\(\int \frac {1}{(d+e x^2)^{5/2} (a+b \text {arcsinh}(c x))^2} \, dx\) [663]

Optimal result

Integrand size = 22, antiderivative size = 22 \[ \int \frac {1}{\left (d+e x^2\right )^{5/2} (a+b \text {arcsinh}(c x))^2} \, dx=\text {Int}\left (\frac {1}{\left (d+e x^2\right )^{5/2} (a+b \text {arcsinh}(c x))^2},x\right ) \]

[Out]

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

Rubi [N/A]

Not integrable

Time = 0.04 (sec) , antiderivative size = 22, 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 = {} \[ \int \frac {1}{\left (d+e x^2\right )^{5/2} (a+b \text {arcsinh}(c x))^2} \, dx=\int \frac {1}{\left (d+e x^2\right )^{5/2} (a+b \text {arcsinh}(c x))^2} \, dx \]

[In]

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

[Out]

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

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

Mathematica [N/A]

Not integrable

Time = 22.71 (sec) , antiderivative size = 24, normalized size of antiderivative = 1.09 \[ \int \frac {1}{\left (d+e x^2\right )^{5/2} (a+b \text {arcsinh}(c x))^2} \, dx=\int \frac {1}{\left (d+e x^2\right )^{5/2} (a+b \text {arcsinh}(c x))^2} \, dx \]

[In]

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

[Out]

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

Maple [N/A] (verified)

Not integrable

Time = 0.67 (sec) , antiderivative size = 20, normalized size of antiderivative = 0.91

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

[In]

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

[Out]

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

Fricas [N/A]

Not integrable

Time = 0.27 (sec) , antiderivative size = 149, normalized size of antiderivative = 6.77 \[ \int \frac {1}{\left (d+e x^2\right )^{5/2} (a+b \text {arcsinh}(c x))^2} \, dx=\int { \frac {1}{{\left (e x^{2} + d\right )}^{\frac {5}{2}} {\left (b \operatorname {arsinh}\left (c x\right ) + a\right )}^{2}} \,d x } \]

[In]

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

[Out]

integral(sqrt(e*x^2 + d)/(a^2*e^3*x^6 + 3*a^2*d*e^2*x^4 + 3*a^2*d^2*e*x^2 + a^2*d^3 + (b^2*e^3*x^6 + 3*b^2*d*e
^2*x^4 + 3*b^2*d^2*e*x^2 + b^2*d^3)*arcsinh(c*x)^2 + 2*(a*b*e^3*x^6 + 3*a*b*d*e^2*x^4 + 3*a*b*d^2*e*x^2 + a*b*
d^3)*arcsinh(c*x)), x)

Sympy [N/A]

Not integrable

Time = 27.89 (sec) , antiderivative size = 22, normalized size of antiderivative = 1.00 \[ \int \frac {1}{\left (d+e x^2\right )^{5/2} (a+b \text {arcsinh}(c x))^2} \, dx=\int \frac {1}{\left (a + b \operatorname {asinh}{\left (c x \right )}\right )^{2} \left (d + e x^{2}\right )^{\frac {5}{2}}}\, dx \]

[In]

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

[Out]

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

Maxima [N/A]

Not integrable

Time = 1.91 (sec) , antiderivative size = 1123, normalized size of antiderivative = 51.05 \[ \int \frac {1}{\left (d+e x^2\right )^{5/2} (a+b \text {arcsinh}(c x))^2} \, dx=\int { \frac {1}{{\left (e x^{2} + d\right )}^{\frac {5}{2}} {\left (b \operatorname {arsinh}\left (c x\right ) + a\right )}^{2}} \,d x } \]

[In]

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

[Out]

