\(\int (c e+d e x)^{5/2} (a+b \text {arccosh}(c+d x)) \, dx\) [116]

Optimal result

Integrand size = 23, antiderivative size = 169 \[ \int (c e+d e x)^{5/2} (a+b \text {arccosh}(c+d x)) \, dx=-\frac {20 b e^2 \sqrt {-1+c+d x} \sqrt {e (c+d x)} \sqrt {1+c+d x}}{147 d}-\frac {4 b \sqrt {-1+c+d x} (e (c+d x))^{5/2} \sqrt {1+c+d x}}{49 d}+\frac {2 (e (c+d x))^{7/2} (a+b \text {arccosh}(c+d x))}{7 d e}-\frac {20 b e^{5/2} \sqrt {1-c-d x} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {e (c+d x)}}{\sqrt {e}}\right ),-1\right )}{147 d \sqrt {-1+c+d x}} \] Output:

-20/147*b*e^2*(d*x+c-1)^(1/2)*(e*(d*x+c))^(1/2)*(d*x+c+1)^(1/2)/d-4/49*b*( 
d*x+c-1)^(1/2)*(e*(d*x+c))^(5/2)*(d*x+c+1)^(1/2)/d+2/7*(e*(d*x+c))^(7/2)*( 
a+b*arccosh(d*x+c))/d/e-20/147*b*e^(5/2)*(-d*x-c+1)^(1/2)*EllipticF((e*(d* 
x+c))^(1/2)/e^(1/2),I)/d/(d*x+c-1)^(1/2)
 

Mathematica [C] (warning: unable to verify)

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

Time = 0.25 (sec) , antiderivative size = 149, normalized size of antiderivative = 0.88 \[ \int (c e+d e x)^{5/2} (a+b \text {arccosh}(c+d x)) \, dx=\frac {2 (e (c+d x))^{5/2} \left ((c+d x)^{7/2} (a+b \text {arccosh}(c+d x))+\frac {2 b \left (5 \left (1-(c+d x)^2\right )+3 (c+d x)^2 \left (1-(c+d x)^2\right )-5 \sqrt {1-(c+d x)^2} \operatorname {Hypergeometric2F1}\left (\frac {1}{4},\frac {1}{2},\frac {5}{4},(c+d x)^2\right )\right )}{21 \sqrt {\frac {-1+c+d x}{c+d x}} \sqrt {1+c+d x}}\right )}{7 d (c+d x)^{5/2}} \] Input:

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

Output:

(2*(e*(c + d*x))^(5/2)*((c + d*x)^(7/2)*(a + b*ArcCosh[c + d*x]) + (2*b*(5 
*(1 - (c + d*x)^2) + 3*(c + d*x)^2*(1 - (c + d*x)^2) - 5*Sqrt[1 - (c + d*x 
)^2]*Hypergeometric2F1[1/4, 1/2, 5/4, (c + d*x)^2]))/(21*Sqrt[(-1 + c + d* 
x)/(c + d*x)]*Sqrt[1 + c + d*x])))/(7*d*(c + d*x)^(5/2))
 

Rubi [A] (verified)

Time = 0.36 (sec) , antiderivative size = 174, normalized size of antiderivative = 1.03, number of steps used = 9, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.348, Rules used = {6411, 6298, 113, 27, 113, 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.

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

\(\Big \downarrow \) 6411

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

\(\Big \downarrow \) 6298

\(\displaystyle \frac {\frac {2 (e (c+d x))^{7/2} (a+b \text {arccosh}(c+d x))}{7 e}-\frac {2 b \int \frac {(e (c+d x))^{7/2}}{\sqrt {c+d x-1} \sqrt {c+d x+1}}d(c+d x)}{7 e}}{d}\)

\(\Big \downarrow \) 113

\(\displaystyle \frac {\frac {2 (e (c+d x))^{7/2} (a+b \text {arccosh}(c+d x))}{7 e}-\frac {2 b \left (\frac {2}{7} \int \frac {5 e^2 (e (c+d x))^{3/2}}{2 \sqrt {c+d x-1} \sqrt {c+d x+1}}d(c+d x)+\frac {2}{7} e \sqrt {c+d x-1} \sqrt {c+d x+1} (e (c+d x))^{5/2}\right )}{7 e}}{d}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {\frac {2 (e (c+d x))^{7/2} (a+b \text {arccosh}(c+d x))}{7 e}-\frac {2 b \left (\frac {5}{7} e^2 \int \frac {(e (c+d x))^{3/2}}{\sqrt {c+d x-1} \sqrt {c+d x+1}}d(c+d x)+\frac {2}{7} e \sqrt {c+d x-1} \sqrt {c+d x+1} (e (c+d x))^{5/2}\right )}{7 e}}{d}\)

