3.184 \(\int \frac{(f x)^m (a+b \text{sech}^{-1}(c x))}{\sqrt{d+e x^2}} \, dx\)

Optimal. Leaf size=27 \[ \text{Unintegrable}\left (\frac{(f x)^m \left (a+b \text{sech}^{-1}(c x)\right )}{\sqrt{d+e x^2}},x\right ) \]

[Out]

Unintegrable[((f*x)^m*(a + b*ArcSech[c*x]))/Sqrt[d + e*x^2], x]

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Rubi [A]  time = 0.0999564, antiderivative size = 0, normalized size of antiderivative = 0., number of steps used = 0, number of rules used = 0, integrand size = 0, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0., Rules used = {} \[ \int \frac{(f x)^m \left (a+b \text{sech}^{-1}(c x)\right )}{\sqrt{d+e x^2}} \, dx \]

Verification is Not applicable to the result.

[In]

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

[Out]

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

Rubi steps

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

Mathematica [A]  time = 1.34371, size = 0, normalized size = 0. \[ \int \frac{(f x)^m \left (a+b \text{sech}^{-1}(c x)\right )}{\sqrt{d+e x^2}} \, dx \]

Verification is Not applicable to the result.

[In]

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

[Out]

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

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Maple [A]  time = 1.279, size = 0, normalized size = 0. \begin{align*} \int{ \left ( fx \right ) ^{m} \left ( a+b{\rm arcsech} \left (cx\right ) \right ){\frac{1}{\sqrt{e{x}^{2}+d}}}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [A]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{{\left (b \operatorname{arsech}\left (c x\right ) + a\right )} \left (f x\right )^{m}}{\sqrt{e x^{2} + d}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((f*x)^m*(a+b*arcsech(c*x))/(e*x^2+d)^(1/2),x, algorithm="maxima")

[Out]

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

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Fricas [A]  time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (\frac{{\left (b \operatorname{arsech}\left (c x\right ) + a\right )} \left (f x\right )^{m}}{\sqrt{e x^{2} + d}}, x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((f*x)^m*(a+b*arcsech(c*x))/(e*x^2+d)^(1/2),x, algorithm="fricas")

[Out]

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

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Sympy [A]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\left (f x\right )^{m} \left (a + b \operatorname{asech}{\left (c x \right )}\right )}{\sqrt{d + e x^{2}}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((f*x)**m*(a+b*asech(c*x))/(e*x**2+d)**(1/2),x)

[Out]

Integral((f*x)**m*(a + b*asech(c*x))/sqrt(d + e*x**2), x)

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Giac [A]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{{\left (b \operatorname{arsech}\left (c x\right ) + a\right )} \left (f x\right )^{m}}{\sqrt{e x^{2} + d}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((f*x)^m*(a+b*arcsech(c*x))/(e*x^2+d)^(1/2),x, algorithm="giac")

[Out]

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