Integrand size = 25, antiderivative size = 412 \[ \int \frac {(c e+d e x)^4}{(a+b \arcsin (c+d x))^{3/2}} \, dx=-\frac {2 e^4 (c+d x)^4 \sqrt {1-(c+d x)^2}}{b d \sqrt {a+b \arcsin (c+d x)}}-\frac {e^4 \sqrt {\frac {\pi }{2}} \cos \left (\frac {a}{b}\right ) \operatorname {FresnelS}\left (\frac {\sqrt {\frac {2}{\pi }} \sqrt {a+b \arcsin (c+d x)}}{\sqrt {b}}\right )}{2 b^{3/2} d}+\frac {3 e^4 \sqrt {\frac {3 \pi }{2}} \cos \left (\frac {3 a}{b}\right ) \operatorname {FresnelS}\left (\frac {\sqrt {\frac {6}{\pi }} \sqrt {a+b \arcsin (c+d x)}}{\sqrt {b}}\right )}{4 b^{3/2} d}-\frac {e^4 \sqrt {\frac {5 \pi }{2}} \cos \left (\frac {5 a}{b}\right ) \operatorname {FresnelS}\left (\frac {\sqrt {\frac {10}{\pi }} \sqrt {a+b \arcsin (c+d x)}}{\sqrt {b}}\right )}{4 b^{3/2} d}+\frac {e^4 \sqrt {\frac {\pi }{2}} \operatorname {FresnelC}\left (\frac {\sqrt {\frac {2}{\pi }} \sqrt {a+b \arcsin (c+d x)}}{\sqrt {b}}\right ) \sin \left (\frac {a}{b}\right )}{2 b^{3/2} d}-\frac {3 e^4 \sqrt {\frac {3 \pi }{2}} \operatorname {FresnelC}\left (\frac {\sqrt {\frac {6}{\pi }} \sqrt {a+b \arcsin (c+d x)}}{\sqrt {b}}\right ) \sin \left (\frac {3 a}{b}\right )}{4 b^{3/2} d}+\frac {e^4 \sqrt {\frac {5 \pi }{2}} \operatorname {FresnelC}\left (\frac {\sqrt {\frac {10}{\pi }} \sqrt {a+b \arcsin (c+d x)}}{\sqrt {b}}\right ) \sin \left (\frac {5 a}{b}\right )}{4 b^{3/2} d} \]
-1/4*e^4*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+1/4*e^4*FresnelC(2^(1/2)/Pi^(1/2)*(a+b*arc sin(d*x+c))^(1/2)/b^(1/2))*sin(a/b)*2^(1/2)*Pi^(1/2)/b^(3/2)/d+3/8*e^4*cos (3*a/b)*FresnelS(6^(1/2)/Pi^(1/2)*(a+b*arcsin(d*x+c))^(1/2)/b^(1/2))*6^(1/ 2)*Pi^(1/2)/b^(3/2)/d-3/8*e^4*FresnelC(6^(1/2)/Pi^(1/2)*(a+b*arcsin(d*x+c) )^(1/2)/b^(1/2))*sin(3*a/b)*6^(1/2)*Pi^(1/2)/b^(3/2)/d-1/8*e^4*cos(5*a/b)* FresnelS(10^(1/2)/Pi^(1/2)*(a+b*arcsin(d*x+c))^(1/2)/b^(1/2))*10^(1/2)*Pi^ (1/2)/b^(3/2)/d+1/8*e^4*FresnelC(10^(1/2)/Pi^(1/2)*(a+b*arcsin(d*x+c))^(1/ 2)/b^(1/2))*sin(5*a/b)*10^(1/2)*Pi^(1/2)/b^(3/2)/d-2*e^4*(d*x+c)^4*(1-(d*x +c)^2)^(1/2)/b/d/(a+b*arcsin(d*x+c))^(1/2)
Result contains complex when optimal does not.
