\(\int x^2 \sqrt {d+c^2 d x^2} (a+b \text {arcsinh}(c x)) \, dx\) [130]

Optimal result

Integrand size = 26, antiderivative size = 181 \[ \int x^2 \sqrt {d+c^2 d x^2} (a+b \text {arcsinh}(c x)) \, dx=-\frac {b x^2 \sqrt {d+c^2 d x^2}}{16 c \sqrt {1+c^2 x^2}}-\frac {b c x^4 \sqrt {d+c^2 d x^2}}{16 \sqrt {1+c^2 x^2}}+\frac {x \sqrt {d+c^2 d x^2} (a+b \text {arcsinh}(c x))}{8 c^2}+\frac {1}{4} x^3 \sqrt {d+c^2 d x^2} (a+b \text {arcsinh}(c x))-\frac {\sqrt {d+c^2 d x^2} (a+b \text {arcsinh}(c x))^2}{16 b c^3 \sqrt {1+c^2 x^2}} \] Output:

-1/16*b*x^2*(c^2*d*x^2+d)^(1/2)/c/(c^2*x^2+1)^(1/2)-1/16*b*c*x^4*(c^2*d*x^ 
2+d)^(1/2)/(c^2*x^2+1)^(1/2)+1/8*x*(c^2*d*x^2+d)^(1/2)*(a+b*arcsinh(c*x))/ 
c^2+1/4*x^3*(c^2*d*x^2+d)^(1/2)*(a+b*arcsinh(c*x))-1/16*(c^2*d*x^2+d)^(1/2 
)*(a+b*arcsinh(c*x))^2/b/c^3/(c^2*x^2+1)^(1/2)
 

Mathematica [A] (verified)

Time = 0.59 (sec) , antiderivative size = 129, normalized size of antiderivative = 0.71 \[ \int x^2 \sqrt {d+c^2 d x^2} (a+b \text {arcsinh}(c x)) \, dx=-\frac {-16 a c x \left (1+2 c^2 x^2\right ) \sqrt {d+c^2 d x^2}+16 a \sqrt {d} \log \left (c d x+\sqrt {d} \sqrt {d+c^2 d x^2}\right )+\frac {b \sqrt {d+c^2 d x^2} \left (8 \text {arcsinh}(c x)^2+\cosh (4 \text {arcsinh}(c x))-4 \text {arcsinh}(c x) \sinh (4 \text {arcsinh}(c x))\right )}{\sqrt {1+c^2 x^2}}}{128 c^3} \] Input:

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

Output:

-1/128*(-16*a*c*x*(1 + 2*c^2*x^2)*Sqrt[d + c^2*d*x^2] + 16*a*Sqrt[d]*Log[c 
*d*x + Sqrt[d]*Sqrt[d + c^2*d*x^2]] + (b*Sqrt[d + c^2*d*x^2]*(8*ArcSinh[c* 
x]^2 + Cosh[4*ArcSinh[c*x]] - 4*ArcSinh[c*x]*Sinh[4*ArcSinh[c*x]]))/Sqrt[1 
 + c^2*x^2])/c^3
 

Rubi [A] (verified)

Time = 0.61 (sec) , antiderivative size = 158, normalized size of antiderivative = 0.87, number of steps used = 5, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.192, Rules used = {6221, 15, 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.

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

\(\Big \downarrow \) 6221

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

\(\Big \downarrow \) 15

\(\displaystyle \frac {\sqrt {c^2 d x^2+d} \int \frac {x^2 (a+b \text {arcsinh}(c x))}{\sqrt {c^2 x^2+1}}dx}{4 \sqrt {c^2 x^2+1}}+\frac {1}{4} x^3 \sqrt {c^2 d x^2+d} (a+b \text {arcsinh}(c x))-\frac {b c x^4 \sqrt {c^2 d x^2+d}}{16 \sqrt {c^2 x^2+1}}\)

\(\Big \downarrow \) 6227

\(\displaystyle \frac {\sqrt {c^2 d x^2+d} \left (-\frac {\int \frac {a+b \text {arcsinh}(c x)}{\sqrt {c^2 x^2+1}}dx}{2 c^2}-\frac {b \int xdx}{2 c}+\frac {x \sqrt {c^2 x^2+1} (a+b \text {arcsinh}(c x))}{2 c^2}\right )}{4 \sqrt {c^2 x^2+1}}+\frac {1}{4} x^3 \sqrt {c^2 d x^2+d} (a+b \text {arcsinh}(c x))-\frac {b c x^4 \sqrt {c^2 d x^2+d}}{16 \sqrt {c^2 x^2+1}}\)

