3.257 \(\int (a+b \sin ^{-1}(c+d x))^{7/2} \, dx\)

Optimal. Leaf size=243 \[ \frac {105 \sqrt {\frac {\pi }{2}} b^{7/2} \cos \left (\frac {a}{b}\right ) C\left (\frac {\sqrt {\frac {2}{\pi }} \sqrt {a+b \sin ^{-1}(c+d x)}}{\sqrt {b}}\right )}{8 d}+\frac {105 \sqrt {\frac {\pi }{2}} b^{7/2} \sin \left (\frac {a}{b}\right ) S\left (\frac {\sqrt {\frac {2}{\pi }} \sqrt {a+b \sin ^{-1}(c+d x)}}{\sqrt {b}}\right )}{8 d}-\frac {105 b^3 \sqrt {1-(c+d x)^2} \sqrt {a+b \sin ^{-1}(c+d x)}}{8 d}-\frac {35 b^2 (c+d x) \left (a+b \sin ^{-1}(c+d x)\right )^{3/2}}{4 d}+\frac {7 b \sqrt {1-(c+d x)^2} \left (a+b \sin ^{-1}(c+d x)\right )^{5/2}}{2 d}+\frac {(c+d x) \left (a+b \sin ^{-1}(c+d x)\right )^{7/2}}{d} \]

[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]  time = 0.41, antiderivative size = 243, normalized size of antiderivative = 1.00, number of steps used = 11, number of rules used = 9, integrand size = 14, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.643, Rules used = {4803, 4619, 4677, 4623, 3306, 3305, 3351, 3304, 3352} \[ \frac {105 \sqrt {\frac {\pi }{2}} b^{7/2} \cos \left (\frac {a}{b}\right ) \text {FresnelC}\left (\frac {\sqrt {\frac {2}{\pi }} \sqrt {a+b \sin ^{-1}(c+d x)}}{\sqrt {b}}\right )}{8 d}+\frac {105 \sqrt {\frac {\pi }{2}} b^{7/2} \sin \left (\frac {a}{b}\right ) S\left (\frac {\sqrt {\frac {2}{\pi }} \sqrt {a+b \sin ^{-1}(c+d x)}}{\sqrt {b}}\right )}{8 d}-\frac {105 b^3 \sqrt {1-(c+d x)^2} \sqrt {a+b \sin ^{-1}(c+d x)}}{8 d}-\frac {35 b^2 (c+d x) \left (a+b \sin ^{-1}(c+d x)\right )^{3/2}}{4 d}+\frac {7 b \sqrt {1-(c+d x)^2} \left (a+b \sin ^{-1}(c+d x)\right )^{5/2}}{2 d}+\frac {(c+d x) \left (a+b \sin ^{-1}(c+d x)\right )^{7/2}}{d} \]

Antiderivative was successfully verified.

[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 3304

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 3305

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 3306

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 3351

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

Rule 3352

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

Rule 4619

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 4623

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 4677

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*d^IntPart[p]*(d + e*x^2)^FracPart[p])/(2*c*(p + 1
)*(1 - c^2*x^2)^FracPart[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 4803

