3.96 \(\int x^7 (d-c^2 d x^2)^{5/2} (a+b \cosh ^{-1}(c x)) \, dx\)

Optimal. Leaf size=458 \[ \frac{\left (d-c^2 d x^2\right )^{13/2} \left (a+b \cosh ^{-1}(c x)\right )}{13 c^8 d^4}-\frac{3 \left (d-c^2 d x^2\right )^{11/2} \left (a+b \cosh ^{-1}(c x)\right )}{11 c^8 d^3}+\frac{\left (d-c^2 d x^2\right )^{9/2} \left (a+b \cosh ^{-1}(c x)\right )}{3 c^8 d^2}-\frac{\left (d-c^2 d x^2\right )^{7/2} \left (a+b \cosh ^{-1}(c x)\right )}{7 c^8 d}-\frac{b c^5 d^2 x^{13} \sqrt{d-c^2 d x^2}}{169 \sqrt{c x-1} \sqrt{c x+1}}+\frac{27 b c^3 d^2 x^{11} \sqrt{d-c^2 d x^2}}{1573 \sqrt{c x-1} \sqrt{c x+1}}-\frac{53 b c d^2 x^9 \sqrt{d-c^2 d x^2}}{3861 \sqrt{c x-1} \sqrt{c x+1}}+\frac{5 b d^2 x^7 \sqrt{d-c^2 d x^2}}{21021 c \sqrt{c x-1} \sqrt{c x+1}}+\frac{2 b d^2 x^5 \sqrt{d-c^2 d x^2}}{5005 c^3 \sqrt{c x-1} \sqrt{c x+1}}+\frac{8 b d^2 x^3 \sqrt{d-c^2 d x^2}}{9009 c^5 \sqrt{c x-1} \sqrt{c x+1}}+\frac{16 b d^2 x \sqrt{d-c^2 d x^2}}{3003 c^7 \sqrt{c x-1} \sqrt{c x+1}} \]

[Out]

(16*b*d^2*x*Sqrt[d - c^2*d*x^2])/(3003*c^7*Sqrt[-1 + c*x]*Sqrt[1 + c*x]) + (8*b*d^2*x^3*Sqrt[d - c^2*d*x^2])/(
9009*c^5*Sqrt[-1 + c*x]*Sqrt[1 + c*x]) + (2*b*d^2*x^5*Sqrt[d - c^2*d*x^2])/(5005*c^3*Sqrt[-1 + c*x]*Sqrt[1 + c
*x]) + (5*b*d^2*x^7*Sqrt[d - c^2*d*x^2])/(21021*c*Sqrt[-1 + c*x]*Sqrt[1 + c*x]) - (53*b*c*d^2*x^9*Sqrt[d - c^2
*d*x^2])/(3861*Sqrt[-1 + c*x]*Sqrt[1 + c*x]) + (27*b*c^3*d^2*x^11*Sqrt[d - c^2*d*x^2])/(1573*Sqrt[-1 + c*x]*Sq
rt[1 + c*x]) - (b*c^5*d^2*x^13*Sqrt[d - c^2*d*x^2])/(169*Sqrt[-1 + c*x]*Sqrt[1 + c*x]) - ((d - c^2*d*x^2)^(7/2
)*(a + b*ArcCosh[c*x]))/(7*c^8*d) + ((d - c^2*d*x^2)^(9/2)*(a + b*ArcCosh[c*x]))/(3*c^8*d^2) - (3*(d - c^2*d*x
^2)^(11/2)*(a + b*ArcCosh[c*x]))/(11*c^8*d^3) + ((d - c^2*d*x^2)^(13/2)*(a + b*ArcCosh[c*x]))/(13*c^8*d^4)

