3.6.57 \(\int (d+e x^2) (a+b \text {arccosh}(c x))^{3/2} \, dx\) [557]

3.6.57.1 Optimal result

Integrand size = 20, antiderivative size = 442 \[ \int \left (d+e x^2\right ) (a+b \text {arccosh}(c x))^{3/2} \, dx=-\frac {3 b d \sqrt {-1+c x} \sqrt {1+c x} \sqrt {a+b \text {arccosh}(c x)}}{2 c}-\frac {b e \sqrt {-1+c x} \sqrt {1+c x} \sqrt {a+b \text {arccosh}(c x)}}{3 c^3}-\frac {b e x^2 \sqrt {-1+c x} \sqrt {1+c x} \sqrt {a+b \text {arccosh}(c x)}}{6 c}+d x (a+b \text {arccosh}(c x))^{3/2}+\frac {1}{3} e x^3 (a+b \text {arccosh}(c x))^{3/2}-\frac {3 b^{3/2} d e^{a/b} \sqrt {\pi } \text {erf}\left (\frac {\sqrt {a+b \text {arccosh}(c x)}}{\sqrt {b}}\right )}{8 c}-\frac {3 b^{3/2} e e^{a/b} \sqrt {\pi } \text {erf}\left (\frac {\sqrt {a+b \text {arccosh}(c x)}}{\sqrt {b}}\right )}{32 c^3}-\frac {b^{3/2} e e^{\frac {3 a}{b}} \sqrt {\frac {\pi }{3}} \text {erf}\left (\frac {\sqrt {3} \sqrt {a+b \text {arccosh}(c x)}}{\sqrt {b}}\right )}{96 c^3}+\frac {3 b^{3/2} d e^{-\frac {a}{b}} \sqrt {\pi } \text {erfi}\left (\frac {\sqrt {a+b \text {arccosh}(c x)}}{\sqrt {b}}\right )}{8 c}+\frac {3 b^{3/2} e e^{-\frac {a}{b}} \sqrt {\pi } \text {erfi}\left (\frac {\sqrt {a+b \text {arccosh}(c x)}}{\sqrt {b}}\right )}{32 c^3}+\frac {b^{3/2} e e^{-\frac {3 a}{b}} \sqrt {\frac {\pi }{3}} \text {erfi}\left (\frac {\sqrt {3} \sqrt {a+b \text {arccosh}(c x)}}{\sqrt {b}}\right )}{96 c^3} \]

output

d*x*(a+b*arccosh(c*x))^(3/2)+1/3*e*x^3*(a+b*arccosh(c*x))^(3/2)-1/288*b^(3 
/2)*e*exp(3*a/b)*erf(3^(1/2)*(a+b*arccosh(c*x))^(1/2)/b^(1/2))*3^(1/2)*Pi^ 
(1/2)/c^3+1/288*b^(3/2)*e*erfi(3^(1/2)*(a+b*arccosh(c*x))^(1/2)/b^(1/2))*3 
^(1/2)*Pi^(1/2)/c^3/exp(3*a/b)-3/8*b^(3/2)*d*exp(a/b)*erf((a+b*arccosh(c*x 
))^(1/2)/b^(1/2))*Pi^(1/2)/c-3/32*b^(3/2)*e*exp(a/b)*erf((a+b*arccosh(c*x) 
)^(1/2)/b^(1/2))*Pi^(1/2)/c^3+3/8*b^(3/2)*d*erfi((a+b*arccosh(c*x))^(1/2)/ 
b^(1/2))*Pi^(1/2)/c/exp(a/b)+3/32*b^(3/2)*e*erfi((a+b*arccosh(c*x))^(1/2)/ 
b^(1/2))*Pi^(1/2)/c^3/exp(a/b)-3/2*b*d*(c*x-1)^(1/2)*(c*x+1)^(1/2)*(a+b*ar 
ccosh(c*x))^(1/2)/c-1/3*b*e*(c*x-1)^(1/2)*(c*x+1)^(1/2)*(a+b*arccosh(c*x)) 
^(1/2)/c^3-1/6*b*e*x^2*(c*x-1)^(1/2)*(c*x+1)^(1/2)*(a+b*arccosh(c*x))^(1/2 
)/c
 

