Integrand size = 16, antiderivative size = 124 \[ \int (d+e x)^2 (a+b \text {arcsinh}(c x)) \, dx=-\frac {b (d+e x)^2 \sqrt {1+c^2 x^2}}{9 c}-\frac {b \left (4 \left (4 c^2 d^2-e^2\right )+5 c^2 d e x\right ) \sqrt {1+c^2 x^2}}{18 c^3}-\frac {b d \left (2 d^2-\frac {3 e^2}{c^2}\right ) \text {arcsinh}(c x)}{6 e}+\frac {(d+e x)^3 (a+b \text {arcsinh}(c x))}{3 e} \]
-1/6*b*d*(2*d^2-3*e^2/c^2)*arcsinh(c*x)/e+1/3*(e*x+d)^3*(a+b*arcsinh(c*x)) /e-1/9*b*(e*x+d)^2*(c^2*x^2+1)^(1/2)/c-1/18*b*(5*c^2*d*e*x+16*c^2*d^2-4*e^ 2)*(c^2*x^2+1)^(1/2)/c^3
Time = 0.08 (sec) , antiderivative size = 121, normalized size of antiderivative = 0.98 \[ \int (d+e x)^2 (a+b \text {arcsinh}(c x)) \, dx=\frac {6 a c^3 x \left (3 d^2+3 d e x+e^2 x^2\right )-b \sqrt {1+c^2 x^2} \left (-4 e^2+c^2 \left (18 d^2+9 d e x+2 e^2 x^2\right )\right )+3 b c \left (6 c^2 d^2 x+2 c^2 e^2 x^3+3 d \left (e+2 c^2 e x^2\right )\right ) \text {arcsinh}(c x)}{18 c^3} \]
(6*a*c^3*x*(3*d^2 + 3*d*e*x + e^2*x^2) - b*Sqrt[1 + c^2*x^2]*(-4*e^2 + c^2 *(18*d^2 + 9*d*e*x + 2*e^2*x^2)) + 3*b*c*(6*c^2*d^2*x + 2*c^2*e^2*x^3 + 3* d*(e + 2*c^2*e*x^2))*ArcSinh[c*x])/(18*c^3)
Time = 0.30 (sec) , antiderivative size = 147, normalized size of antiderivative = 1.19, number of steps used = 4, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.250, Rules used = {6243, 497, 676, 222}
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.
\(\displaystyle \int (d+e x)^2 (a+b \text {arcsinh}(c x)) \, dx\) |
\(\Big \downarrow \) 6243 |
\(\displaystyle \frac {(d+e x)^3 (a+b \text {arcsinh}(c x))}{3 e}-\frac {b c \int \frac {(d+e x)^3}{\sqrt {c^2 x^2+1}}dx}{3 e}\) |
\(\Big \downarrow \) 497 |
\(\displaystyle \frac {(d+e x)^3 (a+b \text {arcsinh}(c x))}{3 e}-\frac {b c \left (\frac {\int \frac {(d+e x) \left (3 d^2 c^2+5 d e x c^2-2 e^2\right )}{\sqrt {c^2 x^2+1}}dx}{3 c^2}+\frac {e \sqrt {c^2 x^2+1} (d+e x)^2}{3 c^2}\right )}{3 e}\) |
\(\Big \downarrow \) 676 |
\(\displaystyle \frac {(d+e x)^3 (a+b \text {arcsinh}(c x))}{3 e}-\frac {b c \left (\frac {\frac {3}{2} d \left (2 c^2 d^2-3 e^2\right ) \int \frac {1}{\sqrt {c^2 x^2+1}}dx+2 e \sqrt {c^2 x^2+1} \left (4 d^2-\frac {e^2}{c^2}\right )+\frac {5}{2} d e^2 x \sqrt {c^2 x^2+1}}{3 c^2}+\frac {e \sqrt {c^2 x^2+1} (d+e x)^2}{3 c^2}\right )}{3 e}\) |
\(\Big \downarrow \) 222 |
