\(\int \frac {(1+c^2 x^2)^{5/2}}{x^3 (a+b \text {arcsinh}(c x))} \, dx\) [378]

Optimal result

Integrand size = 27, antiderivative size = 27 \[ \int \frac {\left (1+c^2 x^2\right )^{5/2}}{x^3 (a+b \text {arcsinh}(c x))} \, dx=\text {Int}\left (\frac {\left (1+c^2 x^2\right )^{5/2}}{x^3 (a+b \text {arcsinh}(c x))},x\right ) \]

[Out]

Unintegrable((c^2*x^2+1)^(5/2)/x^3/(a+b*arcsinh(c*x)),x)

Rubi [N/A]

Not integrable

Time = 0.09 (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 = {} \[ \int \frac {\left (1+c^2 x^2\right )^{5/2}}{x^3 (a+b \text {arcsinh}(c x))} \, dx=\int \frac {\left (1+c^2 x^2\right )^{5/2}}{x^3 (a+b \text {arcsinh}(c x))} \, dx \]

[In]

Int[(1 + c^2*x^2)^(5/2)/(x^3*(a + b*ArcSinh[c*x])),x]

[Out]

Defer[Int][(1 + c^2*x^2)^(5/2)/(x^3*(a + b*ArcSinh[c*x])), x]

Rubi steps \begin{align*} \text {integral}& = \int \frac {\left (1+c^2 x^2\right )^{5/2}}{x^3 (a+b \text {arcsinh}(c x))} \, dx \\ \end{align*}

Mathematica [N/A]

Not integrable

Time = 4.28 (sec) , antiderivative size = 29, normalized size of antiderivative = 1.07 \[ \int \frac {\left (1+c^2 x^2\right )^{5/2}}{x^3 (a+b \text {arcsinh}(c x))} \, dx=\int \frac {\left (1+c^2 x^2\right )^{5/2}}{x^3 (a+b \text {arcsinh}(c x))} \, dx \]

[In]

Integrate[(1 + c^2*x^2)^(5/2)/(x^3*(a + b*ArcSinh[c*x])),x]

[Out]

Integrate[(1 + c^2*x^2)^(5/2)/(x^3*(a + b*ArcSinh[c*x])), x]

Maple [N/A] (verified)

Not integrable

Time = 0.09 (sec) , antiderivative size = 25, normalized size of antiderivative = 0.93

\[\int \frac {\left (c^{2} x^{2}+1\right )^{\frac {5}{2}}}{x^{3} \left (a +b \,\operatorname {arcsinh}\left (c x \right )\right )}d x\]

[In]

int((c^2*x^2+1)^(5/2)/x^3/(a+b*arcsinh(c*x)),x)

[Out]

int((c^2*x^2+1)^(5/2)/x^3/(a+b*arcsinh(c*x)),x)

Fricas [N/A]

Not integrable

Time = 0.26 (sec) , antiderivative size = 48, normalized size of antiderivative = 1.78 \[ \int \frac {\left (1+c^2 x^2\right )^{5/2}}{x^3 (a+b \text {arcsinh}(c x))} \, dx=\int { \frac {{\left (c^{2} x^{2} + 1\right )}^{\frac {5}{2}}}{{\left (b \operatorname {arsinh}\left (c x\right ) + a\right )} x^{3}} \,d x } \]

[In]

integrate((c^2*x^2+1)^(5/2)/x^3/(a+b*arcsinh(c*x)),x, algorithm="fricas")

[Out]

integral((c^4*x^4 + 2*c^2*x^2 + 1)*sqrt(c^2*x^2 + 1)/(b*x^3*arcsinh(c*x) + a*x^3), x)

Sympy [N/A]

Not integrable

Time = 4.05 (sec) , antiderivative size = 24, normalized size of antiderivative = 0.89 \[ \int \frac {\left (1+c^2 x^2\right )^{5/2}}{x^3 (a+b \text {arcsinh}(c x))} \, dx=\int \frac {\left (c^{2} x^{2} + 1\right )^{\frac {5}{2}}}{x^{3} \left (a + b \operatorname {asinh}{\left (c x \right )}\right )}\, dx \]

[In]

integrate((c**2*x**2+1)**(5/2)/x**3/(a+b*asinh(c*x)),x)

[Out]

Integral((c**2*x**2 + 1)**(5/2)/(x**3*(a + b*asinh(c*x))), x)

Maxima [N/A]

Not integrable

Time = 0.27 (sec) , antiderivative size = 27, normalized size of antiderivative = 1.00 \[ \int \frac {\left (1+c^2 x^2\right )^{5/2}}{x^3 (a+b \text {arcsinh}(c x))} \, dx=\int { \frac {{\left (c^{2} x^{2} + 1\right )}^{\frac {5}{2}}}{{\left (b \operatorname {arsinh}\left (c x\right ) + a\right )} x^{3}} \,d x } \]

[In]

integrate((c^2*x^2+1)^(5/2)/x^3/(a+b*arcsinh(c*x)),x, algorithm="maxima")

[Out]

integrate((c^2*x^2 + 1)^(5/2)/((b*arcsinh(c*x) + a)*x^3), x)

Giac [F(-2)]

Exception generated. \[ \int \frac {\left (1+c^2 x^2\right )^{5/2}}{x^3 (a+b \text {arcsinh}(c x))} \, dx=\text {Exception raised: TypeError} \]

[In]

integrate((c^2*x^2+1)^(5/2)/x^3/(a+b*arcsinh(c*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 = 2.72 (sec) , antiderivative size = 27, normalized size of antiderivative = 1.00 \[ \int \frac {\left (1+c^2 x^2\right )^{5/2}}{x^3 (a+b \text {arcsinh}(c x))} \, dx=\int \frac {{\left (c^2\,x^2+1\right )}^{5/2}}{x^3\,\left (a+b\,\mathrm {asinh}\left (c\,x\right )\right )} \,d x \]

[In]

int((c^2*x^2 + 1)^(5/2)/(x^3*(a + b*asinh(c*x))),x)

[Out]

int((c^2*x^2 + 1)^(5/2)/(x^3*(a + b*asinh(c*x))), x)