3.3.55 \(\int (c e+d e x)^2 (a+b \text {ArcSin}(c+d x))^{7/2} \, dx\) [255]

Optimal. Leaf size=518 \[ -\frac {175 b^3 e^2 \sqrt {1-(c+d x)^2} \sqrt {a+b \text {ArcSin}(c+d x)}}{54 d}-\frac {35 b^3 e^2 (c+d x)^2 \sqrt {1-(c+d x)^2} \sqrt {a+b \text {ArcSin}(c+d x)}}{216 d}-\frac {35 b^2 e^2 (c+d x) (a+b \text {ArcSin}(c+d x))^{3/2}}{18 d}-\frac {35 b^2 e^2 (c+d x)^3 (a+b \text {ArcSin}(c+d x))^{3/2}}{108 d}+\frac {7 b e^2 \sqrt {1-(c+d x)^2} (a+b \text {ArcSin}(c+d x))^{5/2}}{9 d}+\frac {7 b e^2 (c+d x)^2 \sqrt {1-(c+d x)^2} (a+b \text {ArcSin}(c+d x))^{5/2}}{18 d}+\frac {e^2 (c+d x)^3 (a+b \text {ArcSin}(c+d x))^{7/2}}{3 d}+\frac {105 b^{7/2} e^2 \sqrt {\frac {\pi }{2}} \cos \left (\frac {a}{b}\right ) \text {FresnelC}\left (\frac {\sqrt {\frac {2}{\pi }} \sqrt {a+b \text {ArcSin}(c+d x)}}{\sqrt {b}}\right )}{32 d}-\frac {35 b^{7/2} e^2 \sqrt {\frac {\pi }{6}} \cos \left (\frac {3 a}{b}\right ) \text {FresnelC}\left (\frac {\sqrt {\frac {6}{\pi }} \sqrt {a+b \text {ArcSin}(c+d x)}}{\sqrt {b}}\right )}{864 d}+\frac {105 b^{7/2} e^2 \sqrt {\frac {\pi }{2}} S\left (\frac {\sqrt {\frac {2}{\pi }} \sqrt {a+b \text {ArcSin}(c+d x)}}{\sqrt {b}}\right ) \sin \left (\frac {a}{b}\right )}{32 d}-\frac {35 b^{7/2} e^2 \sqrt {\frac {\pi }{6}} S\left (\frac {\sqrt {\frac {6}{\pi }} \sqrt {a+b \text {ArcSin}(c+d x)}}{\sqrt {b}}\right ) \sin \left (\frac {3 a}{b}\right )}{864 d} \]

[Out]

-35/18*b^2*e^2*(d*x+c)*(a+b*arcsin(d*x+c))^(3/2)/d-35/108*b^2*e^2*(d*x+c)^3*(a+b*arcsin(d*x+c))^(3/2)/d+1/3*e^
2*(d*x+c)^3*(a+b*arcsin(d*x+c))^(7/2)/d-35/5184*b^(7/2)*e^2*cos(3*a/b)*FresnelC(6^(1/2)/Pi^(1/2)*(a+b*arcsin(d
*x+c))^(1/2)/b^(1/2))*6^(1/2)*Pi^(1/2)/d-35/5184*b^(7/2)*e^2*FresnelS(6^(1/2)/Pi^(1/2)*(a+b*arcsin(d*x+c))^(1/
2)/b^(1/2))*sin(3*a/b)*6^(1/2)*Pi^(1/2)/d+105/64*b^(7/2)*e^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+105/64*b^(7/2)*e^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+7/9*b*e^2*(a+b*arcsin(d*x+c))^(5/2)*(1-(d*x+c)^2)^(1/2)/d+7/18*b*e^2*(d*
x+c)^2*(a+b*arcsin(d*x+c))^(5/2)*(1-(d*x+c)^2)^(1/2)/d-175/54*b^3*e^2*(1-(d*x+c)^2)^(1/2)*(a+b*arcsin(d*x+c))^
(1/2)/d-35/216*b^3*e^2*(d*x+c)^2*(1-(d*x+c)^2)^(1/2)*(a+b*arcsin(d*x+c))^(1/2)/d

