Integrand size = 28, antiderivative size = 227 \[ \int (f+g x) \sqrt {d+c^2 d x^2} (a+b \text {arcsinh}(c x)) \, dx=-\frac {b g x \sqrt {d+c^2 d x^2}}{3 c \sqrt {1+c^2 x^2}}-\frac {b c f x^2 \sqrt {d+c^2 d x^2}}{4 \sqrt {1+c^2 x^2}}-\frac {b c g x^3 \sqrt {d+c^2 d x^2}}{9 \sqrt {1+c^2 x^2}}+\frac {1}{2} f x \sqrt {d+c^2 d x^2} (a+b \text {arcsinh}(c x))+\frac {g \left (1+c^2 x^2\right ) \sqrt {d+c^2 d x^2} (a+b \text {arcsinh}(c x))}{3 c^2}+\frac {f \sqrt {d+c^2 d x^2} (a+b \text {arcsinh}(c x))^2}{4 b c \sqrt {1+c^2 x^2}} \]
1/2*f*x*(a+b*arcsinh(c*x))*(c^2*d*x^2+d)^(1/2)+1/3*g*(c^2*x^2+1)*(a+b*arcs inh(c*x))*(c^2*d*x^2+d)^(1/2)/c^2-1/3*b*g*x*(c^2*d*x^2+d)^(1/2)/c/(c^2*x^2 +1)^(1/2)-1/4*b*c*f*x^2*(c^2*d*x^2+d)^(1/2)/(c^2*x^2+1)^(1/2)-1/9*b*c*g*x^ 3*(c^2*d*x^2+d)^(1/2)/(c^2*x^2+1)^(1/2)+1/4*f*(a+b*arcsinh(c*x))^2*(c^2*d* x^2+d)^(1/2)/b/c/(c^2*x^2+1)^(1/2)
Time = 0.96 (sec) , antiderivative size = 208, normalized size of antiderivative = 0.92 \[ \int (f+g x) \sqrt {d+c^2 d x^2} (a+b \text {arcsinh}(c x)) \, dx=\frac {1}{6} a \sqrt {d+c^2 d x^2} \left (\frac {2 g}{c^2}+x (3 f+2 g x)\right )-\frac {b g \sqrt {d+c^2 d x^2} \left (3 c x+c^3 x^3-3 \left (1+c^2 x^2\right )^{3/2} \text {arcsinh}(c x)\right )}{9 c^2 \sqrt {1+c^2 x^2}}+\frac {a \sqrt {d} f \log \left (c d x+\sqrt {d} \sqrt {d+c^2 d x^2}\right )}{2 c}+\frac {b f \sqrt {d+c^2 d x^2} (-\cosh (2 \text {arcsinh}(c x))+2 \text {arcsinh}(c x) (\text {arcsinh}(c x)+\sinh (2 \text {arcsinh}(c x))))}{8 c \sqrt {1+c^2 x^2}} \]
(a*Sqrt[d + c^2*d*x^2]*((2*g)/c^2 + x*(3*f + 2*g*x)))/6 - (b*g*Sqrt[d + c^ 2*d*x^2]*(3*c*x + c^3*x^3 - 3*(1 + c^2*x^2)^(3/2)*ArcSinh[c*x]))/(9*c^2*Sq rt[1 + c^2*x^2]) + (a*Sqrt[d]*f*Log[c*d*x + Sqrt[d]*Sqrt[d + c^2*d*x^2]])/ (2*c) + (b*f*Sqrt[d + c^2*d*x^2]*(-Cosh[2*ArcSinh[c*x]] + 2*ArcSinh[c*x]*( ArcSinh[c*x] + Sinh[2*ArcSinh[c*x]])))/(8*c*Sqrt[1 + c^2*x^2])
Time = 0.53 (sec) , antiderivative size = 136, normalized size of antiderivative = 0.60, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.107, Rules used = {6260, 6253, 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.
