Integrand size = 29, antiderivative size = 485 \[ \int \frac {x^3 \left (d-c^2 d x^2\right )^2}{(a+b \arcsin (c x))^{3/2}} \, dx=-\frac {2 d^2 x^3 \left (1-c^2 x^2\right )^{5/2}}{b c \sqrt {a+b \arcsin (c x)}}+\frac {d^2 \sqrt {\frac {\pi }{2}} \cos \left (\frac {4 a}{b}\right ) \operatorname {FresnelC}\left (\frac {2 \sqrt {\frac {2}{\pi }} \sqrt {a+b \arcsin (c x)}}{\sqrt {b}}\right )}{8 b^{3/2} c^4}-\frac {d^2 \sqrt {3 \pi } \cos \left (\frac {6 a}{b}\right ) \operatorname {FresnelC}\left (\frac {2 \sqrt {\frac {3}{\pi }} \sqrt {a+b \arcsin (c x)}}{\sqrt {b}}\right )}{16 b^{3/2} c^4}+\frac {3 d^2 \sqrt {\pi } \cos \left (\frac {2 a}{b}\right ) \operatorname {FresnelC}\left (\frac {2 \sqrt {a+b \arcsin (c x)}}{\sqrt {b} \sqrt {\pi }}\right )}{16 b^{3/2} c^4}-\frac {d^2 \sqrt {\pi } \cos \left (\frac {8 a}{b}\right ) \operatorname {FresnelC}\left (\frac {4 \sqrt {a+b \arcsin (c x)}}{\sqrt {b} \sqrt {\pi }}\right )}{16 b^{3/2} c^4}+\frac {3 d^2 \sqrt {\pi } \operatorname {FresnelS}\left (\frac {2 \sqrt {a+b \arcsin (c x)}}{\sqrt {b} \sqrt {\pi }}\right ) \sin \left (\frac {2 a}{b}\right )}{16 b^{3/2} c^4}+\frac {d^2 \sqrt {\frac {\pi }{2}} \operatorname {FresnelS}\left (\frac {2 \sqrt {\frac {2}{\pi }} \sqrt {a+b \arcsin (c x)}}{\sqrt {b}}\right ) \sin \left (\frac {4 a}{b}\right )}{8 b^{3/2} c^4}-\frac {d^2 \sqrt {3 \pi } \operatorname {FresnelS}\left (\frac {2 \sqrt {\frac {3}{\pi }} \sqrt {a+b \arcsin (c x)}}{\sqrt {b}}\right ) \sin \left (\frac {6 a}{b}\right )}{16 b^{3/2} c^4}-\frac {d^2 \sqrt {\pi } \operatorname {FresnelS}\left (\frac {4 \sqrt {a+b \arcsin (c x)}}{\sqrt {b} \sqrt {\pi }}\right ) \sin \left (\frac {8 a}{b}\right )}{16 b^{3/2} c^4} \] Output:
-2*d^2*x^3*(-c^2*x^2+1)^(5/2)/b/c/(a+b*arcsin(c*x))^(1/2)+1/16*d^2*2^(1/2) *Pi^(1/2)*cos(4*a/b)*FresnelC(2*2^(1/2)/Pi^(1/2)*(a+b*arcsin(c*x))^(1/2)/b ^(1/2))/b^(3/2)/c^4-1/16*d^2*3^(1/2)*Pi^(1/2)*cos(6*a/b)*FresnelC(2*3^(1/2 )/Pi^(1/2)*(a+b*arcsin(c*x))^(1/2)/b^(1/2))/b^(3/2)/c^4+3/16*d^2*Pi^(1/2)* cos(2*a/b)*FresnelC(2*(a+b*arcsin(c*x))^(1/2)/b^(1/2)/Pi^(1/2))/b^(3/2)/c^ 4-1/16*d^2*Pi^(1/2)*cos(8*a/b)*FresnelC(4*(a+b*arcsin(c*x))^(1/2)/b^(1/2)/ Pi^(1/2))/b^(3/2)/c^4+3/16*d^2*Pi^(1/2)*FresnelS(2*(a+b*arcsin(c*x))^(1/2) /b^(1/2)/Pi^(1/2))*sin(2*a/b)/b^(3/2)/c^4+1/16*d^2*2^(1/2)*Pi^(1/2)*Fresne lS(2*2^(1/2)/Pi^(1/2)*(a+b*arcsin(c*x))^(1/2)/b^(1/2))*sin(4*a/b)/b^(3/2)/ c^4-1/16*d^2*3^(1/2)*Pi^(1/2)*FresnelS(2*3^(1/2)/Pi^(1/2)*(a+b*arcsin(c*x) )^(1/2)/b^(1/2))*sin(6*a/b)/b^(3/2)/c^4-1/16*d^2*Pi^(1/2)*FresnelS(4*(a+b* arcsin(c*x))^(1/2)/b^(1/2)/Pi^(1/2))*sin(8*a/b)/b^(3/2)/c^4
Result contains complex when optimal does not.
