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

Optimal. Leaf size=25 \[ \text {Int}\left (\frac {(e (c+d x))^m}{a+b \sinh ^{-1}(c+d x)},x\right ) \]

[Out]

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

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Rubi [A]  time = 0.06, 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 = {} \[ \int \frac {(c e+d e x)^m}{a+b \sinh ^{-1}(c+d x)} \, dx \]

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*}

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Mathematica [A]  time = 1.15, size = 0, normalized size = 0.00 \[ \int \frac {(c e+d e x)^m}{a+b \sinh ^{-1}(c+d x)} \, dx \]

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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fricas [A]  time = 0.64, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {{\left (d e x + c e\right )}^{m}}{b \operatorname {arsinh}\left (d x + c\right ) + a}, x\right ) \]

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

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)

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maple [A]  time = 0.79, size = 0, normalized size = 0.00 \[ \int \frac {\left (d e x +c e \right )^{m}}{a +b \arcsinh \left (d x +c \right )}\, dx \]

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

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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mupad [A]  time = 0.00, size = -1, normalized size = -0.04 \[ \int \frac {{\left (c\,e+d\,e\,x\right )}^m}{a+b\,\mathrm {asinh}\left (c+d\,x\right )} \,d x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

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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