Integrand size = 35, antiderivative size = 295 \[ \int \frac {x^2 (a+b \arcsin (c x))^2}{(d+c d x)^{3/2} (e-c e x)^{3/2}} \, dx=\frac {x (a+b \arcsin (c x))^2}{c^2 d e \sqrt {d+c d x} \sqrt {e-c e x}}-\frac {i \sqrt {1-c^2 x^2} (a+b \arcsin (c x))^2}{c^3 d e \sqrt {d+c d x} \sqrt {e-c e x}}-\frac {\sqrt {1-c^2 x^2} (a+b \arcsin (c x))^3}{3 b c^3 d e \sqrt {d+c d x} \sqrt {e-c e x}}+\frac {2 b \sqrt {1-c^2 x^2} (a+b \arcsin (c x)) \log \left (1+e^{2 i \arcsin (c x)}\right )}{c^3 d e \sqrt {d+c d x} \sqrt {e-c e x}}-\frac {i b^2 \sqrt {1-c^2 x^2} \operatorname {PolyLog}\left (2,-e^{2 i \arcsin (c x)}\right )}{c^3 d e \sqrt {d+c d x} \sqrt {e-c e x}} \] Output:
x*(a+b*arcsin(c*x))^2/c^2/d/e/(c*d*x+d)^(1/2)/(-c*e*x+e)^(1/2)-I*(-c^2*x^2 +1)^(1/2)*(a+b*arcsin(c*x))^2/c^3/d/e/(c*d*x+d)^(1/2)/(-c*e*x+e)^(1/2)-1/3 *(-c^2*x^2+1)^(1/2)*(a+b*arcsin(c*x))^3/b/c^3/d/e/(c*d*x+d)^(1/2)/(-c*e*x+ e)^(1/2)+2*b*(-c^2*x^2+1)^(1/2)*(a+b*arcsin(c*x))*ln(1+(I*c*x+(-c^2*x^2+1) ^(1/2))^2)/c^3/d/e/(c*d*x+d)^(1/2)/(-c*e*x+e)^(1/2)-I*b^2*(-c^2*x^2+1)^(1/ 2)*polylog(2,-(I*c*x+(-c^2*x^2+1)^(1/2))^2)/c^3/d/e/(c*d*x+d)^(1/2)/(-c*e* x+e)^(1/2)
Both result and optimal contain complex but leaf count is larger than twice the leaf count of optimal. \(636\) vs. \(2(295)=590\).
Time = 3.15 (sec) , antiderivative size = 636, normalized size of antiderivative = 2.16 \[ \int \frac {x^2 (a+b \arcsin (c x))^2}{(d+c d x)^{3/2} (e-c e x)^{3/2}} \, dx=\frac {3 a^2 c \sqrt {d} e x+3 a^2 \sqrt {e} \sqrt {d+c d x} \sqrt {e-c e x} \arctan \left (\frac {c x \sqrt {d+c d x} \sqrt {e-c e x}}{\sqrt {d} \sqrt {e} \left (-1+c^2 x^2\right )}\right )+3 a b \sqrt {d} e \left (2 c x \arcsin (c x)+\sqrt {1-c^2 x^2} \left (-\arcsin (c x)^2+2 \left (\log \left (\cos \left (\frac {1}{2} \arcsin (c x)\right )-\sin \left (\frac {1}{2} \arcsin (c x)\right )\right )+\log \left (\cos \left (\frac {1}{2} \arcsin (c x)\right )+\sin \left (\frac {1}{2} \arcsin (c x)\right )\right )\right )\right )\right )+b^2 \sqrt {d} e \left (6 i \pi \sqrt {1-c^2 x^2} \arcsin (c x)+3 c x \arcsin (c x)^2-3 i \sqrt {1-c^2 x^2} \arcsin (c x)^2-\sqrt {1-c^2 x^2} \arcsin (c x)^3+12 \pi \sqrt {1-c^2 x^2} \log \left (1+e^{-i \arcsin (c x)}\right )+3 \pi \sqrt {1-c^2 x^2} \log \left (1-i e^{i \arcsin (c x)}\right )+6 \sqrt {1-c^2 x^2} \arcsin (c x) \log \left (1-i e^{i \arcsin (c x)}\right )-3 \pi \sqrt {1-c^2 x^2} \log \left (1+i e^{i \arcsin (c x)}\right )+6 \sqrt {1-c^2 x^2} \arcsin (c x) \log \left (1+i e^{i \arcsin (c x)}\right )-12 \pi \sqrt {1-c^2 x^2} \log \left (\cos \left (\frac {1}{2} \arcsin (c x)\right )\right )+3 \pi \sqrt {1-c^2 x^2} \log \left (-\cos \left (\frac {1}{4} (\pi +2 \arcsin (c x))\right )\right )-3 \pi \sqrt {1-c^2 x^2} \log \left (\sin \left (\frac {1}{4} (\pi +2 \arcsin (c x))\right )\right )-6 i \sqrt {1-c^2 x^2} \operatorname {PolyLog}\left (2,-i e^{i \arcsin (c x)}\right )-6 i \sqrt {1-c^2 x^2} \operatorname {PolyLog}\left (2,i e^{i \arcsin (c x)}\right )\right )}{3 c^3 d^{3/2} e^2 \sqrt {d+c d x} \sqrt {e-c e x}} \] Input:
Integrate[(x^2*(a + b*ArcSin[c*x])^2)/((d + c*d*x)^(3/2)*(e - c*e*x)^(3/2) ),x]
Output:
(3*a^2*c*Sqrt[d]*e*x + 3*a^2*Sqrt[e]*Sqrt[d + c*d*x]*Sqrt[e - c*e*x]*ArcTa n[(c*x*Sqrt[d + c*d*x]*Sqrt[e - c*e*x])/(Sqrt[d]*Sqrt[e]*(-1 + c^2*x^2))] + 3*a*b*Sqrt[d]*e*(2*c*x*ArcSin[c*x] + Sqrt[1 - c^2*x^2]*(-ArcSin[c*x]^2 + 2*(Log[Cos[ArcSin[c*x]/2] - Sin[ArcSin[c*x]/2]] + Log[Cos[ArcSin[c*x]/2] + Sin[ArcSin[c*x]/2]]))) + b^2*Sqrt[d]*e*((6*I)*Pi*Sqrt[1 - c^2*x^2]*ArcSi n[c*x] + 3*c*x*ArcSin[c*x]^2 - (3*I)*Sqrt[1 - c^2*x^2]*ArcSin[c*x]^2 - Sqr t[1 - c^2*x^2]*ArcSin[c*x]^3 + 12*Pi*Sqrt[1 - c^2*x^2]*Log[1 + E^((-I)*Arc Sin[c*x])] + 3*Pi*Sqrt[1 - c^2*x^2]*Log[1 - I*E^(I*ArcSin[c*x])] + 6*Sqrt[ 1 - c^2*x^2]*ArcSin[c*x]*Log[1 - I*E^(I*ArcSin[c*x])] - 3*Pi*Sqrt[1 - c^2* x^2]*Log[1 + I*E^(I*ArcSin[c*x])] + 6*Sqrt[1 - c^2*x^2]*ArcSin[c*x]*Log[1 + I*E^(I*ArcSin[c*x])] - 12*Pi*Sqrt[1 - c^2*x^2]*Log[Cos[ArcSin[c*x]/2]] + 3*Pi*Sqrt[1 - c^2*x^2]*Log[-Cos[(Pi + 2*ArcSin[c*x])/4]] - 3*Pi*Sqrt[1 - c^2*x^2]*Log[Sin[(Pi + 2*ArcSin[c*x])/4]] - (6*I)*Sqrt[1 - c^2*x^2]*PolyLo g[2, (-I)*E^(I*ArcSin[c*x])] - (6*I)*Sqrt[1 - c^2*x^2]*PolyLog[2, I*E^(I*A rcSin[c*x])]))/(3*c^3*d^(3/2)*e^2*Sqrt[d + c*d*x]*Sqrt[e - c*e*x])
Time = 1.61 (sec) , antiderivative size = 169, normalized size of antiderivative = 0.57, number of steps used = 10, number of rules used = 9, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.257, Rules used = {5238, 5206, 5152, 5180, 3042, 4202, 2620, 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.
