\(\int (d+c^2 d x^2)^3 (a+b \text {arcsinh}(c x)) \, dx\) [23]

Optimal result

Integrand size = 21, antiderivative size = 170 \[ \int \left (d+c^2 d x^2\right )^3 (a+b \text {arcsinh}(c x)) \, dx=-\frac {16 b d^3 \sqrt {1+c^2 x^2}}{35 c}-\frac {8 b d^3 \left (1+c^2 x^2\right )^{3/2}}{105 c}-\frac {6 b d^3 \left (1+c^2 x^2\right )^{5/2}}{175 c}-\frac {b d^3 \left (1+c^2 x^2\right )^{7/2}}{49 c}+d^3 x (a+b \text {arcsinh}(c x))+c^2 d^3 x^3 (a+b \text {arcsinh}(c x))+\frac {3}{5} c^4 d^3 x^5 (a+b \text {arcsinh}(c x))+\frac {1}{7} c^6 d^3 x^7 (a+b \text {arcsinh}(c x)) \] Output:

-16/35*b*d^3*(c^2*x^2+1)^(1/2)/c-8/105*b*d^3*(c^2*x^2+1)^(3/2)/c-6/175*b*d 
^3*(c^2*x^2+1)^(5/2)/c-1/49*b*d^3*(c^2*x^2+1)^(7/2)/c+d^3*x*(a+b*arcsinh(c 
*x))+c^2*d^3*x^3*(a+b*arcsinh(c*x))+3/5*c^4*d^3*x^5*(a+b*arcsinh(c*x))+1/7 
*c^6*d^3*x^7*(a+b*arcsinh(c*x))
 

Mathematica [A] (verified)

Time = 0.06 (sec) , antiderivative size = 119, normalized size of antiderivative = 0.70 \[ \int \left (d+c^2 d x^2\right )^3 (a+b \text {arcsinh}(c x)) \, dx=\frac {d^3 \left (105 a c x \left (35+35 c^2 x^2+21 c^4 x^4+5 c^6 x^6\right )-b \sqrt {1+c^2 x^2} \left (2161+757 c^2 x^2+351 c^4 x^4+75 c^6 x^6\right )+105 b c x \left (35+35 c^2 x^2+21 c^4 x^4+5 c^6 x^6\right ) \text {arcsinh}(c x)\right )}{3675 c} \] Input:

Integrate[(d + c^2*d*x^2)^3*(a + b*ArcSinh[c*x]),x]
 

Output:

(d^3*(105*a*c*x*(35 + 35*c^2*x^2 + 21*c^4*x^4 + 5*c^6*x^6) - b*Sqrt[1 + c^ 
2*x^2]*(2161 + 757*c^2*x^2 + 351*c^4*x^4 + 75*c^6*x^6) + 105*b*c*x*(35 + 3 
5*c^2*x^2 + 21*c^4*x^4 + 5*c^6*x^6)*ArcSinh[c*x]))/(3675*c)
 

Rubi [A] (verified)

Time = 0.76 (sec) , antiderivative size = 162, normalized size of antiderivative = 0.95, number of steps used = 6, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.238, Rules used = {6199, 27, 2331, 2389, 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.

\(\displaystyle \int \left (c^2 d x^2+d\right )^3 (a+b \text {arcsinh}(c x)) \, dx\)

\(\Big \downarrow \) 6199

\(\displaystyle -b c \int \frac {d^3 x \left (5 c^6 x^6+21 c^4 x^4+35 c^2 x^2+35\right )}{35 \sqrt {c^2 x^2+1}}dx+\frac {1}{7} c^6 d^3 x^7 (a+b \text {arcsinh}(c x))+\frac {3}{5} c^4 d^3 x^5 (a+b \text {arcsinh}(c x))+c^2 d^3 x^3 (a+b \text {arcsinh}(c x))+d^3 x (a+b \text {arcsinh}(c x))\)

