\(\int x^7 (d-c^2 d x^2)^{3/2} (a+b \text {arccosh}(c x)) \, dx\) [80]

Optimal result

Integrand size = 27, antiderivative size = 399 \[ \int x^7 \left (d-c^2 d x^2\right )^{3/2} (a+b \text {arccosh}(c x)) \, dx=\frac {16 b d x \sqrt {d-c^2 d x^2}}{1155 c^7 \sqrt {-1+c x} \sqrt {1+c x}}+\frac {8 b d x^3 \sqrt {d-c^2 d x^2}}{3465 c^5 \sqrt {-1+c x} \sqrt {1+c x}}+\frac {2 b d x^5 \sqrt {d-c^2 d x^2}}{1925 c^3 \sqrt {-1+c x} \sqrt {1+c x}}+\frac {b d x^7 \sqrt {d-c^2 d x^2}}{1617 c \sqrt {-1+c x} \sqrt {1+c x}}-\frac {4 b c d x^9 \sqrt {d-c^2 d x^2}}{297 \sqrt {-1+c x} \sqrt {1+c x}}+\frac {b c^3 d x^{11} \sqrt {d-c^2 d x^2}}{121 \sqrt {-1+c x} \sqrt {1+c x}}-\frac {\left (d-c^2 d x^2\right )^{5/2} (a+b \text {arccosh}(c x))}{5 c^8 d}+\frac {3 \left (d-c^2 d x^2\right )^{7/2} (a+b \text {arccosh}(c x))}{7 c^8 d^2}-\frac {\left (d-c^2 d x^2\right )^{9/2} (a+b \text {arccosh}(c x))}{3 c^8 d^3}+\frac {\left (d-c^2 d x^2\right )^{11/2} (a+b \text {arccosh}(c x))}{11 c^8 d^4} \]

[Out]

-1/5*(-c^2*d*x^2+d)^(5/2)*(a+b*arccosh(c*x))/c^8/d+3/7*(-c^2*d*x^2+d)^(7/2)*(a+b*arccosh(c*x))/c^8/d^2-1/3*(-c
^2*d*x^2+d)^(9/2)*(a+b*arccosh(c*x))/c^8/d^3+1/11*(-c^2*d*x^2+d)^(11/2)*(a+b*arccosh(c*x))/c^8/d^4+16/1155*b*d
*x*(-c^2*d*x^2+d)^(1/2)/c^7/(c*x-1)^(1/2)/(c*x+1)^(1/2)+8/3465*b*d*x^3*(-c^2*d*x^2+d)^(1/2)/c^5/(c*x-1)^(1/2)/
(c*x+1)^(1/2)+2/1925*b*d*x^5*(-c^2*d*x^2+d)^(1/2)/c^3/(c*x-1)^(1/2)/(c*x+1)^(1/2)+1/1617*b*d*x^7*(-c^2*d*x^2+d
)^(1/2)/c/(c*x-1)^(1/2)/(c*x+1)^(1/2)-4/297*b*c*d*x^9*(-c^2*d*x^2+d)^(1/2)/(c*x-1)^(1/2)/(c*x+1)^(1/2)+1/121*b
*c^3*d*x^11*(-c^2*d*x^2+d)^(1/2)/(c*x-1)^(1/2)/(c*x+1)^(1/2)

Rubi [A] (verified)

Time = 0.19 (sec) , antiderivative size = 399, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.185, Rules used = {272, 45, 5922, 12, 1824} \[ \int x^7 \left (d-c^2 d x^2\right )^{3/2} (a+b \text {arccosh}(c x)) \, dx=\frac {\left (d-c^2 d x^2\right )^{11/2} (a+b \text {arccosh}(c x))}{11 c^8 d^4}-\frac {\left (d-c^2 d x^2\right )^{9/2} (a+b \text {arccosh}(c x))}{3 c^8 d^3}+\frac {3 \left (d-c^2 d x^2\right )^{7/2} (a+b \text {arccosh}(c x))}{7 c^8 d^2}-\frac {\left (d-c^2 d x^2\right )^{5/2} (a+b \text {arccosh}(c x))}{5 c^8 d}-\frac {4 b c d x^9 \sqrt {d-c^2 d x^2}}{297 \sqrt {c x-1} \sqrt {c x+1}}+\frac {b d x^7 \sqrt {d-c^2 d x^2}}{1617 c \sqrt {c x-1} \sqrt {c x+1}}+\frac {16 b d x \sqrt {d-c^2 d x^2}}{1155 c^7 \sqrt {c x-1} \sqrt {c x+1}}+\frac {8 b d x^3 \sqrt {d-c^2 d x^2}}{3465 c^5 \sqrt {c x-1} \sqrt {c x+1}}+\frac {b c^3 d x^{11} \sqrt {d-c^2 d x^2}}{121 \sqrt {c x-1} \sqrt {c x+1}}+\frac {2 b d x^5 \sqrt {d-c^2 d x^2}}{1925 c^3 \sqrt {c x-1} \sqrt {c x+1}} \]

