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

Optimal. Leaf size=301 \[ -\frac {105 \sqrt {\pi } b^{7/2} e \sin \left (\frac {2 a}{b}\right ) C\left (\frac {2 \sqrt {a+b \sin ^{-1}(c+d x)}}{\sqrt {b} \sqrt {\pi }}\right )}{512 d}+\frac {105 \sqrt {\pi } b^{7/2} e \cos \left (\frac {2 a}{b}\right ) S\left (\frac {2 \sqrt {a+b \sin ^{-1}(c+d x)}}{\sqrt {b} \sqrt {\pi }}\right )}{512 d}-\frac {105 b^3 e (c+d x) \sqrt {1-(c+d x)^2} \sqrt {a+b \sin ^{-1}(c+d x)}}{128 d}-\frac {35 b^2 e (c+d x)^2 \left (a+b \sin ^{-1}(c+d x)\right )^{3/2}}{32 d}+\frac {35 b^2 e \left (a+b \sin ^{-1}(c+d x)\right )^{3/2}}{64 d}+\frac {7 b e (c+d x) \sqrt {1-(c+d x)^2} \left (a+b \sin ^{-1}(c+d x)\right )^{5/2}}{8 d}+\frac {e (c+d x)^2 \left (a+b \sin ^{-1}(c+d x)\right )^{7/2}}{2 d}-\frac {e \left (a+b \sin ^{-1}(c+d x)\right )^{7/2}}{4 d} \]

[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

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Rubi [A]  time = 0.76, antiderivative size = 301, normalized size of antiderivative = 1.00, number of steps used = 16, number of rules used = 12, integrand size = 23, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.522, Rules used = {4805, 12, 4629, 4707, 4641, 4635, 4406, 3306, 3305, 3351, 3304, 3352} \[ -\frac {105 \sqrt {\pi } b^{7/2} e \sin \left (\frac {2 a}{b}\right ) \text {FresnelC}\left (\frac {2 \sqrt {a+b \sin ^{-1}(c+d x)}}{\sqrt {\pi } \sqrt {b}}\right )}{512 d}+\frac {105 \sqrt {\pi } b^{7/2} e \cos \left (\frac {2 a}{b}\right ) S\left (\frac {2 \sqrt {a+b \sin ^{-1}(c+d x)}}{\sqrt {b} \sqrt {\pi }}\right )}{512 d}-\frac {105 b^3 e (c+d x) \sqrt {1-(c+d x)^2} \sqrt {a+b \sin ^{-1}(c+d x)}}{128 d}-\frac {35 b^2 e (c+d x)^2 \left (a+b \sin ^{-1}(c+d x)\right )^{3/2}}{32 d}+\frac {35 b^2 e \left (a+b \sin ^{-1}(c+d x)\right )^{3/2}}{64 d}+\frac {7 b e (c+d x) \sqrt {1-(c+d x)^2} \left (a+b \sin ^{-1}(c+d x)\right )^{5/2}}{8 d}+\frac {e (c+d x)^2 \left (a+b \sin ^{-1}(c+d x)\right )^{7/2}}{2 d}-\frac {e \left (a+b \sin ^{-1}(c+d x)\right )^{7/2}}{4 d} \]

Antiderivative was successfully verified.

[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 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 4406

