\(\int x (d-c^2 d x^2)^{5/2} (a+b \text {arccosh}(c x)) \, dx\) [99]

Optimal result

Integrand size = 25, antiderivative size = 218 \[ \int x \left (d-c^2 d x^2\right )^{5/2} (a+b \text {arccosh}(c x)) \, dx=\frac {b d^2 x \sqrt {d-c^2 d x^2}}{7 c \sqrt {-1+c x} \sqrt {1+c x}}-\frac {b c d^2 x^3 \sqrt {d-c^2 d x^2}}{7 \sqrt {-1+c x} \sqrt {1+c x}}+\frac {3 b c^3 d^2 x^5 \sqrt {d-c^2 d x^2}}{35 \sqrt {-1+c x} \sqrt {1+c x}}-\frac {b c^5 d^2 x^7 \sqrt {d-c^2 d x^2}}{49 \sqrt {-1+c x} \sqrt {1+c x}}-\frac {\left (d-c^2 d x^2\right )^{7/2} (a+b \text {arccosh}(c x))}{7 c^2 d} \]

[Out]

-1/7*(-c^2*d*x^2+d)^(7/2)*(a+b*arccosh(c*x))/c^2/d+1/7*b*d^2*x*(-c^2*d*x^2+d)^(1/2)/c/(c*x-1)^(1/2)/(c*x+1)^(1
/2)-1/7*b*c*d^2*x^3*(-c^2*d*x^2+d)^(1/2)/(c*x-1)^(1/2)/(c*x+1)^(1/2)+3/35*b*c^3*d^2*x^5*(-c^2*d*x^2+d)^(1/2)/(
c*x-1)^(1/2)/(c*x+1)^(1/2)-1/49*b*c^5*d^2*x^7*(-c^2*d*x^2+d)^(1/2)/(c*x-1)^(1/2)/(c*x+1)^(1/2)

Rubi [A] (verified)

Time = 0.07 (sec) , antiderivative size = 218, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.120, Rules used = {5914, 41, 200} \[ \int x \left (d-c^2 d x^2\right )^{5/2} (a+b \text {arccosh}(c x)) \, dx=-\frac {\left (d-c^2 d x^2\right )^{7/2} (a+b \text {arccosh}(c x))}{7 c^2 d}+\frac {b d^2 x \sqrt {d-c^2 d x^2}}{7 c \sqrt {c x-1} \sqrt {c x+1}}-\frac {b c d^2 x^3 \sqrt {d-c^2 d x^2}}{7 \sqrt {c x-1} \sqrt {c x+1}}-\frac {b c^5 d^2 x^7 \sqrt {d-c^2 d x^2}}{49 \sqrt {c x-1} \sqrt {c x+1}}+\frac {3 b c^3 d^2 x^5 \sqrt {d-c^2 d x^2}}{35 \sqrt {c x-1} \sqrt {c x+1}} \]

[In]

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

[Out]

(b*d^2*x*Sqrt[d - c^2*d*x^2])/(7*c*Sqrt[-1 + c*x]*Sqrt[1 + c*x]) - (b*c*d^2*x^3*Sqrt[d - c^2*d*x^2])/(7*Sqrt[-
1 + c*x]*Sqrt[1 + c*x]) + (3*b*c^3*d^2*x^5*Sqrt[d - c^2*d*x^2])/(35*Sqrt[-1 + c*x]*Sqrt[1 + c*x]) - (b*c^5*d^2
*x^7*Sqrt[d - c^2*d*x^2])/(49*Sqrt[-1 + c*x]*Sqrt[1 + c*x]) - ((d - c^2*d*x^2)^(7/2)*(a + b*ArcCosh[c*x]))/(7*
c^2*d)

Rule 41

Int[((a_) + (b_.)*(x_))^(m_.)*((c_) + (d_.)*(x_))^(m_.), x_Symbol] :> Int[(a*c + b*d*x^2)^m, x] /; FreeQ[{a, b
, c, d, m}, x] && EqQ[b*c + a*d, 0] && (IntegerQ[m] || (GtQ[a, 0] && GtQ[c, 0]))