| \(\displaystyle \int \frac {x^2 (a+b \arcsin (c x))^2}{(c d x+d)^{3/2} (e-c e x)^{3/2}} \, dx\) |
| \(\Big \downarrow \) 5238 |
| \(\displaystyle \frac {\sqrt {1-c^2 x^2} \int \frac {x^2 (a+b \arcsin (c x))^2}{\left (1-c^2 x^2\right )^{3/2}}dx}{d e \sqrt {c d x+d} \sqrt {e-c e x}}\) |
| \(\Big \downarrow \) 5206 |
| \(\displaystyle \frac {\sqrt {1-c^2 x^2} \left (-\frac {2 b \int \frac {x (a+b \arcsin (c x))}{1-c^2 x^2}dx}{c}-\frac {\int \frac {(a+b \arcsin (c x))^2}{\sqrt {1-c^2 x^2}}dx}{c^2}+\frac {x (a+b \arcsin (c x))^2}{c^2 \sqrt {1-c^2 x^2}}\right )}{d e \sqrt {c d x+d} \sqrt {e-c e x}}\) |
| \(\Big \downarrow \) 5152 |
| \(\displaystyle \frac {\sqrt {1-c^2 x^2} \left (-\frac {2 b \int \frac {x (a+b \arcsin (c x))}{1-c^2 x^2}dx}{c}-\frac {(a+b \arcsin (c x))^3}{3 b c^3}+\frac {x (a+b \arcsin (c x))^2}{c^2 \sqrt {1-c^2 x^2}}\right )}{d e \sqrt {c d x+d} \sqrt {e-c e x}}\) |
| \(\Big \downarrow \) 5180 |
| \(\displaystyle \frac {\sqrt {1-c^2 x^2} \left (-\frac {2 b \int \frac {c x (a+b \arcsin (c x))}{\sqrt {1-c^2 x^2}}d\arcsin (c x)}{c^3}-\frac {(a+b \arcsin (c x))^3}{3 b c^3}+\frac {x (a+b \arcsin (c x))^2}{c^2 \sqrt {1-c^2 x^2}}\right )}{d e \sqrt {c d x+d} \sqrt {e-c e x}}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\sqrt {1-c^2 x^2} \left (-\frac {2 b \int (a+b \arcsin (c x)) \tan (\arcsin (c x))d\arcsin (c x)}{c^3}-\frac {(a+b \arcsin (c x))^3}{3 b c^3}+\frac {x (a+b \arcsin (c x))^2}{c^2 \sqrt {1-c^2 x^2}}\right )}{d e \sqrt {c d x+d} \sqrt {e-c e x}}\) |
| \(\Big \downarrow \) 4202 |
| \(\displaystyle \frac {\sqrt {1-c^2 x^2} \left (-\frac {2 b \left (\frac {i (a+b \arcsin (c x))^2}{2 b}-2 i \int \frac {e^{2 i \arcsin (c x)} (a+b \arcsin (c x))}{1+e^{2 i \arcsin (c x)}}d\arcsin (c x)\right )}{c^3}-\frac {(a+b \arcsin (c x))^3}{3 b c^3}+\frac {x (a+b \arcsin (c x))^2}{c^2 \sqrt {1-c^2 x^2}}\right )}{d e \sqrt {c d x+d} \sqrt {e-c e x}}\) |
| \(\Big \downarrow \) 2620 |
| \(\displaystyle \frac {\sqrt {1-c^2 x^2} \left (-\frac {2 b \left (\frac {i (a+b \arcsin (c x))^2}{2 b}-2 i \left (\frac {1}{2} i b \int \log \left (1+e^{2 i \arcsin (c x)}\right )d\arcsin (c x)-\frac {1}{2} i \log \left (1+e^{2 i \arcsin (c x)}\right ) (a+b \arcsin (c x))\right )\right )}{c^3}-\frac {(a+b \arcsin (c x))^3}{3 b c^3}+\frac {x (a+b \arcsin (c x))^2}{c^2 \sqrt {1-c^2 x^2}}\right )}{d e \sqrt {c d x+d} \sqrt {e-c e x}}\) |
| \(\Big \downarrow \) 2715 |
