3.248 \(\int (a+b \sin ^{-1}(c+d x))^{3/2} \, dx\)

Optimal. Leaf size=175 \[ -\frac{3 \sqrt{\frac{\pi }{2}} b^{3/2} \cos \left (\frac{a}{b}\right ) \text{FresnelC}\left (\frac{\sqrt{\frac{2}{\pi }} \sqrt{a+b \sin ^{-1}(c+d x)}}{\sqrt{b}}\right )}{2 d}-\frac{3 \sqrt{\frac{\pi }{2}} b^{3/2} \sin \left (\frac{a}{b}\right ) S\left (\frac{\sqrt{\frac{2}{\pi }} \sqrt{a+b \sin ^{-1}(c+d x)}}{\sqrt{b}}\right )}{2 d}+\frac{3 b \sqrt{1-(c+d x)^2} \sqrt{a+b \sin ^{-1}(c+d x)}}{2 d}+\frac{(c+d x) \left (a+b \sin ^{-1}(c+d x)\right )^{3/2}}{d} \]

[Out]

(3*b*Sqrt[1 - (c + d*x)^2]*Sqrt[a + b*ArcSin[c + d*x]])/(2*d) + ((c + d*x)*(a + b*ArcSin[c + d*x])^(3/2))/d -
(3*b^(3/2)*Sqrt[Pi/2]*Cos[a/b]*FresnelC[(Sqrt[2/Pi]*Sqrt[a + b*ArcSin[c + d*x]])/Sqrt[b]])/(2*d) - (3*b^(3/2)*
Sqrt[Pi/2]*FresnelS[(Sqrt[2/Pi]*Sqrt[a + b*ArcSin[c + d*x]])/Sqrt[b]]*Sin[a/b])/(2*d)

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Rubi [A]  time = 0.263416, antiderivative size = 175, normalized size of antiderivative = 1., number of steps used = 9, number of rules used = 9, integrand size = 14, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.643, Rules used = {4803, 4619, 4677, 4623, 3306, 3305, 3351, 3304, 3352} \[ -\frac{3 \sqrt{\frac{\pi }{2}} b^{3/2} \cos \left (\frac{a}{b}\right ) \text{FresnelC}\left (\frac{\sqrt{\frac{2}{\pi }} \sqrt{a+b \sin ^{-1}(c+d x)}}{\sqrt{b}}\right )}{2 d}-\frac{3 \sqrt{\frac{\pi }{2}} b^{3/2} \sin \left (\frac{a}{b}\right ) S\left (\frac{\sqrt{\frac{2}{\pi }} \sqrt{a+b \sin ^{-1}(c+d x)}}{\sqrt{b}}\right )}{2 d}+\frac{3 b \sqrt{1-(c+d x)^2} \sqrt{a+b \sin ^{-1}(c+d x)}}{2 d}+\frac{(c+d x) \left (a+b \sin ^{-1}(c+d x)\right )^{3/2}}{d} \]

Antiderivative was successfully verified.

[In]

Int[(a + b*ArcSin[c + d*x])^(3/2),x]

[Out]

(3*b*Sqrt[1 - (c + d*x)^2]*Sqrt[a + b*ArcSin[c + d*x]])/(2*d) + ((c + d*x)*(a + b*ArcSin[c + d*x])^(3/2))/d -
(3*b^(3/2)*Sqrt[Pi/2]*Cos[a/b]*FresnelC[(Sqrt[2/Pi]*Sqrt[a + b*ArcSin[c + d*x]])/Sqrt[b]])/(2*d) - (3*b^(3/2)*
Sqrt[Pi/2]*FresnelS[(Sqrt[2/Pi]*Sqrt[a + b*ArcSin[c + d*x]])/Sqrt[b]]*Sin[a/b])/(2*d)

Rule 4803

Int[((a_.) + ArcSin[(c_) + (d_.)*(x_)]*(b_.))^(n_.), x_Symbol] :> Dist[1/d, Subst[Int[(a + b*ArcSin[x])^n, x],
x, c + d*x], x] /; FreeQ[{a, b, c, d, n}, x]

Rule 4619

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

Rule 4677

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

Rule 4623

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

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 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 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 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 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]

