3.5.95 \(\int \frac {(d-c^2 d x^2)^{5/2} (a+b \text {ArcSin}(c x))^n}{x} \, dx\) [495]

Optimal. Leaf size=827 \[ \frac {11 d^3 e^{-\frac {i a}{b}} \sqrt {1-c^2 x^2} (a+b \text {ArcSin}(c x))^n \left (-\frac {i (a+b \text {ArcSin}(c x))}{b}\right )^{-n} \text {Gamma}\left (1+n,-\frac {i (a+b \text {ArcSin}(c x))}{b}\right )}{16 \sqrt {d-c^2 d x^2}}+\frac {11 d^3 e^{\frac {i a}{b}} \sqrt {1-c^2 x^2} (a+b \text {ArcSin}(c x))^n \left (\frac {i (a+b \text {ArcSin}(c x))}{b}\right )^{-n} \text {Gamma}\left (1+n,\frac {i (a+b \text {ArcSin}(c x))}{b}\right )}{16 \sqrt {d-c^2 d x^2}}-\frac {5\ 3^{-1-n} d^3 e^{-\frac {3 i a}{b}} \sqrt {1-c^2 x^2} (a+b \text {ArcSin}(c x))^n \left (-\frac {i (a+b \text {ArcSin}(c x))}{b}\right )^{-n} \text {Gamma}\left (1+n,-\frac {3 i (a+b \text {ArcSin}(c x))}{b}\right )}{32 \sqrt {d-c^2 d x^2}}+\frac {3^{-n} d^3 e^{-\frac {3 i a}{b}} \sqrt {1-c^2 x^2} (a+b \text {ArcSin}(c x))^n \left (-\frac {i (a+b \text {ArcSin}(c x))}{b}\right )^{-n} \text {Gamma}\left (1+n,-\frac {3 i (a+b \text {ArcSin}(c x))}{b}\right )}{8 \sqrt {d-c^2 d x^2}}-\frac {5\ 3^{-1-n} d^3 e^{\frac {3 i a}{b}} \sqrt {1-c^2 x^2} (a+b \text {ArcSin}(c x))^n \left (\frac {i (a+b \text {ArcSin}(c x))}{b}\right )^{-n} \text {Gamma}\left (1+n,\frac {3 i (a+b \text {ArcSin}(c x))}{b}\right )}{32 \sqrt {d-c^2 d x^2}}+\frac {3^{-n} d^3 e^{\frac {3 i a}{b}} \sqrt {1-c^2 x^2} (a+b \text {ArcSin}(c x))^n \left (\frac {i (a+b \text {ArcSin}(c x))}{b}\right )^{-n} \text {Gamma}\left (1+n,\frac {3 i (a+b \text {ArcSin}(c x))}{b}\right )}{8 \sqrt {d-c^2 d x^2}}+\frac {5^{-1-n} d^3 e^{-\frac {5 i a}{b}} \sqrt {1-c^2 x^2} (a+b \text {ArcSin}(c x))^n \left (-\frac {i (a+b \text {ArcSin}(c x))}{b}\right )^{-n} \text {Gamma}\left (1+n,-\frac {5 i (a+b \text {ArcSin}(c x))}{b}\right )}{32 \sqrt {d-c^2 d x^2}}+\frac {5^{-1-n} d^3 e^{\frac {5 i a}{b}} \sqrt {1-c^2 x^2} (a+b \text {ArcSin}(c x))^n \left (\frac {i (a+b \text {ArcSin}(c x))}{b}\right )^{-n} \text {Gamma}\left (1+n,\frac {5 i (a+b \text {ArcSin}(c x))}{b}\right )}{32 \sqrt {d-c^2 d x^2}}+d^3 \text {Int}\left (\frac {(a+b \text {ArcSin}(c x))^n}{x \sqrt {d-c^2 d x^2}},x\right ) \]

[Out]

11/16*d^3*(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)+11/16*d^3*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)-5/32*3^(-1-n)*d^3*(a+b*arcsin(c*x))^n*GAMMA(
1+n,-3*I*(a+b*arcsin(c*x))/b)*(-c^2*x^2+1)^(1/2)/exp(3*I*a/b)/((-I*(a+b*arcsin(c*x))/b)^n)/(-c^2*d*x^2+d)^(1/2
)+1/8*d^3*(a+b*arcsin(c*x))^n*GAMMA(1+n,-3*I*(a+b*arcsin(c*x))/b)*(-c^2*x^2+1)^(1/2)/(3^n)/exp(3*I*a/b)/((-I*(
a+b*arcsin(c*x))/b)^n)/(-c^2*d*x^2+d)^(1/2)-5/32*3^(-1-n)*d^3*exp(3*I*a/b)*(a+b*arcsin(c*x))^n*GAMMA(1+n,3*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)+1/8*d^3*exp(3*I*a/b)*(
a+b*arcsin(c*x))^n*GAMMA(1+n,3*I*(a+b*arcsin(c*x))/b)*(-c^2*x^2+1)^(1/2)/(3^n)/((I*(a+b*arcsin(c*x))/b)^n)/(-c
^2*d*x^2+d)^(1/2)+1/32*5^(-1-n)*d^3*(a+b*arcsin(c*x))^n*GAMMA(1+n,-5*I*(a+b*arcsin(c*x))/b)*(-c^2*x^2+1)^(1/2)
/exp(5*I*a/b)/((-I*(a+b*arcsin(c*x))/b)^n)/(-c^2*d*x^2+d)^(1/2)+1/32*5^(-1-n)*d^3*exp(5*I*a/b)*(a+b*arcsin(c*x
))^n*GAMMA(1+n,5*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^
3*Unintegrable((a+b*arcsin(c*x))^n/x/(-c^2*d*x^2+d)^(1/2),x)

