3.6.19 \(\int (f x)^m (d+e x^2)^3 (a+b \text {arccosh}(c x)) \, dx\) [519]

3.6.19.1 Optimal result

Integrand size = 23, antiderivative size = 558 \[ \int (f x)^m \left (d+e x^2\right )^3 (a+b \text {arccosh}(c x)) \, dx=\frac {b e \left (3 c^2 d e (7+m)^2 \left (12+7 m+m^2\right )+3 c^4 d^2 \left (35+12 m+m^2\right )^2+e^2 \left (360+342 m+119 m^2+18 m^3+m^4\right )\right ) (f x)^{2+m} \left (1-c^2 x^2\right )}{c^5 f^2 (3+m)^2 (5+m)^2 (7+m)^2 \sqrt {-1+c x} \sqrt {1+c x}}+\frac {b e^2 \left (3 c^2 d (7+m)^2+e \left (30+11 m+m^2\right )\right ) (f x)^{4+m} \left (1-c^2 x^2\right )}{c^3 f^4 (5+m)^2 (7+m)^2 \sqrt {-1+c x} \sqrt {1+c x}}+\frac {b e^3 (f x)^{6+m} \left (1-c^2 x^2\right )}{c f^6 (7+m)^2 \sqrt {-1+c x} \sqrt {1+c x}}+\frac {d^3 (f x)^{1+m} (a+b \text {arccosh}(c x))}{f (1+m)}+\frac {3 d^2 e (f x)^{3+m} (a+b \text {arccosh}(c x))}{f^3 (3+m)}+\frac {3 d e^2 (f x)^{5+m} (a+b \text {arccosh}(c x))}{f^5 (5+m)}+\frac {e^3 (f x)^{7+m} (a+b \text {arccosh}(c x))}{f^7 (7+m)}-\frac {b \left (\frac {c^6 d^3 (3+m) (5+m) (7+m)}{1+m}+\frac {e (2+m) \left (3 c^2 d e (7+m)^2 \left (12+7 m+m^2\right )+3 c^4 d^2 \left (35+12 m+m^2\right )^2+e^2 \left (360+342 m+119 m^2+18 m^3+m^4\right )\right )}{(3+m) (5+m) (7+m)}\right ) (f x)^{2+m} \sqrt {1-c^2 x^2} \operatorname {Hypergeometric2F1}\left (\frac {1}{2},\frac {2+m}{2},\frac {4+m}{2},c^2 x^2\right )}{c^5 f^2 (2+m) (3+m) (5+m) (7+m) \sqrt {-1+c x} \sqrt {1+c x}} \]

output

d^3*(f*x)^(1+m)*(a+b*arccosh(c*x))/f/(1+m)+3*d^2*e*(f*x)^(3+m)*(a+b*arccos 
h(c*x))/f^3/(3+m)+3*d*e^2*(f*x)^(5+m)*(a+b*arccosh(c*x))/f^5/(5+m)+e^3*(f* 
x)^(7+m)*(a+b*arccosh(c*x))/f^7/(7+m)+b*e*(3*c^2*d*e*(7+m)^2*(m^2+7*m+12)+ 
3*c^4*d^2*(m^2+12*m+35)^2+e^2*(m^4+18*m^3+119*m^2+342*m+360))*(f*x)^(2+m)* 
(-c^2*x^2+1)/c^5/f^2/(3+m)^2/(5+m)^2/(7+m)^2/(c*x-1)^(1/2)/(c*x+1)^(1/2)+b 
*e^2*(3*c^2*d*(7+m)^2+e*(m^2+11*m+30))*(f*x)^(4+m)*(-c^2*x^2+1)/c^3/f^4/(5 
+m)^2/(7+m)^2/(c*x-1)^(1/2)/(c*x+1)^(1/2)+b*e^3*(f*x)^(6+m)*(-c^2*x^2+1)/c 
/f^6/(7+m)^2/(c*x-1)^(1/2)/(c*x+1)^(1/2)-b*(c^6*d^3*(3+m)*(5+m)*(7+m)/(1+m 
)+e*(2+m)*(3*c^2*d*e*(7+m)^2*(m^2+7*m+12)+3*c^4*d^2*(m^2+12*m+35)^2+e^2*(m 
^4+18*m^3+119*m^2+342*m+360))/(m^3+15*m^2+71*m+105))*(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^5/f^2/(2+m)/(3+m)/ 
(5+m)/(7+m)/(c*x-1)^(1/2)/(c*x+1)^(1/2)
 