Rubi steps

\begin{align*} \int \left (a+b \sin ^{-1}(c+d x)\right )^{3/2} \, dx &=\frac{\operatorname{Subst}\left (\int \left (a+b \sin ^{-1}(x)\right )^{3/2} \, dx,x,c+d x\right )}{d}\\ &=\frac{(c+d x) \left (a+b \sin ^{-1}(c+d x)\right )^{3/2}}{d}-\frac{(3 b) \operatorname{Subst}\left (\int \frac{x \sqrt{a+b \sin ^{-1}(x)}}{\sqrt{1-x^2}} \, dx,x,c+d x\right )}{2 d}\\ &=\frac{3 b \sqrt{1-(c+d x)^2} \sqrt{a+b \sin ^{-1}(c+d x)}}{2 d}+\frac{(c+d x) \left (a+b \sin ^{-1}(c+d x)\right )^{3/2}}{d}-\frac{\left (3 b^2\right ) \operatorname{Subst}\left (\int \frac{1}{\sqrt{a+b \sin ^{-1}(x)}} \, dx,x,c+d x\right )}{4 d}\\ &=\frac{3 b \sqrt{1-(c+d x)^2} \sqrt{a+b \sin ^{-1}(c+d x)}}{2 d}+\frac{(c+d x) \left (a+b \sin ^{-1}(c+d x)\right )^{3/2}}{d}-\frac{(3 b) \operatorname{Subst}\left (\int \frac{\cos \left (\frac{a}{b}-\frac{x}{b}\right )}{\sqrt{x}} \, dx,x,a+b \sin ^{-1}(c+d x)\right )}{4 d}\\ &=\frac{3 b \sqrt{1-(c+d x)^2} \sqrt{a+b \sin ^{-1}(c+d x)}}{2 d}+\frac{(c+d x) \left (a+b \sin ^{-1}(c+d x)\right )^{3/2}}{d}-\frac{\left (3 b \cos \left (\frac{a}{b}\right )\right ) \operatorname{Subst}\left (\int \frac{\cos \left (\frac{x}{b}\right )}{\sqrt{x}} \, dx,x,a+b \sin ^{-1}(c+d x)\right )}{4 d}-\frac{\left (3 b \sin \left (\frac{a}{b}\right )\right ) \operatorname{Subst}\left (\int \frac{\sin \left (\frac{x}{b}\right )}{\sqrt{x}} \, dx,x,a+b \sin ^{-1}(c+d x)\right )}{4 d}\\ &=\frac{3 b \sqrt{1-(c+d x)^2} \sqrt{a+b \sin ^{-1}(c+d x)}}{2 d}+\frac{(c+d x) \left (a+b \sin ^{-1}(c+d x)\right )^{3/2}}{d}-\frac{\left (3 b \cos \left (\frac{a}{b}\right )\right ) \operatorname{Subst}\left (\int \cos \left (\frac{x^2}{b}\right ) \, dx,x,\sqrt{a+b \sin ^{-1}(c+d x)}\right )}{2 d}-\frac{\left (3 b \sin \left (\frac{a}{b}\right )\right ) \operatorname{Subst}\left (\int \sin \left (\frac{x^2}{b}\right ) \, dx,x,\sqrt{a+b \sin ^{-1}(c+d x)}\right )}{2 d}\\ &=\frac{3 b \sqrt{1-(c+d x)^2} \sqrt{a+b \sin ^{-1}(c+d x)}}{2 d}+\frac{(c+d x) \left (a+b \sin ^{-1}(c+d x)\right )^{3/2}}{d}-\frac{3 b^{3/2} \sqrt{\frac{\pi }{2}} \cos \left (\frac{a}{b}\right ) C\left (\frac{\sqrt{\frac{2}{\pi }} \sqrt{a+b \sin ^{-1}(c+d x)}}{\sqrt{b}}\right )}{2 d}-\frac{3 b^{3/2} \sqrt{\frac{\pi }{2}} S\left (\frac{\sqrt{\frac{2}{\pi }} \sqrt{a+b \sin ^{-1}(c+d x)}}{\sqrt{b}}\right ) \sin \left (\frac{a}{b}\right )}{2 d}\\ \end{align*}

Mathematica [C]  time = 2.90637, size = 313, normalized size = 1.79 \[ \frac{b \left (\frac{2 a e^{-\frac{i a}{b}} \left (\sqrt{-\frac{i \left (a+b \sin ^{-1}(c+d x)\right )}{b}} \text{Gamma}\left (\frac{3}{2},-\frac{i \left (a+b \sin ^{-1}(c+d x)\right )}{b}\right )+e^{\frac{2 i a}{b}} \sqrt{\frac{i \left (a+b \sin ^{-1}(c+d x)\right )}{b}} \text{Gamma}\left (\frac{3}{2},\frac{i \left (a+b \sin ^{-1}(c+d x)\right )}{b}\right )\right )}{\sqrt{a+b \sin ^{-1}(c+d x)}}-\sqrt{2 \pi } \sqrt{\frac{1}{b}} \left (2 a \sin \left (\frac{a}{b}\right )+3 b \cos \left (\frac{a}{b}\right )\right ) \text{FresnelC}\left (\sqrt{\frac{2}{\pi }} \sqrt{\frac{1}{b}} \sqrt{a+b \sin ^{-1}(c+d x)}\right )+\sqrt{2 \pi } \sqrt{\frac{1}{b}} \left (2 a \cos \left (\frac{a}{b}\right )-3 b \sin \left (\frac{a}{b}\right )\right ) S\left (\sqrt{\frac{1}{b}} \sqrt{\frac{2}{\pi }} \sqrt{a+b \sin ^{-1}(c+d x)}\right )+2 \left (3 \sqrt{1-(c+d x)^2}+2 (c+d x) \sin ^{-1}(c+d x)\right ) \sqrt{a+b \sin ^{-1}(c+d x)}\right )}{4 d} \]

