\(\int \frac {a+b \text {arccosh}(c+d x)}{(c e+d e x)^{5/2}} \, dx\) [121]

Optimal result

Integrand size = 23, antiderivative size = 150 \[ \int \frac {a+b \text {arccosh}(c+d x)}{(c e+d e x)^{5/2}} \, dx=\frac {4 b \sqrt {-1+c+d x} \sqrt {1+c+d x}}{3 d e^2 \sqrt {e (c+d x)}}-\frac {2 (a+b \text {arccosh}(c+d x))}{3 d e (e (c+d x))^{3/2}}+\frac {4 b \sqrt {1-c-d x} \sqrt {e (c+d x)} E\left (\left .\arcsin \left (\frac {\sqrt {1-c-d x}}{\sqrt {2}}\right )\right |2\right )}{3 d e^3 \sqrt {-1+c+d x} \sqrt {c+d x}} \] Output:

4/3*b*(d*x+c-1)^(1/2)*(d*x+c+1)^(1/2)/d/e^2/(e*(d*x+c))^(1/2)-2/3*(a+b*arc 
cosh(d*x+c))/d/e/(e*(d*x+c))^(3/2)+4/3*b*(-d*x-c+1)^(1/2)*(e*(d*x+c))^(1/2 
)*EllipticE(1/2*(-d*x-c+1)^(1/2)*2^(1/2),2^(1/2))/d/e^3/(d*x+c-1)^(1/2)/(d 
*x+c)^(1/2)
 

Mathematica [C] (verified)

Result contains higher order function than in optimal. Order 5 vs. order 4 in optimal.

Time = 0.10 (sec) , antiderivative size = 94, normalized size of antiderivative = 0.63 \[ \int \frac {a+b \text {arccosh}(c+d x)}{(c e+d e x)^{5/2}} \, dx=\frac {2 \left (-a-b \text {arccosh}(c+d x)-\frac {2 b (c+d x) \sqrt {1-(c+d x)^2} \operatorname {Hypergeometric2F1}\left (-\frac {1}{4},\frac {1}{2},\frac {3}{4},(c+d x)^2\right )}{\sqrt {-1+c+d x} \sqrt {1+c+d x}}\right )}{3 d e (e (c+d x))^{3/2}} \] Input:

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

Output:

(2*(-a - b*ArcCosh[c + d*x] - (2*b*(c + d*x)*Sqrt[1 - (c + d*x)^2]*Hyperge 
ometric2F1[-1/4, 1/2, 3/4, (c + d*x)^2])/(Sqrt[-1 + c + d*x]*Sqrt[1 + c + 
d*x])))/(3*d*e*(e*(c + d*x))^(3/2))
 

Rubi [A] (verified)

Time = 0.49 (sec) , antiderivative size = 148, normalized size of antiderivative = 0.99, number of steps used = 9, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.348, Rules used = {6411, 6298, 115, 8, 27, 124, 27, 123}

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.

\(\displaystyle \int \frac {a+b \text {arccosh}(c+d x)}{(c e+d e x)^{5/2}} \, dx\)

\(\Big \downarrow \) 6411

\(\displaystyle \frac {\int \frac {a+b \text {arccosh}(c+d x)}{(e (c+d x))^{5/2}}d(c+d x)}{d}\)

\(\Big \downarrow \) 6298

\(\displaystyle \frac {\frac {2 b \int \frac {1}{\sqrt {c+d x-1} (e (c+d x))^{3/2} \sqrt {c+d x+1}}d(c+d x)}{3 e}-\frac {2 (a+b \text {arccosh}(c+d x))}{3 e (e (c+d x))^{3/2}}}{d}\)

\(\Big \downarrow \) 115

\(\displaystyle \frac {\frac {2 b \left (\frac {2 \int -\frac {e (c+d x)}{2 \sqrt {c+d x-1} \sqrt {e (c+d x)} \sqrt {c+d x+1}}d(c+d x)}{e^2}+\frac {2 \sqrt {c+d x-1} \sqrt {c+d x+1}}{e \sqrt {e (c+d x)}}\right )}{3 e}-\frac {2 (a+b \text {arccosh}(c+d x))}{3 e (e (c+d x))^{3/2}}}{d}\)

