Integrand size = 16, antiderivative size = 140 \[ \int (d+e x) (a+b \text {arcsinh}(c x))^2 \, dx=2 b^2 d x+\frac {1}{4} b^2 e x^2-\frac {2 b d \sqrt {1+c^2 x^2} (a+b \text {arcsinh}(c x))}{c}-\frac {b e x \sqrt {1+c^2 x^2} (a+b \text {arcsinh}(c x))}{2 c}-\frac {d^2 (a+b \text {arcsinh}(c x))^2}{2 e}+\frac {e (a+b \text {arcsinh}(c x))^2}{4 c^2}+\frac {(d+e x)^2 (a+b \text {arcsinh}(c x))^2}{2 e} \]
2*b^2*d*x+1/4*b^2*e*x^2-1/2*d^2*(a+b*arcsinh(c*x))^2/e+1/4*e*(a+b*arcsinh( c*x))^2/c^2+1/2*(e*x+d)^2*(a+b*arcsinh(c*x))^2/e-2*b*d*(a+b*arcsinh(c*x))* (c^2*x^2+1)^(1/2)/c-1/2*b*e*x*(a+b*arcsinh(c*x))*(c^2*x^2+1)^(1/2)/c
Time = 1.15 (sec) , antiderivative size = 142, normalized size of antiderivative = 1.01 \[ \int (d+e x) (a+b \text {arcsinh}(c x))^2 \, dx=\frac {c \left (2 a^2 c x (2 d+e x)+b^2 c x (8 d+e x)-2 a b (4 d+e x) \sqrt {1+c^2 x^2}\right )+2 b \left (-b c (4 d+e x) \sqrt {1+c^2 x^2}+a \left (e+4 c^2 d x+2 c^2 e x^2\right )\right ) \text {arcsinh}(c x)+b^2 \left (e+4 c^2 d x+2 c^2 e x^2\right ) \text {arcsinh}(c x)^2}{4 c^2} \]
(c*(2*a^2*c*x*(2*d + e*x) + b^2*c*x*(8*d + e*x) - 2*a*b*(4*d + e*x)*Sqrt[1 + c^2*x^2]) + 2*b*(-(b*c*(4*d + e*x)*Sqrt[1 + c^2*x^2]) + a*(e + 4*c^2*d* x + 2*c^2*e*x^2))*ArcSinh[c*x] + b^2*(e + 4*c^2*d*x + 2*c^2*e*x^2)*ArcSinh [c*x]^2)/(4*c^2)
Time = 0.58 (sec) , antiderivative size = 162, normalized size of antiderivative = 1.16, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.188, Rules used = {6243, 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.
| \(\displaystyle \int (d+e x) (a+b \text {arcsinh}(c x))^2 \, dx\) |
| \(\Big \downarrow \) 6243 |
| \(\displaystyle \frac {(d+e x)^2 (a+b \text {arcsinh}(c x))^2}{2 e}-\frac {b c \int \frac {(d+e x)^2 (a+b \text {arcsinh}(c x))}{\sqrt {c^2 x^2+1}}dx}{e}\) |
| \(\Big \downarrow \) 6253 |
| \(\displaystyle \frac {(d+e x)^2 (a+b \text {arcsinh}(c x))^2}{2 e}-\frac {b c \int \left (\frac {(a+b \text {arcsinh}(c x)) d^2}{\sqrt {c^2 x^2+1}}+\frac {2 e x (a+b \text {arcsinh}(c x)) d}{\sqrt {c^2 x^2+1}}+\frac {e^2 x^2 (a+b \text {arcsinh}(c x))}{\sqrt {c^2 x^2+1}}\right )dx}{e}\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {(d+e x)^2 (a+b \text {arcsinh}(c x))^2}{2 e}-\frac {b c \left (-\frac {e^2 (a+b \text {arcsinh}(c x))^2}{4 b c^3}+\frac {2 d e \sqrt {c^2 x^2+1} (a+b \text {arcsinh}(c x))}{c^2}+\frac {e^2 x \sqrt {c^2 x^2+1} (a+b \text {arcsinh}(c x))}{2 c^2}+\frac {d^2 (a+b \text {arcsinh}(c x))^2}{2 b c}-\frac {2 b d e x}{c}-\frac {b e^2 x^2}{4 c}\right )}{e}\) |
