3.3.47 \(\int (c e+d e x) (a+b \text {ArcSin}(c+d x))^{3/2} \, dx\) [247]

Optimal. Leaf size=199 \[ \frac {3 b e (c+d x) \sqrt {1-(c+d x)^2} \sqrt {a+b \text {ArcSin}(c+d x)}}{8 d}-\frac {e (a+b \text {ArcSin}(c+d x))^{3/2}}{4 d}+\frac {e (c+d x)^2 (a+b \text {ArcSin}(c+d x))^{3/2}}{2 d}-\frac {3 b^{3/2} e \sqrt {\pi } \cos \left (\frac {2 a}{b}\right ) S\left (\frac {2 \sqrt {a+b \text {ArcSin}(c+d x)}}{\sqrt {b} \sqrt {\pi }}\right )}{32 d}+\frac {3 b^{3/2} e \sqrt {\pi } \text {FresnelC}\left (\frac {2 \sqrt {a+b \text {ArcSin}(c+d x)}}{\sqrt {b} \sqrt {\pi }}\right ) \sin \left (\frac {2 a}{b}\right )}{32 d} \]

[Out]

-1/4*e*(a+b*arcsin(d*x+c))^(3/2)/d+1/2*e*(d*x+c)^2*(a+b*arcsin(d*x+c))^(3/2)/d-3/32*b^(3/2)*e*cos(2*a/b)*Fresn
elS(2*(a+b*arcsin(d*x+c))^(1/2)/b^(1/2)/Pi^(1/2))*Pi^(1/2)/d+3/32*b^(3/2)*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+3/8*b*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 = 0.29, antiderivative size = 199, normalized size of antiderivative = 1.00, number of steps used = 13, number of rules used = 12, integrand size = 23, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.522, Rules used = {4889, 12, 4725, 4795, 4737, 4731, 4491, 3387, 3386, 3432, 3385, 3433} \begin {gather*} \frac {3 \sqrt {\pi } b^{3/2} e \sin \left (\frac {2 a}{b}\right ) \text {FresnelC}\left (\frac {2 \sqrt {a+b \text {ArcSin}(c+d x)}}{\sqrt {\pi } \sqrt {b}}\right )}{32 d}-\frac {3 \sqrt {\pi } b^{3/2} e \cos \left (\frac {2 a}{b}\right ) S\left (\frac {2 \sqrt {a+b \text {ArcSin}(c+d x)}}{\sqrt {b} \sqrt {\pi }}\right )}{32 d}+\frac {e (c+d x)^2 (a+b \text {ArcSin}(c+d x))^{3/2}}{2 d}+\frac {3 b e \sqrt {1-(c+d x)^2} (c+d x) \sqrt {a+b \text {ArcSin}(c+d x)}}{8 d}-\frac {e (a+b \text {ArcSin}(c+d x))^{3/2}}{4 d} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(c*e + d*e*x)*(a + b*ArcSin[c + d*x])^(3/2),x]

[Out]

