Integrand size = 30, antiderivative size = 55 \[ \int \frac {a+b \arcsin (c x)}{\sqrt {d+c d x} \sqrt {f-c f x}} \, dx=\frac {\sqrt {1-c^2 x^2} (a+b \arcsin (c x))^2}{2 b c \sqrt {d+c d x} \sqrt {f-c f x}} \] Output:
1/2*(-c^2*x^2+1)^(1/2)*(a+b*arcsin(c*x))^2/b/c/(c*d*x+d)^(1/2)/(-c*f*x+f)^ (1/2)
Time = 1.02 (sec) , antiderivative size = 110, normalized size of antiderivative = 2.00 \[ \int \frac {a+b \arcsin (c x)}{\sqrt {d+c d x} \sqrt {f-c f x}} \, dx=\frac {\frac {b \sqrt {1-c^2 x^2} \arcsin (c x)^2}{\sqrt {d+c d x} \sqrt {f-c f x}}-\frac {2 a \arctan \left (\frac {c x \sqrt {d+c d x} \sqrt {f-c f x}}{\sqrt {d} \sqrt {f} \left (-1+c^2 x^2\right )}\right )}{\sqrt {d} \sqrt {f}}}{2 c} \] Input:
Integrate[(a + b*ArcSin[c*x])/(Sqrt[d + c*d*x]*Sqrt[f - c*f*x]),x]
Output:
((b*Sqrt[1 - c^2*x^2]*ArcSin[c*x]^2)/(Sqrt[d + c*d*x]*Sqrt[f - c*f*x]) - ( 2*a*ArcTan[(c*x*Sqrt[d + c*d*x]*Sqrt[f - c*f*x])/(Sqrt[d]*Sqrt[f]*(-1 + c^ 2*x^2))])/(Sqrt[d]*Sqrt[f]))/(2*c)
Time = 0.37 (sec) , antiderivative size = 55, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.067, Rules used = {5178, 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.
\(\displaystyle \int \frac {a+b \arcsin (c x)}{\sqrt {c d x+d} \sqrt {f-c f x}} \, dx\) |
\(\Big \downarrow \) 5178 |
\(\displaystyle \frac {\sqrt {1-c^2 x^2} \int \frac {a+b \arcsin (c x)}{\sqrt {1-c^2 x^2}}dx}{\sqrt {c d x+d} \sqrt {f-c f x}}\) |
\(\Big \downarrow \) 5152 |
\(\displaystyle \frac {\sqrt {1-c^2 x^2} (a+b \arcsin (c x))^2}{2 b c \sqrt {c d x+d} \sqrt {f-c f x}}\) |
Input:
Int[(a + b*ArcSin[c*x])/(Sqrt[d + c*d*x]*Sqrt[f - c*f*x]),x]
Output:
(Sqrt[1 - c^2*x^2]*(a + b*ArcSin[c*x])^2)/(2*b*c*Sqrt[d + c*d*x]*Sqrt[f - c*f*x])
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_.)*((d_) + (e_.)*(x_))^(p_)*((f_) + (g_.)*(x_))^(q_), x_Symbol] :> Simp[(d + e*x)^q*((f + g*x)^q/(1 - c^2*x^ 2)^q) Int[(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, n}, x] && EqQ[e*f + d*g, 0] && EqQ[c^2*d^2 - e^2, 0] && HalfIntegerQ[p, q] && GeQ[p - q, 0]
Leaf count of result is larger than twice the leaf count of optimal. \(131\) vs. \(2(47)=94\).
