\(\int \frac {a+b \text {arcsinh}(c x)}{(f+g x) \sqrt {d+c^2 d x^2}} \, dx\) [51]

Optimal result

Integrand size = 30, antiderivative size = 325 \[ \int \frac {a+b \text {arcsinh}(c x)}{(f+g x) \sqrt {d+c^2 d x^2}} \, dx=\frac {\sqrt {1+c^2 x^2} (a+b \text {arcsinh}(c x)) \log \left (1+\frac {e^{\text {arcsinh}(c x)} g}{c f-\sqrt {c^2 f^2+g^2}}\right )}{\sqrt {c^2 f^2+g^2} \sqrt {d+c^2 d x^2}}-\frac {\sqrt {1+c^2 x^2} (a+b \text {arcsinh}(c x)) \log \left (1+\frac {e^{\text {arcsinh}(c x)} g}{c f+\sqrt {c^2 f^2+g^2}}\right )}{\sqrt {c^2 f^2+g^2} \sqrt {d+c^2 d x^2}}+\frac {b \sqrt {1+c^2 x^2} \operatorname {PolyLog}\left (2,-\frac {e^{\text {arcsinh}(c x)} g}{c f-\sqrt {c^2 f^2+g^2}}\right )}{\sqrt {c^2 f^2+g^2} \sqrt {d+c^2 d x^2}}-\frac {b \sqrt {1+c^2 x^2} \operatorname {PolyLog}\left (2,-\frac {e^{\text {arcsinh}(c x)} g}{c f+\sqrt {c^2 f^2+g^2}}\right )}{\sqrt {c^2 f^2+g^2} \sqrt {d+c^2 d x^2}} \]

[Out]

(a+b*arcsinh(c*x))*ln(1+(c*x+(c^2*x^2+1)^(1/2))*g/(c*f-(c^2*f^2+g^2)^(1/2)))*(c^2*x^2+1)^(1/2)/(c^2*f^2+g^2)^(
1/2)/(c^2*d*x^2+d)^(1/2)-(a+b*arcsinh(c*x))*ln(1+(c*x+(c^2*x^2+1)^(1/2))*g/(c*f+(c^2*f^2+g^2)^(1/2)))*(c^2*x^2
+1)^(1/2)/(c^2*f^2+g^2)^(1/2)/(c^2*d*x^2+d)^(1/2)+b*polylog(2,-(c*x+(c^2*x^2+1)^(1/2))*g/(c*f-(c^2*f^2+g^2)^(1
/2)))*(c^2*x^2+1)^(1/2)/(c^2*f^2+g^2)^(1/2)/(c^2*d*x^2+d)^(1/2)-b*polylog(2,-(c*x+(c^2*x^2+1)^(1/2))*g/(c*f+(c
^2*f^2+g^2)^(1/2)))*(c^2*x^2+1)^(1/2)/(c^2*f^2+g^2)^(1/2)/(c^2*d*x^2+d)^(1/2)

Rubi [A] (verified)

Time = 0.36 (sec) , antiderivative size = 325, normalized size of antiderivative = 1.00, number of steps used = 10, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.233, Rules used = {5845, 5843, 3403, 2296, 2221, 2317, 2438} \[ \int \frac {a+b \text {arcsinh}(c x)}{(f+g x) \sqrt {d+c^2 d x^2}} \, dx=\frac {\sqrt {c^2 x^2+1} (a+b \text {arcsinh}(c x)) \log \left (\frac {g e^{\text {arcsinh}(c x)}}{c f-\sqrt {c^2 f^2+g^2}}+1\right )}{\sqrt {c^2 d x^2+d} \sqrt {c^2 f^2+g^2}}-\frac {\sqrt {c^2 x^2+1} (a+b \text {arcsinh}(c x)) \log \left (\frac {g e^{\text {arcsinh}(c x)}}{\sqrt {c^2 f^2+g^2}+c f}+1\right )}{\sqrt {c^2 d x^2+d} \sqrt {c^2 f^2+g^2}}+\frac {b \sqrt {c^2 x^2+1} \operatorname {PolyLog}\left (2,-\frac {e^{\text {arcsinh}(c x)} g}{c f-\sqrt {c^2 f^2+g^2}}\right )}{\sqrt {c^2 d x^2+d} \sqrt {c^2 f^2+g^2}}-\frac {b \sqrt {c^2 x^2+1} \operatorname {PolyLog}\left (2,-\frac {e^{\text {arcsinh}(c x)} g}{c f+\sqrt {c^2 f^2+g^2}}\right )}{\sqrt {c^2 d x^2+d} \sqrt {c^2 f^2+g^2}} \]