-(c^3*x^3 + c*x + (c^2*x^2 + 1)^(3/2))/((a*b*c^2*e^2*x^5 + 2*a*b*c^2*d*e*x^3 + a*b*c^2*d^2*x)*sqrt(c^2*x^2 + 1
)*sqrt(e*x^2 + d) + ((b^2*c^2*e^2*x^5 + 2*b^2*c^2*d*e*x^3 + b^2*c^2*d^2*x)*sqrt(c^2*x^2 + 1)*sqrt(e*x^2 + d) +
(b^2*c^3*e^2*x^6 + (2*c^3*d*e + c*e^2)*b^2*x^4 + b^2*c*d^2 + (c^3*d^2 + 2*c*d*e)*b^2*x^2)*sqrt(e*x^2 + d))*lo
g(c*x + sqrt(c^2*x^2 + 1)) + (a*b*c^3*e^2*x^6 + (2*c^3*d*e + c*e^2)*a*b*x^4 + a*b*c*d^2 + (c^3*d^2 + 2*c*d*e)*
a*b*x^2)*sqrt(e*x^2 + d)) - integrate((4*c^5*e*x^6 - (c^5*d - 8*c^3*e)*x^4 - 2*(c^3*d - 2*c*e)*x^2 + (4*c^3*e*
x^4 - (c^3*d - 6*c*e)*x^2 + c*d)*(c^2*x^2 + 1) - c*d + (8*c^4*e*x^5 - 2*(c^4*d - 7*c^2*e)*x^3 - (c^2*d - 5*e)*
x)*sqrt(c^2*x^2 + 1))/((a*b*c^3*e^3*x^8 + 3*a*b*c^3*d*e^2*x^6 + 3*a*b*c^3*d^2*e*x^4 + a*b*c^3*d^3*x^2)*(c^2*x^
2 + 1)*sqrt(e*x^2 + d) + 2*(a*b*c^4*e^3*x^9 + (3*c^4*d*e^2 + c^2*e^3)*a*b*x^7 + a*b*c^2*d^3*x + 3*(c^4*d^2*e +
c^2*d*e^2)*a*b*x^5 + (c^4*d^3 + 3*c^2*d^2*e)*a*b*x^3)*sqrt(c^2*x^2 + 1)*sqrt(e*x^2 + d) + ((b^2*c^3*e^3*x^8 +
3*b^2*c^3*d*e^2*x^6 + 3*b^2*c^3*d^2*e*x^4 + b^2*c^3*d^3*x^2)*(c^2*x^2 + 1)*sqrt(e*x^2 + d) + 2*(b^2*c^4*e^3*x
^9 + (3*c^4*d*e^2 + c^2*e^3)*b^2*x^7 + b^2*c^2*d^3*x + 3*(c^4*d^2*e + c^2*d*e^2)*b^2*x^5 + (c^4*d^3 + 3*c^2*d^
2*e)*b^2*x^3)*sqrt(c^2*x^2 + 1)*sqrt(e*x^2 + d) + (b^2*c^5*e^3*x^10 + (3*c^5*d*e^2 + 2*c^3*e^3)*b^2*x^8 + (3*c
^5*d^2*e + 6*c^3*d*e^2 + c*e^3)*b^2*x^6 + (c^5*d^3 + 6*c^3*d^2*e + 3*c*d*e^2)*b^2*x^4 + b^2*c*d^3 + (2*c^3*d^3
+ 3*c*d^2*e)*b^2*x^2)*sqrt(e*x^2 + d))*log(c*x + sqrt(c^2*x^2 + 1)) + (a*b*c^5*e^3*x^10 + (3*c^5*d*e^2 + 2*c^
3*e^3)*a*b*x^8 + (3*c^5*d^2*e + 6*c^3*d*e^2 + c*e^3)*a*b*x^6 + (c^5*d^3 + 6*c^3*d^2*e + 3*c*d*e^2)*a*b*x^4 + a
*b*c*d^3 + (2*c^3*d^3 + 3*c*d^2*e)*a*b*x^2)*sqrt(e*x^2 + d)), x)

Giac [N/A]

Not integrable

Time = 0.34 (sec) , antiderivative size = 22, normalized size of antiderivative = 1.00 \[ \int \frac {1}{\left (d+e x^2\right )^{5/2} (a+b \text {arcsinh}(c x))^2} \, dx=\int { \frac {1}{{\left (e x^{2} + d\right )}^{\frac {5}{2}} {\left (b \operatorname {arsinh}\left (c x\right ) + a\right )}^{2}} \,d x } \]

[In]

integrate(1/(e*x^2+d)^(5/2)/(a+b*arcsinh(c*x))^2,x, algorithm="giac")

[Out]

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

Mupad [N/A]

Not integrable

Time = 2.63 (sec) , antiderivative size = 22, normalized size of antiderivative = 1.00 \[ \int \frac {1}{\left (d+e x^2\right )^{5/2} (a+b \text {arcsinh}(c x))^2} \, dx=\int \frac {1}{{\left (a+b\,\mathrm {asinh}\left (c\,x\right )\right )}^2\,{\left (e\,x^2+d\right )}^{5/2}} \,d x \]

[In]

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

[Out]

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