Integrand size = 24, antiderivative size = 105 \[ \int x \sqrt {d+c^2 d x^2} (a+b \text {arcsinh}(c x)) \, dx=-\frac {b x \sqrt {d+c^2 d x^2}}{3 c \sqrt {1+c^2 x^2}}-\frac {b c x^3 \sqrt {d+c^2 d x^2}}{9 \sqrt {1+c^2 x^2}}+\frac {\left (d+c^2 d x^2\right )^{3/2} (a+b \text {arcsinh}(c x))}{3 c^2 d} \] Output:
-1/3*b*x*(c^2*d*x^2+d)^(1/2)/c/(c^2*x^2+1)^(1/2)-1/9*b*c*x^3*(c^2*d*x^2+d) ^(1/2)/(c^2*x^2+1)^(1/2)+1/3*(c^2*d*x^2+d)^(3/2)*(a+b*arcsinh(c*x))/c^2/d
Time = 0.11 (sec) , antiderivative size = 92, normalized size of antiderivative = 0.88 \[ \int x \sqrt {d+c^2 d x^2} (a+b \text {arcsinh}(c x)) \, dx=\frac {\sqrt {d+c^2 d x^2} \left (3 a \left (1+c^2 x^2\right )^2-b c x \sqrt {1+c^2 x^2} \left (3+c^2 x^2\right )+3 b \left (1+c^2 x^2\right )^2 \text {arcsinh}(c x)\right )}{9 c^2 \left (1+c^2 x^2\right )} \] Input:
Integrate[x*Sqrt[d + c^2*d*x^2]*(a + b*ArcSinh[c*x]),x]
Output:
(Sqrt[d + c^2*d*x^2]*(3*a*(1 + c^2*x^2)^2 - b*c*x*Sqrt[1 + c^2*x^2]*(3 + c ^2*x^2) + 3*b*(1 + c^2*x^2)^2*ArcSinh[c*x]))/(9*c^2*(1 + c^2*x^2))
Time = 0.30 (sec) , antiderivative size = 80, normalized size of antiderivative = 0.76, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.083, Rules used = {6213, 2009}
Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.
\(\displaystyle \int x \sqrt {c^2 d x^2+d} (a+b \text {arcsinh}(c x)) \, dx\) |
\(\Big \downarrow \) 6213 |
\(\displaystyle \frac {\left (c^2 d x^2+d\right )^{3/2} (a+b \text {arcsinh}(c x))}{3 c^2 d}-\frac {b \sqrt {c^2 d x^2+d} \int \left (c^2 x^2+1\right )dx}{3 c \sqrt {c^2 x^2+1}}\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle \frac {\left (c^2 d x^2+d\right )^{3/2} (a+b \text {arcsinh}(c x))}{3 c^2 d}-\frac {b \left (\frac {c^2 x^3}{3}+x\right ) \sqrt {c^2 d x^2+d}}{3 c \sqrt {c^2 x^2+1}}\) |
Input:
Int[x*Sqrt[d + c^2*d*x^2]*(a + b*ArcSinh[c*x]),x]
Output:
-1/3*(b*Sqrt[d + c^2*d*x^2]*(x + (c^2*x^3)/3))/(c*Sqrt[1 + c^2*x^2]) + ((d + c^2*d*x^2)^(3/2)*(a + b*ArcSinh[c*x]))/(3*c^2*d)
Int[((a_.) + ArcSinh[(c_.)*(x_)]*(b_.))^(n_.)*(x_)*((d_) + (e_.)*(x_)^2)^(p _.), x_Symbol] :> Simp[(d + e*x^2)^(p + 1)*((a + b*ArcSinh[c*x])^n/(2*e*(p + 1))), x] - Simp[b*(n/(2*c*(p + 1)))*Simp[(d + e*x^2)^p/(1 + c^2*x^2)^p] Int[(1 + c^2*x^2)^(p + 1/2)*(a + b*ArcSinh[c*x])^(n - 1), x], x] /; FreeQ[ {a, b, c, d, e, p}, x] && EqQ[e, c^2*d] && GtQ[n, 0] && NeQ[p, -1]
Time = 0.62 (sec) , antiderivative size = 147, normalized size of antiderivative = 1.40
method | result | size |
orering | \(\frac {\left (5 c^{4} x^{4}+13 c^{2} x^{2}+6\right ) \sqrt {c^{2} d \,x^{2}+d}\, \left (a +b \,\operatorname {arcsinh}\left (x c \right )\right )}{9 c^{2} \left (c^{2} x^{2}+1\right )}-\frac {\left (c^{2} x^{2}+3\right ) \left (\sqrt {c^{2} d \,x^{2}+d}\, \left (a +b \,\operatorname {arcsinh}\left (x c \right )\right )+\frac {x^{2} \left (a +b \,\operatorname {arcsinh}\left (x c \right )\right ) c^{2} d}{\sqrt {c^{2} d \,x^{2}+d}}+\frac {b c x \sqrt {c^{2} d \,x^{2}+d}}{\sqrt {c^{2} x^{2}+1}}\right )}{9 c^{2}}\) | \(147\) |
