\(\int \frac {(a+b \text {arcsinh}(c+d x))^4}{(c e+d e x)^3} \, dx\) [153]

Optimal result

Integrand size = 23, antiderivative size = 186 \[ \int \frac {(a+b \text {arcsinh}(c+d x))^4}{(c e+d e x)^3} \, dx=\frac {2 b (a+b \text {arcsinh}(c+d x))^3}{d e^3}-\frac {2 b \sqrt {1+(c+d x)^2} (a+b \text {arcsinh}(c+d x))^3}{d e^3 (c+d x)}-\frac {(a+b \text {arcsinh}(c+d x))^4}{2 d e^3 (c+d x)^2}+\frac {6 b^2 (a+b \text {arcsinh}(c+d x))^2 \log \left (1-e^{-2 \text {arcsinh}(c+d x)}\right )}{d e^3}-\frac {6 b^3 (a+b \text {arcsinh}(c+d x)) \operatorname {PolyLog}\left (2,e^{-2 \text {arcsinh}(c+d x)}\right )}{d e^3}-\frac {3 b^4 \operatorname {PolyLog}\left (3,e^{-2 \text {arcsinh}(c+d x)}\right )}{d e^3} \]

[Out]

2*b*(a+b*arcsinh(d*x+c))^3/d/e^3-1/2*(a+b*arcsinh(d*x+c))^4/d/e^3/(d*x+c)^2+6*b^2*(a+b*arcsinh(d*x+c))^2*ln(1-
1/(d*x+c+(1+(d*x+c)^2)^(1/2))^2)/d/e^3-6*b^3*(a+b*arcsinh(d*x+c))*polylog(2,1/(d*x+c+(1+(d*x+c)^2)^(1/2))^2)/d
/e^3-3*b^4*polylog(3,1/(d*x+c+(1+(d*x+c)^2)^(1/2))^2)/d/e^3-2*b*(a+b*arcsinh(d*x+c))^3*(1+(d*x+c)^2)^(1/2)/d/e
^3/(d*x+c)

Rubi [A] (verified)

Time = 0.29 (sec) , antiderivative size = 186, normalized size of antiderivative = 1.00, number of steps used = 10, number of rules used = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.435, Rules used = {5859, 12, 5776, 5800, 5775, 3797, 2221, 2611, 2320, 6724} \[ \int \frac {(a+b \text {arcsinh}(c+d x))^4}{(c e+d e x)^3} \, dx=-\frac {6 b^3 \operatorname {PolyLog}\left (2,e^{-2 \text {arcsinh}(c+d x)}\right ) (a+b \text {arcsinh}(c+d x))}{d e^3}+\frac {6 b^2 \log \left (1-e^{-2 \text {arcsinh}(c+d x)}\right ) (a+b \text {arcsinh}(c+d x))^2}{d e^3}-\frac {2 b \sqrt {(c+d x)^2+1} (a+b \text {arcsinh}(c+d x))^3}{d e^3 (c+d x)}+\frac {2 b (a+b \text {arcsinh}(c+d x))^3}{d e^3}-\frac {(a+b \text {arcsinh}(c+d x))^4}{2 d e^3 (c+d x)^2}-\frac {3 b^4 \operatorname {PolyLog}\left (3,e^{-2 \text {arcsinh}(c+d x)}\right )}{d e^3} \]

[In]

Int[(a + b*ArcSinh[c + d*x])^4/(c*e + d*e*x)^3,x]

[Out]

(2*b*(a + b*ArcSinh[c + d*x])^3)/(d*e^3) - (2*b*Sqrt[1 + (c + d*x)^2]*(a + b*ArcSinh[c + d*x])^3)/(d*e^3*(c +
d*x)) - (a + b*ArcSinh[c + d*x])^4/(2*d*e^3*(c + d*x)^2) + (6*b^2*(a + b*ArcSinh[c + d*x])^2*Log[1 - E^(-2*Arc
Sinh[c + d*x])])/(d*e^3) - (6*b^3*(a + b*ArcSinh[c + d*x])*PolyLog[2, E^(-2*ArcSinh[c + d*x])])/(d*e^3) - (3*b
^4*PolyLog[3, E^(-2*ArcSinh[c + d*x])])/(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 2320

