Integrand size = 22, antiderivative size = 102 \[ \int x^2 \left (d+c^2 d x^2\right ) (a+b \text {arcsinh}(c x)) \, dx=\frac {2 b d \sqrt {1+c^2 x^2}}{15 c^3}+\frac {b d \left (1+c^2 x^2\right )^{3/2}}{45 c^3}-\frac {b d \left (1+c^2 x^2\right )^{5/2}}{25 c^3}+\frac {1}{3} d x^3 (a+b \text {arcsinh}(c x))+\frac {1}{5} c^2 d x^5 (a+b \text {arcsinh}(c x)) \]
1/45*b*d*(c^2*x^2+1)^(3/2)/c^3-1/25*b*d*(c^2*x^2+1)^(5/2)/c^3+1/3*d*x^3*(a +b*arcsinh(c*x))+1/5*c^2*d*x^5*(a+b*arcsinh(c*x))+2/15*b*d*(c^2*x^2+1)^(1/ 2)/c^3
Time = 0.06 (sec) , antiderivative size = 78, normalized size of antiderivative = 0.76 \[ \int x^2 \left (d+c^2 d x^2\right ) (a+b \text {arcsinh}(c x)) \, dx=\frac {1}{225} d \left (15 a x^3 \left (5+3 c^2 x^2\right )+\frac {b \sqrt {1+c^2 x^2} \left (26-13 c^2 x^2-9 c^4 x^4\right )}{c^3}+15 b x^3 \left (5+3 c^2 x^2\right ) \text {arcsinh}(c x)\right ) \]
(d*(15*a*x^3*(5 + 3*c^2*x^2) + (b*Sqrt[1 + c^2*x^2]*(26 - 13*c^2*x^2 - 9*c ^4*x^4))/c^3 + 15*b*x^3*(5 + 3*c^2*x^2)*ArcSinh[c*x]))/225
Time = 0.30 (sec) , antiderivative size = 102, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.227, Rules used = {6218, 27, 354, 86, 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.
\(\displaystyle \int x^2 \left (c^2 d x^2+d\right ) (a+b \text {arcsinh}(c x)) \, dx\) |
\(\Big \downarrow \) 6218 |
\(\displaystyle -b c \int \frac {d x^3 \left (3 c^2 x^2+5\right )}{15 \sqrt {c^2 x^2+1}}dx+\frac {1}{5} c^2 d x^5 (a+b \text {arcsinh}(c x))+\frac {1}{3} d x^3 (a+b \text {arcsinh}(c x))\) |
\(\Big \downarrow \) 27 |
\(\displaystyle -\frac {1}{15} b c d \int \frac {x^3 \left (3 c^2 x^2+5\right )}{\sqrt {c^2 x^2+1}}dx+\frac {1}{5} c^2 d x^5 (a+b \text {arcsinh}(c x))+\frac {1}{3} d x^3 (a+b \text {arcsinh}(c x))\) |
\(\Big \downarrow \) 354 |
\(\displaystyle -\frac {1}{30} b c d \int \frac {x^2 \left (3 c^2 x^2+5\right )}{\sqrt {c^2 x^2+1}}dx^2+\frac {1}{5} c^2 d x^5 (a+b \text {arcsinh}(c x))+\frac {1}{3} d x^3 (a+b \text {arcsinh}(c x))\) |
\(\Big \downarrow \) 86 |
\(\displaystyle -\frac {1}{30} b c d \int \left (\frac {3 \left (c^2 x^2+1\right )^{3/2}}{c^2}-\frac {\sqrt {c^2 x^2+1}}{c^2}-\frac {2}{c^2 \sqrt {c^2 x^2+1}}\right )dx^2+\frac {1}{5} c^2 d x^5 (a+b \text {arcsinh}(c x))+\frac {1}{3} d x^3 (a+b \text {arcsinh}(c x))\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle \frac {1}{5} c^2 d x^5 (a+b \text {arcsinh}(c x))+\frac {1}{3} d x^3 (a+b \text {arcsinh}(c x))-\frac {1}{30} b c d \left (\frac {6 \left (c^2 x^2+1\right )^{5/2}}{5 c^4}-\frac {2 \left (c^2 x^2+1\right )^{3/2}}{3 c^4}-\frac {4 \sqrt {c^2 x^2+1}}{c^4}\right )\) |
-1/30*(b*c*d*((-4*Sqrt[1 + c^2*x^2])/c^4 - (2*(1 + c^2*x^2)^(3/2))/(3*c^4) + (6*(1 + c^2*x^2)^(5/2))/(5*c^4))) + (d*x^3*(a + b*ArcSinh[c*x]))/3 + (c ^2*d*x^5*(a + b*ArcSinh[c*x]))/5
