\(\int \frac {1}{(a+b \arcsin (c+d x))^{7/2}} \, dx\) [279]

Optimal result

Integrand size = 14, antiderivative size = 218 \[ \int \frac {1}{(a+b \arcsin (c+d x))^{7/2}} \, dx=-\frac {2 \sqrt {1-(c+d x)^2}}{5 b d (a+b \arcsin (c+d x))^{5/2}}+\frac {4 (c+d x)}{15 b^2 d (a+b \arcsin (c+d x))^{3/2}}+\frac {8 \sqrt {1-(c+d x)^2}}{15 b^3 d \sqrt {a+b \arcsin (c+d x)}}+\frac {8 \sqrt {2 \pi } \cos \left (\frac {a}{b}\right ) \operatorname {FresnelS}\left (\frac {\sqrt {\frac {2}{\pi }} \sqrt {a+b \arcsin (c+d x)}}{\sqrt {b}}\right )}{15 b^{7/2} d}-\frac {8 \sqrt {2 \pi } \operatorname {FresnelC}\left (\frac {\sqrt {\frac {2}{\pi }} \sqrt {a+b \arcsin (c+d x)}}{\sqrt {b}}\right ) \sin \left (\frac {a}{b}\right )}{15 b^{7/2} d} \]

[Out]

4/15*(d*x+c)/b^2/d/(a+b*arcsin(d*x+c))^(3/2)+8/15*cos(a/b)*FresnelS(2^(1/2)/Pi^(1/2)*(a+b*arcsin(d*x+c))^(1/2)
/b^(1/2))*2^(1/2)*Pi^(1/2)/b^(7/2)/d-8/15*FresnelC(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)/b^(7/2)/d-2/5*(1-(d*x+c)^2)^(1/2)/b/d/(a+b*arcsin(d*x+c))^(5/2)+8/15*(1-(d*x+c)^2)^(1/2)/b^
3/d/(a+b*arcsin(d*x+c))^(1/2)

Rubi [A] (verified)

Time = 0.28 (sec) , antiderivative size = 218, normalized size of antiderivative = 1.00, number of steps used = 10, number of rules used = 9, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.643, Rules used = {4887, 4717, 4807, 4809, 3387, 3386, 3432, 3385, 3433} \[ \int \frac {1}{(a+b \arcsin (c+d x))^{7/2}} \, dx=-\frac {8 \sqrt {2 \pi } \sin \left (\frac {a}{b}\right ) \operatorname {FresnelC}\left (\frac {\sqrt {\frac {2}{\pi }} \sqrt {a+b \arcsin (c+d x)}}{\sqrt {b}}\right )}{15 b^{7/2} d}+\frac {8 \sqrt {2 \pi } \cos \left (\frac {a}{b}\right ) \operatorname {FresnelS}\left (\frac {\sqrt {\frac {2}{\pi }} \sqrt {a+b \arcsin (c+d x)}}{\sqrt {b}}\right )}{15 b^{7/2} d}+\frac {8 \sqrt {1-(c+d x)^2}}{15 b^3 d \sqrt {a+b \arcsin (c+d x)}}+\frac {4 (c+d x)}{15 b^2 d (a+b \arcsin (c+d x))^{3/2}}-\frac {2 \sqrt {1-(c+d x)^2}}{5 b d (a+b \arcsin (c+d x))^{5/2}} \]

[In]

Int[(a + b*ArcSin[c + d*x])^(-7/2),x]

[Out]

(-2*Sqrt[1 - (c + d*x)^2])/(5*b*d*(a + b*ArcSin[c + d*x])^(5/2)) + (4*(c + d*x))/(15*b^2*d*(a + b*ArcSin[c + d
*x])^(3/2)) + (8*Sqrt[1 - (c + d*x)^2])/(15*b^3*d*Sqrt[a + b*ArcSin[c + d*x]]) + (8*Sqrt[2*Pi]*Cos[a/b]*Fresne
lS[(Sqrt[2/Pi]*Sqrt[a + b*ArcSin[c + d*x]])/Sqrt[b]])/(15*b^(7/2)*d) - (8*Sqrt[2*Pi]*FresnelC[(Sqrt[2/Pi]*Sqrt
[a + b*ArcSin[c + d*x]])/Sqrt[b]]*Sin[a/b])/(15*b^(7/2)*d)

