\(\int x^m (d+c^2 d x^2)^3 (a+b \text {arcsinh}(c x))^2 \, dx\) [327]

Optimal result

Integrand size = 26, antiderivative size = 1311 \[ \int x^m \left (d+c^2 d x^2\right )^3 (a+b \text {arcsinh}(c x))^2 \, dx =\text {Too large to display} \] Output:

-30*b*c*d^3*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)/(7+m)^2/(m^2+5*m+6)-36*b*c*d^3*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/(7+m)/(m^2+5*m+6)-96* 
b*c*d^3*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)/(7+m)/(m^3+6*m^2+11*m+6)+2*b^2*c^2*d^3*x^(3+m)/(3+m)/(7+m)^2+1 
0*b^2*c^2*d^3*x^(3+m)/(7+m)^2/(m^2+8*m+15)+10*b^2*c^4*d^3*x^(5+m)/(5+m)^2/ 
(7+m)^2+4*b^2*c^4*d^3*x^(5+m)/(5+m)/(7+m)^2+12*b^2*c^4*d^3*x^(5+m)/(5+m)^3 
/(7+m)+48*d^3*x^(1+m)*(a+b*arcsinh(c*x))^2/(5+m)/(7+m)/(m^2+4*m+3)+24*d^3* 
x^(1+m)*(c^2*x^2+1)*(a+b*arcsinh(c*x))^2/(7+m)/(m^2+8*m+15)-36*b*c*d^3*x^( 
2+m)*(c^2*x^2+1)^(1/2)*(a+b*arcsinh(c*x))/(3+m)/(5+m)^2/(7+m)-48*b*c*d^3*x 
^(2+m)*(c^2*x^2+1)^(1/2)*(a+b*arcsinh(c*x))/(3+m)^2/(5+m)/(7+m)-10*b*c*d^3 
*x^(2+m)*(c^2*x^2+1)^(3/2)*(a+b*arcsinh(c*x))/(5+m)/(7+m)^2-12*b*c*d^3*x^( 
2+m)*(c^2*x^2+1)^(3/2)*(a+b*arcsinh(c*x))/(5+m)^2/(7+m)-30*b*c*d^3*x^(2+m) 
*(c^2*x^2+1)^(1/2)*(a+b*arcsinh(c*x))/(7+m)^2/(m^2+8*m+15)+d^3*x^(1+m)*(c^ 
2*x^2+1)^3*(a+b*arcsinh(c*x))^2/(7+m)+2*b^2*c^6*d^3*x^(7+m)/(7+m)^3+6*d^3* 
x^(1+m)*(c^2*x^2+1)^2*(a+b*arcsinh(c*x))^2/(5+m)/(7+m)+30*b^2*c^2*d^3*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)/(7+m)^2+36*b^2*c^2*d^3*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/(7+m)+48*b^ 
2*c^2*d^3*x^(3+m)*hypergeom([1, 3/2+1/2*m, 3/2+1/2*m],[2+1/2*m, 5/2+1/2...
 

Mathematica [A] (warning: unable to verify)