\(\Big \downarrow \) 8

\(\displaystyle \frac {\frac {2 b \left (\frac {2 \int -\frac {e \sqrt {e (c+d x)}}{2 \sqrt {c+d x-1} \sqrt {c+d x+1}}d(c+d x)}{e^3}+\frac {2 \sqrt {c+d x-1} \sqrt {c+d x+1}}{e \sqrt {e (c+d x)}}\right )}{3 e}-\frac {2 (a+b \text {arccosh}(c+d x))}{3 e (e (c+d x))^{3/2}}}{d}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {\frac {2 b \left (\frac {2 \sqrt {c+d x-1} \sqrt {c+d x+1}}{e \sqrt {e (c+d x)}}-\frac {\int \frac {\sqrt {e (c+d x)}}{\sqrt {c+d x-1} \sqrt {c+d x+1}}d(c+d x)}{e^2}\right )}{3 e}-\frac {2 (a+b \text {arccosh}(c+d x))}{3 e (e (c+d x))^{3/2}}}{d}\)

\(\Big \downarrow \) 124

\(\displaystyle \frac {\frac {2 b \left (\frac {2 \sqrt {c+d x-1} \sqrt {c+d x+1}}{e \sqrt {e (c+d x)}}-\frac {\sqrt {-c-d x+1} \sqrt {e (c+d x)} \int \frac {\sqrt {2} \sqrt {-c-d x}}{\sqrt {-c-d x+1} \sqrt {c+d x+1}}d(c+d x)}{\sqrt {2} e^2 \sqrt {-c-d x} \sqrt {c+d x-1}}\right )}{3 e}-\frac {2 (a+b \text {arccosh}(c+d x))}{3 e (e (c+d x))^{3/2}}}{d}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {\frac {2 b \left (\frac {2 \sqrt {c+d x-1} \sqrt {c+d x+1}}{e \sqrt {e (c+d x)}}-\frac {\sqrt {-c-d x+1} \sqrt {e (c+d x)} \int \frac {\sqrt {-c-d x}}{\sqrt {-c-d x+1} \sqrt {c+d x+1}}d(c+d x)}{e^2 \sqrt {-c-d x} \sqrt {c+d x-1}}\right )}{3 e}-\frac {2 (a+b \text {arccosh}(c+d x))}{3 e (e (c+d x))^{3/2}}}{d}\)

\(\Big \downarrow \) 123

\(\displaystyle \frac {\frac {2 b \left (\frac {2 \sqrt {c+d x-1} \sqrt {c+d x+1}}{e \sqrt {e (c+d x)}}-\frac {2 \sqrt {-c-d x+1} \sqrt {e (c+d x)} E\left (\left .\arcsin \left (\frac {\sqrt {c+d x+1}}{\sqrt {2}}\right )\right |2\right )}{e^2 \sqrt {-c-d x} \sqrt {c+d x-1}}\right )}{3 e}-\frac {2 (a+b \text {arccosh}(c+d x))}{3 e (e (c+d x))^{3/2}}}{d}\)

Input:

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

Output:

((-2*(a + b*ArcCosh[c + d*x]))/(3*e*(e*(c + d*x))^(3/2)) + (2*b*((2*Sqrt[- 
1 + c + d*x]*Sqrt[1 + c + d*x])/(e*Sqrt[e*(c + d*x)]) - (2*Sqrt[1 - c - d* 
x]*Sqrt[e*(c + d*x)]*EllipticE[ArcSin[Sqrt[1 + c + d*x]/Sqrt[2]], 2])/(e^2 
*Sqrt[-c - d*x]*Sqrt[-1 + c + d*x])))/(3*e))/d
 

