3.168 \(\int \frac {(c e+d e x)^4}{(a+b \sinh ^{-1}(c+d x))^3} \, dx\)

Optimal. Leaf size=320 \[ \frac {e^4 \cosh \left (\frac {a}{b}\right ) \text {Chi}\left (\frac {a+b \sinh ^{-1}(c+d x)}{b}\right )}{16 b^3 d}-\frac {27 e^4 \cosh \left (\frac {3 a}{b}\right ) \text {Chi}\left (\frac {3 \left (a+b \sinh ^{-1}(c+d x)\right )}{b}\right )}{32 b^3 d}+\frac {25 e^4 \cosh \left (\frac {5 a}{b}\right ) \text {Chi}\left (\frac {5 \left (a+b \sinh ^{-1}(c+d x)\right )}{b}\right )}{32 b^3 d}-\frac {e^4 \sinh \left (\frac {a}{b}\right ) \text {Shi}\left (\frac {a+b \sinh ^{-1}(c+d x)}{b}\right )}{16 b^3 d}+\frac {27 e^4 \sinh \left (\frac {3 a}{b}\right ) \text {Shi}\left (\frac {3 \left (a+b \sinh ^{-1}(c+d x)\right )}{b}\right )}{32 b^3 d}-\frac {25 e^4 \sinh \left (\frac {5 a}{b}\right ) \text {Shi}\left (\frac {5 \left (a+b \sinh ^{-1}(c+d x)\right )}{b}\right )}{32 b^3 d}-\frac {5 e^4 (c+d x)^5}{2 b^2 d \left (a+b \sinh ^{-1}(c+d x)\right )}-\frac {2 e^4 (c+d x)^3}{b^2 d \left (a+b \sinh ^{-1}(c+d x)\right )}-\frac {e^4 \sqrt {(c+d x)^2+1} (c+d x)^4}{2 b d \left (a+b \sinh ^{-1}(c+d x)\right )^2} \]

[Out]

-2*e^4*(d*x+c)^3/b^2/d/(a+b*arcsinh(d*x+c))-5/2*e^4*(d*x+c)^5/b^2/d/(a+b*arcsinh(d*x+c))+1/16*e^4*Chi((a+b*arc
sinh(d*x+c))/b)*cosh(a/b)/b^3/d-27/32*e^4*Chi(3*(a+b*arcsinh(d*x+c))/b)*cosh(3*a/b)/b^3/d+25/32*e^4*Chi(5*(a+b
*arcsinh(d*x+c))/b)*cosh(5*a/b)/b^3/d-1/16*e^4*Shi((a+b*arcsinh(d*x+c))/b)*sinh(a/b)/b^3/d+27/32*e^4*Shi(3*(a+
b*arcsinh(d*x+c))/b)*sinh(3*a/b)/b^3/d-25/32*e^4*Shi(5*(a+b*arcsinh(d*x+c))/b)*sinh(5*a/b)/b^3/d-1/2*e^4*(d*x+
c)^4*(1+(d*x+c)^2)^(1/2)/b/d/(a+b*arcsinh(d*x+c))^2

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Rubi [A]  time = 0.89, antiderivative size = 316, normalized size of antiderivative = 0.99, number of steps used = 26, number of rules used = 9, integrand size = 23, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.391, Rules used = {5865, 12, 5667, 5774, 5669, 5448, 3303, 3298, 3301} \[ \frac {e^4 \cosh \left (\frac {a}{b}\right ) \text {Chi}\left (\frac {a}{b}+\sinh ^{-1}(c+d x)\right )}{16 b^3 d}-\frac {27 e^4 \cosh \left (\frac {3 a}{b}\right ) \text {Chi}\left (\frac {3 a}{b}+3 \sinh ^{-1}(c+d x)\right )}{32 b^3 d}+\frac {25 e^4 \cosh \left (\frac {5 a}{b}\right ) \text {Chi}\left (\frac {5 a}{b}+5 \sinh ^{-1}(c+d x)\right )}{32 b^3 d}-\frac {e^4 \sinh \left (\frac {a}{b}\right ) \text {Shi}\left (\frac {a}{b}+\sinh ^{-1}(c+d x)\right )}{16 b^3 d}+\frac {27 e^4 \sinh \left (\frac {3 a}{b}\right ) \text {Shi}\left (\frac {3 a}{b}+3 \sinh ^{-1}(c+d x)\right )}{32 b^3 d}-\frac {25 e^4 \sinh \left (\frac {5 a}{b}\right ) \text {Shi}\left (\frac {5 a}{b}+5 \sinh ^{-1}(c+d x)\right )}{32 b^3 d}-\frac {5 e^4 (c+d x)^5}{2 b^2 d \left (a+b \sinh ^{-1}(c+d x)\right )}-\frac {2 e^4 (c+d x)^3}{b^2 d \left (a+b \sinh ^{-1}(c+d x)\right )}-\frac {e^4 \sqrt {(c+d x)^2+1} (c+d x)^4}{2 b d \left (a+b \sinh ^{-1}(c+d x)\right )^2} \]

Antiderivative was successfully verified.

[In]

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

[Out]

-(e^4*(c + d*x)^4*Sqrt[1 + (c + d*x)^2])/(2*b*d*(a + b*ArcSinh[c + d*x])^2) - (2*e^4*(c + d*x)^3)/(b^2*d*(a +
b*ArcSinh[c + d*x])) - (5*e^4*(c + d*x)^5)/(2*b^2*d*(a + b*ArcSinh[c + d*x])) + (e^4*Cosh[a/b]*CoshIntegral[a/
b + ArcSinh[c + d*x]])/(16*b^3*d) - (27*e^4*Cosh[(3*a)/b]*CoshIntegral[(3*a)/b + 3*ArcSinh[c + d*x]])/(32*b^3*
d) + (25*e^4*Cosh[(5*a)/b]*CoshIntegral[(5*a)/b + 5*ArcSinh[c + d*x]])/(32*b^3*d) - (e^4*Sinh[a/b]*SinhIntegra
l[a/b + ArcSinh[c + d*x]])/(16*b^3*d) + (27*e^4*Sinh[(3*a)/b]*SinhIntegral[(3*a)/b + 3*ArcSinh[c + d*x]])/(32*
b^3*d) - (25*e^4*Sinh[(5*a)/b]*SinhIntegral[(5*a)/b + 5*ArcSinh[c + d*x]])/(32*b^3*d)

Rule 12

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

Rule 3298

Int[sin[(e_.) + (Complex[0, fz_])*(f_.)*(x_)]/((c_.) + (d_.)*(x_)), x_Symbol] :> Simp[(I*SinhIntegral[(c*f*fz)
/d + f*fz*x])/d, x] /; FreeQ[{c, d, e, f, fz}, x] && EqQ[d*e - c*f*fz*I, 0]

Rule 3301

Int[sin[(e_.) + (Complex[0, fz_])*(f_.)*(x_)]/((c_.) + (d_.)*(x_)), x_Symbol] :> Simp[CoshIntegral[(c*f*fz)/d
+ f*fz*x]/d, x] /; FreeQ[{c, d, e, f, fz}, x] && EqQ[d*(e - Pi/2) - c*f*fz*I, 0]

Rule 3303

Int[sin[(e_.) + (f_.)*(x_)]/((c_.) + (d_.)*(x_)), x_Symbol] :> Dist[Cos[(d*e - c*f)/d], Int[Sin[(c*f)/d + f*x]
/(c + d*x), x], x] + Dist[Sin[(d*e - c*f)/d], Int[Cos[(c*f)/d + f*x]/(c + d*x), x], x] /; FreeQ[{c, d, e, f},
x] && NeQ[d*e - c*f, 0]

Rule 5448

Int[Cosh[(a_.) + (b_.)*(x_)]^(p_.)*((c_.) + (d_.)*(x_))^(m_.)*Sinh[(a_.) + (b_.)*(x_)]^(n_.), x_Symbol] :> Int
[ExpandTrigReduce[(c + d*x)^m, Sinh[a + b*x]^n*Cosh[a + b*x]^p, x], x] /; FreeQ[{a, b, c, d, m}, x] && IGtQ[n,
0] && IGtQ[p, 0]

