\(\int \frac {x^3 (a+b \text {arcsinh}(c x))}{(d+c^2 d x^2)^3} \, dx\) [47]

Optimal result

Integrand size = 24, antiderivative size = 97 \[ \int \frac {x^3 (a+b \text {arcsinh}(c x))}{\left (d+c^2 d x^2\right )^3} \, dx=\frac {b x^3}{12 c d^3 \left (1+c^2 x^2\right )^{3/2}}+\frac {b x}{4 c^3 d^3 \sqrt {1+c^2 x^2}}-\frac {b \text {arcsinh}(c x)}{4 c^4 d^3}+\frac {x^4 (a+b \text {arcsinh}(c x))}{4 d^3 \left (1+c^2 x^2\right )^2} \] Output:

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

Mathematica [A] (verified)

Time = 0.13 (sec) , antiderivative size = 79, normalized size of antiderivative = 0.81 \[ \int \frac {x^3 (a+b \text {arcsinh}(c x))}{\left (d+c^2 d x^2\right )^3} \, dx=\frac {-3 a \left (1+2 c^2 x^2\right )+b c x \sqrt {1+c^2 x^2} \left (3+4 c^2 x^2\right )-3 \left (b+2 b c^2 x^2\right ) \text {arcsinh}(c x)}{12 c^4 d^3 \left (1+c^2 x^2\right )^2} \] Input:

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

Output:

(-3*a*(1 + 2*c^2*x^2) + b*c*x*Sqrt[1 + c^2*x^2]*(3 + 4*c^2*x^2) - 3*(b + 2 
*b*c^2*x^2)*ArcSinh[c*x])/(12*c^4*d^3*(1 + c^2*x^2)^2)
 

Rubi [A] (verified)

Time = 0.48 (sec) , antiderivative size = 95, normalized size of antiderivative = 0.98, number of steps used = 4, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.167, Rules used = {6215, 252, 252, 222}

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.

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

\(\Big \downarrow \) 6215

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

\(\Big \downarrow \) 252

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

\(\Big \downarrow \) 252

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

\(\Big \downarrow \) 222

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

Input:

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

Output:

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

Defintions of rubi rules used

rule 222

Int[1/Sqrt[(a_) + (b_.)*(x_)^2], x_Symbol] :> Simp[ArcSinh[Rt[b, 2]*(x/Sqrt 
[a])]/Rt[b, 2], x] /; FreeQ[{a, b}, x] && GtQ[a, 0] && PosQ[b]
 

rule 252

Int[((c_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> Simp[c*(c*x 
)^(m - 1)*((a + b*x^2)^(p + 1)/(2*b*(p + 1))), x] - Simp[c^2*((m - 1)/(2*b* 
(p + 1)))   Int[(c*x)^(m - 2)*(a + b*x^2)^(p + 1), x], x] /; FreeQ[{a, b, c 
}, x] && LtQ[p, -1] && GtQ[m, 1] &&  !ILtQ[(m + 2*p + 3)/2, 0] && IntBinomi 
alQ[a, b, c, 2, m, p, x]
 

rule 6215

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 + 1)*((a + 
b*ArcSinh[c*x])^n/(d*f*(m + 1))), x] - Simp[b*c*(n/(f*(m + 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, p}, x] && EqQ 
[e, c^2*d] && GtQ[n, 0] && EqQ[m + 2*p + 3, 0] && NeQ[m, -1]
 

Maple [A] (verified)

Time = 0.36 (sec) , antiderivative size = 108, normalized size of antiderivative = 1.11

