\(\int (c e+d e x) (a+b \arcsin (c+d x))^{7/2} \, dx\) [256]

Optimal result

Integrand size = 23, antiderivative size = 301 \[ \int (c e+d e x) (a+b \arcsin (c+d x))^{7/2} \, dx=-\frac {105 b^3 e (c+d x) \sqrt {1-(c+d x)^2} \sqrt {a+b \arcsin (c+d x)}}{128 d}+\frac {35 b^2 e (a+b \arcsin (c+d x))^{3/2}}{64 d}-\frac {35 b^2 e (c+d x)^2 (a+b \arcsin (c+d x))^{3/2}}{32 d}+\frac {7 b e (c+d x) \sqrt {1-(c+d x)^2} (a+b \arcsin (c+d x))^{5/2}}{8 d}-\frac {e (a+b \arcsin (c+d x))^{7/2}}{4 d}+\frac {e (c+d x)^2 (a+b \arcsin (c+d x))^{7/2}}{2 d}+\frac {105 b^{7/2} e \sqrt {\pi } \cos \left (\frac {2 a}{b}\right ) \operatorname {FresnelS}\left (\frac {2 \sqrt {a+b \arcsin (c+d x)}}{\sqrt {b} \sqrt {\pi }}\right )}{512 d}-\frac {105 b^{7/2} e \sqrt {\pi } \operatorname {FresnelC}\left (\frac {2 \sqrt {a+b \arcsin (c+d x)}}{\sqrt {b} \sqrt {\pi }}\right ) \sin \left (\frac {2 a}{b}\right )}{512 d} \]

[Out]

35/64*b^2*e*(a+b*arcsin(d*x+c))^(3/2)/d-35/32*b^2*e*(d*x+c)^2*(a+b*arcsin(d*x+c))^(3/2)/d-1/4*e*(a+b*arcsin(d*
x+c))^(7/2)/d+1/2*e*(d*x+c)^2*(a+b*arcsin(d*x+c))^(7/2)/d+105/512*b^(7/2)*e*cos(2*a/b)*FresnelS(2*(a+b*arcsin(
d*x+c))^(1/2)/b^(1/2)/Pi^(1/2))*Pi^(1/2)/d-105/512*b^(7/2)*e*FresnelC(2*(a+b*arcsin(d*x+c))^(1/2)/b^(1/2)/Pi^(
1/2))*sin(2*a/b)*Pi^(1/2)/d+7/8*b*e*(d*x+c)*(a+b*arcsin(d*x+c))^(5/2)*(1-(d*x+c)^2)^(1/2)/d-105/128*b^3*e*(d*x
+c)*(1-(d*x+c)^2)^(1/2)*(a+b*arcsin(d*x+c))^(1/2)/d

Rubi [A] (verified)

Time = 0.52 (sec) , antiderivative size = 301, normalized size of antiderivative = 1.00, number of steps used = 16, number of rules used = 12, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.522, Rules used = {4889, 12, 4725, 4795, 4737, 4731, 4491, 3387, 3386, 3432, 3385, 3433} \[ \int (c e+d e x) (a+b \arcsin (c+d x))^{7/2} \, dx=-\frac {105 \sqrt {\pi } b^{7/2} e \sin \left (\frac {2 a}{b}\right ) \operatorname {FresnelC}\left (\frac {2 \sqrt {a+b \arcsin (c+d x)}}{\sqrt {b} \sqrt {\pi }}\right )}{512 d}+\frac {105 \sqrt {\pi } b^{7/2} e \cos \left (\frac {2 a}{b}\right ) \operatorname {FresnelS}\left (\frac {2 \sqrt {a+b \arcsin (c+d x)}}{\sqrt {b} \sqrt {\pi }}\right )}{512 d}-\frac {105 b^3 e (c+d x) \sqrt {1-(c+d x)^2} \sqrt {a+b \arcsin (c+d x)}}{128 d}-\frac {35 b^2 e (c+d x)^2 (a+b \arcsin (c+d x))^{3/2}}{32 d}+\frac {35 b^2 e (a+b \arcsin (c+d x))^{3/2}}{64 d}+\frac {7 b e (c+d x) \sqrt {1-(c+d x)^2} (a+b \arcsin (c+d x))^{5/2}}{8 d}+\frac {e (c+d x)^2 (a+b \arcsin (c+d x))^{7/2}}{2 d}-\frac {e (a+b \arcsin (c+d x))^{7/2}}{4 d} \]

[In]

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

[Out]

