3.2.85 \(\int \frac {\sqrt {a+b \sinh ^{-1}(c+d x)}}{c e+d e x} \, dx\) [185]

Optimal. Leaf size=29 \[ \frac {\text {Int}\left (\frac {\sqrt {a+b \sinh ^{-1}(c+d x)}}{c+d x},x\right )}{e} \]

[Out]

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

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

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

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Mathematica [A]
time = 1.44, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\sqrt {a+b \sinh ^{-1}(c+d x)}}{c e+d e x} \, dx \end {gather*}

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 = 180.00, size = 0, normalized size = 0.00 \[\int \frac {\sqrt {a +b \arcsinh \left (d x +c \right )}}{d e x +c e}\, dx\]

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 = 0.00, 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((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*x*e + c*e), x)

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Fricas [F(-2)]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: TypeError} \end {gather*}

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: TypeError >>  Error detected within library code:   integrate: implementation incomplete (co
nstant residues)

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

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 = 0.00, 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((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)

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Mupad [A]
time = 0.00, size = -1, normalized size = -0.03 \begin {gather*} \int \frac {\sqrt {a+b\,\mathrm {asinh}\left (c+d\,x\right )}}{c\,e+d\,e\,x} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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