\(\int x^3 (d+e x^2)^3 (a+b \text {arccosh}(c x)) \, dx\) [480]

Optimal result

Integrand size = 21, antiderivative size = 494 \[ \int x^3 \left (d+e x^2\right )^3 (a+b \text {arccosh}(c x)) \, dx=-\frac {b \left (1232 c^8 d^4-2536 c^6 d^3 e-7758 c^4 d^2 e^2-6615 c^2 d e^3-1890 e^4\right ) x \left (1-c^2 x^2\right )}{76800 c^9 e \sqrt {-1+c x} \sqrt {1+c x}}-\frac {b \left (136 c^6 d^3-1096 c^4 d^2 e-1617 c^2 d e^2-630 e^3\right ) x \left (1-c^2 x^2\right ) \left (d+e x^2\right )}{38400 c^7 e \sqrt {-1+c x} \sqrt {1+c x}}+\frac {b \left (26 c^4 d^2+201 c^2 d e+126 e^2\right ) x \left (1-c^2 x^2\right ) \left (d+e x^2\right )^2}{9600 c^5 e \sqrt {-1+c x} \sqrt {1+c x}}+\frac {b \left (11 c^2 d+18 e\right ) x \left (1-c^2 x^2\right ) \left (d+e x^2\right )^3}{1600 c^3 e \sqrt {-1+c x} \sqrt {1+c x}}+\frac {b x \left (1-c^2 x^2\right ) \left (d+e x^2\right )^4}{100 c e \sqrt {-1+c x} \sqrt {1+c x}}-\frac {d \left (d+e x^2\right )^4 (a+b \text {arccosh}(c x))}{8 e^2}+\frac {\left (d+e x^2\right )^5 (a+b \text {arccosh}(c x))}{10 e^2}+\frac {b \left (128 c^{10} d^5-480 c^6 d^3 e^2-800 c^4 d^2 e^3-525 c^2 d e^4-126 e^5\right ) \sqrt {-1+c^2 x^2} \text {arctanh}\left (\frac {c x}{\sqrt {-1+c^2 x^2}}\right )}{5120 c^{10} e^2 \sqrt {-1+c x} \sqrt {1+c x}} \]

[Out]

-1/8*d*(e*x^2+d)^4*(a+b*arccosh(c*x))/e^2+1/10*(e*x^2+d)^5*(a+b*arccosh(c*x))/e^2-1/76800*b*(1232*c^8*d^4-2536
*c^6*d^3*e-7758*c^4*d^2*e^2-6615*c^2*d*e^3-1890*e^4)*x*(-c^2*x^2+1)/c^9/e/(c*x-1)^(1/2)/(c*x+1)^(1/2)-1/38400*
b*(136*c^6*d^3-1096*c^4*d^2*e-1617*c^2*d*e^2-630*e^3)*x*(-c^2*x^2+1)*(e*x^2+d)/c^7/e/(c*x-1)^(1/2)/(c*x+1)^(1/
2)+1/9600*b*(26*c^4*d^2+201*c^2*d*e+126*e^2)*x*(-c^2*x^2+1)*(e*x^2+d)^2/c^5/e/(c*x-1)^(1/2)/(c*x+1)^(1/2)+1/16
00*b*(11*c^2*d+18*e)*x*(-c^2*x^2+1)*(e*x^2+d)^3/c^3/e/(c*x-1)^(1/2)/(c*x+1)^(1/2)+1/100*b*x*(-c^2*x^2+1)*(e*x^
2+d)^4/c/e/(c*x-1)^(1/2)/(c*x+1)^(1/2)+1/5120*b*(128*c^10*d^5-480*c^6*d^3*e^2-800*c^4*d^2*e^3-525*c^2*d*e^4-12
6*e^5)*arctanh(c*x/(c^2*x^2-1)^(1/2))*(c^2*x^2-1)^(1/2)/c^10/e^2/(c*x-1)^(1/2)/(c*x+1)^(1/2)

Rubi [A] (verified)

Time = 0.45 (sec) , antiderivative size = 494, normalized size of antiderivative = 1.00, number of steps used = 10, number of rules used = 9, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.429, Rules used = {272, 45, 5958, 12, 580, 542, 396, 223, 212} \[ \int x^3 \left (d+e x^2\right )^3 (a+b \text {arccosh}(c x)) \, dx=\frac {\left (d+e x^2\right )^5 (a+b \text {arccosh}(c x))}{10 e^2}-\frac {d \left (d+e x^2\right )^4 (a+b \text {arccosh}(c x))}{8 e^2}+\frac {b \sqrt {c^2 x^2-1} \text {arctanh}\left (\frac {c x}{\sqrt {c^2 x^2-1}}\right ) \left (128 c^{10} d^5-480 c^6 d^3 e^2-800 c^4 d^2 e^3-525 c^2 d e^4-126 e^5\right )}{5120 c^{10} e^2 \sqrt {c x-1} \sqrt {c x+1}}+\frac {b x \left (1-c^2 x^2\right ) \left (d+e x^2\right )^4}{100 c e \sqrt {c x-1} \sqrt {c x+1}}+\frac {b x \left (1-c^2 x^2\right ) \left (11 c^2 d+18 e\right ) \left (d+e x^2\right )^3}{1600 c^3 e \sqrt {c x-1} \sqrt {c x+1}}+\frac {b x \left (1-c^2 x^2\right ) \left (26 c^4 d^2+201 c^2 d e+126 e^2\right ) \left (d+e x^2\right )^2}{9600 c^5 e \sqrt {c x-1} \sqrt {c x+1}}-\frac {b x \left (1-c^2 x^2\right ) \left (136 c^6 d^3-1096 c^4 d^2 e-1617 c^2 d e^2-630 e^3\right ) \left (d+e x^2\right )}{38400 c^7 e \sqrt {c x-1} \sqrt {c x+1}}-\frac {b x \left (1-c^2 x^2\right ) \left (1232 c^8 d^4-2536 c^6 d^3 e-7758 c^4 d^2 e^2-6615 c^2 d e^3-1890 e^4\right )}{76800 c^9 e \sqrt {c x-1} \sqrt {c x+1}} \]