Time = 0.58 (sec) , antiderivative size = 546, normalized size of antiderivative = 0.42 \[ \int x^m \left (d+c^2 d x^2\right )^3 (a+b \text {arcsinh}(c x))^2 \, dx=d^3 x^{1+m} \left (\frac {(a+b \text {arcsinh}(c x))^2}{1+m}+\frac {3 c^2 x^2 (a+b \text {arcsinh}(c x))^2}{3+m}+\frac {3 c^4 x^4 (a+b \text {arcsinh}(c x))^2}{5+m}+\frac {c^6 x^6 (a+b \text {arcsinh}(c x))^2}{7+m}+\frac {2 b c x \left (-\left ((3+m) (a+b \text {arcsinh}(c x)) \operatorname {Hypergeometric2F1}\left (\frac {1}{2},\frac {2+m}{2},\frac {4+m}{2},-c^2 x^2\right )\right )+b c x \, _3F_2\left (1,\frac {3}{2}+\frac {m}{2},\frac {3}{2}+\frac {m}{2};2+\frac {m}{2},\frac {5}{2}+\frac {m}{2};-c^2 x^2\right )\right )}{(1+m) (2+m) (3+m)}+\frac {6 b c^3 x^3 \left (-\left ((5+m) (a+b \text {arcsinh}(c x)) \operatorname {Hypergeometric2F1}\left (\frac {1}{2},\frac {4+m}{2},\frac {6+m}{2},-c^2 x^2\right )\right )+b c x \, _3F_2\left (1,\frac {5}{2}+\frac {m}{2},\frac {5}{2}+\frac {m}{2};3+\frac {m}{2},\frac {7}{2}+\frac {m}{2};-c^2 x^2\right )\right )}{(3+m) (4+m) (5+m)}+\frac {6 b c^5 x^5 \left (-\left ((7+m) (a+b \text {arcsinh}(c x)) \operatorname {Hypergeometric2F1}\left (\frac {1}{2},\frac {6+m}{2},\frac {8+m}{2},-c^2 x^2\right )\right )+b c x \, _3F_2\left (1,\frac {7}{2}+\frac {m}{2},\frac {7}{2}+\frac {m}{2};4+\frac {m}{2},\frac {9}{2}+\frac {m}{2};-c^2 x^2\right )\right )}{(5+m) (6+m) (7+m)}+\frac {2 b c^7 x^7 \left (-\left ((9+m) (a+b \text {arcsinh}(c x)) \operatorname {Hypergeometric2F1}\left (\frac {1}{2},4+\frac {m}{2},5+\frac {m}{2},-c^2 x^2\right )\right )+b c x \, _3F_2\left (1,\frac {9}{2}+\frac {m}{2},\frac {9}{2}+\frac {m}{2};5+\frac {m}{2},\frac {11}{2}+\frac {m}{2};-c^2 x^2\right )\right )}{(7+m) (8+m) (9+m)}\right ) \] Input:

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

Output:

d^3*x^(1 + m)*((a + b*ArcSinh[c*x])^2/(1 + m) + (3*c^2*x^2*(a + b*ArcSinh[ 
c*x])^2)/(3 + m) + (3*c^4*x^4*(a + b*ArcSinh[c*x])^2)/(5 + m) + (c^6*x^6*( 
a + b*ArcSinh[c*x])^2)/(7 + 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*Hypergeo 
metricPFQ[{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)) + (6*b*c^3*x^3*(-((5 + m)*(a + b*ArcSinh[c*x])*Hy 
pergeometric2F1[1/2, (4 + m)/2, (6 + m)/2, -(c^2*x^2)]) + b*c*x*Hypergeome 
tricPFQ[{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)) + (6*b*c^5*x^5*(-((7 + m)*(a + b*ArcSinh[c*x])*Hype 
rgeometric2F1[1/2, (6 + m)/2, (8 + m)/2, -(c^2*x^2)]) + b*c*x*Hypergeometr 
icPFQ[{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)) + (2*b*c^7*x^7*(-((9 + m)*(a + b*ArcSinh[c*x])*Hyperg 
eometric2F1[1/2, 4 + m/2, 5 + m/2, -(c^2*x^2)]) + b*c*x*HypergeometricPFQ[ 
{1, 9/2 + m/2, 9/2 + m/2}, {5 + m/2, 11/2 + m/2}, -(c^2*x^2)]))/((7 + m)*( 
8 + m)*(9 + m)))
 

Rubi [A] (verified)

Time = 3.64 (sec) , antiderivative size = 1028, normalized size of antiderivative = 0.78, number of steps used = 12, number of rules used = 12, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.462, Rules used = {6223, 27, 6223, 244, 2009, 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.

