\(\int \frac {x^4 (a+b \arcsin (c x))^2}{\sqrt {d-c^2 d x^2}} \, dx\) [232]

Optimal result

Integrand size = 29, antiderivative size = 323 \[ \int \frac {x^4 (a+b \arcsin (c x))^2}{\sqrt {d-c^2 d x^2}} \, dx=\frac {15 b^2 x \sqrt {d-c^2 d x^2}}{64 c^4 d}+\frac {b^2 x^3 \sqrt {d-c^2 d x^2}}{32 c^2 d}-\frac {15 b^2 \sqrt {1-c^2 x^2} \arcsin (c x)}{64 c^5 \sqrt {d-c^2 d x^2}}+\frac {3 b x^2 \sqrt {1-c^2 x^2} (a+b \arcsin (c x))}{8 c^3 \sqrt {d-c^2 d x^2}}+\frac {b x^4 \sqrt {1-c^2 x^2} (a+b \arcsin (c x))}{8 c \sqrt {d-c^2 d x^2}}-\frac {3 x \sqrt {d-c^2 d x^2} (a+b \arcsin (c x))^2}{8 c^4 d}-\frac {x^3 \sqrt {d-c^2 d x^2} (a+b \arcsin (c x))^2}{4 c^2 d}+\frac {\sqrt {1-c^2 x^2} (a+b \arcsin (c x))^3}{8 b c^5 \sqrt {d-c^2 d x^2}} \] Output:

15/64*b^2*x*(-c^2*d*x^2+d)^(1/2)/c^4/d+1/32*b^2*x^3*(-c^2*d*x^2+d)^(1/2)/c 
^2/d-15/64*b^2*(-c^2*x^2+1)^(1/2)*arcsin(c*x)/c^5/(-c^2*d*x^2+d)^(1/2)+3/8 
*b*x^2*(-c^2*x^2+1)^(1/2)*(a+b*arcsin(c*x))/c^3/(-c^2*d*x^2+d)^(1/2)+1/8*b 
*x^4*(-c^2*x^2+1)^(1/2)*(a+b*arcsin(c*x))/c/(-c^2*d*x^2+d)^(1/2)-3/8*x*(-c 
^2*d*x^2+d)^(1/2)*(a+b*arcsin(c*x))^2/c^4/d-1/4*x^3*(-c^2*d*x^2+d)^(1/2)*( 
a+b*arcsin(c*x))^2/c^2/d+1/8*(-c^2*x^2+1)^(1/2)*(a+b*arcsin(c*x))^3/b/c^5/ 
(-c^2*d*x^2+d)^(1/2)
 

Mathematica [A] (verified)

Time = 1.41 (sec) , antiderivative size = 283, normalized size of antiderivative = 0.88 \[ \int \frac {x^4 (a+b \arcsin (c x))^2}{\sqrt {d-c^2 d x^2}} \, dx=\frac {32 a^2 c \sqrt {d} x \left (-1+c^2 x^2\right ) \left (3+2 c^2 x^2\right )-96 a^2 \sqrt {d-c^2 d x^2} \arctan \left (\frac {c x \sqrt {d-c^2 d x^2}}{\sqrt {d} \left (-1+c^2 x^2\right )}\right )+b^2 \sqrt {d} \sqrt {1-c^2 x^2} \left (32 \arcsin (c x)^3+4 \arcsin (c x) (-16 \cos (2 \arcsin (c x))+\cos (4 \arcsin (c x)))+32 \sin (2 \arcsin (c x))-\sin (4 \arcsin (c x))+8 \arcsin (c x)^2 (-8 \sin (2 \arcsin (c x))+\sin (4 \arcsin (c x)))\right )-4 a b \sqrt {d} \sqrt {1-c^2 x^2} (16 \cos (2 \arcsin (c x))-\cos (4 \arcsin (c x))-4 \arcsin (c x) (6 \arcsin (c x)-8 \sin (2 \arcsin (c x))+\sin (4 \arcsin (c x))))}{256 c^5 \sqrt {d} \sqrt {d-c^2 d x^2}} \] Input:

Integrate[(x^4*(a + b*ArcSin[c*x])^2)/Sqrt[d - c^2*d*x^2],x]
 

Output:

(32*a^2*c*Sqrt[d]*x*(-1 + c^2*x^2)*(3 + 2*c^2*x^2) - 96*a^2*Sqrt[d - c^2*d 
*x^2]*ArcTan[(c*x*Sqrt[d - c^2*d*x^2])/(Sqrt[d]*(-1 + c^2*x^2))] + b^2*Sqr 
t[d]*Sqrt[1 - c^2*x^2]*(32*ArcSin[c*x]^3 + 4*ArcSin[c*x]*(-16*Cos[2*ArcSin 
[c*x]] + Cos[4*ArcSin[c*x]]) + 32*Sin[2*ArcSin[c*x]] - Sin[4*ArcSin[c*x]] 
+ 8*ArcSin[c*x]^2*(-8*Sin[2*ArcSin[c*x]] + Sin[4*ArcSin[c*x]])) - 4*a*b*Sq 
rt[d]*Sqrt[1 - c^2*x^2]*(16*Cos[2*ArcSin[c*x]] - Cos[4*ArcSin[c*x]] - 4*Ar 
cSin[c*x]*(6*ArcSin[c*x] - 8*Sin[2*ArcSin[c*x]] + Sin[4*ArcSin[c*x]])))/(2 
56*c^5*Sqrt[d]*Sqrt[d - c^2*d*x^2])
 

Rubi [A] (verified)

Time = 1.66 (sec) , antiderivative size = 347, normalized size of antiderivative = 1.07, number of steps used = 10, number of rules used = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.345, Rules used = {5210, 5138, 262, 262, 223, 5210, 5138, 262, 223, 5152}

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.

\(\displaystyle \int \frac {x^4 (a+b \arcsin (c x))^2}{\sqrt {d-c^2 d x^2}} \, dx\)

\(\Big \downarrow \) 5210

\(\displaystyle \frac {3 \int \frac {x^2 (a+b \arcsin (c x))^2}{\sqrt {d-c^2 d x^2}}dx}{4 c^2}+\frac {b \sqrt {1-c^2 x^2} \int x^3 (a+b \arcsin (c x))dx}{2 c \sqrt {d-c^2 d x^2}}-\frac {x^3 \sqrt {d-c^2 d x^2} (a+b \arcsin (c x))^2}{4 c^2 d}\)

\(\Big \downarrow \) 5138

\(\displaystyle \frac {3 \int \frac {x^2 (a+b \arcsin (c x))^2}{\sqrt {d-c^2 d x^2}}dx}{4 c^2}+\frac {b \sqrt {1-c^2 x^2} \left (\frac {1}{4} x^4 (a+b \arcsin (c x))-\frac {1}{4} b c \int \frac {x^4}{\sqrt {1-c^2 x^2}}dx\right )}{2 c \sqrt {d-c^2 d x^2}}-\frac {x^3 \sqrt {d-c^2 d x^2} (a+b \arcsin (c x))^2}{4 c^2 d}\)

\(\Big \downarrow \) 262

\(\displaystyle \frac {3 \int \frac {x^2 (a+b \arcsin (c x))^2}{\sqrt {d-c^2 d x^2}}dx}{4 c^2}+\frac {b \sqrt {1-c^2 x^2} \left (\frac {1}{4} x^4 (a+b \arcsin (c x))-\frac {1}{4} b c \left (\frac {3 \int \frac {x^2}{\sqrt {1-c^2 x^2}}dx}{4 c^2}-\frac {x^3 \sqrt {1-c^2 x^2}}{4 c^2}\right )\right )}{2 c \sqrt {d-c^2 d x^2}}-\frac {x^3 \sqrt {d-c^2 d x^2} (a+b \arcsin (c x))^2}{4 c^2 d}\)

\(\Big \downarrow \) 262

\(\displaystyle \frac {3 \int \frac {x^2 (a+b \arcsin (c x))^2}{\sqrt {d-c^2 d x^2}}dx}{4 c^2}+\frac {b \sqrt {1-c^2 x^2} \left (\frac {1}{4} x^4 (a+b \arcsin (c x))-\frac {1}{4} b c \left (\frac {3 \left (\frac {\int \frac {1}{\sqrt {1-c^2 x^2}}dx}{2 c^2}-\frac {x \sqrt {1-c^2 x^2}}{2 c^2}\right )}{4 c^2}-\frac {x^3 \sqrt {1-c^2 x^2}}{4 c^2}\right )\right )}{2 c \sqrt {d-c^2 d x^2}}-\frac {x^3 \sqrt {d-c^2 d x^2} (a+b \arcsin (c x))^2}{4 c^2 d}\)

\(\Big \downarrow \) 223

\(\displaystyle \frac {3 \int \frac {x^2 (a+b \arcsin (c x))^2}{\sqrt {d-c^2 d x^2}}dx}{4 c^2}-\frac {x^3 \sqrt {d-c^2 d x^2} (a+b \arcsin (c x))^2}{4 c^2 d}+\frac {b \sqrt {1-c^2 x^2} \left (\frac {1}{4} x^4 (a+b \arcsin (c x))-\frac {1}{4} b c \left (\frac {3 \left (\frac {\arcsin (c x)}{2 c^3}-\frac {x \sqrt {1-c^2 x^2}}{2 c^2}\right )}{4 c^2}-\frac {x^3 \sqrt {1-c^2 x^2}}{4 c^2}\right )\right )}{2 c \sqrt {d-c^2 d x^2}}\)

