\(\int x^2 (d+c^2 d x^2)^2 (a+b \text {arcsinh}(c x)) \, dx\) [12]

Optimal result

Integrand size = 24, antiderivative size = 157 \[ \int x^2 \left (d+c^2 d x^2\right )^2 (a+b \text {arcsinh}(c x)) \, dx=\frac {8 b d^2 \sqrt {1+c^2 x^2}}{105 c^3}+\frac {4 b d^2 \left (1+c^2 x^2\right )^{3/2}}{315 c^3}+\frac {b d^2 \left (1+c^2 x^2\right )^{5/2}}{175 c^3}-\frac {b d^2 \left (1+c^2 x^2\right )^{7/2}}{49 c^3}+\frac {1}{3} d^2 x^3 (a+b \text {arcsinh}(c x))+\frac {2}{5} c^2 d^2 x^5 (a+b \text {arcsinh}(c x))+\frac {1}{7} c^4 d^2 x^7 (a+b \text {arcsinh}(c x)) \] Output:

8/105*b*d^2*(c^2*x^2+1)^(1/2)/c^3+4/315*b*d^2*(c^2*x^2+1)^(3/2)/c^3+1/175* 
b*d^2*(c^2*x^2+1)^(5/2)/c^3-1/49*b*d^2*(c^2*x^2+1)^(7/2)/c^3+1/3*d^2*x^3*( 
a+b*arcsinh(c*x))+2/5*c^2*d^2*x^5*(a+b*arcsinh(c*x))+1/7*c^4*d^2*x^7*(a+b* 
arcsinh(c*x))
 

Mathematica [A] (verified)

Time = 0.06 (sec) , antiderivative size = 111, normalized size of antiderivative = 0.71 \[ \int x^2 \left (d+c^2 d x^2\right )^2 (a+b \text {arcsinh}(c x)) \, dx=\frac {d^2 \left (105 a c^3 x^3 \left (35+42 c^2 x^2+15 c^4 x^4\right )-b \sqrt {1+c^2 x^2} \left (-818+409 c^2 x^2+612 c^4 x^4+225 c^6 x^6\right )+105 b c^3 x^3 \left (35+42 c^2 x^2+15 c^4 x^4\right ) \text {arcsinh}(c x)\right )}{11025 c^3} \] Input:

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

Output:

(d^2*(105*a*c^3*x^3*(35 + 42*c^2*x^2 + 15*c^4*x^4) - b*Sqrt[1 + c^2*x^2]*( 
-818 + 409*c^2*x^2 + 612*c^4*x^4 + 225*c^6*x^6) + 105*b*c^3*x^3*(35 + 42*c 
^2*x^2 + 15*c^4*x^4)*ArcSinh[c*x]))/(11025*c^3)
 

Rubi [A] (verified)

Time = 0.48 (sec) , antiderivative size = 149, normalized size of antiderivative = 0.95, number of steps used = 6, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.208, Rules used = {6218, 27, 1578, 1195, 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.

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

\(\Big \downarrow \) 6218

\(\displaystyle -b c \int \frac {d^2 x^3 \left (15 c^4 x^4+42 c^2 x^2+35\right )}{105 \sqrt {c^2 x^2+1}}dx+\frac {1}{7} c^4 d^2 x^7 (a+b \text {arcsinh}(c x))+\frac {2}{5} c^2 d^2 x^5 (a+b \text {arcsinh}(c x))+\frac {1}{3} d^2 x^3 (a+b \text {arcsinh}(c x))\)

\(\Big \downarrow \) 27

\(\displaystyle -\frac {1}{105} b c d^2 \int \frac {x^3 \left (15 c^4 x^4+42 c^2 x^2+35\right )}{\sqrt {c^2 x^2+1}}dx+\frac {1}{7} c^4 d^2 x^7 (a+b \text {arcsinh}(c x))+\frac {2}{5} c^2 d^2 x^5 (a+b \text {arcsinh}(c x))+\frac {1}{3} d^2 x^3 (a+b \text {arcsinh}(c x))\)