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 4629

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 4635

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

Rule 4641

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

Rule 4707

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

Rule 4805

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) \left (a+b \sin ^{-1}(c+d x)\right )^{7/2} \, dx &=\frac {\operatorname {Subst}\left (\int e x \left (a+b \sin ^{-1}(x)\right )^{7/2} \, dx,x,c+d x\right )}{d}\\ &=\frac {e \operatorname {Subst}\left (\int x \left (a+b \sin ^{-1}(x)\right )^{7/2} \, dx,x,c+d x\right )}{d}\\ &=\frac {e (c+d x)^2 \left (a+b \sin ^{-1}(c+d x)\right )^{7/2}}{2 d}-\frac {(7 b e) \operatorname {Subst}\left (\int \frac {x^2 \left (a+b \sin ^{-1}(x)\right )^{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} \left (a+b \sin ^{-1}(c+d x)\right )^{5/2}}{8 d}+\frac {e (c+d x)^2 \left (a+b \sin ^{-1}(c+d x)\right )^{7/2}}{2 d}-\frac {(7 b e) \operatorname {Subst}\left (\int \frac {\left (a+b \sin ^{-1}(x)\right )^{5/2}}{\sqrt {1-x^2}} \, dx,x,c+d x\right )}{8 d}-\frac {\left (35 b^2 e\right ) \operatorname {Subst}\left (\int x \left (a+b \sin ^{-1}(x)\right )^{3/2} \, dx,x,c+d x\right )}{16 d}\\ &=-\frac {35 b^2 e (c+d x)^2 \left (a+b \sin ^{-1}(c+d x)\right )^{3/2}}{32 d}+\frac {7 b e (c+d x) \sqrt {1-(c+d x)^2} \left (a+b \sin ^{-1}(c+d x)\right )^{5/2}}{8 d}-\frac {e \left (a+b \sin ^{-1}(c+d x)\right )^{7/2}}{4 d}+\frac {e (c+d x)^2 \left (a+b \sin ^{-1}(c+d x)\right )^{7/2}}{2 d}+\frac {\left (105 b^3 e\right ) \operatorname {Subst}\left (\int \frac {x^2 \sqrt {a+b \sin ^{-1}(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 \sin ^{-1}(c+d x)}}{128 d}-\frac {35 b^2 e (c+d x)^2 \left (a+b \sin ^{-1}(c+d x)\right )^{3/2}}{32 d}+\frac {7 b e (c+d x) \sqrt {1-(c+d x)^2} \left (a+b \sin ^{-1}(c+d x)\right )^{5/2}}{8 d}-\frac {e \left (a+b \sin ^{-1}(c+d x)\right )^{7/2}}{4 d}+\frac {e (c+d x)^2 \left (a+b \sin ^{-1}(c+d x)\right )^{7/2}}{2 d}+\frac {\left (105 b^3 e\right ) \operatorname {Subst}\left (\int \frac {\sqrt {a+b \sin ^{-1}(x)}}{\sqrt {1-x^2}} \, dx,x,c+d x\right )}{128 d}+\frac {\left (105 b^4 e\right ) \operatorname {Subst}\left (\int \frac {x}{\sqrt {a+b \sin ^{-1}(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 \sin ^{-1}(c+d x)}}{128 d}+\frac {35 b^2 e \left (a+b \sin ^{-1}(c+d x)\right )^{3/2}}{64 d}-\frac {35 b^2 e (c+d x)^2 \left (a+b \sin ^{-1}(c+d x)\right )^{3/2}}{32 d}+\frac {7 b e (c+d x) \sqrt {1-(c+d x)^2} \left (a+b \sin ^{-1}(c+d x)\right )^{5/2}}{8 d}-\frac {e \left (a+b \sin ^{-1}(c+d x)\right )^{7/2}}{4 d}+\frac {e (c+d x)^2 \left (a+b \sin ^{-1}(c+d x)\right )^{7/2}}{2 d}+\frac {\left (105 b^4 e\right ) \operatorname {Subst}\left (\int \frac {\cos (x) \sin (x)}{\sqrt {a+b x}} \, dx,x,\sin ^{-1}(c+d x)\right )}{256 d}\\ &=-\frac {105 b^3 e (c+d x) \sqrt {1-(c+d x)^2} \sqrt {a+b \sin ^{-1}(c+d x)}}{128 d}+\frac {35 b^2 e \left (a+b \sin ^{-1}(c+d x)\right )^{3/2}}{64 d}-\frac {35 b^2 e (c+d x)^2 \left (a+b \sin ^{-1}(c+d x)\right )^{3/2}}{32 d}+\frac {7 b e (c+d x) \sqrt {1-(c+d x)^2} \left (a+b \sin ^{-1}(c+d x)\right )^{5/2}}{8 d}-\frac {e \left (a+b \sin ^{-1}(c+d x)\right )^{7/2}}{4 d}+\frac {e (c+d