[In]

Int[x^7*(d - c^2*d*x^2)^(3/2)*(a + b*ArcCosh[c*x]),x]

[Out]

(16*b*d*x*Sqrt[d - c^2*d*x^2])/(1155*c^7*Sqrt[-1 + c*x]*Sqrt[1 + c*x]) + (8*b*d*x^3*Sqrt[d - c^2*d*x^2])/(3465
*c^5*Sqrt[-1 + c*x]*Sqrt[1 + c*x]) + (2*b*d*x^5*Sqrt[d - c^2*d*x^2])/(1925*c^3*Sqrt[-1 + c*x]*Sqrt[1 + c*x]) +
(b*d*x^7*Sqrt[d - c^2*d*x^2])/(1617*c*Sqrt[-1 + c*x]*Sqrt[1 + c*x]) - (4*b*c*d*x^9*Sqrt[d - c^2*d*x^2])/(297*
Sqrt[-1 + c*x]*Sqrt[1 + c*x]) + (b*c^3*d*x^11*Sqrt[d - c^2*d*x^2])/(121*Sqrt[-1 + c*x]*Sqrt[1 + c*x]) - ((d -
c^2*d*x^2)^(5/2)*(a + b*ArcCosh[c*x]))/(5*c^8*d) + (3*(d - c^2*d*x^2)^(7/2)*(a + b*ArcCosh[c*x]))/(7*c^8*d^2)
- ((d - c^2*d*x^2)^(9/2)*(a + b*ArcCosh[c*x]))/(3*c^8*d^3) + ((d - c^2*d*x^2)^(11/2)*(a + b*ArcCosh[c*x]))/(11
*c^8*d^4)

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 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 1824

Int[(Pq_)*((a_) + (b_.)*(x_)^2)^(p_.), x_Symbol] :> Int[ExpandIntegrand[Pq*(a + b*x^2)^p, x], x] /; FreeQ[{a,
b}, x] && PolyQ[Pq, x] && IGtQ[p, -2]

Rule 5922

Int[((a_.) + ArcCosh[(c_.)*(x_)]*(b_.))*(x_)^(m_)*((d_) + (e_.)*(x_)^2)^(p_), x_Symbol] :> With[{u = IntHide[x
^m*(d + e*x^2)^p, x]}, Dist[a + b*ArcCosh[c*x], u, x] - Dist[b*c*Simp[Sqrt[d + e*x^2]/(Sqrt[1 + c*x]*Sqrt[-1 +
c*x])], Int[SimplifyIntegrand[u/Sqrt[d + e*x^2], x], x], x]] /; FreeQ[{a, b, c, d, e}, x] && EqQ[c^2*d + e, 0
] && IntegerQ[p - 1/2] && NeQ[p, -2^(-1)] && (IGtQ[(m + 1)/2, 0] || ILtQ[(m + 2*p + 3)/2, 0])

