3.3.57 \(\int (a+b \arcsin (c+d x))^{7/2} \, dx\) [257]

3.3.57.1 Optimal result

Integrand size = 14, antiderivative size = 243 \[ \int (a+b \arcsin (c+d x))^{7/2} \, dx=-\frac {105 b^3 \sqrt {1-(c+d x)^2} \sqrt {a+b \arcsin (c+d x)}}{8 d}-\frac {35 b^2 (c+d x) (a+b \arcsin (c+d x))^{3/2}}{4 d}+\frac {7 b \sqrt {1-(c+d x)^2} (a+b \arcsin (c+d x))^{5/2}}{2 d}+\frac {(c+d x) (a+b \arcsin (c+d x))^{7/2}}{d}+\frac {105 b^{7/2} \sqrt {\frac {\pi }{2}} \cos \left (\frac {a}{b}\right ) \operatorname {FresnelC}\left (\frac {\sqrt {\frac {2}{\pi }} \sqrt {a+b \arcsin (c+d x)}}{\sqrt {b}}\right )}{8 d}+\frac {105 b^{7/2} \sqrt {\frac {\pi }{2}} \operatorname {FresnelS}\left (\frac {\sqrt {\frac {2}{\pi }} \sqrt {a+b \arcsin (c+d x)}}{\sqrt {b}}\right ) \sin \left (\frac {a}{b}\right )}{8 d} \]

output

-35/4*b^2*(d*x+c)*(a+b*arcsin(d*x+c))^(3/2)/d+(d*x+c)*(a+b*arcsin(d*x+c))^ 
(7/2)/d+105/16*b^(7/2)*cos(a/b)*FresnelC(2^(1/2)/Pi^(1/2)*(a+b*arcsin(d*x+ 
c))^(1/2)/b^(1/2))*2^(1/2)*Pi^(1/2)/d+105/16*b^(7/2)*FresnelS(2^(1/2)/Pi^( 
1/2)*(a+b*arcsin(d*x+c))^(1/2)/b^(1/2))*sin(a/b)*2^(1/2)*Pi^(1/2)/d+7/2*b* 
(a+b*arcsin(d*x+c))^(5/2)*(1-(d*x+c)^2)^(1/2)/d-105/8*b^3*(1-(d*x+c)^2)^(1 
/2)*(a+b*arcsin(d*x+c))^(1/2)/d
 

3.3.57.2 Mathematica [C] (verified)

Result contains complex when optimal does not.

Time = 3.20 (sec) , antiderivative size = 545, normalized size of antiderivative = 2.24 \[ \int (a+b \arcsin (c+d x))^{7/2} \, dx=\frac {e^{-\frac {i a}{b}} \left (\sqrt {b} \left (8 i a^3 \left (-1+e^{\frac {2 i a}{b}}\right )+105 b^3 \left (1+e^{\frac {2 i a}{b}}\right )\right ) \sqrt {\frac {\pi }{2}} \sqrt {a+b \arcsin (c+d x)} \operatorname {FresnelC}\left (\frac {\sqrt {\frac {2}{\pi }} \sqrt {a+b \arcsin (c+d x)}}{\sqrt {b}}\right )+2 b \left (e^{\frac {i a}{b}} (a+b \arcsin (c+d x)) \left (7 \left (-10 a b (c+d x)+4 a^2 \sqrt {1-c^2-2 c d x-d^2 x^2}-15 b^2 \sqrt {1-c^2-2 c d x-d^2 x^2}\right )+\left (24 a^2 (c+d x)-70 b^2 (c+d x)+56 a b \sqrt {1-c^2-2 c d x-d^2 x^2}\right ) \arcsin (c+d x)+4 b \left (6 a (c+d x)+7 b \sqrt {1-c^2-2 c d x-d^2 x^2}\right ) \arcsin (c+d x)^2+8 b^2 (c+d x) \arcsin (c+d x)^3\right )+4 a^3 \sqrt {-\frac {i (a+b \arcsin (c+d x))}{b}} \Gamma \left (\frac {3}{2},-\frac {i (a+b \arcsin (c+d x))}{b}\right )+4 a^3 e^{\frac {2 i a}{b}} \sqrt {\frac {i (a+b \arcsin (c+d x))}{b}} \Gamma \left (\frac {3}{2},\frac {i (a+b \arcsin (c+d x))}{b}\right )\right )+\sqrt {b} \sqrt {2 \pi } \sqrt {a+b \arcsin (c+d x)} \operatorname {FresnelS}\left (\frac {\sqrt {\frac {2}{\pi }} \sqrt {a+b \arcsin (c+d x)}}{\sqrt {b}}\right ) \left (4 a^3 \left (1+e^{\frac {2 i a}{b}}\right )+105 b^3 e^{\frac {i a}{b}} \sin \left (\frac {a}{b}\right )\right )\right )}{16 d \sqrt {a+b \arcsin (c+d x)}} \]