(-105*b^3*e*(c + d*x)*Sqrt[1 - (c + d*x)^2]*Sqrt[a + b*ArcSin[c + d*x]])/(128*d) + (35*b^2*e*(a + b*ArcSin[c +
d*x])^(3/2))/(64*d) - (35*b^2*e*(c + d*x)^2*(a + b*ArcSin[c + d*x])^(3/2))/(32*d) + (7*b*e*(c + d*x)*Sqrt[1 -
(c + d*x)^2]*(a + b*ArcSin[c + d*x])^(5/2))/(8*d) - (e*(a + b*ArcSin[c + d*x])^(7/2))/(4*d) + (e*(c + d*x)^2*
(a + b*ArcSin[c + d*x])^(7/2))/(2*d) + (105*b^(7/2)*e*Sqrt[Pi]*Cos[(2*a)/b]*FresnelS[(2*Sqrt[a + b*ArcSin[c +
d*x]])/(Sqrt[b]*Sqrt[Pi])])/(512*d) - (105*b^(7/2)*e*Sqrt[Pi]*FresnelC[(2*Sqrt[a + b*ArcSin[c + d*x]])/(Sqrt[b
]*Sqrt[Pi])]*Sin[(2*a)/b])/(512*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 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 4737

Int[((a_.) + ArcSin[(c_.)*(x_)]*(b_.))^(n_.)/Sqrt[(d_) + (e_.)*(x_)^2], x_Symbol] :> Simp[(1/(b*c*(n + 1)))*Si
mp[Sqrt[1 - c^2*x^2]/Sqrt[d + e*x^2]]*(a + b*ArcSin[c*x])^(n + 1), x] /; FreeQ[{a, b, c, d, e, n}, x] && EqQ[c
^2*d + e, 0] && NeQ[n, -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*} \text {integral}& = \frac {\text {Subst}\left (\int e x (a+b \arcsin (x))^{7/2} \, dx,x,c+d x\right )}{d} \\ & = \frac {e \text {Subst}\left (\int x (a+b \arcsin (x))^{7/2} \, dx,x,c+d x\right )}{d} \\ & = \frac {e (c+d x)^2 (a+b \arcsin (c+d x))^{7/2}}{2 d}-\frac {(7 b e) \text {Subst}\left (\int \frac {x^2 (a+b \arcsin (x))^{5/2}}{\sqrt {1-x^2}} \, dx,x,c+d x\right )}{4 d} \\ & = \frac {7 b e (c+d x) \sqrt {1-(c+d x)^2} (a+b \arcsin (c+d x))^{5/2}}{8 d}+\frac {e (c+d x)^2 (a+b \arcsin (c+d x))^{7/2}}{2 d}-\frac {(7 b e) \text {Subst}\left (\int \frac {(a+b \arcsin (x))^{5/2}}{\sqrt {1-x^2}} \, dx,x,c+d x\right )}{8 d}-\frac {\left (35 b^2 e\right ) \text {Subst}\left (\int x (a+b \arcsin (x))^{3/2} \, dx,x,c+d x\right )}{16 d} \\ & = -\frac {35 b^2 e (c+d x)^2 (a+b \arcsin (c+d x))^{3/2}}{32 d}+\frac {7 b e (c+d x) \sqrt {1-(c+d x)^2} (a+b \arcsin (c+d x))^{5/2}}{8 d}-\frac {e (a+b \arcsin (c+d x))^{7/2}}{4 d}+\frac {e (c+d x)^2 (a+b \arcsin (c+d x))^{7/2}}{2 d}+\frac {\left (105 b^3 e\right ) \text {Subst}\left (\int \frac {x^2 \sqrt {a+b \arcsin (x)}}{\sqrt {1-x^2}} \, dx,x,c+d x\right )}{64 d} \\ & = -\frac {105 b^3 e (c+d x) \sqrt {1-(c+d x)^2} \sqrt {a+b \arcsin (c+d x)}}{128 d}-\frac {35 b^2 e (c+d x)^2 (a+b \arcsin (c+d x))^{3/2}}{32 d}+\frac {7 b e (c+d x) \sqrt {1-(c+d x)^2} (a+b \arcsin (c+d x))^{5/2}}{8 d}-\frac {e (a+b \arcsin (c+d x))^{7/2}}{4 d}+\frac {e (c+d x)^2 (a+b \arcsin (c+d x))^{7/2}}{2 d}+\frac {\left (105 b^3 e\right ) \text {Subst}\left (\int \frac {\sqrt {a+b \arcsin (x)}}{\sqrt {1-x^2}} \, dx,x,c+d x\right )}{128 d}+\frac {\left (105 b^4 e\right ) \text {Subst}\left (\int \frac {x}{\sqrt {a+b \arcsin (x)}} \, dx,x,c+d x\right )}{256 d} \\ & = -\frac {105 b^3 e (c+d x) \sqrt {1-(c+d x)^2} \sqrt {a+b \arcsin (c+d x)}}{128 d}+\frac {35 b^2 e (a+b \arcsin (c+d x))^{3/2}}{64 d}-\frac {35 b^2 e (c+d x)^2 (a+b \arcsin (c+d x))^{3/2}}{32 d}+\frac {7 b e (c+d x) \sqrt {1-(c+d x)^2} (a+b \arcsin (c+d x))^{5/2}}{8 d}-\frac {e (a+b \arcsin (c+d x))^{7/2}}{4 d}+\frac {e (c+d x)^2 (a+b \arcsin (c+d x))^{7/2}}{2 d}-\frac {\left (105 b^3 e\right ) \text {Subst}\left (\int \frac {\cos \left (\frac {a}{b}-\frac {x}{b}\right ) \sin \left (\frac {a}{b}-\frac {x}{b}\right )}{\sqrt {x}} \, dx,x,a+b \arcsin (c+d x)\right )}{256 d} \\ & = -\frac {105 