c^2 x^2}\right )}{(-c d+e) (c d+e) \sqrt {-c^2 d^2+e^2}}}{2 e^2} \] Input:

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

Output:

((a*(-(e*f) + d*g))/(d + e*x)^2 - (2*a*g)/(d + e*x) - (b*c*e*(e*f - d*g)*S 
qrt[1 - c^2*x^2])/((-(c^2*d^2) + e^2)*(d + e*x)) - (b*(d*g + e*(f + 2*g*x) 
)*ArcSin[c*x])/(d + e*x)^2 + (b*c*(-2*e^2*g + c^2*d*(e*f + d*g))*Log[d + e 
*x])/((c*d - e)*(c*d + e)*Sqrt[-(c^2*d^2) + e^2]) + (b*c*(-2*e^2*g + c^2*d 
*(e*f + d*g))*Log[e + c^2*d*x + Sqrt[-(c^2*d^2) + e^2]*Sqrt[1 - c^2*x^2]]) 
/((-(c*d) + e)*(c*d + e)*Sqrt[-(c^2*d^2) + e^2]))/(2*e^2)
 

Rubi [A] (verified)

Time = 0.53 (sec) , antiderivative size = 238, normalized size of antiderivative = 1.18, number of steps used = 8, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.333, Rules used = {5252, 27, 715, 719, 223, 488, 217}

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.

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

\(\Big \downarrow \) 5252

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 715

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

\(\Big \downarrow \) 719

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

\(\Big \downarrow \) 223

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

\(\Big \downarrow \) 488

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

\(\Big \downarrow \) 217

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

Input:

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

Output:

-1/2*((f + g*x)^2*(a + b*ArcSin[c*x]))/((e*f - d*g)*(d + e*x)^2) + (b*c*(( 
(e*f - d*g)^2*Sqrt[1 - c^2*x^2])/(e*(c^2*d^2 - e^2)*(d + e*x)) + (((c*d - 
e)*(c*d + e)*g^2*ArcSin[c*x])/(c*e^2) - ((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*Sqrt[c^2*d^2 - e^2]))/(c^2*d^2 - e^2)))/(2*(e*f - d*g))
 

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 217

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(-(Rt[-a, 2]*Rt[-b, 2])^( 
-1))*ArcTan[Rt[-b, 2]*(x/Rt[-a, 2])], x] /; FreeQ[{a, b}, x] && PosQ[a/b] & 
& (LtQ[a, 0] || LtQ[b, 0])
 

rule 223

Int[1/Sqrt[(a_) + (b_.)*(x_)^2], x_Symbol] :> Simp[ArcSin[Rt[-b, 2]*(x/Sqrt 
[a])]/Rt[-b, 2], x] /; FreeQ[{a, b}, x] && GtQ[a, 0] && NegQ[b]
 

rule 488

Int[1/(((c_) + (d_.)*(x_))*Sqrt[(a_) + (b_.)*(x_)^2]), x_Symbol] :> -Subst[ 
Int[1/(b*c^2 + a*d^2 - x^2), x], x, (a*d - b*c*x)/Sqrt[a + b*x^2]] /; FreeQ 
[{a, b, c, d}, x]
 

rule 715

Int[((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))^(n_)*((a_) + (c_.)*(x_)^ 
2)^(p_), x_Symbol] :> With[{Q = PolynomialQuotient[(f + g*x)^n, d + e*x, x] 
, R = PolynomialRemainder[(f + g*x)^n, d + e*x, x]}, Simp[(e*R*(d + e*x)^(m 
 + 1)*(a + c*x^2)^(p + 1))/((m + 1)*(c*d^2 + a*e^2)), x] + Simp[1/((m + 1)* 
(c*d^2 + a*e^2))   Int[(d + e*x)^(m + 1)*(a + c*x^2)^p*ExpandToSum[(m + 1)* 
(c*d^2 + a*e^2)*Q + c*d*R*(m + 1) - c*e*R*(m + 2*p + 3)*x, x], x], x]] /; F 
reeQ[{a, c, d, e, f, g, p}, x] && IGtQ[n, 1] && ILtQ[m, -1] && NeQ[c*d^2 + 
a*e^2, 0] && (NeQ[m + n, 0] || EqQ[p, -2^(-1)])
 

