\(\int (d+e x) (a+b \text {arcsinh}(c x)) \, dx\) [6]

Optimal result

Integrand size = 14, antiderivative size = 92 \[ \int (d+e x) (a+b \text {arcsinh}(c x)) \, dx=-\frac {b d \sqrt {1+c^2 x^2}}{c}-\frac {b e x \sqrt {1+c^2 x^2}}{4 c}-\frac {b \left (2 d^2-\frac {e^2}{c^2}\right ) \text {arcsinh}(c x)}{4 e}+\frac {(d+e x)^2 (a+b \text {arcsinh}(c x))}{2 e} \] Output:

-b*d*(c^2*x^2+1)^(1/2)/c-1/4*b*e*x*(c^2*x^2+1)^(1/2)/c-1/4*b*(2*d^2-e^2/c^ 
2)*arcsinh(c*x)/e+1/2*(e*x+d)^2*(a+b*arcsinh(c*x))/e
 

Mathematica [A] (verified)

Time = 0.03 (sec) , antiderivative size = 91, normalized size of antiderivative = 0.99 \[ \int (d+e x) (a+b \text {arcsinh}(c x)) \, dx=a d x+\frac {1}{2} a e x^2-\frac {b d \sqrt {1+c^2 x^2}}{c}-\frac {b e x \sqrt {1+c^2 x^2}}{4 c}+\frac {b e \text {arcsinh}(c x)}{4 c^2}+b d x \text {arcsinh}(c x)+\frac {1}{2} b e x^2 \text {arcsinh}(c x) \] Input:

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

Output:

a*d*x + (a*e*x^2)/2 - (b*d*Sqrt[1 + c^2*x^2])/c - (b*e*x*Sqrt[1 + c^2*x^2] 
)/(4*c) + (b*e*ArcSinh[c*x])/(4*c^2) + b*d*x*ArcSinh[c*x] + (b*e*x^2*ArcSi 
nh[c*x])/2
 

Rubi [A] (verified)

Time = 0.34 (sec) , antiderivative size = 106, normalized size of antiderivative = 1.15, number of steps used = 4, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.286, Rules used = {6243, 497, 455, 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.

\(\displaystyle \int (d+e x) (a+b \text {arcsinh}(c x)) \, dx\)

\(\Big \downarrow \) 6243

\(\displaystyle \frac {(d+e x)^2 (a+b \text {arcsinh}(c x))}{2 e}-\frac {b c \int \frac {(d+e x)^2}{\sqrt {c^2 x^2+1}}dx}{2 e}\)

\(\Big \downarrow \) 497

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

\(\Big \downarrow \) 455

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

\(\Big \downarrow \) 222

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

Input:

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

Output:

((d + e*x)^2*(a + b*ArcSinh[c*x]))/(2*e) - (b*c*((e*(d + e*x)*Sqrt[1 + c^2 
*x^2])/(2*c^2) + (3*d*e*Sqrt[1 + c^2*x^2] + ((2*c^2*d^2 - e^2)*ArcSinh[c*x 
])/c)/(2*c^2)))/(2*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 455

Int[((c_) + (d_.)*(x_))*((a_) + (b_.)*(x_)^2)^(p_.), x_Symbol] :> Simp[d*(( 
a + b*x^2)^(p + 1)/(2*b*(p + 1))), x] + Simp[c   Int[(a + b*x^2)^p, x], x] 
/; FreeQ[{a, b, c, d, p}, x] &&  !LeQ[p, -1]
 

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 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 = 0.34 (sec) , antiderivative size = 84, normalized size of antiderivative = 0.91