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*} \int \left (a+b \sin ^{-1}(c+d x)\right )^{7/2} \, dx &=\frac {\operatorname {Subst}\left (\int \left (a+b \sin ^{-1}(x)\right )^{7/2} \, dx,x,c+d x\right )}{d}\\ &=\frac {(c+d x) \left (a+b \sin ^{-1}(c+d x)\right )^{7/2}}{d}-\frac {(7 b) \operatorname {Subst}\left (\int \frac {x \left (a+b \sin ^{-1}(x)\right )^{5/2}}{\sqrt {1-x^2}} \, dx,x,c+d x\right )}{2 d}\\ &=\frac {7 b \sqrt {1-(c+d x)^2} \left (a+b \sin ^{-1}(c+d x)\right )^{5/2}}{2 d}+\frac {(c+d x) \left (a+b \sin ^{-1}(c+d x)\right )^{7/2}}{d}-\frac {\left (35 b^2\right ) \operatorname {Subst}\left (\int \left (a+b \sin ^{-1}(x)\right )^{3/2} \, dx,x,c+d x\right )}{4 d}\\ &=-\frac {35 b^2 (c+d x) \left (a+b \sin ^{-1}(c+d x)\right )^{3/2}}{4 d}+\frac {7 b \sqrt {1-(c+d x)^2} \left (a+b \sin ^{-1}(c+d x)\right )^{5/2}}{2 d}+\frac {(c+d x) \left (a+b \sin ^{-1}(c+d x)\right )^{7/2}}{d}+\frac {\left (105 b^3\right ) \operatorname {Subst}\left (\int \frac {x \sqrt {a+b \sin ^{-1}(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 \sin ^{-1}(c+d x)}}{8 d}-\frac {35 b^2 (c+d x) \left (a+b \sin ^{-1}(c+d x)\right )^{3/2}}{4 d}+\frac {7 b \sqrt {1-(c+d x)^2} \left (a+b \sin ^{-1}(c+d x)\right )^{5/2}}{2 d}+\frac {(c+d x) \left (a+b \sin ^{-1}(c+d x)\right )^{7/2}}{d}+\frac {\left (105 b^4\right ) \operatorname {Subst}\left (\int \frac {1}{\sqrt {a+b \sin ^{-1}(x)}} \, dx,x,c+d x\right )}{16 d}\\ &=-\frac {105 b^3 \sqrt {1-(c+d x)^2} \sqrt {a+b \sin ^{-1}(c+d x)}}{8 d}-\frac {35 b^2 (c+d x) \left (a+b \sin ^{-1}(c+d x)\right )^{3/2}}{4 d}+\frac {7 b \sqrt {1-(c+d x)^2} \left (a+b \sin ^{-1}(c+d x)\right )^{5/2}}{2 d}+\frac {(c+d x) \left (a+b \sin ^{-1}(c+d x)\right )^{7/2}}{d}+\frac {\left (105 b^3\right ) \operatorname {Subst}\left (\int \frac {\cos \left (\frac {a}{b}-\frac {x}{b}\right )}{\sqrt {x}} \, dx,x,a+b \sin ^{-1}(c+d x)\right )}{16 d}\\ &=-\frac {105 b^3 \sqrt {1-(c+d x)^2} \sqrt {a+b \sin ^{-1}(c+d x)}}{8 d}-\frac {35 b^2 (c+d x) \left (a+b \sin ^{-1}(c+d x)\right )^{3/2}}{4 d}+\frac {7 b \sqrt {1-(c+d x)^2} \left (a+b \sin ^{-1}(c+d x)\right )^{5/2}}{2 d}+\frac {(c+d x) \left (a+b \sin ^{-1}(c+d x)\right )^{7/2}}{d}+\frac {\left (105 b^3 \cos \left (\frac {a}{b}\right )\right ) \operatorname {Subst}\left (\int \frac {\cos \left (\frac {x}{b}\right )}{\sqrt {x}} \, dx,x,a+b \sin ^{-1}(c+d x)\right )}{16 d}+\frac {\left (105 b^3 \sin \left (\frac {a}{b}\right )\right ) \operatorname {Subst}\left (\int \frac {\sin \left (\frac {x}{b}\right )}{\sqrt {x}} \, dx,x,a+b \sin ^{-1}(c+d x)\right )}{16 d}\\ &=-\frac {105 b^3 \sqrt {1-(c+d x)^2} \sqrt {a+b \sin ^{-1}(c+d x)}}{8 d}-\frac {35 b^2 (c+d x) \left (a+b \sin ^{-1}(c+d x)\right )^{3/2}}{4 d}+\frac {7 b \sqrt {1-(c+d x)^2} \left (a+b \sin ^{-1}(c+d x)\right )^{5/2}}{2 d}+\frac {(c+d x) \left (a+b \sin ^{-1}(c+d x)\right )^{7/2}}{d}+\frac {\left (105 b^3 \cos \left (\frac {a}{b}\right )\right ) \operatorname {Subst}\left (\int \cos \left (\frac {x^2}{b}\right ) \, dx,x,\sqrt {a+b \sin ^{-1}(c+d x)}\right )}{8 d}+\frac {\left (105 b^3 \sin \left (\frac {a}{b}\right )\right ) \operatorname {Subst}\left (\int \sin \left (\frac {x^2}{b}\right ) \, dx,x,\sqrt {a+b \sin ^{-1}(c+d x)}\right )}{8 d}\\ &=-\frac {105 b^3 \sqrt {1-(c+d x)^2} \sqrt {a+b \sin ^{-1}(c+d x)}}{8 d}-\frac {35 b^2 (c+d x) \left (a+b \sin ^{-1}(c+d x)\right )^{3/2}}{4 d}+\frac {7 b \sqrt {1-(c+d x)^2} \left (a+b \sin ^{-1}(c+d x)\right )^{5/2}}{2 d}+\frac {(c+d x) \left (a+b \sin ^{-1}(c+d x)\right )^{7/2}}{d}+\frac {105 b^{7/2} \sqrt {\frac {\pi }{2}} \cos \left (\frac {a}{b}\right ) C\left (\frac {\sqrt {\frac {2}{\pi }} \sqrt {a+b \sin ^{-1}(c+d x)}}{\sqrt {b}}\right )}{8 d}+\frac {105 b^{7/2} \sqrt {\frac {\pi }{2}} S\left (\frac {\sqrt {\frac {2}{\pi }} \sqrt {a+b \sin ^{-1}(c+d x)}}{\sqrt {b}}\right ) \sin \left (\frac {a}{b}\right )}{8 d}\\ \end {align*}