Rule 3385

Int[sin[Pi/2 + (e_.) + (f_.)*(x_)]/Sqrt[(c_.) + (d_.)*(x_)], x_Symbol] :> Dist[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 3386

Int[sin[(e_.) + (f_.)*(x_)]/Sqrt[(c_.) + (d_.)*(x_)], x_Symbol] :> Dist[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 3387

Int[sin[(e_.) + (f_.)*(x_)]/Sqrt[(c_.) + (d_.)*(x_)], x_Symbol] :> Dist[Cos[(d*e - c*f)/d], Int[Sin[c*(f/d) +
f*x]/Sqrt[c + d*x], x], x] + Dist[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 3432

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 3433

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 4717

Int[((a_.) + ArcSin[(c_.)*(x_)]*(b_.))^(n_), x_Symbol] :> Simp[Sqrt[1 - c^2*x^2]*((a + b*ArcSin[c*x])^(n + 1)/
(b*c*(n + 1))), x] + Dist[c/(b*(n + 1)), Int[x*((a + b*ArcSin[c*x])^(n + 1)/Sqrt[1 - c^2*x^2]), x], x] /; Free
Q[{a, b, c}, x] && LtQ[n, -1]

Rule 4807

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

Rule 4809

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

Rule 4887

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

Rubi steps \begin{align*} \text {integral}& = \frac {\text {Subst}\left (\int \frac {1}{(a+b \arcsin (x))^{7/2}} \, dx,x,c+d x\right )}{d} \\ & = -\frac {2 \sqrt {1-(c+d x)^2}}{5 b d (a+b \arcsin (c+d x))^{5/2}}-\frac {2 \text {Subst}\left (\int \frac {x}{\sqrt {1-x^2} (a+b \arcsin (x))^{5/2}} \, dx,x,c+d x\right )}{5 b d} \\ & = -\frac {2 \sqrt {1-(c+d x)^2}}{5 b d (a+b \arcsin (c+d x))^{5/2}}+\frac {4 (c+d x)}{15 b^2 d (a+b \arcsin (c+d x))^{3/2}}-\frac {4 \text {Subst}\left (\int \frac {1}{(a+b \arcsin (x))^{3/2}} \, dx,x,c+d x\right )}{15 b^2 d} \\ & = -\frac {2 \sqrt {1-(c+d x)^2}}{5 b d (a+b \arcsin (c+d x))^{5/2}}+\frac {4 (c+d x)}{15 b^2 d (a+b \arcsin (c+d x))^{3/2}}+\frac {8 \sqrt {1-(c+d x)^2}}{15 b^3 d \sqrt {a+b \arcsin (c+d x)}}+\frac {8 \text {Subst}\left (\int \frac {x}{\sqrt {1-x^2} \sqrt {a+b \arcsin (x)}} \, dx,x,c+d x\right )}{15 b^3 d} \\ & = -\frac {2 \sqrt {1-(c+d x)^2}}{5 b d (a+b \arcsin (c+d x))^{5/2}}+\frac {4 (c+d x)}{15 b^2 d (a+b \arcsin (c+d x))^{3/2}}+\frac {8 \sqrt {1-(c+d x)^2}}{15 b^3 d \sqrt {a+b \arcsin (c+d x)}}-\frac {8 \text {Subst}\left (\int \frac {\sin \left (\frac {a}{b}-\frac {x}{b}\right )}{\sqrt {x}} \, dx,x,a+b \arcsin (c+d x)\right )}{15 b^4 d} \\ & = -\frac {2 \sqrt {1-(c+d x)^2}}{5 b d (a+b \arcsin (c+d x))^{5/2}}+\frac {4 (c+d x)}{15 b^2 d (a+b \arcsin (c+d x))^{3/2}}+\frac {8 \sqrt {1-(c+d x)^2}}{15 b^3 d \sqrt {a+b \arcsin (c+d x)}}+\frac {\left (8 \cos \left (\frac {a}{b}\right )\right ) \text {Subst}\left (\int \frac {\sin \left (\frac {x}{b}\right )}{\sqrt {x}} \, dx,x,a+b \arcsin (c+d x)\right )}{15 b^4 d}-\frac {\left (8 \sin \left (\frac {a}{b}\right )\right ) \text {Subst}\left (\int \frac {\cos \left (\frac {x}{b}\right )}{\sqrt {x}} \, dx,x,a+b \arcsin (c+d x)\right )}{15 b^4 d} \\ & = -\frac {2 \sqrt {1-(c+d x)^2}}{5 b d (a+b \arcsin (c+d x))^{5/2}}+\frac {4 (c+d x)}{15 b^2 d (a+b \arcsin (c+d x))^{3/2}}+\frac {8 \sqrt {1-(c+d x)^2}}{15 b^3 d \sqrt {a+b \arcsin (c+d x)}}+\frac {\left (16 \cos \left (\frac {a}{b}\right )\right ) \text {Subst}\left (\int \sin \left (\frac {x^2}{b}\right ) \, dx,x,\sqrt {a+b \arcsin (c+d x)}\right )}{15 b^4 d}-\frac {\left (16 \sin \left (\frac {a}{b}\right )\right ) \text {Subst}\left (\int \cos \left (\frac {x^2}{b}\right ) \, dx,x,\sqrt {a+b \arcsin (c+d x)}\right )}{15 b^4 d} \\ & = -\frac {2 \sqrt {1-(c+d x)^2}}{5 b d (a+b \arcsin (c+d x))^{5/2}}+\frac {4 (c+d x)}{15 b^2 d (a+b \arcsin (c+d x))^{3/2}}+\frac {8 \sqrt {1-(c+d x)^2}}{15 b^3 d \sqrt {a+b \arcsin (c+d x)}}+\frac {8 \sqrt {2 \pi } \cos \left (\frac {a}{b}\right ) \operatorname {FresnelS}\left (\frac {\sqrt {\frac {2}{\pi }} \sqrt {a+b \arcsin (c+d x)}}{\sqrt {b}}\right )}{15 b^{7/2} d}-\frac {8 \sqrt {2 \pi } \operatorname {FresnelC}\left (\frac {\sqrt {\frac {2}{\pi }} \sqrt {a+b \arcsin (c+d x)}}{\sqrt {b}}\right ) \sin \left (\frac {a}{b}\right )}{15 b^{7/2} d} \\ \end{align*}

Mathematica [C] (verified)

Result contains complex when optimal does not.

Time = 0.27 (sec) , antiderivative size = 287, normalized size of antiderivative = 1.32 \[ \int \frac {1}{(a+b \arcsin (c+d x))^{7/2}} \, dx=\frac {-6 b^2 e^{i \arcsin (c+d x)}+4 e^{-\frac {i a}{b}} (a+b \arcsin (c+d x)) \left (e^{\frac {i (a+b \arcsin (c+d x))}{b}} (2 a+b (-i+2 \arcsin (c+d x)))-2 i b \left (-\frac {i (a+b \arcsin (c+d x))}{b}\right )^{3/2} \Gamma \left (\frac {1}{2},-\frac {i (a+b \arcsin (c+d x))}{b}\right )\right )+e^{-i \arcsin (c+d x)} \left (8 a^2+4 a b (i+4 \arcsin (c+d x))+2 b^2 \left (-3+2 i \arcsin (c+d x)+4 \arcsin (c+d x)^2\right )-8 e^{\frac {i (a+b \arcsin (c+d x))}{b}} (a+b \arcsin (c+d x))^2 \sqrt {\frac {i (a+b \arcsin (c+d x))}{b}} \Gamma \left (\frac {1}{2},\frac {i (a+b \arcsin (c+d x))}{b}\right )\right )}{30 b^3 d (a+b \arcsin (c+d x))^{5/2}} \]

