Integrand size = 27, antiderivative size = 293 \[ \int \frac {x^6 (a+b \arcsin (c x))}{\left (d-c^2 d x^2\right )^{5/2}} \, dx=-\frac {b}{6 c^7 d^2 \sqrt {1-c^2 x^2} \sqrt {d-c^2 d x^2}}+\frac {b x^2 \sqrt {1-c^2 x^2}}{4 c^5 d^2 \sqrt {d-c^2 d x^2}}+\frac {x^5 (a+b \arcsin (c x))}{3 c^2 d \left (d-c^2 d x^2\right )^{3/2}}-\frac {5 x^3 (a+b \arcsin (c x))}{3 c^4 d^2 \sqrt {d-c^2 d x^2}}-\frac {5 x \sqrt {d-c^2 d x^2} (a+b \arcsin (c x))}{2 c^6 d^3}+\frac {5 \sqrt {1-c^2 x^2} (a+b \arcsin (c x))^2}{4 b c^7 d^2 \sqrt {d-c^2 d x^2}}-\frac {7 b \sqrt {1-c^2 x^2} \log \left (1-c^2 x^2\right )}{6 c^7 d^2 \sqrt {d-c^2 d x^2}} \]
1/3*x^5*(a+b*arcsin(c*x))/c^2/d/(-c^2*d*x^2+d)^(3/2)-5/3*x^3*(a+b*arcsin(c *x))/c^4/d^2/(-c^2*d*x^2+d)^(1/2)-1/6*b/c^7/d^2/(-c^2*x^2+1)^(1/2)/(-c^2*d *x^2+d)^(1/2)+1/4*b*x^2*(-c^2*x^2+1)^(1/2)/c^5/d^2/(-c^2*d*x^2+d)^(1/2)+5/ 4*(a+b*arcsin(c*x))^2*(-c^2*x^2+1)^(1/2)/b/c^7/d^2/(-c^2*d*x^2+d)^(1/2)-7/ 6*b*ln(-c^2*x^2+1)*(-c^2*x^2+1)^(1/2)/c^7/d^2/(-c^2*d*x^2+d)^(1/2)-5/2*x*( a+b*arcsin(c*x))*(-c^2*d*x^2+d)^(1/2)/c^6/d^3
Time = 0.54 (sec) , antiderivative size = 253, normalized size of antiderivative = 0.86 \[ \int \frac {x^6 (a+b \arcsin (c x))}{\left (d-c^2 d x^2\right )^{5/2}} \, dx=\frac {4 b c \sqrt {d} x \left (15-20 c^2 x^2+3 c^4 x^4\right ) \arcsin (c x)-30 b \sqrt {d} \left (1-c^2 x^2\right )^{3/2} \arcsin (c x)^2-60 a \left (-1+c^2 x^2\right ) \sqrt {d-c^2 d x^2} \arctan \left (\frac {c x \sqrt {d-c^2 d x^2}}{\sqrt {d} \left (-1+c^2 x^2\right )}\right )+\sqrt {d} \left (4 a c x \left (15-20 c^2 x^2+3 c^4 x^4\right )+b \sqrt {1-c^2 x^2} \left (7-9 c^2 x^2+6 c^4 x^4\right )+28 b \left (1-c^2 x^2\right )^{3/2} \log \left (1-c^2 x^2\right )\right )}{24 c^7 d^{5/2} \left (-1+c^2 x^2\right ) \sqrt {d-c^2 d x^2}} \]
(4*b*c*Sqrt[d]*x*(15 - 20*c^2*x^2 + 3*c^4*x^4)*ArcSin[c*x] - 30*b*Sqrt[d]* (1 - c^2*x^2)^(3/2)*ArcSin[c*x]^2 - 60*a*(-1 + c^2*x^2)*Sqrt[d - c^2*d*x^2 ]*ArcTan[(c*x*Sqrt[d - c^2*d*x^2])/(Sqrt[d]*(-1 + c^2*x^2))] + Sqrt[d]*(4* a*c*x*(15 - 20*c^2*x^2 + 3*c^4*x^4) + b*Sqrt[1 - c^2*x^2]*(7 - 9*c^2*x^2 + 6*c^4*x^4) + 28*b*(1 - c^2*x^2)^(3/2)*Log[1 - c^2*x^2]))/(24*c^7*d^(5/2)* (-1 + c^2*x^2)*Sqrt[d - c^2*d*x^2])
Time = 1.05 (sec) , antiderivative size = 358, normalized size of antiderivative = 1.22, number of steps used = 12, number of rules used = 11, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.407, Rules used = {5206, 243, 49, 2009, 5206, 243, 49, 2009, 5210, 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.
