3.242 \(\int (c e+d e x) \sqrt {a+b \sin ^{-1}(c+d x)} \, dx\)

Optimal. Leaf size=156 \[ \frac {\sqrt {\pi } \sqrt {b} e \cos \left (\frac {2 a}{b}\right ) C\left (\frac {2 \sqrt {a+b \sin ^{-1}(c+d x)}}{\sqrt {b} \sqrt {\pi }}\right )}{8 d}+\frac {\sqrt {\pi } \sqrt {b} e \sin \left (\frac {2 a}{b}\right ) S\left (\frac {2 \sqrt {a+b \sin ^{-1}(c+d x)}}{\sqrt {b} \sqrt {\pi }}\right )}{8 d}+\frac {e (c+d x)^2 \sqrt {a+b \sin ^{-1}(c+d x)}}{2 d}-\frac {e \sqrt {a+b \sin ^{-1}(c+d x)}}{4 d} \]

[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 = 0.42, 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 = {4805, 12, 4629, 4723, 3312, 3306, 3305, 3351, 3304, 3352} \[ \frac {\sqrt {\pi } \sqrt {b} e \cos \left (\frac {2 a}{b}\right ) \text {FresnelC}\left (\frac {2 \sqrt {a+b \sin ^{-1}(c+d x)}}{\sqrt {\pi } \sqrt {b}}\right )}{8 d}+\frac {\sqrt {\pi } \sqrt {b} e \sin \left (\frac {2 a}{b}\right ) S\left (\frac {2 \sqrt {a+b \sin ^{-1}(c+d x)}}{\sqrt {b} \sqrt {\pi }}\right )}{8 d}+\frac {e (c+d x)^2 \sqrt {a+b \sin ^{-1}(c+d x)}}{2 d}-\frac {e \sqrt {a+b \sin ^{-1}(c+d x)}}{4 d} \]

Antiderivative was successfully verified.

[In]

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

[Out]

-(e*Sqrt[a + b*ArcSin[c + d*x]])/(4*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 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 3312

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 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 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 4723

Int[((a_.) + ArcSin[(c_.)*(x_)]*(b_.))^(n_.)*(x_)^(m_.)*((d_) + (e_.)*(x_)^2)^(p_.), x_Symbol] :> Dist[d^p/c^(
m + 1), Subst[Int[(a + b*x)^n*Sin[x]^m*Cos[x]^(2*p + 1), x], x, ArcSin[c*x]], x] /; FreeQ[{a, b, c, d, e, n},
x] && EqQ[c^2*d + e, 0] && IntegerQ[2*p] && GtQ[p, -1] && IGtQ[m, 0] && (IntegerQ[p] || GtQ[d, 0])

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) \sqrt {a+b \sin ^{-1}(c+d x)} \, dx &=\frac {\operatorname {Subst}\left (\int e x \sqrt {a+b \sin ^{-1}(x)} \, dx,x,c+d x\right )}{d}\\ &=\frac {e \operatorname {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) \operatorname {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) \operatorname {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) \operatorname {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) \operatorname {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 ) \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 )}{8 d}+\frac {\left (b e \sin \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 )}{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 ) \operatorname {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 ) \operatorname {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]  time = 0.08, size = 154, normalized size = 0.99 \[ -\frac {e e^{-\frac {2 i a}{b}} \sqrt {a+b \sin ^{-1}(c+d x)} \left (\sqrt {\frac {i \left (a+b \sin ^{-1}(c+d x)\right )}{b}} \Gamma \left (\frac {3}{2},-\frac {2 i \left (a+b \sin ^{-1}(c+d x)\right )}{b}\right )+e^{\frac {4 i a}{b}} \sqrt {-\frac {i \left (a+b \sin ^{-1}(c+d x)\right )}{b}} \Gamma \left (\frac {3}{2},\frac {2 i \left (a+b \sin ^{-1}(c+d x)\right )}{b}\right )\right )}{8 \sqrt {2} d \sqrt {\frac {\left (a+b \sin ^{-1}(c+d x)\right )^2}{b^2}}} \]

Warning: Unable to verify antiderivative.

