3.2.37 \(\int (c e+d e x)^4 (a+b \sinh ^{-1}(c+d x))^3 \, dx\) [137]

Optimal. Leaf size=326 \[ \frac {16}{25} a b^2 e^4 x-\frac {298 b^3 e^4 \sqrt {1+(c+d x)^2}}{375 d}+\frac {76 b^3 e^4 \left (1+(c+d x)^2\right )^{3/2}}{1125 d}-\frac {6 b^3 e^4 \left (1+(c+d x)^2\right )^{5/2}}{625 d}+\frac {16 b^3 e^4 (c+d x) \sinh ^{-1}(c+d x)}{25 d}-\frac {8 b^2 e^4 (c+d x)^3 \left (a+b \sinh ^{-1}(c+d x)\right )}{75 d}+\frac {6 b^2 e^4 (c+d x)^5 \left (a+b \sinh ^{-1}(c+d x)\right )}{125 d}-\frac {8 b e^4 \sqrt {1+(c+d x)^2} \left (a+b \sinh ^{-1}(c+d x)\right )^2}{25 d}+\frac {4 b e^4 (c+d x)^2 \sqrt {1+(c+d x)^2} \left (a+b \sinh ^{-1}(c+d x)\right )^2}{25 d}-\frac {3 b e^4 (c+d x)^4 \sqrt {1+(c+d x)^2} \left (a+b \sinh ^{-1}(c+d x)\right )^2}{25 d}+\frac {e^4 (c+d x)^5 \left (a+b \sinh ^{-1}(c+d x)\right )^3}{5 d} \]

[Out]

16/25*a*b^2*e^4*x+76/1125*b^3*e^4*(1+(d*x+c)^2)^(3/2)/d-6/625*b^3*e^4*(1+(d*x+c)^2)^(5/2)/d+16/25*b^3*e^4*(d*x
+c)*arcsinh(d*x+c)/d-8/75*b^2*e^4*(d*x+c)^3*(a+b*arcsinh(d*x+c))/d+6/125*b^2*e^4*(d*x+c)^5*(a+b*arcsinh(d*x+c)
)/d+1/5*e^4*(d*x+c)^5*(a+b*arcsinh(d*x+c))^3/d-298/375*b^3*e^4*(1+(d*x+c)^2)^(1/2)/d-8/25*b*e^4*(a+b*arcsinh(d
*x+c))^2*(1+(d*x+c)^2)^(1/2)/d+4/25*b*e^4*(d*x+c)^2*(a+b*arcsinh(d*x+c))^2*(1+(d*x+c)^2)^(1/2)/d-3/25*b*e^4*(d
*x+c)^4*(a+b*arcsinh(d*x+c))^2*(1+(d*x+c)^2)^(1/2)/d