b^3 e (c+d x) \sqrt {1-(c+d x)^2} \sqrt {a+b \arcsin (c+d x)}}{128 d}+\frac {35 b^2 e (a+b \arcsin (c+d x))^{3/2}}{64 d}-\frac {35 b^2 e (c+d x)^2 (a+b \arcsin (c+d x))^{3/2}}{32 d}+\frac {7 b e (c+d x) \sqrt {1-(c+d x)^2} (a+b \arcsin (c+d x))^{5/2}}{8 d}-\frac {e (a+b \arcsin (c+d x))^{7/2}}{4 d}+\frac {e (c+d x)^2 (a+b \arcsin (c+d x))^{7/2}}{2 d}-\frac {\left (105 b^3 e\right ) \text {Subst}\left (\int \frac {\sin \left (\frac {2 a}{b}-\frac {2 x}{b}\right )}{2 \sqrt {x}} \, dx,x,a+b \arcsin (c+d x)\right )}{256 d} \\ & = -\frac {105 b^3 e (c+d x) \sqrt {1-(c+d x)^2} \sqrt {a+b \arcsin (c+d x)}}{128 d}+\frac {35 b^2 e (a+b \arcsin (c+d x))^{3/2}}{64 d}-\frac {35 b^2 e (c+d x)^2 (a+b \arcsin (c+d x))^{3/2}}{32 d}+\frac {7 b e (c+d x) \sqrt {1-(c+d x)^2} (a+b \arcsin (c+d x))^{5/2}}{8 d}-\frac {e (a+b \arcsin (c+d x))^{7/2}}{4 d}+\frac {e (c+d x)^2 (a+b \arcsin (c+d x))^{7/2}}{2 d}-\frac {\left (105 b^3 e\right ) \text {Subst}\left (\int \frac {\sin \left (\frac {2 a}{b}-\frac {2 x}{b}\right )}{\sqrt {x}} \, dx,x,a+b \arcsin (c+d x)\right )}{512 d} \\ & = -\frac {105 b^3 e (c+d x) \sqrt {1-(c+d x)^2} \sqrt {a+b \arcsin (c+d x)}}{128 d}+\frac {35 b^2 e (a+b \arcsin (c+d x))^{3/2}}{64 d}-\frac {35 b^2 e (c+d x)^2 (a+b \arcsin (c+d x))^{3/2}}{32 d}+\frac {7 b e (c+d x) \sqrt {1-(c+d x)^2} (a+b \arcsin (c+d x))^{5/2}}{8 d}-\frac {e (a+b \arcsin (c+d x))^{7/2}}{4 d}+\frac {e (c+d x)^2 (a+b \arcsin (c+d x))^{7/2}}{2 d}+\frac {\left (105 b^3 e \cos \left (\frac {2 a}{b}\right )\right ) \text {Subst}\left (\int \frac {\sin \left (\frac {2 x}{b}\right )}{\sqrt {x}} \, dx,x,a+b \arcsin (c+d x)\right )}{512 d}-\frac {\left (105 b^3 e \sin \left (\frac {2 a}{b}\right )\right ) \text {Subst}\left (\int \frac {\cos \left (\frac {2 x}{b}\right )}{\sqrt {x}} \, dx,x,a+b \arcsin (c+d x)\right )}{512 d} \\ & = -\frac {105 b^3 e (c+d x) \sqrt {1-(c+d x)^2} \sqrt {a+b \arcsin (c+d x)}}{128 d}+\frac {35 b^2 e (a+b \arcsin (c+d x))^{3/2}}{64 d}-\frac {35 b^2 e (c+d x)^2 (a+b \arcsin (c+d x))^{3/2}}{32 d}+\frac {7 b e (c+d x) \sqrt {1-(c+d x)^2} (a+b \arcsin (c+d x))^{5/2}}{8 d}-\frac {e (a+b \arcsin (c+d x))^{7/2}}{4 d}+\frac {e (c+d x)^2 (a+b \arcsin (c+d x))^{7/2}}{2 d}+\frac {\left (105 b^3 e \cos \left (\frac {2 a}{b}\right )\right ) \text {Subst}\left (\int \sin \left (\frac {2 x^2}{b}\right ) \, dx,x,\sqrt {a+b \arcsin (c+d x)}\right )}{256 d}-\frac {\left (105 b^3 e \sin \left (\frac {2 a}{b}\right )\right ) \text {Subst}\left (\int \cos \left (\frac {2 x^2}{b}\right ) \, dx,x,\sqrt {a+b \arcsin (c+d x)}\right )}{256 d} \\ & = -\frac {105 b^3 e (c+d x) \sqrt {1-(c+d x)^2} \sqrt {a+b \arcsin (c+d x)}}{128 d}+\frac {35 b^2 e (a+b \arcsin (c+d x))^{3/2}}{64 d}-\frac {35 b^2 e (c+d x)^2 (a+b \arcsin (c+d x))^{3/2}}{32 d}+\frac {7 b e (c+d x) \sqrt {1-(c+d x)^2} (a+b \arcsin (c+d x))^{5/2}}{8 d}-\frac {e (a+b \arcsin (c+d x))^{7/2}}{4 d}+\frac {e (c+d x)^2 (a+b \arcsin (c+d x))^{7/2}}{2 d}+\frac {105 b^{7/2} e \sqrt {\pi } \cos \left (\frac {2 a}{b}\right ) \operatorname {FresnelS}\left (\frac {2 \sqrt {a+b \arcsin (c+d x)}}{\sqrt {b} \sqrt {\pi }}\right )}{512 d}-\frac {105 b^{7/2} e \sqrt {\pi } \operatorname {FresnelC}\left (\frac {2 \sqrt {a+b \arcsin (c+d x)}}{\sqrt {b} \sqrt {\pi }}\right ) \sin \left (\frac {2 a}{b}\right )}{512 d} \\ \end{align*}