| \(\displaystyle \int \sqrt {c^2 d x^2+d} (f+g x) (a+b \text {arcsinh}(c x)) \, dx\) |
| \(\Big \downarrow \) 6260 |
| \(\displaystyle \frac {\sqrt {c^2 d x^2+d} \int (f+g x) \sqrt {c^2 x^2+1} (a+b \text {arcsinh}(c x))dx}{\sqrt {c^2 x^2+1}}\) |
| \(\Big \downarrow \) 6253 |
| \(\displaystyle \frac {\sqrt {c^2 d x^2+d} \int \left (f \sqrt {c^2 x^2+1} (a+b \text {arcsinh}(c x))+g x \sqrt {c^2 x^2+1} (a+b \text {arcsinh}(c x))\right )dx}{\sqrt {c^2 x^2+1}}\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {\sqrt {c^2 d x^2+d} \left (\frac {1}{2} f x \sqrt {c^2 x^2+1} (a+b \text {arcsinh}(c x))+\frac {g \left (c^2 x^2+1\right )^{3/2} (a+b \text {arcsinh}(c x))}{3 c^2}+\frac {f (a+b \text {arcsinh}(c x))^2}{4 b c}-\frac {1}{4} b c f x^2-\frac {1}{9} b c g x^3-\frac {b g x}{3 c}\right )}{\sqrt {c^2 x^2+1}}\) |
(Sqrt[d + c^2*d*x^2]*(-1/3*(b*g*x)/c - (b*c*f*x^2)/4 - (b*c*g*x^3)/9 + (f* x*Sqrt[1 + c^2*x^2]*(a + b*ArcSinh[c*x]))/2 + (g*(1 + c^2*x^2)^(3/2)*(a + b*ArcSinh[c*x]))/(3*c^2) + (f*(a + b*ArcSinh[c*x])^2)/(4*b*c)))/Sqrt[1 + c ^2*x^2]
3.1.36.3.1 Defintions of rubi rules used
Int[((a_.) + ArcSinh[(c_.)*(x_)]*(b_.))^(n_.)*((f_) + (g_.)*(x_))^(m_.)*((d _) + (e_.)*(x_)^2)^(p_), x_Symbol] :> Int[ExpandIntegrand[(d + e*x^2)^p*(a + b*ArcSinh[c*x])^n, (f + g*x)^m, x], x] /; FreeQ[{a, b, c, d, e, f, g}, x] && EqQ[e, c^2*d] && IGtQ[m, 0] && IntegerQ[p + 1/2] && GtQ[d, 0] && IGtQ[n , 0] && ((EqQ[n, 1] && GtQ[p, -1]) || GtQ[p, 0] || EqQ[m, 1] || (EqQ[m, 2] && LtQ[p, -2]))
Int[((a_.) + ArcSinh[(c_.)*(x_)]*(b_.))^(n_.)*((f_) + (g_.)*(x_))^(m_.)*((d _) + (e_.)*(x_)^2)^(p_), x_Symbol] :> Simp[Simp[(d + e*x^2)^p/(1 + c^2*x^2) ^p] Int[(f + g*x)^m*(1 + c^2*x^2)^p*(a + b*ArcSinh[c*x])^n, x], x] /; Fre eQ[{a, b, c, d, e, f, g, n}, x] && EqQ[e, c^2*d] && IntegerQ[m] && IntegerQ [p - 1/2] && !GtQ[d, 0]
Leaf count of result is larger than twice the leaf count of optimal. \(581\) vs. \(2(195)=390\).
