\(\int \frac {a+b \text {arccosh}(c+d x)}{(c e+d e x)^{7/2}} \, dx\) [122]

Optimal result

Integrand size = 23, antiderivative size = 130 \[ \int \frac {a+b \text {arccosh}(c+d x)}{(c e+d e x)^{7/2}} \, dx=\frac {4 b \sqrt {-1+c+d x} \sqrt {1+c+d x}}{15 d e^2 (e (c+d x))^{3/2}}-\frac {2 (a+b \text {arccosh}(c+d x))}{5 d e (e (c+d x))^{5/2}}+\frac {4 b \sqrt {1-c-d x} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {e (c+d x)}}{\sqrt {e}}\right ),-1\right )}{15 d e^{7/2} \sqrt {-1+c+d x}} \] Output:

4/15*b*(d*x+c-1)^(1/2)*(d*x+c+1)^(1/2)/d/e^2/(e*(d*x+c))^(3/2)-2/5*(a+b*ar 
ccosh(d*x+c))/d/e/(e*(d*x+c))^(5/2)+4/15*b*(-d*x-c+1)^(1/2)*EllipticF((e*( 
d*x+c))^(1/2)/e^(1/2),I)/d/e^(7/2)/(d*x+c-1)^(1/2)
 

Mathematica [C] (verified)

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

Time = 0.11 (sec) , antiderivative size = 94, normalized size of antiderivative = 0.72 \[ \int \frac {a+b \text {arccosh}(c+d x)}{(c e+d e x)^{7/2}} \, dx=\frac {2 \left (-3 (a+b \text {arccosh}(c+d x))-\frac {2 b (c+d x) \sqrt {1-(c+d x)^2} \operatorname {Hypergeometric2F1}\left (-\frac {3}{4},\frac {1}{2},\frac {1}{4},(c+d x)^2\right )}{\sqrt {-1+c+d x} \sqrt {1+c+d x}}\right )}{15 d e (e (c+d x))^{5/2}} \] Input:

Integrate[(a + b*ArcCosh[c + d*x])/(c*e + d*e*x)^(7/2),x]
 

Output:

(2*(-3*(a + b*ArcCosh[c + d*x]) - (2*b*(c + d*x)*Sqrt[1 - (c + d*x)^2]*Hyp 
ergeometric2F1[-3/4, 1/2, 1/4, (c + d*x)^2])/(Sqrt[-1 + c + d*x]*Sqrt[1 + 
c + d*x])))/(15*d*e*(e*(c + d*x))^(5/2))
 

Rubi [A] (verified)

Time = 0.55 (sec) , antiderivative size = 132, normalized size of antiderivative = 1.02, number of steps used = 8, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.304, Rules used = {6411, 6298, 115, 8, 27, 127, 126}

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.

\(\displaystyle \int \frac {a+b \text {arccosh}(c+d x)}{(c e+d e x)^{7/2}} \, dx\)

\(\Big \downarrow \) 6411

\(\displaystyle \frac {\int \frac {a+b \text {arccosh}(c+d x)}{(e (c+d x))^{7/2}}d(c+d x)}{d}\)

\(\Big \downarrow \) 6298

\(\displaystyle \frac {\frac {2 b \int \frac {1}{\sqrt {c+d x-1} (e (c+d x))^{5/2} \sqrt {c+d x+1}}d(c+d x)}{5 e}-\frac {2 (a+b \text {arccosh}(c+d x))}{5 e (e (c+d x))^{5/2}}}{d}\)

\(\Big \downarrow \) 115

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

\(\Big \downarrow \) 8

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 127

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

\(\Big \downarrow \) 126

\(\displaystyle \frac {\frac {2 b \left (\frac {2 \sqrt {-c-d x+1} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {e (c+d x)}}{\sqrt {e}}\right ),-1\right )}{3 e^{5/2} \sqrt {c+d x-1}}+\frac {2 \sqrt {c+d x-1} \sqrt {c+d x+1}}{3 e (e (c+d x))^{3/2}}\right )}{5 e}-\frac {2 (a+b \text {arccosh}(c+d x))}{5 e (e (c+d x))^{5/2}}}{d}\)

Input:

Int[(a + b*ArcCosh[c + d*x])/(c*e + d*e*x)^(7/2),x]
 

Output:

