Integrand size = 27, antiderivative size = 277 \[ \int \frac {\left (d-c^2 d x^2\right )^{5/2} (a+b \arcsin (c x))}{x^4} \, dx=-\frac {b c d^2 \sqrt {d-c^2 d x^2}}{6 x^2 \sqrt {1-c^2 x^2}}-\frac {b c^5 d^2 x^2 \sqrt {d-c^2 d x^2}}{4 \sqrt {1-c^2 x^2}}+\frac {5}{2} c^4 d^2 x \sqrt {d-c^2 d x^2} (a+b \arcsin (c x))+\frac {5 c^2 d \left (d-c^2 d x^2\right )^{3/2} (a+b \arcsin (c x))}{3 x}-\frac {\left (d-c^2 d x^2\right )^{5/2} (a+b \arcsin (c x))}{3 x^3}+\frac {5 c^3 d^2 \sqrt {d-c^2 d x^2} (a+b \arcsin (c x))^2}{4 b \sqrt {1-c^2 x^2}}-\frac {7 b c^3 d^2 \sqrt {d-c^2 d x^2} \log (x)}{3 \sqrt {1-c^2 x^2}} \]
5/3*c^2*d*(-c^2*d*x^2+d)^(3/2)*(a+b*arcsin(c*x))/x-1/3*(-c^2*d*x^2+d)^(5/2 )*(a+b*arcsin(c*x))/x^3+5/2*c^4*d^2*x*(a+b*arcsin(c*x))*(-c^2*d*x^2+d)^(1/ 2)-1/6*b*c*d^2*(-c^2*d*x^2+d)^(1/2)/x^2/(-c^2*x^2+1)^(1/2)-1/4*b*c^5*d^2*x ^2*(-c^2*d*x^2+d)^(1/2)/(-c^2*x^2+1)^(1/2)+5/4*c^3*d^2*(a+b*arcsin(c*x))^2 *(-c^2*d*x^2+d)^(1/2)/b/(-c^2*x^2+1)^(1/2)-7/3*b*c^3*d^2*ln(x)*(-c^2*d*x^2 +d)^(1/2)/(-c^2*x^2+1)^(1/2)
Time = 1.08 (sec) , antiderivative size = 243, normalized size of antiderivative = 0.88 \[ \int \frac {\left (d-c^2 d x^2\right )^{5/2} (a+b \arcsin (c x))}{x^4} \, dx=\frac {1}{24} d^2 \left (\frac {4 b \sqrt {d-c^2 d x^2} \left (-2+14 c^2 x^2+3 c^4 x^4\right ) \arcsin (c x)}{x^3}+\frac {30 b c^3 \sqrt {d-c^2 d x^2} \arcsin (c x)^2}{\sqrt {1-c^2 x^2}}-60 a c^3 \sqrt {d} \arctan \left (\frac {c x \sqrt {d-c^2 d x^2}}{\sqrt {d} \left (-1+c^2 x^2\right )}\right )+\frac {\sqrt {d-c^2 d x^2} \left (4 a \sqrt {1-c^2 x^2} \left (-2+14 c^2 x^2+3 c^4 x^4\right )+b \left (-4 c x+3 c^3 x^3-6 c^5 x^5\right )-56 b c^3 x^3 \log (c x)\right )}{x^3 \sqrt {1-c^2 x^2}}\right ) \]
(d^2*((4*b*Sqrt[d - c^2*d*x^2]*(-2 + 14*c^2*x^2 + 3*c^4*x^4)*ArcSin[c*x])/ x^3 + (30*b*c^3*Sqrt[d - c^2*d*x^2]*ArcSin[c*x]^2)/Sqrt[1 - c^2*x^2] - 60* a*c^3*Sqrt[d]*ArcTan[(c*x*Sqrt[d - c^2*d*x^2])/(Sqrt[d]*(-1 + c^2*x^2))] + (Sqrt[d - c^2*d*x^2]*(4*a*Sqrt[1 - c^2*x^2]*(-2 + 14*c^2*x^2 + 3*c^4*x^4) + b*(-4*c*x + 3*c^3*x^3 - 6*c^5*x^5) - 56*b*c^3*x^3*Log[c*x]))/(x^3*Sqrt[ 1 - c^2*x^2])))/24
Time = 0.90 (sec) , antiderivative size = 296, normalized size of antiderivative = 1.07, number of steps used = 11, number of rules used = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.370, Rules used = {5200, 243, 49, 2009, 5200, 244, 2009, 5156, 15, 5152}
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.
