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

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

[Out]

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

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Rubi [A]  time = 0.13, 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 = {} \[ \int \frac {1}{x \left (1-c^2 x^2\right )^{5/2} \left (a+b \sin ^{-1}(c x)\right )^2} \, dx \]

Verification is Not applicable to the result.

[In]

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

[Out]

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

Rubi steps

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

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Mathematica [A]  time = 86.25, size = 0, normalized size = 0.00 \[ \int \frac {1}{x \left (1-c^2 x^2\right )^{5/2} \left (a+b \sin ^{-1}(c x)\right )^2} \, dx \]

Verification is Not applicable to the result.

[In]

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

[Out]

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

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fricas [A]  time = 0.41, size = 0, normalized size = 0.00 \[ {\rm integral}\left (-\frac {\sqrt {-c^{2} x^{2} + 1}}{a^{2} c^{6} x^{7} - 3 \, a^{2} c^{4} x^{5} + 3 \, a^{2} c^{2} x^{3} - a^{2} x + {\left (b^{2} c^{6} x^{7} - 3 \, b^{2} c^{4} x^{5} + 3 \, b^{2} c^{2} x^{3} - b^{2} x\right )} \arcsin \left (c x\right )^{2} + 2 \, {\left (a b c^{6} x^{7} - 3 \, a b c^{4} x^{5} + 3 \, a b c^{2} x^{3} - a b x\right )} \arcsin \left (c x\right )}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integral(-sqrt(-c^2*x^2 + 1)/(a^2*c^6*x^7 - 3*a^2*c^4*x^5 + 3*a^2*c^2*x^3 - a^2*x + (b^2*c^6*x^7 - 3*b^2*c^4*x
^5 + 3*b^2*c^2*x^3 - b^2*x)*arcsin(c*x)^2 + 2*(a*b*c^6*x^7 - 3*a*b*c^4*x^5 + 3*a*b*c^2*x^3 - a*b*x)*arcsin(c*x
)), 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(1/x/(-c^2*x^2+1)^(5/2)/(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 = 7.73, size = 0, normalized size = 0.00 \[ \int \frac {1}{x \left (-c^{2} x^{2}+1\right )^{\frac {5}{2}} \left (a +b \arcsin \left (c x \right )\right )^{2}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maxima [F(-1)]  time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Timed out

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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