3.6.20 \(\int (f x)^m (d+e x^2)^2 (a+b \text {arccosh}(c x)) \, dx\) [520]

3.6.20.1 Optimal result

Integrand size = 23, antiderivative size = 353 \[ \int (f x)^m \left (d+e x^2\right )^2 (a+b \text {arccosh}(c x)) \, dx=\frac {b e \left (2 c^2 d (5+m)^2+e \left (12+7 m+m^2\right )\right ) (f x)^{2+m} \left (1-c^2 x^2\right )}{c^3 f^2 (3+m)^2 (5+m)^2 \sqrt {-1+c x} \sqrt {1+c x}}+\frac {b e^2 (f x)^{4+m} \left (1-c^2 x^2\right )}{c f^4 (5+m)^2 \sqrt {-1+c x} \sqrt {1+c x}}+\frac {d^2 (f x)^{1+m} (a+b \text {arccosh}(c x))}{f (1+m)}+\frac {2 d e (f x)^{3+m} (a+b \text {arccosh}(c x))}{f^3 (3+m)}+\frac {e^2 (f x)^{5+m} (a+b \text {arccosh}(c x))}{f^5 (5+m)}-\frac {b \left (\frac {c^4 d^2 (3+m) (5+m)}{1+m}+\frac {e (2+m) \left (2 c^2 d (5+m)^2+e \left (12+7 m+m^2\right )\right )}{(3+m) (5+m)}\right ) (f x)^{2+m} \sqrt {1-c^2 x^2} \operatorname {Hypergeometric2F1}\left (\frac {1}{2},\frac {2+m}{2},\frac {4+m}{2},c^2 x^2\right )}{c^3 f^2 (2+m) (3+m) (5+m) \sqrt {-1+c x} \sqrt {1+c x}} \]

output

d^2*(f*x)^(1+m)*(a+b*arccosh(c*x))/f/(1+m)+2*d*e*(f*x)^(3+m)*(a+b*arccosh( 
c*x))/f^3/(3+m)+e^2*(f*x)^(5+m)*(a+b*arccosh(c*x))/f^5/(5+m)+b*e*(2*c^2*d* 
(5+m)^2+e*(m^2+7*m+12))*(f*x)^(2+m)*(-c^2*x^2+1)/c^3/f^2/(3+m)^2/(5+m)^2/( 
c*x-1)^(1/2)/(c*x+1)^(1/2)+b*e^2*(f*x)^(4+m)*(-c^2*x^2+1)/c/f^4/(5+m)^2/(c 
*x-1)^(1/2)/(c*x+1)^(1/2)-b*(c^4*d^2*(3+m)*(5+m)/(1+m)+e*(2+m)*(2*c^2*d*(5 
+m)^2+e*(m^2+7*m+12))/(3+m)/(5+m))*(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^3/f^2/(2+m)/(3+m)/(5+m)/(c*x-1)^(1/2 
)/(c*x+1)^(1/2)
 

3.6.20.2 Mathematica [A] (verified)

Time = 0.32 (sec) , antiderivative size = 293, normalized size of antiderivative = 0.83 \[ \int (f x)^m \left (d+e x^2\right )^2 (a+b \text {arccosh}(c x)) \, dx=x (f x)^m \left (\frac {d^2 (a+b \text {arccosh}(c x))}{1+m}+\frac {2 d e x^2 (a+b \text {arccosh}(c x))}{3+m}+\frac {e^2 x^4 (a+b \text {arccosh}(c x))}{5+m}-\frac {b c d^2 x \sqrt {1-c^2 x^2} \operatorname {Hypergeometric2F1}\left (\frac {1}{2},\frac {2+m}{2},\frac {4+m}{2},c^2 x^2\right )}{\left (2+3 m+m^2\right ) \sqrt {-1+c x} \sqrt {1+c x}}-\frac {2 b c d e x^3 \sqrt {1-c^2 x^2} \operatorname {Hypergeometric2F1}\left (\frac {1}{2},\frac {4+m}{2},\frac {6+m}{2},c^2 x^2\right )}{\left (12+7 m+m^2\right ) \sqrt {-1+c x} \sqrt {1+c x}}-\frac {b c e^2 x^5 \sqrt {1-c^2 x^2} \operatorname {Hypergeometric2F1}\left (\frac {1}{2},\frac {6+m}{2},\frac {8+m}{2},c^2 x^2\right )}{(5+m) (6+m) \sqrt {-1+c x} \sqrt {1+c x}}\right ) \]