[In]

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

[Out]

-1/76800*(b*(1232*c^8*d^4 - 2536*c^6*d^3*e - 7758*c^4*d^2*e^2 - 6615*c^2*d*e^3 - 1890*e^4)*x*(1 - c^2*x^2))/(c
^9*e*Sqrt[-1 + c*x]*Sqrt[1 + c*x]) - (b*(136*c^6*d^3 - 1096*c^4*d^2*e - 1617*c^2*d*e^2 - 630*e^3)*x*(1 - c^2*x
^2)*(d + e*x^2))/(38400*c^7*e*Sqrt[-1 + c*x]*Sqrt[1 + c*x]) + (b*(26*c^4*d^2 + 201*c^2*d*e + 126*e^2)*x*(1 - c
^2*x^2)*(d + e*x^2)^2)/(9600*c^5*e*Sqrt[-1 + c*x]*Sqrt[1 + c*x]) + (b*(11*c^2*d + 18*e)*x*(1 - c^2*x^2)*(d + e
*x^2)^3)/(1600*c^3*e*Sqrt[-1 + c*x]*Sqrt[1 + c*x]) + (b*x*(1 - c^2*x^2)*(d + e*x^2)^4)/(100*c*e*Sqrt[-1 + c*x]
*Sqrt[1 + c*x]) - (d*(d + e*x^2)^4*(a + b*ArcCosh[c*x]))/(8*e^2) + ((d + e*x^2)^5*(a + b*ArcCosh[c*x]))/(10*e^
2) + (b*(128*c^10*d^5 - 480*c^6*d^3*e^2 - 800*c^4*d^2*e^3 - 525*c^2*d*e^4 - 126*e^5)*Sqrt[-1 + c^2*x^2]*ArcTan
h[(c*x)/Sqrt[-1 + c^2*x^2]])/(5120*c^10*e^2*Sqrt[-1 + c*x]*Sqrt[1 + c*x])

Rule 12

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

Rule 45

Int[((a_.) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_.), x_Symbol] :> Int[ExpandIntegrand[(a + b*x)^m*(c + d
*x)^n, x], x] /; FreeQ[{a, b, c, d, n}, x] && NeQ[b*c - a*d, 0] && IGtQ[m, 0] && ( !IntegerQ[n] || (EqQ[c, 0]
&& LeQ[7*m + 4*n + 4, 0]) || LtQ[9*m + 5*(n + 1), 0] || GtQ[m + n + 2, 0])

Rule 212

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1/(Rt[a, 2]*Rt[-b, 2]))*ArcTanh[Rt[-b, 2]*(x/Rt[a, 2])], x]
/; FreeQ[{a, b}, x] && NegQ[a/b] && (GtQ[a, 0] || LtQ[b, 0])

Rule 223

Int[1/Sqrt[(a_) + (b_.)*(x_)^2], x_Symbol] :> Subst[Int[1/(1 - b*x^2), x], x, x/Sqrt[a + b*x^2]] /; FreeQ[{a,
b}, x] &&  !GtQ[a, 0]

Rule 272

Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Dist[1/n, Subst[Int[x^(Simplify[(m + 1)/n] - 1)*(a
+ b*x)^p, x], x, x^n], x] /; FreeQ[{a, b, m, n, p}, x] && IntegerQ[Simplify[(m + 1)/n]]

Rule 396

Int[((a_) + (b_.)*(x_)^(n_))^(p_)*((c_) + (d_.)*(x_)^(n_)), x_Symbol] :> Simp[d*x*((a + b*x^n)^(p + 1)/(b*(n*(
p + 1) + 1))), x] - Dist[(a*d - b*c*(n*(p + 1) + 1))/(b*(n*(p + 1) + 1)), Int[(a + b*x^n)^p, x], x] /; FreeQ[{
a, b, c, d, n}, x] && NeQ[b*c - a*d, 0] && NeQ[n*(p + 1) + 1, 0]

Rule 542

Int[((a_) + (b_.)*(x_)^(n_))^(p_.)*((c_) + (d_.)*(x_)^(n_))^(q_.)*((e_) + (f_.)*(x_)^(n_)), x_Symbol] :> Simp[
f*x*(a + b*x^n)^(p + 1)*((c + d*x^n)^q/(b*(n*(p + q + 1) + 1))), x] + Dist[1/(b*(n*(p + q + 1) + 1)), Int[(a +
b*x^n)^p*(c + d*x^n)^(q - 1)*Simp[c*(b*e - a*f + b*e*n*(p + q + 1)) + (d*(b*e - a*f) + f*n*q*(b*c - a*d) + b*
d*e*n*(p + q + 1))*x^n, x], x], x] /; FreeQ[{a, b, c, d, e, f, n, p}, x] && GtQ[q, 0] && NeQ[n*(p + q + 1) + 1
, 0]

Rule 580

Int[((e1_) + (f1_.)*(x_)^(n2_.))^(r_.)*((e2_) + (f2_.)*(x_)^(n2_.))^(r_.)*((a_) + (b_.)*(x_)^(n_))^(p_.)*((c_)
+ (d_.)*(x_)^(n_))^(q_.), x_Symbol] :> Dist[(e1 + f1*x^(n/2))^FracPart[r]*((e2 + f2*x^(n/2))^FracPart[r]/(e1*
e2 + f1*f2*x^n)^FracPart[r]), Int[(a + b*x^n)^p*(c + d*x^n)^q*(e1*e2 + f1*f2*x^n)^r, x], x] /; FreeQ[{a, b, c,
d, e1, f1, e2, f2, n, p, q, r}, x] && EqQ[n2, n/2] && EqQ[e2*f1 + e1*f2, 0]