\(\Big \downarrow \) 5210

\(\displaystyle \frac {3 \left (\frac {b \sqrt {1-c^2 x^2} \int x (a+b \arcsin (c x))dx}{c \sqrt {d-c^2 d x^2}}+\frac {\int \frac {(a+b \arcsin (c x))^2}{\sqrt {d-c^2 d x^2}}dx}{2 c^2}-\frac {x \sqrt {d-c^2 d x^2} (a+b \arcsin (c x))^2}{2 c^2 d}\right )}{4 c^2}-\frac {x^3 \sqrt {d-c^2 d x^2} (a+b \arcsin (c x))^2}{4 c^2 d}+\frac {b \sqrt {1-c^2 x^2} \left (\frac {1}{4} x^4 (a+b \arcsin (c x))-\frac {1}{4} b c \left (\frac {3 \left (\frac {\arcsin (c x)}{2 c^3}-\frac {x \sqrt {1-c^2 x^2}}{2 c^2}\right )}{4 c^2}-\frac {x^3 \sqrt {1-c^2 x^2}}{4 c^2}\right )\right )}{2 c \sqrt {d-c^2 d x^2}}\)

\(\Big \downarrow \) 5138

\(\displaystyle \frac {3 \left (\frac {b \sqrt {1-c^2 x^2} \left (\frac {1}{2} x^2 (a+b \arcsin (c x))-\frac {1}{2} b c \int \frac {x^2}{\sqrt {1-c^2 x^2}}dx\right )}{c \sqrt {d-c^2 d x^2}}+\frac {\int \frac {(a+b \arcsin (c x))^2}{\sqrt {d-c^2 d x^2}}dx}{2 c^2}-\frac {x \sqrt {d-c^2 d x^2} (a+b \arcsin (c x))^2}{2 c^2 d}\right )}{4 c^2}-\frac {x^3 \sqrt {d-c^2 d x^2} (a+b \arcsin (c x))^2}{4 c^2 d}+\frac {b \sqrt {1-c^2 x^2} \left (\frac {1}{4} x^4 (a+b \arcsin (c x))-\frac {1}{4} b c \left (\frac {3 \left (\frac {\arcsin (c x)}{2 c^3}-\frac {x \sqrt {1-c^2 x^2}}{2 c^2}\right )}{4 c^2}-\frac {x^3 \sqrt {1-c^2 x^2}}{4 c^2}\right )\right )}{2 c \sqrt {d-c^2 d x^2}}\)

\(\Big \downarrow \) 262

\(\displaystyle \frac {3 \left (\frac {b \sqrt {1-c^2 x^2} \left (\frac {1}{2} x^2 (a+b \arcsin (c x))-\frac {1}{2} b c \left (\frac {\int \frac {1}{\sqrt {1-c^2 x^2}}dx}{2 c^2}-\frac {x \sqrt {1-c^2 x^2}}{2 c^2}\right )\right )}{c \sqrt {d-c^2 d x^2}}+\frac {\int \frac {(a+b \arcsin (c x))^2}{\sqrt {d-c^2 d x^2}}dx}{2 c^2}-\frac {x \sqrt {d-c^2 d x^2} (a+b \arcsin (c x))^2}{2 c^2 d}\right )}{4 c^2}-\frac {x^3 \sqrt {d-c^2 d x^2} (a+b \arcsin (c x))^2}{4 c^2 d}+\frac {b \sqrt {1-c^2 x^2} \left (\frac {1}{4} x^4 (a+b \arcsin (c x))-\frac {1}{4} b c \left (\frac {3 \left (\frac {\arcsin (c x)}{2 c^3}-\frac {x \sqrt {1-c^2 x^2}}{2 c^2}\right )}{4 c^2}-\frac {x^3 \sqrt {1-c^2 x^2}}{4 c^2}\right )\right )}{2 c \sqrt {d-c^2 d x^2}}\)