Mathematica [C] (verified)

Result contains complex when optimal does not.

Time = 0.10 (sec) , antiderivative size = 137, normalized size of antiderivative = 0.46 \[ \int (c e+d e x) (a+b \arcsin (c+d x))^{7/2} \, dx=-\frac {b^4 e e^{-\frac {2 i a}{b}} \left (\sqrt {-\frac {i (a+b \arcsin (c+d x))}{b}} \Gamma \left (\frac {9}{2},-\frac {2 i (a+b \arcsin (c+d x))}{b}\right )+e^{\frac {4 i a}{b}} \sqrt {\frac {i (a+b \arcsin (c+d x))}{b}} \Gamma \left (\frac {9}{2},\frac {2 i (a+b \arcsin (c+d x))}{b}\right )\right )}{64 \sqrt {2} d \sqrt {a+b \arcsin (c+d x)}} \]

[In]

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

[Out]

-1/64*(b^4*e*(Sqrt[((-I)*(a + b*ArcSin[c + d*x]))/b]*Gamma[9/2, ((-2*I)*(a + b*ArcSin[c + d*x]))/b] + E^(((4*I
)*a)/b)*Sqrt[(I*(a + b*ArcSin[c + d*x]))/b]*Gamma[9/2, ((2*I)*(a + b*ArcSin[c + d*x]))/b]))/(Sqrt[2]*d*E^(((2*
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. \(653\) vs. \(2(249)=498\).

Time = 0.92 (sec) , antiderivative size = 654, normalized size of antiderivative = 2.17

