Integrand size = 33, antiderivative size = 136 \[ \int \frac {x (a+b \arcsin (c x))^2}{\sqrt {d+c d x} \sqrt {e-c e x}} \, dx=\frac {2 b^2 \left (1-c^2 x^2\right )}{c^2 \sqrt {d+c d x} \sqrt {e-c e x}}+\frac {2 b x \sqrt {1-c^2 x^2} (a+b \arcsin (c x))}{c \sqrt {d+c d x} \sqrt {e-c e x}}-\frac {\left (1-c^2 x^2\right ) (a+b \arcsin (c x))^2}{c^2 \sqrt {d+c d x} \sqrt {e-c e x}} \] Output:
2*b^2*(-c^2*x^2+1)/c^2/(c*d*x+d)^(1/2)/(-c*e*x+e)^(1/2)+2*b*x*(-c^2*x^2+1) ^(1/2)*(a+b*arcsin(c*x))/c/(c*d*x+d)^(1/2)/(-c*e*x+e)^(1/2)-(-c^2*x^2+1)*( a+b*arcsin(c*x))^2/c^2/(c*d*x+d)^(1/2)/(-c*e*x+e)^(1/2)
Time = 1.59 (sec) , antiderivative size = 150, normalized size of antiderivative = 1.10 \[ \int \frac {x (a+b \arcsin (c x))^2}{\sqrt {d+c d x} \sqrt {e-c e x}} \, dx=-\frac {\sqrt {d+c d x} \sqrt {e-c e x} \left (2 a b c x \sqrt {1-c^2 x^2}+a^2 \left (-1+c^2 x^2\right )-2 b^2 \left (-1+c^2 x^2\right )+2 b \left (b c x \sqrt {1-c^2 x^2}+a \left (-1+c^2 x^2\right )\right ) \arcsin (c x)+b^2 \left (-1+c^2 x^2\right ) \arcsin (c x)^2\right )}{c^2 d e (-1+c x) (1+c x)} \] Input:
Integrate[(x*(a + b*ArcSin[c*x])^2)/(Sqrt[d + c*d*x]*Sqrt[e - c*e*x]),x]
Output:
-((Sqrt[d + c*d*x]*Sqrt[e - c*e*x]*(2*a*b*c*x*Sqrt[1 - c^2*x^2] + a^2*(-1 + c^2*x^2) - 2*b^2*(-1 + c^2*x^2) + 2*b*(b*c*x*Sqrt[1 - c^2*x^2] + a*(-1 + c^2*x^2))*ArcSin[c*x] + b^2*(-1 + c^2*x^2)*ArcSin[c*x]^2))/(c^2*d*e*(-1 + c*x)*(1 + c*x)))
Time = 0.66 (sec) , antiderivative size = 102, normalized size of antiderivative = 0.75, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.091, Rules used = {5238, 5182, 2009}
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.
