\(\int x^4 (d+e x^2)^3 (a+b \text {arccosh}(c x)) \, dx\) [479]

Optimal result

Integrand size = 21, antiderivative size = 435 \[ \int x^4 \left (d+e x^2\right )^3 (a+b \text {arccosh}(c x)) \, dx=\frac {b \left (231 c^6 d^3+495 c^4 d^2 e+385 c^2 d e^2+105 e^3\right ) \left (1-c^2 x^2\right )}{1155 c^{11} \sqrt {-1+c x} \sqrt {1+c x}}-\frac {b \left (462 c^6 d^3+1485 c^4 d^2 e+1540 c^2 d e^2+525 e^3\right ) \left (1-c^2 x^2\right )^2}{3465 c^{11} \sqrt {-1+c x} \sqrt {1+c x}}+\frac {b \left (77 c^6 d^3+495 c^4 d^2 e+770 c^2 d e^2+350 e^3\right ) \left (1-c^2 x^2\right )^3}{1925 c^{11} \sqrt {-1+c x} \sqrt {1+c x}}-\frac {b e \left (99 c^4 d^2+308 c^2 d e+210 e^2\right ) \left (1-c^2 x^2\right )^4}{1617 c^{11} \sqrt {-1+c x} \sqrt {1+c x}}+\frac {b e^2 \left (11 c^2 d+15 e\right ) \left (1-c^2 x^2\right )^5}{297 c^{11} \sqrt {-1+c x} \sqrt {1+c x}}-\frac {b e^3 \left (1-c^2 x^2\right )^6}{121 c^{11} \sqrt {-1+c x} \sqrt {1+c x}}+\frac {1}{5} d^3 x^5 (a+b \text {arccosh}(c x))+\frac {3}{7} d^2 e x^7 (a+b \text {arccosh}(c x))+\frac {1}{3} d e^2 x^9 (a+b \text {arccosh}(c x))+\frac {1}{11} e^3 x^{11} (a+b \text {arccosh}(c x)) \]

[Out]

1/5*d^3*x^5*(a+b*arccosh(c*x))+3/7*d^2*e*x^7*(a+b*arccosh(c*x))+1/3*d*e^2*x^9*(a+b*arccosh(c*x))+1/11*e^3*x^11
*(a+b*arccosh(c*x))+1/1155*b*(231*c^6*d^3+495*c^4*d^2*e+385*c^2*d*e^2+105*e^3)*(-c^2*x^2+1)/c^11/(c*x-1)^(1/2)
/(c*x+1)^(1/2)-1/3465*b*(462*c^6*d^3+1485*c^4*d^2*e+1540*c^2*d*e^2+525*e^3)*(-c^2*x^2+1)^2/c^11/(c*x-1)^(1/2)/
(c*x+1)^(1/2)+1/1925*b*(77*c^6*d^3+495*c^4*d^2*e+770*c^2*d*e^2+350*e^3)*(-c^2*x^2+1)^3/c^11/(c*x-1)^(1/2)/(c*x
+1)^(1/2)-1/1617*b*e*(99*c^4*d^2+308*c^2*d*e+210*e^2)*(-c^2*x^2+1)^4/c^11/(c*x-1)^(1/2)/(c*x+1)^(1/2)+1/297*b*
e^2*(11*c^2*d+15*e)*(-c^2*x^2+1)^5/c^11/(c*x-1)^(1/2)/(c*x+1)^(1/2)-1/121*b*e^3*(-c^2*x^2+1)^6/c^11/(c*x-1)^(1
/2)/(c*x+1)^(1/2)

Rubi [A] (verified)

