Integrand size = 27, antiderivative size = 142 \[ \int \frac {x^3 (a+b \arcsin (c x))}{\left (d-c^2 d x^2\right )^{3/2}} \, dx=-\frac {b x \sqrt {d-c^2 d x^2}}{c^3 d^2 \sqrt {1-c^2 x^2}}+\frac {a+b \arcsin (c x)}{c^4 d \sqrt {d-c^2 d x^2}}+\frac {\sqrt {d-c^2 d x^2} (a+b \arcsin (c x))}{c^4 d^2}-\frac {b \sqrt {d-c^2 d x^2} \text {arctanh}(c x)}{c^4 d^2 \sqrt {1-c^2 x^2}} \]
(a+b*arcsin(c*x))/c^4/d/(-c^2*d*x^2+d)^(1/2)+(a+b*arcsin(c*x))*(-c^2*d*x^2 +d)^(1/2)/c^4/d^2-b*x*(-c^2*d*x^2+d)^(1/2)/c^3/d^2/(-c^2*x^2+1)^(1/2)-b*ar ctanh(c*x)*(-c^2*d*x^2+d)^(1/2)/c^4/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 = 0.17 (sec) , antiderivative size = 136, normalized size of antiderivative = 0.96 \[ \int \frac {x^3 (a+b \arcsin (c x))}{\left (d-c^2 d x^2\right )^{3/2}} \, dx=\frac {\sqrt {d-c^2 d x^2} \left (\sqrt {-c^2} \left (-2 a+a c^2 x^2+b c x \sqrt {1-c^2 x^2}+b \left (-2+c^2 x^2\right ) \arcsin (c x)\right )-i b c \sqrt {1-c^2 x^2} \operatorname {EllipticF}\left (i \text {arcsinh}\left (\sqrt {-c^2} x\right ),1\right )\right )}{c^4 \sqrt {-c^2} d^2 \left (-1+c^2 x^2\right )} \]
(Sqrt[d - c^2*d*x^2]*(Sqrt[-c^2]*(-2*a + a*c^2*x^2 + b*c*x*Sqrt[1 - c^2*x^ 2] + b*(-2 + c^2*x^2)*ArcSin[c*x]) - I*b*c*Sqrt[1 - c^2*x^2]*EllipticF[I*A rcSinh[Sqrt[-c^2]*x], 1]))/(c^4*Sqrt[-c^2]*d^2*(-1 + c^2*x^2))
Time = 0.35 (sec) , antiderivative size = 109, normalized size of antiderivative = 0.77, number of steps used = 4, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.148, Rules used = {5194, 27, 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.
\(\displaystyle \int \frac {x^3 (a+b \arcsin (c x))}{\left (d-c^2 d x^2\right )^{3/2}} \, dx\) |
\(\Big \downarrow \) 5194 |
\(\displaystyle -\frac {b c \sqrt {d-c^2 d x^2} \int \frac {2-c^2 x^2}{c^4 d^2 \left (1-c^2 x^2\right )}dx}{\sqrt {1-c^2 x^2}}+\frac {\sqrt {d-c^2 d x^2} (a+b \arcsin (c x))}{c^4 d^2}+\frac {a+b \arcsin (c x)}{c^4 d \sqrt {d-c^2 d x^2}}\) |
\(\Big \downarrow \) 27 |
\(\displaystyle -\frac {b \sqrt {d-c^2 d x^2} \int \frac {2-c^2 x^2}{1-c^2 x^2}dx}{c^3 d^2 \sqrt {1-c^2 x^2}}+\frac {\sqrt {d-c^2 d x^2} (a+b \arcsin (c x))}{c^4 d^2}+\frac {a+b \arcsin (c x)}{c^4 d \sqrt {d-c^2 d x^2}}\) |
\(\Big \downarrow \) 299 |
\(\displaystyle -\frac {b \sqrt {d-c^2 d x^2} \left (\int \frac {1}{1-c^2 x^2}dx+x\right )}{c^3 d^2 \sqrt {1-c^2 x^2}}+\frac {\sqrt {d-c^2 d x^2} (a+b \arcsin (c x))}{c^4 d^2}+\frac {a+b \arcsin (c x)}{c^4 d \sqrt {d-c^2 d x^2}}\) |
\(\Big \downarrow \) 219 |
\(\displaystyle \frac {\sqrt {d-c^2 d x^2} (a+b \arcsin (c x))}{c^4 d^2}+\frac {a+b \arcsin (c x)}{c^4 d \sqrt {d-c^2 d x^2}}-\frac {b \left (\frac {\text {arctanh}(c x)}{c}+x\right ) \sqrt {d-c^2 d x^2}}{c^3 d^2 \sqrt {1-c^2 x^2}}\) |
(a + b*ArcSin[c*x])/(c^4*d*Sqrt[d - c^2*d*x^2]) + (Sqrt[d - c^2*d*x^2]*(a + b*ArcSin[c*x]))/(c^4*d^2) - (b*Sqrt[d - c^2*d*x^2]*(x + ArcTanh[c*x]/c)) /(c^3*d^2*Sqrt[1 - c^2*x^2])
