\(\int (a+b \arcsin (c+d x))^{7/2} \, dx\) [257]
Optimal result
Integrand size = 14, antiderivative size = 243 \[
\int (a+b \arcsin (c+d x))^{7/2} \, dx=-\frac {105 b^3 \sqrt {1-(c+d x)^2} \sqrt {a+b \arcsin (c+d x)}}{8 d}-\frac {35 b^2 (c+d x) (a+b \arcsin (c+d x))^{3/2}}{4 d}+\frac {7 b \sqrt {1-(c+d x)^2} (a+b \arcsin (c+d x))^{5/2}}{2 d}+\frac {(c+d x) (a+b \arcsin (c+d x))^{7/2}}{d}+\frac {105 b^{7/2} \sqrt {\frac {\pi }{2}} \cos \left (\frac {a}{b}\right ) \operatorname {FresnelC}\left (\frac {\sqrt {\frac {2}{\pi }} \sqrt {a+b \arcsin (c+d x)}}{\sqrt {b}}\right )}{8 d}+\frac {105 b^{7/2} \sqrt {\frac {\pi }{2}} \operatorname {FresnelS}\left (\frac {\sqrt {\frac {2}{\pi }} \sqrt {a+b \arcsin (c+d x)}}{\sqrt {b}}\right ) \sin \left (\frac {a}{b}\right )}{8 d}
\]
[Out]
-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)*Fres
nelC(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
Rubi [A] (verified)
Time = 0.29 (sec) , antiderivative size = 243, normalized size of antiderivative = 1.00, number of
steps used = 11, number of rules used = 9, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.643, Rules used = {4887, 4715, 4767, 4719, 3387,
3386, 3432, 3385, 3433} \[
\int (a+b \arcsin (c+d x))^{7/2} \, dx=\frac {105 \sqrt {\frac {\pi }{2}} b^{7/2} \cos \left (\frac {a}{b}\right ) \operatorname {FresnelC}\left (\frac {\sqrt {\frac {2}{\pi }} \sqrt {a+b \arcsin (c+d x)}}{\sqrt {b}}\right )}{8 d}+\frac {105 \sqrt {\frac {\pi }{2}} b^{7/2} \sin \left (\frac {a}{b}\right ) \operatorname {FresnelS}\left (\frac {\sqrt {\frac {2}{\pi }} \sqrt {a+b \arcsin (c+d x)}}{\sqrt {b}}\right )}{8 d}-\frac {105 b^3 \sqrt {1-(c+d x)^2} \sqrt {a+b \arcsin (c+d x)}}{8 d}-\frac {35 b^2 (c+d x) (a+b \arcsin (c+d x))^{3/2}}{4 d}+\frac {7 b \sqrt {1-(c+d x)^2} (a+b \arcsin (c+d x))^{5/2}}{2 d}+\frac {(c+d x) (a+b \arcsin (c+d x))^{7/2}}{d}
\]
[In]
Int[(a + b*ArcSin[c + d*x])^(7/2),x]
[Out]
(-105*b^3*Sqrt[1 - (c + d*x)^2]*Sqrt[a + b*ArcSin[c + d*x]])/(8*d) - (35*b^2*(c + d*x)*(a + b*ArcSin[c + d*x])
^(3/2))/(4*d) + (7*b*Sqrt[1 - (c + d*x)^2]*(a + b*ArcSin[c + d*x])^(5/2))/(2*d) + ((c + d*x)*(a + b*ArcSin[c +
d*x])^(7/2))/d + (105*b^(7/2)*Sqrt[Pi/2]*Cos[a/b]*FresnelC[(Sqrt[2/Pi]*Sqrt[a + b*ArcSin[c + d*x]])/Sqrt[b]])
/(8*d) + (105*b^(7/2)*Sqrt[Pi/2]*FresnelS[(Sqrt[2/Pi]*Sqrt[a + b*ArcSin[c + d*x]])/Sqrt[b]]*Sin[a/b])/(8*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 4715
Int[((a_.) + ArcSin[(c_.)*(x_)]*(b_.))^(n_.), x_Symbol] :> Simp[x*(a + b*ArcSin[c*x])^n, x] - Dist[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 4719
Int[((a_.) + ArcSin[(c_.)*(x_)]*(b_.))^(n_), x_Symbol] :> Dist[1/(b*c), Subst[Int[x^n*Cos[-a/b + x/b], x], x,
a + b*ArcSin[c*x]], x] /; FreeQ[{a, b, c, n}, x]
Rule 4767
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] + Dist[b*(n/(2*c*(p + 1)))*Simp[(d + e*x^2)^p/(1 - c^2*x^2)^p