x)^2 \left (a+b \sin ^{-1}(c+d x)\right )^{7/2}}{2 d}+\frac {\left (105 b^4 e\right ) \operatorname {Subst}\left (\int \frac {\sin (2 x)}{2 \sqrt {a+b x}} \, dx,x,\sin ^{-1}(c+d x)\right )}{256 d}\\ &=-\frac {105 b^3 e (c+d x) \sqrt {1-(c+d x)^2} \sqrt {a+b \sin ^{-1}(c+d x)}}{128 d}+\frac {35 b^2 e \left (a+b \sin ^{-1}(c+d x)\right )^{3/2}}{64 d}-\frac {35 b^2 e (c+d x)^2 \left (a+b \sin ^{-1}(c+d x)\right )^{3/2}}{32 d}+\frac {7 b e (c+d x) \sqrt {1-(c+d x)^2} \left (a+b \sin ^{-1}(c+d x)\right )^{5/2}}{8 d}-\frac {e \left (a+b \sin ^{-1}(c+d x)\right )^{7/2}}{4 d}+\frac {e (c+d x)^2 \left (a+b \sin ^{-1}(c+d x)\right )^{7/2}}{2 d}+\frac {\left (105 b^4 e\right ) \operatorname {Subst}\left (\int \frac {\sin (2 x)}{\sqrt {a+b x}} \, dx,x,\sin ^{-1}(c+d x)\right )}{512 d}\\ &=-\frac {105 b^3 e (c+d x) \sqrt {1-(c+d x)^2} \sqrt {a+b \sin ^{-1}(c+d x)}}{128 d}+\frac {35 b^2 e \left (a+b \sin ^{-1}(c+d x)\right )^{3/2}}{64 d}-\frac {35 b^2 e (c+d x)^2 \left (a+b \sin ^{-1}(c+d x)\right )^{3/2}}{32 d}+\frac {7 b e (c+d x) \sqrt {1-(c+d x)^2} \left (a+b \sin ^{-1}(c+d x)\right )^{5/2}}{8 d}-\frac {e \left (a+b \sin ^{-1}(c+d x)\right )^{7/2}}{4 d}+\frac {e (c+d x)^2 \left (a+b \sin ^{-1}(c+d x)\right )^{7/2}}{2 d}+\frac {\left (105 b^4 e \cos \left (\frac {2 a}{b}\right )\right ) \operatorname {Subst}\left (\int \frac {\sin \left (\frac {2 a}{b}+2 x\right )}{\sqrt {a+b x}} \, dx,x,\sin ^{-1}(c+d x)\right )}{512 d}-\frac {\left (105 b^4 e \sin \left (\frac {2 a}{b}\right )\right ) \operatorname {Subst}\left (\int \frac {\cos \left (\frac {2 a}{b}+2 x\right )}{\sqrt {a+b x}} \, dx,x,\sin ^{-1}(c+d x)\right )}{512 d}\\ &=-\frac {105 b^3 e (c+d x) \sqrt {1-(c+d x)^2} \sqrt {a+b \sin ^{-1}(c+d x)}}{128 d}+\frac {35 b^2 e \left (a+b \sin ^{-1}(c+d x)\right )^{3/2}}{64 d}-\frac {35 b^2 e (c+d x)^2 \left (a+b \sin ^{-1}(c+d x)\right )^{3/2}}{32 d}+\frac {7 b e (c+d x) \sqrt {1-(c+d x)^2} \left (a+b \sin ^{-1}(c+d x)\right )^{5/2}}{8 d}-\frac {e \left (a+b \sin ^{-1}(c+d x)\right )^{7/2}}{4 d}+\frac {e (c+d x)^2 \left (a+b \sin ^{-1}(c+d x)\right )^{7/2}}{2 d}+\frac {\left (105 b^3 e \cos \left (\frac {2 a}{b}\right )\right ) \operatorname {Subst}\left (\int \sin \left (\frac {2 x^2}{b}\right ) \, dx,x,\sqrt {a+b \sin ^{-1}(c+d x)}\right )}{256 d}-\frac {\left (105 b^3 e \sin \left (\frac {2 a}{b}\right )\right ) \operatorname {Subst}\left (\int \cos \left (\frac {2 x^2}{b}\right ) \, dx,x,\sqrt {a+b \sin ^{-1}(c+d x)}\right )}{256 d}\\ &=-\frac {105 b^3 e (c+d x) \sqrt {1-(c+d x)^2} \sqrt {a+b \sin ^{-1}(c+d x)}}{128 d}+\frac {35 b^2 e \left (a+b \sin ^{-1}(c+d x)\right )^{3/2}}{64 d}-\frac {35 b^2 e (c+d x)^2 \left (a+b \sin ^{-1}(c+d x)\right )^{3/2}}{32 d}+\frac {7 b e (c+d x) \sqrt {1-(c+d x)^2} \left (a+b \sin ^{-1}(c+d x)\right )^{5/2}}{8 d}-\frac {e \left (a+b \sin ^{-1}(c+d x)\right )^{7/2}}{4 d}+\frac {e (c+d x)^2 \left (a+b \sin ^{-1}(c+d x)\right )^{7/2}}{2 d}+\frac {105 b^{7/2} e \sqrt {\pi } \cos \left (\frac {2 a}{b}\right ) S\left (\frac {2 \sqrt {a+b \sin ^{-1}(c+d x)}}{\sqrt {b} \sqrt {\pi }}\right )}{512 d}-\frac {105 b^{7/2} e \sqrt {\pi } C\left (\frac {2 \sqrt {a+b \sin ^{-1}(c+d x)}}{\sqrt {b} \sqrt {\pi }}\right ) \sin \left (\frac {2 a}{b}\right )}{512 d}\\ \end {align*}

