Integrand size = 23, antiderivative size = 252 \[ \int \frac {c e+d e x}{(a+b \arcsin (c+d x))^{7/2}} \, dx=-\frac {2 e (c+d x) \sqrt {1-(c+d x)^2}}{5 b d (a+b \arcsin (c+d x))^{5/2}}-\frac {4 e}{15 b^2 d (a+b \arcsin (c+d x))^{3/2}}+\frac {8 e (c+d x)^2}{15 b^2 d (a+b \arcsin (c+d x))^{3/2}}+\frac {32 e (c+d x) \sqrt {1-(c+d x)^2}}{15 b^3 d \sqrt {a+b \arcsin (c+d x)}}-\frac {32 e \sqrt {\pi } \cos \left (\frac {2 a}{b}\right ) \operatorname {FresnelC}\left (\frac {2 \sqrt {a+b \arcsin (c+d x)}}{\sqrt {b} \sqrt {\pi }}\right )}{15 b^{7/2} d}-\frac {32 e \sqrt {\pi } \operatorname {FresnelS}\left (\frac {2 \sqrt {a+b \arcsin (c+d x)}}{\sqrt {b} \sqrt {\pi }}\right ) \sin \left (\frac {2 a}{b}\right )}{15 b^{7/2} d} \]
-4/15*e/b^2/d/(a+b*arcsin(d*x+c))^(3/2)+8/15*e*(d*x+c)^2/b^2/d/(a+b*arcsin (d*x+c))^(3/2)-32/15*e*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-32/15*e*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-2/5*e*(d*x+c)*(1-(d*x +c)^2)^(1/2)/b/d/(a+b*arcsin(d*x+c))^(5/2)+32/15*e*(d*x+c)*(1-(d*x+c)^2)^( 1/2)/b^3/d/(a+b*arcsin(d*x+c))^(1/2)
Result contains complex when optimal does not.
Time = 0.95 (sec) , antiderivative size = 254, normalized size of antiderivative = 1.01 \[ \int \frac {c e+d e x}{(a+b \arcsin (c+d x))^{7/2}} \, dx=-\frac {e \left ((a+b \arcsin (c+d x)) \left (e^{-\frac {2 i a}{b}} \left (2 e^{\frac {2 i (a+b \arcsin (c+d x))}{b}} (4 i a+b+4 i b \arcsin (c+d x))+8 \sqrt {2} b \left (-\frac {i (a+b \arcsin (c+d x))}{b}\right )^{3/2} \Gamma \left (\frac {1}{2},-\frac {2 i (a+b \arcsin (c+d x))}{b}\right )\right )+2 e^{-2 i \arcsin (c+d x)} \left (-4 i a+b-4 i b \arcsin (c+d x)+4 \sqrt {2} b e^{\frac {2 i (a+b \arcsin (c+d x))}{b}} \left (\frac {i (a+b \arcsin (c+d x))}{b}\right )^{3/2} \Gamma \left (\frac {1}{2},\frac {2 i (a+b \arcsin (c+d x))}{b}\right )\right )\right )+3 b^2 \sin (2 \arcsin (c+d x))\right )}{15 b^3 d (a+b \arcsin (c+d x))^{5/2}} \]
-1/15*(e*((a + b*ArcSin[c + d*x])*((2*E^(((2*I)*(a + b*ArcSin[c + d*x]))/b )*((4*I)*a + b + (4*I)*b*ArcSin[c + d*x]) + 8*Sqrt[2]*b*(((-I)*(a + b*ArcS in[c + d*x]))/b)^(3/2)*Gamma[1/2, ((-2*I)*(a + b*ArcSin[c + d*x]))/b])/E^( ((2*I)*a)/b) + (2*((-4*I)*a + b - (4*I)*b*ArcSin[c + d*x] + 4*Sqrt[2]*b*E^ (((2*I)*(a + b*ArcSin[c + d*x]))/b)*((I*(a + b*ArcSin[c + d*x]))/b)^(3/2)* Gamma[1/2, ((2*I)*(a + b*ArcSin[c + d*x]))/b]))/E^((2*I)*ArcSin[c + d*x])) + 3*b^2*Sin[2*ArcSin[c + d*x]]))/(b^3*d*(a + b*ArcSin[c + d*x])^(5/2))
Time = 1.28 (sec) , antiderivative size = 247, normalized size of antiderivative = 0.98, number of steps used = 15, number of rules used = 14, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.609, Rules used = {5304, 27, 5144, 5152, 5222, 5142, 3042, 3787, 25, 3042, 3785, 3786, 3832, 3833}
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.