], Int[(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 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 (a+b \arcsin (x))^{7/2} \, dx,x,c+d x\right )}{d} \\ & = \frac {(c+d x) (a+b \arcsin (c+d x))^{7/2}}{d}-\frac {(7 b) \text {Subst}\left (\int \frac {x (a+b \arcsin (x))^{5/2}}{\sqrt {1-x^2}} \, dx,x,c+d x\right )}{2 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 {\left (35 b^2\right ) \text {Subst}\left (\int (a+b \arcsin (x))^{3/2} \, dx,x,c+d x\right )}{4 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 {\left (105 b^3\right ) \text {Subst}\left (\int \frac {x \sqrt {a+b \arcsin (x)}}{\sqrt {1-x^2}} \, dx,x,c+d x\right )}{8 d} \\ & = -\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 {\left (105 b^4\right ) \text {Subst}\left (\int \frac {1}{\sqrt {a+b \arcsin (x)}} \, dx,x,c+d x\right )}{16 d} \\ & = -\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 {\left (105 b^3\right ) \text {Subst}\left (\int \frac {\cos \left (\frac {a}{b}-\frac {x}{b}\right )}{\sqrt {x}} \, dx,x,a+b \arcsin (c+d x)\right )}{16 d} \\ & = -\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 {\left (105 b^3 \cos \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 )}{16 d}+\frac {\left (105 b^3 \sin \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 )}{16 d} \\ & = -\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 {\left (105 b^3 \cos \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 )}{8 d}+\frac {\left (105 b^3 \sin \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 )}{8 d} \\ & = -\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} \\
\end{align*}
Mathematica [C] (verified)
Result contains complex when optimal does not.
Time = 3.20 (sec) , antiderivative size = 545, normalized size of antiderivative = 2.24
\[
\int (a+b \arcsin (c+d x))^{7/2} \, dx=\frac {e^{-\frac {i a}{b}} \left (\sqrt {b} \left (8 i a^3 \left (-1+e^{\frac {2 i a}{b}}\right )+105 b^3 \left (1+e^{\frac {2 i a}{b}}\right )\right ) \sqrt {\frac {\pi }{2}} \sqrt {a+b \arcsin (c+d x)} \operatorname {FresnelC}\left (\frac {\sqrt {\frac {2}{\pi }} \sqrt {a+b \arcsin (c+d x)}}{\sqrt {b}}\right )+2 b \left (e^{\frac {i a}{b}} (a+b \arcsin (c+d x)) \left (7 \left (-10 a b (c+d x)+4 a^2 \sqrt {1-c^2-2 c d x-d^2 x^2}-15 b^2 \sqrt {1-c^2-2 c d x-d^2 x^2}\right )+\left (24 a^2 (c+d x)-70 b^2 (c+d x)+56 a b \sqrt {1-c^2-2 c d x-d^2 x^2}\right ) \arcsin (c+d x)+4 b \left (6 a (c+d x)+7 b \sqrt {1-c^2-2 c d x-d^2 x^2}\right ) \arcsin (c+d x)^2+8 b^2 (c+d x) \arcsin (c+d x)^3\right )+4 a^3 \sqrt {-\frac {i (a+b \arcsin (c+d x))}{b}} \Gamma \left (\frac {3}{2},-\frac {i (a+b \arcsin (c+d x))}{b}\right )+4 a^3 e^{\frac {2 i a}{b}} \sqrt {\frac {i (a+b \arcsin (c+d x))}{b}} \Gamma \left (\frac {3}{2},\frac {i (a+b \arcsin (c+d x))}{b}\right )\right )+\sqrt {b} \sqrt {2 \pi } \sqrt {a+b \arcsin (c+d x)} \operatorname {FresnelS}\left (\frac {\sqrt {\frac {2}{\pi }} \sqrt {a+b \arcsin (c+d x)}}{\sqrt {b}}\right ) \left (4 a^3 \left (1+e^{\frac {2 i a}{b}}\right )+105 b^3 e^{\frac {i a}{b}} \sin \left (\frac {a}{b}\right )\right )\right )}{16 d \sqrt {a+b \arcsin (c+d x)}}