Time = 0.43 (sec) , antiderivative size = 435, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.286, Rules used = {276, 5958, 12, 1624, 1813, 1634} \[ \int x^4 \left (d+e x^2\right )^3 (a+b \text {arccosh}(c x)) \, dx=\frac {1}{5} d^3 x^5 (a+b \text {arccosh}(c x))+\frac {3}{7} d^2 e x^7 (a+b \text {arccosh}(c x))+\frac {1}{3} d e^2 x^9 (a+b \text {arccosh}(c x))+\frac {1}{11} e^3 x^{11} (a+b \text {arccosh}(c x))+\frac {b e^2 \left (1-c^2 x^2\right )^5 \left (11 c^2 d+15 e\right )}{297 c^{11} \sqrt {c x-1} \sqrt {c x+1}}-\frac {b e^3 \left (1-c^2 x^2\right )^6}{121 c^{11} \sqrt {c x-1} \sqrt {c x+1}}-\frac {b e \left (1-c^2 x^2\right )^4 \left (99 c^4 d^2+308 c^2 d e+210 e^2\right )}{1617 c^{11} \sqrt {c x-1} \sqrt {c x+1}}+\frac {b \left (1-c^2 x^2\right )^3 \left (77 c^6 d^3+495 c^4 d^2 e+770 c^2 d e^2+350 e^3\right )}{1925 c^{11} \sqrt {c x-1} \sqrt {c x+1}}-\frac {b \left (1-c^2 x^2\right )^2 \left (462 c^6 d^3+1485 c^4 d^2 e+1540 c^2 d e^2+525 e^3\right )}{3465 c^{11} \sqrt {c x-1} \sqrt {c x+1}}+\frac {b \left (1-c^2 x^2\right ) \left (231 c^6 d^3+495 c^4 d^2 e+385 c^2 d e^2+105 e^3\right )}{1155 c^{11} \sqrt {c x-1} \sqrt {c x+1}} \]

[In]

Int[x^4*(d + e*x^2)^3*(a + b*ArcCosh[c*x]),x]

[Out]

(b*(231*c^6*d^3 + 495*c^4*d^2*e + 385*c^2*d*e^2 + 105*e^3)*(1 - c^2*x^2))/(1155*c^11*Sqrt[-1 + c*x]*Sqrt[1 + c
*x]) - (b*(462*c^6*d^3 + 1485*c^4*d^2*e + 1540*c^2*d*e^2 + 525*e^3)*(1 - c^2*x^2)^2)/(3465*c^11*Sqrt[-1 + c*x]
*Sqrt[1 + c*x]) + (b*(77*c^6*d^3 + 495*c^4*d^2*e + 770*c^2*d*e^2 + 350*e^3)*(1 - c^2*x^2)^3)/(1925*c^11*Sqrt[-
1 + c*x]*Sqrt[1 + c*x]) - (b*e*(99*c^4*d^2 + 308*c^2*d*e + 210*e^2)*(1 - c^2*x^2)^4)/(1617*c^11*Sqrt[-1 + c*x]
*Sqrt[1 + c*x]) + (b*e^2*(11*c^2*d + 15*e)*(1 - c^2*x^2)^5)/(297*c^11*Sqrt[-1 + c*x]*Sqrt[1 + c*x]) - (b*e^3*(
1 - c^2*x^2)^6)/(121*c^11*Sqrt[-1 + c*x]*Sqrt[1 + c*x]) + (d^3*x^5*(a + b*ArcCosh[c*x]))/5 + (3*d^2*e*x^7*(a +
b*ArcCosh[c*x]))/7 + (d*e^2*x^9*(a + b*ArcCosh[c*x]))/3 + (e^3*x^11*(a + b*ArcCosh[c*x]))/11

Rule 12

Int[(a_)*(u_), x_Symbol] :> Dist[a, Int[u, x], x] /; FreeQ[a, x] &&  !MatchQ[u, (b_)*(v_) /; FreeQ[b, x]]

Rule 276

Int[((c_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_.), x_Symbol] :> Int[ExpandIntegrand[(c*x)^m*(a + b*x^n)^p,
x], x] /; FreeQ[{a, b, c, m, n}, x] && IGtQ[p, 0]

Rule 1624

Int[(Px_)*((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_)*((e_.) + (f_.)*(x_))^(p_.), x_Symbol] :> Dist[(a
+ b*x)^FracPart[m]*((c + d*x)^FracPart[m]/(a*c + b*d*x^2)^FracPart[m]), Int[Px*(a*c + b*d*x^2)^m*(e + f*x)^p,
x], x] /; FreeQ[{a, b, c, d, e, f, m, n, p}, x] && PolyQ[Px, x] && EqQ[b*c + a*d, 0] && EqQ[m, n] &&  !Intege
rQ[m]