input

Integrate[(a + b*ArcSin[c + d*x])^(7/2),x]
 

output

(Sqrt[b]*((8*I)*a^3*(-1 + E^(((2*I)*a)/b)) + 105*b^3*(1 + E^(((2*I)*a)/b)) 
)*Sqrt[Pi/2]*Sqrt[a + b*ArcSin[c + d*x]]*FresnelC[(Sqrt[2/Pi]*Sqrt[a + b*A 
rcSin[c + d*x]])/Sqrt[b]] + 2*b*(E^((I*a)/b)*(a + b*ArcSin[c + d*x])*(7*(- 
10*a*b*(c + d*x) + 4*a^2*Sqrt[1 - c^2 - 2*c*d*x - d^2*x^2] - 15*b^2*Sqrt[1 
 - c^2 - 2*c*d*x - d^2*x^2]) + (24*a^2*(c + d*x) - 70*b^2*(c + d*x) + 56*a 
*b*Sqrt[1 - c^2 - 2*c*d*x - d^2*x^2])*ArcSin[c + d*x] + 4*b*(6*a*(c + d*x) 
 + 7*b*Sqrt[1 - c^2 - 2*c*d*x - d^2*x^2])*ArcSin[c + d*x]^2 + 8*b^2*(c + d 
*x)*ArcSin[c + d*x]^3) + 4*a^3*Sqrt[((-I)*(a + b*ArcSin[c + d*x]))/b]*Gamm 
a[3/2, ((-I)*(a + b*ArcSin[c + d*x]))/b] + 4*a^3*E^(((2*I)*a)/b)*Sqrt[(I*( 
a + b*ArcSin[c + d*x]))/b]*Gamma[3/2, (I*(a + b*ArcSin[c + d*x]))/b]) + Sq 
rt[b]*Sqrt[2*Pi]*Sqrt[a + b*ArcSin[c + d*x]]*FresnelS[(Sqrt[2/Pi]*Sqrt[a + 
 b*ArcSin[c + d*x]])/Sqrt[b]]*(4*a^3*(1 + E^(((2*I)*a)/b)) + 105*b^3*E^((I 
*a)/b)*Sin[a/b]))/(16*d*E^((I*a)/b)*Sqrt[a + b*ArcSin[c + d*x]])
 

3.3.57.3 Rubi [A] (verified)

Time = 1.07 (sec) , antiderivative size = 228, normalized size of antiderivative = 0.94, number of steps used = 15, number of rules used = 14, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 1.000, Rules used = {5302, 5130, 5182, 5130, 5182, 5134, 3042, 3787, 25, 3042, 3785, 3786, 3832, 3833}

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.

\(\displaystyle \int (a+b \arcsin (c+d x))^{7/2} \, dx\)

\(\Big \downarrow \) 5302

\(\displaystyle \frac {\int (a+b \arcsin (c+d x))^{7/2}d(c+d x)}{d}\)

\(\Big \downarrow \) 5130

\(\displaystyle \frac {(c+d x) (a+b \arcsin (c+d x))^{7/2}-\frac {7}{2} b \int \frac {(c+d x) (a+b \arcsin (c+d x))^{5/2}}{\sqrt {1-(c+d x)^2}}d(c+d x)}{d}\)

\(\Big \downarrow \) 5182

\(\displaystyle \frac {(c+d x) (a+b \arcsin (c+d x))^{7/2}-\frac {7}{2} b \left (\frac {5}{2} b \int (a+b \arcsin (c+d x))^{3/2}d(c+d x)-\sqrt {1-(c+d x)^2} (a+b \arcsin (c+d x))^{5/2}\right )}{d}\)