[In]

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

[Out]

(Sqrt[1 + c^2*x^2]*(a + b*ArcSinh[c*x])*Log[1 + (E^ArcSinh[c*x]*g)/(c*f - Sqrt[c^2*f^2 + g^2])])/(Sqrt[c^2*f^2
+ g^2]*Sqrt[d + c^2*d*x^2]) - (Sqrt[1 + c^2*x^2]*(a + b*ArcSinh[c*x])*Log[1 + (E^ArcSinh[c*x]*g)/(c*f + Sqrt[
c^2*f^2 + g^2])])/(Sqrt[c^2*f^2 + g^2]*Sqrt[d + c^2*d*x^2]) + (b*Sqrt[1 + c^2*x^2]*PolyLog[2, -((E^ArcSinh[c*x
]*g)/(c*f - Sqrt[c^2*f^2 + g^2]))])/(Sqrt[c^2*f^2 + g^2]*Sqrt[d + c^2*d*x^2]) - (b*Sqrt[1 + c^2*x^2]*PolyLog[2
, -((E^ArcSinh[c*x]*g)/(c*f + Sqrt[c^2*f^2 + g^2]))])/(Sqrt[c^2*f^2 + g^2]*Sqrt[d + c^2*d*x^2])

Rule 2221

Int[(((F_)^((g_.)*((e_.) + (f_.)*(x_))))^(n_.)*((c_.) + (d_.)*(x_))^(m_.))/((a_) + (b_.)*((F_)^((g_.)*((e_.) +
(f_.)*(x_))))^(n_.)), x_Symbol] :> Simp[((c + d*x)^m/(b*f*g*n*Log[F]))*Log[1 + b*((F^(g*(e + f*x)))^n/a)], x]
- Dist[d*(m/(b*f*g*n*Log[F])), Int[(c + d*x)^(m - 1)*Log[1 + b*((F^(g*(e + f*x)))^n/a)], x], x] /; FreeQ[{F,
a, b, c, d, e, f, g, n}, x] && IGtQ[m, 0]

Rule 2296

Int[((F_)^(u_)*((f_.) + (g_.)*(x_))^(m_.))/((a_.) + (b_.)*(F_)^(u_) + (c_.)*(F_)^(v_)), x_Symbol] :> With[{q =
Rt[b^2 - 4*a*c, 2]}, Dist[2*(c/q), Int[(f + g*x)^m*(F^u/(b - q + 2*c*F^u)), x], x] - Dist[2*(c/q), Int[(f + g
*x)^m*(F^u/(b + q + 2*c*F^u)), x], x]] /; FreeQ[{F, a, b, c, f, g}, x] && EqQ[v, 2*u] && LinearQ[u, x] && NeQ[
b^2 - 4*a*c, 0] && IGtQ[m, 0]

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 3403

Int[((c_.) + (d_.)*(x_))^(m_.)/((a_) + (b_.)*sin[(e_.) + (Complex[0, fz_])*(f_.)*(x_)]), x_Symbol] :> Dist[2,
Int[(c + d*x)^m*(E^((-I)*e + f*fz*x)/((-I)*b + 2*a*E^((-I)*e + f*fz*x) + I*b*E^(2*((-I)*e + f*fz*x)))), x], x]
/; FreeQ[{a, b, c, d, e, f, fz}, x] && NeQ[a^2 - b^2, 0] && IGtQ[m, 0]

Rule 5843

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

Rule 5845

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

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

Mathematica [A] (verified)

Time = 0.35 (sec) , antiderivative size = 240, normalized size of antiderivative = 0.74 \[ \int \frac {a+b \text {arcsinh}(c x)}{(f+g x) \sqrt {d+c^2 d x^2}} \, dx=\frac {-\frac {a \text {arctanh}\left (\frac {\sqrt {d} \left (g-c^2 f x\right )}{\sqrt {c^2 f^2+g^2} \sqrt {d+c^2 d x^2}}\right )}{\sqrt {d}}+\frac {b \sqrt {1+c^2 x^2} \left (\text {arcsinh}(c x) \left (\log \left (1+\frac {e^{\text {arcsinh}(c x)} g}{c f-\sqrt {c^2 f^2+g^2}}\right )-\log \left (1+\frac {e^{\text {arcsinh}(c x)} g}{c f+\sqrt {c^2 f^2+g^2}}\right )\right )+\operatorname {PolyLog}\left (2,\frac {e^{\text {arcsinh}(c x)} g}{-c f+\sqrt {c^2 f^2+g^2}}\right )-\operatorname {PolyLog}\left (2,-\frac {e^{\text {arcsinh}(c x)} g}{c f+\sqrt {c^2 f^2+g^2}}\right )\right )}{\sqrt {d+c^2 d x^2}}}{\sqrt {c^2 f^2+g^2}} \]

