3.4.74 \(\int \frac {x^3 (d-c^2 d x^2)^2}{(a+b \text {arccosh}(c x))^{3/2}} \, dx\) [374]

3.4.74.1 Optimal result

Integrand size = 29, antiderivative size = 479 \[ \int \frac {x^3 \left (d-c^2 d x^2\right )^2}{(a+b \text {arccosh}(c x))^{3/2}} \, dx=-\frac {2 d^2 x^3 (-1+c x)^{5/2} (1+c x)^{5/2}}{b c \sqrt {a+b \text {arccosh}(c x)}}-\frac {d^2 e^{\frac {4 a}{b}} \sqrt {\pi } \text {erf}\left (\frac {2 \sqrt {a+b \text {arccosh}(c x)}}{\sqrt {b}}\right )}{32 b^{3/2} c^4}+\frac {3 d^2 e^{\frac {2 a}{b}} \sqrt {\frac {\pi }{2}} \text {erf}\left (\frac {\sqrt {2} \sqrt {a+b \text {arccosh}(c x)}}{\sqrt {b}}\right )}{32 b^{3/2} c^4}+\frac {d^2 e^{\frac {8 a}{b}} \sqrt {\frac {\pi }{2}} \text {erf}\left (\frac {2 \sqrt {2} \sqrt {a+b \text {arccosh}(c x)}}{\sqrt {b}}\right )}{32 b^{3/2} c^4}-\frac {d^2 e^{\frac {6 a}{b}} \sqrt {\frac {3 \pi }{2}} \text {erf}\left (\frac {\sqrt {6} \sqrt {a+b \text {arccosh}(c x)}}{\sqrt {b}}\right )}{32 b^{3/2} c^4}-\frac {d^2 e^{-\frac {4 a}{b}} \sqrt {\pi } \text {erfi}\left (\frac {2 \sqrt {a+b \text {arccosh}(c x)}}{\sqrt {b}}\right )}{32 b^{3/2} c^4}+\frac {3 d^2 e^{-\frac {2 a}{b}} \sqrt {\frac {\pi }{2}} \text {erfi}\left (\frac {\sqrt {2} \sqrt {a+b \text {arccosh}(c x)}}{\sqrt {b}}\right )}{32 b^{3/2} c^4}+\frac {d^2 e^{-\frac {8 a}{b}} \sqrt {\frac {\pi }{2}} \text {erfi}\left (\frac {2 \sqrt {2} \sqrt {a+b \text {arccosh}(c x)}}{\sqrt {b}}\right )}{32 b^{3/2} c^4}-\frac {d^2 e^{-\frac {6 a}{b}} \sqrt {\frac {3 \pi }{2}} \text {erfi}\left (\frac {\sqrt {6} \sqrt {a+b \text {arccosh}(c x)}}{\sqrt {b}}\right )}{32 b^{3/2} c^4} \]

output

3/64*d^2*exp(2*a/b)*erf(2^(1/2)*(a+b*arccosh(c*x))^(1/2)/b^(1/2))*2^(1/2)* 
Pi^(1/2)/b^(3/2)/c^4+1/64*d^2*exp(8*a/b)*erf(2*2^(1/2)*(a+b*arccosh(c*x))^ 
(1/2)/b^(1/2))*2^(1/2)*Pi^(1/2)/b^(3/2)/c^4+3/64*d^2*erfi(2^(1/2)*(a+b*arc 
cosh(c*x))^(1/2)/b^(1/2))*2^(1/2)*Pi^(1/2)/b^(3/2)/c^4/exp(2*a/b)+1/64*d^2 
*erfi(2*2^(1/2)*(a+b*arccosh(c*x))^(1/2)/b^(1/2))*2^(1/2)*Pi^(1/2)/b^(3/2) 
/c^4/exp(8*a/b)-1/32*d^2*exp(4*a/b)*erf(2*(a+b*arccosh(c*x))^(1/2)/b^(1/2) 
)*Pi^(1/2)/b^(3/2)/c^4-1/32*d^2*erfi(2*(a+b*arccosh(c*x))^(1/2)/b^(1/2))*P 
i^(1/2)/b^(3/2)/c^4/exp(4*a/b)-1/64*d^2*exp(6*a/b)*erf(6^(1/2)*(a+b*arccos 
h(c*x))^(1/2)/b^(1/2))*6^(1/2)*Pi^(1/2)/b^(3/2)/c^4-1/64*d^2*erfi(6^(1/2)* 
(a+b*arccosh(c*x))^(1/2)/b^(1/2))*6^(1/2)*Pi^(1/2)/b^(3/2)/c^4/exp(6*a/b)- 
2*d^2*x^3*(c*x-1)^(5/2)*(c*x+1)^(5/2)/b/c/(a+b*arccosh(c*x))^(1/2)
 

