\(\int \frac {(d-c^2 d x^2)^{5/2} (a+b \arcsin (c x))^n}{x^2} \, dx\) [496]

Optimal result

Integrand size = 29, antiderivative size = 29 \[ \int \frac {\left (d-c^2 d x^2\right )^{5/2} (a+b \arcsin (c x))^n}{x^2} \, dx=-\frac {15 c d^3 \sqrt {1-c^2 x^2} (a+b \arcsin (c x))^{1+n}}{8 b (1+n) \sqrt {d-c^2 d x^2}}+\frac {i 2^{-2-n} c d^3 e^{-\frac {2 i a}{b}} \sqrt {1-c^2 x^2} (a+b \arcsin (c x))^n \left (-\frac {i (a+b \arcsin (c x))}{b}\right )^{-n} \Gamma \left (1+n,-\frac {2 i (a+b \arcsin (c x))}{b}\right )}{\sqrt {d-c^2 d x^2}}-\frac {i 2^{-2-n} c d^3 e^{\frac {2 i a}{b}} \sqrt {1-c^2 x^2} (a+b \arcsin (c x))^n \left (\frac {i (a+b \arcsin (c x))}{b}\right )^{-n} \Gamma \left (1+n,\frac {2 i (a+b \arcsin (c x))}{b}\right )}{\sqrt {d-c^2 d x^2}}+\frac {i 2^{-2 (3+n)} c d^3 e^{-\frac {4 i a}{b}} \sqrt {1-c^2 x^2} (a+b \arcsin (c x))^n \left (-\frac {i (a+b \arcsin (c x))}{b}\right )^{-n} \Gamma \left (1+n,-\frac {4 i (a+b \arcsin (c x))}{b}\right )}{\sqrt {d-c^2 d x^2}}-\frac {i 2^{-2 (3+n)} c d^3 e^{\frac {4 i a}{b}} \sqrt {1-c^2 x^2} (a+b \arcsin (c x))^n \left (\frac {i (a+b \arcsin (c x))}{b}\right )^{-n} \Gamma \left (1+n,\frac {4 i (a+b \arcsin (c x))}{b}\right )}{\sqrt {d-c^2 d x^2}}+d^3 \text {Int}\left (\frac {(a+b \arcsin (c x))^n}{x^2 \sqrt {d-c^2 d x^2}},x\right ) \]

[Out]

-15/8*c*d^3*(a+b*arcsin(c*x))^(1+n)*(-c^2*x^2+1)^(1/2)/b/(1+n)/(-c^2*d*x^2+d)^(1/2)+I*2^(-2-n)*c*d^3*(a+b*arcs
in(c*x))^n*GAMMA(1+n,-2*I*(a+b*arcsin(c*x))/b)*(-c^2*x^2+1)^(1/2)/exp(2*I*a/b)/((-I*(a+b*arcsin(c*x))/b)^n)/(-
c^2*d*x^2+d)^(1/2)-I*2^(-2-n)*c*d^3*exp(2*I*a/b)*(a+b*arcsin(c*x))^n*GAMMA(1+n,2*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)+I*c*d^3*(a+b*arcsin(c*x))^n*GAMMA(1+n,-4*I*(a+b*
arcsin(c*x))/b)*(-c^2*x^2+1)^(1/2)/(2^(6+2*n))/exp(4*I*a/b)/((-I*(a+b*arcsin(c*x))/b)^n)/(-c^2*d*x^2+d)^(1/2)-
I*c*d^3*exp(4*I*a/b)*(a+b*arcsin(c*x))^n*GAMMA(1+n,4*I*(a+b*arcsin(c*x))/b)*(-c^2*x^2+1)^(1/2)/(2^(6+2*n))/((I
*(a+b*arcsin(c*x))/b)^n)/(-c^2*d*x^2+d)^(1/2)+d^3*Unintegrable((a+b*arcsin(c*x))^n/x^2/(-c^2*d*x^2+d)^(1/2),x)

Rubi [N/A]

Not integrable

Time = 0.81 (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 = {} \[ \int \frac {\left (d-c^2 d x^2\right )^{5/2} (a+b \arcsin (c x))^n}{x^2} \, dx=\int \frac {\left (d-c^2 d x^2\right )^{5/2} (a+b \arcsin (c x))^n}{x^2} \, dx \]

[In]

Int[((d - c^2*d*x^2)^(5/2)*(a + b*ArcSin[c*x])^n)/x^2,x]

[Out]

