Integrand size = 27, antiderivative size = 221 \[ \int \frac {x^5 (a+b \arcsin (c x))}{\left (d-c^2 d x^2\right )^{3/2}} \, dx=-\frac {5 b x \sqrt {d-c^2 d x^2}}{3 c^5 d^2 \sqrt {1-c^2 x^2}}-\frac {b x^3 \sqrt {d-c^2 d x^2}}{9 c^3 d^2 \sqrt {1-c^2 x^2}}+\frac {a+b \arcsin (c x)}{c^6 d \sqrt {d-c^2 d x^2}}+\frac {2 \sqrt {d-c^2 d x^2} (a+b \arcsin (c x))}{c^6 d^2}-\frac {\left (d-c^2 d x^2\right )^{3/2} (a+b \arcsin (c x))}{3 c^6 d^3}-\frac {b \sqrt {d-c^2 d x^2} \text {arctanh}(c x)}{c^6 d^2 \sqrt {1-c^2 x^2}} \]
-1/3*(-c^2*d*x^2+d)^(3/2)*(a+b*arcsin(c*x))/c^6/d^3+(a+b*arcsin(c*x))/c^6/ d/(-c^2*d*x^2+d)^(1/2)+2*(a+b*arcsin(c*x))*(-c^2*d*x^2+d)^(1/2)/c^6/d^2-5/ 3*b*x*(-c^2*d*x^2+d)^(1/2)/c^5/d^2/(-c^2*x^2+1)^(1/2)-1/9*b*x^3*(-c^2*d*x^ 2+d)^(1/2)/c^3/d^2/(-c^2*x^2+1)^(1/2)-b*arctanh(c*x)*(-c^2*d*x^2+d)^(1/2)/ c^6/d^2/(-c^2*x^2+1)^(1/2)
Result contains higher order function than in optimal. Order 4 vs. order 3 in optimal.
Time = 0.23 (sec) , antiderivative size = 166, normalized size of antiderivative = 0.75 \[ \int \frac {x^5 (a+b \arcsin (c x))}{\left (d-c^2 d x^2\right )^{3/2}} \, dx=\frac {\sqrt {d-c^2 d x^2} \left (\sqrt {-c^2} \left (b c x \sqrt {1-c^2 x^2} \left (15+c^2 x^2\right )+3 a \left (-8+4 c^2 x^2+c^4 x^4\right )+3 b \left (-8+4 c^2 x^2+c^4 x^4\right ) \arcsin (c x)\right )-9 i b c \sqrt {1-c^2 x^2} \operatorname {EllipticF}\left (i \text {arcsinh}\left (\sqrt {-c^2} x\right ),1\right )\right )}{9 c^6 \sqrt {-c^2} d^2 \left (-1+c^2 x^2\right )} \]
(Sqrt[d - c^2*d*x^2]*(Sqrt[-c^2]*(b*c*x*Sqrt[1 - c^2*x^2]*(15 + c^2*x^2) + 3*a*(-8 + 4*c^2*x^2 + c^4*x^4) + 3*b*(-8 + 4*c^2*x^2 + c^4*x^4)*ArcSin[c* x]) - (9*I)*b*c*Sqrt[1 - c^2*x^2]*EllipticF[I*ArcSinh[Sqrt[-c^2]*x], 1]))/ (9*c^6*Sqrt[-c^2]*d^2*(-1 + c^2*x^2))
Time = 0.44 (sec) , antiderivative size = 158, normalized size of antiderivative = 0.71, number of steps used = 4, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.148, Rules used = {5194, 27, 1467, 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.