Rubi steps \begin{align*} \text {integral}& = -\frac {\left (d-c^2 d x^2\right )^{5/2} (a+b \text {arccosh}(c x))}{5 c^8 d}+\frac {3 \left (d-c^2 d x^2\right )^{7/2} (a+b \text {arccosh}(c x))}{7 c^8 d^2}-\frac {\left (d-c^2 d x^2\right )^{9/2} (a+b \text {arccosh}(c x))}{3 c^8 d^3}+\frac {\left (d-c^2 d x^2\right )^{11/2} (a+b \text {arccosh}(c x))}{11 c^8 d^4}-\frac {\left (b c \sqrt {d-c^2 d x^2}\right ) \int \frac {d \left (1-c^2 x^2\right )^2 \left (-16-40 c^2 x^2-70 c^4 x^4-105 c^6 x^6\right )}{1155 c^8} \, dx}{\sqrt {-1+c x} \sqrt {1+c x}} \\ & = -\frac {\left (d-c^2 d x^2\right )^{5/2} (a+b \text {arccosh}(c x))}{5 c^8 d}+\frac {3 \left (d-c^2 d x^2\right )^{7/2} (a+b \text {arccosh}(c x))}{7 c^8 d^2}-\frac {\left (d-c^2 d x^2\right )^{9/2} (a+b \text {arccosh}(c x))}{3 c^8 d^3}+\frac {\left (d-c^2 d x^2\right )^{11/2} (a+b \text {arccosh}(c x))}{11 c^8 d^4}-\frac {\left (b d \sqrt {d-c^2 d x^2}\right ) \int \left (1-c^2 x^2\right )^2 \left (-16-40 c^2 x^2-70 c^4 x^4-105 c^6 x^6\right ) \, dx}{1155 c^7 \sqrt {-1+c x} \sqrt {1+c x}} \\ & = -\frac {\left (d-c^2 d x^2\right )^{5/2} (a+b \text {arccosh}(c x))}{5 c^8 d}+\frac {3 \left (d-c^2 d x^2\right )^{7/2} (a+b \text {arccosh}(c x))}{7 c^8 d^2}-\frac {\left (d-c^2 d x^2\right )^{9/2} (a+b \text {arccosh}(c x))}{3 c^8 d^3}+\frac {\left (d-c^2 d x^2\right )^{11/2} (a+b \text {arccosh}(c x))}{11 c^8 d^4}-\frac {\left (b d \sqrt {d-c^2 d x^2}\right ) \int \left (-16-8 c^2 x^2-6 c^4 x^4-5 c^6 x^6+140 c^8 x^8-105 c^{10} x^{10}\right ) \, dx}{1155 c^7 \sqrt {-1+c x} \sqrt {1+c x}} \\ & = \frac {16 b d x \sqrt {d-c^2 d x^2}}{1155 c^7 \sqrt {-1+c x} \sqrt {1+c x}}+\frac {8 b d x^3 \sqrt {d-c^2 d x^2}}{3465 c^5 \sqrt {-1+c x} \sqrt {1+c x}}+\frac {2 b d x^5 \sqrt {d-c^2 d x^2}}{1925 c^3 \sqrt {-1+c x} \sqrt {1+c x}}+\frac {b d x^7 \sqrt {d-c^2 d x^2}}{1617 c \sqrt {-1+c x} \sqrt {1+c x}}-\frac {4 b c d x^9 \sqrt {d-c^2 d x^2}}{297 \sqrt {-1+c x} \sqrt {1+c x}}+\frac {b c^3 d x^{11} \sqrt {d-c^2 d x^2}}{121 \sqrt {-1+c x} \sqrt {1+c x}}-\frac {\left (d-c^2 d x^2\right )^{5/2} (a+b \text {arccosh}(c x))}{5 c^8 d}+\frac {3 \left (d-c^2 d x^2\right )^{7/2} (a+b \text {arccosh}(c x))}{7 c^8 d^2}-\frac {\left (d-c^2 d x^2\right )^{9/2} (a+b \text {arccosh}(c x))}{3 c^8 d^3}+\frac {\left (d-c^2 d x^2\right )^{11/2} (a+b \text {arccosh}(c x))}{11 c^8 d^4} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.18 (sec) , antiderivative size = 209, normalized size of antiderivative = 0.52 \[ \int x^7 \left (d-c^2 d x^2\right )^{3/2} (a+b \text {arccosh}(c x)) \, dx=-\frac {d \sqrt {d-c^2 d x^2} \left (3465 a \sqrt {-1+c x} \sqrt {1+c x} \left (-1+c^2 x^2\right )^2 \left (16+40 c^2 x^2+70 c^4 x^4+105 c^6 x^6\right )-b c x \left (55440+9240 c^2 x^2+4158 c^4 x^4+2475 c^6 x^6-53900 c^8 x^8+33075 c^{10} x^{10}\right )+3465 b \sqrt {-1+c x} \sqrt {1+c x} \left (-1+c^2 x^2\right )^2 \left (16+40 c^2 x^2+70 c^4 x^4+105 c^6 x^6\right ) \text {arccosh}(c x)\right )}{4002075 c^8 \sqrt {-1+c x} \sqrt {1+c x}} \]