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Rubi [A]  time = 0.51565, antiderivative size = 527, normalized size of antiderivative = 1.15, number of steps used = 5, number of rules used = 6, integrand size = 27, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.222, Rules used = {5798, 100, 12, 74, 5733, 1810} \[ -\frac{d^2 x^6 (1-c x)^3 (c x+1)^3 \sqrt{d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )}{13 c^2}-\frac{6 d^2 x^4 (1-c x)^3 (c x+1)^3 \sqrt{d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )}{143 c^4}-\frac{8 d^2 x^2 (1-c x)^3 (c x+1)^3 \sqrt{d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )}{429 c^6}-\frac{16 d^2 (1-c x)^3 (c x+1)^3 \sqrt{d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )}{3003 c^8}-\frac{b c^5 d^2 x^{13} \sqrt{d-c^2 d x^2}}{169 \sqrt{c x-1} \sqrt{c x+1}}+\frac{27 b c^3 d^2 x^{11} \sqrt{d-c^2 d x^2}}{1573 \sqrt{c x-1} \sqrt{c x+1}}-\frac{53 b c d^2 x^9 \sqrt{d-c^2 d x^2}}{3861 \sqrt{c x-1} \sqrt{c x+1}}+\frac{5 b d^2 x^7 \sqrt{d-c^2 d x^2}}{21021 c \sqrt{c x-1} \sqrt{c x+1}}+\frac{2 b d^2 x^5 \sqrt{d-c^2 d x^2}}{5005 c^3 \sqrt{c x-1} \sqrt{c x+1}}+\frac{8 b d^2 x^3 \sqrt{d-c^2 d x^2}}{9009 c^5 \sqrt{c x-1} \sqrt{c x+1}}+\frac{16 b d^2 x \sqrt{d-c^2 d x^2}}{3003 c^7 \sqrt{c x-1} \sqrt{c x+1}} \]

Antiderivative was successfully verified.

[In]

Int[x^7*(d - c^2*d*x^2)^(5/2)*(a + b*ArcCosh[c*x]),x]

[Out]

(16*b*d^2*x*Sqrt[d - c^2*d*x^2])/(3003*c^7*Sqrt[-1 + c*x]*Sqrt[1 + c*x]) + (8*b*d^2*x^3*Sqrt[d - c^2*d*x^2])/(
9009*c^5*Sqrt[-1 + c*x]*Sqrt[1 + c*x]) + (2*b*d^2*x^5*Sqrt[d - c^2*d*x^2])/(5005*c^3*Sqrt[-1 + c*x]*Sqrt[1 + c
*x]) + (5*b*d^2*x^7*Sqrt[d - c^2*d*x^2])/(21021*c*Sqrt[-1 + c*x]*Sqrt[1 + c*x]) - (53*b*c*d^2*x^9*Sqrt[d - c^2
*d*x^2])/(3861*Sqrt[-1 + c*x]*Sqrt[1 + c*x]) + (27*b*c^3*d^2*x^11*Sqrt[d - c^2*d*x^2])/(1573*Sqrt[-1 + c*x]*Sq
rt[1 + c*x]) - (b*c^5*d^2*x^13*Sqrt[d - c^2*d*x^2])/(169*Sqrt[-1 + c*x]*Sqrt[1 + c*x]) - (16*d^2*(1 - c*x)^3*(
1 + c*x)^3*Sqrt[d - c^2*d*x^2]*(a + b*ArcCosh[c*x]))/(3003*c^8) - (8*d^2*x^2*(1 - c*x)^3*(1 + c*x)^3*Sqrt[d -
c^2*d*x^2]*(a + b*ArcCosh[c*x]))/(429*c^6) - (6*d^2*x^4*(1 - c*x)^3*(1 + c*x)^3*Sqrt[d - c^2*d*x^2]*(a + b*Arc
Cosh[c*x]))/(143*c^4) - (d^2*x^6*(1 - c*x)^3*(1 + c*x)^3*Sqrt[d - c^2*d*x^2]*(a + b*ArcCosh[c*x]))/(13*c^2)

Rule 5798

Int[((a_.) + ArcCosh[(c_.)*(x_)]*(b_.))^(n_.)*((f_.)*(x_))^(m_.)*((d_) + (e_.)*(x_)^2)^(p_), x_Symbol] :> Dist
[((-d)^IntPart[p]*(d + e*x^2)^FracPart[p])/((1 + c*x)^FracPart[p]*(-1 + c*x)^FracPart[p]), Int[(f*x)^m*(1 + c*
x)^p*(-1 + c*x)^p*(a + b*ArcCosh[c*x])^n, x], x] /; FreeQ[{a, b, c, d, e, f, m, n, p}, x] && EqQ[c^2*d + e, 0]
&&  !IntegerQ[p]

