\(\int \frac {x (a+b \text {arcsinh}(c x))}{(d+c^2 d x^2)^3} \, dx\) [49]

Optimal result

Integrand size = 22, antiderivative size = 80 \[ \int \frac {x (a+b \text {arcsinh}(c x))}{\left (d+c^2 d x^2\right )^3} \, dx=\frac {b x}{12 c d^3 \left (1+c^2 x^2\right )^{3/2}}+\frac {b x}{6 c d^3 \sqrt {1+c^2 x^2}}-\frac {a+b \text {arcsinh}(c x)}{4 c^2 d^3 \left (1+c^2 x^2\right )^2} \] Output:

1/12*b*x/c/d^3/(c^2*x^2+1)^(3/2)+1/6*b*x/c/d^3/(c^2*x^2+1)^(1/2)-1/4*(a+b* 
arcsinh(c*x))/c^2/d^3/(c^2*x^2+1)^2
 

Mathematica [A] (verified)

Time = 0.11 (sec) , antiderivative size = 56, normalized size of antiderivative = 0.70 \[ \int \frac {x (a+b \text {arcsinh}(c x))}{\left (d+c^2 d x^2\right )^3} \, dx=\frac {-3 a+b c x \sqrt {1+c^2 x^2} \left (3+2 c^2 x^2\right )-3 b \text {arcsinh}(c x)}{12 d^3 \left (c+c^3 x^2\right )^2} \] Input:

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

Output:

(-3*a + b*c*x*Sqrt[1 + c^2*x^2]*(3 + 2*c^2*x^2) - 3*b*ArcSinh[c*x])/(12*d^ 
3*(c + c^3*x^2)^2)
 

Rubi [A] (verified)

Time = 0.25 (sec) , antiderivative size = 78, normalized size of antiderivative = 0.98, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.136, Rules used = {6213, 209, 208}

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.

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

\(\Big \downarrow \) 6213

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

\(\Big \downarrow \) 209

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

\(\Big \downarrow \) 208

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

Input:

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

Output:

(b*(x/(3*(1 + c^2*x^2)^(3/2)) + (2*x)/(3*Sqrt[1 + c^2*x^2])))/(4*c*d^3) - 
(a + b*ArcSinh[c*x])/(4*c^2*d^3*(1 + c^2*x^2)^2)
 

Defintions of rubi rules used

rule 208

Int[((a_) + (b_.)*(x_)^2)^(-3/2), x_Symbol] :> Simp[x/(a*Sqrt[a + b*x^2]), 
x] /; FreeQ[{a, b}, x]
 

rule 209

Int[((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> Simp[(-x)*((a + b*x^2)^(p + 1) 
/(2*a*(p + 1))), x] + Simp[(2*p + 3)/(2*a*(p + 1))   Int[(a + b*x^2)^(p + 1 
), x], x] /; FreeQ[{a, b}, x] && ILtQ[p + 3/2, 0]
 

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 [A] (verified)

Time = 0.41 (sec) , antiderivative size = 76, normalized size of antiderivative = 0.95

method	result	size
derivativedivides	\(\frac {-\frac {a}{4 d^{3} \left (c^{2} x^{2}+1\right )^{2}}+\frac {b \left (-\frac {\operatorname {arcsinh}\left (x c \right )}{4 \left (c^{2} x^{2}+1\right )^{2}}+\frac {x c}{12 \left (c^{2} x^{2}+1\right )^{\frac {3}{2}}}+\frac {x c}{6 \sqrt {c^{2} x^{2}+1}}\right )}{d^{3}}}{c^{2}}\)	\(76\)
default	\(\frac {-\frac {a}{4 d^{3} \left (c^{2} x^{2}+1\right )^{2}}+\frac {b \left (-\frac {\operatorname {arcsinh}\left (x c \right )}{4 \left (c^{2} x^{2}+1\right )^{2}}+\frac {x c}{12 \left (c^{2} x^{2}+1\right )^{\frac {3}{2}}}+\frac {x c}{6 \sqrt {c^{2} x^{2}+1}}\right )}{d^{3}}}{c^{2}}\)	\(76\)
parts	\(-\frac {a}{4 d^{3} c^{2} \left (c^{2} x^{2}+1\right )^{2}}+\frac {b \left (-\frac {\operatorname {arcsinh}\left (x c \right )}{4 \left (c^{2} x^{2}+1\right )^{2}}+\frac {x c}{12 \left (c^{2} x^{2}+1\right )^{\frac {3}{2}}}+\frac {x c}{6 \sqrt {c^{2} x^{2}+1}}\right )}{d^{3} c^{2}}\)	\(78\)
orering	\(\frac {\left (10 c^{6} x^{6}+23 c^{4} x^{4}+7 c^{2} x^{2}-6\right ) \left (a +b \,\operatorname {arcsinh}\left (x c \right )\right )}{12 c^{2} \left (c^{2} d \,x^{2}+d \right )^{3}}+\frac {\left (2 c^{2} x^{2}+3\right ) \left (c^{2} x^{2}+1\right )^{2} \left (\frac {a +b \,\operatorname {arcsinh}\left (x c \right )}{\left (c^{2} d \,x^{2}+d \right )^{3}}+\frac {x b c}{\sqrt {c^{2} x^{2}+1}\, \left (c^{2} d \,x^{2}+d \right )^{3}}-\frac {6 x^{2} \left (a +b \,\operatorname {arcsinh}\left (x c \right )\right ) c^{2} d}{\left (c^{2} d \,x^{2}+d \right )^{4}}\right )}{12 c^{2}}\)	\(157\)

