\(\int (f x)^m (\text {d1}+c \text {d1} x)^{3/2} (\text {d2}-c \text {d2} x)^{3/2} (a+b \text {arccosh}(c x)) \, dx\) [153]

Optimal result

Integrand size = 35, antiderivative size = 546 \[ \int (f x)^m (\text {d1}+c \text {d1} x)^{3/2} (\text {d2}-c \text {d2} x)^{3/2} (a+b \text {arccosh}(c x)) \, dx=-\frac {3 b c (f x)^{2+m} (\text {d1}+c \text {d1} x)^{3/2} (\text {d2}-c \text {d2} x)^{3/2}}{f^2 (2+m)^2 (4+m) \sqrt {-1+c x} \sqrt {1+c x} \left (1-c^2 x^2\right )}-\frac {b c (f x)^{2+m} (\text {d1}+c \text {d1} x)^{3/2} (\text {d2}-c \text {d2} x)^{3/2}}{f^2 \left (8+6 m+m^2\right ) \sqrt {-1+c x} \sqrt {1+c x} \left (1-c^2 x^2\right )}+\frac {b c^3 (f x)^{4+m} (\text {d1}+c \text {d1} x)^{3/2} (\text {d2}-c \text {d2} x)^{3/2}}{f^4 (4+m)^2 \sqrt {-1+c x} \sqrt {1+c x} \left (1-c^2 x^2\right )}+\frac {(f x)^{1+m} (\text {d1}+c \text {d1} x)^{3/2} (\text {d2}-c \text {d2} x)^{3/2} (a+b \text {arccosh}(c x))}{f (4+m)}+\frac {3 (f x)^{1+m} (\text {d1}+c \text {d1} x)^{3/2} (\text {d2}-c \text {d2} x)^{3/2} (a+b \text {arccosh}(c x))}{f \left (8+6 m+m^2\right ) \left (1-c^2 x^2\right )}+\frac {3 (f x)^{1+m} (\text {d1}+c \text {d1} x)^{3/2} (\text {d2}-c \text {d2} x)^{3/2} (a+b \text {arccosh}(c x)) \operatorname {Hypergeometric2F1}\left (\frac {1}{2},\frac {1+m}{2},\frac {3+m}{2},c^2 x^2\right )}{f \left (8+14 m+7 m^2+m^3\right ) \left (1-c^2 x^2\right )^{3/2}}+\frac {3 b c (f x)^{2+m} \sqrt {-1+c x} \sqrt {1+c x} (\text {d1}+c \text {d1} x)^{3/2} (\text {d2}-c \text {d2} x)^{3/2} \, _3F_2\left (1,1+\frac {m}{2},1+\frac {m}{2};\frac {3}{2}+\frac {m}{2},2+\frac {m}{2};c^2 x^2\right )}{f^2 (1+m) (2+m)^2 (4+m) \left (1-c^2 x^2\right )^2} \] Output:

-3*b*c*(f*x)^(2+m)*(c*d1*x+d1)^(3/2)*(-c*d2*x+d2)^(3/2)/f^2/(2+m)^2/(4+m)/ 
(c*x-1)^(1/2)/(c*x+1)^(1/2)/(-c^2*x^2+1)-b*c*(f*x)^(2+m)*(c*d1*x+d1)^(3/2) 
*(-c*d2*x+d2)^(3/2)/f^2/(m^2+6*m+8)/(c*x-1)^(1/2)/(c*x+1)^(1/2)/(-c^2*x^2+ 
1)+b*c^3*(f*x)^(4+m)*(c*d1*x+d1)^(3/2)*(-c*d2*x+d2)^(3/2)/f^4/(4+m)^2/(c*x 
-1)^(1/2)/(c*x+1)^(1/2)/(-c^2*x^2+1)+(f*x)^(1+m)*(c*d1*x+d1)^(3/2)*(-c*d2* 
x+d2)^(3/2)*(a+b*arccosh(c*x))/f/(4+m)+3*(f*x)^(1+m)*(c*d1*x+d1)^(3/2)*(-c 
*d2*x+d2)^(3/2)*(a+b*arccosh(c*x))/f/(m^2+6*m+8)/(-c^2*x^2+1)+3*(f*x)^(1+m 
)*(c*d1*x+d1)^(3/2)*(-c*d2*x+d2)^(3/2)*(a+b*arccosh(c*x))*hypergeom([1/2, 
1/2+1/2*m],[3/2+1/2*m],c^2*x^2)/f/(m^3+7*m^2+14*m+8)/(-c^2*x^2+1)^(3/2)+3* 
b*c*(f*x)^(2+m)*(c*x-1)^(1/2)*(c*x+1)^(1/2)*(c*d1*x+d1)^(3/2)*(-c*d2*x+d2) 
^(3/2)*hypergeom([1, 1+1/2*m, 1+1/2*m],[2+1/2*m, 3/2+1/2*m],c^2*x^2)/f^2/( 
1+m)/(2+m)^2/(4+m)/(-c^2*x^2+1)^2
 