method	result	size
default	\(\frac {e b \left (128 \sqrt {a +b \arcsin \left (d x +c \right )}\, \sqrt {\pi }\, \sqrt {-\frac {1}{b}}\, \arcsin \left (d x +c \right )^{3} \cos \left (-\frac {2 \left (a +b \arcsin \left (d x +c \right )\right )}{b}+\frac {2 a}{b}\right ) b^{3}+384 \sqrt {a +b \arcsin \left (d x +c \right )}\, \sqrt {\pi }\, \sqrt {-\frac {1}{b}}\, \arcsin \left (d x +c \right )^{2} \cos \left (-\frac {2 \left (a +b \arcsin \left (d x +c \right )\right )}{b}+\frac {2 a}{b}\right ) a \,b^{2}+224 \sqrt {a +b \arcsin \left (d x +c \right )}\, \sqrt {\pi }\, \sqrt {-\frac {1}{b}}\, \arcsin \left (d x +c \right )^{2} \sin \left (-\frac {2 \left (a +b \arcsin \left (d x +c \right )\right )}{b}+\frac {2 a}{b}\right ) b^{3}+384 \sqrt {a +b \arcsin \left (d x +c \right )}\, \sqrt {\pi }\, \sqrt {-\frac {1}{b}}\, \arcsin \left (d x +c \right ) \cos \left (-\frac {2 \left (a +b \arcsin \left (d x +c \right )\right )}{b}+\frac {2 a}{b}\right ) a^{2} b -280 \sqrt {a +b \arcsin \left (d x +c \right )}\, \sqrt {\pi }\, \sqrt {-\frac {1}{b}}\, \arcsin \left (d x +c \right ) \cos \left (-\frac {2 \left (a +b \arcsin \left (d x +c \right )\right )}{b}+\frac {2 a}{b}\right ) b^{3}+448 \sqrt {a +b \arcsin \left (d x +c \right )}\, \sqrt {\pi }\, \sqrt {-\frac {1}{b}}\, \arcsin \left (d x +c \right ) \sin \left (-\frac {2 \left (a +b \arcsin \left (d x +c \right )\right )}{b}+\frac {2 a}{b}\right ) a \,b^{2}+128 \sqrt {a +b \arcsin \left (d x +c \right )}\, \sqrt {\pi }\, \sqrt {-\frac {1}{b}}\, \cos \left (-\frac {2 \left (a +b \arcsin \left (d x +c \right )\right )}{b}+\frac {2 a}{b}\right ) a^{3}-280 \sqrt {a +b \arcsin \left (d x +c \right )}\, \sqrt {\pi }\, \sqrt {-\frac {1}{b}}\, \cos \left (-\frac {2 \left (a +b \arcsin \left (d x +c \right )\right )}{b}+\frac {2 a}{b}\right ) a \,b^{2}+224 \sqrt {a +b \arcsin \left (d x +c \right )}\, \sqrt {\pi }\, \sqrt {-\frac {1}{b}}\, \sin \left (-\frac {2 \left (a +b \arcsin \left (d x +c \right )\right )}{b}+\frac {2 a}{b}\right ) a^{2} b -210 \sqrt {a +b \arcsin \left (d x +c \right )}\, \sqrt {\pi }\, \sqrt {-\frac {1}{b}}\, \sin \left (-\frac {2 \left (a +b \arcsin \left (d x +c \right )\right )}{b}+\frac {2 a}{b}\right ) b^{3}-105 \pi \,b^{3} \cos \left (\frac {2 a}{b}\right ) \operatorname {FresnelS}\left (\frac {2 \sqrt {2}\, \sqrt {a +b \arcsin \left (d x +c \right )}}{\sqrt {\pi }\, \sqrt {-\frac {2}{b}}\, b}\right )-105 \pi \,b^{3} \sin \left (\frac {2 a}{b}\right ) \operatorname {FresnelC}\left (\frac {2 \sqrt {2}\, \sqrt {a +b \arcsin \left (d x +c \right )}}{\sqrt {\pi }\, \sqrt {-\frac {2}{b}}\, b}\right )\right ) \sqrt {-\frac {1}{b}}}{512 d \sqrt {\pi }}\)	\(654\)

[In]

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

[Out]