(3*b*e*(c + d*x)*Sqrt[1 - (c + d*x)^2]*Sqrt[a + b*ArcSin[c + d*x]])/(8*d) - (e*(a + b*ArcSin[c + d*x])^(3/2))/
(4*d) + (e*(c + d*x)^2*(a + b*ArcSin[c + d*x])^(3/2))/(2*d) - (3*b^(3/2)*e*Sqrt[Pi]*Cos[(2*a)/b]*FresnelS[(2*S
qrt[a + b*ArcSin[c + d*x]])/(Sqrt[b]*Sqrt[Pi])])/(32*d) + (3*b^(3/2)*e*Sqrt[Pi]*FresnelC[(2*Sqrt[a + b*ArcSin[
c + d*x]])/(Sqrt[b]*Sqrt[Pi])]*Sin[(2*a)/b])/(32*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*} \int (c e+d e x) \left (a+b \sin ^{-1}(c+d x)\right )^{3/2} \, dx &=\frac {\text {Subst}\left (\int e x \left (a+b \sin ^{-1}(x)\right )^{3/2} \, dx,x,c+d x\right )}{d}\\ &=\frac {e \text {Subst}\left (\int x \left (a+b \sin ^{-1}(x)\right )^{3/2} \, dx,x,c+d x\right )}{d}\\ &=\frac {e (c+d x)^2 \left (a+b \sin ^{-1}(c+d x)\right )^{3/2}}{2 d}-\frac {(3 b e) \text {Subst}\left (\int \frac {x^2 \sqrt {a+b \sin ^{-1}(x)}}{\sqrt {1-x^2}} \, dx,x,c+d x\right )}{4 d}\\ &=\frac {3 b e (c+d x) \sqrt {1-(c+d x)^2} \sqrt {a+b \sin ^{-1}(c+d x)}}{8 d}+\frac {e (c+d x)^2 \left (a+b \sin ^{-1}(c+d x)\right )^{3/2}}{2 d}-\frac {(3 b e) \text {Subst}\left (\int \frac {\sqrt {a+b \sin ^{-1}(x)}}{\sqrt {1-x^2}} \, dx,x,c+d x\right )}{8 d}-\frac {\left (3 b^2 e\right ) \text {Subst}\left (\int \frac {x}{\sqrt {a+b \sin ^{-1}(x)}} \, dx,x,c+d x\right )}{16 d}\\ &=\frac {3 b e (c+d x) \sqrt {1-(c+d x)^2} \sqrt {a+b \sin ^{-1}(c+d x)}}{8 d}-\frac {e \left (a+b \sin ^{-1}(c+d x)\right )^{3/2}}{4 d}+\frac {e (c+d x)^2 \left (a+b \sin ^{-1}(c+d x)\right )^{3/2}}{2 d}-\frac {\left (3 b^2 e\right ) \text {Subst}\left (\int \frac {\cos (x) \sin (x)}{\sqrt {a+b x}} \, dx,x,\sin ^{-1}(c+d x)\right )}{16 d}\\ &=\frac {3 b e (c+d x) \sqrt {1-(c+d x)^2} \sqrt {a+b \sin ^{-1}(c+d x)}}{8 d}-\frac {e \left (a+b \sin ^{-1}(c+d x)\right )^{3/2}}{4 d}+\frac {e (c+d x)^2 \left (a+b \sin ^{-1}(c+d x)\right )^{3/2}}{2 d}-\frac {\left (3 b^2 e\right ) \text {Subst}\left (\int \frac {\sin (2 x)}{2 \sqrt {a+b x}} \, dx,x,\sin ^{-1}(c+d x)\right )}{16 d}\\ &=\frac {3 b e (c+d x) \sqrt {1-(c+d x)^2} \sqrt {a+b \sin ^{-1}(c+d x)}}{8 d}-\frac {e \left (a+b \sin ^{-1}(c+d x)\right )^{3/2}}{4 d}+\frac {e (c+d x)^2 \left (a+b \sin ^{-1}(c+d x)\right )^{3/2}}{2 d}-\frac {\left (3 b^2 e\right ) \text {Subst}\left (\int \frac {\sin (2 x)}{\sqrt {a+b x}} \, dx,x,\sin ^{-1}(c+d x)\right )}{32 d}\\ &=\frac {3 b e (c+d x) \sqrt {1-(c+d x)^2} \sqrt {a+b \sin ^{-1}(c+d x)}}{8 d}-\frac {e \left (a+b \sin ^{-1}(c+d x)\right )^{3/2}}{4 d}+\frac {e (c+d x)^2 \left (a+b \sin ^{-1}(c+d x)\right )^{3/2}}{2 d}-\frac {\left (3 b^2 e \cos \left (\frac {2 a}{b}\right )\right ) \text {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 )}{32 d}+\frac {\left (3 b^2 e \sin \left (\frac {2 a}{b}\right )\right ) \text {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 )}{32 d}\\ &=\frac {3 b e (c+d x) \sqrt {1-(c+d x)^2} \sqrt {a+b \sin ^{-1}(c+d x)}}{8 d}-\frac {e \left (a+b \sin ^{-1}(c+d x)\right )^{3/2}}{4 d}+\frac {e (c+d x)^2 \left (a+b \sin ^{-1}(c+d x)\right )^{3/2}}{2 d}-\frac {\left (3 b 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 \sin ^{-1}(c+d x)}\right )}{16 d}+\frac {\left (3 b 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 \sin ^{-1}(c+d x)}\right )}{16 d}\\ &=\frac {3 b e (c+d x) \sqrt {1-(c+d x)^2} \sqrt {a+b \sin ^{-1}(c+d x)}}{8 d}-\frac {e \left (a+b \sin ^{-1}(c+d x)\right )^{3/2}}{4 d}+\frac {e (c+d x)^2 \left (a+b \sin ^{-1}(c+d x)\right )^{3/2}}{2 d}-\frac {3 b^{3/2} 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 )}{32 d}+\frac {3 b^{3/2} 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 )}{32 d}\\ \end {align*}

