3.93 \(\int x^m (a+b \sinh ^{-1}(c+d x))^n \, dx\)

Optimal. Leaf size=18 \[ \text{Unintegrable}\left (x^m \left (a+b \sinh ^{-1}(c+d x)\right )^n,x\right ) \]

[Out]

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

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Rubi [A]  time = 0.0534858, antiderivative size = 0, normalized size of antiderivative = 0., number of steps used = 0, number of rules used = 0, integrand size = 0, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0., Rules used = {} \[ \int x^m \left (a+b \sinh ^{-1}(c+d x)\right )^n \, dx \]

Verification is Not applicable to the result.

[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*} \int x^m \left (a+b \sinh ^{-1}(c+d x)\right )^n \, dx &=\frac{\operatorname{Subst}\left (\int \left (-\frac{c}{d}+\frac{x}{d}\right )^m \left (a+b \sinh ^{-1}(x)\right )^n \, dx,x,c+d x\right )}{d}\\ \end{align*}

Mathematica [A]  time = 0.418615, size = 0, normalized size = 0. \[ \int x^m \left (a+b \sinh ^{-1}(c+d x)\right )^n \, dx \]

Verification is Not applicable to the result.

[In]

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

[Out]

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

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Maple [A]  time = 0.132, size = 0, normalized size = 0. \begin{align*} \int{x}^{m} \left ( a+b{\it Arcsinh} \left ( dx+c \right ) \right ) ^{n}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [A]  time = 0., size = 0, normalized size = 0. \begin{align*} \int{\left (b \operatorname{arsinh}\left (d x + c\right ) + a\right )}^{n} x^{m}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[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)

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Fricas [A]  time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left ({\left (b \operatorname{arsinh}\left (d x + c\right ) + a\right )}^{n} x^{m}, x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[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)

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Sympy [A]  time = 0., size = 0, normalized size = 0. \begin{align*} \int x^{m} \left (a + b \operatorname{asinh}{\left (c + d x \right )}\right )^{n}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Giac [F(-1)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Timed out