3.7.76 \(\int \frac {1}{(d+e x^2)^{5/2} (a+b \text {ArcSin}(c x))} \, dx\) [676]

Optimal. Leaf size=25 \[ \text {Int}\left (\frac {1}{\left (d+e x^2\right )^{5/2} (a+b \text {ArcSin}(c x))},x\right ) \]

[Out]

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

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Rubi [A]
time = 0.03, 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 {1}{\left (d+e x^2\right )^{5/2} (a+b \text {ArcSin}(c x))} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

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

[Out]

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

Rubi steps

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

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Mathematica [A]
time = 2.69, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {1}{\left (d+e x^2\right )^{5/2} (a+b \text {ArcSin}(c x))} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

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

[Out]

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

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Maple [A]
time = 0.42, size = 0, normalized size = 0.00 \[\int \frac {1}{\left (e \,x^{2}+d \right )^{\frac {5}{2}} \left (a +b \arcsin \left (c x \right )\right )}\, dx\]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

int(1/(e*x^2+d)^(5/2)/(a+b*arcsin(c*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(1/(e*x^2+d)^(5/2)/(a+b*arcsin(c*x)),x, algorithm="maxima")

[Out]

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

[Out]

integral(sqrt(x^2*e + d)/(a*x^6*e^3 + 3*a*d*x^4*e^2 + 3*a*d^2*x^2*e + a*d^3 + (b*x^6*e^3 + 3*b*d*x^4*e^2 + 3*b
*d^2*x^2*e + b*d^3)*arcsin(c*x)), x)

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Sympy [A]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {1}{\left (a + b \operatorname {asin}{\left (c x \right )}\right ) \left (d + e x^{2}\right )^{\frac {5}{2}}}\, dx \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Giac [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(1/(e*x^2+d)^(5/2)/(a+b*arcsin(c*x)),x, algorithm="giac")

[Out]

integrate(1/((e*x^2 + d)^(5/2)*(b*arcsin(c*x) + a)), x)

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Mupad [A]
time = 0.00, size = -1, normalized size = -0.04 \begin {gather*} \int \frac {1}{\left (a+b\,\mathrm {asin}\left (c\,x\right )\right )\,{\left (e\,x^2+d\right )}^{5/2}} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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