\(\Big \downarrow \) 15

\(\displaystyle \frac {\sqrt {c^2 d x^2+d} \left (-\frac {\int \frac {a+b \text {arcsinh}(c x)}{\sqrt {c^2 x^2+1}}dx}{2 c^2}+\frac {x \sqrt {c^2 x^2+1} (a+b \text {arcsinh}(c x))}{2 c^2}-\frac {b x^2}{4 c}\right )}{4 \sqrt {c^2 x^2+1}}+\frac {1}{4} x^3 \sqrt {c^2 d x^2+d} (a+b \text {arcsinh}(c x))-\frac {b c x^4 \sqrt {c^2 d x^2+d}}{16 \sqrt {c^2 x^2+1}}\)

\(\Big \downarrow \) 6198

\(\displaystyle \frac {1}{4} x^3 \sqrt {c^2 d x^2+d} (a+b \text {arcsinh}(c x))+\frac {\sqrt {c^2 d x^2+d} \left (-\frac {(a+b \text {arcsinh}(c x))^2}{4 b c^3}+\frac {x \sqrt {c^2 x^2+1} (a+b \text {arcsinh}(c x))}{2 c^2}-\frac {b x^2}{4 c}\right )}{4 \sqrt {c^2 x^2+1}}-\frac {b c x^4 \sqrt {c^2 d x^2+d}}{16 \sqrt {c^2 x^2+1}}\)

Input:

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

Output:

-1/16*(b*c*x^4*Sqrt[d + c^2*d*x^2])/Sqrt[1 + c^2*x^2] + (x^3*Sqrt[d + c^2* 
d*x^2]*(a + b*ArcSinh[c*x]))/4 + (Sqrt[d + c^2*d*x^2]*(-1/4*(b*x^2)/c + (x 
*Sqrt[1 + c^2*x^2]*(a + b*ArcSinh[c*x]))/(2*c^2) - (a + b*ArcSinh[c*x])^2/ 
(4*b*c^3)))/(4*Sqrt[1 + c^2*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 6221

Int[((a_.) + ArcSinh[(c_.)*(x_)]*(b_.))^(n_.)*((f_.)*(x_))^(m_)*Sqrt[(d_) + 
 (e_.)*(x_)^2], x_Symbol] :> Simp[(f*x)^(m + 1)*Sqrt[d + e*x^2]*((a + b*Arc 
Sinh[c*x])^n/(f*(m + 2))), x] + (Simp[(1/(m + 2))*Simp[Sqrt[d + e*x^2]/Sqrt 
[1 + c^2*x^2]]   Int[(f*x)^m*((a + b*ArcSinh[c*x])^n/Sqrt[1 + c^2*x^2]), x] 
, x] - Simp[b*c*(n/(f*(m + 2)))*Simp[Sqrt[d + e*x^2]/Sqrt[1 + c^2*x^2]]   I 
nt[(f*x)^(m + 1)*(a + b*ArcSinh[c*x])^(n - 1), x], x]) /; FreeQ[{a, b, c, d 
, e, f, m}, x] && EqQ[e, c^2*d] && IGtQ[n, 0] && (IGtQ[m, -2] || EqQ[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. \(337\) vs. \(2(155)=310\).

Time = 0.54 (sec) , antiderivative size = 338, normalized size of antiderivative = 1.87

