\(\int \frac {a+b \arcsin (c x)}{(d+e x)^3} \, dx\) [7]

Optimal result

Integrand size = 16, antiderivative size = 135 \[ \int \frac {a+b \arcsin (c x)}{(d+e x)^3} \, dx=\frac {b c \sqrt {1-c^2 x^2}}{2 \left (c^2 d^2-e^2\right ) (d+e x)}-\frac {a+b \arcsin (c x)}{2 e (d+e x)^2}+\frac {b c^3 d \arctan \left (\frac {e+c^2 d x}{\sqrt {c^2 d^2-e^2} \sqrt {1-c^2 x^2}}\right )}{2 e \left (c^2 d^2-e^2\right )^{3/2}} \] Output:

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

Mathematica [C] (verified)

Result contains complex when optimal does not.

Time = 0.23 (sec) , antiderivative size = 207, normalized size of antiderivative = 1.53 \[ \int \frac {a+b \arcsin (c x)}{(d+e x)^3} \, dx=\frac {1}{2} \left (-\frac {a}{e (d+e x)^2}+\frac {b c \sqrt {1-c^2 x^2}}{\left (c^2 d^2-e^2\right ) (d+e x)}-\frac {b \arcsin (c x)}{e (d+e x)^2}-\frac {i b c^3 d \left (\log (4)+\log \left (\frac {e^2 \sqrt {c^2 d^2-e^2} \left (i e+i c^2 d x+\sqrt {c^2 d^2-e^2} \sqrt {1-c^2 x^2}\right )}{b c^3 d (d+e x)}\right )\right )}{(c d-e) e (c d+e) \sqrt {c^2 d^2-e^2}}\right ) \] Input:

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

Output:

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

Rubi [A] (verified)

Time = 0.31 (sec) , antiderivative size = 134, normalized size of antiderivative = 0.99, number of steps used = 5, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.250, Rules used = {5242, 491, 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.

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

\(\Big \downarrow \) 5242

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

\(\Big \downarrow \) 491

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

\(\Big \downarrow \) 488

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

\(\Big \downarrow \) 217

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

Input:

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

Output:

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

Defintions of rubi rules used

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 491

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*(c/(b*c^2 + a*d^2))   Int[(c + d*x)^(n + 1)*(a + b*x^2)^p, x], x] /; 
FreeQ[{a, b, c, d, n, p}, x] && EqQ[n + 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. \(294\) vs. \(2(121)=242\).

Time = 0.59 (sec) , antiderivative size = 295, normalized size of antiderivative = 2.19

