\(\int \frac {(a+b \text {arcsinh}(c+d x))^3}{(c e+d e x)^3} \, dx\) [144]

Optimal result

Integrand size = 23, antiderivative size = 157 \[ \int \frac {(a+b \text {arcsinh}(c+d x))^3}{(c e+d e x)^3} \, dx=\frac {3 b (a+b \text {arcsinh}(c+d x))^2}{2 d e^3}-\frac {3 b \sqrt {1+(c+d x)^2} (a+b \text {arcsinh}(c+d x))^2}{2 d e^3 (c+d x)}-\frac {(a+b \text {arcsinh}(c+d x))^3}{2 d e^3 (c+d x)^2}+\frac {3 b^2 (a+b \text {arcsinh}(c+d x)) \log \left (1-e^{-2 \text {arcsinh}(c+d x)}\right )}{d e^3}-\frac {3 b^3 \operatorname {PolyLog}\left (2,e^{-2 \text {arcsinh}(c+d x)}\right )}{2 d e^3} \]

[Out]

3/2*b*(a+b*arcsinh(d*x+c))^2/d/e^3-1/2*(a+b*arcsinh(d*x+c))^3/d/e^3/(d*x+c)^2+3*b^2*(a+b*arcsinh(d*x+c))*ln(1-
1/(d*x+c+(1+(d*x+c)^2)^(1/2))^2)/d/e^3-3/2*b^3*polylog(2,1/(d*x+c+(1+(d*x+c)^2)^(1/2))^2)/d/e^3-3/2*b*(a+b*arc
sinh(d*x+c))^2*(1+(d*x+c)^2)^(1/2)/d/e^3/(d*x+c)

Rubi [A] (verified)

Time = 0.22 (sec) , antiderivative size = 157, normalized size of antiderivative = 1.00, number of steps used = 9, number of rules used = 9, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.391, Rules used = {5859, 12, 5776, 5800, 5775, 3797, 2221, 2317, 2438} \[ \int \frac {(a+b \text {arcsinh}(c+d x))^3}{(c e+d e x)^3} \, dx=\frac {3 b^2 \log \left (1-e^{-2 \text {arcsinh}(c+d x)}\right ) (a+b \text {arcsinh}(c+d x))}{d e^3}-\frac {3 b \sqrt {(c+d x)^2+1} (a+b \text {arcsinh}(c+d x))^2}{2 d e^3 (c+d x)}+\frac {3 b (a+b \text {arcsinh}(c+d x))^2}{2 d e^3}-\frac {(a+b \text {arcsinh}(c+d x))^3}{2 d e^3 (c+d x)^2}-\frac {3 b^3 \operatorname {PolyLog}\left (2,e^{-2 \text {arcsinh}(c+d x)}\right )}{2 d e^3} \]

[In]

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

[Out]

(3*b*(a + b*ArcSinh[c + d*x])^2)/(2*d*e^3) - (3*b*Sqrt[1 + (c + d*x)^2]*(a + b*ArcSinh[c + d*x])^2)/(2*d*e^3*(
c + d*x)) - (a + b*ArcSinh[c + d*x])^3/(2*d*e^3*(c + d*x)^2) + (3*b^2*(a + b*ArcSinh[c + d*x])*Log[1 - E^(-2*A
rcSinh[c + d*x])])/(d*e^3) - (3*b^3*PolyLog[2, E^(-2*ArcSinh[c + d*x])])/(2*d*e^3)

Rule 12

Int[(a_)*(u_), x_Symbol] :> Dist[a, Int[u, x], x] /; FreeQ[a, x] &&  !MatchQ[u, (b_)*(v_) /; FreeQ[b, x]]

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 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 3797

Int[((c_.) + (d_.)*(x_))^(m_.)*tan[(e_.) + Pi*(k_.) + (Complex[0, fz_])*(f_.)*(x_)], x_Symbol] :> Simp[(-I)*((
c + d*x)^(m + 1)/(d*(m + 1))), x] + Dist[2*I, Int[((c + d*x)^m*(E^(2*((-I)*e + f*fz*x))/(1 + E^(2*((-I)*e + f*
fz*x))/E^(2*I*k*Pi))))/E^(2*I*k*Pi), x], x] /; FreeQ[{c, d, e, f, fz}, x] && IntegerQ[4*k] && IGtQ[m, 0]

