3.405 \(\int \frac {(1-c^2 x^2)^{5/2}}{x^3 (a+b \sin ^{-1}(c x))^2} \, dx\)

Optimal. Leaf size=31 \[ \text {Int}\left (\frac {\left (1-c^2 x^2\right )^{5/2}}{x^3 \left (a+b \sin ^{-1}(c x)\right )^2},x\right ) \]

[Out]

Unintegrable((-c^2*x^2+1)^(5/2)/x^3/(a+b*arcsin(c*x))^2,x)

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Rubi [A]  time = 0.14, 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 = {} \[ \int \frac {\left (1-c^2 x^2\right )^{5/2}}{x^3 \left (a+b \sin ^{-1}(c x)\right )^2} \, dx \]

Verification is Not applicable to the result.

[In]

Int[(1 - c^2*x^2)^(5/2)/(x^3*(a + b*ArcSin[c*x])^2),x]

[Out]

Defer[Int][(1 - c^2*x^2)^(5/2)/(x^3*(a + b*ArcSin[c*x])^2), x]

Rubi steps

\begin {align*} \int \frac {\left (1-c^2 x^2\right )^{5/2}}{x^3 \left (a+b \sin ^{-1}(c x)\right )^2} \, dx &=\int \frac {\left (1-c^2 x^2\right )^{5/2}}{x^3 \left (a+b \sin ^{-1}(c x)\right )^2} \, dx\\ \end {align*}

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Mathematica [A]  time = 18.02, size = 0, normalized size = 0.00 \[ \int \frac {\left (1-c^2 x^2\right )^{5/2}}{x^3 \left (a+b \sin ^{-1}(c x)\right )^2} \, dx \]

Verification is Not applicable to the result.

[In]

Integrate[(1 - c^2*x^2)^(5/2)/(x^3*(a + b*ArcSin[c*x])^2),x]

[Out]

Integrate[(1 - c^2*x^2)^(5/2)/(x^3*(a + b*ArcSin[c*x])^2), x]

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fricas [A]  time = 0.41, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {{\left (c^{4} x^{4} - 2 \, c^{2} x^{2} + 1\right )} \sqrt {-c^{2} x^{2} + 1}}{b^{2} x^{3} \arcsin \left (c x\right )^{2} + 2 \, a b x^{3} \arcsin \left (c x\right ) + a^{2} x^{3}}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-c^2*x^2+1)^(5/2)/x^3/(a+b*arcsin(c*x))^2,x, algorithm="fricas")

[Out]

integral((c^4*x^4 - 2*c^2*x^2 + 1)*sqrt(-c^2*x^2 + 1)/(b^2*x^3*arcsin(c*x)^2 + 2*a*b*x^3*arcsin(c*x) + a^2*x^3
), x)

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giac [F(-2)]  time = 0.00, size = 0, normalized size = 0.00 \[ \text {Exception raised: TypeError} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-c^2*x^2+1)^(5/2)/x^3/(a+b*arcsin(c*x))^2,x, algorithm="giac")

[Out]

Exception raised: TypeError >> An error occurred running a Giac command:INPUT:sage2:=int(sage0,x):;OUTPUT:sym2
poly/r2sym(const gen & e,const index_m & i,const vecteur & l) Error: Bad Argument Value

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maple [A]  time = 5.25, size = 0, normalized size = 0.00 \[ \int \frac {\left (-c^{2} x^{2}+1\right )^{\frac {5}{2}}}{x^{3} \left (a +b \arcsin \left (c x \right )\right )^{2}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((-c^2*x^2+1)^(5/2)/x^3/(a+b*arcsin(c*x))^2,x)

[Out]

int((-c^2*x^2+1)^(5/2)/x^3/(a+b*arcsin(c*x))^2,x)

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maxima [A]  time = 0.00, size = 0, normalized size = 0.00 \[ \frac {c^{6} x^{6} - 3 \, c^{4} x^{4} + 3 \, c^{2} x^{2} - 3 \, {\left (b^{2} c x^{3} \arctan \left (c x, \sqrt {c x + 1} \sqrt {-c x + 1}\right ) + a b c x^{3}\right )} \int \frac {c^{6} x^{6} - c^{4} x^{4} - c^{2} x^{2} + 1}{b^{2} c x^{4} \arctan \left (c x, \sqrt {c x + 1} \sqrt {-c x + 1}\right ) + a b c x^{4}}\,{d x} - 1}{b^{2} c x^{3} \arctan \left (c x, \sqrt {c x + 1} \sqrt {-c x + 1}\right ) + a b c x^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-c^2*x^2+1)^(5/2)/x^3/(a+b*arcsin(c*x))^2,x, algorithm="maxima")

[Out]

(c^6*x^6 - 3*c^4*x^4 + 3*c^2*x^2 - (b^2*c*x^3*arctan2(c*x, sqrt(c*x + 1)*sqrt(-c*x + 1)) + a*b*c*x^3)*integrat
e(3*(c^6*x^6 - c^4*x^4 - c^2*x^2 + 1)/(b^2*c*x^4*arctan2(c*x, sqrt(c*x + 1)*sqrt(-c*x + 1)) + a*b*c*x^4), x) -
1)/(b^2*c*x^3*arctan2(c*x, sqrt(c*x + 1)*sqrt(-c*x + 1)) + a*b*c*x^3)

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mupad [A]  time = 0.00, size = -1, normalized size = -0.03 \[ \int \frac {{\left (1-c^2\,x^2\right )}^{5/2}}{x^3\,{\left (a+b\,\mathrm {asin}\left (c\,x\right )\right )}^2} \,d x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((1 - c^2*x^2)^(5/2)/(x^3*(a + b*asin(c*x))^2),x)

[Out]

int((1 - c^2*x^2)^(5/2)/(x^3*(a + b*asin(c*x))^2), x)

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sympy [A]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\left (- \left (c x - 1\right ) \left (c x + 1\right )\right )^{\frac {5}{2}}}{x^{3} \left (a + b \operatorname {asin}{\left (c x \right )}\right )^{2}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-c**2*x**2+1)**(5/2)/x**3/(a+b*asin(c*x))**2,x)

[Out]

Integral((-(c*x - 1)*(c*x + 1))**(5/2)/(x**3*(a + b*asin(c*x))**2), x)

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