Integrand size = 16, antiderivative size = 81 \[ \int \left (d+e x^2\right ) (a+b \text {arcsinh}(c x)) \, dx=-\frac {b \left (3 c^2 d-e\right ) \sqrt {1+c^2 x^2}}{3 c^3}-\frac {b e \left (1+c^2 x^2\right )^{3/2}}{9 c^3}+d x (a+b \text {arcsinh}(c x))+\frac {1}{3} e x^3 (a+b \text {arcsinh}(c x)) \] Output:
-1/3*b*(3*c^2*d-e)*(c^2*x^2+1)^(1/2)/c^3-1/9*b*e*(c^2*x^2+1)^(3/2)/c^3+d*x *(a+b*arcsinh(c*x))+1/3*e*x^3*(a+b*arcsinh(c*x))
Time = 0.05 (sec) , antiderivative size = 71, normalized size of antiderivative = 0.88 \[ \int \left (d+e x^2\right ) (a+b \text {arcsinh}(c x)) \, dx=\frac {1}{9} \left (3 a x \left (3 d+e x^2\right )-\frac {b \sqrt {1+c^2 x^2} \left (-2 e+c^2 \left (9 d+e x^2\right )\right )}{c^3}+3 b x \left (3 d+e x^2\right ) \text {arcsinh}(c x)\right ) \] Input:
Integrate[(d + e*x^2)*(a + b*ArcSinh[c*x]),x]
Output:
(3*a*x*(3*d + e*x^2) - (b*Sqrt[1 + c^2*x^2]*(-2*e + c^2*(9*d + e*x^2)))/c^ 3 + 3*b*x*(3*d + e*x^2)*ArcSinh[c*x])/9
Time = 0.32 (sec) , antiderivative size = 84, normalized size of antiderivative = 1.04, number of steps used = 6, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.312, Rules used = {6207, 27, 353, 53, 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.
\(\displaystyle \int \left (d+e x^2\right ) (a+b \text {arcsinh}(c x)) \, dx\) |
\(\Big \downarrow \) 6207 |
\(\displaystyle -b c \int \frac {x \left (e x^2+3 d\right )}{3 \sqrt {c^2 x^2+1}}dx+d x (a+b \text {arcsinh}(c x))+\frac {1}{3} e x^3 (a+b \text {arcsinh}(c x))\) |
\(\Big \downarrow \) 27 |
\(\displaystyle -\frac {1}{3} b c \int \frac {x \left (e x^2+3 d\right )}{\sqrt {c^2 x^2+1}}dx+d x (a+b \text {arcsinh}(c x))+\frac {1}{3} e x^3 (a+b \text {arcsinh}(c x))\) |
\(\Big \downarrow \) 353 |
\(\displaystyle -\frac {1}{6} b c \int \frac {e x^2+3 d}{\sqrt {c^2 x^2+1}}dx^2+d x (a+b \text {arcsinh}(c x))+\frac {1}{3} e x^3 (a+b \text {arcsinh}(c x))\) |
\(\Big \downarrow \) 53 |
\(\displaystyle -\frac {1}{6} b c \int \left (\frac {3 c^2 d-e}{c^2 \sqrt {c^2 x^2+1}}+\frac {e \sqrt {c^2 x^2+1}}{c^2}\right )dx^2+d x (a+b \text {arcsinh}(c x))+\frac {1}{3} e x^3 (a+b \text {arcsinh}(c x))\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle d x (a+b \text {arcsinh}(c x))+\frac {1}{3} e x^3 (a+b \text {arcsinh}(c x))-\frac {1}{6} b c \left (\frac {2 \sqrt {c^2 x^2+1} \left (3 c^2 d-e\right )}{c^4}+\frac {2 e \left (c^2 x^2+1\right )^{3/2}}{3 c^4}\right )\) |
Input:
Int[(d + e*x^2)*(a + b*ArcSinh[c*x]),x]
Output:
-1/6*(b*c*((2*(3*c^2*d - e)*Sqrt[1 + c^2*x^2])/c^4 + (2*e*(1 + c^2*x^2)^(3 /2))/(3*c^4))) + d*x*(a + b*ArcSinh[c*x]) + (e*x^3*(a + b*ArcSinh[c*x]))/3
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_))^(m_.)*((c_.) + (d_.)*(x_))^(n_.), x_Symbol] :> Int [ExpandIntegrand[(a + b*x)^m*(c + d*x)^n, x], x] /; FreeQ[{a, b, c, d, n}, x] && IGtQ[m, 0] && ( !IntegerQ[n] || (EqQ[c, 0] && LeQ[7*m + 4*n + 4, 0]) || LtQ[9*m + 5*(n + 1), 0] || GtQ[m + n + 2, 0])