\(\Big \downarrow \) 5130

\(\displaystyle \frac {(c+d x) (a+b \arcsin (c+d x))^{7/2}-\frac {7}{2} b \left (\frac {5}{2} b \left ((c+d x) (a+b \arcsin (c+d x))^{3/2}-\frac {3}{2} b \int \frac {(c+d x) \sqrt {a+b \arcsin (c+d x)}}{\sqrt {1-(c+d x)^2}}d(c+d x)\right )-\sqrt {1-(c+d x)^2} (a+b \arcsin (c+d x))^{5/2}\right )}{d}\)

\(\Big \downarrow \) 5182

\(\displaystyle \frac {(c+d x) (a+b \arcsin (c+d x))^{7/2}-\frac {7}{2} b \left (\frac {5}{2} b \left ((c+d x) (a+b \arcsin (c+d x))^{3/2}-\frac {3}{2} b \left (\frac {1}{2} b \int \frac {1}{\sqrt {a+b \arcsin (c+d x)}}d(c+d x)-\sqrt {1-(c+d x)^2} \sqrt {a+b \arcsin (c+d x)}\right )\right )-\sqrt {1-(c+d x)^2} (a+b \arcsin (c+d x))^{5/2}\right )}{d}\)

\(\Big \downarrow \) 5134

\(\displaystyle \frac {(c+d x) (a+b \arcsin (c+d x))^{7/2}-\frac {7}{2} b \left (\frac {5}{2} b \left ((c+d x) (a+b \arcsin (c+d x))^{3/2}-\frac {3}{2} b \left (\frac {1}{2} \int \frac {\cos \left (\frac {a}{b}-\frac {a+b \arcsin (c+d x)}{b}\right )}{\sqrt {a+b \arcsin (c+d x)}}d(a+b \arcsin (c+d x))-\sqrt {1-(c+d x)^2} \sqrt {a+b \arcsin (c+d x)}\right )\right )-\sqrt {1-(c+d x)^2} (a+b \arcsin (c+d x))^{5/2}\right )}{d}\)

\(\Big \downarrow \) 3042

\(\displaystyle \frac {(c+d x) (a+b \arcsin (c+d x))^{7/2}-\frac {7}{2} b \left (\frac {5}{2} b \left ((c+d x) (a+b \arcsin (c+d x))^{3/2}-\frac {3}{2} b \left (\frac {1}{2} \int \frac {\sin \left (\frac {a}{b}-\frac {a+b \arcsin (c+d x)}{b}+\frac {\pi }{2}\right )}{\sqrt {a+b \arcsin (c+d x)}}d(a+b \arcsin (c+d x))-\sqrt {1-(c+d x)^2} \sqrt {a+b \arcsin (c+d x)}\right )\right )-\sqrt {1-(c+d x)^2} (a+b \arcsin (c+d x))^{5/2}\right )}{d}\)

\(\Big \downarrow \) 3787

\(\displaystyle \frac {(c+d x) (a+b \arcsin (c+d x))^{7/2}-\frac {7}{2} b \left (\frac {5}{2} b \left ((c+d x) (a+b \arcsin (c+d x))^{3/2}-\frac {3}{2} b \left (\frac {1}{2} \left (\cos \left (\frac {a}{b}\right ) \int \frac {\cos \left (\frac {a+b \arcsin (c+d x)}{b}\right )}{\sqrt {a+b \arcsin (c+d x)}}d(a+b \arcsin (c+d x))-\sin \left (\frac {a}{b}\right ) \int -\frac {\sin \left (\frac {a+b \arcsin (c+d x)}{b}\right )}{\sqrt {a+b \arcsin (c+d x)}}d(a+b \arcsin (c+d x))\right )-\sqrt {1-(c+d x)^2} \sqrt {a+b \arcsin (c+d x)}\right )\right )-\sqrt {1-(c+d x)^2} (a+b \arcsin (c+d x))^{5/2}\right )}{d}\)

