3.146 \(\int \frac {x^m (a+b \sin ^{-1}(c x))}{d-c^2 d x^2} \, dx\)

Optimal. Leaf size=28 \[ \text {Int}\left (\frac {x^m \left (a+b \sin ^{-1}(c x)\right )}{d-c^2 d x^2},x\right ) \]

[Out]

Unintegrable(x^m*(a+b*arcsin(c*x))/(-c^2*d*x^2+d),x)

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Rubi [A]  time = 0.06, 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 = {} \[ \int \frac {x^m \left (a+b \sin ^{-1}(c x)\right )}{d-c^2 d x^2} \, dx \]

Verification is Not applicable to the result.

[In]

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

[Out]

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

Rubi steps

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

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Mathematica [A]  time = 4.01, size = 0, normalized size = 0.00 \[ \int \frac {x^m \left (a+b \sin ^{-1}(c x)\right )}{d-c^2 d x^2} \, dx \]

Verification is Not applicable to the result.

[In]

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

[Out]

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

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fricas [A]  time = 0.73, size = 0, normalized size = 0.00 \[ {\rm integral}\left (-\frac {{\left (b \arcsin \left (c x\right ) + a\right )} x^{m}}{c^{2} d x^{2} - d}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maple [A]  time = 0.65, size = 0, normalized size = 0.00 \[ \int \frac {x^{m} \left (a +b \arcsin \left (c x \right )\right )}{-c^{2} d \,x^{2}+d}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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mupad [A]  time = 0.00, size = -1, normalized size = -0.04 \[ \int \frac {x^m\,\left (a+b\,\mathrm {asin}\left (c\,x\right )\right )}{d-c^2\,d\,x^2} \,d x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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