Rule 200

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

Rule 5914

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

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^2 d}-\frac {\left (b d^2 \sqrt {d-c^2 d x^2}\right ) \int (-1+c x)^3 (1+c x)^3 \, dx}{7 c \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^2 d}-\frac {\left (b d^2 \sqrt {d-c^2 d x^2}\right ) \int \left (-1+c^2 x^2\right )^3 \, dx}{7 c \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^2 d}-\frac {\left (b d^2 \sqrt {d-c^2 d x^2}\right ) \int \left (-1+3 c^2 x^2-3 c^4 x^4+c^6 x^6\right ) \, dx}{7 c \sqrt {-1+c x} \sqrt {1+c x}} \\ & = \frac {b d^2 x \sqrt {d-c^2 d x^2}}{7 c \sqrt {-1+c x} \sqrt {1+c x}}-\frac {b c d^2 x^3 \sqrt {d-c^2 d x^2}}{7 \sqrt {-1+c x} \sqrt {1+c x}}+\frac {3 b c^3 d^2 x^5 \sqrt {d-c^2 d x^2}}{35 \sqrt {-1+c x} \sqrt {1+c x}}-\frac {b c^5 d^2 x^7 \sqrt {d-c^2 d x^2}}{49 \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^2 d} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.20 (sec) , antiderivative size = 117, normalized size of antiderivative = 0.54 \[ \int x \left (d-c^2 d x^2\right )^{5/2} (a+b \text {arccosh}(c x)) \, dx=\frac {d^2 \sqrt {d-c^2 d x^2} \left (35 a \left (-1+c^2 x^2\right )^4+b c x \sqrt {-1+c x} \sqrt {1+c x} \left (35-35 c^2 x^2+21 c^4 x^4-5 c^6 x^6\right )+35 b \left (-1+c^2 x^2\right )^4 \text {arccosh}(c x)\right )}{245 c^2 \left (-1+c^2 x^2\right )} \]

[In]

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

[Out]

(d^2*Sqrt[d - c^2*d*x^2]*(35*a*(-1 + c^2*x^2)^4 + b*c*x*Sqrt[-1 + c*x]*Sqrt[1 + c*x]*(35 - 35*c^2*x^2 + 21*c^4
*x^4 - 5*c^6*x^6) + 35*b*(-1 + c^2*x^2)^4*ArcCosh[c*x]))/(245*c^2*(-1 + c^2*x^2))

Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(955\) vs. \(2(182)=364\).

Time = 0.56 (sec) , antiderivative size = 956, normalized size of antiderivative = 4.39