| \(\displaystyle \int \frac {x^5 (a+b \arcsin (c x))}{\left (d-c^2 d x^2\right )^{3/2}} \, dx\) |
| \(\Big \downarrow \) 5194 |
| \(\displaystyle -\frac {b c \sqrt {d-c^2 d x^2} \int \frac {-c^4 x^4-4 c^2 x^2+8}{3 c^6 d^2 \left (1-c^2 x^2\right )}dx}{\sqrt {1-c^2 x^2}}-\frac {\left (d-c^2 d x^2\right )^{3/2} (a+b \arcsin (c x))}{3 c^6 d^3}+\frac {2 \sqrt {d-c^2 d x^2} (a+b \arcsin (c x))}{c^6 d^2}+\frac {a+b \arcsin (c x)}{c^6 d \sqrt {d-c^2 d x^2}}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle -\frac {b \sqrt {d-c^2 d x^2} \int \frac {-c^4 x^4-4 c^2 x^2+8}{1-c^2 x^2}dx}{3 c^5 d^2 \sqrt {1-c^2 x^2}}-\frac {\left (d-c^2 d x^2\right )^{3/2} (a+b \arcsin (c x))}{3 c^6 d^3}+\frac {2 \sqrt {d-c^2 d x^2} (a+b \arcsin (c x))}{c^6 d^2}+\frac {a+b \arcsin (c x)}{c^6 d \sqrt {d-c^2 d x^2}}\) |
| \(\Big \downarrow \) 1467 |
| \(\displaystyle -\frac {b \sqrt {d-c^2 d x^2} \int \left (c^2 x^2+\frac {3}{1-c^2 x^2}+5\right )dx}{3 c^5 d^2 \sqrt {1-c^2 x^2}}-\frac {\left (d-c^2 d x^2\right )^{3/2} (a+b \arcsin (c x))}{3 c^6 d^3}+\frac {2 \sqrt {d-c^2 d x^2} (a+b \arcsin (c x))}{c^6 d^2}+\frac {a+b \arcsin (c x)}{c^6 d \sqrt {d-c^2 d x^2}}\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle -\frac {\left (d-c^2 d x^2\right )^{3/2} (a+b \arcsin (c x))}{3 c^6 d^3}+\frac {2 \sqrt {d-c^2 d x^2} (a+b \arcsin (c x))}{c^6 d^2}+\frac {a+b \arcsin (c x)}{c^6 d \sqrt {d-c^2 d x^2}}-\frac {b \left (\frac {3 \text {arctanh}(c x)}{c}+\frac {c^2 x^3}{3}+5 x\right ) \sqrt {d-c^2 d x^2}}{3 c^5 d^2 \sqrt {1-c^2 x^2}}\) |
(a + b*ArcSin[c*x])/(c^6*d*Sqrt[d - c^2*d*x^2]) + (2*Sqrt[d - c^2*d*x^2]*( a + b*ArcSin[c*x]))/(c^6*d^2) - ((d - c^2*d*x^2)^(3/2)*(a + b*ArcSin[c*x]) )/(3*c^6*d^3) - (b*Sqrt[d - c^2*d*x^2]*(5*x + (c^2*x^3)/3 + (3*ArcTanh[c*x ])/c))/(3*c^5*d^2*Sqrt[1 - c^2*x^2])
3.2.19.3.1 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[((d_) + (e_.)*(x_)^2)^(q_.)*((a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4)^(p_.), x_Symbol] :> Int[ExpandIntegrand[(d + e*x^2)^q*(a + b*x^2 + c*x^4)^p, x], x] /; FreeQ[{a, b, c, d, e}, x] && NeQ[b^2 - 4*a*c, 0] && NeQ[c*d^2 - b*d*e + a*e^2, 0] && IGtQ[p, 0] && IGtQ[q, -2]
Int[((a_.) + ArcSin[(c_.)*(x_)]*(b_.))*(x_)^(m_)*((d_) + (e_.)*(x_)^2)^(p_) , x_Symbol] :> With[{u = IntHide[x^m*(d + e*x^2)^p, x]}, Simp[(a + b*ArcSin [c*x]) u, x] - Simp[b*c*Simp[Sqrt[d + e*x^2]/Sqrt[1 - c^2*x^2]] Int[Sim plifyIntegrand[u/Sqrt[d + e*x^2], x], x], x]] /; FreeQ[{a, b, c, d, e}, x] && EqQ[c^2*d + e, 0] && IntegerQ[p - 1/2] && NeQ[p, -2^(-1)] && (IGtQ[(m + 1)/2, 0] || ILtQ[(m + 2*p + 3)/2, 0])
Result contains complex when optimal does not.
