3.6.21 \(\int (f x)^m (d+e x^2) (a+b \text {arccosh}(c x)) \, dx\) [521]

3.6.21.1 Optimal result

Integrand size = 21, antiderivative size = 198 \[ \int (f x)^m \left (d+e x^2\right ) (a+b \text {arccosh}(c x)) \, dx=-\frac {b e (f x)^{2+m} \sqrt {-1+c x} \sqrt {1+c x}}{c f^2 (3+m)^2}+\frac {d (f x)^{1+m} (a+b \text {arccosh}(c x))}{f (1+m)}+\frac {e (f x)^{3+m} (a+b \text {arccosh}(c x))}{f^3 (3+m)}-\frac {b \left (e (1+m) (2+m)+c^2 d (3+m)^2\right ) (f x)^{2+m} \sqrt {1-c^2 x^2} \operatorname {Hypergeometric2F1}\left (\frac {1}{2},\frac {2+m}{2},\frac {4+m}{2},c^2 x^2\right )}{c f^2 (1+m) (2+m) (3+m)^2 \sqrt {-1+c x} \sqrt {1+c x}} \]

output

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

3.6.21.2 Mathematica [A] (verified)

Time = 0.43 (sec) , antiderivative size = 186, normalized size of antiderivative = 0.94 \[ \int (f x)^m \left (d+e x^2\right ) (a+b \text {arccosh}(c x)) \, dx=x (f x)^m \left (-\frac {b c d x \sqrt {1-c^2 x^2} \operatorname {Hypergeometric2F1}\left (\frac {1}{2},\frac {2+m}{2},\frac {4+m}{2},c^2 x^2\right )}{\left (2+3 m+m^2\right ) \sqrt {-1+c x} \sqrt {1+c x}}+\frac {\frac {\left (d (3+m)+e (1+m) x^2\right ) (a+b \text {arccosh}(c x))}{1+m}-\frac {b c e x^3 \sqrt {1-c^2 x^2} \operatorname {Hypergeometric2F1}\left (\frac {1}{2},\frac {4+m}{2},\frac {6+m}{2},c^2 x^2\right )}{(4+m) \sqrt {-1+c x} \sqrt {1+c x}}}{3+m}\right ) \]

input

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

output

x*(f*x)^m*(-((b*c*d*x*Sqrt[1 - c^2*x^2]*Hypergeometric2F1[1/2, (2 + m)/2, 
(4 + m)/2, c^2*x^2])/((2 + 3*m + m^2)*Sqrt[-1 + c*x]*Sqrt[1 + c*x])) + ((( 
d*(3 + m) + e*(1 + m)*x^2)*(a + b*ArcCosh[c*x]))/(1 + m) - (b*c*e*x^3*Sqrt 
[1 - c^2*x^2]*Hypergeometric2F1[1/2, (4 + m)/2, (6 + m)/2, c^2*x^2])/((4 + 
 m)*Sqrt[-1 + c*x]*Sqrt[1 + c*x]))/(3 + m))
 

3.6.21.3 Rubi [A] (verified)

Time = 0.42 (sec) , antiderivative size = 205, normalized size of antiderivative = 1.04, number of steps used = 5, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.238, Rules used = {6371, 960, 136, 279, 278}

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.

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

\(\Big \downarrow \) 6371

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

\(\Big \downarrow \) 960

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

\(\Big \downarrow \) 136

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

\(\Big \downarrow \) 279

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

\(\Big \downarrow \) 278

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

input

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

output

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

3.6.21.3.1 Defintions of rubi rules used

rule 136

Int[((f_.)*(x_))^(p_)*((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), 
x_] :> Simp[(a + b*x)^FracPart[m]*((c + d*x)^FracPart[m]/(a*c + b*d*x^2)^Fr 
acPart[m])   Int[(a*c + b*d*x^2)^m*(f*x)^p, x], x] /; FreeQ[{a, b, c, d, f, 
 m, n, p}, x] && EqQ[b*c + a*d, 0] && EqQ[n, 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 279

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

rule 960

Int[((e_.)*(x_))^(m_.)*((a1_) + (b1_.)*(x_)^(non2_.))^(p_.)*((a2_) + (b2_.) 
*(x_)^(non2_.))^(p_.)*((c_) + (d_.)*(x_)^(n_)), x_Symbol] :> Simp[d*(e*x)^( 
m + 1)*(a1 + b1*x^(n/2))^(p + 1)*((a2 + b2*x^(n/2))^(p + 1)/(b1*b2*e*(m + n 
*(p + 1) + 1))), x] - Simp[(a1*a2*d*(m + 1) - b1*b2*c*(m + n*(p + 1) + 1))/ 
(b1*b2*(m + n*(p + 1) + 1))   Int[(e*x)^m*(a1 + b1*x^(n/2))^p*(a2 + b2*x^(n 
/2))^p, x], x] /; FreeQ[{a1, b1, a2, b2, c, d, e, m, n, p}, x] && EqQ[non2, 
 n/2] && EqQ[a2*b1 + a1*b2, 0] && NeQ[m + n*(p + 1) + 1, 0]
 

rule 6371

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

3.6.21.4 Maple [F]

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

input

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

output

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

3.6.21.5 Fricas [F]

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

input

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

output

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

3.6.21.6 Sympy [F]

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

input

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

output

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

3.6.21.7 Maxima [F]

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

input

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

output

a*e*f^m*x^3*x^m/(m + 3) + (b*e*f^m*(m + 1)*x^3 + b*d*f^m*(m + 3)*x)*x^m*lo 
g(c*x + sqrt(c*x + 1)*sqrt(c*x - 1))/(m^2 + 4*m + 3) + (f*x)^(m + 1)*a*d/( 
f*(m + 1)) + integrate((b*c*e*f^m*(m + 1)*x^3 + b*c*d*f^m*(m + 3)*x)*x^m/( 
(m^2 + 4*m + 3)*c^3*x^3 - (m^2 + 4*m + 3)*c*x + ((m^2 + 4*m + 3)*c^2*x^2 - 
 m^2 - 4*m - 3)*sqrt(c*x + 1)*sqrt(c*x - 1)), x) - integrate((b*c^2*e*f^m* 
(m + 1)*x^4 + b*c^2*d*f^m*(m + 3)*x^2)*x^m/((m^2 + 4*m + 3)*c^2*x^2 - m^2 
- 4*m - 3), x)
 

3.6.21.8 Giac [F]

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

input

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

output

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

3.6.21.9 Mupad [F(-1)]

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

input

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

output

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