________________________________________________________________________________________

Mathematica [C]  time = 2.29, size = 551, normalized size = 2.27 \[ \frac {e^{-\frac {i a}{b}} \left (\sqrt {2 \pi } \left (8 i a^3 \left (-1+e^{\frac {2 i a}{b}}\right )+105 b^3 \left (1+e^{\frac {2 i a}{b}}\right )\right ) \sqrt {a+b \sin ^{-1}(c+d x)} C\left (\sqrt {\frac {1}{b}} \sqrt {\frac {2}{\pi }} \sqrt {a+b \sin ^{-1}(c+d x)}\right )-i \sqrt {2 \pi } \left (8 i a^3 \left (1+e^{\frac {2 i a}{b}}\right )+105 b^3 \left (-1+e^{\frac {2 i a}{b}}\right )\right ) \sqrt {a+b \sin ^{-1}(c+d x)} S\left (\sqrt {\frac {1}{b}} \sqrt {\frac {2}{\pi }} \sqrt {a+b \sin ^{-1}(c+d x)}\right )+\frac {4 \left (4 a^3 \sqrt {-\frac {i \left (a+b \sin ^{-1}(c+d x)\right )}{b}} \Gamma \left (\frac {3}{2},-\frac {i \left (a+b \sin ^{-1}(c+d x)\right )}{b}\right )+4 a^3 e^{\frac {2 i a}{b}} \sqrt {\frac {i \left (a+b \sin ^{-1}(c+d x)\right )}{b}} \Gamma \left (\frac {3}{2},\frac {i \left (a+b \sin ^{-1}(c+d x)\right )}{b}\right )+e^{\frac {i a}{b}} \left (a+b \sin ^{-1}(c+d x)\right ) \left (7 \left (4 a^2 \sqrt {-c^2-2 c d x-d^2 x^2+1}-10 a b (c+d x)-15 b^2 \sqrt {-c^2-2 c d x-d^2 x^2+1}\right )+\sin ^{-1}(c+d x) \left (24 a^2 (c+d x)+56 a b \sqrt {-c^2-2 c d x-d^2 x^2+1}-70 b^2 (c+d x)\right )+4 b \sin ^{-1}(c+d x)^2 \left (6 a (c+d x)+7 b \sqrt {-c^2-2 c d x-d^2 x^2+1}\right )+8 b^2 (c+d x) \sin ^{-1}(c+d x)^3\right )\right )}{\sqrt {\frac {1}{b}}}\right )}{32 \sqrt {\frac {1}{b}} d \sqrt {a+b \sin ^{-1}(c+d x)}} \]

Warning: Unable to verify antiderivative.