Rule 1634

Int[(Px_)*((a_.) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_.), x_Symbol] :> Int[ExpandIntegrand[Px*(a + b*x)
^m*(c + d*x)^n, x], x] /; FreeQ[{a, b, c, d, m, n}, x] && PolyQ[Px, x] && (IntegersQ[m, n] || IGtQ[m, -2]) &&
GtQ[Expon[Px, x], 2]

Rule 1813

Int[(Pq_)*(x_)^(m_.)*((a_) + (b_.)*(x_)^2)^(p_.), x_Symbol] :> Dist[1/2, Subst[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 5958

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]}, Dist[a + b*ArcCosh[c*x], u, x] - Dist[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] && LeQ[m + p, 0]))

Rubi steps \begin{align*} \text {integral}& = \frac {1}{5} d^3 x^5 (a+b \text {arccosh}(c x))+\frac {3}{7} d^2 e x^7 (a+b \text {arccosh}(c x))+\frac {1}{3} d e^2 x^9 (a+b \text {arccosh}(c x))+\frac {1}{11} e^3 x^{11} (a+b \text {arccosh}(c x))-(b c) \int \frac {x^5 \left (231 d^3+495 d^2 e x^2+385 d e^2 x^4+105 e^3 x^6\right )}{1155 \sqrt {-1+c x} \sqrt {1+c x}} \, dx \\ & = \frac {1}{5} d^3 x^5 (a+b \text {arccosh}(c x))+\frac {3}{7} d^2 e x^7 (a+b \text {arccosh}(c x))+\frac {1}{3} d e^2 x^9 (a+b \text {arccosh}(c x))+\frac {1}{11} e^3 x^{11} (a+b \text {arccosh}(c x))-\frac {(b c) \int \frac {x^5 \left (231 d^3+495 d^2 e x^2+385 d e^2 x^4+105 e^3 x^6\right )}{\sqrt {-1+c x} \sqrt {1+c x}} \, dx}{1155} \\ & = \frac {1}{5} d^3 x^5 (a+b \text {arccosh}(c x))+\frac {3}{7} d^2 e x^7 (a+b \text {arccosh}(c x))+\frac {1}{3} d e^2 x^9 (a+b \text {arccosh}(c x))+\frac {1}{11} e^3 x^{11} (a+b \text {arccosh}(c x))-\frac {\left (b c \sqrt {-1+c^2 x^2}\right ) \int \frac {x^5 \left (231 d^3+495 d^2 e x^2+385 d e^2 x^4+105 e^3 x^6\right )}{\sqrt {-1+c^2 x^2}} \, dx}{1155 \sqrt {-1+c x} \sqrt {1+c x}} \\ & = \frac {1}{5} d^3 x^5 (a+b \text {arccosh}(c x))+\frac {3}{7} d^2 e x^7 (a+b \text {arccosh}(c x))+\frac {1}{3} d e^2 x^9 (a+b \text {arccosh}(c x))+\frac {1}{11} e^3 x^{11} (a+b \text {arccosh}(c x))-\frac {\left (b c \sqrt {-1+c^2 x^2}\right ) \text {Subst}\left (\int \frac {x^2 \left (231 d^3+495 d^2 e x+385 d e^2 x^2+105 e^3 x^3\right )}{\sqrt {-1+c^2 x}} \, dx,x,x^2\right )}{2310 \sqrt {-1+c x} \sqrt {1+c x}} \\ & = \frac {1}{5} d^3 x^5 (a+b \text {arccosh}(c x))+\frac {3}{7} d^2 e x^7 (a+b \text {arccosh}(c x))+\frac {1}{3} d e^2 x^9 (a+b \text {arccosh}(c x))+\frac {1}{11} e^3 x^{11} (a+b \text {arccosh}(c x))-\frac {\left (b c \sqrt {-1+c^2 x^2}\right ) \text {Subst}\left (\int \left (\frac {231 c^6 d^3+495 c^4 d^2 e+385 c^2 d e^2+105 e^3}{c^{10} \sqrt {-1+c^2 x}}+\frac {\left (462 c^6 d^3+1485 c^4 d^2 e+1540 c^2 d e^2+525 e^3\right ) \sqrt {-1+c^2 x}}{c^{10}}+\frac {3 \left (77 c^6 d^3+495 c^4 d^2 e+770 c^2 d e^2+350 e^3\right ) \left (-1+c^2 x\right )^{3/2}}{c^{10}}+\frac {5 e \left (99 c^4 d^2+308 c^2 d e+210 e^2\right ) \left (-1+c^2 x\right )^{5/2}}{c^{10}}+\frac {35 e^2 \left (11 c^2 d+15 e\right ) \left (-1+c^2 x\right )^{7/2}}{c^{10}}+\frac {105 e^3 \left (-1+c^2 x\right )^{9/2}}{c^{10}}\right ) \, dx,x,x^2\right )}{2310 \sqrt {-1+c x} \sqrt {1+c x}} \\ & = \frac {b \left (231 c^6 d^3+495 c^4 d^2 e+385 c^2 d e^2+105 e^3\right ) \left (1-c^2 x^2\right )}{1155 c^{11} \sqrt {-1+c x} \sqrt {1+c x}}-\frac {b \left (462 c^6 d^3+1485 c^4 d^2 e+1540 c^2 d e^2+525 e^3\right ) \left (1-c^2 x^2\right )^2}{3465 c^{11} \sqrt {-1+c x} \sqrt {1+c x}}+\frac {b \left (77 c^6 d^3+495 c^4 d^2 e+770 c^2 d e^2+350 e^3\right ) \left (1-c^2 x^2\right )^3}{1925 c^{11} \sqrt {-1+c x} \sqrt {1+c x}}-\frac {b e \left (99 c^4 d^2+308 c^2 d e+210 e^2\right ) \left (1-c^2 x^2\right )^4}{1617 c^{11} \sqrt {-1+c x} \sqrt {1+c x}}+\frac {b e^2 \left (11 c^2 d+15 e\right ) \left (1-c^2 x^2\right )^5}{297 c^{11} \sqrt {-1+c x} \sqrt {1+c x}}-\frac {b e^3 \left (1-c^2 x^2\right )^6}{121 c^{11} \sqrt {-1+c x} \sqrt {1+c x}}+\frac {1}{5} d^3 x^5 (a+b \text {arccosh}(c x))+\frac {3}{7} d^2 e x^7 (a+b \text {arccosh}(c x))+\frac {1}{3} d e^2 x^9 (a+b \text {arccosh}(c x))+\frac {1}{11} e^3 x^{11} (a+b \text {arccosh}(c x)) \\ \end{align*}

