\(\int \frac {(a+b \arcsin (c x))^2}{(d+c d x)^{3/2} \sqrt {e-c e x}} \, dx\) [88]

Optimal result

Integrand size = 32, antiderivative size = 455 \[ \int \frac {(a+b \arcsin (c x))^2}{(d+c d x)^{3/2} \sqrt {e-c e x}} \, dx=-\frac {e \left (1-c^2 x^2\right ) (a+b \arcsin (c x))^2}{c (d+c d x)^{3/2} (e-c e x)^{3/2}}+\frac {e x \left (1-c^2 x^2\right ) (a+b \arcsin (c x))^2}{(d+c d x)^{3/2} (e-c e x)^{3/2}}-\frac {i e \left (1-c^2 x^2\right )^{3/2} (a+b \arcsin (c x))^2}{c (d+c d x)^{3/2} (e-c e x)^{3/2}}-\frac {4 i b e \left (1-c^2 x^2\right )^{3/2} (a+b \arcsin (c x)) \arctan \left (e^{i \arcsin (c x)}\right )}{c (d+c d x)^{3/2} (e-c e x)^{3/2}}+\frac {2 b e \left (1-c^2 x^2\right )^{3/2} (a+b \arcsin (c x)) \log \left (1+e^{2 i \arcsin (c x)}\right )}{c (d+c d x)^{3/2} (e-c e x)^{3/2}}+\frac {2 i b^2 e \left (1-c^2 x^2\right )^{3/2} \operatorname {PolyLog}\left (2,-i e^{i \arcsin (c x)}\right )}{c (d+c d x)^{3/2} (e-c e x)^{3/2}}-\frac {2 i b^2 e \left (1-c^2 x^2\right )^{3/2} \operatorname {PolyLog}\left (2,i e^{i \arcsin (c x)}\right )}{c (d+c d x)^{3/2} (e-c e x)^{3/2}}-\frac {i b^2 e \left (1-c^2 x^2\right )^{3/2} \operatorname {PolyLog}\left (2,-e^{2 i \arcsin (c x)}\right )}{c (d+c d x)^{3/2} (e-c e x)^{3/2}} \] Output:

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

Mathematica [A] (verified)

Time = 2.28 (sec) , antiderivative size = 217, normalized size of antiderivative = 0.48 \[ \int \frac {(a+b \arcsin (c x))^2}{(d+c d x)^{3/2} \sqrt {e-c e x}} \, dx=-\frac {\sqrt {d+c d x} \sqrt {e-c e x} \left (-b^2 \sqrt {1-c^2 x^2} \arcsin (c x)^2 \left (-i+\cot \left (\frac {1}{4} (\pi +2 \arcsin (c x))\right )\right )+2 b \sqrt {1-c^2 x^2} \arcsin (c x) \left (-a \cot \left (\frac {1}{4} (\pi +2 \arcsin (c x))\right )+2 b \log \left (1+i e^{-i \arcsin (c x)}\right )\right )+a \left (-a+a c x+2 b \sqrt {1-c^2 x^2} \log (1+c x)\right )+4 i b^2 \sqrt {1-c^2 x^2} \operatorname {PolyLog}\left (2,-i e^{-i \arcsin (c x)}\right )\right )}{c d^2 e (-1+c x) (1+c x)} \] Input:

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

Output:

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

Rubi [A] (verified)

Time = 0.89 (sec) , antiderivative size = 240, normalized size of antiderivative = 0.53, number of steps used = 4, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.125, Rules used = {5178, 27, 5262, 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.

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

\(\Big \downarrow \) 5178

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 5262

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

\(\Big \downarrow \) 2009

\(\displaystyle \frac {e \left (1-c^2 x^2\right )^{3/2} \left (-\frac {4 i b \arctan \left (e^{i \arcsin (c x)}\right ) (a+b \arcsin (c x))}{c}+\frac {x (a+b \arcsin (c x))^2}{\sqrt {1-c^2 x^2}}-\frac {(a+b \arcsin (c x))^2}{c \sqrt {1-c^2 x^2}}-\frac {i (a+b \arcsin (c x))^2}{c}+\frac {2 b \log \left (1+e^{2 i \arcsin (c x)}\right ) (a+b \arcsin (c x))}{c}+\frac {2 i b^2 \operatorname {PolyLog}\left (2,-i e^{i \arcsin (c x)}\right )}{c}-\frac {2 i b^2 \operatorname {PolyLog}\left (2,i e^{i \arcsin (c x)}\right )}{c}-\frac {i b^2 \operatorname {PolyLog}\left (2,-e^{2 i \arcsin (c x)}\right )}{c}\right )}{(c d x+d)^{3/2} (e-c e x)^{3/2}}\)

Input:

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

Output:

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

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 5178

Int[((a_.) + ArcSin[(c_.)*(x_)]*(b_.))^(n_.)*((d_) + (e_.)*(x_))^(p_)*((f_) 
 + (g_.)*(x_))^(q_), x_Symbol] :> Simp[(d + e*x)^q*((f + g*x)^q/(1 - c^2*x^ 
2)^q)   Int[(d + e*x)^(p - q)*(1 - c^2*x^2)^q*(a + b*ArcSin[c*x])^n, x], x] 
 /; FreeQ[{a, b, c, d, e, f, g, n}, x] && EqQ[e*f + d*g, 0] && EqQ[c^2*d^2 
- e^2, 0] && HalfIntegerQ[p, q] && GeQ[p - q, 0]
 

rule 5262

Int[((a_.) + ArcSin[(c_.)*(x_)]*(b_.))^(n_.)*((f_) + (g_.)*(x_))^(m_.)*((d_ 
) + (e_.)*(x_)^2)^(p_), x_Symbol] :> Int[ExpandIntegrand[(d + e*x^2)^p*(a + 
 b*ArcSin[c*x])^n, (f + g*x)^m, x], x] /; FreeQ[{a, b, c, d, e, f, g}, x] & 
& EqQ[c^2*d + e, 0] && IGtQ[m, 0] && IntegerQ[p + 1/2] && GtQ[d, 0] && IGtQ 
[n, 0] && (m == 1 || p > 0 || (n == 1 && p > -1) || (m == 2 && p < -2))
 

Maple [A] (verified)

Time = 2.91 (sec) , antiderivative size = 430, normalized size of antiderivative = 0.95

method	result	size
default	\(-\frac {a^{2} \sqrt {-c x e +e}}{c d e \sqrt {c d x +d}}+b^{2} \left (-\frac {\arcsin \left (c x \right )^{2} \sqrt {-e \left (c x -1\right )}\, \sqrt {d \left (c x +1\right )}\, \left (i \sqrt {-c^{2} x^{2}+1}+c x -1\right )}{\left (c x +1\right ) c \,d^{2} e \left (c x -1\right )}+\frac {2 i \sqrt {-c^{2} x^{2}+1}\, \sqrt {d \left (c x +1\right )}\, \sqrt {-e \left (c x -1\right )}\, \left (2 i \arcsin \left (c x \right ) \ln \left (1-i \left (i c x +\sqrt {-c^{2} x^{2}+1}\right )\right )+\arcsin \left (c x \right )^{2}+2 \operatorname {polylog}\left (2, i \left (i c x +\sqrt {-c^{2} x^{2}+1}\right )\right )\right )}{c \,d^{2} e \left (c^{2} x^{2}-1\right )}\right )+2 a b \left (\frac {2 i \sqrt {-c^{2} x^{2}+1}\, \sqrt {d \left (c x +1\right )}\, \sqrt {-e \left (c x -1\right )}\, \arcsin \left (c x \right )}{c \,d^{2} e \left (c^{2} x^{2}-1\right )}-\frac {\arcsin \left (c x \right ) \sqrt {-e \left (c x -1\right )}\, \sqrt {d \left (c x +1\right )}\, \left (i \sqrt {-c^{2} x^{2}+1}+c x -1\right )}{\left (c x +1\right ) c \,d^{2} e \left (c x -1\right )}-\frac {2 \sqrt {-e \left (c x -1\right )}\, \sqrt {d \left (c x +1\right )}\, \sqrt {-c^{2} x^{2}+1}\, \ln \left (i c x +\sqrt {-c^{2} x^{2}+1}+i\right )}{c \,d^{2} e \left (c^{2} x^{2}-1\right )}\right )\)	\(430\)
parts	\(-\frac {a^{2} \sqrt {-c x e +e}}{c d e \sqrt {c d x +d}}+b^{2} \left (-\frac {\arcsin \left (c x \right )^{2} \sqrt {-e \left (c x -1\right )}\, \sqrt {d \left (c x +1\right )}\, \left (i \sqrt {-c^{2} x^{2}+1}+c x -1\right )}{\left (c x +1\right ) c \,d^{2} e \left (c x -1\right )}+\frac {2 i \sqrt {-c^{2} x^{2}+1}\, \sqrt {d \left (c x +1\right )}\, \sqrt {-e \left (c x -1\right )}\, \left (2 i \arcsin \left (c x \right ) \ln \left (1-i \left (i c x +\sqrt {-c^{2} x^{2}+1}\right )\right )+\arcsin \left (c x \right )^{2}+2 \operatorname {polylog}\left (2, i \left (i c x +\sqrt {-c^{2} x^{2}+1}\right )\right )\right )}{c \,d^{2} e \left (c^{2} x^{2}-1\right )}\right )+2 a b \left (\frac {2 i \sqrt {-c^{2} x^{2}+1}\, \sqrt {d \left (c x +1\right )}\, \sqrt {-e \left (c x -1\right )}\, \arcsin \left (c x \right )}{c \,d^{2} e \left (c^{2} x^{2}-1\right )}-\frac {\arcsin \left (c x \right ) \sqrt {-e \left (c x -1\right )}\, \sqrt {d \left (c x +1\right )}\, \left (i \sqrt {-c^{2} x^{2}+1}+c x -1\right )}{\left (c x +1\right ) c \,d^{2} e \left (c x -1\right )}-\frac {2 \sqrt {-e \left (c x -1\right )}\, \sqrt {d \left (c x +1\right )}\, \sqrt {-c^{2} x^{2}+1}\, \ln \left (i c x +\sqrt {-c^{2} x^{2}+1}+i\right )}{c \,d^{2} e \left (c^{2} x^{2}-1\right )}\right )\)	\(430\)