| \(\displaystyle \int \frac {c e+d e x}{(a+b \arcsin (c+d x))^{7/2}} \, dx\) |
| \(\Big \downarrow \) 5304 |
| \(\displaystyle \frac {\int \frac {e (c+d x)}{(a+b \arcsin (c+d x))^{7/2}}d(c+d x)}{d}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {e \int \frac {c+d x}{(a+b \arcsin (c+d x))^{7/2}}d(c+d x)}{d}\) |
| \(\Big \downarrow \) 5144 |
| \(\displaystyle \frac {e \left (\frac {2 \int \frac {1}{\sqrt {1-(c+d x)^2} (a+b \arcsin (c+d x))^{5/2}}d(c+d x)}{5 b}-\frac {4 \int \frac {(c+d x)^2}{\sqrt {1-(c+d x)^2} (a+b \arcsin (c+d x))^{5/2}}d(c+d x)}{5 b}-\frac {2 \sqrt {1-(c+d x)^2} (c+d x)}{5 b (a+b \arcsin (c+d x))^{5/2}}\right )}{d}\) |
| \(\Big \downarrow \) 5152 |
| \(\displaystyle \frac {e \left (-\frac {4 \int \frac {(c+d x)^2}{\sqrt {1-(c+d x)^2} (a+b \arcsin (c+d x))^{5/2}}d(c+d x)}{5 b}-\frac {4}{15 b^2 (a+b \arcsin (c+d x))^{3/2}}-\frac {2 \sqrt {1-(c+d x)^2} (c+d x)}{5 b (a+b \arcsin (c+d x))^{5/2}}\right )}{d}\) |
| \(\Big \downarrow \) 5222 |
| \(\displaystyle \frac {e \left (-\frac {4 \left (\frac {4 \int \frac {c+d x}{(a+b \arcsin (c+d x))^{3/2}}d(c+d x)}{3 b}-\frac {2 (c+d x)^2}{3 b (a+b \arcsin (c+d x))^{3/2}}\right )}{5 b}-\frac {4}{15 b^2 (a+b \arcsin (c+d x))^{3/2}}-\frac {2 \sqrt {1-(c+d x)^2} (c+d x)}{5 b (a+b \arcsin (c+d x))^{5/2}}\right )}{d}\) |
| \(\Big \downarrow \) 5142 |
| \(\displaystyle \frac {e \left (-\frac {4 \left (\frac {4 \left (\frac {2 \int \frac {\cos \left (\frac {2 a}{b}-\frac {2 (a+b \arcsin (c+d x))}{b}\right )}{\sqrt {a+b \arcsin (c+d x)}}d(a+b \arcsin (c+d x))}{b^2}-\frac {2 (c+d x) \sqrt {1-(c+d x)^2}}{b \sqrt {a+b \arcsin (c+d x)}}\right )}{3 b}-\frac {2 (c+d x)^2}{3 b (a+b \arcsin (c+d x))^{3/2}}\right )}{5 b}-\frac {4}{15 b^2 (a+b \arcsin (c+d x))^{3/2}}-\frac {2 \sqrt {1-(c+d x)^2} (c+d x)}{5 b (a+b \arcsin (c+d x))^{5/2}}\right )}{d}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {e \left (-\frac {4 \left (\frac {4 \left (\frac {2 \int \frac {\sin \left (\frac {2 a}{b}-\frac {2 (a+b \arcsin (c+d x))}{b}+\frac {\pi }{2}\right )}{\sqrt {a+b \arcsin (c+d x)}}d(a+b \arcsin (c+d x))}{b^2}-\frac {2 (c+d x) \sqrt {1-(c+d x)^2}}{b \sqrt {a+b \arcsin (c+d x)}}\right )}{3 b}-\frac {2 (c+d x)^2}{3 b (a+b \arcsin (c+d x))^{3/2}}\right )}{5 b}-\frac {4}{15 b^2 (a+b \arcsin (c+d x))^{3/2}}-\frac {2 \sqrt {1-(c+d x)^2} (c+d x)}{5 b (a+b \arcsin (c+d x))^{5/2}}\right )}{d}\) |
| \(\Big \downarrow \) 3787 |
