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

Optimal result

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

[Out]

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)

Rubi [A] (verified)

Time = 0.23 (sec) , antiderivative size = 249, normalized size of antiderivative = 1.00, number of steps used = 10, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.269, Rules used = {5808, 5806, 5816, 4267, 2317, 2438, 8} \[ \int \frac {\left (d+c^2 d x^2\right )^{3/2} (a+b \text {arcsinh}(c x))}{x} \, dx=-\frac {2 d \sqrt {c^2 d x^2+d} \text {arctanh}\left (e^{\text {arcsinh}(c x)}\right ) (a+b \text {arcsinh}(c x))}{\sqrt {c^2 x^2+1}}+\frac {1}{3} \left (c^2 d x^2+d\right )^{3/2} (a+b \text {arcsinh}(c x))+d \sqrt {c^2 d x^2+d} (a+b \text {arcsinh}(c x))-\frac {b d \sqrt {c^2 d x^2+d} \operatorname {PolyLog}\left (2,-e^{\text {arcsinh}(c x)}\right )}{\sqrt {c^2 x^2+1}}+\frac {b d \sqrt {c^2 d x^2+d} \operatorname {PolyLog}\left (2,e^{\text {arcsinh}(c x)}\right )}{\sqrt {c^2 x^2+1}}-\frac {4 b c d x \sqrt {c^2 d x^2+d}}{3 \sqrt {c^2 x^2+1}}-\frac {b c^3 d x^3 \sqrt {c^2 d x^2+d}}{9 \sqrt {c^2 x^2+1}} \]

[In]

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

[Out]

(-4*b*c*d*x*Sqrt[d + c^2*d*x^2])/(3*Sqrt[1 + c^2*x^2]) - (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*ArcSinh[c*x]) + ((d + c^2*d*x^2)^(3/2)*(a + b*ArcSinh[c*x]))/3 - (2*d*Sqrt[d
+ c^2*d*x^2]*(a + b*ArcSinh[c*x])*ArcTanh[E^ArcSinh[c*x]])/Sqrt[1 + c^2*x^2] - (b*d*Sqrt[d + c^2*d*x^2]*PolyL
og[2, -E^ArcSinh[c*x]])/Sqrt[1 + c^2*x^2] + (b*d*Sqrt[d + c^2*d*x^2]*PolyLog[2, E^ArcSinh[c*x]])/Sqrt[1 + c^2*
x^2]

Rule 8

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

Rule 2317

Int[Log[(a_) + (b_.)*((F_)^((e_.)*((c_.) + (d_.)*(x_))))^(n_.)], x_Symbol] :> Dist[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 2438

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 4267

Int[csc[(e_.) + (Complex[0, fz_])*(f_.)*(x_)]*((c_.) + (d_.)*(x_))^(m_.), x_Symbol] :> Simp[-2*(c + d*x)^m*(Ar
cTanh[E^((-I)*e + f*fz*x)]/(f*fz*I)), x] + (-Dist[d*(m/(f*fz*I)), Int[(c + d*x)^(m - 1)*Log[1 - E^((-I)*e + f*
fz*x)], x], x] + Dist[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 5806

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*ArcSinh[c*x])^n/(f*(m + 2))), x] + (Dist[(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] - Dist[b*c*(n/(f*(m + 2)))
*Simp[Sqrt[d + e*x^2]/Sqrt[1 + c^2*x^2]], Int[(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 5808

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*ArcSinh[c*x])^n/(f*(m + 2*p + 1))), x] + (Dist[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] - Dist[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 5816

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

Rubi steps \begin{align*} \text {integral}& = \frac {1}{3} \left (d+c^2 d x^2\right )^{3/2} (a+b \text {arcsinh}(c x))+d \int \frac {\sqrt {d+c^2 d x^2} (a+b \text {arcsinh}(c x))}{x} \, dx-\frac {\left (b c d \sqrt {d+c^2 d x^2}\right ) \int \left (1+c^2 x^2\right ) \, dx}{3 \sqrt {1+c^2 x^2}} \\ & = -\frac {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 {\left (d \sqrt {d+c^2 d x^2}\right ) \int \frac {a+b \text {arcsinh}(c x)}{x \sqrt {1+c^2 x^2}} \, dx}{\sqrt {1+c^2 x^2}}-\frac {\left (b c d \sqrt {d+c^2 d x^2}\right ) \int 1 \, dx}{\sqrt {1+c^2 x^2}} \\ & = -\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 {\left (d \sqrt {d+c^2 d x^2}\right ) \text {Subst}(\int (a+b x) \text {csch}(x) \, dx,x,\text {arcsinh}(c x))}{\sqrt {1+c^2 x^2}} \\ & = -\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 {\left (b d \sqrt {d+c^2 d x^2}\right ) \text {Subst}\left (\int \log \left (1-e^x\right ) \, dx,x,\text {arcsinh}(c x)\right )}{\sqrt {1+c^2 x^2}}+\frac {\left (b d \sqrt {d+c^2 d x^2}\right ) \text {Subst}\left (\int \log \left (1+e^x\right ) \, dx,x,\text {arcsinh}(c x)\right )}{\sqrt {1+c^2 x^2}} \\ & = -\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 {\left (b d \sqrt {d+c^2 d x^2}\right ) \text {Subst}\left (\int \frac {\log (1-x)}{x} \, dx,x,e^{\text {arcsinh}(c x)}\right )}{\sqrt {1+c^2 x^2}}+\frac {\left (b d \sqrt {d+c^2 d x^2}\right ) \text {Subst}\left (\int \frac {\log (1+x)}{x} \, dx,x,e^{\text {arcsinh}(c x)}\right )}{\sqrt {1+c^2 x^2}} \\ & = -\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}} \\ \end{align*}

Mathematica [A] (verified)

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

[In]

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

[Out]

(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] + ArcSinh[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])]))/S
qrt[1 + c^2*x^2]

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

[In]

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

[Out]

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)*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)
*arcsinh(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

Fricas [F]

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

[In]

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

[Out]

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)

Sympy [F]

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

[In]

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

[Out]

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

Maxima [F]

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

[In]

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

[Out]

-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)

Giac [F(-2)]

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

[In]

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

[Out]

Exception raised: TypeError >> an error occurred running a Giac command:INPUT:sage2:=int(sage0,sageVARx):;OUTP
UT:sym2poly/r2sym(const gen & e,const index_m & i,const vecteur & l) Error: Bad Argument Value

Mupad [F(-1)]

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

[In]

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

[Out]

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