Time = 1.32 (sec) , antiderivative size = 132, normalized size of antiderivative = 2.40
method | result | size |
default | \(\frac {a \sqrt {\left (-c f x +f \right ) \left (c d x +d \right )}\, \arctan \left (\frac {\sqrt {c^{2} d f}\, x}{\sqrt {-c^{2} d f \,x^{2}+d f}}\right )}{\sqrt {-c f x +f}\, \sqrt {c d x +d}\, \sqrt {c^{2} d f}}-\frac {b \sqrt {-f \left (c x -1\right )}\, \sqrt {d \left (c x +1\right )}\, \sqrt {-c^{2} x^{2}+1}\, \arcsin \left (c x \right )^{2}}{2 d f c \left (c^{2} x^{2}-1\right )}\) | \(132\) |
parts | \(\frac {a \sqrt {\left (-c f x +f \right ) \left (c d x +d \right )}\, \arctan \left (\frac {\sqrt {c^{2} d f}\, x}{\sqrt {-c^{2} d f \,x^{2}+d f}}\right )}{\sqrt {-c f x +f}\, \sqrt {c d x +d}\, \sqrt {c^{2} d f}}-\frac {b \sqrt {-f \left (c x -1\right )}\, \sqrt {d \left (c x +1\right )}\, \sqrt {-c^{2} x^{2}+1}\, \arcsin \left (c x \right )^{2}}{2 d f c \left (c^{2} x^{2}-1\right )}\) | \(132\) |
Input:
int((a+b*arcsin(c*x))/(c*d*x+d)^(1/2)/(-c*f*x+f)^(1/2),x,method=_RETURNVER BOSE)
Output:
a*((-c*f*x+f)*(c*d*x+d))^(1/2)/(-c*f*x+f)^(1/2)/(c*d*x+d)^(1/2)/(c^2*d*f)^ (1/2)*arctan((c^2*d*f)^(1/2)*x/(-c^2*d*f*x^2+d*f)^(1/2))-1/2*b*(-f*(c*x-1) )^(1/2)*(d*(c*x+1))^(1/2)*(-c^2*x^2+1)^(1/2)/d/f/c/(c^2*x^2-1)*arcsin(c*x) ^2
\[ \int \frac {a+b \arcsin (c x)}{\sqrt {d+c d x} \sqrt {f-c f x}} \, dx=\int { \frac {b \arcsin \left (c x\right ) + a}{\sqrt {c d x + d} \sqrt {-c f x + f}} \,d x } \] Input:
integrate((a+b*arcsin(c*x))/(c*d*x+d)^(1/2)/(-c*f*x+f)^(1/2),x, algorithm= "fricas")
Output:
integral(-sqrt(c*d*x + d)*sqrt(-c*f*x + f)*(b*arcsin(c*x) + a)/(c^2*d*f*x^ 2 - d*f), x)
\[ \int \frac {a+b \arcsin (c x)}{\sqrt {d+c d x} \sqrt {f-c f x}} \, dx=\int \frac {a + b \operatorname {asin}{\left (c x \right )}}{\sqrt {d \left (c x + 1\right )} \sqrt {- f \left (c x - 1\right )}}\, dx \] Input:
integrate((a+b*asin(c*x))/(c*d*x+d)**(1/2)/(-c*f*x+f)**(1/2),x)
Output:
Integral((a + b*asin(c*x))/(sqrt(d*(c*x + 1))*sqrt(-f*(c*x - 1))), x)
Time = 0.12 (sec) , antiderivative size = 32, normalized size of antiderivative = 0.58 \[ \int \frac {a+b \arcsin (c x)}{\sqrt {d+c d x} \sqrt {f-c f x}} \, dx=\frac {b \arcsin \left (c x\right )^{2}}{2 \, \sqrt {d f} c} + \frac {a \arcsin \left (c x\right )}{\sqrt {d f} c} \] Input:
integrate((a+b*arcsin(c*x))/(c*d*x+d)^(1/2)/(-c*f*x+f)^(1/2),x, algorithm= "maxima")
Output:
1/2*b*arcsin(c*x)^2/(sqrt(d*f)*c) + a*arcsin(c*x)/(sqrt(d*f)*c)
\[ \int \frac {a+b \arcsin (c x)}{\sqrt {d+c d x} \sqrt {f-c f x}} \, dx=\int { \frac {b \arcsin \left (c x\right ) + a}{\sqrt {c d x + d} \sqrt {-c f x + f}} \,d x } \] Input:
integrate((a+b*arcsin(c*x))/(c*d*x+d)^(1/2)/(-c*f*x+f)^(1/2),x, algorithm= "giac")
Output:
integrate((b*arcsin(c*x) + a)/(sqrt(c*d*x + d)*sqrt(-c*f*x + f)), x)
Timed out. \[ \int \frac {a+b \arcsin (c x)}{\sqrt {d+c d x} \sqrt {f-c f x}} \, dx=\int \frac {a+b\,\mathrm {asin}\left (c\,x\right )}{\sqrt {d+c\,d\,x}\,\sqrt {f-c\,f\,x}} \,d x \] Input:
int((a + b*asin(c*x))/((d + c*d*x)^(1/2)*(f - c*f*x)^(1/2)),x)
Output:
int((a + b*asin(c*x))/((d + c*d*x)^(1/2)*(f - c*f*x)^(1/2)), x)
\[ \int \frac {a+b \arcsin (c x)}{\sqrt {d+c d x} \sqrt {f-c f x}} \, dx=\frac {-2 \mathit {asin} \left (\frac {\sqrt {-c x +1}}{\sqrt {2}}\right ) a +\left (\int \frac {\mathit {asin} \left (c x \right )}{\sqrt {c x +1}\, \sqrt {-c x +1}}d x \right ) b c}{\sqrt {f}\, \sqrt {d}\, c} \] Input:
int((a+b*asin(c*x))/(c*d*x+d)^(1/2)/(-c*f*x+f)^(1/2),x)
Output:
( - 2*asin(sqrt( - c*x + 1)/sqrt(2))*a + int(asin(c*x)/(sqrt(c*x + 1)*sqrt ( - c*x + 1)),x)*b*c)/(sqrt(f)*sqrt(d)*c)