| \(\displaystyle \frac {e \left (-\frac {4 \left (\frac {4 \left (\frac {2 \left (\cos \left (\frac {2 a}{b}\right ) \int \frac {\cos \left (\frac {2 (a+b \arcsin (c+d x))}{b}\right )}{\sqrt {a+b \arcsin (c+d x)}}d(a+b \arcsin (c+d x))-\sin \left (\frac {2 a}{b}\right ) \int -\frac {\sin \left (\frac {2 (a+b \arcsin (c+d x))}{b}\right )}{\sqrt {a+b \arcsin (c+d x)}}d(a+b \arcsin (c+d x))\right )}{b^2}-\frac {2 (c+d x) \sqrt {1-(c+d x)^2}}{b \sqrt {a+b \arcsin (c+d x)}}\right )}{3 b}-\frac {2 (c+d x)^2}{3 b (a+b \arcsin (c+d x))^{3/2}}\right )}{5 b}-\frac {4}{15 b^2 (a+b \arcsin (c+d x))^{3/2}}-\frac {2 \sqrt {1-(c+d x)^2} (c+d x)}{5 b (a+b \arcsin (c+d x))^{5/2}}\right )}{d}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {e \left (-\frac {4 \left (\frac {4 \left (\frac {2 \left (\sin \left (\frac {2 a}{b}\right ) \int \frac {\sin \left (\frac {2 (a+b \arcsin (c+d x))}{b}\right )}{\sqrt {a+b \arcsin (c+d x)}}d(a+b \arcsin (c+d x))+\cos \left (\frac {2 a}{b}\right ) \int \frac {\cos \left (\frac {2 (a+b \arcsin (c+d x))}{b}\right )}{\sqrt {a+b \arcsin (c+d x)}}d(a+b \arcsin (c+d x))\right )}{b^2}-\frac {2 (c+d x) \sqrt {1-(c+d x)^2}}{b \sqrt {a+b \arcsin (c+d x)}}\right )}{3 b}-\frac {2 (c+d x)^2}{3 b (a+b \arcsin (c+d x))^{3/2}}\right )}{5 b}-\frac {4}{15 b^2 (a+b \arcsin (c+d x))^{3/2}}-\frac {2 \sqrt {1-(c+d x)^2} (c+d x)}{5 b (a+b \arcsin (c+d x))^{5/2}}\right )}{d}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {e \left (-\frac {4 \left (\frac {4 \left (\frac {2 \left (\sin \left (\frac {2 a}{b}\right ) \int \frac {\sin \left (\frac {2 (a+b \arcsin (c+d x))}{b}\right )}{\sqrt {a+b \arcsin (c+d x)}}d(a+b \arcsin (c+d x))+\cos \left (\frac {2 a}{b}\right ) \int \frac {\sin \left (\frac {2 (a+b \arcsin (c+d x))}{b}+\frac {\pi }{2}\right )}{\sqrt {a+b \arcsin (c+d x)}}d(a+b \arcsin (c+d x))\right )}{b^2}-\frac {2 (c+d x) \sqrt {1-(c+d x)^2}}{b \sqrt {a+b \arcsin (c+d x)}}\right )}{3 b}-\frac {2 (c+d x)^2}{3 b (a+b \arcsin (c+d x))^{3/2}}\right )}{5 b}-\frac {4}{15 b^2 (a+b \arcsin (c+d x))^{3/2}}-\frac {2 \sqrt {1-(c+d x)^2} (c+d x)}{5 b (a+b \arcsin (c+d x))^{5/2}}\right )}{d}\) |
| \(\Big \downarrow \) 3785 |