3.2.21.3.1 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[((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 = 0.25 (sec) , antiderivative size = 307, normalized size of antiderivative = 2.16
method | result | size |
default | \(a \left (-\frac {x^{2}}{c^{2} d \sqrt {-c^{2} d \,x^{2}+d}}+\frac {2}{d \,c^{4} \sqrt {-c^{2} d \,x^{2}+d}}\right )+\frac {b \sqrt {-d \left (c^{2} x^{2}-1\right )}\, \sqrt {-c^{2} x^{2}+1}\, x}{c^{3} d^{2} \left (c^{2} x^{2}-1\right )}+\frac {b \sqrt {-d \left (c^{2} x^{2}-1\right )}\, \arcsin \left (c x \right ) x^{2}}{c^{2} d^{2} \left (c^{2} x^{2}-1\right )}-\frac {2 b \sqrt {-d \left (c^{2} x^{2}-1\right )}\, \arcsin \left (c x \right )}{c^{4} d^{2} \left (c^{2} x^{2}-1\right )}+\frac {b \sqrt {-d \left (c^{2} x^{2}-1\right )}\, \sqrt {-c^{2} x^{2}+1}\, \ln \left (i c x +\sqrt {-c^{2} x^{2}+1}+i\right )}{c^{4} d^{2} \left (c^{2} x^{2}-1\right )}-\frac {b \sqrt {-d \left (c^{2} x^{2}-1\right )}\, \sqrt {-c^{2} x^{2}+1}\, \ln \left (i c x +\sqrt {-c^{2} x^{2}+1}-i\right )}{c^{4} d^{2} \left (c^{2} x^{2}-1\right )}\) | \(307\) |
parts | \(a \left (-\frac {x^{2}}{c^{2} d \sqrt {-c^{2} d \,x^{2}+d}}+\frac {2}{d \,c^{4} \sqrt {-c^{2} d \,x^{2}+d}}\right )+\frac {b \sqrt {-d \left (c^{2} x^{2}-1\right )}\, \sqrt {-c^{2} x^{2}+1}\, x}{c^{3} d^{2} \left (c^{2} x^{2}-1\right )}+\frac {b \sqrt {-d \left (c^{2} x^{2}-1\right )}\, \arcsin \left (c x \right ) x^{2}}{c^{2} d^{2} \left (c^{2} x^{2}-1\right )}-\frac {2 b \sqrt {-d \left (c^{2} x^{2}-1\right )}\, \arcsin \left (c x \right )}{c^{4} d^{2} \left (c^{2} x^{2}-1\right )}+\frac {b \sqrt {-d \left (c^{2} x^{2}-1\right )}\, \sqrt {-c^{2} x^{2}+1}\, \ln \left (i c x +\sqrt {-c^{2} x^{2}+1}+i\right )}{c^{4} d^{2} \left (c^{2} x^{2}-1\right )}-\frac {b \sqrt {-d \left (c^{2} x^{2}-1\right )}\, \sqrt {-c^{2} x^{2}+1}\, \ln \left (i c x +\sqrt {-c^{2} x^{2}+1}-i\right )}{c^{4} d^{2} \left (c^{2} x^{2}-1\right )}\) | \(307\) |
a*(-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))+b*(-d*(c^ 2*x^2-1))^(1/2)/c^3/d^2/(c^2*x^2-1)*(-c^2*x^2+1)^(1/2)*x+b*(-d*(c^2*x^2-1) )^(1/2)/c^2/d^2/(c^2*x^2-1)*arcsin(c*x)*x^2-2*b*(-d*(c^2*x^2-1))^(1/2)/c^4 /d^2/(c^2*x^2-1)*arcsin(c*x)+b*(-d*(c^2*x^2-1))^(1/2)*(-c^2*x^2+1)^(1/2)/c ^4/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^4/d^2/(c^2*x^2-1)*ln(I*c*x+(-c^2*x^2+1)^(1/2)-I)