[In]

Integrate[x^7*(d - c^2*d*x^2)^(3/2)*(a + b*ArcCosh[c*x]),x]

[Out]

-1/4002075*(d*Sqrt[d - c^2*d*x^2]*(3465*a*Sqrt[-1 + c*x]*Sqrt[1 + c*x]*(-1 + c^2*x^2)^2*(16 + 40*c^2*x^2 + 70*
c^4*x^4 + 105*c^6*x^6) - b*c*x*(55440 + 9240*c^2*x^2 + 4158*c^4*x^4 + 2475*c^6*x^6 - 53900*c^8*x^8 + 33075*c^1
0*x^10) + 3465*b*Sqrt[-1 + c*x]*Sqrt[1 + c*x]*(-1 + c^2*x^2)^2*(16 + 40*c^2*x^2 + 70*c^4*x^4 + 105*c^6*x^6)*Ar
cCosh[c*x]))/(c^8*Sqrt[-1 + c*x]*Sqrt[1 + c*x])

Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(1845\) vs. \(2(335)=670\).

Time = 0.89 (sec) , antiderivative size = 1846, normalized size of antiderivative = 4.63

method	result	size
default	\(\text {Expression too large to display}\)	\(1846\)
parts	\(\text {Expression too large to display}\)	\(1846\)

[In]

int(x^7*(-c^2*d*x^2+d)^(3/2)*(a+b*arccosh(c*x)),x,method=_RETURNVERBOSE)

[Out]