method	result	size
parts	\(a \left (\frac {1}{2} e \,x^{2}+d x \right )+\frac {b \left (\frac {c \,\operatorname {arcsinh}\left (x c \right ) e \,x^{2}}{2}+\operatorname {arcsinh}\left (x c \right ) x c d -\frac {e \left (\frac {\sqrt {c^{2} x^{2}+1}\, x c}{2}-\frac {\operatorname {arcsinh}\left (x c \right )}{2}\right )+2 d c \sqrt {c^{2} x^{2}+1}}{2 c}\right )}{c}\)	\(84\)
derivativedivides	\(\frac {\frac {a \left (c^{2} d x +\frac {1}{2} c^{2} e \,x^{2}\right )}{c}+\frac {b \left (\operatorname {arcsinh}\left (x c \right ) x \,c^{2} d +\frac {\operatorname {arcsinh}\left (x c \right ) e \,x^{2} c^{2}}{2}-\frac {e \left (\frac {\sqrt {c^{2} x^{2}+1}\, x c}{2}-\frac {\operatorname {arcsinh}\left (x c \right )}{2}\right )}{2}-d c \sqrt {c^{2} x^{2}+1}\right )}{c}}{c}\)	\(96\)
default	\(\frac {\frac {a \left (c^{2} d x +\frac {1}{2} c^{2} e \,x^{2}\right )}{c}+\frac {b \left (\operatorname {arcsinh}\left (x c \right ) x \,c^{2} d +\frac {\operatorname {arcsinh}\left (x c \right ) e \,x^{2} c^{2}}{2}-\frac {e \left (\frac {\sqrt {c^{2} x^{2}+1}\, x c}{2}-\frac {\operatorname {arcsinh}\left (x c \right )}{2}\right )}{2}-d c \sqrt {c^{2} x^{2}+1}\right )}{c}}{c}\)	\(96\)
orering	\(\frac {\left (3 c^{2} e^{2} x^{3}+10 c^{2} d e \,x^{2}+4 c^{2} d^{2} x +2 e^{2} x +5 d e \right ) \left (a +b \,\operatorname {arcsinh}\left (x c \right )\right )}{4 c^{2} \left (e x +d \right )}-\frac {\left (e x +4 d \right ) \left (c^{2} x^{2}+1\right ) \left (e \left (a +b \,\operatorname {arcsinh}\left (x c \right )\right )+\frac {\left (e x +d \right ) b c}{\sqrt {c^{2} x^{2}+1}}\right )}{4 c^{2} \left (e x +d \right )}\)	\(121\)

Input:

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

Output:

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

Fricas [A] (verification not implemented)

Time = 0.10 (sec) , antiderivative size = 87, normalized size of antiderivative = 0.95 \[ \int (d+e x) (a+b \text {arcsinh}(c x)) \, dx=\frac {2 \, a c^{2} e x^{2} + 4 \, a c^{2} d x + {\left (2 \, b c^{2} e x^{2} + 4 \, b c^{2} d x + b e\right )} \log \left (c x + \sqrt {c^{2} x^{2} + 1}\right ) - {\left (b c e x + 4 \, b c d\right )} \sqrt {c^{2} x^{2} + 1}}{4 \, c^{2}} \] Input:

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

Output:

1/4*(2*a*c^2*e*x^2 + 4*a*c^2*d*x + (2*b*c^2*e*x^2 + 4*b*c^2*d*x + b*e)*log 
(c*x + sqrt(c^2*x^2 + 1)) - (b*c*e*x + 4*b*c*d)*sqrt(c^2*x^2 + 1))/c^2
 

Sympy [A] (verification not implemented)

Time = 0.18 (sec) , antiderivative size = 99, normalized size of antiderivative = 1.08 \[ \int (d+e x) (a+b \text {arcsinh}(c x)) \, dx=\begin {cases} a d x + \frac {a e x^{2}}{2} + b d x \operatorname {asinh}{\left (c x \right )} + \frac {b e x^{2} \operatorname {asinh}{\left (c x \right )}}{2} - \frac {b d \sqrt {c^{2} x^{2} + 1}}{c} - \frac {b e x \sqrt {c^{2} x^{2} + 1}}{4 c} + \frac {b e \operatorname {asinh}{\left (c x \right )}}{4 c^{2}} & \text {for}\: c \neq 0 \\a \left (d x + \frac {e x^{2}}{2}\right ) & \text {otherwise} \end {cases} \] Input:

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

Output:

Piecewise((a*d*x + a*e*x**2/2 + b*d*x*asinh(c*x) + b*e*x**2*asinh(c*x)/2 - 
 b*d*sqrt(c**2*x**2 + 1)/c - b*e*x*sqrt(c**2*x**2 + 1)/(4*c) + b*e*asinh(c 
*x)/(4*c**2), Ne(c, 0)), (a*(d*x + e*x**2/2), True))
 

Maxima [A] (verification not implemented)