Mathematica [A] (verified)

Time = 0.28 (sec) , antiderivative size = 276, normalized size of antiderivative = 0.63 \[ \int x^4 \left (d+e x^2\right )^3 (a+b \text {arccosh}(c x)) \, dx=\frac {3465 a x^5 \left (231 d^3+495 d^2 e x^2+385 d e^2 x^4+105 e^3 x^6\right )-\frac {b \sqrt {-1+c x} \sqrt {1+c x} \left (134400 e^3+4480 c^2 e^2 \left (121 d+15 e x^2\right )+80 c^4 e \left (9801 d^2+3388 d e x^2+630 e^2 x^4\right )+24 c^6 \left (17787 d^3+16335 d^2 e x^2+8470 d e^2 x^4+1750 e^3 x^6\right )+c^{10} x^4 \left (160083 d^3+245025 d^2 e x^2+148225 d e^2 x^4+33075 e^3 x^6\right )+2 c^8 \left (106722 d^3 x^2+147015 d^2 e x^4+84700 d e^2 x^6+18375 e^3 x^8\right )\right )}{c^{11}}+3465 b x^5 \left (231 d^3+495 d^2 e x^2+385 d e^2 x^4+105 e^3 x^6\right ) \text {arccosh}(c x)}{4002075} \]

[In]

Integrate[x^4*(d + e*x^2)^3*(a + b*ArcCosh[c*x]),x]

[Out]