Int[u_, x_Symbol] :> With[{v = FunctionOfExponential[u, x]}, Dist[v/D[v, x], Subst[Int[FunctionOfExponentialFu
nction[u, x]/x, x], x, v], x]] /; FunctionOfExponentialQ[u, x] &&  !MatchQ[u, (w_)*((a_.)*(v_)^(n_))^(m_) /; F
reeQ[{a, m, n}, x] && IntegerQ[m*n]] &&  !MatchQ[u, E^((c_.)*((a_.) + (b_.)*x))*(F_)[v_] /; FreeQ[{a, b, c}, x
] && InverseFunctionQ[F[x]]]

Rule 2611

Int[Log[1 + (e_.)*((F_)^((c_.)*((a_.) + (b_.)*(x_))))^(n_.)]*((f_.) + (g_.)*(x_))^(m_.), x_Symbol] :> Simp[(-(
f + g*x)^m)*(PolyLog[2, (-e)*(F^(c*(a + b*x)))^n]/(b*c*n*Log[F])), x] + Dist[g*(m/(b*c*n*Log[F])), Int[(f + g*
x)^(m - 1)*PolyLog[2, (-e)*(F^(c*(a + b*x)))^n], x], x] /; FreeQ[{F, a, b, c, e, f, g, n}, x] && GtQ[m, 0]

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
]

Rule 6724

Int[PolyLog[n_, (c_.)*((a_.) + (b_.)*(x_))^(p_.)]/((d_.) + (e_.)*(x_)), x_Symbol] :> Simp[PolyLog[n + 1, c*(a
+ b*x)^p]/(e*p), x] /; FreeQ[{a, b, c, d, e, n, p}, x] && EqQ[b*d, a*e]

Rubi steps \begin{align*} \text {integral}& = \frac {\text {Subst}\left (\int \frac {(a+b \text {arcsinh}(x))^4}{e^3 x^3} \, dx,x,c+d x\right )}{d} \\ & = \frac {\text {Subst}\left (\int \frac {(a+b \text {arcsinh}(x))^4}{x^3} \, dx,x,c+d x\right )}{d e^3} \\ & = -\frac {(a+b \text {arcsinh}(c+d x))^4}{2 d e^3 (c+d x)^2}+\frac {(2 b) \text {Subst}\left (\int \frac {(a+b \text {arcsinh}(x))^3}{x^2 \sqrt {1+x^2}} \, dx,x,c+d x\right )}{d e^3} \\ & = -\frac {2 b \sqrt {1+(c+d x)^2} (a+b \text {arcsinh}(c+d x))^3}{d e^3 (c+d x)}-\frac {(a+b \text {arcsinh}(c+d x))^4}{2 d e^3 (c+d x)^2}+\frac {\left (6 b^2\right ) \text {Subst}\left (\int \frac {(a+b \text {arcsinh}(x))^2}{x} \, dx,x,c+d x\right )}{d e^3} \\ & = -\frac {2 b \sqrt {1+(c+d x)^2} (a+b \text {arcsinh}(c+d x))^3}{d e^3 (c+d x)}-\frac {(a+b \text {arcsinh}(c+d x))^4}{2 d e^3 (c+d x)^2}-\frac {(6 b) \text {Subst}\left (\int x^2 \coth \left (\frac {a}{b}-\frac {x}{b}\right ) \, dx,x,a+b \text {arcsinh}(c+d x)\right )}{d e^3} \\ & = \frac {2 b (a+b \text {arcsinh}(c+d x))^3}{d e^3}-\frac {2 b \sqrt {1+(c+d x)^2} (a+b \text {arcsinh}(c+d x))^3}{d e^3 (c+d x)}-\frac {(a+b \text {arcsinh}(c+d x))^4}{2 d e^3 (c+d x)^2}+\frac {(12 b) \text {Subst}\left (\int \frac {e^{2 \left (\frac {a}{b}-\frac {x}{b}\right )} x^2}{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 {2 b (a+b \text {arcsinh}(c+d x))^3}{d e^3}-\frac {2 b \sqrt {1+(c+d x)^2} (a+b \text {arcsinh}(c+d x))^3}{d e^3 (c+d x)}-\frac {(a+b \text {arcsinh}(c+d x))^4}{2 d e^3 (c+d x)^2}+\frac {6 b^2 (a+b \text {arcsinh}(c+d x))^2 \log \left (1-e^{-2 \text {arcsinh}(c+d x)}\right )}{d e^3}-\frac {\left (12 b^2\right ) \text {Subst}\left (\int x \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 {2 b (a+b \text {arcsinh}(c+d x))^3}{d e^3}-\frac {2 b \sqrt {1+(c+d x)^2} (a+b \text {arcsinh}(c+d x))^3}{d e^3 (c+d x)}-\frac {(a+b \text {arcsinh}(c+d x))^4}{2 d e^3 (c+d x)^2}+\frac {6 b^2 (a+b \text {arcsinh}(c+d x))^2 \log \left (1-e^{-2 \text {arcsinh}(c+d x)}\right )}{d e^3}-\frac {6 b^3 (a+b \text {arcsinh}(c+d x)) \operatorname {PolyLog}\left (2,e^{-2 \text {arcsinh}(c+d x)}\right )}{d e^3}+\frac {\left (6 b^3\right ) \text {Subst}\left (\int \operatorname {PolyLog}\left (2,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 {2 b (a+b \text {arcsinh}(c+d x))^3}{d e^3}-\frac {2 b \sqrt {1+(c+d x)^2} (a+b \text {arcsinh}(c+d x))^3}{d e^3 (c+d x)}-\frac {(a+b \text {arcsinh}(c+d x))^4}{2 d e^3 (c+d x)^2}+\frac {6 b^2 (a+b \text {arcsinh}(c+d x))^2 \log \left (1-e^{-2 \text {arcsinh}(c+d x)}\right )}{d e^3}-\frac {6 b^3 (a+b \text {arcsinh}(c+d x)) \operatorname {PolyLog}\left (2,e^{-2 \text {arcsinh}(c+d x)}\right )}{d e^3}-\frac {\left (3 b^4\right ) \text {Subst}\left (\int \frac {\operatorname {PolyLog}(2,x)}{x} \, dx,x,e^{2 \left (\frac {a}{b}-\frac {a+b \text {arcsinh}(c+d x)}{b}\right )}\right )}{d e^3} \\ & = \frac {2 b (a+b \text {arcsinh}(c+d x))^3}{d e^3}-\frac {2 b \sqrt {1+(c+d x)^2} (a+b \text {arcsinh}(c+d x))^3}{d e^3 (c+d x)}-\frac {(a+b \text {arcsinh}(c+d x))^4}{2 d e^3 (c+d x)^2}+\frac {6 b^2 (a+b \text {arcsinh}(c+d x))^2 \log \left (1-e^{-2 \text {arcsinh}(c+d x)}\right )}{d e^3}-\frac {6 b^3 (a+b \text {arcsinh}(c+d x)) \operatorname {PolyLog}\left (2,e^{-2 \text {arcsinh}(c+d x)}\right )}{d e^3}-\frac {3 b^4 \operatorname {PolyLog}\left (3,e^{2 \left (\frac {a}{b}-\frac {a+b \text {arcsinh}(c+d x)}{b}\right )}\right )}{d e^3} \\ \end{align*}