\(\Big \downarrow \) 1578

\(\displaystyle -\frac {1}{210} b c d^2 \int \frac {x^2 \left (15 c^4 x^4+42 c^2 x^2+35\right )}{\sqrt {c^2 x^2+1}}dx^2+\frac {1}{7} c^4 d^2 x^7 (a+b \text {arcsinh}(c x))+\frac {2}{5} c^2 d^2 x^5 (a+b \text {arcsinh}(c x))+\frac {1}{3} d^2 x^3 (a+b \text {arcsinh}(c x))\)

\(\Big \downarrow \) 1195

\(\displaystyle -\frac {1}{210} b c d^2 \int \left (\frac {15 \left (c^2 x^2+1\right )^{5/2}}{c^2}-\frac {3 \left (c^2 x^2+1\right )^{3/2}}{c^2}-\frac {4 \sqrt {c^2 x^2+1}}{c^2}-\frac {8}{c^2 \sqrt {c^2 x^2+1}}\right )dx^2+\frac {1}{7} c^4 d^2 x^7 (a+b \text {arcsinh}(c x))+\frac {2}{5} c^2 d^2 x^5 (a+b \text {arcsinh}(c x))+\frac {1}{3} d^2 x^3 (a+b \text {arcsinh}(c x))\)

\(\Big \downarrow \) 2009

\(\displaystyle \frac {1}{7} c^4 d^2 x^7 (a+b \text {arcsinh}(c x))+\frac {2}{5} c^2 d^2 x^5 (a+b \text {arcsinh}(c x))+\frac {1}{3} d^2 x^3 (a+b \text {arcsinh}(c x))-\frac {1}{210} b c d^2 \left (\frac {30 \left (c^2 x^2+1\right )^{7/2}}{7 c^4}-\frac {6 \left (c^2 x^2+1\right )^{5/2}}{5 c^4}-\frac {8 \left (c^2 x^2+1\right )^{3/2}}{3 c^4}-\frac {16 \sqrt {c^2 x^2+1}}{c^4}\right )\)

Input:

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

Output:

-1/210*(b*c*d^2*((-16*Sqrt[1 + c^2*x^2])/c^4 - (8*(1 + c^2*x^2)^(3/2))/(3* 
c^4) - (6*(1 + c^2*x^2)^(5/2))/(5*c^4) + (30*(1 + c^2*x^2)^(7/2))/(7*c^4)) 
) + (d^2*x^3*(a + b*ArcSinh[c*x]))/3 + (2*c^2*d^2*x^5*(a + b*ArcSinh[c*x]) 
)/5 + (c^4*d^2*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 1195

Int[((d_.) + (e_.)*(x_))^(m_.)*((f_.) + (g_.)*(x_))^(n_.)*((a_.) + (b_.)*(x 
_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> Int[ExpandIntegrand[(d + e*x)^m*(f + 
 g*x)^n*(a + b*x + c*x^2)^p, x], x] /; FreeQ[{a, b, c, d, e, f, g, m, n}, x 
] && IGtQ[p, 0]
 

rule 1578

Int[(x_)^(m_.)*((d_) + (e_.)*(x_)^2)^(q_.)*((a_) + (b_.)*(x_)^2 + (c_.)*(x_ 
)^4)^(p_.), x_Symbol] :> Simp[1/2   Subst[Int[x^((m - 1)/2)*(d + e*x)^q*(a 
+ b*x + c*x^2)^p, x], x, x^2], x] /; FreeQ[{a, b, c, d, e, p, q}, x] && Int 
egerQ[(m - 1)/2]
 

rule 2009

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

rule 6218

Int[((a_.) + ArcSinh[(c_.)*(x_)]*(b_.))*((f_.)*(x_))^(m_)*((d_) + (e_.)*(x_ 
)^2)^(p_.), x_Symbol] :> With[{u = IntHide[(f*x)^m*(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, f, m}, x] && EqQ[e, c^2*d] 
&& IGtQ[p, 0]
 

Maple [A] (verified)