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Rubi [A]
time = 1.10, antiderivative size = 0, normalized size of antiderivative = 0.00, number of steps used = 0, number of rules used = 0, integrand size = 0, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.000, Rules used = {} \begin {gather*} \int \frac {\left (d-c^2 d x^2\right )^{5/2} (a+b \text {ArcSin}(c x))^n}{x} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

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

[Out]

(11*d^3*Sqrt[1 - c^2*x^2]*(a + b*ArcSin[c*x])^n*Gamma[1 + n, ((-I)*(a + b*ArcSin[c*x]))/b])/(16*E^((I*a)/b)*Sq
rt[d - c^2*d*x^2]*(((-I)*(a + b*ArcSin[c*x]))/b)^n) + (11*d^3*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])/(16*Sqrt[d - c^2*d*x^2]*((I*(a + b*ArcSin[c*x]))/b)^n) - (5*3^(-1
- n)*d^3*Sqrt[1 - c^2*x^2]*(a + b*ArcSin[c*x])^n*Gamma[1 + n, ((-3*I)*(a + b*ArcSin[c*x]))/b])/(32*E^(((3*I)*
a)/b)*Sqrt[d - c^2*d*x^2]*(((-I)*(a + b*ArcSin[c*x]))/b)^n) + (d^3*Sqrt[1 - c^2*x^2]*(a + b*ArcSin[c*x])^n*Gam
ma[1 + n, ((-3*I)*(a + b*ArcSin[c*x]))/b])/(8*3^n*E^(((3*I)*a)/b)*Sqrt[d - c^2*d*x^2]*(((-I)*(a + b*ArcSin[c*x
]))/b)^n) - (5*3^(-1 - n)*d^3*E^(((3*I)*a)/b)*Sqrt[1 - c^2*x^2]*(a + b*ArcSin[c*x])^n*Gamma[1 + n, ((3*I)*(a +
b*ArcSin[c*x]))/b])/(32*Sqrt[d - c^2*d*x^2]*((I*(a + b*ArcSin[c*x]))/b)^n) + (d^3*E^(((3*I)*a)/b)*Sqrt[1 - c^
2*x^2]*(a + b*ArcSin[c*x])^n*Gamma[1 + n, ((3*I)*(a + b*ArcSin[c*x]))/b])/(8*3^n*Sqrt[d - c^2*d*x^2]*((I*(a +
b*ArcSin[c*x]))/b)^n) + (5^(-1 - n)*d^3*Sqrt[1 - c^2*x^2]*(a + b*ArcSin[c*x])^n*Gamma[1 + n, ((-5*I)*(a + b*Ar
cSin[c*x]))/b])/(32*E^(((5*I)*a)/b)*Sqrt[d - c^2*d*x^2]*(((-I)*(a + b*ArcSin[c*x]))/b)^n) + (5^(-1 - n)*d^3*E^
(((5*I)*a)/b)*Sqrt[1 - c^2*x^2]*(a + b*ArcSin[c*x])^n*Gamma[1 + n, ((5*I)*(a + b*ArcSin[c*x]))/b])/(32*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*Sqrt[d - c^2*d*x^2]), x]

Rubi steps

\begin {align*} \int \frac {\left (d-c^2 d x^2\right )^{5/2} \left (a+b \sin ^{-1}(c x)\right )^n}{x} \, dx &=\int \frac {\left (d-c^2 d x^2\right )^{5/2} \left (a+b \sin ^{-1}(c x)\right )^n}{x} \, dx\\ \end {align*}

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Mathematica [A]
time = 0.16, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\left (d-c^2 d x^2\right )^{5/2} (a+b \text {ArcSin}(c x))^n}{x} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

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

[Out]

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

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Maple [A]
time = 0.19, size = 0, normalized size = 0.00 \[\int \frac {\left (-c^{2} d \,x^{2}+d \right )^{\frac {5}{2}} \left (a +b \arcsin \left (c x \right )\right )^{n}}{x}\, dx\]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [A]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Fricas [A]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

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Sympy [F(-2)]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: SystemError} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Exception raised: SystemError >> excessive stack use: stack is 6190 deep

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Giac [F(-2)]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: TypeError} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-c^2*d*x^2+d)^(5/2)*(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

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Mupad [A]
time = 0.00, size = -1, normalized size = -0.00 \begin {gather*} \int \frac {{\left (a+b\,\mathrm {asin}\left (c\,x\right )\right )}^n\,{\left (d-c^2\,d\,x^2\right )}^{5/2}}{x} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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