\(\Big \downarrow \) 25

\(\displaystyle \frac {(c+d x) (a+b \arcsin (c+d x))^{7/2}-\frac {7}{2} b \left (\frac {5}{2} b \left ((c+d x) (a+b \arcsin (c+d x))^{3/2}-\frac {3}{2} b \left (\frac {1}{2} \left (\sin \left (\frac {a}{b}\right ) \int \frac {\sin \left (\frac {a+b \arcsin (c+d x)}{b}\right )}{\sqrt {a+b \arcsin (c+d x)}}d(a+b \arcsin (c+d x))+\cos \left (\frac {a}{b}\right ) \int \frac {\cos \left (\frac {a+b \arcsin (c+d x)}{b}\right )}{\sqrt {a+b \arcsin (c+d x)}}d(a+b \arcsin (c+d x))\right )-\sqrt {1-(c+d x)^2} \sqrt {a+b \arcsin (c+d x)}\right )\right )-\sqrt {1-(c+d x)^2} (a+b \arcsin (c+d x))^{5/2}\right )}{d}\)

\(\Big \downarrow \) 3042

\(\displaystyle \frac {(c+d x) (a+b \arcsin (c+d x))^{7/2}-\frac {7}{2} b \left (\frac {5}{2} b \left ((c+d x) (a+b \arcsin (c+d x))^{3/2}-\frac {3}{2} b \left (\frac {1}{2} \left (\sin \left (\frac {a}{b}\right ) \int \frac {\sin \left (\frac {a+b \arcsin (c+d x)}{b}\right )}{\sqrt {a+b \arcsin (c+d x)}}d(a+b \arcsin (c+d x))+\cos \left (\frac {a}{b}\right ) \int \frac {\sin \left (\frac {a+b \arcsin (c+d x)}{b}+\frac {\pi }{2}\right )}{\sqrt {a+b \arcsin (c+d x)}}d(a+b \arcsin (c+d x))\right )-\sqrt {1-(c+d x)^2} \sqrt {a+b \arcsin (c+d x)}\right )\right )-\sqrt {1-(c+d x)^2} (a+b \arcsin (c+d x))^{5/2}\right )}{d}\)

\(\Big \downarrow \) 3785

\(\displaystyle \frac {(c+d x) (a+b \arcsin (c+d x))^{7/2}-\frac {7}{2} b \left (\frac {5}{2} b \left ((c+d x) (a+b \arcsin (c+d x))^{3/2}-\frac {3}{2} b \left (\frac {1}{2} \left (\sin \left (\frac {a}{b}\right ) \int \frac {\sin \left (\frac {a+b \arcsin (c+d x)}{b}\right )}{\sqrt {a+b \arcsin (c+d x)}}d(a+b \arcsin (c+d x))+2 \cos \left (\frac {a}{b}\right ) \int \cos \left (\frac {a+b \arcsin (c+d x)}{b}\right )d\sqrt {a+b \arcsin (c+d x)}\right )-\sqrt {1-(c+d x)^2} \sqrt {a+b \arcsin (c+d x)}\right )\right )-\sqrt {1-(c+d x)^2} (a+b \arcsin (c+d x))^{5/2}\right )}{d}\)

\(\Big \downarrow \) 3786

\(\displaystyle \frac {(c+d x) (a+b \arcsin (c+d x))^{7/2}-\frac {7}{2} b \left (\frac {5}{2} b \left ((c+d x) (a+b \arcsin (c+d x))^{3/2}-\frac {3}{2} b \left (\frac {1}{2} \left (2 \sin \left (\frac {a}{b}\right ) \int \sin \left (\frac {a+b \arcsin (c+d x)}{b}\right )d\sqrt {a+b \arcsin (c+d x)}+2 \cos \left (\frac {a}{b}\right ) \int \cos \left (\frac {a+b \arcsin (c+d x)}{b}\right )d\sqrt {a+b \arcsin (c+d x)}\right )-\sqrt {1-(c+d x)^2} \sqrt {a+b \arcsin (c+d x)}\right )\right )-\sqrt {1-(c+d x)^2} (a+b \arcsin (c+d x))^{5/2}\right )}{d}\)

\(\Big \downarrow \) 3832

\(\displaystyle \frac {(c+d x) (a+b \arcsin (c+d x))^{7/2}-\frac {7}{2} b \left (\frac {5}{2} b \left ((c+d x) (a+b \arcsin (c+d x))^{3/2}-\frac {3}{2} b \left (\frac {1}{2} \left (2 \cos \left (\frac {a}{b}\right ) \int \cos \left (\frac {a+b \arcsin (c+d x)}{b}\right )d\sqrt {a+b \arcsin (c+d x)}+\sqrt {2 \pi } \sqrt {b} \sin \left (\frac {a}{b}\right ) \operatorname {FresnelS}\left (\frac {\sqrt {\frac {2}{\pi }} \sqrt {a+b \arcsin (c+d x)}}{\sqrt {b}}\right )\right )-\sqrt {1-(c+d x)^2} \sqrt {a+b \arcsin (c+d x)}\right )\right )-\sqrt {1-(c+d x)^2} (a+b \arcsin (c+d x))^{5/2}\right )}{d}\)