3.4.74.2 Mathematica [A] (warning: unable to verify)

Time = 2.77 (sec) , antiderivative size = 527, normalized size of antiderivative = 1.10 \[ \int \frac {x^3 \left (d-c^2 d x^2\right )^2}{(a+b \text {arccosh}(c x))^{3/2}} \, dx=-\frac {d^2 e^{-\frac {8 a}{b}} \left (128 c^3 e^{\frac {8 a}{b}} x^3 \sqrt {\frac {-1+c x}{1+c x}}+128 c^4 e^{\frac {8 a}{b}} x^4 \sqrt {\frac {-1+c x}{1+c x}}-\sqrt {2} \sqrt {-\frac {a+b \text {arccosh}(c x)}{b}} \Gamma \left (\frac {1}{2},-\frac {8 (a+b \text {arccosh}(c x))}{b}\right )+\sqrt {6} e^{\frac {2 a}{b}} \sqrt {-\frac {a+b \text {arccosh}(c x)}{b}} \Gamma \left (\frac {1}{2},-\frac {6 (a+b \text {arccosh}(c x))}{b}\right )+2 e^{\frac {4 a}{b}} \sqrt {-\frac {a+b \text {arccosh}(c x)}{b}} \Gamma \left (\frac {1}{2},-\frac {4 (a+b \text {arccosh}(c x))}{b}\right )-3 \sqrt {2} e^{\frac {6 a}{b}} \sqrt {-\frac {a+b \text {arccosh}(c x)}{b}} \Gamma \left (\frac {1}{2},-\frac {2 (a+b \text {arccosh}(c x))}{b}\right )+3 \sqrt {2} e^{\frac {10 a}{b}} \sqrt {\frac {a}{b}+\text {arccosh}(c x)} \Gamma \left (\frac {1}{2},\frac {2 (a+b \text {arccosh}(c x))}{b}\right )-2 e^{\frac {12 a}{b}} \sqrt {\frac {a}{b}+\text {arccosh}(c x)} \Gamma \left (\frac {1}{2},\frac {4 (a+b \text {arccosh}(c x))}{b}\right )-\sqrt {6} e^{\frac {14 a}{b}} \sqrt {\frac {a}{b}+\text {arccosh}(c x)} \Gamma \left (\frac {1}{2},\frac {6 (a+b \text {arccosh}(c x))}{b}\right )+\sqrt {2} e^{\frac {16 a}{b}} \sqrt {\frac {a}{b}+\text {arccosh}(c x)} \Gamma \left (\frac {1}{2},\frac {8 (a+b \text {arccosh}(c x))}{b}\right )-26 e^{\frac {8 a}{b}} \sinh (2 \text {arccosh}(c x))-18 e^{\frac {8 a}{b}} \sinh (4 \text {arccosh}(c x))-2 e^{\frac {8 a}{b}} \sinh (6 \text {arccosh}(c x))+e^{\frac {8 a}{b}} \sinh (8 \text {arccosh}(c x))\right )}{64 b c^4 \sqrt {a+b \text {arccosh}(c x)}} \]

