Integrand size = 25, antiderivative size = 288 \[ \int (c e+d e x)^3 \sqrt {a+b \arcsin (c+d x)} \, dx=-\frac {3 e^3 \sqrt {a+b \arcsin (c+d x)}}{32 d}+\frac {e^3 (c+d x)^4 \sqrt {a+b \arcsin (c+d x)}}{4 d}-\frac {\sqrt {b} e^3 \sqrt {\frac {\pi }{2}} \cos \left (\frac {4 a}{b}\right ) \operatorname {FresnelC}\left (\frac {2 \sqrt {\frac {2}{\pi }} \sqrt {a+b \arcsin (c+d x)}}{\sqrt {b}}\right )}{64 d}+\frac {\sqrt {b} e^3 \sqrt {\pi } \cos \left (\frac {2 a}{b}\right ) \operatorname {FresnelC}\left (\frac {2 \sqrt {a+b \arcsin (c+d x)}}{\sqrt {b} \sqrt {\pi }}\right )}{16 d}+\frac {\sqrt {b} e^3 \sqrt {\pi } \operatorname {FresnelS}\left (\frac {2 \sqrt {a+b \arcsin (c+d x)}}{\sqrt {b} \sqrt {\pi }}\right ) \sin \left (\frac {2 a}{b}\right )}{16 d}-\frac {\sqrt {b} e^3 \sqrt {\frac {\pi }{2}} \operatorname {FresnelS}\left (\frac {2 \sqrt {\frac {2}{\pi }} \sqrt {a+b \arcsin (c+d x)}}{\sqrt {b}}\right ) \sin \left (\frac {4 a}{b}\right )}{64 d} \]
-1/128*e^3*cos(4*a/b)*FresnelC(2*2^(1/2)/Pi^(1/2)*(a+b*arcsin(d*x+c))^(1/2 )/b^(1/2))*b^(1/2)*2^(1/2)*Pi^(1/2)/d-1/128*e^3*FresnelS(2*2^(1/2)/Pi^(1/2 )*(a+b*arcsin(d*x+c))^(1/2)/b^(1/2))*sin(4*a/b)*b^(1/2)*2^(1/2)*Pi^(1/2)/d +1/16*e^3*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+1/16*e^3*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-3/32*e^3*(a+b*arcsin(d*x+c))^(1/2 )/d+1/4*e^3*(d*x+c)^4*(a+b*arcsin(d*x+c))^(1/2)/d
Result contains complex when optimal does not.
Time = 0.22 (sec) , antiderivative size = 253, normalized size of antiderivative = 0.88 \[ \int (c e+d e x)^3 \sqrt {a+b \arcsin (c+d x)} \, dx=-\frac {i b e^3 e^{-\frac {4 i a}{b}} \left (4 \sqrt {2} e^{\frac {2 i a}{b}} \sqrt {-\frac {i (a+b \arcsin (c+d x))}{b}} \Gamma \left (\frac {3}{2},-\frac {2 i (a+b \arcsin (c+d x))}{b}\right )-4 \sqrt {2} e^{\frac {6 i a}{b}} \sqrt {\frac {i (a+b \arcsin (c+d x))}{b}} \Gamma \left (\frac {3}{2},\frac {2 i (a+b \arcsin (c+d x))}{b}\right )-\sqrt {-\frac {i (a+b \arcsin (c+d x))}{b}} \Gamma \left (\frac {3}{2},-\frac {4 i (a+b \arcsin (c+d x))}{b}\right )+e^{\frac {8 i a}{b}} \sqrt {\frac {i (a+b \arcsin (c+d x))}{b}} \Gamma \left (\frac {3}{2},\frac {4 i (a+b \arcsin (c+d x))}{b}\right )\right )}{128 d \sqrt {a+b \arcsin (c+d x)}} \]
((-1/128*I)*b*e^3*(4*Sqrt[2]*E^(((2*I)*a)/b)*Sqrt[((-I)*(a + b*ArcSin[c + d*x]))/b]*Gamma[3/2, ((-2*I)*(a + b*ArcSin[c + d*x]))/b] - 4*Sqrt[2]*E^((( 6*I)*a)/b)*Sqrt[(I*(a + b*ArcSin[c + d*x]))/b]*Gamma[3/2, ((2*I)*(a + b*Ar cSin[c + d*x]))/b] - Sqrt[((-I)*(a + b*ArcSin[c + d*x]))/b]*Gamma[3/2, ((- 4*I)*(a + b*ArcSin[c + d*x]))/b] + E^(((8*I)*a)/b)*Sqrt[(I*(a + b*ArcSin[c + d*x]))/b]*Gamma[3/2, ((4*I)*(a + b*ArcSin[c + d*x]))/b]))/(d*E^(((4*I)* a)/b)*Sqrt[a + b*ArcSin[c + d*x]])
Time = 0.78 (sec) , antiderivative size = 264, normalized size of antiderivative = 0.92, number of steps used = 8, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.280, Rules used = {5304, 27, 5140, 5224, 3042, 3793, 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.
