3.2.16 \(\int x^2 \sqrt {d+e x^2} (a+b \sec ^{-1}(c x)) \, dx\) [116]

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

[Out]

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

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Rubi [A]
time = 0.07, 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 x^2 \sqrt {d+e x^2} \left (a+b \sec ^{-1}(c x)\right ) \, dx \end {gather*}

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*}

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Mathematica [A]
time = 7.48, size = 0, normalized size = 0.00 \begin {gather*} \int x^2 \sqrt {d+e x^2} \left (a+b \sec ^{-1}(c x)\right ) \, dx \end {gather*}

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 = 1.76, size = 0, normalized size = 0.00 \[\int x^{2} \left (a +b \,\mathrm {arcsec}\left (c x \right )\right ) \sqrt {e \,x^{2}+d}\, dx\]

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

[Out]

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

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

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]

Integral(x**2*(a + b*asec(c*x))*sqrt(d + e*x**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(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)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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