________________________________________________________________________________________

Rubi [A]
time = 1.11, antiderivative size = 518, normalized size of antiderivative = 1.00, number of steps used = 35, number of rules used = 14, integrand size = 25, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.560, Rules used = {4889, 12, 4725, 4795, 4767, 4715, 4719, 3387, 3386, 3432, 3385, 3433, 4731, 4491} \begin {gather*} \frac {105 \sqrt {\frac {\pi }{2}} b^{7/2} e^2 \cos \left (\frac {a}{b}\right ) \text {FresnelC}\left (\frac {\sqrt {\frac {2}{\pi }} \sqrt {a+b \text {ArcSin}(c+d x)}}{\sqrt {b}}\right )}{32 d}-\frac {35 \sqrt {\frac {\pi }{6}} b^{7/2} e^2 \cos \left (\frac {3 a}{b}\right ) \text {FresnelC}\left (\frac {\sqrt {\frac {6}{\pi }} \sqrt {a+b \text {ArcSin}(c+d x)}}{\sqrt {b}}\right )}{864 d}+\frac {105 \sqrt {\frac {\pi }{2}} b^{7/2} e^2 \sin \left (\frac {a}{b}\right ) S\left (\frac {\sqrt {\frac {2}{\pi }} \sqrt {a+b \text {ArcSin}(c+d x)}}{\sqrt {b}}\right )}{32 d}-\frac {35 \sqrt {\frac {\pi }{6}} b^{7/2} e^2 \sin \left (\frac {3 a}{b}\right ) S\left (\frac {\sqrt {\frac {6}{\pi }} \sqrt {a+b \text {ArcSin}(c+d x)}}{\sqrt {b}}\right )}{864 d}-\frac {175 b^3 e^2 \sqrt {1-(c+d x)^2} \sqrt {a+b \text {ArcSin}(c+d x)}}{54 d}-\frac {35 b^3 e^2 (c+d x)^2 \sqrt {1-(c+d x)^2} \sqrt {a+b \text {ArcSin}(c+d x)}}{216 d}-\frac {35 b^2 e^2 (c+d x)^3 (a+b \text {ArcSin}(c+d x))^{3/2}}{108 d}-\frac {35 b^2 e^2 (c+d x) (a+b \text {ArcSin}(c+d x))^{3/2}}{18 d}+\frac {7 b e^2 \sqrt {1-(c+d x)^2} (a+b \text {ArcSin}(c+d x))^{5/2}}{9 d}+\frac {7 b e^2 (c+d x)^2 \sqrt {1-(c+d x)^2} (a+b \text {ArcSin}(c+d x))^{5/2}}{18 d}+\frac {e^2 (c+d x)^3 (a+b \text {ArcSin}(c+d x))^{7/2}}{3 d} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(-175*b^3*e^2*Sqrt[1 - (c + d*x)^2]*Sqrt[a + b*ArcSin[c + d*x]])/(54*d) - (35*b^3*e^2*(c + d*x)^2*Sqrt[1 - (c
+ d*x)^2]*Sqrt[a + b*ArcSin[c + d*x]])/(216*d) - (35*b^2*e^2*(c + d*x)*(a + b*ArcSin[c + d*x])^(3/2))/(18*d) -
(35*b^2*e^2*(c + d*x)^3*(a + b*ArcSin[c + d*x])^(3/2))/(108*d) + (7*b*e^2*Sqrt[1 - (c + d*x)^2]*(a + b*ArcSin
[c + d*x])^(5/2))/(9*d) + (7*b*e^2*(c + d*x)^2*Sqrt[1 - (c + d*x)^2]*(a + b*ArcSin[c + d*x])^(5/2))/(18*d) + (
e^2*(c + d*x)^3*(a + b*ArcSin[c + d*x])^(7/2))/(3*d) + (105*b^(7/2)*e^2*Sqrt[Pi/2]*Cos[a/b]*FresnelC[(Sqrt[2/P
i]*Sqrt[a + b*ArcSin[c + d*x]])/Sqrt[b]])/(32*d) - (35*b^(7/2)*e^2*Sqrt[Pi/6]*Cos[(3*a)/b]*FresnelC[(Sqrt[6/Pi
]*Sqrt[a + b*ArcSin[c + d*x]])/Sqrt[b]])/(864*d) + (105*b^(7/2)*e^2*Sqrt[Pi/2]*FresnelS[(Sqrt[2/Pi]*Sqrt[a + b
*ArcSin[c + d*x]])/Sqrt[b]]*Sin[a/b])/(32*d) - (35*b^(7/2)*e^2*Sqrt[Pi/6]*FresnelS[(Sqrt[6/Pi]*Sqrt[a + b*ArcS
in[c + d*x]])/Sqrt[b]]*Sin[(3*a)/b])/(864*d)

Rule 12

Int[(a_)*(u_), x_Symbol] :> Dist[a, Int[u, x], x] /; FreeQ[a, x] &&  !MatchQ[u, (b_)*(v_) /; FreeQ[b, x]]

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 4491

Int[Cos[(a_.) + (b_.)*(x_)]^(p_.)*((c_.) + (d_.)*(x_))^(m_.)*Sin[(a_.) + (b_.)*(x_)]^(n_.), x_Symbol] :> Int[E
xpandTrigReduce[(c + d*x)^m, Sin[a + b*x]^n*Cos[a + b*x]^p, x], x] /; FreeQ[{a, b, c, d, m}, x] && IGtQ[n, 0]
&& IGtQ[p, 0]

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 4725

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

Rule 4731

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

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 4795

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] + (Dist[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] + Dist[b*f*(n/(c*(m + 2*p + 1)))*S
imp[(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]

