3.116 \(\int x^2 \sqrt{d+e x^2} (a+b \sec ^{-1}(c x)) \, dx\)

Optimal. Leaf size=25 \[ \text{Unintegrable}\left (x^2 \sqrt{d+e x^2} \left (a+b \sec ^{-1}(c x)\right ),x\right ) \]

[Out]

Unintegrable[x^2*Sqrt[d + e*x^2]*(a + b*ArcSec[c*x]), x]

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Rubi [A]  time = 0.0936754, antiderivative size = 0, normalized size of antiderivative = 0., number of steps used = 0, number of rules used = 0, integrand size = 0, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0., Rules used = {} \[ \int x^2 \sqrt{d+e x^2} \left (a+b \sec ^{-1}(c x)\right ) \, dx \]

Verification is Not applicable to the result.

[In]

Int[x^2*Sqrt[d + e*x^2]*(a + b*ArcSec[c*x]),x]

[Out]

Defer[Int][x^2*Sqrt[d + e*x^2]*(a + b*ArcSec[c*x]), x]

Rubi steps

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

Mathematica [A]  time = 8.82391, size = 0, normalized size = 0. \[ \int x^2 \sqrt{d+e x^2} \left (a+b \sec ^{-1}(c x)\right ) \, dx \]

Verification is Not applicable to the result.

[In]

Integrate[x^2*Sqrt[d + e*x^2]*(a + b*ArcSec[c*x]),x]

[Out]

Integrate[x^2*Sqrt[d + e*x^2]*(a + b*ArcSec[c*x]), x]

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Maple [A]  time = 1.629, size = 0, normalized size = 0. \begin{align*} \int{x}^{2} \left ( a+b{\rm arcsec} \left (cx\right ) \right ) \sqrt{e{x}^{2}+d}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [F(-2)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: ValueError} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^2*(a+b*arcsec(c*x))*(e*x^2+d)^(1/2),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left ({\left (b x^{2} \operatorname{arcsec}\left (c x\right ) + a x^{2}\right )} \sqrt{e x^{2} + d}, x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Timed out

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Giac [A]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \sqrt{e x^{2} + d}{\left (b \operatorname{arcsec}\left (c x\right ) + a\right )} x^{2}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^2*(a+b*arcsec(c*x))*(e*x^2+d)^(1/2),x, algorithm="giac")

[Out]

integrate(sqrt(e*x^2 + d)*(b*arcsec(c*x) + a)*x^2, x)