3.98 \(\int x^3 (d-c^2 d x^2)^{5/2} (a+b \cosh ^{-1}(c x)) \, dx\)

Optimal. Leaf size=298 \[ \frac {\left (d-c^2 d x^2\right )^{9/2} \left (a+b \cosh ^{-1}(c x)\right )}{9 c^4 d^2}-\frac {\left (d-c^2 d x^2\right )^{7/2} \left (a+b \cosh ^{-1}(c x)\right )}{7 c^4 d}-\frac {b c d^2 x^5 \sqrt {d-c^2 d x^2}}{21 \sqrt {c x-1} \sqrt {c x+1}}+\frac {b d^2 x^3 \sqrt {d-c^2 d x^2}}{189 c \sqrt {c x-1} \sqrt {c x+1}}-\frac {b c^5 d^2 x^9 \sqrt {d-c^2 d x^2}}{81 \sqrt {c x-1} \sqrt {c x+1}}+\frac {2 b d^2 x \sqrt {d-c^2 d x^2}}{63 c^3 \sqrt {c x-1} \sqrt {c x+1}}+\frac {19 b c^3 d^2 x^7 \sqrt {d-c^2 d x^2}}{441 \sqrt {c x-1} \sqrt {c x+1}} \]

[Out]

-1/7*(-c^2*d*x^2+d)^(7/2)*(a+b*arccosh(c*x))/c^4/d+1/9*(-c^2*d*x^2+d)^(9/2)*(a+b*arccosh(c*x))/c^4/d^2+2/63*b*
d^2*x*(-c^2*d*x^2+d)^(1/2)/c^3/(c*x-1)^(1/2)/(c*x+1)^(1/2)+1/189*b*d^2*x^3*(-c^2*d*x^2+d)^(1/2)/c/(c*x-1)^(1/2
)/(c*x+1)^(1/2)-1/21*b*c*d^2*x^5*(-c^2*d*x^2+d)^(1/2)/(c*x-1)^(1/2)/(c*x+1)^(1/2)+19/441*b*c^3*d^2*x^7*(-c^2*d
*x^2+d)^(1/2)/(c*x-1)^(1/2)/(c*x+1)^(1/2)-1/81*b*c^5*d^2*x^9*(-c^2*d*x^2+d)^(1/2)/(c*x-1)^(1/2)/(c*x+1)^(1/2)

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Rubi [A]  time = 0.42, antiderivative size = 331, normalized size of antiderivative = 1.11, 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, 373} \[ -\frac {d^2 x^2 (1-c x)^3 (c x+1)^3 \sqrt {d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )}{9 c^2}-\frac {2 d^2 (1-c x)^3 (c x+1)^3 \sqrt {d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )}{63 c^4}-\frac {b c^5 d^2 x^9 \sqrt {d-c^2 d x^2}}{81 \sqrt {c x-1} \sqrt {c x+1}}+\frac {19 b c^3 d^2 x^7 \sqrt {d-c^2 d x^2}}{441 \sqrt {c x-1} \sqrt {c x+1}}-\frac {b c d^2 x^5 \sqrt {d-c^2 d x^2}}{21 \sqrt {c x-1} \sqrt {c x+1}}+\frac {b d^2 x^3 \sqrt {d-c^2 d x^2}}{189 c \sqrt {c x-1} \sqrt {c x+1}}+\frac {2 b d^2 x \sqrt {d-c^2 d x^2}}{63 c^3 \sqrt {c x-1} \sqrt {c x+1}} \]

Antiderivative was successfully verified.