Rule 4889

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

Rubi steps

\begin {align*} \int (c e+d e x)^2 \left (a+b \sin ^{-1}(c+d x)\right )^{7/2} \, dx &=\frac {\text {Subst}\left (\int e^2 x^2 \left (a+b \sin ^{-1}(x)\right )^{7/2} \, dx,x,c+d x\right )}{d}\\ &=\frac {e^2 \text {Subst}\left (\int x^2 \left (a+b \sin ^{-1}(x)\right )^{7/2} \, dx,x,c+d x\right )}{d}\\ &=\frac {e^2 (c+d x)^3 \left (a+b \sin ^{-1}(c+d x)\right )^{7/2}}{3 d}-\frac {\left (7 b e^2\right ) \text {Subst}\left (\int \frac {x^3 \left (a+b \sin ^{-1}(x)\right )^{5/2}}{\sqrt {1-x^2}} \, dx,x,c+d x\right )}{6 d}\\ &=\frac {7 b e^2 (c+d x)^2 \sqrt {1-(c+d x)^2} \left (a+b \sin ^{-1}(c+d x)\right )^{5/2}}{18 d}+\frac {e^2 (c+d x)^3 \left (a+b \sin ^{-1}(c+d x)\right )^{7/2}}{3 d}-\frac {\left (7 b e^2\right ) \text {Subst}\left (\int \frac {x \left (a+b \sin ^{-1}(x)\right )^{5/2}}{\sqrt {1-x^2}} \, dx,x,c+d x\right )}{9 d}-\frac {\left (35 b^2 e^2\right ) \text {Subst}\left (\int x^2 \left (a+b \sin ^{-1}(x)\right )^{3/2} \, dx,x,c+d x\right )}{36 d}\\ &=-\frac {35 b^2 e^2 (c+d x)^3 \left (a+b \sin ^{-1}(c+d x)\right )^{3/2}}{108 d}+\frac {7 b e^2 \sqrt {1-(c+d x)^2} \left (a+b \sin ^{-1}(c+d x)\right )^{5/2}}{9 d}+\frac {7 b e^2 (c+d x)^2 \sqrt {1-(c+d x)^2} \left (a+b \sin ^{-1}(c+d x)\right )^{5/2}}{18 d}+\frac {e^2 (c+d x)^3 \left (a+b \sin ^{-1}(c+d x)\right )^{7/2}}{3 d}-\frac {\left (35 b^2 e^2\right ) \text {Subst}\left (\int \left (a+b \sin ^{-1}(x)\right )^{3/2} \, dx,x,c+d x\right )}{18 d}+\frac {\left (35 b^3 e^2\right ) \text {Subst}\left (\int \frac {x^3 \sqrt {a+b \sin ^{-1}(x)}}{\sqrt {1-x^2}} \, dx,x,c+d x\right )}{72 d}\\ &=-\frac {35 b^3 e^2 (c+d x)^2 \sqrt {1-(c+d x)^2} \sqrt {a+b \sin ^{-1}(c+d x)}}{216 d}-\frac {35 b^2 e^2 (c+d x) \left (a+b \sin ^{-1}(c+d x)\right )^{3/2}}{18 d}-\frac {35 b^2 e^2 (c+d x)^3 \left (a+b \sin ^{-1}(c+d x)\right )^{3/2}}{108 d}+\frac {7 b e^2 \sqrt {1-(c+d x)^2} \left (a+b \sin ^{-1}(c+d x)\right )^{5/2}}{9 d}+\frac {7 b e^2 (c+d x)^2 \sqrt {1-(c+d x)^2} \left (a+b \sin ^{-1}(c+d x)\right )^{5/2}}{18 d}+\frac {e^2 (c+d x)^3 \left (a+b \sin ^{-1}(c+d x)\right )^{7/2}}{3 d}+\frac {\left (35 b^3 e^2\right ) \text {Subst}\left (\int \frac {x \sqrt {a+b \sin ^{-1}(x)}}{\sqrt {1-x^2}} \, dx,x,c+d x\right )}{108 d}+\frac {\left (35 b^3 e^2\right ) \text {Subst}\left (\int \frac {x \sqrt {a+b \sin ^{-1}(x)}}{\sqrt {1-x^2}} \, dx,x,c+d x\right )}{12 d}+\frac {\left (35 b^4 e^2\right ) \text {Subst}\left (\int \frac {x^2}{\sqrt {a+b \sin ^{-1}(x)}} \, dx,x,c+d x\right )}{432 d}\\ &=-\frac {175 b^3 e^2 \sqrt {1-(c+d x)^2} \sqrt {a+b \sin ^{-1}(c+d x)}}{54 d}-\frac {35 b^3 e^2 (c+d x)^2 \sqrt {1-(c+d x)^2} \sqrt {a+b \sin ^{-1}(c+d x)}}{216 d}-\frac {35 b^2 e^2 (c+d x) \left (a+b \sin ^{-1}(c+d x)\right )^{3/2}}{18 d}-\frac {35 b^2 e^2 (c+d x)^3 \left (a+b \sin ^{-1}(c+d x)\right )^{3/2}}{108 d}+\frac {7 b e^2 \sqrt {1-(c+d x)^2} \left (a+b \sin ^{-1}(c+d x)\right )^{5/2}}{9 d}+\frac {7 b e^2 (c+d x)^2 \sqrt {1-(c+d x)^2} \left (a+b \sin ^{-1}(c+d x)\right )^{5/2}}{18 d}+\frac {e^2 (c+d x)^3 \left (a+b \sin ^{-1}(c+d x)\right )^{7/2}}{3 d}+\frac {\left (35 b^4 e^2\right ) \text {Subst}\left (\int \frac {\cos (x) \sin ^2(x)}{\sqrt {a+b x}} \, dx,x,\sin ^{-1}(c+d x)\right )}{432 d}+\frac {\left (35 b^4 e^2\right ) \text {Subst}\left (\int \frac {1}{\sqrt {a+b \sin ^{-1}(x)}} \, dx,x,c+d x\right )}{216 d}+\frac {\left (35 b^4 e^2\right ) \text {Subst}\left (\int \frac {1}{\sqrt {a+b \sin ^{-1}(x)}} \, dx,x,c+d x\right )}{24 d}\\ &=-\frac {175 b^3 e^2 \sqrt {1-(c+d x)^2} \sqrt {a+b \sin ^{-1}(c+d x)}}{54 d}-\frac {35 b^3 e^2 (c+d x)^2 \sqrt {1-(c+d x)^2} \sqrt {a+b \sin ^{-1}(c+d x)}}{216 d}-\frac {35 b^2 e^2 (c+d x) \left (a+b \sin ^{-1}(c+d x)\right )^{3/2}}{18 d}-\frac {35 b^2 e^2 (c+d x)^3 \left (a+b \sin ^{-1}(c+d x)\right )^{3/2}}{108 d}+\frac {7 b e^2 \sqrt {1-(c+d x)^2} \left (a+b \sin ^{-1}(c+d x)\right )^{5/2}}{9 d}+\frac {7 b e^2 (c+d x)^2 \sqrt {1-(c+d x)^2} \left (a+b \sin ^{-1}(c+d x)\right )^{5/2}}{18 d}+\frac {e^2 (c+d x)^3 \left (a+b \sin ^{-1}(c+d x)\right )^{7/2}}{3 d}+\frac {\left (35 b^3 e^2\right ) \text {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 )}{216 d}+\frac {\left (35 b^3 e^2\right ) \text {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 )}{24 d}+\frac {\left (35 b^4 e^2\right ) \text {Subst}\left (\int \left (\frac {\cos (x)}{4 \sqrt {a+b x}}-\frac {\cos (3 x)}{4 \sqrt {a+b x}}\right ) \, dx,x,\sin ^{-1}(c+d x)\right )}{432 d}\\ &=-\frac {175 b^3 e^2 \sqrt {1-(c+d x)^2} \sqrt {a+b \sin ^{-1}(c+d x)}}{54 d}-\frac {35 b^3 e^2 (c+d x)^2 \sqrt {1-(c+d x)^2} \sqrt {a+b \sin ^{-1}(c+d x)}}{216 d}-\frac {35 b^2 e^2 (c+d x) \left (a+b \sin ^{-1}(c+d x)\right )^{3/2}}{18 d}-\frac {35 b^2 e^2 (c+d x)^3 \left (a+b \sin ^{-1}(c+d x)\right )^{3/2}}{108 d}+\frac {7 b e^2 \sqrt {1-(c+d x)^2} \left (a+b \sin ^{-1}(c+d x)\right )^{5/2}}{9 d}+\frac {7 b e^2 (c+d x)^2 \sqrt {1-(c+d x)^2} \left (a+b \sin ^{-1}(c+d x)\right )^{5/2}}{18 d}+\frac {e^2 (c+d x)^3 \left (a+b \sin ^{-1}(c+d x)\right )^{7/2}}{3 d}+\frac {\left (35 b^4 e^2\right ) \text {Subst}\left (\int \frac {\cos (x)}{\sqrt {a+b x}} \, dx,x,\sin ^{-1}(c+d x)\right )}{1728 d}-\frac {\left (35 b^4 e^2\right ) \text {Subst}\left (\int \frac {\cos (3 x)}{\sqrt {a+b x}} \, dx,x,\sin ^{-1}(c+d x)\right )}{1728 d}+\frac {\left (35 b^3 e^2 \cos \left (\frac {a}{b}\right )\right ) \text {Subst}\left (\int \frac {\cos \left (\frac {x}{b}\right )}{\sqrt {x}} \, dx,x,a+b \sin ^{-1}(c+d x)\right )}{216 d}+\frac {\left (35 b^3 e^2 \cos \left (\frac {a}{b}\right )\right ) \text {Subst}\left (\int \frac {\cos \left (\frac {x}{b}\right )}{\sqrt {x}} \, dx,x,a+b \sin ^{-1}(c+d x)\right )}{24 d}+\frac {\left (35 b^3 e^2 \sin \left (\frac {a}{b}\right )\right ) \text {Subst}\left (\int \frac {\sin \left (\frac {x}{b}\right )}{\sqrt {x}} \, dx,x,a+b \sin ^{-1}(c+d x)\right )}{216 d}+\frac {\left (35 b^3 e^2 \sin \left (\frac {a}{b}\right )\right ) \text {Subst}\left (\int \frac {\sin \left (\frac {x}{b}\right )}{\sqrt {x}} \, dx,x,a+b \sin ^{-1}(c+d x)\right )}{24 d}\\ &=-\frac {175 b^3 e^2 \sqrt {1-(c+d x)^2} \sqrt {a+b \sin ^{-1}(c+d x)}}{54 d}-\frac {35 b^3 e^2 (c+d x)^2 \sqrt {1-(c+d x)^2} \sqrt {a+b \sin ^{-1}(c+d x)}}{216 d}-\frac {35 b^2 e^2 (c+d x) \left (a+b \sin ^{-1}(c+d x)\right )^{3/2}}{18 d}-\frac {35 b^2 e^2 (c+d x)^3 \left (a+b \sin ^{-1}(c+d x)\right )^{3/2}}{108 d}+\frac {7 b e^2 \sqrt {1-(c+d x)^2} \left (a+b \sin ^{-1}(c+d x)\right )^{5/2}}{9 d}+\frac {7 b e^2 (c+d x)^2 \sqrt {1-(c+d x)^2} \left (a+b \sin ^{-1}(c+d x)\right )^{5/2}}{18 d}+\frac {e^2 (c+d x)^3 \left (a+b \sin ^{-1}(c+d x)\right )^{7/2}}{3 d}+\frac {\left (35 b^3 e^2 \cos \left (\frac {a}{b}\right )\right ) \text {Subst}\left (\int \cos \left (\frac {x^2}{b}\right ) \, dx,x,\sqrt {a+b \sin ^{-1}(c+d x)}\right )}{108 d}+\frac {\left (35 b^3 e^2 \cos \left (\frac {a}{b}\right )\right ) \text {Subst}\left (\int \cos \left (\frac {x^2}{b}\right ) \, dx,x,\sqrt {a+b \sin ^{-1}(c+d x)}\right )}{12 d}+\frac {\left (35 b^4 e^2 \cos \left (\frac {a}{b}\right )\right ) \text {Subst}\left (\int \frac {\cos \left (\frac {a}{b}+x\right )}{\sqrt {a+b