Rule 5667

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

Rule 5669

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

Rule 5774

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

Rule 5865

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*} \int \frac {(c e+d e x)^4}{\left (a+b \sinh ^{-1}(c+d x)\right )^3} \, dx &=\frac {\operatorname {Subst}\left (\int \frac {e^4 x^4}{\left (a+b \sinh ^{-1}(x)\right )^3} \, dx,x,c+d x\right )}{d}\\ &=\frac {e^4 \operatorname {Subst}\left (\int \frac {x^4}{\left (a+b \sinh ^{-1}(x)\right )^3} \, dx,x,c+d x\right )}{d}\\ &=-\frac {e^4 (c+d x)^4 \sqrt {1+(c+d x)^2}}{2 b d \left (a+b \sinh ^{-1}(c+d x)\right )^2}+\frac {\left (2 e^4\right ) \operatorname {Subst}\left (\int \frac {x^3}{\sqrt {1+x^2} \left (a+b \sinh ^{-1}(x)\right )^2} \, dx,x,c+d x\right )}{b d}+\frac {\left (5 e^4\right ) \operatorname {Subst}\left (\int \frac {x^5}{\sqrt {1+x^2} \left (a+b \sinh ^{-1}(x)\right )^2} \, dx,x,c+d x\right )}{2 b d}\\ &=-\frac {e^4 (c+d x)^4 \sqrt {1+(c+d x)^2}}{2 b d \left (a+b \sinh ^{-1}(c+d x)\right )^2}-\frac {2 e^4 (c+d x)^3}{b^2 d \left (a+b \sinh ^{-1}(c+d x)\right )}-\frac {5 e^4 (c+d x)^5}{2 b^2 d \left (a+b \sinh ^{-1}(c+d x)\right )}+\frac {\left (6 e^4\right ) \operatorname {Subst}\left (\int \frac {x^2}{a+b \sinh ^{-1}(x)} \, dx,x,c+d x\right )}{b^2 d}+\frac {\left (25 e^4\right ) \operatorname {Subst}\left (\int \frac {x^4}{a+b \sinh ^{-1}(x)} \, dx,x,c+d x\right )}{2 b^2 d}\\ &=-\frac {e^4 (c+d x)^4 \sqrt {1+(c+d x)^2}}{2 b d \left (a+b \sinh ^{-1}(c+d x)\right )^2}-\frac {2 e^4 (c+d x)^3}{b^2 d \left (a+b \sinh ^{-1}(c+d x)\right )}-\frac {5 e^4 (c+d x)^5}{2 b^2 d \left (a+b \sinh ^{-1}(c+d x)\right )}+\frac {\left (6 e^4\right ) \operatorname {Subst}\left (\int \frac {\cosh (x) \sinh ^2(x)}{a+b x} \, dx,x,\sinh ^{-1}(c+d x)\right )}{b^2 d}+\frac {\left (25 e^4\right ) \operatorname {Subst}\left (\int \frac {\cosh (x) \sinh ^4(x)}{a+b x} \, dx,x,\sinh ^{-1}(c+d x)\right )}{2 b^2 d}\\ &=-\frac {e^4 (c+d x)^4 \sqrt {1+(c+d x)^2}}{2 b d \left (a+b \sinh ^{-1}(c+d x)\right )^2}-\frac {2 e^4 (c+d x)^3}{b^2 d \left (a+b \sinh ^{-1}(c+d x)\right )}-\frac {5 e^4 (c+d x)^5}{2 b^2 d \left (a+b \sinh ^{-1}(c+d x)\right )}+\frac {\left (6 e^4\right ) \operatorname {Subst}\left (\int \left (-\frac {\cosh (x)}{4 (a+b x)}+\frac {\cosh (3 x)}{4 (a+b x)}\right ) \, dx,x,\sinh ^{-1}(c+d x)\right )}{b^2 d}+\frac {\left (25 e^4\right ) \operatorname {Subst}\left (\int \left (\frac {\cosh (x)}{8 (a+b x)}-\frac {3 \cosh (3 x)}{16 (a+b x)}+\frac {\cosh (5 x)}{16 (a+b x)}\right ) \, dx,x,\sinh ^{-1}(c+d x)\right )}{2 b^2 d}\\ &=-\frac {e^4 (c+d x)^4 \sqrt {1+(c+d x)^2}}{2 b d \left (a+b \sinh ^{-1}(c+d x)\right )^2}-\frac {2 e^4 (c+d x)^3}{b^2 d \left (a+b \sinh ^{-1}(c+d x)\right )}-\frac {5 e^4 (c+d x)^5}{2 b^2 d \left (a+b \sinh ^{-1}(c+d x)\right )}+\frac {\left (25 e^4\right ) \operatorname {Subst}\left (\int \frac {\cosh (5 x)}{a+b x} \, dx,x,\sinh ^{-1}(c+d x)\right )}{32 b^2 d}-\frac {\left (3 e^4\right ) \operatorname {Subst}\left (\int \frac {\cosh (x)}{a+b x} \, dx,x,\sinh ^{-1}(c+d x)\right )}{2 b^2 d}+\frac {\left (3 e^4\right ) \operatorname {Subst}\left (\int \frac {\cosh (3 x)}{a+b x} \, dx,x,\sinh ^{-1}(c+d x)\right )}{2 b^2 d}+\frac {\left (25 e^4\right ) \operatorname {Subst}\left (\int \frac {\cosh (x)}{a+b x} \, dx,x,\sinh ^{-1}(c+d x)\right )}{16 b^2 d}-\frac {\left (75 e^4\right ) \operatorname {Subst}\left (\int \frac {\cosh (3 x)}{a+b x} \, dx,x,\sinh ^{-1}(c+d x)\right )}{32 b^2 d}\\ &=-\frac {e^4 (c+d x)^4 \sqrt {1+(c+d x)^2}}{2 b d \left (a+b \sinh ^{-1}(c+d x)\right )^2}-\frac {2 e^4 (c+d x)^3}{b^2 d \left (a+b \sinh ^{-1}(c+d x)\right )}-\frac {5 e^4 (c+d x)^5}{2 b^2 d \left (a+b \sinh ^{-1}(c+d x)\right )}-\frac {\left (3 e^4 \cosh \left (\frac {a}{b}\right )\right ) \operatorname {Subst}\left (\int \frac {\cosh \left (\frac {a}{b}+x\right )}{a+b x} \, dx,x,\sinh ^{-1}(c+d x)\right )}{2 b^2 d}+\frac {\left (25 e^4 \cosh \left (\frac {a}{b}\right )\right ) \operatorname {Subst}\left (\int \frac {\cosh \left (\frac {a}{b}+x\right )}{a+b x} \, dx,x,\sinh ^{-1}(c+d x)\right )}{16 b^2 d}+\frac {\left (3 e^4 \cosh \left (\frac {3 a}{b}\right )\right ) \operatorname {Subst}\left (\int \frac {\cosh \left (\frac {3 a}{b}+3 x\right )}{a+b x} \, dx,x,\sinh ^{-1}(c+d x)\right )}{2 b^2 d}-\frac {\left (75 e^4 \cosh \left (\frac {3 a}{b}\right )\right ) \operatorname {Subst}\left (\int \frac {\cosh \left (\frac {3 a}{b}+3 x\right )}{a+b x} \, dx,x,\sinh ^{-1}(c+d x)\right )}{32 b^2 d}+\frac {\left (25 e^4 \cosh \left (\frac {5 a}{b}\right )\right ) \operatorname {Subst}\left (\int \frac {\cosh \left (\frac {5 a}{b}+5 x\right )}{a+b x} \, dx,x,\sinh ^{-1}(c+d x)\right )}{32 b^2 d}+\frac {\left (3 e^4 \sinh \left (\frac {a}{b}\right )\right ) \operatorname {Subst}\left (\int \frac {\sinh \left (\frac {a}{b}+x\right )}{a+b x} \, dx,x,\sinh ^{-1}(c+d x)\right )}{2 b^2 d}-\frac {\left (25 e^4 \sinh \left (\frac {a}{b}\right )\right ) \operatorname {Subst}\left (\int \frac {\sinh \left (\frac {a}{b}+x\right )}{a+b x} \, dx,x,\sinh ^{-1}(c+d x)\right )}{16 b^2 d}-\frac {\left (3 e^4 \sinh \left (\frac {3 a}{b}\right )\right ) \operatorname {Subst}\left (\int \frac {\sinh \left (\frac {3 a}{b}+3 x\right )}{a+b x} \, dx,x,\sinh ^{-1}(c+d x)\right )}{2 b^2 d}+\frac {\left (75 e^4 \sinh \left (\frac {3 a}{b}\right )\right ) \operatorname {Subst}\left (\int \frac {\sinh \left (\frac {3 a}{b}+3 x\right )}{a+b x} \, dx,x,\sinh ^{-1}(c+d x)\right )}{32 b^2 d}-\frac {\left (25 e^4 \sinh \left (\frac {5 a}{b}\right )\right ) \operatorname {Subst}\left (\int \frac {\sinh \left (\frac {5 a}{b}+5 x\right )}{a+b x} \, dx,x,\sinh ^{-1}(c+d x)\right )}{32 b^2 d}\\ &=-\frac {e^4 (c+d x)^4 \sqrt {1+(c+d x)^2}}{2 b d \left (a+b \sinh ^{-1}(c+d x)\right )^2}-\frac {2 e^4 (c+d x)^3}{b^2 d \left (a+b \sinh ^{-1}(c+d x)\right )}-\frac {5 e^4 (c+d x)^5}{2 b^2 d \left (a+b \sinh ^{-1}(c+d x)\right )}+\frac {e^4 \cosh \left (\frac {a}{b}\right ) \text {Chi}\left (\frac {a}{b}+\sinh ^{-1}(c+d x)\right )}{16 b^3 d}-\frac {27 e^4 \cosh \left (\frac {3 a}{b}\right ) \text {Chi}\left (\frac {3 a}{b}+3 \sinh ^{-1}(c+d x)\right )}{32 b^3 d}+\frac {25 e^4 \cosh \left (\frac {5 a}{b}\right ) \text {Chi}\left (\frac {5 a}{b}+5 \sinh ^{-1}(c+d x)\right )}{32 b^3 d}-\frac {e^4 \sinh \left (\frac {a}{b}\right ) \text {Shi}\left (\frac {a}{b}+\sinh ^{-1}(c+d x)\right )}{16 b^3 d}+\frac {27 e^4 \sinh \left (\frac {3 a}{b}\right ) \text {Shi}\left (\frac {3 a}{b}+3 \sinh ^{-1}(c+d x)\right )}{32 b^3 d}-\frac {25 e^4 \sinh \left (\frac {5 a}{b}\right ) \text {Shi}\left (\frac {5 a}{b}+5 \sinh ^{-1}(c+d x)\right )}{32 b^3 d}\\ \end {align*}