3.6.19.2 Mathematica [A] (verified)

Time = 0.87 (sec) , antiderivative size = 397, normalized size of antiderivative = 0.71 \[ \int (f x)^m \left (d+e x^2\right )^3 (a+b \text {arccosh}(c x)) \, dx=x (f x)^m \left (\frac {d^3 (a+b \text {arccosh}(c x))}{1+m}+\frac {3 d^2 e x^2 (a+b \text {arccosh}(c x))}{3+m}+\frac {3 d e^2 x^4 (a+b \text {arccosh}(c x))}{5+m}+\frac {e^3 x^6 (a+b \text {arccosh}(c x))}{7+m}-\frac {b c e^3 x^7 \sqrt {1-c^2 x^2} \operatorname {Hypergeometric2F1}\left (\frac {1}{2},4+\frac {m}{2},5+\frac {m}{2},c^2 x^2\right )}{(7+m) (8+m) \sqrt {-1+c x} \sqrt {1+c x}}-\frac {b c d^3 x \sqrt {1-c^2 x^2} \operatorname {Hypergeometric2F1}\left (\frac {1}{2},\frac {2+m}{2},\frac {4+m}{2},c^2 x^2\right )}{\left (2+3 m+m^2\right ) \sqrt {-1+c x} \sqrt {1+c x}}-\frac {3 b c d^2 e x^3 \sqrt {1-c^2 x^2} \operatorname {Hypergeometric2F1}\left (\frac {1}{2},\frac {4+m}{2},\frac {6+m}{2},c^2 x^2\right )}{\left (12+7 m+m^2\right ) \sqrt {-1+c x} \sqrt {1+c x}}-\frac {3 b c d e^2 x^5 \sqrt {1-c^2 x^2} \operatorname {Hypergeometric2F1}\left (\frac {1}{2},\frac {6+m}{2},\frac {8+m}{2},c^2 x^2\right )}{(5+m) (6+m) \sqrt {-1+c x} \sqrt {1+c x}}\right ) \]

input

Integrate[(f*x)^m*(d + e*x^2)^3*(a + b*ArcCosh[c*x]),x]
 

output

x*(f*x)^m*((d^3*(a + b*ArcCosh[c*x]))/(1 + m) + (3*d^2*e*x^2*(a + b*ArcCos 
h[c*x]))/(3 + m) + (3*d*e^2*x^4*(a + b*ArcCosh[c*x]))/(5 + m) + (e^3*x^6*( 
a + b*ArcCosh[c*x]))/(7 + m) - (b*c*e^3*x^7*Sqrt[1 - c^2*x^2]*Hypergeometr 
ic2F1[1/2, 4 + m/2, 5 + m/2, c^2*x^2])/((7 + m)*(8 + m)*Sqrt[-1 + c*x]*Sqr 
t[1 + c*x]) - (b*c*d^3*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]) - ( 
3*b*c*d^2*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]) - (3*b*c*d* 
e^2*x^5*Sqrt[1 - c^2*x^2]*Hypergeometric2F1[1/2, (6 + m)/2, (8 + m)/2, c^2 
*x^2])/((5 + m)*(6 + m)*Sqrt[-1 + c*x]*Sqrt[1 + c*x]))
 

3.6.19.3 Rubi [A] (verified)

Time = 2.47 (sec) , antiderivative size = 544, normalized size of antiderivative = 0.97, number of steps used = 8, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.348, Rules used = {6373, 27, 2113, 2340, 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.

