3.2.69 \(\int \frac {(f x)^m (a+b \sec ^{-1}(c x))}{(d+e x^2)^{3/2}} \, dx\) [169]

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

[Out]

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

________________________________________________________________________________________

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

Verification is not applicable to the result.

[In]

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

[Out]

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

Rubi steps

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

________________________________________________________________________________________

Mathematica [A]
time = 0.86, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {(f x)^m \left (a+b \sec ^{-1}(c x)\right )}{\left (d+e x^2\right )^{3/2}} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

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

[Out]

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

________________________________________________________________________________________

Maple [A]
time = 1.76, size = 0, normalized size = 0.00 \[\int \frac {\left (f x \right )^{m} \left (a +b \,\mathrm {arcsec}\left (c x \right )\right )}{\left (e \,x^{2}+d \right )^{\frac {3}{2}}}\, dx\]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

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

[Out]

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

________________________________________________________________________________________

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

[Out]

integral(sqrt(x^2*e + d)*(b*arcsec(c*x) + a)*(f*x)^m/(x^4*e^2 + 2*d*x^2*e + d^2), x)

________________________________________________________________________________________

Sympy [F(-1)] Timed out
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Timed out

________________________________________________________________________________________

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

[Out]

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

________________________________________________________________________________________

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________