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

Optimal result

Integrand size = 25, antiderivative size = 25 \[ \int (f x)^m \left (d+e x^2\right )^{3/2} \left (a+b \sec ^{-1}(c x)\right ) \, dx=\text {Int}\left ((f x)^m \left (d+e x^2\right )^{3/2} \left (a+b \sec ^{-1}(c x)\right ),x\right ) \]

[Out]

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

Rubi [N/A]

Not integrable

Time = 0.08 (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 = {} \[ \int (f x)^m \left (d+e x^2\right )^{3/2} \left (a+b \sec ^{-1}(c x)\right ) \, dx=\int (f x)^m \left (d+e x^2\right )^{3/2} \left (a+b \sec ^{-1}(c x)\right ) \, dx \]

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

Mathematica [N/A]

Not integrable

Time = 0.96 (sec) , antiderivative size = 27, normalized size of antiderivative = 1.08 \[ \int (f x)^m \left (d+e x^2\right )^{3/2} \left (a+b \sec ^{-1}(c x)\right ) \, dx=\int (f x)^m \left (d+e x^2\right )^{3/2} \left (a+b \sec ^{-1}(c x)\right ) \, dx \]

[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]

Maple [N/A] (verified)

Not integrable

Time = 2.26 (sec) , antiderivative size = 23, normalized size of antiderivative = 0.92

\[\int \left (f x \right )^{m} \left (e \,x^{2}+d \right )^{\frac {3}{2}} \left (a +b \,\operatorname {arcsec}\left (c x \right )\right )d x\]

[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)

Fricas [N/A]

Not integrable

Time = 0.27 (sec) , antiderivative size = 42, normalized size of antiderivative = 1.68 \[ \int (f x)^m \left (d+e x^2\right )^{3/2} \left (a+b \sec ^{-1}(c x)\right ) \, dx=\int { {\left (e x^{2} + d\right )}^{\frac {3}{2}} {\left (b \operatorname {arcsec}\left (c x\right ) + a\right )} \left (f x\right )^{m} \,d x } \]

[In]

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

[Out]

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

Sympy [F(-1)]

Timed out. \[ \int (f x)^m \left (d+e x^2\right )^{3/2} \left (a+b \sec ^{-1}(c x)\right ) \, dx=\text {Timed out} \]

[In]

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

[Out]

Timed out

Maxima [N/A]

Not integrable

Time = 0.52 (sec) , antiderivative size = 25, normalized size of antiderivative = 1.00 \[ \int (f x)^m \left (d+e x^2\right )^{3/2} \left (a+b \sec ^{-1}(c x)\right ) \, dx=\int { {\left (e x^{2} + d\right )}^{\frac {3}{2}} {\left (b \operatorname {arcsec}\left (c x\right ) + a\right )} \left (f x\right )^{m} \,d x } \]

[In]

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

[Out]

integrate((e*x^2 + d)^(3/2)*(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 \[ \int (f x)^m \left (d+e x^2\right )^{3/2} \left (a+b \sec ^{-1}(c x)\right ) \, dx=\int { {\left (e x^{2} + d\right )}^{\frac {3}{2}} {\left (b \operatorname {arcsec}\left (c x\right ) + a\right )} \left (f x\right )^{m} \,d x } \]

[In]

integrate((f*x)^m*(e*x^2+d)^(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)

Mupad [N/A]

Not integrable

Time = 0.95 (sec) , antiderivative size = 29, normalized size of antiderivative = 1.16 \[ \int (f x)^m \left (d+e x^2\right )^{3/2} \left (a+b \sec ^{-1}(c x)\right ) \, dx=\int {\left (f\,x\right )}^m\,{\left (e\,x^2+d\right )}^{3/2}\,\left (a+b\,\mathrm {acos}\left (\frac {1}{c\,x}\right )\right ) \,d x \]

[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)