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

Optimal. Leaf size=19 \[ \text {Int}\left (x^m \left (a+b \sinh ^{-1}(c+d x)\right )^n,x\right ) \]

[Out]

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

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Rubi [A]  time = 0.05, antiderivative size = 0, normalized size of antiderivative = 0.00, number of steps used = 0, number of rules used = 0, integrand size = 0, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.000, 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*}

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Mathematica [A]  time = 0.44, size = 0, normalized size = 0.00 \[ \int x^m \left (a+b \sinh ^{-1}(c+d x)\right )^n \, dx \]

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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fricas [A]  time = 0.57, size = 0, normalized size = 0.00 \[ {\rm integral}\left ({\left (b \operatorname {arsinh}\left (d x + c\right ) + a\right )}^{n} x^{m}, x\right ) \]

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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giac [F(-1)]  time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]

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

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maple [A]  time = 0.19, size = 0, normalized size = 0.00 \[ \int x^{m} \left (a +b \arcsinh \left (d x +c \right )\right )^{n}\, dx \]

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 = 0.00, size = 0, normalized size = 0.00 \[ \int {\left (b \operatorname {arsinh}\left (d x + c\right ) + a\right )}^{n} x^{m}\,{d x} \]

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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mupad [A]  time = 0.00, size = -1, normalized size = -0.05 \[ \int x^m\,{\left (a+b\,\mathrm {asinh}\left (c+d\,x\right )\right )}^n \,d x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^m*(a + b*asinh(c + d*x))^n,x)

[Out]

int(x^m*(a + b*asinh(c + d*x))^n, x)

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sympy [A]  time = 0.00, size = 0, normalized size = 0.00 \[ \int x^{m} \left (a + b \operatorname {asinh}{\left (c + d x \right )}\right )^{n}\, dx \]

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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