Time = 1.82 (sec) , antiderivative size = 540, normalized size of antiderivative = 1.11 \[ \int \frac {x^3 \left (d-c^2 d x^2\right )^2}{(a+b \arcsin (c x))^{3/2}} \, dx=-\frac {i d^2 e^{-\frac {8 i a}{b}} \left (3 \sqrt {2} e^{\frac {6 i a}{b}} \sqrt {-\frac {i (a+b \arcsin (c x))}{b}} \Gamma \left (\frac {1}{2},-\frac {2 i (a+b \arcsin (c x))}{b}\right )-3 \sqrt {2} e^{\frac {10 i a}{b}} \sqrt {\frac {i (a+b \arcsin (c x))}{b}} \Gamma \left (\frac {1}{2},\frac {2 i (a+b \arcsin (c x))}{b}\right )+2 e^{\frac {4 i a}{b}} \sqrt {-\frac {i (a+b \arcsin (c x))}{b}} \Gamma \left (\frac {1}{2},-\frac {4 i (a+b \arcsin (c x))}{b}\right )-2 e^{\frac {12 i a}{b}} \sqrt {\frac {i (a+b \arcsin (c x))}{b}} \Gamma \left (\frac {1}{2},\frac {4 i (a+b \arcsin (c x))}{b}\right )-\sqrt {6} e^{\frac {2 i a}{b}} \sqrt {-\frac {i (a+b \arcsin (c x))}{b}} \Gamma \left (\frac {1}{2},-\frac {6 i (a+b \arcsin (c x))}{b}\right )+\sqrt {6} e^{\frac {14 i a}{b}} \sqrt {\frac {i (a+b \arcsin (c x))}{b}} \Gamma \left (\frac {1}{2},\frac {6 i (a+b \arcsin (c x))}{b}\right )-\sqrt {2} \sqrt {-\frac {i (a+b \arcsin (c x))}{b}} \Gamma \left (\frac {1}{2},-\frac {8 i (a+b \arcsin (c x))}{b}\right )+\sqrt {2} e^{\frac {16 i a}{b}} \sqrt {\frac {i (a+b \arcsin (c x))}{b}} \Gamma \left (\frac {1}{2},\frac {8 i (a+b \arcsin (c x))}{b}\right )-6 i e^{\frac {8 i a}{b}} \sin (2 \arcsin (c x))-2 i e^{\frac {8 i a}{b}} \sin (4 \arcsin (c x))+2 i e^{\frac {8 i a}{b}} \sin (6 \arcsin (c x))+i e^{\frac {8 i a}{b}} \sin (8 \arcsin (c x))\right )}{64 b c^4 \sqrt {a+b \arcsin (c x)}} \] Input:
Integrate[(x^3*(d - c^2*d*x^2)^2)/(a + b*ArcSin[c*x])^(3/2),x]
Output:
((-1/64*I)*d^2*(3*Sqrt[2]*E^(((6*I)*a)/b)*Sqrt[((-I)*(a + b*ArcSin[c*x]))/ b]*Gamma[1/2, ((-2*I)*(a + b*ArcSin[c*x]))/b] - 3*Sqrt[2]*E^(((10*I)*a)/b) *Sqrt[(I*(a + b*ArcSin[c*x]))/b]*Gamma[1/2, ((2*I)*(a + b*ArcSin[c*x]))/b] + 2*E^(((4*I)*a)/b)*Sqrt[((-I)*(a + b*ArcSin[c*x]))/b]*Gamma[1/2, ((-4*I) *(a + b*ArcSin[c*x]))/b] - 2*E^(((12*I)*a)/b)*Sqrt[(I*(a + b*ArcSin[c*x])) /b]*Gamma[1/2, ((4*I)*(a + b*ArcSin[c*x]))/b] - Sqrt[6]*E^(((2*I)*a)/b)*Sq rt[((-I)*(a + b*ArcSin[c*x]))/b]*Gamma[1/2, ((-6*I)*(a + b*ArcSin[c*x]))/b ] + Sqrt[6]*E^(((14*I)*a)/b)*Sqrt[(I*(a + b*ArcSin[c*x]))/b]*Gamma[1/2, (( 6*I)*(a + b*ArcSin[c*x]))/b] - Sqrt[2]*Sqrt[((-I)*(a + b*ArcSin[c*x]))/b]* Gamma[1/2, ((-8*I)*(a + b*ArcSin[c*x]))/b] + Sqrt[2]*E^(((16*I)*a)/b)*Sqrt [(I*(a + b*ArcSin[c*x]))/b]*Gamma[1/2, ((8*I)*(a + b*ArcSin[c*x]))/b] - (6 *I)*E^(((8*I)*a)/b)*Sin[2*ArcSin[c*x]] - (2*I)*E^(((8*I)*a)/b)*Sin[4*ArcSi n[c*x]] + (2*I)*E^(((8*I)*a)/b)*Sin[6*ArcSin[c*x]] + I*E^(((8*I)*a)/b)*Sin [8*ArcSin[c*x]]))/(b*c^4*E^(((8*I)*a)/b)*Sqrt[a + b*ArcSin[c*x]])
Time = 1.80 (sec) , antiderivative size = 605, normalized size of antiderivative = 1.25, number of steps used = 5, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.138, Rules used = {5214, 5224, 4906, 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.
| \(\displaystyle \int \frac {x^3 \left (d-c^2 d x^2\right )^2}{(a+b \arcsin (c x))^{3/2}} \, dx\) |
| \(\Big \downarrow \) 5214 |
| \(\displaystyle \frac {6 d^2 \int \frac {x^2 \left (1-c^2 x^2\right )^{3/2}}{\sqrt {a+b \arcsin (c x)}}dx}{b c}-\frac {16 c d^2 \int \frac {x^4 \left (1-c^2 x^2\right )^{3/2}}{\sqrt {a+b \arcsin (c x)}}dx}{b}-\frac {2 d^2 x^3 \left (1-c^2 x^2\right )^{5/2}}{b c \sqrt {a+b \arcsin (c x)}}\) |
| \(\Big \downarrow \) 5224 |
| \(\displaystyle -\frac {16 d^2 \int \frac {\cos ^4\left (\frac {a}{b}-\frac {a+b \arcsin (c x)}{b}\right ) \sin ^4\left (\frac {a}{b}-\frac {a+b \arcsin (c x)}{b}\right )}{\sqrt {a+b \arcsin (c x)}}d(a+b \arcsin (c x))}{b^2 c^4}+\frac {6 d^2 \int \frac {\cos ^4\left (\frac {a}{b}-\frac {a+b \arcsin (c x)}{b}\right ) \sin ^2\left (\frac {a}{b}-\frac {a+b \arcsin (c x)}{b}\right )}{\sqrt {a+b \arcsin (c x)}}d(a+b \arcsin (c x))}{b^2 c^4}-\frac {2 d^2 x^3 \left (1-c^2 x^2\right )^{5/2}}{b c \sqrt {a+b \arcsin (c x)}}\) |
| \(\Big \downarrow \) 4906 |
| \(\displaystyle -\frac {16 d^2 \int \left (\frac {\cos \left (\frac {8 a}{b}-\frac {8 (a+b \arcsin (c x))}{b}\right )}{128 \sqrt {a+b \arcsin (c x)}}-\frac {\cos \left (\frac {4 a}{b}-\frac {4 (a+b \arcsin (c x))}{b}\right )}{32 \sqrt {a+b \arcsin (c x)}}+\frac {3}{128 \sqrt {a+b \arcsin (c x)}}\right )d(a+b \arcsin (c x))}{b^2 c^4}+\frac {6 d^2 \int \left (-\frac {\cos \left (\frac {6 a}{b}-\frac {6 (a+b \arcsin (c x))}{b}\right )}{32 \sqrt {a+b \arcsin (c x)}}-\frac {\cos \left (\frac {4 a}{b}-\frac {4 (a+b \arcsin (c x))}{b}\right )}{16 \sqrt {a+b \arcsin (c x)}}+\frac {\cos \left (\frac {2 a}{b}-\frac {2 (a+b \arcsin (c x))}{b}\right )}{32 \sqrt {a+b \arcsin (c