\(\int \frac {x^2 (a+b \text {arcsinh}(c x))^2}{\sqrt {d+c^2 d x^2}} \, dx\) [300]

Optimal result

Integrand size = 28, antiderivative size = 198 \[ \int \frac {x^2 (a+b \text {arcsinh}(c x))^2}{\sqrt {d+c^2 d x^2}} \, dx=\frac {b^2 x \sqrt {d+c^2 d x^2}}{4 c^2 d}-\frac {b^2 \sqrt {1+c^2 x^2} \text {arcsinh}(c x)}{4 c^3 \sqrt {d+c^2 d x^2}}-\frac {b x^2 \sqrt {1+c^2 x^2} (a+b \text {arcsinh}(c x))}{2 c \sqrt {d+c^2 d x^2}}+\frac {x \sqrt {d+c^2 d x^2} (a+b \text {arcsinh}(c x))^2}{2 c^2 d}-\frac {\sqrt {1+c^2 x^2} (a+b \text {arcsinh}(c x))^3}{6 b c^3 \sqrt {d+c^2 d x^2}} \] Output:

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

Mathematica [A] (verified)

Time = 1.11 (sec) , antiderivative size = 198, normalized size of antiderivative = 1.00 \[ \int \frac {x^2 (a+b \text {arcsinh}(c x))^2}{\sqrt {d+c^2 d x^2}} \, dx=\frac {12 a^2 c x \left (d+c^2 d x^2\right )-12 a^2 \sqrt {d} \sqrt {d+c^2 d x^2} \log \left (c d x+\sqrt {d} \sqrt {d+c^2 d x^2}\right )-6 a b d \sqrt {1+c^2 x^2} (\cosh (2 \text {arcsinh}(c x))+2 \text {arcsinh}(c x) (\text {arcsinh}(c x)-\sinh (2 \text {arcsinh}(c x))))-b^2 d \sqrt {1+c^2 x^2} \left (4 \text {arcsinh}(c x)^3+6 \text {arcsinh}(c x) \cosh (2 \text {arcsinh}(c x))-3 \left (1+2 \text {arcsinh}(c x)^2\right ) \sinh (2 \text {arcsinh}(c x))\right )}{24 c^3 d \sqrt {d+c^2 d x^2}} \] Input:

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

Output:

(12*a^2*c*x*(d + c^2*d*x^2) - 12*a^2*Sqrt[d]*Sqrt[d + c^2*d*x^2]*Log[c*d*x 
 + Sqrt[d]*Sqrt[d + c^2*d*x^2]] - 6*a*b*d*Sqrt[1 + c^2*x^2]*(Cosh[2*ArcSin 
h[c*x]] + 2*ArcSinh[c*x]*(ArcSinh[c*x] - Sinh[2*ArcSinh[c*x]])) - b^2*d*Sq 
rt[1 + c^2*x^2]*(4*ArcSinh[c*x]^3 + 6*ArcSinh[c*x]*Cosh[2*ArcSinh[c*x]] - 
3*(1 + 2*ArcSinh[c*x]^2)*Sinh[2*ArcSinh[c*x]]))/(24*c^3*d*Sqrt[d + c^2*d*x 
^2])
 

Rubi [A] (verified)

Time = 0.71 (sec) , antiderivative size = 171, normalized size of antiderivative = 0.86, number of steps used = 5, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.179, Rules used = {6227, 6191, 262, 222, 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.

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

\(\Big \downarrow \) 6227

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

\(\Big \downarrow \) 6191

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

\(\Big \downarrow \) 262

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

\(\Big \downarrow \) 222

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

\(\Big \downarrow \) 6198

\(\displaystyle \frac {x \sqrt {c^2 d x^2+d} (a+b \text {arcsinh}(c x))^2}{2 c^2 d}-\frac {\sqrt {c^2 x^2+1} (a+b \text {arcsinh}(c x))^3}{6 b c^3 \sqrt {c^2 d x^2+d}}-\frac {b \sqrt {c^2 x^2+1} \left (\frac {1}{2} x^2 (a+b \text {arcsinh}(c x))-\frac {1}{2} b c \left (\frac {x \sqrt {c^2 x^2+1}}{2 c^2}-\frac {\text {arcsinh}(c x)}{2 c^3}\right )\right )}{c \sqrt {c^2 d x^2+d}}\)

Input:

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

Output:

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

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 262

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

rule 6191

Int[((a_.) + ArcSinh[(c_.)*(x_)]*(b_.))^(n_.)*((d_.)*(x_))^(m_.), x_Symbol] 
 :> Simp[(d*x)^(m + 1)*((a + b*ArcSinh[c*x])^n/(d*(m + 1))), x] - Simp[b*c* 
(n/(d*(m + 1)))   Int[(d*x)^(m + 1)*((a + b*ArcSinh[c*x])^(n - 1)/Sqrt[1 + 
c^2*x^2]), x], x] /; FreeQ[{a, b, c, d, m}, x] && IGtQ[n, 0] && 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. \(505\) vs. \(2(172)=344\).

