Integrand size = 22, antiderivative size = 100 \[ \int \frac {x^m \arcsin (a x)}{\sqrt {1-a^2 x^2}} \, dx=\frac {x^{1+m} \arcsin (a x) \operatorname {Hypergeometric2F1}\left (\frac {1}{2},\frac {1+m}{2},\frac {3+m}{2},a^2 x^2\right )}{1+m}-\frac {a x^{2+m} \, _3F_2\left (1,1+\frac {m}{2},1+\frac {m}{2};\frac {3}{2}+\frac {m}{2},2+\frac {m}{2};a^2 x^2\right )}{2+3 m+m^2} \] Output:
x^(1+m)*arcsin(a*x)*hypergeom([1/2, 1/2+1/2*m],[3/2+1/2*m],a^2*x^2)/(1+m)- a*x^(2+m)*hypergeom([1, 1+1/2*m, 1+1/2*m],[2+1/2*m, 3/2+1/2*m],a^2*x^2)/(m ^2+3*m+2)
Time = 0.02 (sec) , antiderivative size = 95, normalized size of antiderivative = 0.95 \[ \int \frac {x^m \arcsin (a x)}{\sqrt {1-a^2 x^2}} \, dx=\frac {x^{1+m} \left ((2+m) \arcsin (a x) \operatorname {Hypergeometric2F1}\left (\frac {1}{2},\frac {1+m}{2},\frac {3+m}{2},a^2 x^2\right )-a x \, _3F_2\left (1,1+\frac {m}{2},1+\frac {m}{2};\frac {3}{2}+\frac {m}{2},2+\frac {m}{2};a^2 x^2\right )\right )}{(1+m) (2+m)} \] Input:
Integrate[(x^m*ArcSin[a*x])/Sqrt[1 - a^2*x^2],x]
Output:
(x^(1 + m)*((2 + m)*ArcSin[a*x]*Hypergeometric2F1[1/2, (1 + m)/2, (3 + m)/ 2, a^2*x^2] - a*x*HypergeometricPFQ[{1, 1 + m/2, 1 + m/2}, {3/2 + m/2, 2 + m/2}, a^2*x^2]))/((1 + m)*(2 + m))
Time = 0.29 (sec) , antiderivative size = 100, normalized size of antiderivative = 1.00, number of steps used = 1, number of rules used = 1, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.045, Rules used = {5220}
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.
\(\displaystyle \int \frac {x^m \arcsin (a x)}{\sqrt {1-a^2 x^2}} \, dx\) |
\(\Big \downarrow \) 5220 |
\(\displaystyle \frac {x^{m+1} \arcsin (a x) \operatorname {Hypergeometric2F1}\left (\frac {1}{2},\frac {m+1}{2},\frac {m+3}{2},a^2 x^2\right )}{m+1}-\frac {a x^{m+2} \, _3F_2\left (1,\frac {m}{2}+1,\frac {m}{2}+1;\frac {m}{2}+\frac {3}{2},\frac {m}{2}+2;a^2 x^2\right )}{m^2+3 m+2}\) |
Input:
Int[(x^m*ArcSin[a*x])/Sqrt[1 - a^2*x^2],x]
Output:
(x^(1 + m)*ArcSin[a*x]*Hypergeometric2F1[1/2, (1 + m)/2, (3 + m)/2, a^2*x^ 2])/(1 + m) - (a*x^(2 + m)*HypergeometricPFQ[{1, 1 + m/2, 1 + m/2}, {3/2 + m/2, 2 + m/2}, a^2*x^2])/(2 + 3*m + m^2)