[In]

Int[x^3*(d - c^2*d*x^2)^(5/2)*(a + b*ArcCosh[c*x]),x]

[Out]

(2*b*d^2*x*Sqrt[d - c^2*d*x^2])/(63*c^3*Sqrt[-1 + c*x]*Sqrt[1 + c*x]) + (b*d^2*x^3*Sqrt[d - c^2*d*x^2])/(189*c
*Sqrt[-1 + c*x]*Sqrt[1 + c*x]) - (b*c*d^2*x^5*Sqrt[d - c^2*d*x^2])/(21*Sqrt[-1 + c*x]*Sqrt[1 + c*x]) + (19*b*c
^3*d^2*x^7*Sqrt[d - c^2*d*x^2])/(441*Sqrt[-1 + c*x]*Sqrt[1 + c*x]) - (b*c^5*d^2*x^9*Sqrt[d - c^2*d*x^2])/(81*S
qrt[-1 + c*x]*Sqrt[1 + c*x]) - (2*d^2*(1 - c*x)^3*(1 + c*x)^3*Sqrt[d - c^2*d*x^2]*(a + b*ArcCosh[c*x]))/(63*c^
4) - (d^2*x^2*(1 - c*x)^3*(1 + c*x)^3*Sqrt[d - c^2*d*x^2]*(a + b*ArcCosh[c*x]))/(9*c^2)

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 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 373

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

Rubi steps

\begin {align*} \int x^3 \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^3 (-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 {2 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 )}{63 c^4}-\frac {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 )}{9 c^2}-\frac {\left (b c d^2 \sqrt {d-c^2 d x^2}\right ) \int \frac {\left (-2-7 c^2 x^2\right ) \left (1-c^2 x^2\right )^3}{63 c^4} \, dx}{\sqrt {-1+c x} \sqrt {1+c x}}\\ &=-\frac {2 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 )}{63 c^4}-\frac {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 )}{9 c^2}-\frac {\left (b d^2 \sqrt {d-c^2 d x^2}\right ) \int \left (-2-7 c^2 x^2\right ) \left (1-c^2 x^2\right )^3 \, dx}{63 c^3 \sqrt {-1+c x} \sqrt {1+c x}}\\ &=-\frac {2 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 )}{63 c^4}-\frac {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 )}{9 c^2}-\frac {\left (b d^2 \sqrt {d-c^2 d x^2}\right ) \int \left (-2-c^2 x^2+15 c^4 x^4-19 c^6 x^6+7 c^8 x^8\right ) \, dx}{63 c^3 \sqrt {-1+c x} \sqrt {1+c x}}\\ &=\frac {2 b d^2 x \sqrt {d-c^2 d x^2}}{63 c^3 \sqrt {-1+c x} \sqrt {1+c x}}+\frac {b d^2 x^3 \sqrt {d-c^2 d x^2}}{189 c \sqrt {-1+c x} \sqrt {1+c x}}-\frac {b c d^2 x^5 \sqrt {d-c^2 d x^2}}{21 \sqrt {-1+c x} \sqrt {1+c x}}+\frac {19 b c^3 d^2 x^7 \sqrt {d-c^2 d x^2}}{441 \sqrt {-1+c x} \sqrt {1+c x}}-\frac {b c^5 d^2 x^9 \sqrt {d-c^2 d x^2}}{81 \sqrt {-1+c x} \sqrt {1+c x}}-\frac {2 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 )}{63 c^4}-\frac {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 )}{9 c^2}\\ \end {align*}

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Mathematica [A]  time = 0.15, size = 160, normalized size = 0.54 \[ \frac {d^2 \sqrt {d-c^2 d x^2} \left (7 c^2 x^2 (c x-1)^{7/2} (c x+1)^{7/2} \left (a+b \cosh ^{-1}(c x)\right )+2 (c x-1)^{7/2} (c x+1)^{7/2} \left (a+b \cosh ^{-1}(c x)\right )-\frac {7}{9} b c x \left (c^2 x^2-1\right )^4+\frac {25}{9} b c \left (-\frac {1}{7} c^6 x^7+\frac {3 c^4 x^5}{5}-c^2 x^3+x\right )\right )}{63 c^4 \sqrt {c x-1} \sqrt {c x+1}} \]

Antiderivative was successfully verified.