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Mathematica [C]  time = 0.07, size = 137, normalized size = 0.46 \[ -\frac {b^4 e e^{-\frac {2 i a}{b}} \left (\sqrt {-\frac {i \left (a+b \sin ^{-1}(c+d x)\right )}{b}} \Gamma \left (\frac {9}{2},-\frac {2 i \left (a+b \sin ^{-1}(c+d x)\right )}{b}\right )+e^{\frac {4 i a}{b}} \sqrt {\frac {i \left (a+b \sin ^{-1}(c+d x)\right )}{b}} \Gamma \left (\frac {9}{2},\frac {2 i \left (a+b \sin ^{-1}(c+d x)\right )}{b}\right )\right )}{64 \sqrt {2} d \sqrt {a+b \sin ^{-1}(c+d x)}} \]

Warning: Unable to verify antiderivative.

[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]])

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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((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)

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giac [B]  time = 5.56, size = 2496, normalized size = 8.29 \[ \text {result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/4*sqrt(pi)*a^4*b^(3/2)*i*erf(-sqrt(b*arcsin(d*x + c) + a)*sqrt(b)*i/abs(b) - sqrt(b*arcsin(d*x + c) + a)/sqr
t(b))*e^(2*a*i/b + 1)/((b^3*i/abs(b) + b^2)*d) + 9/32*sqrt(pi)*a^2*b^(7/2)*i*erf(-sqrt(b*arcsin(d*x + c) + a)*
sqrt(b)*i/abs(b) - sqrt(b*arcsin(d*x + c) + a)/sqrt(b))*e^(2*a*i/b + 1)/((b^3*i/abs(b) + b^2)*d) + 1/4*sqrt(pi
)*a^4*b^(3/2)*i*erf(sqrt(b*arcsin(d*x + c) + a)*sqrt(b)*i/abs(b) - sqrt(b*arcsin(d*x + c) + a)/sqrt(b))*e^(-2*
a*i/b + 1)/((b^3*i/abs(b) - b^2)*d) + 9/32*sqrt(pi)*a^2*b^(7/2)*i*erf(sqrt(b*arcsin(d*x + c) + a)*sqrt(b)*i/ab
s(b) - sqrt(b*arcsin(d*x + c) + a)/sqrt(b))*e^(-2*a*i/b + 1)/((b^3*i/abs(b) - b^2)*d) - 7/32*sqrt(b*arcsin(d*x
+ c) + a)*b^3*i*arcsin(d*x + c)^2*e^(2*i*arcsin(d*x + c) + 1)/d - 1/8*sqrt(b*arcsin(d*x + c) + a)*b^3*arcsin(
d*x + c)^3*e^(2*i*arcsin(d*x + c) + 1)/d + 7/32*sqrt(b*arcsin(d*x + c) + a)*b^3*i*arcsin(d*x + c)^2*e^(-2*i*ar
csin(d*x + c) + 1)/d - 1/8*sqrt(b*arcsin(d*x + c) + a)*b^3*arcsin(d*x + c)^3*e^(-2*i*arcsin(d*x + c) + 1)/d +
1/16*sqrt(pi)*a^3*b^3*erf(-sqrt(b*arcsin(d*x + c) + a)*sqrt(b)*i/abs(b) - sqrt(b*arcsin(d*x + c) + a)/sqrt(b))