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Rubi [A]
time = 0.32, antiderivative size = 326, normalized size of antiderivative = 1.00, number of steps used = 17, number of rules used = 9, integrand size = 23, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.391, Rules used = {5859, 12, 5776, 5812, 5798, 5772, 267, 272, 45} \begin {gather*} \frac {6 b^2 e^4 (c+d x)^5 \left (a+b \sinh ^{-1}(c+d x)\right )}{125 d}-\frac {8 b^2 e^4 (c+d x)^3 \left (a+b \sinh ^{-1}(c+d x)\right )}{75 d}+\frac {16}{25} a b^2 e^4 x+\frac {e^4 (c+d x)^5 \left (a+b \sinh ^{-1}(c+d x)\right )^3}{5 d}-\frac {3 b e^4 \sqrt {(c+d x)^2+1} (c+d x)^4 \left (a+b \sinh ^{-1}(c+d x)\right )^2}{25 d}+\frac {4 b e^4 \sqrt {(c+d x)^2+1} (c+d x)^2 \left (a+b \sinh ^{-1}(c+d x)\right )^2}{25 d}-\frac {8 b e^4 \sqrt {(c+d x)^2+1} \left (a+b \sinh ^{-1}(c+d x)\right )^2}{25 d}-\frac {6 b^3 e^4 \left ((c+d x)^2+1\right )^{5/2}}{625 d}+\frac {76 b^3 e^4 \left ((c+d x)^2+1\right )^{3/2}}{1125 d}-\frac {298 b^3 e^4 \sqrt {(c+d x)^2+1}}{375 d}+\frac {16 b^3 e^4 (c+d x) \sinh ^{-1}(c+d x)}{25 d} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(16*a*b^2*e^4*x)/25 - (298*b^3*e^4*Sqrt[1 + (c + d*x)^2])/(375*d) + (76*b^3*e^4*(1 + (c + d*x)^2)^(3/2))/(1125
*d) - (6*b^3*e^4*(1 + (c + d*x)^2)^(5/2))/(625*d) + (16*b^3*e^4*(c + d*x)*ArcSinh[c + d*x])/(25*d) - (8*b^2*e^
4*(c + d*x)^3*(a + b*ArcSinh[c + d*x]))/(75*d) + (6*b^2*e^4*(c + d*x)^5*(a + b*ArcSinh[c + d*x]))/(125*d) - (8
*b*e^4*Sqrt[1 + (c + d*x)^2]*(a + b*ArcSinh[c + d*x])^2)/(25*d) + (4*b*e^4*(c + d*x)^2*Sqrt[1 + (c + d*x)^2]*(
a + b*ArcSinh[c + d*x])^2)/(25*d) - (3*b*e^4*(c + d*x)^4*Sqrt[1 + (c + d*x)^2]*(a + b*ArcSinh[c + d*x])^2)/(25
*d) + (e^4*(c + d*x)^5*(a + b*ArcSinh[c + d*x])^3)/(5*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 45

Int[((a_.) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_.), x_Symbol] :> Int[ExpandIntegrand[(a + b*x)^m*(c + d
*x)^n, x], x] /; FreeQ[{a, b, c, d, n}, x] && NeQ[b*c - a*d, 0] && IGtQ[m, 0] && ( !IntegerQ[n] || (EqQ[c, 0]
&& LeQ[7*m + 4*n + 4, 0]) || LtQ[9*m + 5*(n + 1), 0] || GtQ[m + n + 2, 0])

Rule 267

Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[(a + b*x^n)^(p + 1)/(b*n*(p + 1)), x] /; FreeQ
[{a, b, m, n, p}, x] && EqQ[m, n - 1] && NeQ[p, -1]

Rule 272

Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Dist[1/n, Subst[Int[x^(Simplify[(m + 1)/n] - 1)*(a
+ b*x)^p, x], x, x^n], x] /; FreeQ[{a, b, m, n, p}, x] && IntegerQ[Simplify[(m + 1)/n]]

Rule 5772

Int[((a_.) + ArcSinh[(c_.)*(x_)]*(b_.))^(n_.), x_Symbol] :> Simp[x*(a + b*ArcSinh[c*x])^n, x] - Dist[b*c*n, In
t[x*((a + b*ArcSinh[c*x])^(n - 1)/Sqrt[1 + c^2*x^2]), x], x] /; FreeQ[{a, b, c}, x] && GtQ[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 5798