Time = 0.30 (sec) , antiderivative size = 382, normalized size of antiderivative = 2.69 \[ \int \frac {x^3 (a+b \arcsin (c x))}{\left (d-c^2 d x^2\right )^{3/2}} \, dx=\left [\frac {4 \, \sqrt {-c^{2} d x^{2} + d} \sqrt {-c^{2} x^{2} + 1} b c x + {\left (b c^{2} x^{2} - b\right )} \sqrt {d} \log \left (-\frac {c^{6} d x^{6} + 5 \, c^{4} d x^{4} - 5 \, c^{2} d x^{2} + 4 \, {\left (c^{3} x^{3} + c x\right )} \sqrt {-c^{2} d x^{2} + d} \sqrt {-c^{2} x^{2} + 1} \sqrt {d} - d}{c^{6} x^{6} - 3 \, c^{4} x^{4} + 3 \, c^{2} x^{2} - 1}\right ) + 4 \, {\left (a c^{2} x^{2} + {\left (b c^{2} x^{2} - 2 \, b\right )} \arcsin \left (c x\right ) - 2 \, a\right )} \sqrt {-c^{2} d x^{2} + d}}{4 \, {\left (c^{6} d^{2} x^{2} - c^{4} d^{2}\right )}}, \frac {2 \, \sqrt {-c^{2} d x^{2} + d} \sqrt {-c^{2} x^{2} + 1} b c x - {\left (b c^{2} x^{2} - b\right )} \sqrt {-d} \arctan \left (\frac {2 \, \sqrt {-c^{2} d x^{2} + d} \sqrt {-c^{2} x^{2} + 1} c \sqrt {-d} x}{c^{4} d x^{4} - d}\right ) + 2 \, {\left (a c^{2} x^{2} + {\left (b c^{2} x^{2} - 2 \, b\right )} \arcsin \left (c x\right ) - 2 \, a\right )} \sqrt {-c^{2} d x^{2} + d}}{2 \, {\left (c^{6} d^{2} x^{2} - c^{4} d^{2}\right )}}\right ] \]
[1/4*(4*sqrt(-c^2*d*x^2 + d)*sqrt(-c^2*x^2 + 1)*b*c*x + (b*c^2*x^2 - b)*sq rt(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*(a*c^2*x^2 + (b*c^2*x^2 - 2*b)*arcsin(c*x) - 2*a)*sqrt(- c^2*d*x^2 + d))/(c^6*d^2*x^2 - c^4*d^2), 1/2*(2*sqrt(-c^2*d*x^2 + d)*sqrt( -c^2*x^2 + 1)*b*c*x - (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*(a*c^2*x^2 + (b*c^ 2*x^2 - 2*b)*arcsin(c*x) - 2*a)*sqrt(-c^2*d*x^2 + d))/(c^6*d^2*x^2 - c^4*d ^2)]
\[ \int \frac {x^3 (a+b \arcsin (c x))}{\left (d-c^2 d x^2\right )^{3/2}} \, dx=\int \frac {x^{3} \left (a + b \operatorname {asin}{\left (c x \right )}\right )}{\left (- d \left (c x - 1\right ) \left (c x + 1\right )\right )^{\frac {3}{2}}}\, dx \]
Time = 0.31 (sec) , antiderivative size = 142, normalized size of antiderivative = 1.00 \[ \int \frac {x^3 (a+b \arcsin (c x))}{\left (d-c^2 d x^2\right )^{3/2}} \, dx=-\frac {1}{2} \, b c {\left (\frac {2 \, x}{c^{4} d^{\frac {3}{2}}} + \frac {\log \left (c x + 1\right )}{c^{5} d^{\frac {3}{2}}} - \frac {\log \left (c x - 1\right )}{c^{5} d^{\frac {3}{2}}}\right )} - b {\left (\frac {x^{2}}{\sqrt {-c^{2} d x^{2} + d} c^{2} d} - \frac {2}{\sqrt {-c^{2} d x^{2} + d} c^{4} d}\right )} \arcsin \left (c x\right ) - a {\left (\frac {x^{2}}{\sqrt {-c^{2} d x^{2} + d} c^{2} d} - \frac {2}{\sqrt {-c^{2} d x^{2} + d} c^{4} d}\right )} \]
-1/2*b*c*(2*x/(c^4*d^(3/2)) + log(c*x + 1)/(c^5*d^(3/2)) - log(c*x - 1)/(c ^5*d^(3/2))) - b*(x^2/(sqrt(-c^2*d*x^2 + d)*c^2*d) - 2/(sqrt(-c^2*d*x^2 + d)*c^4*d))*arcsin(c*x) - a*(x^2/(sqrt(-c^2*d*x^2 + d)*c^2*d) - 2/(sqrt(-c^ 2*d*x^2 + d)*c^4*d))
Exception generated. \[ \int \frac {x^3 (a+b \arcsin (c x))}{\left (d-c^2 d x^2\right )^{3/2}} \, dx=\text {Exception raised: TypeError} \]
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. \[ \int \frac {x^3 (a+b \arcsin (c x))}{\left (d-c^2 d x^2\right )^{3/2}} \, dx=\int \frac {x^3\,\left (a+b\,\mathrm {asin}\left (c\,x\right )\right )}{{\left (d-c^2\,d\,x^2\right )}^{3/2}} \,d x \]