\(\int x^m (d+c^2 d x^2)^2 (a+b \text {arcsinh}(c x))^2 \, dx\) [328]

Optimal result

Integrand size = 26, antiderivative size = 757 \[ \int x^m \left (d+c^2 d x^2\right )^2 (a+b \text {arcsinh}(c x))^2 \, dx=\frac {6 b^2 c^2 d^2 x^{3+m}}{(3+m)^2 (5+m)^2}+\frac {2 b^2 c^2 d^2 x^{3+m}}{(3+m) (5+m)^2}+\frac {8 b^2 c^2 d^2 x^{3+m}}{(3+m)^3 (5+m)}+\frac {2 b^2 c^4 d^2 x^{5+m}}{(5+m)^3}-\frac {6 b c d^2 x^{2+m} \sqrt {1+c^2 x^2} (a+b \text {arcsinh}(c x))}{(3+m) (5+m)^2}-\frac {8 b c d^2 x^{2+m} \sqrt {1+c^2 x^2} (a+b \text {arcsinh}(c x))}{(3+m)^2 (5+m)}-\frac {2 b c d^2 x^{2+m} \left (1+c^2 x^2\right )^{3/2} (a+b \text {arcsinh}(c x))}{(5+m)^2}+\frac {8 d^2 x^{1+m} (a+b \text {arcsinh}(c x))^2}{(5+m) \left (3+4 m+m^2\right )}+\frac {4 d^2 x^{1+m} \left (1+c^2 x^2\right ) (a+b \text {arcsinh}(c x))^2}{15+8 m+m^2}+\frac {d^2 x^{1+m} \left (1+c^2 x^2\right )^2 (a+b \text {arcsinh}(c x))^2}{5+m}-\frac {8 b c d^2 x^{2+m} (a+b \text {arcsinh}(c x)) \operatorname {Hypergeometric2F1}\left (\frac {1}{2},\frac {2+m}{2},\frac {4+m}{2},-c^2 x^2\right )}{(2+m) (3+m)^2 (5+m)}-\frac {6 b c d^2 x^{2+m} (a+b \text {arcsinh}(c x)) \operatorname {Hypergeometric2F1}\left (\frac {1}{2},\frac {2+m}{2},\frac {4+m}{2},-c^2 x^2\right )}{(5+m)^2 \left (6+5 m+m^2\right )}-\frac {16 b c d^2 x^{2+m} (a+b \text {arcsinh}(c x)) \operatorname {Hypergeometric2F1}\left (\frac {1}{2},\frac {2+m}{2},\frac {4+m}{2},-c^2 x^2\right )}{(5+m) \left (6+11 m+6 m^2+m^3\right )}+\frac {6 b^2 c^2 d^2 x^{3+m} \, _3F_2\left (1,\frac {3}{2}+\frac {m}{2},\frac {3}{2}+\frac {m}{2};2+\frac {m}{2},\frac {5}{2}+\frac {m}{2};-c^2 x^2\right )}{(2+m) (3+m)^2 (5+m)^2}+\frac {8 b^2 c^2 d^2 x^{3+m} \, _3F_2\left (1,\frac {3}{2}+\frac {m}{2},\frac {3}{2}+\frac {m}{2};2+\frac {m}{2},\frac {5}{2}+\frac {m}{2};-c^2 x^2\right )}{(2+m) (3+m)^3 (5+m)}+\frac {16 b^2 c^2 d^2 x^{3+m} \, _3F_2\left (1,\frac {3}{2}+\frac {m}{2},\frac {3}{2}+\frac {m}{2};2+\frac {m}{2},\frac {5}{2}+\frac {m}{2};-c^2 x^2\right )}{(3+m)^2 (5+m) \left (2+3 m+m^2\right )} \] Output:

6*b^2*c^2*d^2*x^(3+m)/(3+m)^2/(5+m)^2+2*b^2*c^2*d^2*x^(3+m)/(3+m)/(5+m)^2+ 
8*b^2*c^2*d^2*x^(3+m)/(3+m)^3/(5+m)+2*b^2*c^4*d^2*x^(5+m)/(5+m)^3-6*b*c*d^ 
2*x^(2+m)*(c^2*x^2+1)^(1/2)*(a+b*arcsinh(c*x))/(3+m)/(5+m)^2-8*b*c*d^2*x^( 
2+m)*(c^2*x^2+1)^(1/2)*(a+b*arcsinh(c*x))/(3+m)^2/(5+m)-2*b*c*d^2*x^(2+m)* 
(c^2*x^2+1)^(3/2)*(a+b*arcsinh(c*x))/(5+m)^2+8*d^2*x^(1+m)*(a+b*arcsinh(c* 
x))^2/(5+m)/(m^2+4*m+3)+4*d^2*x^(1+m)*(c^2*x^2+1)*(a+b*arcsinh(c*x))^2/(m^ 
2+8*m+15)+d^2*x^(1+m)*(c^2*x^2+1)^2*(a+b*arcsinh(c*x))^2/(5+m)-8*b*c*d^2*x 
^(2+m)*(a+b*arcsinh(c*x))*hypergeom([1/2, 1+1/2*m],[2+1/2*m],-c^2*x^2)/(2+ 
m)/(3+m)^2/(5+m)-6*b*c*d^2*x^(2+m)*(a+b*arcsinh(c*x))*hypergeom([1/2, 1+1/ 
2*m],[2+1/2*m],-c^2*x^2)/(5+m)^2/(m^2+5*m+6)-16*b*c*d^2*x^(2+m)*(a+b*arcsi 
nh(c*x))*hypergeom([1/2, 1+1/2*m],[2+1/2*m],-c^2*x^2)/(5+m)/(m^3+6*m^2+11* 
m+6)+6*b^2*c^2*d^2*x^(3+m)*hypergeom([1, 3/2+1/2*m, 3/2+1/2*m],[2+1/2*m, 5 
/2+1/2*m],-c^2*x^2)/(2+m)/(3+m)^2/(5+m)^2+8*b^2*c^2*d^2*x^(3+m)*hypergeom( 
[1, 3/2+1/2*m, 3/2+1/2*m],[2+1/2*m, 5/2+1/2*m],-c^2*x^2)/(2+m)/(3+m)^3/(5+ 
m)+16*b^2*c^2*d^2*x^(3+m)*hypergeom([1, 3/2+1/2*m, 3/2+1/2*m],[2+1/2*m, 5/ 
2+1/2*m],-c^2*x^2)/(3+m)^2/(5+m)/(m^2+3*m+2)
 

Mathematica [A] (warning: unable to verify)

Time = 0.41 (sec) , antiderivative size = 409, normalized size of antiderivative = 0.54 \[ \int x^m \left (d+c^2 d x^2\right )^2 (a+b \text {arcsinh}(c x))^2 \, dx=d^2 x^{1+m} \left (\frac {(a+b \text {arcsinh}(c x))^2}{1+m}+\frac {2 c^2 x^2 (a+b \text {arcsinh}(c x))^2}{3+m}+\frac {c^4 x^4 (a+b \text {arcsinh}(c x))^2}{5+m}+\frac {2 b c x \left (-\left ((3+m) (a+b \text {arcsinh}(c x)) \operatorname {Hypergeometric2F1}\left (\frac {1}{2},\frac {2+m}{2},\frac {4+m}{2},-c^2 x^2\right )\right )+b c x \, _3F_2\left (1,\frac {3}{2}+\frac {m}{2},\frac {3}{2}+\frac {m}{2};2+\frac {m}{2},\frac {5}{2}+\frac {m}{2};-c^2 x^2\right )\right )}{(1+m) (2+m) (3+m)}+\frac {4 b c^3 x^3 \left (-\left ((5+m) (a+b \text {arcsinh}(c x)) \operatorname {Hypergeometric2F1}\left (\frac {1}{2},\frac {4+m}{2},\frac {6+m}{2},-c^2 x^2\right )\right )+b c x \, _3F_2\left (1,\frac {5}{2}+\frac {m}{2},\frac {5}{2}+\frac {m}{2};3+\frac {m}{2},\frac {7}{2}+\frac {m}{2};-c^2 x^2\right )\right )}{(3+m) (4+m) (5+m)}+\frac {2 b c^5 x^5 \left (-\left ((7+m) (a+b \text {arcsinh}(c x)) \operatorname {Hypergeometric2F1}\left (\frac {1}{2},\frac {6+m}{2},\frac {8+m}{2},-c^2 x^2\right )\right )+b c x \, _3F_2\left (1,\frac {7}{2}+\frac {m}{2},\frac {7}{2}+\frac {m}{2};4+\frac {m}{2},\frac {9}{2}+\frac {m}{2};-c^2 x^2\right )\right )}{(5+m) (6+m) (7+m)}\right ) \] Input:

