\(\int x^2 (d+e x^2)^2 (a+b \text {arccosh}(c x)) \, dx\) [371]

Optimal result

Integrand size = 21, antiderivative size = 214 \[ \int x^2 \left (d+e x^2\right )^2 (a+b \text {arccosh}(c x)) \, dx=-\frac {b \left (35 c^4 d^2+42 c^2 d e+15 e^2\right ) \sqrt {-1+c x} \sqrt {1+c x}}{105 c^7}-\frac {b \left (35 c^4 d^2+84 c^2 d e+45 e^2\right ) (-1+c x)^{3/2} (1+c x)^{3/2}}{315 c^7}-\frac {b e \left (14 c^2 d+15 e\right ) (-1+c x)^{5/2} (1+c x)^{5/2}}{175 c^7}-\frac {b e^2 (-1+c x)^{7/2} (1+c x)^{7/2}}{49 c^7}+\frac {1}{3} d^2 x^3 (a+b \text {arccosh}(c x))+\frac {2}{5} d e x^5 (a+b \text {arccosh}(c x))+\frac {1}{7} e^2 x^7 (a+b \text {arccosh}(c x)) \] Output:

-1/105*b*(35*c^4*d^2+42*c^2*d*e+15*e^2)*(c*x-1)^(1/2)*(c*x+1)^(1/2)/c^7-1/ 
315*b*(35*c^4*d^2+84*c^2*d*e+45*e^2)*(c*x-1)^(3/2)*(c*x+1)^(3/2)/c^7-1/175 
*b*e*(14*c^2*d+15*e)*(c*x-1)^(5/2)*(c*x+1)^(5/2)/c^7-1/49*b*e^2*(c*x-1)^(7 
/2)*(c*x+1)^(7/2)/c^7+1/3*d^2*x^3*(a+b*arccosh(c*x))+2/5*d*e*x^5*(a+b*arcc 
osh(c*x))+1/7*e^2*x^7*(a+b*arccosh(c*x))
 

Mathematica [A] (verified)

Time = 0.15 (sec) , antiderivative size = 163, normalized size of antiderivative = 0.76 \[ \int x^2 \left (d+e x^2\right )^2 (a+b \text {arccosh}(c x)) \, dx=\frac {105 a x^3 \left (35 d^2+42 d e x^2+15 e^2 x^4\right )-\frac {b \sqrt {-1+c x} \sqrt {1+c x} \left (720 e^2+24 c^2 e \left (98 d+15 e x^2\right )+2 c^4 \left (1225 d^2+588 d e x^2+135 e^2 x^4\right )+c^6 \left (1225 d^2 x^2+882 d e x^4+225 e^2 x^6\right )\right )}{c^7}+105 b x^3 \left (35 d^2+42 d e x^2+15 e^2 x^4\right ) \text {arccosh}(c x)}{11025} \] Input:

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

Output:

(105*a*x^3*(35*d^2 + 42*d*e*x^2 + 15*e^2*x^4) - (b*Sqrt[-1 + c*x]*Sqrt[1 + 
 c*x]*(720*e^2 + 24*c^2*e*(98*d + 15*e*x^2) + 2*c^4*(1225*d^2 + 588*d*e*x^ 
2 + 135*e^2*x^4) + c^6*(1225*d^2*x^2 + 882*d*e*x^4 + 225*e^2*x^6)))/c^7 + 
105*b*x^3*(35*d^2 + 42*d*e*x^2 + 15*e^2*x^4)*ArcCosh[c*x])/11025
 

Rubi [A] (verified)

Time = 0.87 (sec) , antiderivative size = 226, normalized size of antiderivative = 1.06, number of steps used = 7, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.286, Rules used = {6373, 27, 1905, 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.

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

\(\Big \downarrow \) 6373

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 1905

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

\(\Big \downarrow \) 1578

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

\(\Big \downarrow \) 1195

\(\displaystyle -\frac {b c \sqrt {c^2 x^2-1} \int \left (\frac {15 e^2 \left (c^2 x^2-1\right )^{5/2}}{c^6}+\frac {3 e \left (14 d c^2+15 e\right ) \left (c^2 x^2-1\right )^{3/2}}{c^6}+\frac {\left (35 d^2 c^4+84 d e c^2+45 e^2\right ) \sqrt {c^2 x^2-1}}{c^6}+\frac {35 d^2 c^4+42 d e c^2+15 e^2}{c^6 \sqrt {c^2 x^2-1}}\right )dx^2}{210 \sqrt {c x-1} \sqrt {c x+1}}+\frac {1}{3} d^2 x^3 (a+b \text {arccosh}(c x))+\frac {2}{5} d e x^5 (a+b \text {arccosh}(c x))+\frac {1}{7} e^2 x^7 (a+b \text {arccosh}(c x))\)