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Mathematica [A]  time = 1.25, size = 316, normalized size = 0.99 \[ \frac {e^4 \left (-\frac {16 b^2 \sqrt {(c+d x)^2+1} (c+d x)^4}{\left (a+b \sinh ^{-1}(c+d x)\right )^2}+48 \left (-\cosh \left (\frac {a}{b}\right ) \text {Chi}\left (\frac {a}{b}+\sinh ^{-1}(c+d x)\right )+\cosh \left (\frac {3 a}{b}\right ) \text {Chi}\left (3 \left (\frac {a}{b}+\sinh ^{-1}(c+d x)\right )\right )+\sinh \left (\frac {a}{b}\right ) \text {Shi}\left (\frac {a}{b}+\sinh ^{-1}(c+d x)\right )-\sinh \left (\frac {3 a}{b}\right ) \text {Shi}\left (3 \left (\frac {a}{b}+\sinh ^{-1}(c+d x)\right )\right )\right )+25 \left (2 \cosh \left (\frac {a}{b}\right ) \text {Chi}\left (\frac {a}{b}+\sinh ^{-1}(c+d x)\right )-3 \cosh \left (\frac {3 a}{b}\right ) \text {Chi}\left (3 \left (\frac {a}{b}+\sinh ^{-1}(c+d x)\right )\right )+\cosh \left (\frac {5 a}{b}\right ) \text {Chi}\left (5 \left (\frac {a}{b}+\sinh ^{-1}(c+d x)\right )\right )-2 \sinh \left (\frac {a}{b}\right ) \text {Shi}\left (\frac {a}{b}+\sinh ^{-1}(c+d x)\right )+3 \sinh \left (\frac {3 a}{b}\right ) \text {Shi}\left (3 \left (\frac {a}{b}+\sinh ^{-1}(c+d x)\right )\right )-\sinh \left (\frac {5 a}{b}\right ) \text {Shi}\left (5 \left (\frac {a}{b}+\sinh ^{-1}(c+d x)\right )\right )\right )+\frac {16 b \left (-5 (c+d x)^5-4 (c+d x)^3\right )}{a+b \sinh ^{-1}(c+d x)}\right )}{32 b^3 d} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(e^4*((-16*b^2*(c + d*x)^4*Sqrt[1 + (c + d*x)^2])/(a + b*ArcSinh[c + d*x])^2 + (16*b*(-4*(c + d*x)^3 - 5*(c +
d*x)^5))/(a + b*ArcSinh[c + d*x]) + 48*(-(Cosh[a/b]*CoshIntegral[a/b + ArcSinh[c + d*x]]) + Cosh[(3*a)/b]*Cosh
Integral[3*(a/b + ArcSinh[c + d*x])] + Sinh[a/b]*SinhIntegral[a/b + ArcSinh[c + d*x]] - Sinh[(3*a)/b]*SinhInte
gral[3*(a/b + ArcSinh[c + d*x])]) + 25*(2*Cosh[a/b]*CoshIntegral[a/b + ArcSinh[c + d*x]] - 3*Cosh[(3*a)/b]*Cos
hIntegral[3*(a/b + ArcSinh[c + d*x])] + Cosh[(5*a)/b]*CoshIntegral[5*(a/b + ArcSinh[c + d*x])] - 2*Sinh[a/b]*S
inhIntegral[a/b + ArcSinh[c + d*x]] + 3*Sinh[(3*a)/b]*SinhIntegral[3*(a/b + ArcSinh[c + d*x])] - Sinh[(5*a)/b]
*SinhIntegral[5*(a/b + ArcSinh[c + d*x])])))/(32*b^3*d)

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fricas [F]  time = 0.61, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {d^{4} e^{4} x^{4} + 4 \, c d^{3} e^{4} x^{3} + 6 \, c^{2} d^{2} e^{4} x^{2} + 4 \, c^{3} d e^{4} x + c^{4} e^{4}}{b^{3} \operatorname {arsinh}\left (d x + c\right )^{3} + 3 \, a b^{2} \operatorname {arsinh}\left (d x + c\right )^{2} + 3 \, a^{2} b \operatorname {arsinh}\left (d x + c\right ) + a^{3}}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integral((d^4*e^4*x^4 + 4*c*d^3*e^4*x^3 + 6*c^2*d^2*e^4*x^2 + 4*c^3*d*e^4*x + c^4*e^4)/(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), x)

