Integrand size = 27, antiderivative size = 224 \[ \int \frac {x^5 (a+b \arcsin (c x))}{\sqrt {d-c^2 d x^2}} \, dx=\frac {8 b x \sqrt {1-c^2 x^2}}{15 c^5 \sqrt {d-c^2 d x^2}}+\frac {4 b x^3 \sqrt {1-c^2 x^2}}{45 c^3 \sqrt {d-c^2 d x^2}}+\frac {b x^5 \sqrt {1-c^2 x^2}}{25 c \sqrt {d-c^2 d x^2}}-\frac {8 \sqrt {d-c^2 d x^2} (a+b \arcsin (c x))}{15 c^6 d}-\frac {4 x^2 \sqrt {d-c^2 d x^2} (a+b \arcsin (c x))}{15 c^4 d}-\frac {x^4 \sqrt {d-c^2 d x^2} (a+b \arcsin (c x))}{5 c^2 d} \]
8/15*b*x*(-c^2*x^2+1)^(1/2)/c^5/(-c^2*d*x^2+d)^(1/2)+4/45*b*x^3*(-c^2*x^2+ 1)^(1/2)/c^3/(-c^2*d*x^2+d)^(1/2)+1/25*b*x^5*(-c^2*x^2+1)^(1/2)/c/(-c^2*d* x^2+d)^(1/2)-8/15*(a+b*arcsin(c*x))*(-c^2*d*x^2+d)^(1/2)/c^6/d-4/15*x^2*(a +b*arcsin(c*x))*(-c^2*d*x^2+d)^(1/2)/c^4/d-1/5*x^4*(a+b*arcsin(c*x))*(-c^2 *d*x^2+d)^(1/2)/c^2/d
Time = 0.05 (sec) , antiderivative size = 119, normalized size of antiderivative = 0.53 \[ \int \frac {x^5 (a+b \arcsin (c x))}{\sqrt {d-c^2 d x^2}} \, dx=\frac {b c x \sqrt {1-c^2 x^2} \left (120+20 c^2 x^2+9 c^4 x^4\right )+15 a \left (-8+4 c^2 x^2+c^4 x^4+3 c^6 x^6\right )+15 b \left (-8+4 c^2 x^2+c^4 x^4+3 c^6 x^6\right ) \arcsin (c x)}{225 c^6 \sqrt {d-c^2 d x^2}} \]
(b*c*x*Sqrt[1 - c^2*x^2]*(120 + 20*c^2*x^2 + 9*c^4*x^4) + 15*a*(-8 + 4*c^2 *x^2 + c^4*x^4 + 3*c^6*x^6) + 15*b*(-8 + 4*c^2*x^2 + c^4*x^4 + 3*c^6*x^6)* ArcSin[c*x])/(225*c^6*Sqrt[d - c^2*d*x^2])
Time = 0.61 (sec) , antiderivative size = 235, normalized size of antiderivative = 1.05, number of steps used = 6, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.222, Rules used = {5210, 15, 5210, 15, 5182, 24}
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.
| \(\displaystyle \int \frac {x^5 (a+b \arcsin (c x))}{\sqrt {d-c^2 d x^2}} \, dx\) |
| \(\Big \downarrow \) 5210 |
| \(\displaystyle \frac {4 \int \frac {x^3 (a+b \arcsin (c x))}{\sqrt {d-c^2 d x^2}}dx}{5 c^2}+\frac {b \sqrt {1-c^2 x^2} \int x^4dx}{5 c \sqrt {d-c^2 d x^2}}-\frac {x^4 \sqrt {d-c^2 d x^2} (a+b \arcsin (c x))}{5 c^2 d}\) |
| \(\Big \downarrow \) 15 |
| \(\displaystyle \frac {4 \int \frac {x^3 (a+b \arcsin (c x))}{\sqrt {d-c^2 d x^2}}dx}{5 c^2}-\frac {x^4 \sqrt {d-c^2 d x^2} (a+b \arcsin (c x))}{5 c^2 d}+\frac {b x^5 \sqrt {1-c^2 x^2}}{25 c \sqrt {d-c^2 d x^2}}\) |
| \(\Big \downarrow \) 5210 |
| \(\displaystyle \frac {4 \left (\frac {2 \int \frac {x (a+b \arcsin (c x))}{\sqrt {d-c^2 d x^2}}dx}{3 c^2}+\frac {b \sqrt {1-c^2 x^2} \int x^2dx}{3 c \sqrt {d-c^2 d x^2}}-\frac {x^2 \sqrt {d-c^2 d x^2} (a+b \arcsin (c x))}{3 c^2 d}\right )}{5 c^2}-\frac {x^4 \sqrt {d-c^2 d x^2} (a+b \arcsin (c x))}{5 c^2 d}+\frac {b x^5 \sqrt {1-c^2 x^2}}{25 c \sqrt {d-c^2 d x^2}}\) |
| \(\Big \downarrow \) 15 |
