\(\int x^3 (d-c^2 d x^2)^{5/2} (a+b \text {arccosh}(c x)) \, dx\) [98]

Optimal result

Integrand size = 27, antiderivative size = 298 \[ \int x^3 \left (d-c^2 d x^2\right )^{5/2} (a+b \text {arccosh}(c x)) \, dx=\frac {2 b d^2 x \sqrt {d-c^2 d x^2}}{63 c^3 \sqrt {-1+c x} \sqrt {1+c x}}+\frac {b d^2 x^3 \sqrt {d-c^2 d x^2}}{189 c \sqrt {-1+c x} \sqrt {1+c x}}-\frac {b c d^2 x^5 \sqrt {d-c^2 d x^2}}{21 \sqrt {-1+c x} \sqrt {1+c x}}+\frac {19 b c^3 d^2 x^7 \sqrt {d-c^2 d x^2}}{441 \sqrt {-1+c x} \sqrt {1+c x}}-\frac {b c^5 d^2 x^9 \sqrt {d-c^2 d x^2}}{81 \sqrt {-1+c x} \sqrt {1+c x}}-\frac {\left (d-c^2 d x^2\right )^{7/2} (a+b \text {arccosh}(c x))}{7 c^4 d}+\frac {\left (d-c^2 d x^2\right )^{9/2} (a+b \text {arccosh}(c x))}{9 c^4 d^2} \]

[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)

Rubi [A] (verified)

Time = 0.13 (sec) , antiderivative size = 298, 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, 380} \[ \int x^3 \left (d-c^2 d x^2\right )^{5/2} (a+b \text {arccosh}(c x)) \, dx=\frac {\left (d-c^2 d x^2\right )^{9/2} (a+b \text {arccosh}(c x))}{9 c^4 d^2}-\frac {\left (d-c^2 d x^2\right )^{7/2} (a+b \text {arccosh}(c x))}{7 c^4 d}-\frac {b c d^2 x^5 \sqrt {d-c^2 d x^2}}{21 \sqrt {c x-1} \sqrt {c x+1}}+\frac {b d^2 x^3 \sqrt {d-c^2 d x^2}}{189 c \sqrt {c x-1} \sqrt {c x+1}}-\frac {b c^5 d^2 x^9 \sqrt {d-c^2 d x^2}}{81 \sqrt {c x-1} \sqrt {c x+1}}+\frac {2 b d^2 x \sqrt {d-c^2 d x^2}}{63 c^3 \sqrt {c x-1} \sqrt {c x+1}}+\frac {19 b c^3 d^2 x^7 \sqrt {d-c^2 d x^2}}{441 \sqrt {c x-1} \sqrt {c x+1}} \]

[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]) - ((d - c^2*d*x^2)^(7/2)*(a + b*ArcCosh[c*x]))/(7*c^4*d) + ((d - c^2*d*x^2)^(9/2)
*(a + b*ArcCosh[c*x]))/(9*c^4*d^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 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 380

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 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 )^{7/2} (a+b \text {arccosh}(c x))}{7 c^4 d}+\frac {\left (d-c^2 d x^2\right )^{9/2} (a+b \text {arccosh}(c x))}{9 c^4 d^2}-\frac {\left (b c \sqrt {d-c^2 d x^2}\right ) \int \frac {d^2 \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 {\left (d-c^2 d x^2\right )^{7/2} (a+b \text {arccosh}(c x))}{7 c^4 d}+\frac {\left (d-c^2 d x^2\right )^{9/2} (a+b \text {arccosh}(c x))}{9 c^4 d^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 {\left (d-c^2 d x^2\right )^{7/2} (a+b \text {arccosh}(c x))}{7 c^4 d}+\frac {\left (d-c^2 d x^2\right )^{9/2} (a+b \text {arccosh}(c x))}{9 c^4 d^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 {\left (d-c^2 d x^2\right )^{7/2} (a+b \text {arccosh}(c x))}{7 c^4 d}+\frac {\left (d-c^2 d x^2\right )^{9/2} (a+b \text {arccosh}(c x))}{9 c^4 d^2} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.09 (sec) , antiderivative size = 145, normalized size of antiderivative = 0.49 \[ \int x^3 \left (d-c^2 d x^2\right )^{5/2} (a+b \text {arccosh}(c x)) \, dx=\frac {d^2 \sqrt {d-c^2 d x^2} \left (b c \left (126 x+21 c^2 x^3-189 c^4 x^5+171 c^6 x^7-49 c^8 x^9\right )+126 (-1+c x)^{7/2} (1+c x)^{7/2} (a+b \text {arccosh}(c x))+441 c^2 x^2 (-1+c x)^{7/2} (1+c x)^{7/2} (a+b \text {arccosh}(c x))\right )}{3969 c^4 \sqrt {-1+c x} \sqrt {1+c x}} \]

[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]*(b*c*(126*x + 21*c^2*x^3 - 189*c^4*x^5 + 171*c^6*x^7 - 49*c^8*x^9) + 126*(-1 + c*x)^(
7/2)*(1 + c*x)^(7/2)*(a + b*ArcCosh[c*x]) + 441*c^2*x^2*(-1 + c*x)^(7/2)*(1 + c*x)^(7/2)*(a + b*ArcCosh[c*x]))
)/(3969*c^4*Sqrt[-1 + c*x]*Sqrt[1 + c*x])

Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(1101\) vs. \(2(250)=500\).