\(\Big \downarrow \) 2009

\(\displaystyle \frac {1}{3} d^2 x^3 (a+b \text {arccosh}(c x))+\frac {2}{5} d e x^5 (a+b \text {arccosh}(c x))+\frac {1}{7} e^2 x^7 (a+b \text {arccosh}(c x))-\frac {b c \sqrt {c^2 x^2-1} \left (\frac {6 e \left (c^2 x^2-1\right )^{5/2} \left (14 c^2 d+15 e\right )}{5 c^8}+\frac {30 e^2 \left (c^2 x^2-1\right )^{7/2}}{7 c^8}+\frac {2 \left (c^2 x^2-1\right )^{3/2} \left (35 c^4 d^2+84 c^2 d e+45 e^2\right )}{3 c^8}+\frac {2 \sqrt {c^2 x^2-1} \left (35 c^4 d^2+42 c^2 d e+15 e^2\right )}{c^8}\right )}{210 \sqrt {c x-1} \sqrt {c x+1}}\)

Input:

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

Output:

-1/210*(b*c*Sqrt[-1 + c^2*x^2]*((2*(35*c^4*d^2 + 42*c^2*d*e + 15*e^2)*Sqrt 
[-1 + c^2*x^2])/c^8 + (2*(35*c^4*d^2 + 84*c^2*d*e + 45*e^2)*(-1 + c^2*x^2) 
^(3/2))/(3*c^8) + (6*e*(14*c^2*d + 15*e)*(-1 + c^2*x^2)^(5/2))/(5*c^8) + ( 
30*e^2*(-1 + c^2*x^2)^(7/2))/(7*c^8)))/(Sqrt[-1 + c*x]*Sqrt[1 + c*x]) + (d 
^2*x^3*(a + b*ArcCosh[c*x]))/3 + (2*d*e*x^5*(a + b*ArcCosh[c*x]))/5 + (e^2 
*x^7*(a + b*ArcCosh[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 1905

Int[((f_.)*(x_))^(m_.)*((d1_) + (e1_.)*(x_)^(non2_.))^(q_.)*((d2_) + (e2_.) 
*(x_)^(non2_.))^(q_.)*((a_.) + (b_.)*(x_)^(n_) + (c_.)*(x_)^(n2_))^(p_.), x 
_Symbol] :> Simp[(d1 + e1*x^(n/2))^FracPart[q]*((d2 + e2*x^(n/2))^FracPart[ 
q]/(d1*d2 + e1*e2*x^n)^FracPart[q])   Int[(f*x)^m*(d1*d2 + e1*e2*x^n)^q*(a 
+ b*x^n + c*x^(2*n))^p, x], x] /; FreeQ[{a, b, c, d1, e1, d2, e2, f, n, p, 
q}, x] && EqQ[n2, 2*n] && EqQ[non2, n/2] && EqQ[d2*e1 + d1*e2, 0]
 

rule 2009

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

rule 6373

Int[((a_.) + ArcCosh[(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]}, Sim 
p[(a + b*ArcCosh[c*x])   u, x] - Simp[b*c   Int[SimplifyIntegrand[u/(Sqrt[1 
 + c*x]*Sqrt[-1 + c*x]), x], x], x]] /; FreeQ[{a, b, c, d, e, f, m}, x] && 
NeQ[c^2*d + e, 0] && IntegerQ[p] && (GtQ[p, 0] || (IGtQ[(m - 1)/2, 0] && Le 
Q[m + p, 0]))
 

Maple [A] (verified)

