\(\int \frac {1}{(d+e x^2) (a+b \text {arcsinh}(c x))^{3/2}} \, dx\) [645]

Optimal result

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

[Out]

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

Rubi [N/A]

Not integrable

Time = 0.06 (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 = {} \[ \int \frac {1}{\left (d+e x^2\right ) (a+b \text {arcsinh}(c x))^{3/2}} \, dx=\int \frac {1}{\left (d+e x^2\right ) (a+b \text {arcsinh}(c x))^{3/2}} \, dx \]

[In]

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

[Out]

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

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

Mathematica [N/A]

Not integrable

Time = 0.17 (sec) , antiderivative size = 24, normalized size of antiderivative = 1.09 \[ \int \frac {1}{\left (d+e x^2\right ) (a+b \text {arcsinh}(c x))^{3/2}} \, dx=\int \frac {1}{\left (d+e x^2\right ) (a+b \text {arcsinh}(c x))^{3/2}} \, dx \]

[In]

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

[Out]

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

Maple [N/A] (verified)

Not integrable

Time = 0.58 (sec) , antiderivative size = 20, normalized size of antiderivative = 0.91

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

[In]

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

[Out]

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

Fricas [F(-2)]

Exception generated. \[ \int \frac {1}{\left (d+e x^2\right ) (a+b \text {arcsinh}(c x))^{3/2}} \, dx=\text {Exception raised: TypeError} \]

[In]

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

[Out]

Exception raised: TypeError >>  Error detected within library code:   integrate: implementation incomplete (co
nstant residues)

Sympy [N/A]

Not integrable

Time = 7.80 (sec) , antiderivative size = 20, normalized size of antiderivative = 0.91 \[ \int \frac {1}{\left (d+e x^2\right ) (a+b \text {arcsinh}(c x))^{3/2}} \, dx=\int \frac {1}{\left (a + b \operatorname {asinh}{\left (c x \right )}\right )^{\frac {3}{2}} \left (d + e x^{2}\right )}\, dx \]

[In]

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

[Out]

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

Maxima [N/A]

Not integrable

Time = 0.38 (sec) , antiderivative size = 22, normalized size of antiderivative = 1.00 \[ \int \frac {1}{\left (d+e x^2\right ) (a+b \text {arcsinh}(c x))^{3/2}} \, dx=\int { \frac {1}{{\left (e x^{2} + d\right )} {\left (b \operatorname {arsinh}\left (c x\right ) + a\right )}^{\frac {3}{2}}} \,d x } \]

[In]

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

[Out]

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

Giac [N/A]

Not integrable

Time = 0.39 (sec) , antiderivative size = 22, normalized size of antiderivative = 1.00 \[ \int \frac {1}{\left (d+e x^2\right ) (a+b \text {arcsinh}(c x))^{3/2}} \, dx=\int { \frac {1}{{\left (e x^{2} + d\right )} {\left (b \operatorname {arsinh}\left (c x\right ) + a\right )}^{\frac {3}{2}}} \,d x } \]

[In]

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

[Out]

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

Mupad [N/A]

Not integrable

Time = 2.71 (sec) , antiderivative size = 22, normalized size of antiderivative = 1.00 \[ \int \frac {1}{\left (d+e x^2\right ) (a+b \text {arcsinh}(c x))^{3/2}} \, dx=\int \frac {1}{{\left (a+b\,\mathrm {asinh}\left (c\,x\right )\right )}^{3/2}\,\left (e\,x^2+d\right )} \,d x \]

[In]

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

[Out]

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