Input:

int((a+b*arcsin(c*x))^2/(c*d*x+d)^(3/2)/(-c*e*x+e)^(1/2),x,method=_RETURNV 
ERBOSE)
 

Output:

-a^2/c/d/e/(c*d*x+d)^(1/2)*(-c*e*x+e)^(1/2)+b^2*(-arcsin(c*x)^2*(-e*(c*x-1 
))^(1/2)*(d*(c*x+1))^(1/2)*(I*(-c^2*x^2+1)^(1/2)+c*x-1)/(c*x+1)/c/d^2/e/(c 
*x-1)+2*I*(-c^2*x^2+1)^(1/2)*(d*(c*x+1))^(1/2)*(-e*(c*x-1))^(1/2)*(2*I*ln( 
1-I*(I*c*x+(-c^2*x^2+1)^(1/2)))*arcsin(c*x)+arcsin(c*x)^2+2*polylog(2,I*(I 
*c*x+(-c^2*x^2+1)^(1/2))))/c/d^2/e/(c^2*x^2-1))+2*a*b*(2*I*(-c^2*x^2+1)^(1 
/2)*(d*(c*x+1))^(1/2)*(-e*(c*x-1))^(1/2)/c/d^2/e/(c^2*x^2-1)*arcsin(c*x)-a 
rcsin(c*x)*(-e*(c*x-1))^(1/2)*(d*(c*x+1))^(1/2)*(I*(-c^2*x^2+1)^(1/2)+c*x- 
1)/(c*x+1)/c/d^2/e/(c*x-1)-2*(-e*(c*x-1))^(1/2)*(d*(c*x+1))^(1/2)*(-c^2*x^ 
2+1)^(1/2)/c/d^2/e/(c^2*x^2-1)*ln(I*c*x+(-c^2*x^2+1)^(1/2)+I))
 

Fricas [F]

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

integrate((a+b*arcsin(c*x))^2/(c*d*x+d)^(3/2)/(-c*e*x+e)^(1/2),x, algorith 
m="fricas")
 

Output:

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

Sympy [F]

\[ \int \frac {(a+b \arcsin (c x))^2}{(d+c d x)^{3/2} \sqrt {e-c e x}} \, dx=\int \frac {\left (a + b \operatorname {asin}{\left (c x \right )}\right )^{2}}{\left (d \left (c x + 1\right )\right )^{\frac {3}{2}} \sqrt {- e \left (c x - 1\right )}}\, dx \] Input:

integrate((a+b*asin(c*x))**2/(c*d*x+d)**(3/2)/(-c*e*x+e)**(1/2),x)
 

Output:

Integral((a + b*asin(c*x))**2/((d*(c*x + 1))**(3/2)*sqrt(-e*(c*x - 1))), x 
)
 

Maxima [F(-2)]

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

integrate((a+b*arcsin(c*x))^2/(c*d*x+d)^(3/2)/(-c*e*x+e)^(1/2),x, algorith 
m="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>0)', see `assume?` for more de 
tails)Is e
 

Giac [F]

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

integrate((a+b*arcsin(c*x))^2/(c*d*x+d)^(3/2)/(-c*e*x+e)^(1/2),x, algorith 
m="giac")
 

Output:

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

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

\[ \int \frac {(a+b \arcsin (c x))^2}{(d+c d x)^{3/2} \sqrt {e-c e x}} \, dx=\frac {-\sqrt {-c x +1}\, a^{2}+2 \sqrt {c x +1}\, \left (\int \frac {\mathit {asin} \left (c x \right )}{\sqrt {c x +1}\, \sqrt {-c x +1}\, c x +\sqrt {c x +1}\, \sqrt {-c x +1}}d x \right ) a b c +\sqrt {c x +1}\, \left (\int \frac {\mathit {asin} \left (c x \right )^{2}}{\sqrt {c x +1}\, \sqrt {-c x +1}\, c x +\sqrt {c x +1}\, \sqrt {-c x +1}}d x \right ) b^{2} c}{\sqrt {e}\, \sqrt {d}\, \sqrt {c x +1}\, c d} \] Input:

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

Output:

( - sqrt( - c*x + 1)*a**2 + 2*sqrt(c*x + 1)*int(asin(c*x)/(sqrt(c*x + 1)*s 
qrt( - c*x + 1)*c*x + sqrt(c*x + 1)*sqrt( - c*x + 1)),x)*a*b*c + sqrt(c*x 
+ 1)*int(asin(c*x)**2/(sqrt(c*x + 1)*sqrt( - c*x + 1)*c*x + sqrt(c*x + 1)* 
sqrt( - c*x + 1)),x)*b**2*c)/(sqrt(e)*sqrt(d)*sqrt(c*x + 1)*c*d)