Time = 0.34 (sec) , antiderivative size = 425, normalized size of antiderivative = 1.92
| method | result | size |
| default | \(a \left (-\frac {x^{4}}{3 c^{2} d \sqrt {-c^{2} d \,x^{2}+d}}+\frac {-\frac {4 x^{2}}{3 c^{2} d \sqrt {-c^{2} d \,x^{2}+d}}+\frac {8}{3 d \,c^{4} \sqrt {-c^{2} d \,x^{2}+d}}}{c^{2}}\right )-\frac {65 b \sqrt {-d \left (c^{2} x^{2}-1\right )}\, \arcsin \left (c x \right )}{24 c^{6} d^{2} \left (c^{2} x^{2}-1\right )}-\frac {b \sqrt {-d \left (c^{2} x^{2}-1\right )}\, \sin \left (4 \arcsin \left (c x \right )\right )}{72 c^{6} d^{2} \left (c^{2} x^{2}-1\right )}+\frac {31 b \sqrt {-d \left (c^{2} x^{2}-1\right )}\, \sqrt {-c^{2} x^{2}+1}\, x}{18 c^{5} d^{2} \left (c^{2} x^{2}-1\right )}+\frac {5 b \sqrt {-d \left (c^{2} x^{2}-1\right )}\, \arcsin \left (c x \right ) x^{2}}{3 c^{4} d^{2} \left (c^{2} x^{2}-1\right )}+\frac {b \sqrt {-d \left (c^{2} x^{2}-1\right )}\, \arcsin \left (c x \right ) \cos \left (4 \arcsin \left (c x \right )\right )}{24 c^{6} d^{2} \left (c^{2} x^{2}-1\right )}+\frac {b \sqrt {-d \left (c^{2} x^{2}-1\right )}\, \sqrt {-c^{2} x^{2}+1}\, \ln \left (i c x +\sqrt {-c^{2} x^{2}+1}+i\right )}{c^{6} d^{2} \left (c^{2} x^{2}-1\right )}-\frac {b \sqrt {-d \left (c^{2} x^{2}-1\right )}\, \sqrt {-c^{2} x^{2}+1}\, \ln \left (i c x +\sqrt {-c^{2} x^{2}+1}-i\right )}{c^{6} d^{2} \left (c^{2} x^{2}-1\right )}\) | \(425\) |
| parts | \(a \left (-\frac {x^{4}}{3 c^{2} d \sqrt {-c^{2} d \,x^{2}+d}}+\frac {-\frac {4 x^{2}}{3 c^{2} d \sqrt {-c^{2} d \,x^{2}+d}}+\frac {8}{3 d \,c^{4} \sqrt {-c^{2} d \,x^{2}+d}}}{c^{2}}\right )-\frac {65 b \sqrt {-d \left (c^{2} x^{2}-1\right )}\, \arcsin \left (c x \right )}{24 c^{6} d^{2} \left (c^{2} x^{2}-1\right )}-\frac {b \sqrt {-d \left (c^{2} x^{2}-1\right )}\, \sin \left (4 \arcsin \left (c x \right )\right )}{72 c^{6} d^{2} \left (c^{2} x^{2}-1\right )}+\frac {31 b \sqrt {-d \left (c^{2} x^{2}-1\right )}\, \sqrt {-c^{2} x^{2}+1}\, x}{18 c^{5} d^{2} \left (c^{2} x^{2}-1\right )}+\frac {5 b \sqrt {-d \left (c^{2} x^{2}-1\right )}\, \arcsin \left (c x \right ) x^{2}}{3 c^{4} d^{2} \left (c^{2} x^{2}-1\right )}+\frac {b \sqrt {-d \left (c^{2} x^{2}-1\right )}\, \arcsin \left (c x \right ) \cos \left (4 \arcsin \left (c x \right )\right )}{24 c^{6} d^{2} \left (c^{2} x^{2}-1\right )}+\frac {b \sqrt {-d \left (c^{2} x^{2}-1\right )}\, \sqrt {-c^{2} x^{2}+1}\, \ln \left (i c x +\sqrt {-c^{2} x^{2}+1}+i\right )}{c^{6} d^{2} \left (c^{2} x^{2}-1\right )}-\frac {b \sqrt {-d \left (c^{2} x^{2}-1\right )}\, \sqrt {-c^{2} x^{2}+1}\, \ln \left (i c x +\sqrt {-c^{2} x^{2}+1}-i\right )}{c^{6} d^{2} \left (c^{2} x^{2}-1\right )}\) | \(425\) |
a*(-1/3*x^4/c^2/d/(-c^2*d*x^2+d)^(1/2)+4/3/c^2*(-x^2/c^2/d/(-c^2*d*x^2+d)^ (1/2)+2/d/c^4/(-c^2*d*x^2+d)^(1/2)))-65/24*b*(-d*(c^2*x^2-1))^(1/2)/c^6/d^ 2/(c^2*x^2-1)*arcsin(c*x)-1/72*b*(-d*(c^2*x^2-1))^(1/2)/c^6/d^2/(c^2*x^2-1 )*sin(4*arcsin(c*x))+31/18*b*(-d*(c^2*x^2-1))^(1/2)/c^5/d^2/(c^2*x^2-1)*(- c^2*x^2+1)^(1/2)*x+5/3*b*(-d*(c^2*x^2-1))^(1/2)/c^4/d^2/(c^2*x^2-1)*arcsin (c*x)*x^2+1/24*b*(-d*(c^2*x^2-1))^(1/2)/c^6/d^2/(c^2*x^2-1)*arcsin(c*x)*co s(4*arcsin(c*x))+b*(-d*(c^2*x^2-1))^(1/2)*(-c^2*x^2+1)^(1/2)/c^6/d^2/(c^2* x^2-1)*ln(I*c*x+(-c^2*x^2+1)^(1/2)+I)-b*(-d*(c^2*x^2-1))^(1/2)*(-c^2*x^2+1 )^(1/2)/c^6/d^2/(c^2*x^2-1)*ln(I*c*x+(-c^2*x^2+1)^(1/2)-I)
Time = 0.31 (sec) , antiderivative size = 441, normalized size of antiderivative = 2.00 \[ \int \frac {x^5 (a+b \arcsin (c x))}{\left (d-c^2 d x^2\right )^{3/2}} \, dx=\left [\frac {9 \, {\left (b c^{2} x^{2} - b\right )} \sqrt {d} \log \left (-\frac {c^{6} d x^{6} + 5 \, c^{4} d x^{4} - 5 \, c^{2} d x^{2} + 4 \, {\left (c^{3} x^{3} + c x\right )} \sqrt {-c^{2} d x^{2} + d} \sqrt {-c^{2} x^{2} + 1} \sqrt {d} - d}{c^{6} x^{6} - 3 \, c^{4} x^{4} + 3 \, c^{2} x^{2} - 1}\right ) + 4 \, {\left (b c^{3} x^{3} + 15 \, b c x\right )} \sqrt {-c^{2} d x^{2} + d} \sqrt {-c^{2} x^{2} + 1} + 12 \, {\left (a c^{4} x^{4} + 4 \, a c^{2} x^{2} + {\left (b c^{4} x^{4} + 4 \, b c^{2} x^{2} - 8 \, b\right )} \arcsin \left (c x\right ) - 8 \, a\right )} \sqrt {-c^{2} d x^{2} + d}}{36 \, {\left (c^{8} d^{2} x^{2} - c^{6} d^{2}\right )}}, -\frac {9 \, {\left (b c^{2} x^{2} - b\right )} \sqrt {-d} \arctan \left (\frac {2 \, \sqrt {-c^{2} d x^{2} + d} \sqrt {-c^{2} x^{2} + 1} c \sqrt {-d} x}{c^{4} d x^{4} - d}\right ) - 2 \, {\left (b c^{3} x^{3} + 15 \, b c x\right )} \sqrt {-c^{2} d x^{2} + d} \sqrt {-c^{2} x^{2} + 1} - 6 \, {\left (a c^{4} x^{4} + 4 \, a c^{2} x^{2} + {\left (b c^{4} x^{4} + 4 \, b c^{2} x^{2} - 8 \, b\right )} \arcsin \left (c x\right ) - 8 \, a\right )} \sqrt {-c^{2} d x^{2} + d}}{18 \, {\left (c^{8} d^{2} x^{2} - c^{6} d^{2}\right )}}\right ] \]
