3.2.32 \(\int \frac {(d+c^2 d x^2)^{3/2} (a+b \text {arcsinh}(c x))}{x} \, dx\) [132]

3.2.32.1 Optimal result

Integrand size = 26, antiderivative size = 249 \[ \int \frac {\left (d+c^2 d x^2\right )^{3/2} (a+b \text {arcsinh}(c x))}{x} \, dx=-\frac {4 b c d x \sqrt {d+c^2 d x^2}}{3 \sqrt {1+c^2 x^2}}-\frac {b c^3 d x^3 \sqrt {d+c^2 d x^2}}{9 \sqrt {1+c^2 x^2}}+d \sqrt {d+c^2 d x^2} (a+b \text {arcsinh}(c x))+\frac {1}{3} \left (d+c^2 d x^2\right )^{3/2} (a+b \text {arcsinh}(c x))-\frac {2 d \sqrt {d+c^2 d x^2} (a+b \text {arcsinh}(c x)) \text {arctanh}\left (e^{\text {arcsinh}(c x)}\right )}{\sqrt {1+c^2 x^2}}-\frac {b d \sqrt {d+c^2 d x^2} \operatorname {PolyLog}\left (2,-e^{\text {arcsinh}(c x)}\right )}{\sqrt {1+c^2 x^2}}+\frac {b d \sqrt {d+c^2 d x^2} \operatorname {PolyLog}\left (2,e^{\text {arcsinh}(c x)}\right )}{\sqrt {1+c^2 x^2}} \]

output

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

3.2.32.2 Mathematica [A] (verified)

Time = 0.63 (sec) , antiderivative size = 248, normalized size of antiderivative = 1.00 \[ \int \frac {\left (d+c^2 d x^2\right )^{3/2} (a+b \text {arcsinh}(c x))}{x} \, dx=\frac {1}{3} a d \left (4+c^2 x^2\right ) \sqrt {d+c^2 d x^2}+\frac {b d \sqrt {d+c^2 d x^2} \left (-c x \left (3+c^2 x^2\right )+3 \left (1+c^2 x^2\right )^{3/2} \text {arcsinh}(c x)\right )}{9 \sqrt {1+c^2 x^2}}+a d^{3/2} \log (x)-a d^{3/2} \log \left (d+\sqrt {d} \sqrt {d+c^2 d x^2}\right )+\frac {b d \sqrt {d+c^2 d x^2} \left (-c x+\sqrt {1+c^2 x^2} \text {arcsinh}(c x)+\text {arcsinh}(c x) \log \left (1-e^{-\text {arcsinh}(c x)}\right )-\text {arcsinh}(c x) \log \left (1+e^{-\text {arcsinh}(c x)}\right )+\operatorname {PolyLog}\left (2,-e^{-\text {arcsinh}(c x)}\right )-\operatorname {PolyLog}\left (2,e^{-\text {arcsinh}(c x)}\right )\right )}{\sqrt {1+c^2 x^2}} \]

input

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

output

(a*d*(4 + c^2*x^2)*Sqrt[d + c^2*d*x^2])/3 + (b*d*Sqrt[d + c^2*d*x^2]*(-(c* 
x*(3 + c^2*x^2)) + 3*(1 + c^2*x^2)^(3/2)*ArcSinh[c*x]))/(9*Sqrt[1 + c^2*x^ 
2]) + a*d^(3/2)*Log[x] - a*d^(3/2)*Log[d + Sqrt[d]*Sqrt[d + c^2*d*x^2]] + 
(b*d*Sqrt[d + c^2*d*x^2]*(-(c*x) + Sqrt[1 + c^2*x^2]*ArcSinh[c*x] + ArcSin 
h[c*x]*Log[1 - E^(-ArcSinh[c*x])] - ArcSinh[c*x]*Log[1 + E^(-ArcSinh[c*x]) 
] + PolyLog[2, -E^(-ArcSinh[c*x])] - PolyLog[2, E^(-ArcSinh[c*x])]))/Sqrt[ 
1 + c^2*x^2]
 

3.2.32.3 Rubi [C] (verified)

Result contains complex when optimal does not.

Time = 0.97 (sec) , antiderivative size = 210, normalized size of antiderivative = 0.84, number of steps used = 11, number of rules used = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.385, Rules used = {6223, 2009, 6221, 24, 6231, 3042, 26, 4670, 2715, 2838}

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.