x}} \, dx,x,\sin ^{-1}(c+d x)\right )}{1728 d}-\frac {\left (35 b^4 e^2 \cos \left (\frac {3 a}{b}\right )\right ) \text {Subst}\left (\int \frac {\cos \left (\frac {3 a}{b}+3 x\right )}{\sqrt {a+b x}} \, dx,x,\sin ^{-1}(c+d x)\right )}{1728 d}+\frac {\left (35 b^3 e^2 \sin \left (\frac {a}{b}\right )\right ) \text {Subst}\left (\int \sin \left (\frac {x^2}{b}\right ) \, dx,x,\sqrt {a+b \sin ^{-1}(c+d x)}\right )}{108 d}+\frac {\left (35 b^3 e^2 \sin \left (\frac {a}{b}\right )\right ) \text {Subst}\left (\int \sin \left (\frac {x^2}{b}\right ) \, dx,x,\sqrt {a+b \sin ^{-1}(c+d x)}\right )}{12 d}+\frac {\left (35 b^4 e^2 \sin \left (\frac {a}{b}\right )\right ) \text {Subst}\left (\int \frac {\sin \left (\frac {a}{b}+x\right )}{\sqrt {a+b x}} \, dx,x,\sin ^{-1}(c+d x)\right )}{1728 d}-\frac {\left (35 b^4 e^2 \sin \left (\frac {3 a}{b}\right )\right ) \text {Subst}\left (\int \frac {\sin \left (\frac {3 a}{b}+3 x\right )}{\sqrt {a+b x}} \, dx,x,\sin ^{-1}(c+d x)\right )}{1728 d}\\ &=-\frac {175 b^3 e^2 \sqrt {1-(c+d x)^2} \sqrt {a+b \sin ^{-1}(c+d x)}}{54 d}-\frac {35 b^3 e^2 (c+d x)^2 \sqrt {1-(c+d x)^2} \sqrt {a+b \sin ^{-1}(c+d x)}}{216 d}-\frac {35 b^2 e^2 (c+d x) \left (a+b \sin ^{-1}(c+d x)\right )^{3/2}}{18 d}-\frac {35 b^2 e^2 (c+d x)^3 \left (a+b \sin ^{-1}(c+d x)\right )^{3/2}}{108 d}+\frac {7 b e^2 \sqrt {1-(c+d x)^2} \left (a+b \sin ^{-1}(c+d x)\right )^{5/2}}{9 d}+\frac {7 b e^2 (c+d x)^2 \sqrt {1-(c+d x)^2} \left (a+b \sin ^{-1}(c+d x)\right )^{5/2}}{18 d}+\frac {e^2 (c+d x)^3 \left (a+b \sin ^{-1}(c+d x)\right )^{7/2}}{3 d}+\frac {175 b^{7/2} e^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 )}{54 d}+\frac {175 b^{7/2} e^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 )}{54 d}+\frac {\left (35 b^3 e^2 \cos \left (\frac {a}{b}\right )\right ) \text {Subst}\left (\int \cos \left (\frac {x^2}{b}\right ) \, dx,x,\sqrt {a+b \sin ^{-1}(c+d x)}\right )}{864 d}-\frac {\left (35 b^3 e^2 \cos \left (\frac {3 a}{b}\right )\right ) \text {Subst}\left (\int \cos \left (\frac {3 x^2}{b}\right ) \, dx,x,\sqrt {a+b \sin ^{-1}(c+d x)}\right )}{864 d}+\frac {\left (35 b^3 e^2 \sin \left (\frac {a}{b}\right )\right ) \text {Subst}\left (\int \sin \left (\frac {x^2}{b}\right ) \, dx,x,\sqrt {a+b \sin ^{-1}(c+d x)}\right )}{864 d}-\frac {\left (35 b^3 e^2 \sin \left (\frac {3 a}{b}\right )\right ) \text {Subst}\left (\int \sin \left (\frac {3 x^2}{b}\right ) \, dx,x,\sqrt {a+b \sin ^{-1}(c+d x)}\right )}{864 d}\\ &=-\frac {175 b^3 e^2 \sqrt {1-(c+d x)^2} \sqrt {a+b \sin ^{-1}(c+d x)}}{54 d}-\frac {35 b^3 e^2 (c+d x)^2 \sqrt {1-(c+d x)^2} \sqrt {a+b \sin ^{-1}(c+d x)}}{216 d}-\frac {35 b^2 e^2 (c+d x) \left (a+b \sin ^{-1}(c+d x)\right )^{3/2}}{18 d}-\frac {35 b^2 e^2 (c+d x)^3 \left (a+b \sin ^{-1}(c+d x)\right )^{3/2}}{108 d}+\frac {7 b e^2 \sqrt {1-(c+d x)^2} \left (a+b \sin ^{-1}(c+d x)\right )^{5/2}}{9 d}+\frac {7 b e^2 (c+d x)^2 \sqrt {1-(c+d x)^2} \left (a+b \sin ^{-1}(c+d x)\right )^{5/2}}{18 d}+\frac {e^2 (c+d x)^3 \left (a+b \sin ^{-1}(c+d x)\right )^{7/2}}{3 d}+\frac {105 b^{7/2} e^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 )}{32 d}-\frac {35 b^{7/2} e^2 \sqrt {\frac {\pi }{6}} \cos \left (\frac {3 a}{b}\right ) C\left (\frac {\sqrt {\frac {6}{\pi }} \sqrt {a+b \sin ^{-1}(c+d x)}}{\sqrt {b}}\right )}{864 d}+\frac {105 b^{7/2} e^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 )}{32 d}-\frac {35 b^{7/2} e^2 \sqrt {\frac {\pi }{6}} S\left (\frac {\sqrt {\frac {6}{\pi }} \sqrt {a+b \sin ^{-1}(c+d x)}}{\sqrt {b}}\right ) \sin \left (\frac {3 a}{b}\right )}{864 d}\\ \end {align*}

