Integrand size = 16, antiderivative size = 287 \[ \int \frac {x}{(a+b \arcsin (c+d x))^{3/2}} \, dx=\frac {2 c \sqrt {1-(c+d x)^2}}{b d^2 \sqrt {a+b \arcsin (c+d x)}}-\frac {2 (c+d x) \sqrt {1-(c+d x)^2}}{b d^2 \sqrt {a+b \arcsin (c+d x)}}+\frac {2 \sqrt {\pi } \cos \left (\frac {2 a}{b}\right ) \operatorname {FresnelC}\left (\frac {2 \sqrt {a+b \arcsin (c+d x)}}{\sqrt {b} \sqrt {\pi }}\right )}{b^{3/2} d^2}+\frac {2 c \sqrt {2 \pi } \cos \left (\frac {a}{b}\right ) \operatorname {FresnelS}\left (\frac {\sqrt {\frac {2}{\pi }} \sqrt {a+b \arcsin (c+d x)}}{\sqrt {b}}\right )}{b^{3/2} d^2}-\frac {2 c \sqrt {2 \pi } \operatorname {FresnelC}\left (\frac {\sqrt {\frac {2}{\pi }} \sqrt {a+b \arcsin (c+d x)}}{\sqrt {b}}\right ) \sin \left (\frac {a}{b}\right )}{b^{3/2} d^2}+\frac {2 \sqrt {\pi } \operatorname {FresnelS}\left (\frac {2 \sqrt {a+b \arcsin (c+d x)}}{\sqrt {b} \sqrt {\pi }}\right ) \sin \left (\frac {2 a}{b}\right )}{b^{3/2} d^2} \]
2*cos(2*a/b)*FresnelC(2*(a+b*arcsin(d*x+c))^(1/2)/b^(1/2)/Pi^(1/2))*Pi^(1/ 2)/b^(3/2)/d^2+2*FresnelS(2*(a+b*arcsin(d*x+c))^(1/2)/b^(1/2)/Pi^(1/2))*si n(2*a/b)*Pi^(1/2)/b^(3/2)/d^2+2*c*cos(a/b)*FresnelS(2^(1/2)/Pi^(1/2)*(a+b* arcsin(d*x+c))^(1/2)/b^(1/2))*2^(1/2)*Pi^(1/2)/b^(3/2)/d^2-2*c*FresnelC(2^ (1/2)/Pi^(1/2)*(a+b*arcsin(d*x+c))^(1/2)/b^(1/2))*sin(a/b)*2^(1/2)*Pi^(1/2 )/b^(3/2)/d^2+2*c*(1-(d*x+c)^2)^(1/2)/b/d^2/(a+b*arcsin(d*x+c))^(1/2)-2*(d *x+c)*(1-(d*x+c)^2)^(1/2)/b/d^2/(a+b*arcsin(d*x+c))^(1/2)
Time = 4.14 (sec) , antiderivative size = 251, normalized size of antiderivative = 0.87 \[ \int \frac {x}{(a+b \arcsin (c+d x))^{3/2}} \, dx=\frac {\frac {2 \sqrt {b} c \sqrt {1-(c+d x)^2}}{\sqrt {a+b \arcsin (c+d x)}}+2 \sqrt {\pi } \cos \left (\frac {2 a}{b}\right ) \operatorname {FresnelC}\left (\frac {2 \sqrt {a+b \arcsin (c+d x)}}{\sqrt {b} \sqrt {\pi }}\right )+2 c \sqrt {2 \pi } \cos \left (\frac {a}{b}\right ) \operatorname {FresnelS}\left (\frac {\sqrt {\frac {2}{\pi }} \sqrt {a+b \arcsin (c+d x)}}{\sqrt {b}}\right )-2 c \sqrt {2 \pi } \operatorname {FresnelC}\left (\frac {\sqrt {\frac {2}{\pi }} \sqrt {a+b \arcsin (c+d x)}}{\sqrt {b}}\right ) \sin \left (\frac {a}{b}\right )+2 \sqrt {\pi } \operatorname {FresnelS}\left (\frac {2 \sqrt {a+b \arcsin (c+d x)}}{\sqrt {b} \sqrt {\pi }}\right ) \sin \left (\frac {2 a}{b}\right )-\frac {\sqrt {b} \sin (2 \arcsin (c+d x))}{\sqrt {a+b \arcsin (c+d x)}}}{b^{3/2} d^2} \]
((2*Sqrt[b]*c*Sqrt[1 - (c + d*x)^2])/Sqrt[a + b*ArcSin[c + d*x]] + 2*Sqrt[ Pi]*Cos[(2*a)/b]*FresnelC[(2*Sqrt[a + b*ArcSin[c + d*x]])/(Sqrt[b]*Sqrt[Pi ])] + 2*c*Sqrt[2*Pi]*Cos[a/b]*FresnelS[(Sqrt[2/Pi]*Sqrt[a + b*ArcSin[c + d *x]])/Sqrt[b]] - 2*c*Sqrt[2*Pi]*FresnelC[(Sqrt[2/Pi]*Sqrt[a + b*ArcSin[c + d*x]])/Sqrt[b]]*Sin[a/b] + 2*Sqrt[Pi]*FresnelS[(2*Sqrt[a + b*ArcSin[c + d *x]])/(Sqrt[b]*Sqrt[Pi])]*Sin[(2*a)/b] - (Sqrt[b]*Sin[2*ArcSin[c + d*x]])/ Sqrt[a + b*ArcSin[c + d*x]])/(b^(3/2)*d^2)
Time = 0.72 (sec) , antiderivative size = 274, normalized size of antiderivative = 0.95, number of steps used = 6, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.312, Rules used = {5304, 25, 27, 5244, 2009}
Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.