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Mathematica [C] Result contains complex when optimal does not.
time = 0.04, size = 137, normalized size = 0.69 \begin {gather*} \frac {b^2 e e^{-\frac {2 i a}{b}} \left (\sqrt {-\frac {i (a+b \text {ArcSin}(c+d x))}{b}} \text {Gamma}\left (\frac {5}{2},-\frac {2 i (a+b \text {ArcSin}(c+d x))}{b}\right )+e^{\frac {4 i a}{b}} \sqrt {\frac {i (a+b \text {ArcSin}(c+d x))}{b}} \text {Gamma}\left (\frac {5}{2},\frac {2 i (a+b \text {ArcSin}(c+d x))}{b}\right )\right )}{16 \sqrt {2} d \sqrt {a+b \text {ArcSin}(c+d x)}} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(c*e + d*e*x)*(a + b*ArcSin[c + d*x])^(3/2),x]

[Out]

(b^2*e*(Sqrt[((-I)*(a + b*ArcSin[c + d*x]))/b]*Gamma[5/2, ((-2*I)*(a + b*ArcSin[c + d*x]))/b] + E^(((4*I)*a)/b
)*Sqrt[(I*(a + b*ArcSin[c + d*x]))/b]*Gamma[5/2, ((2*I)*(a + b*ArcSin[c + d*x]))/b]))/(16*Sqrt[2]*d*E^(((2*I)*
a)/b)*Sqrt[a + b*ArcSin[c + d*x]])

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Maple [A]
time = 0.25, size = 314, normalized size = 1.58

method	result	size
default	\(-\frac {e \left (-3 \sqrt {-\frac {2}{b}}\, \sqrt {\pi }\, \sqrt {2}\, \sqrt {a +b \arcsin \left (d x +c \right )}\, \cos \left (\frac {2 a}{b}\right ) \mathrm {S}\left (\frac {2 \sqrt {2}\, \sqrt {a +b \arcsin \left (d x +c \right )}}{\sqrt {\pi }\, \sqrt {-\frac {2}{b}}\, b}\right ) b^{2}-3 \sqrt {-\frac {2}{b}}\, \sqrt {\pi }\, \sqrt {2}\, \sqrt {a +b \arcsin \left (d x +c \right )}\, \sin \left (\frac {2 a}{b}\right ) \FresnelC \left (\frac {2 \sqrt {2}\, \sqrt {a +b \arcsin \left (d x +c \right )}}{\sqrt {\pi }\, \sqrt {-\frac {2}{b}}\, b}\right ) b^{2}+16 \arcsin \left (d x +c \right )^{2} \cos \left (-\frac {2 \left (a +b \arcsin \left (d x +c \right )\right )}{b}+\frac {2 a}{b}\right ) b^{2}+32 \arcsin \left (d x +c \right ) \cos \left (-\frac {2 \left (a +b \arcsin \left (d x +c \right )\right )}{b}+\frac {2 a}{b}\right ) a b +12 \arcsin \left (d x +c \right ) \sin \left (-\frac {2 \left (a +b \arcsin \left (d x +c \right )\right )}{b}+\frac {2 a}{b}\right ) b^{2}+16 \cos \left (-\frac {2 \left (a +b \arcsin \left (d x +c \right )\right )}{b}+\frac {2 a}{b}\right ) a^{2}+12 \sin \left (-\frac {2 \left (a +b \arcsin \left (d x +c \right )\right )}{b}+\frac {2 a}{b}\right ) a b \right )}{64 d \sqrt {a +b \arcsin \left (d x +c \right )}}\)	\(314\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((d*e*x+c*e)*(a+b*arcsin(d*x+c))^(3/2),x,method=_RETURNVERBOSE)