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Mathematica [C] Result contains complex when optimal does not.
time = 0.21, size = 267, normalized size = 0.52 \begin {gather*} \frac {b e^2 e^{-\frac {3 i a}{b}} (a+b \text {ArcSin}(c+d x))^{5/2} \left (-243 e^{\frac {2 i a}{b}} \sqrt {\frac {i (a+b \text {ArcSin}(c+d x))}{b}} \text {Gamma}\left (\frac {9}{2},-\frac {i (a+b \text {ArcSin}(c+d x))}{b}\right )-243 e^{\frac {4 i a}{b}} \sqrt {-\frac {i (a+b \text {ArcSin}(c+d x))}{b}} \text {Gamma}\left (\frac {9}{2},\frac {i (a+b \text {ArcSin}(c+d x))}{b}\right )+\sqrt {3} \left (\sqrt {\frac {i (a+b \text {ArcSin}(c+d x))}{b}} \text {Gamma}\left (\frac {9}{2},-\frac {3 i (a+b \text {ArcSin}(c+d x))}{b}\right )+e^{\frac {6 i a}{b}} \sqrt {-\frac {i (a+b \text {ArcSin}(c+d x))}{b}} \text {Gamma}\left (\frac {9}{2},\frac {3 i (a+b \text {ArcSin}(c+d x))}{b}\right )\right )\right )}{1944 d \left (\frac {(a+b \text {ArcSin}(c+d x))^2}{b^2}\right )^{3/2}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(b*e^2*(a + b*ArcSin[c + d*x])^(5/2)*(-243*E^(((2*I)*a)/b)*Sqrt[(I*(a + b*ArcSin[c + d*x]))/b]*Gamma[9/2, ((-I
)*(a + b*ArcSin[c + d*x]))/b] - 243*E^(((4*I)*a)/b)*Sqrt[((-I)*(a + b*ArcSin[c + d*x]))/b]*Gamma[9/2, (I*(a +
b*ArcSin[c + d*x]))/b] + Sqrt[3]*(Sqrt[(I*(a + b*ArcSin[c + d*x]))/b]*Gamma[9/2, ((-3*I)*(a + b*ArcSin[c + d*x
]))/b] + E^(((6*I)*a)/b)*Sqrt[((-I)*(a + b*ArcSin[c + d*x]))/b]*Gamma[9/2, ((3*I)*(a + b*ArcSin[c + d*x]))/b])
))/(1944*d*E^(((3*I)*a)/b)*((a + b*ArcSin[c + d*x])^2/b^2)^(3/2))

