\(\int \frac {x (a+b \text {arcsinh}(c x))^2}{\sqrt {d+c^2 d x^2}} \, dx\) [301]

Optimal result

Integrand size = 26, antiderivative size = 99 \[ \int \frac {x (a+b \text {arcsinh}(c x))^2}{\sqrt {d+c^2 d x^2}} \, dx=\frac {2 b^2 \sqrt {d+c^2 d x^2}}{c^2 d}-\frac {2 b x \sqrt {1+c^2 x^2} (a+b \text {arcsinh}(c x))}{c \sqrt {d+c^2 d x^2}}+\frac {\sqrt {d+c^2 d x^2} (a+b \text {arcsinh}(c x))^2}{c^2 d} \] Output:

2*b^2*(c^2*d*x^2+d)^(1/2)/c^2/d-2*b*x*(c^2*x^2+1)^(1/2)*(a+b*arcsinh(c*x)) 
/c/(c^2*d*x^2+d)^(1/2)+(c^2*d*x^2+d)^(1/2)*(a+b*arcsinh(c*x))^2/c^2/d
 

Mathematica [A] (verified)

Time = 0.52 (sec) , antiderivative size = 127, normalized size of antiderivative = 1.28 \[ \int \frac {x (a+b \text {arcsinh}(c x))^2}{\sqrt {d+c^2 d x^2}} \, dx=\frac {\sqrt {d+c^2 d x^2} \left (-2 a b c x+a^2 \sqrt {1+c^2 x^2}+2 b^2 \sqrt {1+c^2 x^2}-2 b \left (b c x-a \sqrt {1+c^2 x^2}\right ) \text {arcsinh}(c x)+b^2 \sqrt {1+c^2 x^2} \text {arcsinh}(c x)^2\right )}{c^2 d \sqrt {1+c^2 x^2}} \] Input:

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

Output:

(Sqrt[d + c^2*d*x^2]*(-2*a*b*c*x + a^2*Sqrt[1 + c^2*x^2] + 2*b^2*Sqrt[1 + 
c^2*x^2] - 2*b*(b*c*x - a*Sqrt[1 + c^2*x^2])*ArcSinh[c*x] + b^2*Sqrt[1 + c 
^2*x^2]*ArcSinh[c*x]^2))/(c^2*d*Sqrt[1 + c^2*x^2])
 

Rubi [A] (verified)

Time = 0.33 (sec) , antiderivative size = 95, normalized size of antiderivative = 0.96, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.077, Rules used = {6213, 2009}

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.

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

\(\Big \downarrow \) 6213

\(\displaystyle \frac {\sqrt {c^2 d x^2+d} (a+b \text {arcsinh}(c x))^2}{c^2 d}-\frac {2 b \sqrt {c^2 x^2+1} \int (a+b \text {arcsinh}(c x))dx}{c \sqrt {c^2 d x^2+d}}\)

\(\Big \downarrow \) 2009

\(\displaystyle \frac {\sqrt {c^2 d x^2+d} (a+b \text {arcsinh}(c x))^2}{c^2 d}-\frac {2 b \sqrt {c^2 x^2+1} \left (a x+b x \text {arcsinh}(c x)-\frac {b \sqrt {c^2 x^2+1}}{c}\right )}{c \sqrt {c^2 d x^2+d}}\)

Input:

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

Output:

(Sqrt[d + c^2*d*x^2]*(a + b*ArcSinh[c*x])^2)/(c^2*d) - (2*b*Sqrt[1 + c^2*x 
^2]*(a*x - (b*Sqrt[1 + c^2*x^2])/c + b*x*ArcSinh[c*x]))/(c*Sqrt[d + c^2*d* 
x^2])
 

Defintions of rubi rules used

rule 2009

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

rule 6213

Int[((a_.) + ArcSinh[(c_.)*(x_)]*(b_.))^(n_.)*(x_)*((d_) + (e_.)*(x_)^2)^(p 
_.), x_Symbol] :> Simp[(d + e*x^2)^(p + 1)*((a + b*ArcSinh[c*x])^n/(2*e*(p 
+ 1))), x] - Simp[b*(n/(2*c*(p + 1)))*Simp[(d + e*x^2)^p/(1 + c^2*x^2)^p] 
 Int[(1 + c^2*x^2)^(p + 1/2)*(a + b*ArcSinh[c*x])^(n - 1), x], x] /; FreeQ[ 
{a, b, c, d, e, p}, x] && EqQ[e, c^2*d] && GtQ[n, 0] && NeQ[p, -1]
 

Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(295\) vs. \(2(91)=182\).