input

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

output

-1/64*(d^2*(128*c^3*E^((8*a)/b)*x^3*Sqrt[(-1 + c*x)/(1 + c*x)] + 128*c^4*E 
^((8*a)/b)*x^4*Sqrt[(-1 + c*x)/(1 + c*x)] - Sqrt[2]*Sqrt[-((a + b*ArcCosh[ 
c*x])/b)]*Gamma[1/2, (-8*(a + b*ArcCosh[c*x]))/b] + Sqrt[6]*E^((2*a)/b)*Sq 
rt[-((a + b*ArcCosh[c*x])/b)]*Gamma[1/2, (-6*(a + b*ArcCosh[c*x]))/b] + 2* 
E^((4*a)/b)*Sqrt[-((a + b*ArcCosh[c*x])/b)]*Gamma[1/2, (-4*(a + b*ArcCosh[ 
c*x]))/b] - 3*Sqrt[2]*E^((6*a)/b)*Sqrt[-((a + b*ArcCosh[c*x])/b)]*Gamma[1/ 
2, (-2*(a + b*ArcCosh[c*x]))/b] + 3*Sqrt[2]*E^((10*a)/b)*Sqrt[a/b + ArcCos 
h[c*x]]*Gamma[1/2, (2*(a + b*ArcCosh[c*x]))/b] - 2*E^((12*a)/b)*Sqrt[a/b + 
 ArcCosh[c*x]]*Gamma[1/2, (4*(a + b*ArcCosh[c*x]))/b] - Sqrt[6]*E^((14*a)/ 
b)*Sqrt[a/b + ArcCosh[c*x]]*Gamma[1/2, (6*(a + b*ArcCosh[c*x]))/b] + Sqrt[ 
2]*E^((16*a)/b)*Sqrt[a/b + ArcCosh[c*x]]*Gamma[1/2, (8*(a + b*ArcCosh[c*x] 
))/b] - 26*E^((8*a)/b)*Sinh[2*ArcCosh[c*x]] - 18*E^((8*a)/b)*Sinh[4*ArcCos 
h[c*x]] - 2*E^((8*a)/b)*Sinh[6*ArcCosh[c*x]] + E^((8*a)/b)*Sinh[8*ArcCosh[ 
c*x]]))/(b*c^4*E^((8*a)/b)*Sqrt[a + b*ArcCosh[c*x]])
 

3.4.74.3 Rubi [A] (verified)

Time = 2.27 (sec) , antiderivative size = 571, normalized size of antiderivative = 1.19, number of steps used = 5, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.138, Rules used = {6357, 6368, 5971, 2009}

Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.

\(\displaystyle \int \frac {x^3 \left (d-c^2 d x^2\right )^2}{(a+b \text {arccosh}(c x))^{3/2}} \, dx\)

\(\Big \downarrow \) 6357

\(\displaystyle \frac {16 c d^2 \int \frac {x^4 (c x-1)^{3/2} (c x+1)^{3/2}}{\sqrt {a+b \text {arccosh}(c x)}}dx}{b}-\frac {6 d^2 \int \frac {x^2 (c x-1)^{3/2} (c x+1)^{3/2}}{\sqrt {a+b \text {arccosh}(c x)}}dx}{b c}-\frac {2 d^2 x^3 (c x-1)^{5/2} (c x+1)^{5/2}}{b c \sqrt {a+b \text {arccosh}(c x)}}\)

\(\Big \downarrow \) 6368

\(\displaystyle \frac {16 d^2 \int \frac {\cosh ^4\left (\frac {a}{b}-\frac {a+b \text {arccosh}(c x)}{b}\right ) \sinh ^4\left (\frac {a}{b}-\frac {a+b \text {arccosh}(c x)}{b}\right )}{\sqrt {a+b \text {arccosh}(c x)}}d(a+b \text {arccosh}(c x))}{b^2 c^4}-\frac {6 d^2 \int \frac {\cosh ^2\left (\frac {a}{b}-\frac {a+b \text {arccosh}(c x)}{b}\right ) \sinh ^4\left (\frac {a}{b}-\frac {a+b \text {arccosh}(c x)}{b}\right )}{\sqrt {a+b \text {arccosh}(c x)}}d(a+b \text {arccosh}(c x))}{b^2 c^4}-\frac {2 d^2 x^3 (c x-1)^{5/2} (c x+1)^{5/2}}{b c \sqrt {a+b \text {arccosh}(c x)}}\)