| \(\displaystyle \int \frac {x^6 (a+b \arcsin (c x))}{\left (d-c^2 d x^2\right )^{5/2}} \, dx\) |
| \(\Big \downarrow \) 5206 |
| \(\displaystyle -\frac {5 \int \frac {x^4 (a+b \arcsin (c x))}{\left (d-c^2 d x^2\right )^{3/2}}dx}{3 c^2 d}-\frac {b \sqrt {1-c^2 x^2} \int \frac {x^5}{\left (1-c^2 x^2\right )^2}dx}{3 c d^2 \sqrt {d-c^2 d x^2}}+\frac {x^5 (a+b \arcsin (c x))}{3 c^2 d \left (d-c^2 d x^2\right )^{3/2}}\) |
| \(\Big \downarrow \) 243 |
| \(\displaystyle -\frac {5 \int \frac {x^4 (a+b \arcsin (c x))}{\left (d-c^2 d x^2\right )^{3/2}}dx}{3 c^2 d}-\frac {b \sqrt {1-c^2 x^2} \int \frac {x^4}{\left (1-c^2 x^2\right )^2}dx^2}{6 c d^2 \sqrt {d-c^2 d x^2}}+\frac {x^5 (a+b \arcsin (c x))}{3 c^2 d \left (d-c^2 d x^2\right )^{3/2}}\) |
| \(\Big \downarrow \) 49 |
| \(\displaystyle -\frac {5 \int \frac {x^4 (a+b \arcsin (c x))}{\left (d-c^2 d x^2\right )^{3/2}}dx}{3 c^2 d}-\frac {b \sqrt {1-c^2 x^2} \int \left (\frac {1}{c^4}+\frac {2}{c^4 \left (c^2 x^2-1\right )}+\frac {1}{c^4 \left (c^2 x^2-1\right )^2}\right )dx^2}{6 c d^2 \sqrt {d-c^2 d x^2}}+\frac {x^5 (a+b \arcsin (c x))}{3 c^2 d \left (d-c^2 d x^2\right )^{3/2}}\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle -\frac {5 \int \frac {x^4 (a+b \arcsin (c x))}{\left (d-c^2 d x^2\right )^{3/2}}dx}{3 c^2 d}+\frac {x^5 (a+b \arcsin (c x))}{3 c^2 d \left (d-c^2 d x^2\right )^{3/2}}-\frac {b \sqrt {1-c^2 x^2} \left (\frac {x^2}{c^4}+\frac {1}{c^6 \left (1-c^2 x^2\right )}+\frac {2 \log \left (1-c^2 x^2\right )}{c^6}\right )}{6 c d^2 \sqrt {d-c^2 d x^2}}\) |
| \(\Big \downarrow \) 5206 |
| \(\displaystyle -\frac {5 \left (-\frac {3 \int \frac {x^2 (a+b \arcsin (c x))}{\sqrt {d-c^2 d x^2}}dx}{c^2 d}-\frac {b \sqrt {1-c^2 x^2} \int \frac {x^3}{1-c^2 x^2}dx}{c d \sqrt {d-c^2 d x^2}}+\frac {x^3 (a+b \arcsin (c x))}{c^2 d \sqrt {d-c^2 d x^2}}\right )}{3 c^2 d}+\frac {x^5 (a+b \arcsin (c x))}{3 c^2 d \left (d-c^2 d x^2\right )^{3/2}}-\frac {b \sqrt {1-c^2 x^2} \left (\frac {x^2}{c^4}+\frac {1}{c^6 \left (1-c^2 x^2\right )}+\frac {2 \log \left (1-c^2 x^2\right )}{c^6}\right )}{6 c d^2 \sqrt {d-c^2 d x^2}}\) |
| \(\Big \downarrow \) 243 |
| \(\displaystyle -\frac {5 \left (-\frac {3 \int \frac {x^2 (a+b \arcsin (c x))}{\sqrt {d-c^2 d x^2}}dx}{c^2 d}-\frac {b \sqrt {1-c^2 x^2} \int \frac {x^2}{1-c^2 x^2}dx^2}{2 c d \sqrt {d-c^2 d x^2}}+\frac {x^3 (a+b \arcsin (c x))}{c^2 d \sqrt {d-c^2 d x^2}}\right )}{3 c^2 d}+\frac {x^5 (a+b \arcsin (c x))}{3 c^2 d \left (d-c^2 d x^2\right )^{3/2}}-\frac {b \sqrt {1-c^2 x^2} \left (\frac {x^2}{c^4}+\frac {1}{c^6 \left (1-c^2 x^2\right )}+\frac {2 \log \left (1-c^2 x^2\right )}{c^6}\right )}{6 c d^2 \sqrt {d-c^2 d x^2}}\) |