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 {d \left (d+e x^2\right )^4 (a+b \text {arccosh}(c x))}{8 e^2}+\frac {\left (d+e x^2\right )^5 (a+b \text {arccosh}(c x))}{10 e^2}-(b c) \int \frac {\left (d+e x^2\right )^4 \left (-d+4 e x^2\right )}{40 e^2 \sqrt {-1+c x} \sqrt {1+c x}} \, dx \\ & = -\frac {d \left (d+e x^2\right )^4 (a+b \text {arccosh}(c x))}{8 e^2}+\frac {\left (d+e x^2\right )^5 (a+b \text {arccosh}(c x))}{10 e^2}-\frac {(b c) \int \frac {\left (d+e x^2\right )^4 \left (-d+4 e x^2\right )}{\sqrt {-1+c x} \sqrt {1+c x}} \, dx}{40 e^2} \\ & = -\frac {d \left (d+e x^2\right )^4 (a+b \text {arccosh}(c x))}{8 e^2}+\frac {\left (d+e x^2\right )^5 (a+b \text {arccosh}(c x))}{10 e^2}-\frac {\left (b c \sqrt {-1+c^2 x^2}\right ) \int \frac {\left (d+e x^2\right )^4 \left (-d+4 e x^2\right )}{\sqrt {-1+c^2 x^2}} \, dx}{40 e^2 \sqrt {-1+c x} \sqrt {1+c x}} \\ & = \frac {b x \left (1-c^2 x^2\right ) \left (d+e x^2\right )^4}{100 c e \sqrt {-1+c x} \sqrt {1+c x}}-\frac {d \left (d+e x^2\right )^4 (a+b \text {arccosh}(c x))}{8 e^2}+\frac {\left (d+e x^2\right )^5 (a+b \text {arccosh}(c x))}{10 e^2}-\frac {\left (b \sqrt {-1+c^2 x^2}\right ) \int \frac {\left (d+e x^2\right )^3 \left (-2 d \left (5 c^2 d-2 e\right )+2 e \left (11 c^2 d+18 e\right ) x^2\right )}{\sqrt {-1+c^2 x^2}} \, dx}{400 c e^2 \sqrt {-1+c x} \sqrt {1+c x}} \\ & = \frac {b \left (11 c^2 d+18 e\right ) x \left (1-c^2 x^2\right ) \left (d+e x^2\right )^3}{1600 c^3 e \sqrt {-1+c x} \sqrt {1+c x}}+\frac {b x \left (1-c^2 x^2\right ) \left (d+e x^2\right )^4}{100 c e \sqrt {-1+c x} \sqrt {1+c x}}-\frac {d \left (d+e x^2\right )^4 (a+b \text {arccosh}(c x))}{8 e^2}+\frac {\left (d+e x^2\right )^5 (a+b \text {arccosh}(c x))}{10 e^2}-\frac {\left (b \sqrt {-1+c^2 x^2}\right ) \int \frac {\left (d+e x^2\right )^2 \left (-2 d \left (40 c^4 d^2-27 c^2 d e-18 e^2\right )+2 e \left (26 c^4 d^2+201 c^2 d e+126 e^2\right ) x^2\right )}{\sqrt {-1+c^2 x^2}} \, dx}{3200 c^3 e^2 \sqrt {-1+c x} \sqrt {1+c x}} \\ & = \frac {b \left (26 c^4 d^2+201 c^2 d e+126 e^2\right ) x \left (1-c^2 x^2\right ) \left (d+e x^2\right )^2}{9600 c^5 e \sqrt {-1+c x} \sqrt {1+c x}}+\frac {b \left (11 c^2 d+18 e\right ) x \left (1-c^2 x^2\right ) \left (d+e x^2\right )^3}{1600 c^3 e \sqrt {-1+c x} \sqrt {1+c x}}+\frac {b x \left (1-c^2 x^2\right ) \left (d+e x^2\right )^4}{100 c e \sqrt {-1+c x} \sqrt {1+c x}}-\frac {d \left (d+e x^2\right )^4 (a+b \text {arccosh}(c x))}{8 e^2}+\frac {\left (d+e x^2\right )^5 (a+b \text {arccosh}(c x))}{10 e^2}-\frac {\left (b \sqrt {-1+c^2 x^2}\right ) \int \frac {\left (d+e x^2\right ) \left (-2 d \left (240 c^6 d^3-188 c^4 d^2 e-309 c^2 d e^2-126 e^3\right )-2 e \left (136 c^6 d^3-1096 c^4 d^2 e-1617 c^2 d e^2-630 e^3\right ) x^2\right )}{\sqrt {-1+c^2 x^2}} \, dx}{19200 c^5 e^2 \sqrt {-1+c x} \sqrt {1+c x}} \\ & = -\frac {b \left (136 c^6 d^3-1096 c^4 d^2 e-1617 c^2 d e^2-630 e^3\right ) x \left (1-c^2 x^2\right ) \left (d+e x^2\right )}{38400 c^7 e \sqrt {-1+c x} \sqrt {1+c x}}+\frac {b \left (26 c^4 d^2+201 c^2 d e+126 e^2\right ) x \left (1-c^2 x^2\right ) \left (d+e x^2\right )^2}{9600 c^5 e \sqrt {-1+c x} \sqrt {1+c x}}+\frac {b \left (11 c^2 d+18 e\right ) x \left (1-c^2 x^2\right ) \left (d+e x^2\right )^3}{1600 c^3 e \sqrt {-1+c x} \sqrt {1+c x}}+\frac {b x \left (1-c^2 x^2\right ) \left (d+e x^2\right )^4}{100 c e \sqrt {-1+c x} \sqrt {1+c x}}-\frac {d \left (d+e x^2\right )^4 (a+b \text {arccosh}(c x))}{8 e^2}+\frac {\left (d+e x^2\right )^5 (a+b \text {arccosh}(c x))}{10 e^2}-\frac {\left (b \sqrt {-1+c^2 x^2}\right ) \int \frac {-2 d \left (960 c^8 d^4-616 c^6 d^3 e-2332 c^4 d^2 e^2-2121 c^2 d e^3-630 e^4\right )-2 e \left (1232 c^8 d^4-2536 c^6 d^3 e-7758 c^4 d^2 e^2-6615 c^2 d e^3-1890 e^4\right ) x^2}{\sqrt {-1+c^2 x^2}} \, dx}{76800 c^7 e^2 \sqrt {-1+c x} \sqrt {1+c x}} \\ & = -\frac {b \left (1232 c^8 d^4-2536 c^6 d^3 e-7758 c^4 d^2 e^2-6615 c^2 d e^3-1890 e^4\right ) x \left (1-c^2 x^2\right )}{76800 c^9 e \sqrt {-1+c x} \sqrt {1+c x}}-\frac {b \left (136 c^6 d^3-1096 c^4 d^2 e-1617 c^2 d e^2-630 e^3\right ) x \left (1-c^2 x^2\right ) \left (d+e x^2\right )}{38400 c^7 e \sqrt {-1+c x} \sqrt {1+c x}}+\frac {b \left (26 c^4 d^2+201 c^2 d e+126 e^2\right ) x \left (1-c^2 x^2\right ) \left (d+e x^2\right )^2}{9600 c^5 e \sqrt {-1+c x} \sqrt {1+c x}}+\frac {b \left (11 c^2 d+18 e\right ) x \left (1-c^2 x^2\right ) \left (d+e x^2\right )^3}{1600 c^3 e \sqrt {-1+c x} \sqrt {1+c x}}+\frac {b x \left (1-c^2 x^2\right ) \left (d+e x^2\right )^4}{100 c e \sqrt {-1+c x} \sqrt {1+c x}}-\frac {d \left (d+e x^2\right )^4 (a+b \text {arccosh}(c x))}{8 e^2}+\frac {\left (d+e x^2\right )^5 (a+b \text {arccosh}(c x))}{10 e^2}+\frac {\left (b \left (2 e \left (1232 c^8 d^4-2536 c^6 d^3 e-7758 c^4 d^2 e^2-6615 c^2 d e^3-1890 e^4\right )+4 c^2 d \left (960 c^8 d^4-616 c^6 d^3 e-2332 c^4 d^2 e^2-2121 c^2 d e^3-630 e^4\right )\right ) \sqrt {-1+c^2 x^2}\right ) \int \frac {1}{\sqrt {-1+c^2 x^2}} \, dx}{153600 c^9 e^2 \sqrt {-1+c x} \sqrt {1+c x}} \\ & = -\frac {b \left (1232 c^8 d^4-2536 c^6 d^3 e-7758 c^4 d^2 e^2-6615 c^2 d e^3-1890 e^4\right ) x \left (1-c^2 x^2\right )}{76800 c^9 e \sqrt {-1+c x} \sqrt {1+c x}}-\frac {b \left (136 c^6 d^3-1096 c^4 d^2 e-1617 c^2 d e^2-630 e^3\right ) x \left (1-c^2 x^2\right ) \left (d+e x^2\right )}{38400 c^7 e \sqrt {-1+c x} \sqrt {1+c x}}+\frac {b \left (26 c^4 d^2+201 c^2 d e+126 e^2\right ) x \left (1-c^2 x^2\right ) \left (d+e x^2\right )^2}{9600 c^5 e \sqrt {-1+c x} \sqrt {1+c x}}+\frac {b \left (11 c^2 d+18 e\right ) x \left (1-c^2 x^2\right ) \left (d+e x^2\right )^3}{1600 c^3 e \sqrt {-1+c x} \sqrt {1+c x}}+\frac {b x \left (1-c^2 x^2\right ) \left (d+e x^2\right )^4}{100 c e \sqrt {-1+c x} \sqrt {1+c x}}-\frac {d \left (d+e x^2\right )^4 (a+b \text {arccosh}(c x))}{8 e^2}+\frac {\left (d+e x^2\right )^5 (a+b \text {arccosh}(c x))}{10 e^2}+\frac {\left (b \left (2 e \left (1232 c^8 d^4-2536 c^6 d^3 e-7758 c^4 d^2 e^2-6615 c^2 d e^3-1890 e^4\right )+4 c^2 d \left (960 c^8 d^4-616 c^6 d^3 e-2332 c^4 d^2 e^2-2121 c^2 d e^3-630 e^4\right )\right ) \sqrt {-1+c^2 x^2}\right ) \text {Subst}\left (\int \frac {1}{1-c^2 x^2} \, dx,x,\frac {x}{\sqrt {-1+c^2 x^2}}\right )}{153600 c^9 e^2 \sqrt {-1+c x} \sqrt {1+c x}} \\ & = -\frac {b \left (1232 c^8 d^4-2536 c^6 d^3 e-7758 c^4 d^2 e^2-6615 c^2 d e^3-1890 e^4\right ) x \left (1-c^2 x^2\right )}{76800 c^9 e \sqrt {-1+c x} \sqrt {1+c x}}-\frac {b \left (136 c^6 d^3-1096 c^4 d^2 e-1617 c^2 d e^2-630 e^3\right ) x \left (1-c^2 x^2\right ) \left (d+e x^2\right )}{38400 c^7 e \sqrt {-1+c x} \sqrt {1+c x}}+\frac {b \left (26 c^4 d^2+201 c^2 d e+126 e^2\right ) x \left (1-c^2 x^2\right ) \left (d+e x^2\right )^2}{9600 c^5 e \sqrt {-1+c x} \sqrt {1+c x}}+\frac {b \left (11 c^2 d+18 e\right ) x \left (1-c^2 x^2\right ) \left (d+e x^2\right )^3}{1600 c^3 e \sqrt {-1+c x} \sqrt {1+c x}}+\frac {b x \left (1-c^2 x^2\right ) \left (d+e x^2\right )^4}{100 c e \sqrt {-1+c x} \sqrt {1+c x}}-\frac {d \left (d+e x^2\right )^4 (a+b \text {arccosh}(c x))}{8 e^2}+\frac {\left (d+e x^2\right )^5 (a+b \text {arccosh}(c x))}{10 e^2}+\frac {b \left (128 c^{10} d^5-480 c^6 d^3 e^2-800 c^4 d^2 e^3-525 c^2 d e^4-126 e^5\right ) \sqrt {-1+c^2 x^2} \text {arctanh}\left (\frac {c x}{\sqrt {-1+c^2 x^2}}\right )}{5120 c^{10} e^2 \sqrt {-1+c x} \sqrt {1+c x}} \\ \end{align*}

