\(\int \frac {(f x)^m (a+b \text {arccosh}(c x))}{(\text {d1}+c \text {d1} x)^{3/2} (\text {d2}-c \text {d2} x)^{3/2}} \, dx\) [156]

Optimal result

Integrand size = 35, antiderivative size = 342 \[ \int \frac {(f x)^m (a+b \text {arccosh}(c x))}{(\text {d1}+c \text {d1} x)^{3/2} (\text {d2}-c \text {d2} x)^{3/2}} \, dx=\frac {(f x)^{1+m} \left (1-c^2 x^2\right ) (a+b \text {arccosh}(c x))}{f (\text {d1}+c \text {d1} x)^{3/2} (\text {d2}-c \text {d2} x)^{3/2}}-\frac {m (f x)^{1+m} \left (1-c^2 x^2\right )^{3/2} (a+b \text {arccosh}(c x)) \operatorname {Hypergeometric2F1}\left (\frac {1}{2},\frac {1+m}{2},\frac {3+m}{2},c^2 x^2\right )}{f (1+m) (\text {d1}+c \text {d1} x)^{3/2} (\text {d2}-c \text {d2} x)^{3/2}}+\frac {b c (f x)^{2+m} \sqrt {-1+c x} \sqrt {1+c x} \left (1-c^2 x^2\right ) \operatorname {Hypergeometric2F1}\left (1,\frac {2+m}{2},\frac {4+m}{2},c^2 x^2\right )}{f^2 (2+m) (\text {d1}+c \text {d1} x)^{3/2} (\text {d2}-c \text {d2} x)^{3/2}}-\frac {b c m (f x)^{2+m} \sqrt {-1+c x} \sqrt {1+c x} \left (1-c^2 x^2\right ) \, _3F_2\left (1,1+\frac {m}{2},1+\frac {m}{2};\frac {3}{2}+\frac {m}{2},2+\frac {m}{2};c^2 x^2\right )}{f^2 (1+m) (2+m) (\text {d1}+c \text {d1} x)^{3/2} (\text {d2}-c \text {d2} x)^{3/2}} \] Output:

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

Mathematica [A] (verified)

Time = 0.29 (sec) , antiderivative size = 225, normalized size of antiderivative = 0.66 \[ \int \frac {(f x)^m (a+b \text {arccosh}(c x))}{(\text {d1}+c \text {d1} x)^{3/2} (\text {d2}-c \text {d2} x)^{3/2}} \, dx=\frac {x (f x)^m \left (-m (2+m) \sqrt {1-c^2 x^2} (a+b \text {arccosh}(c x)) \operatorname {Hypergeometric2F1}\left (\frac {1}{2},\frac {1+m}{2},\frac {3+m}{2},c^2 x^2\right )+(1+m) \left ((2+m) (a+b \text {arccosh}(c x))+b c x \sqrt {-1+c x} \sqrt {1+c x} \operatorname {Hypergeometric2F1}\left (1,1+\frac {m}{2},2+\frac {m}{2},c^2 x^2\right )\right )-b c m x \sqrt {-1+c x} \sqrt {1+c x} \, _3F_2\left (1,1+\frac {m}{2},1+\frac {m}{2};\frac {3}{2}+\frac {m}{2},2+\frac {m}{2};c^2 x^2\right )\right )}{\text {d1} \text {d2} (1+m) (2+m) \sqrt {\text {d1}+c \text {d1} x} \sqrt {\text {d2}-c \text {d2} x}} \] Input:

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

Output:

(x*(f*x)^m*(-(m*(2 + m)*Sqrt[1 - c^2*x^2]*(a + b*ArcCosh[c*x])*Hypergeomet 
ric2F1[1/2, (1 + m)/2, (3 + m)/2, c^2*x^2]) + (1 + m)*((2 + m)*(a + b*ArcC 
osh[c*x]) + b*c*x*Sqrt[-1 + c*x]*Sqrt[1 + c*x]*Hypergeometric2F1[1, 1 + m/ 
2, 2 + m/2, c^2*x^2]) - b*c*m*x*Sqrt[-1 + c*x]*Sqrt[1 + c*x]*Hypergeometri 
cPFQ[{1, 1 + m/2, 1 + m/2}, {3/2 + m/2, 2 + m/2}, c^2*x^2]))/(d1*d2*(1 + m 
)*(2 + m)*Sqrt[d1 + c*d1*x]*Sqrt[d2 - c*d2*x])
 

Rubi [A] (verified)

Time = 1.57 (sec) , antiderivative size = 330, normalized size of antiderivative = 0.96, number of steps used = 5, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.143, Rules used = {6352, 25, 82, 278, 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.

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

\(\Big \downarrow \) 6352

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

\(\Big \downarrow \) 25

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

\(\Big \downarrow \) 82

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

\(\Big \downarrow \) 278

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

\(\Big \downarrow \) 6364

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

Input:

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

Output:

((f*x)^(1 + m)*(a + b*ArcCosh[c*x]))/(d1*d2*f*Sqrt[d1 + c*d1*x]*Sqrt[d2 - 
c*d2*x]) + (b*c*(f*x)^(2 + m)*Sqrt[-1 + c*x]*Sqrt[1 + c*x]*Hypergeometric2 
F1[1, (2 + m)/2, (4 + m)/2, c^2*x^2])/(d1*d2*f^2*(2 + m)*Sqrt[d1 + c*d1*x] 
*Sqrt[d2 - c*d2*x]) - (m*(((f*x)^(1 + m)*Sqrt[1 - c^2*x^2]*(a + b*ArcCosh[ 
c*x])*Hypergeometric2F1[1/2, (1 + m)/2, (3 + m)/2, c^2*x^2])/(f*(1 + m)*Sq 
rt[d1 + c*d1*x]*Sqrt[d2 - c*d2*x]) + (b*c*(f*x)^(2 + m)*Sqrt[-1 + c*x]*Sqr 
t[1 + c*x]*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)*Sqrt[d1 + c*d1*x]*Sqrt[d2 - c*d2*x])))/(d1* 
d2)
 

Defintions of rubi rules used

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 278

Int[((c_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> Simp[a^p*(( 
c*x)^(m + 1)/(c*(m + 1)))*Hypergeometric2F1[-p, (m + 1)/2, (m + 1)/2 + 1, ( 
-b)*(x^2/a)], x] /; FreeQ[{a, b, c, m, p}, x] &&  !IGtQ[p, 0] && (ILtQ[p, 0 
] || GtQ[a, 0])
 

rule 6352

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 + 1)*(d2 + e2*x)^(p + 1)*((a + b*ArcCosh[c*x])^n/(2*d1*d 
2*f*(p + 1))), x] + (Simp[(m + 2*p + 3)/(2*d1*d2*(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/(2*f*(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] && LtQ[p, -1] &&  ! 
GtQ[m, 1] && (IntegerQ[m] || 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]

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

Input:

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

Output:

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

Fricas [F]

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

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

Output:

integral(sqrt(c*d1*x + d1)*sqrt(-c*d2*x + d2)*(b*arccosh(c*x) + a)*(f*x)^m 
/(c^4*d1^2*d2^2*x^4 - 2*c^2*d1^2*d2^2*x^2 + d1^2*d2^2), x)
 

Sympy [F(-1)]

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

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

Output:

Timed out
 

Maxima [F]

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

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

Output:

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

Giac [F]

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

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

Output:

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

Mupad [F(-1)]

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

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

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

Output:

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