rule 719

Int[((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))*((a_) + (c_.)*(x_)^2)^(p 
_.), x_Symbol] :> Simp[g/e   Int[(d + e*x)^(m + 1)*(a + c*x^2)^p, x], x] + 
Simp[(e*f - d*g)/e   Int[(d + e*x)^m*(a + c*x^2)^p, x], x] /; FreeQ[{a, c, 
d, e, f, g, m, p}, x] &&  !IGtQ[m, 0]
 

rule 5252

Int[((a_.) + ArcSin[(c_.)*(x_)]*(b_.))*(Px_)*((d_.) + (e_.)*(x_))^(m_.), x_ 
Symbol] :> With[{u = IntHide[Px*(d + e*x)^m, x]}, Simp[(a + b*ArcSin[c*x]) 
  u, x] - Simp[b*c   Int[SimplifyIntegrand[u/Sqrt[1 - c^2*x^2], x], x], x]] 
 /; FreeQ[{a, b, c, d, e, m}, x] && PolynomialQ[Px, x]
 

Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(534\) vs. \(2(186)=372\).

Time = 0.41 (sec) , antiderivative size = 535, normalized size of antiderivative = 2.65

method	result	size
parts	\(a \left (-\frac {-d g +e f}{2 e^{2} \left (e x +d \right )^{2}}-\frac {g}{e^{2} \left (e x +d \right )}\right )+\frac {b \left (\frac {c^{3} \arcsin \left (c x \right ) d g}{2 e^{2} \left (c x e +c d \right )^{2}}-\frac {c^{3} \arcsin \left (c x \right ) f}{2 e \left (c x e +c d \right )^{2}}-\frac {c^{2} \arcsin \left (c x \right ) g}{e^{2} \left (c x e +c d \right )}+\frac {c^{2} \left (-\frac {2 g \ln \left (\frac {-\frac {2 \left (c^{2} d^{2}-e^{2}\right )}{e^{2}}+\frac {2 d c \left (c x +\frac {d c}{e}\right )}{e}+2 \sqrt {-\frac {c^{2} d^{2}-e^{2}}{e^{2}}}\, \sqrt {-\left (c x +\frac {d c}{e}\right )^{2}+\frac {2 d c \left (c x +\frac {d c}{e}\right )}{e}-\frac {c^{2} d^{2}-e^{2}}{e^{2}}}}{c x +\frac {d c}{e}}\right )}{e \sqrt {-\frac {c^{2} d^{2}-e^{2}}{e^{2}}}}-\frac {c \left (d g -e f \right ) \left (\frac {e^{2} \sqrt {-\left (c x +\frac {d c}{e}\right )^{2}+\frac {2 d c \left (c x +\frac {d c}{e}\right )}{e}-\frac {c^{2} d^{2}-e^{2}}{e^{2}}}}{\left (c^{2} d^{2}-e^{2}\right ) \left (c x +\frac {d c}{e}\right )}-\frac {d c e \ln \left (\frac {-\frac {2 \left (c^{2} d^{2}-e^{2}\right )}{e^{2}}+\frac {2 d c \left (c x +\frac {d c}{e}\right )}{e}+2 \sqrt {-\frac {c^{2} d^{2}-e^{2}}{e^{2}}}\, \sqrt {-\left (c x +\frac {d c}{e}\right )^{2}+\frac {2 d c \left (c x +\frac {d c}{e}\right )}{e}-\frac {c^{2} d^{2}-e^{2}}{e^{2}}}}{c x +\frac {d c}{e}}\right )}{\left (c^{2} d^{2}-e^{2}\right ) \sqrt {-\frac {c^{2} d^{2}-e^{2}}{e^{2}}}}\right )}{e^{2}}\right )}{2 e^{2}}\right )}{c}\)	\(535\)