| \(\displaystyle \frac {4 \left (\frac {2 \int \frac {x (a+b \arcsin (c x))}{\sqrt {d-c^2 d x^2}}dx}{3 c^2}-\frac {x^2 \sqrt {d-c^2 d x^2} (a+b \arcsin (c x))}{3 c^2 d}+\frac {b x^3 \sqrt {1-c^2 x^2}}{9 c \sqrt {d-c^2 d x^2}}\right )}{5 c^2}-\frac {x^4 \sqrt {d-c^2 d x^2} (a+b \arcsin (c x))}{5 c^2 d}+\frac {b x^5 \sqrt {1-c^2 x^2}}{25 c \sqrt {d-c^2 d x^2}}\) |
| \(\Big \downarrow \) 5182 |
| \(\displaystyle \frac {4 \left (\frac {2 \left (\frac {b \sqrt {1-c^2 x^2} \int 1dx}{c \sqrt {d-c^2 d x^2}}-\frac {\sqrt {d-c^2 d x^2} (a+b \arcsin (c x))}{c^2 d}\right )}{3 c^2}-\frac {x^2 \sqrt {d-c^2 d x^2} (a+b \arcsin (c x))}{3 c^2 d}+\frac {b x^3 \sqrt {1-c^2 x^2}}{9 c \sqrt {d-c^2 d x^2}}\right )}{5 c^2}-\frac {x^4 \sqrt {d-c^2 d x^2} (a+b \arcsin (c x))}{5 c^2 d}+\frac {b x^5 \sqrt {1-c^2 x^2}}{25 c \sqrt {d-c^2 d x^2}}\) |
| \(\Big \downarrow \) 24 |
| \(\displaystyle -\frac {x^4 \sqrt {d-c^2 d x^2} (a+b \arcsin (c x))}{5 c^2 d}+\frac {4 \left (-\frac {x^2 \sqrt {d-c^2 d x^2} (a+b \arcsin (c x))}{3 c^2 d}+\frac {2 \left (\frac {b x \sqrt {1-c^2 x^2}}{c \sqrt {d-c^2 d x^2}}-\frac {\sqrt {d-c^2 d x^2} (a+b \arcsin (c x))}{c^2 d}\right )}{3 c^2}+\frac {b x^3 \sqrt {1-c^2 x^2}}{9 c \sqrt {d-c^2 d x^2}}\right )}{5 c^2}+\frac {b x^5 \sqrt {1-c^2 x^2}}{25 c \sqrt {d-c^2 d x^2}}\) |
(b*x^5*Sqrt[1 - c^2*x^2])/(25*c*Sqrt[d - c^2*d*x^2]) - (x^4*Sqrt[d - c^2*d *x^2]*(a + b*ArcSin[c*x]))/(5*c^2*d) + (4*((b*x^3*Sqrt[1 - c^2*x^2])/(9*c* Sqrt[d - c^2*d*x^2]) - (x^2*Sqrt[d - c^2*d*x^2]*(a + b*ArcSin[c*x]))/(3*c^ 2*d) + (2*((b*x*Sqrt[1 - c^2*x^2])/(c*Sqrt[d - c^2*d*x^2]) - (Sqrt[d - c^2 *d*x^2]*(a + b*ArcSin[c*x]))/(c^2*d)))/(3*c^2)))/(5*c^2)
3.2.9.3.1 Defintions of rubi rules used
Int[(a_.)*(x_)^(m_.), x_Symbol] :> Simp[a*(x^(m + 1)/(m + 1)), x] /; FreeQ[ {a, m}, x] && NeQ[m, -1]
Int[((a_.) + ArcSin[(c_.)*(x_)]*(b_.))^(n_.)*(x_)*((d_) + (e_.)*(x_)^2)^(p_ .), x_Symbol] :> Simp[(d + e*x^2)^(p + 1)*((a + b*ArcSin[c*x])^n/(2*e*(p + 1))), x] + Simp[b*(n/(2*c*(p + 1)))*Simp[(d + e*x^2)^p/(1 - c^2*x^2)^p] I nt[(1 - c^2*x^2)^(p + 1/2)*(a + b*ArcSin[c*x])^(n - 1), x], x] /; FreeQ[{a, b, c, d, e, p}, x] && EqQ[c^2*d + e, 0] && GtQ[n, 0] && NeQ[p, -1]
Int[((a_.) + ArcSin[(c_.)*(x_)]*(b_.))^(n_.)*((f_.)*(x_))^(m_)*((d_) + (e_. )*(x_)^2)^(p_), x_Symbol] :> Simp[f*(f*x)^(m - 1)*(d + e*x^2)^(p + 1)*((a + b*ArcSin[c*x])^n/(e*(m + 2*p + 1))), x] + (Simp[f^2*((m - 1)/(c^2*(m + 2*p + 1))) Int[(f*x)^(m - 2)*(d + e*x^2)^p*(a + b*ArcSin[c*x])^n, x], x] + S imp[b*f*(n/(c*(m + 2*p + 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, p}, x] && EqQ[c^2*d + e, 0] && GtQ[n, 0] && IGtQ[m , 1] && NeQ[m + 2*p + 1, 0]
Result contains complex when optimal does not.