| \(\displaystyle \int \frac {x (a+b \arcsin (c x))^2}{\sqrt {c d x+d} \sqrt {e-c e x}} \, dx\) |
| \(\Big \downarrow \) 5238 |
| \(\displaystyle \frac {\sqrt {1-c^2 x^2} \int \frac {x (a+b \arcsin (c x))^2}{\sqrt {1-c^2 x^2}}dx}{\sqrt {c d x+d} \sqrt {e-c e x}}\) |
| \(\Big \downarrow \) 5182 |
| \(\displaystyle \frac {\sqrt {1-c^2 x^2} \left (\frac {2 b \int (a+b \arcsin (c x))dx}{c}-\frac {\sqrt {1-c^2 x^2} (a+b \arcsin (c x))^2}{c^2}\right )}{\sqrt {c d x+d} \sqrt {e-c e x}}\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {\sqrt {1-c^2 x^2} \left (\frac {2 b \left (a x+b x \arcsin (c x)+\frac {b \sqrt {1-c^2 x^2}}{c}\right )}{c}-\frac {\sqrt {1-c^2 x^2} (a+b \arcsin (c x))^2}{c^2}\right )}{\sqrt {c d x+d} \sqrt {e-c e x}}\) |
Input:
Int[(x*(a + b*ArcSin[c*x])^2)/(Sqrt[d + c*d*x]*Sqrt[e - c*e*x]),x]
Output:
(Sqrt[1 - c^2*x^2]*(-((Sqrt[1 - c^2*x^2]*(a + b*ArcSin[c*x])^2)/c^2) + (2* b*(a*x + (b*Sqrt[1 - c^2*x^2])/c + b*x*ArcSin[c*x]))/c))/(Sqrt[d + c*d*x]* Sqrt[e - c*e*x])
Int[((a_.) + ArcSin[(c_.)*(x_)]*(b_.))^(n_.)*(x_)*((d_) + (e_.)*(x_)^2)^(p_ .), x_Symbol] :> Simp[(d + e*x^2)^(p + 1)*((a + b*ArcSin[c*x])^n/(2*e*(p + 1))), x] + Simp[b*(n/(2*c*(p + 1)))*Simp[(d + e*x^2)^p/(1 - c^2*x^2)^p] I nt[(1 - c^2*x^2)^(p + 1/2)*(a + b*ArcSin[c*x])^(n - 1), x], x] /; FreeQ[{a, b, c, d, e, p}, x] && EqQ[c^2*d + e, 0] && GtQ[n, 0] && NeQ[p, -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]
Result contains complex when optimal does not.
Time = 2.27 (sec) , antiderivative size = 363, normalized size of antiderivative = 2.67
| method | result | size |
| default | \(-\frac {a^{2} \sqrt {d \left (c x +1\right )}\, \sqrt {-e \left (c x -1\right )}}{c^{2} e d}+b^{2} \left (-\frac {\sqrt {-e \left (c x -1\right )}\, \sqrt {d \left (c x +1\right )}\, \left (c^{2} x^{2}-i \sqrt {-c^{2} x^{2}+1}\, c x -1\right ) \left (\arcsin \left (c x \right )^{2}-2+2 i \arcsin \left (c x \right )\right )}{2 e \left (c x +1\right ) c^{2} d \left (c x -1\right )}-\frac {\sqrt {-e \left (c x -1\right )}\, \sqrt {d \left (c x +1\right )}\, \left (i \sqrt {-c^{2} x^{2}+1}\, c x +c^{2} x^{2}-1\right ) \left (\arcsin \left (c x \right )^{2}-2-2 i \arcsin \left (c x \right )\right )}{2 c^{2} d e \left (c^{2} x^{2}-1\right )}\right )+2 a b \left (-\frac {\sqrt {-e \left (c x -1\right )}\, \sqrt {d \left (c x +1\right )}\, \left (c^{2} x^{2}-i \sqrt {-c^{2} x^{2}+1}\, c x -1\right ) \left (\arcsin \left (c x \right )+i\right )}{2 e \left (c x +1\right ) c^{2} d \left (c x -1\right )}-\frac {\sqrt {-e \left (c x -1\right )}\, \sqrt {d \left (c x +1\right )}\, \left (i \sqrt {-c^{2} x^{2}+1}\, c x +c^{2} x^{2}-1\right ) \left (\arcsin \left (c x \right )-i\right )}{2 c^{2} d e \left (c^{2} x^{2}-1\right )}\right )\) | \(363\) |
