Integrand size = 23, antiderivative size = 166 \[ \int \frac {(a+b \text {arcsinh}(c+d x))^3}{(c e+d e x)^2} \, dx=-\frac {(a+b \text {arcsinh}(c+d x))^3}{d e^2 (c+d x)}-\frac {6 b (a+b \text {arcsinh}(c+d x))^2 \text {arctanh}\left (e^{\text {arcsinh}(c+d x)}\right )}{d e^2}-\frac {6 b^2 (a+b \text {arcsinh}(c+d x)) \operatorname {PolyLog}\left (2,-e^{\text {arcsinh}(c+d x)}\right )}{d e^2}+\frac {6 b^2 (a+b \text {arcsinh}(c+d x)) \operatorname {PolyLog}\left (2,e^{\text {arcsinh}(c+d x)}\right )}{d e^2}+\frac {6 b^3 \operatorname {PolyLog}\left (3,-e^{\text {arcsinh}(c+d x)}\right )}{d e^2}-\frac {6 b^3 \operatorname {PolyLog}\left (3,e^{\text {arcsinh}(c+d x)}\right )}{d e^2} \]
-(a+b*arcsinh(d*x+c))^3/d/e^2/(d*x+c)-6*b*(a+b*arcsinh(d*x+c))^2*arctanh(d *x+c+(1+(d*x+c)^2)^(1/2))/d/e^2-6*b^2*(a+b*arcsinh(d*x+c))*polylog(2,-d*x- c-(1+(d*x+c)^2)^(1/2))/d/e^2+6*b^2*(a+b*arcsinh(d*x+c))*polylog(2,d*x+c+(1 +(d*x+c)^2)^(1/2))/d/e^2+6*b^3*polylog(3,-d*x-c-(1+(d*x+c)^2)^(1/2))/d/e^2 -6*b^3*polylog(3,d*x+c+(1+(d*x+c)^2)^(1/2))/d/e^2
Time = 0.65 (sec) , antiderivative size = 315, normalized size of antiderivative = 1.90 \[ \int \frac {(a+b \text {arcsinh}(c+d x))^3}{(c e+d e x)^2} \, dx=\frac {-\frac {a^3}{c+d x}-\frac {3 a^2 b \text {arcsinh}(c+d x)}{c+d x}+3 a^2 b \log (c+d x)-3 a^2 b \log \left (1+\sqrt {1+c^2+2 c d x+d^2 x^2}\right )+3 a b^2 \left (\text {arcsinh}(c+d x) \left (-\frac {\text {arcsinh}(c+d x)}{c+d x}+2 \log \left (1-e^{-\text {arcsinh}(c+d x)}\right )-2 \log \left (1+e^{-\text {arcsinh}(c+d x)}\right )\right )+2 \operatorname {PolyLog}\left (2,-e^{-\text {arcsinh}(c+d x)}\right )-2 \operatorname {PolyLog}\left (2,e^{-\text {arcsinh}(c+d x)}\right )\right )+b^3 \left (-\frac {\text {arcsinh}(c+d x)^3}{c+d x}+3 \text {arcsinh}(c+d x)^2 \log \left (1-e^{-\text {arcsinh}(c+d x)}\right )-3 \text {arcsinh}(c+d x)^2 \log \left (1+e^{-\text {arcsinh}(c+d x)}\right )+6 \text {arcsinh}(c+d x) \operatorname {PolyLog}\left (2,-e^{-\text {arcsinh}(c+d x)}\right )-6 \text {arcsinh}(c+d x) \operatorname {PolyLog}\left (2,e^{-\text {arcsinh}(c+d x)}\right )+6 \operatorname {PolyLog}\left (3,-e^{-\text {arcsinh}(c+d x)}\right )-6 \operatorname {PolyLog}\left (3,e^{-\text {arcsinh}(c+d x)}\right )\right )}{d e^2} \]
(-(a^3/(c + d*x)) - (3*a^2*b*ArcSinh[c + d*x])/(c + d*x) + 3*a^2*b*Log[c + d*x] - 3*a^2*b*Log[1 + Sqrt[1 + c^2 + 2*c*d*x + d^2*x^2]] + 3*a*b^2*(ArcS inh[c + d*x]*(-(ArcSinh[c + d*x]/(c + d*x)) + 2*Log[1 - E^(-ArcSinh[c + d* x])] - 2*Log[1 + E^(-ArcSinh[c + d*x])]) + 2*PolyLog[2, -E^(-ArcSinh[c + d *x])] - 2*PolyLog[2, E^(-ArcSinh[c + d*x])]) + b^3*(-(ArcSinh[c + d*x]^3/( c + d*x)) + 3*ArcSinh[c + d*x]^2*Log[1 - E^(-ArcSinh[c + d*x])] - 3*ArcSin h[c + d*x]^2*Log[1 + E^(-ArcSinh[c + d*x])] + 6*ArcSinh[c + d*x]*PolyLog[2 , -E^(-ArcSinh[c + d*x])] - 6*ArcSinh[c + d*x]*PolyLog[2, E^(-ArcSinh[c + d*x])] + 6*PolyLog[3, -E^(-ArcSinh[c + d*x])] - 6*PolyLog[3, E^(-ArcSinh[c + d*x])]))/(d*e^2)
Result contains complex when optimal does not.