a*(-1/11*x^6*(-c^2*d*x^2+d)^(5/2)/c^2/d+6/11/c^2*(-1/9*x^4*(-c^2*d*x^2+d)^(5/2)/c^2/d+4/9/c^2*(-1/7*x^2*(-c^2*
d*x^2+d)^(5/2)/c^2/d-2/35/d/c^4*(-c^2*d*x^2+d)^(5/2))))+b*(-1/247808*(-d*(c^2*x^2-1))^(1/2)*(1+4096*c^8*x^8+22
0*(c*x-1)^(1/2)*(c*x+1)^(1/2)*c^3*x^3+2816*(c*x+1)^(1/2)*(c*x-1)^(1/2)*x^7*c^7+620*c^4*x^4-61*c^2*x^2-11*(c*x-
1)^(1/2)*(c*x+1)^(1/2)*c*x-1232*(c*x+1)^(1/2)*(c*x-1)^(1/2)*c^5*x^5+1024*c^12*x^12-2352*c^6*x^6-3328*c^10*x^10
+1024*(c*x+1)^(1/2)*(c*x-1)^(1/2)*x^11*c^11-2816*(c*x+1)^(1/2)*(c*x-1)^(1/2)*x^9*c^9)*(-1+11*arccosh(c*x))*d/(
c*x+1)/c^8/(c*x-1)-1/55296*(-d*(c^2*x^2-1))^(1/2)*(256*c^10*x^10-704*c^8*x^8+256*(c*x+1)^(1/2)*(c*x-1)^(1/2)*x
^9*c^9+688*c^6*x^6-576*(c*x+1)^(1/2)*(c*x-1)^(1/2)*x^7*c^7-280*c^4*x^4+432*(c*x+1)^(1/2)*(c*x-1)^(1/2)*c^5*x^5
+41*c^2*x^2-120*(c*x-1)^(1/2)*(c*x+1)^(1/2)*c^3*x^3+9*(c*x-1)^(1/2)*(c*x+1)^(1/2)*c*x-1)*(-1+9*arccosh(c*x))*d
/(c*x+1)/c^8/(c*x-1)+1/100352*(-d*(c^2*x^2-1))^(1/2)*(64*c^8*x^8-144*c^6*x^6+64*(c*x+1)^(1/2)*(c*x-1)^(1/2)*x^
7*c^7+104*c^4*x^4-112*(c*x+1)^(1/2)*(c*x-1)^(1/2)*c^5*x^5-25*c^2*x^2+56*(c*x-1)^(1/2)*(c*x+1)^(1/2)*c^3*x^3-7*
(c*x-1)^(1/2)*(c*x+1)^(1/2)*c*x+1)*(-1+7*arccosh(c*x))*d/(c*x+1)/c^8/(c*x-1)+11/51200*(-d*(c^2*x^2-1))^(1/2)*(
16*c^6*x^6-28*c^4*x^4+16*(c*x+1)^(1/2)*(c*x-1)^(1/2)*c^5*x^5+13*c^2*x^2-20*(c*x-1)^(1/2)*(c*x+1)^(1/2)*c^3*x^3
+5*(c*x-1)^(1/2)*(c*x+1)^(1/2)*c*x-1)*(-1+5*arccosh(c*x))*d/(c*x+1)/c^8/(c*x-1)+1/3072*(-d*(c^2*x^2-1))^(1/2)*
(4*c^4*x^4-5*c^2*x^2+4*(c*x-1)^(1/2)*(c*x+1)^(1/2)*c^3*x^3-3*(c*x-1)^(1/2)*(c*x+1)^(1/2)*c*x+1)*(-1+3*arccosh(
c*x))*d/(c*x+1)/c^8/(c*x-1)-7/1024*(-d*(c^2*x^2-1))^(1/2)*((c*x-1)^(1/2)*(c*x+1)^(1/2)*c*x+c^2*x^2-1)*(-1+arcc
osh(c*x))*d/(c*x+1)/c^8/(c*x-1)-7/1024*(-d*(c^2*x^2-1))^(1/2)*(-(c*x-1)^(1/2)*(c*x+1)^(1/2)*c*x+c^2*x^2-1)*(1+
arccosh(c*x))*d/(c*x+1)/c^8/(c*x-1)+1/3072*(-d*(c^2*x^2-1))^(1/2)*(-4*(c*x-1)^(1/2)*(c*x+1)^(1/2)*c^3*x^3+4*c^
4*x^4+3*(c*x-1)^(1/2)*(c*x+1)^(1/2)*c*x-5*c^2*x^2+1)*(1+3*arccosh(c*x))*d/(c*x+1)/c^8/(c*x-1)+11/51200*(-d*(c^
2*x^2-1))^(1/2)*(-16*(c*x+1)^(1/2)*(c*x-1)^(1/2)*c^5*x^5+16*c^6*x^6+20*(c*x-1)^(1/2)*(c*x+1)^(1/2)*c^3*x^3-28*
c^4*x^4-5*(c*x-1)^(1/2)*(c*x+1)^(1/2)*c*x+13*c^2*x^2-1)*(1+5*arccosh(c*x))*d/(c*x+1)/c^8/(c*x-1)+1/100352*(-d*
(c^2*x^2-1))^(1/2)*(-64*(c*x+1)^(1/2)*(c*x-1)^(1/2)*x^7*c^7+64*c^8*x^8+112*(c*x+1)^(1/2)*(c*x-1)^(1/2)*c^5*x^5
-144*c^6*x^6-56*(c*x-1)^(1/2)*(c*x+1)^(1/2)*c^3*x^3+104*c^4*x^4+7*(c*x-1)^(1/2)*(c*x+1)^(1/2)*c*x-25*c^2*x^2+1
)*(1+7*arccosh(c*x))*d/(c*x+1)/c^8/(c*x-1)-1/55296*(-d*(c^2*x^2-1))^(1/2)*(-256*(c*x+1)^(1/2)*(c*x-1)^(1/2)*x^
9*c^9+256*c^10*x^10+576*(c*x+1)^(1/2)*(c*x-1)^(1/2)*x^7*c^7-704*c^8*x^8-432*(c*x+1)^(1/2)*(c*x-1)^(1/2)*c^5*x^
5+688*c^6*x^6+120*(c*x-1)^(1/2)*(c*x+1)^(1/2)*c^3*x^3-280*c^4*x^4-9*(c*x-1)^(1/2)*(c*x+1)^(1/2)*c*x+41*c^2*x^2
-1)*(1+9*arccosh(c*x))*d/(c*x+1)/c^8/(c*x-1)-1/247808*(-d*(c^2*x^2-1))^(1/2)*(-1024*(c*x+1)^(1/2)*(c*x-1)^(1/2
)*x^11*c^11+1024*c^12*x^12+2816*(c*x+1)^(1/2)*(c*x-1)^(1/2)*x^9*c^9-3328*c^10*x^10-2816*(c*x+1)^(1/2)*(c*x-1)^
(1/2)*x^7*c^7+4096*c^8*x^8+1232*(c*x+1)^(1/2)*(c*x-1)^(1/2)*c^5*x^5-2352*c^6*x^6-220*(c*x-1)^(1/2)*(c*x+1)^(1/
2)*c^3*x^3+620*c^4*x^4+11*(c*x-1)^(1/2)*(c*x+1)^(1/2)*c*x-61*c^2*x^2+1)*(1+11*arccosh(c*x))*d/(c*x+1)/c^8/(c*x
-1))