1/512*e/d*b*(128*(a+b*arcsin(d*x+c))^(1/2)*Pi^(1/2)*(-1/b)^(1/2)*arcsin(d*x+c)^3*cos(-2*(a+b*arcsin(d*x+c))/b+
2*a/b)*b^3+384*(a+b*arcsin(d*x+c))^(1/2)*Pi^(1/2)*(-1/b)^(1/2)*arcsin(d*x+c)^2*cos(-2*(a+b*arcsin(d*x+c))/b+2*
a/b)*a*b^2+224*(a+b*arcsin(d*x+c))^(1/2)*Pi^(1/2)*(-1/b)^(1/2)*arcsin(d*x+c)^2*sin(-2*(a+b*arcsin(d*x+c))/b+2*
a/b)*b^3+384*(a+b*arcsin(d*x+c))^(1/2)*Pi^(1/2)*(-1/b)^(1/2)*arcsin(d*x+c)*cos(-2*(a+b*arcsin(d*x+c))/b+2*a/b)
*a^2*b-280*(a+b*arcsin(d*x+c))^(1/2)*Pi^(1/2)*(-1/b)^(1/2)*arcsin(d*x+c)*cos(-2*(a+b*arcsin(d*x+c))/b+2*a/b)*b
^3+448*(a+b*arcsin(d*x+c))^(1/2)*Pi^(1/2)*(-1/b)^(1/2)*arcsin(d*x+c)*sin(-2*(a+b*arcsin(d*x+c))/b+2*a/b)*a*b^2
+128*(a+b*arcsin(d*x+c))^(1/2)*Pi^(1/2)*(-1/b)^(1/2)*cos(-2*(a+b*arcsin(d*x+c))/b+2*a/b)*a^3-280*(a+b*arcsin(d
*x+c))^(1/2)*Pi^(1/2)*(-1/b)^(1/2)*cos(-2*(a+b*arcsin(d*x+c))/b+2*a/b)*a*b^2+224*(a+b*arcsin(d*x+c))^(1/2)*Pi^
(1/2)*(-1/b)^(1/2)*sin(-2*(a+b*arcsin(d*x+c))/b+2*a/b)*a^2*b-210*(a+b*arcsin(d*x+c))^(1/2)*Pi^(1/2)*(-1/b)^(1/
2)*sin(-2*(a+b*arcsin(d*x+c))/b+2*a/b)*b^3-105*Pi*b^3*cos(2*a/b)*FresnelS(2*2^(1/2)/Pi^(1/2)/(-2/b)^(1/2)*(a+b
*arcsin(d*x+c))^(1/2)/b)-105*Pi*b^3*sin(2*a/b)*FresnelC(2*2^(1/2)/Pi^(1/2)/(-2/b)^(1/2)*(a+b*arcsin(d*x+c))^(1
/2)/b))*(-1/b)^(1/2)/Pi^(1/2)

Fricas [F(-2)]

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

[In]

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

[In]

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

[Out]

Timed out

Maxima [F]

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

[In]

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

[Out]

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

Giac [C] (verification not implemented)

Result contains complex when optimal does not.

Time = 1.34 (sec) , antiderivative size = 2561, normalized size of antiderivative = 8.51 \[ \int (c e+d e x) (a+b \arcsin (c+d x))^{7/2} \, dx=\text {Too large to display} \]

[In]

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

[Out]