Rule 100

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_.)*((e_.) + (f_.)*(x_))^(p_.), x_Symbol] :> Simp[(b*(a +
b*x)^(m - 1)*(c + d*x)^(n + 1)*(e + f*x)^(p + 1))/(d*f*(m + n + p + 1)), x] + Dist[1/(d*f*(m + n + p + 1)), I
nt[(a + b*x)^(m - 2)*(c + d*x)^n*(e + f*x)^p*Simp[a^2*d*f*(m + n + p + 1) - b*(b*c*e*(m - 1) + a*(d*e*(n + 1)
+ c*f*(p + 1))) + b*(a*d*f*(2*m + n + p) - b*(d*e*(m + n) + c*f*(m + p)))*x, x], x], x] /; FreeQ[{a, b, c, d,
e, f, n, p}, x] && GtQ[m, 1] && NeQ[m + n + p + 1, 0] && IntegerQ[m]

Rule 12

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

Rule 74

Int[((a_.) + (b_.)*(x_))*((c_.) + (d_.)*(x_))^(n_.)*((e_.) + (f_.)*(x_))^(p_.), x_Symbol] :> Simp[(b*(c + d*x)
^(n + 1)*(e + f*x)^(p + 1))/(d*f*(n + p + 2)), x] /; FreeQ[{a, b, c, d, e, f, n, p}, x] && NeQ[n + p + 2, 0] &
& EqQ[a*d*f*(n + p + 2) - b*(d*e*(n + 1) + c*f*(p + 1)), 0]

Rule 5733

Int[((a_.) + ArcCosh[(c_.)*(x_)]*(b_.))*(x_)^(m_)*((d1_) + (e1_.)*(x_))^(p_)*((d2_) + (e2_.)*(x_))^(p_), x_Sym
bol] :> With[{u = IntHide[x^m*(1 + c*x)^p*(-1 + c*x)^p, x]}, Dist[(-(d1*d2))^p*(a + b*ArcCosh[c*x]), u, x] - D
ist[b*c*(-(d1*d2))^p, Int[SimplifyIntegrand[u/(Sqrt[1 + c*x]*Sqrt[-1 + c*x]), x], x], x]] /; FreeQ[{a, b, c, d
1, e1, d2, e2}, x] && EqQ[e1 - c*d1, 0] && EqQ[e2 + c*d2, 0] && IntegerQ[p - 1/2] && (IGtQ[(m + 1)/2, 0] || IL
tQ[(m + 2*p + 3)/2, 0]) && NeQ[p, -2^(-1)] && GtQ[d1, 0] && LtQ[d2, 0]

Rule 1810

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]