| \(\displaystyle \frac {e \left (-\frac {4 \left (\frac {4 \left (\frac {2 \left (\sin \left (\frac {2 a}{b}\right ) \int \frac {\sin \left (\frac {2 (a+b \arcsin (c+d x))}{b}\right )}{\sqrt {a+b \arcsin (c+d x)}}d(a+b \arcsin (c+d x))+2 \cos \left (\frac {2 a}{b}\right ) \int \cos \left (\frac {2 (a+b \arcsin (c+d x))}{b}\right )d\sqrt {a+b \arcsin (c+d x)}\right )}{b^2}-\frac {2 (c+d x) \sqrt {1-(c+d x)^2}}{b \sqrt {a+b \arcsin (c+d x)}}\right )}{3 b}-\frac {2 (c+d x)^2}{3 b (a+b \arcsin (c+d x))^{3/2}}\right )}{5 b}-\frac {4}{15 b^2 (a+b \arcsin (c+d x))^{3/2}}-\frac {2 \sqrt {1-(c+d x)^2} (c+d x)}{5 b (a+b \arcsin (c+d x))^{5/2}}\right )}{d}\) |
| \(\Big \downarrow \) 3786 |
| \(\displaystyle \frac {e \left (-\frac {4 \left (\frac {4 \left (\frac {2 \left (2 \sin \left (\frac {2 a}{b}\right ) \int \sin \left (\frac {2 (a+b \arcsin (c+d x))}{b}\right )d\sqrt {a+b \arcsin (c+d x)}+2 \cos \left (\frac {2 a}{b}\right ) \int \cos \left (\frac {2 (a+b \arcsin (c+d x))}{b}\right )d\sqrt {a+b \arcsin (c+d x)}\right )}{b^2}-\frac {2 (c+d x) \sqrt {1-(c+d x)^2}}{b \sqrt {a+b \arcsin (c+d x)}}\right )}{3 b}-\frac {2 (c+d x)^2}{3 b (a+b \arcsin (c+d x))^{3/2}}\right )}{5 b}-\frac {4}{15 b^2 (a+b \arcsin (c+d x))^{3/2}}-\frac {2 \sqrt {1-(c+d x)^2} (c+d x)}{5 b (a+b \arcsin (c+d x))^{5/2}}\right )}{d}\) |
| \(\Big \downarrow \) 3832 |
| \(\displaystyle \frac {e \left (-\frac {4 \left (\frac {4 \left (\frac {2 \left (2 \cos \left (\frac {2 a}{b}\right ) \int \cos \left (\frac {2 (a+b \arcsin (c+d x))}{b}\right )d\sqrt {a+b \arcsin (c+d x)}+\sqrt {\pi } \sqrt {b} \sin \left (\frac {2 a}{b}\right ) \operatorname {FresnelS}\left (\frac {2 \sqrt {a+b \arcsin (c+d x)}}{\sqrt {b} \sqrt {\pi }}\right )\right )}{b^2}-\frac {2 (c+d x) \sqrt {1-(c+d x)^2}}{b \sqrt {a+b \arcsin (c+d x)}}\right )}{3 b}-\frac {2 (c+d x)^2}{3 b (a+b \arcsin (c+d x))^{3/2}}\right )}{5 b}-\frac {4}{15 b^2 (a+b \arcsin (c+d x))^{3/2}}-\frac {2 \sqrt {1-(c+d x)^2} (c+d x)}{5 b (a+b \arcsin (c+d x))^{5/2}}\right )}{d}\) |
| \(\Big \downarrow \) 3833 |
| \(\displaystyle \frac {e \left (-\frac {4 \left (\frac {4 \left (\frac {2 \left (\sqrt {\pi } \sqrt {b} \cos \left (\frac {2 a}{b}\right ) \operatorname {FresnelC}\left (\frac {2 \sqrt {a+b \arcsin (c+d x)}}{\sqrt {b} \sqrt {\pi }}\right )+\sqrt {\pi } \sqrt {b} \sin \left (\frac {2 a}{b}\right ) \operatorname {FresnelS}\left (\frac {2 \sqrt {a+b \arcsin (c+d x)}}{\sqrt {b} \sqrt {\pi }}\right )\right )}{b^2}-\frac {2 (c+d x) \sqrt {1-(c+d x)^2}}{b \sqrt {a+b \arcsin (c+d x)}}\right )}{3 b}-\frac {2 (c+d x)^2}{3 b (a+b \arcsin (c+d x))^{3/2}}\right )}{5 b}-\frac {4}{15 b^2 (a+b \arcsin (c+d x))^{3/2}}-\frac {2 \sqrt {1-(c+d x)^2} (c+d x)}{5 b (a+b \arcsin (c+d x))^{5/2}}\right )}{d}\) |