Input:

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

Output:

1/c^2*(-1/4*a/d^3/(c^2*x^2+1)^2+b/d^3*(-1/4*arcsinh(x*c)/(c^2*x^2+1)^2+1/1 
2/(c^2*x^2+1)^(3/2)*x*c+1/6*x*c/(c^2*x^2+1)^(1/2)))
 

Fricas [A] (verification not implemented)

Time = 0.09 (sec) , antiderivative size = 98, normalized size of antiderivative = 1.22 \[ \int \frac {x (a+b \text {arcsinh}(c x))}{\left (d+c^2 d x^2\right )^3} \, dx=\frac {3 \, a c^{4} x^{4} + 6 \, a c^{2} x^{2} - 3 \, b \log \left (c x + \sqrt {c^{2} x^{2} + 1}\right ) + {\left (2 \, b c^{3} x^{3} + 3 \, b c x\right )} \sqrt {c^{2} x^{2} + 1}}{12 \, {\left (c^{6} d^{3} x^{4} + 2 \, c^{4} d^{3} x^{2} + c^{2} d^{3}\right )}} \] Input:

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

Output:

1/12*(3*a*c^4*x^4 + 6*a*c^2*x^2 - 3*b*log(c*x + sqrt(c^2*x^2 + 1)) + (2*b* 
c^3*x^3 + 3*b*c*x)*sqrt(c^2*x^2 + 1))/(c^6*d^3*x^4 + 2*c^4*d^3*x^2 + c^2*d 
^3)
 

Sympy [F]

\[ \int \frac {x (a+b \text {arcsinh}(c x))}{\left (d+c^2 d x^2\right )^3} \, dx=\frac {\int \frac {a x}{c^{6} x^{6} + 3 c^{4} x^{4} + 3 c^{2} x^{2} + 1}\, dx + \int \frac {b x \operatorname {asinh}{\left (c x \right )}}{c^{6} x^{6} + 3 c^{4} x^{4} + 3 c^{2} x^{2} + 1}\, dx}{d^{3}} \] Input:

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

Output:

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

Maxima [F]

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

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

Output:

-1/16*b*((4*log(c*x + sqrt(c^2*x^2 + 1)) + 1)/(c^6*d^3*x^4 + 2*c^4*d^3*x^2 
 + c^2*d^3) - 16*integrate(1/4/(c^8*d^3*x^7 + 3*c^6*d^3*x^5 + 3*c^4*d^3*x^ 
3 + c^2*d^3*x + (c^7*d^3*x^6 + 3*c^5*d^3*x^4 + 3*c^3*d^3*x^2 + c*d^3)*sqrt 
(c^2*x^2 + 1)), x)) - 1/4*a/(c^6*d^3*x^4 + 2*c^4*d^3*x^2 + c^2*d^3)
 

Giac [F]

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

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

Output:

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

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

\[ \int \frac {x (a+b \text {arcsinh}(c x))}{\left (d+c^2 d x^2\right )^3} \, dx=\frac {4 \left (\int \frac {\mathit {asinh} \left (c x \right ) x}{c^{6} x^{6}+3 c^{4} x^{4}+3 c^{2} x^{2}+1}d x \right ) b \,c^{6} x^{4}+8 \left (\int \frac {\mathit {asinh} \left (c x \right ) x}{c^{6} x^{6}+3 c^{4} x^{4}+3 c^{2} x^{2}+1}d x \right ) b \,c^{4} x^{2}+4 \left (\int \frac {\mathit {asinh} \left (c x \right ) x}{c^{6} x^{6}+3 c^{4} x^{4}+3 c^{2} x^{2}+1}d x \right ) b \,c^{2}-a}{4 c^{2} d^{3} \left (c^{4} x^{4}+2 c^{2} x^{2}+1\right )} \] Input:

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

Output:

(4*int((asinh(c*x)*x)/(c**6*x**6 + 3*c**4*x**4 + 3*c**2*x**2 + 1),x)*b*c** 
6*x**4 + 8*int((asinh(c*x)*x)/(c**6*x**6 + 3*c**4*x**4 + 3*c**2*x**2 + 1), 
x)*b*c**4*x**2 + 4*int((asinh(c*x)*x)/(c**6*x**6 + 3*c**4*x**4 + 3*c**2*x* 
*2 + 1),x)*b*c**2 - a)/(4*c**2*d**3*(c**4*x**4 + 2*c**2*x**2 + 1))