\(\int \frac {(a+b \text {arcsinh}(c x))^2}{(d+e x)^2} \, dx\) [17]

Optimal result

Integrand size = 18, antiderivative size = 263 \[ \int \frac {(a+b \text {arcsinh}(c x))^2}{(d+e x)^2} \, dx=-\frac {(a+b \text {arcsinh}(c x))^2}{e (d+e x)}+\frac {2 b c (a+b \text {arcsinh}(c x)) \log \left (1+\frac {e e^{\text {arcsinh}(c x)}}{c d-\sqrt {c^2 d^2+e^2}}\right )}{e \sqrt {c^2 d^2+e^2}}-\frac {2 b c (a+b \text {arcsinh}(c x)) \log \left (1+\frac {e e^{\text {arcsinh}(c x)}}{c d+\sqrt {c^2 d^2+e^2}}\right )}{e \sqrt {c^2 d^2+e^2}}+\frac {2 b^2 c \operatorname {PolyLog}\left (2,-\frac {e e^{\text {arcsinh}(c x)}}{c d-\sqrt {c^2 d^2+e^2}}\right )}{e \sqrt {c^2 d^2+e^2}}-\frac {2 b^2 c \operatorname {PolyLog}\left (2,-\frac {e e^{\text {arcsinh}(c x)}}{c d+\sqrt {c^2 d^2+e^2}}\right )}{e \sqrt {c^2 d^2+e^2}} \]

[Out]

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

Rubi [A] (verified)

Time = 0.32 (sec) , antiderivative size = 263, normalized size of antiderivative = 1.00, number of steps used = 10, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.389, Rules used = {5828, 5843, 3403, 2296, 2221, 2317, 2438} \[ \int \frac {(a+b \text {arcsinh}(c x))^2}{(d+e x)^2} \, dx=\frac {2 b c (a+b \text {arcsinh}(c x)) \log \left (\frac {e e^{\text {arcsinh}(c x)}}{c d-\sqrt {c^2 d^2+e^2}}+1\right )}{e \sqrt {c^2 d^2+e^2}}-\frac {2 b c (a+b \text {arcsinh}(c x)) \log \left (\frac {e e^{\text {arcsinh}(c x)}}{\sqrt {c^2 d^2+e^2}+c d}+1\right )}{e \sqrt {c^2 d^2+e^2}}-\frac {(a+b \text {arcsinh}(c x))^2}{e (d+e x)}+\frac {2 b^2 c \operatorname {PolyLog}\left (2,-\frac {e e^{\text {arcsinh}(c x)}}{c d-\sqrt {c^2 d^2+e^2}}\right )}{e \sqrt {c^2 d^2+e^2}}-\frac {2 b^2 c \operatorname {PolyLog}\left (2,-\frac {e e^{\text {arcsinh}(c x)}}{c d+\sqrt {c^2 d^2+e^2}}\right )}{e \sqrt {c^2 d^2+e^2}} \]

[In]

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

[Out]

-((a + b*ArcSinh[c*x])^2/(e*(d + e*x))) + (2*b*c*(a + b*ArcSinh[c*x])*Log[1 + (e*E^ArcSinh[c*x])/(c*d - Sqrt[c
^2*d^2 + e^2])])/(e*Sqrt[c^2*d^2 + e^2]) - (2*b*c*(a + b*ArcSinh[c*x])*Log[1 + (e*E^ArcSinh[c*x])/(c*d + Sqrt[
c^2*d^2 + e^2])])/(e*Sqrt[c^2*d^2 + e^2]) + (2*b^2*c*PolyLog[2, -((e*E^ArcSinh[c*x])/(c*d - Sqrt[c^2*d^2 + e^2
]))])/(e*Sqrt[c^2*d^2 + e^2]) - (2*b^2*c*PolyLog[2, -((e*E^ArcSinh[c*x])/(c*d + Sqrt[c^2*d^2 + e^2]))])/(e*Sqr
t[c^2*d^2 + e^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 5828