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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(1233\) vs. \(2(426)=852\).
time = 0.61, size = 1234, normalized size = 2.38

method	result	size
default	\(\text {Expression too large to display}\)	\(1234\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/5184/d*e^2*(1296*sin(-(a+b*arcsin(d*x+c))/b+a/b)*a^4+5184*arcsin(d*x+c)^3*sin(-(a+b*arcsin(d*x+c))/b+a/b)*a
*b^3+7776*arcsin(d*x+c)^2*sin(-(a+b*arcsin(d*x+c))/b+a/b)*a^2*b^2-13608*arcsin(d*x+c)^2*cos(-(a+b*arcsin(d*x+c
))/b+a/b)*a*b^3+5184*arcsin(d*x+c)*sin(-(a+b*arcsin(d*x+c))/b+a/b)*a^3*b-22680*arcsin(d*x+c)*sin(-(a+b*arcsin(
d*x+c))/b+a/b)*a*b^3-13608*arcsin(d*x+c)*cos(-(a+b*arcsin(d*x+c))/b+a/b)*a^2*b^2+1512*arcsin(d*x+c)^2*cos(-3*(
a+b*arcsin(d*x+c))/b+3*a/b)*a*b^3+1512*arcsin(d*x+c)*cos(-3*(a+b*arcsin(d*x+c))/b+3*a/b)*a^2*b^2-1728*arcsin(d
*x+c)*sin(-3*(a+b*arcsin(d*x+c))/b+3*a/b)*a^3*b-2592*arcsin(d*x+c)^2*sin(-3*(a+b*arcsin(d*x+c))/b+3*a/b)*a^2*b
^2-1728*arcsin(d*x+c)^3*sin(-3*(a+b*arcsin(d*x+c))/b+3*a/b)*a*b^3+840*arcsin(d*x+c)*sin(-3*(a+b*arcsin(d*x+c))
/b+3*a/b)*a*b^3+8505*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)
*Pi^(1/2)*2^(1/2)*(a+b*arcsin(d*x+c))^(1/2)*b^4+1296*arcsin(d*x+c)^4*sin(-(a+b*arcsin(d*x+c))/b+a/b)*b^4-4536*
arcsin(d*x+c)^3*cos(-(a+b*arcsin(d*x+c))/b+a/b)*b^4-11340*arcsin(d*x+c)^2*sin(-(a+b*arcsin(d*x+c))/b+a/b)*b^4+
17010*arcsin(d*x+c)*cos(-(a+b*arcsin(d*x+c))/b+a/b)*b^4-11340*sin(-(a+b*arcsin(d*x+c))/b+a/b)*a^2*b^2-4536*cos
(-(a+b*arcsin(d*x+c))/b+a/b)*a^3*b+17010*cos(-(a+b*arcsin(d*x+c))/b+a/b)*a*b^3+504*cos(-3*(a+b*arcsin(d*x+c))/
b+3*a/b)*a^3*b-210*cos(-3*(a+b*arcsin(d*x+c))/b+3*a/b)*a*b^3-432*arcsin(d*x+c)^4*sin(-3*(a+b*arcsin(d*x+c))/b+
3*a/b)*b^4+504*arcsin(d*x+c)^3*cos(-3*(a+b*arcsin(d*x+c))/b+3*a/b)*b^4+420*arcsin(d*x+c)^2*sin(-3*(a+b*arcsin(
d*x+c))/b+3*a/b)*b^4-210*arcsin(d*x+c)*cos(-3*(a+b*arcsin(d*x+c))/b+3*a/b)*b^4+420*sin(-3*(a+b*arcsin(d*x+c))/
b+3*a/b)*a^2*b^2-432*sin(-3*(a+b*arcsin(d*x+c))/b+3*a/b)*a^4+35*Pi^(1/2)*(a+b*arcsin(d*x+c))^(1/2)*cos(3*a/b)*
FresnelC(3*2^(1/2)/Pi^(1/2)/(-3/b)^(1/2)*(a+b*arcsin(d*x+c))^(1/2)/b)*(-3/b)^(1/2)*2^(1/2)*b^4-35*Pi^(1/2)*(a+
b*arcsin(d*x+c))^(1/2)*sin(3*a/b)*FresnelS(3*2^(1/2)/Pi^(1/2)/(-3/b)^(1/2)*(a+b*arcsin(d*x+c))^(1/2)/b)*(-3/b)
^(1/2)*2^(1/2)*b^4-8505*Pi^(1/2)*(a+b*arcsin(d*x+c))^(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)*(-1/b)^(1/2)*2^(1/2)*b^4)/(a+b*arcsin(d*x+c))^(1/2)

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Maxima [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Fricas [F(-2)]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: TypeError} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((d*e*x+c*e)^2*(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)

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Sympy [F(-2)]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: SystemError} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Exception raised: SystemError >> excessive stack use: stack is 4845 deep