| \(\displaystyle \int (c e+d e x)^3 \sqrt {a+b \arcsin (c+d x)} \, dx\) |
| \(\Big \downarrow \) 5304 |
| \(\displaystyle \frac {\int e^3 (c+d x)^3 \sqrt {a+b \arcsin (c+d x)}d(c+d x)}{d}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {e^3 \int (c+d x)^3 \sqrt {a+b \arcsin (c+d x)}d(c+d x)}{d}\) |
| \(\Big \downarrow \) 5140 |
| \(\displaystyle \frac {e^3 \left (\frac {1}{4} (c+d x)^4 \sqrt {a+b \arcsin (c+d x)}-\frac {1}{8} b \int \frac {(c+d x)^4}{\sqrt {1-(c+d x)^2} \sqrt {a+b \arcsin (c+d x)}}d(c+d x)\right )}{d}\) |
| \(\Big \downarrow \) 5224 |
| \(\displaystyle \frac {e^3 \left (\frac {1}{4} (c+d x)^4 \sqrt {a+b \arcsin (c+d x)}-\frac {1}{8} \int \frac {\sin ^4\left (\frac {a}{b}-\frac {a+b \arcsin (c+d x)}{b}\right )}{\sqrt {a+b \arcsin (c+d x)}}d(a+b \arcsin (c+d x))\right )}{d}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {e^3 \left (\frac {1}{4} (c+d x)^4 \sqrt {a+b \arcsin (c+d x)}-\frac {1}{8} \int \frac {\sin \left (\frac {a}{b}-\frac {a+b \arcsin (c+d x)}{b}\right )^4}{\sqrt {a+b \arcsin (c+d x)}}d(a+b \arcsin (c+d x))\right )}{d}\) |
| \(\Big \downarrow \) 3793 |
| \(\displaystyle \frac {e^3 \left (\frac {1}{4} (c+d x)^4 \sqrt {a+b \arcsin (c+d x)}-\frac {1}{8} \int \left (\frac {\cos \left (\frac {4 a}{b}-\frac {4 (a+b \arcsin (c+d x))}{b}\right )}{8 \sqrt {a+b \arcsin (c+d x)}}-\frac {\cos \left (\frac {2 a}{b}-\frac {2 (a+b \arcsin (c+d x))}{b}\right )}{2 \sqrt {a+b \arcsin (c+d x)}}+\frac {3}{8 \sqrt {a+b \arcsin (c+d x)}}\right )d(a+b \arcsin (c+d x))\right )}{d}\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {e^3 \left (\frac {1}{8} \left (-\frac {1}{8} \sqrt {\frac {\pi }{2}} \sqrt {b} \cos \left (\frac {4 a}{b}\right ) \operatorname {FresnelC}\left (\frac {2 \sqrt {\frac {2}{\pi }} \sqrt {a+b \arcsin (c+d x)}}{\sqrt {b}}\right )+\frac {1}{2} \sqrt {\pi } \sqrt {b} \cos \left (\frac {2 a}{b}\right ) \operatorname {FresnelC}\left (\frac {2 \sqrt {a+b \arcsin (c+d x)}}{\sqrt {b} \sqrt {\pi }}\right )+\frac {1}{2} \sqrt {\pi } \sqrt {b} \sin \left (\frac {2 a}{b}\right ) \operatorname {FresnelS}\left (\frac {2 \sqrt {a+b \arcsin (c+d x)}}{\sqrt {b} \sqrt {\pi }}\right )-\frac {1}{8} \sqrt {\frac {\pi }{2}} \sqrt {b} \sin \left (\frac {4 a}{b}\right ) \operatorname {FresnelS}\left (\frac {2 \sqrt {\frac {2}{\pi }} \sqrt {a+b \arcsin (c+d x)}}{\sqrt {b}}\right )-\frac {3}{4} \sqrt {a+b \arcsin (c+d x)}\right )+\frac {1}{4} (c+d x)^4 \sqrt {a+b \arcsin (c+d x)}\right )}{d}\) |