\(\displaystyle \int \frac {\left (d-c^2 d x^2\right )^{5/2} (a+b \arcsin (c x))}{x^4} \, dx\) |
\(\Big \downarrow \) 5200 |
\(\displaystyle -\frac {5}{3} c^2 d \int \frac {\left (d-c^2 d x^2\right )^{3/2} (a+b \arcsin (c x))}{x^2}dx+\frac {b c d^2 \sqrt {d-c^2 d x^2} \int \frac {\left (1-c^2 x^2\right )^2}{x^3}dx}{3 \sqrt {1-c^2 x^2}}-\frac {\left (d-c^2 d x^2\right )^{5/2} (a+b \arcsin (c x))}{3 x^3}\) |
\(\Big \downarrow \) 243 |
\(\displaystyle -\frac {5}{3} c^2 d \int \frac {\left (d-c^2 d x^2\right )^{3/2} (a+b \arcsin (c x))}{x^2}dx+\frac {b c d^2 \sqrt {d-c^2 d x^2} \int \frac {\left (1-c^2 x^2\right )^2}{x^4}dx^2}{6 \sqrt {1-c^2 x^2}}-\frac {\left (d-c^2 d x^2\right )^{5/2} (a+b \arcsin (c x))}{3 x^3}\) |
\(\Big \downarrow \) 49 |
\(\displaystyle -\frac {5}{3} c^2 d \int \frac {\left (d-c^2 d x^2\right )^{3/2} (a+b \arcsin (c x))}{x^2}dx+\frac {b c d^2 \sqrt {d-c^2 d x^2} \int \left (c^4-\frac {2 c^2}{x^2}+\frac {1}{x^4}\right )dx^2}{6 \sqrt {1-c^2 x^2}}-\frac {\left (d-c^2 d x^2\right )^{5/2} (a+b \arcsin (c x))}{3 x^3}\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle -\frac {5}{3} c^2 d \int \frac {\left (d-c^2 d x^2\right )^{3/2} (a+b \arcsin (c x))}{x^2}dx-\frac {\left (d-c^2 d x^2\right )^{5/2} (a+b \arcsin (c x))}{3 x^3}+\frac {b c d^2 \sqrt {d-c^2 d x^2} \left (c^4 x^2-2 c^2 \log \left (x^2\right )-\frac {1}{x^2}\right )}{6 \sqrt {1-c^2 x^2}}\) |
\(\Big \downarrow \) 5200 |
\(\displaystyle -\frac {5}{3} c^2 d \left (-3 c^2 d \int \sqrt {d-c^2 d x^2} (a+b \arcsin (c x))dx+\frac {b c d \sqrt {d-c^2 d x^2} \int \frac {1-c^2 x^2}{x}dx}{\sqrt {1-c^2 x^2}}-\frac {\left (d-c^2 d x^2\right )^{3/2} (a+b \arcsin (c x))}{x}\right )-\frac {\left (d-c^2 d x^2\right )^{5/2} (a+b \arcsin (c x))}{3 x^3}+\frac {b c d^2 \sqrt {d-c^2 d x^2} \left (c^4 x^2-2 c^2 \log \left (x^2\right )-\frac {1}{x^2}\right )}{6 \sqrt {1-c^2 x^2}}\) |
\(\Big \downarrow \) 244 |
\(\displaystyle -\frac {5}{3} c^2 d \left (-3 c^2 d \int \sqrt {d-c^2 d x^2} (a+b \arcsin (c x))dx+\frac {b c d \sqrt {d-c^2 d x^2} \int \left (\frac {1}{x}-c^2 x\right )dx}{\sqrt {1-c^2 x^2}}-\frac {\left (d-c^2 d x^2\right )^{3/2} (a+b \arcsin (c x))}{x}\right )-\frac {\left (d-c^2 d x^2\right )^{5/2} (a+b \arcsin (c x))}{3 x^3}+\frac {b c d^2 \sqrt {d-c^2 d x^2} \left (c^4 x^2-2 c^2 \log \left (x^2\right )-\frac {1}{x^2}\right )}{6 \sqrt {1-c^2 x^2}}\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle -\frac {5}{3} c^2 d \left (-3 c^2 d \int \sqrt {d-c^2 d x^2} (a+b \arcsin (c x))dx-\frac {\left (d-c^2 d x^2\right )^{3/2} (a+b \arcsin (c x))}{x}+\frac {b c d \sqrt {d-c^2 d x^2} \left (\log (x)-\frac {c^2 x^2}{2}\right )}{\sqrt {1-c^2 x^2}}\right )-\frac {\left (d-c^2 d x^2\right )^{5/2} (a+b \arcsin (c x))}{3 x^3}+\frac {b c d^2 \sqrt {d-c^2 d x^2} \left (c^4 x^2-2 c^2 \log \left (x^2\right )-\frac {1}{x^2}\right )}{6 \sqrt {1-c^2 x^2}}\) |
\(\Big \downarrow \) 5156 |
\(\displaystyle -\frac {5}{3} c^2 d \left (-3 c^2 d \left (\frac {\sqrt {d-c^2 d x^2} \int \frac {a+b \arcsin (c x)}{\sqrt {1-c^2 x^2}}dx}{2 \sqrt {1-c^2 x^2}}-\frac {b c \sqrt {d-c^2 d x^2} \int xdx}{2 \sqrt {1-c^2 x^2}}+\frac {1}{2} x \sqrt {d-c^2 d x^2} (a+b \arcsin (c x))\right )-\frac {\left (d-c^2 d x^2\right )^{3/2} (a+b \arcsin (c x))}{x}+\frac {b c d \sqrt {d-c^2 d x^2} \left (\log (x)-\frac {c^2 x^2}{2}\right )}{\sqrt {1-c^2 x^2}}\right )-\frac {\left (d-c^2 d x^2\right )^{5/2} (a+b \arcsin (c x))}{3 x^3}+\frac {b c d^2 \sqrt {d-c^2 d x^2} \left (c^4 x^2-2 c^2 \log \left (x^2\right )-\frac {1}{x^2}\right )}{6 \sqrt {1-c^2 x^2}}\) |
\(\Big \downarrow \) 15 |
\(\displaystyle -\frac {5}{3} c^2 d \left (-3 c^2 d \left (\frac {\sqrt {d-c^2 d x^2} \int \frac {a+b \arcsin (c x)}{\sqrt {1-c^2 x^2}}dx}{2 \sqrt {1-c^2 x^2}}+\frac {1}{2} x \sqrt {d-c^2 d x^2} (a+b \arcsin (c x))-\frac {b c x^2 \sqrt {d-c^2 d x^2}}{4 \sqrt {1-c^2 x^2}}\right )-\frac {\left (d-c^2 d x^2\right )^{3/2} (a+b \arcsin (c x))}{x}+\frac {b c d \sqrt {d-c^2 d x^2} \left (\log (x)-\frac {c^2 x^2}{2}\right )}{\sqrt {1-c^2 x^2}}\right )-\frac {\left (d-c^2 d x^2\right )^{5/2} (a+b \arcsin (c x))}{3 x^3}+\frac {b c d^2 \sqrt {d-c^2 d x^2} \left (c^4 x^2-2 c^2 \log \left (x^2\right )-\frac {1}{x^2}\right )}{6 \sqrt {1-c^2 x^2}}\) |
\(\Big \downarrow \) 5152 |
\(\displaystyle -\frac {5}{3} c^2 d \left (-3 c^2 d \left (\frac {1}{2} x \sqrt {d-c^2 d x^2} (a+b \arcsin (c x))+\frac {\sqrt {d-c^2 d x^2} (a+b \arcsin (c x))^2}{4 b c \sqrt {1-c^2 x^2}}-\frac {b c x^2 \sqrt {d-c^2 d x^2}}{4 \sqrt {1-c^2 x^2}}\right )-\frac {\left (d-c^2 d x^2\right )^{3/2} (a+b \arcsin (c x))}{x}+\frac {b c d \sqrt {d-c^2 d x^2} \left (\log (x)-\frac {c^2 x^2}{2}\right )}{\sqrt {1-c^2 x^2}}\right )-\frac {\left (d-c^2 d x^2\right )^{5/2} (a+b \arcsin (c x))}{3 x^3}+\frac {b c d^2 \sqrt {d-c^2 d x^2} \left (c^4 x^2-2 c^2 \log \left (x^2\right )-\frac {1}{x^2}\right )}{6 \sqrt {1-c^2 x^2}}\) |