Defintions of rubi rules used

rule 8

Int[(u_.)*(x_)^(m_.)*((a_.)*(x_))^(p_), x_Symbol] :> Simp[1/a^m   Int[u*(a* 
x)^(m + p), x], x] /; FreeQ[{a, m, p}, x] && IntegerQ[m]
 

rule 27

Int[(a_)*(Fx_), x_Symbol] :> Simp[a   Int[Fx, x], x] /; FreeQ[a, x] &&  !Ma 
tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
 

rule 115

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_)*((e_.) + (f_.)*(x_) 
)^(p_), x_] :> Simp[b*(a + b*x)^(m + 1)*(c + d*x)^(n + 1)*((e + f*x)^(p + 1 
)/((m + 1)*(b*c - a*d)*(b*e - a*f))), x] + Simp[1/((m + 1)*(b*c - a*d)*(b*e 
 - a*f))   Int[(a + b*x)^(m + 1)*(c + d*x)^n*(e + f*x)^p*Simp[a*d*f*(m + 1) 
 - b*(d*e*(m + n + 2) + c*f*(m + p + 2)) - b*d*f*(m + n + p + 3)*x, x], x], 
 x] /; FreeQ[{a, b, c, d, e, f, n, p}, x] && LtQ[m, -1] && IntegersQ[2*m, 2 
*n, 2*p]
 

rule 123

Int[Sqrt[(e_.) + (f_.)*(x_)]/(Sqrt[(a_) + (b_.)*(x_)]*Sqrt[(c_) + (d_.)*(x_ 
)]), x_] :> Simp[(2/b)*Rt[-(b*e - a*f)/d, 2]*EllipticE[ArcSin[Sqrt[a + b*x] 
/Rt[-(b*c - a*d)/d, 2]], f*((b*c - a*d)/(d*(b*e - a*f)))], x] /; FreeQ[{a, 
b, c, d, e, f}, x] && GtQ[b/(b*c - a*d), 0] && GtQ[b/(b*e - a*f), 0] &&  !L 
tQ[-(b*c - a*d)/d, 0] &&  !(SimplerQ[c + d*x, a + b*x] && GtQ[-d/(b*c - a*d 
), 0] && GtQ[d/(d*e - c*f), 0] &&  !LtQ[(b*c - a*d)/b, 0])
 

rule 124

Int[Sqrt[(e_.) + (f_.)*(x_)]/(Sqrt[(a_) + (b_.)*(x_)]*Sqrt[(c_) + (d_.)*(x_ 
)]), x_] :> Simp[Sqrt[e + f*x]*(Sqrt[b*((c + d*x)/(b*c - a*d))]/(Sqrt[c + d 
*x]*Sqrt[b*((e + f*x)/(b*e - a*f))]))   Int[Sqrt[b*(e/(b*e - a*f)) + b*f*(x 
/(b*e - a*f))]/(Sqrt[a + b*x]*Sqrt[b*(c/(b*c - a*d)) + b*d*(x/(b*c - a*d))] 
), x], x] /; FreeQ[{a, b, c, d, e, f}, x] &&  !(GtQ[b/(b*c - a*d), 0] && Gt 
Q[b/(b*e - a*f), 0]) &&  !LtQ[-(b*c - a*d)/d, 0]
 

rule 6298

Int[((a_.) + ArcCosh[(c_.)*(x_)]*(b_.))^(n_.)*((d_.)*(x_))^(m_.), x_Symbol] 
 :> Simp[(d*x)^(m + 1)*((a + b*ArcCosh[c*x])^n/(d*(m + 1))), x] - Simp[b*c* 
(n/(d*(m + 1)))   Int[(d*x)^(m + 1)*((a + b*ArcCosh[c*x])^(n - 1)/(Sqrt[1 + 
 c*x]*Sqrt[-1 + c*x])), x], x] /; FreeQ[{a, b, c, d, m}, x] && IGtQ[n, 0] & 
& NeQ[m, -1]
 

rule 6411

Int[((a_.) + ArcCosh[(c_) + (d_.)*(x_)]*(b_.))^(n_.)*((e_.) + (f_.)*(x_))^( 
m_.), x_Symbol] :> Simp[1/d   Subst[Int[((d*e - c*f)/d + f*(x/d))^m*(a + b* 
ArcCosh[x])^n, x], x, c + d*x], x] /; FreeQ[{a, b, c, d, e, f, m, n}, x]
 

Maple [C] (verified)

Result contains complex when optimal does not.