\(\Big \downarrow \) 27

\(\displaystyle -\frac {1}{35} b c d^3 \int \frac {x \left (5 c^6 x^6+21 c^4 x^4+35 c^2 x^2+35\right )}{\sqrt {c^2 x^2+1}}dx+\frac {1}{7} c^6 d^3 x^7 (a+b \text {arcsinh}(c x))+\frac {3}{5} c^4 d^3 x^5 (a+b \text {arcsinh}(c x))+c^2 d^3 x^3 (a+b \text {arcsinh}(c x))+d^3 x (a+b \text {arcsinh}(c x))\)

\(\Big \downarrow \) 2331

\(\displaystyle -\frac {1}{70} b c d^3 \int \frac {5 c^6 x^6+21 c^4 x^4+35 c^2 x^2+35}{\sqrt {c^2 x^2+1}}dx^2+\frac {1}{7} c^6 d^3 x^7 (a+b \text {arcsinh}(c x))+\frac {3}{5} c^4 d^3 x^5 (a+b \text {arcsinh}(c x))+c^2 d^3 x^3 (a+b \text {arcsinh}(c x))+d^3 x (a+b \text {arcsinh}(c x))\)

\(\Big \downarrow \) 2389

\(\displaystyle -\frac {1}{70} b c d^3 \int \left (5 \left (c^2 x^2+1\right )^{5/2}+6 \left (c^2 x^2+1\right )^{3/2}+8 \sqrt {c^2 x^2+1}+\frac {16}{\sqrt {c^2 x^2+1}}\right )dx^2+\frac {1}{7} c^6 d^3 x^7 (a+b \text {arcsinh}(c x))+\frac {3}{5} c^4 d^3 x^5 (a+b \text {arcsinh}(c x))+c^2 d^3 x^3 (a+b \text {arcsinh}(c x))+d^3 x (a+b \text {arcsinh}(c x))\)

\(\Big \downarrow \) 2009

\(\displaystyle \frac {1}{7} c^6 d^3 x^7 (a+b \text {arcsinh}(c x))+\frac {3}{5} c^4 d^3 x^5 (a+b \text {arcsinh}(c x))+c^2 d^3 x^3 (a+b \text {arcsinh}(c x))+d^3 x (a+b \text {arcsinh}(c x))-\frac {1}{70} b c d^3 \left (\frac {10 \left (c^2 x^2+1\right )^{7/2}}{7 c^2}+\frac {12 \left (c^2 x^2+1\right )^{5/2}}{5 c^2}+\frac {16 \left (c^2 x^2+1\right )^{3/2}}{3 c^2}+\frac {32 \sqrt {c^2 x^2+1}}{c^2}\right )\)

Input:

Int[(d + c^2*d*x^2)^3*(a + b*ArcSinh[c*x]),x]
 

Output:

-1/70*(b*c*d^3*((32*Sqrt[1 + c^2*x^2])/c^2 + (16*(1 + c^2*x^2)^(3/2))/(3*c 
^2) + (12*(1 + c^2*x^2)^(5/2))/(5*c^2) + (10*(1 + c^2*x^2)^(7/2))/(7*c^2)) 
) + d^3*x*(a + b*ArcSinh[c*x]) + c^2*d^3*x^3*(a + b*ArcSinh[c*x]) + (3*c^4 
*d^3*x^5*(a + b*ArcSinh[c*x]))/5 + (c^6*d^3*x^7*(a + b*ArcSinh[c*x]))/7
 

Defintions of rubi rules used

rule 27

Int[(a_)*(Fx_), x_Symbol] :> Simp[a   Int[Fx, x], x] /; FreeQ[a, x] &&  !Ma 
tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
 

rule 2009

Int[u_, x_Symbol] :> Simp[IntSum[u, x], x] /; SumQ[u]
 

rule 2331

Int[(Pq_)*(x_)^(m_.)*((a_) + (b_.)*(x_)^2)^(p_.), x_Symbol] :> Simp[1/2   S 
ubst[Int[x^((m - 1)/2)*SubstFor[x^2, Pq, x]*(a + b*x)^p, x], x, x^2], x] /; 
 FreeQ[{a, b, p}, x] && PolyQ[Pq, x^2] && IntegerQ[(m - 1)/2]
 

rule 2389

Int[(Pq_)*((a_) + (b_.)*(x_)^(n_.))^(p_.), x_Symbol] :> Int[ExpandIntegrand 
[Pq*(a + b*x^n)^p, x], x] /; FreeQ[{a, b, n}, x] && PolyQ[Pq, x] && (IGtQ[p 
, 0] || EqQ[n, 1])
 

rule 6199

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] && EqQ[e, c^2*d] && IGtQ[p, 0]
 