-1/3*((d - c^2*d*x^2)^(5/2)*(a + b*ArcSin[c*x]))/x^3 - (5*c^2*d*(-(((d - c ^2*d*x^2)^(3/2)*(a + b*ArcSin[c*x]))/x) - 3*c^2*d*(-1/4*(b*c*x^2*Sqrt[d - c^2*d*x^2])/Sqrt[1 - c^2*x^2] + (x*Sqrt[d - c^2*d*x^2]*(a + b*ArcSin[c*x]) )/2 + (Sqrt[d - c^2*d*x^2]*(a + b*ArcSin[c*x])^2)/(4*b*c*Sqrt[1 - c^2*x^2] )) + (b*c*d*Sqrt[d - c^2*d*x^2]*(-1/2*(c^2*x^2) + Log[x]))/Sqrt[1 - c^2*x^ 2]))/3 + (b*c*d^2*Sqrt[d - c^2*d*x^2]*(-x^(-2) + c^4*x^2 - 2*c^2*Log[x^2]) )/(6*Sqrt[1 - c^2*x^2])
3.1.88.3.1 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_.) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_.), x_Symbol] :> Int [ExpandIntegrand[(a + b*x)^m*(c + d*x)^n, x], x] /; FreeQ[{a, b, c, d}, x] && IGtQ[m, 0] && IGtQ[m + n + 2, 0]
Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> Simp[1/2 Subst[In t[x^((m - 1)/2)*(a + b*x)^p, x], x, x^2], x] /; FreeQ[{a, b, m, p}, x] && I ntegerQ[(m - 1)/2]
Int[((c_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^2)^(p_.), x_Symbol] :> Int[Expand Integrand[(c*x)^m*(a + b*x^2)^p, x], x] /; FreeQ[{a, b, c, m}, x] && IGtQ[p , 0]
Int[((a_.) + ArcSin[(c_.)*(x_)]*(b_.))^(n_.)/Sqrt[(d_) + (e_.)*(x_)^2], x_S ymbol] :> Simp[(1/(b*c*(n + 1)))*Simp[Sqrt[1 - c^2*x^2]/Sqrt[d + e*x^2]]*(a + b*ArcSin[c*x])^(n + 1), x] /; FreeQ[{a, b, c, d, e, n}, x] && EqQ[c^2*d + e, 0] && NeQ[n, -1]
Int[((a_.) + ArcSin[(c_.)*(x_)]*(b_.))^(n_.)*Sqrt[(d_) + (e_.)*(x_)^2], x_S ymbol] :> Simp[x*Sqrt[d + e*x^2]*((a + b*ArcSin[c*x])^n/2), x] + (Simp[(1/2 )*Simp[Sqrt[d + e*x^2]/Sqrt[1 - c^2*x^2]] Int[(a + b*ArcSin[c*x])^n/Sqrt[ 1 - c^2*x^2], x], x] - Simp[b*c*(n/2)*Simp[Sqrt[d + e*x^2]/Sqrt[1 - c^2*x^2 ]] Int[x*(a + b*ArcSin[c*x])^(n - 1), x], x]) /; FreeQ[{a, b, c, d, e}, x ] && EqQ[c^2*d + e, 0] && GtQ[n, 0]
Int[((a_.) + ArcSin[(c_.)*(x_)]*(b_.))^(n_.)*((f_.)*(x_))^(m_)*((d_) + (e_. )*(x_)^2)^(p_.), x_Symbol] :> Simp[(f*x)^(m + 1)*(d + e*x^2)^p*((a + b*ArcS in[c*x])^n/(f*(m + 1))), x] + (-Simp[2*e*(p/(f^2*(m + 1))) Int[(f*x)^(m + 2)*(d + e*x^2)^(p - 1)*(a + b*ArcSin[c*x])^n, x], x] - Simp[b*c*(n/(f*(m + 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] && GtQ[n, 0] && GtQ[p, 0] && LtQ[m, -1]
Result contains complex when optimal does not.
