\(\int x^m (a+b \text {arcsinh}(c+d x))^n \, dx\) [93]

Optimal result

Integrand size = 16, antiderivative size = 16 \[ \int x^m (a+b \text {arcsinh}(c+d x))^n \, dx=\text {Int}\left (x^m (a+b \text {arcsinh}(c+d x))^n,x\right ) \]

[Out]

Unintegrable(x^m*(a+b*arcsinh(d*x+c))^n,x)

Rubi [N/A]

Not integrable

Time = 0.04 (sec) , antiderivative size = 16, 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 = {} \[ \int x^m (a+b \text {arcsinh}(c+d x))^n \, dx=\int x^m (a+b \text {arcsinh}(c+d x))^n \, dx \]

[In]

Int[x^m*(a + b*ArcSinh[c + d*x])^n,x]

[Out]

Defer[Subst][Defer[Int][(-(c/d) + x/d)^m*(a + b*ArcSinh[x])^n, x], x, c + d*x]/d

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

Mathematica [N/A]

Not integrable

Time = 0.34 (sec) , antiderivative size = 18, normalized size of antiderivative = 1.12 \[ \int x^m (a+b \text {arcsinh}(c+d x))^n \, dx=\int x^m (a+b \text {arcsinh}(c+d x))^n \, dx \]

[In]

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

[Out]

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

Maple [N/A] (verified)

Not integrable

Time = 0.32 (sec) , antiderivative size = 16, normalized size of antiderivative = 1.00

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

[In]

int(x^m*(a+b*arcsinh(d*x+c))^n,x)

[Out]

int(x^m*(a+b*arcsinh(d*x+c))^n,x)

Fricas [N/A]

Not integrable

Time = 0.27 (sec) , antiderivative size = 18, normalized size of antiderivative = 1.12 \[ \int x^m (a+b \text {arcsinh}(c+d x))^n \, dx=\int { {\left (b \operatorname {arsinh}\left (d x + c\right ) + a\right )}^{n} x^{m} \,d x } \]

[In]

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

[Out]

integral((b*arcsinh(d*x + c) + a)^n*x^m, x)

Sympy [N/A]

Not integrable

Time = 12.85 (sec) , antiderivative size = 15, normalized size of antiderivative = 0.94 \[ \int x^m (a+b \text {arcsinh}(c+d x))^n \, dx=\int x^{m} \left (a + b \operatorname {asinh}{\left (c + d x \right )}\right )^{n}\, dx \]

[In]

integrate(x**m*(a+b*asinh(d*x+c))**n,x)

[Out]

Integral(x**m*(a + b*asinh(c + d*x))**n, x)

Maxima [N/A]

Not integrable

Time = 0.57 (sec) , antiderivative size = 18, normalized size of antiderivative = 1.12 \[ \int x^m (a+b \text {arcsinh}(c+d x))^n \, dx=\int { {\left (b \operatorname {arsinh}\left (d x + c\right ) + a\right )}^{n} x^{m} \,d x } \]

[In]

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

[Out]

integrate((b*arcsinh(d*x + c) + a)^n*x^m, x)

Giac [F(-1)]

Timed out. \[ \int x^m (a+b \text {arcsinh}(c+d x))^n \, dx=\text {Timed out} \]

[In]

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

[Out]

Timed out

Mupad [N/A]

Not integrable

Time = 2.82 (sec) , antiderivative size = 18, normalized size of antiderivative = 1.12 \[ \int x^m (a+b \text {arcsinh}(c+d x))^n \, dx=\int x^m\,{\left (a+b\,\mathrm {asinh}\left (c+d\,x\right )\right )}^n \,d x \]

[In]

int(x^m*(a + b*asinh(c + d*x))^n,x)

[Out]

int(x^m*(a + b*asinh(c + d*x))^n, x)