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

\(\Big \downarrow \) 6373

\(\displaystyle -b c \int \frac {(f x)^{m+1} \left (\frac {e^3 x^6}{m+7}+\frac {3 d e^2 x^4}{m+5}+\frac {3 d^2 e x^2}{m+3}+\frac {d^3}{m+1}\right )}{f \sqrt {c x-1} \sqrt {c x+1}}dx+\frac {d^3 (f x)^{m+1} (a+b \text {arccosh}(c x))}{f (m+1)}+\frac {3 d^2 e (f x)^{m+3} (a+b \text {arccosh}(c x))}{f^3 (m+3)}+\frac {3 d e^2 (f x)^{m+5} (a+b \text {arccosh}(c x))}{f^5 (m+5)}+\frac {e^3 (f x)^{m+7} (a+b \text {arccosh}(c x))}{f^7 (m+7)}\)

\(\Big \downarrow \) 27

\(\displaystyle -\frac {b c \int \frac {(f x)^{m+1} \left (\frac {e^3 x^6}{m+7}+\frac {3 d e^2 x^4}{m+5}+\frac {3 d^2 e x^2}{m+3}+\frac {d^3}{m+1}\right )}{\sqrt {c x-1} \sqrt {c x+1}}dx}{f}+\frac {d^3 (f x)^{m+1} (a+b \text {arccosh}(c x))}{f (m+1)}+\frac {3 d^2 e (f x)^{m+3} (a+b \text {arccosh}(c x))}{f^3 (m+3)}+\frac {3 d e^2 (f x)^{m+5} (a+b \text {arccosh}(c x))}{f^5 (m+5)}+\frac {e^3 (f x)^{m+7} (a+b \text {arccosh}(c x))}{f^7 (m+7)}\)

\(\Big \downarrow \) 2113

\(\displaystyle -\frac {b c \sqrt {c^2 x^2-1} \int \frac {(f x)^{m+1} \left (\frac {e^3 x^6}{m+7}+\frac {3 d e^2 x^4}{m+5}+\frac {3 d^2 e x^2}{m+3}+\frac {d^3}{m+1}\right )}{\sqrt {c^2 x^2-1}}dx}{f \sqrt {c x-1} \sqrt {c x+1}}+\frac {d^3 (f x)^{m+1} (a+b \text {arccosh}(c x))}{f (m+1)}+\frac {3 d^2 e (f x)^{m+3} (a+b \text {arccosh}(c x))}{f^3 (m+3)}+\frac {3 d e^2 (f x)^{m+5} (a+b \text {arccosh}(c x))}{f^5 (m+5)}+\frac {e^3 (f x)^{m+7} (a+b \text {arccosh}(c x))}{f^7 (m+7)}\)

\(\Big \downarrow \) 2340

\(\displaystyle -\frac {b c \sqrt {c^2 x^2-1} \left (\frac {\int \frac {(f x)^{m+1} \left (\frac {e^2 \left (3 c^2 d (m+7)^2+e \left (m^2+11 m+30\right )\right ) x^4}{(m+5) (m+7)}+\frac {3 c^2 d^2 e (m+7) x^2}{m+3}+\frac {c^2 d^3 (m+7)}{m+1}\right )}{\sqrt {c^2 x^2-1}}dx}{c^2 (m+7)}+\frac {e^3 \sqrt {c^2 x^2-1} (f x)^{m+6}}{c^2 f^5 (m+7)^2}\right )}{f \sqrt {c x-1} \sqrt {c x+1}}+\frac {d^3 (f x)^{m+1} (a+b \text {arccosh}(c x))}{f (m+1)}+\frac {3 d^2 e (f x)^{m+3} (a+b \text {arccosh}(c x))}{f^3 (m+3)}+\frac {3 d e^2 (f x)^{m+5} (a+b \text {arccosh}(c x))}{f^5 (m+5)}+\frac {e^3 (f x)^{m+7} (a+b \text {arccosh}(c x))}{f^7 (m+7)}\)