method	result	size
parts	\(-\frac {a}{2 \left (e x +d \right )^{2} e}-\frac {b \,c^{2} \arcsin \left (c x \right )}{2 \left (c x e +c d \right )^{2} e}+\frac {b \,c^{2} \sqrt {-\left (c x +\frac {d c}{e}\right )^{2}+\frac {2 d c \left (c x +\frac {d c}{e}\right )}{e}-\frac {c^{2} d^{2}-e^{2}}{e^{2}}}}{2 e \left (c^{2} d^{2}-e^{2}\right ) \left (c x +\frac {d c}{e}\right )}-\frac {b \,c^{3} d \ln \left (\frac {-\frac {2 \left (c^{2} d^{2}-e^{2}\right )}{e^{2}}+\frac {2 d c \left (c x +\frac {d c}{e}\right )}{e}+2 \sqrt {-\frac {c^{2} d^{2}-e^{2}}{e^{2}}}\, \sqrt {-\left (c x +\frac {d c}{e}\right )^{2}+\frac {2 d c \left (c x +\frac {d c}{e}\right )}{e}-\frac {c^{2} d^{2}-e^{2}}{e^{2}}}}{c x +\frac {d c}{e}}\right )}{2 e^{2} \left (c^{2} d^{2}-e^{2}\right ) \sqrt {-\frac {c^{2} d^{2}-e^{2}}{e^{2}}}}\)	\(295\)
derivativedivides	\(\frac {-\frac {a \,c^{3}}{2 \left (c x e +c d \right )^{2} e}+b \,c^{3} \left (-\frac {\arcsin \left (c x \right )}{2 \left (c x e +c d \right )^{2} e}+\frac {\frac {e^{2} \sqrt {-\left (c x +\frac {d c}{e}\right )^{2}+\frac {2 d c \left (c x +\frac {d c}{e}\right )}{e}-\frac {c^{2} d^{2}-e^{2}}{e^{2}}}}{\left (c^{2} d^{2}-e^{2}\right ) \left (c x +\frac {d c}{e}\right )}-\frac {d c e \ln \left (\frac {-\frac {2 \left (c^{2} d^{2}-e^{2}\right )}{e^{2}}+\frac {2 d c \left (c x +\frac {d c}{e}\right )}{e}+2 \sqrt {-\frac {c^{2} d^{2}-e^{2}}{e^{2}}}\, \sqrt {-\left (c x +\frac {d c}{e}\right )^{2}+\frac {2 d c \left (c x +\frac {d c}{e}\right )}{e}-\frac {c^{2} d^{2}-e^{2}}{e^{2}}}}{c x +\frac {d c}{e}}\right )}{\left (c^{2} d^{2}-e^{2}\right ) \sqrt {-\frac {c^{2} d^{2}-e^{2}}{e^{2}}}}}{2 e^{3}}\right )}{c}\)	\(303\)
default	\(\frac {-\frac {a \,c^{3}}{2 \left (c x e +c d \right )^{2} e}+b \,c^{3} \left (-\frac {\arcsin \left (c x \right )}{2 \left (c x e +c d \right )^{2} e}+\frac {\frac {e^{2} \sqrt {-\left (c x +\frac {d c}{e}\right )^{2}+\frac {2 d c \left (c x +\frac {d c}{e}\right )}{e}-\frac {c^{2} d^{2}-e^{2}}{e^{2}}}}{\left (c^{2} d^{2}-e^{2}\right ) \left (c x +\frac {d c}{e}\right )}-\frac {d c e \ln \left (\frac {-\frac {2 \left (c^{2} d^{2}-e^{2}\right )}{e^{2}}+\frac {2 d c \left (c x +\frac {d c}{e}\right )}{e}+2 \sqrt {-\frac {c^{2} d^{2}-e^{2}}{e^{2}}}\, \sqrt {-\left (c x +\frac {d c}{e}\right )^{2}+\frac {2 d c \left (c x +\frac {d c}{e}\right )}{e}-\frac {c^{2} d^{2}-e^{2}}{e^{2}}}}{c x +\frac {d c}{e}}\right )}{\left (c^{2} d^{2}-e^{2}\right ) \sqrt {-\frac {c^{2} d^{2}-e^{2}}{e^{2}}}}}{2 e^{3}}\right )}{c}\)	\(303\)

Input:

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

Output:

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

Leaf count of result is larger than twice the leaf count of optimal. 322 vs. \(2 (121) = 242\).