Time = 0.22 (sec) , antiderivative size = 521, normalized size of antiderivative = 2.33
| method | result | size |
| default | \(a \left (-\frac {x^{4} \sqrt {-c^{2} d \,x^{2}+d}}{5 c^{2} d}+\frac {-\frac {4 x^{2} \sqrt {-c^{2} d \,x^{2}+d}}{15 c^{2} d}-\frac {8 \sqrt {-c^{2} d \,x^{2}+d}}{15 d \,c^{4}}}{c^{2}}\right )+b \left (\frac {5 \sqrt {-d \left (c^{2} x^{2}-1\right )}\, \left (2 c^{2} x^{2}-2 i c x \sqrt {-c^{2} x^{2}+1}-1\right ) \left (i+3 \arcsin \left (c x \right )\right )}{576 c^{6} d \left (c^{2} x^{2}-1\right )}-\frac {5 \sqrt {-d \left (c^{2} x^{2}-1\right )}\, \left (c^{2} x^{2}-i c x \sqrt {-c^{2} x^{2}+1}-1\right ) \left (\arcsin \left (c x \right )+i\right )}{16 c^{6} d \left (c^{2} x^{2}-1\right )}-\frac {5 \sqrt {-d \left (c^{2} x^{2}-1\right )}\, \left (i c x \sqrt {-c^{2} x^{2}+1}+c^{2} x^{2}-1\right ) \left (\arcsin \left (c x \right )-i\right )}{16 c^{6} d \left (c^{2} x^{2}-1\right )}+\frac {5 \sqrt {-d \left (c^{2} x^{2}-1\right )}\, \left (2 i c x \sqrt {-c^{2} x^{2}+1}+2 c^{2} x^{2}-1\right ) \left (-i+3 \arcsin \left (c x \right )\right )}{576 c^{6} d \left (c^{2} x^{2}-1\right )}+\frac {\sqrt {-d \left (c^{2} x^{2}-1\right )}\, \arcsin \left (c x \right ) \cos \left (6 \arcsin \left (c x \right )\right )}{160 c^{6} d \left (c^{2} x^{2}-1\right )}-\frac {\sqrt {-d \left (c^{2} x^{2}-1\right )}\, \sin \left (6 \arcsin \left (c x \right )\right )}{800 c^{6} d \left (c^{2} x^{2}-1\right )}-\frac {11 \sqrt {-d \left (c^{2} x^{2}-1\right )}\, \arcsin \left (c x \right ) \cos \left (4 \arcsin \left (c x \right )\right )}{240 c^{6} d \left (c^{2} x^{2}-1\right )}+\frac {29 \sqrt {-d \left (c^{2} x^{2}-1\right )}\, \sin \left (4 \arcsin \left (c x \right )\right )}{1800 c^{6} d \left (c^{2} x^{2}-1\right )}\right )\) | \(521\) |
| parts | \(a \left (-\frac {x^{4} \sqrt {-c^{2} d \,x^{2}+d}}{5 c^{2} d}+\frac {-\frac {4 x^{2} \sqrt {-c^{2} d \,x^{2}+d}}{15 c^{2} d}-\frac {8 \sqrt {-c^{2} d \,x^{2}+d}}{15 d \,c^{4}}}{c^{2}}\right )+b \left (\frac {5 \sqrt {-d \left (c^{2} x^{2}-1\right )}\, \left (2 c^{2} x^{2}-2 i c x \sqrt {-c^{2} x^{2}+1}-1\right ) \left (i+3 \arcsin \left (c x \right )\right )}{576 c^{6} d \left (c^{2} x^{2}-1\right )}-\frac {5 \sqrt {-d \left (c^{2} x^{2}-1\right )}\, \left (c^{2} x^{2}-i c