(3465*a*x^5*(231*d^3 + 495*d^2*e*x^2 + 385*d*e^2*x^4 + 105*e^3*x^6) - (b*Sqrt[-1 + c*x]*Sqrt[1 + c*x]*(134400*
e^3 + 4480*c^2*e^2*(121*d + 15*e*x^2) + 80*c^4*e*(9801*d^2 + 3388*d*e*x^2 + 630*e^2*x^4) + 24*c^6*(17787*d^3 +
16335*d^2*e*x^2 + 8470*d*e^2*x^4 + 1750*e^3*x^6) + c^10*x^4*(160083*d^3 + 245025*d^2*e*x^2 + 148225*d*e^2*x^4
+ 33075*e^3*x^6) + 2*c^8*(106722*d^3*x^2 + 147015*d^2*e*x^4 + 84700*d*e^2*x^6 + 18375*e^3*x^8)))/c^11 + 3465*
b*x^5*(231*d^3 + 495*d^2*e*x^2 + 385*d*e^2*x^4 + 105*e^3*x^6)*ArcCosh[c*x])/4002075

Maple [A] (verified)

Time = 0.74 (sec) , antiderivative size = 319, normalized size of antiderivative = 0.73

method	result	size
parts	\(a \left (\frac {1}{11} e^{3} x^{11}+\frac {1}{3} d \,e^{2} x^{9}+\frac {3}{7} d^{2} e \,x^{7}+\frac {1}{5} d^{3} x^{5}\right )+\frac {b \left (\frac {c^{5} \operatorname {arccosh}\left (c x \right ) e^{3} x^{11}}{11}+\frac {c^{5} \operatorname {arccosh}\left (c x \right ) d \,e^{2} x^{9}}{3}+\frac {3 c^{5} \operatorname {arccosh}\left (c x \right ) d^{2} e \,x^{7}}{7}+\frac {\operatorname {arccosh}\left (c x \right ) c^{5} x^{5} d^{3}}{5}-\frac {\sqrt {c x -1}\, \sqrt {c x +1}\, \left (33075 c^{10} e^{3} x^{10}+148225 c^{10} d \,e^{2} x^{8}+245025 c^{10} d^{2} e \,x^{6}+36750 c^{8} e^{3} x^{8}+160083 c^{10} d^{3} x^{4}+169400 c^{8} d \,e^{2} x^{6}+294030 c^{8} d^{2} e \,x^{4}+42000 c^{6} e^{3} x^{6}+213444 c^{8} d^{3} x^{2}+203280 c^{6} d \,e^{2} x^{4}+392040 c^{6} d^{2} e \,x^{2}+50400 c^{4} x^{4} e^{3}+426888 d^{3} c^{6}+271040 c^{4} d \,e^{2} x^{2}+784080 c^{4} d^{2} e +67200 c^{2} x^{2} e^{3}+542080 c^{2} d \,e^{2}+134400 e^{3}\right )}{4002075 c^{6}}\right )}{c^{5}}\)	\(319\)
derivativedivides	\(\frac {\frac {a \left (\frac {1}{5} c^{11} d^{3} x^{5}+\frac {3}{7} c^{11} d^{2} e \,x^{7}+\frac {1}{3} c^{11} d \,e^{2} x^{9}+\frac {1}{11} e^{3} c^{11} x^{11}\right )}{c^{6}}+\frac {b \left (\frac {\operatorname {arccosh}\left (c x \right ) c^{11} d^{3} x^{5}}{5}+\frac {3 \,\operatorname {arccosh}\left (c x \right ) c^{11} d^{2} e \,x^{7}}{7}+\frac {\operatorname {arccosh}\left (c x \right ) c^{11} d \,e^{2} x^{9}}{3}+\frac {\operatorname {arccosh}\left (c x \right ) e^{3} c^{11} x^{11}}{11}-\frac {\sqrt {c x -1}\, \sqrt {c x +1}\, \left (33075 c^{10} e^{3} x^{10}+148225 c^{10} d \,e^{2} x^{8}+245025 c^{10} d^{2} e \,x^{6}+36750 c^{8} e^{3} x^{8}+160083 c^{10} d^{3} x^{4}+169400 c^{8} d \,e^{2} x^{6}+294030 c^{8} d^{2} e \,x^{4}+42000 c^{6} e^{3} x^{6}+213444 c^{8} d^{3} x^{2}+203280 c^{6} d \,e^{2} x^{4}+392040 c^{6} d^{2} e \,x^{2}+50400 c^{4} x^{4} e^{3}+426888 d^{3} c^{6}+271040 c^{4} d \,e^{2} x^{2}+784080 c^{4} d^{2} e +67200 c^{2} x^{2} e^{3}+542080 c^{2} d \,e^{2}+134400 e^{3}\right )}{4002075}\right )}{c^{6}}}{c^{5}}\)	\(335\)
default	\(\frac {\frac {a \left (\frac {1}{5} c^{11} d^{3} x^{5}+\frac {3}{7} c^{11} d^{2} e \,x^{7}+\frac {1}{3} c^{11} d \,e^{2} x^{9}+\frac {1}{11} e^{3} c^{11} x^{11}\right )}{c^{6}}+\frac {b \left (\frac {\operatorname {arccosh}\left (c x \right ) c^{11} d^{3} x^{5}}{5}+\frac {3 \,\operatorname {arccosh}\left (c x \right ) c^{11} d^{2} e \,x^{7}}{7}+\frac {\operatorname {arccosh}\left (c x \right ) c^{11} d \,e^{2} x^{9}}{3}+\frac {\operatorname {arccosh}\left (c x \right ) e^{3} c^{11} x^{11}}{11}-\frac {\sqrt {c x -1}\, \sqrt {c x +1}\, \left (33075 c^{10} e^{3} x^{10}+148225 c^{10} d \,e^{2} x^{8}+245025 c^{10} d^{2} e \,x^{6}+36750 c^{8} e^{3} x^{8}+160083 c^{10} d^{3} x^{4}+169400 c^{8} d \,e^{2} x^{6}+294030 c^{8} d^{2} e \,x^{4}+42000 c^{6} e^{3} x^{6}+213444 c^{8} d^{3} x^{2}+203280 c^{6} d \,e^{2} x^{4}+392040 c^{6} d^{2} e \,x^{2}+50400 c^{4} x^{4} e^{3}+426888 d^{3} c^{6}+271040 c^{4} d \,e^{2} x^{2}+784080 c^{4} d^{2} e +67200 c^{2} x^{2} e^{3}+542080 c^{2} d \,e^{2}+134400 e^{3}\right )}{4002075}\right )}{c^{6}}}{c^{5}}\)	\(335\)

