3.132 \(\int x^2 (d+e x^2)^{3/2} (a+b \csc ^{-1}(c x)) \, dx\)

Optimal. Leaf size=26 \[ \text {Int}\left (x^2 \left (d+e x^2\right )^{3/2} \left (a+b \csc ^{-1}(c x)\right ),x\right ) \]

[Out]

Unintegrable(x^2*(e*x^2+d)^(3/2)*(a+b*arccsc(c*x)),x)

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Rubi [A]  time = 0.12, 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 = {} \[ \int x^2 \left (d+e x^2\right )^{3/2} \left (a+b \csc ^{-1}(c x)\right ) \, dx \]

Verification is Not applicable to the result.

[In]

Int[x^2*(d + e*x^2)^(3/2)*(a + b*ArcCsc[c*x]),x]

[Out]

Defer[Int][x^2*(d + e*x^2)^(3/2)*(a + b*ArcCsc[c*x]), x]

Rubi steps

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

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Mathematica [A]  time = 9.91, size = 0, normalized size = 0.00 \[ \int x^2 \left (d+e x^2\right )^{3/2} \left (a+b \csc ^{-1}(c x)\right ) \, dx \]

Verification is Not applicable to the result.

[In]

Integrate[x^2*(d + e*x^2)^(3/2)*(a + b*ArcCsc[c*x]),x]

[Out]

Integrate[x^2*(d + e*x^2)^(3/2)*(a + b*ArcCsc[c*x]), x]

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fricas [A]  time = 0.86, size = 0, normalized size = 0.00 \[ {\rm integral}\left ({\left (a e x^{4} + a d x^{2} + {\left (b e x^{4} + b d x^{2}\right )} \operatorname {arccsc}\left (c x\right )\right )} \sqrt {e x^{2} + d}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^2*(e*x^2+d)^(3/2)*(a+b*arccsc(c*x)),x, algorithm="fricas")

[Out]

integral((a*e*x^4 + a*d*x^2 + (b*e*x^4 + b*d*x^2)*arccsc(c*x))*sqrt(e*x^2 + d), x)

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giac [A]  time = 0.00, size = 0, normalized size = 0.00 \[ \int {\left (e x^{2} + d\right )}^{\frac {3}{2}} {\left (b \operatorname {arccsc}\left (c x\right ) + a\right )} x^{2}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^2*(e*x^2+d)^(3/2)*(a+b*arccsc(c*x)),x, algorithm="giac")

[Out]

integrate((e*x^2 + d)^(3/2)*(b*arccsc(c*x) + a)*x^2, x)

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maple [A]  time = 5.86, size = 0, normalized size = 0.00 \[ \int x^{2} \left (e \,x^{2}+d \right )^{\frac {3}{2}} \left (a +b \,\mathrm {arccsc}\left (c x \right )\right )\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^2*(e*x^2+d)^(3/2)*(a+b*arccsc(c*x)),x)

[Out]

int(x^2*(e*x^2+d)^(3/2)*(a+b*arccsc(c*x)),x)

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maxima [A]  time = 0.00, size = 0, normalized size = 0.00 \[ \frac {1}{48} \, {\left (\frac {8 \, {\left (e x^{2} + d\right )}^{\frac {5}{2}} x}{e} - \frac {2 \, {\left (e x^{2} + d\right )}^{\frac {3}{2}} d x}{e} - \frac {3 \, \sqrt {e x^{2} + d} d^{2} x}{e} - \frac {3 \, d^{3} \operatorname {arsinh}\left (\frac {e x}{\sqrt {d e}}\right )}{e^{\frac {3}{2}}}\right )} a + b \int {\left (e x^{4} \arctan \left (1, \sqrt {c x + 1} \sqrt {c x - 1}\right ) + d x^{2} \arctan \left (1, \sqrt {c x + 1} \sqrt {c x - 1}\right )\right )} \sqrt {e x^{2} + d}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^2*(e*x^2+d)^(3/2)*(a+b*arccsc(c*x)),x, algorithm="maxima")

[Out]

1/48*(8*(e*x^2 + d)^(5/2)*x/e - 2*(e*x^2 + d)^(3/2)*d*x/e - 3*sqrt(e*x^2 + d)*d^2*x/e - 3*d^3*arcsinh(e*x/sqrt
(d*e))/e^(3/2))*a + b*integrate((e*x^4*arctan2(1, sqrt(c*x + 1)*sqrt(c*x - 1)) + d*x^2*arctan2(1, sqrt(c*x + 1
)*sqrt(c*x - 1)))*sqrt(e*x^2 + d), x)

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mupad [A]  time = 0.00, size = -1, normalized size = -0.04 \[ \int x^2\,{\left (e\,x^2+d\right )}^{3/2}\,\left (a+b\,\mathrm {asin}\left (\frac {1}{c\,x}\right )\right ) \,d x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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sympy [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(x**2*(e*x**2+d)**(3/2)*(a+b*acsc(c*x)),x)

[Out]

Timed out

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