\(\Big \downarrow \) 5971

\(\displaystyle \frac {16 d^2 \int \left (\frac {\cosh \left (\frac {8 a}{b}-\frac {8 (a+b \text {arccosh}(c x))}{b}\right )}{128 \sqrt {a+b \text {arccosh}(c x)}}-\frac {\cosh \left (\frac {4 a}{b}-\frac {4 (a+b \text {arccosh}(c x))}{b}\right )}{32 \sqrt {a+b \text {arccosh}(c x)}}+\frac {3}{128 \sqrt {a+b \text {arccosh}(c x)}}\right )d(a+b \text {arccosh}(c x))}{b^2 c^4}-\frac {6 d^2 \int \left (\frac {\cosh \left (\frac {6 a}{b}-\frac {6 (a+b \text {arccosh}(c x))}{b}\right )}{32 \sqrt {a+b \text {arccosh}(c x)}}-\frac {\cosh \left (\frac {4 a}{b}-\frac {4 (a+b \text {arccosh}(c x))}{b}\right )}{16 \sqrt {a+b \text {arccosh}(c x)}}-\frac {\cosh \left (\frac {2 a}{b}-\frac {2 (a+b \text {arccosh}(c x))}{b}\right )}{32 \sqrt {a+b \text {arccosh}(c x)}}+\frac {1}{16 \sqrt {a+b \text {arccosh}(c x)}}\right )d(a+b \text {arccosh}(c x))}{b^2 c^4}-\frac {2 d^2 x^3 (c x-1)^{5/2} (c x+1)^{5/2}}{b c \sqrt {a+b \text {arccosh}(c x)}}\)

\(\Big \downarrow \) 2009

\(\displaystyle \frac {16 d^2 \left (-\frac {1}{128} \sqrt {\pi } \sqrt {b} e^{\frac {4 a}{b}} \text {erf}\left (\frac {2 \sqrt {a+b \text {arccosh}(c x)}}{\sqrt {b}}\right )+\frac {1}{512} \sqrt {\frac {\pi }{2}} \sqrt {b} e^{\frac {8 a}{b}} \text {erf}\left (\frac {2 \sqrt {2} \sqrt {a+b \text {arccosh}(c x)}}{\sqrt {b}}\right )-\frac {1}{128} \sqrt {\pi } \sqrt {b} e^{-\frac {4 a}{b}} \text {erfi}\left (\frac {2 \sqrt {a+b \text {arccosh}(c x)}}{\sqrt {b}}\right )+\frac {1}{512} \sqrt {\frac {\pi }{2}} \sqrt {b} e^{-\frac {8 a}{b}} \text {erfi}\left (\frac {2 \sqrt {2} \sqrt {a+b \text {arccosh}(c x)}}{\sqrt {b}}\right )+\frac {3}{64} \sqrt {a+b \text {arccosh}(c x)}\right )}{b^2 c^4}-\frac {6 d^2 \left (-\frac {1}{64} \sqrt {\pi } \sqrt {b} e^{\frac {4 a}{b}} \text {erf}\left (\frac {2 \sqrt {a+b \text {arccosh}(c x)}}{\sqrt {b}}\right )-\frac {1}{64} \sqrt {\frac {\pi }{2}} \sqrt {b} e^{\frac {2 a}{b}} \text {erf}\left (\frac {\sqrt {2} \sqrt {a+b \text {arccosh}(c x)}}{\sqrt {b}}\right )+\frac {1}{64} \sqrt {\frac {\pi }{6}} \sqrt {b} e^{\frac {6 a}{b}} \text {erf}\left (\frac {\sqrt {6} \sqrt {a+b \text {arccosh}(c x)}}{\sqrt {b}}\right )-\frac {1}{64} \sqrt {\pi } \sqrt {b} e^{-\frac {4 a}{b}} \text {erfi}\left (\frac {2 \sqrt {a+b \text {arccosh}(c x)}}{\sqrt {b}}\right )-\frac {1}{64} \sqrt {\frac {\pi }{2}} \sqrt {b} e^{-\frac {2 a}{b}} \text {erfi}\left (\frac {\sqrt {2} \sqrt {a+b \text {arccosh}(c x)}}{\sqrt {b}}\right )+\frac {1}{64} \sqrt {\frac {\pi }{6}} \sqrt {b} e^{-\frac {6 a}{b}} \text {erfi}\left (\frac {\sqrt {6} \sqrt {a+b \text {arccosh}(c x)}}{\sqrt {b}}\right )+\frac {1}{8} \sqrt {a+b \text {arccosh}(c x)}\right )}{b^2 c^4}-\frac {2 d^2 x^3 (c x-1)^{5/2} (c x+1)^{5/2}}{b c \sqrt {a+b \text {arccosh}(c x)}}\)

