Integrand size = 27, antiderivative size = 219 \[ \int \frac {x^5 (a+b \arcsin (c x))}{\left (d-c^2 d x^2\right )^{5/2}} \, dx=-\frac {b x \sqrt {d-c^2 d x^2}}{6 c^5 d^3 \left (1-c^2 x^2\right )^{3/2}}+\frac {b x \sqrt {d-c^2 d x^2}}{c^5 d^3 \sqrt {1-c^2 x^2}}+\frac {a+b \arcsin (c x)}{3 c^6 d \left (d-c^2 d x^2\right )^{3/2}}-\frac {2 (a+b \arcsin (c x))}{c^6 d^2 \sqrt {d-c^2 d x^2}}-\frac {\sqrt {d-c^2 d x^2} (a+b \arcsin (c x))}{c^6 d^3}+\frac {11 b \sqrt {d-c^2 d x^2} \text {arctanh}(c x)}{6 c^6 d^3 \sqrt {1-c^2 x^2}} \]
1/3*(a+b*arcsin(c*x))/c^6/d/(-c^2*d*x^2+d)^(3/2)-2*(a+b*arcsin(c*x))/c^6/d ^2/(-c^2*d*x^2+d)^(1/2)-1/6*b*x*(-c^2*d*x^2+d)^(1/2)/c^5/d^3/(-c^2*x^2+1)^ (3/2)-(a+b*arcsin(c*x))*(-c^2*d*x^2+d)^(1/2)/c^6/d^3+b*x*(-c^2*d*x^2+d)^(1 /2)/c^5/d^3/(-c^2*x^2+1)^(1/2)+11/6*b*arctanh(c*x)*(-c^2*d*x^2+d)^(1/2)/c^ 6/d^3/(-c^2*x^2+1)^(1/2)
Result contains higher order function than in optimal. Order 4 vs. order 3 in optimal.
Time = 0.24 (sec) , antiderivative size = 169, normalized size of antiderivative = 0.77 \[ \int \frac {x^5 (a+b \arcsin (c x))}{\left (d-c^2 d x^2\right )^{5/2}} \, dx=\frac {\sqrt {d-c^2 d x^2} \left (\sqrt {-c^2} \left (b c x \sqrt {1-c^2 x^2} \left (-5+6 c^2 x^2\right )+2 a \left (8-12 c^2 x^2+3 c^4 x^4\right )+2 b \left (8-12 c^2 x^2+3 c^4 x^4\right ) \arcsin (c x)\right )+11 i b c \left (1-c^2 x^2\right )^{3/2} \operatorname {EllipticF}\left (i \text {arcsinh}\left (\sqrt {-c^2} x\right ),1\right )\right )}{6 c^4 \left (-c^2\right )^{3/2} d^3 \left (-1+c^2 x^2\right )^2} \]
(Sqrt[d - c^2*d*x^2]*(Sqrt[-c^2]*(b*c*x*Sqrt[1 - c^2*x^2]*(-5 + 6*c^2*x^2) + 2*a*(8 - 12*c^2*x^2 + 3*c^4*x^4) + 2*b*(8 - 12*c^2*x^2 + 3*c^4*x^4)*Arc Sin[c*x]) + (11*I)*b*c*(1 - c^2*x^2)^(3/2)*EllipticF[I*ArcSinh[Sqrt[-c^2]* x], 1]))/(6*c^4*(-c^2)^(3/2)*d^3*(-1 + c^2*x^2)^2)
Time = 0.44 (sec) , antiderivative size = 171, normalized size of antiderivative = 0.78, number of steps used = 6, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.222, Rules used = {5194, 27, 1471, 25, 299, 219}
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.