-1/1024*(128*sqrt(b*arcsin(d*x + c) + a)*b^3*e*arcsin(d*x + c)^3*e^(2*I*arcsin(d*x + c)) + 128*sqrt(b*arcsin(d
*x + c) + a)*b^3*e*arcsin(d*x + c)^3*e^(-2*I*arcsin(d*x + c)) + 384*sqrt(b*arcsin(d*x + c) + a)*a*b^2*e*arcsin
(d*x + c)^2*e^(2*I*arcsin(d*x + c)) + 224*I*sqrt(b*arcsin(d*x + c) + a)*b^3*e*arcsin(d*x + c)^2*e^(2*I*arcsin(
d*x + c)) + 384*sqrt(b*arcsin(d*x + c) + a)*a*b^2*e*arcsin(d*x + c)^2*e^(-2*I*arcsin(d*x + c)) - 224*I*sqrt(b*
arcsin(d*x + c) + a)*b^3*e*arcsin(d*x + c)^2*e^(-2*I*arcsin(d*x + c)) + 768*I*sqrt(pi)*a^4*b*e*erf(-sqrt(b*arc
sin(d*x + c) + a)/sqrt(b) - I*sqrt(b*arcsin(d*x + c) + a)*sqrt(b)/abs(b))*e^(2*I*a/b)/(b^(3/2) + I*b^(5/2)/abs
(b)) + 192*sqrt(pi)*a^3*b^2*e*erf(-sqrt(b*arcsin(d*x + c) + a)/sqrt(b) - I*sqrt(b*arcsin(d*x + c) + a)*sqrt(b)
/abs(b))*e^(2*I*a/b)/(b^(3/2) + I*b^(5/2)/abs(b)) - 768*I*sqrt(pi)*a^4*b*e*erf(-sqrt(b*arcsin(d*x + c) + a)/sq
rt(b) + I*sqrt(b*arcsin(d*x + c) + a)*sqrt(b)/abs(b))*e^(-2*I*a/b)/(b^(3/2) - I*b^(5/2)/abs(b)) + 192*sqrt(pi)
*a^3*b^2*e*erf(-sqrt(b*arcsin(d*x + c) + a)/sqrt(b) + I*sqrt(b*arcsin(d*x + c) + a)*sqrt(b)/abs(b))*e^(-2*I*a/
b)/(b^(3/2) - I*b^(5/2)/abs(b)) - 256*I*sqrt(pi)*a^4*sqrt(b)*e*erf(-sqrt(b*arcsin(d*x + c) + a)/sqrt(b) - I*sq
rt(b*arcsin(d*x + c) + a)*sqrt(b)/abs(b))*e^(2*I*a/b)/(b + I*b^2/abs(b)) + 832*sqrt(pi)*a^3*b^(3/2)*e*erf(-sqr
t(b*arcsin(d*x + c) + a)/sqrt(b) - I*sqrt(b*arcsin(d*x + c) + a)*sqrt(b)/abs(b))*e^(2*I*a/b)/(b + I*b^2/abs(b)
) - 288*I*sqrt(pi)*a^2*b^(5/2)*e*erf(-sqrt(b*arcsin(d*x + c) + a)/sqrt(b) - I*sqrt(b*arcsin(d*x + c) + a)*sqrt
(b)/abs(b))*e^(2*I*a/b)/(b + I*b^2/abs(b)) + 256*I*sqrt(pi)*a^4*sqrt(b)*e*erf(-sqrt(b*arcsin(d*x + c) + a)/sqr
t(b) + I*sqrt(b*arcsin(d*x + c) + a)*sqrt(b)/abs(b))*e^(-2*I*a/b)/(b - I*b^2/abs(b)) + 832*sqrt(pi)*a^3*b^(3/2
)*e*erf(-sqrt(b*arcsin(d*x + c) + a)/sqrt(b) + I*sqrt(b*arcsin(d*x + c) + a)*sqrt(b)/abs(b))*e^(-2*I*a/b)/(b -
I*b^2/abs(b)) + 288*I*sqrt(pi)*a^2*b^(5/2)*e*erf(-sqrt(b*arcsin(d*x + c) + a)/sqrt(b) + I*sqrt(b*arcsin(d*x +
c) + a)*sqrt(b)/abs(b))*e^(-2*I*a/b)/(b - I*b^2/abs(b)) + 384*sqrt(b*arcsin(d*x + c) + a)*a^2*b*e*arcsin(d*x
+ c)*e^(2*I*arcsin(d*x + c)) + 448*I*sqrt(b*arcsin(d*x + c) + a)*a*b^2*e*arcsin(d*x + c)*e^(2*I*arcsin(d*x + c
)) - 280*sqrt(b*arcsin(d*x + c) + a)*b^3*e*arcsin(d*x + c)*e^(2*I*arcsin(d*x + c)) + 384*sqrt(b*arcsin(d*x + c
) + a)*a^2*b*e*arcsin(d*x + c)*e^(-2*I*arcsin(d*x + c)) - 448*I*sqrt(b*arcsin(d*x + c) + a)*a*b^2*e*arcsin(d*x
+ c)*e^(-2*I*arcsin(d*x + c)) - 280*sqrt(b*arcsin(d*x + c) + a)*b^3*e*arcsin(d*x + c)*e^(-2*I*arcsin(d*x + c)
) - 768*sqrt(pi)*a^3*b*e*erf(-sqrt(b*arcsin(d*x + c) + a)/sqrt(b) - I*sqrt(b*arcsin(d*x + c) + a)*sqrt(b)/abs(
b))*e^(2*I*a/b)/(sqrt(b) + I*b^(3/2)/abs(b)) + 576*I*sqrt(pi)*a^2*b^2*e*erf(-sqrt(b*arcsin(d*x + c) + a)/sqrt(
b) - I*sqrt(b*arcsin(d*x + c) + a)*sqrt(b)/abs(b))*e^(2*I*a/b)/(sqrt(b) + I*b^(3/2)/abs(b)) + 240*sqrt(pi)*a*b