Mathematica [A] (verified)

Time = 1.35 (sec) , antiderivative size = 349, normalized size of antiderivative = 1.88 \[ \int \frac {(a+b \text {arcsinh}(c+d x))^4}{(c e+d e x)^3} \, dx=-\frac {\frac {a^4}{(c+d x)^2}+\frac {4 a^3 b \sqrt {1+(c+d x)^2}}{c+d x}+\frac {4 a^3 b \text {arcsinh}(c+d x)}{(c+d x)^2}+\frac {b^4 \text {arcsinh}(c+d x)^4}{(c+d x)^2}-12 a^2 b^2 \left (-\frac {\sqrt {1+(c+d x)^2} \text {arcsinh}(c+d x)}{c+d x}-\frac {\text {arcsinh}(c+d x)^2}{2 (c+d x)^2}+\log (c+d x)\right )-4 a b^3 \left (\text {arcsinh}(c+d x) \left (3 \text {arcsinh}(c+d x)-\frac {3 \sqrt {1+(c+d x)^2} \text {arcsinh}(c+d x)}{c+d x}-\frac {\text {arcsinh}(c+d x)^2}{(c+d x)^2}+6 \log \left (1-e^{-2 \text {arcsinh}(c+d x)}\right )\right )-3 \operatorname {PolyLog}\left (2,e^{-2 \text {arcsinh}(c+d x)}\right )\right )-2 b^4 \left (2 \text {arcsinh}(c+d x)^2 \left (\text {arcsinh}(c+d x)-\frac {\sqrt {1+(c+d x)^2} \text {arcsinh}(c+d x)}{c+d x}+3 \log \left (1-e^{-2 \text {arcsinh}(c+d x)}\right )\right )-6 \text {arcsinh}(c+d x) \operatorname {PolyLog}\left (2,e^{-2 \text {arcsinh}(c+d x)}\right )-3 \operatorname {PolyLog}\left (3,e^{-2 \text {arcsinh}(c+d x)}\right )\right )}{2 d e^3} \]