\(\Big \downarrow \) 3833

\(\displaystyle \frac {(c+d x) (a+b \arcsin (c+d x))^{7/2}-\frac {7}{2} b \left (\frac {5}{2} b \left ((c+d x) (a+b \arcsin (c+d x))^{3/2}-\frac {3}{2} b \left (\frac {1}{2} \left (\sqrt {2 \pi } \sqrt {b} \cos \left (\frac {a}{b}\right ) \operatorname {FresnelC}\left (\frac {\sqrt {\frac {2}{\pi }} \sqrt {a+b \arcsin (c+d x)}}{\sqrt {b}}\right )+\sqrt {2 \pi } \sqrt {b} \sin \left (\frac {a}{b}\right ) \operatorname {FresnelS}\left (\frac {\sqrt {\frac {2}{\pi }} \sqrt {a+b \arcsin (c+d x)}}{\sqrt {b}}\right )\right )-\sqrt {1-(c+d x)^2} \sqrt {a+b \arcsin (c+d x)}\right )\right )-\sqrt {1-(c+d x)^2} (a+b \arcsin (c+d x))^{5/2}\right )}{d}\)

input

Int[(a + b*ArcSin[c + d*x])^(7/2),x]
 

output

((c + d*x)*(a + b*ArcSin[c + d*x])^(7/2) - (7*b*(-(Sqrt[1 - (c + d*x)^2]*( 
a + b*ArcSin[c + d*x])^(5/2)) + (5*b*((c + d*x)*(a + b*ArcSin[c + d*x])^(3 
/2) - (3*b*(-(Sqrt[1 - (c + d*x)^2]*Sqrt[a + b*ArcSin[c + d*x]]) + (Sqrt[b 
]*Sqrt[2*Pi]*Cos[a/b]*FresnelC[(Sqrt[2/Pi]*Sqrt[a + b*ArcSin[c + d*x]])/Sq 
rt[b]] + Sqrt[b]*Sqrt[2*Pi]*FresnelS[(Sqrt[2/Pi]*Sqrt[a + b*ArcSin[c + d*x 
]])/Sqrt[b]]*Sin[a/b])/2))/2))/2))/2)/d
 

3.3.57.3.1 Defintions of rubi rules used

rule 25

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

rule 3042

Int[u_, x_Symbol] :> Int[DeactivateTrig[u, x], x] /; FunctionOfTrigOfLinear 
Q[u, x]
 

rule 3785

Int[sin[Pi/2 + (e_.) + (f_.)*(x_)]/Sqrt[(c_.) + (d_.)*(x_)], x_Symbol] :> S 
imp[2/d   Subst[Int[Cos[f*(x^2/d)], x], x, Sqrt[c + d*x]], x] /; FreeQ[{c, 
d, e, f}, x] && ComplexFreeQ[f] && EqQ[d*e - c*f, 0]
 

rule 3786

Int[sin[(e_.) + (f_.)*(x_)]/Sqrt[(c_.) + (d_.)*(x_)], x_Symbol] :> Simp[2/d 
   Subst[Int[Sin[f*(x^2/d)], x], x, Sqrt[c + d*x]], x] /; FreeQ[{c, d, e, f 
}, x] && ComplexFreeQ[f] && EqQ[d*e - c*f, 0]
 

rule 3787

Int[sin[(e_.) + (f_.)*(x_)]/Sqrt[(c_.) + (d_.)*(x_)], x_Symbol] :> Simp[Cos 
[(d*e - c*f)/d]   Int[Sin[c*(f/d) + f*x]/Sqrt[c + d*x], x], x] + Simp[Sin[( 
d*e - c*f)/d]   Int[Cos[c*(f/d) + f*x]/Sqrt[c + d*x], x], x] /; FreeQ[{c, d 
, e, f}, x] && ComplexFreeQ[f] && NeQ[d*e - c*f, 0]
 