(-15*c*d^3*Sqrt[1 - c^2*x^2]*(a + b*ArcSin[c*x])^(1 + n))/(8*b*(1 + n)*Sqrt[d - c^2*d*x^2]) + (I*2^(-2 - n)*c*
d^3*Sqrt[1 - c^2*x^2]*(a + b*ArcSin[c*x])^n*Gamma[1 + n, ((-2*I)*(a + b*ArcSin[c*x]))/b])/(E^(((2*I)*a)/b)*Sqr
t[d - c^2*d*x^2]*(((-I)*(a + b*ArcSin[c*x]))/b)^n) - (I*2^(-2 - n)*c*d^3*E^(((2*I)*a)/b)*Sqrt[1 - c^2*x^2]*(a
+ b*ArcSin[c*x])^n*Gamma[1 + n, ((2*I)*(a + b*ArcSin[c*x]))/b])/(Sqrt[d - c^2*d*x^2]*((I*(a + b*ArcSin[c*x]))/
b)^n) + (I*c*d^3*Sqrt[1 - c^2*x^2]*(a + b*ArcSin[c*x])^n*Gamma[1 + n, ((-4*I)*(a + b*ArcSin[c*x]))/b])/(2^(2*(
3 + n))*E^(((4*I)*a)/b)*Sqrt[d - c^2*d*x^2]*(((-I)*(a + b*ArcSin[c*x]))/b)^n) - (I*c*d^3*E^(((4*I)*a)/b)*Sqrt[
1 - c^2*x^2]*(a + b*ArcSin[c*x])^n*Gamma[1 + n, ((4*I)*(a + b*ArcSin[c*x]))/b])/(2^(2*(3 + n))*Sqrt[d - c^2*d*
x^2]*((I*(a + b*ArcSin[c*x]))/b)^n) + d^3*Defer[Int][(a + b*ArcSin[c*x])^n/(x^2*Sqrt[d - c^2*d*x^2]), x]