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giac [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {{\left (d e x + c e\right )}^{4}}{{\left (b \operatorname {arsinh}\left (d x + c\right ) + a\right )}^{3}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maple [B]  time = 0.39, size = 896, normalized size = 2.80 \[ \frac {-\frac {\left (16 \left (d x +c \right )^{5}-16 \left (d x +c \right )^{4} \sqrt {1+\left (d x +c \right )^{2}}+20 \left (d x +c \right )^{3}-12 \left (d x +c \right )^{2} \sqrt {1+\left (d x +c \right )^{2}}+5 d x +5 c -\sqrt {1+\left (d x +c \right )^{2}}\right ) e^{4} \left (5 b \arcsinh \left (d x +c \right )+5 a -b \right )}{64 b^{2} \left (b^{2} \arcsinh \left (d x +c \right )^{2}+2 a b \arcsinh \left (d x +c \right )+a^{2}\right )}-\frac {25 e^{4} {\mathrm e}^{\frac {5 a}{b}} \Ei \left (1, 5 \arcsinh \left (d x +c \right )+\frac {5 a}{b}\right )}{64 b^{3}}+\frac {3 \left (-4 \left (d x +c \right )^{2} \sqrt {1+\left (d x +c \right )^{2}}+4 \left (d x +c \right )^{3}-\sqrt {1+\left (d x +c \right )^{2}}+3 d x +3 c \right ) e^{4} \left (3 b \arcsinh \left (d x +c \right )+3 a -b \right )}{64 b^{2} \left (b^{2} \arcsinh \left (d x +c \right )^{2}+2 a b \arcsinh \left (d x +c \right )+a^{2}\right )}+\frac {27 e^{4} {\mathrm e}^{\frac {3 a}{b}} \Ei \left (1, 3 \arcsinh \left (d x +c \right )+\frac {3 a}{b}\right )}{64 b^{3}}-\frac {\left (-\sqrt {1+\left (d x +c \right )^{2}}+d x +c \right ) e^{4} \left (b \arcsinh \left (d x +c \right )+a -b \right )}{32 b^{2} \left (b^{2} \arcsinh \left (d x +c \right )^{2}+2 a b \arcsinh \left (d x +c \right )+a^{2}\right )}-\frac {e^{4} {\mathrm e}^{\frac {a}{b}} \Ei \left (1, \arcsinh \left (d x +c \right )+\frac {a}{b}\right )}{32 b^{3}}-\frac {e^{4} \left (d x +c +\sqrt {1+\left (d x +c \right )^{2}}\right )}{32 b \left (a +b \arcsinh \left (d x +c \right )\right )^{2}}-\frac {e^{4} \left (d x +c +\sqrt {1+\left (d x +c \right )^{2}}\right )}{32 b^{2} \left (a +b \arcsinh \left (d x +c \right )\right )}-\frac {e^{4} {\mathrm e}^{-\frac {a}{b}} \Ei \left (1, -\arcsinh \left (d x +c \right )-\frac {a}{b}\right )}{32 b^{3}}+\frac {3 e^{4} \left (4 \left (d x +c \right )^{3}+3 d x +3 c +4 \left (d x +c \right )^{2} \sqrt {1+\left (d x +c \right )^{2}}+\sqrt {1+\left (d x +c \right )^{2}}\right )}{64 b \left (a +b \arcsinh \left (d x +c \right )\right )^{2}}+\frac {9 e^{4} \left (4 \left (d x +c \right )^{3}+3 d x +3 c +4 \left (d x +c \right )^{2} \sqrt {1+\left (d x +c \right )^{2}}+\sqrt {1+\left (d x +c \right )^{2}}\right )}{64 b^{2} \left (a +b \arcsinh \left (d x +c \right )\right )}+\frac {27 e^{4} {\mathrm e}^{-\frac {3 a}{b}} \Ei \left (1, -3 \arcsinh \left (d x +c \right )-\frac {3 a}{b}\right )}{64 b^{3}}-\frac {e^{4} \left (16 \left (d x +c \right )^{5}+20 \left (d x +c \right )^{3}+16 \left (d x +c \right )^{4} \sqrt {1+\left (d x +c \right )^{2}}+5 d x +5 c +12 \left (d x +c \right )^{2} \sqrt {1+\left (d x +c \right )^{2}}+\sqrt {1+\left (d x +c \right )^{2}}\right )}{64 b \left (a +b \arcsinh \left (d x +c \right )\right )^{2}}-\frac {5 e^{4} \left (16 \left (d x +c \right )^{5}+20 \left (d x +c \right )^{3}+16 \left (d x +c \right )^{4} \sqrt {1+\left (d x +c \right )^{2}}+5 d x +5 c +12 \left (d x +c \right )^{2} \sqrt {1+\left (d x +c \right )^{2}}+\sqrt {1+\left (d x +c \right )^{2}}\right )}{64 b^{2} \left (a +b \arcsinh \left (d x +c \right )\right )}-\frac {25 e^{4} {\mathrm e}^{-\frac {5 a}{b}} \Ei \left (1, -5 \arcsinh \left (d x +c \right )-\frac {5 a}{b}\right )}{64 b^{3}}}{d} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/d*(-1/64*(16*(d*x+c)^5-16*(d*x+c)^4*(1+(d*x+c)^2)^(1/2)+20*(d*x+c)^3-12*(d*x+c)^2*(1+(d*x+c)^2)^(1/2)+5*d*x+
5*c-(1+(d*x+c)^2)^(1/2))*e^4*(5*b*arcsinh(d*x+c)+5*a-b)/b^2/(b^2*arcsinh(d*x+c)^2+2*a*b*arcsinh(d*x+c)+a^2)-25
/64*e^4/b^3*exp(5*a/b)*Ei(1,5*arcsinh(d*x+c)+5*a/b)+3/64*(-4*(d*x+c)^2*(1+(d*x+c)^2)^(1/2)+4*(d*x+c)^3-(1+(d*x
+c)^2)^(1/2)+3*d*x+3*c)*e^4*(3*b*arcsinh(d*x+c)+3*a-b)/b^2/(b^2*arcsinh(d*x+c)^2+2*a*b*arcsinh(d*x+c)+a^2)+27/
64*e^4/b^3*exp(3*a/b)*Ei(1,3*arcsinh(d*x+c)+3*a/b)-1/32*(-(1+(d*x+c)^2)^(1/2)+d*x+c)*e^4*(b*arcsinh(d*x+c)+a-b
)/b^2/(b^2*arcsinh(d*x+c)^2+2*a*b*arcsinh(d*x+c)+a^2)-1/32*e^4/b^3*exp(a/b)*Ei(1,arcsinh(d*x+c)+a/b)-1/32/b*e^
4*(d*x+c+(1+(d*x+c)^2)^(1/2))/(a+b*arcsinh(d*x+c))^2-1/32/b^2*e^4*(d*x+c+(1+(d*x+c)^2)^(1/2))/(a+b*arcsinh(d*x
+c))-1/32/b^3*e^4*exp(-a/b)*Ei(1,-arcsinh(d*x+c)-a/b)+3/64/b*e^4*(4*(d*x+c)^3+3*d*x+3*c+4*(d*x+c)^2*(1+(d*x+c)
^2)^(1/2)+(1+(d*x+c)^2)^(1/2))/(a+b*arcsinh(d*x+c))^2+9/64/b^2*e^4*(4*(d*x+c)^3+3*d*x+3*c+4*(d*x+c)^2*(1+(d*x+
c)^2)^(1/2)+(1+(d*x+c)^2)^(1/2))/(a+b*arcsinh(d*x+c))+27/64/b^3*e^4*exp(-3*a/b)*Ei(1,-3*arcsinh(d*x+c)-3*a/b)-
1/64/b*e^4*(16*(d*x+c)^5+20*(d*x+c)^3+16*(d*x+c)^4*(1+(d*x+c)^2)^(1/2)+5*d*x+5*c+12*(d*x+c)^2*(1+(d*x+c)^2)^(1
/2)+(1+(d*x+c)^2)^(1/2))/(a+b*arcsinh(d*x+c))^2-5/64/b^2*e^4*(16*(d*x+c)^5+20*(d*x+c)^3+16*(d*x+c)^4*(1+(d*x+c
)^2)^(1/2)+5*d*x+5*c+12*(d*x+c)^2*(1+(d*x+c)^2)^(1/2)+(1+(d*x+c)^2)^(1/2))/(a+b*arcsinh(d*x+c))-25/64/b^3*e^4*
exp(-5*a/b)*Ei(1,-5*arcsinh(d*x+c)-5*a/b))