((d + e*x)^2*(a + b*ArcSinh[c*x])^2)/(2*e) - (b*c*((-2*b*d*e*x)/c - (b*e^2 *x^2)/(4*c) + (2*d*e*Sqrt[1 + c^2*x^2]*(a + b*ArcSinh[c*x]))/c^2 + (e^2*x* Sqrt[1 + c^2*x^2]*(a + b*ArcSinh[c*x]))/(2*c^2) + (d^2*(a + b*ArcSinh[c*x] )^2)/(2*b*c) - (e^2*(a + b*ArcSinh[c*x])^2)/(4*b*c^3)))/e
3.1.14.3.1 Defintions of rubi rules used
Int[((a_.) + ArcSinh[(c_.)*(x_)]*(b_.))^(n_.)*((d_.) + (e_.)*(x_))^(m_.), x _Symbol] :> Simp[(d + e*x)^(m + 1)*((a + b*ArcSinh[c*x])^n/(e*(m + 1))), x] - Simp[b*c*(n/(e*(m + 1))) Int[(d + e*x)^(m + 1)*((a + b*ArcSinh[c*x])^( n - 1)/Sqrt[1 + c^2*x^2]), x], x] /; FreeQ[{a, b, c, d, e, m}, x] && IGtQ[n , 0] && NeQ[m, -1]
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]))
Time = 0.31 (sec) , antiderivative size = 183, normalized size of antiderivative = 1.31
| method | result | size |
| parts | \(a^{2} \left (\frac {1}{2} e \,x^{2}+d x \right )+\frac {b^{2} \left (\frac {e \left (2 \operatorname {arcsinh}\left (c x \right )^{2} x^{2} c^{2}-2 \,\operatorname {arcsinh}\left (c x \right ) c x \sqrt {c^{2} x^{2}+1}+\operatorname {arcsinh}\left (c x \right )^{2}+c^{2} x^{2}+1\right )}{4 c}+d \left (\operatorname {arcsinh}\left (c x \right )^{2} x c -2 \,\operatorname {arcsinh}\left (c x \right ) \sqrt {c^{2} x^{2}+1}+2 c x \right )\right )}{c}+\frac {2 a b \left (\frac {c \,\operatorname {arcsinh}\left (c x \right ) x^{2} e}{2}+\operatorname {arcsinh}\left (c x \right ) d c x -\frac {e \left (\frac {c x \sqrt {c^{2} x^{2}+1}}{2}-\frac {\operatorname {arcsinh}\left (c x \right )}{2}\right )+2 d c \sqrt {c^{2} x^{2}+1}}{2 c}\right )}{c}\) | \(183\) |
| derivativedivides | \(\frac {\frac {a^{2} \left (d \,c^{2} x +\frac {1}{2} c^{2} e \,x^{2}\right )}{c}+\frac {b^{2} \left (\frac {e \left (2 \operatorname {arcsinh}\left (c x \right )^{2} x^{2} c^{2}-2 \,\operatorname {arcsinh}\left (c x \right ) c x \sqrt {c^{2} x^{2}+1}+\operatorname {arcsinh}\left (c x \right )^{2}+c^{2} x^{2}+1\right )}{4}+d c \left (\operatorname {arcsinh}\left (c x \right )^{2} x c -2 \,\operatorname {arcsinh}\left (c x \right ) \sqrt {c^{2} x^{2}+1}+2 c x \right )\right )}{c}+\frac {2 a b \left (\operatorname {arcsinh}\left (c x \right ) d \,c^{2} x +\frac {\operatorname {arcsinh}\left (c x \right ) e \,c^{2} x^{2}}{2}-\frac {e \left (\frac {c x \sqrt {c^{2} x^{2}+1}}{2}-\frac {\operatorname {arcsinh}\left (c x \right )}{2}\right )}{2}-d c \sqrt {c^{2} x^{2}+1}\right )}{c}}{c}\) | \(193\) |
