\(\int \frac {(1+c^2 x^2)^{5/2}}{x^3 (a+b \text {arcsinh}(c x))^2} \, dx\) [433]

Optimal result

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

[Out]

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

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

[In]

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

[Out]

Defer[Int][(1 + c^2*x^2)^(5/2)/(x^3*(a + b*ArcSinh[c*x])^2), 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))^2} \, dx \\ \end{align*}

Mathematica [N/A]

Not integrable

Time = 12.58 (sec) , antiderivative size = 29, normalized size of antiderivative = 1.07 \[ \int \frac {\left (1+c^2 x^2\right )^{5/2}}{x^3 (a+b \text {arcsinh}(c x))^2} \, dx=\int \frac {\left (1+c^2 x^2\right )^{5/2}}{x^3 (a+b \text {arcsinh}(c x))^2} \, dx \]

[In]

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

[Out]

Integrate[(1 + c^2*x^2)^(5/2)/(x^3*(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

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

[In]

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

[Out]

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

Fricas [N/A]

Not integrable

Time = 0.25 (sec) , antiderivative size = 65, normalized size of antiderivative = 2.41 \[ \int \frac {\left (1+c^2 x^2\right )^{5/2}}{x^3 (a+b \text {arcsinh}(c x))^2} \, dx=\int { \frac {{\left (c^{2} x^{2} + 1\right )}^{\frac {5}{2}}}{{\left (b \operatorname {arsinh}\left (c x\right ) + a\right )}^{2} x^{3}} \,d x } \]

[In]

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

[Out]

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

Sympy [N/A]

Not integrable

Time = 7.79 (sec) , antiderivative size = 26, normalized size of antiderivative = 0.96 \[ \int \frac {\left (1+c^2 x^2\right )^{5/2}}{x^3 (a+b \text {arcsinh}(c x))^2} \, dx=\int \frac {\left (c^{2} x^{2} + 1\right )^{\frac {5}{2}}}{x^{3} \left (a + b \operatorname {asinh}{\left (c x \right )}\right )^{2}}\, dx \]

[In]

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

[Out]

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

Maxima [N/A]

Not integrable

Time = 0.69 (sec) , antiderivative size = 492, normalized size of antiderivative = 18.22 \[ \int \frac {\left (1+c^2 x^2\right )^{5/2}}{x^3 (a+b \text {arcsinh}(c x))^2} \, dx=\int { \frac {{\left (c^{2} x^{2} + 1\right )}^{\frac {5}{2}}}{{\left (b \operatorname {arsinh}\left (c x\right ) + a\right )}^{2} x^{3}} \,d x } \]

[In]

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

[Out]

-((c^6*x^6 + 3*c^4*x^4 + 3*c^2*x^2 + 1)*(c^2*x^2 + 1) + (c^7*x^7 + 3*c^5*x^5 + 3*c^3*x^3 + c*x)*sqrt(c^2*x^2 +
1))/(a*b*c^3*x^5 + sqrt(c^2*x^2 + 1)*a*b*c^2*x^4 + a*b*c*x^3 + (b^2*c^3*x^5 + sqrt(c^2*x^2 + 1)*b^2*c^2*x^4 +
b^2*c*x^3)*log(c*x + sqrt(c^2*x^2 + 1))) + integrate(((3*c^7*x^7 + 2*c^5*x^5 - 5*c^3*x^3 - 4*c*x)*(c^2*x^2 +
1)^(3/2) + 3*(2*c^8*x^8 + 3*c^6*x^6 - c^4*x^4 - 3*c^2*x^2 - 1)*(c^2*x^2 + 1) + (3*c^9*x^9 + 7*c^7*x^7 + 3*c^5*
x^5 - 3*c^3*x^3 - 2*c*x)*sqrt(c^2*x^2 + 1))/(a*b*c^5*x^8 + (c^2*x^2 + 1)*a*b*c^3*x^6 + 2*a*b*c^3*x^6 + a*b*c*x
^4 + (b^2*c^5*x^8 + (c^2*x^2 + 1)*b^2*c^3*x^6 + 2*b^2*c^3*x^6 + b^2*c*x^4 + 2*(b^2*c^4*x^7 + b^2*c^2*x^5)*sqrt
(c^2*x^2 + 1))*log(c*x + sqrt(c^2*x^2 + 1)) + 2*(a*b*c^4*x^7 + a*b*c^2*x^5)*sqrt(c^2*x^2 + 1)), x)

Giac [F(-2)]

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

[In]

integrate((c^2*x^2+1)^(5/2)/x^3/(a+b*arcsinh(c*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 = 2.74 (sec) , antiderivative size = 27, normalized size of antiderivative = 1.00 \[ \int \frac {\left (1+c^2 x^2\right )^{5/2}}{x^3 (a+b \text {arcsinh}(c x))^2} \, dx=\int \frac {{\left (c^2\,x^2+1\right )}^{5/2}}{x^3\,{\left (a+b\,\mathrm {asinh}\left (c\,x\right )\right )}^2} \,d x \]

[In]

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

[Out]

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