\(\int \frac {x^2 (a+b \text {arcsinh}(c x))}{\sqrt {d+c^2 d x^2}} \, dx\) [156]

Optimal result

Integrand size = 26, antiderivative size = 119 \[ \int \frac {x^2 (a+b \text {arcsinh}(c x))}{\sqrt {d+c^2 d x^2}} \, dx=-\frac {b x^2 \sqrt {1+c^2 x^2}}{4 c \sqrt {d+c^2 d x^2}}+\frac {x \sqrt {d+c^2 d x^2} (a+b \text {arcsinh}(c x))}{2 c^2 d}-\frac {\sqrt {1+c^2 x^2} (a+b \text {arcsinh}(c x))^2}{4 b c^3 \sqrt {d+c^2 d x^2}} \] Output:

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

Mathematica [A] (verified)

Time = 0.62 (sec) , antiderivative size = 121, normalized size of antiderivative = 1.02 \[ \int \frac {x^2 (a+b \text {arcsinh}(c x))}{\sqrt {d+c^2 d x^2}} \, dx=-\frac {-\frac {4 a c x \sqrt {d+c^2 d x^2}}{d}+\frac {4 a \log \left (c d x+\sqrt {d} \sqrt {d+c^2 d x^2}\right )}{\sqrt {d}}+\frac {b \sqrt {1+c^2 x^2} (\cosh (2 \text {arcsinh}(c x))+2 \text {arcsinh}(c x) (\text {arcsinh}(c x)-\sinh (2 \text {arcsinh}(c x))))}{\sqrt {d+c^2 d x^2}}}{8 c^3} \] Input:

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

Output:

-1/8*((-4*a*c*x*Sqrt[d + c^2*d*x^2])/d + (4*a*Log[c*d*x + Sqrt[d]*Sqrt[d + 
 c^2*d*x^2]])/Sqrt[d] + (b*Sqrt[1 + c^2*x^2]*(Cosh[2*ArcSinh[c*x]] + 2*Arc 
Sinh[c*x]*(ArcSinh[c*x] - Sinh[2*ArcSinh[c*x]])))/Sqrt[d + c^2*d*x^2])/c^3
 

Rubi [A] (verified)

Time = 0.41 (sec) , antiderivative size = 119, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.115, Rules used = {6227, 15, 6198}

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.

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

\(\Big \downarrow \) 6227

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

\(\Big \downarrow \) 15

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

\(\Big \downarrow \) 6198

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

Input:

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

Output:

-1/4*(b*x^2*Sqrt[1 + c^2*x^2])/(c*Sqrt[d + c^2*d*x^2]) + (x*Sqrt[d + c^2*d 
*x^2]*(a + b*ArcSinh[c*x]))/(2*c^2*d) - (Sqrt[1 + c^2*x^2]*(a + b*ArcSinh[ 
c*x])^2)/(4*b*c^3*Sqrt[d + c^2*d*x^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 6198

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

Leaf count of result is larger than twice the leaf count of optimal. \(272\) vs. \(2(103)=206\).

Time = 0.63 (sec) , antiderivative size = 273, normalized size of antiderivative = 2.29

method	result	size
default	\(\frac {a x \sqrt {c^{2} d \,x^{2}+d}}{2 c^{2} d}-\frac {a \ln \left (\frac {x \,c^{2} d}{\sqrt {c^{2} d}}+\sqrt {c^{2} d \,x^{2}+d}\right )}{2 c^{2} \sqrt {c^{2} d}}+b \left (-\frac {\sqrt {d \left (c^{2} x^{2}+1\right )}\, \operatorname {arcsinh}\left (x c \right )^{2}}{4 \sqrt {c^{2} x^{2}+1}\, d \,c^{3}}+\frac {\sqrt {d \left (c^{2} x^{2}+1\right )}\, \left (2 x^{3} c^{3}+2 x^{2} c^{2} \sqrt {c^{2} x^{2}+1}+2 x c +\sqrt {c^{2} x^{2}+1}\right ) \left (-1+2 \,\operatorname {arcsinh}\left (x c \right )\right )}{16 d \,c^{3} \left (c^{2} x^{2}+1\right )}+\frac {\sqrt {d \left (c^{2} x^{2}+1\right )}\, \left (2 x^{3} c^{3}-2 x^{2} c^{2} \sqrt {c^{2} x^{2}+1}+2 x c -\sqrt {c^{2} x^{2}+1}\right ) \left (1+2 \,\operatorname {arcsinh}\left (x c \right )\right )}{16 d \,c^{3} \left (c^{2} x^{2}+1\right )}\right )\)	\(273\)
parts	\(\frac {a x \sqrt {c^{2} d \,x^{2}+d}}{2 c^{2} d}-\frac {a \ln \left (\frac {x \,c^{2} d}{\sqrt {c^{2} d}}+\sqrt {c^{2} d \,x^{2}+d}\right )}{2 c^{2} \sqrt {c^{2} d}}+b \left (-\frac {\sqrt {d \left (c^{2} x^{2}+1\right )}\, \operatorname {arcsinh}\left (x c \right )^{2}}{4 \sqrt {c^{2} x^{2}+1}\, d \,c^{3}}+\frac {\sqrt {d \left (c^{2} x^{2}+1\right )}\, \left (2 x^{3} c^{3}+2 x^{2} c^{2} \sqrt {c^{2} x^{2}+1}+2 x c +\sqrt {c^{2} x^{2}+1}\right ) \left (-1+2 \,\operatorname {arcsinh}\left (x c \right )\right )}{16 d \,c^{3} \left (c^{2} x^{2}+1\right )}+\frac {\sqrt {d \left (c^{2} x^{2}+1\right )}\, \left (2 x^{3} c^{3}-2 x^{2} c^{2} \sqrt {c^{2} x^{2}+1}+2 x c -\sqrt {c^{2} x^{2}+1}\right ) \left (1+2 \,\operatorname {arcsinh}\left (x c \right )\right )}{16 d \,c^{3} \left (c^{2} x^{2}+1\right )}\right )\)	\(273\)

Input:

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

Output:

1/2*a*x/c^2/d*(c^2*d*x^2+d)^(1/2)-1/2*a/c^2*ln(x*c^2*d/(c^2*d)^(1/2)+(c^2* 
d*x^2+d)^(1/2))/(c^2*d)^(1/2)+b*(-1/4*(d*(c^2*x^2+1))^(1/2)/(c^2*x^2+1)^(1 
/2)/d/c^3*arcsinh(x*c)^2+1/16*(d*(c^2*x^2+1))^(1/2)*(2*x^3*c^3+2*x^2*c^2*( 
c^2*x^2+1)^(1/2)+2*x*c+(c^2*x^2+1)^(1/2))*(-1+2*arcsinh(x*c))/d/c^3/(c^2*x 
^2+1)+1/16*(d*(c^2*x^2+1))^(1/2)*(2*x^3*c^3-2*x^2*c^2*(c^2*x^2+1)^(1/2)+2* 
x*c-(c^2*x^2+1)^(1/2))*(1+2*arcsinh(x*c))/d/c^3/(c^2*x^2+1))
 

Fricas [F]

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

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

Output:

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

Sympy [F]

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

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

Output:

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

Maxima [F(-2)]

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

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

Output:

Exception raised: RuntimeError >> ECL says: expt: undefined: 0 to a negati 
ve exponent.
 

Giac [F]

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

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

Output:

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

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

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

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

Output:

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