Rubi steps

\begin{align*} \int x^7 \left (d-c^2 d x^2\right )^{5/2} \left (a+b \cosh ^{-1}(c x)\right ) \, dx &=\frac{\left (d^2 \sqrt{d-c^2 d x^2}\right ) \int x^7 (-1+c x)^{5/2} (1+c x)^{5/2} \left (a+b \cosh ^{-1}(c x)\right ) \, dx}{\sqrt{-1+c x} \sqrt{1+c x}}\\ &=-\frac{16 d^2 (1-c x)^3 (1+c x)^3 \sqrt{d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )}{3003 c^8}-\frac{8 d^2 x^2 (1-c x)^3 (1+c x)^3 \sqrt{d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )}{429 c^6}-\frac{6 d^2 x^4 (1-c x)^3 (1+c x)^3 \sqrt{d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )}{143 c^4}-\frac{d^2 x^6 (1-c x)^3 (1+c x)^3 \sqrt{d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )}{13 c^2}-\frac{\left (b c d^2 \sqrt{d-c^2 d x^2}\right ) \int \frac{\left (1-c^2 x^2\right )^3 \left (-16-56 c^2 x^2-126 c^4 x^4-231 c^6 x^6\right )}{3003 c^8} \, dx}{\sqrt{-1+c x} \sqrt{1+c x}}\\ &=-\frac{16 d^2 (1-c x)^3 (1+c x)^3 \sqrt{d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )}{3003 c^8}-\frac{8 d^2 x^2 (1-c x)^3 (1+c x)^3 \sqrt{d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )}{429 c^6}-\frac{6 d^2 x^4 (1-c x)^3 (1+c x)^3 \sqrt{d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )}{143 c^4}-\frac{d^2 x^6 (1-c x)^3 (1+c x)^3 \sqrt{d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )}{13 c^2}-\frac{\left (b d^2 \sqrt{d-c^2 d x^2}\right ) \int \left (1-c^2 x^2\right )^3 \left (-16-56 c^2 x^2-126 c^4 x^4-231 c^6 x^6\right ) \, dx}{3003 c^7 \sqrt{-1+c x} \sqrt{1+c x}}\\ &=-\frac{16 d^2 (1-c x)^3 (1+c x)^3 \sqrt{d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )}{3003 c^8}-\frac{8 d^2 x^2 (1-c x)^3 (1+c x)^3 \sqrt{d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )}{429 c^6}-\frac{6 d^2 x^4 (1-c x)^3 (1+c x)^3 \sqrt{d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )}{143 c^4}-\frac{d^2 x^6 (1-c x)^3 (1+c x)^3 \sqrt{d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )}{13 c^2}-\frac{\left (b d^2 \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+371 c^8 x^8-567 c^{10} x^{10}+231 c^{12} x^{12}\right ) \, dx}{3003 c^7 \sqrt{-1+c x} \sqrt{1+c x}}\\ &=\frac{16 b d^2 x \sqrt{d-c^2 d x^2}}{3003 c^7 \sqrt{-1+c x} \sqrt{1+c x}}+\frac{8 b d^2 x^3 \sqrt{d-c^2 d x^2}}{9009 c^5 \sqrt{-1+c x} \sqrt{1+c x}}+\frac{2 b d^2 x^5 \sqrt{d-c^2 d x^2}}{5005 c^3 \sqrt{-1+c x} \sqrt{1+c x}}+\frac{5 b d^2 x^7 \sqrt{d-c^2 d x^2}}{21021 c \sqrt{-1+c x} \sqrt{1+c x}}-\frac{53 b c d^2 x^9 \sqrt{d-c^2 d x^2}}{3861 \sqrt{-1+c x} \sqrt{1+c x}}+\frac{27 b c^3 d^2 x^{11} \sqrt{d-c^2 d x^2}}{1573 \sqrt{-1+c x} \sqrt{1+c x}}-\frac{b c^5 d^2 x^{13} \sqrt{d-c^2 d x^2}}{169 \sqrt{-1+c x} \sqrt{1+c x}}-\frac{16 d^2 (1-c x)^3 (1+c x)^3 \sqrt{d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )}{3003 c^8}-\frac{8 d^2 x^2 (1-c x)^3 (1+c x)^3 \sqrt{d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )}{429 c^6}-\frac{6 d^2 x^4 (1-c x)^3 (1+c x)^3 \sqrt{d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )}{143 c^4}-\frac{d^2 x^6 (1-c x)^3 (1+c x)^3 \sqrt{d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )}{13 c^2}\\ \end{align*}

Mathematica [A]  time = 0.246745, size = 193, normalized size = 0.42 \[ \frac{d^2 \sqrt{d-c^2 d x^2} \left (231 c^5 x^6 (c x-1)^{7/2} (c x+1)^{7/2} \left (a+b \cosh ^{-1}(c x)\right )+\frac{2 (c x-1)^{7/2} (c x+1)^{7/2} \left (63 c^4 x^4+28 c^2 x^2+8\right ) \left (a+b \cosh ^{-1}(c x)\right )}{c}+b \left (-\frac{231}{13} c^{12} x^{13}+\frac{567 c^{10} x^{11}}{11}-\frac{371 c^8 x^9}{9}+\frac{5 c^6 x^7}{7}+\frac{6 c^4 x^5}{5}+\frac{8 c^2 x^3}{3}+16 x\right )\right )}{3003 c^7 \sqrt{c x-1} \sqrt{c x+1}} \]

Antiderivative was successfully verified.