| default | \(\frac {\frac {a^{2} \left (d \,c^{2} x +\frac {1}{2} c^{2} e \,x^{2}\right )}{c}+\frac {b^{2} \left (\frac {e \left (2 \operatorname {arcsinh}\left (c x \right )^{2} x^{2} c^{2}-2 \,\operatorname {arcsinh}\left (c x \right ) c x \sqrt {c^{2} x^{2}+1}+\operatorname {arcsinh}\left (c x \right )^{2}+c^{2} x^{2}+1\right )}{4}+d c \left (\operatorname {arcsinh}\left (c x \right )^{2} x c -2 \,\operatorname {arcsinh}\left (c x \right ) \sqrt {c^{2} x^{2}+1}+2 c x \right )\right )}{c}+\frac {2 a b \left (\operatorname {arcsinh}\left (c x \right ) d \,c^{2} x +\frac {\operatorname {arcsinh}\left (c x \right ) e \,c^{2} x^{2}}{2}-\frac {e \left (\frac {c x \sqrt {c^{2} x^{2}+1}}{2}-\frac {\operatorname {arcsinh}\left (c x \right )}{2}\right )}{2}-d c \sqrt {c^{2} x^{2}+1}\right )}{c}}{c}\) | \(193\) |
a^2*(1/2*e*x^2+d*x)+b^2/c*(1/4*e*(2*arcsinh(c*x)^2*x^2*c^2-2*arcsinh(c*x)* c*x*(c^2*x^2+1)^(1/2)+arcsinh(c*x)^2+c^2*x^2+1)/c+d*(arcsinh(c*x)^2*x*c-2* arcsinh(c*x)*(c^2*x^2+1)^(1/2)+2*c*x))+2*a*b/c*(1/2*c*arcsinh(c*x)*x^2*e+a rcsinh(c*x)*d*c*x-1/2/c*(e*(1/2*c*x*(c^2*x^2+1)^(1/2)-1/2*arcsinh(c*x))+2* d*c*(c^2*x^2+1)^(1/2)))
Time = 0.26 (sec) , antiderivative size = 183, normalized size of antiderivative = 1.31 \[ \int (d+e x) (a+b \text {arcsinh}(c x))^2 \, dx=\frac {{\left (2 \, a^{2} + b^{2}\right )} c^{2} e x^{2} + 4 \, {\left (a^{2} + 2 \, b^{2}\right )} c^{2} d x + {\left (2 \, b^{2} c^{2} e x^{2} + 4 \, b^{2} c^{2} d x + b^{2} e\right )} \log \left (c x + \sqrt {c^{2} x^{2} + 1}\right )^{2} + 2 \, {\left (2 \, a b c^{2} e x^{2} + 4 \, a b c^{2} d x + a b e - {\left (b^{2} c e x + 4 \, b^{2} c d\right )} \sqrt {c^{2} x^{2} + 1}\right )} \log \left (c x + \sqrt {c^{2} x^{2} + 1}\right ) - 2 \, {\left (a b c e x + 4 \, a b c d\right )} \sqrt {c^{2} x^{2} + 1}}{4 \, c^{2}} \]
1/4*((2*a^2 + b^2)*c^2*e*x^2 + 4*(a^2 + 2*b^2)*c^2*d*x + (2*b^2*c^2*e*x^2 + 4*b^2*c^2*d*x + b^2*e)*log(c*x + sqrt(c^2*x^2 + 1))^2 + 2*(2*a*b*c^2*e*x ^2 + 4*a*b*c^2*d*x + a*b*e - (b^2*c*e*x + 4*b^2*c*d)*sqrt(c^2*x^2 + 1))*lo g(c*x + sqrt(c^2*x^2 + 1)) - 2*(a*b*c*e*x + 4*a*b*c*d)*sqrt(c^2*x^2 + 1))/ c^2
Time = 0.23 (sec) , antiderivative size = 233, normalized size of antiderivative = 1.66 \[ \int (d+e x) (a+b \text {arcsinh}(c x))^2 \, dx=\begin {cases} a^{2} d x + \frac {a^{2} e x^{2}}{2} + 2 a b d x \operatorname {asinh}{\left (c x \right )} + a b e x^{2} \operatorname {asinh}{\left (c x \right )} - \frac {2 a b d \sqrt {c^{2} x^{2} + 1}}{c} - \frac {a b e x \sqrt {c^{2} x^{2} + 1}}{2 c} + \frac {a b e \operatorname {asinh}{\left (c x \right )}}{2 c^{2}} + b^{2} d x \operatorname {asinh}^{2}{\left (c x \right )} + 2 b^{2} d x + \frac {b^{2} e x^{2} \operatorname {asinh}^{2}{\left (c x \right )}}{2} + \frac {b^{2} e x^{2}}{4} - \frac {2 b^{2} d \sqrt {c^{2} x^{2} + 1} \operatorname {asinh}{\left (c x \right )}}{c} - \frac {b^{2} e x \sqrt {c^{2} x^{2} + 1} \operatorname {asinh}{\left (c x \right )}}{2 c} + \frac {b^{2} e \operatorname {asinh}^{2}{\left (c x \right )}}{4 c^{2}} & \text {for}\: c \neq 0 \\a^{2} \left (d x + \frac {e x^{2}}{2}\right ) & \text {otherwise} \end {cases} \]
