\(\int (f x)^m \sqrt {\text {d1}+c \text {d1} x} \sqrt {\text {d2}-c \text {d2} x} (a+b \text {arccosh}(c x)) \, dx\) [154]

Optimal result

Integrand size = 35, antiderivative size = 308 \[ \int (f x)^m \sqrt {\text {d1}+c \text {d1} x} \sqrt {\text {d2}-c \text {d2} x} (a+b \text {arccosh}(c x)) \, dx=-\frac {b c (f x)^{2+m} \sqrt {\text {d1}+c \text {d1} x} \sqrt {\text {d2}-c \text {d2} x}}{f^2 (2+m)^2 \sqrt {-1+c x} \sqrt {1+c x}}+\frac {(f x)^{1+m} \sqrt {\text {d1}+c \text {d1} x} \sqrt {\text {d2}-c \text {d2} x} (a+b \text {arccosh}(c x))}{f (2+m)}+\frac {(f x)^{1+m} \sqrt {\text {d1}+c \text {d1} x} \sqrt {\text {d2}-c \text {d2} x} (a+b \text {arccosh}(c x)) \operatorname {Hypergeometric2F1}\left (\frac {1}{2},\frac {1+m}{2},\frac {3+m}{2},c^2 x^2\right )}{f \left (2+3 m+m^2\right ) \sqrt {1-c^2 x^2}}+\frac {b c (f x)^{2+m} \sqrt {-1+c x} \sqrt {1+c x} \sqrt {\text {d1}+c \text {d1} x} \sqrt {\text {d2}-c \text {d2} x} \, _3F_2\left (1,1+\frac {m}{2},1+\frac {m}{2};\frac {3}{2}+\frac {m}{2},2+\frac {m}{2};c^2 x^2\right )}{f^2 (1+m) (2+m)^2 \left (1-c^2 x^2\right )} \] Output:

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

Mathematica [A] (verified)

Time = 0.14 (sec) , antiderivative size = 229, normalized size of antiderivative = 0.74 \[ \int (f x)^m \sqrt {\text {d1}+c \text {d1} x} \sqrt {\text {d2}-c \text {d2} x} (a+b \text {arccosh}(c x)) \, dx=\frac {x (f x)^m \sqrt {\text {d1}+c \text {d1} x} \sqrt {\text {d2}-c \text {d2} x} \left ((1+m) \left (-b c x \sqrt {-1+c x} \sqrt {1+c x}+a (2+m) \left (-1+c^2 x^2\right )+b (2+m) \left (-1+c^2 x^2\right ) \text {arccosh}(c x)\right )-(2+m) \sqrt {1-c^2 x^2} (a+b \text {arccosh}(c x)) \operatorname {Hypergeometric2F1}\left (\frac {1}{2},\frac {1+m}{2},\frac {3+m}{2},c^2 x^2\right )-b c x \sqrt {-1+c x} \sqrt {1+c x} \, _3F_2\left (1,1+\frac {m}{2},1+\frac {m}{2};\frac {3}{2}+\frac {m}{2},2+\frac {m}{2};c^2 x^2\right )\right )}{(1+m) (2+m)^2 (-1+c x) (1+c x)} \] Input:

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

Output:

(x*(f*x)^m*Sqrt[d1 + c*d1*x]*Sqrt[d2 - c*d2*x]*((1 + m)*(-(b*c*x*Sqrt[-1 + 
 c*x]*Sqrt[1 + c*x]) + a*(2 + m)*(-1 + c^2*x^2) + b*(2 + m)*(-1 + c^2*x^2) 
*ArcCosh[c*x]) - (2 + m)*Sqrt[1 - c^2*x^2]*(a + b*ArcCosh[c*x])*Hypergeome 
tric2F1[1/2, (1 + m)/2, (3 + m)/2, c^2*x^2] - b*c*x*Sqrt[-1 + c*x]*Sqrt[1 
+ 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*(-1 + c*x)*(1 + c*x))
 

Rubi [A] (warning: unable to verify)

Time = 0.93 (sec) , antiderivative size = 283, normalized size of antiderivative = 0.92, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.086, Rules used = {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.

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

\(\Big \downarrow \) 6342

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

\(\Big \downarrow \) 17

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

\(\Big \downarrow \) 6364

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

Input:

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

Output:

-((b*c*(f*x)^(2 + m)*Sqrt[d1 + c*d1*x]*Sqrt[d2 - c*d2*x])/(f^2*(2 + m)^2*S 
qrt[-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])*Hypergeometric2 
F1[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] 
)
 

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 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 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]

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

Input:

int((f*x)^m*(c*d1*x+d1)^(1/2)*(-c*d2*x+d2)^(1/2)*(a+b*arccosh(c*x)),x)
 

Output:

int((f*x)^m*(c*d1*x+d1)^(1/2)*(-c*d2*x+d2)^(1/2)*(a+b*arccosh(c*x)),x)
 

Fricas [F]

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

integrate((f*x)^m*(c*d1*x+d1)^(1/2)*(-c*d2*x+d2)^(1/2)*(a+b*arccosh(c*x)), 
x, algorithm="fricas")
 

Output:

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

Sympy [F(-1)]

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

integrate((f*x)**m*(c*d1*x+d1)**(1/2)*(-c*d2*x+d2)**(1/2)*(a+b*acosh(c*x)) 
,x)
 

Output:

Timed out
 

Maxima [F]

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

integrate((f*x)^m*(c*d1*x+d1)^(1/2)*(-c*d2*x+d2)^(1/2)*(a+b*arccosh(c*x)), 
x, algorithm="maxima")
 

Output:

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

Giac [F]

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

integrate((f*x)^m*(c*d1*x+d1)^(1/2)*(-c*d2*x+d2)^(1/2)*(a+b*arccosh(c*x)), 
x, algorithm="giac")
 

Output:

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

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

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

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

Output:

f**m*sqrt(d2)*sqrt(d1)*(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)*a)