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

Time = 1.99 (sec) , antiderivative size = 812, normalized size of antiderivative = 1.84 \[ \int \left (d+e x^2\right ) (a+b \text {arccosh}(c x))^{3/2} \, dx=\frac {a d e^{-\frac {a}{b}} \sqrt {a+b \text {arccosh}(c x)} \left (\frac {e^{\frac {2 a}{b}} \Gamma \left (\frac {3}{2},\frac {a}{b}+\text {arccosh}(c x)\right )}{\sqrt {\frac {a}{b}+\text {arccosh}(c x)}}+\frac {\Gamma \left (\frac {3}{2},-\frac {a+b \text {arccosh}(c x)}{b}\right )}{\sqrt {-\frac {a+b \text {arccosh}(c x)}{b}}}\right )}{2 c}+\frac {a e e^{-\frac {3 a}{b}} \sqrt {a+b \text {arccosh}(c x)} \left (9 e^{\frac {4 a}{b}} \sqrt {-\frac {a+b \text {arccosh}(c x)}{b}} \Gamma \left (\frac {3}{2},\frac {a}{b}+\text {arccosh}(c x)\right )+\sqrt {3} \sqrt {\frac {a}{b}+\text {arccosh}(c x)} \Gamma \left (\frac {3}{2},-\frac {3 (a+b \text {arccosh}(c x))}{b}\right )+9 e^{\frac {2 a}{b}} \sqrt {\frac {a}{b}+\text {arccosh}(c x)} \Gamma \left (\frac {3}{2},-\frac {a+b \text {arccosh}(c x)}{b}\right )+\sqrt {3} e^{\frac {6 a}{b}} \sqrt {-\frac {a+b \text {arccosh}(c x)}{b}} \Gamma \left (\frac {3}{2},\frac {3 (a+b \text {arccosh}(c x))}{b}\right )\right )}{72 c^3 \sqrt {-\frac {(a+b \text {arccosh}(c x))^2}{b^2}}}+\frac {b d \left (-12 \sqrt {\frac {-1+c x}{1+c x}} (1+c x) \sqrt {a+b \text {arccosh}(c x)}+8 c x \text {arccosh}(c x) \sqrt {a+b \text {arccosh}(c x)}+\frac {(2 a+3 b) \sqrt {\pi } \text {erfi}\left (\frac {\sqrt {a+b \text {arccosh}(c x)}}{\sqrt {b}}\right ) \left (\cosh \left (\frac {a}{b}\right )-\sinh \left (\frac {a}{b}\right )\right )}{\sqrt {b}}+\frac {(2 a-3 b) \sqrt {\pi } \text {erf}\left (\frac {\sqrt {a+b \text {arccosh}(c x)}}{\sqrt {b}}\right ) \left (\cosh \left (\frac {a}{b}\right )+\sinh \left (\frac {a}{b}\right )\right )}{\sqrt {b}}\right )}{8 c}+\frac {\sqrt {b} e \left (9 \left (-12 \sqrt {b} \sqrt {\frac {-1+c x}{1+c x}} (1+c x) \sqrt {a+b \text {arccosh}(c x)}+8 \sqrt {b} c x \text {arccosh}(c x) \sqrt {a+b \text {arccosh}(c x)}+(2 a+3 b) \sqrt {\pi } \text {erfi}\left (\frac {\sqrt {a+b \text {arccosh}(c x)}}{\sqrt {b}}\right ) \left (\cosh \left (\frac {a}{b}\right )-\sinh \left (\frac {a}{b}\right )\right )+(2 a-3 b) \sqrt {\pi } \text {erf}\left (\frac {\sqrt {a+b \text {arccosh}(c x)}}{\sqrt {b}}\right ) \left (\cosh \left (\frac {a}{b}\right )+\sinh \left (\frac {a}{b}\right )\right )\right )+(2 a+b) \sqrt {3 \pi } \text {erfi}\left (\frac {\sqrt {3} \sqrt {a+b \text {arccosh}(c x)}}{\sqrt {b}}\right ) \left (\cosh \left (\frac {3 a}{b}\right )-\sinh \left (\frac {3 a}{b}\right )\right )+(2 a-b) \sqrt {3 \pi } \text {erf}\left (\frac {\sqrt {3} \sqrt {a+b \text {arccosh}(c x)}}{\sqrt {b}}\right ) \left (\cosh \left (\frac {3 a}{b}\right )+\sinh \left (\frac {3 a}{b}\right )\right )+12 \sqrt {b} \sqrt {a+b \text {arccosh}(c x)} (2 \text {arccosh}(c x) \cosh (3 \text {arccosh}(c x))-\sinh (3 \text {arccosh}(c x)))\right )}{288 c^3} \]