(e^3*(((c + d*x)^4*Sqrt[a + b*ArcSin[c + d*x]])/4 + ((-3*Sqrt[a + b*ArcSin [c + d*x]])/4 - (Sqrt[b]*Sqrt[Pi/2]*Cos[(4*a)/b]*FresnelC[(2*Sqrt[2/Pi]*Sq rt[a + b*ArcSin[c + d*x]])/Sqrt[b]])/8 + (Sqrt[b]*Sqrt[Pi]*Cos[(2*a)/b]*Fr esnelC[(2*Sqrt[a + b*ArcSin[c + d*x]])/(Sqrt[b]*Sqrt[Pi])])/2 + (Sqrt[b]*S qrt[Pi]*FresnelS[(2*Sqrt[a + b*ArcSin[c + d*x]])/(Sqrt[b]*Sqrt[Pi])]*Sin[( 2*a)/b])/2 - (Sqrt[b]*Sqrt[Pi/2]*FresnelS[(2*Sqrt[2/Pi]*Sqrt[a + b*ArcSin[ c + d*x]])/Sqrt[b]]*Sin[(4*a)/b])/8)/8))/d
3.3.40.3.1 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[((c_.) + (d_.)*(x_))^(m_)*sin[(e_.) + (f_.)*(x_)]^(n_), x_Symbol] :> In t[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]))
Int[((a_.) + ArcSin[(c_.)*(x_)]*(b_.))^(n_)*(x_)^(m_.), x_Symbol] :> Simp[x ^(m + 1)*((a + b*ArcSin[c*x])^n/(m + 1)), x] - Simp[b*c*(n/(m + 1)) Int[x ^(m + 1)*((a + b*ArcSin[c*x])^(n - 1)/Sqrt[1 - c^2*x^2]), x], x] /; FreeQ[{ a, b, c}, x] && IGtQ[m, 0] && GtQ[n, 0]
Int[((a_.) + ArcSin[(c_.)*(x_)]*(b_.))^(n_.)*(x_)^(m_.)*((d_) + (e_.)*(x_)^ 2)^(p_.), x_Symbol] :> Simp[(1/(b*c^(m + 1)))*Simp[(d + e*x^2)^p/(1 - c^2*x ^2)^p] Subst[Int[x^n*Sin[-a/b + x/b]^m*Cos[-a/b + x/b]^(2*p + 1), x], x, a + b*ArcSin[c*x]], x] /; FreeQ[{a, b, c, d, e, n}, x] && EqQ[c^2*d + e, 0] && IGtQ[2*p + 2, 0] && IGtQ[m, 0]
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 = 1.14 (sec) , antiderivative size = 395, normalized size of antiderivative = 1.37
| method | result | size |
| default | \(-\frac {e^{3} \left (\cos \left (\frac {4 a}{b}\right ) \operatorname {FresnelC}\left (\frac {2 \sqrt {2}\, \sqrt {a +b \arcsin \left (d x +c \right )}}{\sqrt {\pi }\, \sqrt {-\frac {1}{b}}\, b}\right ) \sqrt {2}\, \sqrt {\pi }\, \sqrt {-\frac {1}{b}}\, \sqrt {a +b \arcsin \left (d x +c \right )}\, b -\sin \left (\frac {4 a}{b}\right ) \operatorname {FresnelS}\left (\frac {2 \sqrt {2}\, \sqrt {a +b \arcsin \left (d x +c \right )}}{\sqrt {\pi }\, \sqrt {-\frac {1}{b}}\, b}\right ) \sqrt {2}\, \sqrt {\pi }\, \sqrt {-\frac {1}{b}}\, \sqrt {a +b \arcsin \left (d x +c \right )}\, b -8 \sqrt {-\frac {1}{b}}\, \sqrt {\pi }\, \sqrt {a +b \arcsin \left (d x +c \right )}\, \cos \left (\frac {2 a}{b}\right ) \operatorname {FresnelC}\left (\frac {2 \sqrt {2}\, \sqrt {a +b \arcsin \left (d x +c \right )}}{\sqrt {\pi }\, \sqrt {-\frac {2}{b}}\, b}\right ) b +8 \sqrt {-\frac {1}{b}}\, \sqrt {\pi }\, \sqrt {a +b \arcsin \left (d x +c \right )}\, \sin \left (\frac {2 a}{b}\right ) \operatorname {FresnelS}\left (\frac {2 \sqrt {2}\, \sqrt {a +b \arcsin \left (d x +c \right )}}{\sqrt {\pi }\, \sqrt {-\frac {2}{b}}\, b}\right ) b +16 \arcsin \left (d x +c \right ) \cos \left (-\frac {2 \left (a +b \arcsin \left (d x +c \right )\right )}{b}+\frac {2 a}{b}\right ) b +16 \cos \left (-\frac {2 \left (a +b \arcsin \left (d x +c \right )\right )}{b}+\frac {2 a}{b}\right ) a -4 \cos \left (-\frac {4 \left (a +b \arcsin \left (d x +c \right )\right )}{b}+\frac {4 a}{b}\right ) \arcsin \left (d x +c \right ) b -4 \cos \left (-\frac {4 \left (a +b \arcsin \left (d x +c \right )\right )}{b}+\frac {4 a}{b}\right ) a \right )}{128 d \sqrt {a +b \arcsin \left (d x +c \right )}}\) | \(395\) |
-1/128*e^3/d/(a+b*arcsin(d*x+c))^(1/2)*(cos(4*a/b)*FresnelC(2*2^(1/2)/Pi^( 1/2)/(-1/b)^(1/2)*(a+b*arcsin(d*x+c))^(1/2)/b)*2^(1/2)*Pi^(1/2)*(-1/b)^(1/ 2)*(a+b*arcsin(d*x+c))^(1/2)*b-sin(4*a/b)*FresnelS(2*2^(1/2)/Pi^(1/2)/(-1/ b)^(1/2)*(a+b*arcsin(d*x+c))^(1/2)/b)*2^(1/2)*Pi^(1/2)*(-1/b)^(1/2)*(a+b*a rcsin(d*x+c))^(1/2)*b-8*(-1/b)^(1/2)*Pi^(1/2)*(a+b*arcsin(d*x+c))^(1/2)*co s(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+8*(-1/b)^(1/2)*Pi^(1/2)*(a+b*arcsin(d*x+c))^(1/2)*sin(2*a/b)*Fresne lS(2*2^(1/2)/Pi^(1/2)/(-2/b)^(1/2)*(a+b*arcsin(d*x+c))^(1/2)/b)*b+16*arcsi n(d*x+c)*cos(-2*(a+b*arcsin(d*x+c))/b+2*a/b)*b+16*cos(-2*(a+b*arcsin(d*x+c ))/b+2*a/b)*a-4*cos(-4*(a+b*arcsin(d*x+c))/b+4*a/b)*arcsin(d*x+c)*b-4*cos( -4*(a+b*arcsin(d*x+c))/b+4*a/b)*a)
Exception generated. \[ \int (c e+d e x)^3 \sqrt {a+b \arcsin (c+d x)} \, dx=\text {Exception raised: TypeError} \]
Exception raised: TypeError >> Error detected within library code: inte grate: implementation incomplete (constant residues)
\[ \int (c e+d e x)^3 \sqrt {a+b \arcsin (c+d x)} \, dx=e^{3} \left (\int c^{3} \sqrt {a + b \operatorname {asin}{\left (c + d x \right )}}\, dx + \int d^{3} x^{3} \sqrt {a + b \operatorname {asin}{\left (c + d x \right )}}\, dx + \int 3 c d^{2} x^{2} \sqrt {a + b \operatorname {asin}{\left (c + d x \right )}}\, dx + \int 3 c^{2} d x \sqrt {a + b \operatorname {asin}{\left (c + d x \right )}}\, dx\right ) \]
e**3*(Integral(c**3*sqrt(a + b*asin(c + d*x)), x) + Integral(d**3*x**3*sqr t(a + b*asin(c + d*x)), x) + Integral(3*c*d**2*x**2*sqrt(a + b*asin(c + d* x)), x) + Integral(3*c**2*d*x*sqrt(a + b*asin(c + d*x)), x))
\[ \int (c e+d e x)^3 \sqrt {a+b \arcsin (c+d x)} \, dx=\int { {\left (d e x + c e\right )}^{3} \sqrt {b \arcsin \left (d x + c\right ) + a} \,d x } \]
Result contains complex when optimal does not.