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

Rule 5812

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

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*} \int (c e+d e x)^4 \left (a+b \sinh ^{-1}(c+d x)\right )^3 \, dx &=\frac {\text {Subst}\left (\int e^4 x^4 \left (a+b \sinh ^{-1}(x)\right )^3 \, dx,x,c+d x\right )}{d}\\ &=\frac {e^4 \text {Subst}\left (\int x^4 \left (a+b \sinh ^{-1}(x)\right )^3 \, dx,x,c+d x\right )}{d}\\ &=\frac {e^4 (c+d x)^5 \left (a+b \sinh ^{-1}(c+d x)\right )^3}{5 d}-\frac {\left (3 b e^4\right ) \text {Subst}\left (\int \frac {x^5 \left (a+b \sinh ^{-1}(x)\right )^2}{\sqrt {1+x^2}} \, dx,x,c+d x\right )}{5 d}\\ &=-\frac {3 b e^4 (c+d x)^4 \sqrt {1+(c+d x)^2} \left (a+b \sinh ^{-1}(c+d x)\right )^2}{25 d}+\frac {e^4 (c+d x)^5 \left (a+b \sinh ^{-1}(c+d x)\right )^3}{5 d}+\frac {\left (12 b e^4\right ) \text {Subst}\left (\int \frac {x^3 \left (a+b \sinh ^{-1}(x)\right )^2}{\sqrt {1+x^2}} \, dx,x,c+d x\right )}{25 d}+\frac {\left (6 b^2 e^4\right ) \text {Subst}\left (\int x^4 \left (a+b \sinh ^{-1}(x)\right ) \, dx,x,c+d x\right )}{25 d}\\ &=\frac {6 b^2 e^4 (c+d x)^5 \left (a+b \sinh ^{-1}(c+d x)\right )}{125 d}+\frac {4 b e^4 (c+d x)^2 \sqrt {1+(c+d x)^2} \left (a+b \sinh ^{-1}(c+d x)\right )^2}{25 d}-\frac {3 b e^4 (c+d x)^4 \sqrt {1+(c+d x)^2} \left (a+b \sinh ^{-1}(c+d x)\right )^2}{25 d}+\frac {e^4 (c+d x)^5 \left (a+b \sinh ^{-1}(c+d x)\right )^3}{5 d}-\frac {\left (8 b e^4\right ) \text {Subst}\left (\int \frac {x \left (a+b \sinh ^{-1}(x)\right )^2}{\sqrt {1+x^2}} \, dx,x,c+d x\right )}{25 d}-\frac {\left (8 b^2 e^4\right ) \text {Subst}\left (\int x^2 \left (a+b \sinh ^{-1}(x)\right ) \, dx,x,c+d x\right )}{25 d}-\frac {\left (6 b^3 e^4\right ) \text {Subst}\left (\int \frac {x^5}{\sqrt {1+x^2}} \, dx,x,c+d x\right )}{125 d}\\ &=-\frac {8 b^2 e^4 (c+d x)^3 \left (a+b \sinh ^{-1}(c+d x)\right )}{75 d}+\frac {6 b^2 e^4 (c+d x)^5 \left (a+b \sinh ^{-1}(c+d x)\right )}{125 d}-\frac {8 b e^4 \sqrt {1+(c+d x)^2} \left (a+b \sinh ^{-1}(c+d x)\right )^2}{25 d}+\frac {4 b e^4 (c+d x)^2 \sqrt {1+(c+d x)^2} \left (a+b \sinh ^{-1}(c+d x)\right )^2}{25 d}-\frac {3 b e^4 (c+d x)^4 \sqrt {1+(c+d x)^2} \left (a+b \sinh ^{-1}(c+d x)\right )^2}{25 d}+\frac {e^4 (c+d x)^5 \left (a+b \sinh ^{-1}(c+d x)\right )^3}{5 d}+\frac {\left (16 b^2 e^4\right ) \text {Subst}\left (\int \left (a+b \sinh ^{-1}(x)\right ) \, dx,x,c+d x\right )}{25 d}-\frac {\left (3 b^3 e^4\right ) \text {Subst}\left (\int \frac {x^2}{\sqrt {1+x}} \, dx,x,(c+d x)^2\right )}{125 d}+\frac {\left (8 b^3 e^4\right ) \text {Subst}\left (\int \frac {x^3}{\sqrt {1+x^2}} \, dx,x,c+d