3.2.55 \(\int \frac {(c e+d e x)^m}{a+b \sinh ^{-1}(c+d x)} \, dx\) [155]

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 = 0.04, 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 = {} \begin {gather*} \int \frac {(c e+d e x)^m}{a+b \sinh ^{-1}(c+d x)} \, dx \end {gather*}

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 {\text {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 = 1.46, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {(c e+d e x)^m}{a+b \sinh ^{-1}(c+d x)} \, dx \end {gather*}

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 = 180.00, 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 \begin {gather*} \text {Failed to integrate} \end {gather*}

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

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Fricas [A]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}

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

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

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 = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}

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

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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