| \(\displaystyle \int \frac {x}{(a+b \arcsin (c+d x))^{3/2}} \, dx\) |
| \(\Big \downarrow \) 5304 |
| \(\displaystyle \frac {\int \frac {x}{(a+b \arcsin (c+d x))^{3/2}}d(c+d x)}{d}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle -\frac {\int -\frac {x}{(a+b \arcsin (c+d x))^{3/2}}d(c+d x)}{d}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle -\frac {\int -\frac {d x}{(a+b \arcsin (c+d x))^{3/2}}d(c+d x)}{d^2}\) |
| \(\Big \downarrow \) 5244 |
| \(\displaystyle -\frac {\int \left (\frac {c}{(a+b \arcsin (c+d x))^{3/2}}-\frac {c+d x}{(a+b \arcsin (c+d x))^{3/2}}\right )d(c+d x)}{d^2}\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle -\frac {\frac {2 \sqrt {2 \pi } c \sin \left (\frac {a}{b}\right ) \operatorname {FresnelC}\left (\frac {\sqrt {\frac {2}{\pi }} \sqrt {a+b \arcsin (c+d x)}}{\sqrt {b}}\right )}{b^{3/2}}-\frac {2 \sqrt {\pi } \cos \left (\frac {2 a}{b}\right ) \operatorname {FresnelC}\left (\frac {2 \sqrt {a+b \arcsin (c+d x)}}{\sqrt {b} \sqrt {\pi }}\right )}{b^{3/2}}-\frac {2 \sqrt {\pi } \sin \left (\frac {2 a}{b}\right ) \operatorname {FresnelS}\left (\frac {2 \sqrt {a+b \arcsin (c+d x)}}{\sqrt {b} \sqrt {\pi }}\right )}{b^{3/2}}-\frac {2 \sqrt {2 \pi } c \cos \left (\frac {a}{b}\right ) \operatorname {FresnelS}\left (\frac {\sqrt {\frac {2}{\pi }} \sqrt {a+b \arcsin (c+d x)}}{\sqrt {b}}\right )}{b^{3/2}}-\frac {2 c \sqrt {1-(c+d x)^2}}{b \sqrt {a+b \arcsin (c+d x)}}+\frac {2 (c+d x) \sqrt {1-(c+d x)^2}}{b \sqrt {a+b \arcsin (c+d x)}}}{d^2}\) |
-(((-2*c*Sqrt[1 - (c + d*x)^2])/(b*Sqrt[a + b*ArcSin[c + d*x]]) + (2*(c + d*x)*Sqrt[1 - (c + d*x)^2])/(b*Sqrt[a + b*ArcSin[c + d*x]]) - (2*Sqrt[Pi]* Cos[(2*a)/b]*FresnelC[(2*Sqrt[a + b*ArcSin[c + d*x]])/(Sqrt[b]*Sqrt[Pi])]) /b^(3/2) - (2*c*Sqrt[2*Pi]*Cos[a/b]*FresnelS[(Sqrt[2/Pi]*Sqrt[a + b*ArcSin [c + d*x]])/Sqrt[b]])/b^(3/2) + (2*c*Sqrt[2*Pi]*FresnelC[(Sqrt[2/Pi]*Sqrt[ a + b*ArcSin[c + d*x]])/Sqrt[b]]*Sin[a/b])/b^(3/2) - (2*Sqrt[Pi]*FresnelS[ (2*Sqrt[a + b*ArcSin[c + d*x]])/(Sqrt[b]*Sqrt[Pi])]*Sin[(2*a)/b])/b^(3/2)) /d^2)
3.2.66.3.1 Defintions of rubi rules used
Int[(a_)*(Fx_), x_Symbol] :> Simp[a Int[Fx, x], x] /; FreeQ[a, x] && !Ma tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
Int[((a_.) + ArcSin[(c_.)*(x_)]*(b_.))^(n_)*((d_) + (e_.)*(x_))^(m_.), x_Sy mbol] :> Int[ExpandIntegrand[(d + e*x)^m*(a + b*ArcSin[c*x])^n, x], x] /; F reeQ[{a, b, c, d, e}, x] && IGtQ[m, 0] && LtQ[n, -1]
Int[((a_.) + ArcSin[(c_) + (d_.)*(x_)]*(b_.))^(n_.)*((e_.) + (f_.)*(x_))^(m _.), x_Symbol] :> Simp[1/d Subst[Int[((d*e - c*f)/d + f*(x/d))^m*(a + b*A rcSin[x])^n, x], x, c + d*x], x] /; FreeQ[{a, b, c, d, e, f, m, n}, x]
Time = 1.01 (sec) , antiderivative size = 321, normalized size of antiderivative = 1.12
| method | result | size |