Mathematica [A] (verified)

Time = 0.64 (sec) , antiderivative size = 288, normalized size of antiderivative = 0.53 \[ \int (f x)^m (\text {d1}+c \text {d1} x)^{3/2} (\text {d2}-c \text {d2} x)^{3/2} (a+b \text {arccosh}(c x)) \, dx=\frac {\text {d1} \text {d2} x (f x)^m \sqrt {\text {d1}+c \text {d1} x} \sqrt {\text {d2}-c \text {d2} x} \left (-\frac {3 b c x}{(2+m)^2 \sqrt {-1+c x} \sqrt {1+c x}}+\frac {b c x \left (-\frac {1}{2+m}+\frac {c^2 x^2}{4+m}\right )}{\sqrt {-1+c x} \sqrt {1+c x}}+\frac {3 (a+b \text {arccosh}(c x))}{2+m}-(-1+c x) (1+c x) (a+b \text {arccosh}(c x))-\frac {3 \sqrt {1-c^2 x^2} (a+b \text {arccosh}(c x)) \operatorname {Hypergeometric2F1}\left (\frac {1}{2},\frac {1+m}{2},\frac {3+m}{2},c^2 x^2\right )}{(1+m) (2+m) (-1+c x) (1+c x)}-\frac {3 b c x \, _3F_2\left (1,1+\frac {m}{2},1+\frac {m}{2};\frac {3}{2}+\frac {m}{2},2+\frac {m}{2};c^2 x^2\right )}{(1+m) (2+m)^2 \sqrt {-1+c x} \sqrt {1+c x}}\right )}{4+m} \] Input:

Integrate[(f*x)^m*(d1 + c*d1*x)^(3/2)*(d2 - c*d2*x)^(3/2)*(a + b*ArcCosh[c 
*x]),x]
 

Output:

(d1*d2*x*(f*x)^m*Sqrt[d1 + c*d1*x]*Sqrt[d2 - c*d2*x]*((-3*b*c*x)/((2 + m)^ 
2*Sqrt[-1 + c*x]*Sqrt[1 + c*x]) + (b*c*x*(-(2 + m)^(-1) + (c^2*x^2)/(4 + m 
)))/(Sqrt[-1 + c*x]*Sqrt[1 + c*x]) + (3*(a + b*ArcCosh[c*x]))/(2 + m) - (- 
1 + c*x)*(1 + c*x)*(a + b*ArcCosh[c*x]) - (3*Sqrt[1 - c^2*x^2]*(a + b*ArcC 
osh[c*x])*Hypergeometric2F1[1/2, (1 + m)/2, (3 + m)/2, c^2*x^2])/((1 + m)* 
(2 + m)*(-1 + c*x)*(1 + c*x)) - (3*b*c*x*HypergeometricPFQ[{1, 1 + m/2, 1 
+ m/2}, {3/2 + m/2, 2 + m/2}, c^2*x^2])/((1 + m)*(2 + m)^2*Sqrt[-1 + c*x]* 
Sqrt[1 + c*x])))/(4 + m)
 

Rubi [A] (warning: unable to verify)

Time = 1.59 (sec) , antiderivative size = 428, normalized size of antiderivative = 0.78, number of steps used = 8, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.229, Rules used = {6346, 25, 82, 244, 2009, 6342, 17, 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.

