\(\int \frac {a+b \arcsin (c x)}{(d+e x)^4} \, dx\) [8]

Optimal result

Integrand size = 16, antiderivative size = 191 \[ \int \frac {a+b \arcsin (c x)}{(d+e x)^4} \, dx=\frac {b c \sqrt {1-c^2 x^2}}{6 \left (c^2 d^2-e^2\right ) (d+e x)^2}+\frac {b c^3 d \sqrt {1-c^2 x^2}}{2 \left (c^2 d^2-e^2\right )^2 (d+e x)}-\frac {a+b \arcsin (c x)}{3 e (d+e x)^3}+\frac {b c^3 \left (2 c^2 d^2+e^2\right ) \arctan \left (\frac {e+c^2 d x}{\sqrt {c^2 d^2-e^2} \sqrt {1-c^2 x^2}}\right )}{6 e \left (c^2 d^2-e^2\right )^{5/2}} \] Output:

1/6*b*c*(-c^2*x^2+1)^(1/2)/(c^2*d^2-e^2)/(e*x+d)^2+1/2*b*c^3*d*(-c^2*x^2+1 
)^(1/2)/(c^2*d^2-e^2)^2/(e*x+d)-1/3*(a+b*arcsin(c*x))/e/(e*x+d)^3+1/6*b*c^ 
3*(2*c^2*d^2+e^2)*arctan((c^2*d*x+e)/(c^2*d^2-e^2)^(1/2)/(-c^2*x^2+1)^(1/2 
))/e/(c^2*d^2-e^2)^(5/2)
 

Mathematica [A] (verified)

Time = 0.28 (sec) , antiderivative size = 241, normalized size of antiderivative = 1.26 \[ \int \frac {a+b \arcsin (c x)}{(d+e x)^4} \, dx=\frac {1}{6} \left (-\frac {2 a}{e (d+e x)^3}+\frac {b \sqrt {1-c^2 x^2} \left (-c e^2+c^3 d (4 d+3 e x)\right )}{\left (-c^2 d^2+e^2\right )^2 (d+e x)^2}-\frac {2 b \arcsin (c x)}{e (d+e x)^3}+\frac {b c^3 \left (2 c^2 d^2+e^2\right ) \log (d+e x)}{e (-c d+e)^2 (c d+e)^2 \sqrt {-c^2 d^2+e^2}}-\frac {b c^3 \left (2 c^2 d^2+e^2\right ) \log \left (e+c^2 d x+\sqrt {-c^2 d^2+e^2} \sqrt {1-c^2 x^2}\right )}{e (-c d+e)^2 (c d+e)^2 \sqrt {-c^2 d^2+e^2}}\right ) \] Input:

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

Output:

((-2*a)/(e*(d + e*x)^3) + (b*Sqrt[1 - c^2*x^2]*(-(c*e^2) + c^3*d*(4*d + 3* 
e*x)))/((-(c^2*d^2) + e^2)^2*(d + e*x)^2) - (2*b*ArcSin[c*x])/(e*(d + e*x) 
^3) + (b*c^3*(2*c^2*d^2 + e^2)*Log[d + e*x])/(e*(-(c*d) + e)^2*(c*d + e)^2 
*Sqrt[-(c^2*d^2) + e^2]) - (b*c^3*(2*c^2*d^2 + e^2)*Log[e + c^2*d*x + Sqrt 
[-(c^2*d^2) + e^2]*Sqrt[1 - c^2*x^2]])/(e*(-(c*d) + e)^2*(c*d + e)^2*Sqrt[ 
-(c^2*d^2) + e^2]))/6
 

Rubi [A] (verified)

Time = 0.41 (sec) , antiderivative size = 208, normalized size of antiderivative = 1.09, number of steps used = 7, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.375, Rules used = {5242, 498, 25, 679, 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.

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

\(\Big \downarrow \) 5242

\(\displaystyle \frac {b c \int \frac {1}{(d+e x)^3 \sqrt {1-c^2 x^2}}dx}{3 e}-\frac {a+b \arcsin (c x)}{3 e (d+e x)^3}\)

\(\Big \downarrow \) 498

\(\displaystyle \frac {b c \left (\frac {e \sqrt {1-c^2 x^2}}{2 \left (c^2 d^2-e^2\right ) (d+e x)^2}-\frac {c^2 \int -\frac {2 d-e x}{(d+e x)^2 \sqrt {1-c^2 x^2}}dx}{2 \left (c^2 d^2-e^2\right )}\right )}{3 e}-\frac {a+b \arcsin (c x)}{3 e (d+e x)^3}\)