*e^(2*a*i/b + 1)/((b^(7/2)*i/abs(b) + b^(5/2))*d) - 9/16*sqrt(pi)*a^2*b^3*i*erf(-sqrt(b*arcsin(d*x + c) + a)*s
qrt(b)*i/abs(b) - sqrt(b*arcsin(d*x + c) + a)/sqrt(b))*e^(2*a*i/b + 1)/((b^(5/2)*i/abs(b) + b^(3/2))*d) - 1/16
*sqrt(pi)*a^3*b^3*erf(sqrt(b*arcsin(d*x + c) + a)*sqrt(b)*i/abs(b) - sqrt(b*arcsin(d*x + c) + a)/sqrt(b))*e^(-
2*a*i/b + 1)/((b^(7/2)*i/abs(b) - b^(5/2))*d) - 1/4*sqrt(pi)*a^4*b*i*erf(sqrt(b*arcsin(d*x + c) + a)*sqrt(b)*i
/abs(b) - sqrt(b*arcsin(d*x + c) + a)/sqrt(b))*e^(-2*a*i/b + 1)/((b^(5/2)*i/abs(b) - b^(3/2))*d) - 9/16*sqrt(p
i)*a^2*b^3*i*erf(sqrt(b*arcsin(d*x + c) + a)*sqrt(b)*i/abs(b) - sqrt(b*arcsin(d*x + c) + a)/sqrt(b))*e^(-2*a*i
/b + 1)/((b^(5/2)*i/abs(b) - b^(3/2))*d) - 13/16*sqrt(pi)*a^3*b^(5/2)*erf(-sqrt(b*arcsin(d*x + c) + a)*sqrt(b)
*i/abs(b) - sqrt(b*arcsin(d*x + c) + a)/sqrt(b))*e^(2*a*i/b + 1)/((b^3*i/abs(b) + b^2)*d) - 1/4*sqrt(pi)*a^4*s
qrt(b)*i*erf(-sqrt(b*arcsin(d*x + c) + a)*sqrt(b)*i/abs(b) - sqrt(b*arcsin(d*x + c) + a)/sqrt(b))*e^(2*a*i/b +
1)/((b^2*i/abs(b) + b)*d) + 9/32*sqrt(pi)*a^2*b^(5/2)*i*erf(-sqrt(b*arcsin(d*x + c) + a)*sqrt(b)*i/abs(b) - s
qrt(b*arcsin(d*x + c) + a)/sqrt(b))*e^(2*a*i/b + 1)/((b^2*i/abs(b) + b)*d) - 105/1024*sqrt(pi)*b^(9/2)*i*erf(-
sqrt(b*arcsin(d*x + c) + a)*sqrt(b)*i/abs(b) - sqrt(b*arcsin(d*x + c) + a)/sqrt(b))*e^(2*a*i/b + 1)/((b^2*i/ab
s(b) + b)*d) + 13/16*sqrt(pi)*a^3*b^(5/2)*erf(sqrt(b*arcsin(d*x + c) + a)*sqrt(b)*i/abs(b) - sqrt(b*arcsin(d*x
+ c) + a)/sqrt(b))*e^(-2*a*i/b + 1)/((b^3*i/abs(b) - b^2)*d) + 9/32*sqrt(pi)*a^2*b^(5/2)*i*erf(sqrt(b*arcsin(
d*x + c) + a)*sqrt(b)*i/abs(b) - sqrt(b*arcsin(d*x + c) + a)/sqrt(b))*e^(-2*a*i/b + 1)/((b^2*i/abs(b) - b)*d)
- 105/1024*sqrt(pi)*b^(9/2)*i*erf(sqrt(b*arcsin(d*x + c) + a)*sqrt(b)*i/abs(b) - sqrt(b*arcsin(d*x + c) + a)/s
qrt(b))*e^(-2*a*i/b + 1)/((b^2*i/abs(b) - b)*d) - 7/16*sqrt(b*arcsin(d*x + c) + a)*a*b^2*i*arcsin(d*x + c)*e^(
2*i*arcsin(d*x + c) + 1)/d - 3/8*sqrt(b*arcsin(d*x + c) + a)*a*b^2*arcsin(d*x + c)^2*e^(2*i*arcsin(d*x + c) +