\(\displaystyle \int (c \text {d1} x+\text {d1})^{3/2} (\text {d2}-c \text {d2} x)^{3/2} (f x)^m (a+b \text {arccosh}(c x)) \, dx\)

\(\Big \downarrow \) 6346

\(\displaystyle \frac {3 \text {d1} \text {d2} \int (f x)^m \sqrt {c x \text {d1}+\text {d1}} \sqrt {\text {d2}-c \text {d2} x} (a+b \text {arccosh}(c x))dx}{m+4}+\frac {b c \text {d1} \text {d2} \sqrt {c \text {d1} x+\text {d1}} \sqrt {\text {d2}-c \text {d2} x} \int -(f x)^{m+1} (1-c x) (c x+1)dx}{f (m+4) \sqrt {c x-1} \sqrt {c x+1}}+\frac {(c \text {d1} x+\text {d1})^{3/2} (\text {d2}-c \text {d2} x)^{3/2} (f x)^{m+1} (a+b \text {arccosh}(c x))}{f (m+4)}\)

\(\Big \downarrow \) 25

\(\displaystyle \frac {3 \text {d1} \text {d2} \int (f x)^m \sqrt {c x \text {d1}+\text {d1}} \sqrt {\text {d2}-c \text {d2} x} (a+b \text {arccosh}(c x))dx}{m+4}-\frac {b c \text {d1} \text {d2} \sqrt {c \text {d1} x+\text {d1}} \sqrt {\text {d2}-c \text {d2} x} \int (f x)^{m+1} (1-c x) (c x+1)dx}{f (m+4) \sqrt {c x-1} \sqrt {c x+1}}+\frac {(c \text {d1} x+\text {d1})^{3/2} (\text {d2}-c \text {d2} x)^{3/2} (f x)^{m+1} (a+b \text {arccosh}(c x))}{f (m+4)}\)

\(\Big \downarrow \) 82

\(\displaystyle \frac {3 \text {d1} \text {d2} \int (f x)^m \sqrt {c x \text {d1}+\text {d1}} \sqrt {\text {d2}-c \text {d2} x} (a+b \text {arccosh}(c x))dx}{m+4}-\frac {b c \text {d1} \text {d2} \sqrt {c \text {d1} x+\text {d1}} \sqrt {\text {d2}-c \text {d2} x} \int (f x)^{m+1} \left (1-c^2 x^2\right )dx}{f (m+4) \sqrt {c x-1} \sqrt {c x+1}}+\frac {(c \text {d1} x+\text {d1})^{3/2} (\text {d2}-c \text {d2} x)^{3/2} (f x)^{m+1} (a+b \text {arccosh}(c x))}{f (m+4)}\)

\(\Big \downarrow \) 244

\(\displaystyle \frac {3 \text {d1} \text {d2} \int (f x)^m \sqrt {c x \text {d1}+\text {d1}} \sqrt {\text {d2}-c \text {d2} x} (a+b \text {arccosh}(c x))dx}{m+4}-\frac {b c \text {d1} \text {d2} \sqrt {c \text {d1} x+\text {d1}} \sqrt {\text {d2}-c \text {d2} x} \int \left ((f x)^{m+1}-\frac {c^2 (f x)^{m+3}}{f^2}\right )dx}{f (m+4) \sqrt {c x-1} \sqrt {c x+1}}+\frac {(c \text {d1} x+\text {d1})^{3/2} (\text {d2}-c \text {d2} x)^{3/2} (f x)^{m+1} (a+b \text {arccosh}(c x))}{f (m+4)}\)

\(\Big \downarrow \) 2009

\(\displaystyle \frac {3 \text {d1} \text {d2} \int (f x)^m \sqrt {c x \text {d1}+\text {d1}} \sqrt {\text {d2}-c \text {d2} x} (a+b \text {arccosh}(c x))dx}{m+4}+\frac {(c \text {d1} x+\text {d1})^{3/2} (\text {d2}-c \text {d2} x)^{3/2} (f x)^{m+1} (a+b \text {arccosh}(c x))}{f (m+4)}-\frac {b c \text {d1} \text {d2} \sqrt {c \text {d1} x+\text {d1}} \sqrt {\text {d2}-c \text {d2} x} \left (\frac {(f x)^{m+2}}{f (m+2)}-\frac {c^2 (f x)^{m+4}}{f^3 (m+4)}\right )}{f (m+4) \sqrt {c x-1} \sqrt {c x+1}}\)