Mathematica [A] (warning: unable to verify)

Time = 0.36 (sec) , antiderivative size = 294, normalized size of antiderivative = 0.60 \[ \int x^3 \left (d+e x^2\right )^3 (a+b \text {arccosh}(c x)) \, dx=\frac {1920 a x^4 \left (10 d^3+20 d^2 e x^2+15 d e^2 x^4+4 e^3 x^6\right )-\frac {b x \sqrt {-1+c x} \sqrt {1+c x} \left (1890 e^3+315 c^2 e^2 \left (25 d+4 e x^2\right )+6 c^4 e \left (2000 d^2+875 d e x^2+168 e^2 x^4\right )+8 c^6 \left (900 d^3+1000 d^2 e x^2+525 d e^2 x^4+108 e^3 x^6\right )+16 c^8 \left (300 d^3 x^2+400 d^2 e x^4+225 d e^2 x^6+48 e^3 x^8\right )\right )}{c^9}+1920 b x^4 \left (10 d^3+20 d^2 e x^2+15 d e^2 x^4+4 e^3 x^6\right ) \text {arccosh}(c x)-\frac {30 b \left (480 c^6 d^3+800 c^4 d^2 e+525 c^2 d e^2+126 e^3\right ) \text {arctanh}\left (\sqrt {\frac {-1+c x}{1+c x}}\right )}{c^{10}}}{76800} \]

[In]

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

[Out]

(1920*a*x^4*(10*d^3 + 20*d^2*e*x^2 + 15*d*e^2*x^4 + 4*e^3*x^6) - (b*x*Sqrt[-1 + c*x]*Sqrt[1 + c*x]*(1890*e^3 +
315*c^2*e^2*(25*d + 4*e*x^2) + 6*c^4*e*(2000*d^2 + 875*d*e*x^2 + 168*e^2*x^4) + 8*c^6*(900*d^3 + 1000*d^2*e*x
^2 + 525*d*e^2*x^4 + 108*e^3*x^6) + 16*c^8*(300*d^3*x^2 + 400*d^2*e*x^4 + 225*d*e^2*x^6 + 48*e^3*x^8)))/c^9 +
1920*b*x^4*(10*d^3 + 20*d^2*e*x^2 + 15*d*e^2*x^4 + 4*e^3*x^6)*ArcCosh[c*x] - (30*b*(480*c^6*d^3 + 800*c^4*d^2*
e + 525*c^2*d*e^2 + 126*e^3)*ArcTanh[Sqrt[(-1 + c*x)/(1 + c*x)]])/c^10)/76800

Maple [A] (verified)

Time = 0.77 (sec) , antiderivative size = 538, normalized size of antiderivative = 1.09