input

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

output

(-2*d^2*x^3*(-1 + c*x)^(5/2)*(1 + c*x)^(5/2))/(b*c*Sqrt[a + b*ArcCosh[c*x] 
]) + (16*d^2*((3*Sqrt[a + b*ArcCosh[c*x]])/64 - (Sqrt[b]*E^((4*a)/b)*Sqrt[ 
Pi]*Erf[(2*Sqrt[a + b*ArcCosh[c*x]])/Sqrt[b]])/128 + (Sqrt[b]*E^((8*a)/b)* 
Sqrt[Pi/2]*Erf[(2*Sqrt[2]*Sqrt[a + b*ArcCosh[c*x]])/Sqrt[b]])/512 - (Sqrt[ 
b]*Sqrt[Pi]*Erfi[(2*Sqrt[a + b*ArcCosh[c*x]])/Sqrt[b]])/(128*E^((4*a)/b)) 
+ (Sqrt[b]*Sqrt[Pi/2]*Erfi[(2*Sqrt[2]*Sqrt[a + b*ArcCosh[c*x]])/Sqrt[b]])/ 
(512*E^((8*a)/b))))/(b^2*c^4) - (6*d^2*(Sqrt[a + b*ArcCosh[c*x]]/8 - (Sqrt 
[b]*E^((4*a)/b)*Sqrt[Pi]*Erf[(2*Sqrt[a + b*ArcCosh[c*x]])/Sqrt[b]])/64 - ( 
Sqrt[b]*E^((2*a)/b)*Sqrt[Pi/2]*Erf[(Sqrt[2]*Sqrt[a + b*ArcCosh[c*x]])/Sqrt 
[b]])/64 + (Sqrt[b]*E^((6*a)/b)*Sqrt[Pi/6]*Erf[(Sqrt[6]*Sqrt[a + b*ArcCosh 
[c*x]])/Sqrt[b]])/64 - (Sqrt[b]*Sqrt[Pi]*Erfi[(2*Sqrt[a + b*ArcCosh[c*x]]) 
/Sqrt[b]])/(64*E^((4*a)/b)) - (Sqrt[b]*Sqrt[Pi/2]*Erfi[(Sqrt[2]*Sqrt[a + b 
*ArcCosh[c*x]])/Sqrt[b]])/(64*E^((2*a)/b)) + (Sqrt[b]*Sqrt[Pi/6]*Erfi[(Sqr 
t[6]*Sqrt[a + b*ArcCosh[c*x]])/Sqrt[b]])/(64*E^((6*a)/b))))/(b^2*c^4)
 

3.4.74.3.1 Defintions of rubi rules used

rule 2009

Int[u_, x_Symbol] :> Simp[IntSum[u, x], x] /; SumQ[u]
 

rule 5971

Int[Cosh[(a_.) + (b_.)*(x_)]^(p_.)*((c_.) + (d_.)*(x_))^(m_.)*Sinh[(a_.) + 
(b_.)*(x_)]^(n_.), x_Symbol] :> Int[ExpandTrigReduce[(c + d*x)^m, Sinh[a + 
b*x]^n*Cosh[a + b*x]^p, x], x] /; FreeQ[{a, b, c, d, m}, x] && IGtQ[n, 0] & 
& IGtQ[p, 0]
 

rule 6357

Int[((a_.) + ArcCosh[(c_.)*(x_)]*(b_.))^(n_)*((f_.)*(x_))^(m_.)*((d_) + (e_ 
.)*(x_)^2)^(p_.), x_Symbol] :> Simp[(f*x)^m*Simp[Sqrt[1 + c*x]*Sqrt[-1 + c* 
x]*(d + e*x^2)^p]*((a + b*ArcCosh[c*x])^(n + 1)/(b*c*(n + 1))), x] + (Simp[ 
f*(m/(b*c*(n + 1)))*Simp[(d + e*x^2)^p/((1 + c*x)^p*(-1 + c*x)^p)]   Int[(f 
*x)^(m - 1)*(1 + c*x)^(p - 1/2)*(-1 + c*x)^(p - 1/2)*(a + b*ArcCosh[c*x])^( 
n + 1), x], x] - Simp[c*((m + 2*p + 1)/(b*f*(n + 1)))*Simp[(d + e*x^2)^p/(( 
1 + c*x)^p*(-1 + c*x)^p)]   Int[(f*x)^(m + 1)*(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, 
f, m, p}, x] && EqQ[c^2*d + e, 0] && LtQ[n, -1] && IGtQ[2*p, 0] && NeQ[m + 
2*p + 1, 0] && IGtQ[m, -3]
 