Time = 0.80 (sec) , antiderivative size = 506, normalized size of antiderivative = 2.56

method	result	size
default	\(\frac {a^{2} x \sqrt {c^{2} d \,x^{2}+d}}{2 c^{2} d}-\frac {a^{2} \ln \left (\frac {x \,c^{2} d}{\sqrt {c^{2} d}}+\sqrt {c^{2} d \,x^{2}+d}\right )}{2 c^{2} \sqrt {c^{2} d}}+b^{2} \left (-\frac {\sqrt {d \left (c^{2} x^{2}+1\right )}\, \operatorname {arcsinh}\left (x c \right )^{3}}{6 \sqrt {c^{2} x^{2}+1}\, d \,c^{3}}+\frac {\sqrt {d \left (c^{2} x^{2}+1\right )}\, \left (2 x^{3} c^{3}+2 x^{2} c^{2} \sqrt {c^{2} x^{2}+1}+2 x c +\sqrt {c^{2} x^{2}+1}\right ) \left (2 \operatorname {arcsinh}\left (x c \right )^{2}-2 \,\operatorname {arcsinh}\left (x c \right )+1\right )}{16 d \,c^{3} \left (c^{2} x^{2}+1\right )}+\frac {\sqrt {d \left (c^{2} x^{2}+1\right )}\, \left (2 x^{3} c^{3}-2 x^{2} c^{2} \sqrt {c^{2} x^{2}+1}+2 x c -\sqrt {c^{2} x^{2}+1}\right ) \left (2 \operatorname {arcsinh}\left (x c \right )^{2}+2 \,\operatorname {arcsinh}\left (x c \right )+1\right )}{16 d \,c^{3} \left (c^{2} x^{2}+1\right )}\right )+2 a b \left (-\frac {\sqrt {d \left (c^{2} x^{2}+1\right )}\, \operatorname {arcsinh}\left (x c \right )^{2}}{4 \sqrt {c^{2} x^{2}+1}\, d \,c^{3}}+\frac {\sqrt {d \left (c^{2} x^{2}+1\right )}\, \left (2 x^{3} c^{3}+2 x^{2} c^{2} \sqrt {c^{2} x^{2}+1}+2 x c +\sqrt {c^{2} x^{2}+1}\right ) \left (-1+2 \,\operatorname {arcsinh}\left (x c \right )\right )}{16 d \,c^{3} \left (c^{2} x^{2}+1\right )}+\frac {\sqrt {d \left (c^{2} x^{2}+1\right )}\, \left (2 x^{3} c^{3}-2 x^{2} c^{2} \sqrt {c^{2} x^{2}+1}+2 x c -\sqrt {c^{2} x^{2}+1}\right ) \left (1+2 \,\operatorname {arcsinh}\left (x c \right )\right )}{16 d \,c^{3} \left (c^{2} x^{2}+1\right )}\right )\)	\(506\)
parts	\(\frac {a^{2} x \sqrt {c^{2} d \,x^{2}+d}}{2 c^{2} d}-\frac {a^{2} \ln \left (\frac {x \,c^{2} d}{\sqrt {c^{2} d}}+\sqrt {c^{2} d \,x^{2}+d}\right )}{2 c^{2} \sqrt {c^{2} d}}+b^{2} \left (-\frac {\sqrt {d \left (c^{2} x^{2}+1\right )}\, \operatorname {arcsinh}\left (x c \right )^{3}}{6 \sqrt {c^{2} x^{2}+1}\, d \,c^{3}}+\frac {\sqrt {d \left (c^{2} x^{2}+1\right )}\, \left (2 x^{3} c^{3}+2 x^{2} c^{2} \sqrt {c^{2} x^{2}+1}+2 x c +\sqrt {c^{2} x^{2}+1}\right ) \left (2 \operatorname {arcsinh}\left (x c \right )^{2}-2 \,\operatorname {arcsinh}\left (x c \right )+1\right )}{16 d \,c^{3} \left (c^{2} x^{2}+1\right )}+\frac {\sqrt {d \left (c^{2} x^{2}+1\right )}\, \left (2 x^{3} c^{3}-2 x^{2} c^{2} \sqrt {c^{2} x^{2}+1}+2 x c -\sqrt {c^{2} x^{2}+1}\right ) \left (2 \operatorname {arcsinh}\left (x c \right )^{2}+2 \,\operatorname {arcsinh}\left (x c \right )+1\right )}{16 d \,c^{3} \left (c^{2} x^{2}+1\right )}\right )+2 a b \left (-\frac {\sqrt {d \left (c^{2} x^{2}+1\right )}\, \operatorname {arcsinh}\left (x c \right )^{2}}{4 \sqrt {c^{2} x^{2}+1}\, d \,c^{3}}+\frac {\sqrt {d \left (c^{2} x^{2}+1\right )}\, \left (2 x^{3} c^{3}+2 x^{2} c^{2} \sqrt {c^{2} x^{2}+1}+2 x c +\sqrt {c^{2} x^{2}+1}\right ) \left (-1+2 \,\operatorname {arcsinh}\left (x c \right )\right )}{16 d \,c^{3} \left (c^{2} x^{2}+1\right )}+\frac {\sqrt {d \left (c^{2} x^{2}+1\right )}\, \left (2 x^{3} c^{3}-2 x^{2} c^{2} \sqrt {c^{2} x^{2}+1}+2 x c -\sqrt {c^{2} x^{2}+1}\right ) \left (1+2 \,\operatorname {arcsinh}\left (x c \right )\right )}{16 d \,c^{3} \left (c^{2} x^{2}+1\right )}\right )\)	\(506\)

Input:

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

Output:

1/2*a^2*x/c^2/d*(c^2*d*x^2+d)^(1/2)-1/2*a^2/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^2*(-1/6*(d*(c^2*x^2+1))^(1/2)/(c^2*x^2 
+1)^(1/2)/d/c^3*arcsinh(x*c)^3+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))*(2*arcsinh(x*c)^2-2*arcsin 
h(x*c)+1)/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))*(2*arcsinh(x*c)^2+2*arcsinh(x 
*c)+1)/d/c^3/(c^2*x^2+1))+2*a*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]

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

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

Output:

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

Sympy [F]

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

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

Output:

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

Maxima [F(-2)]

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

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

Output:

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

Giac [F]

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

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

Output:

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

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

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

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

Output:

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