Maple [A] (verified)

Time = 0.27 (sec) , antiderivative size = 153, normalized size of antiderivative = 0.90

method	result	size
parts	\(d^{3} a \left (\frac {1}{7} c^{6} x^{7}+\frac {3}{5} c^{4} x^{5}+x^{3} c^{2}+x \right )+\frac {d^{3} b \left (\frac {\operatorname {arcsinh}\left (x c \right ) x^{7} c^{7}}{7}+\frac {3 \,\operatorname {arcsinh}\left (x c \right ) x^{5} c^{5}}{5}+\operatorname {arcsinh}\left (x c \right ) x^{3} c^{3}+x c \,\operatorname {arcsinh}\left (x c \right )-\frac {2161 \sqrt {c^{2} x^{2}+1}}{3675}-\frac {757 x^{2} c^{2} \sqrt {c^{2} x^{2}+1}}{3675}-\frac {117 x^{4} c^{4} \sqrt {c^{2} x^{2}+1}}{1225}-\frac {x^{6} c^{6} \sqrt {c^{2} x^{2}+1}}{49}\right )}{c}\)	\(153\)
derivativedivides	\(\frac {d^{3} a \left (\frac {1}{7} x^{7} c^{7}+\frac {3}{5} x^{5} c^{5}+x^{3} c^{3}+x c \right )+d^{3} b \left (\frac {\operatorname {arcsinh}\left (x c \right ) x^{7} c^{7}}{7}+\frac {3 \,\operatorname {arcsinh}\left (x c \right ) x^{5} c^{5}}{5}+\operatorname {arcsinh}\left (x c \right ) x^{3} c^{3}+x c \,\operatorname {arcsinh}\left (x c \right )-\frac {2161 \sqrt {c^{2} x^{2}+1}}{3675}-\frac {757 x^{2} c^{2} \sqrt {c^{2} x^{2}+1}}{3675}-\frac {117 x^{4} c^{4} \sqrt {c^{2} x^{2}+1}}{1225}-\frac {x^{6} c^{6} \sqrt {c^{2} x^{2}+1}}{49}\right )}{c}\)	\(156\)
default	\(\frac {d^{3} a \left (\frac {1}{7} x^{7} c^{7}+\frac {3}{5} x^{5} c^{5}+x^{3} c^{3}+x c \right )+d^{3} b \left (\frac {\operatorname {arcsinh}\left (x c \right ) x^{7} c^{7}}{7}+\frac {3 \,\operatorname {arcsinh}\left (x c \right ) x^{5} c^{5}}{5}+\operatorname {arcsinh}\left (x c \right ) x^{3} c^{3}+x c \,\operatorname {arcsinh}\left (x c \right )-\frac {2161 \sqrt {c^{2} x^{2}+1}}{3675}-\frac {757 x^{2} c^{2} \sqrt {c^{2} x^{2}+1}}{3675}-\frac {117 x^{4} c^{4} \sqrt {c^{2} x^{2}+1}}{1225}-\frac {x^{6} c^{6} \sqrt {c^{2} x^{2}+1}}{49}\right )}{c}\)	\(156\)
orering	\(\frac {x \left (325 c^{6} x^{6}+1437 c^{4} x^{4}+2739 c^{2} x^{2}+5547\right ) \left (c^{2} d \,x^{2}+d \right )^{3} \left (a +b \,\operatorname {arcsinh}\left (x c \right )\right )}{1225 \left (c^{2} x^{2}+1\right )^{3}}-\frac {\left (75 c^{6} x^{6}+351 c^{4} x^{4}+757 c^{2} x^{2}+2161\right ) \left (6 \left (c^{2} d \,x^{2}+d \right )^{2} \left (a +b \,\operatorname {arcsinh}\left (x c \right )\right ) c^{2} d x +\frac {\left (c^{2} d \,x^{2}+d \right )^{3} b c}{\sqrt {c^{2} x^{2}+1}}\right )}{3675 c^{2} \left (c^{2} x^{2}+1\right )^{2}}\)	\(158\)

