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

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

[Out]

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

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Rubi [A]  time = 0.09, 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 = {} \[ \int (f x)^m \sqrt {d+e x^2} \left (a+b \sec ^{-1}(c x)\right ) \, dx \]

Verification is Not applicable to the result.

[In]

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

[Out]

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

Rubi steps

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

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Mathematica [A]  time = 0.11, size = 0, normalized size = 0.00 \[ \int (f x)^m \sqrt {d+e x^2} \left (a+b \sec ^{-1}(c x)\right ) \, dx \]

Verification is Not applicable to the result.

[In]

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

[Out]

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

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fricas [A]  time = 1.02, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\sqrt {e x^{2} + d} {\left (b \operatorname {arcsec}\left (c x\right ) + a\right )} \left (f x\right )^{m}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maple [A]  time = 2.80, size = 0, normalized size = 0.00 \[ \int \left (f x \right )^{m} \sqrt {e \,x^{2}+d}\, \left (a +b \,\mathrm {arcsec}\left (c x \right )\right )\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maxima [A]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \sqrt {e x^{2} + d} {\left (b \operatorname {arcsec}\left (c x\right ) + a\right )} \left (f x\right )^{m}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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sympy [A]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \left (f x\right )^{m} \left (a + b \operatorname {asec}{\left (c x \right )}\right ) \sqrt {d + e x^{2}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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