\(\int (f x)^m \sqrt {d+e x^2} (a+b \sec ^{-1}(c x)) \, dx\) [167]

Optimal result

Integrand size = 25, antiderivative size = 25 \[ \int (f x)^m \sqrt {d+e x^2} \left (a+b \sec ^{-1}(c x)\right ) \, dx=\text {Int}\left ((f x)^m \sqrt {d+e x^2} \left (a+b \sec ^{-1}(c x)\right ),x\right ) \]

[Out]

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

Rubi [N/A]

Not integrable

Time = 0.07 (sec) , antiderivative size = 25, normalized size of antiderivative = 1.00, number of steps used = 0, number of rules used = 0, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.000, Rules used = {} \[ \int (f x)^m \sqrt {d+e x^2} \left (a+b \sec ^{-1}(c x)\right ) \, dx=\int (f x)^m \sqrt {d+e x^2} \left (a+b \sec ^{-1}(c x)\right ) \, dx \]

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

Mathematica [N/A]

Not integrable

Time = 0.12 (sec) , antiderivative size = 27, normalized size of antiderivative = 1.08 \[ \int (f x)^m \sqrt {d+e x^2} \left (a+b \sec ^{-1}(c x)\right ) \, dx=\int (f x)^m \sqrt {d+e x^2} \left (a+b \sec ^{-1}(c x)\right ) \, dx \]

[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]

Maple [N/A] (verified)

Not integrable

Time = 0.90 (sec) , antiderivative size = 23, normalized size of antiderivative = 0.92

\[\int \left (f x \right )^{m} \sqrt {e \,x^{2}+d}\, \left (a +b \,\operatorname {arcsec}\left (c x \right )\right )d x\]

[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)

Fricas [N/A]

Not integrable

Time = 0.28 (sec) , antiderivative size = 25, normalized size of antiderivative = 1.00 \[ \int (f x)^m \sqrt {d+e x^2} \left (a+b \sec ^{-1}(c x)\right ) \, dx=\int { \sqrt {e x^{2} + d} {\left (b \operatorname {arcsec}\left (c x\right ) + a\right )} \left (f x\right )^{m} \,d x } \]

[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)

Sympy [N/A]

Not integrable

Time = 52.55 (sec) , antiderivative size = 24, normalized size of antiderivative = 0.96 \[ \int (f x)^m \sqrt {d+e x^2} \left (a+b \sec ^{-1}(c x)\right ) \, dx=\int \left (f x\right )^{m} \left (a + b \operatorname {asec}{\left (c x \right )}\right ) \sqrt {d + e x^{2}}\, dx \]

[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)

Maxima [N/A]

Not integrable

Time = 0.38 (sec) , antiderivative size = 25, normalized size of antiderivative = 1.00 \[ \int (f x)^m \sqrt {d+e x^2} \left (a+b \sec ^{-1}(c x)\right ) \, dx=\int { \sqrt {e x^{2} + d} {\left (b \operatorname {arcsec}\left (c x\right ) + a\right )} \left (f x\right )^{m} \,d x } \]

[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)

Giac [N/A]

Not integrable

Time = 0.35 (sec) , antiderivative size = 25, normalized size of antiderivative = 1.00 \[ \int (f x)^m \sqrt {d+e x^2} \left (a+b \sec ^{-1}(c x)\right ) \, dx=\int { \sqrt {e x^{2} + d} {\left (b \operatorname {arcsec}\left (c x\right ) + a\right )} \left (f x\right )^{m} \,d x } \]

[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)

Mupad [N/A]

Not integrable

Time = 0.88 (sec) , antiderivative size = 29, normalized size of antiderivative = 1.16 \[ \int (f x)^m \sqrt {d+e x^2} \left (a+b \sec ^{-1}(c x)\right ) \, dx=\int {\left (f\,x\right )}^m\,\sqrt {e\,x^2+d}\,\left (a+b\,\mathrm {acos}\left (\frac {1}{c\,x}\right )\right ) \,d x \]

[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)