Time = 0.96 (sec) , antiderivative size = 296, normalized size of antiderivative = 2.99

method	result	size
default	\(\frac {a^{2} \sqrt {c^{2} d \,x^{2}+d}}{c^{2} d}+b^{2} \left (\frac {\sqrt {d \left (c^{2} x^{2}+1\right )}\, \left (c^{2} x^{2}+\sqrt {c^{2} x^{2}+1}\, x c +1\right ) \left (\operatorname {arcsinh}\left (x c \right )^{2}-2 \,\operatorname {arcsinh}\left (x c \right )+2\right )}{2 c^{2} d \left (c^{2} x^{2}+1\right )}+\frac {\sqrt {d \left (c^{2} x^{2}+1\right )}\, \left (c^{2} x^{2}-\sqrt {c^{2} x^{2}+1}\, x c +1\right ) \left (\operatorname {arcsinh}\left (x c \right )^{2}+2 \,\operatorname {arcsinh}\left (x c \right )+2\right )}{2 c^{2} d \left (c^{2} x^{2}+1\right )}\right )+2 a b \left (\frac {\sqrt {d \left (c^{2} x^{2}+1\right )}\, \left (c^{2} x^{2}+\sqrt {c^{2} x^{2}+1}\, x c +1\right ) \left (\operatorname {arcsinh}\left (x c \right )-1\right )}{2 c^{2} d \left (c^{2} x^{2}+1\right )}+\frac {\sqrt {d \left (c^{2} x^{2}+1\right )}\, \left (c^{2} x^{2}-\sqrt {c^{2} x^{2}+1}\, x c +1\right ) \left (\operatorname {arcsinh}\left (x c \right )+1\right )}{2 c^{2} d \left (c^{2} x^{2}+1\right )}\right )\)	\(296\)
parts	\(\frac {a^{2} \sqrt {c^{2} d \,x^{2}+d}}{c^{2} d}+b^{2} \left (\frac {\sqrt {d \left (c^{2} x^{2}+1\right )}\, \left (c^{2} x^{2}+\sqrt {c^{2} x^{2}+1}\, x c +1\right ) \left (\operatorname {arcsinh}\left (x c \right )^{2}-2 \,\operatorname {arcsinh}\left (x c \right )+2\right )}{2 c^{2} d \left (c^{2} x^{2}+1\right )}+\frac {\sqrt {d \left (c^{2} x^{2}+1\right )}\, \left (c^{2} x^{2}-\sqrt {c^{2} x^{2}+1}\, x c +1\right ) \left (\operatorname {arcsinh}\left (x c \right )^{2}+2 \,\operatorname {arcsinh}\left (x c \right )+2\right )}{2 c^{2} d \left (c^{2} x^{2}+1\right )}\right )+2 a b \left (\frac {\sqrt {d \left (c^{2} x^{2}+1\right )}\, \left (c^{2} x^{2}+\sqrt {c^{2} x^{2}+1}\, x c +1\right ) \left (\operatorname {arcsinh}\left (x c \right )-1\right )}{2 c^{2} d \left (c^{2} x^{2}+1\right )}+\frac {\sqrt {d \left (c^{2} x^{2}+1\right )}\, \left (c^{2} x^{2}-\sqrt {c^{2} x^{2}+1}\, x c +1\right ) \left (\operatorname {arcsinh}\left (x c \right )+1\right )}{2 c^{2} d \left (c^{2} x^{2}+1\right )}\right )\)	\(296\)
orering	\(\frac {\left (c^{4} x^{4}+4 c^{2} x^{2}+2\right ) \left (a +b \,\operatorname {arcsinh}\left (x c \right )\right )^{2}}{c^{4} x^{2} \sqrt {c^{2} d \,x^{2}+d}}-\frac {2 \left (c^{2} x^{2}+1\right ) \left (\frac {\left (a +b \,\operatorname {arcsinh}\left (x c \right )\right )^{2}}{\sqrt {c^{2} d \,x^{2}+d}}+\frac {2 x \left (a +b \,\operatorname {arcsinh}\left (x c \right )\right ) b c}{\sqrt {c^{2} d \,x^{2}+d}\, \sqrt {c^{2} x^{2}+1}}-\frac {x^{2} \left (a +b \,\operatorname {arcsinh}\left (x c \right )\right )^{2} c^{2} d}{\left (c^{2} d \,x^{2}+d \right )^{\frac {3}{2}}}\right )}{c^{4} x^{2}}+\frac {\left (c^{2} x^{2}+1\right )^{2} \left (\frac {4 \left (a +b \,\operatorname {arcsinh}\left (x c \right )\right ) b c}{\sqrt {c^{2} d \,x^{2}+d}\, \sqrt {c^{2} x^{2}+1}}-\frac {3 \left (a +b \,\operatorname {arcsinh}\left (x c \right )\right )^{2} c^{2} d x}{\left (c^{2} d \,x^{2}+d \right )^{\frac {3}{2}}}+\frac {2 x \,b^{2} c^{2}}{\left (c^{2} x^{2}+1\right ) \sqrt {c^{2} d \,x^{2}+d}}-\frac {4 x^{2} \left (a +b \,\operatorname {arcsinh}\left (x c \right )\right ) b \,c^{3} d}{\left (c^{2} d \,x^{2}+d \right )^{\frac {3}{2}} \sqrt {c^{2} x^{2}+1}}-\frac {2 x^{2} \left (a +b \,\operatorname {arcsinh}\left (x c \right )\right ) b \,c^{3}}{\sqrt {c^{2} d \,x^{2}+d}\, \left (c^{2} x^{2}+1\right )^{\frac {3}{2}}}+\frac {3 x^{3} \left (a +b \,\operatorname {arcsinh}\left (x c \right )\right )^{2} c^{4} d^{2}}{\left (c^{2} d \,x^{2}+d \right )^{\frac {5}{2}}}\right )}{c^{4} x}\)	\(385\)

