Integrand size = 18, antiderivative size = 153 \[ \int \left (d+e x^2\right ) (a+b \text {arcsinh}(c x))^2 \, dx=2 b^2 d x-\frac {4 b^2 e x}{9 c^2}+\frac {2}{27} b^2 e x^3-\frac {2 b d \sqrt {1+c^2 x^2} (a+b \text {arcsinh}(c x))}{c}+\frac {4 b e \sqrt {1+c^2 x^2} (a+b \text {arcsinh}(c x))}{9 c^3}-\frac {2 b e x^2 \sqrt {1+c^2 x^2} (a+b \text {arcsinh}(c x))}{9 c}+d x (a+b \text {arcsinh}(c x))^2+\frac {1}{3} e x^3 (a+b \text {arcsinh}(c x))^2 \]
2*b^2*d*x-4/9*b^2*e*x/c^2+2/27*b^2*e*x^3+d*x*(a+b*arcsinh(c*x))^2+1/3*e*x^ 3*(a+b*arcsinh(c*x))^2-2*b*d*(a+b*arcsinh(c*x))*(c^2*x^2+1)^(1/2)/c+4/9*b* e*(a+b*arcsinh(c*x))*(c^2*x^2+1)^(1/2)/c^3-2/9*b*e*x^2*(a+b*arcsinh(c*x))* (c^2*x^2+1)^(1/2)/c
Time = 0.14 (sec) , antiderivative size = 164, normalized size of antiderivative = 1.07 \[ \int \left (d+e x^2\right ) (a+b \text {arcsinh}(c x))^2 \, dx=\frac {9 a^2 c^3 x \left (3 d+e x^2\right )-6 a b \sqrt {1+c^2 x^2} \left (-2 e+c^2 \left (9 d+e x^2\right )\right )+2 b^2 c x \left (-6 e+c^2 \left (27 d+e x^2\right )\right )-6 b \left (-3 a c^3 x \left (3 d+e x^2\right )+b \sqrt {1+c^2 x^2} \left (-2 e+c^2 \left (9 d+e x^2\right )\right )\right ) \text {arcsinh}(c x)+9 b^2 c^3 x \left (3 d+e x^2\right ) \text {arcsinh}(c x)^2}{27 c^3} \]
(9*a^2*c^3*x*(3*d + e*x^2) - 6*a*b*Sqrt[1 + c^2*x^2]*(-2*e + c^2*(9*d + e* x^2)) + 2*b^2*c*x*(-6*e + c^2*(27*d + e*x^2)) - 6*b*(-3*a*c^3*x*(3*d + e*x ^2) + b*Sqrt[1 + c^2*x^2]*(-2*e + c^2*(9*d + e*x^2)))*ArcSinh[c*x] + 9*b^2 *c^3*x*(3*d + e*x^2)*ArcSinh[c*x]^2)/(27*c^3)
Time = 0.47 (sec) , antiderivative size = 153, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.111, Rules used = {6208, 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.
| \(\displaystyle \int \left (d+e x^2\right ) (a+b \text {arcsinh}(c x))^2 \, dx\) |
| \(\Big \downarrow \) 6208 |
| \(\displaystyle \int \left (d (a+b \text {arcsinh}(c x))^2+e x^2 (a+b \text {arcsinh}(c x))^2\right )dx\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle -\frac {2 b d \sqrt {c^2 x^2+1} (a+b \text {arcsinh}(c x))}{c}-\frac {2 b e x^2 \sqrt {c^2 x^2+1} (a+b \text {arcsinh}(c x))}{9 c}+\frac {4 b e \sqrt {c^2 x^2+1} (a+b \text {arcsinh}(c x))}{9 c^3}+d x (a+b \text {arcsinh}(c x))^2+\frac {1}{3} e x^3 (a+b \text {arcsinh}(c x))^2-\frac {4 b^2 e x}{9 c^2}+2 b^2 d x+\frac {2}{27} b^2 e x^3\) |
2*b^2*d*x - (4*b^2*e*x)/(9*c^2) + (2*b^2*e*x^3)/27 - (2*b*d*Sqrt[1 + c^2*x ^2]*(a + b*ArcSinh[c*x]))/c + (4*b*e*Sqrt[1 + c^2*x^2]*(a + b*ArcSinh[c*x] ))/(9*c^3) - (2*b*e*x^2*Sqrt[1 + c^2*x^2]*(a + b*ArcSinh[c*x]))/(9*c) + d* x*(a + b*ArcSinh[c*x])^2 + (e*x^3*(a + b*ArcSinh[c*x])^2)/3
