\(\int (d+e x)^2 (a+b \text {arcsinh}(c x)) \, dx\) [5]

Optimal result

Integrand size = 16, antiderivative size = 133 \[ \int (d+e x)^2 (a+b \text {arcsinh}(c x)) \, dx=-\frac {b \left (3 c^2 d^2-e^2\right ) \sqrt {1+c^2 x^2}}{3 c^3}-\frac {b d e x \sqrt {1+c^2 x^2}}{2 c}-\frac {b e^2 \left (1+c^2 x^2\right )^{3/2}}{9 c^3}-\frac {b d \left (2 d^2-\frac {3 e^2}{c^2}\right ) \text {arcsinh}(c x)}{6 e}+\frac {(d+e x)^3 (a+b \text {arcsinh}(c x))}{3 e} \] Output:

-1/3*b*(3*c^2*d^2-e^2)*(c^2*x^2+1)^(1/2)/c^3-1/2*b*d*e*x*(c^2*x^2+1)^(1/2) 
/c-1/9*b*e^2*(c^2*x^2+1)^(3/2)/c^3-1/6*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
 

Mathematica [A] (verified)

Time = 0.07 (sec) , antiderivative size = 121, normalized size of antiderivative = 0.91 \[ \int (d+e x)^2 (a+b \text {arcsinh}(c x)) \, dx=\frac {6 a c^3 x \left (3 d^2+3 d e x+e^2 x^2\right )-b \sqrt {1+c^2 x^2} \left (-4 e^2+c^2 \left (18 d^2+9 d e x+2 e^2 x^2\right )\right )+3 b c \left (6 c^2 d^2 x+2 c^2 e^2 x^3+3 d \left (e+2 c^2 e x^2\right )\right ) \text {arcsinh}(c x)}{18 c^3} \] Input:

Integrate[(d + e*x)^2*(a + b*ArcSinh[c*x]),x]
 

Output:

(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)
 

Rubi [A] (verified)

Time = 0.52 (sec) , antiderivative size = 147, normalized size of antiderivative = 1.11, 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

\(\displaystyle \frac {(d+e x)^3 (a+b \text {arcsinh}(c x))}{3 e}-\frac {b c \left (\frac {\frac {3 d \text {arcsinh}(c x) \left (2 c^2 d^2-3 e^2\right )}{2 c}+2 e \sqrt {c^2 x^2+1} \left (4 d^2-\frac {e^2}{c^2}\right )+\frac {5}{2} d e^2 x \sqrt {c^2 x^2+1}}{3 c^2}+\frac {e \sqrt {c^2 x^2+1} (d+e x)^2}{3 c^2}\right )}{3 e}\)

Input:

Int[(d + e*x)^2*(a + b*ArcSinh[c*x]),x]
 

Output:

((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)
 

Defintions of rubi rules used

rule 222

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]
 

rule 497

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]
 

rule 676

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]
 

rule 6243

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]
 

Maple [A] (verified)

Time = 0.44 (sec) , antiderivative size = 174, normalized size of antiderivative = 1.31