[1/36*(9*(b*c^2*x^2 - b)*sqrt(d)*log(-(c^6*d*x^6 + 5*c^4*d*x^4 - 5*c^2*d*x ^2 + 4*(c^3*x^3 + c*x)*sqrt(-c^2*d*x^2 + d)*sqrt(-c^2*x^2 + 1)*sqrt(d) - d )/(c^6*x^6 - 3*c^4*x^4 + 3*c^2*x^2 - 1)) + 4*(b*c^3*x^3 + 15*b*c*x)*sqrt(- c^2*d*x^2 + d)*sqrt(-c^2*x^2 + 1) + 12*(a*c^4*x^4 + 4*a*c^2*x^2 + (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^2* x^2 - c^6*d^2), -1/18*(9*(b*c^2*x^2 - b)*sqrt(-d)*arctan(2*sqrt(-c^2*d*x^2 + d)*sqrt(-c^2*x^2 + 1)*c*sqrt(-d)*x/(c^4*d*x^4 - d)) - 2*(b*c^3*x^3 + 15 *b*c*x)*sqrt(-c^2*d*x^2 + d)*sqrt(-c^2*x^2 + 1) - 6*(a*c^4*x^4 + 4*a*c^2*x ^2 + (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^2*x^2 - c^6*d^2)]
\[ \int \frac {x^5 (a+b \arcsin (c x))}{\left (d-c^2 d x^2\right )^{3/2}} \, dx=\int \frac {x^{5} \left (a + b \operatorname {asin}{\left (c x \right )}\right )}{\left (- d \left (c x - 1\right ) \left (c x + 1\right )\right )^{\frac {3}{2}}}\, dx \]
\[ \int \frac {x^5 (a+b \arcsin (c x))}{\left (d-c^2 d x^2\right )^{3/2}} \, dx=\int { \frac {{\left (b \arcsin \left (c x\right ) + a\right )} x^{5}}{{\left (-c^{2} d x^{2} + d\right )}^{\frac {3}{2}}} \,d x } \]
-1/3*a*(x^4/(sqrt(-c^2*d*x^2 + d)*c^2*d) + 4*x^2/(sqrt(-c^2*d*x^2 + d)*c^4 *d) - 8/(sqrt(-c^2*d*x^2 + d)*c^6*d)) - 1/3*(3*sqrt(c*x + 1)*sqrt(-c*x + 1 )*c^6*d^2*integrate(1/3*(c^4*x^6 + 4*c^2*x^4 - 8*x^2)/(c^7*d^2*x^4 - c^5*d ^2*x^2 + (c^5*d^2*x^2 - c^3*d^2)*e^(log(c*x + 1) + log(-c*x + 1))), x) + ( c^4*x^4 + 4*c^2*x^2 - 8)*arctan2(c*x, sqrt(c*x + 1)*sqrt(-c*x + 1)))*b/(sq rt(c*x + 1)*sqrt(-c*x + 1)*c^6*d^(3/2))
Exception generated. \[ \int \frac {x^5 (a+b \arcsin (c x))}{\left (d-c^2 d x^2\right )^{3/2}} \, dx=\text {Exception raised: TypeError} \]
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. \[ \int \frac {x^5 (a+b \arcsin (c x))}{\left (d-c^2 d x^2\right )^{3/2}} \, dx=\int \frac {x^5\,\left (a+b\,\mathrm {asin}\left (c\,x\right )\right )}{{\left (d-c^2\,d\,x^2\right )}^{3/2}} \,d x \]