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

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

[Out]

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

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Rubi [A]  time = 0.088093, 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 = {} \[ \int \frac{\sqrt{a+b \sinh ^{-1}(c+d x)}}{c e+d e x} \, dx \]

Verification is Not applicable to the result.

[In]

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

[Out]

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

Rubi steps

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

Mathematica [A]  time = 2.03031, size = 0, normalized size = 0. \[ \int \frac{\sqrt{a+b \sinh ^{-1}(c+d x)}}{c e+d e x} \, dx \]

Verification is Not applicable to the result.

[In]

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

[Out]

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

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Maple [A]  time = 0.141, size = 0, normalized size = 0. \begin{align*} \int{\frac{1}{dex+ce}\sqrt{a+b{\it Arcsinh} \left ( dx+c \right ) }}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integrate(sqrt(b*arcsinh(d*x + c) + a)/(d*e*x + c*e), 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((a+b*arcsinh(d*x+c))^(1/2)/(d*e*x+c*e),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{\sqrt{a + b \operatorname{asinh}{\left (c + d x \right )}}}{c + d x}\, dx}{e} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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