3.145 \(\int (d+e x^2)^{3/2} (a+b \text{sech}^{-1}(c x)) \, dx\)

Optimal. Leaf size=22 \[ \text{Unintegrable}\left (\left (d+e x^2\right )^{3/2} \left (a+b \text{sech}^{-1}(c x)\right ),x\right ) \]

[Out]

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

________________________________________________________________________________________

Rubi [A]  time = 0.0444486, 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 = {} \[ \int \left (d+e x^2\right )^{3/2} \left (a+b \text{sech}^{-1}(c x)\right ) \, dx \]

Verification is Not applicable to the result.

[In]

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

[Out]

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

Rubi steps

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

Mathematica [A]  time = 4.29346, size = 0, normalized size = 0. \[ \int \left (d+e x^2\right )^{3/2} \left (a+b \text{sech}^{-1}(c x)\right ) \, dx \]

Verification is Not applicable to the result.

[In]

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

[Out]

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

________________________________________________________________________________________

Maple [A]  time = 0.824, size = 0, normalized size = 0. \begin{align*} \int \left ( e{x}^{2}+d \right ) ^{{\frac{3}{2}}} \left ( a+b{\rm arcsech} \left (cx\right ) \right ) \, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

Maxima [F(-2)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: ValueError} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Exception raised: ValueError

________________________________________________________________________________________

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Timed out

________________________________________________________________________________________

Giac [A]  time = 0., size = 0, normalized size = 0. \begin{align*} \int{\left (e x^{2} + d\right )}^{\frac{3}{2}}{\left (b \operatorname{arsech}\left (c x\right ) + a\right )}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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