[In]

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

[Out]

(-((a*ArcTanh[(Sqrt[d]*(g - c^2*f*x))/(Sqrt[c^2*f^2 + g^2]*Sqrt[d + c^2*d*x^2])])/Sqrt[d]) + (b*Sqrt[1 + c^2*x
^2]*(ArcSinh[c*x]*(Log[1 + (E^ArcSinh[c*x]*g)/(c*f - Sqrt[c^2*f^2 + g^2])] - Log[1 + (E^ArcSinh[c*x]*g)/(c*f +
Sqrt[c^2*f^2 + g^2])]) + PolyLog[2, (E^ArcSinh[c*x]*g)/(-(c*f) + Sqrt[c^2*f^2 + g^2])] - PolyLog[2, -((E^ArcS
inh[c*x]*g)/(c*f + Sqrt[c^2*f^2 + g^2]))]))/Sqrt[d + c^2*d*x^2])/Sqrt[c^2*f^2 + g^2]

Maple [A] (verified)

Time = 0.69 (sec) , antiderivative size = 529, normalized size of antiderivative = 1.63

method	result	size
default	\(-\frac {a \ln \left (\frac {\frac {2 d \left (c^{2} f^{2}+g^{2}\right )}{g^{2}}-\frac {2 c^{2} d f \left (x +\frac {f}{g}\right )}{g}+2 \sqrt {\frac {d \left (c^{2} f^{2}+g^{2}\right )}{g^{2}}}\, \sqrt {\left (x +\frac {f}{g}\right )^{2} c^{2} d -\frac {2 c^{2} d f \left (x +\frac {f}{g}\right )}{g}+\frac {d \left (c^{2} f^{2}+g^{2}\right )}{g^{2}}}}{x +\frac {f}{g}}\right )}{g \sqrt {\frac {d \left (c^{2} f^{2}+g^{2}\right )}{g^{2}}}}+b \left (\frac {\sqrt {d \left (c^{2} x^{2}+1\right )}\, \sqrt {c^{2} f^{2}+g^{2}}\, \sqrt {c^{2} x^{2}+1}\, \operatorname {arcsinh}\left (c x \right ) \left (\ln \left (\frac {-\left (c x +\sqrt {c^{2} x^{2}+1}\right ) g -c f +\sqrt {c^{2} f^{2}+g^{2}}}{-c f +\sqrt {c^{2} f^{2}+g^{2}}}\right )-\ln \left (\frac {\left (c x +\sqrt {c^{2} x^{2}+1}\right ) g +c f +\sqrt {c^{2} f^{2}+g^{2}}}{c f +\sqrt {c^{2} f^{2}+g^{2}}}\right )\right )}{d \left (c^{4} f^{2} x^{2}+c^{2} g^{2} x^{2}+c^{2} f^{2}+g^{2}\right )}+\frac {\sqrt {d \left (c^{2} x^{2}+1\right )}\, \sqrt {c^{2} f^{2}+g^{2}}\, \sqrt {c^{2} x^{2}+1}\, \left (\operatorname {dilog}\left (\frac {-\left (c x +\sqrt {c^{2} x^{2}+1}\right ) g -c f +\sqrt {c^{2} f^{2}+g^{2}}}{-c f +\sqrt {c^{2} f^{2}+g^{2}}}\right )-\operatorname {dilog}\left (\frac {\left (c x +\sqrt {c^{2} x^{2}+1}\right ) g +c f +\sqrt {c^{2} f^{2}+g^{2}}}{c f +\sqrt {c^{2} f^{2}+g^{2}}}\right )\right )}{d \left (c^{4} f^{2} x^{2}+c^{2} g^{2} x^{2}+c^{2} f^{2}+g^{2}\right )}\right )\)	\(529\)
parts	\(-\frac {a \ln \left (\frac {\frac {2 d \left (c^{2} f^{2}+g^{2}\right )}{g^{2}}-\frac {2 c^{2} d f \left (x +\frac {f}{g}\right )}{g}+2 \sqrt {\frac {d \left (c^{2} f^{2}+g^{2}\right )}{g^{2}}}\, \sqrt {\left (x +\frac {f}{g}\right )^{2} c^{2} d -\frac {2 c^{2} d f \left (x +\frac {f}{g}\right )}{g}+\frac {d \left (c^{2} f^{2}+g^{2}\right )}{g^{2}}}}{x +\frac {f}{g}}\right )}{g \sqrt {\frac {d \left (c^{2} f^{2}+g^{2}\right )}{g^{2}}}}+b \left (\frac {\sqrt {d \left (c^{2} x^{2}+1\right )}\, \sqrt {c^{2} f^{2}+g^{2}}\, \sqrt {c^{2} x^{2}+1}\, \operatorname {arcsinh}\left (c x \right ) \left (\ln \left (\frac {-\left (c x +\sqrt {c^{2} x^{2}+1}\right ) g -c f +\sqrt {c^{2} f^{2}+g^{2}}}{-c f +\sqrt {c^{2} f^{2}+g^{2}}}\right )-\ln \left (\frac {\left (c x +\sqrt {c^{2} x^{2}+1}\right ) g +c f +\sqrt {c^{2} f^{2}+g^{2}}}{c f +\sqrt {c^{2} f^{2}+g^{2}}}\right )\right )}{d \left (c^{4} f^{2} x^{2}+c^{2} g^{2} x^{2}+c^{2} f^{2}+g^{2}\right )}+\frac {\sqrt {d \left (c^{2} x^{2}+1\right )}\, \sqrt {c^{2} f^{2}+g^{2}}\, \sqrt {c^{2} x^{2}+1}\, \left (\operatorname {dilog}\left (\frac {-\left (c x +\sqrt {c^{2} x^{2}+1}\right ) g -c f +\sqrt {c^{2} f^{2}+g^{2}}}{-c f +\sqrt {c^{2} f^{2}+g^{2}}}\right )-\operatorname {dilog}\left (\frac {\left (c x +\sqrt {c^{2} x^{2}+1}\right ) g +c f +\sqrt {c^{2} f^{2}+g^{2}}}{c f +\sqrt {c^{2} f^{2}+g^{2}}}\right )\right )}{d \left (c^{4} f^{2} x^{2}+c^{2} g^{2} x^{2}+c^{2} f^{2}+g^{2}\right )}\right )\)	\(529\)