Time = 0.25 (sec) , antiderivative size = 182, normalized size of antiderivative = 0.85

method	result	size
parts	\(a \left (\frac {1}{7} e^{2} x^{7}+\frac {2}{5} d e \,x^{5}+\frac {1}{3} d^{2} x^{3}\right )+\frac {b \left (\frac {c^{3} \operatorname {arccosh}\left (c x \right ) e^{2} x^{7}}{7}+\frac {2 c^{3} \operatorname {arccosh}\left (c x \right ) d e \,x^{5}}{5}+\frac {\operatorname {arccosh}\left (c x \right ) c^{3} x^{3} d^{2}}{3}-\frac {\sqrt {c x -1}\, \sqrt {c x +1}\, \left (225 x^{6} c^{6} e^{2}+882 x^{4} c^{6} d e +1225 x^{2} c^{6} d^{2}+270 c^{4} e^{2} x^{4}+1176 c^{4} d e \,x^{2}+2450 c^{4} d^{2}+360 c^{2} e^{2} x^{2}+2352 c^{2} d e +720 e^{2}\right )}{11025 c^{4}}\right )}{c^{3}}\)	\(182\)
derivativedivides	\(\frac {\frac {a \left (\frac {1}{3} d^{2} c^{7} x^{3}+\frac {2}{5} d \,c^{7} e \,x^{5}+\frac {1}{7} e^{2} c^{7} x^{7}\right )}{c^{4}}+\frac {b \left (\frac {\operatorname {arccosh}\left (c x \right ) d^{2} c^{7} x^{3}}{3}+\frac {2 \,\operatorname {arccosh}\left (c x \right ) d \,c^{7} e \,x^{5}}{5}+\frac {\operatorname {arccosh}\left (c x \right ) e^{2} c^{7} x^{7}}{7}-\frac {\sqrt {c x -1}\, \sqrt {c x +1}\, \left (225 x^{6} c^{6} e^{2}+882 x^{4} c^{6} d e +1225 x^{2} c^{6} d^{2}+270 c^{4} e^{2} x^{4}+1176 c^{4} d e \,x^{2}+2450 c^{4} d^{2}+360 c^{2} e^{2} x^{2}+2352 c^{2} d e +720 e^{2}\right )}{11025}\right )}{c^{4}}}{c^{3}}\)	\(195\)
default	\(\frac {\frac {a \left (\frac {1}{3} d^{2} c^{7} x^{3}+\frac {2}{5} d \,c^{7} e \,x^{5}+\frac {1}{7} e^{2} c^{7} x^{7}\right )}{c^{4}}+\frac {b \left (\frac {\operatorname {arccosh}\left (c x \right ) d^{2} c^{7} x^{3}}{3}+\frac {2 \,\operatorname {arccosh}\left (c x \right ) d \,c^{7} e \,x^{5}}{5}+\frac {\operatorname {arccosh}\left (c x \right ) e^{2} c^{7} x^{7}}{7}-\frac {\sqrt {c x -1}\, \sqrt {c x +1}\, \left (225 x^{6} c^{6} e^{2}+882 x^{4} c^{6} d e +1225 x^{2} c^{6} d^{2}+270 c^{4} e^{2} x^{4}+1176 c^{4} d e \,x^{2}+2450 c^{4} d^{2}+360 c^{2} e^{2} x^{2}+2352 c^{2} d e +720 e^{2}\right )}{11025}\right )}{c^{4}}}{c^{3}}\)	\(195\)
orering	\(\frac {\left (2925 c^{8} e^{3} x^{10}+11727 c^{8} d \,e^{2} x^{8}+17199 c^{8} d^{2} e \,x^{6}+270 x^{8} e^{3} c^{6}+6125 c^{8} d^{3} x^{4}+1854 x^{6} e^{2} c^{6} d +7938 x^{4} e \,c^{6} d^{2}+540 x^{6} e^{3} c^{4}+2450 c^{6} d^{3} x^{2}+7236 x^{4} e^{2} c^{4} d -12348 x^{2} e \,c^{4} d^{2}+2160 x^{4} e^{3} c^{2}-4900 c^{4} d^{3}-13392 x^{2} e^{2} c^{2} d -4704 c^{2} d^{2} e -4320 x^{2} e^{3}-1440 d \,e^{2}\right ) \left (a +b \,\operatorname {arccosh}\left (c x \right )\right )}{11025 x \left (e \,x^{2}+d \right ) c^{8}}-\frac {\left (225 x^{6} c^{6} e^{2}+882 x^{4} c^{6} d e +1225 x^{2} c^{6} d^{2}+270 c^{4} e^{2} x^{4}+1176 c^{4} d e \,x^{2}+2450 c^{4} d^{2}+360 c^{2} e^{2} x^{2}+2352 c^{2} d e +720 e^{2}\right ) \left (c x -1\right ) \left (c x +1\right ) \left (2 x \left (e \,x^{2}+d \right )^{2} \left (a +b \,\operatorname {arccosh}\left (c x \right )\right )+4 x^{3} \left (e \,x^{2}+d \right ) \left (a +b \,\operatorname {arccosh}\left (c x \right )\right ) e +\frac {x^{2} \left (e \,x^{2}+d \right )^{2} b c}{\sqrt {c x -1}\, \sqrt {c x +1}}\right )}{11025 c^{8} \left (e \,x^{2}+d \right )^{2} x^{2}}\)	\(392\)