| \(\Big \downarrow \) 49 |
| \(\displaystyle -\frac {5 \left (-\frac {3 \int \frac {x^2 (a+b \arcsin (c x))}{\sqrt {d-c^2 d x^2}}dx}{c^2 d}-\frac {b \sqrt {1-c^2 x^2} \int \left (-\frac {1}{c^2}-\frac {1}{c^2 \left (c^2 x^2-1\right )}\right )dx^2}{2 c d \sqrt {d-c^2 d x^2}}+\frac {x^3 (a+b \arcsin (c x))}{c^2 d \sqrt {d-c^2 d x^2}}\right )}{3 c^2 d}+\frac {x^5 (a+b \arcsin (c x))}{3 c^2 d \left (d-c^2 d x^2\right )^{3/2}}-\frac {b \sqrt {1-c^2 x^2} \left (\frac {x^2}{c^4}+\frac {1}{c^6 \left (1-c^2 x^2\right )}+\frac {2 \log \left (1-c^2 x^2\right )}{c^6}\right )}{6 c d^2 \sqrt {d-c^2 d x^2}}\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle -\frac {5 \left (-\frac {3 \int \frac {x^2 (a+b \arcsin (c x))}{\sqrt {d-c^2 d x^2}}dx}{c^2 d}+\frac {x^3 (a+b \arcsin (c x))}{c^2 d \sqrt {d-c^2 d x^2}}-\frac {b \sqrt {1-c^2 x^2} \left (-\frac {x^2}{c^2}-\frac {\log \left (1-c^2 x^2\right )}{c^4}\right )}{2 c d \sqrt {d-c^2 d x^2}}\right )}{3 c^2 d}+\frac {x^5 (a+b \arcsin (c x))}{3 c^2 d \left (d-c^2 d x^2\right )^{3/2}}-\frac {b \sqrt {1-c^2 x^2} \left (\frac {x^2}{c^4}+\frac {1}{c^6 \left (1-c^2 x^2\right )}+\frac {2 \log \left (1-c^2 x^2\right )}{c^6}\right )}{6 c d^2 \sqrt {d-c^2 d x^2}}\) |
| \(\Big \downarrow \) 5210 |
| \(\displaystyle -\frac {5 \left (-\frac {3 \left (\frac {\int \frac {a+b \arcsin (c x)}{\sqrt {d-c^2 d x^2}}dx}{2 c^2}+\frac {b \sqrt {1-c^2 x^2} \int xdx}{2 c \sqrt {d-c^2 d x^2}}-\frac {x \sqrt {d-c^2 d x^2} (a+b \arcsin (c x))}{2 c^2 d}\right )}{c^2 d}+\frac {x^3 (a+b \arcsin (c x))}{c^2 d \sqrt {d-c^2 d x^2}}-\frac {b \sqrt {1-c^2 x^2} \left (-\frac {x^2}{c^2}-\frac {\log \left (1-c^2 x^2\right )}{c^4}\right )}{2 c d \sqrt {d-c^2 d x^2}}\right )}{3 c^2 d}+\frac {x^5 (a+b \arcsin (c x))}{3 c^2 d \left (d-c^2 d x^2\right )^{3/2}}-\frac {b \sqrt {1-c^2 x^2} \left (\frac {x^2}{c^4}+\frac {1}{c^6 \left (1-c^2 x^2\right )}+\frac {2 \log \left (1-c^2 x^2\right )}{c^6}\right )}{6 c d^2 \sqrt {d-c^2 d x^2}}\) |
| \(\Big \downarrow \) 15 |
