Integrand size = 20, antiderivative size = 369 \[ \int \left (d+e x^2\right ) \sqrt {a+b \arcsin (c x)} \, dx=d x \sqrt {a+b \arcsin (c x)}+\frac {1}{3} e x^3 \sqrt {a+b \arcsin (c x)}-\frac {\sqrt {b} d \sqrt {\frac {\pi }{2}} \cos \left (\frac {a}{b}\right ) \operatorname {FresnelS}\left (\frac {\sqrt {\frac {2}{\pi }} \sqrt {a+b \arcsin (c x)}}{\sqrt {b}}\right )}{c}-\frac {\sqrt {b} e \sqrt {\frac {\pi }{2}} \cos \left (\frac {a}{b}\right ) \operatorname {FresnelS}\left (\frac {\sqrt {\frac {2}{\pi }} \sqrt {a+b \arcsin (c x)}}{\sqrt {b}}\right )}{4 c^3}+\frac {\sqrt {b} e \sqrt {\frac {\pi }{6}} \cos \left (\frac {3 a}{b}\right ) \operatorname {FresnelS}\left (\frac {\sqrt {\frac {6}{\pi }} \sqrt {a+b \arcsin (c x)}}{\sqrt {b}}\right )}{12 c^3}+\frac {\sqrt {b} d \sqrt {\frac {\pi }{2}} \operatorname {FresnelC}\left (\frac {\sqrt {\frac {2}{\pi }} \sqrt {a+b \arcsin (c x)}}{\sqrt {b}}\right ) \sin \left (\frac {a}{b}\right )}{c}+\frac {\sqrt {b} e \sqrt {\frac {\pi }{2}} \operatorname {FresnelC}\left (\frac {\sqrt {\frac {2}{\pi }} \sqrt {a+b \arcsin (c x)}}{\sqrt {b}}\right ) \sin \left (\frac {a}{b}\right )}{4 c^3}-\frac {\sqrt {b} e \sqrt {\frac {\pi }{6}} \operatorname {FresnelC}\left (\frac {\sqrt {\frac {6}{\pi }} \sqrt {a+b \arcsin (c x)}}{\sqrt {b}}\right ) \sin \left (\frac {3 a}{b}\right )}{12 c^3} \]
1/72*e*cos(3*a/b)*FresnelS(6^(1/2)/Pi^(1/2)*(a+b*arcsin(c*x))^(1/2)/b^(1/2 ))*b^(1/2)*6^(1/2)*Pi^(1/2)/c^3-1/72*e*FresnelC(6^(1/2)/Pi^(1/2)*(a+b*arcs in(c*x))^(1/2)/b^(1/2))*sin(3*a/b)*b^(1/2)*6^(1/2)*Pi^(1/2)/c^3-1/2*d*cos( a/b)*FresnelS(2^(1/2)/Pi^(1/2)*(a+b*arcsin(c*x))^(1/2)/b^(1/2))*b^(1/2)*2^ (1/2)*Pi^(1/2)/c-1/8*e*cos(a/b)*FresnelS(2^(1/2)/Pi^(1/2)*(a+b*arcsin(c*x) )^(1/2)/b^(1/2))*b^(1/2)*2^(1/2)*Pi^(1/2)/c^3+1/2*d*FresnelC(2^(1/2)/Pi^(1 /2)*(a+b*arcsin(c*x))^(1/2)/b^(1/2))*sin(a/b)*b^(1/2)*2^(1/2)*Pi^(1/2)/c+1 /8*e*FresnelC(2^(1/2)/Pi^(1/2)*(a+b*arcsin(c*x))^(1/2)/b^(1/2))*sin(a/b)*b ^(1/2)*2^(1/2)*Pi^(1/2)/c^3+d*x*(a+b*arcsin(c*x))^(1/2)+1/3*e*x^3*(a+b*arc sin(c*x))^(1/2)
Result contains complex when optimal does not.