input

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

output

x*(f*x)^m*((d^2*(a + b*ArcCosh[c*x]))/(1 + m) + (2*d*e*x^2*(a + b*ArcCosh[ 
c*x]))/(3 + m) + (e^2*x^4*(a + b*ArcCosh[c*x]))/(5 + m) - (b*c*d^2*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]) - (2*b*c*d*e*x^3*Sqrt[1 - c^2*x^2 
]*Hypergeometric2F1[1/2, (4 + m)/2, (6 + m)/2, c^2*x^2])/((12 + 7*m + m^2) 
*Sqrt[-1 + c*x]*Sqrt[1 + c*x]) - (b*c*e^2*x^5*Sqrt[1 - c^2*x^2]*Hypergeome 
tric2F1[1/2, (6 + m)/2, (8 + m)/2, c^2*x^2])/((5 + m)*(6 + m)*Sqrt[-1 + c* 
x]*Sqrt[1 + c*x]))
 

3.6.20.3 Rubi [A] (verified)

Time = 0.74 (sec) , antiderivative size = 350, normalized size of antiderivative = 0.99, number of steps used = 7, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.304, Rules used = {6373, 27, 1905, 1590, 363, 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.

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

\(\Big \downarrow \) 6373

\(\displaystyle -b c \int \frac {(f x)^{m+1} \left (\frac {e^2 x^4}{m+5}+\frac {2 d e x^2}{m+3}+\frac {d^2}{m+1}\right )}{f \sqrt {c x-1} \sqrt {c x+1}}dx+\frac {d^2 (f x)^{m+1} (a+b \text {arccosh}(c x))}{f (m+1)}+\frac {2 d e (f x)^{m+3} (a+b \text {arccosh}(c x))}{f^3 (m+3)}+\frac {e^2 (f x)^{m+5} (a+b \text {arccosh}(c x))}{f^5 (m+5)}\)

\(\Big \downarrow \) 27

\(\displaystyle -\frac {b c \int \frac {(f x)^{m+1} \left (\frac {e^2 x^4}{m+5}+\frac {2 d e x^2}{m+3}+\frac {d^2}{m+1}\right )}{\sqrt {c x-1} \sqrt {c x+1}}dx}{f}+\frac {d^2 (f x)^{m+1} (a+b \text {arccosh}(c x))}{f (m+1)}+\frac {2 d e (f x)^{m+3} (a+b \text {arccosh}(c x))}{f^3 (m+3)}+\frac {e^2 (f x)^{m+5} (a+b \text {arccosh}(c x))}{f^5 (m+5)}\)

\(\Big \downarrow \) 1905

\(\displaystyle -\frac {b c \sqrt {c^2 x^2-1} \int \frac {(f x)^{m+1} \left (\frac {e^2 x^4}{m+5}+\frac {2 d e x^2}{m+3}+\frac {d^2}{m+1}\right )}{\sqrt {c^2 x^2-1}}dx}{f \sqrt {c x-1} \sqrt {c x+1}}+\frac {d^2 (f x)^{m+1} (a+b \text {arccosh}(c x))}{f (m+1)}+\frac {2 d e (f x)^{m+3} (a+b \text {arccosh}(c x))}{f^3 (m+3)}+\frac {e^2 (f x)^{m+5} (a+b \text {arccosh}(c x))}{f^5 (m+5)}\)

