Integrand size = 27, antiderivative size = 386 \[ \int \frac {\left (d-c^2 d x^2\right )^{5/2} (a+b \arcsin (c x))}{x^3} \, dx=-\frac {b c d^2 \sqrt {d-c^2 d x^2}}{2 x \sqrt {1-c^2 x^2}}+\frac {7 b c^3 d^2 x \sqrt {d-c^2 d x^2}}{3 \sqrt {1-c^2 x^2}}-\frac {b c^5 d^2 x^3 \sqrt {d-c^2 d x^2}}{9 \sqrt {1-c^2 x^2}}-\frac {5}{2} c^2 d^2 \sqrt {d-c^2 d x^2} (a+b \arcsin (c x))-\frac {5}{6} c^2 d \left (d-c^2 d x^2\right )^{3/2} (a+b \arcsin (c x))-\frac {\left (d-c^2 d x^2\right )^{5/2} (a+b \arcsin (c x))}{2 x^2}+\frac {5 c^2 d^2 \sqrt {d-c^2 d x^2} (a+b \arcsin (c x)) \text {arctanh}\left (e^{i \arcsin (c x)}\right )}{\sqrt {1-c^2 x^2}}-\frac {5 i b c^2 d^2 \sqrt {d-c^2 d x^2} \operatorname {PolyLog}\left (2,-e^{i \arcsin (c x)}\right )}{2 \sqrt {1-c^2 x^2}}+\frac {5 i b c^2 d^2 \sqrt {d-c^2 d x^2} \operatorname {PolyLog}\left (2,e^{i \arcsin (c x)}\right )}{2 \sqrt {1-c^2 x^2}} \]
-5/6*c^2*d*(-c^2*d*x^2+d)^(3/2)*(a+b*arcsin(c*x))-1/2*(-c^2*d*x^2+d)^(5/2) *(a+b*arcsin(c*x))/x^2-5/2*c^2*d^2*(a+b*arcsin(c*x))*(-c^2*d*x^2+d)^(1/2)- 1/2*b*c*d^2*(-c^2*d*x^2+d)^(1/2)/x/(-c^2*x^2+1)^(1/2)+7/3*b*c^3*d^2*x*(-c^ 2*d*x^2+d)^(1/2)/(-c^2*x^2+1)^(1/2)-1/9*b*c^5*d^2*x^3*(-c^2*d*x^2+d)^(1/2) /(-c^2*x^2+1)^(1/2)+5*c^2*d^2*(a+b*arcsin(c*x))*arctanh(I*c*x+(-c^2*x^2+1) ^(1/2))*(-c^2*d*x^2+d)^(1/2)/(-c^2*x^2+1)^(1/2)-5/2*I*b*c^2*d^2*polylog(2, -I*c*x-(-c^2*x^2+1)^(1/2))*(-c^2*d*x^2+d)^(1/2)/(-c^2*x^2+1)^(1/2)+5/2*I*b *c^2*d^2*polylog(2,I*c*x+(-c^2*x^2+1)^(1/2))*(-c^2*d*x^2+d)^(1/2)/(-c^2*x^ 2+1)^(1/2)
Time = 3.28 (sec) , antiderivative size = 484, normalized size of antiderivative = 1.25 \[ \int \frac {\left (d-c^2 d x^2\right )^{5/2} (a+b \arcsin (c x))}{x^3} \, dx=\frac {-12 a d^3 \left (-1+c^2 x^2\right ) \left (-3-14 c^2 x^2+2 c^4 x^4\right )-180 a c^2 d^{5/2} x^2 \sqrt {d-c^2 d x^2} \log (x)+180 a c^2 d^{5/2} x^2 \sqrt {d-c^2 d x^2} \log \left (d+\sqrt {d} \sqrt {d-c^2 d x^2}\right )+144 b c^2 d^3 x^2 \sqrt {1-c^2 x^2} \left (c x-\sqrt {1-c^2 x^2} \arcsin (c x)-\arcsin (c x) \left (\log \left (1-e^{i \arcsin (c x)}\right )-\log \left (1+e^{i \arcsin (c x)}\right )\right )-i \left (\operatorname {PolyLog}\left (2,-e^{i \arcsin (c x)}\right )-\operatorname {PolyLog}\left (2,e^{i \arcsin (c x)}\right )\right )\right )+2 b c^2 d^3 x^2 \sqrt {1-c^2 x^2} \left (9 c x-3 \arcsin (c x) \left (3 \sqrt {1-c^2 x^2}+\cos (3 \arcsin (c x))\right )+\sin (3 \arcsin (c x))\right )-9 b c^2 d^3 x^2 \sqrt {1-c^2 x^2} \left (2 \cot \left (\frac {1}{2} \arcsin (c x)\right )+\arcsin (c x) \csc ^2\left (\frac {1}{2} \arcsin (c x)\right )+4 \arcsin (c x) \log \left (1-e^{i \arcsin (c x)}\right )-4 \arcsin (c x) \log \left (1+e^{i \arcsin (c x)}\right )+4 i \operatorname {PolyLog}\left (2,-e^{i \arcsin (c x)}\right )-4 i \operatorname {PolyLog}\left (2,e^{i \arcsin (c x)}\right )-\arcsin (c x) \sec ^2\left (\frac {1}{2} \arcsin (c x)\right )+2 \tan \left (\frac {1}{2} \arcsin (c x)\right )\right )}{72 x^2 \sqrt {d-c^2 d x^2}} \]
(-12*a*d^3*(-1 + c^2*x^2)*(-3 - 14*c^2*x^2 + 2*c^4*x^4) - 180*a*c^2*d^(5/2 )*x^2*Sqrt[d - c^2*d*x^2]*Log[x] + 180*a*c^2*d^(5/2)*x^2*Sqrt[d - c^2*d*x^ 2]*Log[d + Sqrt[d]*Sqrt[d - c^2*d*x^2]] + 144*b*c^2*d^3*x^2*Sqrt[1 - c^2*x ^2]*(c*x - Sqrt[1 - c^2*x^2]*ArcSin[c*x] - ArcSin[c*x]*(Log[1 - E^(I*ArcSi n[c*x])] - Log[1 + E^(I*ArcSin[c*x])]) - I*(PolyLog[2, -E^(I*ArcSin[c*x])] - PolyLog[2, E^(I*ArcSin[c*x])])) + 2*b*c^2*d^3*x^2*Sqrt[1 - c^2*x^2]*(9* c*x - 3*ArcSin[c*x]*(3*Sqrt[1 - c^2*x^2] + Cos[3*ArcSin[c*x]]) + Sin[3*Arc Sin[c*x]]) - 9*b*c^2*d^3*x^2*Sqrt[1 - c^2*x^2]*(2*Cot[ArcSin[c*x]/2] + Arc Sin[c*x]*Csc[ArcSin[c*x]/2]^2 + 4*ArcSin[c*x]*Log[1 - E^(I*ArcSin[c*x])] - 4*ArcSin[c*x]*Log[1 + E^(I*ArcSin[c*x])] + (4*I)*PolyLog[2, -E^(I*ArcSin[ c*x])] - (4*I)*PolyLog[2, E^(I*ArcSin[c*x])] - ArcSin[c*x]*Sec[ArcSin[c*x] /2]^2 + 2*Tan[ArcSin[c*x]/2]))/(72*x^2*Sqrt[d - c^2*d*x^2])
Time = 1.30 (sec) , antiderivative size = 324, normalized size of antiderivative = 0.84, number of steps used = 13, number of rules used = 12, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.444, Rules used = {5200, 244, 2009, 5202, 2009, 5198, 24, 5218, 3042, 4671, 2715, 2838}
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.
