\(\int \frac {1}{(c e+d e x) \sqrt {a+b \text {arcsinh}(c+d x)}} \, dx\) [209]

Optimal result

Integrand size = 25, antiderivative size = 25 \[ \int \frac {1}{(c e+d e x) \sqrt {a+b \text {arcsinh}(c+d x)}} \, dx=\frac {\text {Int}\left (\frac {1}{(c+d x) \sqrt {a+b \text {arcsinh}(c+d x)}},x\right )}{e} \]

[Out]

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

Rubi [N/A]

Not integrable

Time = 0.07 (sec) , antiderivative size = 25, normalized size of antiderivative = 1.00, number of steps used = 0, number of rules used = 0, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.000, Rules used = {} \[ \int \frac {1}{(c e+d e x) \sqrt {a+b \text {arcsinh}(c+d x)}} \, dx=\int \frac {1}{(c e+d e x) \sqrt {a+b \text {arcsinh}(c+d x)}} \, dx \]

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

Mathematica [N/A]

Not integrable

Time = 0.12 (sec) , antiderivative size = 27, normalized size of antiderivative = 1.08 \[ \int \frac {1}{(c e+d e x) \sqrt {a+b \text {arcsinh}(c+d x)}} \, dx=\int \frac {1}{(c e+d e x) \sqrt {a+b \text {arcsinh}(c+d x)}} \, dx \]

[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]

Maple [N/A] (verified)

Not integrable

Time = 0.21 (sec) , antiderivative size = 23, normalized size of antiderivative = 0.92

\[\int \frac {1}{\left (d e x +c e \right ) \sqrt {a +b \,\operatorname {arcsinh}\left (d x +c \right )}}d x\]

[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)

Fricas [F(-2)]

Exception generated. \[ \int \frac {1}{(c e+d e x) \sqrt {a+b \text {arcsinh}(c+d x)}} \, dx=\text {Exception raised: TypeError} \]

[In]

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

[Out]

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

Sympy [N/A]

Not integrable

Time = 0.70 (sec) , antiderivative size = 36, normalized size of antiderivative = 1.44 \[ \int \frac {1}{(c e+d e x) \sqrt {a+b \text {arcsinh}(c+d x)}} \, dx=\frac {\int \frac {1}{c \sqrt {a + b \operatorname {asinh}{\left (c + d x \right )}} + d x \sqrt {a + b \operatorname {asinh}{\left (c + d x \right )}}}\, dx}{e} \]

[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

Maxima [N/A]

Not integrable

Time = 0.80 (sec) , antiderivative size = 25, normalized size of antiderivative = 1.00 \[ \int \frac {1}{(c e+d e x) \sqrt {a+b \text {arcsinh}(c+d x)}} \, dx=\int { \frac {1}{{\left (d e x + c e\right )} \sqrt {b \operatorname {arsinh}\left (d x + c\right ) + a}} \,d x } \]

[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)

Giac [N/A]

Not integrable

Time = 1.85 (sec) , antiderivative size = 25, normalized size of antiderivative = 1.00 \[ \int \frac {1}{(c e+d e x) \sqrt {a+b \text {arcsinh}(c+d x)}} \, dx=\int { \frac {1}{{\left (d e x + c e\right )} \sqrt {b \operatorname {arsinh}\left (d x + c\right ) + a}} \,d x } \]

[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)

Mupad [N/A]

Not integrable

Time = 2.60 (sec) , antiderivative size = 25, normalized size of antiderivative = 1.00 \[ \int \frac {1}{(c e+d e x) \sqrt {a+b \text {arcsinh}(c+d x)}} \, dx=\int \frac {1}{\left (c\,e+d\,e\,x\right )\,\sqrt {a+b\,\mathrm {asinh}\left (c+d\,x\right )}} \,d x \]

[In]

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

[Out]

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