\(\int \frac {x^m (a+b \text {arcsinh}(c x))^2}{(d+c^2 d x^2)^{5/2}} \, dx\) [326]

Optimal result

Integrand size = 28, antiderivative size = 28 \[ \int \frac {x^m (a+b \text {arcsinh}(c x))^2}{\left (d+c^2 d x^2\right )^{5/2}} \, dx=\text {Int}\left (\frac {x^m (a+b \text {arcsinh}(c x))^2}{\left (d+c^2 d x^2\right )^{5/2}},x\right ) \]

[Out]

Unintegrable(x^m*(a+b*arcsinh(c*x))^2/(c^2*d*x^2+d)^(5/2),x)

Rubi [N/A]

Not integrable

Time = 0.11 (sec) , antiderivative size = 28, 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 = {} \[ \int \frac {x^m (a+b \text {arcsinh}(c x))^2}{\left (d+c^2 d x^2\right )^{5/2}} \, dx=\int \frac {x^m (a+b \text {arcsinh}(c x))^2}{\left (d+c^2 d x^2\right )^{5/2}} \, dx \]

[In]

Int[(x^m*(a + b*ArcSinh[c*x])^2)/(d + c^2*d*x^2)^(5/2),x]

[Out]

Defer[Int][(x^m*(a + b*ArcSinh[c*x])^2)/(d + c^2*d*x^2)^(5/2), x]

Rubi steps \begin{align*} \text {integral}& = \int \frac {x^m (a+b \text {arcsinh}(c x))^2}{\left (d+c^2 d x^2\right )^{5/2}} \, dx \\ \end{align*}

Mathematica [N/A]

Not integrable

Time = 4.60 (sec) , antiderivative size = 30, normalized size of antiderivative = 1.07 \[ \int \frac {x^m (a+b \text {arcsinh}(c x))^2}{\left (d+c^2 d x^2\right )^{5/2}} \, dx=\int \frac {x^m (a+b \text {arcsinh}(c x))^2}{\left (d+c^2 d x^2\right )^{5/2}} \, dx \]

[In]

Integrate[(x^m*(a + b*ArcSinh[c*x])^2)/(d + c^2*d*x^2)^(5/2),x]

[Out]

Integrate[(x^m*(a + b*ArcSinh[c*x])^2)/(d + c^2*d*x^2)^(5/2), x]

Maple [N/A] (verified)

Not integrable

Time = 0.46 (sec) , antiderivative size = 26, normalized size of antiderivative = 0.93

\[\int \frac {x^{m} \left (a +b \,\operatorname {arcsinh}\left (c x \right )\right )^{2}}{\left (c^{2} d \,x^{2}+d \right )^{\frac {5}{2}}}d x\]

[In]

int(x^m*(a+b*arcsinh(c*x))^2/(c^2*d*x^2+d)^(5/2),x)

[Out]

int(x^m*(a+b*arcsinh(c*x))^2/(c^2*d*x^2+d)^(5/2),x)

Fricas [N/A]

Not integrable

Time = 0.27 (sec) , antiderivative size = 78, normalized size of antiderivative = 2.79 \[ \int \frac {x^m (a+b \text {arcsinh}(c x))^2}{\left (d+c^2 d x^2\right )^{5/2}} \, dx=\int { \frac {{\left (b \operatorname {arsinh}\left (c x\right ) + a\right )}^{2} x^{m}}{{\left (c^{2} d x^{2} + d\right )}^{\frac {5}{2}}} \,d x } \]

[In]

integrate(x^m*(a+b*arcsinh(c*x))^2/(c^2*d*x^2+d)^(5/2),x, algorithm="fricas")

[Out]

integral(sqrt(c^2*d*x^2 + d)*(b^2*arcsinh(c*x)^2 + 2*a*b*arcsinh(c*x) + a^2)*x^m/(c^6*d^3*x^6 + 3*c^4*d^3*x^4
+ 3*c^2*d^3*x^2 + d^3), x)

Sympy [N/A]

Not integrable

Time = 165.87 (sec) , antiderivative size = 27, normalized size of antiderivative = 0.96 \[ \int \frac {x^m (a+b \text {arcsinh}(c x))^2}{\left (d+c^2 d x^2\right )^{5/2}} \, dx=\int \frac {x^{m} \left (a + b \operatorname {asinh}{\left (c x \right )}\right )^{2}}{\left (d \left (c^{2} x^{2} + 1\right )\right )^{\frac {5}{2}}}\, dx \]

[In]

integrate(x**m*(a+b*asinh(c*x))**2/(c**2*d*x**2+d)**(5/2),x)

[Out]

Integral(x**m*(a + b*asinh(c*x))**2/(d*(c**2*x**2 + 1))**(5/2), x)

Maxima [N/A]

Not integrable

Time = 0.36 (sec) , antiderivative size = 28, normalized size of antiderivative = 1.00 \[ \int \frac {x^m (a+b \text {arcsinh}(c x))^2}{\left (d+c^2 d x^2\right )^{5/2}} \, dx=\int { \frac {{\left (b \operatorname {arsinh}\left (c x\right ) + a\right )}^{2} x^{m}}{{\left (c^{2} d x^{2} + d\right )}^{\frac {5}{2}}} \,d x } \]

[In]

integrate(x^m*(a+b*arcsinh(c*x))^2/(c^2*d*x^2+d)^(5/2),x, algorithm="maxima")

[Out]

integrate((b*arcsinh(c*x) + a)^2*x^m/(c^2*d*x^2 + d)^(5/2), x)

Giac [N/A]

Not integrable

Time = 0.38 (sec) , antiderivative size = 28, normalized size of antiderivative = 1.00 \[ \int \frac {x^m (a+b \text {arcsinh}(c x))^2}{\left (d+c^2 d x^2\right )^{5/2}} \, dx=\int { \frac {{\left (b \operatorname {arsinh}\left (c x\right ) + a\right )}^{2} x^{m}}{{\left (c^{2} d x^{2} + d\right )}^{\frac {5}{2}}} \,d x } \]

[In]

integrate(x^m*(a+b*arcsinh(c*x))^2/(c^2*d*x^2+d)^(5/2),x, algorithm="giac")

[Out]

integrate((b*arcsinh(c*x) + a)^2*x^m/(c^2*d*x^2 + d)^(5/2), x)

Mupad [N/A]

Not integrable

Time = 3.00 (sec) , antiderivative size = 28, normalized size of antiderivative = 1.00 \[ \int \frac {x^m (a+b \text {arcsinh}(c x))^2}{\left (d+c^2 d x^2\right )^{5/2}} \, dx=\int \frac {x^m\,{\left (a+b\,\mathrm {asinh}\left (c\,x\right )\right )}^2}{{\left (d\,c^2\,x^2+d\right )}^{5/2}} \,d x \]

[In]

int((x^m*(a + b*asinh(c*x))^2)/(d + c^2*d*x^2)^(5/2),x)

[Out]

int((x^m*(a + b*asinh(c*x))^2)/(d + c^2*d*x^2)^(5/2), x)