\(\int x^m (d+c^2 d x^2)^{3/2} (a+b \text {arcsinh}(c x))^2 \, dx\) [334]

Optimal result

Integrand size = 28, antiderivative size = 28 \[ \int x^m \left (d+c^2 d x^2\right )^{3/2} (a+b \text {arcsinh}(c x))^2 \, dx=\frac {d \sqrt {d+c^2 d x^2} \text {Int}\left (x^m \left (1+c^2 x^2\right )^{3/2} (a+b \text {arcsinh}(c x))^2,x\right )}{\sqrt {1+c^2 x^2}} \] Output:

d*(c^2*d*x^2+d)^(1/2)*Defer(Int)(x^m*(c^2*x^2+1)^(3/2)*(a+b*arcsinh(c*x))^ 
2,x)/(c^2*x^2+1)^(1/2)
 

Mathematica [N/A]

Not integrable

Time = 0.84 (sec) , antiderivative size = 30, normalized size of antiderivative = 1.07 \[ \int x^m \left (d+c^2 d x^2\right )^{3/2} (a+b \text {arcsinh}(c x))^2 \, dx=\int x^m \left (d+c^2 d x^2\right )^{3/2} (a+b \text {arcsinh}(c x))^2 \, dx \] Input:

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

Output:

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

Rubi [N/A]

Not integrable

Time = 1.73 (sec) , antiderivative size = 28, 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.

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

\(\Big \downarrow \) 6223

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

\(\Big \downarrow \) 6218

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

\(\Big \downarrow \) 363

\(\displaystyle -\frac {2 b c d \sqrt {c^2 d x^2+d} \left (-b c \left (\frac {(3 m+10) \int \frac {x^{m+2}}{\sqrt {c^2 x^2+1}}dx}{(m+2) (m+4)^2}+\frac {\sqrt {c^2 x^2+1} x^{m+3}}{(m+4)^2}\right )+\frac {c^2 x^{m+4} (a+b \text {arcsinh}(c x))}{m+4}+\frac {x^{m+2} (a+b \text {arcsinh}(c x))}{m+2}\right )}{(m+4) \sqrt {c^2 x^2+1}}+\frac {3 d \int x^m \sqrt {c^2 d x^2+d} (a+b \text {arcsinh}(c x))^2dx}{m+4}+\frac {x^{m+1} \left (c^2 d x^2+d\right )^{3/2} (a+b \text {arcsinh}(c x))^2}{m+4}\)

\(\Big \downarrow \) 278

\(\displaystyle \frac {3 d \int x^m \sqrt {c^2 d x^2+d} (a+b \text {arcsinh}(c x))^2dx}{m+4}-\frac {2 b c d \sqrt {c^2 d x^2+d} \left (\frac {c^2 x^{m+4} (a+b \text {arcsinh}(c x))}{m+4}+\frac {x^{m+2} (a+b \text {arcsinh}(c x))}{m+2}-b c \left (\frac {(3 m+10) x^{m+3} \operatorname {Hypergeometric2F1}\left (\frac {1}{2},\frac {m+3}{2},\frac {m+5}{2},-c^2 x^2\right )}{(m+2) (m+3) (m+4)^2}+\frac {\sqrt {c^2 x^2+1} x^{m+3}}{(m+4)^2}\right )\right )}{(m+4) \sqrt {c^2 x^2+1}}+\frac {x^{m+1} \left (c^2 d x^2+d\right )^{3/2} (a+b \text {arcsinh}(c x))^2}{m+4}\)

\(\Big \downarrow \) 6223

\(\displaystyle \frac {3 d \left (-\frac {2 b c \sqrt {c^2 d x^2+d} \int x^{m+1} (a+b \text {arcsinh}(c x))dx}{(m+2) \sqrt {c^2 x^2+1}}+\frac {d \int \frac {x^m (a+b \text {arcsinh}(c x))^2}{\sqrt {c^2 d x^2+d}}dx}{m+2}+\frac {x^{m+1} \sqrt {c^2 d x^2+d} (a+b \text {arcsinh}(c x))^2}{m+2}\right )}{m+4}-\frac {2 b c d \sqrt {c^2 d x^2+d} \left (\frac {c^2 x^{m+4} (a+b \text {arcsinh}(c x))}{m+4}+\frac {x^{m+2} (a+b \text {arcsinh}(c x))}{m+2}-b c \left (\frac {(3 m+10) x^{m+3} \operatorname {Hypergeometric2F1}\left (\frac {1}{2},\frac {m+3}{2},\frac {m+5}{2},-c^2 x^2\right )}{(m+2) (m+3) (m+4)^2}+\frac {\sqrt {c^2 x^2+1} x^{m+3}}{(m+4)^2}\right )\right )}{(m+4) \sqrt {c^2 x^2+1}}+\frac {x^{m+1} \left (c^2 d x^2+d\right )^{3/2} (a+b \text {arcsinh}(c x))^2}{m+4}\)