Time = 0.82 (sec) , antiderivative size = 572, normalized size of antiderivative = 1.39 \[ \int \frac {(c e+d e x)^4}{(a+b \arcsin (c+d x))^{3/2}} \, dx=\frac {e^4 e^{-\frac {5 i (a+b \arcsin (c+d x))}{b}} \left (-e^{\frac {5 i a}{b}}+3 e^{\frac {5 i a}{b}+2 i \arcsin (c+d x)}-2 e^{\frac {5 i a}{b}+4 i \arcsin (c+d x)}-2 e^{\frac {5 i a}{b}+6 i \arcsin (c+d x)}+3 e^{\frac {5 i a}{b}+8 i \arcsin (c+d x)}-e^{\frac {5 i (a+2 b \arcsin (c+d x))}{b}}+2 e^{\frac {4 i a}{b}+5 i \arcsin (c+d x)} \sqrt {-\frac {i (a+b \arcsin (c+d x))}{b}} \Gamma \left (\frac {1}{2},-\frac {i (a+b \arcsin (c+d x))}{b}\right )+2 e^{\frac {6 i a}{b}+5 i \arcsin (c+d x)} \sqrt {\frac {i (a+b \arcsin (c+d x))}{b}} \Gamma \left (\frac {1}{2},\frac {i (a+b \arcsin (c+d x))}{b}\right )-3 \sqrt {3} e^{\frac {2 i a}{b}+5 i \arcsin (c+d x)} \sqrt {-\frac {i (a+b \arcsin (c+d x))}{b}} \Gamma \left (\frac {1}{2},-\frac {3 i (a+b \arcsin (c+d x))}{b}\right )-3 \sqrt {3} e^{\frac {8 i a}{b}+5 i \arcsin (c+d x)} \sqrt {\frac {i (a+b \arcsin (c+d x))}{b}} \Gamma \left (\frac {1}{2},\frac {3 i (a+b \arcsin (c+d x))}{b}\right )+\sqrt {5} e^{5 i \arcsin (c+d x)} \sqrt {-\frac {i (a+b \arcsin (c+d x))}{b}} \Gamma \left (\frac {1}{2},-\frac {5 i (a+b \arcsin (c+d x))}{b}\right )+\sqrt {5} e^{\frac {5 i (2 a+b \arcsin (c+d x))}{b}} \sqrt {\frac {i (a+b \arcsin (c+d x))}{b}} \Gamma \left (\frac {1}{2},\frac {5 i (a+b \arcsin (c+d x))}{b}\right )\right )}{16 b d \sqrt {a+b \arcsin (c+d x)}} \]
(e^4*(-E^(((5*I)*a)/b) + 3*E^(((5*I)*a)/b + (2*I)*ArcSin[c + d*x]) - 2*E^( ((5*I)*a)/b + (4*I)*ArcSin[c + d*x]) - 2*E^(((5*I)*a)/b + (6*I)*ArcSin[c + d*x]) + 3*E^(((5*I)*a)/b + (8*I)*ArcSin[c + d*x]) - E^(((5*I)*(a + 2*b*Ar cSin[c + d*x]))/b) + 2*E^(((4*I)*a)/b + (5*I)*ArcSin[c + d*x])*Sqrt[((-I)* (a + b*ArcSin[c + d*x]))/b]*Gamma[1/2, ((-I)*(a + b*ArcSin[c + d*x]))/b] + 2*E^(((6*I)*a)/b + (5*I)*ArcSin[c + d*x])*Sqrt[(I*(a + b*ArcSin[c + d*x]) )/b]*Gamma[1/2, (I*(a + b*ArcSin[c + d*x]))/b] - 3*Sqrt[3]*E^(((2*I)*a)/b + (5*I)*ArcSin[c + d*x])*Sqrt[((-I)*(a + b*ArcSin[c + d*x]))/b]*Gamma[1/2, ((-3*I)*(a + b*ArcSin[c + d*x]))/b] - 3*Sqrt[3]*E^(((8*I)*a)/b + (5*I)*Ar cSin[c + d*x])*Sqrt[(I*(a + b*ArcSin[c + d*x]))/b]*Gamma[1/2, ((3*I)*(a + b*ArcSin[c + d*x]))/b] + Sqrt[5]*E^((5*I)*ArcSin[c + d*x])*Sqrt[((-I)*(a + b*ArcSin[c + d*x]))/b]*Gamma[1/2, ((-5*I)*(a + b*ArcSin[c + d*x]))/b] + S qrt[5]*E^(((5*I)*(2*a + b*ArcSin[c + d*x]))/b)*Sqrt[(I*(a + b*ArcSin[c + d *x]))/b]*Gamma[1/2, ((5*I)*(a + b*ArcSin[c + d*x]))/b]))/(16*b*d*E^(((5*I) *(a + b*ArcSin[c + d*x]))/b)*Sqrt[a + b*ArcSin[c + d*x]])
Time = 0.66 (sec) , antiderivative size = 383, normalized size of antiderivative = 0.93, number of steps used = 5, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.160, Rules used = {5304, 27, 5142, 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.