[In]

Integrate[x^3*(d - c^2*d*x^2)^(5/2)*(a + b*ArcCosh[c*x]),x]

[Out]

(d^2*Sqrt[d - c^2*d*x^2]*((-7*b*c*x*(-1 + c^2*x^2)^4)/9 + (25*b*c*(x - c^2*x^3 + (3*c^4*x^5)/5 - (c^6*x^7)/7))
/9 + 2*(-1 + c*x)^(7/2)*(1 + c*x)^(7/2)*(a + b*ArcCosh[c*x]) + 7*c^2*x^2*(-1 + c*x)^(7/2)*(1 + c*x)^(7/2)*(a +
b*ArcCosh[c*x])))/(63*c^4*Sqrt[-1 + c*x]*Sqrt[1 + c*x])

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fricas [A]  time = 0.45, size = 281, normalized size = 0.94 \[ \frac {63 \, {\left (7 \, b c^{10} d^{2} x^{10} - 26 \, b c^{8} d^{2} x^{8} + 34 \, b c^{6} d^{2} x^{6} - 16 \, b c^{4} d^{2} x^{4} - b c^{2} d^{2} x^{2} + 2 \, b d^{2}\right )} \sqrt {-c^{2} d x^{2} + d} \log \left (c x + \sqrt {c^{2} x^{2} - 1}\right ) - {\left (49 \, b c^{9} d^{2} x^{9} - 171 \, b c^{7} d^{2} x^{7} + 189 \, b c^{5} d^{2} x^{5} - 21 \, b c^{3} d^{2} x^{3} - 126 \, b c d^{2} x\right )} \sqrt {-c^{2} d x^{2} + d} \sqrt {c^{2} x^{2} - 1} + 63 \, {\left (7 \, a c^{10} d^{2} x^{10} - 26 \, a c^{8} d^{2} x^{8} + 34 \, a c^{6} d^{2} x^{6} - 16 \, a c^{4} d^{2} x^{4} - a c^{2} d^{2} x^{2} + 2 \, a d^{2}\right )} \sqrt {-c^{2} d x^{2} + d}}{3969 \, {\left (c^{6} x^{2} - c^{4}\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^3*(-c^2*d*x^2+d)^(5/2)*(a+b*arccosh(c*x)),x, algorithm="fricas")

[Out]

1/3969*(63*(7*b*c^10*d^2*x^10 - 26*b*c^8*d^2*x^8 + 34*b*c^6*d^2*x^6 - 16*b*c^4*d^2*x^4 - b*c^2*d^2*x^2 + 2*b*d
^2)*sqrt(-c^2*d*x^2 + d)*log(c*x + sqrt(c^2*x^2 - 1)) - (49*b*c^9*d^2*x^9 - 171*b*c^7*d^2*x^7 + 189*b*c^5*d^2*
x^5 - 21*b*c^3*d^2*x^3 - 126*b*c*d^2*x)*sqrt(-c^2*d*x^2 + d)*sqrt(c^2*x^2 - 1) + 63*(7*a*c^10*d^2*x^10 - 26*a*
c^8*d^2*x^8 + 34*a*c^6*d^2*x^6 - 16*a*c^4*d^2*x^4 - a*c^2*d^2*x^2 + 2*a*d^2)*sqrt(-c^2*d*x^2 + d))/(c^6*x^2 -
c^4)

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giac [F(-2)]  time = 0.00, size = 0, normalized size = 0.00 \[ \text {Exception raised: TypeError} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^3*(-c^2*d*x^2+d)^(5/2)*(a+b*arccosh(c*x)),x, algorithm="giac")

[Out]

Exception raised: TypeError >> An error occurred running a Giac command:INPUT:sage2:=int(sage0,x):;OUTPUT:sym2
poly/r2sym(const gen & e,const index_m & i,const vecteur & l) Error: Bad Argument Value