\(\Big \downarrow \) 1590

\(\displaystyle -\frac {b c \sqrt {c^2 x^2-1} \left (\frac {\frac {\int \frac {(f x)^{m+1} \left (\frac {d^3 (m+5) (m+7) c^4}{m+1}+\frac {e \left (3 d^2 \left (m^2+12 m+35\right )^2 c^4+3 d e (m+7)^2 \left (m^2+7 m+12\right ) c^2+e^2 \left (m^4+18 m^3+119 m^2+342 m+360\right )\right ) x^2}{(m+3) (m+5) (m+7)}\right )}{\sqrt {c^2 x^2-1}}dx}{c^2 (m+5)}+\frac {e^2 \sqrt {c^2 x^2-1} (f x)^{m+4} \left (3 c^2 d (m+7)^2+e \left (m^2+11 m+30\right )\right )}{c^2 f^3 (m+5)^2 (m+7)}}{c^2 (m+7)}+\frac {e^3 \sqrt {c^2 x^2-1} (f x)^{m+6}}{c^2 f^5 (m+7)^2}\right )}{f \sqrt {c x-1} \sqrt {c x+1}}+\frac {d^3 (f x)^{m+1} (a+b \text {arccosh}(c x))}{f (m+1)}+\frac {3 d^2 e (f x)^{m+3} (a+b \text {arccosh}(c x))}{f^3 (m+3)}+\frac {3 d e^2 (f x)^{m+5} (a+b \text {arccosh}(c x))}{f^5 (m+5)}+\frac {e^3 (f x)^{m+7} (a+b \text {arccosh}(c x))}{f^7 (m+7)}\)

\(\Big \downarrow \) 363

\(\displaystyle -\frac {b c \sqrt {c^2 x^2-1} \left (\frac {\frac {\left (\frac {c^4 d^3 (m+5) (m+7)}{m+1}+\frac {e (m+2) \left (3 c^4 d^2 \left (m^2+12 m+35\right )^2+3 c^2 d e (m+7)^2 \left (m^2+7 m+12\right )+e^2 \left (m^4+18 m^3+119 m^2+342 m+360\right )\right )}{c^2 (m+3)^2 (m+5) (m+7)}\right ) \int \frac {(f x)^{m+1}}{\sqrt {c^2 x^2-1}}dx+\frac {e \sqrt {c^2 x^2-1} (f x)^{m+2} \left (3 c^4 d^2 \left (m^2+12 m+35\right )^2+3 c^2 d e (m+7)^2 \left (m^2+7 m+12\right )+e^2 \left (m^4+18 m^3+119 m^2+342 m+360\right )\right )}{c^2 f (m+3)^2 (m+5) (m+7)}}{c^2 (m+5)}+\frac {e^2 \sqrt {c^2 x^2-1} (f x)^{m+4} \left (3 c^2 d (m+7)^2+e \left (m^2+11 m+30\right )\right )}{c^2 f^3 (m+5)^2 (m+7)}}{c^2 (m+7)}+\frac {e^3 \sqrt {c^2 x^2-1} (f x)^{m+6}}{c^2 f^5 (m+7)^2}\right )}{f \sqrt {c x-1} \sqrt {c x+1}}+\frac {d^3 (f x)^{m+1} (a+b \text {arccosh}(c x))}{f (m+1)}+\frac {3 d^2 e (f x)^{m+3} (a+b \text {arccosh}(c x))}{f^3 (m+3)}+\frac {3 d e^2 (f x)^{m+5} (a+b \text {arccosh}(c x))}{f^5 (m+5)}+\frac {e^3 (f x)^{m+7} (a+b \text {arccosh}(c x))}{f^7 (m+7)}\)