| \(\displaystyle \int \frac {\left (d-c^2 d x^2\right )^{5/2} (a+b \arcsin (c x))}{x^3} \, dx\) |
| \(\Big \downarrow \) 5200 |
| \(\displaystyle -\frac {5}{2} c^2 d \int \frac {\left (d-c^2 d x^2\right )^{3/2} (a+b \arcsin (c x))}{x}dx+\frac {b c d^2 \sqrt {d-c^2 d x^2} \int \frac {\left (1-c^2 x^2\right )^2}{x^2}dx}{2 \sqrt {1-c^2 x^2}}-\frac {\left (d-c^2 d x^2\right )^{5/2} (a+b \arcsin (c x))}{2 x^2}\) |
| \(\Big \downarrow \) 244 |
| \(\displaystyle -\frac {5}{2} c^2 d \int \frac {\left (d-c^2 d x^2\right )^{3/2} (a+b \arcsin (c x))}{x}dx+\frac {b c d^2 \sqrt {d-c^2 d x^2} \int \left (x^2 c^4-2 c^2+\frac {1}{x^2}\right )dx}{2 \sqrt {1-c^2 x^2}}-\frac {\left (d-c^2 d x^2\right )^{5/2} (a+b \arcsin (c x))}{2 x^2}\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle -\frac {5}{2} c^2 d \int \frac {\left (d-c^2 d x^2\right )^{3/2} (a+b \arcsin (c x))}{x}dx-\frac {\left (d-c^2 d x^2\right )^{5/2} (a+b \arcsin (c x))}{2 x^2}+\frac {b c d^2 \left (\frac {c^4 x^3}{3}-2 c^2 x-\frac {1}{x}\right ) \sqrt {d-c^2 d x^2}}{2 \sqrt {1-c^2 x^2}}\) |
| \(\Big \downarrow \) 5202 |
| \(\displaystyle -\frac {5}{2} c^2 d \left (d \int \frac {\sqrt {d-c^2 d x^2} (a+b \arcsin (c x))}{x}dx-\frac {b c d \sqrt {d-c^2 d x^2} \int \left (1-c^2 x^2\right )dx}{3 \sqrt {1-c^2 x^2}}+\frac {1}{3} \left (d-c^2 d x^2\right )^{3/2} (a+b \arcsin (c x))\right )-\frac {\left (d-c^2 d x^2\right )^{5/2} (a+b \arcsin (c x))}{2 x^2}+\frac {b c d^2 \left (\frac {c^4 x^3}{3}-2 c^2 x-\frac {1}{x}\right ) \sqrt {d-c^2 d x^2}}{2 \sqrt {1-c^2 x^2}}\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle -\frac {5}{2} c^2 d \left (d \int \frac {\sqrt {d-c^2 d x^2} (a+b \arcsin (c x))}{x}dx+\frac {1}{3} \left (d-c^2 d x^2\right )^{3/2} (a+b \arcsin (c x))-\frac {b c d \left (x-\frac {c^2 x^3}{3}\right ) \sqrt {d-c^2 d x^2}}{3 \sqrt {1-c^2 x^2}}\right )-\frac {\left (d-c^2 d x^2\right )^{5/2} (a+b \arcsin (c x))}{2 x^2}+\frac {b c d^2 \left (\frac {c^4 x^3}{3}-2 c^2 x-\frac {1}{x}\right ) \sqrt {d-c^2 d x^2}}{2 \sqrt {1-c^2 x^2}}\) |
| \(\Big \downarrow \) 5198 |
| \(\displaystyle -\frac {5}{2} c^2 d \left (d \left (\frac {\sqrt {d-c^2 d x^2} \int \frac {a+b \arcsin (c x)}{x \sqrt {1-c^2 x^2}}dx}{\sqrt {1-c^2 x^2}}-\frac {b c \sqrt {d-c^2 d x^2} \int 1dx}{\sqrt {1-c^2 x^2}}+\sqrt {d-c^2 d x^2} (a+b \arcsin (c x))\right )+\frac {1}{3} \left (d-c^2 d x^2\right )^{3/2} (a+b \arcsin (c x))-\frac {b c d \left (x-\frac {c^2 x^3}{3}\right ) \sqrt {d-c^2 d x^2}}{3 \sqrt {1-c^2 x^2}}\right )-\frac {\left (d-c^2 d x^2\right )^{5/2} (a+b \arcsin (c x))}{2 x^2}+\frac {b c d^2 \left (\frac {c^4 x^3}{3}-2 c^2 x-\frac {1}{x}\right ) \sqrt {d-c^2 d x^2}}{2 \sqrt {1-c^2 x^2}}\) |