method	result	size
derivativedivides	\(\frac {\frac {a \left (-\frac {1}{2 \left (c^{2} x^{2}+1\right )}+\frac {1}{4 \left (c^{2} x^{2}+1\right )^{2}}\right )}{d^{3}}+\frac {b \left (-\frac {\operatorname {arcsinh}\left (x c \right )}{2 \left (c^{2} x^{2}+1\right )}+\frac {\operatorname {arcsinh}\left (x c \right )}{4 \left (c^{2} x^{2}+1\right )^{2}}-\frac {x c}{12 \left (c^{2} x^{2}+1\right )^{\frac {3}{2}}}+\frac {x c}{3 \sqrt {c^{2} x^{2}+1}}\right )}{d^{3}}}{c^{4}}\)	\(108\)
default	\(\frac {\frac {a \left (-\frac {1}{2 \left (c^{2} x^{2}+1\right )}+\frac {1}{4 \left (c^{2} x^{2}+1\right )^{2}}\right )}{d^{3}}+\frac {b \left (-\frac {\operatorname {arcsinh}\left (x c \right )}{2 \left (c^{2} x^{2}+1\right )}+\frac {\operatorname {arcsinh}\left (x c \right )}{4 \left (c^{2} x^{2}+1\right )^{2}}-\frac {x c}{12 \left (c^{2} x^{2}+1\right )^{\frac {3}{2}}}+\frac {x c}{3 \sqrt {c^{2} x^{2}+1}}\right )}{d^{3}}}{c^{4}}\)	\(108\)
parts	\(\frac {a \left (-\frac {1}{2 c^{4} \left (c^{2} x^{2}+1\right )}+\frac {1}{4 c^{4} \left (c^{2} x^{2}+1\right )^{2}}\right )}{d^{3}}+\frac {b \left (-\frac {\operatorname {arcsinh}\left (x c \right )}{2 \left (c^{2} x^{2}+1\right )}+\frac {\operatorname {arcsinh}\left (x c \right )}{4 \left (c^{2} x^{2}+1\right )^{2}}-\frac {x c}{12 \left (c^{2} x^{2}+1\right )^{\frac {3}{2}}}+\frac {x c}{3 \sqrt {c^{2} x^{2}+1}}\right )}{d^{3} c^{4}}\)	\(113\)
orering	\(\frac {\left (c^{2} x^{2}+1\right ) \left (4 c^{4} x^{4}-3 c^{2} x^{2}-4\right ) \left (a +b \,\operatorname {arcsinh}\left (x c \right )\right )}{4 c^{4} \left (c^{2} d \,x^{2}+d \right )^{3}}+\frac {\left (4 c^{2} x^{2}+3\right ) \left (c^{2} x^{2}+1\right )^{2} \left (\frac {3 x^{2} \left (a +b \,\operatorname {arcsinh}\left (x c \right )\right )}{\left (c^{2} d \,x^{2}+d \right )^{3}}+\frac {x^{3} b c}{\sqrt {c^{2} x^{2}+1}\, \left (c^{2} d \,x^{2}+d \right )^{3}}-\frac {6 x^{4} \left (a +b \,\operatorname {arcsinh}\left (x c \right )\right ) c^{2} d}{\left (c^{2} d \,x^{2}+d \right )^{4}}\right )}{12 x^{2} c^{4}}\)	\(167\)

Input:

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

Output:

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

Fricas [A] (verification not implemented)

Time = 0.09 (sec) , antiderivative size = 99, normalized size of antiderivative = 1.02 \[ \int \frac {x^3 (a+b \text {arcsinh}(c x))}{\left (d+c^2 d x^2\right )^3} \, dx=\frac {3 \, a c^{4} x^{4} - 3 \, {\left (2 \, b c^{2} x^{2} + b\right )} \log \left (c x + \sqrt {c^{2} x^{2} + 1}\right ) + {\left (4 \, b c^{3} x^{3} + 3 \, b c x\right )} \sqrt {c^{2} x^{2} + 1}}{12 \, {\left (c^{8} d^{3} x^{4} + 2 \, c^{6} d^{3} x^{2} + c^{4} d^{3}\right )}} \] Input:

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

Output:

1/12*(3*a*c^4*x^4 - 3*(2*b*c^2*x^2 + b)*log(c*x + sqrt(c^2*x^2 + 1)) + (4* 
b*c^3*x^3 + 3*b*c*x)*sqrt(c^2*x^2 + 1))/(c^8*d^3*x^4 + 2*c^6*d^3*x^2 + c^4 
*d^3)
 

Sympy [F]

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

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

Output:

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

Maxima [F]

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

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

Output:

-1/16*b*((4*c^2*x^2 + 4*(2*c^2*x^2 + 1)*log(c*x + sqrt(c^2*x^2 + 1)) + 3)/ 
(c^8*d^3*x^4 + 2*c^6*d^3*x^2 + c^4*d^3) - 16*integrate(1/4*(2*c^2*x^2 + 1) 
/(c^10*d^3*x^7 + 3*c^8*d^3*x^5 + 3*c^6*d^3*x^3 + c^4*d^3*x + (c^9*d^3*x^6 
+ 3*c^7*d^3*x^4 + 3*c^5*d^3*x^2 + c^3*d^3)*sqrt(c^2*x^2 + 1)), x)) - 1/4*( 
2*c^2*x^2 + 1)*a/(c^8*d^3*x^4 + 2*c^6*d^3*x^2 + c^4*d^3)
 

Giac [F(-2)]

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

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

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

Output:

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

Reduce [F]

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

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

Output:

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