method	result	size
parts	\(a \left (\frac {1}{10} e^{3} x^{10}+\frac {3}{8} d \,e^{2} x^{8}+\frac {1}{2} d^{2} e \,x^{6}+\frac {1}{4} d^{3} x^{4}\right )+\frac {b \left (\frac {c^{4} \operatorname {arccosh}\left (c x \right ) e^{3} x^{10}}{10}+\frac {3 c^{4} \operatorname {arccosh}\left (c x \right ) d \,e^{2} x^{8}}{8}+\frac {c^{4} \operatorname {arccosh}\left (c x \right ) d^{2} e \,x^{6}}{2}+\frac {\operatorname {arccosh}\left (c x \right ) c^{4} x^{4} d^{3}}{4}-\frac {\sqrt {c x -1}\, \sqrt {c x +1}\, \left (4800 c^{9} d^{3} \sqrt {c^{2} x^{2}-1}\, x^{3}+6400 c^{9} d^{2} e \sqrt {c^{2} x^{2}-1}\, x^{5}+3600 c^{9} d \,e^{2} \sqrt {c^{2} x^{2}-1}\, x^{7}+768 e^{3} \sqrt {c^{2} x^{2}-1}\, c^{9} x^{9}+7200 c^{7} d^{3} x \sqrt {c^{2} x^{2}-1}+8000 \sqrt {c^{2} x^{2}-1}\, c^{7} d^{2} e \,x^{3}+4200 \sqrt {c^{2} x^{2}-1}\, c^{7} d \,e^{2} x^{5}+864 e^{3} c^{7} x^{7} \sqrt {c^{2} x^{2}-1}+7200 c^{6} d^{3} \ln \left (c x +\sqrt {c^{2} x^{2}-1}\right )+12000 c^{5} d^{2} e x \sqrt {c^{2} x^{2}-1}+5250 c^{5} d \,e^{2} \sqrt {c^{2} x^{2}-1}\, x^{3}+1008 e^{3} \sqrt {c^{2} x^{2}-1}\, c^{5} x^{5}+12000 c^{4} d^{2} e \ln \left (c x +\sqrt {c^{2} x^{2}-1}\right )+7875 c^{3} d \,e^{2} x \sqrt {c^{2} x^{2}-1}+1260 e^{3} c^{3} x^{3} \sqrt {c^{2} x^{2}-1}+7875 c^{2} d \,e^{2} \ln \left (c x +\sqrt {c^{2} x^{2}-1}\right )+1890 e^{3} c x \sqrt {c^{2} x^{2}-1}+1890 e^{3} \ln \left (c x +\sqrt {c^{2} x^{2}-1}\right )\right )}{76800 c^{6} \sqrt {c^{2} x^{2}-1}}\right )}{c^{4}}\)	\(538\)
derivativedivides	\(\frac {\frac {a \left (\frac {1}{4} c^{10} d^{3} x^{4}+\frac {1}{2} c^{10} d^{2} e \,x^{6}+\frac {3}{8} c^{10} d \,e^{2} x^{8}+\frac {1}{10} c^{10} e^{3} x^{10}\right )}{c^{6}}+\frac {b \left (\frac {\operatorname {arccosh}\left (c x \right ) c^{10} d^{3} x^{4}}{4}+\frac {\operatorname {arccosh}\left (c x \right ) c^{10} d^{2} e \,x^{6}}{2}+\frac {3 \,\operatorname {arccosh}\left (c x \right ) c^{10} d \,e^{2} x^{8}}{8}+\frac {\operatorname {arccosh}\left (c x \right ) e^{3} c^{10} x^{10}}{10}-\frac {\sqrt {c x -1}\, \sqrt {c x +1}\, \left (4800 c^{9} d^{3} \sqrt {c^{2} x^{2}-1}\, x^{3}+6400 c^{9} d^{2} e \sqrt {c^{2} x^{2}-1}\, x^{5}+3600 c^{9} d \,e^{2} \sqrt {c^{2} x^{2}-1}\, x^{7}+768 e^{3} \sqrt {c^{2} x^{2}-1}\, c^{9} x^{9}+7200 c^{7} d^{3} x \sqrt {c^{2} x^{2}-1}+8000 \sqrt {c^{2} x^{2}-1}\, c^{7} d^{2} e \,x^{3}+4200 \sqrt {c^{2} x^{2}-1}\, c^{7} d \,e^{2} x^{5}+864 e^{3} c^{7} x^{7} \sqrt {c^{2} x^{2}-1}+7200 c^{6} d^{3} \ln \left (c x +\sqrt {c^{2} x^{2}-1}\right )+12000 c^{5} d^{2} e x \sqrt {c^{2} x^{2}-1}+5250 c^{5} d \,e^{2} \sqrt {c^{2} x^{2}-1}\, x^{3}+1008 e^{3} \sqrt {c^{2} x^{2}-1}\, c^{5} x^{5}+12000 c^{4} d^{2} e \ln \left (c x +\sqrt {c^{2} x^{2}-1}\right )+7875 c^{3} d \,e^{2} x \sqrt {c^{2} x^{2}-1}+1260 e^{3} c^{3} x^{3} \sqrt {c^{2} x^{2}-1}+7875 c^{2} d \,e^{2} \ln \left (c x +\sqrt {c^{2} x^{2}-1}\right )+1890 e^{3} c x \sqrt {c^{2} x^{2}-1}+1890 e^{3} \ln \left (c x +\sqrt {c^{2} x^{2}-1}\right )\right )}{76800 \sqrt {c^{2} x^{2}-1}}\right )}{c^{6}}}{c^{4}}\)	\(554\)
default	\(\frac {\frac {a \left (\frac {1}{4} c^{10} d^{3} x^{4}+\frac {1}{2} c^{10} d^{2} e \,x^{6}+\frac {3}{8} c^{10} d \,e^{2} x^{8}+\frac {1}{10} c^{10} e^{3} x^{10}\right )}{c^{6}}+\frac {b \left (\frac {\operatorname {arccosh}\left (c x \right ) c^{10} d^{3} x^{4}}{4}+\frac {\operatorname {arccosh}\left (c x \right ) c^{10} d^{2} e \,x^{6}}{2}+\frac {3 \,\operatorname {arccosh}\left (c x \right ) c^{10} d \,e^{2} x^{8}}{8}+\frac {\operatorname {arccosh}\left (c x \right ) e^{3} c^{10} x^{10}}{10}-\frac {\sqrt {c x -1}\, \sqrt {c x +1}\, \left (4800 c^{9} d^{3} \sqrt {c^{2} x^{2}-1}\, x^{3}+6400 c^{9} d^{2} e \sqrt {c^{2} x^{2}-1}\, x^{5}+3600 c^{9} d \,e^{2} \sqrt {c^{2} x^{2}-1}\, x^{7}+768 e^{3} \sqrt {c^{2} x^{2}-1}\, c^{9} x^{9}+7200 c^{7} d^{3} x \sqrt {c^{2} x^{2}-1}+8000 \sqrt {c^{2} x^{2}-1}\, c^{7} d^{2} e \,x^{3}+4200 \sqrt {c^{2} x^{2}-1}\, c^{7} d \,e^{2} x^{5}+864 e^{3} c^{7} x^{7} \sqrt {c^{2} x^{2}-1}+7200 c^{6} d^{3} \ln \left (c x +\sqrt {c^{2} x^{2}-1}\right )+12000 c^{5} d^{2} e x \sqrt {c^{2} x^{2}-1}+5250 c^{5} d \,e^{2} \sqrt {c^{2} x^{2}-1}\, x^{3}+1008 e^{3} \sqrt {c^{2} x^{2}-1}\, c^{5} x^{5}+12000 c^{4} d^{2} e \ln \left (c x +\sqrt {c^{2} x^{2}-1}\right )+7875 c^{3} d \,e^{2} x \sqrt {c^{2} x^{2}-1}+1260 e^{3} c^{3} x^{3} \sqrt {c^{2} x^{2}-1}+7875 c^{2} d \,e^{2} \ln \left (c x +\sqrt {c^{2} x^{2}-1}\right )+1890 e^{3} c x \sqrt {c^{2} x^{2}-1}+1890 e^{3} \ln \left (c x +\sqrt {c^{2} x^{2}-1}\right )\right )}{76800 \sqrt {c^{2} x^{2}-1}}\right )}{c^{6}}}{c^{4}}\)	\(554\)