input

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

output

(a*d*Sqrt[a + b*ArcCosh[c*x]]*((E^((2*a)/b)*Gamma[3/2, a/b + ArcCosh[c*x]] 
)/Sqrt[a/b + ArcCosh[c*x]] + Gamma[3/2, -((a + b*ArcCosh[c*x])/b)]/Sqrt[-( 
(a + b*ArcCosh[c*x])/b)]))/(2*c*E^(a/b)) + (a*e*Sqrt[a + b*ArcCosh[c*x]]*( 
9*E^((4*a)/b)*Sqrt[-((a + b*ArcCosh[c*x])/b)]*Gamma[3/2, a/b + ArcCosh[c*x 
]] + Sqrt[3]*Sqrt[a/b + ArcCosh[c*x]]*Gamma[3/2, (-3*(a + b*ArcCosh[c*x])) 
/b] + 9*E^((2*a)/b)*Sqrt[a/b + ArcCosh[c*x]]*Gamma[3/2, -((a + b*ArcCosh[c 
*x])/b)] + Sqrt[3]*E^((6*a)/b)*Sqrt[-((a + b*ArcCosh[c*x])/b)]*Gamma[3/2, 
(3*(a + b*ArcCosh[c*x]))/b]))/(72*c^3*E^((3*a)/b)*Sqrt[-((a + b*ArcCosh[c* 
x])^2/b^2)]) + (b*d*(-12*Sqrt[(-1 + c*x)/(1 + c*x)]*(1 + c*x)*Sqrt[a + b*A 
rcCosh[c*x]] + 8*c*x*ArcCosh[c*x]*Sqrt[a + b*ArcCosh[c*x]] + ((2*a + 3*b)* 
Sqrt[Pi]*Erfi[Sqrt[a + b*ArcCosh[c*x]]/Sqrt[b]]*(Cosh[a/b] - Sinh[a/b]))/S 
qrt[b] + ((2*a - 3*b)*Sqrt[Pi]*Erf[Sqrt[a + b*ArcCosh[c*x]]/Sqrt[b]]*(Cosh 
[a/b] + Sinh[a/b]))/Sqrt[b]))/(8*c) + (Sqrt[b]*e*(9*(-12*Sqrt[b]*Sqrt[(-1 
+ c*x)/(1 + c*x)]*(1 + c*x)*Sqrt[a + b*ArcCosh[c*x]] + 8*Sqrt[b]*c*x*ArcCo 
sh[c*x]*Sqrt[a + b*ArcCosh[c*x]] + (2*a + 3*b)*Sqrt[Pi]*Erfi[Sqrt[a + b*Ar 
cCosh[c*x]]/Sqrt[b]]*(Cosh[a/b] - Sinh[a/b]) + (2*a - 3*b)*Sqrt[Pi]*Erf[Sq 
rt[a + b*ArcCosh[c*x]]/Sqrt[b]]*(Cosh[a/b] + Sinh[a/b])) + (2*a + b)*Sqrt[ 
3*Pi]*Erfi[(Sqrt[3]*Sqrt[a + b*ArcCosh[c*x]])/Sqrt[b]]*(Cosh[(3*a)/b] - Si 
nh[(3*a)/b]) + (2*a - b)*Sqrt[3*Pi]*Erf[(Sqrt[3]*Sqrt[a + b*ArcCosh[c*x]]) 
/Sqrt[b]]*(Cosh[(3*a)/b] + Sinh[(3*a)/b]) + 12*Sqrt[b]*Sqrt[a + b*ArcCo...
 

3.6.57.3 Rubi [A] (verified)

Time = 2.02 (sec) , antiderivative size = 442, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.100, Rules used = {6324, 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.