Time = 1.26 (sec) , antiderivative size = 1088, normalized size of antiderivative = 3.78 \[ \int (c e+d e x)^3 \sqrt {a+b \arcsin (c+d x)} \, dx=\text {Too large to display} \]
1/16*I*sqrt(pi)*a*b*e^3*erf(-sqrt(2)*sqrt(b*arcsin(d*x + c) + a)/sqrt(b) + I*sqrt(2)*sqrt(b*arcsin(d*x + c) + a)*sqrt(b)/abs(b))*e^(-4*I*a/b)/((sqrt (2)*b^(3/2) - I*sqrt(2)*b^(5/2)/abs(b))*d) + 1/128*sqrt(pi)*b^2*e^3*erf(-s qrt(2)*sqrt(b*arcsin(d*x + c) + a)/sqrt(b) + I*sqrt(2)*sqrt(b*arcsin(d*x + c) + a)*sqrt(b)/abs(b))*e^(-4*I*a/b)/((sqrt(2)*b^(3/2) - I*sqrt(2)*b^(5/2 )/abs(b))*d) - 1/16*I*sqrt(pi)*a*sqrt(b)*e^3*erf(-sqrt(2)*sqrt(b*arcsin(d* x + c) + a)/sqrt(b) - I*sqrt(2)*sqrt(b*arcsin(d*x + c) + a)*sqrt(b)/abs(b) )*e^(4*I*a/b)/((sqrt(2)*b + I*sqrt(2)*b^2/abs(b))*d) + 1/128*sqrt(pi)*b^(3 /2)*e^3*erf(-sqrt(2)*sqrt(b*arcsin(d*x + c) + a)/sqrt(b) - I*sqrt(2)*sqrt( b*arcsin(d*x + c) + a)*sqrt(b)/abs(b))*e^(4*I*a/b)/((sqrt(2)*b + I*sqrt(2) *b^2/abs(b))*d) + 1/8*I*sqrt(pi)*a*sqrt(b)*e^3*erf(-sqrt(b*arcsin(d*x + c) + a)/sqrt(b) - I*sqrt(b*arcsin(d*x + c) + a)*sqrt(b)/abs(b))*e^(2*I*a/b)/ ((b + I*b^2/abs(b))*d) - 1/32*sqrt(pi)*b^(3/2)*e^3*erf(-sqrt(b*arcsin(d*x + c) + a)/sqrt(b) - I*sqrt(b*arcsin(d*x + c) + a)*sqrt(b)/abs(b))*e^(2*I*a /b)/((b + I*b^2/abs(b))*d) - 1/8*I*sqrt(pi)*a*sqrt(b)*e^3*erf(-sqrt(b*arcs in(d*x + c) + a)/sqrt(b) + I*sqrt(b*arcsin(d*x + c) + a)*sqrt(b)/abs(b))*e ^(-2*I*a/b)/((b - I*b^2/abs(b))*d) - 1/32*sqrt(pi)*b^(3/2)*e^3*erf(-sqrt(b *arcsin(d*x + c) + a)/sqrt(b) + I*sqrt(b*arcsin(d*x + c) + a)*sqrt(b)/abs( b))*e^(-2*I*a/b)/((b - I*b^2/abs(b))*d) + 1/16*I*sqrt(pi)*a*e^3*erf(-sqrt( 2)*sqrt(b*arcsin(d*x + c) + a)/sqrt(b) - I*sqrt(2)*sqrt(b*arcsin(d*x + ...
Timed out. \[ \int (c e+d e x)^3 \sqrt {a+b \arcsin (c+d x)} \, dx=\int {\left (c\,e+d\,e\,x\right )}^3\,\sqrt {a+b\,\mathrm {asin}\left (c+d\,x\right )} \,d x \]