\(\int (d+c^2 d x^2)^{3/2} (a+b \text {arcsinh}(c x)) \, dx\) [140]

Optimal result

Integrand size = 23, antiderivative size = 177 \[ \int \left (d+c^2 d x^2\right )^{3/2} (a+b \text {arcsinh}(c x)) \, dx=-\frac {3 b c d x^2 \sqrt {d+c^2 d x^2}}{16 \sqrt {1+c^2 x^2}}-\frac {b d \left (1+c^2 x^2\right )^{3/2} \sqrt {d+c^2 d x^2}}{16 c}+\frac {3}{8} d x \sqrt {d+c^2 d x^2} (a+b \text {arcsinh}(c x))+\frac {1}{4} x \left (d+c^2 d x^2\right )^{3/2} (a+b \text {arcsinh}(c x))+\frac {3 d \sqrt {d+c^2 d x^2} (a+b \text {arcsinh}(c x))^2}{16 b c \sqrt {1+c^2 x^2}} \] Output:

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

Mathematica [A] (verified)

Time = 0.54 (sec) , antiderivative size = 200, normalized size of antiderivative = 1.13 \[ \int \left (d+c^2 d x^2\right )^{3/2} (a+b \text {arcsinh}(c x)) \, dx=\frac {1}{8} a d x \left (5+2 c^2 x^2\right ) \sqrt {d+c^2 d x^2}+\frac {3 a d^{3/2} \log \left (c d x+\sqrt {d} \sqrt {d+c^2 d x^2}\right )}{8 c}+\frac {b d \sqrt {d+c^2 d x^2} (-\cosh (2 \text {arcsinh}(c x))+2 \text {arcsinh}(c x) (\text {arcsinh}(c x)+\sinh (2 \text {arcsinh}(c x))))}{8 c \sqrt {1+c^2 x^2}}-\frac {b d \sqrt {d+c^2 d x^2} \left (8 \text {arcsinh}(c x)^2+\cosh (4 \text {arcsinh}(c x))-4 \text {arcsinh}(c x) \sinh (4 \text {arcsinh}(c x))\right )}{128 c \sqrt {1+c^2 x^2}} \] Input:

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

Output:

(a*d*x*(5 + 2*c^2*x^2)*Sqrt[d + c^2*d*x^2])/8 + (3*a*d^(3/2)*Log[c*d*x + S 
qrt[d]*Sqrt[d + c^2*d*x^2]])/(8*c) + (b*d*Sqrt[d + c^2*d*x^2]*(-Cosh[2*Arc 
Sinh[c*x]] + 2*ArcSinh[c*x]*(ArcSinh[c*x] + Sinh[2*ArcSinh[c*x]])))/(8*c*S 
qrt[1 + c^2*x^2]) - (b*d*Sqrt[d + c^2*d*x^2]*(8*ArcSinh[c*x]^2 + Cosh[4*Ar 
cSinh[c*x]] - 4*ArcSinh[c*x]*Sinh[4*ArcSinh[c*x]]))/(128*c*Sqrt[1 + c^2*x^ 
2])
 

Rubi [A] (verified)

Time = 0.59 (sec) , antiderivative size = 196, normalized size of antiderivative = 1.11, number of steps used = 6, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.261, Rules used = {6201, 244, 2009, 6200, 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.

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

\(\Big \downarrow \) 6201

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

\(\Big \downarrow \) 244

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

\(\Big \downarrow \) 2009

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

\(\Big \downarrow \) 6200

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

\(\Big \downarrow \) 15

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

\(\Big \downarrow \) 6198

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

Input:

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

Output:

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

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 244

Int[((c_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^2)^(p_.), x_Symbol] :> Int[Expand 
Integrand[(c*x)^m*(a + b*x^2)^p, x], x] /; FreeQ[{a, b, c, m}, x] && IGtQ[p 
, 0]
 

rule 2009

Int[u_, x_Symbol] :> Simp[IntSum[u, x], x] /; SumQ[u]
 