\(\displaystyle \frac {(d+e x)^3 (a+b \text {arcsinh}(c x))}{3 e}-\frac {b c \left (\frac {\frac {3 d \text {arcsinh}(c x) \left (2 c^2 d^2-3 e^2\right )}{2 c}+2 e \sqrt {c^2 x^2+1} \left (4 d^2-\frac {e^2}{c^2}\right )+\frac {5}{2} d e^2 x \sqrt {c^2 x^2+1}}{3 c^2}+\frac {e \sqrt {c^2 x^2+1} (d+e x)^2}{3 c^2}\right )}{3 e}\) |
((d + e*x)^3*(a + b*ArcSinh[c*x]))/(3*e) - (b*c*((e*(d + e*x)^2*Sqrt[1 + c ^2*x^2])/(3*c^2) + (2*e*(4*d^2 - e^2/c^2)*Sqrt[1 + c^2*x^2] + (5*d*e^2*x*S qrt[1 + c^2*x^2])/2 + (3*d*(2*c^2*d^2 - 3*e^2)*ArcSinh[c*x])/(2*c))/(3*c^2 )))/(3*e)
3.1.5.3.1 Defintions of rubi rules used
Int[1/Sqrt[(a_) + (b_.)*(x_)^2], x_Symbol] :> Simp[ArcSinh[Rt[b, 2]*(x/Sqrt [a])]/Rt[b, 2], x] /; FreeQ[{a, b}, x] && GtQ[a, 0] && PosQ[b]
Int[((c_) + (d_.)*(x_))^(n_)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> Simp[ d*(c + d*x)^(n - 1)*((a + b*x^2)^(p + 1)/(b*(n + 2*p + 1))), x] + Simp[1/(b *(n + 2*p + 1)) Int[(c + d*x)^(n - 2)*(a + b*x^2)^p*Simp[b*c^2*(n + 2*p + 1) - a*d^2*(n - 1) + 2*b*c*d*(n + p)*x, x], x], x] /; FreeQ[{a, b, c, d, n , p}, x] && If[RationalQ[n], GtQ[n, 1], SumSimplerQ[n, -2]] && NeQ[n + 2*p + 1, 0] && IntQuadraticQ[a, 0, b, c, d, n, p, x]
Int[((d_.) + (e_.)*(x_))*((f_.) + (g_.)*(x_))*((a_) + (c_.)*(x_)^2)^(p_), x _Symbol] :> Simp[(e*f + d*g)*((a + c*x^2)^(p + 1)/(2*c*(p + 1))), x] + (Sim p[e*g*x*((a + c*x^2)^(p + 1)/(c*(2*p + 3))), x] - Simp[(a*e*g - c*d*f*(2*p + 3))/(c*(2*p + 3)) Int[(a + c*x^2)^p, x], x]) /; FreeQ[{a, c, d, e, f, g , p}, x] && !LeQ[p, -1]
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]
Time = 0.32 (sec) , antiderivative size = 174, normalized size of antiderivative = 1.40
method | result | size |
parts | \(\frac {a \left (e x +d \right )^{3}}{3 e}+\frac {b \left (\frac {c \,e^{2} \operatorname {arcsinh}\left (c x \right ) x^{3}}{3}+c \,\operatorname {arcsinh}\left (c x \right ) d e \,x^{2}+\operatorname {arcsinh}\left (c x \right ) c x \,d^{2}+\frac {c \,\operatorname {arcsinh}\left (c x \right ) d^{3}}{3 e}-\frac {c^{3} d^{3} \operatorname {arcsinh}\left (c x \right )+e^{3} \left (\frac {c^{2} x^{2} \sqrt {c^{2} x^{2}+1}}{3}-\frac {2 \sqrt {c^{2} x^{2}+1}}{3}\right )+3 d^{2} c^{2} e \sqrt {c^{2} x^{2}+1}+3 d c \,e^{2} \left (\frac {c x \sqrt {c^{2} x^{2}+1}}{2}-\frac {\operatorname {arcsinh}\left (c x \right )}{2}\right )}{3 c^{2} e}\right )}{c}\) | \(174\) |
