Integrand size = 26, antiderivative size = 175 \[ \int x^3 \sqrt {d+c^2 d x^2} (a+b \text {arcsinh}(c x)) \, dx=\frac {2 b x \sqrt {d+c^2 d x^2}}{15 c^3 \sqrt {1+c^2 x^2}}-\frac {b x^3 \sqrt {d+c^2 d x^2}}{45 c \sqrt {1+c^2 x^2}}-\frac {b c x^5 \sqrt {d+c^2 d x^2}}{25 \sqrt {1+c^2 x^2}}-\frac {\left (d+c^2 d x^2\right )^{3/2} (a+b \text {arcsinh}(c x))}{3 c^4 d}+\frac {\left (d+c^2 d x^2\right )^{5/2} (a+b \text {arcsinh}(c x))}{5 c^4 d^2} \] Output:
2/15*b*x*(c^2*d*x^2+d)^(1/2)/c^3/(c^2*x^2+1)^(1/2)-1/45*b*x^3*(c^2*d*x^2+d )^(1/2)/c/(c^2*x^2+1)^(1/2)-1/25*b*c*x^5*(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^4/d+1/5*(c^2*d*x^2+d)^(5 /2)*(a+b*arcsinh(c*x))/c^4/d^2
Time = 0.12 (sec) , antiderivative size = 120, normalized size of antiderivative = 0.69 \[ \int x^3 \sqrt {d+c^2 d x^2} (a+b \text {arcsinh}(c x)) \, dx=\frac {\sqrt {d+c^2 d x^2} \left (15 a \left (1+c^2 x^2\right )^2 \left (-2+3 c^2 x^2\right )+b c x \sqrt {1+c^2 x^2} \left (30-5 c^2 x^2-9 c^4 x^4\right )+15 b \left (1+c^2 x^2\right )^2 \left (-2+3 c^2 x^2\right ) \text {arcsinh}(c x)\right )}{225 c^4 \left (1+c^2 x^2\right )} \] Input:
Integrate[x^3*Sqrt[d + c^2*d*x^2]*(a + b*ArcSinh[c*x]),x]
Output:
(Sqrt[d + c^2*d*x^2]*(15*a*(1 + c^2*x^2)^2*(-2 + 3*c^2*x^2) + b*c*x*Sqrt[1 + c^2*x^2]*(30 - 5*c^2*x^2 - 9*c^4*x^4) + 15*b*(1 + c^2*x^2)^2*(-2 + 3*c^ 2*x^2)*ArcSinh[c*x]))/(225*c^4*(1 + c^2*x^2))
Time = 0.38 (sec) , antiderivative size = 124, normalized size of antiderivative = 0.71, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.115, Rules used = {6219, 27, 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.
| \(\displaystyle \int x^3 \sqrt {c^2 d x^2+d} (a+b \text {arcsinh}(c x)) \, dx\) |
| \(\Big \downarrow \) 6219 |
| \(\displaystyle -\frac {b c \sqrt {c^2 d x^2+d} \int -\frac {-3 c^4 x^4-c^2 x^2+2}{15 c^4}dx}{\sqrt {c^2 x^2+1}}+\frac {\left (c^2 d x^2+d\right )^{5/2} (a+b \text {arcsinh}(c x))}{5 c^4 d^2}-\frac {\left (c^2 d x^2+d\right )^{3/2} (a+b \text {arcsinh}(c x))}{3 c^4 d}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {b \sqrt {c^2 d x^2+d} \int \left (-3 c^4 x^4-c^2 x^2+2\right )dx}{15 c^3 \sqrt {c^2 x^2+1}}+\frac {\left (c^2 d x^2+d\right )^{5/2} (a+b \text {arcsinh}(c x))}{5 c^4 d^2}-\frac {\left (c^2 d x^2+d\right )^{3/2} (a+b \text {arcsinh}(c x))}{3 c^4 d}\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {\left (c^2 d x^2+d\right )^{5/2} (a+b \text {arcsinh}(c x))}{5 c^4 d^2}-\frac {\left (c^2 d x^2+d\right )^{3/2} (a+b \text {arcsinh}(c x))}{3 c^4 d}+\frac {b \left (-\frac {3}{5} c^4 x^5-\frac {c^2 x^3}{3}+2 x\right ) \sqrt {c^2 d x^2+d}}{15 c^3 \sqrt {c^2 x^2+1}}\) |