\(\Big \downarrow \) 6342

\(\displaystyle \frac {3 \text {d1} \text {d2} \left (-\frac {\sqrt {c \text {d1} x+\text {d1}} \sqrt {\text {d2}-c \text {d2} x} \int \frac {(f x)^m (a+b \text {arccosh}(c x))}{\sqrt {c x-1} \sqrt {c x+1}}dx}{(m+2) \sqrt {c x-1} \sqrt {c x+1}}-\frac {b c \sqrt {c \text {d1} x+\text {d1}} \sqrt {\text {d2}-c \text {d2} x} \int (f x)^{m+1}dx}{f (m+2) \sqrt {c x-1} \sqrt {c x+1}}+\frac {\sqrt {c \text {d1} x+\text {d1}} \sqrt {\text {d2}-c \text {d2} x} (f x)^{m+1} (a+b \text {arccosh}(c x))}{f (m+2)}\right )}{m+4}+\frac {(c \text {d1} x+\text {d1})^{3/2} (\text {d2}-c \text {d2} x)^{3/2} (f x)^{m+1} (a+b \text {arccosh}(c x))}{f (m+4)}-\frac {b c \text {d1} \text {d2} \sqrt {c \text {d1} x+\text {d1}} \sqrt {\text {d2}-c \text {d2} x} \left (\frac {(f x)^{m+2}}{f (m+2)}-\frac {c^2 (f x)^{m+4}}{f^3 (m+4)}\right )}{f (m+4) \sqrt {c x-1} \sqrt {c x+1}}\)

\(\Big \downarrow \) 17

\(\displaystyle \frac {3 \text {d1} \text {d2} \left (-\frac {\sqrt {c \text {d1} x+\text {d1}} \sqrt {\text {d2}-c \text {d2} x} \int \frac {(f x)^m (a+b \text {arccosh}(c x))}{\sqrt {c x-1} \sqrt {c x+1}}dx}{(m+2) \sqrt {c x-1} \sqrt {c x+1}}+\frac {\sqrt {c \text {d1} x+\text {d1}} \sqrt {\text {d2}-c \text {d2} x} (f x)^{m+1} (a+b \text {arccosh}(c x))}{f (m+2)}-\frac {b c \sqrt {c \text {d1} x+\text {d1}} \sqrt {\text {d2}-c \text {d2} x} (f x)^{m+2}}{f^2 (m+2)^2 \sqrt {c x-1} \sqrt {c x+1}}\right )}{m+4}+\frac {(c \text {d1} x+\text {d1})^{3/2} (\text {d2}-c \text {d2} x)^{3/2} (f x)^{m+1} (a+b \text {arccosh}(c x))}{f (m+4)}-\frac {b c \text {d1} \text {d2} \sqrt {c \text {d1} x+\text {d1}} \sqrt {\text {d2}-c \text {d2} x} \left (\frac {(f x)^{m+2}}{f (m+2)}-\frac {c^2 (f x)^{m+4}}{f^3 (m+4)}\right )}{f (m+4) \sqrt {c x-1} \sqrt {c x+1}}\)

\(\Big \downarrow \) 6364

\(\displaystyle \frac {3 \text {d1} \text {d2} \left (-\frac {\sqrt {c \text {d1} x+\text {d1}} \sqrt {\text {d2}-c \text {d2} x} \left (\frac {b c (f x)^{m+2} \, _3F_2\left (1,\frac {m}{2}+1,\frac {m}{2}+1;\frac {m}{2}+\frac {3}{2},\frac {m}{2}+2;c^2 x^2\right )}{f^2 (m+1) (m+2)}+\frac {\sqrt {1-c x} (f x)^{m+1} \operatorname {Hypergeometric2F1}\left (\frac {1}{2},\frac {m+1}{2},\frac {m+3}{2},c^2 x^2\right ) (a+b \text {arccosh}(c x))}{f (m+1) \sqrt {c x-1}}\right )}{(m+2) \sqrt {c x-1} \sqrt {c x+1}}+\frac {\sqrt {c \text {d1} x+\text {d1}} \sqrt {\text {d2}-c \text {d2} x} (f x)^{m+1} (a+b \text {arccosh}(c x))}{f (m+2)}-\frac {b c \sqrt {c \text {d1} x+\text {d1}} \sqrt {\text {d2}-c \text {d2} x} (f x)^{m+2}}{f^2 (m+2)^2 \sqrt {c x-1} \sqrt {c x+1}}\right )}{m+4}+\frac {(c \text {d1} x+\text {d1})^{3/2} (\text {d2}-c \text {d2} x)^{3/2} (f x)^{m+1} (a+b \text {arccosh}(c x))}{f (m+4)}-\frac {b c \text {d1} \text {d2} \sqrt {c \text {d1} x+\text {d1}} \sqrt {\text {d2}-c \text {d2} x} \left (\frac {(f x)^{m+2}}{f (m+2)}-\frac {c^2 (f x)^{m+4}}{f^3 (m+4)}\right )}{f (m+4) \sqrt {c x-1} \sqrt {c x+1}}\)