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maxima [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \text {result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/2*((5*a*d^11*e^4 + b*d^11*e^4)*x^11 + 11*(5*a*c*d^10*e^4 + b*c*d^10*e^4)*x^10 + (5*(55*c^2*d^9*e^4 + 3*d^9*
e^4)*a + (55*c^2*d^9*e^4 + 3*d^9*e^4)*b)*x^9 + 3*(5*(55*c^3*d^8*e^4 + 9*c*d^8*e^4)*a + (55*c^3*d^8*e^4 + 9*c*d
^8*e^4)*b)*x^8 + 3*(5*(110*c^4*d^7*e^4 + 36*c^2*d^7*e^4 + d^7*e^4)*a + (110*c^4*d^7*e^4 + 36*c^2*d^7*e^4 + d^7
*e^4)*b)*x^7 + 21*(5*(22*c^5*d^6*e^4 + 12*c^3*d^6*e^4 + c*d^6*e^4)*a + (22*c^5*d^6*e^4 + 12*c^3*d^6*e^4 + c*d^
6*e^4)*b)*x^6 + (5*(462*c^6*d^5*e^4 + 378*c^4*d^5*e^4 + 63*c^2*d^5*e^4 + d^5*e^4)*a + (462*c^6*d^5*e^4 + 378*c
^4*d^5*e^4 + 63*c^2*d^5*e^4 + d^5*e^4)*b)*x^5 + (5*(330*c^7*d^4*e^4 + 378*c^5*d^4*e^4 + 105*c^3*d^4*e^4 + 5*c*
d^4*e^4)*a + (330*c^7*d^4*e^4 + 378*c^5*d^4*e^4 + 105*c^3*d^4*e^4 + 5*c*d^4*e^4)*b)*x^4 + (5*(165*c^8*d^3*e^4
+ 252*c^6*d^3*e^4 + 105*c^4*d^3*e^4 + 10*c^2*d^3*e^4)*a + (165*c^8*d^3*e^4 + 252*c^6*d^3*e^4 + 105*c^4*d^3*e^4
+ 10*c^2*d^3*e^4)*b)*x^3 + (5*(55*c^9*d^2*e^4 + 108*c^7*d^2*e^4 + 63*c^5*d^2*e^4 + 10*c^3*d^2*e^4)*a + (55*c^
9*d^2*e^4 + 108*c^7*d^2*e^4 + 63*c^5*d^2*e^4 + 10*c^3*d^2*e^4)*b)*x^2 + ((5*a*d^8*e^4 + b*d^8*e^4)*x^8 + 8*(5*
a*c*d^7*e^4 + b*c*d^7*e^4)*x^7 + (4*(35*c^2*d^6*e^4 + 2*d^6*e^4)*a + (28*c^2*d^6*e^4 + d^6*e^4)*b)*x^6 + 2*(4*
(35*c^3*d^5*e^4 + 6*c*d^5*e^4)*a + (28*c^3*d^5*e^4 + 3*c*d^5*e^4)*b)*x^5 + ((350*c^4*d^4*e^4 + 120*c^2*d^4*e^4
+ 3*d^4*e^4)*a + 5*(14*c^4*d^4*e^4 + 3*c^2*d^4*e^4)*b)*x^4 + 4*((70*c^5*d^3*e^4 + 40*c^3*d^3*e^4 + 3*c*d^3*e^
4)*a + (14*c^5*d^3*e^4 + 5*c^3*d^3*e^4)*b)*x^3 + (2*(70*c^6*d^2*e^4 + 60*c^4*d^2*e^4 + 9*c^2*d^2*e^4)*a + (28*
c^6*d^2*e^4 + 15*c^4*d^2*e^4)*b)*x^2 + (5*c^8*e^4 + 8*c^6*e^4 + 3*c^4*e^4)*a + (c^8*e^4 + c^6*e^4)*b + 2*(2*(1
0*c^7*d*e^4 + 12*c^5*d*e^4 + 3*c^3*d*e^4)*a + (4*c^7*d*e^4 + 3*c^5*d*e^4)*b)*x)*(d^2*x^2 + 2*c*d*x + c^2 + 1)^
(3/2) + (3*(5*a*d^9*e^4 + b*d^9*e^4)*x^9 + 27*(5*a*c*d^8*e^4 + b*c*d^8*e^4)*x^8 + ((540*c^2*d^7*e^4 + 31*d^7*e
^4)*a + (108*c^2*d^7*e^4 + 5*d^7*e^4)*b)*x^7 + 7*((180*c^3*d^6*e^4 + 31*c*d^6*e^4)*a + (36*c^3*d^6*e^4 + 5*c*d
^6*e^4)*b)*x^6 + ((1890*c^4*d^5*e^4 + 651*c^2*d^5*e^4 + 20*d^5*e^4)*a + (378*c^4*d^5*e^4 + 105*c^2*d^5*e^4 + 2
*d^5*e^4)*b)*x^5 + (5*(378*c^5*d^4*e^4 + 217*c^3*d^4*e^4 + 20*c*d^4*e^4)*a + (378*c^5*d^4*e^4 + 175*c^3*d^4*e^
4 + 10*c*d^4*e^4)*b)*x^4 + ((1260*c^6*d^3*e^4 + 1085*c^4*d^3*e^4 + 200*c^2*d^3*e^4 + 4*d^3*e^4)*a + (252*c^6*d
^3*e^4 + 175*c^4*d^3*e^4 + 20*c^2*d^3*e^4)*b)*x^3 + ((540*c^7*d^2*e^4 + 651*c^5*d^2*e^4 + 200*c^3*d^2*e^4 + 12
*c*d^2*e^4)*a + (108*c^7*d^2*e^4 + 105*c^5*d^2*e^4 + 20*c^3*d^2*e^4)*b)*x^2 + (15*c^9*e^4 + 31*c^7*e^4 + 20*c^
5*e^4 + 4*c^3*e^4)*a + (3*c^9*e^4 + 5*c^7*e^4 + 2*c^5*e^4)*b + ((135*c^8*d*e^4 + 217*c^6*d*e^4 + 100*c^4*d*e^4
+ 12*c^2*d*e^4)*a + (27*c^8*d*e^4 + 35*c^6*d*e^4 + 10*c^4*d*e^4)*b)*x)*(d^2*x^2 + 2*c*d*x + c^2 + 1) + 5*(c^1
1*e^4 + 3*c^9*e^4 + 3*c^7*e^4 + c^5*e^4)*a + (c^11*e^4 + 3*c^9*e^4 + 3*c^7*e^4 + c^5*e^4)*b + (5*(11*c^10*d*e^
4 + 27*c^8*d*e^4 + 21*c^6*d*e^4 + 5*c^4*d*e^4)*a + (11*c^10*d*e^4 + 27*c^8*d*e^4 + 21*c^6*d*e^4 + 5*c^4*d*e^4)
*b)*x + (5*b*d^11*e^4*x^11 + 55*b*c*d^10*e^4*x^10 + 5*(55*c^2*d^9*e^4 + 3*d^9*e^4)*b*x^9 + 15*(55*c^3*d^8*e^4
+ 9*c*d^8*e^4)*b*x^8 + 15*(110*c^4*d^7*e^4 + 36*c^2*d^7*e^4 + d^7*e^4)*b*x^7 + 105*(22*c^5*d^6*e^4 + 12*c^3*d^
6*e^4 + c*d^6*e^4)*b*x^6 + 5*(462*c^6*d^5*e^4 + 378*c^4*d^5*e^4 + 63*c^2*d^5*e^4 + d^5*e^4)*b*x^5 + 5*(330*c^7
*d^4*e^4 + 378*c^5*d^4*e^4 + 105*c^3*d^4*e^4 + 5*c*d^4*e^4)*b*x^4 + 5*(165*c^8*d^3*e^4 + 252*c^6*d^3*e^4 + 105
*c^4*d^3*e^4 + 10*c^2*d^3*e^4)*b*x^3 + 5*(55*c^9*d^2*e^4 + 108*c^7*d^2*e^4 + 63*c^5*d^2*e^4 + 10*c^3*d^2*e^4)*
b*x^2 + 5*(11*c^10*d*e^4 + 27*c^8*d*e^4 + 21*c^6*d*e^4 + 5*c^4*d*e^4)*b*x + (5*b*d^8*e^4*x^8 + 40*b*c*d^7*e^4*
x^7 + 4*(35*c^2*d^6*e^4 + 2*d^6*e^4)*b*x^6 + 8*(35*c^3*d^5*e^4 + 6*c*d^5*e^4)*b*x^5 + (350*c^4*d^4*e^4 + 120*c
^2*d^4*e^4 + 3*d^4*e^4)*b*x^4 + 4*(70*c^5*d^3*e^4 + 40*c^3*d^3*e^4 + 3*c*d^3*e^4)*b*x^3 + 2*(70*c^6*d^2*e^4 +
60*c^4*d^2*e^4 + 9*c^2*d^2*e^4)*b*x^2 + 4*(10*c^7*d*e^4 + 12*c^5*d*e^4 + 3*c^3*d*e^4)*b*x + (5*c^8*e^4 + 8*c^6
*e^4 + 3*c^4*e^4)*b)*(d^2*x^2 + 2*c*d*x + c^2 + 1)^(3/2) + (15*b*d^9*e^4*x^9 + 135*b*c*d^8*e^4*x^8 + (540*c^2*
d^7*e^4 + 31*d^7*e^4)*b*x^7 + 7*(180*c^3*d^6*e^4 + 31*c*d^6*e^4)*b*x^6 + (1890*c^4*d^5*e^4 + 651*c^2*d^5*e^4 +
20*d^5*e^4)*b*x^5 + 5*(378*c^5*d^4*e^4 + 217*c^3*d^4*e^4 + 20*c*d^4*e^4)*b*x^4 + (1260*c^6*d^3*e^4 + 1085*c^4
*d^3*e^4 + 200*c^2*d^3*e^4 + 4*d^3*e^4)*b*x^3 + (540*c^7*d^2*e^4 + 651*c^5*d^2*e^4 + 200*c^3*d^2*e^4 + 12*c*d^
2*e^4)*b*x^2 + (135*c^8*d*e^4 + 217*c^6*d*e^4 + 100*c^4*d*e^4 + 12*c^2*d*e^4)*b*x + (15*c^9*e^4 + 31*c^7*e^4 +
20*c^5*e^4 + 4*c^3*e^4)*b)*(d^2*x^2 + 2*c*d*x + c^2 + 1) + 5*(c^11*e^4 + 3*c^9*e^4 + 3*c^7*e^4 + c^5*e^4)*b +