method	result	size
default	\(-\frac {a \left (-c^{2} d \,x^{2}+d \right )^{\frac {7}{2}}}{7 c^{2} d}+b \left (\frac {\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}\, c^{5} x^{5}-25 c^{2} x^{2}+56 \sqrt {c x -1}\, \sqrt {c x +1}\, c^{3} x^{3}-7 \sqrt {c x -1}\, \sqrt {c x +1}\, c x +1\right ) \left (-1+7 \,\operatorname {arccosh}\left (c x \right )\right ) d^{2}}{6272 \left (c x +1\right ) c^{2} \left (c x -1\right )}-\frac {\sqrt {-d \left (c^{2} x^{2}-1\right )}\, \left (16 c^{6} x^{6}-28 c^{4} x^{4}+16 \sqrt {c x +1}\, \sqrt {c x -1}\, c^{5} x^{5}+13 c^{2} x^{2}-20 \sqrt {c x -1}\, \sqrt {c x +1}\, c^{3} x^{3}+5 \sqrt {c x -1}\, \sqrt {c x +1}\, c x -1\right ) \left (-1+5 \,\operatorname {arccosh}\left (c x \right )\right ) d^{2}}{640 \left (c x +1\right ) c^{2} \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}\, c^{3} x^{3}-3 \sqrt {c x -1}\, \sqrt {c x +1}\, c x +1\right ) \left (-1+3 \,\operatorname {arccosh}\left (c x \right )\right ) d^{2}}{128 \left (c x +1\right ) c^{2} \left (c x -1\right )}-\frac {5 \sqrt {-d \left (c^{2} x^{2}-1\right )}\, \left (\sqrt {c x -1}\, \sqrt {c x +1}\, c x +c^{2} x^{2}-1\right ) \left (-1+\operatorname {arccosh}\left (c x \right )\right ) d^{2}}{128 \left (c x +1\right ) c^{2} \left (c x -1\right )}-\frac {5 \sqrt {-d \left (c^{2} x^{2}-1\right )}\, \left (-\sqrt {c x -1}\, \sqrt {c x +1}\, c x +c^{2} x^{2}-1\right ) \left (1+\operatorname {arccosh}\left (c x \right )\right ) d^{2}}{128 \left (c x +1\right ) c^{2} \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}\, c^{3} x^{3}+4 c^{4} x^{4}+3 \sqrt {c x -1}\, \sqrt {c x +1}\, c x -5 c^{2} x^{2}+1\right ) \left (1+3 \,\operatorname {arccosh}\left (c x \right )\right ) d^{2}}{128 \left (c x +1\right ) c^{2} \left (c x -1\right )}-\frac {\sqrt {-d \left (c^{2} x^{2}-1\right )}\, \left (-16 \sqrt {c x +1}\, \sqrt {c x -1}\, c^{5} x^{5}+16 c^{6} x^{6}+20 \sqrt {c x -1}\, \sqrt {c x +1}\, c^{3} x^{3}-28 c^{4} x^{4}-5 \sqrt {c x -1}\, \sqrt {c x +1}\, c x +13 c^{2} x^{2}-1\right ) \left (1+5 \,\operatorname {arccosh}\left (c x \right )\right ) d^{2}}{640 \left (c x +1\right ) c^{2} \left (c x -1\right )}+\frac {\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}\, c^{5} x^{5}-144 c^{6} x^{6}-56 \sqrt {c x -1}\, \sqrt {c x +1}\, c^{3} x^{3}+104 c^{4} x^{4}+7 \sqrt {c x -1}\, \sqrt {c x +1}\, c x -25 c^{2} x^{2}+1\right ) \left (1+7 \,\operatorname {arccosh}\left (c x \right )\right ) d^{2}}{6272 \left (c x +1\right ) c^{2} \left (c x -1\right )}\right )\)	\(956\)
parts	\(-\frac {a \left (-c^{2} d \,x^{2}+d \right )^{\frac {7}{2}}}{7 c^{2} d}+b \left (\frac {\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}\, c^{5} x^{5}-25 c^{2} x^{2}+56 \sqrt {c x -1}\, \sqrt {c x +1}\, c^{3} x^{3}-7 \sqrt {c x -1}\, \sqrt {c x +1}\, c x +1\right ) \left (-1+7 \,\operatorname {arccosh}\left (c x \right )\right ) d^{2}}{6272 \left (c x +1\right ) c^{2} \left (c x -1\right )}-\frac {\sqrt {-d \left (c^{2} x^{2}-1\right )}\, \left (16 c^{6} x^{6}-28 c^{4} x^{4}+16 \sqrt {c x +1}\, \sqrt {c x -1}\, c^{5} x^{5}+13 c^{2} x^{2}-20 \sqrt {c x -1}\, \sqrt {c x +1}\, c^{3} x^{3}+5 \sqrt {c x -1}\, \sqrt {c x +1}\, c x -1\right ) \left (-1+5 \,\operatorname {arccosh}\left (c x \right )\right ) d^{2}}{640 \left (c x +1\right ) c^{2} \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}\, c^{3} x^{3}-3 \sqrt {c x -1}\, \sqrt {c x +1}\, c x +1\right ) \left (-1+3 \,\operatorname {arccosh}\left (c x \right )\right ) d^{2}}{128 \left (c x +1\right ) c^{2} \left (c x -1\right )}-\frac {5 \sqrt {-d \left (c^{2} x^{2}-1\right )}\, \left (\sqrt {c x -1}\, \sqrt {c x +1}\, c x +c^{2} x^{2}-1\right ) \left (-1+\operatorname {arccosh}\left (c x \right )\right ) d^{2}}{128 \left (c x +1\right ) c^{2} \left (c x -1\right )}-\frac {5 \sqrt {-d \left (c^{2} x^{2}-1\right )}\, \left (-\sqrt {c x -1}\, \sqrt {c x +1}\, c x +c^{2} x^{2}-1\right ) \left (1+\operatorname {arccosh}\left (c x \right )\right ) d^{2}}{128 \left (c x +1\right ) c^{2} \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}\, c^{3} x^{3}+4 c^{4} x^{4}+3 \sqrt {c x -1}\, \sqrt {c x +1}\, c x -5 c^{2} x^{2}+1\right ) \left (1+3 \,\operatorname {arccosh}\left (c x \right )\right ) d^{2}}{128 \left (c x +1\right ) c^{2} \left (c x -1\right )}-\frac {\sqrt {-d \left (c^{2} x^{2}-1\right )}\, \left (-16 \sqrt {c x +1}\, \sqrt {c x -1}\, c^{5} x^{5}+16 c^{6} x^{6}+20 \sqrt {c x -1}\, \sqrt {c x +1}\, c^{3} x^{3}-28 c^{4} x^{4}-5 \sqrt {c x -1}\, \sqrt {c x +1}\, c x +13 c^{2} x^{2}-1\right ) \left (1+5 \,\operatorname {arccosh}\left (c x \right )\right ) d^{2}}{640 \left (c x +1\right ) c^{2} \left (c x -1\right )}+\frac {\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}\, c^{5} x^{5}-144 c^{6} x^{6}-56 \sqrt {c x -1}\, \sqrt {c x +1}\, c^{3} x^{3}+104 c^{4} x^{4}+7 \sqrt {c x -1}\, \sqrt {c x +1}\, c x -25 c^{2} x^{2}+1\right ) \left (1+7 \,\operatorname {arccosh}\left (c x \right )\right ) d^{2}}{6272 \left (c x +1\right ) c^{2} \left (c x -1\right )}\right )\)	\(956\)