Time = 0.43 (sec) , antiderivative size = 144, normalized size of antiderivative = 0.92

method	result	size
parts	\(a \,d^{2} \left (\frac {1}{7} c^{4} x^{7}+\frac {2}{5} x^{5} c^{2}+\frac {1}{3} x^{3}\right )+\frac {d^{2} b \left (\frac {\operatorname {arcsinh}\left (x c \right ) x^{7} c^{7}}{7}+\frac {2 \,\operatorname {arcsinh}\left (x c \right ) x^{5} c^{5}}{5}+\frac {\operatorname {arcsinh}\left (x c \right ) x^{3} c^{3}}{3}-\frac {409 x^{2} c^{2} \sqrt {c^{2} x^{2}+1}}{11025}+\frac {818 \sqrt {c^{2} x^{2}+1}}{11025}-\frac {68 x^{4} c^{4} \sqrt {c^{2} x^{2}+1}}{1225}-\frac {x^{6} c^{6} \sqrt {c^{2} x^{2}+1}}{49}\right )}{c^{3}}\)	\(144\)
derivativedivides	\(\frac {a \,d^{2} \left (\frac {1}{7} x^{7} c^{7}+\frac {2}{5} x^{5} c^{5}+\frac {1}{3} x^{3} c^{3}\right )+d^{2} b \left (\frac {\operatorname {arcsinh}\left (x c \right ) x^{7} c^{7}}{7}+\frac {2 \,\operatorname {arcsinh}\left (x c \right ) x^{5} c^{5}}{5}+\frac {\operatorname {arcsinh}\left (x c \right ) x^{3} c^{3}}{3}-\frac {409 x^{2} c^{2} \sqrt {c^{2} x^{2}+1}}{11025}+\frac {818 \sqrt {c^{2} x^{2}+1}}{11025}-\frac {68 x^{4} c^{4} \sqrt {c^{2} x^{2}+1}}{1225}-\frac {x^{6} c^{6} \sqrt {c^{2} x^{2}+1}}{49}\right )}{c^{3}}\)	\(148\)
default	\(\frac {a \,d^{2} \left (\frac {1}{7} x^{7} c^{7}+\frac {2}{5} x^{5} c^{5}+\frac {1}{3} x^{3} c^{3}\right )+d^{2} b \left (\frac {\operatorname {arcsinh}\left (x c \right ) x^{7} c^{7}}{7}+\frac {2 \,\operatorname {arcsinh}\left (x c \right ) x^{5} c^{5}}{5}+\frac {\operatorname {arcsinh}\left (x c \right ) x^{3} c^{3}}{3}-\frac {409 x^{2} c^{2} \sqrt {c^{2} x^{2}+1}}{11025}+\frac {818 \sqrt {c^{2} x^{2}+1}}{11025}-\frac {68 x^{4} c^{4} \sqrt {c^{2} x^{2}+1}}{1225}-\frac {x^{6} c^{6} \sqrt {c^{2} x^{2}+1}}{49}\right )}{c^{3}}\)	\(148\)
orering	\(\frac {\left (2925 c^{8} x^{8}+8532 c^{6} x^{6}+7353 c^{4} x^{4}-4090 c^{2} x^{2}-1636\right ) \left (c^{2} d \,x^{2}+d \right )^{2} \left (a +b \,\operatorname {arcsinh}\left (x c \right )\right )}{11025 c^{4} \left (c^{2} x^{2}+1\right )^{2} x}-\frac {\left (225 c^{6} x^{6}+612 c^{4} x^{4}+409 c^{2} x^{2}-818\right ) \left (2 x \left (c^{2} d \,x^{2}+d \right )^{2} \left (a +b \,\operatorname {arcsinh}\left (x c \right )\right )+4 x^{3} \left (c^{2} d \,x^{2}+d \right ) \left (a +b \,\operatorname {arcsinh}\left (x c \right )\right ) c^{2} d +\frac {x^{2} \left (c^{2} d \,x^{2}+d \right )^{2} b c}{\sqrt {c^{2} x^{2}+1}}\right )}{11025 c^{4} \left (c^{2} x^{2}+1\right ) x^{2}}\)	\(200\)

Input:

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

Output:

a*d^2*(1/7*c^4*x^7+2/5*x^5*c^2+1/3*x^3)+d^2*b/c^3*(1/7*arcsinh(x*c)*x^7*c^ 
7+2/5*arcsinh(x*c)*x^5*c^5+1/3*arcsinh(x*c)*x^3*c^3-409/11025*x^2*c^2*(c^2 
*x^2+1)^(1/2)+818/11025*(c^2*x^2+1)^(1/2)-68/1225*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 = 0.11 (sec) , antiderivative size = 153, normalized size of antiderivative = 0.97 \[ \int x^2 \left (d+c^2 d x^2\right )^2 (a+b \text {arcsinh}(c x)) \, dx=\frac {1575 \, a c^{7} d^{2} x^{7} + 4410 \, a c^{5} d^{2} x^{5} + 3675 \, a c^{3} d^{2} x^{3} + 105 \, {\left (15 \, b c^{7} d^{2} x^{7} + 42 \, b c^{5} d^{2} x^{5} + 35 \, b c^{3} d^{2} x^{3}\right )} \log \left (c x + \sqrt {c^{2} x^{2} + 1}\right ) - {\left (225 \, b c^{6} d^{2} x^{6} + 612 \, b c^{4} d^{2} x^{4} + 409 \, b c^{2} d^{2} x^{2} - 818 \, b d^{2}\right )} \sqrt {c^{2} x^{2} + 1}}{11025 \, c^{3}} \] Input:

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

Output:

1/11025*(1575*a*c^7*d^2*x^7 + 4410*a*c^5*d^2*x^5 + 3675*a*c^3*d^2*x^3 + 10 
5*(15*b*c^7*d^2*x^7 + 42*b*c^5*d^2*x^5 + 35*b*c^3*d^2*x^3)*log(c*x + sqrt( 
c^2*x^2 + 1)) - (225*b*c^6*d^2*x^6 + 612*b*c^4*d^2*x^4 + 409*b*c^2*d^2*x^2 
 - 818*b*d^2)*sqrt(c^2*x^2 + 1))/c^3
                                                                                    
                                                                                    
 

Sympy [A] (verification not implemented)

Time = 0.67 (sec) , antiderivative size = 202, normalized size of antiderivative = 1.29 \[ \int x^2 \left (d+c^2 d x^2\right )^2 (a+b \text {arcsinh}(c x)) \, dx=\begin {cases} \frac {a c^{4} d^{2} x^{7}}{7} + \frac {2 a c^{2} d^{2} x^{5}}{5} + \frac {a d^{2} x^{3}}{3} + \frac {b c^{4} d^{2} x^{7} \operatorname {asinh}{\left (c x \right )}}{7} - \frac {b c^{3} d^{2} x^{6} \sqrt {c^{2} x^{2} + 1}}{49} + \frac {2 b c^{2} d^{2} x^{5} \operatorname {asinh}{\left (c x \right )}}{5} - \frac {68 b c d^{2} x^{4} \sqrt {c^{2} x^{2} + 1}}{1225} + \frac {b d^{2} x^{3} \operatorname {asinh}{\left (c x \right )}}{3} - \frac {409 b d^{2} x^{2} \sqrt {c^{2} x^{2} + 1}}{11025 c} + \frac {818 b d^{2} \sqrt {c^{2} x^{2} + 1}}{11025 c^{3}} & \text {for}\: c \neq 0 \\\frac {a d^{2} x^{3}}{3} & \text {otherwise} \end {cases} \] Input:

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

Output:

Piecewise((a*c**4*d**2*x**7/7 + 2*a*c**2*d**2*x**5/5 + a*d**2*x**3/3 + b*c 
**4*d**2*x**7*asinh(c*x)/7 - b*c**3*d**2*x**6*sqrt(c**2*x**2 + 1)/49 + 2*b 
*c**2*d**2*x**5*asinh(c*x)/5 - 68*b*c*d**2*x**4*sqrt(c**2*x**2 + 1)/1225 + 
 b*d**2*x**3*asinh(c*x)/3 - 409*b*d**2*x**2*sqrt(c**2*x**2 + 1)/(11025*c) 
+ 818*b*d**2*sqrt(c**2*x**2 + 1)/(11025*c**3), Ne(c, 0)), (a*d**2*x**3/3, 
True))
 