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

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])

Rubi steps \begin{align*} \text {integral}& = -\frac {(a+b \text {arcsinh}(c x))^2}{e (d+e x)}+\frac {(2 b c) \int \frac {a+b \text {arcsinh}(c x)}{(d+e x) \sqrt {1+c^2 x^2}} \, dx}{e} \\ & = -\frac {(a+b \text {arcsinh}(c x))^2}{e (d+e x)}+\frac {(2 b c) \text {Subst}\left (\int \frac {a+b x}{c d+e \sinh (x)} \, dx,x,\text {arcsinh}(c x)\right )}{e} \\ & = -\frac {(a+b \text {arcsinh}(c x))^2}{e (d+e x)}+\frac {(4 b c) \text {Subst}\left (\int \frac {e^x (a+b x)}{-e+2 c d e^x+e e^{2 x}} \, dx,x,\text {arcsinh}(c x)\right )}{e} \\ & = -\frac {(a+b \text {arcsinh}(c x))^2}{e (d+e x)}+\frac {(4 b c) \text {Subst}\left (\int \frac {e^x (a+b x)}{2 c d-2 \sqrt {c^2 d^2+e^2}+2 e e^x} \, dx,x,\text {arcsinh}(c x)\right )}{\sqrt {c^2 d^2+e^2}}-\frac {(4 b c) \text {Subst}\left (\int \frac {e^x (a+b x)}{2 c d+2 \sqrt {c^2 d^2+e^2}+2 e e^x} \, dx,x,\text {arcsinh}(c x)\right )}{\sqrt {c^2 d^2+e^2}} \\ & = -\frac {(a+b \text {arcsinh}(c x))^2}{e (d+e x)}+\frac {2 b c (a+b \text {arcsinh}(c x)) \log \left (1+\frac {e e^{\text {arcsinh}(c x)}}{c d-\sqrt {c^2 d^2+e^2}}\right )}{e \sqrt {c^2 d^2+e^2}}-\frac {2 b c (a+b \text {arcsinh}(c x)) \log \left (1+\frac {e e^{\text {arcsinh}(c x)}}{c d+\sqrt {c^2 d^2+e^2}}\right )}{e \sqrt {c^2 d^2+e^2}}-\frac {\left (2 b^2 c\right ) \text {Subst}\left (\int \log \left (1+\frac {2 e e^x}{2 c d-2 \sqrt {c^2 d^2+e^2}}\right ) \, dx,x,\text {arcsinh}(c x)\right )}{e \sqrt {c^2 d^2+e^2}}+\frac {\left (2 b^2 c\right ) \text {Subst}\left (\int \log \left (1+\frac {2 e e^x}{2 c d+2 \sqrt {c^2 d^2+e^2}}\right ) \, dx,x,\text {arcsinh}(c x)\right )}{e \sqrt {c^2 d^2+e^2}} \\ & = -\frac {(a+b \text {arcsinh}(c x))^2}{e (d+e x)}+\frac {2 b c (a+b \text {arcsinh}(c x)) \log \left (1+\frac {e e^{\text {arcsinh}(c x)}}{c d-\sqrt {c^2 d^2+e^2}}\right )}{e \sqrt {c^2 d^2+e^2}}-\frac {2 b c (a+b \text {arcsinh}(c x)) \log \left (1+\frac {e e^{\text {arcsinh}(c x)}}{c d+\sqrt {c^2 d^2+e^2}}\right )}{e \sqrt {c^2 d^2+e^2}}-\frac {\left (2 b^2 c\right ) \text {Subst}\left (\int \frac {\log \left (1+\frac {2 e x}{2 c d-2 \sqrt {c^2 d^2+e^2}}\right )}{x} \, dx,x,e^{\text {arcsinh}(c x)}\right )}{e \sqrt {c^2 d^2+e^2}}+\frac {\left (2 b^2 c\right ) \text {Subst}\left (\int \frac {\log \left (1+\frac {2 e x}{2 c d+2 \sqrt {c^2 d^2+e^2}}\right )}{x} \, dx,x,e^{\text {arcsinh}(c x)}\right )}{e \sqrt {c^2 d^2+e^2}} \\ & = -\frac {(a+b \text {arcsinh}(c x))^2}{e (d+e x)}+\frac {2 b c (a+b \text {arcsinh}(c x)) \log \left (1+\frac {e e^{\text {arcsinh}(c x)}}{c d-\sqrt {c^2 d^2+e^2}}\right )}{e \sqrt {c^2 d^2+e^2}}-\frac {2 b c (a+b \text {arcsinh}(c x)) \log \left (1+\frac {e e^{\text {arcsinh}(c x)}}{c d+\sqrt {c^2 d^2+e^2}}\right )}{e \sqrt {c^2 d^2+e^2}}+\frac {2 b^2 c \operatorname {PolyLog}\left (2,-\frac {e e^{\text {arcsinh}(c x)}}{c d-\sqrt {c^2 d^2+e^2}}\right )}{e \sqrt {c^2 d^2+e^2}}-\frac {2 b^2 c \operatorname {PolyLog}\left (2,-\frac {e e^{\text {arcsinh}(c x)}}{c d+\sqrt {c^2 d^2+e^2}}\right )}{e \sqrt {c^2 d^2+e^2}} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.14 (sec) , antiderivative size = 191, normalized size of antiderivative = 0.73 \[ \int \frac {(a+b \text {arcsinh}(c x))^2}{(d+e x)^2} \, dx=\frac {-\frac {(a+b \text {arcsinh}(c x))^2}{d+e x}+\frac {2 b c \left ((a+b \text {arcsinh}(c x)) \left (\log \left (1+\frac {e e^{\text {arcsinh}(c x)}}{c d-\sqrt {c^2 d^2+e^2}}\right )-\log \left (1+\frac {e e^{\text {arcsinh}(c x)}}{c d+\sqrt {c^2 d^2+e^2}}\right )\right )+b \operatorname {PolyLog}\left (2,\frac {e e^{\text {arcsinh}(c x)}}{-c d+\sqrt {c^2 d^2+e^2}}\right )-b \operatorname {PolyLog}\left (2,-\frac {e e^{\text {arcsinh}(c x)}}{c d+\sqrt {c^2 d^2+e^2}}\right )\right )}{\sqrt {c^2 d^2+e^2}}}{e} \]