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

\(\Big \downarrow \) 6223

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

\(\Big \downarrow \) 2009

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

\(\Big \downarrow \) 6221

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

\(\Big \downarrow \) 24

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

\(\Big \downarrow \) 6231

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

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 26

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

\(\Big \downarrow \) 4670

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

\(\Big \downarrow \) 2715

\(\displaystyle d \left (\frac {i \sqrt {c^2 d x^2+d} \left (i b \int e^{-\text {arcsinh}(c x)} \log \left (1-e^{\text {arcsinh}(c x)}\right )de^{\text {arcsinh}(c x)}-i b \int e^{-\text {arcsinh}(c x)} \log \left (1+e^{\text {arcsinh}(c x)}\right )de^{\text {arcsinh}(c x)}+2 i \text {arctanh}\left (e^{\text {arcsinh}(c x)}\right ) (a+b \text {arcsinh}(c x))\right )}{\sqrt {c^2 x^2+1}}+\sqrt {c^2 d x^2+d} (a+b \text {arcsinh}(c x))-\frac {b c x \sqrt {c^2 d x^2+d}}{\sqrt {c^2 x^2+1}}\right )+\frac {1}{3} \left (c^2 d x^2+d\right )^{3/2} (a+b \text {arcsinh}(c x))-\frac {b c d \left (\frac {c^2 x^3}{3}+x\right ) \sqrt {c^2 d x^2+d}}{3 \sqrt {c^2 x^2+1}}\)

\(\Big \downarrow \) 2838

\(\displaystyle d \left (\frac {i \sqrt {c^2 d x^2+d} \left (2 i \text {arctanh}\left (e^{\text {arcsinh}(c x)}\right ) (a+b \text {arcsinh}(c x))+i b \operatorname {PolyLog}\left (2,-e^{\text {arcsinh}(c x)}\right )-i b \operatorname {PolyLog}\left (2,e^{\text {arcsinh}(c x)}\right )\right )}{\sqrt {c^2 x^2+1}}+\sqrt {c^2 d x^2+d} (a+b \text {arcsinh}(c x))-\frac {b c x \sqrt {c^2 d x^2+d}}{\sqrt {c^2 x^2+1}}\right )+\frac {1}{3} \left (c^2 d x^2+d\right )^{3/2} (a+b \text {arcsinh}(c x))-\frac {b c d \left (\frac {c^2 x^3}{3}+x\right ) \sqrt {c^2 d x^2+d}}{3 \sqrt {c^2 x^2+1}}\)

input

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

output

-1/3*(b*c*d*Sqrt[d + c^2*d*x^2]*(x + (c^2*x^3)/3))/Sqrt[1 + c^2*x^2] + ((d 
 + c^2*d*x^2)^(3/2)*(a + b*ArcSinh[c*x]))/3 + d*(-((b*c*x*Sqrt[d + c^2*d*x 
^2])/Sqrt[1 + c^2*x^2]) + Sqrt[d + c^2*d*x^2]*(a + b*ArcSinh[c*x]) + (I*Sq 
rt[d + c^2*d*x^2]*((2*I)*(a + b*ArcSinh[c*x])*ArcTanh[E^ArcSinh[c*x]] + I* 
b*PolyLog[2, -E^ArcSinh[c*x]] - I*b*PolyLog[2, E^ArcSinh[c*x]]))/Sqrt[1 + 
c^2*x^2])
 

3.2.32.3.1 Defintions of rubi rules used

rule 24

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

rule 26

Int[(Complex[0, a_])*(Fx_), x_Symbol] :> Simp[(Complex[Identity[0], a])   I 
nt[Fx, x], x] /; FreeQ[a, x] && EqQ[a^2, 1]
 

rule 2009

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

rule 2715

Int[Log[(a_) + (b_.)*((F_)^((e_.)*((c_.) + (d_.)*(x_))))^(n_.)], x_Symbol] 
:> Simp[1/(d*e*n*Log[F])   Subst[Int[Log[a + b*x]/x, x], x, (F^(e*(c + d*x) 
))^n], x] /; FreeQ[{F, a, b, c, d, e, n}, x] && GtQ[a, 0]
 

rule 2838

Int[Log[(c_.)*((d_) + (e_.)*(x_)^(n_.))]/(x_), x_Symbol] :> Simp[-PolyLog[2 
, (-c)*e*x^n]/n, x] /; FreeQ[{c, d, e, n}, x] && EqQ[c*d, 1]
 

rule 3042

Int[u_, x_Symbol] :> Int[DeactivateTrig[u, x], x] /; FunctionOfTrigOfLinear 
Q[u, x]
 