| \(\displaystyle -\frac {5 \left (-\frac {3 \left (\frac {\int \frac {a+b \arcsin (c x)}{\sqrt {d-c^2 d x^2}}dx}{2 c^2}-\frac {x \sqrt {d-c^2 d x^2} (a+b \arcsin (c x))}{2 c^2 d}+\frac {b x^2 \sqrt {1-c^2 x^2}}{4 c \sqrt {d-c^2 d x^2}}\right )}{c^2 d}+\frac {x^3 (a+b \arcsin (c x))}{c^2 d \sqrt {d-c^2 d x^2}}-\frac {b \sqrt {1-c^2 x^2} \left (-\frac {x^2}{c^2}-\frac {\log \left (1-c^2 x^2\right )}{c^4}\right )}{2 c d \sqrt {d-c^2 d x^2}}\right )}{3 c^2 d}+\frac {x^5 (a+b \arcsin (c x))}{3 c^2 d \left (d-c^2 d x^2\right )^{3/2}}-\frac {b \sqrt {1-c^2 x^2} \left (\frac {x^2}{c^4}+\frac {1}{c^6 \left (1-c^2 x^2\right )}+\frac {2 \log \left (1-c^2 x^2\right )}{c^6}\right )}{6 c d^2 \sqrt {d-c^2 d x^2}}\) |
| \(\Big \downarrow \) 5152 |
| \(\displaystyle \frac {x^5 (a+b \arcsin (c x))}{3 c^2 d \left (d-c^2 d x^2\right )^{3/2}}-\frac {5 \left (\frac {x^3 (a+b \arcsin (c x))}{c^2 d \sqrt {d-c^2 d x^2}}-\frac {3 \left (-\frac {x \sqrt {d-c^2 d x^2} (a+b \arcsin (c x))}{2 c^2 d}+\frac {\sqrt {1-c^2 x^2} (a+b \arcsin (c x))^2}{4 b c^3 \sqrt {d-c^2 d x^2}}+\frac {b x^2 \sqrt {1-c^2 x^2}}{4 c \sqrt {d-c^2 d x^2}}\right )}{c^2 d}-\frac {b \sqrt {1-c^2 x^2} \left (-\frac {x^2}{c^2}-\frac {\log \left (1-c^2 x^2\right )}{c^4}\right )}{2 c d \sqrt {d-c^2 d x^2}}\right )}{3 c^2 d}-\frac {b \sqrt {1-c^2 x^2} \left (\frac {x^2}{c^4}+\frac {1}{c^6 \left (1-c^2 x^2\right )}+\frac {2 \log \left (1-c^2 x^2\right )}{c^6}\right )}{6 c d^2 \sqrt {d-c^2 d x^2}}\) |
(x^5*(a + b*ArcSin[c*x]))/(3*c^2*d*(d - c^2*d*x^2)^(3/2)) - (b*Sqrt[1 - c^ 2*x^2]*(x^2/c^4 + 1/(c^6*(1 - c^2*x^2)) + (2*Log[1 - c^2*x^2])/c^6))/(6*c* d^2*Sqrt[d - c^2*d*x^2]) - (5*((x^3*(a + b*ArcSin[c*x]))/(c^2*d*Sqrt[d - c ^2*d*x^2]) - (3*((b*x^2*Sqrt[1 - c^2*x^2])/(4*c*Sqrt[d - c^2*d*x^2]) - (x* Sqrt[d - c^2*d*x^2]*(a + b*ArcSin[c*x]))/(2*c^2*d) + (Sqrt[1 - c^2*x^2]*(a + b*ArcSin[c*x])^2)/(4*b*c^3*Sqrt[d - c^2*d*x^2])))/(c^2*d) - (b*Sqrt[1 - c^2*x^2]*(-(x^2/c^2) - Log[1 - c^2*x^2]/c^4))/(2*c*d*Sqrt[d - c^2*d*x^2]) ))/(3*c^2*d)
3.2.29.3.1 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[((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_.)*((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/(2*e*(p + 1))), x] + (-Simp[f^2*((m - 1)/(2*e*(p + 1))) Int[(f*x)^(m - 2)*(d + e*x^2)^(p + 1)*(a + b*ArcSin[c*x])^n, x], x] + Simp [b*f*(n/(2*c*(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}, x] && EqQ[c^2*d + e, 0] && GtQ[n, 0] && LtQ[p, -1] && IG tQ[m, 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 = 0.23 (sec) , antiderivative size = 426, normalized size of antiderivative = 1.45
| method | result | size |