Rubi steps \begin{align*} \text {integral}& = \int \left (-\frac {3 c^2 d^3 (a+b \arcsin (c x))^n}{\sqrt {d-c^2 d x^2}}+\frac {d^3 (a+b \arcsin (c x))^n}{x^2 \sqrt {d-c^2 d x^2}}+\frac {3 c^4 d^3 x^2 (a+b \arcsin (c x))^n}{\sqrt {d-c^2 d x^2}}-\frac {c^6 d^3 x^4 (a+b \arcsin (c x))^n}{\sqrt {d-c^2 d x^2}}\right ) \, dx \\ & = d^3 \int \frac {(a+b \arcsin (c x))^n}{x^2 \sqrt {d-c^2 d x^2}} \, dx-\left (3 c^2 d^3\right ) \int \frac {(a+b \arcsin (c x))^n}{\sqrt {d-c^2 d x^2}} \, dx+\left (3 c^4 d^3\right ) \int \frac {x^2 (a+b \arcsin (c x))^n}{\sqrt {d-c^2 d x^2}} \, dx-\left (c^6 d^3\right ) \int \frac {x^4 (a+b \arcsin (c x))^n}{\sqrt {d-c^2 d x^2}} \, dx \\ & = -\frac {3 c d^3 \sqrt {1-c^2 x^2} (a+b \arcsin (c x))^{1+n}}{b (1+n) \sqrt {d-c^2 d x^2}}+d^3 \int \frac {(a+b \arcsin (c x))^n}{x^2 \sqrt {d-c^2 d x^2}} \, dx-\frac {\left (c d^3 \sqrt {1-c^2 x^2}\right ) \text {Subst}\left (\int x^n \sin ^4\left (\frac {a}{b}-\frac {x}{b}\right ) \, dx,x,a+b \arcsin (c x)\right )}{b \sqrt {d-c^2 d x^2}}+\frac {\left (3 c d^3 \sqrt {1-c^2 x^2}\right ) \text {Subst}\left (\int x^n \sin ^2\left (\frac {a}{b}-\frac {x}{b}\right ) \, dx,x,a+b \arcsin (c x)\right )}{b \sqrt {d-c^2 d x^2}} \\ & = -\frac {3 c d^3 \sqrt {1-c^2 x^2} (a+b \arcsin (c x))^{1+n}}{b (1+n) \sqrt {d-c^2 d x^2}}+d^3 \int \frac {(a+b \arcsin (c x))^n}{x^2 \sqrt {d-c^2 d x^2}} \, dx-\frac {\left (c d^3 \sqrt {1-c^2 x^2}\right ) \text {Subst}\left (\int \left (\frac {3 x^n}{8}+\frac {1}{8} x^n \cos \left (\frac {4 a}{b}-\frac {4 x}{b}\right )-\frac {1}{2} x^n \cos \left (\frac {2 a}{b}-\frac {2 x}{b}\right )\right ) \, dx,x,a+b \arcsin (c x)\right )}{b \sqrt {d-c^2 d x^2}}+\frac {\left (3 c d^3 \sqrt {1-c^2 x^2}\right ) \text {Subst}\left (\int \left (\frac {x^n}{2}-\frac {1}{2} x^n \cos \left (\frac {2 a}{b}-\frac {2 x}{b}\right )\right ) \, dx,x,a+b \arcsin (c x)\right )}{b \sqrt {d-c^2 d x^2}} \\ & = -\frac {15 c d^3 \sqrt {1-c^2 x^2} (a+b \arcsin (c x))^{1+n}}{8 b (1+n) \sqrt {d-c^2 d x^2}}+d^3 \int \frac {(a+b \arcsin (c x))^n}{x^2 \sqrt {d-c^2 d x^2}} \, dx-\frac {\left (c d^3 \sqrt {1-c^2 x^2}\right ) \text {Subst}\left (\int x^n \cos \left (\frac {4 a}{b}-\frac {4 x}{b}\right ) \, dx,x,a+b \arcsin (c x)\right )}{8 b \sqrt {d-c^2 d x^2}}+\frac {\left (c d^3 \sqrt {1-c^2 x^2}\right ) \text {Subst}\left (\int x^n \cos \left (\frac {2 a}{b}-\frac {2 x}{b}\right ) \, dx,x,a+b \arcsin (c x)\right )}{2 b \sqrt {d-c^2 d x^2}}-\frac {\left (3 c d^3 \sqrt {1-c^2 x^2}\right ) \text {Subst}\left (\int x^n \cos \left (\frac {2 a}{b}-\frac {2 x}{b}\right ) \, dx,x,a+b \arcsin (c x)\right )}{2 b \sqrt {d-c^2 d x^2}} \\ & = -\frac {15 c d^3 \sqrt {1-c^2 x^2} (a+b \arcsin (c x))^{1+n}}{8 b (1+n) \sqrt {d-c^2 d x^2}}+d^3 \int \frac {(a+b \arcsin (c x))^n}{x^2 \sqrt {d-c^2 d x^2}} \, dx-\frac {\left (c d^3 \sqrt {1-c^2 x^2}\right ) \text {Subst}\left (\int e^{-i \left (\frac {4 a}{b}-\frac {4 x}{b}\right )} x^n \, dx,x,a+b \arcsin (c x)\right )}{16 b \sqrt {d-c^2 d x^2}}-\frac {\left (c d^3 \sqrt {1-c^2 x^2}\right ) \text {Subst}\left (\int e^{i \left (\frac {4 a}{b}-\frac {4 x}{b}\right )} x^n \, dx,x,a+b \arcsin (c x)\right )}{16 b \sqrt {d-c^2 d x^2}}+\frac {\left (c d^3 \sqrt {1-c^2 x^2}\right ) \text {Subst}\left (\int e^{-i \left (\frac {2 a}{b}-\frac {2 x}{b}\right )} x^n \, dx,x,a+b \arcsin (c x)\right )}{4 b \sqrt {d-c^2 d x^2}}+\frac {\left (c d^3 \sqrt {1-c^2 x^2}\right ) \text {Subst}\left (\int e^{i \left (\frac {2 a}{b}-\frac {2 x}{b}\right )} x^n \, dx,x,a+b \arcsin (c x)\right )}{4 b \sqrt {d-c^2 d x^2}}-\frac {\left (3 c d^3 \sqrt {1-c^2 x^2}\right ) \text {Subst}\left (\int e^{-i \left (\frac {2 a}{b}-\frac {2 x}{b}\right )} x^n \, dx,x,a+b \arcsin (c x)\right )}{4 b \sqrt {d-c^2 d x^2}}-\frac {\left (3 c d^3 \sqrt {1-c^2 x^2}\right ) \text {Subst}\left (\int e^{i \left (\frac {2 a}{b}-\frac {2 x}{b}\right )} x^n \, dx,x,a+b \arcsin (c x)\right )}{4 b \sqrt {d-c^2 d x^2}} \\ & = -\frac {15 c d^3 \sqrt {1-c^2 x^2} (a+b \arcsin (c x))^{1+n}}{8 b (1+n) \sqrt {d-c^2 d x^2}}+\frac {i 2^{-2-n} c d^3 e^{-\frac {2 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 {2 i (a+b \arcsin (c x))}{b}\right )}{\sqrt {d-c^2 d x^2}}-\frac {i 2^{-2-n} c d^3 e^{\frac {2 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 {2 i (a+b \arcsin (c x))}{b}\right )}{\sqrt {d-c^2 d x^2}}+\frac {i 4^{-3-n} c d^3 e^{-\frac {4 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 {4 i (a+b \arcsin (c x))}{b}\right )}{\sqrt {d-c^2 d x^2}}-\frac {i 4^{-3-n} c d^3 e^{\frac {4 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 {4 i (a+b \arcsin (c x))}{b}\right )}{\sqrt {d-c^2 d x^2}}+d^3 \int \frac {(a+b \arcsin (c x))^n}{x^2 \sqrt {d-c^2 d x^2}} \, dx \\ \end{align*}