Rule 5775

Int[((a_.) + ArcSinh[(c_.)*(x_)]*(b_.))^(n_.)/(x_), x_Symbol] :> Dist[1/b, Subst[Int[x^n*Coth[-a/b + x/b], x],
x, a + b*ArcSinh[c*x]], x] /; FreeQ[{a, b, c}, x] && IGtQ[n, 0]

Rule 5776

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

Rule 5800

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 + 1)*((a + b*ArcSinh[c*x])^n/(d*f*(m + 1))), x] - Dist[b*c*(n/(f*(m + 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, p}, x] && EqQ[e, c^2*d] && GtQ[n, 0] && EqQ[m + 2*p + 3, 0] && NeQ[m, -1]

Rule 5859

Int[((a_.) + ArcSinh[(c_) + (d_.)*(x_)]*(b_.))^(n_.)*((e_.) + (f_.)*(x_))^(m_.), x_Symbol] :> Dist[1/d, Subst[
Int[((d*e - c*f)/d + f*(x/d))^m*(a + b*ArcSinh[x])^n, x], x, c + d*x], x] /; FreeQ[{a, b, c, d, e, f, m, n}, x
]

Rubi steps \begin{align*} \text {integral}& = \frac {\text {Subst}\left (\int \frac {(a+b \text {arcsinh}(x))^3}{e^3 x^3} \, dx,x,c+d x\right )}{d} \\ & = \frac {\text {Subst}\left (\int \frac {(a+b \text {arcsinh}(x))^3}{x^3} \, dx,x,c+d x\right )}{d e^3} \\ & = -\frac {(a+b \text {arcsinh}(c+d x))^3}{2 d e^3 (c+d x)^2}+\frac {(3 b) \text {Subst}\left (\int \frac {(a+b \text {arcsinh}(x))^2}{x^2 \sqrt {1+x^2}} \, dx,x,c+d x\right )}{2 d e^3} \\ & = -\frac {3 b \sqrt {1+(c+d x)^2} (a+b \text {arcsinh}(c+d x))^2}{2 d e^3 (c+d x)}-\frac {(a+b \text {arcsinh}(c+d x))^3}{2 d e^3 (c+d x)^2}+\frac {\left (3 b^2\right ) \text {Subst}\left (\int \frac {a+b \text {arcsinh}(x)}{x} \, dx,x,c+d x\right )}{d e^3} \\ & = -\frac {3 b \sqrt {1+(c+d x)^2} (a+b \text {arcsinh}(c+d x))^2}{2 d e^3 (c+d x)}-\frac {(a+b \text {arcsinh}(c+d x))^3}{2 d e^3 (c+d x)^2}-\frac {(3 b) \text {Subst}\left (\int x \coth \left (\frac {a}{b}-\frac {x}{b}\right ) \, dx,x,a+b \text {arcsinh}(c+d x)\right )}{d e^3} \\ & = \frac {3 b (a+b \text {arcsinh}(c+d x))^2}{2 d e^3}-\frac {3 b \sqrt {1+(c+d x)^2} (a+b \text {arcsinh}(c+d x))^2}{2 d e^3 (c+d x)}-\frac {(a+b \text {arcsinh}(c+d x))^3}{2 d e^3 (c+d x)^2}+\frac {(6 b) \text {Subst}\left (\int \frac {e^{2 \left (\frac {a}{b}-\frac {x}{b}\right )} x}{1-e^{2 \left (\frac {a}{b}-\frac {x}{b}\right )}} \, dx,x,a+b \text {arcsinh}(c+d x)\right )}{d e^3} \\ & = \frac {3 b (a+b \text {arcsinh}(c+d x))^2}{2 d e^3}-\frac {3 b \sqrt {1+(c+d x)^2} (a+b \text {arcsinh}(c+d x))^2}{2 d e^3 (c+d x)}-\frac {(a+b \text {arcsinh}(c+d x))^3}{2 d e^3 (c+d x)^2}+\frac {3 b^2 (a+b \text {arcsinh}(c+d x)) \log \left (1-e^{-2 \text {arcsinh}(c+d x)}\right )}{d e^3}-\frac {\left (3 b^2\right ) \text {Subst}\left (\int \log \left (1-e^{2 \left (\frac {a}{b}-\frac {x}{b}\right )}\right ) \, dx,x,a+b \text {arcsinh}(c+d x)\right )}{d e^3} \\ & = \frac {3 b (a+b \text {arcsinh}(c+d x))^2}{2 d e^3}-\frac {3 b \sqrt {1+(c+d x)^2} (a+b \text {arcsinh}(c+d x))^2}{2 d e^3 (c+d x)}-\frac {(a+b \text {arcsinh}(c+d x))^3}{2 d e^3 (c+d x)^2}+\frac {3 b^2 (a+b \text {arcsinh}(c+d x)) \log \left (1-e^{-2 \text {arcsinh}(c+d x)}\right )}{d e^3}+\frac {\left (3 b^3\right ) \text {Subst}\left (\int \frac {\log (1-x)}{x} \, dx,x,e^{2 \left (\frac {a}{b}-\frac {a+b \text {arcsinh}(c+d x)}{b}\right )}\right )}{2 d e^3} \\ & = \frac {3 b (a+b \text {arcsinh}(c+d x))^2}{2 d e^3}-\frac {3 b \sqrt {1+(c+d x)^2} (a+b \text {arcsinh}(c+d x))^2}{2 d e^3 (c+d x)}-\frac {(a+b \text {arcsinh}(c+d x))^3}{2 d e^3 (c+d x)^2}+\frac {3 b^2 (a+b \text {arcsinh}(c+d x)) \log \left (1-e^{-2 \text {arcsinh}(c+d x)}\right )}{d e^3}-\frac {3 b^3 \operatorname {PolyLog}\left (2,e^{2 \left (\frac {a}{b}-\frac {a+b \text {arcsinh}(c+d x)}{b}\right )}\right )}{2 d e^3} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.72 (sec) , antiderivative size = 229, normalized size of antiderivative = 1.46 \[ \int \frac {(a+b \text {arcsinh}(c+d x))^3}{(c e+d e x)^3} \, dx=-\frac {3 b^2 \left (a+b (c+d x) \left (-c-d x+\sqrt {1+c^2+2 c d x+d^2 x^2}\right )\right ) \text {arcsinh}(c+d x)^2+b^3 \text {arcsinh}(c+d x)^3+3 b \text {arcsinh}(c+d x) \left (a \left (a+2 b (c+d x) \sqrt {1+c^2+2 c d x+d^2 x^2}\right )-2 b^2 (c+d x)^2 \log \left (1-e^{-2 \text {arcsinh}(c+d x)}\right )\right )+a \left (a \left (a+3 b (c+d x) \sqrt {1+c^2+2 c d x+d^2 x^2}\right )-6 b^2 (c+d x)^2 \log (c+d x)\right )+3 b^3 (c+d x)^2 \operatorname {PolyLog}\left (2,e^{-2 \text {arcsinh}(c+d x)}\right )}{2 d e^3 (c+d x)^2} \]

