\(\int \frac {x^m}{\sqrt {1+c^2 x^2} (a+b \text {arcsinh}(c x))^2} \, dx\) [458]

Optimal result

Integrand size = 27, antiderivative size = 27 \[ \int \frac {x^m}{\sqrt {1+c^2 x^2} (a+b \text {arcsinh}(c x))^2} \, dx=-\frac {x^m}{b c (a+b \text {arcsinh}(c x))}+\frac {m \text {Int}\left (\frac {x^{-1+m}}{a+b \text {arcsinh}(c x)},x\right )}{b c} \]

[Out]

-x^m/b/c/(a+b*arcsinh(c*x))+m*Unintegrable(x^(-1+m)/(a+b*arcsinh(c*x)),x)/b/c

Rubi [N/A]

Not integrable

Time = 0.10 (sec) , antiderivative size = 27, 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 = {} \[ \int \frac {x^m}{\sqrt {1+c^2 x^2} (a+b \text {arcsinh}(c x))^2} \, dx=\int \frac {x^m}{\sqrt {1+c^2 x^2} (a+b \text {arcsinh}(c x))^2} \, dx \]

[In]

Int[x^m/(Sqrt[1 + c^2*x^2]*(a + b*ArcSinh[c*x])^2),x]

[Out]

-(x^m/(b*c*(a + b*ArcSinh[c*x]))) + (m*Defer[Int][x^(-1 + m)/(a + b*ArcSinh[c*x]), x])/(b*c)

Rubi steps \begin{align*} \text {integral}& = -\frac {x^m}{b c (a+b \text {arcsinh}(c x))}+\frac {m \int \frac {x^{-1+m}}{a+b \text {arcsinh}(c x)} \, dx}{b c} \\ \end{align*}

Mathematica [N/A]

Not integrable

Time = 0.59 (sec) , antiderivative size = 29, normalized size of antiderivative = 1.07 \[ \int \frac {x^m}{\sqrt {1+c^2 x^2} (a+b \text {arcsinh}(c x))^2} \, dx=\int \frac {x^m}{\sqrt {1+c^2 x^2} (a+b \text {arcsinh}(c x))^2} \, dx \]

[In]

Integrate[x^m/(Sqrt[1 + c^2*x^2]*(a + b*ArcSinh[c*x])^2),x]

[Out]

Integrate[x^m/(Sqrt[1 + c^2*x^2]*(a + b*ArcSinh[c*x])^2), x]

Maple [N/A] (verified)

Not integrable

Time = 0.14 (sec) , antiderivative size = 25, normalized size of antiderivative = 0.93

\[\int \frac {x^{m}}{\sqrt {c^{2} x^{2}+1}\, \left (a +b \,\operatorname {arcsinh}\left (c x \right )\right )^{2}}d x\]

[In]

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

[Out]

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

Fricas [N/A]

Not integrable

Time = 0.25 (sec) , antiderivative size = 73, normalized size of antiderivative = 2.70 \[ \int \frac {x^m}{\sqrt {1+c^2 x^2} (a+b \text {arcsinh}(c x))^2} \, dx=\int { \frac {x^{m}}{\sqrt {c^{2} x^{2} + 1} {\left (b \operatorname {arsinh}\left (c x\right ) + a\right )}^{2}} \,d x } \]

[In]

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

[Out]

integral(sqrt(c^2*x^2 + 1)*x^m/(a^2*c^2*x^2 + (b^2*c^2*x^2 + b^2)*arcsinh(c*x)^2 + a^2 + 2*(a*b*c^2*x^2 + a*b)
*arcsinh(c*x)), x)

Sympy [N/A]

Not integrable

Time = 2.84 (sec) , antiderivative size = 26, normalized size of antiderivative = 0.96 \[ \int \frac {x^m}{\sqrt {1+c^2 x^2} (a+b \text {arcsinh}(c x))^2} \, dx=\int \frac {x^{m}}{\left (a + b \operatorname {asinh}{\left (c x \right )}\right )^{2} \sqrt {c^{2} x^{2} + 1}}\, dx \]

[In]

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

[Out]

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

Maxima [N/A]

Not integrable

Time = 0.53 (sec) , antiderivative size = 441, normalized size of antiderivative = 16.33 \[ \int \frac {x^m}{\sqrt {1+c^2 x^2} (a+b \text {arcsinh}(c x))^2} \, dx=\int { \frac {x^{m}}{\sqrt {c^{2} x^{2} + 1} {\left (b \operatorname {arsinh}\left (c x\right ) + a\right )}^{2}} \,d x } \]

[In]

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

[Out]

-((c^2*x^2 + 1)^(3/2)*x^m + (c^3*x^3 + c*x)*x^m)/((c^2*x^2 + 1)*a*b*c^2*x + ((c^2*x^2 + 1)*b^2*c^2*x + (b^2*c^
3*x^2 + b^2*c)*sqrt(c^2*x^2 + 1))*log(c*x + sqrt(c^2*x^2 + 1)) + (a*b*c^3*x^2 + a*b*c)*sqrt(c^2*x^2 + 1)) + in
tegrate(((c^3*m*x^3 + c*(m - 1)*x)*(c^2*x^2 + 1)*x^m + (2*c^4*m*x^4 + 3*c^2*m*x^2 + m)*sqrt(c^2*x^2 + 1)*x^m +
(c^5*m*x^5 + c^3*(2*m + 1)*x^3 + c*(m + 1)*x)*x^m)/((c^2*x^2 + 1)^(3/2)*a*b*c^3*x^3 + 2*(a*b*c^4*x^4 + a*b*c^
2*x^2)*(c^2*x^2 + 1) + ((c^2*x^2 + 1)^(3/2)*b^2*c^3*x^3 + 2*(b^2*c^4*x^4 + b^2*c^2*x^2)*(c^2*x^2 + 1) + (b^2*c
^5*x^5 + 2*b^2*c^3*x^3 + b^2*c*x)*sqrt(c^2*x^2 + 1))*log(c*x + sqrt(c^2*x^2 + 1)) + (a*b*c^5*x^5 + 2*a*b*c^3*x
^3 + a*b*c*x)*sqrt(c^2*x^2 + 1)), x)

Giac [N/A]

Not integrable

Time = 0.32 (sec) , antiderivative size = 27, normalized size of antiderivative = 1.00 \[ \int \frac {x^m}{\sqrt {1+c^2 x^2} (a+b \text {arcsinh}(c x))^2} \, dx=\int { \frac {x^{m}}{\sqrt {c^{2} x^{2} + 1} {\left (b \operatorname {arsinh}\left (c x\right ) + a\right )}^{2}} \,d x } \]

[In]

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

[Out]

integrate(x^m/(sqrt(c^2*x^2 + 1)*(b*arcsinh(c*x) + a)^2), x)

Mupad [N/A]

Not integrable

Time = 2.66 (sec) , antiderivative size = 27, normalized size of antiderivative = 1.00 \[ \int \frac {x^m}{\sqrt {1+c^2 x^2} (a+b \text {arcsinh}(c x))^2} \, dx=\int \frac {x^m}{{\left (a+b\,\mathrm {asinh}\left (c\,x\right )\right )}^2\,\sqrt {c^2\,x^2+1}} \,d x \]

[In]

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

[Out]

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