Time = 0.76 (sec) , antiderivative size = 142, normalized size of antiderivative = 0.86, number of steps used = 11, number of rules used = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.435, Rules used = {6274, 27, 6191, 6231, 3042, 26, 4670, 3011, 2720, 7143}
Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.
| \(\displaystyle \int \frac {(a+b \text {arcsinh}(c+d x))^3}{(c e+d e x)^2} \, dx\) |
| \(\Big \downarrow \) 6274 |
| \(\displaystyle \frac {\int \frac {(a+b \text {arcsinh}(c+d x))^3}{e^2 (c+d x)^2}d(c+d x)}{d}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\int \frac {(a+b \text {arcsinh}(c+d x))^3}{(c+d x)^2}d(c+d x)}{d e^2}\) |
| \(\Big \downarrow \) 6191 |
| \(\displaystyle \frac {3 b \int \frac {(a+b \text {arcsinh}(c+d x))^2}{(c+d x) \sqrt {(c+d x)^2+1}}d(c+d x)-\frac {(a+b \text {arcsinh}(c+d x))^3}{c+d x}}{d e^2}\) |
| \(\Big \downarrow \) 6231 |
| \(\displaystyle \frac {3 b \int \frac {(a+b \text {arcsinh}(c+d x))^2}{c+d x}d\text {arcsinh}(c+d x)-\frac {(a+b \text {arcsinh}(c+d x))^3}{c+d x}}{d e^2}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {-\frac {(a+b \text {arcsinh}(c+d x))^3}{c+d x}+3 b \int i (a+b \text {arcsinh}(c+d x))^2 \csc (i \text {arcsinh}(c+d x))d\text {arcsinh}(c+d x)}{d e^2}\) |
| \(\Big \downarrow \) 26 |
| \(\displaystyle \frac {-\frac {(a+b \text {arcsinh}(c+d x))^3}{c+d x}+3 i b \int (a+b \text {arcsinh}(c+d x))^2 \csc (i \text {arcsinh}(c+d x))d\text {arcsinh}(c+d x)}{d e^2}\) |
| \(\Big \downarrow \) 4670 |
| \(\displaystyle \frac {-\frac {(a+b \text {arcsinh}(c+d x))^3}{c+d x}+3 i b \left (2 i b \int (a+b \text {arcsinh}(c+d x)) \log \left (1-e^{\text {arcsinh}(c+d x)}\right )d\text {arcsinh}(c+d x)-2 i b \int (a+b \text {arcsinh}(c+d x)) \log \left (1+e^{\text {arcsinh}(c+d x)}\right )d\text {arcsinh}(c+d x)+2 i \text {arctanh}\left (e^{\text {arcsinh}(c+d x)}\right ) (a+b \text {arcsinh}(c+d x))^2\right )}{d e^2}\) |
| \(\Big \downarrow \) 3011 |
| \(\displaystyle \frac {-\frac {(a+b \text {arcsinh}(c+d x))^3}{c+d x}+3 i b \left (-2 i b \left (b \int \operatorname {PolyLog}\left (2,-e^{\text {arcsinh}(c+d x)}\right )d\text {arcsinh}(c+d x)-\operatorname {PolyLog}\left (2,-e^{\text {arcsinh}(c+d x)}\right ) (a+b \text {arcsinh}(c+d x))\right )+2 i b \left (b \int \operatorname {PolyLog}\left (2,e^{\text {arcsinh}(c+d x)}\right )d\text {arcsinh}(c+d x)-\operatorname {PolyLog}\left (2,e^{\text {arcsinh}(c+d x)}\right ) (a+b \text {arcsinh}(c+d x))\right )+2 i \text {arctanh}\left (e^{\text {arcsinh}(c+d x)}\right ) (a+b \text {arcsinh}(c+d x))^2\right )}{d e^2}\) |
| \(\Big \downarrow \) 2720 |