| default | \(\frac {-2 \sqrt {-\frac {1}{b}}\, \sqrt {\pi }\, \sqrt {2}\, \sqrt {a +b \arcsin \left (d x +c \right )}\, \cos \left (\frac {a}{b}\right ) \operatorname {FresnelS}\left (\frac {\sqrt {2}\, \sqrt {a +b \arcsin \left (d x +c \right )}}{\sqrt {\pi }\, \sqrt {-\frac {1}{b}}\, b}\right ) c -2 \sqrt {-\frac {1}{b}}\, \sqrt {\pi }\, \sqrt {2}\, \sqrt {a +b \arcsin \left (d x +c \right )}\, \sin \left (\frac {a}{b}\right ) \operatorname {FresnelC}\left (\frac {\sqrt {2}\, \sqrt {a +b \arcsin \left (d x +c \right )}}{\sqrt {\pi }\, \sqrt {-\frac {1}{b}}\, b}\right ) c +2 \sqrt {-\frac {1}{b}}\, \sqrt {\pi }\, \sqrt {a +b \arcsin \left (d x +c \right )}\, \cos \left (\frac {2 a}{b}\right ) \operatorname {FresnelC}\left (\frac {2 \sqrt {2}\, \sqrt {a +b \arcsin \left (d x +c \right )}}{\sqrt {\pi }\, \sqrt {-\frac {2}{b}}\, b}\right )-2 \sqrt {-\frac {1}{b}}\, \sqrt {\pi }\, \sqrt {a +b \arcsin \left (d x +c \right )}\, \sin \left (\frac {2 a}{b}\right ) \operatorname {FresnelS}\left (\frac {2 \sqrt {2}\, \sqrt {a +b \arcsin \left (d x +c \right )}}{\sqrt {\pi }\, \sqrt {-\frac {2}{b}}\, b}\right )+2 \cos \left (-\frac {a +b \arcsin \left (d x +c \right )}{b}+\frac {a}{b}\right ) c +\sin \left (-\frac {2 \left (a +b \arcsin \left (d x +c \right )\right )}{b}+\frac {2 a}{b}\right )}{d^{2} b \sqrt {a +b \arcsin \left (d x +c \right )}}\) | \(321\) |
1/d^2/b/(a+b*arcsin(d*x+c))^(1/2)*(-2*(-1/b)^(1/2)*Pi^(1/2)*2^(1/2)*(a+b*a rcsin(d*x+c))^(1/2)*cos(a/b)*FresnelS(2^(1/2)/Pi^(1/2)/(-1/b)^(1/2)*(a+b*a rcsin(d*x+c))^(1/2)/b)*c-2*(-1/b)^(1/2)*Pi^(1/2)*2^(1/2)*(a+b*arcsin(d*x+c ))^(1/2)*sin(a/b)*FresnelC(2^(1/2)/Pi^(1/2)/(-1/b)^(1/2)*(a+b*arcsin(d*x+c ))^(1/2)/b)*c+2*(-1/b)^(1/2)*Pi^(1/2)*(a+b*arcsin(d*x+c))^(1/2)*cos(2*a/b) *FresnelC(2*2^(1/2)/Pi^(1/2)/(-2/b)^(1/2)*(a+b*arcsin(d*x+c))^(1/2)/b)-2*( -1/b)^(1/2)*Pi^(1/2)*(a+b*arcsin(d*x+c))^(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)+2*cos(-(a+b*arcsin(d *x+c))/b+a/b)*c+sin(-2*(a+b*arcsin(d*x+c))/b+2*a/b))
Exception generated. \[ \int \frac {x}{(a+b \arcsin (c+d x))^{3/2}} \, dx=\text {Exception raised: TypeError} \]
Exception raised: TypeError >> Error detected within library code: inte grate: implementation incomplete (constant residues)
\[ \int \frac {x}{(a+b \arcsin (c+d x))^{3/2}} \, dx=\int \frac {x}{\left (a + b \operatorname {asin}{\left (c + d x \right )}\right )^{\frac {3}{2}}}\, dx \]
\[ \int \frac {x}{(a+b \arcsin (c+d x))^{3/2}} \, dx=\int { \frac {x}{{\left (b \arcsin \left (d x + c\right ) + a\right )}^{\frac {3}{2}}} \,d x } \]
\[ \int \frac {x}{(a+b \arcsin (c+d x))^{3/2}} \, dx=\int { \frac {x}{{\left (b \arcsin \left (d x + c\right ) + a\right )}^{\frac {3}{2}}} \,d x } \]
Timed out. \[ \int \frac {x}{(a+b \arcsin (c+d x))^{3/2}} \, dx=\int \frac {x}{{\left (a+b\,\mathrm {asin}\left (c+d\,x\right )\right )}^{3/2}} \,d x \]