\(\int \frac {x^3 (a+b \text {arcsinh}(c x))}{\sqrt {d+c^2 d x^2}} \, dx\) [155]

Optimal result

Integrand size = 26, antiderivative size = 142 \[ \int \frac {x^3 (a+b \text {arcsinh}(c x))}{\sqrt {d+c^2 d x^2}} \, dx=\frac {2 b x \sqrt {1+c^2 x^2}}{3 c^3 \sqrt {d+c^2 d x^2}}-\frac {b x^3 \sqrt {1+c^2 x^2}}{9 c \sqrt {d+c^2 d x^2}}-\frac {2 \sqrt {d+c^2 d x^2} (a+b \text {arcsinh}(c x))}{3 c^4 d}+\frac {x^2 \sqrt {d+c^2 d x^2} (a+b \text {arcsinh}(c x))}{3 c^2 d} \] Output:

2/3*b*x*(c^2*x^2+1)^(1/2)/c^3/(c^2*d*x^2+d)^(1/2)-1/9*b*x^3*(c^2*x^2+1)^(1 
/2)/c/(c^2*d*x^2+d)^(1/2)-2/3*(c^2*d*x^2+d)^(1/2)*(a+b*arcsinh(c*x))/c^4/d 
+1/3*x^2*(c^2*d*x^2+d)^(1/2)*(a+b*arcsinh(c*x))/c^2/d
 

Mathematica [A] (verified)

Time = 0.13 (sec) , antiderivative size = 93, normalized size of antiderivative = 0.65 \[ \int \frac {x^3 (a+b \text {arcsinh}(c x))}{\sqrt {d+c^2 d x^2}} \, dx=\frac {b c x \left (6-c^2 x^2\right ) \sqrt {1+c^2 x^2}+3 a \left (-2-c^2 x^2+c^4 x^4\right )+3 b \left (-2-c^2 x^2+c^4 x^4\right ) \text {arcsinh}(c x)}{9 c^4 \sqrt {d+c^2 d x^2}} \] Input:

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

Output:

(b*c*x*(6 - c^2*x^2)*Sqrt[1 + c^2*x^2] + 3*a*(-2 - c^2*x^2 + c^4*x^4) + 3* 
b*(-2 - c^2*x^2 + c^4*x^4)*ArcSinh[c*x])/(9*c^4*Sqrt[d + c^2*d*x^2])
 

Rubi [A] (verified)

Time = 0.46 (sec) , antiderivative size = 145, normalized size of antiderivative = 1.02, number of steps used = 4, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.154, Rules used = {6227, 15, 6213, 24}

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.

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

\(\Big \downarrow \) 6227

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

\(\Big \downarrow \) 15

\(\displaystyle -\frac {2 \int \frac {x (a+b \text {arcsinh}(c x))}{\sqrt {c^2 d x^2+d}}dx}{3 c^2}+\frac {x^2 \sqrt {c^2 d x^2+d} (a+b \text {arcsinh}(c x))}{3 c^2 d}-\frac {b x^3 \sqrt {c^2 x^2+1}}{9 c \sqrt {c^2 d x^2+d}}\)

\(\Big \downarrow \) 6213

\(\displaystyle -\frac {2 \left (\frac {\sqrt {c^2 d x^2+d} (a+b \text {arcsinh}(c x))}{c^2 d}-\frac {b \sqrt {c^2 x^2+1} \int 1dx}{c \sqrt {c^2 d x^2+d}}\right )}{3 c^2}+\frac {x^2 \sqrt {c^2 d x^2+d} (a+b \text {arcsinh}(c x))}{3 c^2 d}-\frac {b x^3 \sqrt {c^2 x^2+1}}{9 c \sqrt {c^2 d x^2+d}}\)

\(\Big \downarrow \) 24

\(\displaystyle \frac {x^2 \sqrt {c^2 d x^2+d} (a+b \text {arcsinh}(c x))}{3 c^2 d}-\frac {2 \left (\frac {\sqrt {c^2 d x^2+d} (a+b \text {arcsinh}(c x))}{c^2 d}-\frac {b x \sqrt {c^2 x^2+1}}{c \sqrt {c^2 d x^2+d}}\right )}{3 c^2}-\frac {b x^3 \sqrt {c^2 x^2+1}}{9 c \sqrt {c^2 d x^2+d}}\)

Input:

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

Output:

-1/9*(b*x^3*Sqrt[1 + c^2*x^2])/(c*Sqrt[d + c^2*d*x^2]) + (x^2*Sqrt[d + c^2 
*d*x^2]*(a + b*ArcSinh[c*x]))/(3*c^2*d) - (2*(-((b*x*Sqrt[1 + c^2*x^2])/(c 
*Sqrt[d + c^2*d*x^2])) + (Sqrt[d + c^2*d*x^2]*(a + b*ArcSinh[c*x]))/(c^2*d 
)))/(3*c^2)
 