[In]

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

[Out]

(((8*I)*a^3*(-1 + E^(((2*I)*a)/b)) + 105*b^3*(1 + E^(((2*I)*a)/b)))*Sqrt[2*Pi]*Sqrt[a + b*ArcSin[c + d*x]]*Fre
snelC[Sqrt[b^(-1)]*Sqrt[2/Pi]*Sqrt[a + b*ArcSin[c + d*x]]] - I*(105*b^3*(-1 + E^(((2*I)*a)/b)) + (8*I)*a^3*(1
+ E^(((2*I)*a)/b)))*Sqrt[2*Pi]*Sqrt[a + b*ArcSin[c + d*x]]*FresnelS[Sqrt[b^(-1)]*Sqrt[2/Pi]*Sqrt[a + b*ArcSin[
c + d*x]]] + (4*(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]*Gamma[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]))/Sqrt[b^(-1)])/(32*Sqrt[b^(-1)]*d*E^((I*a)/b)*Sqrt[a + b*ArcSin[c + d*x]])

________________________________________________________________________________________

fricas [F(-2)]  time = 0.00, size = 0, normalized size = 0.00 \[ \text {Exception raised: TypeError} \]

Verification of antiderivative is not currently implemented for this CAS.

[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)

________________________________________________________________________________________

giac [B]  time = 5.07, size = 2541, normalized size = 10.46 \[ \text {result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/2*sqrt(2)*sqrt(pi)*a^4*b^4*erf(-1/2*sqrt(2)*sqrt(b*arcsin(d*x + c) + a)*i/sqrt(abs(b)) - 1/2*sqrt(2)*sqrt(b
*arcsin(d*x + c) + a)*sqrt(abs(b))/b)*e^(a*i/b)/((b^5*i/sqrt(abs(b)) + b^4*sqrt(abs(b)))*d) - sqrt(2)*sqrt(pi)
*a^3*b^4*i*erf(-1/2*sqrt(2)*sqrt(b*arcsin(d*x + c) + a)*i/sqrt(abs(b)) - 1/2*sqrt(2)*sqrt(b*arcsin(d*x + c) +
a)*sqrt(abs(b))/b)*e^(a*i/b)/((b^4*i/sqrt(abs(b)) + b^3*sqrt(abs(b)))*d) + 1/2*sqrt(2)*sqrt(pi)*a^4*b^4*erf(1/
2*sqrt(2)*sqrt(b*arcsin(d*x + c) + a)*i/sqrt(abs(b)) - 1/2*sqrt(2)*sqrt(b*arcsin(d*x + c) + a)*sqrt(abs(b))/b)
*e^(-a*i/b)/((b^5*i/sqrt(abs(b)) - b^4*sqrt(abs(b)))*d) - sqrt(2)*sqrt(pi)*a^3*b^4*i*erf(1/2*sqrt(2)*sqrt(b*ar
csin(d*x + c) + a)*i/sqrt(abs(b)) - 1/2*sqrt(2)*sqrt(b*arcsin(d*x + c) + a)*sqrt(abs(b))/b)*e^(-a*i/b)/((b^4*i
/sqrt(abs(b)) - b^3*sqrt(abs(b)))*d) + 2*sqrt(2)*sqrt(pi)*a^4*b^3*erf(-1/2*sqrt(2)*sqrt(b*arcsin(d*x + c) + a)
*i/sqrt(abs(b)) - 1/2*sqrt(2)*sqrt(b*arcsin(d*x + c) + a)*sqrt(abs(b))/b)*e^(a*i/b)/((b^4*i/sqrt(abs(b)) + b^3
*sqrt(abs(b)))*d) + 4*sqrt(2)*sqrt(pi)*a^3*b^3*i*erf(-1/2*sqrt(2)*sqrt(b*arcsin(d*x + c) + a)*i/sqrt(abs(b)) -