Time = 0.22 (sec) , antiderivative size = 673, normalized size of antiderivative = 4.99 \[ \int \frac {a+b \arcsin (c x)}{(d+e x)^3} \, dx=\left [-\frac {2 \, a c^{4} d^{4} - 4 \, a c^{2} d^{2} e^{2} + 2 \, a e^{4} - {\left (b c^{3} d e^{2} x^{2} + 2 \, b c^{3} d^{2} e x + b c^{3} d^{3}\right )} \sqrt {-c^{2} d^{2} + e^{2}} \log \left (\frac {2 \, c^{2} d e x - c^{2} d^{2} + {\left (2 \, c^{4} d^{2} - c^{2} e^{2}\right )} x^{2} + 2 \, \sqrt {-c^{2} d^{2} + e^{2}} {\left (c^{2} d x + e\right )} \sqrt {-c^{2} x^{2} + 1} + 2 \, e^{2}}{e^{2} x^{2} + 2 \, d e x + d^{2}}\right ) + 2 \, {\left (b c^{4} d^{4} - 2 \, b c^{2} d^{2} e^{2} + b e^{4}\right )} \arcsin \left (c x\right ) - 2 \, {\left (b c^{3} d^{3} e - b c d e^{3} + {\left (b c^{3} d^{2} e^{2} - b c e^{4}\right )} x\right )} \sqrt {-c^{2} x^{2} + 1}}{4 \, {\left (c^{4} d^{6} e - 2 \, c^{2} d^{4} e^{3} + d^{2} e^{5} + {\left (c^{4} d^{4} e^{3} - 2 \, c^{2} d^{2} e^{5} + e^{7}\right )} x^{2} + 2 \, {\left (c^{4} d^{5} e^{2} - 2 \, c^{2} d^{3} e^{4} + d e^{6}\right )} x\right )}}, -\frac {a c^{4} d^{4} - 2 \, a c^{2} d^{2} e^{2} + a e^{4} - {\left (b c^{3} d e^{2} x^{2} + 2 \, b c^{3} d^{2} e x + b c^{3} d^{3}\right )} \sqrt {c^{2} d^{2} - e^{2}} \arctan \left (\frac {\sqrt {c^{2} d^{2} - e^{2}} {\left (c^{2} d x + e\right )} \sqrt {-c^{2} x^{2} + 1}}{c^{2} d^{2} - {\left (c^{4} d^{2} - c^{2} e^{2}\right )} x^{2} - e^{2}}\right ) + {\left (b c^{4} d^{4} - 2 \, b c^{2} d^{2} e^{2} + b e^{4}\right )} \arcsin \left (c x\right ) - {\left (b c^{3} d^{3} e - b c d e^{3} + {\left (b c^{3} d^{2} e^{2} - b c e^{4}\right )} x\right )} \sqrt {-c^{2} x^{2} + 1}}{2 \, {\left (c^{4} d^{6} e - 2 \, c^{2} d^{4} e^{3} + d^{2} e^{5} + {\left (c^{4} d^{4} e^{3} - 2 \, c^{2} d^{2} e^{5} + e^{7}\right )} x^{2} + 2 \, {\left (c^{4} d^{5} e^{2} - 2 \, c^{2} d^{3} e^{4} + d e^{6}\right )} x\right )}}\right ] \] Input:

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

Output:

[-1/4*(2*a*c^4*d^4 - 4*a*c^2*d^2*e^2 + 2*a*e^4 - (b*c^3*d*e^2*x^2 + 2*b*c^ 
3*d^2*e*x + b*c^3*d^3)*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*sqrt(-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*(b*c^4*d^4 - 2*b*c^2*d^2* 
e^2 + b*e^4)*arcsin(c*x) - 2*(b*c^3*d^3*e - b*c*d*e^3 + (b*c^3*d^2*e^2 - b 
*c*e^4)*x)*sqrt(-c^2*x^2 + 1))/(c^4*d^6*e - 2*c^2*d^4*e^3 + d^2*e^5 + (c^4 
*d^4*e^3 - 2*c^2*d^2*e^5 + e^7)*x^2 + 2*(c^4*d^5*e^2 - 2*c^2*d^3*e^4 + d*e 
^6)*x), -1/2*(a*c^4*d^4 - 2*a*c^2*d^2*e^2 + a*e^4 - (b*c^3*d*e^2*x^2 + 2*b 
*c^3*d^2*e*x + b*c^3*d^3)*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)) 
 + (b*c^4*d^4 - 2*b*c^2*d^2*e^2 + b*e^4)*arcsin(c*x) - (b*c^3*d^3*e - b*c* 
d*e^3 + (b*c^3*d^2*e^2 - b*c*e^4)*x)*sqrt(-c^2*x^2 + 1))/(c^4*d^6*e - 2*c^ 
2*d^4*e^3 + d^2*e^5 + (c^4*d^4*e^3 - 2*c^2*d^2*e^5 + e^7)*x^2 + 2*(c^4*d^5 
*e^2 - 2*c^2*d^3*e^4 + d*e^6)*x)]
 

Sympy [F]

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

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

Output:

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

Maxima [F(-2)]

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

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

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

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

Output:

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

Reduce [F]

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

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

Output:

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