x\right )}{75 d}\\ &=\frac {16}{25} a b^2 e^4 x-\frac {8 b^2 e^4 (c+d x)^3 \left (a+b \sinh ^{-1}(c+d x)\right )}{75 d}+\frac {6 b^2 e^4 (c+d x)^5 \left (a+b \sinh ^{-1}(c+d x)\right )}{125 d}-\frac {8 b e^4 \sqrt {1+(c+d x)^2} \left (a+b \sinh ^{-1}(c+d x)\right )^2}{25 d}+\frac {4 b e^4 (c+d x)^2 \sqrt {1+(c+d x)^2} \left (a+b \sinh ^{-1}(c+d x)\right )^2}{25 d}-\frac {3 b e^4 (c+d x)^4 \sqrt {1+(c+d x)^2} \left (a+b \sinh ^{-1}(c+d x)\right )^2}{25 d}+\frac {e^4 (c+d x)^5 \left (a+b \sinh ^{-1}(c+d x)\right )^3}{5 d}-\frac {\left (3 b^3 e^4\right ) \text {Subst}\left (\int \left (\frac {1}{\sqrt {1+x}}-2 \sqrt {1+x}+(1+x)^{3/2}\right ) \, dx,x,(c+d x)^2\right )}{125 d}+\frac {\left (4 b^3 e^4\right ) \text {Subst}\left (\int \frac {x}{\sqrt {1+x}} \, dx,x,(c+d x)^2\right )}{75 d}+\frac {\left (16 b^3 e^4\right ) \text {Subst}\left (\int \sinh ^{-1}(x) \, dx,x,c+d x\right )}{25 d}\\ &=\frac {16}{25} a b^2 e^4 x-\frac {6 b^3 e^4 \sqrt {1+(c+d x)^2}}{125 d}+\frac {4 b^3 e^4 \left (1+(c+d x)^2\right )^{3/2}}{125 d}-\frac {6 b^3 e^4 \left (1+(c+d x)^2\right )^{5/2}}{625 d}+\frac {16 b^3 e^4 (c+d x) \sinh ^{-1}(c+d x)}{25 d}-\frac {8 b^2 e^4 (c+d x)^3 \left (a+b \sinh ^{-1}(c+d x)\right )}{75 d}+\frac {6 b^2 e^4 (c+d x)^5 \left (a+b \sinh ^{-1}(c+d x)\right )}{125 d}-\frac {8 b e^4 \sqrt {1+(c+d x)^2} \left (a+b \sinh ^{-1}(c+d x)\right )^2}{25 d}+\frac {4 b e^4 (c+d x)^2 \sqrt {1+(c+d x)^2} \left (a+b \sinh ^{-1}(c+d x)\right )^2}{25 d}-\frac {3 b e^4 (c+d x)^4 \sqrt {1+(c+d x)^2} \left (a+b \sinh ^{-1}(c+d x)\right )^2}{25 d}+\frac {e^4 (c+d x)^5 \left (a+b \sinh ^{-1}(c+d x)\right )^3}{5 d}+\frac {\left (4 b^3 e^4\right ) \text {Subst}\left (\int \left (-\frac {1}{\sqrt {1+x}}+\sqrt {1+x}\right ) \, dx,x,(c+d x)^2\right )}{75 d}-\frac {\left (16 b^3 e^4\right ) \text {Subst}\left (\int \frac {x}{\sqrt {1+x^2}} \, dx,x,c+d x\right )}{25 d}\\ &=\frac {16}{25} a b^2 e^4 x-\frac {298 b^3 e^4 \sqrt {1+(c+d x)^2}}{375 d}+\frac {76 b^3 e^4 \left (1+(c+d x)^2\right )^{3/2}}{1125 d}-\frac {6 b^3 e^4 \left (1+(c+d x)^2\right )^{5/2}}{625 d}+\frac {16 b^3 e^4 (c+d x) \sinh ^{-1}(c+d x)}{25 d}-\frac {8 b^2 e^4 (c+d x)^3 \left (a+b \sinh ^{-1}(c+d x)\right )}{75 d}+\frac {6 b^2 e^4 (c+d x)^5 \left (a+b \sinh ^{-1}(c+d x)\right )}{125 d}-\frac {8 b e^4 \sqrt {1+(c+d x)^2} \left (a+b \sinh ^{-1}(c+d x)\right )^2}{25 d}+\frac {4 b e^4 (c+d x)^2 \sqrt {1+(c+d x)^2} \left (a+b \sinh ^{-1}(c+d x)\right )^2}{25 d}-\frac {3 b e^4 (c+d x)^4 \sqrt {1+(c+d x)^2} \left (a+b \sinh ^{-1}(c+d x)\right )^2}{25 d}+\frac {e^4 (c+d x)^5 \left (a+b \sinh ^{-1}(c+d x)\right )^3}{5 d}\\ \end {align*}