\(\Big \downarrow \) 6191

\(\displaystyle \frac {3 d \left (-\frac {2 b c \sqrt {c^2 d x^2+d} \left (\frac {x^{m+2} (a+b \text {arcsinh}(c x))}{m+2}-\frac {b c \int \frac {x^{m+2}}{\sqrt {c^2 x^2+1}}dx}{m+2}\right )}{(m+2) \sqrt {c^2 x^2+1}}+\frac {d \int \frac {x^m (a+b \text {arcsinh}(c x))^2}{\sqrt {c^2 d x^2+d}}dx}{m+2}+\frac {x^{m+1} \sqrt {c^2 d x^2+d} (a+b \text {arcsinh}(c x))^2}{m+2}\right )}{m+4}-\frac {2 b c d \sqrt {c^2 d x^2+d} \left (\frac {c^2 x^{m+4} (a+b \text {arcsinh}(c x))}{m+4}+\frac {x^{m+2} (a+b \text {arcsinh}(c x))}{m+2}-b c \left (\frac {(3 m+10) x^{m+3} \operatorname {Hypergeometric2F1}\left (\frac {1}{2},\frac {m+3}{2},\frac {m+5}{2},-c^2 x^2\right )}{(m+2) (m+3) (m+4)^2}+\frac {\sqrt {c^2 x^2+1} x^{m+3}}{(m+4)^2}\right )\right )}{(m+4) \sqrt {c^2 x^2+1}}+\frac {x^{m+1} \left (c^2 d x^2+d\right )^{3/2} (a+b \text {arcsinh}(c x))^2}{m+4}\)

\(\Big \downarrow \) 278

\(\displaystyle \frac {3 d \left (\frac {d \int \frac {x^m (a+b \text {arcsinh}(c x))^2}{\sqrt {c^2 d x^2+d}}dx}{m+2}-\frac {2 b c \sqrt {c^2 d x^2+d} \left (\frac {x^{m+2} (a+b \text {arcsinh}(c x))}{m+2}-\frac {b c x^{m+3} \operatorname {Hypergeometric2F1}\left (\frac {1}{2},\frac {m+3}{2},\frac {m+5}{2},-c^2 x^2\right )}{(m+2) (m+3)}\right )}{(m+2) \sqrt {c^2 x^2+1}}+\frac {x^{m+1} \sqrt {c^2 d x^2+d} (a+b \text {arcsinh}(c x))^2}{m+2}\right )}{m+4}-\frac {2 b c d \sqrt {c^2 d x^2+d} \left (\frac {c^2 x^{m+4} (a+b \text {arcsinh}(c x))}{m+4}+\frac {x^{m+2} (a+b \text {arcsinh}(c x))}{m+2}-b c \left (\frac {(3 m+10) x^{m+3} \operatorname {Hypergeometric2F1}\left (\frac {1}{2},\frac {m+3}{2},\frac {m+5}{2},-c^2 x^2\right )}{(m+2) (m+3) (m+4)^2}+\frac {\sqrt {c^2 x^2+1} x^{m+3}}{(m+4)^2}\right )\right )}{(m+4) \sqrt {c^2 x^2+1}}+\frac {x^{m+1} \left (c^2 d x^2+d\right )^{3/2} (a+b \text {arcsinh}(c x))^2}{m+4}\)

\(\Big \downarrow \) 6239

\(\displaystyle \frac {3 d \left (\frac {d \int \frac {x^m (a+b \text {arcsinh}(c x))^2}{\sqrt {c^2 d x^2+d}}dx}{m+2}-\frac {2 b c \sqrt {c^2 d x^2+d} \left (\frac {x^{m+2} (a+b \text {arcsinh}(c x))}{m+2}-\frac {b c x^{m+3} \operatorname {Hypergeometric2F1}\left (\frac {1}{2},\frac {m+3}{2},\frac {m+5}{2},-c^2 x^2\right )}{(m+2) (m+3)}\right )}{(m+2) \sqrt {c^2 x^2+1}}+\frac {x^{m+1} \sqrt {c^2 d x^2+d} (a+b \text {arcsinh}(c x))^2}{m+2}\right )}{m+4}-\frac {2 b c d \sqrt {c^2 d x^2+d} \left (\frac {c^2 x^{m+4} (a+b \text {arcsinh}(c x))}{m+4}+\frac {x^{m+2} (a+b \text {arcsinh}(c x))}{m+2}-b c \left (\frac {(3 m+10) x^{m+3} \operatorname {Hypergeometric2F1}\left (\frac {1}{2},\frac {m+3}{2},\frac {m+5}{2},-c^2 x^2\right )}{(m+2) (m+3) (m+4)^2}+\frac {\sqrt {c^2 x^2+1} x^{m+3}}{(m+4)^2}\right )\right )}{(m+4) \sqrt {c^2 x^2+1}}+\frac {x^{m+1} \left (c^2 d x^2+d\right )^{3/2} (a+b \text {arcsinh}(c x))^2}{m+4}\)