| \(\displaystyle \int \frac {(c e+d e x)^4}{(a+b \arcsin (c+d x))^{3/2}} \, dx\) |
| \(\Big \downarrow \) 5304 |
| \(\displaystyle \frac {\int \frac {e^4 (c+d x)^4}{(a+b \arcsin (c+d x))^{3/2}}d(c+d x)}{d}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {e^4 \int \frac {(c+d x)^4}{(a+b \arcsin (c+d x))^{3/2}}d(c+d x)}{d}\) |
| \(\Big \downarrow \) 5142 |
| \(\displaystyle \frac {e^4 \left (\frac {2 \int \left (\frac {5 \sin \left (\frac {5 a}{b}-\frac {5 (a+b \arcsin (c+d x))}{b}\right )}{16 \sqrt {a+b \arcsin (c+d x)}}-\frac {9 \sin \left (\frac {3 a}{b}-\frac {3 (a+b \arcsin (c+d x))}{b}\right )}{16 \sqrt {a+b \arcsin (c+d x)}}+\frac {\sin \left (\frac {a}{b}-\frac {a+b \arcsin (c+d x)}{b}\right )}{8 \sqrt {a+b \arcsin (c+d x)}}\right )d(a+b \arcsin (c+d x))}{b^2}-\frac {2 (c+d x)^4 \sqrt {1-(c+d x)^2}}{b \sqrt {a+b \arcsin (c+d x)}}\right )}{d}\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {e^4 \left (\frac {2 \left (\frac {1}{4} \sqrt {\frac {\pi }{2}} \sqrt {b} \sin \left (\frac {a}{b}\right ) \operatorname {FresnelC}\left (\frac {\sqrt {\frac {2}{\pi }} \sqrt {a+b \arcsin (c+d x)}}{\sqrt {b}}\right )-\frac {3}{8} \sqrt {\frac {3 \pi }{2}} \sqrt {b} \sin \left (\frac {3 a}{b}\right ) \operatorname {FresnelC}\left (\frac {\sqrt {\frac {6}{\pi }} \sqrt {a+b \arcsin (c+d x)}}{\sqrt {b}}\right )+\frac {1}{8} \sqrt {\frac {5 \pi }{2}} \sqrt {b} \sin \left (\frac {5 a}{b}\right ) \operatorname {FresnelC}\left (\frac {\sqrt {\frac {10}{\pi }} \sqrt {a+b \arcsin (c+d x)}}{\sqrt {b}}\right )-\frac {1}{4} \sqrt {\frac {\pi }{2}} \sqrt {b} \cos \left (\frac {a}{b}\right ) \operatorname {FresnelS}\left (\frac {\sqrt {\frac {2}{\pi }} \sqrt {a+b \arcsin (c+d x)}}{\sqrt {b}}\right )+\frac {3}{8} \sqrt {\frac {3 \pi }{2}} \sqrt {b} \cos \left (\frac {3 a}{b}\right ) \operatorname {FresnelS}\left (\frac {\sqrt {\frac {6}{\pi }} \sqrt {a+b \arcsin (c+d x)}}{\sqrt {b}}\right )-\frac {1}{8} \sqrt {\frac {5 \pi }{2}} \sqrt {b} \cos \left (\frac {5 a}{b}\right ) \operatorname {FresnelS}\left (\frac {\sqrt {\frac {10}{\pi }} \sqrt {a+b \arcsin (c+d x)}}{\sqrt {b}}\right )\right )}{b^2}-\frac {2 (c+d x)^4 \sqrt {1-(c+d x)^2}}{b \sqrt {a+b \arcsin (c+d x)}}\right )}{d}\) |
(e^4*((-2*(c + d*x)^4*Sqrt[1 - (c + d*x)^2])/(b*Sqrt[a + b*ArcSin[c + d*x] ]) + (2*(-1/4*(Sqrt[b]*Sqrt[Pi/2]*Cos[a/b]*FresnelS[(Sqrt[2/Pi]*Sqrt[a + b *ArcSin[c + d*x]])/Sqrt[b]]) + (3*Sqrt[b]*Sqrt[(3*Pi)/2]*Cos[(3*a)/b]*Fres nelS[(Sqrt[6/Pi]*Sqrt[a + b*ArcSin[c + d*x]])/Sqrt[b]])/8 - (Sqrt[b]*Sqrt[ (5*Pi)/2]*Cos[(5*a)/b]*FresnelS[(Sqrt[10/Pi]*Sqrt[a + b*ArcSin[c + d*x]])/ Sqrt[b]])/8 + (Sqrt[b]*Sqrt[Pi/2]*FresnelC[(Sqrt[2/Pi]*Sqrt[a + b*ArcSin[c + d*x]])/Sqrt[b]]*Sin[a/b])/4 - (3*Sqrt[b]*Sqrt[(3*Pi)/2]*FresnelC[(Sqrt[ 6/Pi]*Sqrt[a + b*ArcSin[c + d*x]])/Sqrt[b]]*Sin[(3*a)/b])/8 + (Sqrt[b]*Sqr t[(5*Pi)/2]*FresnelC[(Sqrt[10/Pi]*Sqrt[a + b*ArcSin[c + d*x]])/Sqrt[b]]*Si n[(5*a)/b])/8))/b^2))/d