\(\Big \downarrow \) 1590

\(\displaystyle -\frac {b c \sqrt {c^2 x^2-1} \left (\frac {\int \frac {(f x)^{m+1} \left (\frac {c^2 (m+5) d^2}{m+1}+\frac {e \left (2 c^2 d (m+5)^2+e \left (m^2+7 m+12\right )\right ) x^2}{(m+3) (m+5)}\right )}{\sqrt {c^2 x^2-1}}dx}{c^2 (m+5)}+\frac {e^2 \sqrt {c^2 x^2-1} (f x)^{m+4}}{c^2 f^3 (m+5)^2}\right )}{f \sqrt {c x-1} \sqrt {c x+1}}+\frac {d^2 (f x)^{m+1} (a+b \text {arccosh}(c x))}{f (m+1)}+\frac {2 d e (f x)^{m+3} (a+b \text {arccosh}(c x))}{f^3 (m+3)}+\frac {e^2 (f x)^{m+5} (a+b \text {arccosh}(c x))}{f^5 (m+5)}\)

\(\Big \downarrow \) 363

\(\displaystyle -\frac {b c \sqrt {c^2 x^2-1} \left (\frac {\left (\frac {c^2 d^2 (m+5)}{m+1}+\frac {e (m+2) \left (2 c^2 d (m+5)^2+e \left (m^2+7 m+12\right )\right )}{c^2 (m+3)^2 (m+5)}\right ) \int \frac {(f x)^{m+1}}{\sqrt {c^2 x^2-1}}dx+\frac {e \sqrt {c^2 x^2-1} (f x)^{m+2} \left (2 c^2 d (m+5)^2+e \left (m^2+7 m+12\right )\right )}{c^2 f (m+3)^2 (m+5)}}{c^2 (m+5)}+\frac {e^2 \sqrt {c^2 x^2-1} (f x)^{m+4}}{c^2 f^3 (m+5)^2}\right )}{f \sqrt {c x-1} \sqrt {c x+1}}+\frac {d^2 (f x)^{m+1} (a+b \text {arccosh}(c x))}{f (m+1)}+\frac {2 d e (f x)^{m+3} (a+b \text {arccosh}(c x))}{f^3 (m+3)}+\frac {e^2 (f x)^{m+5} (a+b \text {arccosh}(c x))}{f^5 (m+5)}\)

\(\Big \downarrow \) 279

\(\displaystyle -\frac {b c \sqrt {c^2 x^2-1} \left (\frac {\frac {\sqrt {1-c^2 x^2} \left (\frac {c^2 d^2 (m+5)}{m+1}+\frac {e (m+2) \left (2 c^2 d (m+5)^2+e \left (m^2+7 m+12\right )\right )}{c^2 (m+3)^2 (m+5)}\right ) \int \frac {(f x)^{m+1}}{\sqrt {1-c^2 x^2}}dx}{\sqrt {c^2 x^2-1}}+\frac {e \sqrt {c^2 x^2-1} (f x)^{m+2} \left (2 c^2 d (m+5)^2+e \left (m^2+7 m+12\right )\right )}{c^2 f (m+3)^2 (m+5)}}{c^2 (m+5)}+\frac {e^2 \sqrt {c^2 x^2-1} (f x)^{m+4}}{c^2 f^3 (m+5)^2}\right )}{f \sqrt {c x-1} \sqrt {c x+1}}+\frac {d^2 (f x)^{m+1} (a+b \text {arccosh}(c x))}{f (m+1)}+\frac {2 d e (f x)^{m+3} (a+b \text {arccosh}(c x))}{f^3 (m+3)}+\frac {e^2 (f x)^{m+5} (a+b \text {arccosh}(c x))}{f^5 (m+5)}\)