[In]

Integrate[x^7*(d - c^2*d*x^2)^(5/2)*(a + b*ArcCosh[c*x]),x]

[Out]

(d^2*Sqrt[d - c^2*d*x^2]*(b*(16*x + (8*c^2*x^3)/3 + (6*c^4*x^5)/5 + (5*c^6*x^7)/7 - (371*c^8*x^9)/9 + (567*c^1
0*x^11)/11 - (231*c^12*x^13)/13) + 231*c^5*x^6*(-1 + c*x)^(7/2)*(1 + c*x)^(7/2)*(a + b*ArcCosh[c*x]) + (2*(-1
+ c*x)^(7/2)*(1 + c*x)^(7/2)*(8 + 28*c^2*x^2 + 63*c^4*x^4)*(a + b*ArcCosh[c*x]))/c))/(3003*c^7*Sqrt[-1 + c*x]*
Sqrt[1 + c*x])

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Maple [B]  time = 0.5, size = 2374, normalized size = 5.2 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^7*(-c^2*d*x^2+d)^(5/2)*(a+b*arccosh(c*x)),x)

[Out]

a*(-1/13*x^6*(-c^2*d*x^2+d)^(7/2)/c^2/d+6/13/c^2*(-1/11*x^4*(-c^2*d*x^2+d)^(7/2)/c^2/d+4/11/c^2*(-1/9*x^2*(-c^
2*d*x^2+d)^(7/2)/c^2/d-2/63/d/c^4*(-c^2*d*x^2+d)^(7/2))))+b*(1/1384448*(-d*(c^2*x^2-1))^(1/2)*(-1-16896*c^8*x^
8+6496*c^6*x^6-1204*c^4*x^4+85*c^2*x^2+4096*x^14*c^14-15360*x^12*c^12+22784*x^10*c^10-364*(c*x+1)^(1/2)*(c*x-1
)^(1/2)*x^3*c^3+13*(c*x+1)^(1/2)*(c*x-1)^(1/2)*x*c+4096*(c*x+1)^(1/2)*(c*x-1)^(1/2)*x^13*c^13-13312*(c*x+1)^(1
/2)*(c*x-1)^(1/2)*x^11*c^11+16640*(c*x+1)^(1/2)*(c*x-1)^(1/2)*x^9*c^9-9984*(c*x+1)^(1/2)*(c*x-1)^(1/2)*x^7*c^7
+2912*(c*x+1)^(1/2)*(c*x-1)^(1/2)*x^5*c^5)*(-1+13*arccosh(c*x))*d^2/(c*x+1)/c^8/(c*x-1)+1/991232*(-d*(c^2*x^2-
1))^(1/2)*(1+4096*c^8*x^8-2352*c^6*x^6+620*c^4*x^4-61*c^2*x^2+1024*x^12*c^12-3328*x^10*c^10+220*(c*x+1)^(1/2)*
(c*x-1)^(1/2)*x^3*c^3-11*(c*x+1)^(1/2)*(c*x-1)^(1/2)*x*c+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+2816*(c*x+1)^(1/2)*(c*x-1)^(1/2)*x^7*c^7-1232*(c*x+1)^(1/2)*(c*x-1)^(1/2)*x^5*c
^5)*(-1+11*arccosh(c*x))*d^2/(c*x+1)/c^8/(c*x-1)-1/110592*(-d*(c^2*x^2-1))^(1/2)*(256*x^10*c^10-704*c^8*x^8+25
6*(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)*x^5*c^5+41*c^2*x^2-120*(c*x+1)^(1/2)*(c*x-1)^(1/2)*x^3*c^3+9*(c*x+1)^(1/2)*(c*x-1)^(1/
2)*x*c-1)*(-1+9*arccosh(c*x))*d^2/(c*x+1)/c^8/(c*x-1)-3/200704*(-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)*x^5*c^5-25*c^2*x^2+56*(c*x+
1)^(1/2)*(c*x-1)^(1/2)*x^3*c^3-7*(c*x+1)^(1/2)*(c*x-1)^(1/2)*x*c+1)*(-1+7*arccosh(c*x))*d^2/(c*x+1)/c^8/(c*x-1
)+3/40960*(-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)*x^5*c^5+13*c^2*x^2-20*(
