Integrand size = 16, antiderivative size = 468 \[ \int \frac {x}{(a+b \arcsin (c+d x))^{7/2}} \, dx=\frac {2 c \sqrt {1-(c+d x)^2}}{5 b d^2 (a+b \arcsin (c+d x))^{5/2}}-\frac {2 (c+d x) \sqrt {1-(c+d x)^2}}{5 b d^2 (a+b \arcsin (c+d x))^{5/2}}-\frac {4}{15 b^2 d^2 (a+b \arcsin (c+d x))^{3/2}}-\frac {4 c (c+d x)}{15 b^2 d^2 (a+b \arcsin (c+d x))^{3/2}}+\frac {8 (c+d x)^2}{15 b^2 d^2 (a+b \arcsin (c+d x))^{3/2}}-\frac {8 c \sqrt {1-(c+d x)^2}}{15 b^3 d^2 \sqrt {a+b \arcsin (c+d x)}}+\frac {32 (c+d x) \sqrt {1-(c+d x)^2}}{15 b^3 d^2 \sqrt {a+b \arcsin (c+d x)}}-\frac {32 \sqrt {\pi } \cos \left (\frac {2 a}{b}\right ) \operatorname {FresnelC}\left (\frac {2 \sqrt {a+b \arcsin (c+d x)}}{\sqrt {b} \sqrt {\pi }}\right )}{15 b^{7/2} d^2}-\frac {8 c \sqrt {2 \pi } \cos \left (\frac {a}{b}\right ) \operatorname {FresnelS}\left (\frac {\sqrt {\frac {2}{\pi }} \sqrt {a+b \arcsin (c+d x)}}{\sqrt {b}}\right )}{15 b^{7/2} d^2}+\frac {8 c \sqrt {2 \pi } \operatorname {FresnelC}\left (\frac {\sqrt {\frac {2}{\pi }} \sqrt {a+b \arcsin (c+d x)}}{\sqrt {b}}\right ) \sin \left (\frac {a}{b}\right )}{15 b^{7/2} d^2}-\frac {32 \sqrt {\pi } \operatorname {FresnelS}\left (\frac {2 \sqrt {a+b \arcsin (c+d x)}}{\sqrt {b} \sqrt {\pi }}\right ) \sin \left (\frac {2 a}{b}\right )}{15 b^{7/2} d^2} \]
-4/15/b^2/d^2/(a+b*arcsin(d*x+c))^(3/2)-4/15*c*(d*x+c)/b^2/d^2/(a+b*arcsin (d*x+c))^(3/2)+8/15*(d*x+c)^2/b^2/d^2/(a+b*arcsin(d*x+c))^(3/2)-32/15*cos( 2*a/b)*FresnelC(2*(a+b*arcsin(d*x+c))^(1/2)/b^(1/2)/Pi^(1/2))*Pi^(1/2)/b^( 7/2)/d^2-32/15*FresnelS(2*(a+b*arcsin(d*x+c))^(1/2)/b^(1/2)/Pi^(1/2))*sin( 2*a/b)*Pi^(1/2)/b^(7/2)/d^2-8/15*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^(7/2)/d^2+8/15*c*Fresnel C(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^(7/2)/d^2+2/5*c*(1-(d*x+c)^2)^(1/2)/b/d^2/(a+b*arcsin(d*x+c))^(5/2 )-2/5*(d*x+c)*(1-(d*x+c)^2)^(1/2)/b/d^2/(a+b*arcsin(d*x+c))^(5/2)-8/15*c*( 1-(d*x+c)^2)^(1/2)/b^3/d^2/(a+b*arcsin(d*x+c))^(1/2)+32/15*(d*x+c)*(1-(d*x +c)^2)^(1/2)/b^3/d^2/(a+b*arcsin(d*x+c))^(1/2)