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maple [B]  time = 0.54, size = 1102, normalized size = 3.70 \[ a \left (-\frac {x^{2} \left (-c^{2} d \,x^{2}+d \right )^{\frac {7}{2}}}{9 c^{2} d}-\frac {2 \left (-c^{2} d \,x^{2}+d \right )^{\frac {7}{2}}}{63 d \,c^{4}}\right )+b \left (\frac {\sqrt {-d \left (c^{2} x^{2}-1\right )}\, \left (256 c^{10} x^{10}-704 c^{8} x^{8}+256 \sqrt {c x +1}\, \sqrt {c x -1}\, x^{9} c^{9}+688 c^{6} x^{6}-576 \sqrt {c x +1}\, \sqrt {c x -1}\, x^{7} c^{7}-280 c^{4} x^{4}+432 \sqrt {c x +1}\, \sqrt {c x -1}\, x^{5} c^{5}+41 c^{2} x^{2}-120 \sqrt {c x +1}\, \sqrt {c x -1}\, x^{3} c^{3}+9 \sqrt {c x +1}\, \sqrt {c x -1}\, x c -1\right ) \left (-1+9 \,\mathrm {arccosh}\left (c x \right )\right ) d^{2}}{41472 \left (c x +1\right ) c^{4} \left (c x -1\right )}-\frac {3 \sqrt {-d \left (c^{2} x^{2}-1\right )}\, \left (64 c^{8} x^{8}-144 c^{6} x^{6}+64 \sqrt {c x +1}\, \sqrt {c x -1}\, x^{7} c^{7}+104 c^{4} x^{4}-112 \sqrt {c x +1}\, \sqrt {c x -1}\, x^{5} c^{5}-25 c^{2} x^{2}+56 \sqrt {c x +1}\, \sqrt {c x -1}\, x^{3} c^{3}-7 \sqrt {c x +1}\, \sqrt {c x -1}\, x c +1\right ) \left (-1+7 \,\mathrm {arccosh}\left (c x \right )\right ) d^{2}}{25088 \left (c x +1\right ) c^{4} \left (c x -1\right )}+\frac {\sqrt {-d \left (c^{2} x^{2}-1\right )}\, \left (4 c^{4} x^{4}-5 c^{2} x^{2}+4 \sqrt {c x +1}\, \sqrt {c x -1}\, x^{3} c^{3}-3 \sqrt {c x +1}\, \sqrt {c x -1}\, x c +1\right ) \left (-1+3 \,\mathrm {arccosh}\left (c x \right )\right ) d^{2}}{576 \left (c x +1\right ) c^{4} \left (c x -1\right )}-\frac {3 \sqrt {-d \left (c^{2} x^{2}-1\right )}\, \left (\sqrt {c x +1}\, \sqrt {c x -1}\, x c +c^{2} x^{2}-1\right ) \left (-1+\mathrm {arccosh}\left (c x \right )\right ) d^{2}}{256 \left (c x +1\right ) c^{4} \left (c x -1\right )}-\frac {3 \sqrt {-d \left (c^{2} x^{2}-1\right )}\, \left (-\sqrt {c x +1}\, \sqrt {c x -1}\, x c +c^{2} x^{2}-1\right ) \left (1+\mathrm {arccosh}\left (c x \right )\right ) d^{2}}{256 \left (c x +1\right ) c^{4} \left (c x -1\right )}+\frac {\sqrt {-d \left (c^{2} x^{2}-1\right )}\, \left (-4 \sqrt {c x +1}\, \sqrt {c x -1}\, x^{3} c^{3}+4 c^{4} x^{4}+3 \sqrt {c x +1}\, \sqrt {c x -1}\, x c -5 c^{2} x^{2}+1\right ) \left (1+3 \,\mathrm {arccosh}\left (c x \right )\right ) d^{2}}{576 \left (c x +1\right ) c^{4} \left (c x -1\right )}-\frac {3 \sqrt {-d \left (c^{2} x^{2}-1\right )}\, \left (-64 \sqrt {c x +1}\, \sqrt {c x -1}\, x^{7} c^{7}+64 c^{8} x^{8}+112 \sqrt {c x +1}\, \sqrt {c x -1}\, x^{5} c^{5}-144 c^{6} x^{6}-56 \sqrt {c x +1}\, \sqrt {c x -1}\, x^{3} c^{3}+104 c^{4} x^{4}+7 \sqrt {c x +1}\, \sqrt {c x -1}\, x c -25 c^{2} x^{2}+1\right ) \left (1+7 \,\mathrm {arccosh}\left (c x \right )\right ) d^{2}}{25088 \left (c x +1\right ) c^{4} \left (c x -1\right )}+\frac {\sqrt {-d \left (c^{2} x^{2}-1\right )}\, \left (-256 \sqrt {c x +1}\, \sqrt {c x -1}\, x^{9} c^{9}+256 c^{10} x^{10}+576 \sqrt {c x +1}\, \sqrt {c x -1}\, x^{7} c^{7}-704 c^{8} x^{8}-432 \sqrt {c x +1}\, \sqrt {c x -1}\, x^{5} c^{5}+688 c^{6} x^{6}+120 \sqrt {c x +1}\, \sqrt {c x -1}\, x^{3} c^{3}-280 c^{4} x^{4}-9 \sqrt {c x +1}\, \sqrt {c x -1}\, x c +41 c^{2} x^{2}-1\right ) \left (1+9 \,\mathrm {arccosh}\left (c x \right )\right ) d^{2}}{41472 \left (c x +1\right ) c^{4} \left (c x -1\right )}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^3*(-c^2*d*x^2+d)^(5/2)*(a+b*arccosh(c*x)),x)