\(\Big \downarrow \) 278

\(\displaystyle \frac {d^2 (f x)^{m+1} (a+b \text {arccosh}(c x))}{f (m+1)}+\frac {2 d e (f x)^{m+3} (a+b \text {arccosh}(c x))}{f^3 (m+3)}+\frac {e^2 (f x)^{m+5} (a+b \text {arccosh}(c x))}{f^5 (m+5)}-\frac {b c \sqrt {c^2 x^2-1} \left (\frac {\frac {\sqrt {1-c^2 x^2} (f x)^{m+2} \left (\frac {c^2 d^2 (m+5)}{m+1}+\frac {e (m+2) \left (2 c^2 d (m+5)^2+e \left (m^2+7 m+12\right )\right )}{c^2 (m+3)^2 (m+5)}\right ) \operatorname {Hypergeometric2F1}\left (\frac {1}{2},\frac {m+2}{2},\frac {m+4}{2},c^2 x^2\right )}{f (m+2) \sqrt {c^2 x^2-1}}+\frac {e \sqrt {c^2 x^2-1} (f x)^{m+2} \left (2 c^2 d (m+5)^2+e \left (m^2+7 m+12\right )\right )}{c^2 f (m+3)^2 (m+5)}}{c^2 (m+5)}+\frac {e^2 \sqrt {c^2 x^2-1} (f x)^{m+4}}{c^2 f^3 (m+5)^2}\right )}{f \sqrt {c x-1} \sqrt {c x+1}}\)

input

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

output

(d^2*(f*x)^(1 + m)*(a + b*ArcCosh[c*x]))/(f*(1 + m)) + (2*d*e*(f*x)^(3 + m 
)*(a + b*ArcCosh[c*x]))/(f^3*(3 + m)) + (e^2*(f*x)^(5 + m)*(a + b*ArcCosh[ 
c*x]))/(f^5*(5 + m)) - (b*c*Sqrt[-1 + c^2*x^2]*((e^2*(f*x)^(4 + m)*Sqrt[-1 
 + c^2*x^2])/(c^2*f^3*(5 + m)^2) + ((e*(2*c^2*d*(5 + m)^2 + e*(12 + 7*m + 
m^2))*(f*x)^(2 + m)*Sqrt[-1 + c^2*x^2])/(c^2*f*(3 + m)^2*(5 + m)) + (((c^2 
*d^2*(5 + m))/(1 + m) + (e*(2 + m)*(2*c^2*d*(5 + m)^2 + e*(12 + 7*m + m^2) 
))/(c^2*(3 + m)^2*(5 + m)))*(f*x)^(2 + m)*Sqrt[1 - c^2*x^2]*Hypergeometric 
2F1[1/2, (2 + m)/2, (4 + m)/2, c^2*x^2])/(f*(2 + m)*Sqrt[-1 + c^2*x^2]))/( 
c^2*(5 + m))))/(f*Sqrt[-1 + c*x]*Sqrt[1 + c*x])
 

3.6.20.3.1 Defintions of rubi rules used

rule 27

Int[(a_)*(Fx_), x_Symbol] :> Simp[a   Int[Fx, x], x] /; FreeQ[a, x] &&  !Ma 
tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
 

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 363

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

rule 1590

Int[((f_.)*(x_))^(m_.)*((d_) + (e_.)*(x_)^2)^(q_.)*((a_) + (b_.)*(x_)^2 + ( 
c_.)*(x_)^4)^(p_.), x_Symbol] :> Simp[c^p*(f*x)^(m + 4*p - 1)*((d + e*x^2)^ 
(q + 1)/(e*f^(4*p - 1)*(m + 4*p + 2*q + 1))), x] + Simp[1/(e*(m + 4*p + 2*q 
 + 1))   Int[(f*x)^m*(d + e*x^2)^q*ExpandToSum[e*(m + 4*p + 2*q + 1)*((a + 
b*x^2 + c*x^4)^p - c^p*x^(4*p)) - d*c^p*(m + 4*p - 1)*x^(4*p - 2), x], x], 
x] /; FreeQ[{a, b, c, d, e, f, m, q}, x] && NeQ[b^2 - 4*a*c, 0] && IGtQ[p, 
0] &&  !IntegerQ[q] && NeQ[m + 4*p + 2*q + 1, 0]
 