Time = 5.38 (sec) , antiderivative size = 521, normalized size of antiderivative = 1.11 \[ \int \frac {x}{(a+b \arcsin (c+d x))^{7/2}} \, dx=-\frac {4 a b^{3/2} c (c+d x)+8 a^2 \sqrt {b} c \sqrt {1-(c+d x)^2}-6 b^{5/2} c \sqrt {1-(c+d x)^2}+4 b^{5/2} c (c+d x) \arcsin (c+d x)+16 a b^{3/2} c \sqrt {1-(c+d x)^2} \arcsin (c+d x)+8 b^{5/2} c \sqrt {1-(c+d x)^2} \arcsin (c+d x)^2+4 a b^{3/2} \cos (2 \arcsin (c+d x))+4 b^{5/2} \arcsin (c+d x) \cos (2 \arcsin (c+d x))+32 \sqrt {\pi } (a+b \arcsin (c+d x))^{5/2} \cos \left (\frac {2 a}{b}\right ) \operatorname {FresnelC}\left (\frac {2 \sqrt {a+b \arcsin (c+d x)}}{\sqrt {b} \sqrt {\pi }}\right )+8 c \sqrt {2 \pi } (a+b \arcsin (c+d x))^{5/2} \cos \left (\frac {a}{b}\right ) \operatorname {FresnelS}\left (\frac {\sqrt {\frac {2}{\pi }} \sqrt {a+b \arcsin (c+d x)}}{\sqrt {b}}\right )-8 c \sqrt {2 \pi } (a+b \arcsin (c+d x))^{5/2} \operatorname {FresnelC}\left (\frac {\sqrt {\frac {2}{\pi }} \sqrt {a+b \arcsin (c+d x)}}{\sqrt {b}}\right ) \sin \left (\frac {a}{b}\right )+32 \sqrt {\pi } (a+b \arcsin (c+d x))^{5/2} \operatorname {FresnelS}\left (\frac {2 \sqrt {a+b \arcsin (c+d x)}}{\sqrt {b} \sqrt {\pi }}\right ) \sin \left (\frac {2 a}{b}\right )-16 a^2 \sqrt {b} \sin (2 \arcsin (c+d x))+3 b^{5/2} \sin (2 \arcsin (c+d x))-32 a b^{3/2} \arcsin (c+d x) \sin (2 \arcsin (c+d x))-16 b^{5/2} \arcsin (c+d x)^2 \sin (2 \arcsin (c+d x))}{15 b^{7/2} d^2 (a+b \arcsin (c+d x))^{5/2}} \]
-1/15*(4*a*b^(3/2)*c*(c + d*x) + 8*a^2*Sqrt[b]*c*Sqrt[1 - (c + d*x)^2] - 6 *b^(5/2)*c*Sqrt[1 - (c + d*x)^2] + 4*b^(5/2)*c*(c + d*x)*ArcSin[c + d*x] + 16*a*b^(3/2)*c*Sqrt[1 - (c + d*x)^2]*ArcSin[c + d*x] + 8*b^(5/2)*c*Sqrt[1 - (c + d*x)^2]*ArcSin[c + d*x]^2 + 4*a*b^(3/2)*Cos[2*ArcSin[c + d*x]] + 4 *b^(5/2)*ArcSin[c + d*x]*Cos[2*ArcSin[c + d*x]] + 32*Sqrt[Pi]*(a + b*ArcSi n[c + d*x])^(5/2)*Cos[(2*a)/b]*FresnelC[(2*Sqrt[a + b*ArcSin[c + d*x]])/(S qrt[b]*Sqrt[Pi])] + 8*c*Sqrt[2*Pi]*(a + b*ArcSin[c + d*x])^(5/2)*Cos[a/b]* FresnelS[(Sqrt[2/Pi]*Sqrt[a + b*ArcSin[c + d*x]])/Sqrt[b]] - 8*c*Sqrt[2*Pi ]*(a + b*ArcSin[c + d*x])^(5/2)*FresnelC[(Sqrt[2/Pi]*Sqrt[a + b*ArcSin[c + d*x]])/Sqrt[b]]*Sin[a/b] + 32*Sqrt[Pi]*(a + b*ArcSin[c + d*x])^(5/2)*Fres nelS[(2*Sqrt[a + b*ArcSin[c + d*x]])/(Sqrt[b]*Sqrt[Pi])]*Sin[(2*a)/b] - 16 *a^2*Sqrt[b]*Sin[2*ArcSin[c + d*x]] + 3*b^(5/2)*Sin[2*ArcSin[c + d*x]] - 3 2*a*b^(3/2)*ArcSin[c + d*x]*Sin[2*ArcSin[c + d*x]] - 16*b^(5/2)*ArcSin[c + d*x]^2*Sin[2*ArcSin[c + d*x]])/(b^(7/2)*d^2*(a + b*ArcSin[c + d*x])^(5/2) )
Time = 1.10 (sec) , antiderivative size = 440, normalized size of antiderivative = 0.94, 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.