rule 3832

Int[Sin[(d_.)*((e_.) + (f_.)*(x_))^2], x_Symbol] :> Simp[(Sqrt[Pi/2]/(f*Rt[ 
d, 2]))*FresnelS[Sqrt[2/Pi]*Rt[d, 2]*(e + f*x)], x] /; FreeQ[{d, e, f}, x]
 

rule 3833

Int[Cos[(d_.)*((e_.) + (f_.)*(x_))^2], x_Symbol] :> Simp[(Sqrt[Pi/2]/(f*Rt[ 
d, 2]))*FresnelC[Sqrt[2/Pi]*Rt[d, 2]*(e + f*x)], x] /; FreeQ[{d, e, f}, x]
 

rule 5130

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

rule 5134

Int[((a_.) + ArcSin[(c_.)*(x_)]*(b_.))^(n_), x_Symbol] :> Simp[1/(b*c)   Su 
bst[Int[x^n*Cos[-a/b + x/b], x], x, a + b*ArcSin[c*x]], x] /; FreeQ[{a, b, 
c, n}, x]
 

rule 5182

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

rule 5302

Int[((a_.) + ArcSin[(c_) + (d_.)*(x_)]*(b_.))^(n_.), x_Symbol] :> Simp[1/d 
  Subst[Int[(a + b*ArcSin[x])^n, x], x, c + d*x], x] /; FreeQ[{a, b, c, d, 
n}, x]
 

3.3.57.4 Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(615\) vs. \(2(197)=394\).

Time = 0.36 (sec) , antiderivative size = 616, normalized size of antiderivative = 2.53

method	result	size
default	\(-\frac {-105 \sqrt {a +b \arcsin \left (d x +c \right )}\, \sqrt {\pi }\, \cos \left (\frac {a}{b}\right ) \operatorname {FresnelC}\left (\frac {\sqrt {2}\, \sqrt {a +b \arcsin \left (d x +c \right )}}{\sqrt {\pi }\, \sqrt {-\frac {1}{b}}\, b}\right ) \sqrt {2}\, \sqrt {-\frac {1}{b}}\, b^{4}+105 \sqrt {a +b \arcsin \left (d x +c \right )}\, \sqrt {\pi }\, \sin \left (\frac {a}{b}\right ) \operatorname {FresnelS}\left (\frac {\sqrt {2}\, \sqrt {a +b \arcsin \left (d x +c \right )}}{\sqrt {\pi }\, \sqrt {-\frac {1}{b}}\, b}\right ) \sqrt {2}\, \sqrt {-\frac {1}{b}}\, b^{4}+16 \arcsin \left (d x +c \right )^{4} \sin \left (-\frac {a +b \arcsin \left (d x +c \right )}{b}+\frac {a}{b}\right ) b^{4}+64 \arcsin \left (d x +c \right )^{3} \sin \left (-\frac {a +b \arcsin \left (d x +c \right )}{b}+\frac {a}{b}\right ) a \,b^{3}-56 \arcsin \left (d x +c \right )^{3} \cos \left (-\frac {a +b \arcsin \left (d x +c \right )}{b}+\frac {a}{b}\right ) b^{4}+96 \arcsin \left (d x +c \right )^{2} \sin \left (-\frac {a +b \arcsin \left (d x +c \right )}{b}+\frac {a}{b}\right ) a^{2} b^{2}-140 \arcsin \left (d x +c \right )^{2} \sin \left (-\frac {a +b \arcsin \left (d x +c \right )}{b}+\frac {a}{b}\right ) b^{4}-168 \arcsin \left (d x +c \right )^{2} \cos \left (-\frac {a +b \arcsin \left (d x +c \right )}{b}+\frac {a}{b}\right ) a \,b^{3}+64 \arcsin \left (d x +c \right ) \sin \left (-\frac {a +b \arcsin \left (d x +c \right )}{b}+\frac {a}{b}\right ) a^{3} b -280 \arcsin \left (d x +c \right ) \sin \left (-\frac {a +b \arcsin \left (d x +c \right )}{b}+\frac {a}{b}\right ) a \,b^{3}-168 \arcsin \left (d x +c \right ) \cos \left (-\frac {a +b \arcsin \left (d x +c \right )}{b}+\frac {a}{b}\right ) a^{2} b^{2}+210 \arcsin \left (d x +c \right ) \cos \left (-\frac {a +b \arcsin \left (d x +c \right )}{b}+\frac {a}{b}\right ) b^{4}+16 \sin \left (-\frac {a +b \arcsin \left (d x +c \right )}{b}+\frac {a}{b}\right ) a^{4}-140 \sin \left (-\frac {a +b \arcsin \left (d x +c \right )}{b}+\frac {a}{b}\right ) a^{2} b^{2}-56 \cos \left (-\frac {a +b \arcsin \left (d x +c \right )}{b}+\frac {a}{b}\right ) a^{3} b +210 \cos \left (-\frac {a +b \arcsin \left (d x +c \right )}{b}+\frac {a}{b}\right ) a \,b^{3}}{16 d \sqrt {a +b \arcsin \left (d x +c \right )}}\)	\(616\)