Maxima [A] (verification not implemented)

Time = 0.03 (sec) , antiderivative size = 261, normalized size of antiderivative = 1.66 \[ \int x^2 \left (d+c^2 d x^2\right )^2 (a+b \text {arcsinh}(c x)) \, dx=\frac {1}{7} \, a c^{4} d^{2} x^{7} + \frac {2}{5} \, a c^{2} d^{2} x^{5} + \frac {1}{245} \, {\left (35 \, x^{7} \operatorname {arsinh}\left (c x\right ) - {\left (\frac {5 \, \sqrt {c^{2} x^{2} + 1} x^{6}}{c^{2}} - \frac {6 \, \sqrt {c^{2} x^{2} + 1} x^{4}}{c^{4}} + \frac {8 \, \sqrt {c^{2} x^{2} + 1} x^{2}}{c^{6}} - \frac {16 \, \sqrt {c^{2} x^{2} + 1}}{c^{8}}\right )} c\right )} b c^{4} d^{2} + \frac {2}{75} \, {\left (15 \, x^{5} \operatorname {arsinh}\left (c x\right ) - {\left (\frac {3 \, \sqrt {c^{2} x^{2} + 1} x^{4}}{c^{2}} - \frac {4 \, \sqrt {c^{2} x^{2} + 1} x^{2}}{c^{4}} + \frac {8 \, \sqrt {c^{2} x^{2} + 1}}{c^{6}}\right )} c\right )} b c^{2} d^{2} + \frac {1}{3} \, a d^{2} x^{3} + \frac {1}{9} \, {\left (3 \, x^{3} \operatorname {arsinh}\left (c x\right ) - c {\left (\frac {\sqrt {c^{2} x^{2} + 1} x^{2}}{c^{2}} - \frac {2 \, \sqrt {c^{2} x^{2} + 1}}{c^{4}}\right )}\right )} b d^{2} \] Input:

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

Output:

1/7*a*c^4*d^2*x^7 + 2/5*a*c^2*d^2*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^4*d^2 + 2/75*(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^2*d^2 + 1/3*a*d^2*x^3 + 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*d^2
 

Giac [F(-2)]

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

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

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

Output:

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

Reduce [B] (verification not implemented)

Time = 0.20 (sec) , antiderivative size = 145, normalized size of antiderivative = 0.92 \[ \int x^2 \left (d+c^2 d x^2\right )^2 (a+b \text {arcsinh}(c x)) \, dx=\frac {d^{2} \left (1575 \mathit {asinh} \left (c x \right ) b \,c^{7} x^{7}+4410 \mathit {asinh} \left (c x \right ) b \,c^{5} x^{5}+3675 \mathit {asinh} \left (c x \right ) b \,c^{3} x^{3}-225 \sqrt {c^{2} x^{2}+1}\, b \,c^{6} x^{6}-612 \sqrt {c^{2} x^{2}+1}\, b \,c^{4} x^{4}-409 \sqrt {c^{2} x^{2}+1}\, b \,c^{2} x^{2}+818 \sqrt {c^{2} x^{2}+1}\, b +1575 a \,c^{7} x^{7}+4410 a \,c^{5} x^{5}+3675 a \,c^{3} x^{3}\right )}{11025 c^{3}} \] Input:

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

Output:

(d**2*(1575*asinh(c*x)*b*c**7*x**7 + 4410*asinh(c*x)*b*c**5*x**5 + 3675*as 
inh(c*x)*b*c**3*x**3 - 225*sqrt(c**2*x**2 + 1)*b*c**6*x**6 - 612*sqrt(c**2 
*x**2 + 1)*b*c**4*x**4 - 409*sqrt(c**2*x**2 + 1)*b*c**2*x**2 + 818*sqrt(c* 
*2*x**2 + 1)*b + 1575*a*c**7*x**7 + 4410*a*c**5*x**5 + 3675*a*c**3*x**3))/ 
(11025*c**3)