\(\Big \downarrow \) 113

\(\displaystyle \frac {\frac {2 (e (c+d x))^{7/2} (a+b \text {arccosh}(c+d x))}{7 e}-\frac {2 b \left (\frac {5}{7} e^2 \left (\frac {2}{3} \int \frac {e^2}{2 \sqrt {c+d x-1} \sqrt {e (c+d x)} \sqrt {c+d x+1}}d(c+d x)+\frac {2}{3} e \sqrt {c+d x-1} \sqrt {c+d x+1} \sqrt {e (c+d x)}\right )+\frac {2}{7} e \sqrt {c+d x-1} \sqrt {c+d x+1} (e (c+d x))^{5/2}\right )}{7 e}}{d}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {\frac {2 (e (c+d x))^{7/2} (a+b \text {arccosh}(c+d x))}{7 e}-\frac {2 b \left (\frac {5}{7} e^2 \left (\frac {1}{3} e^2 \int \frac {1}{\sqrt {c+d x-1} \sqrt {e (c+d x)} \sqrt {c+d x+1}}d(c+d x)+\frac {2}{3} e \sqrt {c+d x-1} \sqrt {c+d x+1} \sqrt {e (c+d x)}\right )+\frac {2}{7} e \sqrt {c+d x-1} \sqrt {c+d x+1} (e (c+d x))^{5/2}\right )}{7 e}}{d}\)

\(\Big \downarrow \) 127

\(\displaystyle \frac {\frac {2 (e (c+d x))^{7/2} (a+b \text {arccosh}(c+d x))}{7 e}-\frac {2 b \left (\frac {5}{7} e^2 \left (\frac {e^2 \sqrt {-c-d x+1} \int \frac {1}{\sqrt {-c-d x+1} \sqrt {e (c+d x)} \sqrt {c+d x+1}}d(c+d x)}{3 \sqrt {c+d x-1}}+\frac {2}{3} e \sqrt {c+d x-1} \sqrt {c+d x+1} \sqrt {e (c+d x)}\right )+\frac {2}{7} e \sqrt {c+d x-1} \sqrt {c+d x+1} (e (c+d x))^{5/2}\right )}{7 e}}{d}\)

\(\Big \downarrow \) 126

\(\displaystyle \frac {\frac {2 (e (c+d x))^{7/2} (a+b \text {arccosh}(c+d x))}{7 e}-\frac {2 b \left (\frac {5}{7} e^2 \left (\frac {2 e^{3/2} \sqrt {-c-d x+1} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {e (c+d x)}}{\sqrt {e}}\right ),-1\right )}{3 \sqrt {c+d x-1}}+\frac {2}{3} e \sqrt {c+d x-1} \sqrt {c+d x+1} \sqrt {e (c+d x)}\right )+\frac {2}{7} e \sqrt {c+d x-1} \sqrt {c+d x+1} (e (c+d x))^{5/2}\right )}{7 e}}{d}\)

Input:

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

Output:

((2*(e*(c + d*x))^(7/2)*(a + b*ArcCosh[c + d*x]))/(7*e) - (2*b*((2*e*Sqrt[ 
-1 + c + d*x]*(e*(c + d*x))^(5/2)*Sqrt[1 + c + d*x])/7 + (5*e^2*((2*e*Sqrt 
[-1 + c + d*x]*Sqrt[e*(c + d*x)]*Sqrt[1 + c + d*x])/3 + (2*e^(3/2)*Sqrt[1 
- c - d*x]*EllipticF[ArcSin[Sqrt[e*(c + d*x)]/Sqrt[e]], -1])/(3*Sqrt[-1 + 
c + d*x])))/7))/(7*e))/d
 