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 6200

Int[((a_.) + ArcSinh[(c_.)*(x_)]*(b_.))^(n_.)*Sqrt[(d_) + (e_.)*(x_)^2], x_ 
Symbol] :> Simp[x*Sqrt[d + e*x^2]*((a + b*ArcSinh[c*x])^n/2), x] + (Simp[(1 
/2)*Simp[Sqrt[d + e*x^2]/Sqrt[1 + c^2*x^2]]   Int[(a + b*ArcSinh[c*x])^n/Sq 
rt[1 + c^2*x^2], x], x] - Simp[b*c*(n/2)*Simp[Sqrt[d + e*x^2]/Sqrt[1 + c^2* 
x^2]]   Int[x*(a + b*ArcSinh[c*x])^(n - 1), x], x]) /; FreeQ[{a, b, c, d, e 
}, x] && EqQ[e, c^2*d] && GtQ[n, 0]
 

rule 6201

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

Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(495\) vs. \(2(151)=302\).

Time = 0.65 (sec) , antiderivative size = 496, normalized size of antiderivative = 2.80

method	result	size
default	\(\frac {x \left (c^{2} d \,x^{2}+d \right )^{\frac {3}{2}} a}{4}+\frac {3 a d x \sqrt {c^{2} d \,x^{2}+d}}{8}+\frac {3 a \,d^{2} \ln \left (\frac {x \,c^{2} d}{\sqrt {c^{2} d}}+\sqrt {c^{2} d \,x^{2}+d}\right )}{8 \sqrt {c^{2} d}}+b \left (\frac {3 \sqrt {d \left (c^{2} x^{2}+1\right )}\, \operatorname {arcsinh}\left (x c \right )^{2} d}{16 \sqrt {c^{2} x^{2}+1}\, c}+\frac {\sqrt {d \left (c^{2} x^{2}+1\right )}\, \left (8 x^{5} c^{5}+8 x^{4} c^{4} \sqrt {c^{2} x^{2}+1}+12 x^{3} c^{3}+8 x^{2} c^{2} \sqrt {c^{2} x^{2}+1}+4 x c +\sqrt {c^{2} x^{2}+1}\right ) \left (-1+4 \,\operatorname {arcsinh}\left (x c \right )\right ) d}{256 \left (c^{2} x^{2}+1\right ) c}+\frac {\sqrt {d \left (c^{2} x^{2}+1\right )}\, \left (2 x^{3} c^{3}+2 x^{2} c^{2} \sqrt {c^{2} x^{2}+1}+2 x c +\sqrt {c^{2} x^{2}+1}\right ) \left (-1+2 \,\operatorname {arcsinh}\left (x c \right )\right ) d}{16 \left (c^{2} x^{2}+1\right ) c}+\frac {\sqrt {d \left (c^{2} x^{2}+1\right )}\, \left (2 x^{3} c^{3}-2 x^{2} c^{2} \sqrt {c^{2} x^{2}+1}+2 x c -\sqrt {c^{2} x^{2}+1}\right ) \left (1+2 \,\operatorname {arcsinh}\left (x c \right )\right ) d}{16 \left (c^{2} x^{2}+1\right ) c}+\frac {\sqrt {d \left (c^{2} x^{2}+1\right )}\, \left (8 x^{5} c^{5}-8 x^{4} c^{4} \sqrt {c^{2} x^{2}+1}+12 x^{3} c^{3}-8 x^{2} c^{2} \sqrt {c^{2} x^{2}+1}+4 x c -\sqrt {c^{2} x^{2}+1}\right ) \left (1+4 \,\operatorname {arcsinh}\left (x c \right )\right ) d}{256 \left (c^{2} x^{2}+1\right ) c}\right )\)	\(496\)
parts	\(\frac {x \left (c^{2} d \,x^{2}+d \right )^{\frac {3}{2}} a}{4}+\frac {3 a d x \sqrt {c^{2} d \,x^{2}+d}}{8}+\frac {3 a \,d^{2} \ln \left (\frac {x \,c^{2} d}{\sqrt {c^{2} d}}+\sqrt {c^{2} d \,x^{2}+d}\right )}{8 \sqrt {c^{2} d}}+b \left (\frac {3 \sqrt {d \left (c^{2} x^{2}+1\right )}\, \operatorname {arcsinh}\left (x c \right )^{2} d}{16 \sqrt {c^{2} x^{2}+1}\, c}+\frac {\sqrt {d \left (c^{2} x^{2}+1\right )}\, \left (8 x^{5} c^{5}+8 x^{4} c^{4} \sqrt {c^{2} x^{2}+1}+12 x^{3} c^{3}+8 x^{2} c^{2} \sqrt {c^{2} x^{2}+1}+4 x c +\sqrt {c^{2} x^{2}+1}\right ) \left (-1+4 \,\operatorname {arcsinh}\left (x c \right )\right ) d}{256 \left (c^{2} x^{2}+1\right ) c}+\frac {\sqrt {d \left (c^{2} x^{2}+1\right )}\, \left (2 x^{3} c^{3}+2 x^{2} c^{2} \sqrt {c^{2} x^{2}+1}+2 x c +\sqrt {c^{2} x^{2}+1}\right ) \left (-1+2 \,\operatorname {arcsinh}\left (x c \right )\right ) d}{16 \left (c^{2} x^{2}+1\right ) c}+\frac {\sqrt {d \left (c^{2} x^{2}+1\right )}\, \left (2 x^{3} c^{3}-2 x^{2} c^{2} \sqrt {c^{2} x^{2}+1}+2 x c -\sqrt {c^{2} x^{2}+1}\right ) \left (1+2 \,\operatorname {arcsinh}\left (x c \right )\right ) d}{16 \left (c^{2} x^{2}+1\right ) c}+\frac {\sqrt {d \left (c^{2} x^{2}+1\right )}\, \left (8 x^{5} c^{5}-8 x^{4} c^{4} \sqrt {c^{2} x^{2}+1}+12 x^{3} c^{3}-8 x^{2} c^{2} \sqrt {c^{2} x^{2}+1}+4 x c -\sqrt {c^{2} x^{2}+1}\right ) \left (1+4 \,\operatorname {arcsinh}\left (x c \right )\right ) d}{256 \left (c^{2} x^{2}+1\right ) c}\right )\)	\(496\)

