\(\int \frac {\sqrt {d+c^2 d x^2} (a+b \text {arcsinh}(c x))^n}{x} \, dx\) [515]

Optimal result

Integrand size = 28, antiderivative size = 28 \[ \int \frac {\sqrt {d+c^2 d x^2} (a+b \text {arcsinh}(c x))^n}{x} \, dx=\frac {d e^{-\frac {a}{b}} \sqrt {1+c^2 x^2} (a+b \text {arcsinh}(c x))^n \left (-\frac {a+b \text {arcsinh}(c x)}{b}\right )^{-n} \Gamma \left (1+n,-\frac {a+b \text {arcsinh}(c x)}{b}\right )}{2 \sqrt {d+c^2 d x^2}}+\frac {d e^{a/b} \sqrt {1+c^2 x^2} (a+b \text {arcsinh}(c x))^n \left (\frac {a+b \text {arcsinh}(c x)}{b}\right )^{-n} \Gamma \left (1+n,\frac {a+b \text {arcsinh}(c x)}{b}\right )}{2 \sqrt {d+c^2 d x^2}}+d \text {Int}\left (\frac {(a+b \text {arcsinh}(c x))^n}{x \sqrt {d+c^2 d x^2}},x\right ) \]

[Out]

1/2*d*(a+b*arcsinh(c*x))^n*GAMMA(1+n,(-a-b*arcsinh(c*x))/b)*(c^2*x^2+1)^(1/2)/exp(a/b)/(((-a-b*arcsinh(c*x))/b
)^n)/(c^2*d*x^2+d)^(1/2)+1/2*d*exp(a/b)*(a+b*arcsinh(c*x))^n*GAMMA(1+n,(a+b*arcsinh(c*x))/b)*(c^2*x^2+1)^(1/2)
/(((a+b*arcsinh(c*x))/b)^n)/(c^2*d*x^2+d)^(1/2)+d*Unintegrable((a+b*arcsinh(c*x))^n/x/(c^2*d*x^2+d)^(1/2),x)

Rubi [N/A]

Not integrable

Time = 0.37 (sec) , antiderivative size = 28, 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 = {} \[ \int \frac {\sqrt {d+c^2 d x^2} (a+b \text {arcsinh}(c x))^n}{x} \, dx=\int \frac {\sqrt {d+c^2 d x^2} (a+b \text {arcsinh}(c x))^n}{x} \, dx \]

[In]

Int[(Sqrt[d + c^2*d*x^2]*(a + b*ArcSinh[c*x])^n)/x,x]

[Out]

(d*Sqrt[1 + c^2*x^2]*(a + b*ArcSinh[c*x])^n*Gamma[1 + n, -((a + b*ArcSinh[c*x])/b)])/(2*E^(a/b)*Sqrt[d + c^2*d
*x^2]*(-((a + b*ArcSinh[c*x])/b))^n) + (d*E^(a/b)*Sqrt[1 + c^2*x^2]*(a + b*ArcSinh[c*x])^n*Gamma[1 + n, (a + b
*ArcSinh[c*x])/b])/(2*Sqrt[d + c^2*d*x^2]*((a + b*ArcSinh[c*x])/b)^n) + d*Defer[Int][(a + b*ArcSinh[c*x])^n/(x
*Sqrt[d + c^2*d*x^2]), x]

Rubi steps \begin{align*} \text {integral}& = \int \left (\frac {d (a+b \text {arcsinh}(c x))^n}{x \sqrt {d+c^2 d x^2}}+\frac {c^2 d x (a+b \text {arcsinh}(c x))^n}{\sqrt {d+c^2 d x^2}}\right ) \, dx \\ & = d \int \frac {(a+b \text {arcsinh}(c x))^n}{x \sqrt {d+c^2 d x^2}} \, dx+\left (c^2 d\right ) \int \frac {x (a+b \text {arcsinh}(c x))^n}{\sqrt {d+c^2 d x^2}} \, dx \\ & = d \int \frac {(a+b \text {arcsinh}(c x))^n}{x \sqrt {d+c^2 d x^2}} \, dx-\frac {\left (d \sqrt {1+c^2 x^2}\right ) \text {Subst}\left (\int x^n \sinh \left (\frac {a}{b}-\frac {x}{b}\right ) \, dx,x,a+b \text {arcsinh}(c x)\right )}{b \sqrt {d+c^2 d x^2}} \\ & = d \int \frac {(a+b \text {arcsinh}(c x))^n}{x \sqrt {d+c^2 d x^2}} \, dx-\frac {\left (d \sqrt {1+c^2 x^2}\right ) \text {Subst}\left (\int e^{-i \left (\frac {i a}{b}-\frac {i x}{b}\right )} x^n \, dx,x,a+b \text {arcsinh}(c x)\right )}{2 b \sqrt {d+c^2 d x^2}}+\frac {\left (d \sqrt {1+c^2 x^2}\right ) \text {Subst}\left (\int e^{i \left (\frac {i a}{b}-\frac {i x}{b}\right )} x^n \, dx,x,a+b \text {arcsinh}(c x)\right )}{2 b \sqrt {d+c^2 d x^2}} \\ & = \frac {d e^{-\frac {a}{b}} \sqrt {1+c^2 x^2} (a+b \text {arcsinh}(c x))^n \left (-\frac {a+b \text {arcsinh}(c x)}{b}\right )^{-n} \Gamma \left (1+n,-\frac {a+b \text {arcsinh}(c x)}{b}\right )}{2 \sqrt {d+c^2 d x^2}}+\frac {d e^{a/b} \sqrt {1+c^2 x^2} (a+b \text {arcsinh}(c x))^n \left (\frac {a+b \text {arcsinh}(c x)}{b}\right )^{-n} \Gamma \left (1+n,\frac {a+b \text {arcsinh}(c x)}{b}\right )}{2 \sqrt {d+c^2 d x^2}}+d \int \frac {(a+b \text {arcsinh}(c x))^n}{x \sqrt {d+c^2 d x^2}} \, dx \\ \end{align*}