\(\Big \downarrow \) 25

\(\displaystyle \frac {b c \left (\frac {c^2 \int \frac {2 d-e x}{(d+e x)^2 \sqrt {1-c^2 x^2}}dx}{2 \left (c^2 d^2-e^2\right )}+\frac {e \sqrt {1-c^2 x^2}}{2 \left (c^2 d^2-e^2\right ) (d+e x)^2}\right )}{3 e}-\frac {a+b \arcsin (c x)}{3 e (d+e x)^3}\)

\(\Big \downarrow \) 679

\(\displaystyle \frac {b c \left (\frac {c^2 \left (\frac {\left (2 c^2 d^2+e^2\right ) \int \frac {1}{(d+e x) \sqrt {1-c^2 x^2}}dx}{c^2 d^2-e^2}+\frac {3 d e \sqrt {1-c^2 x^2}}{\left (c^2 d^2-e^2\right ) (d+e x)}\right )}{2 \left (c^2 d^2-e^2\right )}+\frac {e \sqrt {1-c^2 x^2}}{2 \left (c^2 d^2-e^2\right ) (d+e x)^2}\right )}{3 e}-\frac {a+b \arcsin (c x)}{3 e (d+e x)^3}\)

\(\Big \downarrow \) 488

\(\displaystyle \frac {b c \left (\frac {c^2 \left (\frac {3 d e \sqrt {1-c^2 x^2}}{\left (c^2 d^2-e^2\right ) (d+e x)}-\frac {\left (2 c^2 d^2+e^2\right ) \int \frac {1}{-c^2 d^2+e^2-\frac {\left (d x c^2+e\right )^2}{1-c^2 x^2}}d\frac {d x c^2+e}{\sqrt {1-c^2 x^2}}}{c^2 d^2-e^2}\right )}{2 \left (c^2 d^2-e^2\right )}+\frac {e \sqrt {1-c^2 x^2}}{2 \left (c^2 d^2-e^2\right ) (d+e x)^2}\right )}{3 e}-\frac {a+b \arcsin (c x)}{3 e (d+e x)^3}\)

\(\Big \downarrow \) 217

\(\displaystyle \frac {b c \left (\frac {c^2 \left (\frac {\left (2 c^2 d^2+e^2\right ) \arctan \left (\frac {c^2 d x+e}{\sqrt {1-c^2 x^2} \sqrt {c^2 d^2-e^2}}\right )}{\left (c^2 d^2-e^2\right )^{3/2}}+\frac {3 d e \sqrt {1-c^2 x^2}}{\left (c^2 d^2-e^2\right ) (d+e x)}\right )}{2 \left (c^2 d^2-e^2\right )}+\frac {e \sqrt {1-c^2 x^2}}{2 \left (c^2 d^2-e^2\right ) (d+e x)^2}\right )}{3 e}-\frac {a+b \arcsin (c x)}{3 e (d+e x)^3}\)

Input:

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

Output:

-1/3*(a + b*ArcSin[c*x])/(e*(d + e*x)^3) + (b*c*((e*Sqrt[1 - c^2*x^2])/(2* 
(c^2*d^2 - e^2)*(d + e*x)^2) + (c^2*((3*d*e*Sqrt[1 - c^2*x^2])/((c^2*d^2 - 
 e^2)*(d + e*x)) + ((2*c^2*d^2 + e^2)*ArcTan[(e + c^2*d*x)/(Sqrt[c^2*d^2 - 
 e^2]*Sqrt[1 - c^2*x^2])])/(c^2*d^2 - e^2)^(3/2)))/(2*(c^2*d^2 - e^2))))/( 
3*e)
 

Defintions of rubi rules used

rule 25

Int[-(Fx_), x_Symbol] :> Simp[Identity[-1]   Int[Fx, x], 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 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 498

Int[((c_) + (d_.)*(x_))^(n_)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> Simp[ 
d*(c + d*x)^(n + 1)*((a + b*x^2)^(p + 1)/((n + 1)*(b*c^2 + a*d^2))), x] + S 
imp[b/((n + 1)*(b*c^2 + a*d^2))   Int[(c + d*x)^(n + 1)*(a + b*x^2)^p*(c*(n 
 + 1) - d*(n + 2*p + 3)*x), x], x] /; FreeQ[{a, b, c, d, n, p}, x] && NeQ[n 
, -1] && ((LtQ[n, -1] && IntQuadraticQ[a, 0, b, c, d, n, p, x]) || (SumSimp 
lerQ[n, 1] && IntegerQ[p]) || ILtQ[Simplify[n + 2*p + 3], 0])
 

rule 679

Int[((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))*((a_) + (c_.)*(x_)^2)^(p 
_.), x_Symbol] :> Simp[(-(e*f - d*g))*(d + e*x)^(m + 1)*((a + c*x^2)^(p + 1 
)/(2*(p + 1)*(c*d^2 + a*e^2))), x] + Simp[(c*d*f + a*e*g)/(c*d^2 + a*e^2) 
 Int[(d + e*x)^(m + 1)*(a + c*x^2)^p, x], x] /; FreeQ[{a, c, d, e, f, g, m, 
 p}, x] && EqQ[Simplify[m + 2*p + 3], 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]
 

Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(553\) vs. \(2(173)=346\).

