\(\int (c e+d e x)^{3/2} (a+b \text {arccosh}(c+d x))^2 \, dx\) [124]

Optimal result

Integrand size = 25, antiderivative size = 153 \[ \int (c e+d e x)^{3/2} (a+b \text {arccosh}(c+d x))^2 \, dx=\frac {2 (e (c+d x))^{5/2} (a+b \text {arccosh}(c+d x))^2}{5 d e}-\frac {8 b \sqrt {1-c-d x} (e (c+d x))^{7/2} (a+b \text {arccosh}(c+d x)) \operatorname {Hypergeometric2F1}\left (\frac {1}{2},\frac {7}{4},\frac {11}{4},(c+d x)^2\right )}{35 d e^2 \sqrt {-1+c+d x}}-\frac {16 b^2 (e (c+d x))^{9/2} \, _3F_2\left (1,\frac {9}{4},\frac {9}{4};\frac {11}{4},\frac {13}{4};(c+d x)^2\right )}{315 d e^3} \] Output:

2/5*(e*(d*x+c))^(5/2)*(a+b*arccosh(d*x+c))^2/d/e-8/35*b*(-d*x-c+1)^(1/2)*( 
e*(d*x+c))^(7/2)*(a+b*arccosh(d*x+c))*hypergeom([1/2, 7/4],[11/4],(d*x+c)^ 
2)/d/e^2/(d*x+c-1)^(1/2)-16/315*b^2*(e*(d*x+c))^(9/2)*hypergeom([1, 9/4, 9 
/4],[11/4, 13/4],(d*x+c)^2)/d/e^3
 

Mathematica [A] (verified)

Time = 0.33 (sec) , antiderivative size = 140, normalized size of antiderivative = 0.92 \[ \int (c e+d e x)^{3/2} (a+b \text {arccosh}(c+d x))^2 \, dx=\frac {2 (e (c+d x))^{5/2} \left (63 (a+b \text {arccosh}(c+d x))^2-4 b (c+d x) \left (\frac {9 \sqrt {1-(c+d x)^2} (a+b \text {arccosh}(c+d x)) \operatorname {Hypergeometric2F1}\left (\frac {1}{2},\frac {7}{4},\frac {11}{4},(c+d x)^2\right )}{\sqrt {-1+c+d x} \sqrt {1+c+d x}}+2 b (c+d x) \, _3F_2\left (1,\frac {9}{4},\frac {9}{4};\frac {11}{4},\frac {13}{4};(c+d x)^2\right )\right )\right )}{315 d e} \] Input:

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

Output:

(2*(e*(c + d*x))^(5/2)*(63*(a + b*ArcCosh[c + d*x])^2 - 4*b*(c + d*x)*((9* 
Sqrt[1 - (c + d*x)^2]*(a + b*ArcCosh[c + d*x])*Hypergeometric2F1[1/2, 7/4, 
 11/4, (c + d*x)^2])/(Sqrt[-1 + c + d*x]*Sqrt[1 + c + d*x]) + 2*b*(c + d*x 
)*HypergeometricPFQ[{1, 9/4, 9/4}, {11/4, 13/4}, (c + d*x)^2])))/(315*d*e)
 

Rubi [A] (verified)

Time = 0.70 (sec) , antiderivative size = 154, normalized size of antiderivative = 1.01, 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.

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

\(\Big \downarrow \) 6411

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

\(\Big \downarrow \) 6298

\(\displaystyle \frac {\frac {2 (e (c+d x))^{5/2} (a+b \text {arccosh}(c+d x))^2}{5 e}-\frac {4 b \int \frac {(e (c+d x))^{5/2} (a+b \text {arccosh}(c+d x))}{\sqrt {c+d x-1} \sqrt {c+d x+1}}d(c+d x)}{5 e}}{d}\)

\(\Big \downarrow \) 6364

\(\displaystyle \frac {\frac {2 (e (c+d x))^{5/2} (a+b \text {arccosh}(c+d x))^2}{5 e}-\frac {4 b \left (\frac {4 b (e (c+d x))^{9/2} \, _3F_2\left (1,\frac {9}{4},\frac {9}{4};\frac {11}{4},\frac {13}{4};(c+d x)^2\right )}{63 e^2}+\frac {2 \sqrt {-c-d x+1} (e (c+d x))^{7/2} \operatorname {Hypergeometric2F1}\left (\frac {1}{2},\frac {7}{4},\frac {11}{4},(c+d x)^2\right ) (a+b \text {arccosh}(c+d x))}{7 e \sqrt {c+d x-1}}\right )}{5 e}}{d}\)

Input:

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

Output:

((2*(e*(c + d*x))^(5/2)*(a + b*ArcCosh[c + d*x])^2)/(5*e) - (4*b*((2*Sqrt[ 
1 - c - d*x]*(e*(c + d*x))^(7/2)*(a + b*ArcCosh[c + d*x])*Hypergeometric2F 
1[1/2, 7/4, 11/4, (c + d*x)^2])/(7*e*Sqrt[-1 + c + d*x]) + (4*b*(e*(c + d* 
x))^(9/2)*HypergeometricPFQ[{1, 9/4, 9/4}, {11/4, 13/4}, (c + d*x)^2])/(63 
*e^2)))/(5*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]

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

Input:

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

Output:

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

Fricas [F]

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

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

Output:

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

Sympy [F]

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

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

Output:

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

Maxima [F(-2)]

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

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

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

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

Output:

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

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

\[ \int (c e+d e x)^{3/2} (a+b \text {arccosh}(c+d x))^2 \, dx=\frac {\sqrt {e}\, e \left (2 \sqrt {d x +c}\, a^{2} c^{2}+4 \sqrt {d x +c}\, a^{2} c d x +2 \sqrt {d x +c}\, a^{2} d^{2} x^{2}+10 \left (\int \sqrt {d x +c}\, \mathit {acosh} \left (d x +c \right ) x d x \right ) a b \,d^{2}+10 \left (\int \sqrt {d x +c}\, \mathit {acosh} \left (d x +c \right )d x \right ) a b c d +5 \left (\int \sqrt {d x +c}\, \mathit {acosh} \left (d x +c \right )^{2} x d x \right ) b^{2} d^{2}+5 \left (\int \sqrt {d x +c}\, \mathit {acosh} \left (d x +c \right )^{2}d x \right ) b^{2} c d \right )}{5 d} \] Input:

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

Output:

(sqrt(e)*e*(2*sqrt(c + d*x)*a**2*c**2 + 4*sqrt(c + d*x)*a**2*c*d*x + 2*sqr 
t(c + d*x)*a**2*d**2*x**2 + 10*int(sqrt(c + d*x)*acosh(c + d*x)*x,x)*a*b*d 
**2 + 10*int(sqrt(c + d*x)*acosh(c + d*x),x)*a*b*c*d + 5*int(sqrt(c + d*x) 
*acosh(c + d*x)**2*x,x)*b**2*d**2 + 5*int(sqrt(c + d*x)*acosh(c + d*x)**2, 
x)*b**2*c*d))/(5*d)