3.1.3.3.1 Defintions of rubi rules used
Int[(a_)*(Fx_), x_Symbol] :> Simp[a Int[Fx, x], x] /; FreeQ[a, x] && !Ma tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
Int[((a_.) + (b_.)*(x_))*((c_) + (d_.)*(x_))^(n_.)*((e_.) + (f_.)*(x_))^(p_ .), x_] :> Int[ExpandIntegrand[(a + b*x)*(c + d*x)^n*(e + f*x)^p, x], x] /; FreeQ[{a, b, c, d, e, f, n}, x] && ((ILtQ[n, 0] && ILtQ[p, 0]) || EqQ[p, 1 ] || (IGtQ[p, 0] && ( !IntegerQ[n] || LeQ[9*p + 5*(n + 2), 0] || GeQ[n + p + 1, 0] || (GeQ[n + p + 2, 0] && RationalQ[a, b, c, d, e, f]))))
Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^2)^(p_.)*((c_) + (d_.)*(x_)^2)^(q_.), x_S ymbol] :> Simp[1/2 Subst[Int[x^((m - 1)/2)*(a + b*x)^p*(c + d*x)^q, x], x , x^2], x] /; FreeQ[{a, b, c, d, p, q}, x] && NeQ[b*c - a*d, 0] && IntegerQ [(m - 1)/2]
Int[((a_.) + ArcSinh[(c_.)*(x_)]*(b_.))*((f_.)*(x_))^(m_)*((d_) + (e_.)*(x_ )^2)^(p_.), x_Symbol] :> With[{u = IntHide[(f*x)^m*(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, f, m}, x] && EqQ[e, c^2*d] && IGtQ[p, 0]
Time = 0.04 (sec) , antiderivative size = 101, normalized size of antiderivative = 0.99
method | result | size |
parts | \(d a \left (\frac {1}{5} c^{2} x^{5}+\frac {1}{3} x^{3}\right )+\frac {d b \left (\frac {\operatorname {arcsinh}\left (c x \right ) c^{5} x^{5}}{5}+\frac {\operatorname {arcsinh}\left (c x \right ) c^{3} x^{3}}{3}-\frac {c^{4} x^{4} \sqrt {c^{2} x^{2}+1}}{25}-\frac {13 c^{2} x^{2} \sqrt {c^{2} x^{2}+1}}{225}+\frac {26 \sqrt {c^{2} x^{2}+1}}{225}\right )}{c^{3}}\) | \(101\) |
derivativedivides | \(\frac {d a \left (\frac {1}{5} c^{5} x^{5}+\frac {1}{3} c^{3} x^{3}\right )+d b \left (\frac {\operatorname {arcsinh}\left (c x \right ) c^{5} x^{5}}{5}+\frac {\operatorname {arcsinh}\left (c x \right ) c^{3} x^{3}}{3}-\frac {c^{4} x^{4} \sqrt {c^{2} x^{2}+1}}{25}-\frac {13 c^{2} x^{2} \sqrt {c^{2} x^{2}+1}}{225}+\frac {26 \sqrt {c^{2} x^{2}+1}}{225}\right )}{c^{3}}\) | \(105\) |
default | \(\frac {d a \left (\frac {1}{5} c^{5} x^{5}+\frac {1}{3} c^{3} x^{3}\right )+d b \left (\frac {\operatorname {arcsinh}\left (c x \right ) c^{5} x^{5}}{5}+\frac {\operatorname {arcsinh}\left (c x \right ) c^{3} x^{3}}{3}-\frac {c^{4} x^{4} \sqrt {c^{2} x^{2}+1}}{25}-\frac {13 c^{2} x^{2} \sqrt {c^{2} x^{2}+1}}{225}+\frac {26 \sqrt {c^{2} x^{2}+1}}{225}\right )}{c^{3}}\) | \(105\) |
d*a*(1/5*c^2*x^5+1/3*x^3)+d*b/c^3*(1/5*arcsinh(c*x)*c^5*x^5+1/3*arcsinh(c* x)*c^3*x^3-1/25*c^4*x^4*(c^2*x^2+1)^(1/2)-13/225*c^2*x^2*(c^2*x^2+1)^(1/2) +26/225*(c^2*x^2+1)^(1/2))