^3*e*erf(-sqrt(b*arcsin(d*x + c) + a)/sqrt(b) - I*sqrt(b*arcsin(d*x + c) + a)*sqrt(b)/abs(b))*e^(2*I*a/b)/(sqr
t(b) + I*b^(3/2)/abs(b)) - 256*I*sqrt(pi)*a^4*e*erf(-sqrt(b*arcsin(d*x + c) + a)/sqrt(b) + I*sqrt(b*arcsin(d*x
+ c) + a)*sqrt(b)/abs(b))*e^(-2*I*a/b)/(sqrt(b) - I*b^(3/2)/abs(b)) - 768*sqrt(pi)*a^3*b*e*erf(-sqrt(b*arcsin
(d*x + c) + a)/sqrt(b) + I*sqrt(b*arcsin(d*x + c) + a)*sqrt(b)/abs(b))*e^(-2*I*a/b)/(sqrt(b) - I*b^(3/2)/abs(b
)) - 576*I*sqrt(pi)*a^2*b^2*e*erf(-sqrt(b*arcsin(d*x + c) + a)/sqrt(b) + I*sqrt(b*arcsin(d*x + c) + a)*sqrt(b)
/abs(b))*e^(-2*I*a/b)/(sqrt(b) - I*b^(3/2)/abs(b)) + 240*sqrt(pi)*a*b^3*e*erf(-sqrt(b*arcsin(d*x + c) + a)/sqr
t(b) + I*sqrt(b*arcsin(d*x + c) + a)*sqrt(b)/abs(b))*e^(-2*I*a/b)/(sqrt(b) - I*b^(3/2)/abs(b)) - 512*I*sqrt(pi
)*a^4*e*erf(-sqrt(b*arcsin(d*x + c) + a)/sqrt(b) - I*sqrt(b*arcsin(d*x + c) + a)*sqrt(b)/abs(b))*e^(2*I*a/b)/(
sqrt(b)*(I*b/abs(b) + 1)) - 256*sqrt(pi)*a^3*sqrt(b)*e*erf(-sqrt(b*arcsin(d*x + c) + a)/sqrt(b) - I*sqrt(b*arc
sin(d*x + c) + a)*sqrt(b)/abs(b))*e^(2*I*a/b)/(I*b/abs(b) + 1) - 288*I*sqrt(pi)*a^2*b^(3/2)*e*erf(-sqrt(b*arcs
in(d*x + c) + a)/sqrt(b) - I*sqrt(b*arcsin(d*x + c) + a)*sqrt(b)/abs(b))*e^(2*I*a/b)/(I*b/abs(b) + 1) - 240*sq
rt(pi)*a*b^(5/2)*e*erf(-sqrt(b*arcsin(d*x + c) + a)/sqrt(b) - I*sqrt(b*arcsin(d*x + c) + a)*sqrt(b)/abs(b))*e^
(2*I*a/b)/(I*b/abs(b) + 1) + 105*I*sqrt(pi)*b^(7/2)*e*erf(-sqrt(b*arcsin(d*x + c) + a)/sqrt(b) - I*sqrt(b*arcs
in(d*x + c) + a)*sqrt(b)/abs(b))*e^(2*I*a/b)/(I*b/abs(b) + 1) + 768*I*sqrt(pi)*a^4*e*erf(-sqrt(b*arcsin(d*x +
c) + a)/sqrt(b) + I*sqrt(b*arcsin(d*x + c) + a)*sqrt(b)/abs(b))*e^(-2*I*a/b)/(sqrt(b)*(-I*b/abs(b) + 1)) - 256
*sqrt(pi)*a^3*sqrt(b)*e*erf(-sqrt(b*arcsin(d*x + c) + a)/sqrt(b) + I*sqrt(b*arcsin(d*x + c) + a)*sqrt(b)/abs(b
))*e^(-2*I*a/b)/(-I*b/abs(b) + 1) + 288*I*sqrt(pi)*a^2*b^(3/2)*e*erf(-sqrt(b*arcsin(d*x + c) + a)/sqrt(b) + I*
sqrt(b*arcsin(d*x + c) + a)*sqrt(b)/abs(b))*e^(-2*I*a/b)/(-I*b/abs(b) + 1) - 240*sqrt(pi)*a*b^(5/2)*e*erf(-sqr
t(b*arcsin(d*x + c) + a)/sqrt(b) + I*sqrt(b*arcsin(d*x + c) + a)*sqrt(b)/abs(b))*e^(-2*I*a/b)/(-I*b/abs(b) + 1
) - 105*I*sqrt(pi)*b^(7/2)*e*erf(-sqrt(b*arcsin(d*x + c) + a)/sqrt(b) + I*sqrt(b*arcsin(d*x + c) + a)*sqrt(b)/
abs(b))*e^(-2*I*a/b)/(-I*b/abs(b) + 1) + 128*sqrt(b*arcsin(d*x + c) + a)*a^3*e*e^(2*I*arcsin(d*x + c)) + 224*I
*sqrt(b*arcsin(d*x + c) + a)*a^2*b*e*e^(2*I*arcsin(d*x + c)) - 280*sqrt(b*arcsin(d*x + c) + a)*a*b^2*e*e^(2*I*
arcsin(d*x + c)) - 210*I*sqrt(b*arcsin(d*x + c) + a)*b^3*e*e^(2*I*arcsin(d*x + c)) + 128*sqrt(b*arcsin(d*x + c
) + a)*a^3*e*e^(-2*I*arcsin(d*x + c)) - 224*I*sqrt(b*arcsin(d*x + c) + a)*a^2*b*e*e^(-2*I*arcsin(d*x + c)) - 2
80*sqrt(b*arcsin(d*x + c) + a)*a*b^2*e*e^(-2*I*arcsin(d*x + c)) + 210*I*sqrt(b*arcsin(d*x + c) + a)*b^3*e*e^(-
2*I*arcsin(d*x + c)))/d

Mupad [F(-1)]

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

[In]

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

[Out]

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