((-2*(a + b*ArcCosh[c + d*x]))/(5*e*(e*(c + d*x))^(5/2)) + (2*b*((2*Sqrt[- 
1 + c + d*x]*Sqrt[1 + c + d*x])/(3*e*(e*(c + d*x))^(3/2)) + (2*Sqrt[1 - c 
- d*x]*EllipticF[ArcSin[Sqrt[e*(c + d*x)]/Sqrt[e]], -1])/(3*e^(5/2)*Sqrt[- 
1 + c + d*x])))/(5*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 126

Int[1/(Sqrt[(b_.)*(x_)]*Sqrt[(c_) + (d_.)*(x_)]*Sqrt[(e_) + (f_.)*(x_)]), x 
_] :> Simp[(2/(b*Sqrt[e]))*Rt[-b/d, 2]*EllipticF[ArcSin[Sqrt[b*x]/(Sqrt[c]* 
Rt[-b/d, 2])], c*(f/(d*e))], x] /; FreeQ[{b, c, d, e, f}, x] && GtQ[c, 0] & 
& GtQ[e, 0] && (PosQ[-b/d] || NegQ[-b/f])
 

rule 127

Int[1/(Sqrt[(b_.)*(x_)]*Sqrt[(c_) + (d_.)*(x_)]*Sqrt[(e_) + (f_.)*(x_)]), x 
_] :> Simp[Sqrt[1 + d*(x/c)]*(Sqrt[1 + f*(x/e)]/(Sqrt[c + d*x]*Sqrt[e + f*x 
]))   Int[1/(Sqrt[b*x]*Sqrt[1 + d*(x/c)]*Sqrt[1 + f*(x/e)]), x], x] /; Free 
Q[{b, c, d, e, f}, x] &&  !(GtQ[c, 0] && GtQ[e, 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 [A] (verified)

Time = 4.73 (sec) , antiderivative size = 201, normalized size of antiderivative = 1.55

method	result	size
derivativedivides	\(\frac {-\frac {2 a}{5 \left (d e x +c e \right )^{\frac {5}{2}}}+2 b \left (-\frac {\operatorname {arccosh}\left (\frac {d e x +c e}{e}\right )}{5 \left (d e x +c e \right )^{\frac {5}{2}}}+\frac {\frac {2 \sqrt {\frac {d e x +c e +e}{e}}\, \sqrt {\frac {-d e x -c e +e}{e}}\, \operatorname {EllipticF}\left (\sqrt {d e x +c e}\, \sqrt {-\frac {1}{e}}, i\right ) \left (d e x +c e \right )^{\frac {3}{2}}}{15}+\frac {2 \sqrt {-\frac {1}{e}}\, \left (d e x +c e \right )^{2}}{15}-\frac {2 \sqrt {-\frac {1}{e}}\, e^{2}}{15}}{e^{3} \sqrt {-\frac {1}{e}}\, \left (d e x +c e \right )^{\frac {3}{2}} \sqrt {\frac {d e x +c e +e}{e}}\, \sqrt {-\frac {-d e x -c e +e}{e}}}\right )}{d e}\)	\(201\)
default	\(\frac {-\frac {2 a}{5 \left (d e x +c e \right )^{\frac {5}{2}}}+2 b \left (-\frac {\operatorname {arccosh}\left (\frac {d e x +c e}{e}\right )}{5 \left (d e x +c e \right )^{\frac {5}{2}}}+\frac {\frac {2 \sqrt {\frac {d e x +c e +e}{e}}\, \sqrt {\frac {-d e x -c e +e}{e}}\, \operatorname {EllipticF}\left (\sqrt {d e x +c e}\, \sqrt {-\frac {1}{e}}, i\right ) \left (d e x +c e \right )^{\frac {3}{2}}}{15}+\frac {2 \sqrt {-\frac {1}{e}}\, \left (d e x +c e \right )^{2}}{15}-\frac {2 \sqrt {-\frac {1}{e}}\, e^{2}}{15}}{e^{3} \sqrt {-\frac {1}{e}}\, \left (d e x +c e \right )^{\frac {3}{2}} \sqrt {\frac {d e x +c e +e}{e}}\, \sqrt {-\frac {-d e x -c e +e}{e}}}\right )}{d e}\)	\(201\)
parts	\(-\frac {2 a}{5 \left (d e x +c e \right )^{\frac {5}{2}} d e}+\frac {2 b \left (-\frac {\operatorname {arccosh}\left (\frac {d e x +c e}{e}\right )}{5 \left (d e x +c e \right )^{\frac {5}{2}}}+\frac {\frac {2 \sqrt {\frac {d e x +c e +e}{e}}\, \sqrt {-\frac {d e x +c e -e}{e}}\, \operatorname {EllipticF}\left (\sqrt {d e x +c e}\, \sqrt {-\frac {1}{e}}, i\right ) \left (d e x +c e \right )^{\frac {3}{2}}}{15}+\frac {2 \sqrt {-\frac {1}{e}}\, \left (d e x +c e \right )^{2}}{15}-\frac {2 \sqrt {-\frac {1}{e}}\, e^{2}}{15}}{e^{3} \sqrt {-\frac {1}{e}}\, \left (d e x +c e \right )^{\frac {3}{2}} \sqrt {\frac {d e x +c e +e}{e}}\, \sqrt {\frac {d e x +c e -e}{e}}}\right )}{d e}\)	\(206\)