c*x+1)^(1/2)*(c*x-1)^(1/2)*x^3*c^3+5*(c*x+1)^(1/2)*(c*x-1)^(1/2)*x*c-1)*(-1+5*arccosh(c*x))*d^2/(c*x+1)/c^8/(c
*x-1)+5/24576*(-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)*x^3*c^3-3*(c*x+1)^(1/2
)*(c*x-1)^(1/2)*x*c+1)*(-1+3*arccosh(c*x))*d^2/(c*x+1)/c^8/(c*x-1)-5/2048*(-d*(c^2*x^2-1))^(1/2)*((c*x+1)^(1/2
)*(c*x-1)^(1/2)*x*c+c^2*x^2-1)*(-1+arccosh(c*x))*d^2/(c*x+1)/c^8/(c*x-1)-5/2048*(-d*(c^2*x^2-1))^(1/2)*(-(c*x+
1)^(1/2)*(c*x-1)^(1/2)*x*c+c^2*x^2-1)*(1+arccosh(c*x))*d^2/(c*x+1)/c^8/(c*x-1)+5/24576*(-d*(c^2*x^2-1))^(1/2)*
(-4*(c*x+1)^(1/2)*(c*x-1)^(1/2)*x^3*c^3+4*c^4*x^4+3*(c*x+1)^(1/2)*(c*x-1)^(1/2)*x*c-5*c^2*x^2+1)*(1+3*arccosh(
c*x))*d^2/(c*x+1)/c^8/(c*x-1)+3/40960*(-d*(c^2*x^2-1))^(1/2)*(-16*(c*x+1)^(1/2)*(c*x-1)^(1/2)*x^5*c^5+16*c^6*x
^6+20*(c*x+1)^(1/2)*(c*x-1)^(1/2)*x^3*c^3-28*c^4*x^4-5*(c*x+1)^(1/2)*(c*x-1)^(1/2)*x*c+13*c^2*x^2-1)*(1+5*arcc
osh(c*x))*d^2/(c*x+1)/c^8/(c*x-1)-3/200704*(-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)*x^5*c^5-144*c^6*x^6-56*(c*x+1)^(1/2)*(c*x-1)^(1/2)*x^3*c^3+104*c^4*x^4
+7*(c*x+1)^(1/2)*(c*x-1)^(1/2)*x*c-25*c^2*x^2+1)*(1+7*arccosh(c*x))*d^2/(c*x+1)/c^8/(c*x-1)-1/110592*(-d*(c^2*
x^2-1))^(1/2)*(-256*(c*x+1)^(1/2)*(c*x-1)^(1/2)*x^9*c^9+256*x^10*c^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)*x^5*c^5+688*c^6*x^6+120*(c*x+1)^(1/2)*(c*x-1)^(1/2)*x^3*c^3-280*c^
4*x^4-9*(c*x+1)^(1/2)*(c*x-1)^(1/2)*x*c+41*c^2*x^2-1)*(1+9*arccosh(c*x))*d^2/(c*x+1)/c^8/(c*x-1)+1/991232*(-d*
(c^2*x^2-1))^(1/2)*(-1024*(c*x+1)^(1/2)*(c*x-1)^(1/2)*x^11*c^11+1024*x^12*c^12+2816*(c*x+1)^(1/2)*(c*x-1)^(1/2
)*x^9*c^9-3328*x^10*c^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
)*x^5*c^5-2352*c^6*x^6-220*(c*x+1)^(1/2)*(c*x-1)^(1/2)*x^3*c^3+620*c^4*x^4+11*(c*x+1)^(1/2)*(c*x-1)^(1/2)*x*c-
61*c^2*x^2+1)*(1+11*arccosh(c*x))*d^2/(c*x+1)/c^8/(c*x-1)+1/1384448*(-d*(c^2*x^2-1))^(1/2)*(-4096*(c*x+1)^(1/2
)*(c*x-1)^(1/2)*x^13*c^13+4096*x^14*c^14+13312*(c*x+1)^(1/2)*(c*x-1)^(1/2)*x^11*c^11-15360*x^12*c^12-16640*(c*
x+1)^(1/2)*(c*x-1)^(1/2)*x^9*c^9+22784*x^10*c^10+9984*(c*x+1)^(1/2)*(c*x-1)^(1/2)*x^7*c^7-16896*c^8*x^8-2912*(
c*x+1)^(1/2)*(c*x-1)^(1/2)*x^5*c^5+6496*c^6*x^6+364*(c*x+1)^(1/2)*(c*x-1)^(1/2)*x^3*c^3-1204*c^4*x^4-13*(c*x+1
)^(1/2)*(c*x-1)^(1/2)*x*c+85*c^2*x^2-1)*(1+13*arccosh(c*x))*d^2/(c*x+1)/c^8/(c*x-1))