rule 4670

Int[csc[(e_.) + (Complex[0, fz_])*(f_.)*(x_)]*((c_.) + (d_.)*(x_))^(m_.), x 
_Symbol] :> Simp[-2*(c + d*x)^m*(ArcTanh[E^((-I)*e + f*fz*x)]/(f*fz*I)), x] 
 + (-Simp[d*(m/(f*fz*I))   Int[(c + d*x)^(m - 1)*Log[1 - E^((-I)*e + f*fz*x 
)], x], x] + Simp[d*(m/(f*fz*I))   Int[(c + d*x)^(m - 1)*Log[1 + E^((-I)*e 
+ f*fz*x)], x], x]) /; FreeQ[{c, d, e, f, fz}, x] && IGtQ[m, 0]
 

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 6223

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

rule 6231

Int[(((a_.) + ArcSinh[(c_.)*(x_)]*(b_.))^(n_.)*(x_)^(m_))/Sqrt[(d_) + (e_.) 
*(x_)^2], x_Symbol] :> Simp[(1/c^(m + 1))*Simp[Sqrt[1 + c^2*x^2]/Sqrt[d + e 
*x^2]]   Subst[Int[(a + b*x)^n*Sinh[x]^m, x], x, ArcSinh[c*x]], x] /; FreeQ 
[{a, b, c, d, e}, x] && EqQ[e, c^2*d] && IGtQ[n, 0] && IntegerQ[m]
 

3.2.32.4 Maple [A] (verified)

Time = 0.27 (sec) , antiderivative size = 428, normalized size of antiderivative = 1.72

method	result	size
default	\(\frac {\left (c^{2} d \,x^{2}+d \right )^{\frac {3}{2}} a}{3}-a \,d^{\frac {3}{2}} \ln \left (\frac {2 d +2 \sqrt {d}\, \sqrt {c^{2} d \,x^{2}+d}}{x}\right )+a d \sqrt {c^{2} d \,x^{2}+d}+\frac {b \sqrt {d \left (c^{2} x^{2}+1\right )}\, \operatorname {polylog}\left (2, c x +\sqrt {c^{2} x^{2}+1}\right ) d}{\sqrt {c^{2} x^{2}+1}}+\frac {4 b \sqrt {d \left (c^{2} x^{2}+1\right )}\, d \,\operatorname {arcsinh}\left (c x \right )}{3 \left (c^{2} x^{2}+1\right )}-\frac {b \sqrt {d \left (c^{2} x^{2}+1\right )}\, \operatorname {polylog}\left (2, -c x -\sqrt {c^{2} x^{2}+1}\right ) d}{\sqrt {c^{2} x^{2}+1}}-\frac {4 b \sqrt {d \left (c^{2} x^{2}+1\right )}\, d c x}{3 \sqrt {c^{2} x^{2}+1}}-\frac {b \sqrt {d \left (c^{2} x^{2}+1\right )}\, \operatorname {arcsinh}\left (c x \right ) \ln \left (1+c x +\sqrt {c^{2} x^{2}+1}\right ) d}{\sqrt {c^{2} x^{2}+1}}+\frac {b \sqrt {d \left (c^{2} x^{2}+1\right )}\, d \,\operatorname {arcsinh}\left (c x \right ) x^{4} c^{4}}{3 c^{2} x^{2}+3}-\frac {b \sqrt {d \left (c^{2} x^{2}+1\right )}\, d \,c^{3} x^{3}}{9 \sqrt {c^{2} x^{2}+1}}+\frac {5 b \sqrt {d \left (c^{2} x^{2}+1\right )}\, d \,\operatorname {arcsinh}\left (c x \right ) x^{2} c^{2}}{3 \left (c^{2} x^{2}+1\right )}+\frac {b \sqrt {d \left (c^{2} x^{2}+1\right )}\, \operatorname {arcsinh}\left (c x \right ) \ln \left (1-c x -\sqrt {c^{2} x^{2}+1}\right ) d}{\sqrt {c^{2} x^{2}+1}}\)	\(428\)
parts	\(\frac {\left (c^{2} d \,x^{2}+d \right )^{\frac {3}{2}} a}{3}-a \,d^{\frac {3}{2}} \ln \left (\frac {2 d +2 \sqrt {d}\, \sqrt {c^{2} d \,x^{2}+d}}{x}\right )+a d \sqrt {c^{2} d \,x^{2}+d}+\frac {b \sqrt {d \left (c^{2} x^{2}+1\right )}\, \operatorname {polylog}\left (2, c x +\sqrt {c^{2} x^{2}+1}\right ) d}{\sqrt {c^{2} x^{2}+1}}+\frac {4 b \sqrt {d \left (c^{2} x^{2}+1\right )}\, d \,\operatorname {arcsinh}\left (c x \right )}{3 \left (c^{2} x^{2}+1\right )}-\frac {b \sqrt {d \left (c^{2} x^{2}+1\right )}\, \operatorname {polylog}\left (2, -c x -\sqrt {c^{2} x^{2}+1}\right ) d}{\sqrt {c^{2} x^{2}+1}}-\frac {4 b \sqrt {d \left (c^{2} x^{2}+1\right )}\, d c x}{3 \sqrt {c^{2} x^{2}+1}}-\frac {b \sqrt {d \left (c^{2} x^{2}+1\right )}\, \operatorname {arcsinh}\left (c x \right ) \ln \left (1+c x +\sqrt {c^{2} x^{2}+1}\right ) d}{\sqrt {c^{2} x^{2}+1}}+\frac {b \sqrt {d \left (c^{2} x^{2}+1\right )}\, d \,\operatorname {arcsinh}\left (c x \right ) x^{4} c^{4}}{3 c^{2} x^{2}+3}-\frac {b \sqrt {d \left (c^{2} x^{2}+1\right )}\, d \,c^{3} x^{3}}{9 \sqrt {c^{2} x^{2}+1}}+\frac {5 b \sqrt {d \left (c^{2} x^{2}+1\right )}\, d \,\operatorname {arcsinh}\left (c x \right ) x^{2} c^{2}}{3 \left (c^{2} x^{2}+1\right )}+\frac {b \sqrt {d \left (c^{2} x^{2}+1\right )}\, \operatorname {arcsinh}\left (c x \right ) \ln \left (1-c x -\sqrt {c^{2} x^{2}+1}\right ) d}{\sqrt {c^{2} x^{2}+1}}\)	\(428\)