\(\Big \downarrow \) 279

\(\displaystyle -\frac {b c \sqrt {c^2 x^2-1} \left (\frac {\frac {\frac {\sqrt {1-c^2 x^2} \left (\frac {c^4 d^3 (m+5) (m+7)}{m+1}+\frac {e (m+2) \left (3 c^4 d^2 \left (m^2+12 m+35\right )^2+3 c^2 d e (m+7)^2 \left (m^2+7 m+12\right )+e^2 \left (m^4+18 m^3+119 m^2+342 m+360\right )\right )}{c^2 (m+3)^2 (m+5) (m+7)}\right ) \int \frac {(f x)^{m+1}}{\sqrt {1-c^2 x^2}}dx}{\sqrt {c^2 x^2-1}}+\frac {e \sqrt {c^2 x^2-1} (f x)^{m+2} \left (3 c^4 d^2 \left (m^2+12 m+35\right )^2+3 c^2 d e (m+7)^2 \left (m^2+7 m+12\right )+e^2 \left (m^4+18 m^3+119 m^2+342 m+360\right )\right )}{c^2 f (m+3)^2 (m+5) (m+7)}}{c^2 (m+5)}+\frac {e^2 \sqrt {c^2 x^2-1} (f x)^{m+4} \left (3 c^2 d (m+7)^2+e \left (m^2+11 m+30\right )\right )}{c^2 f^3 (m+5)^2 (m+7)}}{c^2 (m+7)}+\frac {e^3 \sqrt {c^2 x^2-1} (f x)^{m+6}}{c^2 f^5 (m+7)^2}\right )}{f \sqrt {c x-1} \sqrt {c x+1}}+\frac {d^3 (f x)^{m+1} (a+b \text {arccosh}(c x))}{f (m+1)}+\frac {3 d^2 e (f x)^{m+3} (a+b \text {arccosh}(c x))}{f^3 (m+3)}+\frac {3 d e^2 (f x)^{m+5} (a+b \text {arccosh}(c x))}{f^5 (m+5)}+\frac {e^3 (f x)^{m+7} (a+b \text {arccosh}(c x))}{f^7 (m+7)}\)

\(\Big \downarrow \) 278

\(\displaystyle \frac {d^3 (f x)^{m+1} (a+b \text {arccosh}(c x))}{f (m+1)}+\frac {3 d^2 e (f x)^{m+3} (a+b \text {arccosh}(c x))}{f^3 (m+3)}+\frac {3 d e^2 (f x)^{m+5} (a+b \text {arccosh}(c x))}{f^5 (m+5)}+\frac {e^3 (f x)^{m+7} (a+b \text {arccosh}(c x))}{f^7 (m+7)}-\frac {b c \sqrt {c^2 x^2-1} \left (\frac {e^3 \sqrt {c^2 x^2-1} (f x)^{m+6}}{c^2 f^5 (m+7)^2}+\frac {\frac {e^2 \sqrt {c^2 x^2-1} (f x)^{m+4} \left (3 c^2 d (m+7)^2+e \left (m^2+11 m+30\right )\right )}{c^2 f^3 (m+5)^2 (m+7)}+\frac {\frac {e \sqrt {c^2 x^2-1} (f x)^{m+2} \left (3 c^4 d^2 \left (m^2+12 m+35\right )^2+3 c^2 d e (m+7)^2 \left (m^2+7 m+12\right )+e^2 \left (m^4+18 m^3+119 m^2+342 m+360\right )\right )}{c^2 f (m+3)^2 (m+5) (m+7)}+\frac {\sqrt {1-c^2 x^2} (f x)^{m+2} \left (\frac {c^4 d^3 (m+5) (m+7)}{m+1}+\frac {e (m+2) \left (3 c^4 d^2 \left (m^2+12 m+35\right )^2+3 c^2 d e (m+7)^2 \left (m^2+7 m+12\right )+e^2 \left (m^4+18 m^3+119 m^2+342 m+360\right )\right )}{c^2 (m+3)^2 (m+5) (m+7)}\right ) \operatorname {Hypergeometric2F1}\left (\frac {1}{2},\frac {m+2}{2},\frac {m+4}{2},c^2 x^2\right )}{f (m+2) \sqrt {c^2 x^2-1}}}{c^2 (m+5)}}{c^2 (m+7)}\right )}{f \sqrt {c x-1} \sqrt {c x+1}}\)