[Out]

-1/64/d*e/(a+b*arcsin(d*x+c))^(1/2)*(-3*(-2/b)^(1/2)*Pi^(1/2)*2^(1/2)*(a+b*arcsin(d*x+c))^(1/2)*cos(2*a/b)*Fre
snelS(2*2^(1/2)/Pi^(1/2)/(-2/b)^(1/2)*(a+b*arcsin(d*x+c))^(1/2)/b)*b^2-3*(-2/b)^(1/2)*Pi^(1/2)*2^(1/2)*(a+b*ar
csin(d*x+c))^(1/2)*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)*b^2+16*arc
sin(d*x+c)^2*cos(-2*(a+b*arcsin(d*x+c))/b+2*a/b)*b^2+32*arcsin(d*x+c)*cos(-2*(a+b*arcsin(d*x+c))/b+2*a/b)*a*b+
12*arcsin(d*x+c)*sin(-2*(a+b*arcsin(d*x+c))/b+2*a/b)*b^2+16*cos(-2*(a+b*arcsin(d*x+c))/b+2*a/b)*a^2+12*sin(-2*
(a+b*arcsin(d*x+c))/b+2*a/b)*a*b)

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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)*(a+b*arcsin(d*x+c))^(3/2),x, algorithm="maxima")

[Out]

integrate((d*x*e + c*e)*(b*arcsin(d*x + c) + a)^(3/2), 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)*(a+b*arcsin(d*x+c))^(3/2),x, algorithm="fricas")

[Out]

Exception raised: TypeError >>  Error detected within library code:   integrate: implementation incomplete (co
nstant residues)

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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} e \left (\int a c \sqrt {a + b \operatorname {asin}{\left (c + d x \right )}}\, dx + \int a d x \sqrt {a + b \operatorname {asin}{\left (c + d x \right )}}\, dx + \int b c \sqrt {a + b \operatorname {asin}{\left (c + d x \right )}} \operatorname {asin}{\left (c + d x \right )}\, dx + \int b d x \sqrt {a + b \operatorname {asin}{\left (c + d x \right )}} \operatorname {asin}{\left (c + d x \right )}\, dx\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((d*e*x+c*e)*(a+b*asin(d*x+c))**(3/2),x)

[Out]

e*(Integral(a*c*sqrt(a + b*asin(c + d*x)), x) + Integral(a*d*x*sqrt(a + b*asin(c + d*x)), x) + Integral(b*c*sq
rt(a + b*asin(c + d*x))*asin(c + d*x), x) + Integral(b*d*x*sqrt(a + b*asin(c + d*x))*asin(c + d*x), x))

