Integrand size = 21, antiderivative size = 128 \[ \int \left (d+c^2 d x^2\right )^2 (a+b \text {arcsinh}(c x)) \, dx=-\frac {8 b d^2 \sqrt {1+c^2 x^2}}{15 c}-\frac {4 b d^2 \left (1+c^2 x^2\right )^{3/2}}{45 c}-\frac {b d^2 \left (1+c^2 x^2\right )^{5/2}}{25 c}+d^2 x (a+b \text {arcsinh}(c x))+\frac {2}{3} c^2 d^2 x^3 (a+b \text {arcsinh}(c x))+\frac {1}{5} c^4 d^2 x^5 (a+b \text {arcsinh}(c x)) \] Output:
-8/15*b*d^2*(c^2*x^2+1)^(1/2)/c-4/45*b*d^2*(c^2*x^2+1)^(3/2)/c-1/25*b*d^2* (c^2*x^2+1)^(5/2)/c+d^2*x*(a+b*arcsinh(c*x))+2/3*c^2*d^2*x^3*(a+b*arcsinh( c*x))+1/5*c^4*d^2*x^5*(a+b*arcsinh(c*x))
Time = 0.05 (sec) , antiderivative size = 95, normalized size of antiderivative = 0.74 \[ \int \left (d+c^2 d x^2\right )^2 (a+b \text {arcsinh}(c x)) \, dx=\frac {d^2 \left (15 a c x \left (15+10 c^2 x^2+3 c^4 x^4\right )-b \sqrt {1+c^2 x^2} \left (149+38 c^2 x^2+9 c^4 x^4\right )+15 b c x \left (15+10 c^2 x^2+3 c^4 x^4\right ) \text {arcsinh}(c x)\right )}{225 c} \] Input:
Integrate[(d + c^2*d*x^2)^2*(a + b*ArcSinh[c*x]),x]
Output:
(d^2*(15*a*c*x*(15 + 10*c^2*x^2 + 3*c^4*x^4) - b*Sqrt[1 + c^2*x^2]*(149 + 38*c^2*x^2 + 9*c^4*x^4) + 15*b*c*x*(15 + 10*c^2*x^2 + 3*c^4*x^4)*ArcSinh[c *x]))/(225*c)
Time = 0.38 (sec) , antiderivative size = 124, normalized size of antiderivative = 0.97, number of steps used = 6, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.238, Rules used = {6199, 27, 1576, 1140, 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.
\(\displaystyle \int \left (c^2 d x^2+d\right )^2 (a+b \text {arcsinh}(c x)) \, dx\) |
\(\Big \downarrow \) 6199 |
\(\displaystyle -b c \int \frac {d^2 x \left (3 c^4 x^4+10 c^2 x^2+15\right )}{15 \sqrt {c^2 x^2+1}}dx+\frac {1}{5} c^4 d^2 x^5 (a+b \text {arcsinh}(c x))+\frac {2}{3} c^2 d^2 x^3 (a+b \text {arcsinh}(c x))+d^2 x (a+b \text {arcsinh}(c x))\) |
\(\Big \downarrow \) 27 |
\(\displaystyle -\frac {1}{15} b c d^2 \int \frac {x \left (3 c^4 x^4+10 c^2 x^2+15\right )}{\sqrt {c^2 x^2+1}}dx+\frac {1}{5} c^4 d^2 x^5 (a+b \text {arcsinh}(c x))+\frac {2}{3} c^2 d^2 x^3 (a+b \text {arcsinh}(c x))+d^2 x (a+b \text {arcsinh}(c x))\) |
\(\Big \downarrow \) 1576 |
\(\displaystyle -\frac {1}{30} b c d^2 \int \frac {3 c^4 x^4+10 c^2 x^2+15}{\sqrt {c^2 x^2+1}}dx^2+\frac {1}{5} c^4 d^2 x^5 (a+b \text {arcsinh}(c x))+\frac {2}{3} c^2 d^2 x^3 (a+b \text {arcsinh}(c x))+d^2 x (a+b \text {arcsinh}(c x))\) |
\(\Big \downarrow \) 1140 |
\(\displaystyle -\frac {1}{30} b c d^2 \int \left (3 \left (c^2 x^2+1\right )^{3/2}+4 \sqrt {c^2 x^2+1}+\frac {8}{\sqrt {c^2 x^2+1}}\right )dx^2+\frac {1}{5} c^4 d^2 x^5 (a+b \text {arcsinh}(c x))+\frac {2}{3} c^2 d^2 x^3 (a+b \text {arcsinh}(c x))+d^2 x (a+b \text {arcsinh}(c x))\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle \frac {1}{5} c^4 d^2 x^5 (a+b \text {arcsinh}(c x))+\frac {2}{3} c^2 d^2 x^3 (a+b \text {arcsinh}(c x))+d^2 x (a+b \text {arcsinh}(c x))-\frac {1}{30} b c d^2 \left (\frac {6 \left (c^2 x^2+1\right )^{5/2}}{5 c^2}+\frac {8 \left (c^2 x^2+1\right )^{3/2}}{3 c^2}+\frac {16 \sqrt {c^2 x^2+1}}{c^2}\right )\) |