\(\Big \downarrow \) 223

\(\displaystyle \frac {3 \left (\frac {\int \frac {(a+b \arcsin (c x))^2}{\sqrt {d-c^2 d x^2}}dx}{2 c^2}-\frac {x \sqrt {d-c^2 d x^2} (a+b \arcsin (c x))^2}{2 c^2 d}+\frac {b \sqrt {1-c^2 x^2} \left (\frac {1}{2} x^2 (a+b \arcsin (c x))-\frac {1}{2} b c \left (\frac {\arcsin (c x)}{2 c^3}-\frac {x \sqrt {1-c^2 x^2}}{2 c^2}\right )\right )}{c \sqrt {d-c^2 d x^2}}\right )}{4 c^2}-\frac {x^3 \sqrt {d-c^2 d x^2} (a+b \arcsin (c x))^2}{4 c^2 d}+\frac {b \sqrt {1-c^2 x^2} \left (\frac {1}{4} x^4 (a+b \arcsin (c x))-\frac {1}{4} b c \left (\frac {3 \left (\frac {\arcsin (c x)}{2 c^3}-\frac {x \sqrt {1-c^2 x^2}}{2 c^2}\right )}{4 c^2}-\frac {x^3 \sqrt {1-c^2 x^2}}{4 c^2}\right )\right )}{2 c \sqrt {d-c^2 d x^2}}\)

\(\Big \downarrow \) 5152

\(\displaystyle -\frac {x^3 \sqrt {d-c^2 d x^2} (a+b \arcsin (c x))^2}{4 c^2 d}+\frac {3 \left (-\frac {x \sqrt {d-c^2 d x^2} (a+b \arcsin (c x))^2}{2 c^2 d}+\frac {\sqrt {1-c^2 x^2} (a+b \arcsin (c x))^3}{6 b c^3 \sqrt {d-c^2 d x^2}}+\frac {b \sqrt {1-c^2 x^2} \left (\frac {1}{2} x^2 (a+b \arcsin (c x))-\frac {1}{2} b c \left (\frac {\arcsin (c x)}{2 c^3}-\frac {x \sqrt {1-c^2 x^2}}{2 c^2}\right )\right )}{c \sqrt {d-c^2 d x^2}}\right )}{4 c^2}+\frac {b \sqrt {1-c^2 x^2} \left (\frac {1}{4} x^4 (a+b \arcsin (c x))-\frac {1}{4} b c \left (\frac {3 \left (\frac {\arcsin (c x)}{2 c^3}-\frac {x \sqrt {1-c^2 x^2}}{2 c^2}\right )}{4 c^2}-\frac {x^3 \sqrt {1-c^2 x^2}}{4 c^2}\right )\right )}{2 c \sqrt {d-c^2 d x^2}}\)

Input:

Int[(x^4*(a + b*ArcSin[c*x])^2)/Sqrt[d - c^2*d*x^2],x]
 

Output:

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

Defintions of rubi rules used

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 262

Int[((c_.)*(x_))^(m_)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> Simp[c*(c*x) 
^(m - 1)*((a + b*x^2)^(p + 1)/(b*(m + 2*p + 1))), x] - Simp[a*c^2*((m - 1)/ 
(b*(m + 2*p + 1)))   Int[(c*x)^(m - 2)*(a + b*x^2)^p, x], x] /; FreeQ[{a, b 
, c, p}, x] && GtQ[m, 2 - 1] && NeQ[m + 2*p + 1, 0] && IntBinomialQ[a, b, c 
, 2, m, p, x]
 

rule 5138

Int[((a_.) + ArcSin[(c_.)*(x_)]*(b_.))^(n_.)*((d_.)*(x_))^(m_.), x_Symbol] 
:> Simp[(d*x)^(m + 1)*((a + b*ArcSin[c*x])^n/(d*(m + 1))), x] - Simp[b*c*(n 
/(d*(m + 1)))   Int[(d*x)^(m + 1)*((a + b*ArcSin[c*x])^(n - 1)/Sqrt[1 - c^2 
*x^2]), x], x] /; FreeQ[{a, b, c, d, m}, x] && IGtQ[n, 0] && NeQ[m, -1]
 

rule 5152

Int[((a_.) + ArcSin[(c_.)*(x_)]*(b_.))^(n_.)/Sqrt[(d_) + (e_.)*(x_)^2], x_S 
ymbol] :> Simp[(1/(b*c*(n + 1)))*Simp[Sqrt[1 - c^2*x^2]/Sqrt[d + e*x^2]]*(a 
 + b*ArcSin[c*x])^(n + 1), x] /; FreeQ[{a, b, c, d, e, n}, x] && EqQ[c^2*d 
+ e, 0] && NeQ[n, -1]
 

rule 5210

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

Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(721\) vs. \(2(283)=566\).

