\(\int \sqrt {d+e x^2} (a+b \text {arcsinh}(c x)) \, dx\) [647]

Optimal result

Integrand size = 20, antiderivative size = 20 \[ \int \sqrt {d+e x^2} (a+b \text {arcsinh}(c x)) \, dx=\text {Int}\left (\sqrt {d+e x^2} (a+b \text {arcsinh}(c x)),x\right ) \]

[Out]

Unintegrable((a+b*arcsinh(c*x))*(e*x^2+d)^(1/2),x)

Rubi [N/A]

Not integrable

Time = 0.02 (sec) , antiderivative size = 20, 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 = {} \[ \int \sqrt {d+e x^2} (a+b \text {arcsinh}(c x)) \, dx=\int \sqrt {d+e x^2} (a+b \text {arcsinh}(c x)) \, dx \]

[In]

Int[Sqrt[d + e*x^2]*(a + b*ArcSinh[c*x]),x]

[Out]

Defer[Int][Sqrt[d + e*x^2]*(a + b*ArcSinh[c*x]), x]

Rubi steps \begin{align*} \text {integral}& = \int \sqrt {d+e x^2} (a+b \text {arcsinh}(c x)) \, dx \\ \end{align*}

Mathematica [N/A]

Not integrable

Time = 3.63 (sec) , antiderivative size = 22, normalized size of antiderivative = 1.10 \[ \int \sqrt {d+e x^2} (a+b \text {arcsinh}(c x)) \, dx=\int \sqrt {d+e x^2} (a+b \text {arcsinh}(c x)) \, dx \]

[In]

Integrate[Sqrt[d + e*x^2]*(a + b*ArcSinh[c*x]),x]

[Out]

Integrate[Sqrt[d + e*x^2]*(a + b*ArcSinh[c*x]), x]

Maple [N/A] (verified)

Not integrable

Time = 0.55 (sec) , antiderivative size = 18, normalized size of antiderivative = 0.90

\[\int \left (a +b \,\operatorname {arcsinh}\left (c x \right )\right ) \sqrt {e \,x^{2}+d}d x\]

[In]

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

[Out]

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

Fricas [N/A]

Not integrable

Time = 0.25 (sec) , antiderivative size = 20, normalized size of antiderivative = 1.00 \[ \int \sqrt {d+e x^2} (a+b \text {arcsinh}(c x)) \, dx=\int { \sqrt {e x^{2} + d} {\left (b \operatorname {arsinh}\left (c x\right ) + a\right )} \,d x } \]

[In]

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

[Out]

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

Sympy [N/A]

Not integrable

Time = 1.44 (sec) , antiderivative size = 19, normalized size of antiderivative = 0.95 \[ \int \sqrt {d+e x^2} (a+b \text {arcsinh}(c x)) \, dx=\int \left (a + b \operatorname {asinh}{\left (c x \right )}\right ) \sqrt {d + e x^{2}}\, dx \]

[In]

integrate((a+b*asinh(c*x))*(e*x**2+d)**(1/2),x)

[Out]

Integral((a + b*asinh(c*x))*sqrt(d + e*x**2), x)

Maxima [F(-2)]

Exception generated. \[ \int \sqrt {d+e x^2} (a+b \text {arcsinh}(c x)) \, dx=\text {Exception raised: ValueError} \]

[In]

integrate((a+b*arcsinh(c*x))*(e*x^2+d)^(1/2),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 = 0.33 (sec) , antiderivative size = 20, normalized size of antiderivative = 1.00 \[ \int \sqrt {d+e x^2} (a+b \text {arcsinh}(c x)) \, dx=\int { \sqrt {e x^{2} + d} {\left (b \operatorname {arsinh}\left (c x\right ) + a\right )} \,d x } \]

[In]

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

[Out]

integrate(sqrt(e*x^2 + d)*(b*arcsinh(c*x) + a), x)

Mupad [N/A]

Not integrable

Time = 2.97 (sec) , antiderivative size = 20, normalized size of antiderivative = 1.00 \[ \int \sqrt {d+e x^2} (a+b \text {arcsinh}(c x)) \, dx=\int \left (a+b\,\mathrm {asinh}\left (c\,x\right )\right )\,\sqrt {e\,x^2+d} \,d x \]

[In]

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

[Out]

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