Piecewise((a**2*d*x + a**2*e*x**2/2 + 2*a*b*d*x*asinh(c*x) + a*b*e*x**2*as inh(c*x) - 2*a*b*d*sqrt(c**2*x**2 + 1)/c - a*b*e*x*sqrt(c**2*x**2 + 1)/(2* c) + a*b*e*asinh(c*x)/(2*c**2) + b**2*d*x*asinh(c*x)**2 + 2*b**2*d*x + b** 2*e*x**2*asinh(c*x)**2/2 + b**2*e*x**2/4 - 2*b**2*d*sqrt(c**2*x**2 + 1)*as inh(c*x)/c - b**2*e*x*sqrt(c**2*x**2 + 1)*asinh(c*x)/(2*c) + b**2*e*asinh( c*x)**2/(4*c**2), Ne(c, 0)), (a**2*(d*x + e*x**2/2), True))
Time = 0.19 (sec) , antiderivative size = 219, normalized size of antiderivative = 1.56 \[ \int (d+e x) (a+b \text {arcsinh}(c x))^2 \, dx=\frac {1}{2} \, b^{2} e x^{2} \operatorname {arsinh}\left (c x\right )^{2} + b^{2} d x \operatorname {arsinh}\left (c x\right )^{2} + \frac {1}{2} \, a^{2} e x^{2} + \frac {1}{2} \, {\left (2 \, x^{2} \operatorname {arsinh}\left (c x\right ) - c {\left (\frac {\sqrt {c^{2} x^{2} + 1} x}{c^{2}} - \frac {\operatorname {arsinh}\left (c x\right )}{c^{3}}\right )}\right )} a b e + \frac {1}{4} \, {\left (c^{2} {\left (\frac {x^{2}}{c^{2}} - \frac {\log \left (c x + \sqrt {c^{2} x^{2} + 1}\right )^{2}}{c^{4}}\right )} - 2 \, c {\left (\frac {\sqrt {c^{2} x^{2} + 1} x}{c^{2}} - \frac {\operatorname {arsinh}\left (c x\right )}{c^{3}}\right )} \operatorname {arsinh}\left (c x\right )\right )} b^{2} e + 2 \, b^{2} d {\left (x - \frac {\sqrt {c^{2} x^{2} + 1} \operatorname {arsinh}\left (c x\right )}{c}\right )} + a^{2} d x + \frac {2 \, {\left (c x \operatorname {arsinh}\left (c x\right ) - \sqrt {c^{2} x^{2} + 1}\right )} a b d}{c} \]
1/2*b^2*e*x^2*arcsinh(c*x)^2 + b^2*d*x*arcsinh(c*x)^2 + 1/2*a^2*e*x^2 + 1/ 2*(2*x^2*arcsinh(c*x) - c*(sqrt(c^2*x^2 + 1)*x/c^2 - arcsinh(c*x)/c^3))*a* b*e + 1/4*(c^2*(x^2/c^2 - log(c*x + sqrt(c^2*x^2 + 1))^2/c^4) - 2*c*(sqrt( c^2*x^2 + 1)*x/c^2 - arcsinh(c*x)/c^3)*arcsinh(c*x))*b^2*e + 2*b^2*d*(x - sqrt(c^2*x^2 + 1)*arcsinh(c*x)/c) + a^2*d*x + 2*(c*x*arcsinh(c*x) - sqrt(c ^2*x^2 + 1))*a*b*d/c
Exception generated. \[ \int (d+e x) (a+b \text {arcsinh}(c x))^2 \, dx=\text {Exception raised: RuntimeError} \]
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. \[ \int (d+e x) (a+b \text {arcsinh}(c x))^2 \, dx=\int {\left (a+b\,\mathrm {asinh}\left (c\,x\right )\right )}^2\,\left (d+e\,x\right ) \,d x \]