Input:

Int[(f*x)^m*(d1 + c*d1*x)^(3/2)*(d2 - c*d2*x)^(3/2)*(a + b*ArcCosh[c*x]),x 
]
 

Output:

-((b*c*d1*d2*Sqrt[d1 + c*d1*x]*Sqrt[d2 - c*d2*x]*((f*x)^(2 + m)/(f*(2 + m) 
) - (c^2*(f*x)^(4 + m))/(f^3*(4 + m))))/(f*(4 + m)*Sqrt[-1 + c*x]*Sqrt[1 + 
 c*x])) + ((f*x)^(1 + m)*(d1 + c*d1*x)^(3/2)*(d2 - c*d2*x)^(3/2)*(a + b*Ar 
cCosh[c*x]))/(f*(4 + m)) + (3*d1*d2*(-((b*c*(f*x)^(2 + m)*Sqrt[d1 + c*d1*x 
]*Sqrt[d2 - c*d2*x])/(f^2*(2 + m)^2*Sqrt[-1 + c*x]*Sqrt[1 + c*x])) + ((f*x 
)^(1 + m)*Sqrt[d1 + c*d1*x]*Sqrt[d2 - c*d2*x]*(a + b*ArcCosh[c*x]))/(f*(2 
+ m)) - (Sqrt[d1 + c*d1*x]*Sqrt[d2 - c*d2*x]*(((f*x)^(1 + m)*Sqrt[1 - c*x] 
*(a + b*ArcCosh[c*x])*Hypergeometric2F1[1/2, (1 + m)/2, (3 + m)/2, c^2*x^2 
])/(f*(1 + m)*Sqrt[-1 + c*x]) + (b*c*(f*x)^(2 + m)*HypergeometricPFQ[{1, 1 
 + m/2, 1 + m/2}, {3/2 + m/2, 2 + m/2}, c^2*x^2])/(f^2*(1 + m)*(2 + m))))/ 
((2 + m)*Sqrt[-1 + c*x]*Sqrt[1 + c*x])))/(4 + m)
 

Defintions of rubi rules used

rule 17

Int[(c_.)*((a_.) + (b_.)*(x_))^(m_.), x_Symbol] :> Simp[c*((a + b*x)^(m + 1 
)/(b*(m + 1))), x] /; FreeQ[{a, b, c, m}, x] && NeQ[m, -1]
 

rule 25

Int[-(Fx_), x_Symbol] :> Simp[Identity[-1]   Int[Fx, x], x]
 

rule 82

Int[((a_) + (b_.)*(x_))^(m_.)*((c_) + (d_.)*(x_))^(n_.)*((e_.) + (f_.)*(x_) 
)^(p_.), x_] :> Int[(a*c + b*d*x^2)^m*(e + f*x)^p, x] /; FreeQ[{a, b, c, d, 
 e, f, m, n, p}, x] && EqQ[b*c + a*d, 0] && EqQ[n, m] && IntegerQ[m]
 

rule 244

Int[((c_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^2)^(p_.), x_Symbol] :> Int[Expand 
Integrand[(c*x)^m*(a + b*x^2)^p, x], x] /; FreeQ[{a, b, c, m}, x] && IGtQ[p 
, 0]
 

rule 2009

Int[u_, x_Symbol] :> Simp[IntSum[u, x], x] /; SumQ[u]
 

rule 6342

Int[((a_.) + ArcCosh[(c_.)*(x_)]*(b_.))^(n_.)*((f_.)*(x_))^(m_)*Sqrt[(d1_) 
+ (e1_.)*(x_)]*Sqrt[(d2_) + (e2_.)*(x_)], x_Symbol] :> Simp[(f*x)^(m + 1)*S 
qrt[d1 + e1*x]*Sqrt[d2 + e2*x]*((a + b*ArcCosh[c*x])^n/(f*(m + 2))), x] + ( 
-Simp[(1/(m + 2))*Simp[Sqrt[d1 + e1*x]/Sqrt[1 + c*x]]*Simp[Sqrt[d2 + e2*x]/ 
Sqrt[-1 + c*x]]   Int[(f*x)^m*((a + b*ArcCosh[c*x])^n/(Sqrt[1 + c*x]*Sqrt[- 
1 + c*x])), x], x] - Simp[b*c*(n/(f*(m + 2)))*Simp[Sqrt[d1 + e1*x]/Sqrt[1 + 
 c*x]]*Simp[Sqrt[d2 + e2*x]/Sqrt[-1 + c*x]]   Int[(f*x)^(m + 1)*(a + b*ArcC 
osh[c*x])^(n - 1), x], x]) /; FreeQ[{a, b, c, d1, e1, d2, e2, f, m}, x] && 
EqQ[e1, c*d1] && EqQ[e2, (-c)*d2] && IGtQ[n, 0] && (IGtQ[m, -2] || EqQ[n, 1 
])
 