Input:
Int[(d + c^2*d*x^2)^2*(a + b*ArcSinh[c*x]),x]
Output:
-1/30*(b*c*d^2*((16*Sqrt[1 + c^2*x^2])/c^2 + (8*(1 + c^2*x^2)^(3/2))/(3*c^ 2) + (6*(1 + c^2*x^2)^(5/2))/(5*c^2))) + d^2*x*(a + b*ArcSinh[c*x]) + (2*c ^2*d^2*x^3*(a + b*ArcSinh[c*x]))/3 + (c^4*d^2*x^5*(a + b*ArcSinh[c*x]))/5
Int[(a_)*(Fx_), x_Symbol] :> Simp[a Int[Fx, x], x] /; FreeQ[a, x] && !Ma tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
Int[((d_.) + (e_.)*(x_))^(m_.)*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_.), x _Symbol] :> Int[ExpandIntegrand[(d + e*x)^m*(a + b*x + c*x^2)^p, x], x] /; FreeQ[{a, b, c, d, e, m}, x] && IGtQ[p, 0]
Int[(x_)*((d_) + (e_.)*(x_)^2)^(q_.)*((a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4)^( p_.), x_Symbol] :> Simp[1/2 Subst[Int[(d + e*x)^q*(a + b*x + c*x^2)^p, x] , x, x^2], x] /; FreeQ[{a, b, c, d, e, p, q}, x]
Int[((a_.) + ArcSinh[(c_.)*(x_)]*(b_.))*((d_) + (e_.)*(x_)^2)^(p_.), x_Symb ol] :> With[{u = IntHide[(d + e*x^2)^p, x]}, Simp[(a + b*ArcSinh[c*x]) u, x] - Simp[b*c Int[SimplifyIntegrand[u/Sqrt[1 + c^2*x^2], x], x], x]] /; FreeQ[{a, b, c, d, e}, x] && EqQ[e, c^2*d] && IGtQ[p, 0]
Time = 0.26 (sec) , antiderivative size = 116, normalized size of antiderivative = 0.91
method | result | size |
parts | \(a \,d^{2} \left (\frac {1}{5} c^{4} x^{5}+\frac {2}{3} x^{3} c^{2}+x \right )+\frac {d^{2} b \left (\frac {\operatorname {arcsinh}\left (x c \right ) x^{5} c^{5}}{5}+\frac {2 \,\operatorname {arcsinh}\left (x c \right ) x^{3} c^{3}}{3}+x c \,\operatorname {arcsinh}\left (x c \right )-\frac {149 \sqrt {c^{2} x^{2}+1}}{225}-\frac {38 x^{2} c^{2} \sqrt {c^{2} x^{2}+1}}{225}-\frac {x^{4} c^{4} \sqrt {c^{2} x^{2}+1}}{25}\right )}{c}\) | \(116\) |
derivativedivides | \(\frac {d^{2} a \left (\frac {1}{5} x^{5} c^{5}+\frac {2}{3} x^{3} c^{3}+x c \right )+d^{2} b \left (\frac {\operatorname {arcsinh}\left (x c \right ) x^{5} c^{5}}{5}+\frac {2 \,\operatorname {arcsinh}\left (x c \right ) x^{3} c^{3}}{3}+x c \,\operatorname {arcsinh}\left (x c \right )-\frac {149 \sqrt {c^{2} x^{2}+1}}{225}-\frac {38 x^{2} c^{2} \sqrt {c^{2} x^{2}+1}}{225}-\frac {x^{4} c^{4} \sqrt {c^{2} x^{2}+1}}{25}\right )}{c}\) | \(119\) |
default | \(\frac {d^{2} a \left (\frac {1}{5} x^{5} c^{5}+\frac {2}{3} x^{3} c^{3}+x c \right )+d^{2} b \left (\frac {\operatorname {arcsinh}\left (x c \right ) x^{5} c^{5}}{5}+\frac {2 \,\operatorname {arcsinh}\left (x c \right ) x^{3} c^{3}}{3}+x c \,\operatorname {arcsinh}\left (x c \right )-\frac {149 \sqrt {c^{2} x^{2}+1}}{225}-\frac {38 x^{2} c^{2} \sqrt {c^{2} x^{2}+1}}{225}-\frac {x^{4} c^{4} \sqrt {c^{2} x^{2}+1}}{25}\right )}{c}\) | \(119\) |