| \(\displaystyle \int \frac {x^5 (a+b \arcsin (c x))}{\left (d-c^2 d x^2\right )^{5/2}} \, dx\) |
| \(\Big \downarrow \) 5194 |
| \(\displaystyle -\frac {b c \sqrt {d-c^2 d x^2} \int -\frac {3 c^4 x^4-12 c^2 x^2+8}{3 c^6 d^3 \left (1-c^2 x^2\right )^2}dx}{\sqrt {1-c^2 x^2}}-\frac {\sqrt {d-c^2 d x^2} (a+b \arcsin (c x))}{c^6 d^3}-\frac {2 (a+b \arcsin (c x))}{c^6 d^2 \sqrt {d-c^2 d x^2}}+\frac {a+b \arcsin (c x)}{3 c^6 d \left (d-c^2 d x^2\right )^{3/2}}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {b \sqrt {d-c^2 d x^2} \int \frac {3 c^4 x^4-12 c^2 x^2+8}{\left (1-c^2 x^2\right )^2}dx}{3 c^5 d^3 \sqrt {1-c^2 x^2}}-\frac {\sqrt {d-c^2 d x^2} (a+b \arcsin (c x))}{c^6 d^3}-\frac {2 (a+b \arcsin (c x))}{c^6 d^2 \sqrt {d-c^2 d x^2}}+\frac {a+b \arcsin (c x)}{3 c^6 d \left (d-c^2 d x^2\right )^{3/2}}\) |
| \(\Big \downarrow \) 1471 |
| \(\displaystyle \frac {b \sqrt {d-c^2 d x^2} \left (-\frac {1}{2} \int -\frac {17-6 c^2 x^2}{1-c^2 x^2}dx-\frac {x}{2 \left (1-c^2 x^2\right )}\right )}{3 c^5 d^3 \sqrt {1-c^2 x^2}}-\frac {\sqrt {d-c^2 d x^2} (a+b \arcsin (c x))}{c^6 d^3}-\frac {2 (a+b \arcsin (c x))}{c^6 d^2 \sqrt {d-c^2 d x^2}}+\frac {a+b \arcsin (c x)}{3 c^6 d \left (d-c^2 d x^2\right )^{3/2}}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {b \sqrt {d-c^2 d x^2} \left (\frac {1}{2} \int \frac {17-6 c^2 x^2}{1-c^2 x^2}dx-\frac {x}{2 \left (1-c^2 x^2\right )}\right )}{3 c^5 d^3 \sqrt {1-c^2 x^2}}-\frac {\sqrt {d-c^2 d x^2} (a+b \arcsin (c x))}{c^6 d^3}-\frac {2 (a+b \arcsin (c x))}{c^6 d^2 \sqrt {d-c^2 d x^2}}+\frac {a+b \arcsin (c x)}{3 c^6 d \left (d-c^2 d x^2\right )^{3/2}}\) |
| \(\Big \downarrow \) 299 |
| \(\displaystyle \frac {b \sqrt {d-c^2 d x^2} \left (\frac {1}{2} \left (11 \int \frac {1}{1-c^2 x^2}dx+6 x\right )-\frac {x}{2 \left (1-c^2 x^2\right )}\right )}{3 c^5 d^3 \sqrt {1-c^2 x^2}}-\frac {\sqrt {d-c^2 d x^2} (a+b \arcsin (c x))}{c^6 d^3}-\frac {2 (a+b \arcsin (c x))}{c^6 d^2 \sqrt {d-c^2 d x^2}}+\frac {a+b \arcsin (c x)}{3 c^6 d \left (d-c^2 d x^2\right )^{3/2}}\) |
| \(\Big \downarrow \) 219 |
| \(\displaystyle -\frac {\sqrt {d-c^2 d x^2} (a+b \arcsin (c x))}{c^6 d^3}-\frac {2 (a+b \arcsin (c x))}{c^6 d^2 \sqrt {d-c^2 d x^2}}+\frac {a+b \arcsin (c x)}{3 c^6 d \left (d-c^2 d x^2\right )^{3/2}}+\frac {b \left (\frac {1}{2} \left (\frac {11 \text {arctanh}(c x)}{c}+6 x\right )-\frac {x}{2 \left (1-c^2 x^2\right )}\right ) \sqrt {d-c^2 d x^2}}{3 c^5 d^3 \sqrt {1-c^2 x^2}}\) |
(a + b*ArcSin[c*x])/(3*c^6*d*(d - c^2*d*x^2)^(3/2)) - (2*(a + b*ArcSin[c*x ]))/(c^6*d^2*Sqrt[d - c^2*d*x^2]) - (Sqrt[d - c^2*d*x^2]*(a + b*ArcSin[c*x ]))/(c^6*d^3) + (b*Sqrt[d - c^2*d*x^2]*(-1/2*x/(1 - c^2*x^2) + (6*x + (11* ArcTanh[c*x])/c)/2))/(3*c^5*d^3*Sqrt[1 - c^2*x^2])