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

Output:

d^2*x^(1 + m)*((a + b*ArcSinh[c*x])^2/(1 + m) + (2*c^2*x^2*(a + b*ArcSinh[ 
c*x])^2)/(3 + m) + (c^4*x^4*(a + b*ArcSinh[c*x])^2)/(5 + m) + (2*b*c*x*(-( 
(3 + m)*(a + b*ArcSinh[c*x])*Hypergeometric2F1[1/2, (2 + m)/2, (4 + m)/2, 
-(c^2*x^2)]) + b*c*x*HypergeometricPFQ[{1, 3/2 + m/2, 3/2 + m/2}, {2 + m/2 
, 5/2 + m/2}, -(c^2*x^2)]))/((1 + m)*(2 + m)*(3 + m)) + (4*b*c^3*x^3*(-((5 
 + m)*(a + b*ArcSinh[c*x])*Hypergeometric2F1[1/2, (4 + m)/2, (6 + m)/2, -( 
c^2*x^2)]) + b*c*x*HypergeometricPFQ[{1, 5/2 + m/2, 5/2 + m/2}, {3 + m/2, 
7/2 + m/2}, -(c^2*x^2)]))/((3 + m)*(4 + m)*(5 + m)) + (2*b*c^5*x^5*(-((7 + 
 m)*(a + b*ArcSinh[c*x])*Hypergeometric2F1[1/2, (6 + m)/2, (8 + m)/2, -(c^ 
2*x^2)]) + b*c*x*HypergeometricPFQ[{1, 7/2 + m/2, 7/2 + m/2}, {4 + m/2, 9/ 
2 + m/2}, -(c^2*x^2)]))/((5 + m)*(6 + m)*(7 + m)))
 

Rubi [A] (verified)

Time = 2.88 (sec) , antiderivative size = 650, normalized size of antiderivative = 0.86, number of steps used = 9, number of rules used = 9, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.346, Rules used = {6223, 27, 6223, 244, 2009, 6191, 6221, 15, 6232}

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.