default | \(\frac {a \left (c^{2} d \,x^{2}+d \right )^{\frac {3}{2}}}{3 c^{2} d}+b \left (\frac {\sqrt {d \left (c^{2} x^{2}+1\right )}\, \left (4 c^{4} x^{4}+4 \sqrt {c^{2} x^{2}+1}\, c^{3} x^{3}+5 c^{2} x^{2}+3 \sqrt {c^{2} x^{2}+1}\, x c +1\right ) \left (-1+3 \,\operatorname {arcsinh}\left (x c \right )\right )}{72 c^{2} \left (c^{2} x^{2}+1\right )}+\frac {\sqrt {d \left (c^{2} x^{2}+1\right )}\, \left (c^{2} x^{2}+\sqrt {c^{2} x^{2}+1}\, x c +1\right ) \left (\operatorname {arcsinh}\left (x c \right )-1\right )}{8 c^{2} \left (c^{2} x^{2}+1\right )}+\frac {\sqrt {d \left (c^{2} x^{2}+1\right )}\, \left (c^{2} x^{2}-\sqrt {c^{2} x^{2}+1}\, x c +1\right ) \left (\operatorname {arcsinh}\left (x c \right )+1\right )}{8 c^{2} \left (c^{2} x^{2}+1\right )}+\frac {\sqrt {d \left (c^{2} x^{2}+1\right )}\, \left (4 c^{4} x^{4}-4 \sqrt {c^{2} x^{2}+1}\, c^{3} x^{3}+5 c^{2} x^{2}-3 \sqrt {c^{2} x^{2}+1}\, x c +1\right ) \left (1+3 \,\operatorname {arcsinh}\left (x c \right )\right )}{72 c^{2} \left (c^{2} x^{2}+1\right )}\right )\) | \(321\) |
parts | \(\frac {a \left (c^{2} d \,x^{2}+d \right )^{\frac {3}{2}}}{3 c^{2} d}+b \left (\frac {\sqrt {d \left (c^{2} x^{2}+1\right )}\, \left (4 c^{4} x^{4}+4 \sqrt {c^{2} x^{2}+1}\, c^{3} x^{3}+5 c^{2} x^{2}+3 \sqrt {c^{2} x^{2}+1}\, x c +1\right ) \left (-1+3 \,\operatorname {arcsinh}\left (x c \right )\right )}{72 c^{2} \left (c^{2} x^{2}+1\right )}+\frac {\sqrt {d \left (c^{2} x^{2}+1\right )}\, \left (c^{2} x^{2}+\sqrt {c^{2} x^{2}+1}\, x c +1\right ) \left (\operatorname {arcsinh}\left (x c \right )-1\right )}{8 c^{2} \left (c^{2} x^{2}+1\right )}+\frac {\sqrt {d \left (c^{2} x^{2}+1\right )}\, \left (c^{2} x^{2}-\sqrt {c^{2} x^{2}+1}\, x c +1\right ) \left (\operatorname {arcsinh}\left (x c \right )+1\right )}{8 c^{2} \left (c^{2} x^{2}+1\right )}+\frac {\sqrt {d \left (c^{2} x^{2}+1\right )}\, \left (4 c^{4} x^{4}-4 \sqrt {c^{2} x^{2}+1}\, c^{3} x^{3}+5 c^{2} x^{2}-3 \sqrt {c^{2} x^{2}+1}\, x c +1\right ) \left (1+3 \,\operatorname {arcsinh}\left (x c \right )\right )}{72 c^{2} \left (c^{2} x^{2}+1\right )}\right )\) | \(321\) |
Input:
int(x*(c^2*d*x^2+d)^(1/2)*(a+b*arcsinh(x*c)),x,method=_RETURNVERBOSE)
Output:
1/9*(5*c^4*x^4+13*c^2*x^2+6)/c^2/(c^2*x^2+1)*(c^2*d*x^2+d)^(1/2)*(a+b*arcs inh(x*c))-1/9*(c^2*x^2+3)/c^2*((c^2*d*x^2+d)^(1/2)*(a+b*arcsinh(x*c))+x^2/ (c^2*d*x^2+d)^(1/2)*(a+b*arcsinh(x*c))*c^2*d+b*c*x*(c^2*d*x^2+d)^(1/2)/(c^ 2*x^2+1)^(1/2))