(15*b*d^10*e^4*x^10 + 150*b*c*d^9*e^4*x^9 + (675*c^2*d^8*e^4 + 38*d^8*e^4)*b*x^8 + 8*(225*c^3*d^7*e^4 + 38*c*
d^7*e^4)*b*x^7 + 2*(1575*c^4*d^6*e^4 + 532*c^2*d^6*e^4 + 16*d^6*e^4)*b*x^6 + 4*(945*c^5*d^5*e^4 + 532*c^3*d^5*
e^4 + 48*c*d^5*e^4)*b*x^5 + (3150*c^6*d^4*e^4 + 2660*c^4*d^4*e^4 + 480*c^2*d^4*e^4 + 9*d^4*e^4)*b*x^4 + 4*(450
*c^7*d^3*e^4 + 532*c^5*d^3*e^4 + 160*c^3*d^3*e^4 + 9*c*d^3*e^4)*b*x^3 + (675*c^8*d^2*e^4 + 1064*c^6*d^2*e^4 +
480*c^4*d^2*e^4 + 54*c^2*d^2*e^4)*b*x^2 + 2*(75*c^9*d*e^4 + 152*c^7*d*e^4 + 96*c^5*d*e^4 + 18*c^3*d*e^4)*b*x +
(15*c^10*e^4 + 38*c^8*e^4 + 32*c^6*e^4 + 9*c^4*e^4)*b)*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*(5*a*d^10*e^4 + b*d^10*e^4)*x^10 + 30*(5*a*c*d^9*e^4 + b*c*d^9*e^4)*x^9 + (
(675*c^2*d^8*e^4 + 38*d^8*e^4)*a + (135*c^2*d^8*e^4 + 7*d^8*e^4)*b)*x^8 + 8*((225*c^3*d^7*e^4 + 38*c*d^7*e^4)*
a + (45*c^3*d^7*e^4 + 7*c*d^7*e^4)*b)*x^7 + (2*(1575*c^4*d^6*e^4 + 532*c^2*d^6*e^4 + 16*d^6*e^4)*a + (630*c^4*
d^6*e^4 + 196*c^2*d^6*e^4 + 5*d^6*e^4)*b)*x^6 + 2*(2*(945*c^5*d^5*e^4 + 532*c^3*d^5*e^4 + 48*c*d^5*e^4)*a + (3
78*c^5*d^5*e^4 + 196*c^3*d^5*e^4 + 15*c*d^5*e^4)*b)*x^5 + ((3150*c^6*d^4*e^4 + 2660*c^4*d^4*e^4 + 480*c^2*d^4*
e^4 + 9*d^4*e^4)*a + (630*c^6*d^4*e^4 + 490*c^4*d^4*e^4 + 75*c^2*d^4*e^4 + d^4*e^4)*b)*x^4 + 4*((450*c^7*d^3*e
^4 + 532*c^5*d^3*e^4 + 160*c^3*d^3*e^4 + 9*c*d^3*e^4)*a + (90*c^7*d^3*e^4 + 98*c^5*d^3*e^4 + 25*c^3*d^3*e^4 +
c*d^3*e^4)*b)*x^3 + ((675*c^8*d^2*e^4 + 1064*c^6*d^2*e^4 + 480*c^4*d^2*e^4 + 54*c^2*d^2*e^4)*a + (135*c^8*d^2*
e^4 + 196*c^6*d^2*e^4 + 75*c^4*d^2*e^4 + 6*c^2*d^2*e^4)*b)*x^2 + (15*c^10*e^4 + 38*c^8*e^4 + 32*c^6*e^4 + 9*c^
4*e^4)*a + (3*c^10*e^4 + 7*c^8*e^4 + 5*c^6*e^4 + c^4*e^4)*b + 2*((75*c^9*d*e^4 + 152*c^7*d*e^4 + 96*c^5*d*e^4
+ 18*c^3*d*e^4)*a + (15*c^9*d*e^4 + 28*c^7*d*e^4 + 15*c^5*d*e^4 + 2*c^3*d*e^4)*b)*x)*sqrt(d^2*x^2 + 2*c*d*x +
c^2 + 1))/(a^2*b^2*d^7*x^6 + 6*a^2*b^2*c*d^6*x^5 + 3*(5*c^2*d^5 + d^5)*a^2*b^2*x^4 + 4*(5*c^3*d^4 + 3*c*d^4)*a
^2*b^2*x^3 + 3*(5*c^4*d^3 + 6*c^2*d^3 + d^3)*a^2*b^2*x^2 + 6*(c^5*d^2 + 2*c^3*d^2 + c*d^2)*a^2*b^2*x + (c^6*d
+ 3*c^4*d + 3*c^2*d + d)*a^2*b^2 + (b^4*d^7*x^6 + 6*b^4*c*d^6*x^5 + 3*(5*c^2*d^5 + d^5)*b^4*x^4 + 4*(5*c^3*d^4
+ 3*c*d^4)*b^4*x^3 + 3*(5*c^4*d^3 + 6*c^2*d^3 + d^3)*b^4*x^2 + 6*(c^5*d^2 + 2*c^3*d^2 + c*d^2)*b^4*x + (c^6*d
+ 3*c^4*d + 3*c^2*d + d)*b^4 + (b^4*d^4*x^3 + 3*b^4*c*d^3*x^2 + 3*b^4*c^2*d^2*x + b^4*c^3*d)*(d^2*x^2 + 2*c*d
*x + c^2 + 1)^(3/2) + 3*(b^4*d^5*x^4 + 4*b^4*c*d^4*x^3 + (6*c^2*d^3 + d^3)*b^4*x^2 + 2*(2*c^3*d^2 + c*d^2)*b^4
*x + (c^4*d + c^2*d)*b^4)*(d^2*x^2 + 2*c*d*x + c^2 + 1) + 3*(b^4*d^6*x^5 + 5*b^4*c*d^5*x^4 + 2*(5*c^2*d^4 + d^
4)*b^4*x^3 + 2*(5*c^3*d^3 + 3*c*d^3)*b^4*x^2 + (5*c^4*d^2 + 6*c^2*d^2 + d^2)*b^4*x + (c^5*d + 2*c^3*d + c*d)*b
^4)*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))^2 + (a^2*b^2*d^4*x^3 +
3*a^2*b^2*c*d^3*x^2 + 3*a^2*b^2*c^2*d^2*x + a^2*b^2*c^3*d)*(d^2*x^2 + 2*c*d*x + c^2 + 1)^(3/2) + 3*(a^2*b^2*d
^5*x^4 + 4*a^2*b^2*c*d^4*x^3 + (6*c^2*d^3 + d^3)*a^2*b^2*x^2 + 2*(2*c^3*d^2 + c*d^2)*a^2*b^2*x + (c^4*d + c^2*
d)*a^2*b^2)*(d^2*x^2 + 2*c*d*x + c^2 + 1) + 2*(a*b^3*d^7*x^6 + 6*a*b^3*c*d^6*x^5 + 3*(5*c^2*d^5 + d^5)*a*b^3*x
^4 + 4*(5*c^3*d^4 + 3*c*d^4)*a*b^3*x^3 + 3*(5*c^4*d^3 + 6*c^2*d^3 + d^3)*a*b^3*x^2 + 6*(c^5*d^2 + 2*c^3*d^2 +
c*d^2)*a*b^3*x + (c^6*d + 3*c^4*d + 3*c^2*d + d)*a*b^3 + (a*b^3*d^4*x^3 + 3*a*b^3*c*d^3*x^2 + 3*a*b^3*c^2*d^2*
x + a*b^3*c^3*d)*(d^2*x^2 + 2*c*d*x + c^2 + 1)^(3/2) + 3*(a*b^3*d^5*x^4 + 4*a*b^3*c*d^4*x^3 + (6*c^2*d^3 + d^3
)*a*b^3*x^2 + 2*(2*c^3*d^2 + c*d^2)*a*b^3*x + (c^4*d + c^2*d)*a*b^3)*(d^2*x^2 + 2*c*d*x + c^2 + 1) + 3*(a*b^3*
d^6*x^5 + 5*a*b^3*c*d^5*x^4 + 2*(5*c^2*d^4 + d^4)*a*b^3*x^3 + 2*(5*c^3*d^3 + 3*c*d^3)*a*b^3*x^2 + (5*c^4*d^2 +
6*c^2*d^2 + d^2)*a*b^3*x + (c^5*d + 2*c^3*d + c*d)*a*b^3)*sqrt(d^2*x^2 + 2*c*d*x + c^2 + 1))*log(d*x + c + sq
rt(d^2*x^2 + 2*c*d*x + c^2 + 1)) + 3*(a^2*b^2*d^6*x^5 + 5*a^2*b^2*c*d^5*x^4 + 2*(5*c^2*d^4 + d^4)*a^2*b^2*x^3
+ 2*(5*c^3*d^3 + 3*c*d^3)*a^2*b^2*x^2 + (5*c^4*d^2 + 6*c^2*d^2 + d^2)*a^2*b^2*x + (c^5*d + 2*c^3*d + c*d)*a^2*
b^2)*sqrt(d^2*x^2 + 2*c*d*x + c^2 + 1)) + integrate(1/2*(25*d^12*e^4*x^12 + 300*c*d^11*e^4*x^11 + 25*c^12*e^4
+ 100*c^10*e^4 + 150*c^8*e^4 + 50*(33*c^2*d^10*e^4 + 2*d^10*e^4)*x^10 + 100*c^6*e^4 + 500*(11*c^3*d^9*e^4 + 2*
c*d^9*e^4)*x^9 + 75*(165*c^4*d^8*e^4 + 60*c^2*d^8*e^4 + 2*d^8*e^4)*x^8 + 25*c^4*e^4 + 600*(33*c^5*d^7*e^4 + 20
*c^3*d^7*e^4 + 2*c*d^7*e^4)*x^7 + 100*(231*c^6*d^6*e^4 + 210*c^4*d^6*e^4 + 42*c^2*d^6*e^4 + d^6*e^4)*x^6 + 600
*(33*c^7*d^5*e^4 + 42*c^5*d^5*e^4 + 14*c^3*d^5*e^4 + c*d^5*e^4)*x^5 + 25*(495*c^8*d^4*e^4 + 840*c^6*d^4*e^4 +
420*c^4*d^4*e^4 + 60*c^2*d^4*e^4 + d^4*e^4)*x^4 + 100*(55*c^9*d^3*e^4 + 120*c^7*d^3*e^4 + 84*c^5*d^3*e^4 + 20*
c^3*d^3*e^4 + c*d^3*e^4)*x^3 + (25*d^8*e^4*x^8 + 200*c*d^7*e^4*x^7 + 25*c^8*e^4 + 24*c^6*e^4 + 3*c^4*e^4 + 4*(
175*c^2*d^6*e^4 + 6*d^6*e^4)*x^6 + 8*(175*c^3*d^5*e^4 + 18*c*d^5*e^4)*x^5 + (1750*c^4*d^4*e^4 + 360*c^2*d^4*e^