Time = 0.41 (sec) , antiderivative size = 244, normalized size of antiderivative = 0.66 \[ \int \left (d+e x^2\right ) \sqrt {a+b \arcsin (c x)} \, dx=\frac {b e^{-\frac {3 i a}{b}} \left (9 \left (4 c^2 d+e\right ) e^{\frac {2 i a}{b}} \sqrt {-\frac {i (a+b \arcsin (c x))}{b}} \Gamma \left (\frac {3}{2},-\frac {i (a+b \arcsin (c x))}{b}\right )+9 \left (4 c^2 d+e\right ) e^{\frac {4 i a}{b}} \sqrt {\frac {i (a+b \arcsin (c x))}{b}} \Gamma \left (\frac {3}{2},\frac {i (a+b \arcsin (c x))}{b}\right )-\sqrt {3} e \left (\sqrt {-\frac {i (a+b \arcsin (c x))}{b}} \Gamma \left (\frac {3}{2},-\frac {3 i (a+b \arcsin (c x))}{b}\right )+e^{\frac {6 i a}{b}} \sqrt {\frac {i (a+b \arcsin (c x))}{b}} \Gamma \left (\frac {3}{2},\frac {3 i (a+b \arcsin (c x))}{b}\right )\right )\right )}{72 c^3 \sqrt {a+b \arcsin (c x)}} \]
(b*(9*(4*c^2*d + e)*E^(((2*I)*a)/b)*Sqrt[((-I)*(a + b*ArcSin[c*x]))/b]*Gam ma[3/2, ((-I)*(a + b*ArcSin[c*x]))/b] + 9*(4*c^2*d + e)*E^(((4*I)*a)/b)*Sq rt[(I*(a + b*ArcSin[c*x]))/b]*Gamma[3/2, (I*(a + b*ArcSin[c*x]))/b] - Sqrt [3]*e*(Sqrt[((-I)*(a + b*ArcSin[c*x]))/b]*Gamma[3/2, ((-3*I)*(a + b*ArcSin [c*x]))/b] + E^(((6*I)*a)/b)*Sqrt[(I*(a + b*ArcSin[c*x]))/b]*Gamma[3/2, (( 3*I)*(a + b*ArcSin[c*x]))/b])))/(72*c^3*E^(((3*I)*a)/b)*Sqrt[a + b*ArcSin[ c*x]])
Time = 1.11 (sec) , antiderivative size = 369, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.100, Rules used = {5172, 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.
| \(\displaystyle \int \left (d+e x^2\right ) \sqrt {a+b \arcsin (c x)} \, dx\) |
| \(\Big \downarrow \) 5172 |
| \(\displaystyle \int \left (d \sqrt {a+b \arcsin (c x)}+e x^2 \sqrt {a+b \arcsin (c x)}\right )dx\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {\sqrt {\frac {\pi }{2}} \sqrt {b} e \sin \left (\frac {a}{b}\right ) \operatorname {FresnelC}\left (\frac {\sqrt {\frac {2}{\pi }} \sqrt {a+b \arcsin (c x)}}{\sqrt {b}}\right )}{4 c^3}-\frac {\sqrt {\frac {\pi }{6}} \sqrt {b} e \sin \left (\frac {3 a}{b}\right ) \operatorname {FresnelC}\left (\frac {\sqrt {\frac {6}{\pi }} \sqrt {a+b \arcsin (c x)}}{\sqrt {b}}\right )}{12 c^3}-\frac {\sqrt {\frac {\pi }{2}} \sqrt {b} e \cos \left (\frac {a}{b}\right ) \operatorname {FresnelS}\left (\frac {\sqrt {\frac {2}{\pi }} \sqrt {a+b \arcsin (c x)}}{\sqrt {b}}\right )}{4 c^3}+\frac {\sqrt {\frac {\pi }{6}} \sqrt {b} e \cos \left (\frac {3 a}{b}\right ) \operatorname {FresnelS}\left (\frac {\sqrt {\frac {6}{\pi }} \sqrt {a+b \arcsin (c x)}}{\sqrt {b}}\right )}{12 c^3}+\frac {\sqrt {\frac {\pi }{2}} \sqrt {b} d \sin \left (\frac {a}{b}\right ) \operatorname {FresnelC}\left (\frac {\sqrt {\frac {2}{\pi }} \sqrt {a+b \arcsin (c x)}}{\sqrt {b}}\right )}{c}-\frac {\sqrt {\frac {\pi }{2}} \sqrt {b} d \cos \left (\frac {a}{b}\right ) \operatorname {FresnelS}\left (\frac {\sqrt {\frac {2}{\pi }} \sqrt {a+b \arcsin (c x)}}{\sqrt {b}}\right )}{c}+d x \sqrt {a+b \arcsin (c x)}+\frac {1}{3} e x^3 \sqrt {a+b \arcsin (c x)}\) |