Time = 0.03 (sec) , antiderivative size = 82, normalized size of antiderivative = 0.89 \[ \int (d+e x) (a+b \text {arcsinh}(c x)) \, dx=\frac {1}{2} \, a e x^{2} + \frac {1}{4} \, {\left (2 \, x^{2} \operatorname {arsinh}\left (c x\right ) - c {\left (\frac {\sqrt {c^{2} x^{2} + 1} x}{c^{2}} - \frac {\operatorname {arsinh}\left (c x\right )}{c^{3}}\right )}\right )} b e + a d x + \frac {{\left (c x \operatorname {arsinh}\left (c x\right ) - \sqrt {c^{2} x^{2} + 1}\right )} b d}{c} \] Input:

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

Output:

1/2*a*e*x^2 + 1/4*(2*x^2*arcsinh(c*x) - c*(sqrt(c^2*x^2 + 1)*x/c^2 - arcsi 
nh(c*x)/c^3))*b*e + a*d*x + (c*x*arcsinh(c*x) - sqrt(c^2*x^2 + 1))*b*d/c
 

Giac [A] (verification not implemented)

Time = 0.18 (sec) , antiderivative size = 122, normalized size of antiderivative = 1.33 \[ \int (d+e x) (a+b \text {arcsinh}(c x)) \, dx=\frac {1}{2} \, a e x^{2} + {\left (x \log \left (c x + \sqrt {c^{2} x^{2} + 1}\right ) - \frac {\sqrt {c^{2} x^{2} + 1}}{c}\right )} b d + \frac {1}{4} \, {\left (2 \, x^{2} \log \left (c x + \sqrt {c^{2} x^{2} + 1}\right ) - c {\left (\frac {\sqrt {c^{2} x^{2} + 1} x}{c^{2}} + \frac {\log \left (-x {\left | c \right |} + \sqrt {c^{2} x^{2} + 1}\right )}{c^{2} {\left | c \right |}}\right )}\right )} b e + a d x \] Input:

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

Output:

1/2*a*e*x^2 + (x*log(c*x + sqrt(c^2*x^2 + 1)) - sqrt(c^2*x^2 + 1)/c)*b*d + 
 1/4*(2*x^2*log(c*x + sqrt(c^2*x^2 + 1)) - c*(sqrt(c^2*x^2 + 1)*x/c^2 + lo 
g(-x*abs(c) + sqrt(c^2*x^2 + 1))/(c^2*abs(c))))*b*e + a*d*x
 

Mupad [B] (verification not implemented)

Time = 2.91 (sec) , antiderivative size = 78, normalized size of antiderivative = 0.85 \[ \int (d+e x) (a+b \text {arcsinh}(c x)) \, dx=\frac {a\,x\,\left (2\,d+e\,x\right )}{2}-\frac {b\,d\,\left (\sqrt {c^2\,x^2+1}-c\,x\,\mathrm {asinh}\left (c\,x\right )\right )}{c}-\frac {b\,e\,x\,\sqrt {c^2\,x^2+1}}{4\,c}+b\,e\,x\,\mathrm {asinh}\left (c\,x\right )\,\left (\frac {x}{2}+\frac {1}{4\,c^2\,x}\right ) \] Input:

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

Output:

(a*x*(2*d + e*x))/2 - (b*d*((c^2*x^2 + 1)^(1/2) - c*x*asinh(c*x)))/c - (b* 
e*x*(c^2*x^2 + 1)^(1/2))/(4*c) + b*e*x*asinh(c*x)*(x/2 + 1/(4*c^2*x))
 

Reduce [B] (verification not implemented)

Time = 0.17 (sec) , antiderivative size = 108, normalized size of antiderivative = 1.17 \[ \int (d+e x) (a+b \text {arcsinh}(c x)) \, dx=\frac {4 \mathit {asinh} \left (c x \right ) b \,c^{2} d x +2 \mathit {asinh} \left (c x \right ) b \,c^{2} e \,x^{2}+2 \mathit {asinh} \left (c x \right ) b e -4 \sqrt {c^{2} x^{2}+1}\, b c d -\sqrt {c^{2} x^{2}+1}\, b c e x -\mathrm {log}\left (\sqrt {c^{2} x^{2}+1}+c x \right ) b e +4 a \,c^{2} d x +2 a \,c^{2} e \,x^{2}}{4 c^{2}} \] Input:

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

Output:

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