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

\(\Big \downarrow \) 6223

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 6223

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

\(\Big \downarrow \) 244

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

\(\Big \downarrow \) 2009

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

\(\Big \downarrow \) 6191

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

\(\Big \downarrow \) 6221

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

\(\Big \downarrow \) 15

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

\(\Big \downarrow \) 6232

\(\displaystyle \frac {4 d^2 \left (\frac {2 \left (\frac {x^{m+1} (a+b \text {arcsinh}(c x))^2}{m+1}-\frac {2 b c \left (\frac {x^{m+2} \operatorname {Hypergeometric2F1}\left (\frac {1}{2},\frac {m+2}{2},\frac {m+4}{2},-c^2 x^2\right ) (a+b \text {arcsinh}(c x))}{m+2}-\frac {b c x^{m+3} \, _3F_2\left (1,\frac {m}{2}+\frac {3}{2},\frac {m}{2}+\frac {3}{2};\frac {m}{2}+2,\frac {m}{2}+\frac {5}{2};-c^2 x^2\right )}{m^2+5 m+6}\right )}{m+1}\right )}{m+3}-\frac {2 b c \left (\frac {\frac {x^{m+2} \operatorname {Hypergeometric2F1}\left (\frac {1}{2},\frac {m+2}{2},\frac {m+4}{2},-c^2 x^2\right ) (a+b \text {arcsinh}(c x))}{m+2}-\frac {b c x^{m+3} \, _3F_2\left (1,\frac {m}{2}+\frac {3}{2},\frac {m}{2}+\frac {3}{2};\frac {m}{2}+2,\frac {m}{2}+\frac {5}{2};-c^2 x^2\right )}{m^2+5 m+6}}{m+3}+\frac {\sqrt {c^2 x^2+1} x^{m+2} (a+b \text {arcsinh}(c x))}{m+3}-\frac {b c x^{m+3}}{(m+3)^2}\right )}{m+3}+\frac {\left (c^2 x^2+1\right ) x^{m+1} (a+b \text {arcsinh}(c x))^2}{m+3}\right )}{m+5}-\frac {2 b c d^2 \left (\frac {3 \left (\frac {\frac {x^{m+2} \operatorname {Hypergeometric2F1}\left (\frac {1}{2},\frac {m+2}{2},\frac {m+4}{2},-c^2 x^2\right ) (a+b \text {arcsinh}(c x))}{m+2}-\frac {b c x^{m+3} \, _3F_2\left (1,\frac {m}{2}+\frac {3}{2},\frac {m}{2}+\frac {3}{2};\frac {m}{2}+2,\frac {m}{2}+\frac {5}{2};-c^2 x^2\right )}{m^2+5 m+6}}{m+3}+\frac {\sqrt {c^2 x^2+1} x^{m+2} (a+b \text {arcsinh}(c x))}{m+3}-\frac {b c x^{m+3}}{(m+3)^2}\right )}{m+5}+\frac {\left (c^2 x^2+1\right )^{3/2} x^{m+2} (a+b \text {arcsinh}(c x))}{m+5}-\frac {b c \left (\frac {c^2 x^{m+5}}{m+5}+\frac {x^{m+3}}{m+3}\right )}{m+5}\right )}{m+5}+\frac {d^2 \left (c^2 x^2+1\right )^2 x^{m+1} (a+b \text {arcsinh}(c x))^2}{m+5}\)

Input:

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

Output:

(d^2*x^(1 + m)*(1 + c^2*x^2)^2*(a + b*ArcSinh[c*x])^2)/(5 + m) + (4*d^2*(( 
x^(1 + m)*(1 + c^2*x^2)*(a + b*ArcSinh[c*x])^2)/(3 + m) + (2*((x^(1 + m)*( 
a + b*ArcSinh[c*x])^2)/(1 + m) - (2*b*c*((x^(2 + m)*(a + b*ArcSinh[c*x])*H 
ypergeometric2F1[1/2, (2 + m)/2, (4 + m)/2, -(c^2*x^2)])/(2 + m) - (b*c*x^ 
(3 + m)*HypergeometricPFQ[{1, 3/2 + m/2, 3/2 + m/2}, {2 + m/2, 5/2 + m/2}, 
 -(c^2*x^2)])/(6 + 5*m + m^2)))/(1 + m)))/(3 + m) - (2*b*c*(-((b*c*x^(3 + 
m))/(3 + m)^2) + (x^(2 + m)*Sqrt[1 + c^2*x^2]*(a + b*ArcSinh[c*x]))/(3 + m 
) + ((x^(2 + m)*(a + b*ArcSinh[c*x])*Hypergeometric2F1[1/2, (2 + m)/2, (4 
+ m)/2, -(c^2*x^2)])/(2 + m) - (b*c*x^(3 + m)*HypergeometricPFQ[{1, 3/2 + 
m/2, 3/2 + m/2}, {2 + m/2, 5/2 + m/2}, -(c^2*x^2)])/(6 + 5*m + m^2))/(3 + 
m)))/(3 + m)))/(5 + m) - (2*b*c*d^2*(-((b*c*(x^(3 + m)/(3 + m) + (c^2*x^(5 
 + m))/(5 + m)))/(5 + m)) + (x^(2 + m)*(1 + c^2*x^2)^(3/2)*(a + b*ArcSinh[ 
c*x]))/(5 + m) + (3*(-((b*c*x^(3 + m))/(3 + m)^2) + (x^(2 + m)*Sqrt[1 + c^ 
2*x^2]*(a + b*ArcSinh[c*x]))/(3 + m) + ((x^(2 + m)*(a + b*ArcSinh[c*x])*Hy 
pergeometric2F1[1/2, (2 + m)/2, (4 + m)/2, -(c^2*x^2)])/(2 + m) - (b*c*x^( 
3 + m)*HypergeometricPFQ[{1, 3/2 + m/2, 3/2 + m/2}, {2 + m/2, 5/2 + m/2}, 
-(c^2*x^2)])/(6 + 5*m + m^2))/(3 + m)))/(5 + m)))/(5 + m)
 

