\(\int \frac {(a+b \arcsin (c x))^2}{\sqrt {d+c d x} (e-c e x)^{3/2}} \, dx\) [90]

Optimal result

Integrand size = 32, antiderivative size = 454 \[ \int \frac {(a+b \arcsin (c x))^2}{\sqrt {d+c d x} (e-c e x)^{3/2}} \, dx=\frac {d \left (1-c^2 x^2\right ) (a+b \arcsin (c x))^2}{c (d+c d x)^{3/2} (e-c e x)^{3/2}}+\frac {d x \left (1-c^2 x^2\right ) (a+b \arcsin (c x))^2}{(d+c d x)^{3/2} (e-c e x)^{3/2}}-\frac {i d \left (1-c^2 x^2\right )^{3/2} (a+b \arcsin (c x))^2}{c (d+c d x)^{3/2} (e-c e x)^{3/2}}+\frac {4 i b d \left (1-c^2 x^2\right )^{3/2} (a+b \arcsin (c x)) \arctan \left (e^{i \arcsin (c x)}\right )}{c (d+c d x)^{3/2} (e-c e x)^{3/2}}+\frac {2 b d \left (1-c^2 x^2\right )^{3/2} (a+b \arcsin (c x)) \log \left (1+e^{2 i \arcsin (c x)}\right )}{c (d+c d x)^{3/2} (e-c e x)^{3/2}}-\frac {2 i b^2 d \left (1-c^2 x^2\right )^{3/2} \operatorname {PolyLog}\left (2,-i e^{i \arcsin (c x)}\right )}{c (d+c d x)^{3/2} (e-c e x)^{3/2}}+\frac {2 i b^2 d \left (1-c^2 x^2\right )^{3/2} \operatorname {PolyLog}\left (2,i e^{i \arcsin (c x)}\right )}{c (d+c d x)^{3/2} (e-c e x)^{3/2}}-\frac {i b^2 d \left (1-c^2 x^2\right )^{3/2} \operatorname {PolyLog}\left (2,-e^{2 i \arcsin (c x)}\right )}{c (d+c d x)^{3/2} (e-c e x)^{3/2}} \] Output:

d*(-c^2*x^2+1)*(a+b*arcsin(c*x))^2/c/(c*d*x+d)^(3/2)/(-c*e*x+e)^(3/2)+d*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*d*(-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*d*(-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*d*(-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*d*(-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*d*(-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*d*(-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 = 2.24 (sec) , antiderivative size = 221, normalized size of antiderivative = 0.49 \[ \int \frac {(a+b \arcsin (c x))^2}{\sqrt {d+c d x} (e-c e x)^{3/2}} \, dx=-\frac {\sqrt {d+c d x} \sqrt {e-c e x} \left (a \left (a+a c x+4 b \sqrt {1-c^2 x^2} \log \left (\cos \left (\frac {1}{4} (\pi +2 \arcsin (c x))\right )\right )\right )-4 i b^2 \sqrt {1-c^2 x^2} \operatorname {PolyLog}\left (2,-i e^{i \arcsin (c x)}\right )+b^2 \sqrt {1-c^2 x^2} \arcsin (c x)^2 \left (-i+\tan \left (\frac {1}{4} (\pi +2 \arcsin (c x))\right )\right )+2 b \sqrt {1-c^2 x^2} \arcsin (c x) \left (2 b \log \left (1+i e^{i \arcsin (c x)}\right )+a \tan \left (\frac {1}{4} (\pi +2 \arcsin (c x))\right )\right )\right )}{c d e^2 (-1+c x) (1+c x)} \] Input:

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

Output:

-((Sqrt[d + c*d*x]*Sqrt[e - c*e*x]*(a*(a + a*c*x + 4*b*Sqrt[1 - c^2*x^2]*L 
og[Cos[(Pi + 2*ArcSin[c*x])/4]]) - (4*I)*b^2*Sqrt[1 - c^2*x^2]*PolyLog[2, 
(-I)*E^(I*ArcSin[c*x])] + b^2*Sqrt[1 - c^2*x^2]*ArcSin[c*x]^2*(-I + Tan[(P 
i + 2*ArcSin[c*x])/4]) + 2*b*Sqrt[1 - c^2*x^2]*ArcSin[c*x]*(2*b*Log[1 + I* 
E^(I*ArcSin[c*x])] + a*Tan[(Pi + 2*ArcSin[c*x])/4])))/(c*d*e^2*(-1 + c*x)* 
(1 + c*x)))
 

Rubi [A] (verified)

Time = 0.91 (sec) , antiderivative size = 239, 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.