Input:

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

Output:

$Aborted
 

Maple [N/A]

Not integrable

Time = 2.11 (sec) , antiderivative size = 26, normalized size of antiderivative = 0.93

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

Input:

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

Output:

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

Fricas [N/A]

Not integrable

Time = 0.11 (sec) , antiderivative size = 80, normalized size of antiderivative = 2.86 \[ \int x^m \left (d+c^2 d x^2\right )^{3/2} (a+b \text {arcsinh}(c x))^2 \, dx=\int { {\left (c^{2} d x^{2} + d\right )}^{\frac {3}{2}} {\left (b \operatorname {arsinh}\left (c x\right ) + a\right )}^{2} x^{m} \,d x } \] Input:

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

Output:

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

Sympy [F(-1)]

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

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

Output:

Timed out
 

Maxima [N/A]

Not integrable

Time = 0.16 (sec) , antiderivative size = 28, normalized size of antiderivative = 1.00 \[ \int x^m \left (d+c^2 d x^2\right )^{3/2} (a+b \text {arcsinh}(c x))^2 \, dx=\int { {\left (c^{2} d x^{2} + d\right )}^{\frac {3}{2}} {\left (b \operatorname {arsinh}\left (c x\right ) + a\right )}^{2} x^{m} \,d x } \] Input:

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

Output:

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

Giac [F(-2)]

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

integrate(x^m*(c^2*d*x^2+d)^(3/2)*(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 [N/A]

Not integrable

Time = 3.09 (sec) , antiderivative size = 28, normalized size of antiderivative = 1.00 \[ \int x^m \left (d+c^2 d x^2\right )^{3/2} (a+b \text {arcsinh}(c x))^2 \, dx=\int x^m\,{\left (a+b\,\mathrm {asinh}\left (c\,x\right )\right )}^2\,{\left (d\,c^2\,x^2+d\right )}^{3/2} \,d x \] Input:

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

Output:

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

Reduce [N/A]

Not integrable

Time = 0.25 (sec) , antiderivative size = 163, normalized size of antiderivative = 5.82 \[ \int x^m \left (d+c^2 d x^2\right )^{3/2} (a+b \text {arcsinh}(c x))^2 \, dx=\sqrt {d}\, d \left (2 \left (\int x^{m} \sqrt {c^{2} x^{2}+1}\, \mathit {asinh} \left (c x \right ) x^{2}d x \right ) a b \,c^{2}+2 \left (\int x^{m} \sqrt {c^{2} x^{2}+1}\, \mathit {asinh} \left (c x \right )d x \right ) a b +\left (\int x^{m} \sqrt {c^{2} x^{2}+1}\, \mathit {asinh} \left (c x \right )^{2} x^{2}d x \right ) b^{2} c^{2}+\left (\int x^{m} \sqrt {c^{2} x^{2}+1}\, \mathit {asinh} \left (c x \right )^{2}d x \right ) b^{2}+\left (\int x^{m} \sqrt {c^{2} x^{2}+1}\, x^{2}d x \right ) a^{2} c^{2}+\left (\int x^{m} \sqrt {c^{2} x^{2}+1}d x \right ) a^{2}\right ) \] Input:

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

Output:

sqrt(d)*d*(2*int(x**m*sqrt(c**2*x**2 + 1)*asinh(c*x)*x**2,x)*a*b*c**2 + 2* 
int(x**m*sqrt(c**2*x**2 + 1)*asinh(c*x),x)*a*b + int(x**m*sqrt(c**2*x**2 + 
 1)*asinh(c*x)**2*x**2,x)*b**2*c**2 + int(x**m*sqrt(c**2*x**2 + 1)*asinh(c 
*x)**2,x)*b**2 + int(x**m*sqrt(c**2*x**2 + 1)*x**2,x)*a**2*c**2 + int(x**m 
*sqrt(c**2*x**2 + 1),x)*a**2)