Time = 0.37 (sec) , antiderivative size = 554, normalized size of antiderivative = 2.90

method	result	size
parts	\(-\frac {a}{3 \left (e x +d \right )^{3} e}-\frac {b \,c^{3} \arcsin \left (c x \right )}{3 \left (c x e +c d \right )^{3} e}+\frac {b \,c^{3} \sqrt {-\left (c x +\frac {d c}{e}\right )^{2}+\frac {2 d c \left (c x +\frac {d c}{e}\right )}{e}-\frac {c^{2} d^{2}-e^{2}}{e^{2}}}}{6 e^{2} \left (c^{2} d^{2}-e^{2}\right ) \left (c x +\frac {d c}{e}\right )^{2}}+\frac {b \,c^{4} d \sqrt {-\left (c x +\frac {d c}{e}\right )^{2}+\frac {2 d c \left (c x +\frac {d c}{e}\right )}{e}-\frac {c^{2} d^{2}-e^{2}}{e^{2}}}}{2 e \left (c^{2} d^{2}-e^{2}\right )^{2} \left (c x +\frac {d c}{e}\right )}-\frac {b \,c^{5} d^{2} \ln \left (\frac {-\frac {2 \left (c^{2} d^{2}-e^{2}\right )}{e^{2}}+\frac {2 d c \left (c x +\frac {d c}{e}\right )}{e}+2 \sqrt {-\frac {c^{2} d^{2}-e^{2}}{e^{2}}}\, \sqrt {-\left (c x +\frac {d c}{e}\right )^{2}+\frac {2 d c \left (c x +\frac {d c}{e}\right )}{e}-\frac {c^{2} d^{2}-e^{2}}{e^{2}}}}{c x +\frac {d c}{e}}\right )}{2 e^{2} \left (c^{2} d^{2}-e^{2}\right )^{2} \sqrt {-\frac {c^{2} d^{2}-e^{2}}{e^{2}}}}+\frac {b \,c^{3} \ln \left (\frac {-\frac {2 \left (c^{2} d^{2}-e^{2}\right )}{e^{2}}+\frac {2 d c \left (c x +\frac {d c}{e}\right )}{e}+2 \sqrt {-\frac {c^{2} d^{2}-e^{2}}{e^{2}}}\, \sqrt {-\left (c x +\frac {d c}{e}\right )^{2}+\frac {2 d c \left (c x +\frac {d c}{e}\right )}{e}-\frac {c^{2} d^{2}-e^{2}}{e^{2}}}}{c x +\frac {d c}{e}}\right )}{6 e^{2} \left (c^{2} d^{2}-e^{2}\right ) \sqrt {-\frac {c^{2} d^{2}-e^{2}}{e^{2}}}}\)	\(554\)
derivativedivides	\(\frac {-\frac {a \,c^{4}}{3 \left (c x e +c d \right )^{3} e}-\frac {b \,c^{4} \arcsin \left (c x \right )}{3 \left (c x e +c d \right )^{3} e}+\frac {b \,c^{4} \sqrt {-\left (c x +\frac {d c}{e}\right )^{2}+\frac {2 d c \left (c x +\frac {d c}{e}\right )}{e}-\frac {c^{2} d^{2}-e^{2}}{e^{2}}}}{6 e^{2} \left (c^{2} d^{2}-e^{2}\right ) \left (c x +\frac {d c}{e}\right )^{2}}+\frac {b \,c^{5} d \sqrt {-\left (c x +\frac {d c}{e}\right )^{2}+\frac {2 d c \left (c x +\frac {d c}{e}\right )}{e}-\frac {c^{2} d^{2}-e^{2}}{e^{2}}}}{2 e \left (c^{2} d^{2}-e^{2}\right )^{2} \left (c x +\frac {d c}{e}\right )}-\frac {b \,c^{6} d^{2} \ln \left (\frac {-\frac {2 \left (c^{2} d^{2}-e^{2}\right )}{e^{2}}+\frac {2 d c \left (c x +\frac {d c}{e}\right )}{e}+2 \sqrt {-\frac {c^{2} d^{2}-e^{2}}{e^{2}}}\, \sqrt {-\left (c x +\frac {d c}{e}\right )^{2}+\frac {2 d c \left (c x +\frac {d c}{e}\right )}{e}-\frac {c^{2} d^{2}-e^{2}}{e^{2}}}}{c x +\frac {d