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

\(\Big \downarrow \) 6223

\(\displaystyle -\frac {2 b c d^3 \int x^{m+1} \left (c^2 x^2+1\right )^{5/2} (a+b \text {arcsinh}(c x))dx}{m+7}+\frac {6 d \int d^2 x^m \left (c^2 x^2+1\right )^2 (a+b \text {arcsinh}(c x))^2dx}{m+7}+\frac {d^3 \left (c^2 x^2+1\right )^3 x^{m+1} (a+b \text {arcsinh}(c x))^2}{m+7}\)

\(\Big \downarrow \) 27

\(\displaystyle -\frac {2 b c d^3 \int x^{m+1} \left (c^2 x^2+1\right )^{5/2} (a+b \text {arcsinh}(c x))dx}{m+7}+\frac {6 d^3 \int x^m \left (c^2 x^2+1\right )^2 (a+b \text {arcsinh}(c x))^2dx}{m+7}+\frac {d^3 \left (c^2 x^2+1\right )^3 x^{m+1} (a+b \text {arcsinh}(c x))^2}{m+7}\)

\(\Big \downarrow \) 6223

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

\(\Big \downarrow \) 244

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

\(\Big \downarrow \) 2009

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

\(\Big \downarrow \) 6223

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

\(\Big \downarrow \) 244

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

\(\Big \downarrow \) 2009

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

\(\Big \downarrow \) 6191

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

\(\Big \downarrow \) 6221

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

\(\Big \downarrow \) 15

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

\(\Big \downarrow \) 6232

\(\displaystyle \frac {d^3 \left (c^2 x^2+1\right )^3 (a+b \text {arcsinh}(c x))^2 x^{m+1}}{m+7}+\frac {6 d^3 \left (\frac {\left (c^2 x^2+1\right )^2 (a+b \text {arcsinh}(c x))^2 x^{m+1}}{m+5}+\frac {4 \left (\frac {\left (c^2 x^2+1\right ) (a+b \text {arcsinh}(c x))^2 x^{m+1}}{m+3}+\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} (a+b \text {arcsinh}(c x)) \operatorname {Hypergeometric2F1}\left (\frac {1}{2},\frac {m+2}{2},\frac {m+4}{2},-c^2 x^2\right )}{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 {\sqrt {c^2 x^2+1} (a+b \text {arcsinh}(c x)) x^{m+2}}{m+3}-\frac {b c x^{m+3}}{(m+3)^2}+\frac {\frac {x^{m+2} (a+b \text {arcsinh}(c x)) \operatorname {Hypergeometric2F1}\left (\frac {1}{2},\frac {m+2}{2},\frac {m+4}{2},-c^2 x^2\right )}{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}\right )}{m+3}\right )}{m+5}-\frac {2 b c \left (\frac {\left (c^2 x^2+1\right )^{3/2} (a+b \text {arcsinh}(c x)) x^{m+2}}{m+5}-\frac {b c \left (\frac {x^{m+3}}{m+3}+\frac {c^2 x^{m+5}}{m+5}\right )}{m+5}+\frac {3 \left (\frac {\sqrt {c^2 x^2+1} (a+b \text {arcsinh}(c x)) x^{m+2}}{m+3}-\frac {b c x^{m+3}}{(m+3)^2}+\frac {\frac {x^{m+2} (a+b \text {arcsinh}(c x)) \operatorname {Hypergeometric2F1}\left (\frac {1}{2},\frac {m+2}{2},\frac {m+4}{2},-c^2 x^2\right )}{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}\right )}{m+5}\right )}{m+5}\right )}{m+7}-\frac {2 b c d^3 \left (\frac {\left (c^2 x^2+1\right )^{5/2} (a+b \text {arcsinh}(c x)) x^{m+2}}{m+7}-\frac {b c \left (\frac {x^{m+3}}{m+3}+\frac {2 c^2 x^{m+5}}{m+5}+\frac {c^4 x^{m+7}}{m+7}\right )}{m+7}+\frac {5 \left (\frac {\left (c^2 x^2+1\right )^{3/2} (a+b \text {arcsinh}(c x)) x^{m+2}}{m+5}-\frac {b c \left (\frac {x^{m+3}}{m+3}+\frac {c^2 x^{m+5}}{m+5}\right )}{m+5}+\frac {3 \left (\frac {\sqrt {c^2 x^2+1} (a+b \text {arcsinh}(c x)) x^{m+2}}{m+3}-\frac {b c x^{m+3}}{(m+3)^2}+\frac {\frac {x^{m+2} (a+b \text {arcsinh}(c x)) \operatorname {Hypergeometric2F1}\left (\frac {1}{2},\frac {m+2}{2},\frac {m+4}{2},-c^2 x^2\right )}{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}\right )}{m+5}\right )}{m+7}\right )}{m+7}\)