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Giac [C] Result contains complex when optimal does not.
time = 2.67, size = 8028, normalized size = 15.50 \begin {gather*} \text {Too large to display} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/3456*(1296*sqrt(2)*sqrt(pi)*a^4*b^2*e^2*erf(-1/2*I*sqrt(2)*sqrt(b*arcsin(d*x + c) + a)/sqrt(abs(b)) - 1/2*sq
rt(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))) + 1296*sqr
t(2)*sqrt(pi)*a^4*b^2*e^2*erf(1/2*I*sqrt(2)*sqrt(b*arcsin(d*x + c) + a)/sqrt(abs(b)) - 1/2*sqrt(2)*sqrt(b*arcs
in(d*x + c) + a)*sqrt(abs(b))/b)*e^(-I*a/b)/(-I*b^3/sqrt(abs(b)) + b^2*sqrt(abs(b))) - 864*sqrt(2)*sqrt(pi)*a^
4*b*e^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))) + 2808*I*sqrt(2)*sqrt(pi)*a^3*b^2*e^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))) - 864*sqrt(2)*sqrt(pi)*a^4*b*e^2*erf(1/2*I*sqrt(2)*sqrt(b*arcs
in(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/sqr
t(abs(b)) + b*sqrt(abs(b))) - 2808*I*sqrt(2)*sqrt(pi)*a^3*b^2*e^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*s
qrt(abs(b))) + 432*I*sqrt(b*arcsin(d*x + c) + a)*a*b^2*e^2*arcsin(d*x + c)^2*e^(3*I*arcsin(d*x + c)) - 1296*I*
sqrt(b*arcsin(d*x + c) + a)*a*b^2*e^2*arcsin(d*x + c)^2*e^(I*arcsin(d*x + c)) + 1296*I*sqrt(b*arcsin(d*x + c)
+ a)*a*b^2*e^2*arcsin(d*x + c)^2*e^(-I*arcsin(d*x + c)) - 432*I*sqrt(b*arcsin(d*x + c) + a)*a*b^2*e^2*arcsin(d
*x + c)^2*e^(-3*I*arcsin(d*x + c)) - 864*sqrt(pi)*a^4*sqrt(b)*e^2*erf(-1/2*sqrt(6)*sqrt(b*arcsin(d*x + c) + a)
/sqrt(b) - 1/2*I*sqrt(6)*sqrt(b*arcsin(d*x + c) + a)*sqrt(b)/abs(b))*e^(3*I*a/b)/(sqrt(6)*b + I*sqrt(6)*b^2/ab
s(b)) - 1872*I*sqrt(pi)*a^3*b^(3/2)*e^2*erf(-1/2*sqrt(6)*sqrt(b*arcsin(d*x + c) + a)/sqrt(b) - 1/2*I*sqrt(6)*s
qrt(b*arcsin(d*x + c) + a)*sqrt(b)/abs(b))*e^(3*I*a/b)/(sqrt(6)*b + I*sqrt(6)*b^2/abs(b)) - 2592*I*sqrt(2)*sqr
t(pi)*a^3*b*e^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))) - 972*sqrt(2)*sqrt(pi)*a^2*b^2*e^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))) - 2430*I*sqrt(2)*sqrt(pi)*a*b^3*e^2*erf(-1/2*I*sqrt(2)*sqrt(b*arc
sin(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(a
bs(b)) + sqrt(abs(b))) + 2592*I*sqrt(2)*sqrt(pi)*a^3*b*e^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))
) - 972*sqrt(2)*sqrt(pi)*a^2*b^2*e^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))) + 2430*I*sqrt(2)*sqr
t(pi)*a*b^3*e^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))) - 864*sqrt(pi)*a^4*sqrt(b)*e^2*erf(-1/2*s
qrt(6)*sqrt(b*arcsin(d*x + c) + a)/sqrt(b) + 1/2*I*sqrt(6)*sqrt(b*arcsin(d*x + c) + a)*sqrt(b)/abs(b))*e^(-3*I
*a/b)/(sqrt(6)*b - I*sqrt(6)*b^2/abs(b)) + 1872*I*sqrt(pi)*a^3*b^(3/2)*e^2*erf(-1/2*sqrt(6)*sqrt(b*arcsin(d*x
+ c) + a)/sqrt(b) + 1/2*I*sqrt(6)*sqrt(b*arcsin(d*x + c) + a)*sqrt(b)/abs(b))*e^(-3*I*a/b)/(sqrt(6)*b - I*sqrt
(6)*b^2/abs(b)) + 432*I*sqrt(b*arcsin(d*x + c) + a)*a^2*b*e^2*arcsin(d*x + c)*e^(3*I*arcsin(d*x + c)) - 360*sq
rt(b*arcsin(d*x + c) + a)*a*b^2*e^2*arcsin(d*x + c)*e^(3*I*arcsin(d*x + c)) - 1296*I*sqrt(b*arcsin(d*x + c) +
a)*a^2*b*e^2*arcsin(d*x + c)*e^(I*arcsin(d*x + c)) + 3240*sqrt(b*arcsin(d*x + c) + a)*a*b^2*e^2*arcsin(d*x + c
)*e^(I*arcsin(d*x + c)) + 1296*I*sqrt(b*arcsin(d*x + c) + a)*a^2*b*e^2*arcsin(d*x + c)*e^(-I*arcsin(d*x + c))
+ 3240*sqrt(b*arcsin(d*x + c) + a)*a*b^2*e^2*arcsin(d*x + c)*e^(-I*arcsin(d*x + c)) - 432*I*sqrt(b*arcsin(d*x
+ c) + a)*a^2*b*e^2*arcsin(d*x + c)*e^(-3*I*arcsin(d*x + c)) - 360*sqrt(b*arcsin(d*x + c) + a)*a*b^2*e^2*arcsi
n(d*x + c)*e^(-3*I*arcsin(d*x + c)) + 864*sqrt(pi)*a^4*e^2*erf(-1/2*sqrt(6)*sqrt(b*arcsin(d*x + c) + a)/sqrt(b
) - 1/2*I*sqrt(6)*sqrt(b*arcsin(d*x + c) + a)*sqrt(b)/abs(b))*e^(3*I*a/b)/(sqrt(6)*sqrt(b) + I*sqrt(6)*b^(3/2)
/abs(b)) + 1728*I*sqrt(pi)*a^3*b*e^2*erf(-1/2*sqrt(6)*sqrt(b*arcsin(d*x + c) + a)/sqrt(b) - 1/2*I*sqrt(6)*sqrt
(b*arcsin(d*x + c) + a)*sqrt(b)/abs(b))*e^(3*I*a/b)/(sqrt(6)*sqrt(b) + I*sqrt(6)*b^(3/2)/abs(b)) + 648*sqrt(pi
)*a^2*b^2*e^2*erf(-1/2*sqrt(6)*sqrt(b*arcsin(d*x + c) + a)/sqrt(b) - 1/2*I*sqrt(6)*sqrt(b*arcsin(d*x + c) + a)
*sqrt(b)/abs(b))*e^(3*I*a/b)/(sqrt(6)*sqrt(b) + I*sqrt(6)*b^(3/2)/abs(b)) - 864*sqrt(pi)*a^4*e^2*erf(-1/2*I*sq
rt(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(a...

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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