4 + 3*d^4*e^4)*x^4 + 4*(350*c^5*d^3*e^4 + 120*c^3*d^3*e^4 + 3*c*d^3*e^4)*x^3 + 2*(350*c^6*d^2*e^4 + 180*c^4*d^
2*e^4 + 9*c^2*d^2*e^4)*x^2 + 4*(50*c^7*d*e^4 + 36*c^5*d*e^4 + 3*c^3*d*e^4)*x)*(d^2*x^2 + 2*c*d*x + c^2 + 1)^2
+ 150*(11*c^10*d^2*e^4 + 30*c^8*d^2*e^4 + 28*c^6*d^2*e^4 + 10*c^4*d^2*e^4 + c^2*d^2*e^4)*x^2 + (100*d^9*e^4*x^
9 + 900*c*d^8*e^4*x^8 + 100*c^9*e^4 + 172*c^7*e^4 + 87*c^5*e^4 + 4*(900*c^2*d^7*e^4 + 43*d^7*e^4)*x^7 + 12*c^3
*e^4 + 28*(300*c^3*d^6*e^4 + 43*c*d^6*e^4)*x^6 + 3*(4200*c^4*d^5*e^4 + 1204*c^2*d^5*e^4 + 29*d^5*e^4)*x^5 + 5*
(2520*c^5*d^4*e^4 + 1204*c^3*d^4*e^4 + 87*c*d^4*e^4)*x^4 + 2*(4200*c^6*d^3*e^4 + 3010*c^4*d^3*e^4 + 435*c^2*d^
3*e^4 + 6*d^3*e^4)*x^3 + 6*(600*c^7*d^2*e^4 + 602*c^5*d^2*e^4 + 145*c^3*d^2*e^4 + 6*c*d^2*e^4)*x^2 + (900*c^8*
d*e^4 + 1204*c^6*d*e^4 + 435*c^4*d*e^4 + 36*c^2*d*e^4)*x)*(d^2*x^2 + 2*c*d*x + c^2 + 1)^(3/2) + 3*(50*d^10*e^4
*x^10 + 500*c*d^9*e^4*x^9 + 50*c^10*e^4 + 124*c^8*e^4 + 105*c^6*e^4 + 2*(1125*c^2*d^8*e^4 + 62*d^8*e^4)*x^8 +
35*c^4*e^4 + 16*(375*c^3*d^7*e^4 + 62*c*d^7*e^4)*x^7 + 7*(1500*c^4*d^6*e^4 + 496*c^2*d^6*e^4 + 15*d^6*e^4)*x^6
+ 4*c^2*e^4 + 14*(900*c^5*d^5*e^4 + 496*c^3*d^5*e^4 + 45*c*d^5*e^4)*x^5 + 35*(300*c^6*d^4*e^4 + 248*c^4*d^4*e
^4 + 45*c^2*d^4*e^4 + d^4*e^4)*x^4 + 4*(1500*c^7*d^3*e^4 + 1736*c^5*d^3*e^4 + 525*c^3*d^3*e^4 + 35*c*d^3*e^4)*
x^3 + (2250*c^8*d^2*e^4 + 3472*c^6*d^2*e^4 + 1575*c^4*d^2*e^4 + 210*c^2*d^2*e^4 + 4*d^2*e^4)*x^2 + 2*(250*c^9*
d*e^4 + 496*c^7*d*e^4 + 315*c^5*d*e^4 + 70*c^3*d*e^4 + 4*c*d*e^4)*x)*(d^2*x^2 + 2*c*d*x + c^2 + 1) + 100*(3*c^
11*d*e^4 + 10*c^9*d*e^4 + 12*c^7*d*e^4 + 6*c^5*d*e^4 + c^3*d*e^4)*x + (100*d^11*e^4*x^11 + 1100*c*d^10*e^4*x^1
0 + 100*c^11*e^4 + 324*c^9*e^4 + 381*c^7*e^4 + 4*(1375*c^2*d^9*e^4 + 81*d^9*e^4)*x^9 + 193*c^5*e^4 + 12*(1375*
c^3*d^8*e^4 + 243*c*d^8*e^4)*x^8 + 3*(11000*c^4*d^7*e^4 + 3888*c^2*d^7*e^4 + 127*d^7*e^4)*x^7 + 36*c^3*e^4 + 2
1*(2200*c^5*d^6*e^4 + 1296*c^3*d^6*e^4 + 127*c*d^6*e^4)*x^6 + (46200*c^6*d^5*e^4 + 40824*c^4*d^5*e^4 + 8001*c^
2*d^5*e^4 + 193*d^5*e^4)*x^5 + (33000*c^7*d^4*e^4 + 40824*c^5*d^4*e^4 + 13335*c^3*d^4*e^4 + 965*c*d^4*e^4)*x^4
+ (16500*c^8*d^3*e^4 + 27216*c^6*d^3*e^4 + 13335*c^4*d^3*e^4 + 1930*c^2*d^3*e^4 + 36*d^3*e^4)*x^3 + (5500*c^9
*d^2*e^4 + 11664*c^7*d^2*e^4 + 8001*c^5*d^2*e^4 + 1930*c^3*d^2*e^4 + 108*c*d^2*e^4)*x^2 + (1100*c^10*d*e^4 + 2
916*c^8*d*e^4 + 2667*c^6*d*e^4 + 965*c^4*d*e^4 + 108*c^2*d*e^4)*x)*sqrt(d^2*x^2 + 2*c*d*x + c^2 + 1))/(a*b^2*d
^8*x^8 + 8*a*b^2*c*d^7*x^7 + 4*(7*c^2*d^6 + d^6)*a*b^2*x^6 + 8*(7*c^3*d^5 + 3*c*d^5)*a*b^2*x^5 + 2*(35*c^4*d^4
+ 30*c^2*d^4 + 3*d^4)*a*b^2*x^4 + 8*(7*c^5*d^3 + 10*c^3*d^3 + 3*c*d^3)*a*b^2*x^3 + 4*(7*c^6*d^2 + 15*c^4*d^2
+ 9*c^2*d^2 + d^2)*a*b^2*x^2 + 8*(c^7*d + 3*c^5*d + 3*c^3*d + c*d)*a*b^2*x + (c^8 + 4*c^6 + 6*c^4 + 4*c^2 + 1)
*a*b^2 + (a*b^2*d^4*x^4 + 4*a*b^2*c*d^3*x^3 + 6*a*b^2*c^2*d^2*x^2 + 4*a*b^2*c^3*d*x + a*b^2*c^4)*(d^2*x^2 + 2*
c*d*x + c^2 + 1)^2 + 4*(a*b^2*d^5*x^5 + 5*a*b^2*c*d^4*x^4 + (10*c^2*d^3 + d^3)*a*b^2*x^3 + (10*c^3*d^2 + 3*c*d
^2)*a*b^2*x^2 + (5*c^4*d + 3*c^2*d)*a*b^2*x + (c^5 + c^3)*a*b^2)*(d^2*x^2 + 2*c*d*x + c^2 + 1)^(3/2) + 6*(a*b^
2*d^6*x^6 + 6*a*b^2*c*d^5*x^5 + (15*c^2*d^4 + 2*d^4)*a*b^2*x^4 + 4*(5*c^3*d^3 + 2*c*d^3)*a*b^2*x^3 + (15*c^4*d
^2 + 12*c^2*d^2 + d^2)*a*b^2*x^2 + 2*(3*c^5*d + 4*c^3*d + c*d)*a*b^2*x + (c^6 + 2*c^4 + c^2)*a*b^2)*(d^2*x^2 +
2*c*d*x + c^2 + 1) + (b^3*d^8*x^8 + 8*b^3*c*d^7*x^7 + 4*(7*c^2*d^6 + d^6)*b^3*x^6 + 8*(7*c^3*d^5 + 3*c*d^5)*b
^3*x^5 + 2*(35*c^4*d^4 + 30*c^2*d^4 + 3*d^4)*b^3*x^4 + 8*(7*c^5*d^3 + 10*c^3*d^3 + 3*c*d^3)*b^3*x^3 + 4*(7*c^6
*d^2 + 15*c^4*d^2 + 9*c^2*d^2 + d^2)*b^3*x^2 + 8*(c^7*d + 3*c^5*d + 3*c^3*d + c*d)*b^3*x + (c^8 + 4*c^6 + 6*c^
4 + 4*c^2 + 1)*b^3 + (b^3*d^4*x^4 + 4*b^3*c*d^3*x^3 + 6*b^3*c^2*d^2*x^2 + 4*b^3*c^3*d*x + b^3*c^4)*(d^2*x^2 +
2*c*d*x + c^2 + 1)^2 + 4*(b^3*d^5*x^5 + 5*b^3*c*d^4*x^4 + (10*c^2*d^3 + d^3)*b^3*x^3 + (10*c^3*d^2 + 3*c*d^2)*
b^3*x^2 + (5*c^4*d + 3*c^2*d)*b^3*x + (c^5 + c^3)*b^3)*(d^2*x^2 + 2*c*d*x + c^2 + 1)^(3/2) + 6*(b^3*d^6*x^6 +
6*b^3*c*d^5*x^5 + (15*c^2*d^4 + 2*d^4)*b^3*x^4 + 4*(5*c^3*d^3 + 2*c*d^3)*b^3*x^3 + (15*c^4*d^2 + 12*c^2*d^2 +
d^2)*b^3*x^2 + 2*(3*c^5*d + 4*c^3*d + c*d)*b^3*x + (c^6 + 2*c^4 + c^2)*b^3)*(d^2*x^2 + 2*c*d*x + c^2 + 1) + 4*
(b^3*d^7*x^7 + 7*b^3*c*d^6*x^6 + 3*(7*c^2*d^5 + d^5)*b^3*x^5 + 5*(7*c^3*d^4 + 3*c*d^4)*b^3*x^4 + (35*c^4*d^3 +
30*c^2*d^3 + 3*d^3)*b^3*x^3 + 3*(7*c^5*d^2 + 10*c^3*d^2 + 3*c*d^2)*b^3*x^2 + (7*c^6*d + 15*c^4*d + 9*c^2*d +
d)*b^3*x + (c^7 + 3*c^5 + 3*c^3 + c)*b^3)*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)) + 4*(a*b^2*d^7*x^7 + 7*a*b^2*c*d^6*x^6 + 3*(7*c^2*d^5 + d^5)*a*b^2*x^5 + 5*(7*c^3*d^4 + 3*c*d^
4)*a*b^2*x^4 + (35*c^4*d^3 + 30*c^2*d^3 + 3*d^3)*a*b^2*x^3 + 3*(7*c^5*d^2 + 10*c^3*d^2 + 3*c*d^2)*a*b^2*x^2 +
(7*c^6*d + 15*c^4*d + 9*c^2*d + d)*a*b^2*x + (c^7 + 3*c^5 + 3*c^3 + c)*a*b^2)*sqrt(d^2*x^2 + 2*c*d*x + c^2 + 1
)), x)