3.7.15.3.1 Defintions of rubi rules used
Int[((a_.) + ArcSinh[(c_.)*(x_)]*(b_.))^(n_.)*((d_) + (e_.)*(x_)^2)^(p_.), x_Symbol] :> Int[ExpandIntegrand[(a + b*ArcSinh[c*x])^n, (d + e*x^2)^p, x], x] /; FreeQ[{a, b, c, d, e, n}, x] && NeQ[e, c^2*d] && IntegerQ[p] && (p > 0 || IGtQ[n, 0])
Time = 0.22 (sec) , antiderivative size = 215, normalized size of antiderivative = 1.41
| method | result | size |
| parts | \(a^{2} \left (\frac {1}{3} x^{3} e +d x \right )+\frac {b^{2} \left (\frac {e \left (9 \operatorname {arcsinh}\left (c x \right )^{2} x^{3} c^{3}-6 \,\operatorname {arcsinh}\left (c x \right ) \sqrt {c^{2} x^{2}+1}\, x^{2} c^{2}+2 c^{3} x^{3}+12 \,\operatorname {arcsinh}\left (c x \right ) \sqrt {c^{2} x^{2}+1}-12 c x \right )}{27 c^{2}}+d \left (\operatorname {arcsinh}\left (c x \right )^{2} x c -2 \,\operatorname {arcsinh}\left (c x \right ) \sqrt {c^{2} x^{2}+1}+2 c x \right )\right )}{c}+\frac {2 a b \left (\frac {c \,\operatorname {arcsinh}\left (c x \right ) x^{3} e}{3}+\operatorname {arcsinh}\left (c x \right ) d c x -\frac {e \left (\frac {c^{2} x^{2} \sqrt {c^{2} x^{2}+1}}{3}-\frac {2 \sqrt {c^{2} x^{2}+1}}{3}\right )+3 d \,c^{2} \sqrt {c^{2} x^{2}+1}}{3 c^{2}}\right )}{c}\) | \(215\) |
| derivativedivides | \(\frac {\frac {a^{2} \left (d \,c^{3} x +\frac {1}{3} e \,c^{3} x^{3}\right )}{c^{2}}+\frac {b^{2} \left (d \,c^{2} \left (\operatorname {arcsinh}\left (c x \right )^{2} x c -2 \,\operatorname {arcsinh}\left (c x \right ) \sqrt {c^{2} x^{2}+1}+2 c x \right )+\frac {e \left (9 \operatorname {arcsinh}\left (c x \right )^{2} x^{3} c^{3}-6 \,\operatorname {arcsinh}\left (c x \right ) \sqrt {c^{2} x^{2}+1}\, x^{2} c^{2}+2 c^{3} x^{3}+12 \,\operatorname {arcsinh}\left (c x \right ) \sqrt {c^{2} x^{2}+1}-12 c x \right )}{27}\right )}{c^{2}}+\frac {2 a b \left (\operatorname {arcsinh}\left (c x \right ) d \,c^{3} x +\frac {\operatorname {arcsinh}\left (c x \right ) e \,c^{3} x^{3}}{3}-\frac {e \left (\frac {c^{2} x^{2} \sqrt {c^{2} x^{2}+1}}{3}-\frac {2 \sqrt {c^{2} x^{2}+1}}{3}\right )}{3}-d \,c^{2} \sqrt {c^{2} x^{2}+1}\right )}{c^{2}}}{c}\) | \(227\) |
| default | \(\frac {\frac {a^{2} \left (d \,c^{3} x +\frac {1}{3} e \,c^{3} x^{3}\right )}{c^{2}}+\frac {b^{2} \left (d \,c^{2} \left (\operatorname {arcsinh}\left (c x \right )^{2} x c -2 \,\operatorname {arcsinh}\left (c x \right ) \sqrt {c^{2} x^{2}+1}+2 c x \right )+\frac {e \left (9 \operatorname {arcsinh}\left (c x \right )^{2} x^{3} c^{3}-6 \,\operatorname {arcsinh}\left (c x \right ) \sqrt {c^{2} x^{2}+1}\, x^{2} c^{2}+2 c^{3} x^{3}+12 \,\operatorname {arcsinh}\left (c x \right ) \sqrt {c^{2} x^{2}+1}-12 c x \right )}{27}\right )}{c^{2}}+\frac {2 a b \left (\operatorname {arcsinh}\left (c x \right ) d \,c^{3} x +\frac {\operatorname {arcsinh}\left (c x \right ) e \,c^{3} x^{3}}{3}-\frac {e \left (\frac {c^{2} x^{2} \sqrt {c^{2} x^{2}+1}}{3}-\frac {2 \sqrt {c^{2} x^{2}+1}}{3}\right )}{3}-d \,c^{2} \sqrt {c^{2} x^{2}+1}\right )}{c^{2}}}{c}\) | \(227\) |
a^2*(1/3*x^3*e+d*x)+b^2/c*(1/27*e*(9*arcsinh(c*x)^2*x^3*c^3-6*arcsinh(c*x) *(c^2*x^2+1)^(1/2)*x^2*c^2+2*c^3*x^3+12*arcsinh(c*x)*(c^2*x^2+1)^(1/2)-12* c*x)/c^2+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*(1/3*c*arcsinh(c*x)*x^3*e+arcsinh(c*x)*d*c*x-1/3/c^2*(e*(1/3*c^2*x^2 *(c^2*x^2+1)^(1/2)-2/3*(c^2*x^2+1)^(1/2))+3*d*c^2*(c^2*x^2+1)^(1/2)))
Time = 0.28 (sec) , antiderivative size = 209, normalized size of antiderivative = 1.37 \[ \int \left (d+e x^2\right ) (a+b \text {arcsinh}(c x))^2 \, dx=\frac {{\left (9 \, a^{2} + 2 \, b^{2}\right )} c^{3} e x^{3} + 9 \, {\left (b^{2} c^{3} e x^{3} + 3 \, b^{2} c^{3} d x\right )} \log \left (c x + \sqrt {c^{2} x^{2} + 1}\right )^{2} + 3 \, {\left (9 \, {\left (a^{2} + 2 \, b^{2}\right )} c^{3} d - 4 \, b^{2} c e\right )} x + 6 \, {\left (3 \, a b c^{3} e x^{3} + 9 \, a b c^{3} d x - {\left (b^{2} c^{2} e x^{2} + 9 \, b^{2} c^{2} d - 2 \, b^{2} e\right )} \sqrt {c^{2} x^{2} + 1}\right )} \log \left (c x + \sqrt {c^{2} x^{2} + 1}\right ) - 6 \, {\left (a b c^{2} e x^{2} + 9 \, a b c^{2} d - 2 \, a b e\right )} \sqrt {c^{2} x^{2} + 1}}{27 \, c^{3}} \]
1/27*((9*a^2 + 2*b^2)*c^3*e*x^3 + 9*(b^2*c^3*e*x^3 + 3*b^2*c^3*d*x)*log(c* x + sqrt(c^2*x^2 + 1))^2 + 3*(9*(a^2 + 2*b^2)*c^3*d - 4*b^2*c*e)*x + 6*(3* a*b*c^3*e*x^3 + 9*a*b*c^3*d*x - (b^2*c^2*e*x^2 + 9*b^2*c^2*d - 2*b^2*e)*sq rt(c^2*x^2 + 1))*log(c*x + sqrt(c^2*x^2 + 1)) - 6*(a*b*c^2*e*x^2 + 9*a*b*c ^2*d - 2*a*b*e)*sqrt(c^2*x^2 + 1))/c^3
Time = 0.31 (sec) , antiderivative size = 279, normalized size of antiderivative = 1.82 \[ \int \left (d+e x^2\right ) (a+b \text {arcsinh}(c x))^2 \, dx=\begin {cases} a^{2} d x + \frac {a^{2} e x^{3}}{3} + 2 a b d x \operatorname {asinh}{\left (c x \right )} + \frac {2 a b e x^{3} \operatorname {asinh}{\left (c x \right )}}{3} - \frac {2 a b d \sqrt {c^{2} x^{2} + 1}}{c} - \frac {2 a b e x^{2} \sqrt {c^{2} x^{2} + 1}}{9 c} + \frac {4 a b e \sqrt {c^{2} x^{2} + 1}}{9 c^{3}} + b^{2} d x \operatorname {asinh}^{2}{\left (c x \right )} + 2 b^{2} d x + \frac {b^{2} e x^{3} \operatorname {asinh}^{2}{\left (c x \right )}}{3} + \frac {2 b^{2} e