Input:

int((a+b*arccosh(d*x+c))/(d*e*x+c*e)^(7/2),x,method=_RETURNVERBOSE)
 

Output:

2/d/e*(-1/5*a/(d*e*x+c*e)^(5/2)+b*(-1/5/(d*e*x+c*e)^(5/2)*arccosh((d*e*x+c 
*e)/e)+2/15/e^3*(((d*e*x+c*e+e)/e)^(1/2)*((-d*e*x-c*e+e)/e)^(1/2)*Elliptic 
F((d*e*x+c*e)^(1/2)*(-1/e)^(1/2),I)*(d*e*x+c*e)^(3/2)+(-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)^(3/2)/((d*e*x+c*e+e)/e)^ 
(1/2)/(-(-d*e*x-c*e+e)/e)^(1/2)))
 

Fricas [A] (verification not implemented)

Time = 0.12 (sec) , antiderivative size = 208, normalized size of antiderivative = 1.60 \[ \int \frac {a+b \text {arccosh}(c+d x)}{(c e+d e x)^{7/2}} \, dx=-\frac {2 \, {\left (3 \, \sqrt {d e x + c e} b d^{2} \log \left (d x + c + \sqrt {d^{2} x^{2} + 2 \, c d x + c^{2} - 1}\right ) + 3 \, \sqrt {d e x + c e} a d^{2} - 2 \, {\left (b d^{3} x^{3} + 3 \, b c d^{2} x^{2} + 3 \, b c^{2} d x + b c^{3}\right )} \sqrt {d^{3} e} {\rm weierstrassPInverse}\left (\frac {4}{d^{2}}, 0, \frac {d x + c}{d}\right ) - 2 \, {\left (b d^{3} x + b c d^{2}\right )} \sqrt {d^{2} x^{2} + 2 \, c d x + c^{2} - 1} \sqrt {d e x + c e}\right )}}{15 \, {\left (d^{6} e^{4} x^{3} + 3 \, c d^{5} e^{4} x^{2} + 3 \, c^{2} d^{4} e^{4} x + c^{3} d^{3} e^{4}\right )}} \] Input:

integrate((a+b*arccosh(d*x+c))/(d*e*x+c*e)^(7/2),x, algorithm="fricas")
 

Output:

-2/15*(3*sqrt(d*e*x + c*e)*b*d^2*log(d*x + c + sqrt(d^2*x^2 + 2*c*d*x + c^ 
2 - 1)) + 3*sqrt(d*e*x + c*e)*a*d^2 - 2*(b*d^3*x^3 + 3*b*c*d^2*x^2 + 3*b*c 
^2*d*x + b*c^3)*sqrt(d^3*e)*weierstrassPInverse(4/d^2, 0, (d*x + c)/d) - 2 
*(b*d^3*x + b*c*d^2)*sqrt(d^2*x^2 + 2*c*d*x + c^2 - 1)*sqrt(d*e*x + c*e))/ 
(d^6*e^4*x^3 + 3*c*d^5*e^4*x^2 + 3*c^2*d^4*e^4*x + c^3*d^3*e^4)
 

Sympy [F(-1)]

Timed out. \[ \int \frac {a+b \text {arccosh}(c+d x)}{(c e+d e x)^{7/2}} \, dx=\text {Timed out} \] Input:

integrate((a+b*acosh(d*x+c))/(d*e*x+c*e)**(7/2),x)
 