[In]

Integrate[(a + b*ArcSinh[c + d*x])^4/(c*e + d*e*x)^3,x]

[Out]

-1/2*(a^4/(c + d*x)^2 + (4*a^3*b*Sqrt[1 + (c + d*x)^2])/(c + d*x) + (4*a^3*b*ArcSinh[c + d*x])/(c + d*x)^2 + (
b^4*ArcSinh[c + d*x]^4)/(c + d*x)^2 - 12*a^2*b^2*(-((Sqrt[1 + (c + d*x)^2]*ArcSinh[c + d*x])/(c + d*x)) - ArcS
inh[c + d*x]^2/(2*(c + d*x)^2) + Log[c + d*x]) - 4*a*b^3*(ArcSinh[c + d*x]*(3*ArcSinh[c + d*x] - (3*Sqrt[1 + (
c + d*x)^2]*ArcSinh[c + d*x])/(c + d*x) - ArcSinh[c + d*x]^2/(c + d*x)^2 + 6*Log[1 - E^(-2*ArcSinh[c + d*x])])
- 3*PolyLog[2, E^(-2*ArcSinh[c + d*x])]) - 2*b^4*(2*ArcSinh[c + d*x]^2*(ArcSinh[c + d*x] - (Sqrt[1 + (c + d*x
)^2]*ArcSinh[c + d*x])/(c + d*x) + 3*Log[1 - E^(-2*ArcSinh[c + d*x])]) - 6*ArcSinh[c + d*x]*PolyLog[2, E^(-2*A
rcSinh[c + d*x])] - 3*PolyLog[3, E^(-2*ArcSinh[c + d*x])]))/(d*e^3)

Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(560\) vs. \(2(206)=412\).

Time = 0.52 (sec) , antiderivative size = 561, normalized size of antiderivative = 3.02

method	result	size
derivativedivides	\(\frac {-\frac {a^{4}}{2 e^{3} \left (d x +c \right )^{2}}+\frac {b^{4} \left (-\frac {\operatorname {arcsinh}\left (d x +c \right )^{3} \left (-4 \left (d x +c \right )^{2}+4 \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}}-4 \operatorname {arcsinh}\left (d x +c \right )^{3}+6 \operatorname {arcsinh}\left (d x +c \right )^{2} \ln \left (1+d x +c +\sqrt {1+\left (d x +c \right )^{2}}\right )+12 \,\operatorname {arcsinh}\left (d x +c \right ) \operatorname {polylog}\left (2, -d x -c -\sqrt {1+\left (d x +c \right )^{2}}\right )-12 \operatorname {polylog}\left (3, -d x -c -\sqrt {1+\left (d x +c \right )^{2}}\right )+6 \operatorname {arcsinh}\left (d x +c \right )^{2} \ln \left (1-d x -c -\sqrt {1+\left (d x +c \right )^{2}}\right )+12 \,\operatorname {arcsinh}\left (d x +c \right ) \operatorname {polylog}\left (2, d x +c +\sqrt {1+\left (d x +c \right )^{2}}\right )-12 \operatorname {polylog}\left (3, d x +c +\sqrt {1+\left (d x +c \right )^{2}}\right )\right )}{e^{3}}+\frac {4 a \,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 {6 a^{2} 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 {4 b \,a^{3} \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}\)	\(561\)
default	\(\frac {-\frac {a^{4}}{2 e^{3} \left (d x +c \right )^{2}}+\frac {b^{4} \left (-\frac {\operatorname {arcsinh}\left (d x +c \right )^{3} \left (-4 \left (d x +c \right )^{2}+4 \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}}-4 \operatorname {arcsinh}\left (d x +c \right )^{3}+6 \operatorname {arcsinh}\left (d x +c \right )^{2} \ln \left (1+d x +c +\sqrt {1+\left (d x +c \right )^{2}}\right )+12 \,\operatorname {arcsinh}\left (d x +c \right ) \operatorname {polylog}\left (2, -d x -c -\sqrt {1+\left (d x +c \right )^{2}}\right )-12 \operatorname {polylog}\left (3, -d x -c -\sqrt {1+\left (d x +c \right )^{2}}\right )+6 \operatorname {arcsinh}\left (d x +c \right )^{2} \ln \left (1-d x -c -\sqrt {1+\left (d x +c \right )^{2}}\right )+12 \,\operatorname {arcsinh}\left (d x +c \right ) \operatorname {polylog}\left (2, d x +c +\sqrt {1+\left (d x +c \right )^{2}}\right )-12 \operatorname {polylog}\left (3, d x +c +\sqrt {1+\left (d x +c \right )^{2}}\right )\right )}{e^{3}}+\frac {4 a \,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 {6 a^{2} 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 {4 b \,a^{3} \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}\)	\(561\)
parts	\(-\frac {a^{4}}{2 e^{3} \left (d x +c \right )^{2} d}+\frac {b^{4} \left (-\frac {\operatorname {arcsinh}\left (d x +c \right )^{3} \left (-4 \left (d x +c \right )^{2}+4 \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}}-4 \operatorname {arcsinh}\left (d x +c \right )^{3}+6 \operatorname {arcsinh}\left (d x +c \right )^{2} \ln \left (1+d x +c +\sqrt {1+\left (d x +c \right )^{2}}\right )+12 \,\operatorname {arcsinh}\left (d x +c \right ) \operatorname {polylog}\left (2, -d x -c -\sqrt {1+\left (d x +c \right )^{2}}\right )-12 \operatorname {polylog}\left (3, -d x -c -\sqrt {1+\left (d x +c \right )^{2}}\right )+6 \operatorname {arcsinh}\left (d x +c \right )^{2} \ln \left (1-d x -c -\sqrt {1+\left (d x +c \right )^{2}}\right )+12 \,\operatorname {arcsinh}\left (d x +c \right ) \operatorname {polylog}\left (2, d x +c +\sqrt {1+\left (d x +c \right )^{2}}\right )-12 \operatorname {polylog}\left (3, d x +c +\sqrt {1+\left (d x +c \right )^{2}}\right )\right )}{e^{3} d}+\frac {4 a \,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 {6 a^{2} 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 {4 b \,a^{3} \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}\)	\(572\)