derivativedivides | \(\frac {\frac {a \left (e c x +c d \right )^{3}}{3 c^{2} e}+\frac {b \left (\frac {\operatorname {arcsinh}\left (c x \right ) c^{3} d^{3}}{3 e}+\operatorname {arcsinh}\left (c x \right ) c^{3} d^{2} x +e \,\operatorname {arcsinh}\left (c x \right ) c^{3} d \,x^{2}+\frac {e^{2} \operatorname {arcsinh}\left (c x \right ) c^{3} x^{3}}{3}-\frac {c^{3} d^{3} \operatorname {arcsinh}\left (c x \right )+e^{3} \left (\frac {c^{2} x^{2} \sqrt {c^{2} x^{2}+1}}{3}-\frac {2 \sqrt {c^{2} x^{2}+1}}{3}\right )+3 d^{2} c^{2} e \sqrt {c^{2} x^{2}+1}+3 d c \,e^{2} \left (\frac {c x \sqrt {c^{2} x^{2}+1}}{2}-\frac {\operatorname {arcsinh}\left (c x \right )}{2}\right )}{3 e}\right )}{c^{2}}}{c}\) | \(189\) |
default | \(\frac {\frac {a \left (e c x +c d \right )^{3}}{3 c^{2} e}+\frac {b \left (\frac {\operatorname {arcsinh}\left (c x \right ) c^{3} d^{3}}{3 e}+\operatorname {arcsinh}\left (c x \right ) c^{3} d^{2} x +e \,\operatorname {arcsinh}\left (c x \right ) c^{3} d \,x^{2}+\frac {e^{2} \operatorname {arcsinh}\left (c x \right ) c^{3} x^{3}}{3}-\frac {c^{3} d^{3} \operatorname {arcsinh}\left (c x \right )+e^{3} \left (\frac {c^{2} x^{2} \sqrt {c^{2} x^{2}+1}}{3}-\frac {2 \sqrt {c^{2} x^{2}+1}}{3}\right )+3 d^{2} c^{2} e \sqrt {c^{2} x^{2}+1}+3 d c \,e^{2} \left (\frac {c x \sqrt {c^{2} x^{2}+1}}{2}-\frac {\operatorname {arcsinh}\left (c x \right )}{2}\right )}{3 e}\right )}{c^{2}}}{c}\) | \(189\) |
1/3*a*(e*x+d)^3/e+b/c*(1/3*c*e^2*arcsinh(c*x)*x^3+c*arcsinh(c*x)*d*e*x^2+a rcsinh(c*x)*c*x*d^2+1/3*c/e*arcsinh(c*x)*d^3-1/3/c^2/e*(c^3*d^3*arcsinh(c* x)+e^3*(1/3*c^2*x^2*(c^2*x^2+1)^(1/2)-2/3*(c^2*x^2+1)^(1/2))+3*d^2*c^2*e*( c^2*x^2+1)^(1/2)+3*d*c*e^2*(1/2*c*x*(c^2*x^2+1)^(1/2)-1/2*arcsinh(c*x))))
Time = 0.25 (sec) , antiderivative size = 147, normalized size of antiderivative = 1.19 \[ \int (d+e x)^2 (a+b \text {arcsinh}(c x)) \, dx=\frac {6 \, a c^{3} e^{2} x^{3} + 18 \, a c^{3} d e x^{2} + 18 \, a c^{3} d^{2} x + 3 \, {\left (2 \, b c^{3} e^{2} x^{3} + 6 \, b c^{3} d e x^{2} + 6 \, b c^{3} d^{2} x + 3 \, b c d e\right )} \log \left (c x + \sqrt {c^{2} x^{2} + 1}\right ) - {\left (2 \, b c^{2} e^{2} x^{2} + 9 \, b c^{2} d e x + 18 \, b c^{2} d^{2} - 4 \, b e^{2}\right )} \sqrt {c^{2} x^{2} + 1}}{18 \, c^{3}} \]
1/18*(6*a*c^3*e^2*x^3 + 18*a*c^3*d*e*x^2 + 18*a*c^3*d^2*x + 3*(2*b*c^3*e^2 *x^3 + 6*b*c^3*d*e*x^2 + 6*b*c^3*d^2*x + 3*b*c*d*e)*log(c*x + sqrt(c^2*x^2 + 1)) - (2*b*c^2*e^2*x^2 + 9*b*c^2*d*e*x + 18*b*c^2*d^2 - 4*b*e^2)*sqrt(c ^2*x^2 + 1))/c^3
