\(\int \frac {\sqrt {d+c^2 d x^2} (a+b \text {arcsinh}(c x))^n}{x^2} \, dx\) [516]

Optimal result

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

[Out]

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

Rubi [N/A]

Not integrable

Time = 0.24 (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 = {} \[ \int \frac {\sqrt {d+c^2 d x^2} (a+b \text {arcsinh}(c x))^n}{x^2} \, dx=\int \frac {\sqrt {d+c^2 d x^2} (a+b \text {arcsinh}(c x))^n}{x^2} \, dx \]

[In]

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

[Out]

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

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

Mathematica [N/A]

Not integrable

Time = 0.33 (sec) , antiderivative size = 30, normalized size of antiderivative = 1.07 \[ \int \frac {\sqrt {d+c^2 d x^2} (a+b \text {arcsinh}(c x))^n}{x^2} \, dx=\int \frac {\sqrt {d+c^2 d x^2} (a+b \text {arcsinh}(c x))^n}{x^2} \, dx \]

[In]

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

[Out]

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

Maple [N/A] (verified)

Not integrable

Time = 0.23 (sec) , antiderivative size = 26, normalized size of antiderivative = 0.93

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

[In]

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

[Out]

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

Fricas [N/A]

Not integrable

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

[In]

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

[Out]

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

Sympy [N/A]

Not integrable

Time = 4.09 (sec) , antiderivative size = 27, normalized size of antiderivative = 0.96 \[ \int \frac {\sqrt {d+c^2 d x^2} (a+b \text {arcsinh}(c x))^n}{x^2} \, dx=\int \frac {\sqrt {d \left (c^{2} x^{2} + 1\right )} \left (a + b \operatorname {asinh}{\left (c x \right )}\right )^{n}}{x^{2}}\, dx \]

[In]

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

[Out]

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

Maxima [N/A]

Not integrable

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

[In]

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

[Out]

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

Giac [F(-2)]

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

[In]

integrate((a+b*arcsinh(c*x))^n*(c^2*d*x^2+d)^(1/2)/x^2,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 = 2.60 (sec) , antiderivative size = 28, normalized size of antiderivative = 1.00 \[ \int \frac {\sqrt {d+c^2 d x^2} (a+b \text {arcsinh}(c x))^n}{x^2} \, dx=\int \frac {{\left (a+b\,\mathrm {asinh}\left (c\,x\right )\right )}^n\,\sqrt {d\,c^2\,x^2+d}}{x^2} \,d x \]

[In]

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

[Out]

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