d*x*Sqrt[a + b*ArcSin[c*x]] + (e*x^3*Sqrt[a + b*ArcSin[c*x]])/3 - (Sqrt[b] *d*Sqrt[Pi/2]*Cos[a/b]*FresnelS[(Sqrt[2/Pi]*Sqrt[a + b*ArcSin[c*x]])/Sqrt[ b]])/c - (Sqrt[b]*e*Sqrt[Pi/2]*Cos[a/b]*FresnelS[(Sqrt[2/Pi]*Sqrt[a + b*Ar cSin[c*x]])/Sqrt[b]])/(4*c^3) + (Sqrt[b]*e*Sqrt[Pi/6]*Cos[(3*a)/b]*Fresnel S[(Sqrt[6/Pi]*Sqrt[a + b*ArcSin[c*x]])/Sqrt[b]])/(12*c^3) + (Sqrt[b]*d*Sqr t[Pi/2]*FresnelC[(Sqrt[2/Pi]*Sqrt[a + b*ArcSin[c*x]])/Sqrt[b]]*Sin[a/b])/c + (Sqrt[b]*e*Sqrt[Pi/2]*FresnelC[(Sqrt[2/Pi]*Sqrt[a + b*ArcSin[c*x]])/Sqr t[b]]*Sin[a/b])/(4*c^3) - (Sqrt[b]*e*Sqrt[Pi/6]*FresnelC[(Sqrt[6/Pi]*Sqrt[ a + b*ArcSin[c*x]])/Sqrt[b]]*Sin[(3*a)/b])/(12*c^3)
3.7.87.3.1 Defintions of rubi rules used
Int[((a_.) + ArcSin[(c_.)*(x_)]*(b_.))^(n_.)*((d_) + (e_.)*(x_)^2)^(p_.), x _Symbol] :> Int[ExpandIntegrand[(a + b*ArcSin[c*x])^n, (d + e*x^2)^p, x], x ] /; FreeQ[{a, b, c, d, e, n}, x] && NeQ[c^2*d + e, 0] && IntegerQ[p] && (G tQ[p, 0] || IGtQ[n, 0])
Time = 0.40 (sec) , antiderivative size = 557, normalized size of antiderivative = 1.51
| method | result | size |
| default | \(\frac {36 \sqrt {-\frac {1}{b}}\, \cos \left (\frac {a}{b}\right ) \operatorname {FresnelS}\left (\frac {\sqrt {2}\, \sqrt {a +b \arcsin \left (c x \right )}}{\sqrt {\pi }\, \sqrt {-\frac {1}{b}}\, b}\right ) \sqrt {a +b \arcsin \left (c x \right )}\, \sqrt {2}\, \sqrt {\pi }\, b \,c^{2} d +36 \sqrt {-\frac {1}{b}}\, \sin \left (\frac {a}{b}\right ) \operatorname {FresnelC}\left (\frac {\sqrt {2}\, \sqrt {a +b \arcsin \left (c x \right )}}{\sqrt {\pi }\, \sqrt {-\frac {1}{b}}\, b}\right ) \sqrt {a +b \arcsin \left (c x \right )}\, \sqrt {2}\, \sqrt {\pi }\, b \,c^{2} d +9 \sqrt {-\frac {1}{b}}\, \cos \left (\frac {a}{b}\right ) \operatorname {FresnelS}\left (\frac {\sqrt {2}\, \sqrt {a +b \arcsin \left (c x \right )}}{\sqrt {\pi }\, \sqrt {-\frac {1}{b}}\, b}\right ) \sqrt {a +b \arcsin \left (c x \right )}\, \sqrt {2}\, \sqrt {\pi }\, b e +9 \sqrt {-\frac {1}{b}}\, \sin \left (\frac {a}{b}\right ) \operatorname {FresnelC}\left (\frac {\sqrt {2}\, \sqrt {a +b \arcsin \left (c x \right )}}{\sqrt {\pi }\, \sqrt {-\frac {1}{b}}\, b}\right ) \sqrt {a +b \arcsin \left (c x \right )}\, \sqrt {2}\, \sqrt {\pi }\, b e -\cos \left (\frac {3 a}{b}\right ) \operatorname {FresnelS}\left (\frac {3 \sqrt {2}\, \sqrt {a +b \arcsin \left (c x \right )}}{\sqrt {\pi }\, \sqrt {-\frac {3}{b}}\, b}\right ) \sqrt {a +b \arcsin \left (c x \right )}\, \sqrt {2}\, \sqrt {\pi }\, \sqrt {-\frac {3}{b}}\, b e -\sin \left (\frac {3 