3.2.30.3.1 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_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1/(Rt[a, 2]*Rt[-b, 2]))* ArcTanh[Rt[-b, 2]*(x/Rt[a, 2])], x] /; FreeQ[{a, b}, x] && NegQ[a/b] && (Gt Q[a, 0] || LtQ[b, 0])
Int[((a_) + (b_.)*(x_)^2)^(p_)*((c_) + (d_.)*(x_)^2), x_Symbol] :> Simp[d*x *((a + b*x^2)^(p + 1)/(b*(2*p + 3))), x] - Simp[(a*d - b*c*(2*p + 3))/(b*(2 *p + 3)) Int[(a + b*x^2)^p, x], x] /; FreeQ[{a, b, c, d}, x] && NeQ[b*c - a*d, 0] && NeQ[2*p + 3, 0]
Int[((d_) + (e_.)*(x_)^2)^(q_)*((a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4)^(p_.), x_Symbol] :> With[{Qx = PolynomialQuotient[(a + b*x^2 + c*x^4)^p, d + e*x^2 , x], R = Coeff[PolynomialRemainder[(a + b*x^2 + c*x^4)^p, d + e*x^2, x], x , 0]}, Simp[(-R)*x*((d + e*x^2)^(q + 1)/(2*d*(q + 1))), x] + Simp[1/(2*d*(q + 1)) Int[(d + e*x^2)^(q + 1)*ExpandToSum[2*d*(q + 1)*Qx + R*(2*q + 3), x], 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] && LtQ[q, -1]
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 = 414, normalized size of antiderivative = 1.89
| method | result | size |
| default | \(a \left (-\frac {x^{4}}{c^{2} d \left (-c^{2} d \,x^{2}+d \right )^{\frac {3}{2}}}+\frac {\frac {4 x^{2}}{c^{2} d \left (-c^{2} d \,x^{2}+d \right )^{\frac {3}{2}}}-\frac {8}{3 d \,c^{4} \left (-c^{2} d \,x^{2}+d \right )^{\frac {3}{2}}}}{c^{2}}\right )+b \left (-\frac {\sqrt {-d \left (c^{2} x^{2}-1\right )}\, \left (c^{2} x^{2}-i c x \sqrt {-c^{2} x^{2}+1}-1\right ) \left (\arcsin \left (c x \right )+i\right )}{2 c^{6} d^{3} \left (c^{2} x^{2}-1\right )}-\frac {\sqrt {-d \left (c^{2} x^{2}-1\right )}\, \left (i c x \sqrt {-c^{2} x^{2}+1}+c^{2} x^{2}-1\right ) \left (\arcsin \left (c x \right )-i\right )}{2 c^{6} d^{3} \left (c^{2} x^{2}-1\right )}+\frac {\sqrt {-d \left (c^{2} x^{2}-1\right )}\, \left (12 c^{2} x^{2} \arcsin \left (c x \right )-c x \sqrt {-c^{2} x^{2}+1}-10 \arcsin \left (c x \right )\right )}{6 c^{6} \left (c^{2} x^{2}-1\right )^{2} d^{3}}+\frac {11 \sqrt {-d \left (c^{2} x^{2}-1\right )}\, \sqrt {-c^{2} x^{2}+1}\, \ln \left (i c x +\sqrt {-c^{2} x^{2}+1}-i\right )}{6 c^{6} d^{3} \left (c^{2} x^{2}-1\right )}-\frac {11 \sqrt {-d \left (c^{2} x^{2}-1\right )}\, \sqrt {-c^{2} x^{2}+1}\, \ln \left (i c x +\sqrt {-c^{2} x^{2}+1}+i\right )}{6 c^{6} d^{3} \left (c^{2} x^{2}-1\right )}\right )\) | \(414\) |