Time = 0.54 (sec) , antiderivative size = 722, normalized size of antiderivative = 2.24

method	result	size
default	\(-\frac {a^{2} x^{3} \sqrt {-c^{2} d \,x^{2}+d}}{4 c^{2} d}-\frac {3 a^{2} x \sqrt {-c^{2} d \,x^{2}+d}}{8 c^{4} d}+\frac {3 a^{2} \arctan \left (\frac {\sqrt {c^{2} d}\, x}{\sqrt {-c^{2} d \,x^{2}+d}}\right )}{8 c^{4} \sqrt {c^{2} d}}+b^{2} \left (-\frac {\sqrt {-d \left (c^{2} x^{2}-1\right )}\, \sqrt {-c^{2} x^{2}+1}\, \arcsin \left (c x \right )^{3}}{8 c^{5} d \left (c^{2} x^{2}-1\right )}+\frac {\sqrt {-d \left (c^{2} x^{2}-1\right )}\, \sqrt {-c^{2} x^{2}+1}\, \arcsin \left (c x \right )}{8 c^{5} d \left (c^{2} x^{2}-1\right )}+\frac {\sqrt {-d \left (c^{2} x^{2}-1\right )}\, \left (2 \arcsin \left (c x \right )^{2}-1\right ) x}{16 c^{4} d \left (c^{2} x^{2}-1\right )}-\frac {\sqrt {-d \left (c^{2} x^{2}-1\right )}\, \arcsin \left (c x \right ) \cos \left (5 \arcsin \left (c x \right )\right )}{128 c^{5} d \left (c^{2} x^{2}-1\right )}-\frac {\sqrt {-d \left (c^{2} x^{2}-1\right )}\, \left (8 \arcsin \left (c x \right )^{2}-1\right ) \sin \left (5 \arcsin \left (c x \right )\right )}{512 c^{5} d \left (c^{2} x^{2}-1\right )}+\frac {15 \sqrt {-d \left (c^{2} x^{2}-1\right )}\, \arcsin \left (c x \right ) \cos \left (3 \arcsin \left (c x \right )\right )}{128 c^{5} d \left (c^{2} x^{2}-1\right )}+\frac {\sqrt {-d \left (c^{2} x^{2}-1\right )}\, \left (56 \arcsin \left (c x \right )^{2}-31\right ) \sin \left (3 \arcsin \left (c x \right )\right )}{512 c^{5} d \left (c^{2} x^{2}-1\right )}\right )+2 a b \left (-\frac {3 \sqrt {-d \left (c^{2} x^{2}-1\right )}\, \sqrt {-c^{2} x^{2}+1}\, \arcsin \left (c x \right )^{2}}{16 c^{5} d \left (c^{2} x^{2}-1\right )}-\frac {\sqrt {-c^{2} x^{2}+1}}{16 c^{5} \sqrt {-d \left (c^{2} x^{2}-1\right )}}+\frac {\sqrt {-d \left (c^{2} x^{2}-1\right )}\, \arcsin \left (c x \right ) x}{8 c^{4} d \left (c^{2} x^{2}-1\right )}-\frac {\sqrt {-d \left (c^{2} x^{2}-1\right )}\, \cos \left (5 \arcsin \left (c x \right )\right )}{256 c^{5} d \left (c^{2} x^{2}-1\right )}-\frac {\sqrt {-d \left (c^{2} x^{2}-1\right )}\, \arcsin \left (c x \right ) \sin \left (5 \arcsin \left (c x \right )\right )}{64 c^{5} d \left (c^{2} x^{2}-1\right )}+\frac {15 \sqrt {-d \left (c^{2} x^{2}-1\right )}\, \cos \left (3 \arcsin \left (c x \right )\right )}{256 c^{5} d \left (c^{2} x^{2}-1\right )}+\frac {7 \sqrt {-d \left (c^{2} x^{2}-1\right )}\, \arcsin \left (c x \right ) \sin \left (3 \arcsin \left (c x \right )\right )}{64 c^{5} d \left (c^{2} x^{2}-1\right )}\right )\)	\(722\)