Input:

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

Output:

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

Fricas [F]

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

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

Output:

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

Sympy [F]

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

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

Output:

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

Maxima [F(-2)]

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

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

Output:

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

Giac [F(-2)]

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

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

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

Output:

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

Reduce [F]

\[ \int \left (d+c^2 d x^2\right )^{3/2} (a+b \text {arcsinh}(c x)) \, dx=\frac {\sqrt {d}\, d \left (2 \sqrt {c^{2} x^{2}+1}\, a \,c^{3} x^{3}+5 \sqrt {c^{2} x^{2}+1}\, a c x +8 \left (\int \sqrt {c^{2} x^{2}+1}\, \mathit {asinh} \left (c x \right ) x^{2}d x \right ) b \,c^{3}+8 \left (\int \sqrt {c^{2} x^{2}+1}\, \mathit {asinh} \left (c x \right )d x \right ) b c +3 \,\mathrm {log}\left (\sqrt {c^{2} x^{2}+1}+c x \right ) a \right )}{8 c} \] Input:

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

Output:

(sqrt(d)*d*(2*sqrt(c**2*x**2 + 1)*a*c**3*x**3 + 5*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 + 8*int(sqrt(c**2* 
x**2 + 1)*asinh(c*x),x)*b*c + 3*log(sqrt(c**2*x**2 + 1) + c*x)*a))/(8*c)