Time = 0.60 (sec) , antiderivative size = 1102, normalized size of antiderivative = 3.70

method	result	size
default	\(\text {Expression too large to display}\)	\(1102\)
parts	\(\text {Expression too large to display}\)	\(1102\)

[In]

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

[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)*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^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)*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^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)*c^3*
x^3-3*(c*x-1)^(1/2)*(c*x+1)^(1/2)*c*x+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)*c*x+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)*c*x+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)*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^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)*c^5*x^5-144*c^6*x^6-56*(c*x-1)^(1/2)*(c*x+1)^(1/2)*c^3*x^3+10
4*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^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)*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^2/(c*x+1)/c^4/(c*x-1))

Fricas [A] (verification not implemented)

none

Time = 0.25 (sec) , antiderivative size = 281, normalized size of antiderivative = 0.94 \[ \int x^3 \left (d-c^2 d x^2\right )^{5/2} (a+b \text {arccosh}(c x)) \, dx=\frac {63 \, {\left (7 \, b c^{10} d^{2} x^{10} - 26 \, b c^{8} d^{2} x^{8} + 34 \, b c^{6} d^{2} x^{6} - 16 \, b c^{4} d^{2} x^{4} - b c^{2} d^{2} x^{2} + 2 \, b d^{2}\right )} \sqrt {-c^{2} d x^{2} + d} \log \left (c x + \sqrt {c^{2} x^{2} - 1}\right ) - {\left (49 \, b c^{9} d^{2} x^{9} - 171 \, b c^{7} d^{2} x^{7} + 189 \, b c^{5} d^{2} x^{5} - 21 \, b c^{3} d^{2} x^{3} - 126 \, b c d^{2} x\right )} \sqrt {-c^{2} d x^{2} + d} \sqrt {c^{2} x^{2} - 1} + 63 \, {\left (7 \, a c^{10} d^{2} x^{10} - 26 \, a c^{8} d^{2} x^{8} + 34 \, a c^{6} d^{2} x^{6} - 16 \, a c^{4} d^{2} x^{4} - a c^{2} d^{2} x^{2} + 2 \, a d^{2}\right )} \sqrt {-c^{2} d x^{2} + d}}{3969 \, {\left (c^{6} x^{2} - c^{4}\right )}} \]

[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)

Sympy [F(-1)]

Timed out. \[ \int x^3 \left (d-c^2 d x^2\right )^{5/2} (a+b \text {arccosh}(c x)) \, dx=\text {Timed out} \]

[In]

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

[Out]

Timed out

Maxima [A] (verification not implemented)

none

Time = 0.29 (sec) , antiderivative size = 185, normalized size of antiderivative = 0.62 \[ \int x^3 \left (d-c^2 d x^2\right )^{5/2} (a+b \text {arccosh}(c x)) \, dx=-\frac {1}{63} \, {\left (\frac {7 \, {\left (-c^{2} d x^{2} + d\right )}^{\frac {7}{2}} x^{2}}{c^{2} d} + \frac {2 \, {\left (-c^{2} d x^{2} + d\right )}^{\frac {7}{2}}}{c^{4} d}\right )} b \operatorname {arcosh}\left (c x\right ) - \frac {1}{63} \, {\left (\frac {7 \, {\left (-c^{2} d x^{2} + d\right )}^{\frac {7}{2}} x^{2}}{c^{2} d} + \frac {2 \, {\left (-c^{2} d x^{2} + d\right )}^{\frac {7}{2}}}{c^{4} d}\right )} a - \frac {{\left (49 \, c^{8} \sqrt {-d} d^{2} x^{9} - 171 \, c^{6} \sqrt {-d} d^{2} x^{7} + 189 \, c^{4} \sqrt {-d} d^{2} x^{5} - 21 \, c^{2} \sqrt {-d} d^{2} x^{3} - 126 \, \sqrt {-d} d^{2} x\right )} b}{3969 \, c^{3}} \]

[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

Giac [F(-2)]

Exception generated. \[ \int x^3 \left (d-c^2 d x^2\right )^{5/2} (a+b \text {arccosh}(c x)) \, dx=\text {Exception raised: TypeError} \]

[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,sageVARx):;OUTP
UT:sym2poly/r2sym(const gen & e,const index_m & i,const vecteur & l) Error: Bad Argument Value

Mupad [F(-1)]

Timed out. \[ \int x^3 \left (d-c^2 d x^2\right )^{5/2} (a+b \text {arccosh}(c x)) \, dx=\int x^3\,\left (a+b\,\mathrm {acosh}\left (c\,x\right )\right )\,{\left (d-c^2\,d\,x^2\right )}^{5/2} \,d x \]

[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)