(e*((-2*(c + d*x)*Sqrt[1 - (c + d*x)^2])/(5*b*(a + b*ArcSin[c + d*x])^(5/2 )) - 4/(15*b^2*(a + b*ArcSin[c + d*x])^(3/2)) - (4*((-2*(c + d*x)^2)/(3*b* (a + b*ArcSin[c + d*x])^(3/2)) + (4*((-2*(c + d*x)*Sqrt[1 - (c + d*x)^2])/ (b*Sqrt[a + b*ArcSin[c + d*x]]) + (2*(Sqrt[b]*Sqrt[Pi]*Cos[(2*a)/b]*Fresne lC[(2*Sqrt[a + b*ArcSin[c + d*x]])/(Sqrt[b]*Sqrt[Pi])] + Sqrt[b]*Sqrt[Pi]* FresnelS[(2*Sqrt[a + b*ArcSin[c + d*x]])/(Sqrt[b]*Sqrt[Pi])]*Sin[(2*a)/b]) )/b^2))/(3*b)))/(5*b)))/d
3.3.78.3.1 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[sin[Pi/2 + (e_.) + (f_.)*(x_)]/Sqrt[(c_.) + (d_.)*(x_)], x_Symbol] :> S imp[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]
Int[sin[(e_.) + (f_.)*(x_)]/Sqrt[(c_.) + (d_.)*(x_)], x_Symbol] :> Simp[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]
Int[sin[(e_.) + (f_.)*(x_)]/Sqrt[(c_.) + (d_.)*(x_)], x_Symbol] :> Simp[Cos [(d*e - c*f)/d] Int[Sin[c*(f/d) + f*x]/Sqrt[c + d*x], x], x] + Simp[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]
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]
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]
Int[((a_.) + ArcSin[(c_.)*(x_)]*(b_.))^(n_)*(x_)^(m_.), x_Symbol] :> Simp[x ^m*Sqrt[1 - c^2*x^2]*((a + b*ArcSin[c*x])^(n + 1)/(b*c*(n + 1))), x] - Simp [1/(b^2*c^(m + 1)*(n + 1)) Subst[Int[ExpandTrigReduce[x^(n + 1), Sin[-a/b + x/b]^(m - 1)*(m - (m + 1)*Sin[-a/b + x/b]^2), x], x], x, a + b*ArcSin[c* x]], x] /; FreeQ[{a, b, c}, x] && IGtQ[m, 0] && GeQ[n, -2] && LtQ[n, -1]
Int[((a_.) + ArcSin[(c_.)*(x_)]*(b_.))^(n_)*(x_)^(m_.), x_Symbol] :> Simp[x ^m*Sqrt[1 - c^2*x^2]*((a + b*ArcSin[c*x])^(n + 1)/(b*c*(n + 1))), x] + (Sim p[c*((m + 1)/(b*(n + 1))) Int[x^(m + 1)*((a + b*ArcSin[c*x])^(n + 1)/Sqrt [1 - c^2*x^2]), x], x] - Simp[m/(b*c*(n + 1)) Int[x^(m - 1)*((a + b*ArcSi n[c*x])^(n + 1)/Sqrt[1 - c^2*x^2]), x], x]) /; FreeQ[{a, b, c}, x] && IGtQ[ m, 0] && LtQ[n, -2]
Int[((a_.) + ArcSin[(c_.)*(x_)]*(b_.))^(n_.)/Sqrt[(d_) + (e_.)*(x_)^2], x_S ymbol] :> Simp[(1/(b*c*(n + 1)))*Simp[Sqrt[1 - c^2*x^2]/Sqrt[d + e*x^2]]*(a + b*ArcSin[c*x])^(n + 1), x] /; FreeQ[{a, b, c, d, e, n}, x] && EqQ[c^2*d + e, 0] && NeQ[n, -1]