| \(\displaystyle \frac {\sqrt {1-c^2 x^2} \left (-\frac {2 b \left (\frac {i (a+b \arcsin (c x))^2}{2 b}-2 i \left (\frac {1}{4} b \int e^{-2 i \arcsin (c x)} \log \left (1+e^{2 i \arcsin (c x)}\right )de^{2 i \arcsin (c x)}-\frac {1}{2} i \log \left (1+e^{2 i \arcsin (c x)}\right ) (a+b \arcsin (c x))\right )\right )}{c^3}-\frac {(a+b \arcsin (c x))^3}{3 b c^3}+\frac {x (a+b \arcsin (c x))^2}{c^2 \sqrt {1-c^2 x^2}}\right )}{d e \sqrt {c d x+d} \sqrt {e-c e x}}\) |
| \(\Big \downarrow \) 2838 |
| \(\displaystyle \frac {\sqrt {1-c^2 x^2} \left (-\frac {2 b \left (\frac {i (a+b \arcsin (c x))^2}{2 b}-2 i \left (-\frac {1}{2} i \log \left (1+e^{2 i \arcsin (c x)}\right ) (a+b \arcsin (c x))-\frac {1}{4} b \operatorname {PolyLog}\left (2,-e^{2 i \arcsin (c x)}\right )\right )\right )}{c^3}-\frac {(a+b \arcsin (c x))^3}{3 b c^3}+\frac {x (a+b \arcsin (c x))^2}{c^2 \sqrt {1-c^2 x^2}}\right )}{d e \sqrt {c d x+d} \sqrt {e-c e x}}\) |
Input:
Int[(x^2*(a + b*ArcSin[c*x])^2)/((d + c*d*x)^(3/2)*(e - c*e*x)^(3/2)),x]
Output:
(Sqrt[1 - c^2*x^2]*((x*(a + b*ArcSin[c*x])^2)/(c^2*Sqrt[1 - c^2*x^2]) - (a + b*ArcSin[c*x])^3/(3*b*c^3) - (2*b*(((I/2)*(a + b*ArcSin[c*x])^2)/b - (2 *I)*((-1/2*I)*(a + b*ArcSin[c*x])*Log[1 + E^((2*I)*ArcSin[c*x])] - (b*Poly Log[2, -E^((2*I)*ArcSin[c*x])])/4)))/c^3))/(d*e*Sqrt[d + c*d*x]*Sqrt[e - c *e*x])
Int[(((F_)^((g_.)*((e_.) + (f_.)*(x_))))^(n_.)*((c_.) + (d_.)*(x_))^(m_.))/ ((a_) + (b_.)*((F_)^((g_.)*((e_.) + (f_.)*(x_))))^(n_.)), x_Symbol] :> Simp [((c + d*x)^m/(b*f*g*n*Log[F]))*Log[1 + b*((F^(g*(e + f*x)))^n/a)], x] - Si mp[d*(m/(b*f*g*n*Log[F])) Int[(c + d*x)^(m - 1)*Log[1 + b*((F^(g*(e + f*x )))^n/a)], x], x] /; FreeQ[{F, a, b, c, d, e, f, g, n}, x] && IGtQ[m, 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[((c_.) + (d_.)*(x_))^(m_.)*tan[(e_.) + (f_.)*(x_)], x_Symbol] :> Simp[I *((c + d*x)^(m + 1)/(d*(m + 1))), x] - Simp[2*I Int[(c + d*x)^m*(E^(2*I*( e + f*x))/(1 + E^(2*I*(e + f*x)))), x], x] /; FreeQ[{c, d, e, f}, x] && IGt Q[m, 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_.)*(x_))/((d_) + (e_.)*(x_)^2), x_Symbol] :> Simp[-e^(-1) Subst[Int[(a + b*x)^n*Tan[x], x], x, ArcSin[c*x ]], x] /; FreeQ[{a, b, c, d, e}, x] && EqQ[c^2*d + e, 0] && IGtQ[n, 0]
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_.)*((h_.)*(x_))^(m_.)*((d_) + (e_ .)*(x_))^(p_)*((f_) + (g_.)*(x_))^(q_), x_Symbol] :> Simp[((-d^2)*(g/e))^In tPart[q]*(d + e*x)^FracPart[q]*((f + g*x)^FracPart[q]/(1 - c^2*x^2)^FracPar t[q]) Int[(h*x)^m*(d + e*x)^(p - q)*(1 - c^2*x^2)^q*(a + b*ArcSin[c*x])^n , x], x] /; FreeQ[{a, b, c, d, e, f, g, h, m, n}, x] && EqQ[e*f + d*g, 0] & & EqQ[c^2*d^2 - e^2, 0] && HalfIntegerQ[p, q] && GeQ[p - q, 0]
Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 724 vs. \(2 (283 ) = 566\).