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Giac [C] Result contains complex when optimal does not.
time = 0.87, size = 929, normalized size = 4.67 \begin {gather*} \frac {i \, \sqrt {\pi } a^{2} b^{\frac {3}{2}} e \operatorname {erf}\left (-\frac {\sqrt {b \arcsin \left (d x + c\right ) + a}}{\sqrt {b}} - \frac {i \, \sqrt {b \arcsin \left (d x + c\right ) + a} \sqrt {b}}{{\left | b \right |}}\right ) e^{\left (\frac {2 i \, a}{b}\right )}}{4 \, {\left (b^{2} + \frac {i \, b^{3}}{{\left | b \right |}}\right )} d} - \frac {\sqrt {\pi } a b^{\frac {5}{2}} e \operatorname {erf}\left (-\frac {\sqrt {b \arcsin \left (d x + c\right ) + a}}{\sqrt {b}} - \frac {i \, \sqrt {b \arcsin \left (d x + c\right ) + a} \sqrt {b}}{{\left | b \right |}}\right ) e^{\left (\frac {2 i \, a}{b}\right )}}{8 \, {\left (b^{2} + \frac {i \, b^{3}}{{\left | b \right |}}\right )} d} - \frac {i \, \sqrt {\pi } a^{2} b^{\frac {3}{2}} e \operatorname {erf}\left (-\frac {\sqrt {b \arcsin \left (d x + c\right ) + a}}{\sqrt {b}} + \frac {i \, \sqrt {b \arcsin \left (d x + c\right ) + a} \sqrt {b}}{{\left | b \right |}}\right ) e^{\left (-\frac {2 i \, a}{b}\right )}}{4 \, {\left (b^{2} - \frac {i \, b^{3}}{{\left | b \right |}}\right )} d} - \frac {\sqrt {\pi } a b^{\frac {5}{2}} e \operatorname {erf}\left (-\frac {\sqrt {b \arcsin \left (d x + c\right ) + a}}{\sqrt {b}} + \frac {i \, \sqrt {b \arcsin \left (d x + c\right ) + a} \sqrt {b}}{{\left | b \right |}}\right ) e^{\left (-\frac {2 i \, a}{b}\right )}}{8 \, {\left (b^{2} - \frac {i \, b^{3}}{{\left | b \right |}}\right )} d} + \frac {\sqrt {\pi } a b^{2} e \operatorname {erf}\left (-\frac {\sqrt {b \arcsin \left (d x + c\right ) + a}}{\sqrt {b}} - \frac {i \, \sqrt {b \arcsin \left (d x + c\right ) + a} \sqrt {b}}{{\left | b \right |}}\right ) e^{\left (\frac {2 i \, a}{b}\right )}}{8 \, {\left (b^{\frac {3}{2}} + \frac {i \, b^{\frac {5}{2}}}{{\left | b \right |}}\right )} d} + \frac {i \, \sqrt {\pi } a^{2} b e \operatorname {erf}\left (-\frac {\sqrt {b \arcsin \left (d x + c\right ) + a}}{\sqrt {b}} + \frac {i \, \sqrt {b \arcsin \left (d x + c\right ) + a} \sqrt {b}}{{\left | b \right |}}\right ) e^{\left (-\frac {2 i \, a}{b}\right )}}{4 \, {\left (b^{\frac {3}{2}} - \frac {i \, b^{\frac {5}{2}}}{{\left | b \right |}}\right )} d} + \frac {\sqrt {\pi } a b^{2} e \operatorname {erf}\left (-\frac {\sqrt {b \arcsin \left (d x + c\right ) + a}}{\sqrt {b}} + \frac {i \, \sqrt {b \arcsin \left (d x + c\right ) + a} \sqrt {b}}{{\left | b \right |}}\right ) e^{\left (-\frac {2 i \, a}{b}\right )}}{8 \, {\left (b^{\frac {3}{2}} - \frac {i \, b^{\frac {5}{2}}}{{\left | b \right |}}\right )} d} - \frac {i \, \sqrt {\pi } a^{2} \sqrt {b} e \operatorname {erf}\left (-\frac {\sqrt {b \arcsin \left (d x + c\right ) + a}}{\sqrt {b}} - \frac {i \, \sqrt {b \arcsin \left (d x + c\right ) + a} \sqrt {b}}{{\left | b \right |}}\right ) e^{\left (\frac {2 i \, a}{b}\right )}}{4 \, {\left (b + \frac {i \, b^{2}}{{\left | b \right |}}\right )} d} + \frac {3 i \, \sqrt {\pi } b^{\frac {5}{2}} e \operatorname {erf}\left (-\frac {\sqrt {b \arcsin \left (d x + c\right ) + a}}{\sqrt {b}} - \frac {i \, \sqrt {b \arcsin \left (d x + c\right ) + a} \sqrt {b}}{{\left | b \right |}}\right ) e^{\left (\frac {2 i \, a}{b}\right )}}{64 \, {\left (b + \frac {i \, b^{2}}{{\left | b \right |}}\right )} d} - \frac {3 i \, \sqrt {\pi } b^{\frac {5}{2}} e \operatorname {erf}\left (-\frac {\sqrt {b \arcsin \left (d x + c\right ) + a}}{\sqrt {b}} + \frac {i \, \sqrt {b \arcsin \left (d x + c\right ) + a} \sqrt {b}}{{\left | b \right |}}\right ) e^{\left (-\frac {2 i \, a}{b}\right )}}{64 \, {\left (b - \frac {i \, b^{2}}{{\left | b \right |}}\right )} d} - \frac {\sqrt {b \arcsin \left (d x + c\right ) + a} b e \arcsin \left (d x + c\right ) e^{\left (2 i \, \arcsin \left (d x + c\right )\right )}}{8 \, d} - \frac {\sqrt {b \arcsin \left (d x + c\right ) + a} b e \arcsin \left (d x + c\right ) e^{\left (-2 i \, \arcsin \left (d x + c\right )\right )}}{8 \, d} - \frac {\sqrt {b \arcsin \left (d x + c\right ) + a} a e e^{\left (2 i \, \arcsin \left (d x + c\right )\right )}}{8 \, d} - \frac {3 i \, \sqrt {b \arcsin \left (d x + c\right ) + a} b e e^{\left (2 i \, \arcsin \left (d x + c\right )\right )}}{32 \, d} - \frac {\sqrt {b \arcsin \left (d x + c\right ) + a} a e e^{\left (-2 i \, \arcsin \left (d x + c\right )\right )}}{8 \, d} + \frac {3 i \, \sqrt {b \arcsin \left (d x + c\right ) + a} b e e^{\left (-2 i \, \arcsin \left (d x + c\right )\right )}}{32 \, d} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((d*e*x+c*e)*(a+b*arcsin(d*x+c))^(3/2),x, algorithm="giac")