[In]

int((a+b*arcsinh(d*x+c))^4/(d*e*x+c*e)^3,x,method=_RETURNVERBOSE)

[Out]

1/d*(-1/2*a^4/e^3/(d*x+c)^2+b^4/e^3*(-1/2*arcsinh(d*x+c)^3*(-4*(d*x+c)^2+4*(d*x+c)*(1+(d*x+c)^2)^(1/2)+arcsinh
(d*x+c))/(d*x+c)^2-4*arcsinh(d*x+c)^3+6*arcsinh(d*x+c)^2*ln(1+d*x+c+(1+(d*x+c)^2)^(1/2))+12*arcsinh(d*x+c)*pol
ylog(2,-d*x-c-(1+(d*x+c)^2)^(1/2))-12*polylog(3,-d*x-c-(1+(d*x+c)^2)^(1/2))+6*arcsinh(d*x+c)^2*ln(1-d*x-c-(1+(
d*x+c)^2)^(1/2))+12*arcsinh(d*x+c)*polylog(2,d*x+c+(1+(d*x+c)^2)^(1/2))-12*polylog(3,d*x+c+(1+(d*x+c)^2)^(1/2)
))+4*a*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*a
rcsinh(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*arc
sinh(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)))+6*a^2*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))+4*b*a^3/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]

\[ \int \frac {(a+b \text {arcsinh}(c+d x))^4}{(c e+d e x)^3} \, dx=\int { \frac {{\left (b \operatorname {arsinh}\left (d x + c\right ) + a\right )}^{4}}{{\left (d e x + c e\right )}^{3}} \,d x } \]

[In]

integrate((a+b*arcsinh(d*x+c))^4/(d*e*x+c*e)^3,x, algorithm="fricas")

[Out]

integral((b^4*arcsinh(d*x + c)^4 + 4*a*b^3*arcsinh(d*x + c)^3 + 6*a^2*b^2*arcsinh(d*x + c)^2 + 4*a^3*b*arcsinh
(d*x + c) + a^4)/(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]

\[ \int \frac {(a+b \text {arcsinh}(c+d x))^4}{(c e+d e x)^3} \, dx=\frac {\int \frac {a^{4}}{c^{3} + 3 c^{2} d x + 3 c d^{2} x^{2} + d^{3} x^{3}}\, dx + \int \frac {b^{4} \operatorname {asinh}^{4}{\left (c + d x \right )}}{c^{3} + 3 c^{2} d x + 3 c d^{2} x^{2} + d^{3} x^{3}}\, dx + \int \frac {4 a b^{3} \operatorname {asinh}^{3}{\left (c + d x \right )}}{c^{3} + 3 c^{2} d x + 3 c d^{2} x^{2} + d^{3} x^{3}}\, dx + \int \frac {6 a^{2} b^{2} \operatorname {asinh}^{2}{\left (c + d x \right )}}{c^{3} + 3 c^{2} d x + 3 c d^{2} x^{2} + d^{3} x^{3}}\, dx + \int \frac {4 a^{3} b \operatorname {asinh}{\left (c + d x \right )}}{c^{3} + 3 c^{2} d x + 3 c d^{2} x^{2} + d^{3} x^{3}}\, dx}{e^{3}} \]