[In]

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

[Out]

-1/2*(3*b^2*(a + b*(c + d*x)*(-c - d*x + Sqrt[1 + c^2 + 2*c*d*x + d^2*x^2]))*ArcSinh[c + d*x]^2 + b^3*ArcSinh[
c + d*x]^3 + 3*b*ArcSinh[c + d*x]*(a*(a + 2*b*(c + d*x)*Sqrt[1 + c^2 + 2*c*d*x + d^2*x^2]) - 2*b^2*(c + d*x)^2
*Log[1 - E^(-2*ArcSinh[c + d*x])]) + a*(a*(a + 3*b*(c + d*x)*Sqrt[1 + c^2 + 2*c*d*x + d^2*x^2]) - 6*b^2*(c + d
*x)^2*Log[c + d*x]) + 3*b^3*(c + d*x)^2*PolyLog[2, E^(-2*ArcSinh[c + d*x])])/(d*e^3*(c + d*x)^2)

Maple [A] (verified)

Time = 0.47 (sec) , antiderivative size = 325, normalized size of antiderivative = 2.07

method	result	size
derivativedivides	\(\frac {-\frac {a^{3}}{2 e^{3} \left (d x +c \right )^{2}}+\frac {b^{3} \left (-\frac {\operatorname {arcsinh}\left (d x +c \right )^{2} \left (3 \left (d x +c \right ) \sqrt {1+\left (d x +c \right )^{2}}-3 \left (d x +c \right )^{2}+\operatorname {arcsinh}\left (d x +c \right )\right )}{2 \left (d x +c \right )^{2}}-3 \operatorname {arcsinh}\left (d x +c \right )^{2}+3 \,\operatorname {arcsinh}\left (d x +c \right ) \ln \left (1+d x +c +\sqrt {1+\left (d x +c \right )^{2}}\right )+3 \operatorname {polylog}\left (2, -d x -c -\sqrt {1+\left (d x +c \right )^{2}}\right )+3 \,\operatorname {arcsinh}\left (d x +c \right ) \ln \left (1-d x -c -\sqrt {1+\left (d x +c \right )^{2}}\right )+3 \operatorname {polylog}\left (2, d x +c +\sqrt {1+\left (d x +c \right )^{2}}\right )\right )}{e^{3}}+\frac {3 a \,b^{2} \left (-2 \,\operatorname {arcsinh}\left (d x +c \right )-\frac {\operatorname {arcsinh}\left (d x +c \right ) \left (-2 \left (d x +c \right )^{2}+2 \left (d x +c \right ) \sqrt {1+\left (d x +c \right )^{2}}+\operatorname {arcsinh}\left (d x +c \right )\right )}{2 \left (d x +c \right )^{2}}+\ln \left (\left (d x +c +\sqrt {1+\left (d x +c \right )^{2}}\right )^{2}-1\right )\right )}{e^{3}}+\frac {3 a^{2} b \left (-\frac {\operatorname {arcsinh}\left (d x +c \right )}{2 \left (d x +c \right )^{2}}-\frac {\sqrt {1+\left (d x +c \right )^{2}}}{2 \left (d x +c \right )}\right )}{e^{3}}}{d}\)	\(325\)
default	\(\frac {-\frac {a^{3}}{2 e^{3} \left (d x +c \right )^{2}}+\frac {b^{3} \left (-\frac {\operatorname {arcsinh}\left (d x +c \right )^{2} \left (3 \left (d x +c \right ) \sqrt {1+\left (d x +c \right )^{2}}-3 \left (d x +c \right )^{2}+\operatorname {arcsinh}\left (d x +c \right )\right )}{2 \left (d x +c \right )^{2}}-3 \operatorname {arcsinh}\left (d x +c \right )^{2}+3 \,\operatorname {arcsinh}\left (d x +c \right ) \ln \left (1+d x +c +\sqrt {1+\left (d x +c \right )^{2}}\right )+3 \operatorname {polylog}\left (2, -d x -c -\sqrt {1+\left (d x +c \right )^{2}}\right )+3 \,\operatorname {arcsinh}\left (d x +c \right ) \ln \left (1-d x -c -\sqrt {1+\left (d x +c \right )^{2}}\right )+3 \operatorname {polylog}\left (2, d x +c +\sqrt {1+\left (d x +c \right )^{2}}\right )\right )}{e^{3}}+\frac {3 a \,b^{2} \left (-2 \,\operatorname {arcsinh}\left (d x +c \right )-\frac {\operatorname {arcsinh}\left (d x +c \right ) \left (-2 \left (d x +c \right )^{2}+2 \left (d x +c \right ) \sqrt {1+\left (d x +c \right )^{2}}+\operatorname {arcsinh}\left (d x +c \right )\right )}{2 \left (d x +c \right )^{2}}+\ln \left (\left (d x +c +\sqrt {1+\left (d x +c \right )^{2}}\right )^{2}-1\right )\right )}{e^{3}}+\frac {3 a^{2} b \left (-\frac {\operatorname {arcsinh}\left (d x +c \right )}{2 \left (d x +c \right )^{2}}-\frac {\sqrt {1+\left (d x +c \right )^{2}}}{2 \left (d x +c \right )}\right )}{e^{3}}}{d}\)	\(325\)
parts	\(-\frac {a^{3}}{2 e^{3} \left (d x +c \right )^{2} d}+\frac {b^{3} \left (-\frac {\operatorname {arcsinh}\left (d x +c \right )^{2} \left (3 \left (d x +c \right ) \sqrt {1+\left (d x +c \right )^{2}}-3 \left (d x +c \right )^{2}+\operatorname {arcsinh}\left (d x +c \right )\right )}{2 \left (d x +c \right )^{2}}-3 \operatorname {arcsinh}\left (d x +c \right )^{2}+3 \,\operatorname {arcsinh}\left (d x +c \right ) \ln \left (1+d x +c +\sqrt {1+\left (d x +c \right )^{2}}\right )+3 \operatorname {polylog}\left (2, -d x -c -\sqrt {1+\left (d x +c \right )^{2}}\right )+3 \,\operatorname {arcsinh}\left (d x +c \right ) \ln \left (1-d x -c -\sqrt {1+\left (d x +c \right )^{2}}\right )+3 \operatorname {polylog}\left (2, d x +c +\sqrt {1+\left (d x +c \right )^{2}}\right )\right )}{e^{3} d}+\frac {3 a \,b^{2} \left (-2 \,\operatorname {arcsinh}\left (d x +c \right )-\frac {\operatorname {arcsinh}\left (d x +c \right ) \left (-2 \left (d x +c \right )^{2}+2 \left (d x +c \right ) \sqrt {1+\left (d x +c \right )^{2}}+\operatorname {arcsinh}\left (d x +c \right )\right )}{2 \left (d x +c \right )^{2}}+\ln \left (\left (d x +c +\sqrt {1+\left (d x +c \right )^{2}}\right )^{2}-1\right )\right )}{e^{3} d}+\frac {3 a^{2} b \left (-\frac {\operatorname {arcsinh}\left (d x +c \right )}{2 \left (d x +c \right )^{2}}-\frac {\sqrt {1+\left (d x +c \right )^{2}}}{2 \left (d x +c \right )}\right )}{e^{3} d}\)	\(333\)