Fricas [A] (verification not implemented)

none

Time = 0.26 (sec) , antiderivative size = 275, normalized size of antiderivative = 0.69 \[ \int x^7 \left (d-c^2 d x^2\right )^{3/2} (a+b \text {arccosh}(c x)) \, dx=-\frac {3465 \, {\left (105 \, b c^{12} d x^{12} - 245 \, b c^{10} d x^{10} + 145 \, b c^{8} d x^{8} + b c^{6} d x^{6} + 2 \, b c^{4} d x^{4} + 8 \, b c^{2} d x^{2} - 16 \, b d\right )} \sqrt {-c^{2} d x^{2} + d} \log \left (c x + \sqrt {c^{2} x^{2} - 1}\right ) - {\left (33075 \, b c^{11} d x^{11} - 53900 \, b c^{9} d x^{9} + 2475 \, b c^{7} d x^{7} + 4158 \, b c^{5} d x^{5} + 9240 \, b c^{3} d x^{3} + 55440 \, b c d x\right )} \sqrt {-c^{2} d x^{2} + d} \sqrt {c^{2} x^{2} - 1} + 3465 \, {\left (105 \, a c^{12} d x^{12} - 245 \, a c^{10} d x^{10} + 145 \, a c^{8} d x^{8} + a c^{6} d x^{6} + 2 \, a c^{4} d x^{4} + 8 \, a c^{2} d x^{2} - 16 \, a d\right )} \sqrt {-c^{2} d x^{2} + d}}{4002075 \, {\left (c^{10} x^{2} - c^{8}\right )}} \]

[In]

integrate(x^7*(-c^2*d*x^2+d)^(3/2)*(a+b*arccosh(c*x)),x, algorithm="fricas")

[Out]

-1/4002075*(3465*(105*b*c^12*d*x^12 - 245*b*c^10*d*x^10 + 145*b*c^8*d*x^8 + b*c^6*d*x^6 + 2*b*c^4*d*x^4 + 8*b*
c^2*d*x^2 - 16*b*d)*sqrt(-c^2*d*x^2 + d)*log(c*x + sqrt(c^2*x^2 - 1)) - (33075*b*c^11*d*x^11 - 53900*b*c^9*d*x
^9 + 2475*b*c^7*d*x^7 + 4158*b*c^5*d*x^5 + 9240*b*c^3*d*x^3 + 55440*b*c*d*x)*sqrt(-c^2*d*x^2 + d)*sqrt(c^2*x^2
- 1) + 3465*(105*a*c^12*d*x^12 - 245*a*c^10*d*x^10 + 145*a*c^8*d*x^8 + a*c^6*d*x^6 + 2*a*c^4*d*x^4 + 8*a*c^2*
d*x^2 - 16*a*d)*sqrt(-c^2*d*x^2 + d))/(c^10*x^2 - c^8)

Sympy [F(-1)]

Timed out. \[ \int x^7 \left (d-c^2 d x^2\right )^{3/2} (a+b \text {arccosh}(c x)) \, dx=\text {Timed out} \]

[In]

integrate(x**7*(-c**2*d*x**2+d)**(3/2)*(a+b*acosh(c*x)),x)

[Out]