Defintions of rubi rules used

rule 15

Int[(a_.)*(x_)^(m_.), x_Symbol] :> Simp[a*(x^(m + 1)/(m + 1)), x] /; FreeQ[ 
{a, m}, x] && NeQ[m, -1]
 

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 244

Int[((c_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^2)^(p_.), x_Symbol] :> Int[Expand 
Integrand[(c*x)^m*(a + b*x^2)^p, x], x] /; FreeQ[{a, b, c, m}, x] && IGtQ[p 
, 0]
 

rule 2009

Int[u_, x_Symbol] :> Simp[IntSum[u, x], x] /; SumQ[u]
 

rule 6191

Int[((a_.) + ArcSinh[(c_.)*(x_)]*(b_.))^(n_.)*((d_.)*(x_))^(m_.), x_Symbol] 
 :> Simp[(d*x)^(m + 1)*((a + b*ArcSinh[c*x])^n/(d*(m + 1))), x] - Simp[b*c* 
(n/(d*(m + 1)))   Int[(d*x)^(m + 1)*((a + b*ArcSinh[c*x])^(n - 1)/Sqrt[1 + 
c^2*x^2]), x], x] /; FreeQ[{a, b, c, d, m}, x] && IGtQ[n, 0] && NeQ[m, -1]
 

rule 6221

Int[((a_.) + ArcSinh[(c_.)*(x_)]*(b_.))^(n_.)*((f_.)*(x_))^(m_)*Sqrt[(d_) + 
 (e_.)*(x_)^2], x_Symbol] :> Simp[(f*x)^(m + 1)*Sqrt[d + e*x^2]*((a + b*Arc 
Sinh[c*x])^n/(f*(m + 2))), x] + (Simp[(1/(m + 2))*Simp[Sqrt[d + e*x^2]/Sqrt 
[1 + c^2*x^2]]   Int[(f*x)^m*((a + b*ArcSinh[c*x])^n/Sqrt[1 + c^2*x^2]), x] 
, x] - Simp[b*c*(n/(f*(m + 2)))*Simp[Sqrt[d + e*x^2]/Sqrt[1 + c^2*x^2]]   I 
nt[(f*x)^(m + 1)*(a + b*ArcSinh[c*x])^(n - 1), x], x]) /; FreeQ[{a, b, c, d 
, e, f, m}, x] && EqQ[e, c^2*d] && IGtQ[n, 0] && (IGtQ[m, -2] || EqQ[n, 1])
 

rule 6223

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

rule 6232

Int[(((a_.) + ArcSinh[(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*ArcSinh[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^2*x^2]/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[e, c^2*d] &&  !IntegerQ[m]
 

Maple [F]

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

Input:

int(x^m*(c^2*d*x^2+d)^2*(a+b*arcsinh(x*c))^2,x)
 

Output:

int(x^m*(c^2*d*x^2+d)^2*(a+b*arcsinh(x*c))^2,x)
 

Fricas [F]

\[ \int x^m \left (d+c^2 d x^2\right )^2 (a+b \text {arcsinh}(c x))^2 \, dx=\int { {\left (c^{2} d x^{2} + d\right )}^{2} {\left (b \operatorname {arsinh}\left (c x\right ) + a\right )}^{2} x^{m} \,d x } \] Input:

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

Output:

integral((a^2*c^4*d^2*x^4 + 2*a^2*c^2*d^2*x^2 + a^2*d^2 + (b^2*c^4*d^2*x^4 
 + 2*b^2*c^2*d^2*x^2 + b^2*d^2)*arcsinh(c*x)^2 + 2*(a*b*c^4*d^2*x^4 + 2*a* 
b*c^2*d^2*x^2 + a*b*d^2)*arcsinh(c*x))*x^m, x)
 