| parts | \(-\frac {a^{2} \sqrt {d \left (c x +1\right )}\, \sqrt {-e \left (c x -1\right )}}{c^{2} e d}+b^{2} \left (-\frac {\sqrt {-e \left (c x -1\right )}\, \sqrt {d \left (c x +1\right )}\, \left (c^{2} x^{2}-i \sqrt {-c^{2} x^{2}+1}\, c x -1\right ) \left (\arcsin \left (c x \right )^{2}-2+2 i \arcsin \left (c x \right )\right )}{2 e \left (c x +1\right ) c^{2} d \left (c x -1\right )}-\frac {\sqrt {-e \left (c x -1\right )}\, \sqrt {d \left (c x +1\right )}\, \left (i \sqrt {-c^{2} x^{2}+1}\, c x +c^{2} x^{2}-1\right ) \left (\arcsin \left (c x \right )^{2}-2-2 i \arcsin \left (c x \right )\right )}{2 c^{2} d e \left (c^{2} x^{2}-1\right )}\right )+2 a b \left (-\frac {\sqrt {-e \left (c x -1\right )}\, \sqrt {d \left (c x +1\right )}\, \left (c^{2} x^{2}-i \sqrt {-c^{2} x^{2}+1}\, c x -1\right ) \left (\arcsin \left (c x \right )+i\right )}{2 e \left (c x +1\right ) c^{2} d \left (c x -1\right )}-\frac {\sqrt {-e \left (c x -1\right )}\, \sqrt {d \left (c x +1\right )}\, \left (i \sqrt {-c^{2} x^{2}+1}\, c x +c^{2} x^{2}-1\right ) \left (\arcsin \left (c x \right )-i\right )}{2 c^{2} d e \left (c^{2} x^{2}-1\right )}\right )\) | \(363\) |
| orering | \(\frac {\left (c^{4} x^{4}-4 c^{2} x^{2}+2\right ) \left (a +b \arcsin \left (c x \right )\right )^{2}}{c^{4} x^{2} \sqrt {c d x +d}\, \sqrt {-c x e +e}}+\frac {2 \left (c x -1\right ) \left (c x +1\right ) \left (\frac {\left (a +b \arcsin \left (c x \right )\right )^{2}}{\sqrt {c d x +d}\, \sqrt {-c x e +e}}+\frac {2 x \left (a +b \arcsin \left (c x \right )\right ) b c}{\sqrt {c d x +d}\, \sqrt {-c x e +e}\, \sqrt {-c^{2} x^{2}+1}}-\frac {x \left (a +b \arcsin \left (c x \right )\right )^{2} c d}{2 \left (c d x +d \right )^{\frac {3}{2}} \sqrt {-c x e +e}}+\frac {x \left (a +b \arcsin \left (c x \right )\right )^{2} c e}{2 \sqrt {c d x +d}\, \left (-c x e +e \right )^{\frac {3}{2}}}\right )}{c^{4} x^{2}}+\frac {\left (c x -1\right )^{2} \left (c x +1\right )^{2} \left (\frac {4 \left (a +b \arcsin \left (c x \right )\right ) b c}{\sqrt {c d x +d}\, \sqrt {-c x e +e}\, \sqrt {-c^{2} x^{2}+1}}-\frac {\left (a +b \arcsin \left (c x \right )\right )^{2} c