Mathematica [N/A]

Not integrable

Time = 0.68 (sec) , antiderivative size = 31, normalized size of antiderivative = 1.07 \[ \int \frac {\left (d-c^2 d x^2\right )^{5/2} (a+b \arcsin (c x))^n}{x^2} \, dx=\int \frac {\left (d-c^2 d x^2\right )^{5/2} (a+b \arcsin (c x))^n}{x^2} \, dx \]

[In]

Integrate[((d - c^2*d*x^2)^(5/2)*(a + b*ArcSin[c*x])^n)/x^2,x]

[Out]

Integrate[((d - c^2*d*x^2)^(5/2)*(a + b*ArcSin[c*x])^n)/x^2, x]

Maple [N/A] (verified)

Not integrable

Time = 0.20 (sec) , antiderivative size = 27, normalized size of antiderivative = 0.93

\[\int \frac {\left (-c^{2} d \,x^{2}+d \right )^{\frac {5}{2}} \left (a +b \arcsin \left (c x \right )\right )^{n}}{x^{2}}d x\]

[In]

int((-c^2*d*x^2+d)^(5/2)*(a+b*arcsin(c*x))^n/x^2,x)

[Out]

int((-c^2*d*x^2+d)^(5/2)*(a+b*arcsin(c*x))^n/x^2,x)

Fricas [N/A]

Not integrable

Time = 0.26 (sec) , antiderivative size = 54, normalized size of antiderivative = 1.86 \[ \int \frac {\left (d-c^2 d x^2\right )^{5/2} (a+b \arcsin (c x))^n}{x^2} \, dx=\int { \frac {{\left (-c^{2} d x^{2} + d\right )}^{\frac {5}{2}} {\left (b \arcsin \left (c x\right ) + a\right )}^{n}}{x^{2}} \,d x } \]

[In]

integrate((-c^2*d*x^2+d)^(5/2)*(a+b*arcsin(c*x))^n/x^2,x, algorithm="fricas")

[Out]

integral((c^4*d^2*x^4 - 2*c^2*d^2*x^2 + d^2)*sqrt(-c^2*d*x^2 + d)*(b*arcsin(c*x) + a)^n/x^2, x)

Sympy [F(-1)]

Timed out. \[ \int \frac {\left (d-c^2 d x^2\right )^{5/2} (a+b \arcsin (c x))^n}{x^2} \, dx=\text {Timed out} \]

[In]

integrate((-c**2*d*x**2+d)**(5/2)*(a+b*asin(c*x))**n/x**2,x)

[Out]

Timed out

Maxima [N/A]

Not integrable

Time = 0.90 (sec) , antiderivative size = 29, normalized size of antiderivative = 1.00 \[ \int \frac {\left (d-c^2 d x^2\right )^{5/2} (a+b \arcsin (c x))^n}{x^2} \, dx=\int { \frac {{\left (-c^{2} d x^{2} + d\right )}^{\frac {5}{2}} {\left (b \arcsin \left (c x\right ) + a\right )}^{n}}{x^{2}} \,d x } \]

[In]

integrate((-c^2*d*x^2+d)^(5/2)*(a+b*arcsin(c*x))^n/x^2,x, algorithm="maxima")

[Out]

integrate((-c^2*d*x^2 + d)^(5/2)*(b*arcsin(c*x) + a)^n/x^2, x)

Giac [F(-2)]

Exception generated. \[ \int \frac {\left (d-c^2 d x^2\right )^{5/2} (a+b \arcsin (c x))^n}{x^2} \, dx=\text {Exception raised: TypeError} \]

[In]

integrate((-c^2*d*x^2+d)^(5/2)*(a+b*arcsin(c*x))^n/x^2,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 \[ \int \frac {\left (d-c^2 d x^2\right )^{5/2} (a+b \arcsin (c x))^n}{x^2} \, dx=\int \frac {{\left (a+b\,\mathrm {asin}\left (c\,x\right )\right )}^n\,{\left (d-c^2\,d\,x^2\right )}^{5/2}}{x^2} \,d x \]

[In]

int(((a + b*asin(c*x))^n*(d - c^2*d*x^2)^(5/2))/x^2,x)

[Out]

int(((a + b*asin(c*x))^n*(d - c^2*d*x^2)^(5/2))/x^2, x)