| \(\displaystyle \frac {-\frac {(a+b \text {arcsinh}(c+d x))^3}{c+d x}+3 i b \left (2 i b \left (b \int e^{-\text {arcsinh}(c+d x)} \operatorname {PolyLog}\left (2,e^{\text {arcsinh}(c+d x)}\right )de^{\text {arcsinh}(c+d x)}-\operatorname {PolyLog}\left (2,e^{\text {arcsinh}(c+d x)}\right ) (a+b \text {arcsinh}(c+d x))\right )-2 i b \left (b \int e^{-\text {arcsinh}(c+d x)} \operatorname {PolyLog}(2,-c-d x)de^{\text {arcsinh}(c+d x)}-\operatorname {PolyLog}\left (2,-e^{\text {arcsinh}(c+d x)}\right ) (a+b \text {arcsinh}(c+d x))\right )+2 i \text {arctanh}\left (e^{\text {arcsinh}(c+d x)}\right ) (a+b \text {arcsinh}(c+d x))^2\right )}{d e^2}\) |
| \(\Big \downarrow \) 7143 |
| \(\displaystyle \frac {-\frac {(a+b \text {arcsinh}(c+d x))^3}{c+d x}+3 i b \left (2 i \text {arctanh}\left (e^{\text {arcsinh}(c+d x)}\right ) (a+b \text {arcsinh}(c+d x))^2+2 i b \left (b \operatorname {PolyLog}\left (3,e^{\text {arcsinh}(c+d x)}\right )-\operatorname {PolyLog}\left (2,e^{\text {arcsinh}(c+d x)}\right ) (a+b \text {arcsinh}(c+d x))\right )-2 i b \left (b \operatorname {PolyLog}(3,-c-d x)-\operatorname {PolyLog}\left (2,-e^{\text {arcsinh}(c+d x)}\right ) (a+b \text {arcsinh}(c+d x))\right )\right )}{d e^2}\) |
(-((a + b*ArcSinh[c + d*x])^3/(c + d*x)) + (3*I)*b*((2*I)*(a + b*ArcSinh[c + d*x])^2*ArcTanh[E^ArcSinh[c + d*x]] + (2*I)*b*(-((a + b*ArcSinh[c + d*x ])*PolyLog[2, E^ArcSinh[c + d*x]]) + b*PolyLog[3, E^ArcSinh[c + d*x]]) - ( 2*I)*b*(-((a + b*ArcSinh[c + d*x])*PolyLog[2, -E^ArcSinh[c + d*x]]) + b*Po lyLog[3, -c - d*x])))/(d*e^2)
3.2.43.3.1 Defintions of rubi rules used
Int[(Complex[0, a_])*(Fx_), x_Symbol] :> Simp[(Complex[Identity[0], a]) I nt[Fx, x], x] /; FreeQ[a, x] && EqQ[a^2, 1]
Int[(a_)*(Fx_), x_Symbol] :> Simp[a Int[Fx, x], x] /; FreeQ[a, x] && !Ma tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
Int[u_, x_Symbol] :> With[{v = FunctionOfExponential[u, x]}, Simp[v/D[v, x] Subst[Int[FunctionOfExponentialFunction[u, x]/x, x], x, v], x]] /; Funct ionOfExponentialQ[u, x] && !MatchQ[u, (w_)*((a_.)*(v_)^(n_))^(m_) /; FreeQ [{a, m, n}, x] && IntegerQ[m*n]] && !MatchQ[u, E^((c_.)*((a_.) + (b_.)*x)) *(F_)[v_] /; FreeQ[{a, b, c}, x] && InverseFunctionQ[F[x]]]
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] + Simp[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]
Int[csc[(e_.) + (Complex[0, fz_])*(f_.)*(x_)]*((c_.) + (d_.)*(x_))^(m_.), x _Symbol] :> Simp[-2*(c + d*x)^m*(ArcTanh[E^((-I)*e + f*fz*x)]/(f*fz*I)), x] + (-Simp[d*(m/(f*fz*I)) Int[(c + d*x)^(m - 1)*Log[1 - E^((-I)*e + f*fz*x )], x], x] + Simp[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]