\]
[In]
Integrate[(a + b*ArcSin[c + d*x])^(7/2),x]
[Out]
(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*ArcSin[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*S
qrt[((-I)*(a + b*ArcSin[c + d*x]))/b]*Gamma[3/2, ((-I)*(a + b*ArcSin[c + d*x]))/b] + 4*a^3*E^(((2*I)*a)/b)*Sqr
t[(I*(a + b*ArcSin[c + d*x]))/b]*Gamma[3/2, (I*(a + b*ArcSin[c + d*x]))/b]) + Sqrt[b]*Sqrt[2*Pi]*Sqrt[a + b*Ar
cSin[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]])
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
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| method | result | size |
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| 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 )}}\) |
\(616\) |
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[In]
int((a+b*arcsin(d*x+c))^(7/2),x,method=_RETURNVERBOSE)
[Out]
-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*arcsin(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*ar
csin(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*a
rcsin(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)
Fricas [F(-2)]
Exception generated. \[
\int (a+b \arcsin (c+d x))^{7/2} \, dx=\text {Exception raised: TypeError}
\]
[In]
integrate((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(-1)]
Timed out. \[
\int (a+b \arcsin (c+d x))^{7/2} \, dx=\text {Timed out}
\]
[In]
integrate((a+b*asin(d*x+c))**(7/2),x)
[Out]
Timed out
Maxima [F]
\[
\int (a+b \arcsin (c+d x))^{7/2} \, dx=\int { {\left (b \arcsin \left (d x + c\right ) + a\right )}^{\frac {7}{2}} \,d x }
\]
[In]
integrate((a+b*arcsin(d*x+c))^(7/2),x, algorithm="maxima")
[Out]
integrate((b*arcsin(d*x + c) + a)^(7/2), x)
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
\[
\int (a+b \arcsin (c+d x))^{7/2} \, dx=\text {Too large to display}
\]
[In]
integrate((a+b*arcsin(d*x+c))^(7/2),x, algorithm="giac")
[Out]
-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)*s
qrt(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
(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))) - 64*sqrt(2)*sqrt(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*ar
csin(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/sqr
t(abs(b)) + b^2*sqrt(abs(b))) - 64*sqrt(2)*sqrt(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*sqr
t(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*ar
csin(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*x + c) + a)*sqrt(abs(b))/b)*e^(I*a/b)/(I*b^2/sqrt(abs(b)) + b*s
qrt(abs(b))) - 128*I*sqrt(2)*sqrt(pi)*a^3*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^2/sqrt(abs(b)) + b*sqrt(abs(b))) - 72*sqr
t(2)*sqrt(pi)*a^2*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^2/sqrt(abs(b)) + b*sqrt(abs(b))) + 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*x + c) + a)*sqrt(abs(b))/