[In]

int(x^4*(e*x^2+d)^3*(a+b*arccosh(c*x)),x,method=_RETURNVERBOSE)

[Out]

a*(1/11*e^3*x^11+1/3*d*e^2*x^9+3/7*d^2*e*x^7+1/5*d^3*x^5)+b/c^5*(1/11*c^5*arccosh(c*x)*e^3*x^11+1/3*c^5*arccos
h(c*x)*d*e^2*x^9+3/7*c^5*arccosh(c*x)*d^2*e*x^7+1/5*arccosh(c*x)*c^5*x^5*d^3-1/4002075/c^6*(c*x-1)^(1/2)*(c*x+
1)^(1/2)*(33075*c^10*e^3*x^10+148225*c^10*d*e^2*x^8+245025*c^10*d^2*e*x^6+36750*c^8*e^3*x^8+160083*c^10*d^3*x^
4+169400*c^8*d*e^2*x^6+294030*c^8*d^2*e*x^4+42000*c^6*e^3*x^6+213444*c^8*d^3*x^2+203280*c^6*d*e^2*x^4+392040*c
^6*d^2*e*x^2+50400*c^4*e^3*x^4+426888*c^6*d^3+271040*c^4*d*e^2*x^2+784080*c^4*d^2*e+67200*c^2*e^3*x^2+542080*c
^2*d*e^2+134400*e^3))

Fricas [A] (verification not implemented)