d}{\left (c d x +d \right )^{\frac {3}{2}} \sqrt {-c x e +e}}+\frac {\left (a +b \arcsin \left (c x \right )\right )^{2} c e}{\sqrt {c d x +d}\, \left (-c x e +e \right )^{\frac {3}{2}}}+\frac {2 x \,b^{2} c^{2}}{\left (-c^{2} x^{2}+1\right ) \sqrt {c d x +d}\, \sqrt {-c x e +e}}-\frac {2 x \left (a +b \arcsin \left (c x \right )\right ) b \,c^{2} d}{\left (c d x +d \right )^{\frac {3}{2}} \sqrt {-c x e +e}\, \sqrt {-c^{2} x^{2}+1}}+\frac {2 x \left (a +b \arcsin \left (c x \right )\right ) b \,c^{2} e}{\sqrt {c d x +d}\, \left (-c x e +e \right )^{\frac {3}{2}} \sqrt {-c^{2} x^{2}+1}}+\frac {2 x^{2} \left (a +b \arcsin \left (c x \right )\right ) b \,c^{3}}{\sqrt {c d x +d}\, \sqrt {-c x e +e}\, \left (-c^{2} x^{2}+1\right )^{\frac {3}{2}}}+\frac {3 x \left (a +b \arcsin \left (c x \right )\right )^{2} c^{2} d^{2}}{4 \left (c d x +d \right )^{\frac {5}{2}} \sqrt {-c x e +e}}-\frac {x \left (a +b \arcsin \left (c x \right )\right )^{2} c^{2} d e}{2 \left (c d x +d \right )^{\frac {3}{2}} \left (-c x e +e \right )^{\frac {3}{2}}}+\frac {3 x \left (a +b \arcsin \left (c x \right )\right )^{2} c^{2} e^{2}}{4 \sqrt {c d x +d}\, \left (-c x e +e \right )^{\frac {5}{2}}}\right )}{c^{4} x}\) | \(611\) |
Input:
int(x*(a+b*arcsin(c*x))^2/(c*d*x+d)^(1/2)/(-c*e*x+e)^(1/2),x,method=_RETUR NVERBOSE)
Output:
-a^2*(d*(c*x+1))^(1/2)*(-e*(c*x-1))^(1/2)/c^2/e/d+b^2*(-1/2*(-e*(c*x-1))^( 1/2)*(d*(c*x+1))^(1/2)*(c^2*x^2-I*c*x*(-c^2*x^2+1)^(1/2)-1)*(arcsin(c*x)^2 -2+2*I*arcsin(c*x))/e/(c*x+1)/c^2/d/(c*x-1)-1/2*(-e*(c*x-1))^(1/2)*(d*(c*x +1))^(1/2)*(I*(-c^2*x^2+1)^(1/2)*c*x+c^2*x^2-1)*(arcsin(c*x)^2-2-2*I*arcsi n(c*x))/c^2/d/e/(c^2*x^2-1))+2*a*b*(-1/2*(-e*(c*x-1))^(1/2)*(d*(c*x+1))^(1 /2)*(c^2*x^2-I*c*x*(-c^2*x^2+1)^(1/2)-1)*(arcsin(c*x)+I)/e/(c*x+1)/c^2/d/( c*x-1)-1/2*(-e*(c*x-1))^(1/2)*(d*(c*x+1))^(1/2)*(I*(-c^2*x^2+1)^(1/2)*c*x+ c^2*x^2-1)*(arcsin(c*x)-I)/c^2/d/e/(c^2*x^2-1))
Time = 0.12 (sec) , antiderivative size = 138, normalized size of antiderivative = 1.01 \[ \int \frac {x (a+b \arcsin (c x))^2}{\sqrt {d+c d x} \sqrt {e-c e x}} \, dx=-\frac {{\left ({\left (a^{2} - 2 \, b^{2}\right )} c^{2} x^{2} + {\left (b^{2} c^{2} x^{2} - b^{2}\right )} \arcsin \left (c x\right )^{2} - a^{2} + 2 \, b^{2} + 2 \, {\left (a b c^{2} x^{2} - a b\right )} \arcsin \left (c x\right ) + 2 \, {\left (b^{2} c x \arcsin \left (c x\right ) + a b c x\right )} \sqrt {-c^{2} x^{2} + 1}\right )} \sqrt {c d x + d} \sqrt {-c e x + e}}{c^{4} d e x^{2} - c^{2} d e} \] Input:
integrate(x*(a+b*arcsin(c*x))^2/(c*d*x+d)^(1/2)/(-c*e*x+e)^(1/2),x, algori thm="fricas")
Output:
-((a^2 - 2*b^2)*c^2*x^2 + (b^2*c^2*x^2 - b^2)*arcsin(c*x)^2 - a^2 + 2*b^2 + 2*(a*b*c^2*x^2 - a*b)*arcsin(c*x) + 2*(b^2*c*x*arcsin(c*x) + a*b*c*x)*sq rt(-c^2*x^2 + 1))*sqrt(c*d*x + d)*sqrt(-c*e*x + e)/(c^4*d*e*x^2 - c^2*d*e)
\[ \int \frac {x (a+b \arcsin (c x))^2}{\sqrt {d+c d x} \sqrt {e-c e x}} \, dx=\int \frac {x \left (a + b \operatorname {asin}{\left (c x \right )}\right )^{2}}{\sqrt {d \left (c x + 1\right )} \sqrt {- e \left (c x - 1\right )}}\, dx \] Input:
integrate(x*(a+b*asin(c*x))**2/(c*d*x+d)**(1/2)/(-c*e*x+e)**(1/2),x)
Output:
Integral(x*(a + b*asin(c*x))**2/(sqrt(d*(c*x + 1))*sqrt(-e*(c*x - 1))), x)
Exception generated. \[ \int \frac {x (a+b \arcsin (c x))^2}{\sqrt {d+c d x} \sqrt {e-c e x}} \, dx=\text {Exception raised: ValueError} \] Input:
integrate(x*(a+b*arcsin(c*x))^2/(c*d*x+d)^(1/2)/(-c*e*x+e)^(1/2),x, algori thm="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
\[ \int \frac {x (a+b \arcsin (c x))^2}{\sqrt {d+c d x} \sqrt {e-c e x}} \, dx=\int { \frac {{\left (b \arcsin \left (c x\right ) + a\right )}^{2} x}{\sqrt {c d x + d} \sqrt {-c e x + e}} \,d x } \] Input:
integrate(x*(a+b*arcsin(c*x))^2/(c*d*x+d)^(1/2)/(-c*e*x+e)^(1/2),x, algori thm="giac")
Output:
integrate((b*arcsin(c*x) + a)^2*x/(sqrt(c*d*x + d)*sqrt(-c*e*x + e)), x)
Timed out. \[ \int \frac {x (a+b \arcsin (c x))^2}{\sqrt {d+c d x} \sqrt {e-c e x}} \, dx=\int \frac {x\,{\left (a+b\,\mathrm {asin}\left (c\,x\right )\right )}^2}{\sqrt {d+c\,d\,x}\,\sqrt {e-c\,e\,x}} \,d x \] Input:
int((x*(a + b*asin(c*x))^2)/((d + c*d*x)^(1/2)*(e - c*e*x)^(1/2)),x)
Output:
int((x*(a + b*asin(c*x))^2)/((d + c*d*x)^(1/2)*(e - c*e*x)^(1/2)), x)
\[ \int \frac {x (a+b \arcsin (c x))^2}{\sqrt {d+c d x} \sqrt {e-c e x}} \, dx=\frac {-\sqrt {c x +1}\, \sqrt {-c x +1}\, a^{2}+2 \left (\int \frac {\mathit {asin} \left (c x \right ) x}{\sqrt {c x +1}\, \sqrt {-c x +1}}d x \right ) a b \,c^{2}+\left (\int \frac {\mathit {asin} \left (c x \right )^{2} x}{\sqrt {c x +1}\, \sqrt {-c x +1}}d x \right ) b^{2} c^{2}}{\sqrt {e}\, \sqrt {d}\, c^{2}} \] Input:
int(x*(a+b*asin(c*x))^2/(c*d*x+d)^(1/2)/(-c*e*x+e)^(1/2),x)
Output:
( - sqrt(c*x + 1)*sqrt( - c*x + 1)*a**2 + 2*int((asin(c*x)*x)/(sqrt(c*x + 1)*sqrt( - c*x + 1)),x)*a*b*c**2 + int((asin(c*x)**2*x)/(sqrt(c*x + 1)*sqr t( - c*x + 1)),x)*b**2*c**2)/(sqrt(e)*sqrt(d)*c**2)