[In]

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

[Out]

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

Maple [A] (verified)

Time = 0.54 (sec) , antiderivative size = 525, normalized size of antiderivative = 2.00

method	result	size
derivativedivides	\(\frac {-\frac {a^{2} c^{2}}{\left (e c x +c d \right ) e}+b^{2} c^{2} \left (-\frac {\operatorname {arcsinh}\left (c x \right )^{2}}{e \left (e c x +c d \right )}+\frac {2 \,\operatorname {arcsinh}\left (c x \right ) \ln \left (\frac {-c d -e \left (c x +\sqrt {c^{2} x^{2}+1}\right )+\sqrt {c^{2} d^{2}+e^{2}}}{-c d +\sqrt {c^{2} d^{2}+e^{2}}}\right )}{e \sqrt {c^{2} d^{2}+e^{2}}}-\frac {2 \,\operatorname {arcsinh}\left (c x \right ) \ln \left (\frac {c d +e \left (c x +\sqrt {c^{2} x^{2}+1}\right )+\sqrt {c^{2} d^{2}+e^{2}}}{c d +\sqrt {c^{2} d^{2}+e^{2}}}\right )}{e \sqrt {c^{2} d^{2}+e^{2}}}+\frac {2 \operatorname {dilog}\left (\frac {-c d -e \left (c x +\sqrt {c^{2} x^{2}+1}\right )+\sqrt {c^{2} d^{2}+e^{2}}}{-c d +\sqrt {c^{2} d^{2}+e^{2}}}\right )}{e \sqrt {c^{2} d^{2}+e^{2}}}-\frac {2 \operatorname {dilog}\left (\frac {c d +e \left (c x +\sqrt {c^{2} x^{2}+1}\right )+\sqrt {c^{2} d^{2}+e^{2}}}{c d +\sqrt {c^{2} d^{2}+e^{2}}}\right )}{e \sqrt {c^{2} d^{2}+e^{2}}}\right )+2 a b \,c^{2} \left (-\frac {\operatorname {arcsinh}\left (c x \right )}{\left (e c x +c d \right ) e}-\frac {\ln \left (\frac {\frac {2 c^{2} d^{2}+2 e^{2}}{e^{2}}-\frac {2 d c \left (c x +\frac {d c}{e}\right )}{e}+2 \sqrt {\frac {c^{2} d^{2}+e^{2}}{e^{2}}}\, \sqrt {\left (c x +\frac {d c}{e}\right )^{2}-\frac {2 d c \left (c x +\frac {d c}{e}\right )}{e}+\frac {c^{2} d^{2}+e^{2}}{e^{2}}}}{c x +\frac {d c}{e}}\right )}{e^{2} \sqrt {\frac {c^{2} d^{2}+e^{2}}{e^{2}}}}\right )}{c}\)	\(525\)
default	\(\frac {-\frac {a^{2} c^{2}}{\left (e c x +c d \right ) e}+b^{2} c^{2} \left (-\frac {\operatorname {arcsinh}\left (c x \right )^{2}}{e \left (e c x +c d \right )}+\frac {2 \,\operatorname {arcsinh}\left (c x \right ) \ln \left (\frac {-c d -e \left (c x +\sqrt {c^{2} x^{2}+1}\right )+\sqrt {c^{2} d^{2}+e^{2}}}{-c d +\sqrt {c^{2} d^{2}+e^{2}}}\right )}{e \sqrt {c^{2} d^{2}+e^{2}}}-\frac {2 \,\operatorname {arcsinh}\left (c x \right ) \ln \left (\frac {c d +e \left (c x +\sqrt {c^{2} x^{2}+1}\right )+\sqrt {c^{2} d^{2}+e^{2}}}{c d +\sqrt {c^{2} d^{2}+e^{2}}}\right )}{e \sqrt {c^{2} d^{2}+e^{2}}}+\frac {2 \operatorname {dilog}\left (\frac {-c d -e \left (c x +\sqrt {c^{2} x^{2}+1}\right )+\sqrt {c^{2} d^{2}+e^{2}}}{-c d +\sqrt {c^{2} d^{2}+e^{2}}}\right )}{e \sqrt {c^{2} d^{2}+e^{2}}}-\frac {2 \operatorname {dilog}\left (\frac {c d +e \left (c x +\sqrt {c^{2} x^{2}+1}\right )+\sqrt {c^{2} d^{2}+e^{2}}}{c d +\sqrt {c^{2} d^{2}+e^{2}}}\right )}{e \sqrt {c^{2} d^{2}+e^{2}}}\right )+2 a b \,c^{2} \left (-\frac {\operatorname {arcsinh}\left (c x \right )}{\left (e c x +c d \right ) e}-\frac {\ln \left (\frac {\frac {2 c^{2} d^{2}+2 e^{2}}{e^{2}}-\frac {2 d c \left (c x +\frac {d c}{e}\right )}{e}+2 \sqrt {\frac {c^{2} d^{2}+e^{2}}{e^{2}}}\, \sqrt {\left (c x +\frac {d c}{e}\right )^{2}-\frac {2 d c \left (c x +\frac {d c}{e}\right )}{e}+\frac {c^{2} d^{2}+e^{2}}{e^{2}}}}{c x +\frac {d c}{e}}\right )}{e^{2} \sqrt {\frac {c^{2} d^{2}+e^{2}}{e^{2}}}}\right )}{c}\)	\(525\)