Time = 0.09 (sec) , antiderivative size = 127, normalized size of antiderivative = 1.21 \[ \int x \sqrt {d+c^2 d x^2} (a+b \text {arcsinh}(c x)) \, dx=\frac {3 \, {\left (b c^{4} x^{4} + 2 \, b c^{2} x^{2} + b\right )} \sqrt {c^{2} d x^{2} + d} \log \left (c x + \sqrt {c^{2} x^{2} + 1}\right ) + {\left (3 \, a c^{4} x^{4} + 6 \, a c^{2} x^{2} - {\left (b c^{3} x^{3} + 3 \, b c x\right )} \sqrt {c^{2} x^{2} + 1} + 3 \, a\right )} \sqrt {c^{2} d x^{2} + d}}{9 \, {\left (c^{4} x^{2} + c^{2}\right )}} \] Input:
integrate(x*(c^2*d*x^2+d)^(1/2)*(a+b*arcsinh(c*x)),x, algorithm="fricas")
Output:
1/9*(3*(b*c^4*x^4 + 2*b*c^2*x^2 + b)*sqrt(c^2*d*x^2 + d)*log(c*x + sqrt(c^ 2*x^2 + 1)) + (3*a*c^4*x^4 + 6*a*c^2*x^2 - (b*c^3*x^3 + 3*b*c*x)*sqrt(c^2* x^2 + 1) + 3*a)*sqrt(c^2*d*x^2 + d))/(c^4*x^2 + c^2)
\[ \int x \sqrt {d+c^2 d x^2} (a+b \text {arcsinh}(c x)) \, dx=\int x \sqrt {d \left (c^{2} x^{2} + 1\right )} \left (a + b \operatorname {asinh}{\left (c x \right )}\right )\, dx \] Input:
integrate(x*(c**2*d*x**2+d)**(1/2)*(a+b*asinh(c*x)),x)
Output:
Integral(x*sqrt(d*(c**2*x**2 + 1))*(a + b*asinh(c*x)), x)
Time = 0.04 (sec) , antiderivative size = 73, normalized size of antiderivative = 0.70 \[ \int x \sqrt {d+c^2 d x^2} (a+b \text {arcsinh}(c x)) \, dx=\frac {{\left (c^{2} d x^{2} + d\right )}^{\frac {3}{2}} b \operatorname {arsinh}\left (c x\right )}{3 \, c^{2} d} - \frac {{\left (c^{2} d^{\frac {3}{2}} x^{3} + 3 \, d^{\frac {3}{2}} x\right )} b}{9 \, c d} + \frac {{\left (c^{2} d x^{2} + d\right )}^{\frac {3}{2}} a}{3 \, c^{2} d} \] Input:
integrate(x*(c^2*d*x^2+d)^(1/2)*(a+b*arcsinh(c*x)),x, algorithm="maxima")
Output:
1/3*(c^2*d*x^2 + d)^(3/2)*b*arcsinh(c*x)/(c^2*d) - 1/9*(c^2*d^(3/2)*x^3 + 3*d^(3/2)*x)*b/(c*d) + 1/3*(c^2*d*x^2 + d)^(3/2)*a/(c^2*d)
Exception generated. \[ \int x \sqrt {d+c^2 d x^2} (a+b \text {arcsinh}(c x)) \, dx=\text {Exception raised: TypeError} \] Input:
integrate(x*(c^2*d*x^2+d)^(1/2)*(a+b*arcsinh(c*x)),x, algorithm="giac")
Output:
Exception raised: TypeError >> an error occurred running a Giac command:IN PUT:sage2:=int(sage0,sageVARx):;OUTPUT:sym2poly/r2sym(const gen & e,const index_m & i,const vecteur & l) Error: Bad Argument Value
Timed out. \[ \int x \sqrt {d+c^2 d x^2} (a+b \text {arcsinh}(c x)) \, dx=\int x\,\left (a+b\,\mathrm {asinh}\left (c\,x\right )\right )\,\sqrt {d\,c^2\,x^2+d} \,d x \] Input:
int(x*(a + b*asinh(c*x))*(d + c^2*d*x^2)^(1/2),x)
Output:
int(x*(a + b*asinh(c*x))*(d + c^2*d*x^2)^(1/2), x)
\[ \int x \sqrt {d+c^2 d x^2} (a+b \text {arcsinh}(c x)) \, dx=\frac {\sqrt {d}\, \left (\sqrt {c^{2} x^{2}+1}\, a \,c^{2} x^{2}+\sqrt {c^{2} x^{2}+1}\, a +3 \left (\int \sqrt {c^{2} x^{2}+1}\, \mathit {asinh} \left (c x \right ) x d x \right ) b \,c^{2}\right )}{3 c^{2}} \] Input:
int(x*(c^2*d*x^2+d)^(1/2)*(a+b*asinh(c*x)),x)
Output:
(sqrt(d)*(sqrt(c**2*x**2 + 1)*a*c**2*x**2 + sqrt(c**2*x**2 + 1)*a + 3*int( sqrt(c**2*x**2 + 1)*asinh(c*x)*x,x)*b*c**2))/(3*c**2)