| \(\Big \downarrow \) 24 |
| \(\displaystyle -\frac {5}{2} c^2 d \left (d \left (\frac {\sqrt {d-c^2 d x^2} \int \frac {a+b \arcsin (c x)}{x \sqrt {1-c^2 x^2}}dx}{\sqrt {1-c^2 x^2}}+\sqrt {d-c^2 d x^2} (a+b \arcsin (c x))-\frac {b c x \sqrt {d-c^2 d x^2}}{\sqrt {1-c^2 x^2}}\right )+\frac {1}{3} \left (d-c^2 d x^2\right )^{3/2} (a+b \arcsin (c x))-\frac {b c d \left (x-\frac {c^2 x^3}{3}\right ) \sqrt {d-c^2 d x^2}}{3 \sqrt {1-c^2 x^2}}\right )-\frac {\left (d-c^2 d x^2\right )^{5/2} (a+b \arcsin (c x))}{2 x^2}+\frac {b c d^2 \left (\frac {c^4 x^3}{3}-2 c^2 x-\frac {1}{x}\right ) \sqrt {d-c^2 d x^2}}{2 \sqrt {1-c^2 x^2}}\) |
| \(\Big \downarrow \) 5218 |
| \(\displaystyle -\frac {5}{2} c^2 d \left (d \left (\frac {\sqrt {d-c^2 d x^2} \int \frac {a+b \arcsin (c x)}{c x}d\arcsin (c x)}{\sqrt {1-c^2 x^2}}+\sqrt {d-c^2 d x^2} (a+b \arcsin (c x))-\frac {b c x \sqrt {d-c^2 d x^2}}{\sqrt {1-c^2 x^2}}\right )+\frac {1}{3} \left (d-c^2 d x^2\right )^{3/2} (a+b \arcsin (c x))-\frac {b c d \left (x-\frac {c^2 x^3}{3}\right ) \sqrt {d-c^2 d x^2}}{3 \sqrt {1-c^2 x^2}}\right )-\frac {\left (d-c^2 d x^2\right )^{5/2} (a+b \arcsin (c x))}{2 x^2}+\frac {b c d^2 \left (\frac {c^4 x^3}{3}-2 c^2 x-\frac {1}{x}\right ) \sqrt {d-c^2 d x^2}}{2 \sqrt {1-c^2 x^2}}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle -\frac {5}{2} c^2 d \left (d \left (\frac {\sqrt {d-c^2 d x^2} \int (a+b \arcsin (c x)) \csc (\arcsin (c x))d\arcsin (c x)}{\sqrt {1-c^2 x^2}}+\sqrt {d-c^2 d x^2} (a+b \arcsin (c x))-\frac {b c x \sqrt {d-c^2 d x^2}}{\sqrt {1-c^2 x^2}}\right )+\frac {1}{3} \left (d-c^2 d x^2\right )^{3/2} (a+b \arcsin (c x))-\frac {b c d \left (x-\frac {c^2 x^3}{3}\right ) \sqrt {d-c^2 d x^2}}{3 \sqrt {1-c^2 x^2}}\right )-\frac {\left (d-c^2 d x^2\right )^{5/2} (a+b \arcsin (c x))}{2 x^2}+\frac {b c d^2 \left (\frac {c^4 x^3}{3}-2 c^2 x-\frac {1}{x}\right ) \sqrt {d-c^2 d x^2}}{2 \sqrt {1-c^2 x^2}}\) |
| \(\Big \downarrow \) 4671 |
| \(\displaystyle -\frac {5}{2} c^2 d \left (d \left (\frac {\sqrt {d-c^2 d x^2} \left (-b \int \log \left (1-e^{i \arcsin (c x)}\right )d\arcsin (c x)+b \int \log \left (1+e^{i \arcsin (c x)}\right )d\arcsin (c x)-2 \text {arctanh}\left (e^{i \arcsin (c x)}\right ) (a+b \arcsin (c x))\right )}{\sqrt {1-c^2 x^2}}+\sqrt {d-c^2 d x^2} (a+b \arcsin (c x))-\frac {b c x \sqrt {d-c^2 d x^2}}{\sqrt {1-c^2 x^2}}\right )+\frac {1}{3} \left (d-c^2 d x^2\right )^{3/2} (a+b \arcsin (c x))-\frac {b c d \left (x-\frac {c^2 x^3}{3}\right ) \sqrt {d-c^2 d x^2}}{3 \sqrt {1-c^2 x^2}}\right )-\frac {\left (d-c^2 d x^2\right )^{5/2} (a+b \arcsin (c x))}{2 x^2}+\frac {b c d^2 \left (\frac {c^4 x^3}{3}-2 c^2 x-\frac {1}{x}\right ) \sqrt {d-c^2 d x^2}}{2 \sqrt {1-c^2 x^2}}\) |