x \sqrt {-c^{2} x^{2}+1}-1\right ) \left (\arcsin \left (c x \right )+i\right )}{16 c^{6} d \left (c^{2} x^{2}-1\right )}-\frac {5 \sqrt {-d \left (c^{2} x^{2}-1\right )}\, \left (i c x \sqrt {-c^{2} x^{2}+1}+c^{2} x^{2}-1\right ) \left (\arcsin \left (c x \right )-i\right )}{16 c^{6} d \left (c^{2} x^{2}-1\right )}+\frac {5 \sqrt {-d \left (c^{2} x^{2}-1\right )}\, \left (2 i c x \sqrt {-c^{2} x^{2}+1}+2 c^{2} x^{2}-1\right ) \left (-i+3 \arcsin \left (c x \right )\right )}{576 c^{6} d \left (c^{2} x^{2}-1\right )}+\frac {\sqrt {-d \left (c^{2} x^{2}-1\right )}\, \arcsin \left (c x \right ) \cos \left (6 \arcsin \left (c x \right )\right )}{160 c^{6} d \left (c^{2} x^{2}-1\right )}-\frac {\sqrt {-d \left (c^{2} x^{2}-1\right )}\, \sin \left (6 \arcsin \left (c x \right )\right )}{800 c^{6} d \left (c^{2} x^{2}-1\right )}-\frac {11 \sqrt {-d \left (c^{2} x^{2}-1\right )}\, \arcsin \left (c x \right ) \cos \left (4 \arcsin \left (c x \right )\right )}{240 c^{6} d \left (c^{2} x^{2}-1\right )}+\frac {29 \sqrt {-d \left (c^{2} x^{2}-1\right )}\, \sin \left (4 \arcsin \left (c x \right )\right )}{1800 c^{6} d \left (c^{2} x^{2}-1\right )}\right )\) | \(521\) |
a*(-1/5*x^4/c^2/d*(-c^2*d*x^2+d)^(1/2)+4/5/c^2*(-1/3*x^2/c^2/d*(-c^2*d*x^2 +d)^(1/2)-2/3/d/c^4*(-c^2*d*x^2+d)^(1/2)))+b*(5/576*(-d*(c^2*x^2-1))^(1/2) *(2*c^2*x^2-2*I*c*x*(-c^2*x^2+1)^(1/2)-1)*(I+3*arcsin(c*x))/c^6/d/(c^2*x^2 -1)-5/16*(-d*(c^2*x^2-1))^(1/2)*(c^2*x^2-I*(-c^2*x^2+1)^(1/2)*x*c-1)*(arcs in(c*x)+I)/c^6/d/(c^2*x^2-1)-5/16*(-d*(c^2*x^2-1))^(1/2)*(I*(-c^2*x^2+1)^( 1/2)*x*c+c^2*x^2-1)*(arcsin(c*x)-I)/c^6/d/(c^2*x^2-1)+5/576*(-d*(c^2*x^2-1 ))^(1/2)*(2*I*c*x*(-c^2*x^2+1)^(1/2)+2*c^2*x^2-1)*(-I+3*arcsin(c*x))/c^6/d /(c^2*x^2-1)+1/160*(-d*(c^2*x^2-1))^(1/2)/c^6/d/(c^2*x^2-1)*arcsin(c*x)*co s(6*arcsin(c*x))-1/800*(-d*(c^2*x^2-1))^(1/2)/c^6/d/(c^2*x^2-1)*sin(6*arcs in(c*x))-11/240*(-d*(c^2*x^2-1))^(1/2)/c^6/d/(c^2*x^2-1)*arcsin(c*x)*cos(4 *arcsin(c*x))+29/1800*(-d*(c^2*x^2-1))^(1/2)/c^6/d/(c^2*x^2-1)*sin(4*arcsi n(c*x)))