Input:

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

Output:

(d^3*x^(1 + m)*(1 + c^2*x^2)^3*(a + b*ArcSinh[c*x])^2)/(7 + m) + (6*d^3*(( 
x^(1 + m)*(1 + c^2*x^2)^2*(a + b*ArcSinh[c*x])^2)/(5 + m) + (4*((x^(1 + m) 
*(1 + c^2*x^2)*(a + b*ArcSinh[c*x])^2)/(3 + m) + (2*((x^(1 + m)*(a + b*Arc 
Sinh[c*x])^2)/(1 + m) - (2*b*c*((x^(2 + m)*(a + b*ArcSinh[c*x])*Hypergeome 
tric2F1[1/2, (2 + m)/2, (4 + m)/2, -(c^2*x^2)])/(2 + m) - (b*c*x^(3 + m)*H 
ypergeometricPFQ[{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*(-((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])*Hypergeometric2 
F1[1/2, (2 + m)/2, (4 + m)/2, -(c^2*x^2)])/(2 + m) - (b*c*x^(3 + m)*Hyperg 
eometricPFQ[{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)))/(7 + m) - (2*b*c*d^3*(-((b* 
c*(x^(3 + m)/(3 + m) + (2*c^2*x^(5 + m))/(5 + m) + (c^4*x^(7 + m))/(7 + m) 
))/(7 + m)) + (x^(2 + m)*(1 + c^2*x^2)^(5/2)*(a + b*ArcSinh[c*x]))/(7 +...
 

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]

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

Input:

int(x^m*(c^2*d*x^2+d)^3*(a+b*arcsinh(x*c))^2,x)
 

Output:

int(x^m*(c^2*d*x^2+d)^3*(a+b*arcsinh(x*c))^2,x)
 

Fricas [F]

\[ \int x^m \left (d+c^2 d x^2\right )^3 (a+b \text {arcsinh}(c x))^2 \, dx=\int { {\left (c^{2} d x^{2} + d\right )}^{3} {\left (b \operatorname {arsinh}\left (c x\right ) + a\right )}^{2} x^{m} \,d x } \] Input:

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

Output:

integral((a^2*c^6*d^3*x^6 + 3*a^2*c^4*d^3*x^4 + 3*a^2*c^2*d^3*x^2 + a^2*d^ 
3 + (b^2*c^6*d^3*x^6 + 3*b^2*c^4*d^3*x^4 + 3*b^2*c^2*d^3*x^2 + b^2*d^3)*ar 
csinh(c*x)^2 + 2*(a*b*c^6*d^3*x^6 + 3*a*b*c^4*d^3*x^4 + 3*a*b*c^2*d^3*x^2 
+ a*b*d^3)*arcsinh(c*x))*x^m, x)
 

Sympy [F(-1)]

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

integrate(x**m*(c**2*d*x**2+d)**3*(a+b*asinh(c*x))**2,x)
 

Output:

Timed out
 

Maxima [F]

\[ \int x^m \left (d+c^2 d x^2\right )^3 (a+b \text {arcsinh}(c x))^2 \, dx=\int { {\left (c^{2} d x^{2} + d\right )}^{3} {\left (b \operatorname {arsinh}\left (c x\right ) + a\right )}^{2} x^{m} \,d x } \] Input:

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

Output:

a^2*c^6*d^3*x^(m + 7)/(m + 7) + 3*a^2*c^4*d^3*x^(m + 5)/(m + 5) + 3*a^2*c^ 
2*d^3*x^(m + 3)/(m + 3) + a^2*d^3*x^(m + 1)/(m + 1) + ((m^3 + 9*m^2 + 23*m 
 + 15)*b^2*c^6*d^3*x^7 + 3*(m^3 + 11*m^2 + 31*m + 21)*b^2*c^4*d^3*x^5 + 3* 
(m^3 + 13*m^2 + 47*m + 35)*b^2*c^2*d^3*x^3 + (m^3 + 15*m^2 + 71*m + 105)*b 
^2*d^3*x)*x^m*log(c*x + sqrt(c^2*x^2 + 1))^2/(m^4 + 16*m^3 + 86*m^2 + 176* 
m + 105) + integrate(2*((((m^4 + 16*m^3 + 86*m^2 + 176*m + 105)*a*b*c^8*d^ 
3 - (m^3 + 9*m^2 + 23*m + 15)*b^2*c^8*d^3)*x^8 + (4*(m^4 + 16*m^3 + 86*m^2 
 + 176*m + 105)*a*b*c^6*d^3 - 3*(m^3 + 11*m^2 + 31*m + 21)*b^2*c^6*d^3)*x^ 
6 + (m^4 + 16*m^3 + 86*m^2 + 176*m + 105)*a*b*d^3 + 3*(2*(m^4 + 16*m^3 + 8 
6*m^2 + 176*m + 105)*a*b*c^4*d^3 - (m^3 + 13*m^2 + 47*m + 35)*b^2*c^4*d^3) 
*x^4 + (4*(m^4 + 16*m^3 + 86*m^2 + 176*m + 105)*a*b*c^2*d^3 - (m^3 + 15*m^ 
2 + 71*m + 105)*b^2*c^2*d^3)*x^2)*sqrt(c^2*x^2 + 1)*x^m + (((m^4 + 16*m^3 
+ 86*m^2 + 176*m + 105)*a*b*c^9*d^3 - (m^3 + 9*m^2 + 23*m + 15)*b^2*c^9*d^ 
3)*x^9 + 2*(2*(m^4 + 16*m^3 + 86*m^2 + 176*m + 105)*a*b*c^7*d^3 - (2*m^3 + 
 21*m^2 + 58*m + 39)*b^2*c^7*d^3)*x^7 + 6*((m^4 + 16*m^3 + 86*m^2 + 176*m 
+ 105)*a*b*c^5*d^3 - (m^3 + 12*m^2 + 39*m + 28)*b^2*c^5*d^3)*x^5 + 2*(2*(m 
^4 + 16*m^3 + 86*m^2 + 176*m + 105)*a*b*c^3*d^3 - (2*m^3 + 27*m^2 + 106*m 
+ 105)*b^2*c^3*d^3)*x^3 + ((m^4 + 16*m^3 + 86*m^2 + 176*m + 105)*a*b*c*d^3 
 - (m^3 + 15*m^2 + 71*m + 105)*b^2*c*d^3)*x)*x^m)*log(c*x + sqrt(c^2*x^2 + 
 1))/((m^4 + 16*m^3 + 86*m^2 + 176*m + 105)*c^3*x^3 + (m^4 + 16*m^3 + 8...
                                                                                    
                                                                                    
 

Giac [F(-2)]