Time = 0.23 (sec) , antiderivative size = 190, normalized size of antiderivative = 1.53 \[ \int (d+e x)^2 (a+b \text {arcsinh}(c x)) \, dx=\begin {cases} a d^{2} x + a d e x^{2} + \frac {a e^{2} x^{3}}{3} + b d^{2} x \operatorname {asinh}{\left (c x \right )} + b d e x^{2} \operatorname {asinh}{\left (c x \right )} + \frac {b e^{2} x^{3} \operatorname {asinh}{\left (c x \right )}}{3} - \frac {b d^{2} \sqrt {c^{2} x^{2} + 1}}{c} - \frac {b d e x \sqrt {c^{2} x^{2} + 1}}{2 c} - \frac {b e^{2} x^{2} \sqrt {c^{2} x^{2} + 1}}{9 c} + \frac {b d e \operatorname {asinh}{\left (c x \right )}}{2 c^{2}} + \frac {2 b e^{2} \sqrt {c^{2} x^{2} + 1}}{9 c^{3}} & \text {for}\: c \neq 0 \\a \left (d^{2} x + d e x^{2} + \frac {e^{2} x^{3}}{3}\right ) & \text {otherwise} \end {cases} \]
Piecewise((a*d**2*x + a*d*e*x**2 + a*e**2*x**3/3 + b*d**2*x*asinh(c*x) + b *d*e*x**2*asinh(c*x) + b*e**2*x**3*asinh(c*x)/3 - b*d**2*sqrt(c**2*x**2 + 1)/c - b*d*e*x*sqrt(c**2*x**2 + 1)/(2*c) - b*e**2*x**2*sqrt(c**2*x**2 + 1) /(9*c) + b*d*e*asinh(c*x)/(2*c**2) + 2*b*e**2*sqrt(c**2*x**2 + 1)/(9*c**3) , Ne(c, 0)), (a*(d**2*x + d*e*x**2 + e**2*x**3/3), True))
Time = 0.18 (sec) , antiderivative size = 150, normalized size of antiderivative = 1.21 \[ \int (d+e x)^2 (a+b \text {arcsinh}(c x)) \, dx=\frac {1}{3} \, a e^{2} x^{3} + a d e x^{2} + \frac {1}{2} \, {\left (2 \, x^{2} \operatorname {arsinh}\left (c x\right ) - c {\left (\frac {\sqrt {c^{2} x^{2} + 1} x}{c^{2}} - \frac {\operatorname {arsinh}\left (c x\right )}{c^{3}}\right )}\right )} b d e + \frac {1}{9} \, {\left (3 \, x^{3} \operatorname {arsinh}\left (c x\right ) - c {\left (\frac {\sqrt {c^{2} x^{2} + 1} x^{2}}{c^{2}} - \frac {2 \, \sqrt {c^{2} x^{2} + 1}}{c^{4}}\right )}\right )} b e^{2} + a d^{2} x + \frac {{\left (c x \operatorname {arsinh}\left (c x\right ) - \sqrt {c^{2} x^{2} + 1}\right )} b d^{2}}{c} \]
1/3*a*e^2*x^3 + a*d*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))*b*d*e + 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*e^2 + a*d^2*x + (c*x*arcsin h(c*x) - sqrt(c^2*x^2 + 1))*b*d^2/c
Exception generated. \[ \int (d+e x)^2 (a+b \text {arcsinh}(c x)) \, dx=\text {Exception raised: RuntimeError} \]
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. \[ \int (d+e x)^2 (a+b \text {arcsinh}(c x)) \, dx=\int \left (a+b\,\mathrm {asinh}\left (c\,x\right )\right )\,{\left (d+e\,x\right )}^2 \,d x \]