1/2*sqrt(2)*sqrt(b*arcsin(d*x + c) + a)*sqrt(abs(b))/b)*e^(a*i/b)/((b^3*i/sqrt(abs(b)) + b^2*sqrt(abs(b)))*d)
- 2*sqrt(2)*sqrt(pi)*a^4*b^3*erf(1/2*sqrt(2)*sqrt(b*arcsin(d*x + c) + a)*i/sqrt(abs(b)) - 1/2*sqrt(2)*sqrt(b*
arcsin(d*x + c) + a)*sqrt(abs(b))/b)*e^(-a*i/b)/((b^4*i/sqrt(abs(b)) - b^3*sqrt(abs(b)))*d) + 4*sqrt(2)*sqrt(p
i)*a^3*b^3*i*erf(1/2*sqrt(2)*sqrt(b*arcsin(d*x + c) + a)*i/sqrt(abs(b)) - 1/2*sqrt(2)*sqrt(b*arcsin(d*x + c) +
a)*sqrt(abs(b))/b)*e^(-a*i/b)/((b^3*i/sqrt(abs(b)) - b^2*sqrt(abs(b)))*d) - 1/2*sqrt(b*arcsin(d*x + c) + a)*b
^3*i*arcsin(d*x + c)^3*e^(i*arcsin(d*x + c))/d + 1/2*sqrt(b*arcsin(d*x + c) + a)*b^3*i*arcsin(d*x + c)^3*e^(-i
*arcsin(d*x + c))/d - sqrt(2)*sqrt(pi)*a^4*b^2*erf(-1/2*sqrt(2)*sqrt(b*arcsin(d*x + c) + a)*i/sqrt(abs(b)) - 1
/2*sqrt(2)*sqrt(b*arcsin(d*x + c) + a)*sqrt(abs(b))/b)*e^(a*i/b)/((b^3*i/sqrt(abs(b)) + b^2*sqrt(abs(b)))*d) +
9/4*sqrt(2)*sqrt(pi)*a^2*b^4*erf(-1/2*sqrt(2)*sqrt(b*arcsin(d*x + c) + a)*i/sqrt(abs(b)) - 1/2*sqrt(2)*sqrt(b
*arcsin(d*x + c) + a)*sqrt(abs(b))/b)*e^(a*i/b)/((b^3*i/sqrt(abs(b)) + b^2*sqrt(abs(b)))*d) - 3*sqrt(2)*sqrt(p
i)*a^3*b^2*i*erf(-1/2*sqrt(2)*sqrt(b*arcsin(d*x + c) + a)*i/sqrt(abs(b)) - 1/2*sqrt(2)*sqrt(b*arcsin(d*x + c)
+ a)*sqrt(abs(b))/b)*e^(a*i/b)/((b^2*i/sqrt(abs(b)) + b*sqrt(abs(b)))*d) + sqrt(2)*sqrt(pi)*a^4*b^2*erf(1/2*sq
rt(2)*sqrt(b*arcsin(d*x + c) + a)*i/sqrt(abs(b)) - 1/2*sqrt(2)*sqrt(b*arcsin(d*x + c) + a)*sqrt(abs(b))/b)*e^(
-a*i/b)/((b^3*i/sqrt(abs(b)) - b^2*sqrt(abs(b)))*d) - 9/4*sqrt(2)*sqrt(pi)*a^2*b^4*erf(1/2*sqrt(2)*sqrt(b*arcs
in(d*x + c) + a)*i/sqrt(abs(b)) - 1/2*sqrt(2)*sqrt(b*arcsin(d*x + c) + a)*sqrt(abs(b))/b)*e^(-a*i/b)/((b^3*i/s
qrt(abs(b)) - b^2*sqrt(abs(b)))*d) - 3*sqrt(2)*sqrt(pi)*a^3*b^2*i*erf(1/2*sqrt(2)*sqrt(b*arcsin(d*x + c) + a)*
i/sqrt(abs(b)) - 1/2*sqrt(2)*sqrt(b*arcsin(d*x + c) + a)*sqrt(abs(b))/b)*e^(-a*i/b)/((b^2*i/sqrt(abs(b)) - b*s
qrt(abs(b)))*d) - 3/2*sqrt(b*arcsin(d*x + c) + a)*a*b^2*i*arcsin(d*x + c)^2*e^(i*arcsin(d*x + c))/d + 3/2*sqrt
(b*arcsin(d*x + c) + a)*a*b^2*i*arcsin(d*x + c)^2*e^(-i*arcsin(d*x + c))/d - 9/4*sqrt(2)*sqrt(pi)*a^2*b^3*erf(
-1/2*sqrt(2)*sqrt(b*arcsin(d*x + c) + a)*i/sqrt(abs(b)) - 1/2*sqrt(2)*sqrt(b*arcsin(d*x + c) + a)*sqrt(abs(b))