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Mathematica [A]
time = 0.29, size = 355, normalized size = 1.09 \begin {gather*} \frac {e^4 \left (240 a b^2 (c+d x)-40 a b^2 (c+d x)^3+3 a \left (25 a^2+6 b^2\right ) (c+d x)^5+\frac {1}{15} b \sqrt {1+(c+d x)^2} \left (-8 \left (225 a^2+518 b^2\right )+4 \left (225 a^2+68 b^2\right ) (c+d x)^2-27 \left (25 a^2+2 b^2\right ) (c+d x)^4\right )-b \left (-240 b^2 (c+d x)+40 b^2 (c+d x)^3-225 a^2 (c+d x)^5-18 b^2 (c+d x)^5+240 a b \sqrt {1+(c+d x)^2}-120 a b (c+d x)^2 \sqrt {1+(c+d x)^2}+90 a b (c+d x)^4 \sqrt {1+(c+d x)^2}\right ) \sinh ^{-1}(c+d x)-15 b^2 \left (-15 a (c+d x)^5+8 b \sqrt {1+(c+d x)^2}-4 b (c+d x)^2 \sqrt {1+(c+d x)^2}+3 b (c+d x)^4 \sqrt {1+(c+d x)^2}\right ) \sinh ^{-1}(c+d x)^2+75 b^3 (c+d x)^5 \sinh ^{-1}(c+d x)^3\right )}{375 d} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(e^4*(240*a*b^2*(c + d*x) - 40*a*b^2*(c + d*x)^3 + 3*a*(25*a^2 + 6*b^2)*(c + d*x)^5 + (b*Sqrt[1 + (c + d*x)^2]
*(-8*(225*a^2 + 518*b^2) + 4*(225*a^2 + 68*b^2)*(c + d*x)^2 - 27*(25*a^2 + 2*b^2)*(c + d*x)^4))/15 - b*(-240*b
^2*(c + d*x) + 40*b^2*(c + d*x)^3 - 225*a^2*(c + d*x)^5 - 18*b^2*(c + d*x)^5 + 240*a*b*Sqrt[1 + (c + d*x)^2] -
120*a*b*(c + d*x)^2*Sqrt[1 + (c + d*x)^2] + 90*a*b*(c + d*x)^4*Sqrt[1 + (c + d*x)^2])*ArcSinh[c + d*x] - 15*b
^2*(-15*a*(c + d*x)^5 + 8*b*Sqrt[1 + (c + d*x)^2] - 4*b*(c + d*x)^2*Sqrt[1 + (c + d*x)^2] + 3*b*(c + d*x)^4*Sq
rt[1 + (c + d*x)^2])*ArcSinh[c + d*x]^2 + 75*b^3*(c + d*x)^5*ArcSinh[c + d*x]^3))/(375*d)

