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

Optimal. Leaf size=28 \[ \frac{\text{Unintegrable}\left (\frac{1}{(c+d x) \sqrt{a+b \sinh ^{-1}(c+d x)}},x\right )}{e} \]

[Out]

Unintegrable[1/((c + d*x)*Sqrt[a + b*ArcSinh[c + d*x]]), x]/e

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Rubi [A]  time = 0.0979978, 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 = {} \[ \int \frac{1}{(c e+d e x) \sqrt{a+b \sinh ^{-1}(c+d x)}} \, dx \]

Verification is Not applicable to the result.

[In]

Int[1/((c*e + d*e*x)*Sqrt[a + b*ArcSinh[c + d*x]]),x]

[Out]

Defer[Subst][Defer[Int][1/(x*Sqrt[a + b*ArcSinh[x]]), x], x, c + d*x]/(d*e)

Rubi steps

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

Mathematica [A]  time = 0.0645254, size = 0, normalized size = 0. \[ \int \frac{1}{(c e+d e x) \sqrt{a+b \sinh ^{-1}(c+d x)}} \, dx \]

Verification is Not applicable to the result.

[In]

Integrate[1/((c*e + d*e*x)*Sqrt[a + b*ArcSinh[c + d*x]]),x]

[Out]

Integrate[1/((c*e + d*e*x)*Sqrt[a + b*ArcSinh[c + d*x]]), x]

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Maple [A]  time = 0.171, size = 0, normalized size = 0. \begin{align*} \int{\frac{1}{dex+ce}{\frac{1}{\sqrt{a+b{\it Arcsinh} \left ( dx+c \right ) }}}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(d*e*x+c*e)/(a+b*arcsinh(d*x+c))^(1/2),x)

[Out]

int(1/(d*e*x+c*e)/(a+b*arcsinh(d*x+c))^(1/2),x)

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Maxima [A]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{1}{{\left (d e x + c e\right )} \sqrt{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(1/(d*e*x+c*e)/(a+b*arcsinh(d*x+c))^(1/2),x, algorithm="maxima")

[Out]

integrate(1/((d*e*x + c*e)*sqrt(b*arcsinh(d*x + c) + a)), x)

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Fricas [F(-2)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: UnboundLocalError} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(d*e*x+c*e)/(a+b*arcsinh(d*x+c))^(1/2),x, algorithm="fricas")

[Out]

Exception raised: UnboundLocalError

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Sympy [A]  time = 0., size = 0, normalized size = 0. \begin{align*} \frac{\int \frac{1}{c \sqrt{a + b \operatorname{asinh}{\left (c + d x \right )}} + d x \sqrt{a + b \operatorname{asinh}{\left (c + d x \right )}}}\, dx}{e} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(d*e*x+c*e)/(a+b*asinh(d*x+c))**(1/2),x)

[Out]

Integral(1/(c*sqrt(a + b*asinh(c + d*x)) + d*x*sqrt(a + b*asinh(c + d*x))), x)/e

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Giac [A]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{1}{{\left (d e x + c e\right )} \sqrt{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(1/(d*e*x+c*e)/(a+b*arcsinh(d*x+c))^(1/2),x, algorithm="giac")

[Out]

integrate(1/((d*e*x + c*e)*sqrt(b*arcsinh(d*x + c) + a)), x)