method	result	size
parts	\(\frac {a \left (e x +d \right )^{3}}{3 e}+\frac {b \left (\frac {c \,e^{2} \operatorname {arcsinh}\left (x c \right ) x^{3}}{3}+c e \,\operatorname {arcsinh}\left (x c \right ) x^{2} d +\operatorname {arcsinh}\left (x c \right ) x c \,d^{2}+\frac {c \,\operatorname {arcsinh}\left (x c \right ) d^{3}}{3 e}-\frac {c^{3} d^{3} \operatorname {arcsinh}\left (x c \right )+e^{3} \left (\frac {x^{2} c^{2} \sqrt {c^{2} x^{2}+1}}{3}-\frac {2 \sqrt {c^{2} x^{2}+1}}{3}\right )+3 d c \,e^{2} \left (\frac {\sqrt {c^{2} x^{2}+1}\, x c}{2}-\frac {\operatorname {arcsinh}\left (x c \right )}{2}\right )+3 d^{2} c^{2} e \sqrt {c^{2} x^{2}+1}}{3 c^{2} e}\right )}{c}\)	\(174\)
derivativedivides	\(\frac {\frac {a \left (c e x +c d \right )^{3}}{3 c^{2} e}+\frac {b \left (\frac {\operatorname {arcsinh}\left (x c \right ) c^{3} d^{3}}{3 e}+\operatorname {arcsinh}\left (x c \right ) c^{3} d^{2} x +e \,\operatorname {arcsinh}\left (x c \right ) c^{3} d \,x^{2}+\frac {e^{2} \operatorname {arcsinh}\left (x c \right ) x^{3} c^{3}}{3}-\frac {c^{3} d^{3} \operatorname {arcsinh}\left (x c \right )+e^{3} \left (\frac {x^{2} c^{2} \sqrt {c^{2} x^{2}+1}}{3}-\frac {2 \sqrt {c^{2} x^{2}+1}}{3}\right )+3 d c \,e^{2} \left (\frac {\sqrt {c^{2} x^{2}+1}\, x c}{2}-\frac {\operatorname {arcsinh}\left (x c \right )}{2}\right )+3 d^{2} c^{2} e \sqrt {c^{2} x^{2}+1}}{3 e}\right )}{c^{2}}}{c}\)	\(189\)
default	\(\frac {\frac {a \left (c e x +c d \right )^{3}}{3 c^{2} e}+\frac {b \left (\frac {\operatorname {arcsinh}\left (x c \right ) c^{3} d^{3}}{3 e}+\operatorname {arcsinh}\left (x c \right ) c^{3} d^{2} x +e \,\operatorname {arcsinh}\left (x c \right ) c^{3} d \,x^{2}+\frac {e^{2} \operatorname {arcsinh}\left (x c \right ) x^{3} c^{3}}{3}-\frac {c^{3} d^{3} \operatorname {arcsinh}\left (x c \right )+e^{3} \left (\frac {x^{2} c^{2} \sqrt {c^{2} x^{2}+1}}{3}-\frac {2 \sqrt {c^{2} x^{2}+1}}{3}\right )+3 d c \,e^{2} \left (\frac {\sqrt {c^{2} x^{2}+1}\, x c}{2}-\frac {\operatorname {arcsinh}\left (x c \right )}{2}\right )+3 d^{2} c^{2} e \sqrt {c^{2} x^{2}+1}}{3 e}\right )}{c^{2}}}{c}\)	\(189\)
orering	\(\frac {\left (10 c^{4} e^{3} x^{4}+42 c^{4} d \,e^{2} x^{3}+72 c^{4} d^{2} e \,x^{2}+18 c^{4} d^{3} x -4 c^{2} e^{3} x^{2}+27 c^{2} d \,e^{2} x +45 c^{2} d^{2} e -8 e^{3}\right ) \left (a +b \,\operatorname {arcsinh}\left (x c \right )\right )}{18 c^{4} \left (e x +d \right )}-\frac {\left (2 c^{2} e^{2} x^{2}+9 c^{2} d e x +18 c^{2} d^{2}-4 e^{2}\right ) \left (c^{2} x^{2}+1\right ) \left (2 \left (e x +d \right ) \left (a +b \,\operatorname {arcsinh}\left (x c \right )\right ) e +\frac {\left (e x +d \right )^{2} b c}{\sqrt {c^{2} x^{2}+1}}\right )}{18 c^{4} \left (e x +d \right )^{2}}\)	\(194\)

Input:

int((e*x+d)^2*(a+b*arcsinh(x*c)),x,method=_RETURNVERBOSE)
 

Output:

1/3*a*(e*x+d)^3/e+b/c*(1/3*c*e^2*arcsinh(x*c)*x^3+c*e*arcsinh(x*c)*x^2*d+a 
rcsinh(x*c)*x*c*d^2+1/3*c/e*arcsinh(x*c)*d^3-1/3/c^2/e*(c^3*d^3*arcsinh(x* 
c)+e^3*(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*e^2*(1/ 
2*(c^2*x^2+1)^(1/2)*x*c-1/2*arcsinh(x*c))+3*d^2*c^2*e*(c^2*x^2+1)^(1/2)))
 

Fricas [A] (verification not implemented)

Time = 0.09 (sec) , antiderivative size = 147, normalized size of antiderivative = 1.11 \[ \int (d+e x)^2 (a+b \text {arcsinh}(c x)) \, dx=\frac {6 \, a c^{3} e^{2} x^{3} + 18 \, a c^{3} d e x^{2} + 18 \, a c^{3} d^{2} x + 3 \, {\left (2 \, b c^{3} e^{2} x^{3} + 6 \, b c^{3} d e x^{2} + 6 \, b c^{3} d^{2} x + 3 \, b c d e\right )} \log \left (c x + \sqrt {c^{2} x^{2} + 1}\right ) - {\left (2 \, b c^{2} e^{2} x^{2} + 9 \, b c^{2} d e x + 18 \, b c^{2} d^{2} - 4 \, b e^{2}\right )} \sqrt {c^{2} x^{2} + 1}}{18 \, c^{3}} \] Input:

integrate((e*x+d)^2*(a+b*arcsinh(c*x)),x, algorithm="fricas")
 

Output:

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
 

Sympy [A] (verification not implemented)