x)}}+\frac {1}{16 \sqrt {a+b \arcsin (c x)}}\right )d(a+b \arcsin (c x))}{b^2 c^4}-\frac {2 d^2 x^3 \left (1-c^2 x^2\right )^{5/2}}{b c \sqrt {a+b \arcsin (c x)}}\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {6 d^2 \left (-\frac {1}{16} \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 x)}}{\sqrt {b}}\right )-\frac {1}{32} \sqrt {\frac {\pi }{3}} \sqrt {b} \cos \left (\frac {6 a}{b}\right ) \operatorname {FresnelC}\left (\frac {2 \sqrt {\frac {3}{\pi }} \sqrt {a+b \arcsin (c x)}}{\sqrt {b}}\right )+\frac {1}{32} \sqrt {\pi } \sqrt {b} \cos \left (\frac {2 a}{b}\right ) \operatorname {FresnelC}\left (\frac {2 \sqrt {a+b \arcsin (c x)}}{\sqrt {b} \sqrt {\pi }}\right )+\frac {1}{32} \sqrt {\pi } \sqrt {b} \sin \left (\frac {2 a}{b}\right ) \operatorname {FresnelS}\left (\frac {2 \sqrt {a+b \arcsin (c x)}}{\sqrt {b} \sqrt {\pi }}\right )-\frac {1}{16} \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 x)}}{\sqrt {b}}\right )-\frac {1}{32} \sqrt {\frac {\pi }{3}} \sqrt {b} \sin \left (\frac {6 a}{b}\right ) \operatorname {FresnelS}\left (\frac {2 \sqrt {\frac {3}{\pi }} \sqrt {a+b \arcsin (c x)}}{\sqrt {b}}\right )+\frac {1}{8} \sqrt {a+b \arcsin (c x)}\right )}{b^2 c^4}-\frac {16 d^2 \left (-\frac {1}{32} \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 x)}}{\sqrt {b}}\right )+\frac {1}{256} \sqrt {\pi } \sqrt {b} \cos \left (\frac {8 a}{b}\right ) \operatorname {FresnelC}\left (\frac {4 \sqrt {a+b \arcsin (c x)}}{\sqrt {b} \sqrt {\pi }}\right )-\frac {1}{32} \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 x)}}{\sqrt {b}}\right )+\frac {1}{256} \sqrt {\pi } \sqrt {b} \sin \left (\frac {8 a}{b}\right ) \operatorname {FresnelS}\left (\frac {4 \sqrt {a+b \arcsin (c x)}}{\sqrt {b} \sqrt {\pi }}\right )+\frac {3}{64} \sqrt {a+b \arcsin (c x)}\right )}{b^2 c^4}-\frac {2 d^2 x^3 \left (1-c^2 x^2\right )^{5/2}}{b c \sqrt {a+b \arcsin (c x)}}\) |
Input:
Int[(x^3*(d - c^2*d*x^2)^2)/(a + b*ArcSin[c*x])^(3/2),x]
Output:
(-2*d^2*x^3*(1 - c^2*x^2)^(5/2))/(b*c*Sqrt[a + b*ArcSin[c*x]]) + (6*d^2*(S qrt[a + b*ArcSin[c*x]]/8 - (Sqrt[b]*Sqrt[Pi/2]*Cos[(4*a)/b]*FresnelC[(2*Sq rt[2/Pi]*Sqrt[a + b*ArcSin[c*x]])/Sqrt[b]])/16 - (Sqrt[b]*Sqrt[Pi/3]*Cos[( 6*a)/b]*FresnelC[(2*Sqrt[3/Pi]*Sqrt[a + b*ArcSin[c*x]])/Sqrt[b]])/32 + (Sq rt[b]*Sqrt[Pi]*Cos[(2*a)/b]*FresnelC[(2*Sqrt[a + b*ArcSin[c*x]])/(Sqrt[b]* Sqrt[Pi])])/32 + (Sqrt[b]*Sqrt[Pi]*FresnelS[(2*Sqrt[a + b*ArcSin[c*x]])/(S qrt[b]*Sqrt[Pi])]*Sin[(2*a)/b])/32 - (Sqrt[b]*Sqrt[Pi/2]*FresnelS[(2*Sqrt[ 2/Pi]*Sqrt[a + b*ArcSin[c*x]])/Sqrt[b]]*Sin[(4*a)/b])/16 - (Sqrt[b]*Sqrt[P i/3]*FresnelS[(2*Sqrt[3/Pi]*Sqrt[a + b*ArcSin[c*x]])/Sqrt[b]]*Sin[(6*a)/b] )/32))/(b^2*c^4) - (16*d^2*((3*Sqrt[a + b*ArcSin[c*x]])/64 - (Sqrt[b]*Sqrt [Pi/2]*Cos[(4*a)/b]*FresnelC[(2*Sqrt[2/Pi]*Sqrt[a + b*ArcSin[c*x]])/Sqrt[b ]])/32 + (Sqrt[b]*Sqrt[Pi]*Cos[(8*a)/b]*FresnelC[(4*Sqrt[a + b*ArcSin[c*x] ])/(Sqrt[b]*Sqrt[Pi])])/256 - (Sqrt[b]*Sqrt[Pi/2]*FresnelS[(2*Sqrt[2/Pi]*S qrt[a + b*ArcSin[c*x]])/Sqrt[b]]*Sin[(4*a)/b])/32 + (Sqrt[b]*Sqrt[Pi]*Fres nelS[(4*Sqrt[a + b*ArcSin[c*x]])/(Sqrt[b]*Sqrt[Pi])]*Sin[(8*a)/b])/256))/( b^2*c^4)
Int[Cos[(a_.) + (b_.)*(x_)]^(p_.)*((c_.) + (d_.)*(x_))^(m_.)*Sin[(a_.) + (b _.)*(x_)]^(n_.), x_Symbol] :> Int[ExpandTrigReduce[(c + d*x)^m, Sin[a + b*x ]^n*Cos[a + b*x]^p, x], x] /; FreeQ[{a, b, c, d, m}, x] && IGtQ[n, 0] && IG tQ[p, 0]
Int[((a_.) + ArcSin[(c_.)*(x_)]*(b_.))^(n_)*((f_.)*(x_))^(m_.)*((d_) + (e_. )*(x_)^2)^(p_.), x_Symbol] :> Simp[(f*x)^m*Sqrt[1 - c^2*x^2]*(d + e*x^2)^p* ((a + b*ArcSin[c*x])^(n + 1)/(b*c*(n + 1))), x] + (-Simp[f*(m/(b*c*(n + 1)) )*Simp[(d + e*x^2)^p/(1 - c^2*x^2)^p] Int[(f*x)^(m - 1)*(1 - c^2*x^2)^(p - 1/2)*(a + b*ArcSin[c*x])^(n + 1), x], x] + Simp[c*((m + 2*p + 1)/(b*f*(n + 1)))*Simp[(d + e*x^2)^p/(1 - c^2*x^2)^p] Int[(f*x)^(m + 1)*(1 - c^2*x^2 )^(p - 1/2)*(a + b*ArcSin[c*x])^(n + 1), x], x]) /; FreeQ[{a, b, c, d, e, f }, x] && EqQ[c^2*d + e, 0] && LtQ[n, -1] && IGtQ[2*p, 0] && NeQ[m + 2*p + 1 , 0] && IGtQ[m, -3]
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]
Time = 1.34 (sec) , antiderivative size = 591, normalized size of antiderivative = 1.22
| method | result | size |
| default | \(\frac {d^{2} \left (4 \sqrt {-\frac {1}{b}}\, \sqrt {2}\, \sqrt {\pi }\, \sqrt {a +b \arcsin \left (c x \right )}\, \cos \left (\frac {4 a}{b}\right ) \operatorname {FresnelC}\left (\frac {2 \sqrt {2}\, \sqrt {a +b \arcsin \left (c x \right )}}{\sqrt {\pi }\, \sqrt {-\frac {1}{b}}\, b}\right )-4 \sqrt {-\frac {1}{b}}\, \sqrt {2}\, \sqrt {\pi }\, \sqrt {a +b \arcsin \left (c x \right )}\, \sin \left (\frac {4 a}{b}\right ) \operatorname {FresnelS}\left (\frac {2 \sqrt {2}\, \sqrt {a +b \arcsin \left (c x \right )}}{\sqrt {\pi }\, \sqrt {-\frac {1}{b}}\, b}\right )-2 \sqrt {-\frac {6}{b}}\, \operatorname {FresnelC}\left (\frac {6 \sqrt {2}\, \sqrt {a +b \arcsin \left (c x \right )}}{\sqrt {\pi }\, \sqrt {-\frac {6}{b}}\, b}\right ) \sqrt {a +b \arcsin \left (c x \right )}\, \sqrt {2}\, \cos \left (\frac {6 a}{b}\right ) \sqrt {\pi }+2 \sqrt {-\frac {6}{b}}\, \operatorname {FresnelS}\left (\frac {6 \sqrt {2}\, \sqrt {a +b \arcsin \left (c x \right )}}{\sqrt {\pi }\, \sqrt {-\frac {6}{b}}\, b}\right ) \sqrt {a +b \arcsin \left (c x \right )}\, \sqrt {2}\, \sin \left (\frac {6 a}{b}\right ) \sqrt {\pi }+12 \sqrt {-\frac {1}{b}}\, \sqrt {\pi }\, \sqrt {a +b \arcsin \left (c x \right )}\, \cos \left (\frac {2 a}{b}\right ) \operatorname {FresnelC}\left (\frac {2 \sqrt {2}\, \sqrt {a +b \arcsin \left (c x \right )}}{\sqrt {\pi }\, \sqrt {-\frac {2}{b}}\, b}\right )-12 \sqrt {-\frac {1}{b}}\, \sqrt {\pi }\, \sqrt {a +b \arcsin \left (c x \right )}\, \sin \left (\frac {2 a}{b}\right ) \operatorname {FresnelS}\left (\frac {2 \sqrt {2}\, \sqrt {a +b \arcsin \left (c x \right )}}{\sqrt {\pi }\, \sqrt {-\frac {2}{b}}\, b}\right )-4 \cos \left (\frac {8 a}{b}\right ) \operatorname {FresnelC}\left (\frac {4 \sqrt {2}\, \sqrt {a +b \arcsin \left (c x \right )}}{\sqrt {\pi }\, \sqrt {-\frac {2}{b}}\, b}\right ) \sqrt {a +b \arcsin \left (c x \right )}\, \sqrt {\pi }\, \sqrt {-\frac {1}{b}}+4 \sin \left (\frac {8 a}{b}\right ) \operatorname {FresnelS}\left (\frac {4 \sqrt {2}\, \sqrt {a +b \arcsin \left (c x \right )}}{\sqrt {\pi }\, \sqrt {-\frac {2}{b}}\, b}\right ) \sqrt {a +b \arcsin \left (c x \right )}\, \sqrt {\pi }\, \sqrt {-\frac {1}{b}}+6 \sin \left (-\frac {2 \left (a +b \arcsin \left (c x \right )\right )}{b}+\frac {2 a}{b}\right )+2 \sin \left (-\frac {4 \left (a +b \arcsin \left (c x \right )\right )}{b}+\frac {4 a}{b}\right )-2 \sin \left (-\frac {6 \left (a +b \arcsin \left (c x \right )\right )}{b}+\frac {6 a}{b}\right )-\sin \left (-\frac {8 \left (a +b \arcsin \left (c x \right )\right )}{b}+\frac {8 a}{b}\right )\right )}{64 c^{4} b \sqrt {a +b \arcsin \left (c x \right )}}\) | \(591\) |
Input:
int(x^3*(-c^2*d*x^2+d)^2/(a+b*arcsin(c*x))^(3/2),x,method=_RETURNVERBOSE)
Output:
1/64*d^2/c^4/b*(4*(-1/b)^(1/2)*2^(1/2)*Pi^(1/2)*(a+b*arcsin(c*x))^(1/2)*co s(4*a/b)*FresnelC(2*2^(1/2)/Pi^(1/2)/(-1/b)^(1/2)*(a+b*arcsin(c*x))^(1/2)/ b)-4*(-1/b)^(1/2)*2^(1/2)*Pi^(1/2)*(a+b*arcsin(c*x))^(1/2)*sin(4*a/b)*Fres nelS(2*2^(1/2)/Pi^(1/2)/(-1/b)^(1/2)*(a+b*arcsin(c*x))^(1/2)/b)-2*(-6/b)^( 1/2)*FresnelC(6*2^(1/2)/Pi^(1/2)/(-6/b)^(1/2)*(a+b*arcsin(c*x))^(1/2)/b)*( a+b*arcsin(c*x))^(1/2)*2^(1/2)*cos(6*a/b)*Pi^(1/2)+2*(-6/b)^(1/2)*FresnelS (6*2^(1/2)/Pi^(1/2)/(-6/b)^(1/2)*(a+b*arcsin(c*x))^(1/2)/b)*(a+b*arcsin(c* x))^(1/2)*2^(1/2)*sin(6*a/b)*Pi^(1/2)+12*(-1/b)^(1/2)*Pi^(1/2)*(a+b*arcsin (c*x))^(1/2)*cos(2*a/b)*FresnelC(2*2^(1/2)/Pi^(1/2)/(-2/b)^(1/2)*(a+b*arcs in(c*x))^(1/2)/b)-12*(-1/b)^(1/2)*Pi^(1/2)*(a+b*arcsin(c*x))^(1/2)*sin(2*a /b)*FresnelS(2*2^(1/2)/Pi^(1/2)/(-2/b)^(1/2)*(a+b*arcsin(c*x))^(1/2)/b)-4* cos(8*a/b)*FresnelC(4*2^(1/2)/Pi^(1/2)/(-2/b)^(1/2)*(a+b*arcsin(c*x))^(1/2 )/b)*(a+b*arcsin(c*x))^(1/2)*Pi^(1/2)*(-1/b)^(1/2)+4*sin(8*a/b)*FresnelS(4 *2^(1/2)/Pi^(1/2)/(-2/b)^(1/2)*(a+b*arcsin(c*x))^(1/2)/b)*(a+b*arcsin(c*x) )^(1/2)*Pi^(1/2)*(-1/b)^(1/2)+6*sin(-2*(a+b*arcsin(c*x))/b+2*a/b)+2*sin(-4 *(a+b*arcsin(c*x))/b+4*a/b)-2*sin(-6*(a+b*arcsin(c*x))/b+6*a/b)-sin(-8*(a+ b*arcsin(c*x))/b+8*a/b))/(a+b*arcsin(c*x))^(1/2)
Exception generated. \[ \int \frac {x^3 \left (d-c^2 d x^2\right )^2}{(a+b \arcsin (c x))^{3/2}} \, dx=\text {Exception raised: TypeError} \] Input:
integrate(x^3*(-c^2*d*x^2+d)^2/(a+b*arcsin(c*x))^(3/2),x, algorithm="frica s")
Output:
Exception raised: TypeError >> Error detected within library code: inte grate: implementation incomplete (constant residues)
\[ \int \frac {x^3 \left (d-c^2 d x^2\right )^2}{(a+b \arcsin (c x))^{3/2}} \, dx=d^{2} \left (\int \frac {x^{3}}{a \sqrt {a + b \operatorname {asin}{\left (c x \right )}} + b \sqrt {a + b \operatorname {asin}{\left (c x \right )}} \operatorname {asin}{\left (c x \right )}}\, dx + \int \left (- \frac {2 c^{2} x^{5}}{a \sqrt {a + b \operatorname {asin}{\left (c x \right )}} + b \sqrt {a + b \operatorname {asin}{\left (c x \right )}} \operatorname {asin}{\left (c x \right )}}\right )\, dx + \int \frac {c^{4} x^{7}}{a \sqrt {a + b \operatorname {asin}{\left (c x \right )}} + b \sqrt {a + b \operatorname {asin}{\left (c x \right )}} \operatorname {asin}{\left (c x \right )}}\, dx\right ) \] Input:
integrate(x**3*(-c**2*d*x**2+d)**2/(a+b*asin(c*x))**(3/2),x)
Output:
d**2*(Integral(x**3/(a*sqrt(a + b*asin(c*x)) + b*sqrt(a + b*asin(c*x))*asi n(c*x)), x) + Integral(-2*c**2*x**5/(a*sqrt(a + b*asin(c*x)) + b*sqrt(a + b*asin(c*x))*asin(c*x)), x) + Integral(c**4*x**7/(a*sqrt(a + b*asin(c*x)) + b*sqrt(a + b*asin(c*x))*asin(c*x)), x))
\[ \int \frac {x^3 \left (d-c^2 d x^2\right )^2}{(a+b \arcsin (c x))^{3/2}} \, dx=\int { \frac {{\left (c^{2} d x^{2} - d\right )}^{2} x^{3}}{{\left (b \arcsin \left (c x\right ) + a\right )}^{\frac {3}{2}}} \,d x } \] Input:
integrate(x^3*(-c^2*d*x^2+d)^2/(a+b*arcsin(c*x))^(3/2),x, algorithm="maxim a")
Output:
integrate((c^2*d*x^2 - d)^2*x^3/(b*arcsin(c*x) + a)^(3/2), x)
\[ \int \frac {x^3 \left (d-c^2 d x^2\right )^2}{(a+b \arcsin (c x))^{3/2}} \, dx=\int { \frac {{\left (c^{2} d x^{2} - d\right )}^{2} x^{3}}{{\left (b \arcsin \left (c x\right ) + a\right )}^{\frac {3}{2}}} \,d x } \] Input:
integrate(x^3*(-c^2*d*x^2+d)^2/(a+b*arcsin(c*x))^(3/2),x, algorithm="giac" )
Output:
integrate((c^2*d*x^2 - d)^2*x^3/(b*arcsin(c*x) + a)^(3/2), x)
Timed out. \[ \int \frac {x^3 \left (d-c^2 d x^2\right )^2}{(a+b \arcsin (c x))^{3/2}} \, dx=\int \frac {x^3\,{\left (d-c^2\,d\,x^2\right )}^2}{{\left (a+b\,\mathrm {asin}\left (c\,x\right )\right )}^{3/2}} \,d x \] Input:
int((x^3*(d - c^2*d*x^2)^2)/(a + b*asin(c*x))^(3/2),x)
Output:
int((x^3*(d - c^2*d*x^2)^2)/(a + b*asin(c*x))^(3/2), x)
\[ \int \frac {x^3 \left (d-c^2 d x^2\right )^2}{(a+b \arcsin (c x))^{3/2}} \, dx=d^{2} \left (\left (\int \frac {\sqrt {\mathit {asin} \left (c x \right ) b +a}\, x^{7}}{\mathit {asin} \left (c x \right )^{2} b^{2}+2 \mathit {asin} \left (c x \right ) a b +a^{2}}d x \right ) c^{4}-2 \left (\int \frac {\sqrt {\mathit {asin} \left (c x \right ) b +a}\, x^{5}}{\mathit {asin} \left (c x \right )^{2} b^{2}+2 \mathit {asin} \left (c x \right ) a b +a^{2}}d x \right ) c^{2}+\int \frac {\sqrt {\mathit {asin} \left (c x \right ) b +a}\, x^{3}}{\mathit {asin} \left (c x \right )^{2} b^{2}+2 \mathit {asin} \left (c x \right ) a b +a^{2}}d x \right ) \] Input:
int(x^3*(-c^2*d*x^2+d)^2/(a+b*asin(c*x))^(3/2),x)
Output:
d**2*(int((sqrt(asin(c*x)*b + a)*x**7)/(asin(c*x)**2*b**2 + 2*asin(c*x)*a* b + a**2),x)*c**4 - 2*int((sqrt(asin(c*x)*b + a)*x**5)/(asin(c*x)**2*b**2 + 2*asin(c*x)*a*b + a**2),x)*c**2 + int((sqrt(asin(c*x)*b + a)*x**3)/(asin (c*x)**2*b**2 + 2*asin(c*x)*a*b + a**2),x))