Int[((a_.) + ArcSinh[(c_.)*(x_)]*(b_.))^(n_.)*((d_.)*(x_))^(m_.), x_Symbol] :> Simp[(d*x)^(m + 1)*((a + b*ArcSinh[c*x])^n/(d*(m + 1))), x] - Simp[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]
Int[(((a_.) + ArcSinh[(c_.)*(x_)]*(b_.))^(n_.)*(x_)^(m_))/Sqrt[(d_) + (e_.) *(x_)^2], x_Symbol] :> Simp[(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] /; FreeQ [{a, b, c, d, e}, x] && EqQ[e, c^2*d] && IGtQ[n, 0] && IntegerQ[m]
Int[((a_.) + ArcSinh[(c_) + (d_.)*(x_)]*(b_.))^(n_.)*((e_.) + (f_.)*(x_))^( m_.), x_Symbol] :> Simp[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]
Int[PolyLog[n_, (c_.)*((a_.) + (b_.)*(x_))^(p_.)]/((d_.) + (e_.)*(x_)), x_S ymbol] :> 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]
Time = 0.39 (sec) , antiderivative size = 377, normalized size of antiderivative = 2.27
| method | result | size |
| derivativedivides | \(\frac {-\frac {a^{3}}{e^{2} \left (d x +c \right )}+\frac {b^{3} \left (-\frac {\operatorname {arcsinh}\left (d x +c \right )^{3}}{d x +c}-3 \operatorname {arcsinh}\left (d x +c \right )^{2} \ln \left (1+d x +c +\sqrt {1+\left (d x +c \right )^{2}}\right )-6 \,\operatorname {arcsinh}\left (d x +c \right ) \operatorname {polylog}\left (2, -d x -c -\sqrt {1+\left (d x +c \right )^{2}}\right )+6 \operatorname {polylog}\left (3, -d x -c -\sqrt {1+\left (d x +c \right )^{2}}\right )+3 \operatorname {arcsinh}\left (d x +c \right )^{2} \ln \left (1-d x -c -\sqrt {1+\left (d x +c \right )^{2}}\right )+6 \,\operatorname {arcsinh}\left (d x +c \right ) \operatorname {polylog}\left (2, d x +c +\sqrt {1+\left (d x +c \right )^{2}}\right )-6 \operatorname {polylog}\left (3, d x +c +\sqrt {1+\left (d x +c \right )^{2}}\right )\right )}{e^{2}}+\frac {3 a \,b^{2} \left (-\frac {\operatorname {arcsinh}\left (d x +c \right )^{2}}{d x +c}-2 \,\operatorname {arcsinh}\left (d x +c \right ) \ln \left (1+d x +c +\sqrt {1+\left (d x +c \right )^{2}}\right )-2 \operatorname {polylog}\left (2, -d x -c -\sqrt {1+\left (d x +c \right )^{2}}\right )+2 \,\operatorname {arcsinh}\left (d x +c \right ) \ln \left (1-d x -c -\sqrt {1+\left (d x +c \right )^{2}}\right )+2 \operatorname {polylog}\left (2, d x +c +\sqrt {1+\left (d x +c \right )^{2}}\right )\right )}{e^{2}}+\frac {3 a^{2} b \left (-\frac {\operatorname {arcsinh}\left (d x +c \right )}{d x +c}-\operatorname {arctanh}\left (\frac {1}{\sqrt {1+\left (d x +c \right )^{2}}}\right )\right )}{e^{2}}}{d}\) | \(377\) |
