\(\int (c e+d e x)^m (a+b \text {arccosh}(c+d x))^2 \, dx\) [136]

Optimal result

Integrand size = 23, antiderivative size = 206 \[ \int (c e+d e x)^m (a+b \text {arccosh}(c+d x))^2 \, dx=\frac {(e (c+d x))^{1+m} (a+b \text {arccosh}(c+d x))^2}{d e (1+m)}-\frac {2 b \sqrt {1-c-d x} (e (c+d x))^{2+m} (a+b \text {arccosh}(c+d x)) \operatorname {Hypergeometric2F1}\left (\frac {1}{2},\frac {2+m}{2},\frac {4+m}{2},(c+d x)^2\right )}{d e^2 (1+m) (2+m) \sqrt {-1+c+d x}}-\frac {2 b^2 (e (c+d x))^{3+m} \, _3F_2\left (1,\frac {3}{2}+\frac {m}{2},\frac {3}{2}+\frac {m}{2};2+\frac {m}{2},\frac {5}{2}+\frac {m}{2};(c+d x)^2\right )}{d e^3 (1+m) \left (6+5 m+m^2\right )} \] Output:

(e*(d*x+c))^(1+m)*(a+b*arccosh(d*x+c))^2/d/e/(1+m)-2*b*(-d*x-c+1)^(1/2)*(e 
*(d*x+c))^(2+m)*(a+b*arccosh(d*x+c))*hypergeom([1/2, 1+1/2*m],[2+1/2*m],(d 
*x+c)^2)/d/e^2/(1+m)/(2+m)/(d*x+c-1)^(1/2)-2*b^2*(e*(d*x+c))^(3+m)*hyperge 
om([1, 3/2+1/2*m, 3/2+1/2*m],[2+1/2*m, 5/2+1/2*m],(d*x+c)^2)/d/e^3/(1+m)/( 
m^2+5*m+6)
 

Mathematica [A] (verified)

Time = 0.29 (sec) , antiderivative size = 178, normalized size of antiderivative = 0.86 \[ \int (c e+d e x)^m (a+b \text {arccosh}(c+d x))^2 \, dx=\frac {(c+d x) (e (c+d x))^m \left ((a+b \text {arccosh}(c+d x))^2-\frac {2 b (c+d x) \left (\frac {\sqrt {1-(c+d x)^2} (a+b \text {arccosh}(c+d x)) \operatorname {Hypergeometric2F1}\left (\frac {1}{2},\frac {2+m}{2},\frac {4+m}{2},(c+d x)^2\right )}{\sqrt {-1+c+d x} \sqrt {1+c+d x}}+\frac {b (c+d x) \, _3F_2\left (1,\frac {3}{2}+\frac {m}{2},\frac {3}{2}+\frac {m}{2};2+\frac {m}{2},\frac {5}{2}+\frac {m}{2};(c+d x)^2\right )}{3+m}\right )}{2+m}\right )}{d (1+m)} \] Input:

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

Output:

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

Rubi [A] (verified)

Time = 0.60 (sec) , antiderivative size = 198, normalized size of antiderivative = 0.96, number of steps used = 4, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.130, 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.

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

\(\Big \downarrow \) 6411

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

\(\Big \downarrow \) 6298

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

\(\Big \downarrow \) 6364

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

Input:

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

Output:

(((e*(c + d*x))^(1 + m)*(a + b*ArcCosh[c + d*x])^2)/(e*(1 + m)) - (2*b*((S 
qrt[1 - c - d*x]*(e*(c + d*x))^(2 + m)*(a + b*ArcCosh[c + d*x])*Hypergeome 
tric2F1[1/2, (2 + m)/2, (4 + m)/2, (c + d*x)^2])/(e*(2 + m)*Sqrt[-1 + c + 
d*x]) + (b*(e*(c + d*x))^(3 + m)*HypergeometricPFQ[{1, 3/2 + m/2, 3/2 + m/ 
2}, {2 + m/2, 5/2 + m/2}, (c + d*x)^2])/(e^2*(2 + m)*(3 + m))))/(e*(1 + m) 
))/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]

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

Input:

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

Output:

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

Fricas [F]

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

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

Output:

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

Sympy [F]

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

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

Output:

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

Maxima [F]

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

integrate((d*e*x+c*e)^m*(a+b*arccosh(d*x+c))^2,x, algorithm="maxima")
 

Output:

(b^2*d*e^m*x + b^2*c*e^m)*(d*x + c)^m*log(d*x + sqrt(d*x + c + 1)*sqrt(d*x 
 + c - 1) + c)^2/(d*(m + 1)) + (d*e*x + c*e)^(m + 1)*a^2/(d*e*(m + 1)) + i 
ntegrate(-2*((b^2*c^2*e^m - (c^2*e^m*(m + 1) - e^m*(m + 1))*a*b - (a*b*d^2 
*e^m*(m + 1) - b^2*d^2*e^m)*x^2 - 2*(a*b*c*d*e^m*(m + 1) - b^2*c*d*e^m)*x) 
*sqrt(d*x + c + 1)*sqrt(d*x + c - 1)*(d*x + c)^m - ((a*b*d^3*e^m*(m + 1) - 
 b^2*d^3*e^m)*x^3 + (c^3*e^m*(m + 1) - c*e^m*(m + 1))*a*b - (c^3*e^m - c*e 
^m)*b^2 + 3*(a*b*c*d^2*e^m*(m + 1) - b^2*c*d^2*e^m)*x^2 + ((3*c^2*d*e^m*(m 
 + 1) - d*e^m*(m + 1))*a*b - (3*c^2*d*e^m - d*e^m)*b^2)*x)*(d*x + c)^m)*lo 
g(d*x + sqrt(d*x + c + 1)*sqrt(d*x + c - 1) + c)/(d^3*(m + 1)*x^3 + 3*c*d^ 
2*(m + 1)*x^2 + c^3*(m + 1) + (d^2*(m + 1)*x^2 + 2*c*d*(m + 1)*x + c^2*(m 
+ 1) - m - 1)*sqrt(d*x + c + 1)*sqrt(d*x + c - 1) - c*(m + 1) + (3*c^2*d*( 
m + 1) - d*(m + 1))*x), x)
 

Giac [F]

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

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

Output:

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

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

\[ \int (c e+d e x)^m (a+b \text {arccosh}(c+d x))^2 \, dx=\frac {\left (d e x +c e \right )^{m} a^{2} c +\left (d e x +c e \right )^{m} a^{2} d x +2 \left (\int \left (d e x +c e \right )^{m} \mathit {acosh} \left (d x +c \right )d x \right ) a b d m +2 \left (\int \left (d e x +c e \right )^{m} \mathit {acosh} \left (d x +c \right )d x \right ) a b d +\left (\int \left (d e x +c e \right )^{m} \mathit {acosh} \left (d x +c \right )^{2}d x \right ) b^{2} d m +\left (\int \left (d e x +c e \right )^{m} \mathit {acosh} \left (d x +c \right )^{2}d x \right ) b^{2} d}{d \left (m +1\right )} \] Input:

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

Output:

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