| parts | \(a \left (-\frac {x^{4}}{c^{2} d \left (-c^{2} d \,x^{2}+d \right )^{\frac {3}{2}}}+\frac {\frac {4 x^{2}}{c^{2} d \left (-c^{2} d \,x^{2}+d \right )^{\frac {3}{2}}}-\frac {8}{3 d \,c^{4} \left (-c^{2} d \,x^{2}+d \right )^{\frac {3}{2}}}}{c^{2}}\right )+b \left (-\frac {\sqrt {-d \left (c^{2} x^{2}-1\right )}\, \left (c^{2} x^{2}-i c x \sqrt {-c^{2} x^{2}+1}-1\right ) \left (\arcsin \left (c x \right )+i\right )}{2 c^{6} d^{3} \left (c^{2} x^{2}-1\right )}-\frac {\sqrt {-d \left (c^{2} x^{2}-1\right )}\, \left (i c x \sqrt {-c^{2} x^{2}+1}+c^{2} x^{2}-1\right ) \left (\arcsin \left (c x \right )-i\right )}{2 c^{6} d^{3} \left (c^{2} x^{2}-1\right )}+\frac {\sqrt {-d \left (c^{2} x^{2}-1\right )}\, \left (12 c^{2} x^{2} \arcsin \left (c x \right )-c x \sqrt {-c^{2} x^{2}+1}-10 \arcsin \left (c x \right )\right )}{6 c^{6} \left (c^{2} x^{2}-1\right )^{2} d^{3}}+\frac {11 \sqrt {-d \left (c^{2} x^{2}-1\right )}\, \sqrt {-c^{2} x^{2}+1}\, \ln \left (i c x +\sqrt {-c^{2} x^{2}+1}-i\right )}{6 c^{6} d^{3} \left (c^{2} x^{2}-1\right )}-\frac {11 \sqrt {-d \left (c^{2} x^{2}-1\right )}\, \sqrt {-c^{2} x^{2}+1}\, \ln \left (i c x +\sqrt {-c^{2} x^{2}+1}+i\right )}{6 c^{6} d^{3} \left (c^{2} x^{2}-1\right )}\right )\) | \(414\) |
a*(-x^4/c^2/d/(-c^2*d*x^2+d)^(3/2)+4/c^2*(x^2/c^2/d/(-c^2*d*x^2+d)^(3/2)-2 /3/d/c^4/(-c^2*d*x^2+d)^(3/2)))+b*(-1/2*(-d*(c^2*x^2-1))^(1/2)*(c^2*x^2-I* (-c^2*x^2+1)^(1/2)*x*c-1)*(arcsin(c*x)+I)/c^6/d^3/(c^2*x^2-1)-1/2*(-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^ 3/(c^2*x^2-1)+1/6*(-d*(c^2*x^2-1))^(1/2)*(12*c^2*x^2*arcsin(c*x)-c*x*(-c^2 *x^2+1)^(1/2)-10*arcsin(c*x))/c^6/(c^2*x^2-1)^2/d^3+11/6*(-d*(c^2*x^2-1))^ (1/2)*(-c^2*x^2+1)^(1/2)/c^6/d^3/(c^2*x^2-1)*ln(I*c*x+(-c^2*x^2+1)^(1/2)-I )-11/6*(-d*(c^2*x^2-1))^(1/2)*(-c^2*x^2+1)^(1/2)/c^6/d^3/(c^2*x^2-1)*ln(I* c*x+(-c^2*x^2+1)^(1/2)+I))
Time = 0.32 (sec) , antiderivative size = 481, normalized size of antiderivative = 2.20 \[ \int \frac {x^5 (a+b \arcsin (c x))}{\left (d-c^2 d x^2\right )^{5/2}} \, dx=\left [\frac {11 \, {\left (b c^{4} x^{4} - 2 \, b c^{2} x^{2} + b\right )} \sqrt {d} \log \left (-\frac {c^{6} d x^{6} + 5 \, c^{4} d x^{4} - 5 \, c^{2} d x^{2} - 4 \, {\left (c^{3} x^{3} + c x\right )} \sqrt {-c^{2} d x^{2} + d} \sqrt {-c^{2} x^{2} + 1} \sqrt {d} - d}{c^{6} x^{6} - 3 \, c^{4} x^{4} + 3 \, c^{2} x^{2} - 1}\right ) - 4 \, {\left (6 \, b c^{3} x^{3} - 5 \, b c x\right )} \sqrt {-c^{2} d x^{2} + d} \sqrt {-c^{2} x^{2} + 1} - 8 \, {\left (3 \, a c^{4} x^{4} - 12 \, a c^{2} x^{2} + {\left (3 \, b c^{4} x^{4} - 12 \, b c^{2} x^{2} + 8 \, b\right )} \arcsin \left (c x\right ) + 8 \, a\right )} \sqrt {-c^{2} d x^{2} + d}}{24 \, {\left (c^{10} d^{3} x^{4} - 2 \, c^{8} d^{3} x^{2} + c^{6} d^{3}\right )}}, \frac {11 \, {\left (b c^{4} x^{4} - 2 \, b c^{2} x^{2} + b\right )} \sqrt {-d} \arctan \left (\frac {2 \, \sqrt {-c^{2} d x^{2} + d} \sqrt {-c^{2} x^{2} + 1} c \sqrt {-d} x}{c^{4} d x^{4} - d}\right ) - 2 \, {\left (6 \, b c^{3} x^{3} - 5 \, b c x\right )} \sqrt {-c^{2} d x^{2} + d} \sqrt {-c^{2} x^{2} + 1} - 4 \, {\left (3 \, a c^{4} x^{4} - 12 \, a c^{2} x^{2} + {\left (3 \, b c^{4} x^{4} - 12 \, b c^{2} x^{2} + 8 \, b\right )} \arcsin \left (c x\right ) + 8 \, a\right )} \sqrt {-c^{2} d x^{2} + d}}{12 \, {\left (c^{10} d^{3} x^{4} - 2 \, c^{8} d^{3} x^{2} + c^{6} d^{3}\right )}}\right ] \]
[1/24*(11*(b*c^4*x^4 - 2*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*(6*b*c^3*x^3 - 5*b*c*x)*sqrt(-c^2*d*x^2 + d)*sqrt(-c^2*x^2 + 1) - 8*(3*a*c^4*x^4 - 12*a* c^2*x^2 + (3*b*c^4*x^4 - 12*b*c^2*x^2 + 8*b)*arcsin(c*x) + 8*a)*sqrt(-c^2* d*x^2 + d))/(c^10*d^3*x^4 - 2*c^8*d^3*x^2 + c^6*d^3), 1/12*(11*(b*c^4*x^4 - 2*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*(6*b*c^3*x^3 - 5*b*c*x)*sqrt(-c^2*d*x ^2 + d)*sqrt(-c^2*x^2 + 1) - 4*(3*a*c^4*x^4 - 12*a*c^2*x^2 + (3*b*c^4*x^4 - 12*b*c^2*x^2 + 8*b)*arcsin(c*x) + 8*a)*sqrt(-c^2*d*x^2 + d))/(c^10*d^3*x ^4 - 2*c^8*d^3*x^2 + c^6*d^3)]
\[ \int \frac {x^5 (a+b \arcsin (c x))}{\left (d-c^2 d x^2\right )^{5/2}} \, dx=\int \frac {x^{5} \left (a + b \operatorname {asin}{\left (c x \right )}\right )}{\left (- d \left (c x - 1\right ) \left (c x + 1\right )\right )^{\frac {5}{2}}}\, dx \]
\[ \int \frac {x^5 (a+b \arcsin (c x))}{\left (d-c^2 d x^2\right )^{5/2}} \, dx=\int { \frac {{\left (b \arcsin \left (c x\right ) + a\right )} x^{5}}{{\left (-c^{2} d x^{2} + d\right )}^{\frac {5}{2}}} \,d x } \]
-1/3*a*(3*x^4/((-c^2*d*x^2 + d)^(3/2)*c^2*d) - 12*x^2/((-c^2*d*x^2 + d)^(3 /2)*c^4*d) + 8/((-c^2*d*x^2 + d)^(3/2)*c^6*d)) + 1/3*(3*(c^8*d^3*x^2 - c^6 *d^3)*sqrt(c*x + 1)*sqrt(-c*x + 1)*sqrt(d)*integrate(1/3*(3*c^4*x^6 - 12*c ^2*x^4 + 8*x^2)/(c^9*d^3*x^6 - 2*c^7*d^3*x^4 + c^5*d^3*x^2 + (c^7*d^3*x^4 - 2*c^5*d^3*x^2 + c^3*d^3)*e^(log(c*x + 1) + log(-c*x + 1))), x) + (3*c^4* x^4 - 12*c^2*x^2 + 8)*sqrt(d)*arctan2(c*x, sqrt(c*x + 1)*sqrt(-c*x + 1)))* b/((c^8*d^3*x^2 - c^6*d^3)*sqrt(c*x + 1)*sqrt(-c*x + 1))
Exception generated. \[ \int \frac {x^5 (a+b \arcsin (c x))}{\left (d-c^2 d x^2\right )^{5/2}} \, dx=\text {Exception raised: TypeError} \]
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. \[ \int \frac {x^5 (a+b \arcsin (c x))}{\left (d-c^2 d x^2\right )^{5/2}} \, dx=\int \frac {x^5\,\left (a+b\,\mathrm {asin}\left (c\,x\right )\right )}{{\left (d-c^2\,d\,x^2\right )}^{5/2}} \,d x \]