3.2.55 \(\int \frac {(f x)^m (a+b \text {arccosh}(c x))}{(d-c^2 d x^2)^{3/2}} \, dx\) [155]

3.2.55.1 Optimal result

Integrand size = 29, antiderivative size = 300 \[ \int \frac {(f x)^m (a+b \text {arccosh}(c x))}{\left (d-c^2 d x^2\right )^{3/2}} \, dx=\frac {(f x)^{1+m} (a+b \text {arccosh}(c x))}{d f \sqrt {d-c^2 d x^2}}-\frac {m (f x)^{1+m} \sqrt {1-c^2 x^2} (a+b \text {arccosh}(c x)) \operatorname {Hypergeometric2F1}\left (\frac {1}{2},\frac {1+m}{2},\frac {3+m}{2},c^2 x^2\right )}{d f (1+m) \sqrt {d-c^2 d x^2}}+\frac {b c (f x)^{2+m} \sqrt {-1+c x} \sqrt {1+c x} \operatorname {Hypergeometric2F1}\left (1,\frac {2+m}{2},\frac {4+m}{2},c^2 x^2\right )}{d f^2 (2+m) \sqrt {d-c^2 d x^2}}-\frac {b c m (f x)^{2+m} \sqrt {-1+c x} \sqrt {1+c x} \, _3F_2\left (1,1+\frac {m}{2},1+\frac {m}{2};\frac {3}{2}+\frac {m}{2},2+\frac {m}{2};c^2 x^2\right )}{d f^2 (1+m) (2+m) \sqrt {d-c^2 d x^2}} \]

output

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

3.2.55.2 Mathematica [A] (verified)

Time = 0.18 (sec) , antiderivative size = 216, normalized size of antiderivative = 0.72 \[ \int \frac {(f x)^m (a+b \text {arccosh}(c x))}{\left (d-c^2 d x^2\right )^{3/2}} \, dx=\frac {x (f x)^m \left (-m (2+m) \sqrt {1-c^2 x^2} (a+b \text {arccosh}(c x)) \operatorname {Hypergeometric2F1}\left (\frac {1}{2},\frac {1+m}{2},\frac {3+m}{2},c^2 x^2\right )+(1+m) \left ((2+m) (a+b \text {arccosh}(c x))+b c x \sqrt {-1+c x} \sqrt {1+c x} \operatorname {Hypergeometric2F1}\left (1,1+\frac {m}{2},2+\frac {m}{2},c^2 x^2\right )\right )-b c m x \sqrt {-1+c x} \sqrt {1+c x} \, _3F_2\left (1,1+\frac {m}{2},1+\frac {m}{2};\frac {3}{2}+\frac {m}{2},2+\frac {m}{2};c^2 x^2\right )\right )}{d (1+m) (2+m) \sqrt {d-c^2 d x^2}} \]

input

Integrate[((f*x)^m*(a + b*ArcCosh[c*x]))/(d - c^2*d*x^2)^(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]))/(d*(1 + m)*(2 
 + m)*Sqrt[d - c^2*d*x^2])
 

3.2.55.3 Rubi [A] (verified)

Time = 0.64 (sec) , antiderivative size = 297, normalized size of antiderivative = 0.99, number of steps used = 5, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.172, Rules used = {6351, 25, 82, 278, 6363}

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.

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

\(\Big \downarrow \) 6351

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

\(\Big \downarrow \) 25

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

\(\Big \downarrow \) 82

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

\(\Big \downarrow \) 278

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

\(\Big \downarrow \) 6363

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

input

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

output

((f*x)^(1 + m)*(a + b*ArcCosh[c*x]))/(d*f*Sqrt[d - c^2*d*x^2]) + (b*c*(f*x 
)^(2 + m)*Sqrt[-1 + c*x]*Sqrt[1 + c*x]*Hypergeometric2F1[1, (2 + m)/2, (4 
+ m)/2, c^2*x^2])/(d*f^2*(2 + m)*Sqrt[d - c^2*d*x^2]) - (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)*Sqrt[d - c^2*d*x^2]) + (b*c*(f*x)^(2 + m)* 
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])/(f^2*(1 + m)*(2 + m)*Sqrt[d - c^2*d*x^2])))/d
 

3.2.55.3.1 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 6351

Int[((a_.) + ArcCosh[(c_.)*(x_)]*(b_.))^(n_.)*((f_.)*(x_))^(m_)*((d_) + (e_ 
.)*(x_)^2)^(p_), x_Symbol] :> Simp[(-(f*x)^(m + 1))*(d + e*x^2)^(p + 1)*((a 
 + b*ArcCosh[c*x])^n/(2*d*f*(p + 1))), x] + (Simp[(m + 2*p + 3)/(2*d*(p + 1 
))   Int[(f*x)^m*(d + e*x^2)^(p + 1)*(a + b*ArcCosh[c*x])^n, x], x] - Simp[ 
b*c*(n/(2*f*(p + 1)))*Simp[(d + e*x^2)^p/((1 + c*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, d, e, f, m}, x] && EqQ[c^2*d + e, 0] & 
& GtQ[n, 0] && LtQ[p, -1] &&  !GtQ[m, 1] && (IntegerQ[m] || IntegerQ[p] || 
EqQ[n, 1])
 

rule 6363

Int[(((a_.) + ArcCosh[(c_.)*(x_)]*(b_.))*((f_.)*(x_))^(m_))/Sqrt[(d_) + (e_ 
.)*(x_)^2], x_Symbol] :> Simp[((f*x)^(m + 1)/(f*(m + 1)))*Simp[Sqrt[1 - c^2 
*x^2]/Sqrt[d + e*x^2]]*(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[-1 + c*x]/Sqrt[d + e*x^2])]*HypergeometricPFQ[{1, 
 1 + m/2, 1 + m/2}, {3/2 + m/2, 2 + m/2}, c^2*x^2], x] /; FreeQ[{a, b, c, d 
, e, f, m}, x] && EqQ[c^2*d + e, 0] &&  !IntegerQ[m]
 

3.2.55.4 Maple [F]

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

input

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

output

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

3.2.55.5 Fricas [F]

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

input

integrate((f*x)^m*(a+b*arccosh(c*x))/(-c^2*d*x^2+d)^(3/2),x, algorithm="fr 
icas")
 

output

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

3.2.55.6 Sympy [F]

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

input

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

output

Integral((f*x)**m*(a + b*acosh(c*x))/(-d*(c*x - 1)*(c*x + 1))**(3/2), x)
 

3.2.55.7 Maxima [F]

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

input

integrate((f*x)^m*(a+b*arccosh(c*x))/(-c^2*d*x^2+d)^(3/2),x, algorithm="ma 
xima")
 

output

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

3.2.55.8 Giac [F]

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

input

integrate((f*x)^m*(a+b*arccosh(c*x))/(-c^2*d*x^2+d)^(3/2),x, algorithm="gi 
ac")
 

output

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

3.2.55.9 Mupad [F(-1)]

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

input

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

output

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