| default | \(-\frac {a \,x^{5}}{2 c^{2} d \left (-c^{2} d \,x^{2}+d \right )^{\frac {3}{2}}}+\frac {5 a \,x^{3}}{6 c^{4} d \left (-c^{2} d \,x^{2}+d \right )^{\frac {3}{2}}}-\frac {5 a x}{2 c^{6} d^{2} \sqrt {-c^{2} d \,x^{2}+d}}+\frac {5 a \arctan \left (\frac {\sqrt {c^{2} d}\, x}{\sqrt {-c^{2} d \,x^{2}+d}}\right )}{2 c^{6} d^{2} \sqrt {c^{2} d}}-\frac {b \sqrt {-d \left (c^{2} x^{2}-1\right )}\, \sqrt {-c^{2} x^{2}+1}\, \left (-12 \sqrt {-c^{2} x^{2}+1}\, \arcsin \left (c x \right ) c^{5} x^{5}+6 c^{6} x^{6}+30 \arcsin \left (c x \right )^{2} x^{4} c^{4}+56 i \arcsin \left (c x \right ) x^{4} c^{4}-56 \ln \left (1+\left (i c x +\sqrt {-c^{2} x^{2}+1}\right )^{2}\right ) x^{4} c^{4}+80 \sqrt {-c^{2} x^{2}+1}\, \arcsin \left (c x \right ) c^{3} x^{3}-15 c^{4} x^{4}-60 \arcsin \left (c x \right )^{2} x^{2} c^{2}+56 i \arcsin \left (c x \right )+112 \ln \left (1+\left (i c x +\sqrt {-c^{2} x^{2}+1}\right )^{2}\right ) x^{2} c^{2}-60 \sqrt {-c^{2} x^{2}+1}\, \arcsin \left (c x \right ) x c +16 c^{2} x^{2}+30 \arcsin \left (c x \right )^{2}-112 i \arcsin \left (c x \right ) x^{2} c^{2}-56 \ln \left (1+\left (i c x +\sqrt {-c^{2} x^{2}+1}\right )^{2}\right )-7\right )}{24 d^{3} \left (c^{6} x^{6}-3 c^{4} x^{4}+3 c^{2} x^{2}-1\right ) c^{7}}\) | \(426\) |
| parts | \(-\frac {a \,x^{5}}{2 c^{2} d \left (-c^{2} d \,x^{2}+d \right )^{\frac {3}{2}}}+\frac {5 a \,x^{3}}{6 c^{4} d \left (-c^{2} d \,x^{2}+d \right )^{\frac {3}{2}}}-\frac {5 a x}{2 c^{6} d^{2} \sqrt {-c^{2} d \,x^{2}+d}}+\frac {5 a \arctan \left (\frac {\sqrt {c^{2} d}\, x}{\sqrt {-c^{2} d \,x^{2}+d}}\right )}{2 c^{6} d^{2} \sqrt {c^{2} d}}-\frac {b \sqrt {-d \left (c^{2} x^{2}-1\right )}\, \sqrt {-c^{2} x^{2}+1}\, \left (-12 \sqrt {-c^{2} x^{2}+1}\, \arcsin \left (c x \right ) c^{5} x^{5}+6 c^{6} x^{6}+30 \arcsin \left (c x \right )^{2} x^{4} c^{4}+56 i \arcsin \left (c x \right ) x^{4} c^{4}-56 \ln \left (1+\left (i c x +\sqrt {-c^{2} x^{2}+1}\right )^{2}\right ) x^{4} c^{4}+80 \sqrt {-c^{2} x^{2}+1}\, \arcsin \left (c x \right ) c^{3} x^{3}-15 c^{4} x^{4}-60 \arcsin \left (c x \right )^{2} x^{2} c^{2}+56 i \arcsin \left (c x \right )+112 \ln \left (1+\left (i c x +\sqrt {-c^{2} x^{2}+1}\right )^{2}\right ) x^{2} c^{2}-60 \sqrt {-c^{2} x^{2}+1}\, \arcsin \left (c x \right ) x c +16 c^{2} x^{2}+30 \arcsin \left (c x \right )^{2}-112 i \arcsin \left (c x \right ) x^{2} c^{2}-56 \ln \left (1+\left (i c x +\sqrt {-c^{2} x^{2}+1}\right )^{2}\right )-7\right )}{24 d^{3} \left (c^{6} x^{6}-3 c^{4} x^{4}+3 c^{2} x^{2}-1\right ) c^{7}}\) | \(426\) |