Input:

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

Output:

a*(1/7*e^2*x^7+2/5*d*e*x^5+1/3*d^2*x^3)+b/c^3*(1/7*c^3*arccosh(c*x)*e^2*x^ 
7+2/5*c^3*arccosh(c*x)*d*e*x^5+1/3*arccosh(c*x)*c^3*x^3*d^2-1/11025/c^4*(c 
*x-1)^(1/2)*(c*x+1)^(1/2)*(225*c^6*e^2*x^6+882*c^6*d*e*x^4+1225*c^6*d^2*x^ 
2+270*c^4*e^2*x^4+1176*c^4*d*e*x^2+2450*c^4*d^2+360*c^2*e^2*x^2+2352*c^2*d 
*e+720*e^2))
 

Fricas [A] (verification not implemented)

Time = 0.10 (sec) , antiderivative size = 198, normalized size of antiderivative = 0.93 \[ \int x^2 \left (d+e x^2\right )^2 (a+b \text {arccosh}(c x)) \, dx=\frac {1575 \, a c^{7} e^{2} x^{7} + 4410 \, a c^{7} d e x^{5} + 3675 \, a c^{7} d^{2} x^{3} + 105 \, {\left (15 \, b c^{7} e^{2} x^{7} + 42 \, b c^{7} d e x^{5} + 35 \, b c^{7} d^{2} x^{3}\right )} \log \left (c x + \sqrt {c^{2} x^{2} - 1}\right ) - {\left (225 \, b c^{6} e^{2} x^{6} + 2450 \, b c^{4} d^{2} + 2352 \, b c^{2} d e + 18 \, {\left (49 \, b c^{6} d e + 15 \, b c^{4} e^{2}\right )} x^{4} + 720 \, b e^{2} + {\left (1225 \, b c^{6} d^{2} + 1176 \, b c^{4} d e + 360 \, b c^{2} e^{2}\right )} x^{2}\right )} \sqrt {c^{2} x^{2} - 1}}{11025 \, c^{7}} \] Input:

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

Output:

1/11025*(1575*a*c^7*e^2*x^7 + 4410*a*c^7*d*e*x^5 + 3675*a*c^7*d^2*x^3 + 10 
5*(15*b*c^7*e^2*x^7 + 42*b*c^7*d*e*x^5 + 35*b*c^7*d^2*x^3)*log(c*x + sqrt( 
c^2*x^2 - 1)) - (225*b*c^6*e^2*x^6 + 2450*b*c^4*d^2 + 2352*b*c^2*d*e + 18* 
(49*b*c^6*d*e + 15*b*c^4*e^2)*x^4 + 720*b*e^2 + (1225*b*c^6*d^2 + 1176*b*c 
^4*d*e + 360*b*c^2*e^2)*x^2)*sqrt(c^2*x^2 - 1))/c^7
 

Sympy [F]

\[ \int x^2 \left (d+e x^2\right )^2 (a+b \text {arccosh}(c x)) \, dx=\int x^{2} \left (a + b \operatorname {acosh}{\left (c x \right )}\right ) \left (d + e x^{2}\right )^{2}\, dx \] Input:

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

Output:

Integral(x**2*(a + b*acosh(c*x))*(d + e*x**2)**2, x)
 

Maxima [A] (verification not implemented)