1)/d + 7/16*sqrt(b*arcsin(d*x + c) + a)*a*b^2*i*arcsin(d*x + c)*e^(-2*i*arcsin(d*x + c) + 1)/d - 3/8*sqrt(b*ar
csin(d*x + c) + a)*a*b^2*arcsin(d*x + c)^2*e^(-2*i*arcsin(d*x + c) + 1)/d + 3/4*sqrt(pi)*a^3*b^2*erf(-sqrt(b*a
rcsin(d*x + c) + a)*sqrt(b)*i/abs(b) - sqrt(b*arcsin(d*x + c) + a)/sqrt(b))*e^(2*a*i/b + 1)/((b^(5/2)*i/abs(b)
+ b^(3/2))*d) - 15/64*sqrt(pi)*a*b^4*erf(-sqrt(b*arcsin(d*x + c) + a)*sqrt(b)*i/abs(b) - sqrt(b*arcsin(d*x +
c) + a)/sqrt(b))*e^(2*a*i/b + 1)/((b^(5/2)*i/abs(b) + b^(3/2))*d) - 3/4*sqrt(pi)*a^3*b^2*erf(sqrt(b*arcsin(d*x
+ c) + a)*sqrt(b)*i/abs(b) - sqrt(b*arcsin(d*x + c) + a)/sqrt(b))*e^(-2*a*i/b + 1)/((b^(5/2)*i/abs(b) - b^(3/
2))*d) + 15/64*sqrt(pi)*a*b^4*erf(sqrt(b*arcsin(d*x + c) + a)*sqrt(b)*i/abs(b) - sqrt(b*arcsin(d*x + c) + a)/s
qrt(b))*e^(-2*a*i/b + 1)/((b^(5/2)*i/abs(b) - b^(3/2))*d) + 15/64*sqrt(pi)*a*b^(7/2)*erf(-sqrt(b*arcsin(d*x +
c) + a)*sqrt(b)*i/abs(b) - sqrt(b*arcsin(d*x + c) + a)/sqrt(b))*e^(2*a*i/b + 1)/((b^2*i/abs(b) + b)*d) - 15/64
*sqrt(pi)*a*b^(7/2)*erf(sqrt(b*arcsin(d*x + c) + a)*sqrt(b)*i/abs(b) - sqrt(b*arcsin(d*x + c) + a)/sqrt(b))*e^
(-2*a*i/b + 1)/((b^2*i/abs(b) - b)*d) - 7/32*sqrt(b*arcsin(d*x + c) + a)*a^2*b*i*e^(2*i*arcsin(d*x + c) + 1)/d
+ 105/512*sqrt(b*arcsin(d*x + c) + a)*b^3*i*e^(2*i*arcsin(d*x + c) + 1)/d - 3/8*sqrt(b*arcsin(d*x + c) + a)*a
^2*b*arcsin(d*x + c)*e^(2*i*arcsin(d*x + c) + 1)/d + 35/128*sqrt(b*arcsin(d*x + c) + a)*b^3*arcsin(d*x + c)*e^
(2*i*arcsin(d*x + c) + 1)/d + 7/32*sqrt(b*arcsin(d*x + c) + a)*a^2*b*i*e^(-2*i*arcsin(d*x + c) + 1)/d - 105/51
2*sqrt(b*arcsin(d*x + c) + a)*b^3*i*e^(-2*i*arcsin(d*x + c) + 1)/d - 3/8*sqrt(b*arcsin(d*x + c) + a)*a^2*b*arc
sin(d*x + c)*e^(-2*i*arcsin(d*x + c) + 1)/d + 35/128*sqrt(b*arcsin(d*x + c) + a)*b^3*arcsin(d*x + c)*e^(-2*i*a
rcsin(d*x + c) + 1)/d - 1/8*sqrt(b*arcsin(d*x + c) + a)*a^3*e^(2*i*arcsin(d*x + c) + 1)/d + 35/128*sqrt(b*arcs
in(d*x + c) + a)*a*b^2*e^(2*i*arcsin(d*x + c) + 1)/d - 1/8*sqrt(b*arcsin(d*x + c) + a)*a^3*e^(-2*i*arcsin(d*x
+ c) + 1)/d + 35/128*sqrt(b*arcsin(d*x + c) + a)*a*b^2*e^(-2*i*arcsin(d*x + c) + 1)/d