| \(\Big \downarrow \) 2715 |
| \(\displaystyle -\frac {5}{2} c^2 d \left (d \left (\frac {\sqrt {d-c^2 d x^2} \left (i b \int e^{-i \arcsin (c x)} \log \left (1-e^{i \arcsin (c x)}\right )de^{i \arcsin (c x)}-i b \int e^{-i \arcsin (c x)} \log \left (1+e^{i \arcsin (c x)}\right )de^{i \arcsin (c x)}-2 \text {arctanh}\left (e^{i \arcsin (c x)}\right ) (a+b \arcsin (c x))\right )}{\sqrt {1-c^2 x^2}}+\sqrt {d-c^2 d x^2} (a+b \arcsin (c x))-\frac {b c x \sqrt {d-c^2 d x^2}}{\sqrt {1-c^2 x^2}}\right )+\frac {1}{3} \left (d-c^2 d x^2\right )^{3/2} (a+b \arcsin (c x))-\frac {b c d \left (x-\frac {c^2 x^3}{3}\right ) \sqrt {d-c^2 d x^2}}{3 \sqrt {1-c^2 x^2}}\right )-\frac {\left (d-c^2 d x^2\right )^{5/2} (a+b \arcsin (c x))}{2 x^2}+\frac {b c d^2 \left (\frac {c^4 x^3}{3}-2 c^2 x-\frac {1}{x}\right ) \sqrt {d-c^2 d x^2}}{2 \sqrt {1-c^2 x^2}}\) |
| \(\Big \downarrow \) 2838 |
| \(\displaystyle -\frac {5}{2} c^2 d \left (d \left (\frac {\sqrt {d-c^2 d x^2} \left (-2 \text {arctanh}\left (e^{i \arcsin (c x)}\right ) (a+b \arcsin (c x))+i b \operatorname {PolyLog}\left (2,-e^{i \arcsin (c x)}\right )-i b \operatorname {PolyLog}\left (2,e^{i \arcsin (c x)}\right )\right )}{\sqrt {1-c^2 x^2}}+\sqrt {d-c^2 d x^2} (a+b \arcsin (c x))-\frac {b c x \sqrt {d-c^2 d x^2}}{\sqrt {1-c^2 x^2}}\right )+\frac {1}{3} \left (d-c^2 d x^2\right )^{3/2} (a+b \arcsin (c x))-\frac {b c d \left (x-\frac {c^2 x^3}{3}\right ) \sqrt {d-c^2 d x^2}}{3 \sqrt {1-c^2 x^2}}\right )-\frac {\left (d-c^2 d x^2\right )^{5/2} (a+b \arcsin (c x))}{2 x^2}+\frac {b c d^2 \left (\frac {c^4 x^3}{3}-2 c^2 x-\frac {1}{x}\right ) \sqrt {d-c^2 d x^2}}{2 \sqrt {1-c^2 x^2}}\) |
(b*c*d^2*Sqrt[d - c^2*d*x^2]*(-x^(-1) - 2*c^2*x + (c^4*x^3)/3))/(2*Sqrt[1 - c^2*x^2]) - ((d - c^2*d*x^2)^(5/2)*(a + b*ArcSin[c*x]))/(2*x^2) - (5*c^2 *d*(-1/3*(b*c*d*Sqrt[d - c^2*d*x^2]*(x - (c^2*x^3)/3))/Sqrt[1 - c^2*x^2] + ((d - c^2*d*x^2)^(3/2)*(a + b*ArcSin[c*x]))/3 + d*(-((b*c*x*Sqrt[d - c^2* d*x^2])/Sqrt[1 - c^2*x^2]) + Sqrt[d - c^2*d*x^2]*(a + b*ArcSin[c*x]) + (Sq rt[d - c^2*d*x^2]*(-2*(a + b*ArcSin[c*x])*ArcTanh[E^(I*ArcSin[c*x])] + I*b *PolyLog[2, -E^(I*ArcSin[c*x])] - I*b*PolyLog[2, E^(I*ArcSin[c*x])]))/Sqrt [1 - c^2*x^2])))/2
3.1.97.3.1 Defintions of rubi rules used
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[Log[(a_) + (b_.)*((F_)^((e_.)*((c_.) + (d_.)*(x_))))^(n_.)], x_Symbol] :> Simp[1/(d*e*n*Log[F]) Subst[Int[Log[a + b*x]/x, x], x, (F^(e*(c + d*x) ))^n], x] /; FreeQ[{F, a, b, c, d, e, n}, x] && GtQ[a, 0]
Int[Log[(c_.)*((d_) + (e_.)*(x_)^(n_.))]/(x_), x_Symbol] :> Simp[-PolyLog[2 , (-c)*e*x^n]/n, x] /; FreeQ[{c, d, e, n}, x] && EqQ[c*d, 1]
