\(\int \frac {\sqrt {d-c^2 d x^2} (a+b \arcsin (c x))^n}{x} \, dx\) [485]

Optimal result

Integrand size = 29, antiderivative size = 29 \[ \int \frac {\sqrt {d-c^2 d x^2} (a+b \arcsin (c x))^n}{x} \, dx=\frac {d e^{-\frac {i a}{b}} \sqrt {1-c^2 x^2} (a+b \arcsin (c x))^n \left (-\frac {i (a+b \arcsin (c x))}{b}\right )^{-n} \Gamma \left (1+n,-\frac {i (a+b \arcsin (c x))}{b}\right )}{2 \sqrt {d-c^2 d x^2}}+\frac {d e^{\frac {i a}{b}} \sqrt {1-c^2 x^2} (a+b \arcsin (c x))^n \left (\frac {i (a+b \arcsin (c x))}{b}\right )^{-n} \Gamma \left (1+n,\frac {i (a+b \arcsin (c x))}{b}\right )}{2 \sqrt {d-c^2 d x^2}}+d \text {Int}\left (\frac {(a+b \arcsin (c x))^n}{x \sqrt {d-c^2 d x^2}},x\right ) \]

[Out]

1/2*d*(a+b*arcsin(c*x))^n*GAMMA(1+n,-I*(a+b*arcsin(c*x))/b)*(-c^2*x^2+1)^(1/2)/exp(I*a/b)/((-I*(a+b*arcsin(c*x
))/b)^n)/(-c^2*d*x^2+d)^(1/2)+1/2*d*exp(I*a/b)*(a+b*arcsin(c*x))^n*GAMMA(1+n,I*(a+b*arcsin(c*x))/b)*(-c^2*x^2+
1)^(1/2)/((I*(a+b*arcsin(c*x))/b)^n)/(-c^2*d*x^2+d)^(1/2)+d*Unintegrable((a+b*arcsin(c*x))^n/x/(-c^2*d*x^2+d)^
(1/2),x)

Rubi [N/A]

Not integrable

Time = 0.34 (sec) , antiderivative size = 29, normalized size of antiderivative = 1.00, number of steps used = 0, number of rules used = 0, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.000, Rules used = {} \[ \int \frac {\sqrt {d-c^2 d x^2} (a+b \arcsin (c x))^n}{x} \, dx=\int \frac {\sqrt {d-c^2 d x^2} (a+b \arcsin (c x))^n}{x} \, dx \]

[In]

Int[(Sqrt[d - c^2*d*x^2]*(a + b*ArcSin[c*x])^n)/x,x]

[Out]

(d*Sqrt[1 - c^2*x^2]*(a + b*ArcSin[c*x])^n*Gamma[1 + n, ((-I)*(a + b*ArcSin[c*x]))/b])/(2*E^((I*a)/b)*Sqrt[d -
c^2*d*x^2]*(((-I)*(a + b*ArcSin[c*x]))/b)^n) + (d*E^((I*a)/b)*Sqrt[1 - c^2*x^2]*(a + b*ArcSin[c*x])^n*Gamma[1
+ n, (I*(a + b*ArcSin[c*x]))/b])/(2*Sqrt[d - c^2*d*x^2]*((I*(a + b*ArcSin[c*x]))/b)^n) + d*Defer[Int][(a + b*
ArcSin[c*x])^n/(x*Sqrt[d - c^2*d*x^2]), x]

Rubi steps \begin{align*} \text {integral}& = \int \left (\frac {d (a+b \arcsin (c x))^n}{x \sqrt {d-c^2 d x^2}}-\frac {c^2 d x (a+b \arcsin (c x))^n}{\sqrt {d-c^2 d x^2}}\right ) \, dx \\ & = d \int \frac {(a+b \arcsin (c x))^n}{x \sqrt {d-c^2 d x^2}} \, dx-\left (c^2 d\right ) \int \frac {x (a+b \arcsin (c x))^n}{\sqrt {d-c^2 d x^2}} \, dx \\ & = d \int \frac {(a+b \arcsin (c x))^n}{x \sqrt {d-c^2 d x^2}} \, dx+\frac {\left (d \sqrt {1-c^2 x^2}\right ) \text {Subst}\left (\int x^n \sin \left (\frac {a}{b}-\frac {x}{b}\right ) \, dx,x,a+b \arcsin (c x)\right )}{b \sqrt {d-c^2 d x^2}} \\ & = d \int \frac {(a+b \arcsin (c x))^n}{x \sqrt {d-c^2 d x^2}} \, dx+\frac {\left (i d \sqrt {1-c^2 x^2}\right ) \text {Subst}\left (\int e^{-i \left (\frac {a}{b}-\frac {x}{b}\right )} x^n \, dx,x,a+b \arcsin (c x)\right )}{2 b \sqrt {d-c^2 d x^2}}-\frac {\left (i d \sqrt {1-c^2 x^2}\right ) \text {Subst}\left (\int e^{i \left (\frac {a}{b}-\frac {x}{b}\right )} x^n \, dx,x,a+b \arcsin (c x)\right )}{2 b \sqrt {d-c^2 d x^2}} \\ & = \frac {d e^{-\frac {i a}{b}} \sqrt {1-c^2 x^2} (a+b \arcsin (c x))^n \left (-\frac {i (a+b \arcsin (c x))}{b}\right )^{-n} \Gamma \left (1+n,-\frac {i (a+b \arcsin (c x))}{b}\right )}{2 \sqrt {d-c^2 d x^2}}+\frac {d e^{\frac {i a}{b}} \sqrt {1-c^2 x^2} (a+b \arcsin (c x))^n \left (\frac {i (a+b \arcsin (c x))}{b}\right )^{-n} \Gamma \left (1+n,\frac {i (a+b \arcsin (c x))}{b}\right )}{2 \sqrt {d-c^2 d x^2}}+d \int \frac {(a+b \arcsin (c x))^n}{x \sqrt {d-c^2 d x^2}} \, dx \\ \end{align*}