| \(\displaystyle \int \frac {x}{(a+b \arcsin (c+d x))^{7/2}} \, dx\) |
| \(\Big \downarrow \) 5304 |
| \(\displaystyle \frac {\int \frac {x}{(a+b \arcsin (c+d x))^{7/2}}d(c+d x)}{d}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle -\frac {\int -\frac {x}{(a+b \arcsin (c+d x))^{7/2}}d(c+d x)}{d}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle -\frac {\int -\frac {d x}{(a+b \arcsin (c+d x))^{7/2}}d(c+d x)}{d^2}\) |
| \(\Big \downarrow \) 5244 |
| \(\displaystyle -\frac {\int \left (\frac {c}{(a+b \arcsin (c+d x))^{7/2}}-\frac {c+d x}{(a+b \arcsin (c+d x))^{7/2}}\right )d(c+d x)}{d^2}\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle -\frac {-\frac {8 \sqrt {2 \pi } c \sin \left (\frac {a}{b}\right ) \operatorname {FresnelC}\left (\frac {\sqrt {\frac {2}{\pi }} \sqrt {a+b \arcsin (c+d x)}}{\sqrt {b}}\right )}{15 b^{7/2}}+\frac {32 \sqrt {\pi } \cos \left (\frac {2 a}{b}\right ) \operatorname {FresnelC}\left (\frac {2 \sqrt {a+b \arcsin (c+d x)}}{\sqrt {b} \sqrt {\pi }}\right )}{15 b^{7/2}}+\frac {32 \sqrt {\pi } \sin \left (\frac {2 a}{b}\right ) \operatorname {FresnelS}\left (\frac {2 \sqrt {a+b \arcsin (c+d x)}}{\sqrt {b} \sqrt {\pi }}\right )}{15 b^{7/2}}+\frac {8 \sqrt {2 \pi } c \cos \left (\frac {a}{b}\right ) \operatorname {FresnelS}\left (\frac {\sqrt {\frac {2}{\pi }} \sqrt {a+b \arcsin (c+d x)}}{\sqrt {b}}\right )}{15 b^{7/2}}-\frac {32 \sqrt {1-(c+d x)^2} (c+d x)}{15 b^3 \sqrt {a+b \arcsin (c+d x)}}+\frac {8 c \sqrt {1-(c+d x)^2}}{15 b^3 \sqrt {a+b \arcsin (c+d x)}}-\frac {8 (c+d x)^2}{15 b^2 (a+b \arcsin (c+d x))^{3/2}}+\frac {4 c (c+d x)}{15 b^2 (a+b \arcsin (c+d x))^{3/2}}+\frac {4}{15 b^2 (a+b \arcsin (c+d x))^{3/2}}+\frac {2 \sqrt {1-(c+d x)^2} (c+d x)}{5 b (a+b \arcsin (c+d x))^{5/2}}-\frac {2 c \sqrt {1-(c+d x)^2}}{5 b (a+b \arcsin (c+d x))^{5/2}}}{d^2}\) |
-(((-2*c*Sqrt[1 - (c + d*x)^2])/(5*b*(a + b*ArcSin[c + d*x])^(5/2)) + (2*( c + d*x)*Sqrt[1 - (c + d*x)^2])/(5*b*(a + b*ArcSin[c + d*x])^(5/2)) + 4/(1 5*b^2*(a + b*ArcSin[c + d*x])^(3/2)) + (4*c*(c + d*x))/(15*b^2*(a + b*ArcS in[c + d*x])^(3/2)) - (8*(c + d*x)^2)/(15*b^2*(a + b*ArcSin[c + d*x])^(3/2 )) + (8*c*Sqrt[1 - (c + d*x)^2])/(15*b^3*Sqrt[a + b*ArcSin[c + d*x]]) - (3 2*(c + d*x)*Sqrt[1 - (c + d*x)^2])/(15*b^3*Sqrt[a + b*ArcSin[c + d*x]]) + (32*Sqrt[Pi]*Cos[(2*a)/b]*FresnelC[(2*Sqrt[a + b*ArcSin[c + d*x]])/(Sqrt[b ]*Sqrt[Pi])])/(15*b^(7/2)) + (8*c*Sqrt[2*Pi]*Cos[a/b]*FresnelS[(Sqrt[2/Pi] *Sqrt[a + b*ArcSin[c + d*x]])/Sqrt[b]])/(15*b^(7/2)) - (8*c*Sqrt[2*Pi]*Fre snelC[(Sqrt[2/Pi]*Sqrt[a + b*ArcSin[c + d*x]])/Sqrt[b]]*Sin[a/b])/(15*b^(7 /2)) + (32*Sqrt[Pi]*FresnelS[(2*Sqrt[a + b*ArcSin[c + d*x]])/(Sqrt[b]*Sqrt [Pi])]*Sin[(2*a)/b])/(15*b^(7/2)))/d^2)
3.2.70.3.1 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]
Leaf count of result is larger than twice the leaf count of optimal. \(1237\) vs. \(2(384)=768\).
