\(\int \frac {x^4 (a+b \sec ^{-1}(c x))}{(d+e x^2)^{3/2}} \, dx\) [146]

Optimal result

Integrand size = 23, antiderivative size = 23 \[ \int \frac {x^4 \left (a+b \sec ^{-1}(c x)\right )}{\left (d+e x^2\right )^{3/2}} \, dx=\text {Int}\left (\frac {x^4 \left (a+b \sec ^{-1}(c x)\right )}{\left (d+e x^2\right )^{3/2}},x\right ) \]

[Out]

Unintegrable(x^4*(a+b*arcsec(c*x))/(e*x^2+d)^(3/2),x)

Rubi [N/A]

Not integrable

Time = 0.08 (sec) , antiderivative size = 23, 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 = {} \[ \int \frac {x^4 \left (a+b \sec ^{-1}(c x)\right )}{\left (d+e x^2\right )^{3/2}} \, dx=\int \frac {x^4 \left (a+b \sec ^{-1}(c x)\right )}{\left (d+e x^2\right )^{3/2}} \, dx \]

[In]

Int[(x^4*(a + b*ArcSec[c*x]))/(d + e*x^2)^(3/2),x]

[Out]

Defer[Int][(x^4*(a + b*ArcSec[c*x]))/(d + e*x^2)^(3/2), x]

Rubi steps \begin{align*} \text {integral}& = \int \frac {x^4 \left (a+b \sec ^{-1}(c x)\right )}{\left (d+e x^2\right )^{3/2}} \, dx \\ \end{align*}

Mathematica [N/A]

Not integrable

Time = 14.41 (sec) , antiderivative size = 25, normalized size of antiderivative = 1.09 \[ \int \frac {x^4 \left (a+b \sec ^{-1}(c x)\right )}{\left (d+e x^2\right )^{3/2}} \, dx=\int \frac {x^4 \left (a+b \sec ^{-1}(c x)\right )}{\left (d+e x^2\right )^{3/2}} \, dx \]

[In]

Integrate[(x^4*(a + b*ArcSec[c*x]))/(d + e*x^2)^(3/2),x]

[Out]

Integrate[(x^4*(a + b*ArcSec[c*x]))/(d + e*x^2)^(3/2), x]

Maple [N/A] (verified)

Not integrable

Time = 1.35 (sec) , antiderivative size = 21, normalized size of antiderivative = 0.91

\[\int \frac {x^{4} \left (a +b \,\operatorname {arcsec}\left (c x \right )\right )}{\left (e \,x^{2}+d \right )^{\frac {3}{2}}}d x\]

[In]

int(x^4*(a+b*arcsec(c*x))/(e*x^2+d)^(3/2),x)

[Out]

int(x^4*(a+b*arcsec(c*x))/(e*x^2+d)^(3/2),x)

Fricas [N/A]

Not integrable

Time = 0.26 (sec) , antiderivative size = 47, normalized size of antiderivative = 2.04 \[ \int \frac {x^4 \left (a+b \sec ^{-1}(c x)\right )}{\left (d+e x^2\right )^{3/2}} \, dx=\int { \frac {{\left (b \operatorname {arcsec}\left (c x\right ) + a\right )} x^{4}}{{\left (e x^{2} + d\right )}^{\frac {3}{2}}} \,d x } \]

[In]

integrate(x^4*(a+b*arcsec(c*x))/(e*x^2+d)^(3/2),x, algorithm="fricas")

[Out]

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

Sympy [N/A]

Not integrable

Time = 144.52 (sec) , antiderivative size = 22, normalized size of antiderivative = 0.96 \[ \int \frac {x^4 \left (a+b \sec ^{-1}(c x)\right )}{\left (d+e x^2\right )^{3/2}} \, dx=\int \frac {x^{4} \left (a + b \operatorname {asec}{\left (c x \right )}\right )}{\left (d + e x^{2}\right )^{\frac {3}{2}}}\, dx \]

[In]

integrate(x**4*(a+b*asec(c*x))/(e*x**2+d)**(3/2),x)

[Out]

Integral(x**4*(a + b*asec(c*x))/(d + e*x**2)**(3/2), x)

Maxima [F(-2)]

Exception generated. \[ \int \frac {x^4 \left (a+b \sec ^{-1}(c x)\right )}{\left (d+e x^2\right )^{3/2}} \, dx=\text {Exception raised: ValueError} \]

[In]

integrate(x^4*(a+b*arcsec(c*x))/(e*x^2+d)^(3/2),x, algorithm="maxima")

[Out]

Exception raised: ValueError >> Computation failed since Maxima requested additional constraints; using the 'a
ssume' command before evaluation *may* help (example of legal syntax is 'assume(e>0)', see `assume?` for more
details)Is e

Giac [N/A]

Not integrable

Time = 0.39 (sec) , antiderivative size = 23, normalized size of antiderivative = 1.00 \[ \int \frac {x^4 \left (a+b \sec ^{-1}(c x)\right )}{\left (d+e x^2\right )^{3/2}} \, dx=\int { \frac {{\left (b \operatorname {arcsec}\left (c x\right ) + a\right )} x^{4}}{{\left (e x^{2} + d\right )}^{\frac {3}{2}}} \,d x } \]

[In]

integrate(x^4*(a+b*arcsec(c*x))/(e*x^2+d)^(3/2),x, algorithm="giac")

[Out]

integrate((b*arcsec(c*x) + a)*x^4/(e*x^2 + d)^(3/2), x)

Mupad [N/A]

Not integrable

Time = 1.50 (sec) , antiderivative size = 27, normalized size of antiderivative = 1.17 \[ \int \frac {x^4 \left (a+b \sec ^{-1}(c x)\right )}{\left (d+e x^2\right )^{3/2}} \, dx=\int \frac {x^4\,\left (a+b\,\mathrm {acos}\left (\frac {1}{c\,x}\right )\right )}{{\left (e\,x^2+d\right )}^{3/2}} \,d x \]

[In]

int((x^4*(a + b*acos(1/(c*x))))/(d + e*x^2)^(3/2),x)

[Out]

int((x^4*(a + b*acos(1/(c*x))))/(d + e*x^2)^(3/2), x)