input

int((a+b*arcsin(d*x+c))^(7/2),x,method=_RETURNVERBOSE)
 

output

-1/16/d*(-105*(a+b*arcsin(d*x+c))^(1/2)*Pi^(1/2)*cos(a/b)*FresnelC(2^(1/2) 
/Pi^(1/2)/(-1/b)^(1/2)*(a+b*arcsin(d*x+c))^(1/2)/b)*2^(1/2)*(-1/b)^(1/2)*b 
^4+105*(a+b*arcsin(d*x+c))^(1/2)*Pi^(1/2)*sin(a/b)*FresnelS(2^(1/2)/Pi^(1/ 
2)/(-1/b)^(1/2)*(a+b*arcsin(d*x+c))^(1/2)/b)*2^(1/2)*(-1/b)^(1/2)*b^4+16*a 
rcsin(d*x+c)^4*sin(-(a+b*arcsin(d*x+c))/b+a/b)*b^4+64*arcsin(d*x+c)^3*sin( 
-(a+b*arcsin(d*x+c))/b+a/b)*a*b^3-56*arcsin(d*x+c)^3*cos(-(a+b*arcsin(d*x+ 
c))/b+a/b)*b^4+96*arcsin(d*x+c)^2*sin(-(a+b*arcsin(d*x+c))/b+a/b)*a^2*b^2- 
140*arcsin(d*x+c)^2*sin(-(a+b*arcsin(d*x+c))/b+a/b)*b^4-168*arcsin(d*x+c)^ 
2*cos(-(a+b*arcsin(d*x+c))/b+a/b)*a*b^3+64*arcsin(d*x+c)*sin(-(a+b*arcsin( 
d*x+c))/b+a/b)*a^3*b-280*arcsin(d*x+c)*sin(-(a+b*arcsin(d*x+c))/b+a/b)*a*b 
^3-168*arcsin(d*x+c)*cos(-(a+b*arcsin(d*x+c))/b+a/b)*a^2*b^2+210*arcsin(d* 
x+c)*cos(-(a+b*arcsin(d*x+c))/b+a/b)*b^4+16*sin(-(a+b*arcsin(d*x+c))/b+a/b 
)*a^4-140*sin(-(a+b*arcsin(d*x+c))/b+a/b)*a^2*b^2-56*cos(-(a+b*arcsin(d*x+ 
c))/b+a/b)*a^3*b+210*cos(-(a+b*arcsin(d*x+c))/b+a/b)*a*b^3)/(a+b*arcsin(d* 
x+c))^(1/2)
 

3.3.57.5 Fricas [F(-2)]

Exception generated. \[ \int (a+b \arcsin (c+d x))^{7/2} \, dx=\text {Exception raised: TypeError} \]

input

integrate((a+b*arcsin(d*x+c))^(7/2),x, algorithm="fricas")
 

output

Exception raised: TypeError >>  Error detected within library code:   inte 
grate: implementation incomplete (constant residues)
 

3.3.57.6 Sympy [F(-1)]

Timed out. \[ \int (a+b \arcsin (c+d x))^{7/2} \, dx=\text {Timed out} \]

input

integrate((a+b*asin(d*x+c))**(7/2),x)
 

output

Timed out
 

3.3.57.7 Maxima [F]

\[ \int (a+b \arcsin (c+d x))^{7/2} \, dx=\int { {\left (b \arcsin \left (d x + c\right ) + a\right )}^{\frac {7}{2}} \,d x } \]

input

integrate((a+b*arcsin(d*x+c))^(7/2),x, algorithm="maxima")
 

output

integrate((b*arcsin(d*x + c) + a)^(7/2), x)
 

3.3.57.8 Giac [C] (verification not implemented)

Result contains complex when optimal does not.