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Maxima [F(-2)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: ValueError} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^7*(-c^2*d*x^2+d)^(5/2)*(a+b*arccosh(c*x)),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 2.29234, size = 833, normalized size = 1.82 \begin{align*} \frac{45045 \,{\left (231 \, b c^{14} d^{2} x^{14} - 798 \, b c^{12} d^{2} x^{12} + 938 \, b c^{10} d^{2} x^{10} - 376 \, b c^{8} d^{2} x^{8} - b c^{6} d^{2} x^{6} - 2 \, b c^{4} d^{2} x^{4} - 8 \, b c^{2} d^{2} x^{2} + 16 \, b d^{2}\right )} \sqrt{-c^{2} d x^{2} + d} \log \left (c x + \sqrt{c^{2} x^{2} - 1}\right ) -{\left (800415 \, b c^{13} d^{2} x^{13} - 2321865 \, b c^{11} d^{2} x^{11} + 1856855 \, b c^{9} d^{2} x^{9} - 32175 \, b c^{7} d^{2} x^{7} - 54054 \, b c^{5} d^{2} x^{5} - 120120 \, b c^{3} d^{2} x^{3} - 720720 \, b c d^{2} x\right )} \sqrt{-c^{2} d x^{2} + d} \sqrt{c^{2} x^{2} - 1} + 45045 \,{\left (231 \, a c^{14} d^{2} x^{14} - 798 \, a c^{12} d^{2} x^{12} + 938 \, a c^{10} d^{2} x^{10} - 376 \, a c^{8} d^{2} x^{8} - a c^{6} d^{2} x^{6} - 2 \, a c^{4} d^{2} x^{4} - 8 \, a c^{2} d^{2} x^{2} + 16 \, a d^{2}\right )} \sqrt{-c^{2} d x^{2} + d}}{135270135 \,{\left (c^{10} x^{2} - c^{8}\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^7*(-c^2*d*x^2+d)^(5/2)*(a+b*arccosh(c*x)),x, algorithm="fricas")

[Out]

1/135270135*(45045*(231*b*c^14*d^2*x^14 - 798*b*c^12*d^2*x^12 + 938*b*c^10*d^2*x^10 - 376*b*c^8*d^2*x^8 - b*c^
6*d^2*x^6 - 2*b*c^4*d^2*x^4 - 8*b*c^2*d^2*x^2 + 16*b*d^2)*sqrt(-c^2*d*x^2 + d)*log(c*x + sqrt(c^2*x^2 - 1)) -
(800415*b*c^13*d^2*x^13 - 2321865*b*c^11*d^2*x^11 + 1856855*b*c^9*d^2*x^9 - 32175*b*c^7*d^2*x^7 - 54054*b*c^5*
d^2*x^5 - 120120*b*c^3*d^2*x^3 - 720720*b*c*d^2*x)*sqrt(-c^2*d*x^2 + d)*sqrt(c^2*x^2 - 1) + 45045*(231*a*c^14*
d^2*x^14 - 798*a*c^12*d^2*x^12 + 938*a*c^10*d^2*x^10 - 376*a*c^8*d^2*x^8 - a*c^6*d^2*x^6 - 2*a*c^4*d^2*x^4 - 8
*a*c^2*d^2*x^2 + 16*a*d^2)*sqrt(-c^2*d*x^2 + d))/(c^10*x^2 - c^8)

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Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x**7*(-c**2*d*x**2+d)**(5/2)*(a+b*acosh(c*x)),x)

[Out]

Timed out

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Giac [F(-2)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: NotImplementedError} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^7*(-c^2*d*x^2+d)^(5/2)*(a+b*arccosh(c*x)),x, algorithm="giac")

[Out]

Exception raised: NotImplementedError