parts	\(-\frac {a^{2} x^{3} \sqrt {-c^{2} d \,x^{2}+d}}{4 c^{2} d}-\frac {3 a^{2} x \sqrt {-c^{2} d \,x^{2}+d}}{8 c^{4} d}+\frac {3 a^{2} \arctan \left (\frac {\sqrt {c^{2} d}\, x}{\sqrt {-c^{2} d \,x^{2}+d}}\right )}{8 c^{4} \sqrt {c^{2} d}}+b^{2} \left (-\frac {\sqrt {-d \left (c^{2} x^{2}-1\right )}\, \sqrt {-c^{2} x^{2}+1}\, \arcsin \left (c x \right )^{3}}{8 c^{5} d \left (c^{2} x^{2}-1\right )}+\frac {\sqrt {-d \left (c^{2} x^{2}-1\right )}\, \sqrt {-c^{2} x^{2}+1}\, \arcsin \left (c x \right )}{8 c^{5} d \left (c^{2} x^{2}-1\right )}+\frac {\sqrt {-d \left (c^{2} x^{2}-1\right )}\, \left (2 \arcsin \left (c x \right )^{2}-1\right ) x}{16 c^{4} d \left (c^{2} x^{2}-1\right )}-\frac {\sqrt {-d \left (c^{2} x^{2}-1\right )}\, \arcsin \left (c x \right ) \cos \left (5 \arcsin \left (c x \right )\right )}{128 c^{5} d \left (c^{2} x^{2}-1\right )}-\frac {\sqrt {-d \left (c^{2} x^{2}-1\right )}\, \left (8 \arcsin \left (c x \right )^{2}-1\right ) \sin \left (5 \arcsin \left (c x \right )\right )}{512 c^{5} d \left (c^{2} x^{2}-1\right )}+\frac {15 \sqrt {-d \left (c^{2} x^{2}-1\right )}\, \arcsin \left (c x \right ) \cos \left (3 \arcsin \left (c x \right )\right )}{128 c^{5} d \left (c^{2} x^{2}-1\right )}+\frac {\sqrt {-d \left (c^{2} x^{2}-1\right )}\, \left (56 \arcsin \left (c x \right )^{2}-31\right ) \sin \left (3 \arcsin \left (c x \right )\right )}{512 c^{5} d \left (c^{2} x^{2}-1\right )}\right )+2 a b \left (-\frac {3 \sqrt {-d \left (c^{2} x^{2}-1\right )}\, \sqrt {-c^{2} x^{2}+1}\, \arcsin \left (c x \right )^{2}}{16 c^{5} d \left (c^{2} x^{2}-1\right )}-\frac {\sqrt {-c^{2} x^{2}+1}}{16 c^{5} \sqrt {-d \left (c^{2} x^{2}-1\right )}}+\frac {\sqrt {-d \left (c^{2} x^{2}-1\right )}\, \arcsin \left (c x \right ) x}{8 c^{4} d \left (c^{2} x^{2}-1\right )}-\frac {\sqrt {-d \left (c^{2} x^{2}-1\right )}\, \cos \left (5 \arcsin \left (c x \right )\right )}{256 c^{5} d \left (c^{2} x^{2}-1\right )}-\frac {\sqrt {-d \left (c^{2} x^{2}-1\right )}\, \arcsin \left (c x \right ) \sin \left (5 \arcsin \left (c x \right )\right )}{64 c^{5} d \left (c^{2} x^{2}-1\right )}+\frac {15 \sqrt {-d \left (c^{2} x^{2}-1\right )}\, \cos \left (3 \arcsin \left (c x \right )\right )}{256 c^{5} d \left (c^{2} x^{2}-1\right )}+\frac {7 \sqrt {-d \left (c^{2} x^{2}-1\right )}\, \arcsin \left (c x \right ) \sin \left (3 \arcsin \left (c x \right )\right )}{64 c^{5} d \left (c^{2} x^{2}-1\right )}\right )\)	\(722\)