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

\(\Big \downarrow \) 6324

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

\(\Big \downarrow \) 2009

\(\displaystyle -\frac {3 \sqrt {\pi } b^{3/2} e e^{a/b} \text {erf}\left (\frac {\sqrt {a+b \text {arccosh}(c x)}}{\sqrt {b}}\right )}{32 c^3}-\frac {\sqrt {\frac {\pi }{3}} b^{3/2} e e^{\frac {3 a}{b}} \text {erf}\left (\frac {\sqrt {3} \sqrt {a+b \text {arccosh}(c x)}}{\sqrt {b}}\right )}{96 c^3}+\frac {3 \sqrt {\pi } b^{3/2} e e^{-\frac {a}{b}} \text {erfi}\left (\frac {\sqrt {a+b \text {arccosh}(c x)}}{\sqrt {b}}\right )}{32 c^3}+\frac {\sqrt {\frac {\pi }{3}} b^{3/2} e e^{-\frac {3 a}{b}} \text {erfi}\left (\frac {\sqrt {3} \sqrt {a+b \text {arccosh}(c x)}}{\sqrt {b}}\right )}{96 c^3}-\frac {3 \sqrt {\pi } b^{3/2} d e^{a/b} \text {erf}\left (\frac {\sqrt {a+b \text {arccosh}(c x)}}{\sqrt {b}}\right )}{8 c}+\frac {3 \sqrt {\pi } b^{3/2} d e^{-\frac {a}{b}} \text {erfi}\left (\frac {\sqrt {a+b \text {arccosh}(c x)}}{\sqrt {b}}\right )}{8 c}-\frac {b e \sqrt {c x-1} \sqrt {c x+1} \sqrt {a+b \text {arccosh}(c x)}}{3 c^3}+d x (a+b \text {arccosh}(c x))^{3/2}-\frac {3 b d \sqrt {c x-1} \sqrt {c x+1} \sqrt {a+b \text {arccosh}(c x)}}{2 c}+\frac {1}{3} e x^3 (a+b \text {arccosh}(c x))^{3/2}-\frac {b e x^2 \sqrt {c x-1} \sqrt {c x+1} \sqrt {a+b \text {arccosh}(c x)}}{6 c}\)

input

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

output

(-3*b*d*Sqrt[-1 + c*x]*Sqrt[1 + c*x]*Sqrt[a + b*ArcCosh[c*x]])/(2*c) - (b* 
e*Sqrt[-1 + c*x]*Sqrt[1 + c*x]*Sqrt[a + b*ArcCosh[c*x]])/(3*c^3) - (b*e*x^ 
2*Sqrt[-1 + c*x]*Sqrt[1 + c*x]*Sqrt[a + b*ArcCosh[c*x]])/(6*c) + d*x*(a + 
b*ArcCosh[c*x])^(3/2) + (e*x^3*(a + b*ArcCosh[c*x])^(3/2))/3 - (3*b^(3/2)* 
d*E^(a/b)*Sqrt[Pi]*Erf[Sqrt[a + b*ArcCosh[c*x]]/Sqrt[b]])/(8*c) - (3*b^(3/ 
2)*e*E^(a/b)*Sqrt[Pi]*Erf[Sqrt[a + b*ArcCosh[c*x]]/Sqrt[b]])/(32*c^3) - (b 
^(3/2)*e*E^((3*a)/b)*Sqrt[Pi/3]*Erf[(Sqrt[3]*Sqrt[a + b*ArcCosh[c*x]])/Sqr 
t[b]])/(96*c^3) + (3*b^(3/2)*d*Sqrt[Pi]*Erfi[Sqrt[a + b*ArcCosh[c*x]]/Sqrt 
[b]])/(8*c*E^(a/b)) + (3*b^(3/2)*e*Sqrt[Pi]*Erfi[Sqrt[a + b*ArcCosh[c*x]]/ 
Sqrt[b]])/(32*c^3*E^(a/b)) + (b^(3/2)*e*Sqrt[Pi/3]*Erfi[(Sqrt[3]*Sqrt[a + 
b*ArcCosh[c*x]])/Sqrt[b]])/(96*c^3*E^((3*a)/b))
 

3.6.57.3.1 Defintions of rubi rules used

rule 2009

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

rule 6324

Int[((a_.) + ArcCosh[(c_.)*(x_)]*(b_.))^(n_.)*((d_) + (e_.)*(x_)^2)^(p_.), 
x_Symbol] :> Int[ExpandIntegrand[(a + b*ArcCosh[c*x])^n, (d + e*x^2)^p, x], 
 x] /; FreeQ[{a, b, c, d, e, n}, x] && NeQ[c^2*d + e, 0] && IntegerQ[p] && 
(p > 0 || IGtQ[n, 0])
 

3.6.57.4 Maple [F]

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

input

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

output

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

3.6.57.5 Fricas [F(-2)]

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

input

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

output

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

3.6.57.6 Sympy [F]

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

input

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

output

Integral((a + b*acosh(c*x))**(3/2)*(d + e*x**2), x)
 

3.6.57.7 Maxima [F]

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

input

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

output

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

3.6.57.8 Giac [F(-2)]

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

input

integrate((e*x^2+d)*(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
 

3.6.57.9 Mupad [F(-1)]

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

input

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

output

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