input

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

output

(d^3*(f*x)^(1 + m)*(a + b*ArcCosh[c*x]))/(f*(1 + m)) + (3*d^2*e*(f*x)^(3 + 
 m)*(a + b*ArcCosh[c*x]))/(f^3*(3 + m)) + (3*d*e^2*(f*x)^(5 + m)*(a + b*Ar 
cCosh[c*x]))/(f^5*(5 + m)) + (e^3*(f*x)^(7 + m)*(a + b*ArcCosh[c*x]))/(f^7 
*(7 + m)) - (b*c*Sqrt[-1 + c^2*x^2]*((e^3*(f*x)^(6 + m)*Sqrt[-1 + c^2*x^2] 
)/(c^2*f^5*(7 + m)^2) + ((e^2*(3*c^2*d*(7 + m)^2 + e*(30 + 11*m + m^2))*(f 
*x)^(4 + m)*Sqrt[-1 + c^2*x^2])/(c^2*f^3*(5 + m)^2*(7 + m)) + ((e*(3*c^2*d 
*e*(7 + m)^2*(12 + 7*m + m^2) + 3*c^4*d^2*(35 + 12*m + m^2)^2 + e^2*(360 + 
 342*m + 119*m^2 + 18*m^3 + m^4))*(f*x)^(2 + m)*Sqrt[-1 + c^2*x^2])/(c^2*f 
*(3 + m)^2*(5 + m)*(7 + m)) + (((c^4*d^3*(5 + m)*(7 + m))/(1 + m) + (e*(2 
+ m)*(3*c^2*d*e*(7 + m)^2*(12 + 7*m + m^2) + 3*c^4*d^2*(35 + 12*m + m^2)^2 
 + e^2*(360 + 342*m + 119*m^2 + 18*m^3 + m^4)))/(c^2*(3 + m)^2*(5 + m)*(7 
+ 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^2*x^2]))/(c^2*(5 + m)))/(c^2*(7 
 + m))))/(f*Sqrt[-1 + c*x]*Sqrt[1 + c*x])
 

3.6.19.3.1 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 2113

Int[(Px_)*((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_)*((e_.) + (f_. 
)*(x_))^(p_.), x_Symbol] :> Simp[(a + b*x)^FracPart[m]*((c + d*x)^FracPart[ 
m]/(a*c + b*d*x^2)^FracPart[m])   Int[Px*(a*c + b*d*x^2)^m*(e + f*x)^p, x], 
 x] /; FreeQ[{a, b, c, d, e, f, m, n, p}, x] && PolyQ[Px, x] && EqQ[b*c + a 
*d, 0] && EqQ[m, n] &&  !IntegerQ[m]
 

rule 2340

Int[(Pq_)*((c_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> With[ 
{q = Expon[Pq, x], f = Coeff[Pq, x, Expon[Pq, x]]}, Simp[f*(c*x)^(m + q - 1 
)*((a + b*x^2)^(p + 1)/(b*c^(q - 1)*(m + q + 2*p + 1))), x] + Simp[1/(b*(m 
+ q + 2*p + 1))   Int[(c*x)^m*(a + b*x^2)^p*ExpandToSum[b*(m + q + 2*p + 1) 
*Pq - b*f*(m + q + 2*p + 1)*x^q - a*f*(m + q - 1)*x^(q - 2), x], x], x] /; 
GtQ[q, 1] && NeQ[m + q + 2*p + 1, 0]] /; FreeQ[{a, b, c, m, p}, x] && PolyQ 
[Pq, x] && ( !IGtQ[m, 0] || IGtQ[p + 1/2, -1])
 