Int[(((a_.) + ArcSin[(c_.)*(x_)]*(b_.))^(n_)*((f_.)*(x_))^(m_.))/Sqrt[(d_) + (e_.)*(x_)^2], x_Symbol] :> Simp[((f*x)^m/(b*c*(n + 1)))*Simp[Sqrt[1 - c^ 2*x^2]/Sqrt[d + e*x^2]]*(a + b*ArcSin[c*x])^(n + 1), x] - Simp[f*(m/(b*c*(n + 1)))*Simp[Sqrt[1 - c^2*x^2]/Sqrt[d + e*x^2]] Int[(f*x)^(m - 1)*(a + b* ArcSin[c*x])^(n + 1), x], x] /; FreeQ[{a, b, c, d, e, f, m}, x] && EqQ[c^2* d + e, 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. \(624\) vs. \(2(208)=416\).
Time = 0.85 (sec) , antiderivative size = 625, normalized size of antiderivative = 2.48
| method | result | size |
| default | \(-\frac {e \left (32 \arcsin \left (d x +c \right )^{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 ) b^{2}-32 \arcsin \left (d x +c \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 ) b^{2}+64 \arcsin \left (d x +c \right ) \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 ) a b -64 \arcsin \left (d x +c \right ) \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 ) a b +32 \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 ) a^{2}-32 \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 ) a^{2}+16 \arcsin \left (d x +c \right )^{2} \sin \left (-\frac {2 \left (a +b \arcsin \left (d x +c \right )\right )}{b}+\frac {2 a}{b}\right ) b^{2}+32 \arcsin \left (d x +c \right ) \sin \left (-\frac {2 \left (a +b \arcsin \left (d x +c \right )\right )}{b}+\frac {2 a}{b}\right ) a 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^{2}+16 \sin \left (-\frac {2 \left (a +b \arcsin \left (d x +c \right )\right )}{b}+\frac {2 a}{b}\right ) a^{2}-3 \sin \left (-\frac {2 \left (a +b \arcsin \left (d x +c \right )\right )}{b}+\frac {2 a}{b}\right ) b^{2}+4 \cos \left (-\frac {2 \left (a +b \arcsin \left (d x +c \right )\right )}{b}+\frac {2 a}{b}\right ) a b \right )}{15 d \,b^{3} \left (a +b \arcsin \left (d x +c \right )\right )^{\frac {5}{2}}}\) | \(625\) |
-1/15*e/d/b^3*(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)*sin(2*a/b)*FresnelS(2*2^(1/2)/Pi^(1/2)/(-2/b)^(1/2)*(a+b*arcsi n(d*x+c))^(1/2)/b)*b^2+64*arcsin(d*x+c)*(-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*arc sin(d*x+c))^(1/2)/b)*a*b-64*arcsin(d*x+c)*(-1/b)^(1/2)*Pi^(1/2)*(a+b*arcsi n(d*x+c))^(1/2)*sin(2*a/b)*FresnelS(2*2^(1/2)/Pi^(1/2)/(-2/b)^(1/2)*(a+b*a rcsin(d*x+c))^(1/2)/b)*a*b+32*(-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)*a^2-32*(-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)*a ^2+16*arcsin(d*x+c)^2*sin(-2*(a+b*arcsin(d*x+c))/b+2*a/b)*b^2+32*arcsin(d* x+c)*sin(-2*(a+b*arcsin(d*x+c))/b+2*a/b)*a*b+4*arcsin(d*x+c)*cos(-2*(a+b*a rcsin(d*x+c))/b+2*a/b)*b^2+16*sin(-2*(a+b*arcsin(d*x+c))/b+2*a/b)*a^2-3*si n(-2*(a+b*arcsin(d*x+c))/b+2*a/b)*b^2+4*cos(-2*(a+b*arcsin(d*x+c))/b+2*a/b )*a*b)/(a+b*arcsin(d*x+c))^(5/2)