x^{3}}{27} - \frac {2 b^{2} d \sqrt {c^{2} x^{2} + 1} \operatorname {asinh}{\left (c x \right )}}{c} - \frac {2 b^{2} e x^{2} \sqrt {c^{2} x^{2} + 1} \operatorname {asinh}{\left (c x \right )}}{9 c} - \frac {4 b^{2} e x}{9 c^{2}} + \frac {4 b^{2} e \sqrt {c^{2} x^{2} + 1} \operatorname {asinh}{\left (c x \right )}}{9 c^{3}} & \text {for}\: c \neq 0 \\a^{2} \left (d x + \frac {e x^{3}}{3}\right ) & \text {otherwise} \end {cases} \]
Piecewise((a**2*d*x + a**2*e*x**3/3 + 2*a*b*d*x*asinh(c*x) + 2*a*b*e*x**3* asinh(c*x)/3 - 2*a*b*d*sqrt(c**2*x**2 + 1)/c - 2*a*b*e*x**2*sqrt(c**2*x**2 + 1)/(9*c) + 4*a*b*e*sqrt(c**2*x**2 + 1)/(9*c**3) + b**2*d*x*asinh(c*x)** 2 + 2*b**2*d*x + b**2*e*x**3*asinh(c*x)**2/3 + 2*b**2*e*x**3/27 - 2*b**2*d *sqrt(c**2*x**2 + 1)*asinh(c*x)/c - 2*b**2*e*x**2*sqrt(c**2*x**2 + 1)*asin h(c*x)/(9*c) - 4*b**2*e*x/(9*c**2) + 4*b**2*e*sqrt(c**2*x**2 + 1)*asinh(c* x)/(9*c**3), Ne(c, 0)), (a**2*(d*x + e*x**3/3), True))
Time = 0.20 (sec) , antiderivative size = 218, normalized size of antiderivative = 1.42 \[ \int \left (d+e x^2\right ) (a+b \text {arcsinh}(c x))^2 \, dx=\frac {1}{3} \, b^{2} e x^{3} \operatorname {arsinh}\left (c x\right )^{2} + \frac {1}{3} \, a^{2} e x^{3} + b^{2} d x \operatorname {arsinh}\left (c x\right )^{2} + \frac {2}{9} \, {\left (3 \, x^{3} \operatorname {arsinh}\left (c x\right ) - c {\left (\frac {\sqrt {c^{2} x^{2} + 1} x^{2}}{c^{2}} - \frac {2 \, \sqrt {c^{2} x^{2} + 1}}{c^{4}}\right )}\right )} a b e - \frac {2}{27} \, {\left (3 \, c {\left (\frac {\sqrt {c^{2} x^{2} + 1} x^{2}}{c^{2}} - \frac {2 \, \sqrt {c^{2} x^{2} + 1}}{c^{4}}\right )} \operatorname {arsinh}\left (c x\right ) - \frac {c^{2} x^{3} - 6 \, x}{c^{2}}\right )} b^{2} e + 2 \, b^{2} d {\left (x - \frac {\sqrt {c^{2} x^{2} + 1} \operatorname {arsinh}\left (c x\right )}{c}\right )} + a^{2} d x + \frac {2 \, {\left (c x \operatorname {arsinh}\left (c x\right ) - \sqrt {c^{2} x^{2} + 1}\right )} a b d}{c} \]
1/3*b^2*e*x^3*arcsinh(c*x)^2 + 1/3*a^2*e*x^3 + b^2*d*x*arcsinh(c*x)^2 + 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))*a*b*e - 2/27*(3*c*(sqrt(c^2*x^2 + 1)*x^2/c^2 - 2*sqrt(c^2*x^2 + 1)/ c^4)*arcsinh(c*x) - (c^2*x^3 - 6*x)/c^2)*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. \[ \int \left (d+e x^2\right ) (a+b \text {arcsinh}(c x))^2 \, dx=\text {Exception raised: RuntimeError} \]
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. \[ \int \left (d+e x^2\right ) (a+b \text {arcsinh}(c x))^2 \, dx=\int {\left (a+b\,\mathrm {asinh}\left (c\,x\right )\right )}^2\,\left (e\,x^2+d\right ) \,d x \]