[In]

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

[Out]

(-6*b^2*E^(I*ArcSin[c + d*x]) + (4*(a + b*ArcSin[c + d*x])*(E^((I*(a + b*ArcSin[c + d*x]))/b)*(2*a + b*(-I + 2
*ArcSin[c + d*x])) - (2*I)*b*(((-I)*(a + b*ArcSin[c + d*x]))/b)^(3/2)*Gamma[1/2, ((-I)*(a + b*ArcSin[c + d*x])
)/b]))/E^((I*a)/b) + (8*a^2 + 4*a*b*(I + 4*ArcSin[c + d*x]) + 2*b^2*(-3 + (2*I)*ArcSin[c + d*x] + 4*ArcSin[c +
d*x]^2) - 8*E^((I*(a + b*ArcSin[c + d*x]))/b)*(a + b*ArcSin[c + d*x])^2*Sqrt[(I*(a + b*ArcSin[c + d*x]))/b]*G
amma[1/2, (I*(a + b*ArcSin[c + d*x]))/b])/E^(I*ArcSin[c + d*x]))/(30*b^3*d*(a + b*ArcSin[c + d*x])^(5/2))

Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(623\) vs. \(2(178)=356\).

Time = 0.36 (sec) , antiderivative size = 624, normalized size of antiderivative = 2.86

method	result	size
default	\(\frac {-\frac {8 \arcsin \left (d x +c \right )^{2} \sqrt {a +b \arcsin \left (d x +c \right )}\, \cos \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 {\pi }\, \sqrt {-\frac {1}{b}}\, b^{2}}{15}-\frac {8 \arcsin \left (d x +c \right )^{2} \sqrt {a +b \arcsin \left (d x +c \right )}\, \sin \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 {\pi }\, \sqrt {-\frac {1}{b}}\, b^{2}}{15}-\frac {16 \arcsin \left (d x +c \right ) \sqrt {a +b \arcsin \left (d x +c \right )}\, \cos \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 {\pi }\, \sqrt {-\frac {1}{b}}\, a b}{15}-\frac {16 \arcsin \left (d x +c \right ) \sqrt {a +b \arcsin \left (d x +c \right )}\, \sin \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 {\pi }\, \sqrt {-\frac {1}{b}}\, a b}{15}-\frac {8 \sqrt {a +b \arcsin \left (d x +c \right )}\, \cos \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 {\pi }\, \sqrt {-\frac {1}{b}}\, a^{2}}{15}-\frac {8 \sqrt {a +b \arcsin \left (d x +c \right )}\, \sin \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 {\pi }\, \sqrt {-\frac {1}{b}}\, a^{2}}{15}+\frac {8 \arcsin \left (d x +c \right )^{2} \cos \left (-\frac {a +b \arcsin \left (d x +c \right )}{b}+\frac {a}{b}\right ) b^{2}}{15}+\frac {16 \arcsin \left (d x +c \right ) \cos \left (-\frac {a +b \arcsin \left (d x +c \right )}{b}+\frac {a}{b}\right ) a b}{15}-\frac {4 \arcsin \left (d x +c \right ) \sin \left (-\frac {a +b \arcsin \left (d x +c \right )}{b}+\frac {a}{b}\right ) b^{2}}{15}+\frac {8 \cos \left (-\frac {a +b \arcsin \left (d x +c \right )}{b}+\frac {a}{b}\right ) a^{2}}{15}-\frac {2 \cos \left (-\frac {a +b \arcsin \left (d x +c \right )}{b}+\frac {a}{b}\right ) b^{2}}{5}-\frac {4 \sin \left (-\frac {a +b \arcsin \left (d x +c \right )}{b}+\frac {a}{b}\right ) a b}{15}}{d \,b^{3} \left (a +b \arcsin \left (d x +c \right )\right )^{\frac {5}{2}}}\)	\(624\)