Output:

Timed out
 

Maxima [F(-2)]

Exception generated. \[ \int \frac {a+b \text {arccosh}(c+d x)}{(c e+d e x)^{7/2}} \, dx=\text {Exception raised: ValueError} \] Input:

integrate((a+b*arccosh(d*x+c))/(d*e*x+c*e)^(7/2),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]

\[ \int \frac {a+b \text {arccosh}(c+d x)}{(c e+d e x)^{7/2}} \, dx=\int { \frac {b \operatorname {arcosh}\left (d x + c\right ) + a}{{\left (d e x + c e\right )}^{\frac {7}{2}}} \,d x } \] Input:

integrate((a+b*arccosh(d*x+c))/(d*e*x+c*e)^(7/2),x, algorithm="giac")
 

Output:

integrate((b*arccosh(d*x + c) + a)/(d*e*x + c*e)^(7/2), x)
 

Mupad [F(-1)]

Timed out. \[ \int \frac {a+b \text {arccosh}(c+d x)}{(c e+d e x)^{7/2}} \, dx=\int \frac {a+b\,\mathrm {acosh}\left (c+d\,x\right )}{{\left (c\,e+d\,e\,x\right )}^{7/2}} \,d x \] Input:

int((a + b*acosh(c + d*x))/(c*e + d*e*x)^(7/2),x)
 

Output:

int((a + b*acosh(c + d*x))/(c*e + d*e*x)^(7/2), x)
 

Reduce [F]

\[ \int \frac {a+b \text {arccosh}(c+d x)}{(c e+d e x)^{7/2}} \, dx=\frac {5 \sqrt {d x +c}\, \left (\int \frac {\mathit {acosh} \left (d x +c \right )}{\sqrt {d x +c}\, c^{3}+3 \sqrt {d x +c}\, c^{2} d x +3 \sqrt {d x +c}\, c \,d^{2} x^{2}+\sqrt {d x +c}\, d^{3} x^{3}}d x \right ) b \,c^{2} d +10 \sqrt {d x +c}\, \left (\int \frac {\mathit {acosh} \left (d x +c \right )}{\sqrt {d x +c}\, c^{3}+3 \sqrt {d x +c}\, c^{2} d x +3 \sqrt {d x +c}\, c \,d^{2} x^{2}+\sqrt {d x +c}\, d^{3} x^{3}}d x \right ) b c \,d^{2} x +5 \sqrt {d x +c}\, \left (\int \frac {\mathit {acosh} \left (d x +c \right )}{\sqrt {d x +c}\, c^{3}+3 \sqrt {d x +c}\, c^{2} d x +3 \sqrt {d x +c}\, c \,d^{2} x^{2}+\sqrt {d x +c}\, d^{3} x^{3}}d x \right ) b \,d^{3} x^{2}-2 a}{5 \sqrt {e}\, \sqrt {d x +c}\, d \,e^{3} \left (d^{2} x^{2}+2 c d x +c^{2}\right )} \] Input:

int((a+b*acosh(d*x+c))/(d*e*x+c*e)^(7/2),x)
 

Output:

(5*sqrt(c + d*x)*int(acosh(c + d*x)/(sqrt(c + d*x)*c**3 + 3*sqrt(c + d*x)* 
c**2*d*x + 3*sqrt(c + d*x)*c*d**2*x**2 + sqrt(c + d*x)*d**3*x**3),x)*b*c** 
2*d + 10*sqrt(c + d*x)*int(acosh(c + d*x)/(sqrt(c + d*x)*c**3 + 3*sqrt(c + 
 d*x)*c**2*d*x + 3*sqrt(c + d*x)*c*d**2*x**2 + sqrt(c + d*x)*d**3*x**3),x) 
*b*c*d**2*x + 5*sqrt(c + d*x)*int(acosh(c + d*x)/(sqrt(c + d*x)*c**3 + 3*s 
qrt(c + d*x)*c**2*d*x + 3*sqrt(c + d*x)*c*d**2*x**2 + sqrt(c + d*x)*d**3*x 
**3),x)*b*d**3*x**2 - 2*a)/(5*sqrt(e)*sqrt(c + d*x)*d*e**3*(c**2 + 2*c*d*x 
 + d**2*x**2))