/b)*e^(a*i/b)/((b^2*i/sqrt(abs(b)) + b*sqrt(abs(b)))*d) - 105/32*sqrt(2)*sqrt(pi)*b^5*erf(-1/2*sqrt(2)*sqrt(b*
arcsin(d*x + c) + a)*i/sqrt(abs(b)) - 1/2*sqrt(2)*sqrt(b*arcsin(d*x + c) + a)*sqrt(abs(b))/b)*e^(a*i/b)/((b^2*
i/sqrt(abs(b)) + b*sqrt(abs(b)))*d) + 9/4*sqrt(2)*sqrt(pi)*a^2*b^3*erf(1/2*sqrt(2)*sqrt(b*arcsin(d*x + c) + a)
*i/sqrt(abs(b)) - 1/2*sqrt(2)*sqrt(b*arcsin(d*x + c) + a)*sqrt(abs(b))/b)*e^(-a*i/b)/((b^2*i/sqrt(abs(b)) - b*
sqrt(abs(b)))*d) + 105/32*sqrt(2)*sqrt(pi)*b^5*erf(1/2*sqrt(2)*sqrt(b*arcsin(d*x + c) + a)*i/sqrt(abs(b)) - 1/
2*sqrt(2)*sqrt(b*arcsin(d*x + c) + a)*sqrt(abs(b))/b)*e^(-a*i/b)/((b^2*i/sqrt(abs(b)) - b*sqrt(abs(b)))*d) - 3
/2*sqrt(b*arcsin(d*x + c) + a)*a^2*b*i*arcsin(d*x + c)*e^(i*arcsin(d*x + c))/d + 35/8*sqrt(b*arcsin(d*x + c) +
a)*b^3*i*arcsin(d*x + c)*e^(i*arcsin(d*x + c))/d + 7/4*sqrt(b*arcsin(d*x + c) + a)*b^3*arcsin(d*x + c)^2*e^(i
*arcsin(d*x + c))/d + 3/2*sqrt(b*arcsin(d*x + c) + a)*a^2*b*i*arcsin(d*x + c)*e^(-i*arcsin(d*x + c))/d - 35/8*
sqrt(b*arcsin(d*x + c) + a)*b^3*i*arcsin(d*x + c)*e^(-i*arcsin(d*x + c))/d + 7/4*sqrt(b*arcsin(d*x + c) + a)*b
^3*arcsin(d*x + c)^2*e^(-i*arcsin(d*x + c))/d - sqrt(pi)*a^4*b*erf(-1/2*sqrt(2)*sqrt(b*arcsin(d*x + c) + a)*i/
sqrt(abs(b)) - 1/2*sqrt(2)*sqrt(b*arcsin(d*x + c) + a)*sqrt(abs(b))/b)*e^(a*i/b)/((sqrt(2)*b^2*i/sqrt(abs(b))
+ sqrt(2)*b*sqrt(abs(b)))*d) + sqrt(pi)*a^4*b*erf(1/2*sqrt(2)*sqrt(b*arcsin(d*x + c) + a)*i/sqrt(abs(b)) - 1/2
*sqrt(2)*sqrt(b*arcsin(d*x + c) + a)*sqrt(abs(b))/b)*e^(-a*i/b)/((sqrt(2)*b^2*i/sqrt(abs(b)) - sqrt(2)*b*sqrt(
abs(b)))*d) - 1/2*sqrt(b*arcsin(d*x + c) + a)*a^3*i*e^(i*arcsin(d*x + c))/d + 35/8*sqrt(b*arcsin(d*x + c) + a)
*a*b^2*i*e^(i*arcsin(d*x + c))/d + 7/2*sqrt(b*arcsin(d*x + c) + a)*a*b^2*arcsin(d*x + c)*e^(i*arcsin(d*x + c))
/d + 1/2*sqrt(b*arcsin(d*x + c) + a)*a^3*i*e^(-i*arcsin(d*x + c))/d - 35/8*sqrt(b*arcsin(d*x + c) + a)*a*b^2*i
*e^(-i*arcsin(d*x + c))/d + 7/2*sqrt(b*arcsin(d*x + c) + a)*a*b^2*arcsin(d*x + c)*e^(-i*arcsin(d*x + c))/d + 7
/4*sqrt(b*arcsin(d*x + c) + a)*a^2*b*e^(i*arcsin(d*x + c))/d - 105/16*sqrt(b*arcsin(d*x + c) + a)*b^3*e^(i*arc
sin(d*x + c))/d + 7/4*sqrt(b*arcsin(d*x + c) + a)*a^2*b*e^(-i*arcsin(d*x + c))/d - 105/16*sqrt(b*arcsin(d*x +
c) + a)*b^3*e^(-i*arcsin(d*x + c))/d