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mupad [F]  time = 0.00, size = -1, normalized size = -0.00 \[ \int \frac {{\left (c\,e+d\,e\,x\right )}^4}{{\left (a+b\,\mathrm {asinh}\left (c+d\,x\right )\right )}^3} \,d x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \[ e^{4} \left (\int \frac {c^{4}}{a^{3} + 3 a^{2} b \operatorname {asinh}{\left (c + d x \right )} + 3 a b^{2} \operatorname {asinh}^{2}{\left (c + d x \right )} + b^{3} \operatorname {asinh}^{3}{\left (c + d x \right )}}\, dx + \int \frac {d^{4} x^{4}}{a^{3} + 3 a^{2} b \operatorname {asinh}{\left (c + d x \right )} + 3 a b^{2} \operatorname {asinh}^{2}{\left (c + d x \right )} + b^{3} \operatorname {asinh}^{3}{\left (c + d x \right )}}\, dx + \int \frac {4 c d^{3} x^{3}}{a^{3} + 3 a^{2} b \operatorname {asinh}{\left (c + d x \right )} + 3 a b^{2} \operatorname {asinh}^{2}{\left (c + d x \right )} + b^{3} \operatorname {asinh}^{3}{\left (c + d x \right )}}\, dx + \int \frac {6 c^{2} d^{2} x^{2}}{a^{3} + 3 a^{2} b \operatorname {asinh}{\left (c + d x \right )} + 3 a b^{2} \operatorname {asinh}^{2}{\left (c + d x \right )} + b^{3} \operatorname {asinh}^{3}{\left (c + d x \right )}}\, dx + \int \frac {4 c^{3} d x}{a^{3} + 3 a^{2} b \operatorname {asinh}{\left (c + d x \right )} + 3 a b^{2} \operatorname {asinh}^{2}{\left (c + d x \right )} + b^{3} \operatorname {asinh}^{3}{\left (c + d x \right )}}\, dx\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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