[In]

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

[Out]

1/d*(-1/2*a^3/e^3/(d*x+c)^2+b^3/e^3*(-1/2*arcsinh(d*x+c)^2*(3*(d*x+c)*(1+(d*x+c)^2)^(1/2)-3*(d*x+c)^2+arcsinh(
d*x+c))/(d*x+c)^2-3*arcsinh(d*x+c)^2+3*arcsinh(d*x+c)*ln(1+d*x+c+(1+(d*x+c)^2)^(1/2))+3*polylog(2,-d*x-c-(1+(d
*x+c)^2)^(1/2))+3*arcsinh(d*x+c)*ln(1-d*x-c-(1+(d*x+c)^2)^(1/2))+3*polylog(2,d*x+c+(1+(d*x+c)^2)^(1/2)))+3*a*b
^2/e^3*(-2*arcsinh(d*x+c)-1/2*arcsinh(d*x+c)*(-2*(d*x+c)^2+2*(d*x+c)*(1+(d*x+c)^2)^(1/2)+arcsinh(d*x+c))/(d*x+
c)^2+ln((d*x+c+(1+(d*x+c)^2)^(1/2))^2-1))+3*a^2*b/e^3*(-1/2/(d*x+c)^2*arcsinh(d*x+c)-1/2/(d*x+c)*(1+(d*x+c)^2)
^(1/2)))

Fricas [F]

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

[In]

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

[Out]

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

Sympy [F]

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

[In]

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

[Out]

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

Maxima [F]

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

[In]

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

[Out]

-3*(sqrt(d^2*x^2 + 2*c*d*x + c^2 + 1)*d*arcsinh(d*x + c)/(d^3*e^3*x + c*d^2*e^3) - log(d*x + c)/(d*e^3))*a*b^2
- 1/2*(log(d*x + c + sqrt(d^2*x^2 + 2*c*d*x + c^2 + 1))^3/(d^3*e^3*x^2 + 2*c*d^2*e^3*x + c^2*d*e^3) - 2*integ
rate(3/2*(d^2*x^2 + 2*c*d*x + c^2 + sqrt(d^2*x^2 + 2*c*d*x + c^2 + 1)*(d*x + c) + 1)*log(d*x + c + sqrt(d^2*x^
2 + 2*c*d*x + c^2 + 1))^2/(d^5*e^3*x^5 + 5*c*d^4*e^3*x^4 + c^5*e^3 + c^3*e^3 + (10*c^2*d^3*e^3 + d^3*e^3)*x^3
+ (10*c^3*d^2*e^3 + 3*c*d^2*e^3)*x^2 + (5*c^4*d*e^3 + 3*c^2*d*e^3)*x + (d^4*e^3*x^4 + 4*c*d^3*e^3*x^3 + c^4*e^
3 + c^2*e^3 + (6*c^2*d^2*e^3 + d^2*e^3)*x^2 + 2*(2*c^3*d*e^3 + c*d*e^3)*x)*sqrt(d^2*x^2 + 2*c*d*x + c^2 + 1)),
x))*b^3 - 3/2*a^2*b*(sqrt(d^2*x^2 + 2*c*d*x + c^2 + 1)*d/(d^3*e^3*x + c*d^2*e^3) + arcsinh(d*x + c)/(d^3*e^3*
x^2 + 2*c*d^2*e^3*x + c^2*d*e^3)) - 3/2*a*b^2*arcsinh(d*x + c)^2/(d^3*e^3*x^2 + 2*c*d^2*e^3*x + c^2*d*e^3) - 1
/2*a^3/(d^3*e^3*x^2 + 2*c*d^2*e^3*x + c^2*d*e^3)

Giac [F]

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

[In]

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

[Out]

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

Mupad [F(-1)]

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

[In]

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

[Out]

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