[Out]

a*(-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*(1/41472*(-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)*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^4/(c*x-1)-3/25088*(-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^4/(c*x-1)+1/576*(-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^4/(c*x-1)-3/256*(-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^4/(c*x-1)-3/256*(-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^4/(c*x-1)+1/576*(-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^4/(c*x-1)-3/25088*(-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+10
4*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^4/(c*x-1)+1/41472*(
-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)*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^4/(c*x-1))

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maxima [A]  time = 0.95, size = 185, normalized size = 0.62 \[ -\frac {1}{63} \, {\left (\frac {7 \, {\left (-c^{2} d x^{2} + d\right )}^{\frac {7}{2}} x^{2}}{c^{2} d} + \frac {2 \, {\left (-c^{2} d x^{2} + d\right )}^{\frac {7}{2}}}{c^{4} d}\right )} b \operatorname {arcosh}\left (c x\right ) - \frac {1}{63} \, {\left (\frac {7 \, {\left (-c^{2} d x^{2} + d\right )}^{\frac {7}{2}} x^{2}}{c^{2} d} + \frac {2 \, {\left (-c^{2} d x^{2} + d\right )}^{\frac {7}{2}}}{c^{4} d}\right )} a - \frac {{\left (49 \, c^{8} \sqrt {-d} d^{2} x^{9} - 171 \, c^{6} \sqrt {-d} d^{2} x^{7} + 189 \, c^{4} \sqrt {-d} d^{2} x^{5} - 21 \, c^{2} \sqrt {-d} d^{2} x^{3} - 126 \, \sqrt {-d} d^{2} x\right )} b}{3969 \, c^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^3*(-c^2*d*x^2+d)^(5/2)*(a+b*arccosh(c*x)),x, algorithm="maxima")

[Out]

-1/63*(7*(-c^2*d*x^2 + d)^(7/2)*x^2/(c^2*d) + 2*(-c^2*d*x^2 + d)^(7/2)/(c^4*d))*b*arccosh(c*x) - 1/63*(7*(-c^2
*d*x^2 + d)^(7/2)*x^2/(c^2*d) + 2*(-c^2*d*x^2 + d)^(7/2)/(c^4*d))*a - 1/3969*(49*c^8*sqrt(-d)*d^2*x^9 - 171*c^
6*sqrt(-d)*d^2*x^7 + 189*c^4*sqrt(-d)*d^2*x^5 - 21*c^2*sqrt(-d)*d^2*x^3 - 126*sqrt(-d)*d^2*x)*b/c^3

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mupad [F]  time = 0.00, size = -1, normalized size = -0.00 \[ \int x^3\,\left (a+b\,\mathrm {acosh}\left (c\,x\right )\right )\,{\left (d-c^2\,d\,x^2\right )}^{5/2} \,d x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^3*(a + b*acosh(c*x))*(d - c^2*d*x^2)^(5/2),x)

[Out]

int(x^3*(a + b*acosh(c*x))*(d - c^2*d*x^2)^(5/2), x)

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sympy [F(-1)]  time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x**3*(-c**2*d*x**2+d)**(5/2)*(a+b*acosh(c*x)),x)

[Out]

Timed out

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