Input:

int(x^4*(a+b*arcsin(c*x))^2/(-c^2*d*x^2+d)^(1/2),x,method=_RETURNVERBOSE)
 

Output:

-1/4*a^2*x^3/c^2/d*(-c^2*d*x^2+d)^(1/2)-3/8*a^2/c^4*x/d*(-c^2*d*x^2+d)^(1/ 
2)+3/8*a^2/c^4/(c^2*d)^(1/2)*arctan((c^2*d)^(1/2)*x/(-c^2*d*x^2+d)^(1/2))+ 
b^2*(-1/8*(-d*(c^2*x^2-1))^(1/2)*(-c^2*x^2+1)^(1/2)/c^5/d/(c^2*x^2-1)*arcs 
in(c*x)^3+1/8*(-d*(c^2*x^2-1))^(1/2)*(-c^2*x^2+1)^(1/2)/c^5/d/(c^2*x^2-1)* 
arcsin(c*x)+1/16*(-d*(c^2*x^2-1))^(1/2)/c^4/d/(c^2*x^2-1)*(2*arcsin(c*x)^2 
-1)*x-1/128*(-d*(c^2*x^2-1))^(1/2)/c^5/d/(c^2*x^2-1)*arcsin(c*x)*cos(5*arc 
sin(c*x))-1/512*(-d*(c^2*x^2-1))^(1/2)/c^5/d/(c^2*x^2-1)*(8*arcsin(c*x)^2- 
1)*sin(5*arcsin(c*x))+15/128*(-d*(c^2*x^2-1))^(1/2)/c^5/d/(c^2*x^2-1)*arcs 
in(c*x)*cos(3*arcsin(c*x))+1/512*(-d*(c^2*x^2-1))^(1/2)/c^5/d/(c^2*x^2-1)* 
(56*arcsin(c*x)^2-31)*sin(3*arcsin(c*x)))+2*a*b*(-3/16*(-d*(c^2*x^2-1))^(1 
/2)*(-c^2*x^2+1)^(1/2)/c^5/d/(c^2*x^2-1)*arcsin(c*x)^2-1/16/c^5/(-d*(c^2*x 
^2-1))^(1/2)*(-c^2*x^2+1)^(1/2)+1/8*(-d*(c^2*x^2-1))^(1/2)/c^4/d/(c^2*x^2- 
1)*arcsin(c*x)*x-1/256*(-d*(c^2*x^2-1))^(1/2)/c^5/d/(c^2*x^2-1)*cos(5*arcs 
in(c*x))-1/64*(-d*(c^2*x^2-1))^(1/2)/c^5/d/(c^2*x^2-1)*arcsin(c*x)*sin(5*a 
rcsin(c*x))+15/256*(-d*(c^2*x^2-1))^(1/2)/c^5/d/(c^2*x^2-1)*cos(3*arcsin(c 
*x))+7/64*(-d*(c^2*x^2-1))^(1/2)/c^5/d/(c^2*x^2-1)*arcsin(c*x)*sin(3*arcsi 
n(c*x)))
 

Fricas [F]

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

integrate(x^4*(a+b*arcsin(c*x))^2/(-c^2*d*x^2+d)^(1/2),x, algorithm="frica 
s")
 

Output:

integral(-(b^2*x^4*arcsin(c*x)^2 + 2*a*b*x^4*arcsin(c*x) + a^2*x^4)*sqrt(- 
c^2*d*x^2 + d)/(c^2*d*x^2 - d), x)
                                                                                    
                                                                                    
 

Sympy [F]

\[ \int \frac {x^4 (a+b \arcsin (c x))^2}{\sqrt {d-c^2 d x^2}} \, dx=\int \frac {x^{4} \left (a + b \operatorname {asin}{\left (c x \right )}\right )^{2}}{\sqrt {- d \left (c x - 1\right ) \left (c x + 1\right )}}\, dx \] Input:

integrate(x**4*(a+b*asin(c*x))**2/(-c**2*d*x**2+d)**(1/2),x)
 

Output:

Integral(x**4*(a + b*asin(c*x))**2/sqrt(-d*(c*x - 1)*(c*x + 1)), x)
 

Maxima [F]

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

integrate(x^4*(a+b*arcsin(c*x))^2/(-c^2*d*x^2+d)^(1/2),x, algorithm="maxim 
a")
 

Output:

-1/8*a^2*(2*sqrt(-c^2*d*x^2 + d)*x^3/(c^2*d) + 3*sqrt(-c^2*d*x^2 + d)*x/(c 
^4*d) - 3*arcsin(c*x)/(c^5*sqrt(d))) - sqrt(d)*integrate((b^2*x^4*arctan2( 
c*x, sqrt(c*x + 1)*sqrt(-c*x + 1))^2 + 2*a*b*x^4*arctan2(c*x, sqrt(c*x + 1 
)*sqrt(-c*x + 1)))*sqrt(c*x + 1)*sqrt(-c*x + 1)/(c^2*d*x^2 - d), x)
 

Giac [A] (verification not implemented)