________________________________________________________________________________________

maple [B]  time = 0.00, size = 608, normalized size = 2.50 \[ \frac {105 \cos \left (\frac {a}{b}\right ) \FresnelC \left (\frac {\sqrt {2}\, \sqrt {a +b \arcsin \left (d x +c \right )}}{\sqrt {\pi }\, \sqrt {\frac {1}{b}}\, b}\right ) \sqrt {\frac {1}{b}}\, \sqrt {a +b \arcsin \left (d x +c \right )}\, \sqrt {\pi }\, \sqrt {2}\, b^{4}+105 \sin \left (\frac {a}{b}\right ) \mathrm {S}\left (\frac {\sqrt {2}\, \sqrt {a +b \arcsin \left (d x +c \right )}}{\sqrt {\pi }\, \sqrt {\frac {1}{b}}\, b}\right ) \sqrt {\frac {1}{b}}\, \sqrt {a +b \arcsin \left (d x +c \right )}\, \sqrt {\pi }\, \sqrt {2}\, b^{4}+16 \sin \left (\frac {a +b \arcsin \left (d x +c \right )}{b}-\frac {a}{b}\right ) \arcsin \left (d x +c \right )^{4} b^{4}+64 \sin \left (\frac {a +b \arcsin \left (d x +c \right )}{b}-\frac {a}{b}\right ) \arcsin \left (d x +c \right )^{3} a \,b^{3}+56 \cos \left (\frac {a +b \arcsin \left (d x +c \right )}{b}-\frac {a}{b}\right ) \arcsin \left (d x +c \right )^{3} b^{4}+96 \sin \left (\frac {a +b \arcsin \left (d x +c \right )}{b}-\frac {a}{b}\right ) \arcsin \left (d x +c \right )^{2} a^{2} b^{2}-140 \sin \left (\frac {a +b \arcsin \left (d x +c \right )}{b}-\frac {a}{b}\right ) \arcsin \left (d x +c \right )^{2} b^{4}+168 \cos \left (\frac {a +b \arcsin \left (d x +c \right )}{b}-\frac {a}{b}\right ) \arcsin \left (d x +c \right )^{2} a \,b^{3}+64 \sin \left (\frac {a +b \arcsin \left (d x +c \right )}{b}-\frac {a}{b}\right ) \arcsin \left (d x +c \right ) a^{3} b -280 \sin \left (\frac {a +b \arcsin \left (d x +c \right )}{b}-\frac {a}{b}\right ) \arcsin \left (d x +c \right ) a \,b^{3}+168 \cos \left (\frac {a +b \arcsin \left (d x +c \right )}{b}-\frac {a}{b}\right ) \arcsin \left (d x +c \right ) a^{2} b^{2}-210 \cos \left (\frac {a +b \arcsin \left (d x +c \right )}{b}-\frac {a}{b}\right ) \arcsin \left (d x +c \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 )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

maxima [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int {\left (b \arcsin \left (d x + c\right ) + a\right )}^{\frac {7}{2}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

mupad [F]  time = 0.00, size = -1, normalized size = -0.00 \[ \int {\left (a+b\,\mathrm {asin}\left (c+d\,x\right )\right )}^{7/2} \,d x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a + b*asin(c + d*x))^(7/2),x)

[Out]

int((a + b*asin(c + d*x))^(7/2), x)

________________________________________________________________________________________

sympy [F(-1)]  time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Timed out

________________________________________________________________________________________