Time = 3.47 (sec) , antiderivative size = 268, normalized size of antiderivative = 1.79

method	result	size
derivativedivides	\(\frac {-\frac {2 a}{3 \left (d e x +c e \right )^{\frac {3}{2}}}+2 b \left (-\frac {\operatorname {arccosh}\left (\frac {d e x +c e}{e}\right )}{3 \left (d e x +c e \right )^{\frac {3}{2}}}+\frac {-\frac {2 \sqrt {\frac {d e x +c e +e}{e}}\, \sqrt {\frac {-d e x -c e +e}{e}}\, \sqrt {d e x +c e}\, \operatorname {EllipticF}\left (\sqrt {d e x +c e}\, \sqrt {-\frac {1}{e}}, i\right ) e}{3}+\frac {2 \sqrt {\frac {d e x +c e +e}{e}}\, \sqrt {\frac {-d e x -c e +e}{e}}\, \sqrt {d e x +c e}\, \operatorname {EllipticE}\left (\sqrt {d e x +c e}\, \sqrt {-\frac {1}{e}}, i\right ) e}{3}+\frac {2 \sqrt {-\frac {1}{e}}\, \left (d e x +c e \right )^{2}}{3}-\frac {2 \sqrt {-\frac {1}{e}}\, e^{2}}{3}}{e^{3} \sqrt {-\frac {1}{e}}\, \sqrt {d e x +c e}\, \sqrt {\frac {d e x +c e +e}{e}}\, \sqrt {-\frac {-d e x -c e +e}{e}}}\right )}{d e}\)	\(268\)
default	\(\frac {-\frac {2 a}{3 \left (d e x +c e \right )^{\frac {3}{2}}}+2 b \left (-\frac {\operatorname {arccosh}\left (\frac {d e x +c e}{e}\right )}{3 \left (d e x +c e \right )^{\frac {3}{2}}}+\frac {-\frac {2 \sqrt {\frac {d e x +c e +e}{e}}\, \sqrt {\frac {-d e x -c e +e}{e}}\, \sqrt {d e x +c e}\, \operatorname {EllipticF}\left (\sqrt {d e x +c e}\, \sqrt {-\frac {1}{e}}, i\right ) e}{3}+\frac {2 \sqrt {\frac {d e x +c e +e}{e}}\, \sqrt {\frac {-d e x -c e +e}{e}}\, \sqrt {d e x +c e}\, \operatorname {EllipticE}\left (\sqrt {d e x +c e}\, \sqrt {-\frac {1}{e}}, i\right ) e}{3}+\frac {2 \sqrt {-\frac {1}{e}}\, \left (d e x +c e \right )^{2}}{3}-\frac {2 \sqrt {-\frac {1}{e}}\, e^{2}}{3}}{e^{3} \sqrt {-\frac {1}{e}}\, \sqrt {d e x +c e}\, \sqrt {\frac {d e x +c e +e}{e}}\, \sqrt {-\frac {-d e x -c e +e}{e}}}\right )}{d e}\)	\(268\)
parts	\(-\frac {2 a}{3 \left (d e x +c e \right )^{\frac {3}{2}} d e}+\frac {2 b \left (-\frac {\operatorname {arccosh}\left (\frac {d e x +c e}{e}\right )}{3 \left (d e x +c e \right )^{\frac {3}{2}}}+\frac {-\frac {2 \sqrt {\frac {d e x +c e +e}{e}}\, \sqrt {-\frac {d e x +c e -e}{e}}\, \sqrt {d e x +c e}\, \operatorname {EllipticF}\left (\sqrt {d e x +c e}\, \sqrt {-\frac {1}{e}}, i\right ) e}{3}+\frac {2 \sqrt {\frac {d e x +c e +e}{e}}\, \sqrt {-\frac {d e x +c e -e}{e}}\, \sqrt {d e x +c e}\, \operatorname {EllipticE}\left (\sqrt {d e x +c e}\, \sqrt {-\frac {1}{e}}, i\right ) e}{3}+\frac {2 \sqrt {-\frac {1}{e}}\, \left (d e x +c e \right )^{2}}{3}-\frac {2 \sqrt {-\frac {1}{e}}\, e^{2}}{3}}{e^{3} \sqrt {-\frac {1}{e}}\, \sqrt {d e x +c e}\, \sqrt {\frac {d e x +c e +e}{e}}\, \sqrt {\frac {d e x +c e -e}{e}}}\right )}{d e}\)	\(274\)

Input:

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

Output:

2/d/e*(-1/3*a/(d*e*x+c*e)^(3/2)+b*(-1/3/(d*e*x+c*e)^(3/2)*arccosh((d*e*x+c 
*e)/e)+2/3/e^3*(-((d*e*x+c*e+e)/e)^(1/2)*((-d*e*x-c*e+e)/e)^(1/2)*(d*e*x+c 
*e)^(1/2)*EllipticF((d*e*x+c*e)^(1/2)*(-1/e)^(1/2),I)*e+((d*e*x+c*e+e)/e)^ 
(1/2)*((-d*e*x-c*e+e)/e)^(1/2)*(d*e*x+c*e)^(1/2)*EllipticE((d*e*x+c*e)^(1/ 
2)*(-1/e)^(1/2),I)*e+(-1/e)^(1/2)*(d*e*x+c*e)^2-(-1/e)^(1/2)*e^2)/(-1/e)^( 
1/2)/(d*e*x+c*e)^(1/2)/((d*e*x+c*e+e)/e)^(1/2)/(-(-d*e*x-c*e+e)/e)^(1/2)))
 

Fricas [A] (verification not implemented)

Time = 0.11 (sec) , antiderivative size = 181, normalized size of antiderivative = 1.21 \[ \int \frac {a+b \text {arccosh}(c+d x)}{(c e+d e x)^{5/2}} \, dx=-\frac {2 \, {\left (\sqrt {d e x + c e} b d \log \left (d x + c + \sqrt {d^{2} x^{2} + 2 \, c d x + c^{2} - 1}\right ) + \sqrt {d e x + c e} a d - 2 \, {\left (b d^{2} x^{2} + 2 \, b c d x + b c^{2}\right )} \sqrt {d^{3} e} {\rm weierstrassZeta}\left (\frac {4}{d^{2}}, 0, {\rm weierstrassPInverse}\left (\frac {4}{d^{2}}, 0, \frac {d x + c}{d}\right )\right ) - 2 \, {\left (b d^{2} x + b c d\right )} \sqrt {d^{2} x^{2} + 2 \, c d x + c^{2} - 1} \sqrt {d e x + c e}\right )}}{3 \, {\left (d^{4} e^{3} x^{2} + 2 \, c d^{3} e^{3} x + c^{2} d^{2} e^{3}\right )}} \] Input:

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

Output:

-2/3*(sqrt(d*e*x + c*e)*b*d*log(d*x + c + sqrt(d^2*x^2 + 2*c*d*x + c^2 - 1 
)) + sqrt(d*e*x + c*e)*a*d - 2*(b*d^2*x^2 + 2*b*c*d*x + b*c^2)*sqrt(d^3*e) 
*weierstrassZeta(4/d^2, 0, weierstrassPInverse(4/d^2, 0, (d*x + c)/d)) - 2 
*(b*d^2*x + b*c*d)*sqrt(d^2*x^2 + 2*c*d*x + c^2 - 1)*sqrt(d*e*x + c*e))/(d 
^4*e^3*x^2 + 2*c*d^3*e^3*x + c^2*d^2*e^3)
 

Sympy [F]

\[ \int \frac {a+b \text {arccosh}(c+d x)}{(c e+d e x)^{5/2}} \, dx=\int \frac {a + b \operatorname {acosh}{\left (c + d x \right )}}{\left (e \left (c + d x\right )\right )^{\frac {5}{2}}}\, dx \] Input:

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

Output:

Integral((a + b*acosh(c + d*x))/(e*(c + d*x))**(5/2), x)
 

Maxima [F(-2)]

Exception generated. \[ \int \frac {a+b \text {arccosh}(c+d x)}{(c e+d e x)^{5/2}} \, dx=\text {Exception raised: ValueError} \] Input:

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

Output:

Exception raised: ValueError >> Computation failed since Maxima requested 
additional constraints; using the 'assume' command before evaluation *may* 
 help (example of legal syntax is 'assume(e>0)', see `assume?` for more de 
tails)Is e
 

Giac [F(-2)]

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

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

Output:

Exception raised: TypeError >> an error occurred running a Giac command:IN 
PUT:sage2:=int(sage0,sageVARx):;OUTPUT:sym2poly/r2sym(const gen & e,const 
index_m & i,const vecteur & l) Error: Bad Argument Value
 

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

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

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

Output:

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