Input:

int((c^2*d*x^2+d)^3*(a+b*arcsinh(x*c)),x,method=_RETURNVERBOSE)
 

Output:

d^3*a*(1/7*c^6*x^7+3/5*c^4*x^5+x^3*c^2+x)+d^3*b/c*(1/7*arcsinh(x*c)*x^7*c^ 
7+3/5*arcsinh(x*c)*x^5*c^5+arcsinh(x*c)*x^3*c^3+x*c*arcsinh(x*c)-2161/3675 
*(c^2*x^2+1)^(1/2)-757/3675*x^2*c^2*(c^2*x^2+1)^(1/2)-117/1225*x^4*c^4*(c^ 
2*x^2+1)^(1/2)-1/49*x^6*c^6*(c^2*x^2+1)^(1/2))
 

Fricas [A] (verification not implemented)

Time = 0.09 (sec) , antiderivative size = 169, normalized size of antiderivative = 0.99 \[ \int \left (d+c^2 d x^2\right )^3 (a+b \text {arcsinh}(c x)) \, dx=\frac {525 \, a c^{7} d^{3} x^{7} + 2205 \, a c^{5} d^{3} x^{5} + 3675 \, a c^{3} d^{3} x^{3} + 3675 \, a c d^{3} x + 105 \, {\left (5 \, b c^{7} d^{3} x^{7} + 21 \, b c^{5} d^{3} x^{5} + 35 \, b c^{3} d^{3} x^{3} + 35 \, b c d^{3} x\right )} \log \left (c x + \sqrt {c^{2} x^{2} + 1}\right ) - {\left (75 \, b c^{6} d^{3} x^{6} + 351 \, b c^{4} d^{3} x^{4} + 757 \, b c^{2} d^{3} x^{2} + 2161 \, b d^{3}\right )} \sqrt {c^{2} x^{2} + 1}}{3675 \, c} \] Input:

integrate((c^2*d*x^2+d)^3*(a+b*arcsinh(c*x)),x, algorithm="fricas")
 

Output:

1/3675*(525*a*c^7*d^3*x^7 + 2205*a*c^5*d^3*x^5 + 3675*a*c^3*d^3*x^3 + 3675 
*a*c*d^3*x + 105*(5*b*c^7*d^3*x^7 + 21*b*c^5*d^3*x^5 + 35*b*c^3*d^3*x^3 + 
35*b*c*d^3*x)*log(c*x + sqrt(c^2*x^2 + 1)) - (75*b*c^6*d^3*x^6 + 351*b*c^4 
*d^3*x^4 + 757*b*c^2*d^3*x^2 + 2161*b*d^3)*sqrt(c^2*x^2 + 1))/c
                                                                                    
                                                                                    
 

Sympy [A] (verification not implemented)

