\(\int \frac {(a+b \text {arccosh}(c+d x))^2}{(c e+d e x)^{3/2}} \, dx\) [127]

Optimal result

Integrand size = 25, antiderivative size = 149 \[ \int \frac {(a+b \text {arccosh}(c+d x))^2}{(c e+d e x)^{3/2}} \, dx=-\frac {2 (a+b \text {arccosh}(c+d x))^2}{d e \sqrt {e (c+d x)}}+\frac {8 b \sqrt {1-c-d x} \sqrt {e (c+d x)} (a+b \text {arccosh}(c+d x)) \operatorname {Hypergeometric2F1}\left (\frac {1}{4},\frac {1}{2},\frac {5}{4},(c+d x)^2\right )}{d e^2 \sqrt {-1+c+d x}}+\frac {16 b^2 (e (c+d x))^{3/2} \, _3F_2\left (\frac {3}{4},\frac {3}{4},1;\frac {5}{4},\frac {7}{4};(c+d x)^2\right )}{3 d e^3} \] Output:

-2*(a+b*arccosh(d*x+c))^2/d/e/(e*(d*x+c))^(1/2)+8*b*(-d*x-c+1)^(1/2)*(e*(d 
*x+c))^(1/2)*(a+b*arccosh(d*x+c))*hypergeom([1/4, 1/2],[5/4],(d*x+c)^2)/d/ 
e^2/(d*x+c-1)^(1/2)+16/3*b^2*(e*(d*x+c))^(3/2)*hypergeom([3/4, 3/4, 1],[5/ 
4, 7/4],(d*x+c)^2)/d/e^3
 

Mathematica [A] (verified)

Time = 0.20 (sec) , antiderivative size = 140, normalized size of antiderivative = 0.94 \[ \int \frac {(a+b \text {arccosh}(c+d x))^2}{(c e+d e x)^{3/2}} \, dx=\frac {2 \left (-3 (a+b \text {arccosh}(c+d x))^2+4 b (c+d x) \left (\frac {3 \sqrt {1-(c+d x)^2} (a+b \text {arccosh}(c+d x)) \operatorname {Hypergeometric2F1}\left (\frac {1}{4},\frac {1}{2},\frac {5}{4},(c+d x)^2\right )}{\sqrt {-1+c+d x} \sqrt {1+c+d x}}+2 b (c+d x) \, _3F_2\left (\frac {3}{4},\frac {3}{4},1;\frac {5}{4},\frac {7}{4};(c+d x)^2\right )\right )\right )}{3 d e \sqrt {e (c+d x)}} \] Input:

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

Output:

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

Rubi [A] (verified)

Time = 0.56 (sec) , antiderivative size = 148, normalized size of antiderivative = 0.99, number of steps used = 4, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.120, Rules used = {6411, 6298, 6364}

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.

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

\(\Big \downarrow \) 6411

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

\(\Big \downarrow \) 6298

\(\displaystyle \frac {\frac {4 b \int \frac {a+b \text {arccosh}(c+d x)}{\sqrt {c+d x-1} \sqrt {e (c+d x)} \sqrt {c+d x+1}}d(c+d x)}{e}-\frac {2 (a+b \text {arccosh}(c+d x))^2}{e \sqrt {e (c+d x)}}}{d}\)

\(\Big \downarrow \) 6364

\(\displaystyle \frac {\frac {4 b \left (\frac {4 b (e (c+d x))^{3/2} \, _3F_2\left (\frac {3}{4},\frac {3}{4},1;\frac {5}{4},\frac {7}{4};(c+d x)^2\right )}{3 e^2}+\frac {2 \sqrt {-c-d x+1} \sqrt {e (c+d x)} \operatorname {Hypergeometric2F1}\left (\frac {1}{4},\frac {1}{2},\frac {5}{4},(c+d x)^2\right ) (a+b \text {arccosh}(c+d x))}{e \sqrt {c+d x-1}}\right )}{e}-\frac {2 (a+b \text {arccosh}(c+d x))^2}{e \sqrt {e (c+d x)}}}{d}\)

Input:

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

Output:

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

Defintions of rubi rules used

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 6364

Int[(((a_.) + ArcCosh[(c_.)*(x_)]*(b_.))*((f_.)*(x_))^(m_))/(Sqrt[(d1_) + ( 
e1_.)*(x_)]*Sqrt[(d2_) + (e2_.)*(x_)]), x_Symbol] :> Simp[((f*x)^(m + 1)/(f 
*(m + 1)))*Simp[Sqrt[1 - c^2*x^2]/(Sqrt[d1 + e1*x]*Sqrt[d2 + e2*x])]*(a + b 
*ArcCosh[c*x])*Hypergeometric2F1[1/2, (1 + m)/2, (3 + m)/2, c^2*x^2], x] + 
Simp[b*c*((f*x)^(m + 2)/(f^2*(m + 1)*(m + 2)))*Simp[Sqrt[1 + c*x]/Sqrt[d1 + 
 e1*x]]*Simp[Sqrt[-1 + c*x]/Sqrt[d2 + e2*x]]*HypergeometricPFQ[{1, 1 + m/2, 
 1 + m/2}, {3/2 + m/2, 2 + m/2}, c^2*x^2], x] /; FreeQ[{a, b, c, d1, e1, d2 
, e2, f, m}, x] && EqQ[e1, c*d1] && EqQ[e2, (-c)*d2] &&  !IntegerQ[m]
 

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 [F]

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

Input:

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

Output:

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

Fricas [F]

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

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

Output:

integral((b^2*arccosh(d*x + c)^2 + 2*a*b*arccosh(d*x + c) + a^2)*sqrt(d*e* 
x + c*e)/(d^2*e^2*x^2 + 2*c*d*e^2*x + c^2*e^2), x)
 

Sympy [F]

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

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

Output:

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

Maxima [F(-2)]

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

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

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

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

Output:

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

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

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

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

Output:

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