Int[(((a_.) + ArcSin[(c_.)*(x_)]*(b_.))*((f_.)*(x_))^(m_))/Sqrt[(d_) + (e_. )*(x_)^2], x_Symbol] :> Simp[((f*x)^(m + 1)/(f*(m + 1)))*Simp[Sqrt[1 - c^2* x^2]/Sqrt[d + e*x^2]]*(a + b*ArcSin[c*x])*Hypergeometric2F1[1/2, (1 + m)/2, (3 + m)/2, c^2*x^2], x] - Simp[b*c*((f*x)^(m + 2)/(f^2*(m + 1)*(m + 2)))*S imp[Sqrt[1 - c^2*x^2]/Sqrt[d + e*x^2]]*HypergeometricPFQ[{1, 1 + m/2, 1 + m /2}, {3/2 + m/2, 2 + m/2}, c^2*x^2], x] /; FreeQ[{a, b, c, d, e, f, m}, x] && EqQ[c^2*d + e, 0] && !IntegerQ[m]
\[\int \frac {x^{m} \arcsin \left (a x \right )}{\sqrt {-a^{2} x^{2}+1}}d x\]
Input:
int(x^m*arcsin(a*x)/(-a^2*x^2+1)^(1/2),x)
Output:
int(x^m*arcsin(a*x)/(-a^2*x^2+1)^(1/2),x)
\[ \int \frac {x^m \arcsin (a x)}{\sqrt {1-a^2 x^2}} \, dx=\int { \frac {x^{m} \arcsin \left (a x\right )}{\sqrt {-a^{2} x^{2} + 1}} \,d x } \] Input:
integrate(x^m*arcsin(a*x)/(-a^2*x^2+1)^(1/2),x, algorithm="fricas")
Output:
integral(-sqrt(-a^2*x^2 + 1)*x^m*arcsin(a*x)/(a^2*x^2 - 1), x)
\[ \int \frac {x^m \arcsin (a x)}{\sqrt {1-a^2 x^2}} \, dx=\int \frac {x^{m} \operatorname {asin}{\left (a x \right )}}{\sqrt {- \left (a x - 1\right ) \left (a x + 1\right )}}\, dx \] Input:
integrate(x**m*asin(a*x)/(-a**2*x**2+1)**(1/2),x)
Output:
Integral(x**m*asin(a*x)/sqrt(-(a*x - 1)*(a*x + 1)), x)
\[ \int \frac {x^m \arcsin (a x)}{\sqrt {1-a^2 x^2}} \, dx=\int { \frac {x^{m} \arcsin \left (a x\right )}{\sqrt {-a^{2} x^{2} + 1}} \,d x } \] Input:
integrate(x^m*arcsin(a*x)/(-a^2*x^2+1)^(1/2),x, algorithm="maxima")
Output:
integrate(x^m*arcsin(a*x)/sqrt(-a^2*x^2 + 1), x)
\[ \int \frac {x^m \arcsin (a x)}{\sqrt {1-a^2 x^2}} \, dx=\int { \frac {x^{m} \arcsin \left (a x\right )}{\sqrt {-a^{2} x^{2} + 1}} \,d x } \] Input:
integrate(x^m*arcsin(a*x)/(-a^2*x^2+1)^(1/2),x, algorithm="giac")
Output:
integrate(x^m*arcsin(a*x)/sqrt(-a^2*x^2 + 1), x)
Timed out. \[ \int \frac {x^m \arcsin (a x)}{\sqrt {1-a^2 x^2}} \, dx=\int \frac {x^m\,\mathrm {asin}\left (a\,x\right )}{\sqrt {1-a^2\,x^2}} \,d x \] Input:
int((x^m*asin(a*x))/(1 - a^2*x^2)^(1/2),x)
Output:
int((x^m*asin(a*x))/(1 - a^2*x^2)^(1/2), x)
\[ \int \frac {x^m \arcsin (a x)}{\sqrt {1-a^2 x^2}} \, dx=\int \frac {x^{m} \mathit {asin} \left (a x \right )}{\sqrt {-a^{2} x^{2}+1}}d x \] Input:
int(x^m*asin(a*x)/(-a^2*x^2+1)^(1/2),x)
Output:
int((x**m*asin(a*x))/sqrt( - a**2*x**2 + 1),x)