3.155 \(\int \frac{(c e+d e x)^m}{a+b \sinh ^{-1}(c+d x)} \, dx\)

Optimal. Leaf size=24 \[ \text{Unintegrable}\left (\frac{(e (c+d x))^m}{a+b \sinh ^{-1}(c+d x)},x\right ) \]

[Out]

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

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Rubi [A]  time = 0.0603417, 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 = {} \[ \int \frac{(c e+d e x)^m}{a+b \sinh ^{-1}(c+d x)} \, dx \]

Verification is Not applicable to the result.

[In]

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

[Out]

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

Rubi steps

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

Mathematica [A]  time = 1.11493, size = 0, normalized size = 0. \[ \int \frac{(c e+d e x)^m}{a+b \sinh ^{-1}(c+d x)} \, dx \]

Verification is Not applicable to the result.

[In]

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

[Out]

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

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Maple [A]  time = 0.622, size = 0, normalized size = 0. \begin{align*} \int{\frac{ \left ( dex+ce \right ) ^{m}}{a+b{\it Arcsinh} \left ( dx+c \right ) }}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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