Time = 6.33 (sec) , antiderivative size = 725, normalized size of antiderivative = 2.46
| method | result | size |
| default | \(\frac {a^{2} \left (-\arctan \left (\frac {\sqrt {c^{2} d e}\, x}{\sqrt {-d e \left (c x -1\right ) \left (c x +1\right )}}\right ) x^{2} c^{2} d e +\arctan \left (\frac {\sqrt {c^{2} d e}\, x}{\sqrt {-d e \left (c x -1\right ) \left (c x +1\right )}}\right ) e d -\sqrt {c^{2} d e}\, \sqrt {-d e \left (c^{2} x^{2}-1\right )}\, x \right ) \sqrt {d \left (c x +1\right )}\, \sqrt {-e \left (c x -1\right )}}{d^{2} e^{2} \left (c x +1\right ) \sqrt {c^{2} d e}\, \left (c x -1\right ) \sqrt {-d e \left (c^{2} x^{2}-1\right )}\, c^{2}}+\frac {b^{2} \left (\arcsin \left (c x \right )^{3} c^{2} x^{2}+3 i \arcsin \left (c x \right )^{2} x^{2} c^{2}-6 \arcsin \left (c x \right ) \ln \left (1+i \left (i c x +\sqrt {-c^{2} x^{2}+1}\right )\right ) x^{2} c^{2}-6 \arcsin \left (c x \right ) \ln \left (1-i \left (i c x +\sqrt {-c^{2} x^{2}+1}\right )\right ) x^{2} c^{2}+6 i \operatorname {polylog}\left (2, -i \left (i c x +\sqrt {-c^{2} x^{2}+1}\right )\right ) x^{2} c^{2}+6 i \operatorname {polylog}\left (2, i \left (i c x +\sqrt {-c^{2} x^{2}+1}\right )\right ) x^{2} c^{2}+3 \arcsin \left (c x \right )^{2} \sqrt {-c^{2} x^{2}+1}\, c x -\arcsin \left (c x \right )^{3}-3 i \arcsin \left (c x \right )^{2}+6 \arcsin \left (c x \right ) \ln \left (1+i \left (i c x +\sqrt {-c^{2} x^{2}+1}\right )\right )+6 \arcsin \left (c x \right ) \ln \left (1-i \left (i c x +\sqrt {-c^{2} x^{2}+1}\right )\right )-6 i \operatorname {polylog}\left (2, -i \left (i c x +\sqrt {-c^{2} x^{2}+1}\right )\right )-6 i \operatorname {polylog}\left (2, i \left (i c x +\sqrt {-c^{2} x^{2}+1}\right )\right )\right ) \sqrt {-e \left (c x -1\right )}\, \sqrt {d \left (c x +1\right )}\, \sqrt {-c^{2} x^{2}+1}}{3 \left (c^{4} x^{4}-2 c^{2} x^{2}+1\right ) c^{3} d^{2} e^{2}}+\frac {a b \left (\arcsin \left (c x \right )^{2} x^{2} c^{2}+2 i \arcsin \left (c x \right ) x^{2} c^{2}-2 \ln \left (1+\left (i c x +\sqrt {-c^{2} x^{2}+1}\right )^{2}\right ) x^{2} c^{2}+2 \arcsin \left (c x \right ) \sqrt {-c^{2} x^{2}+1}\, c x -\arcsin \left (c x \right )^{2}-2 i \arcsin \left (c x \right )+2 \ln \left (1+\left (i c x +\sqrt {-c^{2} x^{2}+1}\right )^{2}\right )\right ) \sqrt {-e \left (c x -1\right )}\, \sqrt {d \left (c x +1\right )}\, \sqrt {-c^{2} x^{2}+1}}{\left (c^{4} x^{4}-2 c^{2} x^{2}+1\right ) c^{3} d^{2} e^{2}}\) | \(725\) |