orering | \(\frac {x \left (81 c^{4} x^{4}+302 c^{2} x^{2}+821\right ) \left (c^{2} d \,x^{2}+d \right )^{2} \left (a +b \,\operatorname {arcsinh}\left (x c \right )\right )}{225 \left (c^{2} x^{2}+1\right )^{2}}-\frac {\left (9 c^{4} x^{4}+38 c^{2} x^{2}+149\right ) \left (4 \left (c^{2} d \,x^{2}+d \right ) \left (a +b \,\operatorname {arcsinh}\left (x c \right )\right ) c^{2} d x +\frac {\left (c^{2} d \,x^{2}+d \right )^{2} b c}{\sqrt {c^{2} x^{2}+1}}\right )}{225 c^{2} \left (c^{2} x^{2}+1\right )}\) | \(140\) |
Input:
int((c^2*d*x^2+d)^2*(a+b*arcsinh(x*c)),x,method=_RETURNVERBOSE)
Output:
a*d^2*(1/5*c^4*x^5+2/3*x^3*c^2+x)+d^2*b/c*(1/5*arcsinh(x*c)*x^5*c^5+2/3*ar csinh(x*c)*x^3*c^3+x*c*arcsinh(x*c)-149/225*(c^2*x^2+1)^(1/2)-38/225*x^2*c ^2*(c^2*x^2+1)^(1/2)-1/25*x^4*c^4*(c^2*x^2+1)^(1/2))
Time = 0.09 (sec) , antiderivative size = 133, normalized size of antiderivative = 1.04 \[ \int \left (d+c^2 d x^2\right )^2 (a+b \text {arcsinh}(c x)) \, dx=\frac {45 \, a c^{5} d^{2} x^{5} + 150 \, a c^{3} d^{2} x^{3} + 225 \, a c d^{2} x + 15 \, {\left (3 \, b c^{5} d^{2} x^{5} + 10 \, b c^{3} d^{2} x^{3} + 15 \, b c d^{2} x\right )} \log \left (c x + \sqrt {c^{2} x^{2} + 1}\right ) - {\left (9 \, b c^{4} d^{2} x^{4} + 38 \, b c^{2} d^{2} x^{2} + 149 \, b d^{2}\right )} \sqrt {c^{2} x^{2} + 1}}{225 \, c} \] Input:
integrate((c^2*d*x^2+d)^2*(a+b*arcsinh(c*x)),x, algorithm="fricas")
Output:
1/225*(45*a*c^5*d^2*x^5 + 150*a*c^3*d^2*x^3 + 225*a*c*d^2*x + 15*(3*b*c^5* d^2*x^5 + 10*b*c^3*d^2*x^3 + 15*b*c*d^2*x)*log(c*x + sqrt(c^2*x^2 + 1)) - (9*b*c^4*d^2*x^4 + 38*b*c^2*d^2*x^2 + 149*b*d^2)*sqrt(c^2*x^2 + 1))/c
Time = 0.32 (sec) , antiderivative size = 165, normalized size of antiderivative = 1.29 \[ \int \left (d+c^2 d x^2\right )^2 (a+b \text {arcsinh}(c x)) \, dx=\begin {cases} \frac {a c^{4} d^{2} x^{5}}{5} + \frac {2 a c^{2} d^{2} x^{3}}{3} + a d^{2} x + \frac {b c^{4} d^{2} x^{5} \operatorname {asinh}{\left (c x \right )}}{5} - \frac {b c^{3} d^{2} x^{4} \sqrt {c^{2} x^{2} + 1}}{25} + \frac {2 b c^{2} d^{2} x^{3} \operatorname {asinh}{\left (c x \right )}}{3} - \frac {38 b c d^{2} x^{2} \sqrt {c^{2} x^{2} + 1}}{225} + b d^{2} x \operatorname {asinh}{\left (c x \right )} - \frac {149 b d^{2} \sqrt {c^{2} x^{2} + 1}}{225 c} & \text {for}\: c \neq 0 \\a d^{2} x & \text {otherwise} \end {cases} \] Input:
integrate((c**2*d*x**2+d)**2*(a+b*asinh(c*x)),x)
Output:
Piecewise((a*c**4*d**2*x**5/5 + 2*a*c**2*d**2*x**3/3 + a*d**2*x + b*c**4*d **2*x**5*asinh(c*x)/5 - b*c**3*d**2*x**4*sqrt(c**2*x**2 + 1)/25 + 2*b*c**2 *d**2*x**3*asinh(c*x)/3 - 38*b*c*d**2*x**2*sqrt(c**2*x**2 + 1)/225 + b*d** 2*x*asinh(c*x) - 149*b*d**2*sqrt(c**2*x**2 + 1)/(225*c), Ne(c, 0)), (a*d** 2*x, True))
Time = 0.03 (sec) , antiderivative size = 194, normalized size of antiderivative = 1.52 \[ \int \left (d+c^2 d x^2\right )^2 (a+b \text {arcsinh}(c x)) \, dx=\frac {1}{5} \, a c^{4} d^{2} x^{5} + \frac {1}{75} \, {\left (15 \, x^{5} \operatorname {arsinh}\left (c x\right ) - {\left (\frac {3 \, \sqrt {c^{2} x^{2} + 1} x^{4}}{c^{2}} - \frac {4 \, \sqrt {c^{2} x^{2} + 1} x^{2}}{c^{4}} + \frac {8 \, \sqrt {c^{2} x^{2} + 1}}{c^{6}}\right )} c\right )} b c^{4} d^{2} + \frac {2}{3} \, a c^{2} d^{2} x^{3} + \frac {2}{9} \, {\left (3 \, x^{3} \operatorname {arsinh}\left (c x\right ) - c {\left (\frac {\sqrt {c^{2} x^{2} + 1} x^{2}}{c^{2}} - \frac {2 \, \sqrt {c^{2} x^{2} + 1}}{c^{4}}\right )}\right )} b c^{2} d^{2} + a d^{2} x + \frac {{\left (c x \operatorname {arsinh}\left (c x\right ) - \sqrt {c^{2} x^{2} + 1}\right )} b d^{2}}{c} \] Input:
integrate((c^2*d*x^2+d)^2*(a+b*arcsinh(c*x)),x, algorithm="maxima")
Output:
1/5*a*c^4*d^2*x^5 + 1/75*(15*x^5*arcsinh(c*x) - (3*sqrt(c^2*x^2 + 1)*x^4/c ^2 - 4*sqrt(c^2*x^2 + 1)*x^2/c^4 + 8*sqrt(c^2*x^2 + 1)/c^6)*c)*b*c^4*d^2 + 2/3*a*c^2*d^2*x^3 + 2/9*(3*x^3*arcsinh(c*x) - c*(sqrt(c^2*x^2 + 1)*x^2/c^ 2 - 2*sqrt(c^2*x^2 + 1)/c^4))*b*c^2*d^2 + a*d^2*x + (c*x*arcsinh(c*x) - sq rt(c^2*x^2 + 1))*b*d^2/c
Exception generated. \[ \int \left (d+c^2 d x^2\right )^2 (a+b \text {arcsinh}(c x)) \, dx=\text {Exception raised: TypeError} \] Input:
integrate((c^2*d*x^2+d)^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. \[ \int \left (d+c^2 d x^2\right )^2 (a+b \text {arcsinh}(c x)) \, dx=\int \left (a+b\,\mathrm {asinh}\left (c\,x\right )\right )\,{\left (d\,c^2\,x^2+d\right )}^2 \,d x \] Input:
int((a + b*asinh(c*x))*(d + c^2*d*x^2)^2,x)
Output:
int((a + b*asinh(c*x))*(d + c^2*d*x^2)^2, x)
Time = 0.19 (sec) , antiderivative size = 118, normalized size of antiderivative = 0.92 \[ \int \left (d+c^2 d x^2\right )^2 (a+b \text {arcsinh}(c x)) \, dx=\frac {d^{2} \left (45 \mathit {asinh} \left (c x \right ) b \,c^{5} x^{5}+150 \mathit {asinh} \left (c x \right ) b \,c^{3} x^{3}+225 \mathit {asinh} \left (c x \right ) b c x -9 \sqrt {c^{2} x^{2}+1}\, b \,c^{4} x^{4}-38 \sqrt {c^{2} x^{2}+1}\, b \,c^{2} x^{2}-149 \sqrt {c^{2} x^{2}+1}\, b +45 a \,c^{5} x^{5}+150 a \,c^{3} x^{3}+225 a c x \right )}{225 c} \] Input:
int((c^2*d*x^2+d)^2*(a+b*asinh(c*x)),x)
Output:
(d**2*(45*asinh(c*x)*b*c**5*x**5 + 150*asinh(c*x)*b*c**3*x**3 + 225*asinh( c*x)*b*c*x - 9*sqrt(c**2*x**2 + 1)*b*c**4*x**4 - 38*sqrt(c**2*x**2 + 1)*b* c**2*x**2 - 149*sqrt(c**2*x**2 + 1)*b + 45*a*c**5*x**5 + 150*a*c**3*x**3 + 225*a*c*x))/(225*c)