[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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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))^(1/2),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 = 1.85, size = 517, normalized size = 3.31 \[ \frac {\sqrt {\pi } a \sqrt {b} i \operatorname {erf}\left (-\frac {\sqrt {b \arcsin \left (d x + c\right ) + a} \sqrt {b} i}{{\left | b \right |}} - \frac {\sqrt {b \arcsin \left (d x + c\right ) + a}}{\sqrt {b}}\right ) e^{\left (\frac {2 \, a i}{b} + 1\right )}}{4 \, {\left (\frac {b^{2} i}{{\left | b \right |}} + b\right )} d} + \frac {\sqrt {\pi } a \sqrt {b} i \operatorname {erf}\left (\frac {\sqrt {b \arcsin \left (d x + c\right ) + a} \sqrt {b} i}{{\left | b \right |}} - \frac {\sqrt {b \arcsin \left (d x + c\right ) + a}}{\sqrt {b}}\right ) e^{\left (-\frac {2 \, a i}{b} + 1\right )}}{4 \, {\left (\frac {b^{2} i}{{\left | b \right |}} - b\right )} d} - \frac {\sqrt {\pi } a i \operatorname {erf}\left (\frac {\sqrt {b \arcsin \left (d x + c\right ) + a} \sqrt {b} i}{{\left | b \right |}} - \frac {\sqrt {b \arcsin \left (d x + c\right ) + a}}{\sqrt {b}}\right ) e^{\left (-\frac {2 \, a i}{b} + 1\right )}}{4 \, {\left (\frac {b^{\frac {3}{2}} i}{{\left | b \right |}} - \sqrt {b}\right )} d} - \frac {\sqrt {\pi } b^{\frac {3}{2}} \operatorname {erf}\left (-\frac {\sqrt {b \arcsin \left (d x + c\right ) + a} \sqrt {b} i}{{\left | b \right |}} - \frac {\sqrt {b \arcsin \left (d x + c\right ) + a}}{\sqrt {b}}\right ) e^{\left (\frac {2 \, a i}{b} + 1\right )}}{16 \, {\left (\frac {b^{2} i}{{\left | b \right |}} + b\right )} d} - \frac {\sqrt {\pi } a i \operatorname {erf}\left (-\frac {\sqrt {b \arcsin \left (d x + c\right ) + a} \sqrt {b} i}{{\left | b \right |}} - \frac {\sqrt {b \arcsin \left (d x + c\right ) + a}}{\sqrt {b}}\right ) e^{\left (\frac {2 \, a i}{b} + 1\right )}}{4 \, \sqrt {b} d {\left (\frac {b i}{{\left | b \right |}} + 1\right )}} + \frac {\sqrt {\pi } b^{\frac {3}{2}} \operatorname {erf}\left (\frac {\sqrt {b \arcsin \left (d x + c\right ) + a} \sqrt {b} i}{{\left | b \right |}} - \frac {\sqrt {b \arcsin \left (d x + c\right ) + a}}{\sqrt {b}}\right ) e^{\left (-\frac {2 \, a i}{b} + 1\right )}}{16 \, {\left (\frac {b^{2} i}{{\left | b \right |}} - b\right )} d} - \frac {\sqrt {b \arcsin \left (d x + c\right ) + a} e^{\left (2 \, i \arcsin \left (d x + c\right ) + 1\right )}}{8 \, d} - \frac {\sqrt {b \arcsin \left (d x + c\right ) + a} e^{\left (-2 \, i \arcsin \left (d x + c\right ) + 1\right )}}{8 \, d} \]

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*sqrt(pi)*a*sqrt(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) + 1/4*sqrt(pi)*a*sqrt(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) - 1/4*sqrt(pi)*a*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^(3/2
)*i/abs(b) - sqrt(b))*d) - 1/16*sqrt(pi)*b^(3/2)*erf(-sqrt(b*arcsin(d*x + c) + a)*sqrt(b)*i/abs(b) - sqrt(b*ar
csin(d*x + c) + a)/sqrt(b))*e^(2*a*i/b + 1)/((b^2*i/abs(b) + b)*d) - 1/4*sqrt(pi)*a*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)/(sqrt(b)*d*(b*i/abs(b) + 1))
+ 1/16*sqrt(pi)*b^(3/2)*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) - 1/8*sqrt(b*arcsin(d*x + c) + a)*e^(2*i*arcsin(d*x + c) + 1)/d - 1/
8*sqrt(b*arcsin(d*x + c) + a)*e^(-2*i*arcsin(d*x + c) + 1)/d

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maple [A]  time = 0.21, size = 190, normalized size = 1.22 \[ -\frac {e \left (-\cos \left (\frac {2 a}{b}\right ) \FresnelC \left (\frac {2 \sqrt {a +b \arcsin \left (d x +c \right )}}{\sqrt {\pi }\, \sqrt {\frac {1}{b}}\, b}\right ) \sqrt {a +b \arcsin \left (d x +c \right )}\, \sqrt {\frac {1}{b}}\, \sqrt {\pi }\, b -\sin \left (\frac {2 a}{b}\right ) \mathrm {S}\left (\frac {2 \sqrt {a +b \arcsin \left (d x +c \right )}}{\sqrt {\pi }\, \sqrt {\frac {1}{b}}\, b}\right ) \sqrt {a +b \arcsin \left (d x +c \right )}\, \sqrt {\frac {1}{b}}\, \sqrt {\pi }\, b +2 \arcsin \left (d x +c \right ) \cos \left (\frac {2 a +2 b \arcsin \left (d x +c \right )}{b}-\frac {2 a}{b}\right ) b +2 \cos \left (\frac {2 a +2 b \arcsin \left (d x +c \right )}{b}-\frac {2 a}{b}\right ) a \right )}{8 d \sqrt {a +b \arcsin \left (d x +c \right )}} \]

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)

[Out]

-1/8*e/d/(a+b*arcsin(d*x+c))^(1/2)*(-cos(2*a/b)*FresnelC(2/Pi^(1/2)/(1/b)^(1/2)*(a+b*arcsin(d*x+c))^(1/2)/b)*(
a+b*arcsin(d*x+c))^(1/2)*(1/b)^(1/2)*Pi^(1/2)*b-sin(2*a/b)*FresnelS(2/Pi^(1/2)/(1/b)^(1/2)*(a+b*arcsin(d*x+c))
^(1/2)/b)*(a+b*arcsin(d*x+c))^(1/2)*(1/b)^(1/2)*Pi^(1/2)*b+2*arcsin(d*x+c)*cos(2*(a+b*arcsin(d*x+c))/b-2*a/b)*
b+2*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 \[ \int {\left (d e x + c e\right )} \sqrt {b \arcsin \left (d x + c\right ) + a}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((d*e*x+c*e)*(a+b*arcsin(d*x+c))^(1/2),x, algorithm="maxima")

[Out]

integrate((d*e*x + c*e)*sqrt(b*arcsin(d*x + c) + a), x)

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mupad [F]  time = 0.00, size = -1, normalized size = -0.01 \[ \int \left (c\,e+d\,e\,x\right )\,\sqrt {a+b\,\mathrm {asin}\left (c+d\,x\right )} \,d x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

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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