Time = 0.27 (sec) , antiderivative size = 150, normalized size of antiderivative = 0.67 \[ \int \frac {x^5 (a+b \arcsin (c x))}{\sqrt {d-c^2 d x^2}} \, dx=-\frac {{\left (9 \, b c^{5} x^{5} + 20 \, b c^{3} x^{3} + 120 \, b c x\right )} \sqrt {-c^{2} d x^{2} + d} \sqrt {-c^{2} x^{2} + 1} + 15 \, {\left (3 \, a c^{6} x^{6} + a c^{4} x^{4} + 4 \, a c^{2} x^{2} + {\left (3 \, b c^{6} x^{6} + b c^{4} x^{4} + 4 \, b c^{2} x^{2} - 8 \, b\right )} \arcsin \left (c x\right ) - 8 \, a\right )} \sqrt {-c^{2} d x^{2} + d}}{225 \, {\left (c^{8} d x^{2} - c^{6} d\right )}} \]
-1/225*((9*b*c^5*x^5 + 20*b*c^3*x^3 + 120*b*c*x)*sqrt(-c^2*d*x^2 + d)*sqrt (-c^2*x^2 + 1) + 15*(3*a*c^6*x^6 + a*c^4*x^4 + 4*a*c^2*x^2 + (3*b*c^6*x^6 + b*c^4*x^4 + 4*b*c^2*x^2 - 8*b)*arcsin(c*x) - 8*a)*sqrt(-c^2*d*x^2 + d))/ (c^8*d*x^2 - c^6*d)
\[ \int \frac {x^5 (a+b \arcsin (c x))}{\sqrt {d-c^2 d x^2}} \, dx=\int \frac {x^{5} \left (a + b \operatorname {asin}{\left (c x \right )}\right )}{\sqrt {- d \left (c x - 1\right ) \left (c x + 1\right )}}\, dx \]
Time = 0.29 (sec) , antiderivative size = 180, normalized size of antiderivative = 0.80 \[ \int \frac {x^5 (a+b \arcsin (c x))}{\sqrt {d-c^2 d x^2}} \, dx=-\frac {1}{15} \, {\left (\frac {3 \, \sqrt {-c^{2} d x^{2} + d} x^{4}}{c^{2} d} + \frac {4 \, \sqrt {-c^{2} d x^{2} + d} x^{2}}{c^{4} d} + \frac {8 \, \sqrt {-c^{2} d x^{2} + d}}{c^{6} d}\right )} b \arcsin \left (c x\right ) - \frac {1}{15} \, {\left (\frac {3 \, \sqrt {-c^{2} d x^{2} + d} x^{4}}{c^{2} d} + \frac {4 \, \sqrt {-c^{2} d x^{2} + d} x^{2}}{c^{4} d} + \frac {8 \, \sqrt {-c^{2} d x^{2} + d}}{c^{6} d}\right )} a + \frac {{\left (9 \, c^{4} x^{5} + 20 \, c^{2} x^{3} + 120 \, x\right )} b}{225 \, c^{5} \sqrt {d}} \]
-1/15*(3*sqrt(-c^2*d*x^2 + d)*x^4/(c^2*d) + 4*sqrt(-c^2*d*x^2 + d)*x^2/(c^ 4*d) + 8*sqrt(-c^2*d*x^2 + d)/(c^6*d))*b*arcsin(c*x) - 1/15*(3*sqrt(-c^2*d *x^2 + d)*x^4/(c^2*d) + 4*sqrt(-c^2*d*x^2 + d)*x^2/(c^4*d) + 8*sqrt(-c^2*d *x^2 + d)/(c^6*d))*a + 1/225*(9*c^4*x^5 + 20*c^2*x^3 + 120*x)*b/(c^5*sqrt( d))
Exception generated. \[ \int \frac {x^5 (a+b \arcsin (c x))}{\sqrt {d-c^2 d x^2}} \, dx=\text {Exception raised: TypeError} \]
Exception raised: TypeError >> an error occurred running a Giac command:IN PUT:sage2:=int(sage0,sageVARx):;OUTPUT:sym2poly/r2sym(const gen & e,const index_m & i,const vecteur & l) Error: Bad Argument Value
Timed out. \[ \int \frac {x^5 (a+b \arcsin (c x))}{\sqrt {d-c^2 d x^2}} \, dx=\int \frac {x^5\,\left (a+b\,\mathrm {asin}\left (c\,x\right )\right )}{\sqrt {d-c^2\,d\,x^2}} \,d x \]