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

Optimal. Leaf size=26 \[ \text {Int}\left (\frac {a+b \sec ^{-1}(c x)}{x \left (d+e x^2\right )^{3/2}},x\right ) \]

[Out]

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

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Rubi [A]  time = 0.11, 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 = {} \[ \int \frac {a+b \sec ^{-1}(c x)}{x \left (d+e x^2\right )^{3/2}} \, dx \]

Verification is Not applicable to the result.

[In]

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

[Out]

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

Rubi steps

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

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Mathematica [A]  time = 15.67, size = 0, normalized size = 0.00 \[ \int \frac {a+b \sec ^{-1}(c x)}{x \left (d+e x^2\right )^{3/2}} \, dx \]

Verification is Not applicable to the result.

[In]

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

[Out]

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

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fricas [A]  time = 0.61, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {\sqrt {e x^{2} + d} {\left (b \operatorname {arcsec}\left (c x\right ) + a\right )}}{e^{2} x^{5} + 2 \, d e x^{3} + d^{2} x}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integral(sqrt(e*x^2 + d)*(b*arcsec(c*x) + a)/(e^2*x^5 + 2*d*e*x^3 + d^2*x), x)

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giac [A]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {b \operatorname {arcsec}\left (c x\right ) + a}{{\left (e x^{2} + d\right )}^{\frac {3}{2}} x}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maple [A]  time = 2.14, size = 0, normalized size = 0.00 \[ \int \frac {a +b \,\mathrm {arcsec}\left (c x \right )}{x \left (e \,x^{2}+d \right )^{\frac {3}{2}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maxima [A]  time = 0.00, size = 0, normalized size = 0.00 \[ -a {\left (\frac {\operatorname {arsinh}\left (\frac {d}{\sqrt {d e} {\left | x \right |}}\right )}{d^{\frac {3}{2}}} - \frac {1}{\sqrt {e x^{2} + d} d}\right )} + b \int \frac {\arctan \left (\sqrt {c x + 1} \sqrt {c x - 1}\right )}{{\left (e x^{3} + d x\right )} \sqrt {e x^{2} + d}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-a*(arcsinh(d/(sqrt(d*e)*abs(x)))/d^(3/2) - 1/(sqrt(e*x^2 + d)*d)) + b*integrate(arctan(sqrt(c*x + 1)*sqrt(c*x
- 1))/((e*x^3 + d*x)*sqrt(e*x^2 + d)), x)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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sympy [F(-1)]  time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Timed out

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