Time = 0.25 (sec) , antiderivative size = 190, normalized size of antiderivative = 1.43 \[ \int (d+e x)^2 (a+b \text {arcsinh}(c x)) \, dx=\begin {cases} a d^{2} x + a d e x^{2} + \frac {a e^{2} x^{3}}{3} + b d^{2} x \operatorname {asinh}{\left (c x \right )} + b d e x^{2} \operatorname {asinh}{\left (c x \right )} + \frac {b e^{2} x^{3} \operatorname {asinh}{\left (c x \right )}}{3} - \frac {b d^{2} \sqrt {c^{2} x^{2} + 1}}{c} - \frac {b d e x \sqrt {c^{2} x^{2} + 1}}{2 c} - \frac {b e^{2} x^{2} \sqrt {c^{2} x^{2} + 1}}{9 c} + \frac {b d e \operatorname {asinh}{\left (c x \right )}}{2 c^{2}} + \frac {2 b e^{2} \sqrt {c^{2} x^{2} + 1}}{9 c^{3}} & \text {for}\: c \neq 0 \\a \left (d^{2} x + d e x^{2} + \frac {e^{2} x^{3}}{3}\right ) & \text {otherwise} \end {cases} \] Input:

integrate((e*x+d)**2*(a+b*asinh(c*x)),x)
 

Output:

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))
 

Maxima [A] (verification not implemented)

Time = 0.04 (sec) , antiderivative size = 150, normalized size of antiderivative = 1.13 \[ \int (d+e x)^2 (a+b \text {arcsinh}(c x)) \, dx=\frac {1}{3} \, a e^{2} x^{3} + a d e x^{2} + \frac {1}{2} \, {\left (2 \, x^{2} \operatorname {arsinh}\left (c x\right ) - c {\left (\frac {\sqrt {c^{2} x^{2} + 1} x}{c^{2}} - \frac {\operatorname {arsinh}\left (c x\right )}{c^{3}}\right )}\right )} b d e + \frac {1}{9} \, {\left (3 \, x^{3} \operatorname {arsinh}\left (c x\right ) - c {\left (\frac {\sqrt {c^{2} x^{2} + 1} x^{2}}{c^{2}} - \frac {2 \, \sqrt {c^{2} x^{2} + 1}}{c^{4}}\right )}\right )} b e^{2} + a d^{2} x + \frac {{\left (c x \operatorname {arsinh}\left (c x\right ) - \sqrt {c^{2} x^{2} + 1}\right )} b d^{2}}{c} \] Input:

integrate((e*x+d)^2*(a+b*arcsinh(c*x)),x, algorithm="maxima")
 

Output:

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
 

Giac [F(-2)]

Exception generated. \[ \int (d+e x)^2 (a+b \text {arcsinh}(c x)) \, dx=\text {Exception raised: RuntimeError} \] Input:

integrate((e*x+d)^2*(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
 

Mupad [F(-1)]

Timed out. \[ \int (d+e x)^2 (a+b \text {arcsinh}(c x)) \, dx=\int \left (a+b\,\mathrm {asinh}\left (c\,x\right )\right )\,{\left (d+e\,x\right )}^2 \,d x \] Input:

int((a + b*asinh(c*x))*(d + e*x)^2,x)
 

Output:

int((a + b*asinh(c*x))*(d + e*x)^2, x)
 

Reduce [B] (verification not implemented)

Time = 0.17 (sec) , antiderivative size = 181, normalized size of antiderivative = 1.36 \[ \int (d+e x)^2 (a+b \text {arcsinh}(c x)) \, dx=\frac {18 \mathit {asinh} \left (c x \right ) b \,c^{3} d^{2} x +18 \mathit {asinh} \left (c x \right ) b \,c^{3} d e \,x^{2}+6 \mathit {asinh} \left (c x \right ) b \,c^{3} e^{2} x^{3}-18 \sqrt {c^{2} x^{2}+1}\, b \,c^{2} d^{2}-9 \sqrt {c^{2} x^{2}+1}\, b \,c^{2} d e x -2 \sqrt {c^{2} x^{2}+1}\, b \,c^{2} e^{2} x^{2}+4 \sqrt {c^{2} x^{2}+1}\, b \,e^{2}+9 \,\mathrm {log}\left (\sqrt {c^{2} x^{2}+1}+c x \right ) b c d e +18 a \,c^{3} d^{2} x +18 a \,c^{3} d e \,x^{2}+6 a \,c^{3} e^{2} x^{3}}{18 c^{3}} \] Input:

int((e*x+d)^2*(a+b*asinh(c*x)),x)
 

Output:

(18*asinh(c*x)*b*c**3*d**2*x + 18*asinh(c*x)*b*c**3*d*e*x**2 + 6*asinh(c*x 
)*b*c**3*e**2*x**3 - 18*sqrt(c**2*x**2 + 1)*b*c**2*d**2 - 9*sqrt(c**2*x**2 
 + 1)*b*c**2*d*e*x - 2*sqrt(c**2*x**2 + 1)*b*c**2*e**2*x**2 + 4*sqrt(c**2* 
x**2 + 1)*b*e**2 + 9*log(sqrt(c**2*x**2 + 1) + c*x)*b*c*d*e + 18*a*c**3*d* 
*2*x + 18*a*c**3*d*e*x**2 + 6*a*c**3*e**2*x**3)/(18*c**3)