Time = 0.57 (sec) , antiderivative size = 221, normalized size of antiderivative = 1.30 \[ \int \left (d+c^2 d x^2\right )^3 (a+b \text {arcsinh}(c x)) \, dx=\begin {cases} \frac {a c^{6} d^{3} x^{7}}{7} + \frac {3 a c^{4} d^{3} x^{5}}{5} + a c^{2} d^{3} x^{3} + a d^{3} x + \frac {b c^{6} d^{3} x^{7} \operatorname {asinh}{\left (c x \right )}}{7} - \frac {b c^{5} d^{3} x^{6} \sqrt {c^{2} x^{2} + 1}}{49} + \frac {3 b c^{4} d^{3} x^{5} \operatorname {asinh}{\left (c x \right )}}{5} - \frac {117 b c^{3} d^{3} x^{4} \sqrt {c^{2} x^{2} + 1}}{1225} + b c^{2} d^{3} x^{3} \operatorname {asinh}{\left (c x \right )} - \frac {757 b c d^{3} x^{2} \sqrt {c^{2} x^{2} + 1}}{3675} + b d^{3} x \operatorname {asinh}{\left (c x \right )} - \frac {2161 b d^{3} \sqrt {c^{2} x^{2} + 1}}{3675 c} & \text {for}\: c \neq 0 \\a d^{3} x & \text {otherwise} \end {cases} \] Input:

integrate((c**2*d*x**2+d)**3*(a+b*asinh(c*x)),x)
 

Output:

Piecewise((a*c**6*d**3*x**7/7 + 3*a*c**4*d**3*x**5/5 + a*c**2*d**3*x**3 + 
a*d**3*x + b*c**6*d**3*x**7*asinh(c*x)/7 - b*c**5*d**3*x**6*sqrt(c**2*x**2 
 + 1)/49 + 3*b*c**4*d**3*x**5*asinh(c*x)/5 - 117*b*c**3*d**3*x**4*sqrt(c** 
2*x**2 + 1)/1225 + b*c**2*d**3*x**3*asinh(c*x) - 757*b*c*d**3*x**2*sqrt(c* 
*2*x**2 + 1)/3675 + b*d**3*x*asinh(c*x) - 2161*b*d**3*sqrt(c**2*x**2 + 1)/ 
(3675*c), Ne(c, 0)), (a*d**3*x, True))
 

Maxima [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 301 vs. \(2 (150) = 300\).

Time = 0.04 (sec) , antiderivative size = 301, normalized size of antiderivative = 1.77 \[ \int \left (d+c^2 d x^2\right )^3 (a+b \text {arcsinh}(c x)) \, dx=\frac {1}{7} \, a c^{6} d^{3} x^{7} + \frac {3}{5} \, a c^{4} d^{3} x^{5} + \frac {1}{245} \, {\left (35 \, x^{7} \operatorname {arsinh}\left (c x\right ) - {\left (\frac {5 \, \sqrt {c^{2} x^{2} + 1} x^{6}}{c^{2}} - \frac {6 \, \sqrt {c^{2} x^{2} + 1} x^{4}}{c^{4}} + \frac {8 \, \sqrt {c^{2} x^{2} + 1} x^{2}}{c^{6}} - \frac {16 \, \sqrt {c^{2} x^{2} + 1}}{c^{8}}\right )} c\right )} b c^{6} d^{3} + \frac {1}{25} \, {\left (15 \, x^{5} \operatorname {arsinh}\left (c x\right ) - {\left (\frac {3 \, \sqrt {c^{2} x^{2} + 1} x^{4}}{c^{2}} - \frac {4 \, \sqrt {c^{2} x^{2} + 1} x^{2}}{c^{4}} + \frac {8 \, \sqrt {c^{2} x^{2} + 1}}{c^{6}}\right )} c\right )} b c^{4} d^{3} + a c^{2} d^{3} x^{3} + \frac {1}{3} \, {\left (3 \, x^{3} \operatorname {arsinh}\left (c x\right ) - c {\left (\frac {\sqrt {c^{2} x^{2} + 1} x^{2}}{c^{2}} - \frac {2 \, \sqrt {c^{2} x^{2} + 1}}{c^{4}}\right )}\right )} b c^{2} d^{3} + a d^{3} x + \frac {{\left (c x \operatorname {arsinh}\left (c x\right ) - \sqrt {c^{2} x^{2} + 1}\right )} b d^{3}}{c} \] Input:

integrate((c^2*d*x^2+d)^3*(a+b*arcsinh(c*x)),x, algorithm="maxima")
 

Output:

1/7*a*c^6*d^3*x^7 + 3/5*a*c^4*d^3*x^5 + 1/245*(35*x^7*arcsinh(c*x) - (5*sq 
rt(c^2*x^2 + 1)*x^6/c^2 - 6*sqrt(c^2*x^2 + 1)*x^4/c^4 + 8*sqrt(c^2*x^2 + 1 
)*x^2/c^6 - 16*sqrt(c^2*x^2 + 1)/c^8)*c)*b*c^6*d^3 + 1/25*(15*x^5*arcsinh( 
c*x) - (3*sqrt(c^2*x^2 + 1)*x^4/c^2 - 4*sqrt(c^2*x^2 + 1)*x^2/c^4 + 8*sqrt 
(c^2*x^2 + 1)/c^6)*c)*b*c^4*d^3 + a*c^2*d^3*x^3 + 1/3*(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*c^2*d^3 + a*d 
^3*x + (c*x*arcsinh(c*x) - sqrt(c^2*x^2 + 1))*b*d^3/c
 

Giac [F(-2)]

Exception generated. \[ \int \left (d+c^2 d x^2\right )^3 (a+b \text {arcsinh}(c x)) \, dx=\text {Exception raised: TypeError} \] Input:

integrate((c^2*d*x^2+d)^3*(a+b*arcsinh(c*x)),x, algorithm="giac")
 

Output:

Exception raised: TypeError >> an error occurred running a Giac command:IN 
PUT:sage2:=int(sage0,sageVARx):;OUTPUT:sym2poly/r2sym(const gen & e,const 
index_m & i,const vecteur & l) Error: Bad Argument Value
 

Mupad [F(-1)]

Timed out. \[ \int \left (d+c^2 d x^2\right )^3 (a+b \text {arcsinh}(c x)) \, dx=\int \left (a+b\,\mathrm {asinh}\left (c\,x\right )\right )\,{\left (d\,c^2\,x^2+d\right )}^3 \,d x \] Input:

int((a + b*asinh(c*x))*(d + c^2*d*x^2)^3,x)
 

Output:

int((a + b*asinh(c*x))*(d + c^2*d*x^2)^3, x)
 

Reduce [B] (verification not implemented)

Time = 0.21 (sec) , antiderivative size = 159, normalized size of antiderivative = 0.94 \[ \int \left (d+c^2 d x^2\right )^3 (a+b \text {arcsinh}(c x)) \, dx=\frac {d^{3} \left (525 \mathit {asinh} \left (c x \right ) b \,c^{7} x^{7}+2205 \mathit {asinh} \left (c x \right ) b \,c^{5} x^{5}+3675 \mathit {asinh} \left (c x \right ) b \,c^{3} x^{3}+3675 \mathit {asinh} \left (c x \right ) b c x -75 \sqrt {c^{2} x^{2}+1}\, b \,c^{6} x^{6}-351 \sqrt {c^{2} x^{2}+1}\, b \,c^{4} x^{4}-757 \sqrt {c^{2} x^{2}+1}\, b \,c^{2} x^{2}-2161 \sqrt {c^{2} x^{2}+1}\, b +525 a \,c^{7} x^{7}+2205 a \,c^{5} x^{5}+3675 a \,c^{3} x^{3}+3675 a c x \right )}{3675 c} \] Input:

int((c^2*d*x^2+d)^3*(a+b*asinh(c*x)),x)
 

Output:

(d**3*(525*asinh(c*x)*b*c**7*x**7 + 2205*asinh(c*x)*b*c**5*x**5 + 3675*asi 
nh(c*x)*b*c**3*x**3 + 3675*asinh(c*x)*b*c*x - 75*sqrt(c**2*x**2 + 1)*b*c** 
6*x**6 - 351*sqrt(c**2*x**2 + 1)*b*c**4*x**4 - 757*sqrt(c**2*x**2 + 1)*b*c 
**2*x**2 - 2161*sqrt(c**2*x**2 + 1)*b + 525*a*c**7*x**7 + 2205*a*c**5*x**5 
 + 3675*a*c**3*x**3 + 3675*a*c*x))/(3675*c)