a}{b}\right ) \operatorname {FresnelC}\left (\frac {3 \sqrt {2}\, \sqrt {a +b \arcsin \left (c x \right )}}{\sqrt {\pi }\, \sqrt {-\frac {3}{b}}\, b}\right ) \sqrt {a +b \arcsin \left (c x \right )}\, \sqrt {2}\, \sqrt {\pi }\, \sqrt {-\frac {3}{b}}\, b e -72 \arcsin \left (c x \right ) \sin \left (-\frac {a +b \arcsin \left (c x \right )}{b}+\frac {a}{b}\right ) b \,c^{2} d -72 \sin \left (-\frac {a +b \arcsin \left (c x \right )}{b}+\frac {a}{b}\right ) a \,c^{2} d +6 \arcsin \left (c x \right ) \sin \left (-\frac {3 \left (a +b \arcsin \left (c x \right )\right )}{b}+\frac {3 a}{b}\right ) b e -18 \arcsin \left (c x \right ) \sin \left (-\frac {a +b \arcsin \left (c x \right )}{b}+\frac {a}{b}\right ) b e +6 \sin \left (-\frac {3 \left (a +b \arcsin \left (c x \right )\right )}{b}+\frac {3 a}{b}\right ) a e -18 \sin \left (-\frac {a +b \arcsin \left (c x \right )}{b}+\frac {a}{b}\right ) a e}{72 c^{3} \sqrt {a +b \arcsin \left (c x \right )}}\) | \(557\) |
1/72/c^3*(36*(-1/b)^(1/2)*cos(a/b)*FresnelS(2^(1/2)/Pi^(1/2)/(-1/b)^(1/2)* (a+b*arcsin(c*x))^(1/2)/b)*(a+b*arcsin(c*x))^(1/2)*2^(1/2)*Pi^(1/2)*b*c^2* d+36*(-1/b)^(1/2)*sin(a/b)*FresnelC(2^(1/2)/Pi^(1/2)/(-1/b)^(1/2)*(a+b*arc sin(c*x))^(1/2)/b)*(a+b*arcsin(c*x))^(1/2)*2^(1/2)*Pi^(1/2)*b*c^2*d+9*(-1/ b)^(1/2)*cos(a/b)*FresnelS(2^(1/2)/Pi^(1/2)/(-1/b)^(1/2)*(a+b*arcsin(c*x)) ^(1/2)/b)*(a+b*arcsin(c*x))^(1/2)*2^(1/2)*Pi^(1/2)*b*e+9*(-1/b)^(1/2)*sin( a/b)*FresnelC(2^(1/2)/Pi^(1/2)/(-1/b)^(1/2)*(a+b*arcsin(c*x))^(1/2)/b)*(a+ b*arcsin(c*x))^(1/2)*2^(1/2)*Pi^(1/2)*b*e-cos(3*a/b)*FresnelS(3*2^(1/2)/Pi ^(1/2)/(-3/b)^(1/2)*(a+b*arcsin(c*x))^(1/2)/b)*(a+b*arcsin(c*x))^(1/2)*2^( 1/2)*Pi^(1/2)*(-3/b)^(1/2)*b*e-sin(3*a/b)*FresnelC(3*2^(1/2)/Pi^(1/2)/(-3/ b)^(1/2)*(a+b*arcsin(c*x))^(1/2)/b)*(a+b*arcsin(c*x))^(1/2)*2^(1/2)*Pi^(1/ 2)*(-3/b)^(1/2)*b*e-72*arcsin(c*x)*sin(-(a+b*arcsin(c*x))/b+a/b)*b*c^2*d-7 2*sin(-(a+b*arcsin(c*x))/b+a/b)*a*c^2*d+6*arcsin(c*x)*sin(-3*(a+b*arcsin(c *x))/b+3*a/b)*b*e-18*arcsin(c*x)*sin(-(a+b*arcsin(c*x))/b+a/b)*b*e+6*sin(- 3*(a+b*arcsin(c*x))/b+3*a/b)*a*e-18*sin(-(a+b*arcsin(c*x))/b+a/b)*a*e)/(a+ b*arcsin(c*x))^(1/2)
Exception generated. \[ \int \left (d+e x^2\right ) \sqrt {a+b \arcsin (c x)} \, dx=\text {Exception raised: TypeError} \]
Exception raised: TypeError >> Error detected within library code: inte grate: implementation incomplete (constant residues)
\[ \int \left (d+e x^2\right ) \sqrt {a+b \arcsin (c x)} \, dx=\int \sqrt {a + b \operatorname {asin}{\left (c x \right )}} \left (d + e x^{2}\right )\, dx \]
\[ \int \left (d+e x^2\right ) \sqrt {a+b \arcsin (c x)} \, dx=\int { {\left (e x^{2} + d\right )} \sqrt {b \arcsin \left (c x\right ) + a} \,d x } \]
Result contains complex when optimal does not.