3.3.65.3.1 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_)*(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_) + (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 = 1.53 (sec) , antiderivative size = 483, normalized size of antiderivative = 1.17
| method | result | size |
| default | \(-\frac {e^{4} \left (-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 ) \sqrt {-\frac {1}{b}}\, \sqrt {\pi }\, \sqrt {2}-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 ) \sqrt {-\frac {1}{b}}\, \sqrt {\pi }\, \sqrt {2}+3 \cos \left (\frac {3 a}{b}\right ) \operatorname {FresnelS}\left (\frac {3 \sqrt {2}\, \sqrt {a +b \arcsin \left (d x +c \right )}}{\sqrt {\pi }\, \sqrt {-\frac {3}{b}}\, b}\right ) \sqrt {2}\, \sqrt {\pi }\, \sqrt {-\frac {3}{b}}\, \sqrt {a +b \arcsin \left (d x +c \right )}+3 \sin \left (\frac {3 a}{b}\right ) \operatorname {FresnelC}\left (\frac {3 \sqrt {2}\, \sqrt {a +b \arcsin \left (d x +c \right )}}{\sqrt {\pi }\, \sqrt {-\frac {3}{b}}\, b}\right ) \sqrt {2}\, \sqrt {\pi }\, \sqrt {-\frac {3}{b}}\, \sqrt {a +b \arcsin \left (d x +c \right )}-\cos \left (\frac {5 a}{b}\right ) \operatorname {FresnelS}\left (\frac {5 \sqrt {2}\, \sqrt {a +b \arcsin \left (d x +c \right )}}{\sqrt {\pi }\, \sqrt {-\frac {5}{b}}\, b}\right ) \sqrt {2}\, \sqrt {-\frac {5}{b}}\, \sqrt {\pi }\, \sqrt {a +b \arcsin \left (d x +c \right )}-\sin \left (\frac {5 a}{b}\right ) \operatorname {FresnelC}\left (\frac {5 \sqrt {2}\, \sqrt {a +b \arcsin \left (d x +c \right )}}{\sqrt {\pi }\, \sqrt {-\frac {5}{b}}\, b}\right ) \sqrt {2}\, \sqrt {-\frac {5}{b}}\, \sqrt {\pi }\, \sqrt {a +b \arcsin \left (d x +c \right )}+2 \cos \left (-\frac {a +b \arcsin \left (d x +c \right )}{b}+\frac {a}{b}\right )-3 \cos \left (-\frac {3 \left (a +b \arcsin \left (d x +c \right )\right )}{b}+\frac {3 a}{b}\right )+\cos \left (-\frac {5 \left (a +b \arcsin \left (d x +c \right )\right )}{b}+\frac {5 a}{b}\right )\right )}{8 d b \sqrt {a +b \arcsin \left (d x +c \right )}}\) | \(483\) |
-1/8*e^4/d/b*(-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)*(-1/b)^(1/2)*Pi^(1/2)*2^(1/2 )-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)*(-1/b)^(1/2)*Pi^(1/2)*2^(1/2)+3*cos(3*a/b )*FresnelS(3*2^(1/2)/Pi^(1/2)/(-3/b)^(1/2)*(a+b*arcsin(d*x+c))^(1/2)/b)*2^ (1/2)*Pi^(1/2)*(-3/b)^(1/2)*(a+b*arcsin(d*x+c))^(1/2)+3*sin(3*a/b)*Fresnel C(3*2^(1/2)/Pi^(1/2)/(-3/b)^(1/2)*(a+b*arcsin(d*x+c))^(1/2)/b)*2^(1/2)*Pi^ (1/2)*(-3/b)^(1/2)*(a+b*arcsin(d*x+c))^(1/2)-cos(5*a/b)*FresnelS(5*2^(1/2) /Pi^(1/2)/(-5/b)^(1/2)*(a+b*arcsin(d*x+c))^(1/2)/b)*2^(1/2)*(-5/b)^(1/2)*P i^(1/2)*(a+b*arcsin(d*x+c))^(1/2)-sin(5*a/b)*FresnelC(5*2^(1/2)/Pi^(1/2)/( -5/b)^(1/2)*(a+b*arcsin(d*x+c))^(1/2)/b)*2^(1/2)*(-5/b)^(1/2)*Pi^(1/2)*(a+ b*arcsin(d*x+c))^(1/2)+2*cos(-(a+b*arcsin(d*x+c))/b+a/b)-3*cos(-3*(a+b*arc sin(d*x+c))/b+3*a/b)+cos(-5*(a+b*arcsin(d*x+c))/b+5*a/b))/(a+b*arcsin(d*x+ c))^(1/2)