derivativedivides	\(\frac {a \,c^{2} \left (-\frac {g}{e^{2} \left (c x e +c d \right )}+\frac {c \left (d g -e f \right )}{2 e^{2} \left (c x e +c d \right )^{2}}\right )+b \,c^{2} \left (-\frac {\arcsin \left (c x \right ) g}{e^{2} \left (c x e +c d \right )}+\frac {\arcsin \left (c x \right ) c d g}{2 e^{2} \left (c x e +c d \right )^{2}}-\frac {\arcsin \left (c x \right ) c f}{2 e \left (c x e +c d \right )^{2}}+\frac {-\frac {2 g \ln \left (\frac {-\frac {2 \left (c^{2} d^{2}-e^{2}\right )}{e^{2}}+\frac {2 d c \left (c x +\frac {d c}{e}\right )}{e}+2 \sqrt {-\frac {c^{2} d^{2}-e^{2}}{e^{2}}}\, \sqrt {-\left (c x +\frac {d c}{e}\right )^{2}+\frac {2 d c \left (c x +\frac {d c}{e}\right )}{e}-\frac {c^{2} d^{2}-e^{2}}{e^{2}}}}{c x +\frac {d c}{e}}\right )}{e \sqrt {-\frac {c^{2} d^{2}-e^{2}}{e^{2}}}}-\frac {c \left (d g -e f \right ) \left (\frac {e^{2} \sqrt {-\left (c x +\frac {d c}{e}\right )^{2}+\frac {2 d c \left (c x +\frac {d c}{e}\right )}{e}-\frac {c^{2} d^{2}-e^{2}}{e^{2}}}}{\left (c^{2} d^{2}-e^{2}\right ) \left (c x +\frac {d c}{e}\right )}-\frac {d c e \ln \left (\frac {-\frac {2 \left (c^{2} d^{2}-e^{2}\right )}{e^{2}}+\frac {2 d c \left (c x +\frac {d c}{e}\right )}{e}+2 \sqrt {-\frac {c^{2} d^{2}-e^{2}}{e^{2}}}\, \sqrt {-\left (c x +\frac {d c}{e}\right )^{2}+\frac {2 d c \left (c x +\frac {d c}{e}\right )}{e}-\frac {c^{2} d^{2}-e^{2}}{e^{2}}}}{c x +\frac {d c}{e}}\right )}{\left (c^{2} d^{2}-e^{2}\right ) \sqrt {-\frac {c^{2} d^{2}-e^{2}}{e^{2}}}}\right )}{e^{2}}}{2 e^{2}}\right )}{c}\)	\(539\)
default	\(\frac {a \,c^{2} \left (-\frac {g}{e^{2} \left (c x e +c d \right )}+\frac {c \left (d g -e f \right )}{2 e^{2} \left (c x e +c d \right )^{2}}\right )+b \,c^{2} \left (-\frac {\arcsin \left (c x \right ) g}{e^{2} \left (c x e +c d \right )}+\frac {\arcsin \left (c x \right ) c d g}{2 e^{2} \left (c x e +c d \right )^{2}}-\frac {\arcsin \left (c x \right ) c f}{2 e \left (c x e +c d \right )^{2}}+\frac {-\frac {2 g \ln \left (\frac {-\frac {2 \left (c^{2} d^{2}-e^{2}\right )}{e^{2}}+\frac {2 d c \left (c x +\frac {d c}{e}\right )}{e}+2 \sqrt {-\frac {c^{2} d^{2}-e^{2}}{e^{2}}}\, \sqrt {-\left (c x +\frac {d c}{e}\right )^{2}+\frac {2 d c \left (c x +\frac {d c}{e}\right )}{e}-\frac {c^{2} d^{2}-e^{2}}{e^{2}}}}{c x +\frac {d c}{e}}\right )}{e \sqrt {-\frac {c^{2} d^{2}-e^{2}}{e^{2}}}}-\frac {c \left (d g -e f \right ) \left (\frac {e^{2} \sqrt {-\left (c x +\frac {d c}{e}\right )^{2}+\frac {2 d c \left (c x +\frac {d c}{e}\right )}{e}-\frac {c^{2} d^{2}-e^{2}}{e^{2}}}}{\left (c^{2} d^{2}-e^{2}\right ) \left (c x +\frac {d c}{e}\right )}-\frac {d c e \ln \left (\frac {-\frac {2 \left (c^{2} d^{2}-e^{2}\right )}{e^{2}}+\frac {2 d c \left (c x +\frac {d c}{e}\right )}{e}+2 \sqrt {-\frac {c^{2} d^{2}-e^{2}}{e^{2}}}\, \sqrt {-\left (c x +\frac {d c}{e}\right )^{2}+\frac {2 d c \left (c x +\frac {d c}{e}\right )}{e}-\frac {c^{2} d^{2}-e^{2}}{e^{2}}}}{c x +\frac {d c}{e}}\right )}{\left (c^{2} d^{2}-e^{2}\right ) \sqrt {-\frac {c^{2} d^{2}-e^{2}}{e^{2}}}}\right )}{e^{2}}}{2 e^{2}}\right )}{c}\)	\(539\)

