\(\int \frac {x^m (a+b \text {arcsinh}(c x))^2}{(d+c^2 d x^2)^3} \, dx\) [332]

Optimal result

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

Defer(Int)(x^m*(a+b*arcsinh(c*x))^2/(c^2*d*x^2+d)^3,x)
 

Mathematica [N/A]

Not integrable

Time = 6.66 (sec) , antiderivative size = 28, normalized size of antiderivative = 1.08 \[ \int \frac {x^m (a+b \text {arcsinh}(c x))^2}{\left (d+c^2 d x^2\right )^3} \, dx=\int \frac {x^m (a+b \text {arcsinh}(c x))^2}{\left (d+c^2 d x^2\right )^3} \, dx \] Input:

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

Output:

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

Rubi [N/A]

Not integrable

Time = 2.23 (sec) , antiderivative size = 26, normalized size of antiderivative = 1.00, number of steps used = 8, number of rules used = 0, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.000, Rules used = {}

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.

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

\(\Big \downarrow \) 6226

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 6226

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

\(\Big \downarrow \) 278

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

\(\Big \downarrow \) 6226

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

\(\Big \downarrow \) 278

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

\(\Big \downarrow \) 6232

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

\(\Big \downarrow \) 6239

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

Input:

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

Output:

$Aborted
 

Maple [N/A]

Not integrable

Time = 1.46 (sec) , antiderivative size = 26, normalized size of antiderivative = 1.00

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

Input:

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

Output:

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

Fricas [N/A]

Not integrable

Time = 0.09 (sec) , antiderivative size = 66, normalized size of antiderivative = 2.54 \[ \int \frac {x^m (a+b \text {arcsinh}(c x))^2}{\left (d+c^2 d x^2\right )^3} \, dx=\int { \frac {{\left (b \operatorname {arsinh}\left (c x\right ) + a\right )}^{2} x^{m}}{{\left (c^{2} d x^{2} + d\right )}^{3}} \,d x } \] Input:

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

Output:

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

Sympy [F(-1)]

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

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

Output:

Timed out
 

Maxima [N/A]

Not integrable

Time = 0.14 (sec) , antiderivative size = 28, normalized size of antiderivative = 1.08 \[ \int \frac {x^m (a+b \text {arcsinh}(c x))^2}{\left (d+c^2 d x^2\right )^3} \, dx=\int { \frac {{\left (b \operatorname {arsinh}\left (c x\right ) + a\right )}^{2} x^{m}}{{\left (c^{2} d x^{2} + d\right )}^{3}} \,d x } \] Input:

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

Output:

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

Giac [N/A]

Not integrable

Time = 0.17 (sec) , antiderivative size = 28, normalized size of antiderivative = 1.08 \[ \int \frac {x^m (a+b \text {arcsinh}(c x))^2}{\left (d+c^2 d x^2\right )^3} \, dx=\int { \frac {{\left (b \operatorname {arsinh}\left (c x\right ) + a\right )}^{2} x^{m}}{{\left (c^{2} d x^{2} + d\right )}^{3}} \,d x } \] Input:

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

Output:

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

Mupad [N/A]

Not integrable

Time = 3.08 (sec) , antiderivative size = 28, normalized size of antiderivative = 1.08 \[ \int \frac {x^m (a+b \text {arcsinh}(c x))^2}{\left (d+c^2 d x^2\right )^3} \, dx=\int \frac {x^m\,{\left (a+b\,\mathrm {asinh}\left (c\,x\right )\right )}^2}{{\left (d\,c^2\,x^2+d\right )}^3} \,d x \] 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 [N/A]

Not integrable

Time = 0.22 (sec) , antiderivative size = 126, normalized size of antiderivative = 4.85 \[ \int \frac {x^m (a+b \text {arcsinh}(c x))^2}{\left (d+c^2 d x^2\right )^3} \, dx=\frac {\left (\int \frac {x^{m}}{c^{6} x^{6}+3 c^{4} x^{4}+3 c^{2} x^{2}+1}d x \right ) a^{2}+2 \left (\int \frac {x^{m} \mathit {asinh} \left (c x \right )}{c^{6} x^{6}+3 c^{4} x^{4}+3 c^{2} x^{2}+1}d x \right ) a b +\left (\int \frac {x^{m} \mathit {asinh} \left (c x \right )^{2}}{c^{6} x^{6}+3 c^{4} x^{4}+3 c^{2} x^{2}+1}d x \right ) b^{2}}{d^{3}} \] Input:

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

Output:

(int(x**m/(c**6*x**6 + 3*c**4*x**4 + 3*c**2*x**2 + 1),x)*a**2 + 2*int((x** 
m*asinh(c*x))/(c**6*x**6 + 3*c**4*x**4 + 3*c**2*x**2 + 1),x)*a*b + int((x* 
*m*asinh(c*x)**2)/(c**6*x**6 + 3*c**4*x**4 + 3*c**2*x**2 + 1),x)*b**2)/d** 
3