parts	\(-\frac {a^{2}}{\left (e x +d \right ) e}+\frac {b^{2} \left (-\frac {c^{2} \operatorname {arcsinh}\left (c x \right )^{2}}{e \left (e c x +c d \right )}+\frac {2 c^{2} \operatorname {arcsinh}\left (c x \right ) \ln \left (\frac {-c d -e \left (c x +\sqrt {c^{2} x^{2}+1}\right )+\sqrt {c^{2} d^{2}+e^{2}}}{-c d +\sqrt {c^{2} d^{2}+e^{2}}}\right )}{e \sqrt {c^{2} d^{2}+e^{2}}}-\frac {2 c^{2} \operatorname {arcsinh}\left (c x \right ) \ln \left (\frac {c d +e \left (c x +\sqrt {c^{2} x^{2}+1}\right )+\sqrt {c^{2} d^{2}+e^{2}}}{c d +\sqrt {c^{2} d^{2}+e^{2}}}\right )}{e \sqrt {c^{2} d^{2}+e^{2}}}+\frac {2 c^{2} \operatorname {dilog}\left (\frac {-c d -e \left (c x +\sqrt {c^{2} x^{2}+1}\right )+\sqrt {c^{2} d^{2}+e^{2}}}{-c d +\sqrt {c^{2} d^{2}+e^{2}}}\right )}{e \sqrt {c^{2} d^{2}+e^{2}}}-\frac {2 c^{2} \operatorname {dilog}\left (\frac {c d +e \left (c x +\sqrt {c^{2} x^{2}+1}\right )+\sqrt {c^{2} d^{2}+e^{2}}}{c d +\sqrt {c^{2} d^{2}+e^{2}}}\right )}{e \sqrt {c^{2} d^{2}+e^{2}}}\right )}{c}-\frac {2 a b c \,\operatorname {arcsinh}\left (c x \right )}{\left (e c x +c d \right ) e}-\frac {2 a b c \ln \left (\frac {\frac {2 c^{2} d^{2}+2 e^{2}}{e^{2}}-\frac {2 d c \left (c x +\frac {d c}{e}\right )}{e}+2 \sqrt {\frac {c^{2} d^{2}+e^{2}}{e^{2}}}\, \sqrt {\left (c x +\frac {d c}{e}\right )^{2}-\frac {2 d c \left (c x +\frac {d c}{e}\right )}{e}+\frac {c^{2} d^{2}+e^{2}}{e^{2}}}}{c x +\frac {d c}{e}}\right )}{e^{2} \sqrt {\frac {c^{2} d^{2}+e^{2}}{e^{2}}}}\)	\(528\)

[In]

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

[Out]

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

Fricas [F]

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

[In]

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

[Out]

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

Sympy [F]

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

[In]

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

[Out]

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

Maxima [F]

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

[In]

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

[Out]

-b^2*(log(c*x + sqrt(c^2*x^2 + 1))^2/(e^2*x + d*e) - integrate(2*(c^3*x^2 + sqrt(c^2*x^2 + 1)*c^2*x + c)*log(c
*x + sqrt(c^2*x^2 + 1))/(c^3*e^2*x^4 + c^3*d*e*x^3 + c*e^2*x^2 + c*d*e*x + (c^2*e^2*x^3 + c^2*d*e*x^2 + e^2*x
+ d*e)*sqrt(c^2*x^2 + 1)), x)) - 2*a*b*(arcsinh(c*x)/(e^2*x + d*e) - c*arcsinh(c*d*sqrt(e^4)*x/(e*abs(e^2*x +
d*e)) - sqrt(e^4)/(c*abs(e^2*x + d*e)))/(sqrt(c^2*d^2/e^2 + 1)*e^2)) - a^2/(e^2*x + d*e)

Giac [F]

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

[In]

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

[Out]

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

Mupad [F(-1)]

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

[In]

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

[Out]

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