Time = 1.31 (sec) , antiderivative size = 1661, normalized size of antiderivative = 4.50 \[ \int \left (d+e x^2\right ) \sqrt {a+b \arcsin (c x)} \, dx=\text {Too large to display} \]
1/2*sqrt(2)*sqrt(pi)*a*b^2*d*erf(-1/2*I*sqrt(2)*sqrt(b*arcsin(c*x) + a)/sq rt(abs(b)) - 1/2*sqrt(2)*sqrt(b*arcsin(c*x) + a)*sqrt(abs(b))/b)*e^(I*a/b) /((I*b^3/sqrt(abs(b)) + b^2*sqrt(abs(b)))*c) + 1/4*I*sqrt(2)*sqrt(pi)*b^3* d*erf(-1/2*I*sqrt(2)*sqrt(b*arcsin(c*x) + a)/sqrt(abs(b)) - 1/2*sqrt(2)*sq rt(b*arcsin(c*x) + a)*sqrt(abs(b))/b)*e^(I*a/b)/((I*b^3/sqrt(abs(b)) + b^2 *sqrt(abs(b)))*c) + 1/2*sqrt(2)*sqrt(pi)*a*b^2*d*erf(1/2*I*sqrt(2)*sqrt(b* arcsin(c*x) + a)/sqrt(abs(b)) - 1/2*sqrt(2)*sqrt(b*arcsin(c*x) + a)*sqrt(a bs(b))/b)*e^(-I*a/b)/((-I*b^3/sqrt(abs(b)) + b^2*sqrt(abs(b)))*c) - 1/4*I* sqrt(2)*sqrt(pi)*b^3*d*erf(1/2*I*sqrt(2)*sqrt(b*arcsin(c*x) + a)/sqrt(abs( b)) - 1/2*sqrt(2)*sqrt(b*arcsin(c*x) + a)*sqrt(abs(b))/b)*e^(-I*a/b)/((-I* b^3/sqrt(abs(b)) + b^2*sqrt(abs(b)))*c) - sqrt(pi)*a*b*d*erf(-1/2*I*sqrt(2 )*sqrt(b*arcsin(c*x) + a)/sqrt(abs(b)) - 1/2*sqrt(2)*sqrt(b*arcsin(c*x) + a)*sqrt(abs(b))/b)*e^(I*a/b)/((I*sqrt(2)*b^2/sqrt(abs(b)) + sqrt(2)*b*sqrt (abs(b)))*c) - sqrt(pi)*a*b*d*erf(1/2*I*sqrt(2)*sqrt(b*arcsin(c*x) + a)/sq rt(abs(b)) - 1/2*sqrt(2)*sqrt(b*arcsin(c*x) + a)*sqrt(abs(b))/b)*e^(-I*a/b )/((-I*sqrt(2)*b^2/sqrt(abs(b)) + sqrt(2)*b*sqrt(abs(b)))*c) + 1/8*sqrt(2) *sqrt(pi)*a*b^2*e*erf(-1/2*I*sqrt(2)*sqrt(b*arcsin(c*x) + a)/sqrt(abs(b)) - 1/2*sqrt(2)*sqrt(b*arcsin(c*x) + a)*sqrt(abs(b))/b)*e^(I*a/b)/((I*b^3/sq rt(abs(b)) + b^2*sqrt(abs(b)))*c^3) + 1/16*I*sqrt(2)*sqrt(pi)*b^3*e*erf(-1 /2*I*sqrt(2)*sqrt(b*arcsin(c*x) + a)/sqrt(abs(b)) - 1/2*sqrt(2)*sqrt(b*...
Timed out. \[ \int \left (d+e x^2\right ) \sqrt {a+b \arcsin (c x)} \, dx=\int \sqrt {a+b\,\mathrm {asin}\left (c\,x\right )}\,\left (e\,x^2+d\right ) \,d x \]