[In]

int(1/(a+b*arcsin(d*x+c))^(7/2),x,method=_RETURNVERBOSE)

[Out]

2/15/d/b^3*(-4*arcsin(d*x+c)^2*(a+b*arcsin(d*x+c))^(1/2)*cos(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)*Pi^(1/2)*(-1/b)^(1/2)*b^2-4*arcsin(d*x+c)^2*(a+b*arcsin(d*x+c))^(1/2)*sin(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)*Pi^(1/2)*(-1/b)^(1/2)*b^2-8*arcsi
n(d*x+c)*(a+b*arcsin(d*x+c))^(1/2)*cos(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)*Pi^(1/2)*(-1/b)^(1/2)*a*b-8*arcsin(d*x+c)*(a+b*arcsin(d*x+c))^(1/2)*sin(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)*Pi^(1/2)*(-1/b)^(1/2)*a*b-4*(a+b*arcsin(d*x+c))^(1/2)*cos
(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)*Pi^(1/2)*(-1/b)^(1/2)*a^2-4*
(a+b*arcsin(d*x+c))^(1/2)*sin(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)
*Pi^(1/2)*(-1/b)^(1/2)*a^2+4*arcsin(d*x+c)^2*cos(-(a+b*arcsin(d*x+c))/b+a/b)*b^2+8*arcsin(d*x+c)*cos(-(a+b*arc
sin(d*x+c))/b+a/b)*a*b-2*arcsin(d*x+c)*sin(-(a+b*arcsin(d*x+c))/b+a/b)*b^2+4*cos(-(a+b*arcsin(d*x+c))/b+a/b)*a
^2-3*cos(-(a+b*arcsin(d*x+c))/b+a/b)*b^2-2*sin(-(a+b*arcsin(d*x+c))/b+a/b)*a*b)/(a+b*arcsin(d*x+c))^(5/2)

Fricas [F(-2)]

Exception generated. \[ \int \frac {1}{(a+b \arcsin (c+d x))^{7/2}} \, dx=\text {Exception raised: TypeError} \]

[In]

integrate(1/(a+b*arcsin(d*x+c))^(7/2),x, algorithm="fricas")

[Out]

Exception raised: TypeError >>  Error detected within library code:   integrate: implementation incomplete (co
nstant residues)

Sympy [F]

\[ \int \frac {1}{(a+b \arcsin (c+d x))^{7/2}} \, dx=\int \frac {1}{\left (a + b \operatorname {asin}{\left (c + d x \right )}\right )^{\frac {7}{2}}}\, dx \]

[In]

integrate(1/(a+b*asin(d*x+c))**(7/2),x)

[Out]

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

Maxima [F]

\[ \int \frac {1}{(a+b \arcsin (c+d x))^{7/2}} \, dx=\int { \frac {1}{{\left (b \arcsin \left (d x + c\right ) + a\right )}^{\frac {7}{2}}} \,d x } \]

[In]

integrate(1/(a+b*arcsin(d*x+c))^(7/2),x, algorithm="maxima")

[Out]

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

Giac [F]

\[ \int \frac {1}{(a+b \arcsin (c+d x))^{7/2}} \, dx=\int { \frac {1}{{\left (b \arcsin \left (d x + c\right ) + a\right )}^{\frac {7}{2}}} \,d x } \]

[In]

integrate(1/(a+b*arcsin(d*x+c))^(7/2),x, algorithm="giac")

[Out]

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

Mupad [F(-1)]

Timed out. \[ \int \frac {1}{(a+b \arcsin (c+d x))^{7/2}} \, dx=\int \frac {1}{{\left (a+b\,\mathrm {asin}\left (c+d\,x\right )\right )}^{7/2}} \,d x \]

[In]

int(1/(a + b*asin(c + d*x))^(7/2),x)

[Out]

int(1/(a + b*asin(c + d*x))^(7/2), x)