[In]

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

[Out]

a*(1/10*e^3*x^10+3/8*d*e^2*x^8+1/2*d^2*e*x^6+1/4*d^3*x^4)+b/c^4*(1/10*c^4*arccosh(c*x)*e^3*x^10+3/8*c^4*arccos
h(c*x)*d*e^2*x^8+1/2*c^4*arccosh(c*x)*d^2*e*x^6+1/4*arccosh(c*x)*c^4*x^4*d^3-1/76800/c^6*(c*x-1)^(1/2)*(c*x+1)
^(1/2)*(4800*c^9*d^3*(c^2*x^2-1)^(1/2)*x^3+6400*c^9*d^2*e*(c^2*x^2-1)^(1/2)*x^5+3600*c^9*d*e^2*(c^2*x^2-1)^(1/
2)*x^7+768*e^3*(c^2*x^2-1)^(1/2)*c^9*x^9+7200*c^7*d^3*x*(c^2*x^2-1)^(1/2)+8000*(c^2*x^2-1)^(1/2)*c^7*d^2*e*x^3
+4200*(c^2*x^2-1)^(1/2)*c^7*d*e^2*x^5+864*e^3*c^7*x^7*(c^2*x^2-1)^(1/2)+7200*c^6*d^3*ln(c*x+(c^2*x^2-1)^(1/2))
+12000*c^5*d^2*e*x*(c^2*x^2-1)^(1/2)+5250*c^5*d*e^2*(c^2*x^2-1)^(1/2)*x^3+1008*e^3*(c^2*x^2-1)^(1/2)*c^5*x^5+1
2000*c^4*d^2*e*ln(c*x+(c^2*x^2-1)^(1/2))+7875*c^3*d*e^2*x*(c^2*x^2-1)^(1/2)+1260*e^3*c^3*x^3*(c^2*x^2-1)^(1/2)
+7875*c^2*d*e^2*ln(c*x+(c^2*x^2-1)^(1/2))+1890*e^3*c*x*(c^2*x^2-1)^(1/2)+1890*e^3*ln(c*x+(c^2*x^2-1)^(1/2)))/(
c^2*x^2-1)^(1/2))

Fricas [A] (verification not implemented)

none

Time = 0.27 (sec) , antiderivative size = 330, normalized size of antiderivative = 0.67 \[ \int x^3 \left (d+e x^2\right )^3 (a+b \text {arccosh}(c x)) \, dx=\frac {7680 \, a c^{10} e^{3} x^{10} + 28800 \, a c^{10} d e^{2} x^{8} + 38400 \, a c^{10} d^{2} e x^{6} + 19200 \, a c^{10} d^{3} x^{4} + 15 \, {\left (512 \, b c^{10} e^{3} x^{10} + 1920 \, b c^{10} d e^{2} x^{8} + 2560 \, b c^{10} d^{2} e x^{6} + 1280 \, b c^{10} d^{3} x^{4} - 480 \, b c^{6} d^{3} - 800 \, b c^{4} d^{2} e - 525 \, b c^{2} d e^{2} - 126 \, b e^{3}\right )} \log \left (c x + \sqrt {c^{2} x^{2} - 1}\right ) - {\left (768 \, b c^{9} e^{3} x^{9} + 144 \, {\left (25 \, b c^{9} d e^{2} + 6 \, b c^{7} e^{3}\right )} x^{7} + 8 \, {\left (800 \, b c^{9} d^{2} e + 525 \, b c^{7} d e^{2} + 126 \, b c^{5} e^{3}\right )} x^{5} + 10 \, {\left (480 \, b c^{9} d^{3} + 800 \, b c^{7} d^{2} e + 525 \, b c^{5} d e^{2} + 126 \, b c^{3} e^{3}\right )} x^{3} + 15 \, {\left (480 \, b c^{7} d^{3} + 800 \, b c^{5} d^{2} e + 525 \, b c^{3} d e^{2} + 126 \, b c e^{3}\right )} x\right )} \sqrt {c^{2} x^{2} - 1}}{76800 \, c^{10}} \]

[In]

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

[Out]

1/76800*(7680*a*c^10*e^3*x^10 + 28800*a*c^10*d*e^2*x^8 + 38400*a*c^10*d^2*e*x^6 + 19200*a*c^10*d^3*x^4 + 15*(5
12*b*c^10*e^3*x^10 + 1920*b*c^10*d*e^2*x^8 + 2560*b*c^10*d^2*e*x^6 + 1280*b*c^10*d^3*x^4 - 480*b*c^6*d^3 - 800
*b*c^4*d^2*e - 525*b*c^2*d*e^2 - 126*b*e^3)*log(c*x + sqrt(c^2*x^2 - 1)) - (768*b*c^9*e^3*x^9 + 144*(25*b*c^9*
d*e^2 + 6*b*c^7*e^3)*x^7 + 8*(800*b*c^9*d^2*e + 525*b*c^7*d*e^2 + 126*b*c^5*e^3)*x^5 + 10*(480*b*c^9*d^3 + 800
*b*c^7*d^2*e + 525*b*c^5*d*e^2 + 126*b*c^3*e^3)*x^3 + 15*(480*b*c^7*d^3 + 800*b*c^5*d^2*e + 525*b*c^3*d*e^2 +
126*b*c*e^3)*x)*sqrt(c^2*x^2 - 1))/c^10