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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(1388\) vs. \(2(292)=584\).
time = 4.43, size = 1389, normalized size = 4.26

method	result	size
default	\(\text {Expression too large to display}\)	\(1389\)

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,method=_RETURNVERBOSE)

[Out]

1/5*e^4*(d*x+c)^5*a^3/d+1/5625*e^4*b^3*(-1800*(d^2*x^2+2*c*d*x+c^2+1)^(1/2)*arcsinh(d*x+c)^2+1125*arcsinh(d*x+
c)^3*c^5+270*arcsinh(d*x+c)*c^5-600*arcsinh(d*x+c)*c^3+3600*arcsinh(d*x+c)*c-54*(d^2*x^2+2*c*d*x+c^2+1)^(1/2)*
c^4+272*(d^2*x^2+2*c*d*x+c^2+1)^(1/2)*c^2-4144*(d^2*x^2+2*c*d*x+c^2+1)^(1/2)-675*arcsinh(d*x+c)^2*(d^2*x^2+2*c
*d*x+c^2+1)^(1/2)*c^4-675*arcsinh(d*x+c)^2*(d^2*x^2+2*c*d*x+c^2+1)^(1/2)*x^4*d^4+900*arcsinh(d*x+c)^2*(d^2*x^2
+2*c*d*x+c^2+1)^(1/2)*x^2*d^2+5625*arcsinh(d*x+c)^3*x^4*c*d^4+11250*arcsinh(d*x+c)^3*x^3*c^2*d^3+11250*arcsinh
(d*x+c)^3*x^2*c^3*d^2+1350*arcsinh(d*x+c)*x^4*c*d^4+5625*arcsinh(d*x+c)^3*x*c^4*d+2700*arcsinh(d*x+c)*x^3*c^2*
d^3+2700*arcsinh(d*x+c)*x^2*c^3*d^2+1350*arcsinh(d*x+c)*x*c^4*d-1800*arcsinh(d*x+c)*x^2*c*d^2-1800*arcsinh(d*x
+c)*x*c^2*d+900*arcsinh(d*x+c)^2*(d^2*x^2+2*c*d*x+c^2+1)^(1/2)*c^2+1125*arcsinh(d*x+c)^3*x^5*d^5+270*arcsinh(d
*x+c)*x^5*d^5-600*arcsinh(d*x+c)*x^3*d^3+3600*arcsinh(d*x+c)*x*d-54*(d^2*x^2+2*c*d*x+c^2+1)^(1/2)*x^4*d^4+272*
(d^2*x^2+2*c*d*x+c^2+1)^(1/2)*x^2*d^2-2700*arcsinh(d*x+c)^2*(d^2*x^2+2*c*d*x+c^2+1)^(1/2)*x*c^3*d+1800*arcsinh
(d*x+c)^2*(d^2*x^2+2*c*d*x+c^2+1)^(1/2)*x*c*d-2700*arcsinh(d*x+c)^2*(d^2*x^2+2*c*d*x+c^2+1)^(1/2)*x^3*c*d^3-40
50*arcsinh(d*x+c)^2*(d^2*x^2+2*c*d*x+c^2+1)^(1/2)*x^2*c^2*d^2-216*(d^2*x^2+2*c*d*x+c^2+1)^(1/2)*x*c^3*d+544*(d
^2*x^2+2*c*d*x+c^2+1)^(1/2)*x*c*d-216*(d^2*x^2+2*c*d*x+c^2+1)^(1/2)*x^3*c*d^3-324*(d^2*x^2+2*c*d*x+c^2+1)^(1/2
)*x^2*c^2*d^2)/d+1/375*e^4*a*b^2*(-240*arcsinh(d*x+c)*(d^2*x^2+2*c*d*x+c^2+1)^(1/2)+225*arcsinh(d*x+c)^2*c^5+1
8*x^5*d^5+240*c+90*x*c^4*d-40*c^3-40*d^3*x^3+18*c^5+240*d*x+180*x^2*c^3*d^2-90*arcsinh(d*x+c)*(d^2*x^2+2*c*d*x
+c^2+1)^(1/2)*c^4+120*arcsinh(d*x+c)*(d^2*x^2+2*c*d*x+c^2+1)^(1/2)*c^2+225*arcsinh(d*x+c)^2*x^5*d^5-120*x^2*c*
d^2-540*arcsinh(d*x+c)*(d^2*x^2+2*c*d*x+c^2+1)^(1/2)*x^2*c^2*d^2-360*arcsinh(d*x+c)*(d^2*x^2+2*c*d*x+c^2+1)^(1
/2)*x*c^3*d+240*arcsinh(d*x+c)*(d^2*x^2+2*c*d*x+c^2+1)^(1/2)*x*c*d-360*arcsinh(d*x+c)*(d^2*x^2+2*c*d*x+c^2+1)^
(1/2)*x^3*c*d^3-90*arcsinh(d*x+c)*(d^2*x^2+2*c*d*x+c^2+1)^(1/2)*x^4*d^4+120*arcsinh(d*x+c)*(d^2*x^2+2*c*d*x+c^
2+1)^(1/2)*x^2*d^2+1125*arcsinh(d*x+c)^2*x^4*c*d^4+2250*arcsinh(d*x+c)^2*x^3*c^2*d^3+2250*arcsinh(d*x+c)^2*x^2
*c^3*d^2+1125*arcsinh(d*x+c)^2*x*c^4*d-120*x*c^2*d+180*x^3*c^2*d^3+90*x^4*c*d^4)/d+3*e^4*a^2*b/d*(1/5*(d*x+c)^
5*arcsinh(d*x+c)-1/25*(d*x+c)^4*(1+(d*x+c)^2)^(1/2)+4/75*(d*x+c)^2*(1+(d*x+c)^2)^(1/2)-8/75*(1+(d*x+c)^2)^(1/2
))