c}{e}}\right )}{2 e^{2} \left (c^{2} d^{2}-e^{2}\right )^{2} \sqrt {-\frac {c^{2} d^{2}-e^{2}}{e^{2}}}}+\frac {b \,c^{4} \ln \left (\frac {-\frac {2 \left (c^{2} d^{2}-e^{2}\right )}{e^{2}}+\frac {2 d c \left (c x +\frac {d c}{e}\right )}{e}+2 \sqrt {-\frac {c^{2} d^{2}-e^{2}}{e^{2}}}\, \sqrt {-\left (c x +\frac {d c}{e}\right )^{2}+\frac {2 d c \left (c x +\frac {d c}{e}\right )}{e}-\frac {c^{2} d^{2}-e^{2}}{e^{2}}}}{c x +\frac {d c}{e}}\right )}{6 e^{2} \left (c^{2} d^{2}-e^{2}\right ) \sqrt {-\frac {c^{2} d^{2}-e^{2}}{e^{2}}}}}{c}\)	\(564\)
default	\(\frac {-\frac {a \,c^{4}}{3 \left (c x e +c d \right )^{3} e}-\frac {b \,c^{4} \arcsin \left (c x \right )}{3 \left (c x e +c d \right )^{3} e}+\frac {b \,c^{4} \sqrt {-\left (c x +\frac {d c}{e}\right )^{2}+\frac {2 d c \left (c x +\frac {d c}{e}\right )}{e}-\frac {c^{2} d^{2}-e^{2}}{e^{2}}}}{6 e^{2} \left (c^{2} d^{2}-e^{2}\right ) \left (c x +\frac {d c}{e}\right )^{2}}+\frac {b \,c^{5} d \sqrt {-\left (c x +\frac {d c}{e}\right )^{2}+\frac {2 d c \left (c x +\frac {d c}{e}\right )}{e}-\frac {c^{2} d^{2}-e^{2}}{e^{2}}}}{2 e \left (c^{2} d^{2}-e^{2}\right )^{2} \left (c x +\frac {d c}{e}\right )}-\frac {b \,c^{6} d^{2} \ln \left (\frac {-\frac {2 \left (c^{2} d^{2}-e^{2}\right )}{e^{2}}+\frac {2 d c \left (c x +\frac {d c}{e}\right )}{e}+2 \sqrt {-\frac {c^{2} d^{2}-e^{2}}{e^{2}}}\, \sqrt {-\left (c x +\frac {d c}{e}\right )^{2}+\frac {2 d c \left (c x +\frac {d c}{e}\right )}{e}-\frac {c^{2} d^{2}-e^{2}}{e^{2}}}}{c x +\frac {d c}{e}}\right )}{2 e^{2} \left (c^{2} d^{2}-e^{2}\right )^{2} \sqrt {-\frac {c^{2} d^{2}-e^{2}}{e^{2}}}}+\frac {b \,c^{4} \ln \left (\frac {-\frac {2 \left (c^{2} d^{2}-e^{2}\right )}{e^{2}}+\frac {2 d c \left (c x +\frac {d c}{e}\right )}{e}+2 \sqrt {-\frac {c^{2} d^{2}-e^{2}}{e^{2}}}\, \sqrt {-\left (c x +\frac {d c}{e}\right )^{2}+\frac {2 d c \left (c x +\frac {d c}{e}\right )}{e}-\frac {c^{2} d^{2}-e^{2}}{e^{2}}}}{c x +\frac {d c}{e}}\right )}{6 e^{2} \left (c^{2} d^{2}-e^{2}\right ) \sqrt {-\frac {c^{2} d^{2}-e^{2}}{e^{2}}}}}{c}\)	\(564\)

Input:

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

Output:

-1/3*a/(e*x+d)^3/e-1/3*b*c^3/(c*e*x+c*d)^3/e*arcsin(c*x)+1/6*b*c^3/e^2/(c^ 
2*d^2-e^2)/(c*x+d*c/e)^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)+1/2*b*c^4/e*d/(c^2*d^2-e^2)^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)-1/2*b*c^5/e^2*d^2/(c^2*d^2-e^2)^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))+1/6*b*c^3/e^2/(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. 550 vs. \(2 (173) = 346\).