Warning: Unable to verify antiderivative.

[In]

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

[Out]

(b*(2*Sqrt[a + b*ArcSin[c + d*x]]*(3*Sqrt[1 - (c + d*x)^2] + 2*(c + d*x)*ArcSin[c + d*x]) + (2*a*(Sqrt[((-I)*(
a + b*ArcSin[c + d*x]))/b]*Gamma[3/2, ((-I)*(a + b*ArcSin[c + d*x]))/b] + E^(((2*I)*a)/b)*Sqrt[(I*(a + b*ArcSi
n[c + d*x]))/b]*Gamma[3/2, (I*(a + b*ArcSin[c + d*x]))/b]))/(E^((I*a)/b)*Sqrt[a + b*ArcSin[c + d*x]]) - Sqrt[b
^(-1)]*Sqrt[2*Pi]*FresnelC[Sqrt[b^(-1)]*Sqrt[2/Pi]*Sqrt[a + b*ArcSin[c + d*x]]]*(3*b*Cos[a/b] + 2*a*Sin[a/b])
+ Sqrt[b^(-1)]*Sqrt[2*Pi]*FresnelS[Sqrt[b^(-1)]*Sqrt[2/Pi]*Sqrt[a + b*ArcSin[c + d*x]]]*(2*a*Cos[a/b] - 3*b*Si
n[a/b])))/(4*d)

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Maple [B]  time = 0., size = 296, normalized size = 1.7 \begin{align*}{\frac{1}{4\,d} \left ( -3\,\sqrt{{b}^{-1}}\sqrt{\pi }\sqrt{2}\sqrt{a+b\arcsin \left ( dx+c \right ) }\cos \left ({\frac{a}{b}} \right ){\it FresnelC} \left ({\frac{\sqrt{2}\sqrt{a+b\arcsin \left ( dx+c \right ) }}{\sqrt{{b}^{-1}}\sqrt{\pi }b}} \right ){b}^{2}-3\,\sqrt{{b}^{-1}}\sqrt{\pi }\sqrt{2}\sqrt{a+b\arcsin \left ( dx+c \right ) }\sin \left ({\frac{a}{b}} \right ){\it FresnelS} \left ({\frac{\sqrt{2}\sqrt{a+b\arcsin \left ( dx+c \right ) }}{\sqrt{{b}^{-1}}\sqrt{\pi }b}} \right ){b}^{2}+4\, \left ( \arcsin \left ( dx+c \right ) \right ) ^{2}\sin \left ({\frac{a+b\arcsin \left ( dx+c \right ) }{b}}-{\frac{a}{b}} \right ){b}^{2}+8\,\arcsin \left ( dx+c \right ) \sin \left ({\frac{a+b\arcsin \left ( dx+c \right ) }{b}}-{\frac{a}{b}} \right ) ab+6\,\arcsin \left ( dx+c \right ) \cos \left ({\frac{a+b\arcsin \left ( dx+c \right ) }{b}}-{\frac{a}{b}} \right ){b}^{2}+4\,\sin \left ({\frac{a+b\arcsin \left ( dx+c \right ) }{b}}-{\frac{a}{b}} \right ){a}^{2}+6\,\cos \left ({\frac{a+b\arcsin \left ( dx+c \right ) }{b}}-{\frac{a}{b}} \right ) ab \right ){\frac{1}{\sqrt{a+b\arcsin \left ( dx+c \right ) }}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/4/d/(a+b*arcsin(d*x+c))^(1/2)*(-3*(1/b)^(1/2)*Pi^(1/2)*2^(1/2)*(a+b*arcsin(d*x+c))^(1/2)*cos(a/b)*FresnelC(2
^(1/2)/Pi^(1/2)/(1/b)^(1/2)*(a+b*arcsin(d*x+c))^(1/2)/b)*b^2-3*(1/b)^(1/2)*Pi^(1/2)*2^(1/2)*(a+b*arcsin(d*x+c)
)^(1/2)*sin(a/b)*FresnelS(2^(1/2)/Pi^(1/2)/(1/b)^(1/2)*(a+b*arcsin(d*x+c))^(1/2)/b)*b^2+4*arcsin(d*x+c)^2*sin(
(a+b*arcsin(d*x+c))/b-a/b)*b^2+8*arcsin(d*x+c)*sin((a+b*arcsin(d*x+c))/b-a/b)*a*b+6*arcsin(d*x+c)*cos((a+b*arc
sin(d*x+c))/b-a/b)*b^2+4*sin((a+b*arcsin(d*x+c))/b-a/b)*a^2+6*cos((a+b*arcsin(d*x+c))/b-a/b)*a*b)