-1/2*a*x^5/c^2/d/(-c^2*d*x^2+d)^(3/2)+5/6*a/c^4*x^3/d/(-c^2*d*x^2+d)^(3/2) -5/2*a/c^6/d^2*x/(-c^2*d*x^2+d)^(1/2)+5/2*a/c^6/d^2/(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)*(-12*(-c^2*x^2+1)^(1/2)*arcsin(c*x)*c^5*x^5+6*c^6*x^6+30*arcsi n(c*x)^2*x^4*c^4+56*I*arcsin(c*x)*x^4*c^4-56*ln(1+(I*c*x+(-c^2*x^2+1)^(1/2 ))^2)*x^4*c^4+80*(-c^2*x^2+1)^(1/2)*arcsin(c*x)*c^3*x^3-15*c^4*x^4-60*arcs in(c*x)^2*x^2*c^2+56*I*arcsin(c*x)+112*ln(1+(I*c*x+(-c^2*x^2+1)^(1/2))^2)* x^2*c^2-60*(-c^2*x^2+1)^(1/2)*arcsin(c*x)*x*c+16*c^2*x^2+30*arcsin(c*x)^2- 112*I*arcsin(c*x)*x^2*c^2-56*ln(1+(I*c*x+(-c^2*x^2+1)^(1/2))^2)-7)/d^3/(c^ 6*x^6-3*c^4*x^4+3*c^2*x^2-1)/c^7
\[ \int \frac {x^6 (a+b \arcsin (c x))}{\left (d-c^2 d x^2\right )^{5/2}} \, dx=\int { \frac {{\left (b \arcsin \left (c x\right ) + a\right )} x^{6}}{{\left (-c^{2} d x^{2} + d\right )}^{\frac {5}{2}}} \,d x } \]
integral(-(b*x^6*arcsin(c*x) + a*x^6)*sqrt(-c^2*d*x^2 + d)/(c^6*d^3*x^6 - 3*c^4*d^3*x^4 + 3*c^2*d^3*x^2 - d^3), x)
\[ \int \frac {x^6 (a+b \arcsin (c x))}{\left (d-c^2 d x^2\right )^{5/2}} \, dx=\int \frac {x^{6} \left (a + b \operatorname {asin}{\left (c x \right )}\right )}{\left (- d \left (c x - 1\right ) \left (c x + 1\right )\right )^{\frac {5}{2}}}\, dx \]
\[ \int \frac {x^6 (a+b \arcsin (c x))}{\left (d-c^2 d x^2\right )^{5/2}} \, dx=\int { \frac {{\left (b \arcsin \left (c x\right ) + a\right )} x^{6}}{{\left (-c^{2} d x^{2} + d\right )}^{\frac {5}{2}}} \,d x } \]
-1/6*a*(3*x^5/((-c^2*d*x^2 + d)^(3/2)*c^2*d) - 5*x*(3*x^2/((-c^2*d*x^2 + d )^(3/2)*c^2*d) - 2/((-c^2*d*x^2 + d)^(3/2)*c^4*d))/c^2 + 5*x/(sqrt(-c^2*d* x^2 + d)*c^6*d^2) - 15*arcsin(c*x)/(c^7*d^(5/2))) + b*integrate(x^6*arctan 2(c*x, sqrt(c*x + 1)*sqrt(-c*x + 1))/((c^4*d^2*x^4 - 2*c^2*d^2*x^2 + d^2)* sqrt(c*x + 1)*sqrt(-c*x + 1)), x)/sqrt(d)
Exception generated. \[ \int \frac {x^6 (a+b \arcsin (c x))}{\left (d-c^2 d x^2\right )^{5/2}} \, dx=\text {Exception raised: TypeError} \]
Exception raised: TypeError >> an error occurred running a Giac command:IN PUT:sage2:=int(sage0,sageVARx):;OUTPUT:index.cc index_m i_lex_is_greater E rror: Bad Argument Value
Timed out. \[ \int \frac {x^6 (a+b \arcsin (c x))}{\left (d-c^2 d x^2\right )^{5/2}} \, dx=\int \frac {x^6\,\left (a+b\,\mathrm {asin}\left (c\,x\right )\right )}{{\left (d-c^2\,d\,x^2\right )}^{5/2}} \,d x \]