3.3.42 \(\int (c e+d e x) \sqrt {a+b \text {ArcSin}(c+d x)} \, dx\) [242]

Optimal. Leaf size=156 \[ -\frac {e \sqrt {a+b \text {ArcSin}(c+d x)}}{4 d}+\frac {e (c+d x)^2 \sqrt {a+b \text {ArcSin}(c+d x)}}{2 d}+\frac {\sqrt {b} e \sqrt {\pi } \cos \left (\frac {2 a}{b}\right ) \text {FresnelC}\left (\frac {2 \sqrt {a+b \text {ArcSin}(c+d x)}}{\sqrt {b} \sqrt {\pi }}\right )}{8 d}+\frac {\sqrt {b} e \sqrt {\pi } S\left (\frac {2 \sqrt {a+b \text {ArcSin}(c+d x)}}{\sqrt {b} \sqrt {\pi }}\right ) \sin \left (\frac {2 a}{b}\right )}{8 d} \]

[Out]

1/8*e*cos(2*a/b)*FresnelC(2*(a+b*arcsin(d*x+c))^(1/2)/b^(1/2)/Pi^(1/2))*b^(1/2)*Pi^(1/2)/d+1/8*e*FresnelS(2*(a
+b*arcsin(d*x+c))^(1/2)/b^(1/2)/Pi^(1/2))*sin(2*a/b)*b^(1/2)*Pi^(1/2)/d-1/4*e*(a+b*arcsin(d*x+c))^(1/2)/d+1/2*
e*(d*x+c)^2*(a+b*arcsin(d*x+c))^(1/2)/d

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Rubi [A]
time = 0.26, antiderivative size = 156, normalized size of antiderivative = 1.00, number of steps used = 11, number of rules used = 10, integrand size = 23, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.435, Rules used = {4889, 12, 4725, 4809, 3393, 3387, 3386, 3432, 3385, 3433} \begin {gather*} \frac {\sqrt {\pi } \sqrt {b} e \cos \left (\frac {2 a}{b}\right ) \text {FresnelC}\left (\frac {2 \sqrt {a+b \text {ArcSin}(c+d x)}}{\sqrt {\pi } \sqrt {b}}\right )}{8 d}+\frac {\sqrt {\pi } \sqrt {b} e \sin \left (\frac {2 a}{b}\right ) S\left (\frac {2 \sqrt {a+b \text {ArcSin}(c+d x)}}{\sqrt {b} \sqrt {\pi }}\right )}{8 d}+\frac {e (c+d x)^2 \sqrt {a+b \text {ArcSin}(c+d x)}}{2 d}-\frac {e \sqrt {a+b \text {ArcSin}(c+d x)}}{4 d} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(c*e + d*e*x)*Sqrt[a + b*ArcSin[c + d*x]],x]

[Out]

-1/4*(e*Sqrt[a + b*ArcSin[c + d*x]])/d + (e*(c + d*x)^2*Sqrt[a + b*ArcSin[c + d*x]])/(2*d) + (Sqrt[b]*e*Sqrt[P
i]*Cos[(2*a)/b]*FresnelC[(2*Sqrt[a + b*ArcSin[c + d*x]])/(Sqrt[b]*Sqrt[Pi])])/(8*d) + (Sqrt[b]*e*Sqrt[Pi]*Fres
nelS[(2*Sqrt[a + b*ArcSin[c + d*x]])/(Sqrt[b]*Sqrt[Pi])]*Sin[(2*a)/b])/(8*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 3393

Int[((c_.) + (d_.)*(x_))^(m_)*sin[(e_.) + (f_.)*(x_)]^(n_), x_Symbol] :> Int[ExpandTrigReduce[(c + d*x)^m, Sin
[e + f*x]^n, x], x] /; FreeQ[{c, d, e, f, m}, x] && IGtQ[n, 1] && ( !RationalQ[m] || (GeQ[m, -1] && LtQ[m, 1])
)

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 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 4809