Time = 0.04 (sec) , antiderivative size = 247, normalized size of antiderivative = 1.15 \[ \int x^2 \left (d+e x^2\right )^2 (a+b \text {arccosh}(c x)) \, dx=\frac {1}{7} \, a e^{2} x^{7} + \frac {2}{5} \, a d e x^{5} + \frac {1}{3} \, a d^{2} x^{3} + \frac {1}{9} \, {\left (3 \, x^{3} \operatorname {arcosh}\left (c x\right ) - c {\left (\frac {\sqrt {c^{2} x^{2} - 1} x^{2}}{c^{2}} + \frac {2 \, \sqrt {c^{2} x^{2} - 1}}{c^{4}}\right )}\right )} b d^{2} + \frac {2}{75} \, {\left (15 \, x^{5} \operatorname {arcosh}\left (c x\right ) - {\left (\frac {3 \, \sqrt {c^{2} x^{2} - 1} x^{4}}{c^{2}} + \frac {4 \, \sqrt {c^{2} x^{2} - 1} x^{2}}{c^{4}} + \frac {8 \, \sqrt {c^{2} x^{2} - 1}}{c^{6}}\right )} c\right )} b d e + \frac {1}{245} \, {\left (35 \, x^{7} \operatorname {arcosh}\left (c x\right ) - {\left (\frac {5 \, \sqrt {c^{2} x^{2} - 1} x^{6}}{c^{2}} + \frac {6 \, \sqrt {c^{2} x^{2} - 1} x^{4}}{c^{4}} + \frac {8 \, \sqrt {c^{2} x^{2} - 1} x^{2}}{c^{6}} + \frac {16 \, \sqrt {c^{2} x^{2} - 1}}{c^{8}}\right )} c\right )} b e^{2} \] Input:

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

Output:

1/7*a*e^2*x^7 + 2/5*a*d*e*x^5 + 1/3*a*d^2*x^3 + 1/9*(3*x^3*arccosh(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 + 2/75*(15* 
x^5*arccosh(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*d*e + 1/245*(35*x^7*arccosh(c*x) - (5* 
sqrt(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*e^2
 

Giac [F(-2)]

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

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

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

Output:

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

Reduce [B] (verification not implemented)

Time = 0.19 (sec) , antiderivative size = 271, normalized size of antiderivative = 1.27 \[ \int x^2 \left (d+e x^2\right )^2 (a+b \text {arccosh}(c x)) \, dx=\frac {3675 \mathit {acosh} \left (c x \right ) b \,c^{7} d^{2} x^{3}+4410 \mathit {acosh} \left (c x \right ) b \,c^{7} d e \,x^{5}+1575 \mathit {acosh} \left (c x \right ) b \,c^{7} e^{2} x^{7}-1225 \sqrt {c^{2} x^{2}-1}\, b \,c^{6} d^{2} x^{2}-882 \sqrt {c^{2} x^{2}-1}\, b \,c^{6} d e \,x^{4}-225 \sqrt {c^{2} x^{2}-1}\, b \,c^{6} e^{2} x^{6}-2450 \sqrt {c^{2} x^{2}-1}\, b \,c^{4} d^{2}-1176 \sqrt {c^{2} x^{2}-1}\, b \,c^{4} d e \,x^{2}-270 \sqrt {c^{2} x^{2}-1}\, b \,c^{4} e^{2} x^{4}-2352 \sqrt {c^{2} x^{2}-1}\, b \,c^{2} d e -360 \sqrt {c^{2} x^{2}-1}\, b \,c^{2} e^{2} x^{2}-720 \sqrt {c^{2} x^{2}-1}\, b \,e^{2}+3675 a \,c^{7} d^{2} x^{3}+4410 a \,c^{7} d e \,x^{5}+1575 a \,c^{7} e^{2} x^{7}}{11025 c^{7}} \] Input:

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

Output:

(3675*acosh(c*x)*b*c**7*d**2*x**3 + 4410*acosh(c*x)*b*c**7*d*e*x**5 + 1575 
*acosh(c*x)*b*c**7*e**2*x**7 - 1225*sqrt(c**2*x**2 - 1)*b*c**6*d**2*x**2 - 
 882*sqrt(c**2*x**2 - 1)*b*c**6*d*e*x**4 - 225*sqrt(c**2*x**2 - 1)*b*c**6* 
e**2*x**6 - 2450*sqrt(c**2*x**2 - 1)*b*c**4*d**2 - 1176*sqrt(c**2*x**2 - 1 
)*b*c**4*d*e*x**2 - 270*sqrt(c**2*x**2 - 1)*b*c**4*e**2*x**4 - 2352*sqrt(c 
**2*x**2 - 1)*b*c**2*d*e - 360*sqrt(c**2*x**2 - 1)*b*c**2*e**2*x**2 - 720* 
sqrt(c**2*x**2 - 1)*b*e**2 + 3675*a*c**7*d**2*x**3 + 4410*a*c**7*d*e*x**5 
+ 1575*a*c**7*e**2*x**7)/(11025*c**7)