Mathematica [N/A]

Not integrable

Time = 0.30 (sec) , antiderivative size = 31, normalized size of antiderivative = 1.07 \[ \int \frac {\sqrt {d-c^2 d x^2} (a+b \arcsin (c x))^n}{x} \, dx=\int \frac {\sqrt {d-c^2 d x^2} (a+b \arcsin (c x))^n}{x} \, dx \]

[In]

Integrate[(Sqrt[d - c^2*d*x^2]*(a + b*ArcSin[c*x])^n)/x,x]

[Out]

Integrate[(Sqrt[d - c^2*d*x^2]*(a + b*ArcSin[c*x])^n)/x, x]

Maple [N/A] (verified)

Not integrable

Time = 0.34 (sec) , antiderivative size = 27, normalized size of antiderivative = 0.93

\[\int \frac {\sqrt {-c^{2} d \,x^{2}+d}\, \left (a +b \arcsin \left (c x \right )\right )^{n}}{x}d x\]

[In]

int((-c^2*d*x^2+d)^(1/2)*(a+b*arcsin(c*x))^n/x,x)

[Out]

int((-c^2*d*x^2+d)^(1/2)*(a+b*arcsin(c*x))^n/x,x)

Fricas [N/A]

Not integrable

Time = 0.26 (sec) , antiderivative size = 29, normalized size of antiderivative = 1.00 \[ \int \frac {\sqrt {d-c^2 d x^2} (a+b \arcsin (c x))^n}{x} \, dx=\int { \frac {\sqrt {-c^{2} d x^{2} + d} {\left (b \arcsin \left (c x\right ) + a\right )}^{n}}{x} \,d x } \]

[In]

integrate((-c^2*d*x^2+d)^(1/2)*(a+b*arcsin(c*x))^n/x,x, algorithm="fricas")

[Out]

integral(sqrt(-c^2*d*x^2 + d)*(b*arcsin(c*x) + a)^n/x, x)

Sympy [N/A]

Not integrable

Time = 2.38 (sec) , antiderivative size = 29, normalized size of antiderivative = 1.00 \[ \int \frac {\sqrt {d-c^2 d x^2} (a+b \arcsin (c x))^n}{x} \, dx=\int \frac {\sqrt {- d \left (c x - 1\right ) \left (c x + 1\right )} \left (a + b \operatorname {asin}{\left (c x \right )}\right )^{n}}{x}\, dx \]

[In]

integrate((-c**2*d*x**2+d)**(1/2)*(a+b*asin(c*x))**n/x,x)

[Out]

Integral(sqrt(-d*(c*x - 1)*(c*x + 1))*(a + b*asin(c*x))**n/x, x)

Maxima [N/A]

Not integrable

Time = 0.69 (sec) , antiderivative size = 29, normalized size of antiderivative = 1.00 \[ \int \frac {\sqrt {d-c^2 d x^2} (a+b \arcsin (c x))^n}{x} \, dx=\int { \frac {\sqrt {-c^{2} d x^{2} + d} {\left (b \arcsin \left (c x\right ) + a\right )}^{n}}{x} \,d x } \]

[In]

integrate((-c^2*d*x^2+d)^(1/2)*(a+b*arcsin(c*x))^n/x,x, algorithm="maxima")

[Out]

integrate(sqrt(-c^2*d*x^2 + d)*(b*arcsin(c*x) + a)^n/x, x)

Giac [F(-2)]

Exception generated. \[ \int \frac {\sqrt {d-c^2 d x^2} (a+b \arcsin (c x))^n}{x} \, dx=\text {Exception raised: TypeError} \]

[In]

integrate((-c^2*d*x^2+d)^(1/2)*(a+b*arcsin(c*x))^n/x,x, algorithm="giac")

[Out]

Exception raised: TypeError >> an error occurred running a Giac command:INPUT:sage2:=int(sage0,sageVARx):;OUTP
UT:sym2poly/r2sym(const gen & e,const index_m & i,const vecteur & l) Error: Bad Argument Value

Mupad [N/A]

Not integrable

Time = 0.14 (sec) , antiderivative size = 29, normalized size of antiderivative = 1.00 \[ \int \frac {\sqrt {d-c^2 d x^2} (a+b \arcsin (c x))^n}{x} \, dx=\int \frac {{\left (a+b\,\mathrm {asin}\left (c\,x\right )\right )}^n\,\sqrt {d-c^2\,d\,x^2}}{x} \,d x \]

[In]

int(((a + b*asin(c*x))^n*(d - c^2*d*x^2)^(1/2))/x,x)

[Out]

int(((a + b*asin(c*x))^n*(d - c^2*d*x^2)^(1/2))/x, x)