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Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int{\left (b \arcsin \left (d x + c\right ) + a\right )}^{\frac{3}{2}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integrate((b*arcsin(d*x + c) + a)^(3/2), x)

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Fricas [F(-2)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: UnboundLocalError} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Exception raised: UnboundLocalError

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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \left (a + b \operatorname{asin}{\left (c + d x \right )}\right )^{\frac{3}{2}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Giac [B]  time = 1.87629, size = 980, normalized size = 5.6 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/4*sqrt(2)*sqrt(pi)*a*b^3*i*erf(-1/2*sqrt(2)*sqrt(b*arcsin(d*x + c) + a)*i/sqrt(abs(b)) - 1/2*sqrt(2)*sqrt(b
*arcsin(d*x + c) + a)*sqrt(abs(b))/b)*e^(a*i/b)/((b^3*i/sqrt(abs(b)) + b^2*sqrt(abs(b)))*d) - 1/4*sqrt(2)*sqrt
(pi)*a*b^3*i*erf(1/2*sqrt(2)*sqrt(b*arcsin(d*x + c) + a)*i/sqrt(abs(b)) - 1/2*sqrt(2)*sqrt(b*arcsin(d*x + c) +
a)*sqrt(abs(b))/b)*e^(-a*i/b)/((b^3*i/sqrt(abs(b)) - b^2*sqrt(abs(b)))*d) + 3/8*sqrt(2)*sqrt(pi)*b^4*erf(-1/2
*sqrt(2)*sqrt(b*arcsin(d*x + c) + a)*i/sqrt(abs(b)) - 1/2*sqrt(2)*sqrt(b*arcsin(d*x + c) + a)*sqrt(abs(b))/b)*
e^(a*i/b)/((b^3*i/sqrt(abs(b)) + b^2*sqrt(abs(b)))*d) + 1/4*sqrt(2)*sqrt(pi)*a*b^2*i*erf(-1/2*sqrt(2)*sqrt(b*a
rcsin(d*x + c) + a)*i/sqrt(abs(b)) - 1/2*sqrt(2)*sqrt(b*arcsin(d*x + c) + a)*sqrt(abs(b))/b)*e^(a*i/b)/((b^2*i
/sqrt(abs(b)) + b*sqrt(abs(b)))*d) - 3/8*sqrt(2)*sqrt(pi)*b^4*erf(1/2*sqrt(2)*sqrt(b*arcsin(d*x + c) + a)*i/sq
rt(abs(b)) - 1/2*sqrt(2)*sqrt(b*arcsin(d*x + c) + a)*sqrt(abs(b))/b)*e^(-a*i/b)/((b^3*i/sqrt(abs(b)) - b^2*sqr
t(abs(b)))*d) + 1/4*sqrt(2)*sqrt(pi)*a*b^2*i*erf(1/2*sqrt(2)*sqrt(b*arcsin(d*x + c) + a)*i/sqrt(abs(b)) - 1/2*
sqrt(2)*sqrt(b*arcsin(d*x + c) + a)*sqrt(abs(b))/b)*e^(-a*i/b)/((b^2*i/sqrt(abs(b)) - b*sqrt(abs(b)))*d) - 1/2
*sqrt(b*arcsin(d*x + c) + a)*b*i*arcsin(d*x + c)*e^(i*arcsin(d*x + c))/d + 1/2*sqrt(b*arcsin(d*x + c) + a)*b*i
*arcsin(d*x + c)*e^(-i*arcsin(d*x + c))/d - 1/2*sqrt(b*arcsin(d*x + c) + a)*a*i*e^(i*arcsin(d*x + c))/d + 1/2*
sqrt(b*arcsin(d*x + c) + a)*a*i*e^(-i*arcsin(d*x + c))/d + 3/4*sqrt(b*arcsin(d*x + c) + a)*b*e^(i*arcsin(d*x +
c))/d + 3/4*sqrt(b*arcsin(d*x + c) + a)*b*e^(-i*arcsin(d*x + c))/d