| default | \(\frac {-\frac {a^{3}}{e^{2} \left (d x +c \right )}+\frac {b^{3} \left (-\frac {\operatorname {arcsinh}\left (d x +c \right )^{3}}{d x +c}-3 \operatorname {arcsinh}\left (d x +c \right )^{2} \ln \left (1+d x +c +\sqrt {1+\left (d x +c \right )^{2}}\right )-6 \,\operatorname {arcsinh}\left (d x +c \right ) \operatorname {polylog}\left (2, -d x -c -\sqrt {1+\left (d x +c \right )^{2}}\right )+6 \operatorname {polylog}\left (3, -d x -c -\sqrt {1+\left (d x +c \right )^{2}}\right )+3 \operatorname {arcsinh}\left (d x +c \right )^{2} \ln \left (1-d x -c -\sqrt {1+\left (d x +c \right )^{2}}\right )+6 \,\operatorname {arcsinh}\left (d x +c \right ) \operatorname {polylog}\left (2, d x +c +\sqrt {1+\left (d x +c \right )^{2}}\right )-6 \operatorname {polylog}\left (3, d x +c +\sqrt {1+\left (d x +c \right )^{2}}\right )\right )}{e^{2}}+\frac {3 a \,b^{2} \left (-\frac {\operatorname {arcsinh}\left (d x +c \right )^{2}}{d x +c}-2 \,\operatorname {arcsinh}\left (d x +c \right ) \ln \left (1+d x +c +\sqrt {1+\left (d x +c \right )^{2}}\right )-2 \operatorname {polylog}\left (2, -d x -c -\sqrt {1+\left (d x +c \right )^{2}}\right )+2 \,\operatorname {arcsinh}\left (d x +c \right ) \ln \left (1-d x -c -\sqrt {1+\left (d x +c \right )^{2}}\right )+2 \operatorname {polylog}\left (2, d x +c +\sqrt {1+\left (d x +c \right )^{2}}\right )\right )}{e^{2}}+\frac {3 a^{2} b \left (-\frac {\operatorname {arcsinh}\left (d x +c \right )}{d x +c}-\operatorname {arctanh}\left (\frac {1}{\sqrt {1+\left (d x +c \right )^{2}}}\right )\right )}{e^{2}}}{d}\) | \(377\) |
| parts | \(-\frac {a^{3}}{e^{2} \left (d x +c \right ) d}+\frac {b^{3} \left (-\frac {\operatorname {arcsinh}\left (d x +c \right )^{3}}{d x +c}-3 \operatorname {arcsinh}\left (d x +c \right )^{2} \ln \left (1+d x +c +\sqrt {1+\left (d x +c \right )^{2}}\right )-6 \,\operatorname {arcsinh}\left (d x +c \right ) \operatorname {polylog}\left (2, -d x -c -\sqrt {1+\left (d x +c \right )^{2}}\right )+6 \operatorname {polylog}\left (3, -d x -c -\sqrt {1+\left (d x +c \right )^{2}}\right )+3 \operatorname {arcsinh}\left (d x +c \right )^{2} \ln \left (1-d x -c -\sqrt {1+\left (d x +c \right )^{2}}\right )+6 \,\operatorname {arcsinh}\left (d x +c \right ) \operatorname {polylog}\left (2, d x +c +\sqrt {1+\left (d x +c \right )^{2}}\right )-6 \operatorname {polylog}\left (3, d x +c +\sqrt {1+\left (d x +c \right )^{2}}\right )\right )}{e^{2} d}+\frac {3 a \,b^{2} \left (-\frac {\operatorname {arcsinh}\left (d x +c \right )^{2}}{d x +c}-2 \,\operatorname {arcsinh}\left (d x +c \right ) \ln \left (1+d x +c +\sqrt {1+\left (d x +c \right )^{2}}\right )-2 \operatorname {polylog}\left (2, -d x -c -\sqrt {1+\left (d x +c \right )^{2}}\right )+2 \,\operatorname {arcsinh}\left (d x +c \right ) \ln \left (1-d x -c -\sqrt {1+\left (d x +c \right )^{2}}\right )+2 \operatorname {polylog}\left (2, d x +c +\sqrt {1+\left (d x +c \right )^{2}}\right )\right )}{e^{2} d}+\frac {3 a^{2} b \left (-\frac {\operatorname {arcsinh}\left (d x +c \right )}{d x +c}-\operatorname {arctanh}\left (\frac {1}{\sqrt {1+\left (d x +c \right )^{2}}}\right )\right )}{e^{2} d}\) | \(385\) |