none

Time = 0.26 (sec) , antiderivative size = 334, normalized size of antiderivative = 0.77 \[ \int x^4 \left (d+e x^2\right )^3 (a+b \text {arccosh}(c x)) \, dx=\frac {363825 \, a c^{11} e^{3} x^{11} + 1334025 \, a c^{11} d e^{2} x^{9} + 1715175 \, a c^{11} d^{2} e x^{7} + 800415 \, a c^{11} d^{3} x^{5} + 3465 \, {\left (105 \, b c^{11} e^{3} x^{11} + 385 \, b c^{11} d e^{2} x^{9} + 495 \, b c^{11} d^{2} e x^{7} + 231 \, b c^{11} d^{3} x^{5}\right )} \log \left (c x + \sqrt {c^{2} x^{2} - 1}\right ) - {\left (33075 \, b c^{10} e^{3} x^{10} + 426888 \, b c^{6} d^{3} + 1225 \, {\left (121 \, b c^{10} d e^{2} + 30 \, b c^{8} e^{3}\right )} x^{8} + 784080 \, b c^{4} d^{2} e + 25 \, {\left (9801 \, b c^{10} d^{2} e + 6776 \, b c^{8} d e^{2} + 1680 \, b c^{6} e^{3}\right )} x^{6} + 542080 \, b c^{2} d e^{2} + 3 \, {\left (53361 \, b c^{10} d^{3} + 98010 \, b c^{8} d^{2} e + 67760 \, b c^{6} d e^{2} + 16800 \, b c^{4} e^{3}\right )} x^{4} + 134400 \, b e^{3} + 4 \, {\left (53361 \, b c^{8} d^{3} + 98010 \, b c^{6} d^{2} e + 67760 \, b c^{4} d e^{2} + 16800 \, b c^{2} e^{3}\right )} x^{2}\right )} \sqrt {c^{2} x^{2} - 1}}{4002075 \, c^{11}} \]

[In]

integrate(x^4*(e*x^2+d)^3*(a+b*arccosh(c*x)),x, algorithm="fricas")

[Out]

1/4002075*(363825*a*c^11*e^3*x^11 + 1334025*a*c^11*d*e^2*x^9 + 1715175*a*c^11*d^2*e*x^7 + 800415*a*c^11*d^3*x^
5 + 3465*(105*b*c^11*e^3*x^11 + 385*b*c^11*d*e^2*x^9 + 495*b*c^11*d^2*e*x^7 + 231*b*c^11*d^3*x^5)*log(c*x + sq
rt(c^2*x^2 - 1)) - (33075*b*c^10*e^3*x^10 + 426888*b*c^6*d^3 + 1225*(121*b*c^10*d*e^2 + 30*b*c^8*e^3)*x^8 + 78
4080*b*c^4*d^2*e + 25*(9801*b*c^10*d^2*e + 6776*b*c^8*d*e^2 + 1680*b*c^6*e^3)*x^6 + 542080*b*c^2*d*e^2 + 3*(53
361*b*c^10*d^3 + 98010*b*c^8*d^2*e + 67760*b*c^6*d*e^2 + 16800*b*c^4*e^3)*x^4 + 134400*b*e^3 + 4*(53361*b*c^8*
d^3 + 98010*b*c^6*d^2*e + 67760*b*c^4*d*e^2 + 16800*b*c^2*e^3)*x^2)*sqrt(c^2*x^2 - 1))/c^11

Sympy [F(-1)]

Timed out. \[ \int x^4 \left (d+e x^2\right )^3 (a+b \text {arccosh}(c x)) \, dx=\text {Timed out} \]

[In]

integrate(x**4*(e*x**2+d)**3*(a+b*acosh(c*x)),x)

[Out]

Timed out

Maxima [A] (verification not implemented)