Time = 1.54 (sec) , antiderivative size = 2308, normalized size of antiderivative = 9.50 \[ \int (a+b \arcsin (c+d x))^{7/2} \, dx=\text {Too large to display} \]

input

integrate((a+b*arcsin(d*x+c))^(7/2),x, algorithm="giac")
 

output

-1/32*(16*sqrt(2)*sqrt(pi)*a^4*b^3*erf(-1/2*I*sqrt(2)*sqrt(b*arcsin(d*x + 
c) + a)/sqrt(abs(b)) - 1/2*sqrt(2)*sqrt(b*arcsin(d*x + c) + a)*sqrt(abs(b) 
)/b)*e^(I*a/b)/(I*b^4/sqrt(abs(b)) + b^3*sqrt(abs(b))) + 16*sqrt(2)*sqrt(p 
i)*a^4*b^3*erf(1/2*I*sqrt(2)*sqrt(b*arcsin(d*x + c) + a)/sqrt(abs(b)) - 1/ 
2*sqrt(2)*sqrt(b*arcsin(d*x + c) + a)*sqrt(abs(b))/b)*e^(-I*a/b)/(-I*b^4/s 
qrt(abs(b)) + b^3*sqrt(abs(b))) - 64*sqrt(2)*sqrt(pi)*a^4*b^2*erf(-1/2*I*s 
qrt(2)*sqrt(b*arcsin(d*x + c) + a)/sqrt(abs(b)) - 1/2*sqrt(2)*sqrt(b*arcsi 
n(d*x + c) + a)*sqrt(abs(b))/b)*e^(I*a/b)/(I*b^3/sqrt(abs(b)) + b^2*sqrt(a 
bs(b))) + 32*I*sqrt(2)*sqrt(pi)*a^3*b^3*erf(-1/2*I*sqrt(2)*sqrt(b*arcsin(d 
*x + c) + a)/sqrt(abs(b)) - 1/2*sqrt(2)*sqrt(b*arcsin(d*x + c) + a)*sqrt(a 
bs(b))/b)*e^(I*a/b)/(I*b^3/sqrt(abs(b)) + b^2*sqrt(abs(b))) - 64*sqrt(2)*s 
qrt(pi)*a^4*b^2*erf(1/2*I*sqrt(2)*sqrt(b*arcsin(d*x + c) + a)/sqrt(abs(b)) 
 - 1/2*sqrt(2)*sqrt(b*arcsin(d*x + c) + a)*sqrt(abs(b))/b)*e^(-I*a/b)/(-I* 
b^3/sqrt(abs(b)) + b^2*sqrt(abs(b))) - 32*I*sqrt(2)*sqrt(pi)*a^3*b^3*erf(1 
/2*I*sqrt(2)*sqrt(b*arcsin(d*x + c) + a)/sqrt(abs(b)) - 1/2*sqrt(2)*sqrt(b 
*arcsin(d*x + c) + a)*sqrt(abs(b))/b)*e^(-I*a/b)/(-I*b^3/sqrt(abs(b)) + b^ 
2*sqrt(abs(b))) + 16*I*sqrt(b*arcsin(d*x + c) + a)*b^3*arcsin(d*x + c)^3*e 
^(I*arcsin(d*x + c)) - 16*I*sqrt(b*arcsin(d*x + c) + a)*b^3*arcsin(d*x + c 
)^3*e^(-I*arcsin(d*x + c)) + 32*sqrt(2)*sqrt(pi)*a^4*b*erf(-1/2*I*sqrt(2)* 
sqrt(b*arcsin(d*x + c) + a)/sqrt(abs(b)) - 1/2*sqrt(2)*sqrt(b*arcsin(d*...
 

3.3.57.9 Mupad [F(-1)]

Timed out. \[ \int (a+b \arcsin (c+d x))^{7/2} \, dx=\int {\left (a+b\,\mathrm {asin}\left (c+d\,x\right )\right )}^{7/2} \,d x \]

input

int((a + b*asin(c + d*x))^(7/2),x)
 

output

int((a + b*asin(c + d*x))^(7/2), x)