[Out]

1/4*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)/((b^2 + I*b^3/abs(b))*d) - 1/8*sqrt(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)/((b^2 + I*b^3/abs(b))*d) - 1/4*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)
/((b^2 - I*b^3/abs(b))*d) - 1/8*sqrt(pi)*a*b^(5/2)*e*erf(-sqrt(b*arcsin(d*x + c) + a)/sqrt(b) + I*sqrt(b*arcsi
n(d*x + c) + a)*sqrt(b)/abs(b))*e^(-2*I*a/b)/((b^2 - I*b^3/abs(b))*d) + 1/8*sqrt(pi)*a*b^2*e*erf(-sqrt(b*arcsi
n(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))*d) + 1/4*I*sqrt(pi)*a^2*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)/((b^(3/2) - I*b^(5/2)/abs(b))*d) + 1/8*sqrt(pi)*a*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))*d) - 1/4
*I*sqrt(pi)*a^2*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)/((b + I*b^2/abs(b))*d) + 3/64*I*sqrt(pi)*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))*d) - 3/64*I*sqrt(pi)*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))*d) - 1/8*sqrt(b*arcsin(d*x + c) + a)*b*e*arcsin(d*x + c)*e^(2*I*arcsin(d*x + c))/d - 1/8*sqrt(b*ar
csin(d*x + c) + a)*b*e*arcsin(d*x + c)*e^(-2*I*arcsin(d*x + c))/d - 1/8*sqrt(b*arcsin(d*x + c) + a)*a*e*e^(2*I
*arcsin(d*x + c))/d - 3/32*I*sqrt(b*arcsin(d*x + c) + a)*b*e*e^(2*I*arcsin(d*x + c))/d - 1/8*sqrt(b*arcsin(d*x
+ c) + a)*a*e*e^(-2*I*arcsin(d*x + c))/d + 3/32*I*sqrt(b*arcsin(d*x + c) + a)*b*e*e^(-2*I*arcsin(d*x + c))/d

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Mupad [F]
time = 0.00, size = -1, normalized size = -0.01 \begin {gather*} \int \left (c\,e+d\,e\,x\right )\,{\left (a+b\,\mathrm {asin}\left (c+d\,x\right )\right )}^{3/2} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((c*e + d*e*x)*(a + b*asin(c + d*x))^(3/2),x)

[Out]

int((c*e + d*e*x)*(a + b*asin(c + d*x))^(3/2), x)

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