[In]

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

[Out]

-a/g/(d*(c^2*f^2+g^2)/g^2)^(1/2)*ln((2*d*(c^2*f^2+g^2)/g^2-2*c^2*d*f/g*(x+f/g)+2*(d*(c^2*f^2+g^2)/g^2)^(1/2)*(
(x+f/g)^2*c^2*d-2*c^2*d*f/g*(x+f/g)+d*(c^2*f^2+g^2)/g^2)^(1/2))/(x+f/g))+b*((d*(c^2*x^2+1))^(1/2)*(c^2*f^2+g^2
)^(1/2)*(c^2*x^2+1)^(1/2)/d/(c^4*f^2*x^2+c^2*g^2*x^2+c^2*f^2+g^2)*arcsinh(c*x)*(ln((-(c*x+(c^2*x^2+1)^(1/2))*g
-c*f+(c^2*f^2+g^2)^(1/2))/(-c*f+(c^2*f^2+g^2)^(1/2)))-ln(((c*x+(c^2*x^2+1)^(1/2))*g+c*f+(c^2*f^2+g^2)^(1/2))/(
c*f+(c^2*f^2+g^2)^(1/2))))+(d*(c^2*x^2+1))^(1/2)*(c^2*f^2+g^2)^(1/2)*(c^2*x^2+1)^(1/2)/d/(c^4*f^2*x^2+c^2*g^2*
x^2+c^2*f^2+g^2)*(dilog((-(c*x+(c^2*x^2+1)^(1/2))*g-c*f+(c^2*f^2+g^2)^(1/2))/(-c*f+(c^2*f^2+g^2)^(1/2)))-dilog
(((c*x+(c^2*x^2+1)^(1/2))*g+c*f+(c^2*f^2+g^2)^(1/2))/(c*f+(c^2*f^2+g^2)^(1/2)))))

Fricas [F]

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

[In]

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

[Out]

integral(sqrt(c^2*d*x^2 + d)*(b*arcsinh(c*x) + a)/(c^2*d*g*x^3 + c^2*d*f*x^2 + d*g*x + d*f), x)

Sympy [F]

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

[In]

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

[Out]

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

Maxima [F]

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

[In]

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

[Out]

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

Giac [F]

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

[In]

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

[Out]

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

Mupad [F(-1)]

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

[In]

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

[Out]

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