rule 6346

Int[((a_.) + ArcCosh[(c_.)*(x_)]*(b_.))^(n_.)*((f_.)*(x_))^(m_)*((d1_) + (e 
1_.)*(x_))^(p_)*((d2_) + (e2_.)*(x_))^(p_), x_Symbol] :> Simp[(f*x)^(m + 1) 
*(d1 + e1*x)^p*(d2 + e2*x)^p*((a + b*ArcCosh[c*x])^n/(f*(m + 2*p + 1))), x] 
 + (Simp[2*d1*d2*(p/(m + 2*p + 1))   Int[(f*x)^m*(d1 + e1*x)^(p - 1)*(d2 + 
e2*x)^(p - 1)*(a + b*ArcCosh[c*x])^n, x], x] - Simp[b*c*(n/(f*(m + 2*p + 1) 
))*Simp[(d1 + e1*x)^p/(1 + c*x)^p]*Simp[(d2 + e2*x)^p/(-1 + c*x)^p]   Int[( 
f*x)^(m + 1)*(1 + c*x)^(p - 1/2)*(-1 + c*x)^(p - 1/2)*(a + b*ArcCosh[c*x])^ 
(n - 1), x], x]) /; FreeQ[{a, b, c, d1, e1, d2, e2, f, m}, x] && EqQ[e1, c* 
d1] && EqQ[e2, (-c)*d2] && GtQ[n, 0] && GtQ[p, 0] &&  !LtQ[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]
 

Maple [F]

\[\int \left (f x \right )^{m} \left (c \operatorname {d1} x +\operatorname {d1} \right )^{\frac {3}{2}} \left (-c \operatorname {d2} x +\operatorname {d2} \right )^{\frac {3}{2}} \left (a +b \,\operatorname {arccosh}\left (c x \right )\right )d x\]

Input:

int((f*x)^m*(c*d1*x+d1)^(3/2)*(-c*d2*x+d2)^(3/2)*(a+b*arccosh(c*x)),x)
 

Output:

int((f*x)^m*(c*d1*x+d1)^(3/2)*(-c*d2*x+d2)^(3/2)*(a+b*arccosh(c*x)),x)
 

Fricas [F]

\[ \int (f x)^m (\text {d1}+c \text {d1} x)^{3/2} (\text {d2}-c \text {d2} x)^{3/2} (a+b \text {arccosh}(c x)) \, dx=\int { {\left (c d_{1} x + d_{1}\right )}^{\frac {3}{2}} {\left (-c d_{2} x + d_{2}\right )}^{\frac {3}{2}} {\left (b \operatorname {arcosh}\left (c x\right ) + a\right )} \left (f x\right )^{m} \,d x } \] Input:

integrate((f*x)^m*(c*d1*x+d1)^(3/2)*(-c*d2*x+d2)^(3/2)*(a+b*arccosh(c*x)), 
x, algorithm="fricas")
 

Output:

integral(-(a*c^2*d1*d2*x^2 - a*d1*d2 + (b*c^2*d1*d2*x^2 - b*d1*d2)*arccosh 
(c*x))*sqrt(c*d1*x + d1)*sqrt(-c*d2*x + d2)*(f*x)^m, x)
 

Sympy [F(-1)]

Timed out. \[ \int (f x)^m (\text {d1}+c \text {d1} x)^{3/2} (\text {d2}-c \text {d2} x)^{3/2} (a+b \text {arccosh}(c x)) \, dx=\text {Timed out} \] Input:

integrate((f*x)**m*(c*d1*x+d1)**(3/2)*(-c*d2*x+d2)**(3/2)*(a+b*acosh(c*x)) 
,x)
 

Output:

Timed out
 

Maxima [F]

\[ \int (f x)^m (\text {d1}+c \text {d1} x)^{3/2} (\text {d2}-c \text {d2} x)^{3/2} (a+b \text {arccosh}(c x)) \, dx=\int { {\left (c d_{1} x + d_{1}\right )}^{\frac {3}{2}} {\left (-c d_{2} x + d_{2}\right )}^{\frac {3}{2}} {\left (b \operatorname {arcosh}\left (c x\right ) + a\right )} \left (f x\right )^{m} \,d x } \] Input:

integrate((f*x)^m*(c*d1*x+d1)^(3/2)*(-c*d2*x+d2)^(3/2)*(a+b*arccosh(c*x)), 
x, algorithm="maxima")
 

Output:

integrate((c*d1*x + d1)^(3/2)*(-c*d2*x + d2)^(3/2)*(b*arccosh(c*x) + a)*(f 
*x)^m, x)
                                                                                    
                                                                                    
 

Giac [F]

\[ \int (f x)^m (\text {d1}+c \text {d1} x)^{3/2} (\text {d2}-c \text {d2} x)^{3/2} (a+b \text {arccosh}(c x)) \, dx=\int { {\left (c d_{1} x + d_{1}\right )}^{\frac {3}{2}} {\left (-c d_{2} x + d_{2}\right )}^{\frac {3}{2}} {\left (b \operatorname {arcosh}\left (c x\right ) + a\right )} \left (f x\right )^{m} \,d x } \] Input:

integrate((f*x)^m*(c*d1*x+d1)^(3/2)*(-c*d2*x+d2)^(3/2)*(a+b*arccosh(c*x)), 
x, algorithm="giac")
 

Output:

integrate((c*d1*x + d1)^(3/2)*(-c*d2*x + d2)^(3/2)*(b*arccosh(c*x) + a)*(f 
*x)^m, x)
 

Mupad [F(-1)]

Timed out. \[ \int (f x)^m (\text {d1}+c \text {d1} x)^{3/2} (\text {d2}-c \text {d2} x)^{3/2} (a+b \text {arccosh}(c x)) \, dx=\int \left (a+b\,\mathrm {acosh}\left (c\,x\right )\right )\,{\left (f\,x\right )}^m\,{\left (d_{1}+c\,d_{1}\,x\right )}^{3/2}\,{\left (d_{2}-c\,d_{2}\,x\right )}^{3/2} \,d x \] Input:

int((a + b*acosh(c*x))*(f*x)^m*(d1 + c*d1*x)^(3/2)*(d2 - c*d2*x)^(3/2),x)
 

Output:

int((a + b*acosh(c*x))*(f*x)^m*(d1 + c*d1*x)^(3/2)*(d2 - c*d2*x)^(3/2), x)
 

Reduce [F]

\[ \int (f x)^m (\text {d1}+c \text {d1} x)^{3/2} (\text {d2}-c \text {d2} x)^{3/2} (a+b \text {arccosh}(c x)) \, dx=f^{m} \sqrt {\mathit {d2}}\, \sqrt {\mathit {d1}}\, \mathit {d1} \mathit {d2} \left (-\left (\int x^{m} \sqrt {c x +1}\, \sqrt {-c x +1}\, \mathit {acosh} \left (c x \right ) x^{2}d x \right ) b \,c^{2}+\left (\int x^{m} \sqrt {c x +1}\, \sqrt {-c x +1}\, \mathit {acosh} \left (c x \right )d x \right ) b -\left (\int x^{m} \sqrt {c x +1}\, \sqrt {-c x +1}\, x^{2}d x \right ) a \,c^{2}+\left (\int x^{m} \sqrt {c x +1}\, \sqrt {-c x +1}d x \right ) a \right ) \] Input:

int((f*x)^m*(c*d1*x+d1)^(3/2)*(-c*d2*x+d2)^(3/2)*(a+b*acosh(c*x)),x)
 

Output:

f**m*sqrt(d2)*sqrt(d1)*d1*d2*( - int(x**m*sqrt(c*x + 1)*sqrt( - c*x + 1)*a 
cosh(c*x)*x**2,x)*b*c**2 + int(x**m*sqrt(c*x + 1)*sqrt( - c*x + 1)*acosh(c 
*x),x)*b - int(x**m*sqrt(c*x + 1)*sqrt( - c*x + 1)*x**2,x)*a*c**2 + int(x* 
*m*sqrt(c*x + 1)*sqrt( - c*x + 1),x)*a)