Input:

int(x*(a+b*arcsinh(x*c))^2/(c^2*d*x^2+d)^(1/2),x,method=_RETURNVERBOSE)
 

Output:

a^2/c^2/d*(c^2*d*x^2+d)^(1/2)+b^2*(1/2*(d*(c^2*x^2+1))^(1/2)*(c^2*x^2+(c^2 
*x^2+1)^(1/2)*x*c+1)*(arcsinh(x*c)^2-2*arcsinh(x*c)+2)/c^2/d/(c^2*x^2+1)+1 
/2*(d*(c^2*x^2+1))^(1/2)*(c^2*x^2-(c^2*x^2+1)^(1/2)*x*c+1)*(arcsinh(x*c)^2 
+2*arcsinh(x*c)+2)/c^2/d/(c^2*x^2+1))+2*a*b*(1/2*(d*(c^2*x^2+1))^(1/2)*(c^ 
2*x^2+(c^2*x^2+1)^(1/2)*x*c+1)*(arcsinh(x*c)-1)/c^2/d/(c^2*x^2+1)+1/2*(d*( 
c^2*x^2+1))^(1/2)*(c^2*x^2-(c^2*x^2+1)^(1/2)*x*c+1)*(arcsinh(x*c)+1)/c^2/d 
/(c^2*x^2+1))
 

Fricas [A] (verification not implemented)

Time = 0.10 (sec) , antiderivative size = 179, normalized size of antiderivative = 1.81 \[ \int \frac {x (a+b \text {arcsinh}(c x))^2}{\sqrt {d+c^2 d x^2}} \, dx=\frac {{\left (b^{2} c^{2} x^{2} + b^{2}\right )} \sqrt {c^{2} d x^{2} + d} \log \left (c x + \sqrt {c^{2} x^{2} + 1}\right )^{2} + 2 \, {\left (a b c^{2} x^{2} - \sqrt {c^{2} x^{2} + 1} b^{2} c x + a b\right )} \sqrt {c^{2} d x^{2} + d} \log \left (c x + \sqrt {c^{2} x^{2} + 1}\right ) + {\left ({\left (a^{2} + 2 \, b^{2}\right )} c^{2} x^{2} - 2 \, \sqrt {c^{2} x^{2} + 1} a b c x + a^{2} + 2 \, b^{2}\right )} \sqrt {c^{2} d x^{2} + d}}{c^{4} d x^{2} + c^{2} d} \] Input:

integrate(x*(a+b*arcsinh(c*x))^2/(c^2*d*x^2+d)^(1/2),x, algorithm="fricas" 
)
 

Output:

((b^2*c^2*x^2 + b^2)*sqrt(c^2*d*x^2 + d)*log(c*x + sqrt(c^2*x^2 + 1))^2 + 
2*(a*b*c^2*x^2 - sqrt(c^2*x^2 + 1)*b^2*c*x + a*b)*sqrt(c^2*d*x^2 + d)*log( 
c*x + sqrt(c^2*x^2 + 1)) + ((a^2 + 2*b^2)*c^2*x^2 - 2*sqrt(c^2*x^2 + 1)*a* 
b*c*x + a^2 + 2*b^2)*sqrt(c^2*d*x^2 + d))/(c^4*d*x^2 + c^2*d)
 

Sympy [F]

\[ \int \frac {x (a+b \text {arcsinh}(c x))^2}{\sqrt {d+c^2 d x^2}} \, dx=\int \frac {x \left (a + b \operatorname {asinh}{\left (c x \right )}\right )^{2}}{\sqrt {d \left (c^{2} x^{2} + 1\right )}}\, dx \] Input:

integrate(x*(a+b*asinh(c*x))**2/(c**2*d*x**2+d)**(1/2),x)
 

Output:

Integral(x*(a + b*asinh(c*x))**2/sqrt(d*(c**2*x**2 + 1)), x)
 

Maxima [A] (verification not implemented)

Time = 0.04 (sec) , antiderivative size = 125, normalized size of antiderivative = 1.26 \[ \int \frac {x (a+b \text {arcsinh}(c x))^2}{\sqrt {d+c^2 d x^2}} \, dx=-2 \, b^{2} {\left (\frac {x \operatorname {arsinh}\left (c x\right )}{c \sqrt {d}} - \frac {\sqrt {c^{2} x^{2} + 1}}{c^{2} \sqrt {d}}\right )} - \frac {2 \, a b x}{c \sqrt {d}} + \frac {\sqrt {c^{2} d x^{2} + d} b^{2} \operatorname {arsinh}\left (c x\right )^{2}}{c^{2} d} + \frac {2 \, \sqrt {c^{2} d x^{2} + d} a b \operatorname {arsinh}\left (c x\right )}{c^{2} d} + \frac {\sqrt {c^{2} d x^{2} + d} a^{2}}{c^{2} d} \] Input:

integrate(x*(a+b*arcsinh(c*x))^2/(c^2*d*x^2+d)^(1/2),x, algorithm="maxima" 
)
 

Output:

-2*b^2*(x*arcsinh(c*x)/(c*sqrt(d)) - sqrt(c^2*x^2 + 1)/(c^2*sqrt(d))) - 2* 
a*b*x/(c*sqrt(d)) + sqrt(c^2*d*x^2 + d)*b^2*arcsinh(c*x)^2/(c^2*d) + 2*sqr 
t(c^2*d*x^2 + d)*a*b*arcsinh(c*x)/(c^2*d) + sqrt(c^2*d*x^2 + d)*a^2/(c^2*d 
)
 

Giac [F]

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

integrate(x*(a+b*arcsinh(c*x))^2/(c^2*d*x^2+d)^(1/2),x, algorithm="giac")
 

Output:

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

Mupad [F(-1)]

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

int((x*(a + b*asinh(c*x))^2)/(d + c^2*d*x^2)^(1/2),x)
 

Output:

int((x*(a + b*asinh(c*x))^2)/(d + c^2*d*x^2)^(1/2), x)
 

Reduce [B] (verification not implemented)

Time = 0.21 (sec) , antiderivative size = 94, normalized size of antiderivative = 0.95 \[ \int \frac {x (a+b \text {arcsinh}(c x))^2}{\sqrt {d+c^2 d x^2}} \, dx=\frac {\sqrt {d}\, \left (\sqrt {c^{2} x^{2}+1}\, \mathit {asinh} \left (c x \right )^{2} b^{2}+2 \sqrt {c^{2} x^{2}+1}\, \mathit {asinh} \left (c x \right ) a b -2 \mathit {asinh} \left (c x \right ) b^{2} c x +\sqrt {c^{2} x^{2}+1}\, a^{2}+2 \sqrt {c^{2} x^{2}+1}\, b^{2}-2 a b c x \right )}{c^{2} d} \] Input:

int(x*(a+b*asinh(c*x))^2/(c^2*d*x^2+d)^(1/2),x)
 

Output:

(sqrt(d)*(sqrt(c**2*x**2 + 1)*asinh(c*x)**2*b**2 + 2*sqrt(c**2*x**2 + 1)*a 
sinh(c*x)*a*b - 2*asinh(c*x)*b**2*c*x + sqrt(c**2*x**2 + 1)*a**2 + 2*sqrt( 
c**2*x**2 + 1)*b**2 - 2*a*b*c*x))/(c**2*d)