Int[csc[(e_.) + (f_.)*(x_)]*((c_.) + (d_.)*(x_))^(m_.), x_Symbol] :> Simp[- 2*(c + d*x)^m*(ArcTanh[E^(I*(e + f*x))]/f), x] + (-Simp[d*(m/f) Int[(c + d*x)^(m - 1)*Log[1 - E^(I*(e + f*x))], x], x] + Simp[d*(m/f) Int[(c + d*x )^(m - 1)*Log[1 + E^(I*(e + f*x))], x], x]) /; FreeQ[{c, d, e, f}, x] && IG tQ[m, 0]
Int[((a_.) + ArcSin[(c_.)*(x_)]*(b_.))^(n_.)*((f_.)*(x_))^(m_)*Sqrt[(d_) + (e_.)*(x_)^2], x_Symbol] :> Simp[(f*x)^(m + 1)*Sqrt[d + e*x^2]*((a + b*ArcS in[c*x])^n/(f*(m + 2))), x] + (Simp[(1/(m + 2))*Simp[Sqrt[d + e*x^2]/Sqrt[1 - c^2*x^2]] Int[(f*x)^m*((a + b*ArcSin[c*x])^n/Sqrt[1 - c^2*x^2]), x], x ] - Simp[b*c*(n/(f*(m + 2)))*Simp[Sqrt[d + e*x^2]/Sqrt[1 - c^2*x^2]] Int[ (f*x)^(m + 1)*(a + b*ArcSin[c*x])^(n - 1), x], x]) /; FreeQ[{a, b, c, d, e, f, m}, x] && EqQ[c^2*d + e, 0] && GtQ[n, 0] && (IGtQ[m, -2] || EqQ[n, 1])
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]
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 + 2*p + 1))), x] + (Simp[2*d*(p/(m + 2*p + 1)) Int[(f*x) ^m*(d + e*x^2)^(p - 1)*(a + b*ArcSin[c*x])^n, x], x] - Simp[b*c*(n/(f*(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, m}, x] && EqQ[c^2*d + e, 0] && GtQ[n, 0] && GtQ[p, 0] && !LtQ[m, -1]
Int[(((a_.) + ArcSin[(c_.)*(x_)]*(b_.))^(n_.)*(x_)^(m_))/Sqrt[(d_) + (e_.)* (x_)^2], x_Symbol] :> Simp[(1/c^(m + 1))*Simp[Sqrt[1 - c^2*x^2]/Sqrt[d + e* x^2]] Subst[Int[(a + b*x)^n*Sin[x]^m, x], x, ArcSin[c*x]], x] /; FreeQ[{a , b, c, d, e}, x] && EqQ[c^2*d + e, 0] && IGtQ[n, 0] && IntegerQ[m]
Time = 0.20 (sec) , antiderivative size = 659, normalized size of antiderivative = 1.71
| method | result | size |
| default | \(a \left (-\frac {\left (-c^{2} d \,x^{2}+d \right )^{\frac {7}{2}}}{2 d \,x^{2}}-\frac {5 c^{2} \left (\frac {\left (-c^{2} d \,x^{2}+d \right )^{\frac {5}{2}}}{5}+d \left (\frac {\left (-c^{2} d \,x^{2}+d \right )^{\frac {3}{2}}}{3}+d \left (\sqrt {-c^{2} d \,x^{2}+d}-\sqrt {d}\, \ln \left (\frac {2 d +2 \sqrt {d}\, \sqrt {-c^{2} d \,x^{2}+d}}{x}\right )\right )\right )\right )}{2}\right )+b \left (\frac {\sqrt {-d \left (c^{2} x^{2}-1\right )}\, \left (4 c^{4} x^{4}-5 c^{2} x^{2}-4 i \sqrt {-c^{2} x^{2}+1}\, x^{3} c^{3}+3 i c x \sqrt {-c^{2} x^{2}+1}+1\right ) \left (i+3 \arcsin \left (c x \right )\right ) d^{2} c^{2}}{72 c^{2} x^{2}-72}-\frac {9 \sqrt {-d \left (c^{2} x^{2}-1\right )}\, \left (c^{2} x^{2}-i c x \sqrt {-c^{2} x^{2}+1}-1\right ) \left (\arcsin \left (c x \right )+i\right ) d^{2} c^{2}}{8 \left (c^{2} x^{2}-1\right )}-\frac {9 \sqrt {-d \left (c^{2} x^{2}-1\right )}\, \left (i c x \sqrt {-c^{2} x^{2}+1}+c^{2} x^{2}-1\right ) \left (\arcsin \left (c x \right )-i\right ) d^{2} c^{2}}{8 \left (c^{2} x^{2}-1\right )}+\frac {\sqrt {-d \left (c^{2} x^{2}-1\right )}\, \left (4 i \sqrt {-c^{2} x^{2}+1}\, x^{3} c^{3}+4 c^{4} x^{4}-3 i c x \sqrt {-c^{2} x^{2}+1}-5 c^{2} x^{2}+1\right ) \left (-i+3 \arcsin \left (c x \right )\right ) d^{2} c^{2}}{72 c^{2} x^{2}-72}-\frac {d^{2} \left (c^{2} x^{2} \arcsin \left (c x \right )-c x \sqrt {-c^{2} x^{2}+1}-\arcsin \left (c x \right )\right ) \sqrt {-d \left (c^{2} x^{2}-1\right )}}{2 x^{2} \left (c^{2} x^{2}-1\right )}+\frac {5 i \sqrt {-d \left (c^{2} x^{2}-1\right )}\, \sqrt {-c^{2} x^{2}+1}\, \left (i \arcsin \left (c x \right ) \ln \left (1+i c x +\sqrt {-c^{2} x^{2}+1}\right )-i \arcsin \left (c x \right ) \ln \left (1-i c x -\sqrt {-c^{2} x^{2}+1}\right )-\operatorname {polylog}\left (2, i c x +\sqrt {-c^{2} x^{2}+1}\right )+\operatorname {polylog}\left (2, -i c x -\sqrt {-c^{2} x^{2}+1}\right )\right ) d^{2} c^{2}}{2 c^{2} x^{2}-2}\right )\) | \(659\) |
| parts | \(a \left (-\frac {\left (-c^{2} d \,x^{2}+d \right )^{\frac {7}{2}}}{2 d \,x^{2}}-\frac {5 c^{2} \left (\frac {\left (-c^{2} d \,x^{2}+d \right )^{\frac {5}{2}}}{5}+d \left (\frac {\left (-c^{2} d \,x^{2}+d \right )^{\frac {3}{2}}}{3}+d \left (\sqrt {-c^{2} d \,x^{2}+d}-\sqrt {d}\, \ln \left (\frac {2 d +2 \sqrt {d}\, \sqrt {-c^{2} d \,x^{2}+d}}{x}\right )\right )\right )\right )}{2}\right )+b \left (\frac {\sqrt {-d \left (c^{2} x^{2}-1\right )}\, \left (4 c^{4} x^{4}-5 c^{2} x^{2}-4 i \sqrt {-c^{2} x^{2}+1}\, x^{3} c^{3}+3 i c x \sqrt {-c^{2} x^{2}+1}+1\right ) \left (i+3 \arcsin \left (c x \right )\right ) d^{2} c^{2}}{72 c^{2} x^{2}-72}-\frac {9 \sqrt {-d \left (c^{2} x^{2}-1\right )}\, \left (c^{2} x^{2}-i c x \sqrt {-c^{2} x^{2}+1}-1\right ) \left (\arcsin \left (c x \right )+i\right ) d^{2} c^{2}}{8 \left (c^{2} x^{2}-1\right )}-\frac {9 \sqrt {-d \left (c^{2} x^{2}-1\right )}\, \left (i c x \sqrt {-c^{2} x^{2}+1}+c^{2} x^{2}-1\right ) \left (\arcsin \left (c x \right )-i\right ) d^{2} c^{2}}{8 \left (c^{2} x^{2}-1\right )}+\frac {\sqrt {-d \left (c^{2} x^{2}-1\right )}\, \left (4 i \sqrt {-c^{2} x^{2}+1}\, x^{3} c^{3}+4 c^{4} x^{4}-3 i c x \sqrt {-c^{2} x^{2}+1}-5 c^{2} x^{2}+1\right ) \left (-i+3 \arcsin \left (c x \right )\right ) d^{2} c^{2}}{72 c^{2} x^{2}-72}-\frac {d^{2} \left (c^{2} x^{2} \arcsin \left (c x \right )-c x \sqrt {-c^{2} x^{2}+1}-\arcsin \left (c x \right )\right ) \sqrt {-d \left (c^{2} x^{2}-1\right )}}{2 x^{2} \left (c^{2} x^{2}-1\right )}+\frac {5 i \sqrt {-d \left (c^{2} x^{2}-1\right )}\, \sqrt {-c^{2} x^{2}+1}\, \left (i \arcsin \left (c x \right ) \ln \left (1+i c x +\sqrt {-c^{2} x^{2}+1}\right )-i \arcsin \left (c x \right ) \ln \left (1-i c x -\sqrt {-c^{2} x^{2}+1}\right )-\operatorname {polylog}\left (2, i c x +\sqrt {-c^{2} x^{2}+1}\right )+\operatorname {polylog}\left (2, -i c x -\sqrt {-c^{2} x^{2}+1}\right )\right ) d^{2} c^{2}}{2 c^{2} x^{2}-2}\right )\) | \(659\) |