1/d*(-a^3/e^2/(d*x+c)+b^3/e^2*(-1/(d*x+c)*arcsinh(d*x+c)^3-3*arcsinh(d*x+c )^2*ln(1+d*x+c+(1+(d*x+c)^2)^(1/2))-6*arcsinh(d*x+c)*polylog(2,-d*x-c-(1+( d*x+c)^2)^(1/2))+6*polylog(3,-d*x-c-(1+(d*x+c)^2)^(1/2))+3*arcsinh(d*x+c)^ 2*ln(1-d*x-c-(1+(d*x+c)^2)^(1/2))+6*arcsinh(d*x+c)*polylog(2,d*x+c+(1+(d*x +c)^2)^(1/2))-6*polylog(3,d*x+c+(1+(d*x+c)^2)^(1/2)))+3*a*b^2/e^2*(-1/(d*x +c)*arcsinh(d*x+c)^2-2*arcsinh(d*x+c)*ln(1+d*x+c+(1+(d*x+c)^2)^(1/2))-2*po lylog(2,-d*x-c-(1+(d*x+c)^2)^(1/2))+2*arcsinh(d*x+c)*ln(1-d*x-c-(1+(d*x+c) ^2)^(1/2))+2*polylog(2,d*x+c+(1+(d*x+c)^2)^(1/2)))+3*a^2*b/e^2*(-1/(d*x+c) *arcsinh(d*x+c)-arctanh(1/(1+(d*x+c)^2)^(1/2))))
\[ \int \frac {(a+b \text {arcsinh}(c+d x))^3}{(c e+d e x)^2} \, dx=\int { \frac {{\left (b \operatorname {arsinh}\left (d x + c\right ) + a\right )}^{3}}{{\left (d e x + c e\right )}^{2}} \,d x } \]
integral((b^3*arcsinh(d*x + c)^3 + 3*a*b^2*arcsinh(d*x + c)^2 + 3*a^2*b*ar csinh(d*x + c) + a^3)/(d^2*e^2*x^2 + 2*c*d*e^2*x + c^2*e^2), x)
\[ \int \frac {(a+b \text {arcsinh}(c+d x))^3}{(c e+d e x)^2} \, dx=\frac {\int \frac {a^{3}}{c^{2} + 2 c d x + d^{2} x^{2}}\, dx + \int \frac {b^{3} \operatorname {asinh}^{3}{\left (c + d x \right )}}{c^{2} + 2 c d x + d^{2} x^{2}}\, dx + \int \frac {3 a b^{2} \operatorname {asinh}^{2}{\left (c + d x \right )}}{c^{2} + 2 c d x + d^{2} x^{2}}\, dx + \int \frac {3 a^{2} b \operatorname {asinh}{\left (c + d x \right )}}{c^{2} + 2 c d x + d^{2} x^{2}}\, dx}{e^{2}} \]
(Integral(a**3/(c**2 + 2*c*d*x + d**2*x**2), x) + Integral(b**3*asinh(c + d*x)**3/(c**2 + 2*c*d*x + d**2*x**2), x) + Integral(3*a*b**2*asinh(c + d*x )**2/(c**2 + 2*c*d*x + d**2*x**2), x) + Integral(3*a**2*b*asinh(c + d*x)/( c**2 + 2*c*d*x + d**2*x**2), x))/e**2
Exception generated. \[ \int \frac {(a+b \text {arcsinh}(c+d x))^3}{(c e+d e x)^2} \, dx=\text {Exception raised: ValueError} \]
Exception raised: ValueError >> Computation failed since Maxima requested additional constraints; using the 'assume' command before evaluation *may* help (example of legal syntax is 'assume(e>0)', see `assume?` for more de tails)Is e
\[ \int \frac {(a+b \text {arcsinh}(c+d x))^3}{(c e+d e x)^2} \, dx=\int { \frac {{\left (b \operatorname {arsinh}\left (d x + c\right ) + a\right )}^{3}}{{\left (d e x + c e\right )}^{2}} \,d x } \]
Timed out. \[ \int \frac {(a+b \text {arcsinh}(c+d x))^3}{(c e+d e x)^2} \, dx=\int \frac {{\left (a+b\,\mathrm {asinh}\left (c+d\,x\right )\right )}^3}{{\left (c\,e+d\,e\,x\right )}^2} \,d x \]