Defintions of rubi rules used

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 113

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 
)/(d*f*(m + n + p + 1))), x] + Simp[1/(d*f*(m + n + p + 1))   Int[(a + b*x) 
^(m - 2)*(c + d*x)^n*(e + f*x)^p*Simp[a^2*d*f*(m + n + p + 1) - b*(b*c*e*(m 
 - 1) + a*(d*e*(n + 1) + c*f*(p + 1))) + b*(a*d*f*(2*m + n + p) - b*(d*e*(m 
 + n) + c*f*(m + p)))*x, x], x], x] /; FreeQ[{a, b, c, d, e, f, n, p}, x] & 
& GtQ[m, 1] && NeQ[m + n + p + 1, 0] && 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 = 6.45 (sec) , antiderivative size = 218, normalized size of antiderivative = 1.29

method	result	size
derivativedivides	\(\frac {\frac {2 a \left (d e x +c e \right )^{\frac {7}{2}}}{7}+2 b \left (\frac {\left (d e x +c e \right )^{\frac {7}{2}} \operatorname {arccosh}\left (\frac {d e x +c e}{e}\right )}{7}-\frac {2 \left (3 \sqrt {-\frac {1}{e}}\, \left (d e x +c e \right )^{\frac {9}{2}}+2 \sqrt {-\frac {1}{e}}\, e^{2} \left (d e x +c e \right )^{\frac {5}{2}}+5 e^{4} \sqrt {\frac {d e x +c e +e}{e}}\, \sqrt {\frac {-d e x -c e +e}{e}}\, \operatorname {EllipticF}\left (\sqrt {d e x +c e}\, \sqrt {-\frac {1}{e}}, i\right )-5 \sqrt {-\frac {1}{e}}\, e^{4} \sqrt {d e x +c e}\right )}{147 e \sqrt {-\frac {1}{e}}\, \sqrt {\frac {d e x +c e +e}{e}}\, \sqrt {-\frac {-d e x -c e +e}{e}}}\right )}{d e}\)	\(218\)
default	\(\frac {\frac {2 a \left (d e x +c e \right )^{\frac {7}{2}}}{7}+2 b \left (\frac {\left (d e x +c e \right )^{\frac {7}{2}} \operatorname {arccosh}\left (\frac {d e x +c e}{e}\right )}{7}-\frac {2 \left (3 \sqrt {-\frac {1}{e}}\, \left (d e x +c e \right )^{\frac {9}{2}}+2 \sqrt {-\frac {1}{e}}\, e^{2} \left (d e x +c e \right )^{\frac {5}{2}}+5 e^{4} \sqrt {\frac {d e x +c e +e}{e}}\, \sqrt {\frac {-d e x -c e +e}{e}}\, \operatorname {EllipticF}\left (\sqrt {d e x +c e}\, \sqrt {-\frac {1}{e}}, i\right )-5 \sqrt {-\frac {1}{e}}\, e^{4} \sqrt {d e x +c e}\right )}{147 e \sqrt {-\frac {1}{e}}\, \sqrt {\frac {d e x +c e +e}{e}}\, \sqrt {-\frac {-d e x -c e +e}{e}}}\right )}{d e}\)	\(218\)
parts	\(\frac {2 a \left (d e x +c e \right )^{\frac {7}{2}}}{7 d e}+\frac {2 b \left (\frac {\left (d e x +c e \right )^{\frac {7}{2}} \operatorname {arccosh}\left (\frac {d e x +c e}{e}\right )}{7}-\frac {2 \left (3 \sqrt {-\frac {1}{e}}\, \left (d e x +c e \right )^{\frac {9}{2}}+2 \sqrt {-\frac {1}{e}}\, e^{2} \left (d e x +c e \right )^{\frac {5}{2}}+5 e^{4} \sqrt {\frac {d e x +c e +e}{e}}\, \sqrt {-\frac {d e x +c e -e}{e}}\, \operatorname {EllipticF}\left (\sqrt {d e x +c e}\, \sqrt {-\frac {1}{e}}, i\right )-5 \sqrt {-\frac {1}{e}}\, e^{4} \sqrt {d e x +c e}\right )}{147 e \sqrt {-\frac {1}{e}}\, \sqrt {\frac {d e x +c e +e}{e}}\, \sqrt {\frac {d e x +c e -e}{e}}}\right )}{d e}\)	\(223\)

Input:

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

Output:

2/d/e*(1/7*a*(d*e*x+c*e)^(7/2)+b*(1/7*(d*e*x+c*e)^(7/2)*arccosh((d*e*x+c*e 
)/e)-2/147/e*(3*(-1/e)^(1/2)*(d*e*x+c*e)^(9/2)+2*(-1/e)^(1/2)*e^2*(d*e*x+c 
*e)^(5/2)+5*e^4*((d*e*x+c*e+e)/e)^(1/2)*((-d*e*x-c*e+e)/e)^(1/2)*EllipticF 
((d*e*x+c*e)^(1/2)*(-1/e)^(1/2),I)-5*(-1/e)^(1/2)*e^4*(d*e*x+c*e)^(1/2))/( 
-1/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 = 0.16 (sec) , antiderivative size = 254, normalized size of antiderivative = 1.50 \[ \int (c e+d e x)^{5/2} (a+b \text {arccosh}(c+d x)) \, dx=-\frac {2 \, {\left (10 \, \sqrt {d^{3} e} b e^{2} {\rm weierstrassPInverse}\left (\frac {4}{d^{2}}, 0, \frac {d x + c}{d}\right ) - 21 \, {\left (b d^{5} e^{2} x^{3} + 3 \, b c d^{4} e^{2} x^{2} + 3 \, b c^{2} d^{3} e^{2} x + b c^{3} d^{2} e^{2}\right )} \sqrt {d e x + c e} \log \left (d x + c + \sqrt {d^{2} x^{2} + 2 \, c d x + c^{2} - 1}\right ) + 2 \, {\left (3 \, b d^{4} e^{2} x^{2} + 6 \, b c d^{3} e^{2} x + {\left (3 \, b c^{2} + 5 \, b\right )} d^{2} e^{2}\right )} \sqrt {d^{2} x^{2} + 2 \, c d x + c^{2} - 1} \sqrt {d e x + c e} - 21 \, {\left (a d^{5} e^{2} x^{3} + 3 \, a c d^{4} e^{2} x^{2} + 3 \, a c^{2} d^{3} e^{2} x + a c^{3} d^{2} e^{2}\right )} \sqrt {d e x + c e}\right )}}{147 \, d^{3}} \] Input:

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

Output:

-2/147*(10*sqrt(d^3*e)*b*e^2*weierstrassPInverse(4/d^2, 0, (d*x + c)/d) - 
21*(b*d^5*e^2*x^3 + 3*b*c*d^4*e^2*x^2 + 3*b*c^2*d^3*e^2*x + b*c^3*d^2*e^2) 
*sqrt(d*e*x + c*e)*log(d*x + c + sqrt(d^2*x^2 + 2*c*d*x + c^2 - 1)) + 2*(3 
*b*d^4*e^2*x^2 + 6*b*c*d^3*e^2*x + (3*b*c^2 + 5*b)*d^2*e^2)*sqrt(d^2*x^2 + 
 2*c*d*x + c^2 - 1)*sqrt(d*e*x + c*e) - 21*(a*d^5*e^2*x^3 + 3*a*c*d^4*e^2* 
x^2 + 3*a*c^2*d^3*e^2*x + a*c^3*d^2*e^2)*sqrt(d*e*x + c*e))/d^3
 

Sympy [F(-1)]

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

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

Output:

Timed out
 

Maxima [F(-2)]

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

integrate((d*e*x+c*e)^(5/2)*(a+b*arccosh(d*x+c)),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]

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

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

Output:

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

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

\[ \int (c e+d e x)^{5/2} (a+b \text {arccosh}(c+d x)) \, dx=\frac {\sqrt {e}\, e^{2} \left (2 \sqrt {d x +c}\, a \,c^{3}+6 \sqrt {d x +c}\, a \,c^{2} d x +6 \sqrt {d x +c}\, a c \,d^{2} x^{2}+2 \sqrt {d x +c}\, a \,d^{3} x^{3}+7 \left (\int \sqrt {d x +c}\, \mathit {acosh} \left (d x +c \right ) x^{2}d x \right ) b \,d^{3}+14 \left (\int \sqrt {d x +c}\, \mathit {acosh} \left (d x +c \right ) x d x \right ) b c \,d^{2}+7 \left (\int \sqrt {d x +c}\, \mathit {acosh} \left (d x +c \right )d x \right ) b \,c^{2} d \right )}{7 d} \] Input:

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

Output:

(sqrt(e)*e**2*(2*sqrt(c + d*x)*a*c**3 + 6*sqrt(c + d*x)*a*c**2*d*x + 6*sqr 
t(c + d*x)*a*c*d**2*x**2 + 2*sqrt(c + d*x)*a*d**3*x**3 + 7*int(sqrt(c + d* 
x)*acosh(c + d*x)*x**2,x)*b*d**3 + 14*int(sqrt(c + d*x)*acosh(c + d*x)*x,x 
)*b*c*d**2 + 7*int(sqrt(c + d*x)*acosh(c + d*x),x)*b*c**2*d))/(7*d)