Defintions of rubi rules used

rule 15

Int[(a_.)*(x_)^(m_.), x_Symbol] :> Simp[a*(x^(m + 1)/(m + 1)), x] /; FreeQ[ 
{a, m}, x] && NeQ[m, -1]
 

rule 24

Int[a_, x_Symbol] :> Simp[a*x, x] /; FreeQ[a, x]
 

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]
 

rule 6227

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

Maple [A] (verified)

Time = 1.05 (sec) , antiderivative size = 155, normalized size of antiderivative = 1.09

method	result	size
orering	\(\frac {\left (5 c^{4} x^{4}-12 c^{2} x^{2}-24\right ) \left (a +b \,\operatorname {arcsinh}\left (x c \right )\right )}{9 c^{4} \sqrt {c^{2} d \,x^{2}+d}}-\frac {\left (c^{2} x^{2}-6\right ) \left (c^{2} x^{2}+1\right ) \left (\frac {3 x^{2} \left (a +b \,\operatorname {arcsinh}\left (x c \right )\right )}{\sqrt {c^{2} d \,x^{2}+d}}+\frac {x^{3} b c}{\sqrt {c^{2} x^{2}+1}\, \sqrt {c^{2} d \,x^{2}+d}}-\frac {x^{4} \left (a +b \,\operatorname {arcsinh}\left (x c \right )\right ) c^{2} d}{\left (c^{2} d \,x^{2}+d \right )^{\frac {3}{2}}}\right )}{9 x^{2} c^{4}}\)	\(155\)
default	\(a \left (\frac {x^{2} \sqrt {c^{2} d \,x^{2}+d}}{3 c^{2} d}-\frac {2 \sqrt {c^{2} d \,x^{2}+d}}{3 d \,c^{4}}\right )+b \left (\frac {\sqrt {d \left (c^{2} x^{2}+1\right )}\, \left (4 c^{4} x^{4}+4 \sqrt {c^{2} x^{2}+1}\, c^{3} x^{3}+5 c^{2} x^{2}+3 \sqrt {c^{2} x^{2}+1}\, x c +1\right ) \left (-1+3 \,\operatorname {arcsinh}\left (x c \right )\right )}{72 c^{4} d \left (c^{2} x^{2}+1\right )}-\frac {3 \sqrt {d \left (c^{2} x^{2}+1\right )}\, \left (c^{2} x^{2}+\sqrt {c^{2} x^{2}+1}\, x c +1\right ) \left (\operatorname {arcsinh}\left (x c \right )-1\right )}{8 c^{4} d \left (c^{2} x^{2}+1\right )}-\frac {3 \sqrt {d \left (c^{2} x^{2}+1\right )}\, \left (c^{2} x^{2}-\sqrt {c^{2} x^{2}+1}\, x c +1\right ) \left (\operatorname {arcsinh}\left (x c \right )+1\right )}{8 c^{4} d \left (c^{2} x^{2}+1\right )}+\frac {\sqrt {d \left (c^{2} x^{2}+1\right )}\, \left (4 c^{4} x^{4}-4 \sqrt {c^{2} x^{2}+1}\, c^{3} x^{3}+5 c^{2} x^{2}-3 \sqrt {c^{2} x^{2}+1}\, x c +1\right ) \left (1+3 \,\operatorname {arcsinh}\left (x c \right )\right )}{72 c^{4} d \left (c^{2} x^{2}+1\right )}\right )\)	\(358\)
parts	\(a \left (\frac {x^{2} \sqrt {c^{2} d \,x^{2}+d}}{3 c^{2} d}-\frac {2 \sqrt {c^{2} d \,x^{2}+d}}{3 d \,c^{4}}\right )+b \left (\frac {\sqrt {d \left (c^{2} x^{2}+1\right )}\, \left (4 c^{4} x^{4}+4 \sqrt {c^{2} x^{2}+1}\, c^{3} x^{3}+5 c^{2} x^{2}+3 \sqrt {c^{2} x^{2}+1}\, x c +1\right ) \left (-1+3 \,\operatorname {arcsinh}\left (x c \right )\right )}{72 c^{4} d \left (c^{2} x^{2}+1\right )}-\frac {3 \sqrt {d \left (c^{2} x^{2}+1\right )}\, \left (c^{2} x^{2}+\sqrt {c^{2} x^{2}+1}\, x c +1\right ) \left (\operatorname {arcsinh}\left (x c \right )-1\right )}{8 c^{4} d \left (c^{2} x^{2}+1\right )}-\frac {3 \sqrt {d \left (c^{2} x^{2}+1\right )}\, \left (c^{2} x^{2}-\sqrt {c^{2} x^{2}+1}\, x c +1\right ) \left (\operatorname {arcsinh}\left (x c \right )+1\right )}{8 c^{4} d \left (c^{2} x^{2}+1\right )}+\frac {\sqrt {d \left (c^{2} x^{2}+1\right )}\, \left (4 c^{4} x^{4}-4 \sqrt {c^{2} x^{2}+1}\, c^{3} x^{3}+5 c^{2} x^{2}-3 \sqrt {c^{2} x^{2}+1}\, x c +1\right ) \left (1+3 \,\operatorname {arcsinh}\left (x c \right )\right )}{72 c^{4} d \left (c^{2} x^{2}+1\right )}\right )\)	\(358\)