Int[((a_.) + ArcSin[(c_.)*(x_)]*(b_.))^(n_.)*(x_)^(m_.)*((d_) + (e_.)*(x_)^2)^(p_.), x_Symbol] :> Dist[(1/(b*c
^(m + 1)))*Simp[(d + e*x^2)^p/(1 - c^2*x^2)^p], Subst[Int[x^n*Sin[-a/b + x/b]^m*Cos[-a/b + x/b]^(2*p + 1), x],
x, a + b*ArcSin[c*x]], x] /; FreeQ[{a, b, c, d, e, n}, x] && EqQ[c^2*d + e, 0] && IGtQ[2*p + 2, 0] && IGtQ[m,
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) \sqrt {a+b \sin ^{-1}(c+d x)} \, dx &=\frac {\text {Subst}\left (\int e x \sqrt {a+b \sin ^{-1}(x)} \, dx,x,c+d x\right )}{d}\\ &=\frac {e \text {Subst}\left (\int x \sqrt {a+b \sin ^{-1}(x)} \, dx,x,c+d x\right )}{d}\\ &=\frac {e (c+d x)^2 \sqrt {a+b \sin ^{-1}(c+d x)}}{2 d}-\frac {(b e) \text {Subst}\left (\int \frac {x^2}{\sqrt {1-x^2} \sqrt {a+b \sin ^{-1}(x)}} \, dx,x,c+d x\right )}{4 d}\\ &=\frac {e (c+d x)^2 \sqrt {a+b \sin ^{-1}(c+d x)}}{2 d}-\frac {(b e) \text {Subst}\left (\int \frac {\sin ^2(x)}{\sqrt {a+b x}} \, dx,x,\sin ^{-1}(c+d x)\right )}{4 d}\\ &=\frac {e (c+d x)^2 \sqrt {a+b \sin ^{-1}(c+d x)}}{2 d}-\frac {(b e) \text {Subst}\left (\int \left (\frac {1}{2 \sqrt {a+b x}}-\frac {\cos (2 x)}{2 \sqrt {a+b x}}\right ) \, dx,x,\sin ^{-1}(c+d x)\right )}{4 d}\\ &=-\frac {e \sqrt {a+b \sin ^{-1}(c+d x)}}{4 d}+\frac {e (c+d x)^2 \sqrt {a+b \sin ^{-1}(c+d x)}}{2 d}+\frac {(b e) \text {Subst}\left (\int \frac {\cos (2 x)}{\sqrt {a+b x}} \, dx,x,\sin ^{-1}(c+d x)\right )}{8 d}\\ &=-\frac {e \sqrt {a+b \sin ^{-1}(c+d x)}}{4 d}+\frac {e (c+d x)^2 \sqrt {a+b \sin ^{-1}(c+d x)}}{2 d}+\frac {\left (b e \cos \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 )}{8 d}+\frac {\left (b e \sin \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 )}{8 d}\\ &=-\frac {e \sqrt {a+b \sin ^{-1}(c+d x)}}{4 d}+\frac {e (c+d x)^2 \sqrt {a+b \sin ^{-1}(c+d x)}}{2 d}+\frac {\left (e \cos \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 )}{4 d}+\frac {\left (e \sin \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 )}{4 d}\\ &=-\frac {e \sqrt {a+b \sin ^{-1}(c+d x)}}{4 d}+\frac {e (c+d x)^2 \sqrt {a+b \sin ^{-1}(c+d x)}}{2 d}+\frac {\sqrt {b} e \sqrt {\pi } \cos \left (\frac {2 a}{b}\right ) C\left (\frac {2 \sqrt {a+b \sin ^{-1}(c+d x)}}{\sqrt {b} \sqrt {\pi }}\right )}{8 d}+\frac {\sqrt {b} e \sqrt {\pi } S\left (\frac {2 \sqrt {a+b \sin ^{-1}(c+d x)}}{\sqrt {b} \sqrt {\pi }}\right ) \sin \left (\frac {2 a}{b}\right )}{8 d}\\ \end {align*}

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Mathematica [C] Result contains complex when optimal does not.
time = 0.06, size = 154, normalized size = 0.99 \begin {gather*} -\frac {e e^{-\frac {2 i a}{b}} \sqrt {a+b \text {ArcSin}(c+d x)} \left (\sqrt {\frac {i (a+b \text {ArcSin}(c+d x))}{b}} \text {Gamma}\left (\frac {3}{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 {3}{2},\frac {2 i (a+b \text {ArcSin}(c+d x))}{b}\right )\right )}{8 \sqrt {2} d \sqrt {\frac {(a+b \text {ArcSin}(c+d x))^2}{b^2}}} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(c*e + d*e*x)*Sqrt[a + b*ArcSin[c + d*x]],x]

[Out]

-1/8*(e*Sqrt[a + b*ArcSin[c + d*x]]*(Sqrt[(I*(a + b*ArcSin[c + d*x]))/b]*Gamma[3/2, ((-2*I)*(a + b*ArcSin[c +
d*x]))/b] + E^(((4*I)*a)/b)*Sqrt[((-I)*(a + b*ArcSin[c + d*x]))/b]*Gamma[3/2, ((2*I)*(a + b*ArcSin[c + d*x]))/
b]))/(Sqrt[2]*d*E^(((2*I)*a)/b)*Sqrt[(a + b*ArcSin[c + d*x])^2/b^2])