a*(-1/2/d/x^2*(-c^2*d*x^2+d)^(7/2)-5/2*c^2*(1/5*(-c^2*d*x^2+d)^(5/2)+d*(1/ 3*(-c^2*d*x^2+d)^(3/2)+d*((-c^2*d*x^2+d)^(1/2)-d^(1/2)*ln((2*d+2*d^(1/2)*( -c^2*d*x^2+d)^(1/2))/x)))))+b*(1/72*(-d*(c^2*x^2-1))^(1/2)*(4*c^4*x^4-5*c^ 2*x^2-4*I*c^3*x^3*(-c^2*x^2+1)^(1/2)+3*I*(-c^2*x^2+1)^(1/2)*x*c+1)*(I+3*ar csin(c*x))*d^2*c^2/(c^2*x^2-1)-9/8*(-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)*d^2*c^2/(c^2*x^2-1)-9/8*(-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)*d^2*c^2/(c^ 2*x^2-1)+1/72*(-d*(c^2*x^2-1))^(1/2)*(4*I*c^3*x^3*(-c^2*x^2+1)^(1/2)+4*c^4 *x^4-3*I*(-c^2*x^2+1)^(1/2)*x*c-5*c^2*x^2+1)*(-I+3*arcsin(c*x))*d^2*c^2/(c ^2*x^2-1)-1/2*d^2*(c^2*x^2*arcsin(c*x)-c*x*(-c^2*x^2+1)^(1/2)-arcsin(c*x)) *(-d*(c^2*x^2-1))^(1/2)/x^2/(c^2*x^2-1)+5*I*(-d*(c^2*x^2-1))^(1/2)*(-c^2*x ^2+1)^(1/2)*(I*arcsin(c*x)*ln(1+I*c*x+(-c^2*x^2+1)^(1/2))-I*arcsin(c*x)*ln (1-I*c*x-(-c^2*x^2+1)^(1/2))-polylog(2,I*c*x+(-c^2*x^2+1)^(1/2))+polylog(2 ,-I*c*x-(-c^2*x^2+1)^(1/2)))*d^2*c^2/(2*c^2*x^2-2))
\[ \int \frac {\left (d-c^2 d x^2\right )^{5/2} (a+b \arcsin (c x))}{x^3} \, dx=\int { \frac {{\left (-c^{2} d x^{2} + d\right )}^{\frac {5}{2}} {\left (b \arcsin \left (c x\right ) + a\right )}}{x^{3}} \,d x } \]
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^3, x)
\[ \int \frac {\left (d-c^2 d x^2\right )^{5/2} (a+b \arcsin (c x))}{x^3} \, dx=\int \frac {\left (- d \left (c x - 1\right ) \left (c x + 1\right )\right )^{\frac {5}{2}} \left (a + b \operatorname {asin}{\left (c x \right )}\right )}{x^{3}}\, dx \]
\[ \int \frac {\left (d-c^2 d x^2\right )^{5/2} (a+b \arcsin (c x))}{x^3} \, dx=\int { \frac {{\left (-c^{2} d x^{2} + d\right )}^{\frac {5}{2}} {\left (b \arcsin \left (c x\right ) + a\right )}}{x^{3}} \,d x } \]
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^3, x) + 1/6*(15*c^ 2*d^(5/2)*log(2*sqrt(-c^2*d*x^2 + d)*sqrt(d)/abs(x) + 2*d/abs(x)) - 3*(-c^ 2*d*x^2 + d)^(5/2)*c^2 - 5*(-c^2*d*x^2 + d)^(3/2)*c^2*d - 15*sqrt(-c^2*d*x ^2 + d)*c^2*d^2 - 3*(-c^2*d*x^2 + d)^(7/2)/(d*x^2))*a
Exception generated. \[ \int \frac {\left (d-c^2 d x^2\right )^{5/2} (a+b \arcsin (c x))}{x^3} \, dx=\text {Exception raised: TypeError} \]
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. \[ \int \frac {\left (d-c^2 d x^2\right )^{5/2} (a+b \arcsin (c x))}{x^3} \, dx=\int \frac {\left (a+b\,\mathrm {asin}\left (c\,x\right )\right )\,{\left (d-c^2\,d\,x^2\right )}^{5/2}}{x^3} \,d x \]