Sympy [F]

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

[In]

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

[Out]

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

Maxima [A] (verification not implemented)

none

Time = 0.23 (sec) , antiderivative size = 487, normalized size of antiderivative = 0.99 \[ \int x^3 \left (d+e x^2\right )^3 (a+b \text {arccosh}(c x)) \, dx=\frac {1}{10} \, a e^{3} x^{10} + \frac {3}{8} \, a d e^{2} x^{8} + \frac {1}{2} \, a d^{2} e x^{6} + \frac {1}{4} \, a d^{3} x^{4} + \frac {1}{32} \, {\left (8 \, x^{4} \operatorname {arcosh}\left (c x\right ) - {\left (\frac {2 \, \sqrt {c^{2} x^{2} - 1} x^{3}}{c^{2}} + \frac {3 \, \sqrt {c^{2} x^{2} - 1} x}{c^{4}} + \frac {3 \, \log \left (2 \, c^{2} x + 2 \, \sqrt {c^{2} x^{2} - 1} c\right )}{c^{5}}\right )} c\right )} b d^{3} + \frac {1}{96} \, {\left (48 \, x^{6} \operatorname {arcosh}\left (c x\right ) - {\left (\frac {8 \, \sqrt {c^{2} x^{2} - 1} x^{5}}{c^{2}} + \frac {10 \, \sqrt {c^{2} x^{2} - 1} x^{3}}{c^{4}} + \frac {15 \, \sqrt {c^{2} x^{2} - 1} x}{c^{6}} + \frac {15 \, \log \left (2 \, c^{2} x + 2 \, \sqrt {c^{2} x^{2} - 1} c\right )}{c^{7}}\right )} c\right )} b d^{2} e + \frac {1}{1024} \, {\left (384 \, x^{8} \operatorname {arcosh}\left (c x\right ) - {\left (\frac {48 \, \sqrt {c^{2} x^{2} - 1} x^{7}}{c^{2}} + \frac {56 \, \sqrt {c^{2} x^{2} - 1} x^{5}}{c^{4}} + \frac {70 \, \sqrt {c^{2} x^{2} - 1} x^{3}}{c^{6}} + \frac {105 \, \sqrt {c^{2} x^{2} - 1} x}{c^{8}} + \frac {105 \, \log \left (2 \, c^{2} x + 2 \, \sqrt {c^{2} x^{2} - 1} c\right )}{c^{9}}\right )} c\right )} b d e^{2} + \frac {1}{12800} \, {\left (1280 \, x^{10} \operatorname {arcosh}\left (c x\right ) - {\left (\frac {128 \, \sqrt {c^{2} x^{2} - 1} x^{9}}{c^{2}} + \frac {144 \, \sqrt {c^{2} x^{2} - 1} x^{7}}{c^{4}} + \frac {168 \, \sqrt {c^{2} x^{2} - 1} x^{5}}{c^{6}} + \frac {210 \, \sqrt {c^{2} x^{2} - 1} x^{3}}{c^{8}} + \frac {315 \, \sqrt {c^{2} x^{2} - 1} x}{c^{10}} + \frac {315 \, \log \left (2 \, c^{2} x + 2 \, \sqrt {c^{2} x^{2} - 1} c\right )}{c^{11}}\right )} c\right )} b e^{3} \]

[In]

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

[Out]

1/10*a*e^3*x^10 + 3/8*a*d*e^2*x^8 + 1/2*a*d^2*e*x^6 + 1/4*a*d^3*x^4 + 1/32*(8*x^4*arccosh(c*x) - (2*sqrt(c^2*x
^2 - 1)*x^3/c^2 + 3*sqrt(c^2*x^2 - 1)*x/c^4 + 3*log(2*c^2*x + 2*sqrt(c^2*x^2 - 1)*c)/c^5)*c)*b*d^3 + 1/96*(48*
x^6*arccosh(c*x) - (8*sqrt(c^2*x^2 - 1)*x^5/c^2 + 10*sqrt(c^2*x^2 - 1)*x^3/c^4 + 15*sqrt(c^2*x^2 - 1)*x/c^6 +
15*log(2*c^2*x + 2*sqrt(c^2*x^2 - 1)*c)/c^7)*c)*b*d^2*e + 1/1024*(384*x^8*arccosh(c*x) - (48*sqrt(c^2*x^2 - 1)
*x^7/c^2 + 56*sqrt(c^2*x^2 - 1)*x^5/c^4 + 70*sqrt(c^2*x^2 - 1)*x^3/c^6 + 105*sqrt(c^2*x^2 - 1)*x/c^8 + 105*log
(2*c^2*x + 2*sqrt(c^2*x^2 - 1)*c)/c^9)*c)*b*d*e^2 + 1/12800*(1280*x^10*arccosh(c*x) - (128*sqrt(c^2*x^2 - 1)*x
^9/c^2 + 144*sqrt(c^2*x^2 - 1)*x^7/c^4 + 168*sqrt(c^2*x^2 - 1)*x^5/c^6 + 210*sqrt(c^2*x^2 - 1)*x^3/c^8 + 315*s
qrt(c^2*x^2 - 1)*x/c^10 + 315*log(2*c^2*x + 2*sqrt(c^2*x^2 - 1)*c)/c^11)*c)*b*e^3

Giac [F(-2)]

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

[In]

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

[Out]

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

Mupad [F(-1)]

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

[In]

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

[Out]

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