[In]

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

[Out]

-1/7*a*(-c^2*d*x^2+d)^(7/2)/c^2/d+b*(1/6272*(-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^2/(c*x-1)-1/640*(-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^2/(c*x+1)/c^2/(c*x-1)+1/128*(-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^2/(c*x-1)-5/128*(-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^2/(c*x-1)-5/128*(-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^2/(c*x-1)+1/128*(-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^2/(c*x-
1)-1/640*(-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^2/(c*x+1)/c^2/(c*
x-1)+1/6272*(-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^2/(c*x+1)/c^2/(c*x-1))

Fricas [A] (verification not implemented)

none

Time = 0.26 (sec) , antiderivative size = 241, normalized size of antiderivative = 1.11 \[ \int x \left (d-c^2 d x^2\right )^{5/2} (a+b \text {arccosh}(c x)) \, dx=\frac {35 \, {\left (b c^{8} d^{2} x^{8} - 4 \, b c^{6} d^{2} x^{6} + 6 \, b c^{4} d^{2} x^{4} - 4 \, b c^{2} d^{2} x^{2} + b d^{2}\right )} \sqrt {-c^{2} d x^{2} + d} \log \left (c x + \sqrt {c^{2} x^{2} - 1}\right ) - {\left (5 \, b c^{7} d^{2} x^{7} - 21 \, b c^{5} d^{2} x^{5} + 35 \, b c^{3} d^{2} x^{3} - 35 \, b c d^{2} x\right )} \sqrt {-c^{2} d x^{2} + d} \sqrt {c^{2} x^{2} - 1} + 35 \, {\left (a c^{8} d^{2} x^{8} - 4 \, a c^{6} d^{2} x^{6} + 6 \, a c^{4} d^{2} x^{4} - 4 \, a c^{2} d^{2} x^{2} + a d^{2}\right )} \sqrt {-c^{2} d x^{2} + d}}{245 \, {\left (c^{4} x^{2} - c^{2}\right )}} \]

[In]

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

[Out]

1/245*(35*(b*c^8*d^2*x^8 - 4*b*c^6*d^2*x^6 + 6*b*c^4*d^2*x^4 - 4*b*c^2*d^2*x^2 + b*d^2)*sqrt(-c^2*d*x^2 + d)*l
og(c*x + sqrt(c^2*x^2 - 1)) - (5*b*c^7*d^2*x^7 - 21*b*c^5*d^2*x^5 + 35*b*c^3*d^2*x^3 - 35*b*c*d^2*x)*sqrt(-c^2
*d*x^2 + d)*sqrt(c^2*x^2 - 1) + 35*(a*c^8*d^2*x^8 - 4*a*c^6*d^2*x^6 + 6*a*c^4*d^2*x^4 - 4*a*c^2*d^2*x^2 + a*d^
2)*sqrt(-c^2*d*x^2 + d))/(c^4*x^2 - c^2)

Sympy [F(-1)]

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

[In]

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

[Out]

Timed out

Maxima [A] (verification not implemented)

none

Time = 0.20 (sec) , antiderivative size = 118, normalized size of antiderivative = 0.54 \[ \int x \left (d-c^2 d x^2\right )^{5/2} (a+b \text {arccosh}(c x)) \, dx=-\frac {{\left (-c^{2} d x^{2} + d\right )}^{\frac {7}{2}} b \operatorname {arcosh}\left (c x\right )}{7 \, c^{2} d} - \frac {{\left (-c^{2} d x^{2} + d\right )}^{\frac {7}{2}} a}{7 \, c^{2} d} - \frac {{\left (5 \, c^{6} \sqrt {-d} d^{3} x^{7} - 21 \, c^{4} \sqrt {-d} d^{3} x^{5} + 35 \, c^{2} \sqrt {-d} d^{3} x^{3} - 35 \, \sqrt {-d} d^{3} x\right )} b}{245 \, c d} \]

[In]

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

[Out]

-1/7*(-c^2*d*x^2 + d)^(7/2)*b*arccosh(c*x)/(c^2*d) - 1/7*(-c^2*d*x^2 + d)^(7/2)*a/(c^2*d) - 1/245*(5*c^6*sqrt(
-d)*d^3*x^7 - 21*c^4*sqrt(-d)*d^3*x^5 + 35*c^2*sqrt(-d)*d^3*x^3 - 35*sqrt(-d)*d^3*x)*b/(c*d)

Giac [F(-2)]

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

[In]

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

[In]

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

[Out]

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