none

Time = 0.22 (sec) , antiderivative size = 451, normalized size of antiderivative = 1.04 \[ \int x^4 \left (d+e x^2\right )^3 (a+b \text {arccosh}(c x)) \, dx=\frac {1}{11} \, a e^{3} x^{11} + \frac {1}{3} \, a d e^{2} x^{9} + \frac {3}{7} \, a d^{2} e x^{7} + \frac {1}{5} \, a d^{3} x^{5} + \frac {1}{75} \, {\left (15 \, x^{5} \operatorname {arcosh}\left (c x\right ) - {\left (\frac {3 \, \sqrt {c^{2} x^{2} - 1} x^{4}}{c^{2}} + \frac {4 \, \sqrt {c^{2} x^{2} - 1} x^{2}}{c^{4}} + \frac {8 \, \sqrt {c^{2} x^{2} - 1}}{c^{6}}\right )} c\right )} b d^{3} + \frac {3}{245} \, {\left (35 \, x^{7} \operatorname {arcosh}\left (c x\right ) - {\left (\frac {5 \, \sqrt {c^{2} x^{2} - 1} x^{6}}{c^{2}} + \frac {6 \, \sqrt {c^{2} x^{2} - 1} x^{4}}{c^{4}} + \frac {8 \, \sqrt {c^{2} x^{2} - 1} x^{2}}{c^{6}} + \frac {16 \, \sqrt {c^{2} x^{2} - 1}}{c^{8}}\right )} c\right )} b d^{2} e + \frac {1}{945} \, {\left (315 \, x^{9} \operatorname {arcosh}\left (c x\right ) - {\left (\frac {35 \, \sqrt {c^{2} x^{2} - 1} x^{8}}{c^{2}} + \frac {40 \, \sqrt {c^{2} x^{2} - 1} x^{6}}{c^{4}} + \frac {48 \, \sqrt {c^{2} x^{2} - 1} x^{4}}{c^{6}} + \frac {64 \, \sqrt {c^{2} x^{2} - 1} x^{2}}{c^{8}} + \frac {128 \, \sqrt {c^{2} x^{2} - 1}}{c^{10}}\right )} c\right )} b d e^{2} + \frac {1}{7623} \, {\left (693 \, x^{11} \operatorname {arcosh}\left (c x\right ) - {\left (\frac {63 \, \sqrt {c^{2} x^{2} - 1} x^{10}}{c^{2}} + \frac {70 \, \sqrt {c^{2} x^{2} - 1} x^{8}}{c^{4}} + \frac {80 \, \sqrt {c^{2} x^{2} - 1} x^{6}}{c^{6}} + \frac {96 \, \sqrt {c^{2} x^{2} - 1} x^{4}}{c^{8}} + \frac {128 \, \sqrt {c^{2} x^{2} - 1} x^{2}}{c^{10}} + \frac {256 \, \sqrt {c^{2} x^{2} - 1}}{c^{12}}\right )} c\right )} b e^{3} \]

[In]

integrate(x^4*(e*x^2+d)^3*(a+b*arccosh(c*x)),x, algorithm="maxima")

[Out]

1/11*a*e^3*x^11 + 1/3*a*d*e^2*x^9 + 3/7*a*d^2*e*x^7 + 1/5*a*d^3*x^5 + 1/75*(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^3 + 3/245*(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*d^2*e + 1/945*(315*x^9*arccosh(c*x) - (35*sqrt(c^2*x^2 - 1)*x^8/c^2 + 40*sqrt(c^2*x^2 - 1)*x^6
/c^4 + 48*sqrt(c^2*x^2 - 1)*x^4/c^6 + 64*sqrt(c^2*x^2 - 1)*x^2/c^8 + 128*sqrt(c^2*x^2 - 1)/c^10)*c)*b*d*e^2 +
1/7623*(693*x^11*arccosh(c*x) - (63*sqrt(c^2*x^2 - 1)*x^10/c^2 + 70*sqrt(c^2*x^2 - 1)*x^8/c^4 + 80*sqrt(c^2*x^
2 - 1)*x^6/c^6 + 96*sqrt(c^2*x^2 - 1)*x^4/c^8 + 128*sqrt(c^2*x^2 - 1)*x^2/c^10 + 256*sqrt(c^2*x^2 - 1)/c^12)*c
)*b*e^3

Giac [F(-2)]

Exception generated. \[ \int x^4 \left (d+e x^2\right )^3 (a+b \text {arccosh}(c x)) \, dx=\text {Exception raised: RuntimeError} \]

[In]

integrate(x^4*(e*x^2+d)^3*(a+b*arccosh(c*x)),x, algorithm="giac")

[Out]

Exception raised: RuntimeError >> an error occurred running a Giac command:INPUT:sage2OUTPUT:sym2poly/r2sym(co
nst gen & e,const index_m & i,const vecteur & l) Error: Bad Argument Value

Mupad [F(-1)]

Timed out. \[ \int x^4 \left (d+e x^2\right )^3 (a+b \text {arccosh}(c x)) \, dx=\int x^4\,\left (a+b\,\mathrm {acosh}\left (c\,x\right )\right )\,{\left (e\,x^2+d\right )}^3 \,d x \]

[In]

int(x^4*(a + b*acosh(c*x))*(d + e*x^2)^3,x)

[Out]

int(x^4*(a + b*acosh(c*x))*(d + e*x^2)^3, x)