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Maple [A]
time = 0.24, size = 209, normalized size = 1.34

method	result	size
default	\(-\frac {e \left (-\sqrt {a +b \arcsin \left (d x +c \right )}\, \sqrt {-\frac {2}{b}}\, \sqrt {\pi }\, \sqrt {2}\, \cos \left (\frac {2 a}{b}\right ) \FresnelC \left (\frac {2 \sqrt {2}\, \sqrt {a +b \arcsin \left (d x +c \right )}}{\sqrt {\pi }\, \sqrt {-\frac {2}{b}}\, b}\right ) b +\sqrt {a +b \arcsin \left (d x +c \right )}\, \sqrt {-\frac {2}{b}}\, \sqrt {\pi }\, \sqrt {2}\, \sin \left (\frac {2 a}{b}\right ) \mathrm {S}\left (\frac {2 \sqrt {2}\, \sqrt {a +b \arcsin \left (d x +c \right )}}{\sqrt {\pi }\, \sqrt {-\frac {2}{b}}\, b}\right ) b +4 \arcsin \left (d x +c \right ) \cos \left (-\frac {2 \left (a +b \arcsin \left (d x +c \right )\right )}{b}+\frac {2 a}{b}\right ) b +4 \cos \left (-\frac {2 \left (a +b \arcsin \left (d x +c \right )\right )}{b}+\frac {2 a}{b}\right ) a \right )}{16 d \sqrt {a +b \arcsin \left (d x +c \right )}}\)	\(209\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((d*e*x+c*e)*(a+b*arcsin(d*x+c))^(1/2),x,method=_RETURNVERBOSE)

[Out]

-1/16/d*e/(a+b*arcsin(d*x+c))^(1/2)*(-(a+b*arcsin(d*x+c))^(1/2)*(-2/b)^(1/2)*Pi^(1/2)*2^(1/2)*cos(2*a/b)*Fresn
elC(2*2^(1/2)/Pi^(1/2)/(-2/b)^(1/2)*(a+b*arcsin(d*x+c))^(1/2)/b)*b+(a+b*arcsin(d*x+c))^(1/2)*(-2/b)^(1/2)*Pi^(
1/2)*2^(1/2)*sin(2*a/b)*FresnelS(2*2^(1/2)/Pi^(1/2)/(-2/b)^(1/2)*(a+b*arcsin(d*x+c))^(1/2)/b)*b+4*arcsin(d*x+c
)*cos(-2*(a+b*arcsin(d*x+c))/b+2*a/b)*b+4*cos(-2*(a+b*arcsin(d*x+c))/b+2*a/b)*a)

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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))^(1/2),x, algorithm="maxima")

[Out]

integrate((d*x*e + c*e)*sqrt(b*arcsin(d*x + c) + a), 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))^(1/2),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 c \sqrt {a + b \operatorname {asin}{\left (c + d x \right )}}\, dx + \int d x \sqrt {a + b \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))**(1/2),x)

[Out]

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

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Giac [C] Result contains complex when optimal does not.
time = 1.10, size = 488, normalized size = 3.13 \begin {gather*} \frac {i \, \sqrt {\pi } a \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 {\sqrt {\pi } 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 )}}{16 \, {\left (b + \frac {i \, b^{2}}{{\left | b \right |}}\right )} d} - \frac {i \, \sqrt {\pi } a \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 {\sqrt {\pi } 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 )}}{16 \, {\left (b - \frac {i \, b^{2}}{{\left | b \right |}}\right )} d} + \frac {i \, \sqrt {\pi } a 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 \, d {\left (\sqrt {b} - \frac {i \, b^{\frac {3}{2}}}{{\left | b \right |}}\right )}} - \frac {i \, \sqrt {\pi } a 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 \, \sqrt {b} d {\left (\frac {i \, b}{{\left | b \right |}} + 1\right )}} - \frac {\sqrt {b \arcsin \left (d x + c\right ) + a} e e^{\left (2 i \, \arcsin \left (d x + c\right )\right )}}{8 \, d} - \frac {\sqrt {b \arcsin \left (d x + c\right ) + a} e e^{\left (-2 i \, \arcsin \left (d x + c\right )\right )}}{8 \, 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))^(1/2),x, algorithm="giac")

[Out]

1/4*I*sqrt(pi)*a*sqrt(b)*e*erf(-sqrt(b*arcsin(d*x + c) + a)/sqrt(b) - I*sqrt(b*arcsin(d*x + c) + a)*sqrt(b)/ab
s(b))*e^(2*I*a/b)/((b + I*b^2/abs(b))*d) - 1/16*sqrt(pi)*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))*d) - 1/4*I*sqrt(pi)*a*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) - 1/16*sqrt(pi)*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))*d) + 1/4*I*sqrt(pi)*a*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)/(d*(sqrt(b) - I*b^(3/2)/abs(b))) - 1/4*I*
sqrt(pi)*a*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)*d*(I*b/abs(b) + 1)) - 1/8*sqrt(b*arcsin(d*x + c) + a)*e*e^(2*I*arcsin(d*x + c))/d - 1/8*sqrt(b*ar
csin(d*x + c) + a)*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 )\,\sqrt {a+b\,\mathrm {asin}\left (c+d\,x\right )} \,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))^(1/2),x)

[Out]

int((c*e + d*e*x)*(a + b*asin(c + d*x))^(1/2), x)

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