Time = 0.82 (sec) , antiderivative size = 1125, normalized size of antiderivative = 5.89 \[ \int \frac {a+b \arcsin (c x)}{(d+e x)^4} \, dx =\text {Too large to display} \] Input:

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

Output:

[-1/12*(4*a*c^6*d^6 - 12*a*c^4*d^4*e^2 + 12*a*c^2*d^2*e^4 - 4*a*e^6 + (2*b 
*c^5*d^5 + b*c^3*d^3*e^2 + (2*b*c^5*d^2*e^3 + b*c^3*e^5)*x^3 + 3*(2*b*c^5* 
d^3*e^2 + b*c^3*d*e^4)*x^2 + 3*(2*b*c^5*d^4*e + b*c^3*d^2*e^3)*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*sq 
rt(-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)) + 4*(b*c^6*d^6 - 3*b*c^4*d^4*e^2 + 3*b*c^2*d^2*e^4 - b*e^6)* 
arcsin(c*x) - 2*(4*b*c^5*d^5*e - 5*b*c^3*d^3*e^3 + b*c*d*e^5 + 3*(b*c^5*d^ 
3*e^3 - b*c^3*d*e^5)*x^2 + (7*b*c^5*d^4*e^2 - 8*b*c^3*d^2*e^4 + b*c*e^6)*x 
)*sqrt(-c^2*x^2 + 1))/(c^6*d^9*e - 3*c^4*d^7*e^3 + 3*c^2*d^5*e^5 - d^3*e^7 
 + (c^6*d^6*e^4 - 3*c^4*d^4*e^6 + 3*c^2*d^2*e^8 - e^10)*x^3 + 3*(c^6*d^7*e 
^3 - 3*c^4*d^5*e^5 + 3*c^2*d^3*e^7 - d*e^9)*x^2 + 3*(c^6*d^8*e^2 - 3*c^4*d 
^6*e^4 + 3*c^2*d^4*e^6 - d^2*e^8)*x), -1/6*(2*a*c^6*d^6 - 6*a*c^4*d^4*e^2 
+ 6*a*c^2*d^2*e^4 - 2*a*e^6 - (2*b*c^5*d^5 + b*c^3*d^3*e^2 + (2*b*c^5*d^2* 
e^3 + b*c^3*e^5)*x^3 + 3*(2*b*c^5*d^3*e^2 + b*c^3*d*e^4)*x^2 + 3*(2*b*c^5* 
d^4*e + b*c^3*d^2*e^3)*x)*sqrt(c^2*d^2 - e^2)*arctan(sqrt(c^2*d^2 - e^2)*( 
c^2*d*x + e)*sqrt(-c^2*x^2 + 1)/(c^2*d^2 - (c^4*d^2 - c^2*e^2)*x^2 - e^2)) 
 + 2*(b*c^6*d^6 - 3*b*c^4*d^4*e^2 + 3*b*c^2*d^2*e^4 - b*e^6)*arcsin(c*x) - 
 (4*b*c^5*d^5*e - 5*b*c^3*d^3*e^3 + b*c*d*e^5 + 3*(b*c^5*d^3*e^3 - b*c^3*d 
*e^5)*x^2 + (7*b*c^5*d^4*e^2 - 8*b*c^3*d^2*e^4 + b*c*e^6)*x)*sqrt(-c^2*x^2 
 + 1))/(c^6*d^9*e - 3*c^4*d^7*e^3 + 3*c^2*d^5*e^5 - d^3*e^7 + (c^6*d^6*...
 

Sympy [F]

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

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

Output:

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

Maxima [F]

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

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

Output:

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

Giac [F(-2)]

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

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

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

Output:

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

Reduce [F]

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

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

Output:

(3*int(asin(c*x)/(d**4 + 4*d**3*e*x + 6*d**2*e**2*x**2 + 4*d*e**3*x**3 + e 
**4*x**4),x)*b*d**3*e + 9*int(asin(c*x)/(d**4 + 4*d**3*e*x + 6*d**2*e**2*x 
**2 + 4*d*e**3*x**3 + e**4*x**4),x)*b*d**2*e**2*x + 9*int(asin(c*x)/(d**4 
+ 4*d**3*e*x + 6*d**2*e**2*x**2 + 4*d*e**3*x**3 + e**4*x**4),x)*b*d*e**3*x 
**2 + 3*int(asin(c*x)/(d**4 + 4*d**3*e*x + 6*d**2*e**2*x**2 + 4*d*e**3*x** 
3 + e**4*x**4),x)*b*e**4*x**3 - a)/(3*e*(d**3 + 3*d**2*e*x + 3*d*e**2*x**2 
 + e**3*x**3))