Exception generated. \[ \int \frac {(c e+d e x)^4}{(a+b \arcsin (c+d x))^{3/2}} \, dx=\text {Exception raised: TypeError} \]
Exception raised: TypeError >> Error detected within library code: inte grate: implementation incomplete (constant residues)
\[ \int \frac {(c e+d e x)^4}{(a+b \arcsin (c+d x))^{3/2}} \, dx=e^{4} \left (\int \frac {c^{4}}{a \sqrt {a + b \operatorname {asin}{\left (c + d x \right )}} + b \sqrt {a + b \operatorname {asin}{\left (c + d x \right )}} \operatorname {asin}{\left (c + d x \right )}}\, dx + \int \frac {d^{4} x^{4}}{a \sqrt {a + b \operatorname {asin}{\left (c + d x \right )}} + b \sqrt {a + b \operatorname {asin}{\left (c + d x \right )}} \operatorname {asin}{\left (c + d x \right )}}\, dx + \int \frac {4 c d^{3} x^{3}}{a \sqrt {a + b \operatorname {asin}{\left (c + d x \right )}} + b \sqrt {a + b \operatorname {asin}{\left (c + d x \right )}} \operatorname {asin}{\left (c + d x \right )}}\, dx + \int \frac {6 c^{2} d^{2} x^{2}}{a \sqrt {a + b \operatorname {asin}{\left (c + d x \right )}} + b \sqrt {a + b \operatorname {asin}{\left (c + d x \right )}} \operatorname {asin}{\left (c + d x \right )}}\, dx + \int \frac {4 c^{3} d x}{a \sqrt {a + b \operatorname {asin}{\left (c + d x \right )}} + b \sqrt {a + b \operatorname {asin}{\left (c + d x \right )}} \operatorname {asin}{\left (c + d x \right )}}\, dx\right ) \]
e**4*(Integral(c**4/(a*sqrt(a + b*asin(c + d*x)) + b*sqrt(a + b*asin(c + d *x))*asin(c + d*x)), x) + Integral(d**4*x**4/(a*sqrt(a + b*asin(c + d*x)) + b*sqrt(a + b*asin(c + d*x))*asin(c + d*x)), x) + Integral(4*c*d**3*x**3/ (a*sqrt(a + b*asin(c + d*x)) + b*sqrt(a + b*asin(c + d*x))*asin(c + d*x)), x) + Integral(6*c**2*d**2*x**2/(a*sqrt(a + b*asin(c + d*x)) + b*sqrt(a + b*asin(c + d*x))*asin(c + d*x)), x) + Integral(4*c**3*d*x/(a*sqrt(a + b*as in(c + d*x)) + b*sqrt(a + b*asin(c + d*x))*asin(c + d*x)), x))
\[ \int \frac {(c e+d e x)^4}{(a+b \arcsin (c+d x))^{3/2}} \, dx=\int { \frac {{\left (d e x + c e\right )}^{4}}{{\left (b \arcsin \left (d x + c\right ) + a\right )}^{\frac {3}{2}}} \,d x } \]
\[ \int \frac {(c e+d e x)^4}{(a+b \arcsin (c+d x))^{3/2}} \, dx=\int { \frac {{\left (d e x + c e\right )}^{4}}{{\left (b \arcsin \left (d x + c\right ) + a\right )}^{\frac {3}{2}}} \,d x } \]
Timed out. \[ \int \frac {(c e+d e x)^4}{(a+b \arcsin (c+d x))^{3/2}} \, dx=\int \frac {{\left (c\,e+d\,e\,x\right )}^4}{{\left (a+b\,\mathrm {asin}\left (c+d\,x\right )\right )}^{3/2}} \,d x \]