Mathematica [N/A]

Not integrable

Time = 0.30 (sec) , antiderivative size = 30, normalized size of antiderivative = 1.07 \[ \int \frac {\sqrt {d+c^2 d x^2} (a+b \text {arcsinh}(c x))^n}{x} \, dx=\int \frac {\sqrt {d+c^2 d x^2} (a+b \text {arcsinh}(c x))^n}{x} \, dx \]

[In]

Integrate[(Sqrt[d + c^2*d*x^2]*(a + b*ArcSinh[c*x])^n)/x,x]

[Out]

Integrate[(Sqrt[d + c^2*d*x^2]*(a + b*ArcSinh[c*x])^n)/x, x]

Maple [N/A] (verified)

Not integrable

Time = 0.22 (sec) , antiderivative size = 26, normalized size of antiderivative = 0.93

\[\int \frac {\left (a +b \,\operatorname {arcsinh}\left (c x \right )\right )^{n} \sqrt {c^{2} d \,x^{2}+d}}{x}d x\]

[In]

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

[Out]

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

Fricas [N/A]

Not integrable

Time = 0.26 (sec) , antiderivative size = 28, normalized size of antiderivative = 1.00 \[ \int \frac {\sqrt {d+c^2 d x^2} (a+b \text {arcsinh}(c x))^n}{x} \, dx=\int { \frac {\sqrt {c^{2} d x^{2} + d} {\left (b \operatorname {arsinh}\left (c x\right ) + a\right )}^{n}}{x} \,d x } \]

[In]

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

[Out]

integral(sqrt(c^2*d*x^2 + d)*(b*arcsinh(c*x) + a)^n/x, x)

Sympy [N/A]

Not integrable

Time = 2.23 (sec) , antiderivative size = 26, normalized size of antiderivative = 0.93 \[ \int \frac {\sqrt {d+c^2 d x^2} (a+b \text {arcsinh}(c x))^n}{x} \, dx=\int \frac {\sqrt {d \left (c^{2} x^{2} + 1\right )} \left (a + b \operatorname {asinh}{\left (c x \right )}\right )^{n}}{x}\, dx \]

[In]

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

[Out]

Integral(sqrt(d*(c**2*x**2 + 1))*(a + b*asinh(c*x))**n/x, x)

Maxima [N/A]

Not integrable

Time = 0.41 (sec) , antiderivative size = 28, normalized size of antiderivative = 1.00 \[ \int \frac {\sqrt {d+c^2 d x^2} (a+b \text {arcsinh}(c x))^n}{x} \, dx=\int { \frac {\sqrt {c^{2} d x^{2} + d} {\left (b \operatorname {arsinh}\left (c x\right ) + a\right )}^{n}}{x} \,d x } \]

[In]

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

[Out]

integrate(sqrt(c^2*d*x^2 + d)*(b*arcsinh(c*x) + a)^n/x, x)

Giac [F(-2)]

Exception generated. \[ \int \frac {\sqrt {d+c^2 d x^2} (a+b \text {arcsinh}(c x))^n}{x} \, dx=\text {Exception raised: TypeError} \]

[In]

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

[Out]

Exception raised: TypeError >> an error occurred running a Giac command:INPUT:sage2:=int(sage0,sageVARx):;OUTP
UT:sym2poly/r2sym(const gen & e,const index_m & i,const vecteur & l) Error: Bad Argument Value

Mupad [N/A]

Not integrable

Time = 2.62 (sec) , antiderivative size = 28, normalized size of antiderivative = 1.00 \[ \int \frac {\sqrt {d+c^2 d x^2} (a+b \text {arcsinh}(c x))^n}{x} \, dx=\int \frac {{\left (a+b\,\mathrm {asinh}\left (c\,x\right )\right )}^n\,\sqrt {d\,c^2\,x^2+d}}{x} \,d x \]

[In]

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

[Out]

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