Time = 0.29 (sec) , antiderivative size = 103, normalized size of antiderivative = 1.01 \[ \int x^2 \left (d+c^2 d x^2\right ) (a+b \text {arcsinh}(c x)) \, dx=\frac {45 \, a c^{5} d x^{5} + 75 \, a c^{3} d x^{3} + 15 \, {\left (3 \, b c^{5} d x^{5} + 5 \, b c^{3} d x^{3}\right )} \log \left (c x + \sqrt {c^{2} x^{2} + 1}\right ) - {\left (9 \, b c^{4} d x^{4} + 13 \, b c^{2} d x^{2} - 26 \, b d\right )} \sqrt {c^{2} x^{2} + 1}}{225 \, c^{3}} \]
1/225*(45*a*c^5*d*x^5 + 75*a*c^3*d*x^3 + 15*(3*b*c^5*d*x^5 + 5*b*c^3*d*x^3 )*log(c*x + sqrt(c^2*x^2 + 1)) - (9*b*c^4*d*x^4 + 13*b*c^2*d*x^2 - 26*b*d) *sqrt(c^2*x^2 + 1))/c^3
Time = 0.37 (sec) , antiderivative size = 126, normalized size of antiderivative = 1.24 \[ \int x^2 \left (d+c^2 d x^2\right ) (a+b \text {arcsinh}(c x)) \, dx=\begin {cases} \frac {a c^{2} d x^{5}}{5} + \frac {a d x^{3}}{3} + \frac {b c^{2} d x^{5} \operatorname {asinh}{\left (c x \right )}}{5} - \frac {b c d x^{4} \sqrt {c^{2} x^{2} + 1}}{25} + \frac {b d x^{3} \operatorname {asinh}{\left (c x \right )}}{3} - \frac {13 b d x^{2} \sqrt {c^{2} x^{2} + 1}}{225 c} + \frac {26 b d \sqrt {c^{2} x^{2} + 1}}{225 c^{3}} & \text {for}\: c \neq 0 \\\frac {a d x^{3}}{3} & \text {otherwise} \end {cases} \]
Piecewise((a*c**2*d*x**5/5 + a*d*x**3/3 + b*c**2*d*x**5*asinh(c*x)/5 - b*c *d*x**4*sqrt(c**2*x**2 + 1)/25 + b*d*x**3*asinh(c*x)/3 - 13*b*d*x**2*sqrt( c**2*x**2 + 1)/(225*c) + 26*b*d*sqrt(c**2*x**2 + 1)/(225*c**3), Ne(c, 0)), (a*d*x**3/3, True))
Time = 0.20 (sec) , antiderivative size = 145, normalized size of antiderivative = 1.42 \[ \int x^2 \left (d+c^2 d x^2\right ) (a+b \text {arcsinh}(c x)) \, dx=\frac {1}{5} \, a c^{2} d x^{5} + \frac {1}{75} \, {\left (15 \, x^{5} \operatorname {arsinh}\left (c x\right ) - {\left (\frac {3 \, \sqrt {c^{2} x^{2} + 1} x^{4}}{c^{2}} - \frac {4 \, \sqrt {c^{2} x^{2} + 1} x^{2}}{c^{4}} + \frac {8 \, \sqrt {c^{2} x^{2} + 1}}{c^{6}}\right )} c\right )} b c^{2} d + \frac {1}{3} \, a d x^{3} + \frac {1}{9} \, {\left (3 \, x^{3} \operatorname {arsinh}\left (c x\right ) - c {\left (\frac {\sqrt {c^{2} x^{2} + 1} x^{2}}{c^{2}} - \frac {2 \, \sqrt {c^{2} x^{2} + 1}}{c^{4}}\right )}\right )} b d \]
1/5*a*c^2*d*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^2*d + 1/3 *a*d*x^3 + 1/9*(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*d
Exception generated. \[ \int x^2 \left (d+c^2 d x^2\right ) (a+b \text {arcsinh}(c x)) \, dx=\text {Exception raised: RuntimeError} \]
Exception raised: RuntimeError >> an error occurred running a Giac command :INPUT:sage2OUTPUT:sym2poly/r2sym(const gen & e,const index_m & i,const ve cteur & l) Error: Bad Argument Value
Timed out. \[ \int x^2 \left (d+c^2 d x^2\right ) (a+b \text {arcsinh}(c x)) \, dx=\int x^2\,\left (a+b\,\mathrm {asinh}\left (c\,x\right )\right )\,\left (d\,c^2\,x^2+d\right ) \,d x \]