| parts | \(\frac {a^{2} \left (-\arctan \left (\frac {\sqrt {c^{2} d e}\, x}{\sqrt {-d e \left (c x -1\right ) \left (c x +1\right )}}\right ) x^{2} c^{2} d e +\arctan \left (\frac {\sqrt {c^{2} d e}\, x}{\sqrt {-d e \left (c x -1\right ) \left (c x +1\right )}}\right ) e d -\sqrt {c^{2} d e}\, \sqrt {-d e \left (c^{2} x^{2}-1\right )}\, x \right ) \sqrt {d \left (c x +1\right )}\, \sqrt {-e \left (c x -1\right )}}{d^{2} e^{2} \left (c x +1\right ) \sqrt {c^{2} d e}\, \left (c x -1\right ) \sqrt {-d e \left (c^{2} x^{2}-1\right )}\, c^{2}}+\frac {b^{2} \left (\arcsin \left (c x \right )^{3} c^{2} x^{2}+3 i \arcsin \left (c x \right )^{2} x^{2} c^{2}-6 \arcsin \left (c x \right ) \ln \left (1+i \left (i c x +\sqrt {-c^{2} x^{2}+1}\right )\right ) x^{2} c^{2}-6 \arcsin \left (c x \right ) \ln \left (1-i \left (i c x +\sqrt {-c^{2} x^{2}+1}\right )\right ) x^{2} c^{2}+6 i \operatorname {polylog}\left (2, -i \left (i c x +\sqrt {-c^{2} x^{2}+1}\right )\right ) x^{2} c^{2}+6 i \operatorname {polylog}\left (2, i \left (i c x +\sqrt {-c^{2} x^{2}+1}\right )\right ) x^{2} c^{2}+3 \arcsin \left (c x \right )^{2} \sqrt {-c^{2} x^{2}+1}\, c x -\arcsin \left (c x \right )^{3}-3 i \arcsin \left (c x \right )^{2}+6 \arcsin \left (c x \right ) \ln \left (1+i \left (i c x +\sqrt {-c^{2} x^{2}+1}\right )\right )+6 \arcsin \left (c x \right ) \ln \left (1-i \left (i c x +\sqrt {-c^{2} x^{2}+1}\right )\right )-6 i \operatorname {polylog}\left (2, -i \left (i c x +\sqrt {-c^{2} x^{2}+1}\right )\right )-6 i \operatorname {polylog}\left (2, i \left (i c x +\sqrt {-c^{2} x^{2}+1}\right )\right )\right ) \sqrt {-e \left (c x -1\right )}\, \sqrt {d \left (c x +1\right )}\, \sqrt {-c^{2} x^{2}+1}}{3 \left (c^{4} x^{4}-2 c^{2} x^{2}+1\right ) c^{3} d^{2} e^{2}}+\frac {a b \left (\arcsin \left (c x \right )^{2} x^{2} c^{2}+2 i \arcsin \left (c x \right ) x^{2} c^{2}-2 \ln \left (1+\left (i c x +\sqrt {-c^{2} x^{2}+1}\right )^{2}\right ) x^{2} c^{2}+2 \arcsin \left (c x \right ) \sqrt {-c^{2} x^{2}+1}\, c x -\arcsin \left (c x \right )^{2}-2 i \arcsin \left (c x \right )+2 \ln \left (1+\left (i c x +\sqrt {-c^{2} x^{2}+1}\right )^{2}\right )\right ) \sqrt {-e \left (c x -1\right )}\, \sqrt {d \left (c x +1\right )}\, \sqrt {-c^{2} x^{2}+1}}{\left (c^{4} x^{4}-2 c^{2} x^{2}+1\right ) c^{3} d^{2} e^{2}}\) | \(725\) |
Input:
int(x^2*(a+b*arcsin(c*x))^2/(c*d*x+d)^(3/2)/(-c*e*x+e)^(3/2),x,method=_RET URNVERBOSE)
Output:
a^2*(-arctan((c^2*d*e)^(1/2)*x/(-d*e*(c*x-1)*(c*x+1))^(1/2))*x^2*c^2*d*e+a rctan((c^2*d*e)^(1/2)*x/(-d*e*(c*x-1)*(c*x+1))^(1/2))*e*d-(c^2*d*e)^(1/2)* (-d*e*(c^2*x^2-1))^(1/2)*x)*(d*(c*x+1))^(1/2)*(-e*(c*x-1))^(1/2)/d^2/e^2/( c*x+1)/(c^2*d*e)^(1/2)/(c*x-1)/(-d*e*(c^2*x^2-1))^(1/2)/c^2+1/3*b^2*(arcsi n(c*x)^3*c^2*x^2+3*I*arcsin(c*x)^2*x^2*c^2-6*arcsin(c*x)*ln(1+I*(I*c*x+(-c ^2*x^2+1)^(1/2)))*x^2*c^2-6*arcsin(c*x)*ln(1-I*(I*c*x+(-c^2*x^2+1)^(1/2))) *x^2*c^2+6*I*polylog(2,-I*(I*c*x+(-c^2*x^2+1)^(1/2)))*x^2*c^2+6*I*polylog( 2,I*(I*c*x+(-c^2*x^2+1)^(1/2)))*x^2*c^2+3*arcsin(c*x)^2*(-c^2*x^2+1)^(1/2) *c*x-arcsin(c*x)^3-3*I*arcsin(c*x)^2+6*arcsin(c*x)*ln(1+I*(I*c*x+(-c^2*x^2 +1)^(1/2)))+6*arcsin(c*x)*ln(1-I*(I*c*x+(-c^2*x^2+1)^(1/2)))-6*I*polylog(2 ,-I*(I*c*x+(-c^2*x^2+1)^(1/2)))-6*I*polylog(2,I*(I*c*x+(-c^2*x^2+1)^(1/2)) ))*(-e*(c*x-1))^(1/2)*(d*(c*x+1))^(1/2)*(-c^2*x^2+1)^(1/2)/(c^4*x^4-2*c^2* x^2+1)/c^3/d^2/e^2+a*b*(arcsin(c*x)^2*x^2*c^2+2*I*arcsin(c*x)*x^2*c^2-2*ln (1+(I*c*x+(-c^2*x^2+1)^(1/2))^2)*x^2*c^2+2*arcsin(c*x)*(-c^2*x^2+1)^(1/2)* c*x-arcsin(c*x)^2-2*I*arcsin(c*x)+2*ln(1+(I*c*x+(-c^2*x^2+1)^(1/2))^2))*(- e*(c*x-1))^(1/2)*(d*(c*x+1))^(1/2)*(-c^2*x^2+1)^(1/2)/(c^4*x^4-2*c^2*x^2+1 )/c^3/d^2/e^2
\[ \int \frac {x^2 (a+b \arcsin (c x))^2}{(d+c d x)^{3/2} (e-c e x)^{3/2}} \, dx=\int { \frac {{\left (b \arcsin \left (c x\right ) + a\right )}^{2} x^{2}}{{\left (c d x + d\right )}^{\frac {3}{2}} {\left (-c e x + e\right )}^{\frac {3}{2}}} \,d x } \] Input:
integrate(x^2*(a+b*arcsin(c*x))^2/(c*d*x+d)^(3/2)/(-c*e*x+e)^(3/2),x, algo rithm="fricas")
Output:
integral((b^2*x^2*arcsin(c*x)^2 + 2*a*b*x^2*arcsin(c*x) + a^2*x^2)*sqrt(c* d*x + d)*sqrt(-c*e*x + e)/(c^4*d^2*e^2*x^4 - 2*c^2*d^2*e^2*x^2 + d^2*e^2), x)
\[ \int \frac {x^2 (a+b \arcsin (c x))^2}{(d+c d x)^{3/2} (e-c e x)^{3/2}} \, dx=\int \frac {x^{2} \left (a + b \operatorname {asin}{\left (c x \right )}\right )^{2}}{\left (d \left (c x + 1\right )\right )^{\frac {3}{2}} \left (- e \left (c x - 1\right )\right )^{\frac {3}{2}}}\, dx \] Input:
integrate(x**2*(a+b*asin(c*x))**2/(c*d*x+d)**(3/2)/(-c*e*x+e)**(3/2),x)
Output:
Integral(x**2*(a + b*asin(c*x))**2/((d*(c*x + 1))**(3/2)*(-e*(c*x - 1))**( 3/2)), x)
Exception generated. \[ \int \frac {x^2 (a+b \arcsin (c x))^2}{(d+c d x)^{3/2} (e-c e x)^{3/2}} \, dx=\text {Exception raised: ValueError} \] Input:
integrate(x^2*(a+b*arcsin(c*x))^2/(c*d*x+d)^(3/2)/(-c*e*x+e)^(3/2),x, algo rithm="maxima")
Output:
Exception raised: ValueError >> Computation failed since Maxima requested additional constraints; using the 'assume' command before evaluation *may* help (example of legal syntax is 'assume(e>0)', see `assume?` for more de tails)Is e
\[ \int \frac {x^2 (a+b \arcsin (c x))^2}{(d+c d x)^{3/2} (e-c e x)^{3/2}} \, dx=\int { \frac {{\left (b \arcsin \left (c x\right ) + a\right )}^{2} x^{2}}{{\left (c d x + d\right )}^{\frac {3}{2}} {\left (-c e x + e\right )}^{\frac {3}{2}}} \,d x } \] Input:
integrate(x^2*(a+b*arcsin(c*x))^2/(c*d*x+d)^(3/2)/(-c*e*x+e)^(3/2),x, algo rithm="giac")
Output:
integrate((b*arcsin(c*x) + a)^2*x^2/((c*d*x + d)^(3/2)*(-c*e*x + e)^(3/2)) , x)
Timed out. \[ \int \frac {x^2 (a+b \arcsin (c x))^2}{(d+c d x)^{3/2} (e-c e x)^{3/2}} \, dx=\int \frac {x^2\,{\left (a+b\,\mathrm {asin}\left (c\,x\right )\right )}^2}{{\left (d+c\,d\,x\right )}^{3/2}\,{\left (e-c\,e\,x\right )}^{3/2}} \,d x \] Input:
int((x^2*(a + b*asin(c*x))^2)/((d + c*d*x)^(3/2)*(e - c*e*x)^(3/2)),x)
Output:
int((x^2*(a + b*asin(c*x))^2)/((d + c*d*x)^(3/2)*(e - c*e*x)^(3/2)), x)
\[ \int \frac {x^2 (a+b \arcsin (c x))^2}{(d+c d x)^{3/2} (e-c e x)^{3/2}} \, dx=\frac {2 \sqrt {c x +1}\, \sqrt {-c x +1}\, \mathit {asin} \left (\frac {\sqrt {-c x +1}}{\sqrt {2}}\right ) a^{2}-2 \sqrt {c x +1}\, \sqrt {-c x +1}\, \left (\int \frac {\mathit {asin} \left (c x \right ) x^{2}}{\sqrt {c x +1}\, \sqrt {-c x +1}\, c^{2} x^{2}-\sqrt {c x +1}\, \sqrt {-c x +1}}d x \right ) a b \,c^{3}-\sqrt {c x +1}\, \sqrt {-c x +1}\, \left (\int \frac {\mathit {asin} \left (c x \right )^{2} x^{2}}{\sqrt {c x +1}\, \sqrt {-c x +1}\, c^{2} x^{2}-\sqrt {c x +1}\, \sqrt {-c x +1}}d x \right ) b^{2} c^{3}+a^{2} c x}{\sqrt {e}\, \sqrt {d}\, \sqrt {c x +1}\, \sqrt {-c x +1}\, c^{3} d e} \] Input:
int(x^2*(a+b*asin(c*x))^2/(c*d*x+d)^(3/2)/(-c*e*x+e)^(3/2),x)
Output:
(2*sqrt(c*x + 1)*sqrt( - c*x + 1)*asin(sqrt( - c*x + 1)/sqrt(2))*a**2 - 2* sqrt(c*x + 1)*sqrt( - c*x + 1)*int((asin(c*x)*x**2)/(sqrt(c*x + 1)*sqrt( - c*x + 1)*c**2*x**2 - sqrt(c*x + 1)*sqrt( - c*x + 1)),x)*a*b*c**3 - sqrt(c *x + 1)*sqrt( - c*x + 1)*int((asin(c*x)**2*x**2)/(sqrt(c*x + 1)*sqrt( - c* x + 1)*c**2*x**2 - sqrt(c*x + 1)*sqrt( - c*x + 1)),x)*b**2*c**3 + a**2*c*x )/(sqrt(e)*sqrt(d)*sqrt(c*x + 1)*sqrt( - c*x + 1)*c**3*d*e)