rule 6368

Int[((a_.) + ArcCosh[(c_.)*(x_)]*(b_.))^(n_.)*(x_)^(m_.)*((d1_) + (e1_.)*(x 
_))^(p_.)*((d2_) + (e2_.)*(x_))^(p_.), x_Symbol] :> Simp[(1/(b*c^(m + 1)))* 
Simp[(d1 + e1*x)^p/(1 + c*x)^p]*Simp[(d2 + e2*x)^p/(-1 + c*x)^p]   Subst[In 
t[x^n*Cosh[-a/b + x/b]^m*Sinh[-a/b + x/b]^(2*p + 1), x], x, a + b*ArcCosh[c 
*x]], x] /; FreeQ[{a, b, c, d1, e1, d2, e2, n}, x] && EqQ[e1, c*d1] && EqQ[ 
e2, (-c)*d2] && IGtQ[p + 3/2, 0] && IGtQ[m, 0]
 

3.4.74.4 Maple [F]

\[\int \frac {x^{3} \left (-c^{2} d \,x^{2}+d \right )^{2}}{\left (a +b \,\operatorname {arccosh}\left (c x \right )\right )^{\frac {3}{2}}}d x\]

input

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

output

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

3.4.74.5 Fricas [F(-2)]

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

input

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

output

Exception raised: TypeError >>  Error detected within library code:   inte 
grate: implementation incomplete (constant residues)
 

3.4.74.6 Sympy [F]

\[ \int \frac {x^3 \left (d-c^2 d x^2\right )^2}{(a+b \text {arccosh}(c x))^{3/2}} \, dx=d^{2} \left (\int \frac {x^{3}}{a \sqrt {a + b \operatorname {acosh}{\left (c x \right )}} + b \sqrt {a + b \operatorname {acosh}{\left (c x \right )}} \operatorname {acosh}{\left (c x \right )}}\, dx + \int \left (- \frac {2 c^{2} x^{5}}{a \sqrt {a + b \operatorname {acosh}{\left (c x \right )}} + b \sqrt {a + b \operatorname {acosh}{\left (c x \right )}} \operatorname {acosh}{\left (c x \right )}}\right )\, dx + \int \frac {c^{4} x^{7}}{a \sqrt {a + b \operatorname {acosh}{\left (c x \right )}} + b \sqrt {a + b \operatorname {acosh}{\left (c x \right )}} \operatorname {acosh}{\left (c x \right )}}\, dx\right ) \]

input

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

output

d**2*(Integral(x**3/(a*sqrt(a + b*acosh(c*x)) + b*sqrt(a + b*acosh(c*x))*a 
cosh(c*x)), x) + Integral(-2*c**2*x**5/(a*sqrt(a + b*acosh(c*x)) + b*sqrt( 
a + b*acosh(c*x))*acosh(c*x)), x) + Integral(c**4*x**7/(a*sqrt(a + b*acosh 
(c*x)) + b*sqrt(a + b*acosh(c*x))*acosh(c*x)), x))
 

3.4.74.7 Maxima [F]

\[ \int \frac {x^3 \left (d-c^2 d x^2\right )^2}{(a+b \text {arccosh}(c x))^{3/2}} \, dx=\int { \frac {{\left (c^{2} d x^{2} - d\right )}^{2} x^{3}}{{\left (b \operatorname {arcosh}\left (c x\right ) + a\right )}^{\frac {3}{2}}} \,d x } \]

input

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

output

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

3.4.74.8 Giac [F(-2)]

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

input

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

output

Exception raised: RuntimeError >> an error occurred running a Giac command 
:INPUT:sage2OUTPUT:sym2poly/r2sym(const gen & e,const index_m & i,const ve 
cteur & l) Error: Bad Argument Value
 

3.4.74.9 Mupad [F(-1)]

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

input

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

output

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