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]))
 

3.6.19.4 Maple [F]

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

input

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

output

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

3.6.19.5 Fricas [F]

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

input

integrate((f*x)^m*(e*x^2+d)^3*(a+b*arccosh(c*x)),x, algorithm="fricas")
 

output

integral((a*e^3*x^6 + 3*a*d*e^2*x^4 + 3*a*d^2*e*x^2 + a*d^3 + (b*e^3*x^6 + 
 3*b*d*e^2*x^4 + 3*b*d^2*e*x^2 + b*d^3)*arccosh(c*x))*(f*x)^m, x)
 

3.6.19.6 Sympy [F(-1)]

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

input

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

output

Timed out
 

3.6.19.7 Maxima [F]

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

input

integrate((f*x)^m*(e*x^2+d)^3*(a+b*arccosh(c*x)),x, algorithm="maxima")
 

output

a*e^3*f^m*x^7*x^m/(m + 7) + 3*a*d*e^2*f^m*x^5*x^m/(m + 5) + 3*a*d^2*e*f^m* 
x^3*x^m/(m + 3) + (f*x)^(m + 1)*a*d^3/(f*(m + 1)) + ((m^3 + 9*m^2 + 23*m + 
 15)*b*e^3*f^m*x^7 + 3*(m^3 + 11*m^2 + 31*m + 21)*b*d*e^2*f^m*x^5 + 3*(m^3 
 + 13*m^2 + 47*m + 35)*b*d^2*e*f^m*x^3 + (m^3 + 15*m^2 + 71*m + 105)*b*d^3 
*f^m*x)*x^m*log(c*x + sqrt(c*x + 1)*sqrt(c*x - 1))/(m^4 + 16*m^3 + 86*m^2 
+ 176*m + 105) + integrate(((m^3 + 9*m^2 + 23*m + 15)*b*c*e^3*f^m*x^7 + 3* 
(m^3 + 11*m^2 + 31*m + 21)*b*c*d*e^2*f^m*x^5 + 3*(m^3 + 13*m^2 + 47*m + 35 
)*b*c*d^2*e*f^m*x^3 + (m^3 + 15*m^2 + 71*m + 105)*b*c*d^3*f^m*x)*x^m/((m^4 
 + 16*m^3 + 86*m^2 + 176*m + 105)*c^3*x^3 - (m^4 + 16*m^3 + 86*m^2 + 176*m 
 + 105)*c*x + ((m^4 + 16*m^3 + 86*m^2 + 176*m + 105)*c^2*x^2 - m^4 - 16*m^ 
3 - 86*m^2 - 176*m - 105)*sqrt(c*x + 1)*sqrt(c*x - 1)), x) - integrate(((m 
^3 + 9*m^2 + 23*m + 15)*b*c^2*e^3*f^m*x^8 + 3*(m^3 + 11*m^2 + 31*m + 21)*b 
*c^2*d*e^2*f^m*x^6 + 3*(m^3 + 13*m^2 + 47*m + 35)*b*c^2*d^2*e*f^m*x^4 + (m 
^3 + 15*m^2 + 71*m + 105)*b*c^2*d^3*f^m*x^2)*x^m/((m^4 + 16*m^3 + 86*m^2 + 
 176*m + 105)*c^2*x^2 - m^4 - 16*m^3 - 86*m^2 - 176*m - 105), x)
 

3.6.19.8 Giac [F]

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

input

integrate((f*x)^m*(e*x^2+d)^3*(a+b*arccosh(c*x)),x, algorithm="giac")
 

output

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

3.6.19.9 Mupad [F(-1)]

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

input

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

output

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