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maple [B]  time = 0.27, size = 622, normalized size = 2.07 \[ -\frac {e b \left (128 \arcsin \left (d x +c \right )^{3} \sqrt {a +b \arcsin \left (d x +c \right )}\, \cos \left (\frac {2 a +2 b \arcsin \left (d x +c \right )}{b}-\frac {2 a}{b}\right ) \sqrt {\frac {1}{b}}\, \sqrt {\pi }\, b^{3}+384 \arcsin \left (d x +c \right )^{2} \sqrt {a +b \arcsin \left (d x +c \right )}\, \cos \left (\frac {2 a +2 b \arcsin \left (d x +c \right )}{b}-\frac {2 a}{b}\right ) \sqrt {\frac {1}{b}}\, \sqrt {\pi }\, a \,b^{2}-224 \arcsin \left (d x +c \right )^{2} \sqrt {a +b \arcsin \left (d x +c \right )}\, \sin \left (\frac {2 a +2 b \arcsin \left (d x +c \right )}{b}-\frac {2 a}{b}\right ) \sqrt {\frac {1}{b}}\, \sqrt {\pi }\, b^{3}+384 \arcsin \left (d x +c \right ) \sqrt {a +b \arcsin \left (d x +c \right )}\, \cos \left (\frac {2 a +2 b \arcsin \left (d x +c \right )}{b}-\frac {2 a}{b}\right ) \sqrt {\frac {1}{b}}\, \sqrt {\pi }\, a^{2} b -280 \arcsin \left (d x +c \right ) \sqrt {a +b \arcsin \left (d x +c \right )}\, \cos \left (\frac {2 a +2 b \arcsin \left (d x +c \right )}{b}-\frac {2 a}{b}\right ) \sqrt {\frac {1}{b}}\, \sqrt {\pi }\, b^{3}-448 \arcsin \left (d x +c \right ) \sqrt {a +b \arcsin \left (d x +c \right )}\, \sin \left (\frac {2 a +2 b \arcsin \left (d x +c \right )}{b}-\frac {2 a}{b}\right ) \sqrt {\frac {1}{b}}\, \sqrt {\pi }\, a \,b^{2}+128 \sqrt {a +b \arcsin \left (d x +c \right )}\, \cos \left (\frac {2 a +2 b \arcsin \left (d x +c \right )}{b}-\frac {2 a}{b}\right ) \sqrt {\frac {1}{b}}\, \sqrt {\pi }\, a^{3}-280 \sqrt {a +b \arcsin \left (d x +c \right )}\, \cos \left (\frac {2 a +2 b \arcsin \left (d x +c \right )}{b}-\frac {2 a}{b}\right ) \sqrt {\frac {1}{b}}\, \sqrt {\pi }\, a \,b^{2}-224 \sqrt {a +b \arcsin \left (d x +c \right )}\, \sin \left (\frac {2 a +2 b \arcsin \left (d x +c \right )}{b}-\frac {2 a}{b}\right ) \sqrt {\frac {1}{b}}\, \sqrt {\pi }\, a^{2} b +210 \sqrt {a +b \arcsin \left (d x +c \right )}\, \sin \left (\frac {2 a +2 b \arcsin \left (d x +c \right )}{b}-\frac {2 a}{b}\right ) \sqrt {\frac {1}{b}}\, \sqrt {\pi }\, b^{3}-105 \pi \,b^{3} \cos \left (\frac {2 a}{b}\right ) \mathrm {S}\left (\frac {2 \sqrt {a +b \arcsin \left (d x +c \right )}}{\sqrt {\pi }\, \sqrt {\frac {1}{b}}\, b}\right )+105 \pi \,b^{3} \sin \left (\frac {2 a}{b}\right ) \FresnelC \left (\frac {2 \sqrt {a +b \arcsin \left (d x +c \right )}}{\sqrt {\pi }\, \sqrt {\frac {1}{b}}\, b}\right )\right ) \sqrt {\frac {1}{b}}}{512 d \sqrt {\pi }} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((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)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((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)

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

[Out]

Timed out

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