\(\int \sqrt {c e+d e x} (a+b \text {arccosh}(c+d x)) \, dx\) [118]

Optimal result

Integrand size = 23, antiderivative size = 127 \[ \int \sqrt {c e+d e x} (a+b \text {arccosh}(c+d x)) \, dx=-\frac {4 b \sqrt {-1+c+d x} \sqrt {e (c+d x)} \sqrt {1+c+d x}}{9 d}+\frac {2 (e (c+d x))^{3/2} (a+b \text {arccosh}(c+d x))}{3 d e}-\frac {4 b \sqrt {e} \sqrt {1-c-d x} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {e (c+d x)}}{\sqrt {e}}\right ),-1\right )}{9 d \sqrt {-1+c+d x}} \] Output:

-4/9*b*(d*x+c-1)^(1/2)*(e*(d*x+c))^(1/2)*(d*x+c+1)^(1/2)/d+2/3*(e*(d*x+c)) 
^(3/2)*(a+b*arccosh(d*x+c))/d/e-4/9*b*e^(1/2)*(-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] (verified)

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

Time = 0.30 (sec) , antiderivative size = 131, normalized size of antiderivative = 1.03 \[ \int \sqrt {c e+d e x} (a+b \text {arccosh}(c+d x)) \, dx=\frac {\sqrt {e (c+d x)} \left (\frac {2}{3} (c+d x)^{3/2} (a+b \text {arccosh}(c+d x))-\frac {4 b \left (-1+c^2+2 c d x+d^2 x^2+\sqrt {1-(c+d x)^2} \operatorname {Hypergeometric2F1}\left (\frac {1}{4},\frac {1}{2},\frac {5}{4},(c+d x)^2\right )\right )}{9 \sqrt {\frac {-1+c+d x}{c+d x}} \sqrt {1+c+d x}}\right )}{d \sqrt {c+d x}} \] Input:

Integrate[Sqrt[c*e + d*e*x]*(a + b*ArcCosh[c + d*x]),x]
 

Output:

(Sqrt[e*(c + d*x)]*((2*(c + d*x)^(3/2)*(a + b*ArcCosh[c + d*x]))/3 - (4*b* 
(-1 + c^2 + 2*c*d*x + d^2*x^2 + Sqrt[1 - (c + d*x)^2]*Hypergeometric2F1[1/ 
4, 1/2, 5/4, (c + d*x)^2]))/(9*Sqrt[(-1 + c + d*x)/(c + d*x)]*Sqrt[1 + c + 
 d*x])))/(d*Sqrt[c + d*x])
 

Rubi [A] (verified)

Time = 0.34 (sec) , antiderivative size = 130, normalized size of antiderivative = 1.02, number of steps used = 7, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.261, Rules used = {6411, 6298, 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.

\(\displaystyle \int \sqrt {c e+d e x} (a+b \text {arccosh}(c+d x)) \, dx\)

\(\Big \downarrow \) 6411

\(\displaystyle \frac {\int \sqrt {e (c+d x)} (a+b \text {arccosh}(c+d x))d(c+d x)}{d}\)

\(\Big \downarrow \) 6298

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

\(\Big \downarrow \) 113

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 127

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

\(\Big \downarrow \) 126

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

Input:

Int[Sqrt[c*e + d*e*x]*(a + b*ArcCosh[c + d*x]),x]
 

Output:

((2*(e*(c + d*x))^(3/2)*(a + b*ArcCosh[c + d*x]))/(3*e) - (2*b*((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])))/(3*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 = 2.22 (sec) , antiderivative size = 194, normalized size of antiderivative = 1.53