Time = 0.82 (sec) , antiderivative size = 307, normalized size of antiderivative = 0.95 \[ \int \frac {x^4 (a+b \arcsin (c x))^2}{\sqrt {d-c^2 d x^2}} \, dx=\frac {16 \, {\left (-c^{2} x^{2} + 1\right )}^{\frac {3}{2}} b^{2} x \arcsin \left (c x\right )^{2} + 32 \, {\left (-c^{2} x^{2} + 1\right )}^{\frac {3}{2}} a b x \arcsin \left (c x\right ) - 40 \, \sqrt {-c^{2} x^{2} + 1} b^{2} x \arcsin \left (c x\right )^{2} + 16 \, {\left (-c^{2} x^{2} + 1\right )}^{\frac {3}{2}} a^{2} x - 2 \, {\left (-c^{2} x^{2} + 1\right )}^{\frac {3}{2}} b^{2} x - 80 \, \sqrt {-c^{2} x^{2} + 1} a b x \arcsin \left (c x\right ) + \frac {8 \, {\left (c^{2} x^{2} - 1\right )}^{2} b^{2} \arcsin \left (c x\right )}{c} + \frac {8 \, b^{2} \arcsin \left (c x\right )^{3}}{c} - 40 \, \sqrt {-c^{2} x^{2} + 1} a^{2} x + 17 \, \sqrt {-c^{2} x^{2} + 1} b^{2} x + \frac {8 \, {\left (c^{2} x^{2} - 1\right )}^{2} a b}{c} + \frac {40 \, {\left (c^{2} x^{2} - 1\right )} b^{2} \arcsin \left (c x\right )}{c} + \frac {24 \, a b \arcsin \left (c x\right )^{2}}{c} + \frac {40 \, {\left (c^{2} x^{2} - 1\right )} a b}{c} + \frac {24 \, a^{2} \arcsin \left (c x\right )}{c} + \frac {17 \, b^{2} \arcsin \left (c x\right )}{c} + \frac {17 \, a b}{c}}{64 \, c^{4} \sqrt {d}} \] Input:

integrate(x^4*(a+b*arcsin(c*x))^2/(-c^2*d*x^2+d)^(1/2),x, algorithm="giac" 
)
 

Output:

1/64*(16*(-c^2*x^2 + 1)^(3/2)*b^2*x*arcsin(c*x)^2 + 32*(-c^2*x^2 + 1)^(3/2 
)*a*b*x*arcsin(c*x) - 40*sqrt(-c^2*x^2 + 1)*b^2*x*arcsin(c*x)^2 + 16*(-c^2 
*x^2 + 1)^(3/2)*a^2*x - 2*(-c^2*x^2 + 1)^(3/2)*b^2*x - 80*sqrt(-c^2*x^2 + 
1)*a*b*x*arcsin(c*x) + 8*(c^2*x^2 - 1)^2*b^2*arcsin(c*x)/c + 8*b^2*arcsin( 
c*x)^3/c - 40*sqrt(-c^2*x^2 + 1)*a^2*x + 17*sqrt(-c^2*x^2 + 1)*b^2*x + 8*( 
c^2*x^2 - 1)^2*a*b/c + 40*(c^2*x^2 - 1)*b^2*arcsin(c*x)/c + 24*a*b*arcsin( 
c*x)^2/c + 40*(c^2*x^2 - 1)*a*b/c + 24*a^2*arcsin(c*x)/c + 17*b^2*arcsin(c 
*x)/c + 17*a*b/c)/(c^4*sqrt(d))
 

Mupad [F(-1)]

Timed out. \[ \int \frac {x^4 (a+b \arcsin (c x))^2}{\sqrt {d-c^2 d x^2}} \, dx=\int \frac {x^4\,{\left (a+b\,\mathrm {asin}\left (c\,x\right )\right )}^2}{\sqrt {d-c^2\,d\,x^2}} \,d x \] Input:

int((x^4*(a + b*asin(c*x))^2)/(d - c^2*d*x^2)^(1/2),x)
 

Output:

int((x^4*(a + b*asin(c*x))^2)/(d - c^2*d*x^2)^(1/2), x)
 

Reduce [F]

\[ \int \frac {x^4 (a+b \arcsin (c x))^2}{\sqrt {d-c^2 d x^2}} \, dx=\frac {3 \mathit {asin} \left (c x \right ) a^{2}-2 \sqrt {-c^{2} x^{2}+1}\, a^{2} c^{3} x^{3}-3 \sqrt {-c^{2} x^{2}+1}\, a^{2} c x +16 \left (\int \frac {\mathit {asin} \left (c x \right ) x^{4}}{\sqrt {-c^{2} x^{2}+1}}d x \right ) a b \,c^{5}+8 \left (\int \frac {\mathit {asin} \left (c x \right )^{2} x^{4}}{\sqrt {-c^{2} x^{2}+1}}d x \right ) b^{2} c^{5}}{8 \sqrt {d}\, c^{5}} \] Input:

int(x^4*(a+b*asin(c*x))^2/(-c^2*d*x^2+d)^(1/2),x)
 

Output:

(3*asin(c*x)*a**2 - 2*sqrt( - c**2*x**2 + 1)*a**2*c**3*x**3 - 3*sqrt( - c* 
*2*x**2 + 1)*a**2*c*x + 16*int((asin(c*x)*x**4)/sqrt( - c**2*x**2 + 1),x)* 
a*b*c**5 + 8*int((asin(c*x)**2*x**4)/sqrt( - c**2*x**2 + 1),x)*b**2*c**5)/ 
(8*sqrt(d)*c**5)