Timed out

Maxima [A] (verification not implemented)

none

Time = 0.41 (sec) , antiderivative size = 285, normalized size of antiderivative = 0.71 \[ \int x^7 \left (d-c^2 d x^2\right )^{3/2} (a+b \text {arccosh}(c x)) \, dx=-\frac {1}{1155} \, {\left (\frac {105 \, {\left (-c^{2} d x^{2} + d\right )}^{\frac {5}{2}} x^{6}}{c^{2} d} + \frac {70 \, {\left (-c^{2} d x^{2} + d\right )}^{\frac {5}{2}} x^{4}}{c^{4} d} + \frac {40 \, {\left (-c^{2} d x^{2} + d\right )}^{\frac {5}{2}} x^{2}}{c^{6} d} + \frac {16 \, {\left (-c^{2} d x^{2} + d\right )}^{\frac {5}{2}}}{c^{8} d}\right )} b \operatorname {arcosh}\left (c x\right ) - \frac {1}{1155} \, {\left (\frac {105 \, {\left (-c^{2} d x^{2} + d\right )}^{\frac {5}{2}} x^{6}}{c^{2} d} + \frac {70 \, {\left (-c^{2} d x^{2} + d\right )}^{\frac {5}{2}} x^{4}}{c^{4} d} + \frac {40 \, {\left (-c^{2} d x^{2} + d\right )}^{\frac {5}{2}} x^{2}}{c^{6} d} + \frac {16 \, {\left (-c^{2} d x^{2} + d\right )}^{\frac {5}{2}}}{c^{8} d}\right )} a + \frac {{\left (33075 \, c^{10} \sqrt {-d} d x^{11} - 53900 \, c^{8} \sqrt {-d} d x^{9} + 2475 \, c^{6} \sqrt {-d} d x^{7} + 4158 \, c^{4} \sqrt {-d} d x^{5} + 9240 \, c^{2} \sqrt {-d} d x^{3} + 55440 \, \sqrt {-d} d x\right )} b}{4002075 \, c^{7}} \]

[In]

integrate(x^7*(-c^2*d*x^2+d)^(3/2)*(a+b*arccosh(c*x)),x, algorithm="maxima")

[Out]

-1/1155*(105*(-c^2*d*x^2 + d)^(5/2)*x^6/(c^2*d) + 70*(-c^2*d*x^2 + d)^(5/2)*x^4/(c^4*d) + 40*(-c^2*d*x^2 + d)^
(5/2)*x^2/(c^6*d) + 16*(-c^2*d*x^2 + d)^(5/2)/(c^8*d))*b*arccosh(c*x) - 1/1155*(105*(-c^2*d*x^2 + d)^(5/2)*x^6
/(c^2*d) + 70*(-c^2*d*x^2 + d)^(5/2)*x^4/(c^4*d) + 40*(-c^2*d*x^2 + d)^(5/2)*x^2/(c^6*d) + 16*(-c^2*d*x^2 + d)
^(5/2)/(c^8*d))*a + 1/4002075*(33075*c^10*sqrt(-d)*d*x^11 - 53900*c^8*sqrt(-d)*d*x^9 + 2475*c^6*sqrt(-d)*d*x^7
+ 4158*c^4*sqrt(-d)*d*x^5 + 9240*c^2*sqrt(-d)*d*x^3 + 55440*sqrt(-d)*d*x)*b/c^7

Giac [F(-2)]

Exception generated. \[ \int x^7 \left (d-c^2 d x^2\right )^{3/2} (a+b \text {arccosh}(c x)) \, dx=\text {Exception raised: TypeError} \]

[In]

integrate(x^7*(-c^2*d*x^2+d)^(3/2)*(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. \[ \int x^7 \left (d-c^2 d x^2\right )^{3/2} (a+b \text {arccosh}(c x)) \, dx=\int x^7\,\left (a+b\,\mathrm {acosh}\left (c\,x\right )\right )\,{\left (d-c^2\,d\,x^2\right )}^{3/2} \,d x \]

[In]

int(x^7*(a + b*acosh(c*x))*(d - c^2*d*x^2)^(3/2),x)

[Out]

int(x^7*(a + b*acosh(c*x))*(d - c^2*d*x^2)^(3/2), x)