Exception generated. \[ \int x^m \left (d+c^2 d x^2\right )^3 (a+b \text {arcsinh}(c x))^2 \, dx=\text {Exception raised: TypeError} \] Input:

integrate(x^m*(c^2*d*x^2+d)^3*(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. \[ \int x^m \left (d+c^2 d x^2\right )^3 (a+b \text {arcsinh}(c x))^2 \, dx=\int x^m\,{\left (a+b\,\mathrm {asinh}\left (c\,x\right )\right )}^2\,{\left (d\,c^2\,x^2+d\right )}^3 \,d x \] Input:

int(x^m*(a + b*asinh(c*x))^2*(d + c^2*d*x^2)^3,x)
 

Output:

int(x^m*(a + b*asinh(c*x))^2*(d + c^2*d*x^2)^3, x)
 

Reduce [F]

\[ \int x^m \left (d+c^2 d x^2\right )^3 (a+b \text {arcsinh}(c x))^2 \, dx =\text {Too large to display} \] Input:

int(x^m*(c^2*d*x^2+d)^3*(a+b*asinh(c*x))^2,x)
 

Output:

(d**3*(x**m*a**2*c**6*m**3*x**7 + 9*x**m*a**2*c**6*m**2*x**7 + 23*x**m*a** 
2*c**6*m*x**7 + 15*x**m*a**2*c**6*x**7 + 3*x**m*a**2*c**4*m**3*x**5 + 33*x 
**m*a**2*c**4*m**2*x**5 + 93*x**m*a**2*c**4*m*x**5 + 63*x**m*a**2*c**4*x** 
5 + 3*x**m*a**2*c**2*m**3*x**3 + 39*x**m*a**2*c**2*m**2*x**3 + 141*x**m*a* 
*2*c**2*m*x**3 + 105*x**m*a**2*c**2*x**3 + x**m*a**2*m**3*x + 15*x**m*a**2 
*m**2*x + 71*x**m*a**2*m*x + 105*x**m*a**2*x + 2*int(x**m*asinh(c*x)*x**6, 
x)*a*b*c**6*m**4 + 32*int(x**m*asinh(c*x)*x**6,x)*a*b*c**6*m**3 + 172*int( 
x**m*asinh(c*x)*x**6,x)*a*b*c**6*m**2 + 352*int(x**m*asinh(c*x)*x**6,x)*a* 
b*c**6*m + 210*int(x**m*asinh(c*x)*x**6,x)*a*b*c**6 + 6*int(x**m*asinh(c*x 
)*x**4,x)*a*b*c**4*m**4 + 96*int(x**m*asinh(c*x)*x**4,x)*a*b*c**4*m**3 + 5 
16*int(x**m*asinh(c*x)*x**4,x)*a*b*c**4*m**2 + 1056*int(x**m*asinh(c*x)*x* 
*4,x)*a*b*c**4*m + 630*int(x**m*asinh(c*x)*x**4,x)*a*b*c**4 + 6*int(x**m*a 
sinh(c*x)*x**2,x)*a*b*c**2*m**4 + 96*int(x**m*asinh(c*x)*x**2,x)*a*b*c**2* 
m**3 + 516*int(x**m*asinh(c*x)*x**2,x)*a*b*c**2*m**2 + 1056*int(x**m*asinh 
(c*x)*x**2,x)*a*b*c**2*m + 630*int(x**m*asinh(c*x)*x**2,x)*a*b*c**2 + 2*in 
t(x**m*asinh(c*x),x)*a*b*m**4 + 32*int(x**m*asinh(c*x),x)*a*b*m**3 + 172*i 
nt(x**m*asinh(c*x),x)*a*b*m**2 + 352*int(x**m*asinh(c*x),x)*a*b*m + 210*in 
t(x**m*asinh(c*x),x)*a*b + int(x**m*asinh(c*x)**2*x**6,x)*b**2*c**6*m**4 + 
 16*int(x**m*asinh(c*x)**2*x**6,x)*b**2*c**6*m**3 + 86*int(x**m*asinh(c*x) 
**2*x**6,x)*b**2*c**6*m**2 + 176*int(x**m*asinh(c*x)**2*x**6,x)*b**2*c*...