method	result	size
derivativedivides	\(\frac {\frac {2 \left (d e x +c e \right )^{\frac {3}{2}} a}{3}+2 b \left (\frac {\left (d e x +c e \right )^{\frac {3}{2}} \operatorname {arccosh}\left (\frac {d e x +c e}{e}\right )}{3}-\frac {2 \left (\sqrt {-\frac {1}{e}}\, \left (d e x +c e \right )^{\frac {5}{2}}+\sqrt {\frac {d e x +c e +e}{e}}\, \sqrt {\frac {-d e x -c e +e}{e}}\, \operatorname {EllipticF}\left (\sqrt {d e x +c e}\, \sqrt {-\frac {1}{e}}, i\right ) e^{2}-\sqrt {-\frac {1}{e}}\, e^{2} \sqrt {d e x +c e}\right )}{9 e \sqrt {-\frac {1}{e}}\, \sqrt {\frac {d e x +c e +e}{e}}\, \sqrt {-\frac {-d e x -c e +e}{e}}}\right )}{d e}\)	\(194\)
default	\(\frac {\frac {2 \left (d e x +c e \right )^{\frac {3}{2}} a}{3}+2 b \left (\frac {\left (d e x +c e \right )^{\frac {3}{2}} \operatorname {arccosh}\left (\frac {d e x +c e}{e}\right )}{3}-\frac {2 \left (\sqrt {-\frac {1}{e}}\, \left (d e x +c e \right )^{\frac {5}{2}}+\sqrt {\frac {d e x +c e +e}{e}}\, \sqrt {\frac {-d e x -c e +e}{e}}\, \operatorname {EllipticF}\left (\sqrt {d e x +c e}\, \sqrt {-\frac {1}{e}}, i\right ) e^{2}-\sqrt {-\frac {1}{e}}\, e^{2} \sqrt {d e x +c e}\right )}{9 e \sqrt {-\frac {1}{e}}\, \sqrt {\frac {d e x +c e +e}{e}}\, \sqrt {-\frac {-d e x -c e +e}{e}}}\right )}{d e}\)	\(194\)
parts	\(\frac {2 a \left (d e x +c e \right )^{\frac {3}{2}}}{3 d e}+\frac {2 b \left (\frac {\left (d e x +c e \right )^{\frac {3}{2}} \operatorname {arccosh}\left (\frac {d e x +c e}{e}\right )}{3}-\frac {2 \left (\sqrt {-\frac {1}{e}}\, \left (d e x +c e \right )^{\frac {5}{2}}+\sqrt {\frac {d e x +c e +e}{e}}\, \sqrt {-\frac {d e x +c e -e}{e}}\, \operatorname {EllipticF}\left (\sqrt {d e x +c e}\, \sqrt {-\frac {1}{e}}, i\right ) e^{2}-\sqrt {-\frac {1}{e}}\, e^{2} \sqrt {d e x +c e}\right )}{9 e \sqrt {-\frac {1}{e}}\, \sqrt {\frac {d e x +c e +e}{e}}\, \sqrt {\frac {d e x +c e -e}{e}}}\right )}{d e}\)	\(199\)

Input:

int((d*e*x+c*e)^(1/2)*(a+b*arccosh(d*x+c)),x,method=_RETURNVERBOSE)
 

Output:

2/d/e*(1/3*(d*e*x+c*e)^(3/2)*a+b*(1/3*(d*e*x+c*e)^(3/2)*arccosh((d*e*x+c*e 
)/e)-2/9/e*((-1/e)^(1/2)*(d*e*x+c*e)^(5/2)+((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)*e^2-(-1/e)^( 
1/2)*e^2*(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 = 0.11 (sec) , antiderivative size = 142, normalized size of antiderivative = 1.12 \[ \int \sqrt {c e+d e x} (a+b \text {arccosh}(c+d x)) \, dx=-\frac {2 \, {\left (2 \, \sqrt {d^{2} x^{2} + 2 \, c d x + c^{2} - 1} \sqrt {d e x + c e} b d^{2} - 3 \, {\left (b d^{3} x + b c d^{2}\right )} \sqrt {d e x + c e} \log \left (d x + c + \sqrt {d^{2} x^{2} + 2 \, c d x + c^{2} - 1}\right ) + 2 \, \sqrt {d^{3} e} b {\rm weierstrassPInverse}\left (\frac {4}{d^{2}}, 0, \frac {d x + c}{d}\right ) - 3 \, {\left (a d^{3} x + a c d^{2}\right )} \sqrt {d e x + c e}\right )}}{9 \, d^{3}} \] Input:

integrate((d*e*x+c*e)^(1/2)*(a+b*arccosh(d*x+c)),x, algorithm="fricas")
 

Output:

-2/9*(2*sqrt(d^2*x^2 + 2*c*d*x + c^2 - 1)*sqrt(d*e*x + c*e)*b*d^2 - 3*(b*d 
^3*x + b*c*d^2)*sqrt(d*e*x + c*e)*log(d*x + c + sqrt(d^2*x^2 + 2*c*d*x + c 
^2 - 1)) + 2*sqrt(d^3*e)*b*weierstrassPInverse(4/d^2, 0, (d*x + c)/d) - 3* 
(a*d^3*x + a*c*d^2)*sqrt(d*e*x + c*e))/d^3
 

Sympy [F]

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

integrate((d*e*x+c*e)**(1/2)*(a+b*acosh(d*x+c)),x)
 

Output:

Integral(sqrt(e*(c + d*x))*(a + b*acosh(c + d*x)), x)
 

Maxima [F(-2)]

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

integrate((d*e*x+c*e)^(1/2)*(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]

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

integrate((d*e*x+c*e)^(1/2)*(a+b*arccosh(d*x+c)),x, algorithm="giac")
 

Output:

integrate(sqrt(d*e*x + c*e)*(b*arccosh(d*x + c) + a), x)
 

Mupad [F(-1)]

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

int((c*e + d*e*x)^(1/2)*(a + b*acosh(c + d*x)),x)
 

Output:

int((c*e + d*e*x)^(1/2)*(a + b*acosh(c + d*x)), x)
 

Reduce [F]

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

int((d*e*x+c*e)^(1/2)*(a+b*acosh(d*x+c)),x)
 

Output:

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