Time = 0.20 (sec) , antiderivative size = 343, normalized size of antiderivative = 1.24
method | result | size |
default | \(-\frac {a \left (-c^{2} d \,x^{2}+d \right )^{\frac {7}{2}}}{3 d \,x^{3}}+\frac {4 a \,c^{2} \left (-c^{2} d \,x^{2}+d \right )^{\frac {7}{2}}}{3 d x}+\frac {4 a \,c^{4} x \left (-c^{2} d \,x^{2}+d \right )^{\frac {5}{2}}}{3}+\frac {5 a \,c^{4} d x \left (-c^{2} d \,x^{2}+d \right )^{\frac {3}{2}}}{3}+\frac {5 a \,c^{4} d^{2} x \sqrt {-c^{2} d \,x^{2}+d}}{2}+\frac {5 a \,c^{4} d^{3} \arctan \left (\frac {\sqrt {c^{2} d}\, x}{\sqrt {-c^{2} d \,x^{2}+d}}\right )}{2 \sqrt {c^{2} d}}-\frac {b \sqrt {-d \left (c^{2} x^{2}-1\right )}\, \sqrt {-c^{2} x^{2}+1}\, \left (12 \sqrt {-c^{2} x^{2}+1}\, \arcsin \left (c x \right ) x^{4} c^{4}-6 c^{5} x^{5}+30 c^{3} x^{3} \arcsin \left (c x \right )^{2}+56 i \arcsin \left (c x \right ) x^{3} c^{3}-56 \ln \left (\left (i c x +\sqrt {-c^{2} x^{2}+1}\right )^{2}-1\right ) x^{3} c^{3}+56 \sqrt {-c^{2} x^{2}+1}\, \arcsin \left (c x \right ) x^{2} c^{2}+3 c^{3} x^{3}-8 \arcsin \left (c x \right ) \sqrt {-c^{2} x^{2}+1}-4 c x \right ) d^{2}}{24 \left (c^{2} x^{2}-1\right ) x^{3}}\) | \(343\) |
parts | \(-\frac {a \left (-c^{2} d \,x^{2}+d \right )^{\frac {7}{2}}}{3 d \,x^{3}}+\frac {4 a \,c^{2} \left (-c^{2} d \,x^{2}+d \right )^{\frac {7}{2}}}{3 d x}+\frac {4 a \,c^{4} x \left (-c^{2} d \,x^{2}+d \right )^{\frac {5}{2}}}{3}+\frac {5 a \,c^{4} d x \left (-c^{2} d \,x^{2}+d \right )^{\frac {3}{2}}}{3}+\frac {5 a \,c^{4} d^{2} x \sqrt {-c^{2} d \,x^{2}+d}}{2}+\frac {5 a \,c^{4} d^{3} \arctan \left (\frac {\sqrt {c^{2} d}\, x}{\sqrt {-c^{2} d \,x^{2}+d}}\right )}{2 \sqrt {c^{2} d}}-\frac {b \sqrt {-d \left (c^{2} x^{2}-1\right )}\, \sqrt {-c^{2} x^{2}+1}\, \left (12 \sqrt {-c^{2} x^{2}+1}\, \arcsin \left (c x \right ) x^{4} c^{4}-6 c^{5} x^{5}+30 c^{3} x^{3} \arcsin \left (c x \right )^{2}+56 i \arcsin \left (c x \right ) x^{3} c^{3}-56 \ln \left (\left (i c x +\sqrt {-c^{2} x^{2}+1}\right )^{2}-1\right ) x^{3} c^{3}+56 \sqrt {-c^{2} x^{2}+1}\, \arcsin \left (c x \right ) x^{2} c^{2}+3 c^{3} x^{3}-8 \arcsin \left (c x \right ) \sqrt {-c^{2} x^{2}+1}-4 c x \right ) d^{2}}{24 \left (c^{2} x^{2}-1\right ) x^{3}}\) | \(343\) |