input

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

output

1/3*(c^2*d*x^2+d)^(3/2)*a-a*d^(3/2)*ln((2*d+2*d^(1/2)*(c^2*d*x^2+d)^(1/2)) 
/x)+a*d*(c^2*d*x^2+d)^(1/2)+b*(d*(c^2*x^2+1))^(1/2)/(c^2*x^2+1)^(1/2)*poly 
log(2,c*x+(c^2*x^2+1)^(1/2))*d+4/3*b*(d*(c^2*x^2+1))^(1/2)*d/(c^2*x^2+1)*a 
rcsinh(c*x)-b*(d*(c^2*x^2+1))^(1/2)/(c^2*x^2+1)^(1/2)*polylog(2,-c*x-(c^2* 
x^2+1)^(1/2))*d-4/3*b*(d*(c^2*x^2+1))^(1/2)*d/(c^2*x^2+1)^(1/2)*c*x-b*(d*( 
c^2*x^2+1))^(1/2)/(c^2*x^2+1)^(1/2)*arcsinh(c*x)*ln(1+c*x+(c^2*x^2+1)^(1/2 
))*d+1/3*b*(d*(c^2*x^2+1))^(1/2)*d/(c^2*x^2+1)*arcsinh(c*x)*x^4*c^4-1/9*b* 
(d*(c^2*x^2+1))^(1/2)*d/(c^2*x^2+1)^(1/2)*c^3*x^3+5/3*b*(d*(c^2*x^2+1))^(1 
/2)*d/(c^2*x^2+1)*arcsinh(c*x)*x^2*c^2+b*(d*(c^2*x^2+1))^(1/2)/(c^2*x^2+1) 
^(1/2)*arcsinh(c*x)*ln(1-c*x-(c^2*x^2+1)^(1/2))*d
 

3.2.32.5 Fricas [F]

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

input

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

3.2.32.6 Sympy [F]

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

input

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

output

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

3.2.32.7 Maxima [F]

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

input

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

output

-1/3*(3*d^(3/2)*arcsinh(1/(c*abs(x))) - (c^2*d*x^2 + d)^(3/2) - 3*sqrt(c^2 
*d*x^2 + d)*d)*a + b*integrate((c^2*d*x^2 + d)^(3/2)*log(c*x + sqrt(c^2*x^ 
2 + 1))/x, x)
 

3.2.32.8 Giac [F(-2)]

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

input

integrate((c^2*d*x^2+d)^(3/2)*(a+b*arcsinh(c*x))/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
 

3.2.32.9 Mupad [F(-1)]

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

input

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

output

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