\(\int \frac {1}{(d+e x^2)^{3/2} (a+b \text {arcsinh}(c x))} \, dx\) [658]

Optimal result

Integrand size = 22, antiderivative size = 22 \[ \int \frac {1}{\left (d+e x^2\right )^{3/2} (a+b \text {arcsinh}(c x))} \, dx=\text {Int}\left (\frac {1}{\left (d+e x^2\right )^{3/2} (a+b \text {arcsinh}(c x))},x\right ) \]

[Out]

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

Rubi [N/A]

Not integrable

Time = 0.04 (sec) , antiderivative size = 22, 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 = {} \[ \int \frac {1}{\left (d+e x^2\right )^{3/2} (a+b \text {arcsinh}(c x))} \, dx=\int \frac {1}{\left (d+e x^2\right )^{3/2} (a+b \text {arcsinh}(c x))} \, dx \]

[In]

Int[1/((d + e*x^2)^(3/2)*(a + b*ArcSinh[c*x])),x]

[Out]

Defer[Int][1/((d + e*x^2)^(3/2)*(a + b*ArcSinh[c*x])), x]

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

Mathematica [N/A]

Not integrable

Time = 1.40 (sec) , antiderivative size = 24, normalized size of antiderivative = 1.09 \[ \int \frac {1}{\left (d+e x^2\right )^{3/2} (a+b \text {arcsinh}(c x))} \, dx=\int \frac {1}{\left (d+e x^2\right )^{3/2} (a+b \text {arcsinh}(c x))} \, dx \]

[In]

Integrate[1/((d + e*x^2)^(3/2)*(a + b*ArcSinh[c*x])),x]

[Out]

Integrate[1/((d + e*x^2)^(3/2)*(a + b*ArcSinh[c*x])), x]

Maple [N/A] (verified)

Not integrable

Time = 0.66 (sec) , antiderivative size = 20, normalized size of antiderivative = 0.91

\[\int \frac {1}{\left (e \,x^{2}+d \right )^{\frac {3}{2}} \left (a +b \,\operatorname {arcsinh}\left (c x \right )\right )}d x\]

[In]

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

[Out]

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

Fricas [N/A]

Not integrable

Time = 0.24 (sec) , antiderivative size = 63, normalized size of antiderivative = 2.86 \[ \int \frac {1}{\left (d+e x^2\right )^{3/2} (a+b \text {arcsinh}(c x))} \, dx=\int { \frac {1}{{\left (e x^{2} + d\right )}^{\frac {3}{2}} {\left (b \operatorname {arsinh}\left (c x\right ) + a\right )}} \,d x } \]

[In]

integrate(1/(e*x^2+d)^(3/2)/(a+b*arcsinh(c*x)),x, algorithm="fricas")

[Out]

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

Sympy [N/A]

Not integrable

Time = 1.70 (sec) , antiderivative size = 20, normalized size of antiderivative = 0.91 \[ \int \frac {1}{\left (d+e x^2\right )^{3/2} (a+b \text {arcsinh}(c x))} \, dx=\int \frac {1}{\left (a + b \operatorname {asinh}{\left (c x \right )}\right ) \left (d + e x^{2}\right )^{\frac {3}{2}}}\, dx \]

[In]

integrate(1/(e*x**2+d)**(3/2)/(a+b*asinh(c*x)),x)

[Out]

Integral(1/((a + b*asinh(c*x))*(d + e*x**2)**(3/2)), x)

Maxima [N/A]

Not integrable

Time = 0.26 (sec) , antiderivative size = 22, normalized size of antiderivative = 1.00 \[ \int \frac {1}{\left (d+e x^2\right )^{3/2} (a+b \text {arcsinh}(c x))} \, dx=\int { \frac {1}{{\left (e x^{2} + d\right )}^{\frac {3}{2}} {\left (b \operatorname {arsinh}\left (c x\right ) + a\right )}} \,d x } \]

[In]

integrate(1/(e*x^2+d)^(3/2)/(a+b*arcsinh(c*x)),x, algorithm="maxima")

[Out]

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

Giac [N/A]

Not integrable

Time = 0.31 (sec) , antiderivative size = 22, normalized size of antiderivative = 1.00 \[ \int \frac {1}{\left (d+e x^2\right )^{3/2} (a+b \text {arcsinh}(c x))} \, dx=\int { \frac {1}{{\left (e x^{2} + d\right )}^{\frac {3}{2}} {\left (b \operatorname {arsinh}\left (c x\right ) + a\right )}} \,d x } \]

[In]

integrate(1/(e*x^2+d)^(3/2)/(a+b*arcsinh(c*x)),x, algorithm="giac")

[Out]

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

Mupad [N/A]

Not integrable

Time = 2.53 (sec) , antiderivative size = 22, normalized size of antiderivative = 1.00 \[ \int \frac {1}{\left (d+e x^2\right )^{3/2} (a+b \text {arcsinh}(c x))} \, dx=\int \frac {1}{\left (a+b\,\mathrm {asinh}\left (c\,x\right )\right )\,{\left (e\,x^2+d\right )}^{3/2}} \,d x \]

[In]

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

[Out]

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