Input:

int(x^3*(a+b*arcsinh(x*c))/(c^2*d*x^2+d)^(1/2),x,method=_RETURNVERBOSE)
 

Output:

1/9*(5*c^4*x^4-12*c^2*x^2-24)/c^4*(a+b*arcsinh(x*c))/(c^2*d*x^2+d)^(1/2)-1 
/9/x^2*(c^2*x^2-6)/c^4*(c^2*x^2+1)*(3*x^2*(a+b*arcsinh(x*c))/(c^2*d*x^2+d) 
^(1/2)+x^3*b*c/(c^2*x^2+1)^(1/2)/(c^2*d*x^2+d)^(1/2)-x^4*(a+b*arcsinh(x*c) 
)/(c^2*d*x^2+d)^(3/2)*c^2*d)
 

Fricas [A] (verification not implemented)

Time = 0.10 (sec) , antiderivative size = 132, normalized size of antiderivative = 0.93 \[ \int \frac {x^3 (a+b \text {arcsinh}(c x))}{\sqrt {d+c^2 d x^2}} \, dx=\frac {3 \, {\left (b c^{4} x^{4} - b c^{2} x^{2} - 2 \, b\right )} \sqrt {c^{2} d x^{2} + d} \log \left (c x + \sqrt {c^{2} x^{2} + 1}\right ) + {\left (3 \, a c^{4} x^{4} - 3 \, a c^{2} x^{2} - {\left (b c^{3} x^{3} - 6 \, b c x\right )} \sqrt {c^{2} x^{2} + 1} - 6 \, a\right )} \sqrt {c^{2} d x^{2} + d}}{9 \, {\left (c^{6} d x^{2} + c^{4} d\right )}} \] Input:

integrate(x^3*(a+b*arcsinh(c*x))/(c^2*d*x^2+d)^(1/2),x, algorithm="fricas" 
)
 

Output:

1/9*(3*(b*c^4*x^4 - b*c^2*x^2 - 2*b)*sqrt(c^2*d*x^2 + d)*log(c*x + sqrt(c^ 
2*x^2 + 1)) + (3*a*c^4*x^4 - 3*a*c^2*x^2 - (b*c^3*x^3 - 6*b*c*x)*sqrt(c^2* 
x^2 + 1) - 6*a)*sqrt(c^2*d*x^2 + d))/(c^6*d*x^2 + c^4*d)
 

Sympy [F]

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

integrate(x**3*(a+b*asinh(c*x))/(c**2*d*x**2+d)**(1/2),x)
 

Output:

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

Maxima [A] (verification not implemented)

Time = 0.04 (sec) , antiderivative size = 117, normalized size of antiderivative = 0.82 \[ \int \frac {x^3 (a+b \text {arcsinh}(c x))}{\sqrt {d+c^2 d x^2}} \, dx=\frac {1}{3} \, b {\left (\frac {\sqrt {c^{2} d x^{2} + d} x^{2}}{c^{2} d} - \frac {2 \, \sqrt {c^{2} d x^{2} + d}}{c^{4} d}\right )} \operatorname {arsinh}\left (c x\right ) + \frac {1}{3} \, a {\left (\frac {\sqrt {c^{2} d x^{2} + d} x^{2}}{c^{2} d} - \frac {2 \, \sqrt {c^{2} d x^{2} + d}}{c^{4} d}\right )} - \frac {{\left (c^{2} x^{3} - 6 \, x\right )} b}{9 \, c^{3} \sqrt {d}} \] Input:

integrate(x^3*(a+b*arcsinh(c*x))/(c^2*d*x^2+d)^(1/2),x, algorithm="maxima" 
)
 

Output:

1/3*b*(sqrt(c^2*d*x^2 + d)*x^2/(c^2*d) - 2*sqrt(c^2*d*x^2 + d)/(c^4*d))*ar 
csinh(c*x) + 1/3*a*(sqrt(c^2*d*x^2 + d)*x^2/(c^2*d) - 2*sqrt(c^2*d*x^2 + d 
)/(c^4*d)) - 1/9*(c^2*x^3 - 6*x)*b/(c^3*sqrt(d))
 

Giac [F(-2)]

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

integrate(x^3*(a+b*arcsinh(c*x))/(c^2*d*x^2+d)^(1/2),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 [F(-1)]

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

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

Output:

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

Reduce [F]

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

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

Output:

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