method	result	size
default	\(\frac {a x \left (c^{2} d \,x^{2}+d \right )^{\frac {3}{2}}}{4 c^{2} d}-\frac {a x \sqrt {c^{2} d \,x^{2}+d}}{8 c^{2}}-\frac {a d \ln \left (\frac {x \,c^{2} d}{\sqrt {c^{2} d}}+\sqrt {c^{2} d \,x^{2}+d}\right )}{8 c^{2} \sqrt {c^{2} d}}+b \left (-\frac {\sqrt {d \left (c^{2} x^{2}+1\right )}\, \operatorname {arcsinh}\left (x c \right )^{2}}{16 \sqrt {c^{2} x^{2}+1}\, c^{3}}+\frac {\sqrt {d \left (c^{2} x^{2}+1\right )}\, \left (8 x^{5} c^{5}+8 x^{4} c^{4} \sqrt {c^{2} x^{2}+1}+12 x^{3} c^{3}+8 x^{2} c^{2} \sqrt {c^{2} x^{2}+1}+4 x c +\sqrt {c^{2} x^{2}+1}\right ) \left (-1+4 \,\operatorname {arcsinh}\left (x c \right )\right )}{256 c^{3} \left (c^{2} x^{2}+1\right )}+\frac {\sqrt {d \left (c^{2} x^{2}+1\right )}\, \left (8 x^{5} c^{5}-8 x^{4} c^{4} \sqrt {c^{2} x^{2}+1}+12 x^{3} c^{3}-8 x^{2} c^{2} \sqrt {c^{2} x^{2}+1}+4 x c -\sqrt {c^{2} x^{2}+1}\right ) \left (1+4 \,\operatorname {arcsinh}\left (x c \right )\right )}{256 c^{3} \left (c^{2} x^{2}+1\right )}\right )\)	\(338\)
parts	\(\frac {a x \left (c^{2} d \,x^{2}+d \right )^{\frac {3}{2}}}{4 c^{2} d}-\frac {a x \sqrt {c^{2} d \,x^{2}+d}}{8 c^{2}}-\frac {a d \ln \left (\frac {x \,c^{2} d}{\sqrt {c^{2} d}}+\sqrt {c^{2} d \,x^{2}+d}\right )}{8 c^{2} \sqrt {c^{2} d}}+b \left (-\frac {\sqrt {d \left (c^{2} x^{2}+1\right )}\, \operatorname {arcsinh}\left (x c \right )^{2}}{16 \sqrt {c^{2} x^{2}+1}\, c^{3}}+\frac {\sqrt {d \left (c^{2} x^{2}+1\right )}\, \left (8 x^{5} c^{5}+8 x^{4} c^{4} \sqrt {c^{2} x^{2}+1}+12 x^{3} c^{3}+8 x^{2} c^{2} \sqrt {c^{2} x^{2}+1}+4 x c +\sqrt {c^{2} x^{2}+1}\right ) \left (-1+4 \,\operatorname {arcsinh}\left (x c \right )\right )}{256 c^{3} \left (c^{2} x^{2}+1\right )}+\frac {\sqrt {d \left (c^{2} x^{2}+1\right )}\, \left (8 x^{5} c^{5}-8 x^{4} c^{4} \sqrt {c^{2} x^{2}+1}+12 x^{3} c^{3}-8 x^{2} c^{2} \sqrt {c^{2} x^{2}+1}+4 x c -\sqrt {c^{2} x^{2}+1}\right ) \left (1+4 \,\operatorname {arcsinh}\left (x c \right )\right )}{256 c^{3} \left (c^{2} x^{2}+1\right )}\right )\)	\(338\)

Input:

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

Output:

1/4*a*x*(c^2*d*x^2+d)^(3/2)/c^2/d-1/8*a/c^2*x*(c^2*d*x^2+d)^(1/2)-1/8*a/c^ 
2*d*ln(x*c^2*d/(c^2*d)^(1/2)+(c^2*d*x^2+d)^(1/2))/(c^2*d)^(1/2)+b*(-1/16*( 
d*(c^2*x^2+1))^(1/2)/(c^2*x^2+1)^(1/2)/c^3*arcsinh(x*c)^2+1/256*(d*(c^2*x^ 
2+1))^(1/2)*(8*x^5*c^5+8*x^4*c^4*(c^2*x^2+1)^(1/2)+12*x^3*c^3+8*x^2*c^2*(c 
^2*x^2+1)^(1/2)+4*x*c+(c^2*x^2+1)^(1/2))*(-1+4*arcsinh(x*c))/c^3/(c^2*x^2+ 
1)+1/256*(d*(c^2*x^2+1))^(1/2)*(8*x^5*c^5-8*x^4*c^4*(c^2*x^2+1)^(1/2)+12*x 
^3*c^3-8*x^2*c^2*(c^2*x^2+1)^(1/2)+4*x*c-(c^2*x^2+1)^(1/2))*(1+4*arcsinh(x 
*c))/c^3/(c^2*x^2+1))
 

Fricas [F]

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

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

Output:

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

Sympy [F]

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

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

Output:

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

Maxima [F(-2)]

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

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

Output:

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

Giac [F]

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

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

Output:

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

Mupad [F(-1)]

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

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

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

Output:

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