Input:

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

Output:

a*(-1/2*(-d*g+e*f)/e^2/(e*x+d)^2-g/e^2/(e*x+d))+b/c*(1/2*c^3*arcsin(c*x)/e 
^2/(c*e*x+c*d)^2*d*g-1/2*c^3*arcsin(c*x)/e/(c*e*x+c*d)^2*f-c^2*arcsin(c*x) 
*g/e^2/(c*e*x+c*d)+1/2*c^2/e^2*(-2*g/e/(-(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))-c*(d*g- 
e*f)/e^2*(1/(c^2*d^2-e^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)-d*c*e/(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 [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 578 vs. \(2 (185) = 370\).

Time = 3.39 (sec) , antiderivative size = 1184, normalized size of antiderivative = 5.86 \[ \int \frac {(f+g x) (a+b \arcsin (c x))}{(d+e x)^3} \, dx=\text {Too large to display} \] Input:

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

Output:

[-1/4*(4*(a*c^4*d^4*e - 2*a*c^2*d^2*e^3 + a*e^5)*g*x + (b*c^3*d^3*e*f + (b 
*c^3*d*e^3*f + (b*c^3*d^2*e^2 - 2*b*c*e^4)*g)*x^2 + (b*c^3*d^4 - 2*b*c*d^2 
*e^2)*g + 2*(b*c^3*d^2*e^2*f + (b*c^3*d^3*e - 2*b*c*d*e^3)*g)*x)*sqrt(-c^2 
*d^2 + e^2)*log((2*c^2*d*e*x - c^2*d^2 + (2*c^4*d^2 - c^2*e^2)*x^2 - 2*sqr 
t(-c^2*d^2 + e^2)*(c^2*d*x + e)*sqrt(-c^2*x^2 + 1) + 2*e^2)/(e^2*x^2 + 2*d 
*e*x + d^2)) + 2*(a*c^4*d^4*e - 2*a*c^2*d^2*e^3 + a*e^5)*f + 2*(a*c^4*d^5 
- 2*a*c^2*d^3*e^2 + a*d*e^4)*g + 2*(2*(b*c^4*d^4*e - 2*b*c^2*d^2*e^3 + b*e 
^5)*g*x + (b*c^4*d^4*e - 2*b*c^2*d^2*e^3 + b*e^5)*f + (b*c^4*d^5 - 2*b*c^2 
*d^3*e^2 + b*d*e^4)*g)*arcsin(c*x) - 2*sqrt(-c^2*x^2 + 1)*((b*c^3*d^3*e^2 
- b*c*d*e^4)*f - (b*c^3*d^4*e - b*c*d^2*e^3)*g + ((b*c^3*d^2*e^3 - b*c*e^5 
)*f - (b*c^3*d^3*e^2 - b*c*d*e^4)*g)*x))/(c^4*d^6*e^2 - 2*c^2*d^4*e^4 + d^ 
2*e^6 + (c^4*d^4*e^4 - 2*c^2*d^2*e^6 + e^8)*x^2 + 2*(c^4*d^5*e^3 - 2*c^2*d 
^3*e^5 + d*e^7)*x), -1/2*(2*(a*c^4*d^4*e - 2*a*c^2*d^2*e^3 + a*e^5)*g*x - 
(b*c^3*d^3*e*f + (b*c^3*d*e^3*f + (b*c^3*d^2*e^2 - 2*b*c*e^4)*g)*x^2 + (b* 
c^3*d^4 - 2*b*c*d^2*e^2)*g + 2*(b*c^3*d^2*e^2*f + (b*c^3*d^3*e - 2*b*c*d*e 
^3)*g)*x)*sqrt(c^2*d^2 - e^2)*arctan(sqrt(c^2*d^2 - e^2)*(c^2*d*x + e)*sqr 
t(-c^2*x^2 + 1)/(c^2*d^2 - (c^4*d^2 - c^2*e^2)*x^2 - e^2)) + (a*c^4*d^4*e 
- 2*a*c^2*d^2*e^3 + a*e^5)*f + (a*c^4*d^5 - 2*a*c^2*d^3*e^2 + a*d*e^4)*g + 
 (2*(b*c^4*d^4*e - 2*b*c^2*d^2*e^3 + b*e^5)*g*x + (b*c^4*d^4*e - 2*b*c^2*d 
^2*e^3 + b*e^5)*f + (b*c^4*d^5 - 2*b*c^2*d^3*e^2 + b*d*e^4)*g)*arcsin(c...
 