Input:
Int[x^3*Sqrt[d + c^2*d*x^2]*(a + b*ArcSinh[c*x]),x]
Output:
(b*Sqrt[d + c^2*d*x^2]*(2*x - (c^2*x^3)/3 - (3*c^4*x^5)/5))/(15*c^3*Sqrt[1 + c^2*x^2]) - ((d + c^2*d*x^2)^(3/2)*(a + b*ArcSinh[c*x]))/(3*c^4*d) + (( d + c^2*d*x^2)^(5/2)*(a + b*ArcSinh[c*x]))/(5*c^4*d^2)
Int[(a_)*(Fx_), x_Symbol] :> Simp[a Int[Fx, x], x] /; FreeQ[a, x] && !Ma tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
Int[((a_.) + ArcSinh[(c_.)*(x_)]*(b_.))*(x_)^(m_)*((d_) + (e_.)*(x_)^2)^(p_ ), x_Symbol] :> With[{u = IntHide[x^m*(d + e*x^2)^p, x]}, Simp[(a + b*ArcSi nh[c*x]) u, x] - Simp[b*c*Simp[Sqrt[d + e*x^2]/Sqrt[1 + c^2*x^2]] Int[S implifyIntegrand[u/Sqrt[d + e*x^2], x], x], x]] /; FreeQ[{a, b, c, d, e}, x ] && EqQ[e, c^2*d] && IntegerQ[p - 1/2] && NeQ[p, -2^(-1)] && (IGtQ[(m + 1) /2, 0] || ILtQ[(m + 2*p + 3)/2, 0])
Time = 1.46 (sec) , antiderivative size = 173, normalized size of antiderivative = 0.99
| method | result | size |
| orering | \(\frac {\left (81 c^{6} x^{6}+107 c^{4} x^{4}-120 c^{2} x^{2}-120\right ) \sqrt {c^{2} d \,x^{2}+d}\, \left (a +b \,\operatorname {arcsinh}\left (x c \right )\right )}{225 c^{4} \left (c^{2} x^{2}+1\right )}-\frac {\left (9 c^{4} x^{4}+5 c^{2} x^{2}-30\right ) \left (3 x^{2} \sqrt {c^{2} d \,x^{2}+d}\, \left (a +b \,\operatorname {arcsinh}\left (x c \right )\right )+\frac {x^{4} \left (a +b \,\operatorname {arcsinh}\left (x c \right )\right ) c^{2} d}{\sqrt {c^{2} d \,x^{2}+d}}+\frac {b c \,x^{3} \sqrt {c^{2} d \,x^{2}+d}}{\sqrt {c^{2} x^{2}+1}}\right )}{225 x^{2} c^{4}}\) | \(173\) |
| default | \(a \left (\frac {x^{2} \left (c^{2} d \,x^{2}+d \right )^{\frac {3}{2}}}{5 c^{2} d}-\frac {2 \left (c^{2} d \,x^{2}+d \right )^{\frac {3}{2}}}{15 d \,c^{4}}\right )+b \left (\frac {\sqrt {d \left (c^{2} x^{2}+1\right )}\, \left (16 c^{6} x^{6}+16 \sqrt {c^{2} x^{2}+1}\, x^{5} c^{5}+28 c^{4} x^{4}+20 \sqrt {c^{2} x^{2}+1}\, c^{3} x^{3}+13 c^{2} x^{2}+5 \sqrt {c^{2} x^{2}+1}\, x c +1\right ) \left (-1+5 \,\operatorname {arcsinh}\left (x c \right )\right )}{800 c^{4} \left (c^{2} x^{2}+1\right )}-\frac {\sqrt {d \left (c^{2} x^{2}+1\right )}\, \left (4 c^{4} x^{4}+4 \sqrt {c^{2} x^{2}+1}\, c^{3} x^{3}+5 c^{2} x^{2}+3 \sqrt {c^{2} x^{2}+1}\, x c +1\right ) \left (-1+3 \,\operatorname {arcsinh}\left (x c \right )\right )}{288 c^{4} \left (c^{2} x^{2}+1\right )}-\frac {\sqrt {d \left (c^{2} x^{2}+1\right )}\, \left (c^{2} x^{2}+\sqrt {c^{2} x^{2}+1}\, x c +1\right ) \left (\operatorname {arcsinh}\left (x c \right )-1\right )}{16 c^{4} \left (c^{2} x^{2}+1\right )}-\frac {\sqrt {d \left (c^{2} x^{2}+1\right )}\, \left (c^{2} x^{2}-\sqrt {c^{2} x^{2}+1}\, x c +1\right ) \left (\operatorname {arcsinh}\left (x c \right )+1\right )}{16 c^{4} \left (c^{2} x^{2}+1\right )}-\frac {\sqrt {d \left (c^{2} x^{2}+1\right )}\, \left (4 c^{4} x^{4}-4 \sqrt {c^{2} x^{2}+1}\, c^{3} x^{3}+5 c^{2} x^{2}-3 \sqrt {c^{2} x^{2}+1}\, x c +1\right ) \left (1+3 \,\operatorname {arcsinh}\left (x c \right )\right )}{288 c^{4} \left (c^{2} x^{2}+1\right )}+\frac {\sqrt {d \left (c^{2} x^{2}+1\right )}\, \left (16 c^{6} x^{6}-16 \sqrt {c^{2} x^{2}+1}\, x^{5} c^{5}+28 c^{4} x^{4}-20 \sqrt {c^{2} x^{2}+1}\, c^{3} x^{3}+13 c^{2} x^{2}-5 \sqrt {c^{2} x^{2}+1}\, x c +1\right ) \left (1+5 \,\operatorname {arcsinh}\left (x c \right )\right )}{800 c^{4} \left (c^{2} x^{2}+1\right )}\right )\) | \(578\) |
| parts | \(a \left (\frac {x^{2} \left (c^{2} d \,x^{2}+d \right )^{\frac {3}{2}}}{5 c^{2} d}-\frac {2 \left (c^{2} d \,x^{2}+d \right )^{\frac {3}{2}}}{15 d \,c^{4}}\right )+b \left (\frac {\sqrt {d \left (c^{2} x^{2}+1\right )}\, \left (16 c^{6} x^{6}+16 \sqrt {c^{2} x^{2}+1}\, x^{5} c^{5}+28 c^{4} x^{4}+20 \sqrt {c^{2} x^{2}+1}\, c^{3} x^{3}+13 c^{2} x^{2}+5 \sqrt {c^{2} x^{2}+1}\, x c +1\right ) \left (-1+5 \,\operatorname {arcsinh}\left (x c \right )\right )}{800 c^{4} \left (c^{2} x^{2}+1\right )}-\frac {\sqrt {d \left (c^{2} x^{2}+1\right )}\, \left (4 c^{4} x^{4}+4 \sqrt {c^{2} x^{2}+1}\, c^{3} x^{3}+5 c^{2} x^{2}+3 \sqrt {c^{2} x^{2}+1}\, x c +1\right ) \left (-1+3 \,\operatorname {arcsinh}\left (x c \right )\right )}{288 c^{4} \left (c^{2} x^{2}+1\right )}-\frac {\sqrt {d \left (c^{2} x^{2}+1\right )}\, \left (c^{2} x^{2}+\sqrt {c^{2} x^{2}+1}\, x c +1\right ) \left (\operatorname {arcsinh}\left (x c \right )-1\right )}{16 c^{4} \left (c^{2} x^{2}+1\right )}-\frac {\sqrt {d \left (c^{2} x^{2}+1\right )}\, \left (c^{2} x^{2}-\sqrt {c^{2} x^{2}+1}\, x c +1\right ) \left (\operatorname {arcsinh}\left (x c \right )+1\right )}{16 c^{4} \left (c^{2} x^{2}+1\right )}-\frac {\sqrt {d \left (c^{2} x^{2}+1\right )}\, \left (4 c^{4} x^{4}-4 \sqrt {c^{2} x^{2}+1}\, c^{3} x^{3}+5 c^{2} x^{2}-3 \sqrt {c^{2} x^{2}+1}\, x c +1\right ) \left (1+3 \,\operatorname {arcsinh}\left (x c \right )\right )}{288 c^{4} \left (c^{2} x^{2}+1\right )}+\frac {\sqrt {d \left (c^{2} x^{2}+1\right )}\, \left (16 c^{6} x^{6}-16 \sqrt {c^{2} x^{2}+1}\, x^{5} c^{5}+28 c^{4} x^{4}-20 \sqrt {c^{2} x^{2}+1}\, c^{3} x^{3}+13 c^{2} x^{2}-5 \sqrt {c^{2} x^{2}+1}\, x c +1\right ) \left (1+5 \,\operatorname {arcsinh}\left (x c \right )\right )}{800 c^{4} \left (c^{2} x^{2}+1\right )}\right )\) | \(578\) |
Input:
int(x^3*(c^2*d*x^2+d)^(1/2)*(a+b*arcsinh(x*c)),x,method=_RETURNVERBOSE)
Output:
1/225*(81*c^6*x^6+107*c^4*x^4-120*c^2*x^2-120)/c^4/(c^2*x^2+1)*(c^2*d*x^2+ d)^(1/2)*(a+b*arcsinh(x*c))-1/225/x^2*(9*c^4*x^4+5*c^2*x^2-30)/c^4*(3*x^2* (c^2*d*x^2+d)^(1/2)*(a+b*arcsinh(x*c))+x^4/(c^2*d*x^2+d)^(1/2)*(a+b*arcsin h(x*c))*c^2*d+b*c*x^3*(c^2*d*x^2+d)^(1/2)/(c^2*x^2+1)^(1/2))
Time = 0.10 (sec) , antiderivative size = 158, normalized size of antiderivative = 0.90 \[ \int x^3 \sqrt {d+c^2 d x^2} (a+b \text {arcsinh}(c x)) \, dx=\frac {15 \, {\left (3 \, b c^{6} x^{6} + 4 \, b c^{4} x^{4} - b c^{2} x^{2} - 2 \, b\right )} \sqrt {c^{2} d x^{2} + d} \log \left (c x + \sqrt {c^{2} x^{2} + 1}\right ) + {\left (45 \, a c^{6} x^{6} + 60 \, a c^{4} x^{4} - 15 \, a c^{2} x^{2} - {\left (9 \, b c^{5} x^{5} + 5 \, b c^{3} x^{3} - 30 \, b c x\right )} \sqrt {c^{2} x^{2} + 1} - 30 \, a\right )} \sqrt {c^{2} d x^{2} + d}}{225 \, {\left (c^{6} x^{2} + c^{4}\right )}} \] Input:
integrate(x^3*(c^2*d*x^2+d)^(1/2)*(a+b*arcsinh(c*x)),x, algorithm="fricas" )
Output:
1/225*(15*(3*b*c^6*x^6 + 4*b*c^4*x^4 - b*c^2*x^2 - 2*b)*sqrt(c^2*d*x^2 + d )*log(c*x + sqrt(c^2*x^2 + 1)) + (45*a*c^6*x^6 + 60*a*c^4*x^4 - 15*a*c^2*x ^2 - (9*b*c^5*x^5 + 5*b*c^3*x^3 - 30*b*c*x)*sqrt(c^2*x^2 + 1) - 30*a)*sqrt (c^2*d*x^2 + d))/(c^6*x^2 + c^4)
\[ \int x^3 \sqrt {d+c^2 d x^2} (a+b \text {arcsinh}(c x)) \, dx=\int x^{3} \sqrt {d \left (c^{2} x^{2} + 1\right )} \left (a + b \operatorname {asinh}{\left (c x \right )}\right )\, dx \] Input:
integrate(x**3*(c**2*d*x**2+d)**(1/2)*(a+b*asinh(c*x)),x)
Output:
Integral(x**3*sqrt(d*(c**2*x**2 + 1))*(a + b*asinh(c*x)), x)
Time = 0.04 (sec) , antiderivative size = 134, normalized size of antiderivative = 0.77 \[ \int x^3 \sqrt {d+c^2 d x^2} (a+b \text {arcsinh}(c x)) \, dx=\frac {1}{15} \, b {\left (\frac {3 \, {\left (c^{2} d x^{2} + d\right )}^{\frac {3}{2}} x^{2}}{c^{2} d} - \frac {2 \, {\left (c^{2} d x^{2} + d\right )}^{\frac {3}{2}}}{c^{4} d}\right )} \operatorname {arsinh}\left (c x\right ) + \frac {1}{15} \, a {\left (\frac {3 \, {\left (c^{2} d x^{2} + d\right )}^{\frac {3}{2}} x^{2}}{c^{2} d} - \frac {2 \, {\left (c^{2} d x^{2} + d\right )}^{\frac {3}{2}}}{c^{4} d}\right )} - \frac {{\left (9 \, c^{4} \sqrt {d} x^{5} + 5 \, c^{2} \sqrt {d} x^{3} - 30 \, \sqrt {d} x\right )} b}{225 \, c^{3}} \] Input:
integrate(x^3*(c^2*d*x^2+d)^(1/2)*(a+b*arcsinh(c*x)),x, algorithm="maxima" )
Output:
1/15*b*(3*(c^2*d*x^2 + d)^(3/2)*x^2/(c^2*d) - 2*(c^2*d*x^2 + d)^(3/2)/(c^4 *d))*arcsinh(c*x) + 1/15*a*(3*(c^2*d*x^2 + d)^(3/2)*x^2/(c^2*d) - 2*(c^2*d *x^2 + d)^(3/2)/(c^4*d)) - 1/225*(9*c^4*sqrt(d)*x^5 + 5*c^2*sqrt(d)*x^3 - 30*sqrt(d)*x)*b/c^3
Exception generated. \[ \int x^3 \sqrt {d+c^2 d x^2} (a+b \text {arcsinh}(c x)) \, dx=\text {Exception raised: TypeError} \] Input:
integrate(x^3*(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. \[ \int x^3 \sqrt {d+c^2 d x^2} (a+b \text {arcsinh}(c x)) \, dx=\int x^3\,\left (a+b\,\mathrm {asinh}\left (c\,x\right )\right )\,\sqrt {d\,c^2\,x^2+d} \,d x \] Input:
int(x^3*(a + b*asinh(c*x))*(d + c^2*d*x^2)^(1/2),x)
Output:
int(x^3*(a + b*asinh(c*x))*(d + c^2*d*x^2)^(1/2), x)
\[ \int x^3 \sqrt {d+c^2 d x^2} (a+b \text {arcsinh}(c x)) \, dx=\frac {\sqrt {d}\, \left (3 \sqrt {c^{2} x^{2}+1}\, a \,c^{4} x^{4}+\sqrt {c^{2} x^{2}+1}\, a \,c^{2} x^{2}-2 \sqrt {c^{2} x^{2}+1}\, a +15 \left (\int \sqrt {c^{2} x^{2}+1}\, \mathit {asinh} \left (c x \right ) x^{3}d x \right ) b \,c^{4}\right )}{15 c^{4}} \] Input:
int(x^3*(c^2*d*x^2+d)^(1/2)*(a+b*asinh(c*x)),x)
Output:
(sqrt(d)*(3*sqrt(c**2*x**2 + 1)*a*c**4*x**4 + sqrt(c**2*x**2 + 1)*a*c**2*x **2 - 2*sqrt(c**2*x**2 + 1)*a + 15*int(sqrt(c**2*x**2 + 1)*asinh(c*x)*x**3 ,x)*b*c**4))/(15*c**4)