Time = 1.18 (sec) , antiderivative size = 1238, normalized size of antiderivative = 2.65
-1/15/d^2/b^3/(a+b*arcsin(d*x+c))^(5/2)*(-8*arcsin(d*x+c)^2*(-1/b)^(1/2)*P i^(1/2)*2^(1/2)*(a+b*arcsin(d*x+c))^(1/2)*cos(a/b)*FresnelS(2^(1/2)/Pi^(1/ 2)/(-1/b)^(1/2)*(a+b*arcsin(d*x+c))^(1/2)/b)*b^2*c-8*arcsin(d*x+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)*b^2*c-16*arcsin(d*x+ c)*(-1/b)^(1/2)*Pi^(1/2)*2^(1/2)*(a+b*arcsin(d*x+c))^(1/2)*cos(a/b)*Fresne lS(2^(1/2)/Pi^(1/2)/(-1/b)^(1/2)*(a+b*arcsin(d*x+c))^(1/2)/b)*a*b*c-16*arc sin(d*x+c)*(-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)*a*b* c+32*arcsin(d*x+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 )*b^2-32*arcsin(d*x+c)^2*(-1/b)^(1/2)*Pi^(1/2)*(a+b*arcsin(d*x+c))^(1/2)*s in(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^2-8*(-1/b)^(1/2)*Pi^(1/2)*2^(1/2)*(a+b*arcsin(d*x+c))^(1/2)*cos(a/ b)*FresnelS(2^(1/2)/Pi^(1/2)/(-1/b)^(1/2)*(a+b*arcsin(d*x+c))^(1/2)/b)*a^2 *c-8*(-1/b)^(1/2)*Pi^(1/2)*2^(1/2)*(a+b*arcsin(d*x+c))^(1/2)*sin(a/b)*Fres nelC(2^(1/2)/Pi^(1/2)/(-1/b)^(1/2)*(a+b*arcsin(d*x+c))^(1/2)/b)*a^2*c+64*a rcsin(d*x+c)*(-1/b)^(1/2)*Pi^(1/2)*(a+b*arcsin(d*x+c))^(1/2)*cos(2*a/b)*Fr esnelC(2*2^(1/2)/Pi^(1/2)/(-2/b)^(1/2)*(a+b*arcsin(d*x+c))^(1/2)/b)*a*b-64 *arcsin(d*x+c)*(-1/b)^(1/2)*Pi^(1/2)*(a+b*arcsin(d*x+c))^(1/2)*sin(2*a/...
Exception generated. \[ \int \frac {x}{(a+b \arcsin (c+d x))^{7/2}} \, dx=\text {Exception raised: TypeError} \]
Exception raised: TypeError >> Error detected within library code: inte grate: implementation incomplete (constant residues)
\[ \int \frac {x}{(a+b \arcsin (c+d x))^{7/2}} \, dx=\int \frac {x}{\left (a + b \operatorname {asin}{\left (c + d x \right )}\right )^{\frac {7}{2}}}\, dx \]
\[ \int \frac {x}{(a+b \arcsin (c+d x))^{7/2}} \, dx=\int { \frac {x}{{\left (b \arcsin \left (d x + c\right ) + a\right )}^{\frac {7}{2}}} \,d x } \]
\[ \int \frac {x}{(a+b \arcsin (c+d x))^{7/2}} \, dx=\int { \frac {x}{{\left (b \arcsin \left (d x + c\right ) + a\right )}^{\frac {7}{2}}} \,d x } \]
Timed out. \[ \int \frac {x}{(a+b \arcsin (c+d x))^{7/2}} \, dx=\int \frac {x}{{\left (a+b\,\mathrm {asin}\left (c+d\,x\right )\right )}^{7/2}} \,d x \]