[In]

integrate((a+b*asinh(d*x+c))**4/(d*e*x+c*e)**3,x)

[Out]

(Integral(a**4/(c**3 + 3*c**2*d*x + 3*c*d**2*x**2 + d**3*x**3), x) + Integral(b**4*asinh(c + d*x)**4/(c**3 + 3
*c**2*d*x + 3*c*d**2*x**2 + d**3*x**3), x) + Integral(4*a*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(6*a**2*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(4*a**3*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]

\[ \int \frac {(a+b \text {arcsinh}(c+d x))^4}{(c e+d e x)^3} \, dx=\int { \frac {{\left (b \operatorname {arsinh}\left (d x + c\right ) + a\right )}^{4}}{{\left (d e x + c e\right )}^{3}} \,d x } \]

[In]

integrate((a+b*arcsinh(d*x+c))^4/(d*e*x+c*e)^3,x, algorithm="maxima")

[Out]

-1/2*b^4*log(d*x + c + sqrt(d^2*x^2 + 2*c*d*x + c^2 + 1))^4/(d^3*e^3*x^2 + 2*c*d^2*e^3*x + c^2*d*e^3) - 6*(sqr
t(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^2*b^2 - 2*
a^3*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*a^2*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^4/(d^3*e^3
*x^2 + 2*c*d^2*e^3*x + c^2*d*e^3) + integrate(2*(2*(c^3 + c)*a*b^3 + (c^3 + c)*b^4 + (2*a*b^3*d^3 + b^4*d^3)*x
^3 + 3*(2*a*b^3*c*d^2 + b^4*c*d^2)*x^2 + (2*(3*c^2*d + d)*a*b^3 + (3*c^2*d + d)*b^4)*x + (b^4*c^2 + 2*(c^2 + 1
)*a*b^3 + (2*a*b^3*d^2 + b^4*d^2)*x^2 + 2*(2*a*b^3*c*d + b^4*c*d)*x)*sqrt(d^2*x^2 + 2*c*d*x + c^2 + 1))*log(d*
x + c + sqrt(d^2*x^2 + 2*c*d*x + c^2 + 1))^3/(d^6*e^3*x^6 + 6*c*d^5*e^3*x^5 + c^6*e^3 + c^4*e^3 + (15*c^2*d^4*
e^3 + d^4*e^3)*x^4 + 4*(5*c^3*d^3*e^3 + c*d^3*e^3)*x^3 + 3*(5*c^4*d^2*e^3 + 2*c^2*d^2*e^3)*x^2 + 2*(3*c^5*d*e^
3 + 2*c^3*d*e^3)*x + (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)*sqrt(d^2*x^2 + 2*c*d*x + c^2 + 1)), x)

Giac [F]

\[ \int \frac {(a+b \text {arcsinh}(c+d x))^4}{(c e+d e x)^3} \, dx=\int { \frac {{\left (b \operatorname {arsinh}\left (d x + c\right ) + a\right )}^{4}}{{\left (d e x + c e\right )}^{3}} \,d x } \]

[In]

integrate((a+b*arcsinh(d*x+c))^4/(d*e*x+c*e)^3,x, algorithm="giac")

[Out]

integrate((b*arcsinh(d*x + c) + a)^4/(d*e*x + c*e)^3, x)

Mupad [F(-1)]

Timed out. \[ \int \frac {(a+b \text {arcsinh}(c+d x))^4}{(c e+d e x)^3} \, dx=\int \frac {{\left (a+b\,\mathrm {asinh}\left (c+d\,x\right )\right )}^4}{{\left (c\,e+d\,e\,x\right )}^3} \,d x \]

[In]

int((a + b*asinh(c + d*x))^4/(c*e + d*e*x)^3,x)

[Out]

int((a + b*asinh(c + d*x))^4/(c*e + d*e*x)^3, x)