Sympy [F]

\[ \int x^m \left (d+c^2 d x^2\right )^2 (a+b \text {arcsinh}(c x))^2 \, dx=d^{2} \left (\int a^{2} x^{m}\, dx + \int b^{2} x^{m} \operatorname {asinh}^{2}{\left (c x \right )}\, dx + \int 2 a b x^{m} \operatorname {asinh}{\left (c x \right )}\, dx + \int 2 a^{2} c^{2} x^{2} x^{m}\, dx + \int a^{2} c^{4} x^{4} x^{m}\, dx + \int 2 b^{2} c^{2} x^{2} x^{m} \operatorname {asinh}^{2}{\left (c x \right )}\, dx + \int b^{2} c^{4} x^{4} x^{m} \operatorname {asinh}^{2}{\left (c x \right )}\, dx + \int 4 a b c^{2} x^{2} x^{m} \operatorname {asinh}{\left (c x \right )}\, dx + \int 2 a b c^{4} x^{4} x^{m} \operatorname {asinh}{\left (c x \right )}\, dx\right ) \] Input:

integrate(x**m*(c**2*d*x**2+d)**2*(a+b*asinh(c*x))**2,x)
 

Output:

d**2*(Integral(a**2*x**m, x) + Integral(b**2*x**m*asinh(c*x)**2, x) + Inte 
gral(2*a*b*x**m*asinh(c*x), x) + Integral(2*a**2*c**2*x**2*x**m, x) + Inte 
gral(a**2*c**4*x**4*x**m, x) + Integral(2*b**2*c**2*x**2*x**m*asinh(c*x)** 
2, x) + Integral(b**2*c**4*x**4*x**m*asinh(c*x)**2, x) + Integral(4*a*b*c* 
*2*x**2*x**m*asinh(c*x), x) + Integral(2*a*b*c**4*x**4*x**m*asinh(c*x), x) 
)
 

Maxima [F]

\[ \int x^m \left (d+c^2 d x^2\right )^2 (a+b \text {arcsinh}(c x))^2 \, dx=\int { {\left (c^{2} d x^{2} + d\right )}^{2} {\left (b \operatorname {arsinh}\left (c x\right ) + a\right )}^{2} x^{m} \,d x } \] Input:

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

Output:

a^2*c^4*d^2*x^(m + 5)/(m + 5) + 2*a^2*c^2*d^2*x^(m + 3)/(m + 3) + a^2*d^2* 
x^(m + 1)/(m + 1) + ((m^2 + 4*m + 3)*b^2*c^4*d^2*x^5 + 2*(m^2 + 6*m + 5)*b 
^2*c^2*d^2*x^3 + (m^2 + 8*m + 15)*b^2*d^2*x)*x^m*log(c*x + sqrt(c^2*x^2 + 
1))^2/(m^3 + 9*m^2 + 23*m + 15) + integrate(2*((((m^3 + 9*m^2 + 23*m + 15) 
*a*b*c^6*d^2 - (m^2 + 4*m + 3)*b^2*c^6*d^2)*x^6 + (m^3 + 9*m^2 + 23*m + 15 
)*a*b*d^2 + (3*(m^3 + 9*m^2 + 23*m + 15)*a*b*c^4*d^2 - 2*(m^2 + 6*m + 5)*b 
^2*c^4*d^2)*x^4 + (3*(m^3 + 9*m^2 + 23*m + 15)*a*b*c^2*d^2 - (m^2 + 8*m + 
15)*b^2*c^2*d^2)*x^2)*sqrt(c^2*x^2 + 1)*x^m + (((m^3 + 9*m^2 + 23*m + 15)* 
a*b*c^7*d^2 - (m^2 + 4*m + 3)*b^2*c^7*d^2)*x^7 + (3*(m^3 + 9*m^2 + 23*m + 
15)*a*b*c^5*d^2 - (3*m^2 + 16*m + 13)*b^2*c^5*d^2)*x^5 + (3*(m^3 + 9*m^2 + 
 23*m + 15)*a*b*c^3*d^2 - (3*m^2 + 20*m + 25)*b^2*c^3*d^2)*x^3 + ((m^3 + 9 
*m^2 + 23*m + 15)*a*b*c*d^2 - (m^2 + 8*m + 15)*b^2*c*d^2)*x)*x^m)*log(c*x 
+ sqrt(c^2*x^2 + 1))/((m^3 + 9*m^2 + 23*m + 15)*c^3*x^3 + (m^3 + 9*m^2 + 2 
3*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^2*x^2 + 1)), x)
                                                                                    