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

\(\Big \downarrow \) 5178

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 5262

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

\(\Big \downarrow \) 2009

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

Input:

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

Output:

(d*(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 = 3.00 (sec) , antiderivative size = 429, normalized size of antiderivative = 0.94

method	result	size
default	\(\frac {a^{2} \sqrt {c d x +d}}{c d e \sqrt {-c x e +e}}+b^{2} \left (-\frac {\arcsin \left (c x \right )^{2} \left (i \sqrt {-c^{2} x^{2}+1}+c x +1\right ) \sqrt {-e \left (c x -1\right )}\, \sqrt {d \left (c x +1\right )}}{\left (c x -1\right ) c d \,e^{2} \left (c x +1\right )}+\frac {2 i \sqrt {-c^{2} x^{2}+1}\, \sqrt {d \left (c x +1\right )}\, \sqrt {-e \left (c x -1\right )}\, \left (2 i \arcsin \left (c x \right ) \ln \left (1+i \left (i c x +\sqrt {-c^{2} x^{2}+1}\right )\right )+\arcsin \left (c x \right )^{2}+2 \operatorname {polylog}\left (2, -i \left (i c x +\sqrt {-c^{2} x^{2}+1}\right )\right )\right )}{c d \,e^{2} \left (c^{2} x^{2}-1\right )}\right )+2 a b \left (\frac {2 i \sqrt {-c^{2} x^{2}+1}\, \sqrt {d \left (c x +1\right )}\, \sqrt {-e \left (c x -1\right )}\, \arcsin \left (c x \right )}{c d \,e^{2} \left (c^{2} x^{2}-1\right )}-\frac {\arcsin \left (c x \right ) \left (i \sqrt {-c^{2} x^{2}+1}+c x +1\right ) \sqrt {-e \left (c x -1\right )}\, \sqrt {d \left (c x +1\right )}}{\left (c x -1\right ) c d \,e^{2} \left (c x +1\right )}-\frac {2 \sqrt {-e \left (c x -1\right )}\, \sqrt {d \left (c x +1\right )}\, \sqrt {-c^{2} x^{2}+1}\, \ln \left (i c x +\sqrt {-c^{2} x^{2}+1}-i\right )}{c d \,e^{2} \left (c^{2} x^{2}-1\right )}\right )\)	\(429\)
parts	\(\frac {a^{2} \sqrt {c d x +d}}{c d e \sqrt {-c x e +e}}+b^{2} \left (-\frac {\arcsin \left (c x \right )^{2} \left (i \sqrt {-c^{2} x^{2}+1}+c x +1\right ) \sqrt {-e \left (c x -1\right )}\, \sqrt {d \left (c x +1\right )}}{\left (c x -1\right ) c d \,e^{2} \left (c x +1\right )}+\frac {2 i \sqrt {-c^{2} x^{2}+1}\, \sqrt {d \left (c x +1\right )}\, \sqrt {-e \left (c x -1\right )}\, \left (2 i \arcsin \left (c x \right ) \ln \left (1+i \left (i c x +\sqrt {-c^{2} x^{2}+1}\right )\right )+\arcsin \left (c x \right )^{2}+2 \operatorname {polylog}\left (2, -i \left (i c x +\sqrt {-c^{2} x^{2}+1}\right )\right )\right )}{c d \,e^{2} \left (c^{2} x^{2}-1\right )}\right )+2 a b \left (\frac {2 i \sqrt {-c^{2} x^{2}+1}\, \sqrt {d \left (c x +1\right )}\, \sqrt {-e \left (c x -1\right )}\, \arcsin \left (c x \right )}{c d \,e^{2} \left (c^{2} x^{2}-1\right )}-\frac {\arcsin \left (c x \right ) \left (i \sqrt {-c^{2} x^{2}+1}+c x +1\right ) \sqrt {-e \left (c x -1\right )}\, \sqrt {d \left (c x +1\right )}}{\left (c x -1\right ) c d \,e^{2} \left (c x +1\right )}-\frac {2 \sqrt {-e \left (c x -1\right )}\, \sqrt {d \left (c x +1\right )}\, \sqrt {-c^{2} x^{2}+1}\, \ln \left (i c x +\sqrt {-c^{2} x^{2}+1}-i\right )}{c d \,e^{2} \left (c^{2} x^{2}-1\right )}\right )\)	\(429\)

Input:

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

Output:

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

Fricas [F]

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

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

Sympy [F]

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

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

Output:

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

Maxima [F(-2)]

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

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

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

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

Output:

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

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

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

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

Output:

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