Int[(x_)*((a_) + (b_.)*(x_)^2)^(p_.)*((c_) + (d_.)*(x_)^2)^(q_.), x_Symbol] :> Simp[1/2 Subst[Int[(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]
Int[((a_.) + ArcSinh[(c_.)*(x_)]*(b_.))*((d_) + (e_.)*(x_)^2)^(p_.), x_Symb ol] :> With[{u = IntHide[(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}, x] && NeQ[e, c^2*d] && (IGtQ[p, 0] || ILtQ[p + 1/2, 0])
Time = 0.36 (sec) , antiderivative size = 97, normalized size of antiderivative = 1.20
method | result | size |
parts | \(a \left (\frac {1}{3} x^{3} e +d x \right )+\frac {b \left (\frac {c \,\operatorname {arcsinh}\left (x c \right ) x^{3} e}{3}+\operatorname {arcsinh}\left (x c \right ) x c d -\frac {e \left (\frac {x^{2} c^{2} \sqrt {c^{2} x^{2}+1}}{3}-\frac {2 \sqrt {c^{2} x^{2}+1}}{3}\right )+3 d \,c^{2} \sqrt {c^{2} x^{2}+1}}{3 c^{2}}\right )}{c}\) | \(97\) |
derivativedivides | \(\frac {\frac {a \left (c^{3} d x +\frac {1}{3} e \,x^{3} c^{3}\right )}{c^{2}}+\frac {b \left (\operatorname {arcsinh}\left (x c \right ) x \,c^{3} d +\frac {\operatorname {arcsinh}\left (x c \right ) e \,x^{3} c^{3}}{3}-\frac {e \left (\frac {x^{2} c^{2} \sqrt {c^{2} x^{2}+1}}{3}-\frac {2 \sqrt {c^{2} x^{2}+1}}{3}\right )}{3}-d \,c^{2} \sqrt {c^{2} x^{2}+1}\right )}{c^{2}}}{c}\) | \(109\) |
default | \(\frac {\frac {a \left (c^{3} d x +\frac {1}{3} e \,x^{3} c^{3}\right )}{c^{2}}+\frac {b \left (\operatorname {arcsinh}\left (x c \right ) x \,c^{3} d +\frac {\operatorname {arcsinh}\left (x c \right ) e \,x^{3} c^{3}}{3}-\frac {e \left (\frac {x^{2} c^{2} \sqrt {c^{2} x^{2}+1}}{3}-\frac {2 \sqrt {c^{2} x^{2}+1}}{3}\right )}{3}-d \,c^{2} \sqrt {c^{2} x^{2}+1}\right )}{c^{2}}}{c}\) | \(109\) |
orering | \(\frac {x \left (5 e^{2} x^{4} c^{4}+30 c^{4} d e \,x^{2}+9 c^{4} d^{2}-2 c^{2} e^{2} x^{2}+18 c^{2} d e -4 e^{2}\right ) \left (a +b \,\operatorname {arcsinh}\left (x c \right )\right )}{9 c^{4} \left (e \,x^{2}+d \right )}-\frac {\left (c^{2} e \,x^{2}+9 c^{2} d -2 e \right ) \left (c^{2} x^{2}+1\right ) \left (2 e x \left (a +b \,\operatorname {arcsinh}\left (x c \right )\right )+\frac {\left (e \,x^{2}+d \right ) b c}{\sqrt {c^{2} x^{2}+1}}\right )}{9 c^{4} \left (e \,x^{2}+d \right )}\) | \(153\) |
Input:
int((e*x^2+d)*(a+b*arcsinh(x*c)),x,method=_RETURNVERBOSE)
Output:
a*(1/3*x^3*e+d*x)+b/c*(1/3*c*arcsinh(x*c)*x^3*e+arcsinh(x*c)*x*c*d-1/3/c^2 *(e*(1/3*x^2*c^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 = 0.09 (sec) , antiderivative size = 94, normalized size of antiderivative = 1.16 \[ \int \left (d+e x^2\right ) (a+b \text {arcsinh}(c x)) \, dx=\frac {3 \, a c^{3} e x^{3} + 9 \, a c^{3} d x + 3 \, {\left (b c^{3} e x^{3} + 3 \, b c^{3} d x\right )} \log \left (c x + \sqrt {c^{2} x^{2} + 1}\right ) - {\left (b c^{2} e x^{2} + 9 \, b c^{2} d - 2 \, b e\right )} \sqrt {c^{2} x^{2} + 1}}{9 \, c^{3}} \] Input:
integrate((e*x^2+d)*(a+b*arcsinh(c*x)),x, algorithm="fricas")
Output:
1/9*(3*a*c^3*e*x^3 + 9*a*c^3*d*x + 3*(b*c^3*e*x^3 + 3*b*c^3*d*x)*log(c*x + sqrt(c^2*x^2 + 1)) - (b*c^2*e*x^2 + 9*b*c^2*d - 2*b*e)*sqrt(c^2*x^2 + 1)) /c^3
Time = 0.21 (sec) , antiderivative size = 109, normalized size of antiderivative = 1.35 \[ \int \left (d+e x^2\right ) (a+b \text {arcsinh}(c x)) \, dx=\begin {cases} a d x + \frac {a e x^{3}}{3} + b d x \operatorname {asinh}{\left (c x \right )} + \frac {b e x^{3} \operatorname {asinh}{\left (c x \right )}}{3} - \frac {b d \sqrt {c^{2} x^{2} + 1}}{c} - \frac {b e x^{2} \sqrt {c^{2} x^{2} + 1}}{9 c} + \frac {2 b e \sqrt {c^{2} x^{2} + 1}}{9 c^{3}} & \text {for}\: c \neq 0 \\a \left (d x + \frac {e x^{3}}{3}\right ) & \text {otherwise} \end {cases} \] Input:
integrate((e*x**2+d)*(a+b*asinh(c*x)),x)
Output:
Piecewise((a*d*x + a*e*x**3/3 + b*d*x*asinh(c*x) + b*e*x**3*asinh(c*x)/3 - b*d*sqrt(c**2*x**2 + 1)/c - b*e*x**2*sqrt(c**2*x**2 + 1)/(9*c) + 2*b*e*sq rt(c**2*x**2 + 1)/(9*c**3), Ne(c, 0)), (a*(d*x + e*x**3/3), True))
Time = 0.03 (sec) , antiderivative size = 91, normalized size of antiderivative = 1.12 \[ \int \left (d+e x^2\right ) (a+b \text {arcsinh}(c x)) \, dx=\frac {1}{3} \, a e x^{3} + \frac {1}{9} \, {\left (3 \, x^{3} \operatorname {arsinh}\left (c x\right ) - c {\left (\frac {\sqrt {c^{2} x^{2} + 1} x^{2}}{c^{2}} - \frac {2 \, \sqrt {c^{2} x^{2} + 1}}{c^{4}}\right )}\right )} b e + a d x + \frac {{\left (c x \operatorname {arsinh}\left (c x\right ) - \sqrt {c^{2} x^{2} + 1}\right )} b d}{c} \] Input:
integrate((e*x^2+d)*(a+b*arcsinh(c*x)),x, algorithm="maxima")
Output:
1/3*a*e*x^3 + 1/9*(3*x^3*arcsinh(c*x) - c*(sqrt(c^2*x^2 + 1)*x^2/c^2 - 2*s qrt(c^2*x^2 + 1)/c^4))*b*e + a*d*x + (c*x*arcsinh(c*x) - sqrt(c^2*x^2 + 1) )*b*d/c
Exception generated. \[ \int \left (d+e x^2\right ) (a+b \text {arcsinh}(c x)) \, dx=\text {Exception raised: RuntimeError} \] Input:
integrate((e*x^2+d)*(a+b*arcsinh(c*x)),x, algorithm="giac")
Output:
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. \[ \int \left (d+e x^2\right ) (a+b \text {arcsinh}(c x)) \, dx=\int \left (a+b\,\mathrm {asinh}\left (c\,x\right )\right )\,\left (e\,x^2+d\right ) \,d x \] Input:
int((a + b*asinh(c*x))*(d + e*x^2),x)
Output:
int((a + b*asinh(c*x))*(d + e*x^2), x)
Time = 0.19 (sec) , antiderivative size = 101, normalized size of antiderivative = 1.25 \[ \int \left (d+e x^2\right ) (a+b \text {arcsinh}(c x)) \, dx=\frac {9 \mathit {asinh} \left (c x \right ) b \,c^{3} d x +3 \mathit {asinh} \left (c x \right ) b \,c^{3} e \,x^{3}-9 \sqrt {c^{2} x^{2}+1}\, b \,c^{2} d -\sqrt {c^{2} x^{2}+1}\, b \,c^{2} e \,x^{2}+2 \sqrt {c^{2} x^{2}+1}\, b e +9 a \,c^{3} d x +3 a \,c^{3} e \,x^{3}}{9 c^{3}} \] Input:
int((e*x^2+d)*(a+b*asinh(c*x)),x)
Output:
(9*asinh(c*x)*b*c**3*d*x + 3*asinh(c*x)*b*c**3*e*x**3 - 9*sqrt(c**2*x**2 + 1)*b*c**2*d - sqrt(c**2*x**2 + 1)*b*c**2*e*x**2 + 2*sqrt(c**2*x**2 + 1)*b *e + 9*a*c**3*d*x + 3*a*c**3*e*x**3)/(9*c**3)