                                                                                    
 

Giac [F(-2)]

Exception generated. \[ \int x^m \left (d+c^2 d x^2\right )^2 (a+b \text {arcsinh}(c x))^2 \, dx=\text {Exception raised: TypeError} \] Input:

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

Output:

Exception raised: TypeError >> an error occurred running a Giac command:IN 
PUT:sage2:=int(sage0,sageVARx):;OUTPUT:sym2poly/r2sym(const gen & e,const 
index_m & i,const vecteur & l) Error: Bad Argument Value
 

Mupad [F(-1)]

Timed out. \[ \int x^m \left (d+c^2 d x^2\right )^2 (a+b \text {arcsinh}(c x))^2 \, dx=\int x^m\,{\left (a+b\,\mathrm {asinh}\left (c\,x\right )\right )}^2\,{\left (d\,c^2\,x^2+d\right )}^2 \,d x \] Input:

int(x^m*(a + b*asinh(c*x))^2*(d + c^2*d*x^2)^2,x)
 

Output:

int(x^m*(a + b*asinh(c*x))^2*(d + c^2*d*x^2)^2, x)
 

Reduce [F]

\[ \int x^m \left (d+c^2 d x^2\right )^2 (a+b \text {arcsinh}(c x))^2 \, dx =\text {Too large to display} \] Input:

int(x^m*(c^2*d*x^2+d)^2*(a+b*asinh(c*x))^2,x)
 

Output:

(d**2*(x**m*a**2*c**4*m**2*x**5 + 4*x**m*a**2*c**4*m*x**5 + 3*x**m*a**2*c* 
*4*x**5 + 2*x**m*a**2*c**2*m**2*x**3 + 12*x**m*a**2*c**2*m*x**3 + 10*x**m* 
a**2*c**2*x**3 + x**m*a**2*m**2*x + 8*x**m*a**2*m*x + 15*x**m*a**2*x + 2*i 
nt(x**m*asinh(c*x)*x**4,x)*a*b*c**4*m**3 + 18*int(x**m*asinh(c*x)*x**4,x)* 
a*b*c**4*m**2 + 46*int(x**m*asinh(c*x)*x**4,x)*a*b*c**4*m + 30*int(x**m*as 
inh(c*x)*x**4,x)*a*b*c**4 + 4*int(x**m*asinh(c*x)*x**2,x)*a*b*c**2*m**3 + 
36*int(x**m*asinh(c*x)*x**2,x)*a*b*c**2*m**2 + 92*int(x**m*asinh(c*x)*x**2 
,x)*a*b*c**2*m + 60*int(x**m*asinh(c*x)*x**2,x)*a*b*c**2 + 2*int(x**m*asin 
h(c*x),x)*a*b*m**3 + 18*int(x**m*asinh(c*x),x)*a*b*m**2 + 46*int(x**m*asin 
h(c*x),x)*a*b*m + 30*int(x**m*asinh(c*x),x)*a*b + int(x**m*asinh(c*x)**2*x 
**4,x)*b**2*c**4*m**3 + 9*int(x**m*asinh(c*x)**2*x**4,x)*b**2*c**4*m**2 + 
23*int(x**m*asinh(c*x)**2*x**4,x)*b**2*c**4*m + 15*int(x**m*asinh(c*x)**2* 
x**4,x)*b**2*c**4 + 2*int(x**m*asinh(c*x)**2*x**2,x)*b**2*c**2*m**3 + 18*i 
nt(x**m*asinh(c*x)**2*x**2,x)*b**2*c**2*m**2 + 46*int(x**m*asinh(c*x)**2*x 
**2,x)*b**2*c**2*m + 30*int(x**m*asinh(c*x)**2*x**2,x)*b**2*c**2 + int(x** 
m*asinh(c*x)**2,x)*b**2*m**3 + 9*int(x**m*asinh(c*x)**2,x)*b**2*m**2 + 23* 
int(x**m*asinh(c*x)**2,x)*b**2*m + 15*int(x**m*asinh(c*x)**2,x)*b**2))/(m* 
*3 + 9*m**2 + 23*m + 15)