\(\int \frac {1}{(1+a^2+2 a b x+b^2 x^2)^{3/2} \text {arcsinh}(a+b x)} \, dx\) [281]

Optimal result

Integrand size = 30, antiderivative size = 30 \[ \int \frac {1}{\left (1+a^2+2 a b x+b^2 x^2\right )^{3/2} \text {arcsinh}(a+b x)} \, dx=\text {Int}\left (\frac {1}{\left (1+(a+b x)^2\right )^{3/2} \text {arcsinh}(a+b x)},x\right ) \]

[Out]

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

Rubi [N/A]

Not integrable

Time = 0.06 (sec) , antiderivative size = 30, 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 = {} \[ \int \frac {1}{\left (1+a^2+2 a b x+b^2 x^2\right )^{3/2} \text {arcsinh}(a+b x)} \, dx=\int \frac {1}{\left (1+a^2+2 a b x+b^2 x^2\right )^{3/2} \text {arcsinh}(a+b x)} \, dx \]

[In]

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

[Out]

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

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

Mathematica [N/A]

Not integrable

Time = 0.51 (sec) , antiderivative size = 32, normalized size of antiderivative = 1.07 \[ \int \frac {1}{\left (1+a^2+2 a b x+b^2 x^2\right )^{3/2} \text {arcsinh}(a+b x)} \, dx=\int \frac {1}{\left (1+a^2+2 a b x+b^2 x^2\right )^{3/2} \text {arcsinh}(a+b x)} \, dx \]

[In]

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

[Out]

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

Maple [N/A] (verified)

Not integrable

Time = 0.37 (sec) , antiderivative size = 28, normalized size of antiderivative = 0.93

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

[In]

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

[Out]

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

Fricas [N/A]

Not integrable

Time = 0.24 (sec) , antiderivative size = 82, normalized size of antiderivative = 2.73 \[ \int \frac {1}{\left (1+a^2+2 a b x+b^2 x^2\right )^{3/2} \text {arcsinh}(a+b x)} \, dx=\int { \frac {1}{{\left (b^{2} x^{2} + 2 \, a b x + a^{2} + 1\right )}^{\frac {3}{2}} \operatorname {arsinh}\left (b x + a\right )} \,d x } \]

[In]

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

[Out]

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

Sympy [N/A]

Not integrable

Time = 1.61 (sec) , antiderivative size = 31, normalized size of antiderivative = 1.03 \[ \int \frac {1}{\left (1+a^2+2 a b x+b^2 x^2\right )^{3/2} \text {arcsinh}(a+b x)} \, dx=\int \frac {1}{\left (a^{2} + 2 a b x + b^{2} x^{2} + 1\right )^{\frac {3}{2}} \operatorname {asinh}{\left (a + b x \right )}}\, dx \]

[In]

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

[Out]

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

Maxima [N/A]

Not integrable

Time = 0.33 (sec) , antiderivative size = 30, normalized size of antiderivative = 1.00 \[ \int \frac {1}{\left (1+a^2+2 a b x+b^2 x^2\right )^{3/2} \text {arcsinh}(a+b x)} \, dx=\int { \frac {1}{{\left (b^{2} x^{2} + 2 \, a b x + a^{2} + 1\right )}^{\frac {3}{2}} \operatorname {arsinh}\left (b x + a\right )} \,d x } \]

[In]

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

[Out]

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

Giac [N/A]

Not integrable

Time = 0.30 (sec) , antiderivative size = 30, normalized size of antiderivative = 1.00 \[ \int \frac {1}{\left (1+a^2+2 a b x+b^2 x^2\right )^{3/2} \text {arcsinh}(a+b x)} \, dx=\int { \frac {1}{{\left (b^{2} x^{2} + 2 \, a b x + a^{2} + 1\right )}^{\frac {3}{2}} \operatorname {arsinh}\left (b x + a\right )} \,d x } \]

[In]

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

[Out]

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

Mupad [N/A]

Not integrable

Time = 2.65 (sec) , antiderivative size = 30, normalized size of antiderivative = 1.00 \[ \int \frac {1}{\left (1+a^2+2 a b x+b^2 x^2\right )^{3/2} \text {arcsinh}(a+b x)} \, dx=\int \frac {1}{\mathrm {asinh}\left (a+b\,x\right )\,{\left (a^2+2\,a\,b\,x+b^2\,x^2+1\right )}^{3/2}} \,d x \]

[In]

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

[Out]

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