-1/3*a/d/x^3*(-c^2*d*x^2+d)^(7/2)+4/3*a*c^2/d/x*(-c^2*d*x^2+d)^(7/2)+4/3*a *c^4*x*(-c^2*d*x^2+d)^(5/2)+5/3*a*c^4*d*x*(-c^2*d*x^2+d)^(3/2)+5/2*a*c^4*d ^2*x*(-c^2*d*x^2+d)^(1/2)+5/2*a*c^4*d^3/(c^2*d)^(1/2)*arctan((c^2*d)^(1/2) *x/(-c^2*d*x^2+d)^(1/2))-1/24*b*(-d*(c^2*x^2-1))^(1/2)*(-c^2*x^2+1)^(1/2)/ (c^2*x^2-1)/x^3*(12*(-c^2*x^2+1)^(1/2)*arcsin(c*x)*x^4*c^4-6*c^5*x^5+30*c^ 3*x^3*arcsin(c*x)^2+56*I*arcsin(c*x)*x^3*c^3-56*ln((I*c*x+(-c^2*x^2+1)^(1/ 2))^2-1)*x^3*c^3+56*(-c^2*x^2+1)^(1/2)*arcsin(c*x)*x^2*c^2+3*c^3*x^3-8*arc sin(c*x)*(-c^2*x^2+1)^(1/2)-4*c*x)*d^2
\[ \int \frac {\left (d-c^2 d x^2\right )^{5/2} (a+b \arcsin (c x))}{x^4} \, dx=\int { \frac {{\left (-c^{2} d x^{2} + d\right )}^{\frac {5}{2}} {\left (b \arcsin \left (c x\right ) + a\right )}}{x^{4}} \,d x } \]
integral((a*c^4*d^2*x^4 - 2*a*c^2*d^2*x^2 + a*d^2 + (b*c^4*d^2*x^4 - 2*b*c ^2*d^2*x^2 + b*d^2)*arcsin(c*x))*sqrt(-c^2*d*x^2 + d)/x^4, x)
\[ \int \frac {\left (d-c^2 d x^2\right )^{5/2} (a+b \arcsin (c x))}{x^4} \, dx=\int \frac {\left (- d \left (c x - 1\right ) \left (c x + 1\right )\right )^{\frac {5}{2}} \left (a + b \operatorname {asin}{\left (c x \right )}\right )}{x^{4}}\, dx \]
\[ \int \frac {\left (d-c^2 d x^2\right )^{5/2} (a+b \arcsin (c x))}{x^4} \, dx=\int { \frac {{\left (-c^{2} d x^{2} + d\right )}^{\frac {5}{2}} {\left (b \arcsin \left (c x\right ) + a\right )}}{x^{4}} \,d x } \]
b*sqrt(d)*integrate((c^4*d^2*x^4 - 2*c^2*d^2*x^2 + d^2)*sqrt(c*x + 1)*sqrt (-c*x + 1)*arctan2(c*x, sqrt(c*x + 1)*sqrt(-c*x + 1))/x^4, x) + 1/6*(10*(- c^2*d*x^2 + d)^(3/2)*c^4*d*x + 15*sqrt(-c^2*d*x^2 + d)*c^4*d^2*x + 15*c^3* d^(5/2)*arcsin(c*x) + 8*(-c^2*d*x^2 + d)^(5/2)*c^2/x - 2*(-c^2*d*x^2 + d)^ (7/2)/(d*x^3))*a
Exception generated. \[ \int \frac {\left (d-c^2 d x^2\right )^{5/2} (a+b \arcsin (c x))}{x^4} \, dx=\text {Exception raised: TypeError} \]
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. \[ \int \frac {\left (d-c^2 d x^2\right )^{5/2} (a+b \arcsin (c x))}{x^4} \, dx=\int \frac {\left (a+b\,\mathrm {asin}\left (c\,x\right )\right )\,{\left (d-c^2\,d\,x^2\right )}^{5/2}}{x^4} \,d x \]