b)*e^(-I*a/b)/(-I*b^2/sqrt(abs(b)) + b*sqrt(abs(b))) + 128*I*sqrt(2)*sqrt(pi)*a^3*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^
2/sqrt(abs(b)) + b*sqrt(abs(b))) - 72*sqrt(2)*sqrt(pi)*a^2*b^3*erf(1/2*I*sqrt(2)*sqrt(b*arcsin(d*x + c) + a)/s
qrt(abs(b)) - 1/2*sqrt(2)*sqrt(b*arcsin(d*x + c) + a)*sqrt(abs(b))/b)*e^(-I*a/b)/(-I*b^2/sqrt(abs(b)) + b*sqrt
(abs(b))) + 48*I*sqrt(b*arcsin(d*x + c) + a)*a*b^2*arcsin(d*x + c)^2*e^(I*arcsin(d*x + c)) - 56*sqrt(b*arcsin(
d*x + c) + a)*b^3*arcsin(d*x + c)^2*e^(I*arcsin(d*x + c)) - 48*I*sqrt(b*arcsin(d*x + c) + a)*a*b^2*arcsin(d*x
+ c)^2*e^(-I*arcsin(d*x + c)) - 56*sqrt(b*arcsin(d*x + c) + a)*b^3*arcsin(d*x + c)^2*e^(-I*arcsin(d*x + c)) +
96*I*sqrt(2)*sqrt(pi)*a^3*b*erf(-1/2*I*sqrt(2)*sqrt(b*arcsin(d*x + c) + a)/sqrt(abs(b)) - 1/2*sqrt(2)*sqrt(b*a
rcsin(d*x + c) + a)*sqrt(abs(b))/b)*e^(I*a/b)/(I*b/sqrt(abs(b)) + sqrt(abs(b))) + 72*sqrt(2)*sqrt(pi)*a^2*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/sqrt(abs(b)) + sqrt(abs(b))) + 105*sqrt(2)*sqrt(pi)*b^4*erf(-1/2*I*sqrt(2)*sqrt(b*arcsi
n(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/sqrt(abs
(b)) + sqrt(abs(b))) - 96*I*sqrt(2)*sqrt(pi)*a^3*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*x + c) + a)*sqrt(abs(b))/b)*e^(-I*a/b)/(-I*b/sqrt(abs(b)) + sqrt(abs(b))) + 72*s
qrt(2)*sqrt(pi)*a^2*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/sqrt(abs(b)) + sqrt(abs(b))) + 105*sqrt(2)*sqrt(pi)*b^4*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/sqrt(abs(b)) + sqrt(abs(b))) + 48*I*sqrt(b*arcsin(d*x + c) + a)*a^2*b*arcsin(d*x + c)*e^(I*a
rcsin(d*x + c)) - 112*sqrt(b*arcsin(d*x + c) + a)*a*b^2*arcsin(d*x + c)*e^(I*arcsin(d*x + c)) - 140*I*sqrt(b*a
rcsin(d*x + c) + a)*b^3*arcsin(d*x + c)*e^(I*arcsin(d*x + c)) - 48*I*sqrt(b*arcsin(d*x + c) + a)*a^2*b*arcsin(
d*x + c)*e^(-I*arcsin(d*x + c)) - 112*sqrt(b*arcsin(d*x + c) + a)*a*b^2*arcsin(d*x + c)*e^(-I*arcsin(d*x + c))
+ 140*I*sqrt(b*arcsin(d*x + c) + a)*b^3*arcsin(d*x + c)*e^(-I*arcsin(d*x + c)) + 32*sqrt(pi)*a^4*erf(-1/2*I*s
qrt(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*sqrt(2)*b/sqrt(abs(b)) + sqrt(2)*sqrt(abs(b))) + 32*sqrt(pi)*a^4*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*sqrt(2)*b/sqrt
(abs(b)) + sqrt(2)*sqrt(abs(b))) + 16*I*sqrt(b*arcsin(d*x + c) + a)*a^3*e^(I*arcsin(d*x + c)) - 56*sqrt(b*arcs
in(d*x + c) + a)*a^2*b*e^(I*arcsin(d*x + c)) - 140*I*sqrt(b*arcsin(d*x + c) + a)*a*b^2*e^(I*arcsin(d*x + c)) +
210*sqrt(b*arcsin(d*x + c) + a)*b^3*e^(I*arcsin(d*x + c)) - 16*I*sqrt(b*arcsin(d*x + c) + a)*a^3*e^(-I*arcsin
(d*x + c)) - 56*sqrt(b*arcsin(d*x + c) + a)*a^2*b*e^(-I*arcsin(d*x + c)) + 140*I*sqrt(b*arcsin(d*x + c) + a)*a
*b^2*e^(-I*arcsin(d*x + c)) + 210*sqrt(b*arcsin(d*x + c) + a)*b^3*e^(-I*arcsin(d*x + c)))/d
Mupad [F(-1)]
Timed out. \[
\int (a+b \arcsin (c+d x))^{7/2} \, dx=\int {\left (a+b\,\mathrm {asin}\left (c+d\,x\right )\right )}^{7/2} \,d x
\]
[In]
int((a + b*asin(c + d*x))^(7/2),x)
[Out]
int((a + b*asin(c + d*x))^(7/2), x)