Time = 0.78 (sec) , antiderivative size = 582, normalized size of antiderivative = 2.56
| method | result | size |
| default | \(\frac {a f x \sqrt {c^{2} d \,x^{2}+d}}{2}+\frac {a f d \ln \left (\frac {c^{2} d x}{\sqrt {c^{2} d}}+\sqrt {c^{2} d \,x^{2}+d}\right )}{2 \sqrt {c^{2} d}}+\frac {a g \left (c^{2} d \,x^{2}+d \right )^{\frac {3}{2}}}{3 c^{2} d}+b \left (\frac {\sqrt {d \left (c^{2} x^{2}+1\right )}\, f \operatorname {arcsinh}\left (c x \right )^{2}}{4 \sqrt {c^{2} x^{2}+1}\, c}+\frac {\sqrt {d \left (c^{2} x^{2}+1\right )}\, \left (4 c^{4} x^{4}+4 c^{3} x^{3} \sqrt {c^{2} x^{2}+1}+5 c^{2} x^{2}+3 c x \sqrt {c^{2} x^{2}+1}+1\right ) g \left (-1+3 \,\operatorname {arcsinh}\left (c x \right )\right )}{72 c^{2} \left (c^{2} x^{2}+1\right )}+\frac {\sqrt {d \left (c^{2} x^{2}+1\right )}\, \left (2 c^{3} x^{3}+2 c^{2} x^{2} \sqrt {c^{2} x^{2}+1}+2 c x +\sqrt {c^{2} x^{2}+1}\right ) f \left (-1+2 \,\operatorname {arcsinh}\left (c x \right )\right )}{16 c \left (c^{2} x^{2}+1\right )}+\frac {\sqrt {d \left (c^{2} x^{2}+1\right )}\, \left (c^{2} x^{2}+c x \sqrt {c^{2} x^{2}+1}+1\right ) g \left (-1+\operatorname {arcsinh}\left (c x \right )\right )}{8 c^{2} \left (c^{2} x^{2}+1\right )}+\frac {\sqrt {d \left (c^{2} x^{2}+1\right )}\, \left (c^{2} x^{2}-c x \sqrt {c^{2} x^{2}+1}+1\right ) g \left (\operatorname {arcsinh}\left (c x \right )+1\right )}{8 c^{2} \left (c^{2} x^{2}+1\right )}+\frac {\sqrt {d \left (c^{2} x^{2}+1\right )}\, \left (2 c^{3} x^{3}-2 c^{2} x^{2} \sqrt {c^{2} x^{2}+1}+2 c x -\sqrt {c^{2} x^{2}+1}\right ) f \left (1+2 \,\operatorname {arcsinh}\left (c x \right )\right )}{16 c \left (c^{2} x^{2}+1\right )}+\frac {\sqrt {d \left (c^{2} x^{2}+1\right )}\, \left (4 c^{4} x^{4}-4 c^{3} x^{3} \sqrt {c^{2} x^{2}+1}+5 c^{2} x^{2}-3 c x \sqrt {c^{2} x^{2}+1}+1\right ) g \left (3 \,\operatorname {arcsinh}\left (c x \right )+1\right )}{72 c^{2} \left (c^{2} x^{2}+1\right )}\right )\) | \(582\) |
| parts | \(\frac {a f x \sqrt {c^{2} d \,x^{2}+d}}{2}+\frac {a f d \ln \left (\frac {c^{2} d x}{\sqrt {c^{2} d}}+\sqrt {c^{2} d \,x^{2}+d}\right )}{2 \sqrt {c^{2} d}}+\frac {a g \left (c^{2} d \,x^{2}+d \right )^{\frac {3}{2}}}{3 c^{2} d}+b \left (\frac {\sqrt {d \left (c^{2} x^{2}+1\right )}\, f \operatorname {arcsinh}\left (c x \right )^{2}}{4 \sqrt {c^{2} x^{2}+1}\, c}+\frac {\sqrt {d \left (c^{2} x^{2}+1\right )}\, \left (4 c^{4} x^{4}+4 c^{3} x^{3} \sqrt {c^{2} x^{2}+1}+5 c^{2} x^{2}+3 c x \sqrt {c^{2} x^{2}+1}+1\right ) g \left (-1+3 \,\operatorname {arcsinh}\left (c x \right )\right )}{72 c^{2} \left (c^{2} x^{2}+1\right )}+\frac {\sqrt {d \left (c^{2} x^{2}+1\right )}\, \left (2 c^{3} x^{3}+2 c^{2} x^{2} \sqrt {c^{2} x^{2}+1}+2 c x +\sqrt {c^{2} x^{2}+1}\right ) f \left (-1+2 \,\operatorname {arcsinh}\left (c x \right )\right )}{16 c \left (c^{2} x^{2}+1\right )}+\frac {\sqrt {d \left (c^{2} x^{2}+1\right )}\, \left (c^{2} x^{2}+c x \sqrt {c^{2} x^{2}+1}+1\right ) g \left (-1+\operatorname {arcsinh}\left (c x \right )\right )}{8 c^{2} \left (c^{2} x^{2}+1\right )}+\frac {\sqrt {d \left (c^{2} x^{2}+1\right )}\, \left (c^{2} x^{2}-c x \sqrt {c^{2} x^{2}+1}+1\right ) g \left (\operatorname {arcsinh}\left (c x \right )+1\right )}{8 c^{2} \left (c^{2} x^{2}+1\right )}+\frac {\sqrt {d \left (c^{2} x^{2}+1\right )}\, \left (2 c^{3} x^{3}-2 c^{2} x^{2} \sqrt {c^{2} x^{2}+1}+2 c x -\sqrt {c^{2} x^{2}+1}\right ) f \left (1+2 \,\operatorname {arcsinh}\left (c x \right )\right )}{16 c \left (c^{2} x^{2}+1\right )}+\frac {\sqrt {d \left (c^{2} x^{2}+1\right )}\, \left (4 c^{4} x^{4}-4 c^{3} x^{3} \sqrt {c^{2} x^{2}+1}+5 c^{2} x^{2}-3 c x \sqrt {c^{2} x^{2}+1}+1\right ) g \left (3 \,\operatorname {arcsinh}\left (c x \right )+1\right )}{72 c^{2} \left (c^{2} x^{2}+1\right )}\right )\) | \(582\) |
1/2*a*f*x*(c^2*d*x^2+d)^(1/2)+1/2*a*f*d*ln(c^2*d*x/(c^2*d)^(1/2)+(c^2*d*x^ 2+d)^(1/2))/(c^2*d)^(1/2)+1/3*a*g/c^2/d*(c^2*d*x^2+d)^(3/2)+b*(1/4*(d*(c^2 *x^2+1))^(1/2)/(c^2*x^2+1)^(1/2)/c*f*arcsinh(c*x)^2+1/72*(d*(c^2*x^2+1))^( 1/2)*(4*c^4*x^4+4*c^3*x^3*(c^2*x^2+1)^(1/2)+5*c^2*x^2+3*c*x*(c^2*x^2+1)^(1 /2)+1)*g*(-1+3*arcsinh(c*x))/c^2/(c^2*x^2+1)+1/16*(d*(c^2*x^2+1))^(1/2)*(2 *c^3*x^3+2*c^2*x^2*(c^2*x^2+1)^(1/2)+2*c*x+(c^2*x^2+1)^(1/2))*f*(-1+2*arcs inh(c*x))/c/(c^2*x^2+1)+1/8*(d*(c^2*x^2+1))^(1/2)*(c^2*x^2+c*x*(c^2*x^2+1) ^(1/2)+1)*g*(-1+arcsinh(c*x))/c^2/(c^2*x^2+1)+1/8*(d*(c^2*x^2+1))^(1/2)*(c ^2*x^2-c*x*(c^2*x^2+1)^(1/2)+1)*g*(arcsinh(c*x)+1)/c^2/(c^2*x^2+1)+1/16*(d *(c^2*x^2+1))^(1/2)*(2*c^3*x^3-2*c^2*x^2*(c^2*x^2+1)^(1/2)+2*c*x-(c^2*x^2+ 1)^(1/2))*f*(1+2*arcsinh(c*x))/c/(c^2*x^2+1)+1/72*(d*(c^2*x^2+1))^(1/2)*(4 *c^4*x^4-4*c^3*x^3*(c^2*x^2+1)^(1/2)+5*c^2*x^2-3*c*x*(c^2*x^2+1)^(1/2)+1)* g*(3*arcsinh(c*x)+1)/c^2/(c^2*x^2+1))
\[ \int (f+g x) \sqrt {d+c^2 d x^2} (a+b \text {arcsinh}(c x)) \, dx=\int { \sqrt {c^{2} d x^{2} + d} {\left (g x + f\right )} {\left (b \operatorname {arsinh}\left (c x\right ) + a\right )} \,d x } \]
\[ \int (f+g x) \sqrt {d+c^2 d x^2} (a+b \text {arcsinh}(c x)) \, dx=\int \sqrt {d \left (c^{2} x^{2} + 1\right )} \left (a + b \operatorname {asinh}{\left (c x \right )}\right ) \left (f + g x\right )\, dx \]
Exception generated. \[ \int (f+g x) \sqrt {d+c^2 d x^2} (a+b \text {arcsinh}(c x)) \, dx=\text {Exception raised: RuntimeError} \]
Exception generated. \[ \int (f+g x) \sqrt {d+c^2 d x^2} (a+b \text {arcsinh}(c x)) \, dx=\text {Exception raised: RuntimeError} \]
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. \[ \int (f+g x) \sqrt {d+c^2 d x^2} (a+b \text {arcsinh}(c x)) \, dx=\int \left (f+g\,x\right )\,\left (a+b\,\mathrm {asinh}\left (c\,x\right )\right )\,\sqrt {d\,c^2\,x^2+d} \,d x \]