Exception generated. \[ \int \frac {c e+d e x}{(a+b \arcsin (c+d x))^{7/2}} \, dx=\text {Exception raised: TypeError} \]
Exception raised: TypeError >> Error detected within library code: inte grate: implementation incomplete (constant residues)
\[ \int \frac {c e+d e x}{(a+b \arcsin (c+d x))^{7/2}} \, dx=e \left (\int \frac {c}{a^{3} \sqrt {a + b \operatorname {asin}{\left (c + d x \right )}} + 3 a^{2} b \sqrt {a + b \operatorname {asin}{\left (c + d x \right )}} \operatorname {asin}{\left (c + d x \right )} + 3 a b^{2} \sqrt {a + b \operatorname {asin}{\left (c + d x \right )}} \operatorname {asin}^{2}{\left (c + d x \right )} + b^{3} \sqrt {a + b \operatorname {asin}{\left (c + d x \right )}} \operatorname {asin}^{3}{\left (c + d x \right )}}\, dx + \int \frac {d x}{a^{3} \sqrt {a + b \operatorname {asin}{\left (c + d x \right )}} + 3 a^{2} b \sqrt {a + b \operatorname {asin}{\left (c + d x \right )}} \operatorname {asin}{\left (c + d x \right )} + 3 a b^{2} \sqrt {a + b \operatorname {asin}{\left (c + d x \right )}} \operatorname {asin}^{2}{\left (c + d x \right )} + b^{3} \sqrt {a + b \operatorname {asin}{\left (c + d x \right )}} \operatorname {asin}^{3}{\left (c + d x \right )}}\, dx\right ) \]
e*(Integral(c/(a**3*sqrt(a + b*asin(c + d*x)) + 3*a**2*b*sqrt(a + b*asin(c + d*x))*asin(c + d*x) + 3*a*b**2*sqrt(a + b*asin(c + d*x))*asin(c + d*x)* *2 + b**3*sqrt(a + b*asin(c + d*x))*asin(c + d*x)**3), x) + Integral(d*x/( a**3*sqrt(a + b*asin(c + d*x)) + 3*a**2*b*sqrt(a + b*asin(c + d*x))*asin(c + d*x) + 3*a*b**2*sqrt(a + b*asin(c + d*x))*asin(c + d*x)**2 + b**3*sqrt( a + b*asin(c + d*x))*asin(c + d*x)**3), x))
\[ \int \frac {c e+d e x}{(a+b \arcsin (c+d x))^{7/2}} \, dx=\int { \frac {d e x + c e}{{\left (b \arcsin \left (d x + c\right ) + a\right )}^{\frac {7}{2}}} \,d x } \]
\[ \int \frac {c e+d e x}{(a+b \arcsin (c+d x))^{7/2}} \, dx=\int { \frac {d e x + c e}{{\left (b \arcsin \left (d x + c\right ) + a\right )}^{\frac {7}{2}}} \,d x } \]
Timed out. \[ \int \frac {c e+d e x}{(a+b \arcsin (c+d x))^{7/2}} \, dx=\int \frac {c\,e+d\,e\,x}{{\left (a+b\,\mathrm {asin}\left (c+d\,x\right )\right )}^{7/2}} \,d x \]