Sympy [F]

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

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

Output:

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

Maxima [F(-2)]

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

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

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

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

Output:

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

Reduce [F]

\[ \int \frac {(f+g x) (a+b \arcsin (c x))}{(d+e x)^3} \, dx =\text {Too large to display} \] Input:

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

Output:

(6*int(asin(c*x)/(3*c**2*d**5 + 9*c**2*d**4*e*x + 9*c**2*d**3*e**2*x**2 + 
3*c**2*d**2*e**3*x**3 + d**3*e**2 + 3*d**2*e**3*x + 3*d*e**4*x**2 + e**5*x 
**3),x)*b*c**2*d**5*e*f + 12*int(asin(c*x)/(3*c**2*d**5 + 9*c**2*d**4*e*x 
+ 9*c**2*d**3*e**2*x**2 + 3*c**2*d**2*e**3*x**3 + d**3*e**2 + 3*d**2*e**3* 
x + 3*d*e**4*x**2 + e**5*x**3),x)*b*c**2*d**4*e**2*f*x + 6*int(asin(c*x)/( 
3*c**2*d**5 + 9*c**2*d**4*e*x + 9*c**2*d**3*e**2*x**2 + 3*c**2*d**2*e**3*x 
**3 + d**3*e**2 + 3*d**2*e**3*x + 3*d*e**4*x**2 + e**5*x**3),x)*b*c**2*d** 
3*e**3*f*x**2 + 2*int(asin(c*x)/(3*c**2*d**5 + 9*c**2*d**4*e*x + 9*c**2*d* 
*3*e**2*x**2 + 3*c**2*d**2*e**3*x**3 + d**3*e**2 + 3*d**2*e**3*x + 3*d*e** 
4*x**2 + e**5*x**3),x)*b*d**3*e**3*f + 4*int(asin(c*x)/(3*c**2*d**5 + 9*c* 
*2*d**4*e*x + 9*c**2*d**3*e**2*x**2 + 3*c**2*d**2*e**3*x**3 + d**3*e**2 + 
3*d**2*e**3*x + 3*d*e**4*x**2 + e**5*x**3),x)*b*d**2*e**4*f*x + 2*int(asin 
(c*x)/(3*c**2*d**5 + 9*c**2*d**4*e*x + 9*c**2*d**3*e**2*x**2 + 3*c**2*d**2 
*e**3*x**3 + d**3*e**2 + 3*d**2*e**3*x + 3*d*e**4*x**2 + e**5*x**3),x)*b*d 
*e**5*f*x**2 + 6*int((asin(c*x)*x)/(3*c**2*d**5 + 9*c**2*d**4*e*x + 9*c**2 
*d**3*e**2*x**2 + 3*c**2*d**2*e**3*x**3 + d**3*e**2 + 3*d**2*e**3*x + 3*d* 
e**4*x**2 + e**5*x**3),x)*b*c**2*d**5*e*g + 12*int((asin(c*x)*x)/(3*c**2*d 
**5 + 9*c**2*d**4*e*x + 9*c**2*d**3*e**2*x**2 + 3*c**2*d**2*e**3*x**3 + d* 
*3*e**2 + 3*d**2*e**3*x + 3*d*e**4*x**2 + e**5*x**3),x)*b*c**2*d**4*e**2*g 
*x + 6*int((asin(c*x)*x)/(3*c**2*d**5 + 9*c**2*d**4*e*x + 9*c**2*d**3*e...