rule 1905

Int[((f_.)*(x_))^(m_.)*((d1_) + (e1_.)*(x_)^(non2_.))^(q_.)*((d2_) + (e2_.) 
*(x_)^(non2_.))^(q_.)*((a_.) + (b_.)*(x_)^(n_) + (c_.)*(x_)^(n2_))^(p_.), x 
_Symbol] :> Simp[(d1 + e1*x^(n/2))^FracPart[q]*((d2 + e2*x^(n/2))^FracPart[ 
q]/(d1*d2 + e1*e2*x^n)^FracPart[q])   Int[(f*x)^m*(d1*d2 + e1*e2*x^n)^q*(a 
+ b*x^n + c*x^(2*n))^p, x], x] /; FreeQ[{a, b, c, d1, e1, d2, e2, f, n, p, 
q}, x] && EqQ[n2, 2*n] && EqQ[non2, n/2] && EqQ[d2*e1 + d1*e2, 0]
 

rule 6373

Int[((a_.) + ArcCosh[(c_.)*(x_)]*(b_.))*((f_.)*(x_))^(m_.)*((d_) + (e_.)*(x 
_)^2)^(p_.), x_Symbol] :> With[{u = IntHide[(f*x)^m*(d + e*x^2)^p, x]}, Sim 
p[(a + b*ArcCosh[c*x])   u, x] - Simp[b*c   Int[SimplifyIntegrand[u/(Sqrt[1 
 + c*x]*Sqrt[-1 + c*x]), x], x], x]] /; FreeQ[{a, b, c, d, e, f, m}, x] && 
NeQ[c^2*d + e, 0] && IntegerQ[p] && (GtQ[p, 0] || (IGtQ[(m - 1)/2, 0] && Le 
Q[m + p, 0]))
 

3.6.20.4 Maple [F]

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

input

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

output

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

3.6.20.5 Fricas [F]

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

input

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

output

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

3.6.20.6 Sympy [F]

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

input

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

output

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

3.6.20.7 Maxima [F]

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

input

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

output

a*e^2*f^m*x^5*x^m/(m + 5) + 2*a*d*e*f^m*x^3*x^m/(m + 3) + (f*x)^(m + 1)*a* 
d^2/(f*(m + 1)) + ((m^2 + 4*m + 3)*b*e^2*f^m*x^5 + 2*(m^2 + 6*m + 5)*b*d*e 
*f^m*x^3 + (m^2 + 8*m + 15)*b*d^2*f^m*x)*x^m*log(c*x + sqrt(c*x + 1)*sqrt( 
c*x - 1))/(m^3 + 9*m^2 + 23*m + 15) + integrate(((m^2 + 4*m + 3)*b*c*e^2*f 
^m*x^5 + 2*(m^2 + 6*m + 5)*b*c*d*e*f^m*x^3 + (m^2 + 8*m + 15)*b*c*d^2*f^m* 
x)*x^m/((m^3 + 9*m^2 + 23*m + 15)*c^3*x^3 - (m^3 + 9*m^2 + 23*m + 15)*c*x 
+ ((m^3 + 9*m^2 + 23*m + 15)*c^2*x^2 - m^3 - 9*m^2 - 23*m - 15)*sqrt(c*x + 
 1)*sqrt(c*x - 1)), x) - integrate(((m^2 + 4*m + 3)*b*c^2*e^2*f^m*x^6 + 2* 
(m^2 + 6*m + 5)*b*c^2*d*e*f^m*x^4 + (m^2 + 8*m + 15)*b*c^2*d^2*f^m*x^2)*x^ 
m/((m^3 + 9*m^2 + 23*m + 15)*c^2*x^2 - m^3 - 9*m^2 - 23*m - 15), x)
 

3.6.20.8 Giac [F]

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

input

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

output

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

3.6.20.9 Mupad [F(-1)]

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

input

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

output

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