3.2.66 \(\int (f x)^m (d+e x^2)^{3/2} (a+b \sec ^{-1}(c x)) \, dx\) [166]

Optimal. Leaf size=28 \[ \text {Int}\left ((f x)^m \left (d+e x^2\right )^{3/2} \left (a+b \sec ^{-1}(c x)\right ),x\right ) \]

[Out]

Unintegrable((f*x)^m*(e*x^2+d)^(3/2)*(a+b*arcsec(c*x)),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 (f x)^m \left (d+e x^2\right )^{3/2} \left (a+b \sec ^{-1}(c x)\right ) \, dx \end {gather*}

Verification is not applicable to the result.

[In]

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

[Out]

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

Rubi steps

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

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Mathematica [A]
time = 0.50, size = 0, normalized size = 0.00 \begin {gather*} \int (f x)^m \left (d+e x^2\right )^{3/2} \left (a+b \sec ^{-1}(c x)\right ) \, dx \end {gather*}

Verification is not applicable to the result.

[In]

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

[Out]

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

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Maple [A]
time = 1.35, size = 0, normalized size = 0.00 \[\int \left (f x \right )^{m} \left (e \,x^{2}+d \right )^{\frac {3}{2}} \left (a +b \,\mathrm {arcsec}\left (c x \right )\right )\, dx\]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

int((f*x)^m*(e*x^2+d)^(3/2)*(a+b*arcsec(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((f*x)^m*(e*x^2+d)^(3/2)*(a+b*arcsec(c*x)),x, algorithm="maxima")

[Out]

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

[Out]

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

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Sympy [F(-2)]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: SystemError} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((f*x)**m*(e*x**2+d)**(3/2)*(a+b*asec(c*x)),x)

[Out]

Exception raised: SystemError >> excessive stack use: stack is 4369 deep

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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((f*x)^m*(e*x^2+d)^(3/2)*(a+b*arcsec(c*x)),x, algorithm="giac")

[Out]

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

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Mupad [A]
time = 0.00, size = -1, normalized size = -0.04 \begin {gather*} \int {\left (f\,x\right )}^m\,{\left (e\,x^2+d\right )}^{3/2}\,\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((f*x)^m*(d + e*x^2)^(3/2)*(a + b*acos(1/(c*x))),x)

[Out]

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

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