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Maxima [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \end {gather*}

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]

1/5*a^3*d^4*x^5*e^4 + a^3*c*d^3*x^4*e^4 + 2*a^3*c^2*d^2*x^3*e^4 + 2*a^3*c^3*d*x^2*e^4 + 3*(2*x^2*arcsinh(d*x +
c) - d*(3*c^2*arcsinh(2*(d^2*x + c*d)/sqrt(-4*c^2*d^2 + 4*(c^2 + 1)*d^2))/d^3 + sqrt(d^2*x^2 + 2*c*d*x + c^2
+ 1)*x/d^2 - (c^2 + 1)*arcsinh(2*(d^2*x + c*d)/sqrt(-4*c^2*d^2 + 4*(c^2 + 1)*d^2))/d^3 - 3*sqrt(d^2*x^2 + 2*c*
d*x + c^2 + 1)*c/d^3))*a^2*b*c^3*d*e^4 + (6*x^3*arcsinh(d*x + c) - d*(2*sqrt(d^2*x^2 + 2*c*d*x + c^2 + 1)*x^2/
d^2 - 15*c^3*arcsinh(2*(d^2*x + c*d)/sqrt(-4*c^2*d^2 + 4*(c^2 + 1)*d^2))/d^4 - 5*sqrt(d^2*x^2 + 2*c*d*x + c^2
+ 1)*c*x/d^3 + 9*(c^2 + 1)*c*arcsinh(2*(d^2*x + c*d)/sqrt(-4*c^2*d^2 + 4*(c^2 + 1)*d^2))/d^4 + 15*sqrt(d^2*x^2
+ 2*c*d*x + c^2 + 1)*c^2/d^4 - 4*sqrt(d^2*x^2 + 2*c*d*x + c^2 + 1)*(c^2 + 1)/d^4))*a^2*b*c^2*d^2*e^4 + 1/8*(2
4*x^4*arcsinh(d*x + c) - (6*sqrt(d^2*x^2 + 2*c*d*x + c^2 + 1)*x^3/d^2 - 14*sqrt(d^2*x^2 + 2*c*d*x + c^2 + 1)*c
*x^2/d^3 + 105*c^4*arcsinh(2*(d^2*x + c*d)/sqrt(-4*c^2*d^2 + 4*(c^2 + 1)*d^2))/d^5 + 35*sqrt(d^2*x^2 + 2*c*d*x
+ c^2 + 1)*c^2*x/d^4 - 90*(c^2 + 1)*c^2*arcsinh(2*(d^2*x + c*d)/sqrt(-4*c^2*d^2 + 4*(c^2 + 1)*d^2))/d^5 - 105
*sqrt(d^2*x^2 + 2*c*d*x + c^2 + 1)*c^3/d^5 - 9*sqrt(d^2*x^2 + 2*c*d*x + c^2 + 1)*(c^2 + 1)*x/d^4 + 9*(c^2 + 1)
^2*arcsinh(2*(d^2*x + c*d)/sqrt(-4*c^2*d^2 + 4*(c^2 + 1)*d^2))/d^5 + 55*sqrt(d^2*x^2 + 2*c*d*x + c^2 + 1)*(c^2
+ 1)*c/d^5)*d)*a^2*b*c*d^3*e^4 + 1/200*(120*x^5*arcsinh(d*x + c) - (24*sqrt(d^2*x^2 + 2*c*d*x + c^2 + 1)*x^4/
d^2 - 54*sqrt(d^2*x^2 + 2*c*d*x + c^2 + 1)*c*x^3/d^3 + 126*sqrt(d^2*x^2 + 2*c*d*x + c^2 + 1)*c^2*x^2/d^4 - 945
*c^5*arcsinh(2*(d^2*x + c*d)/sqrt(-4*c^2*d^2 + 4*(c^2 + 1)*d^2))/d^6 - 315*sqrt(d^2*x^2 + 2*c*d*x + c^2 + 1)*c
^3*x/d^5 - 32*sqrt(d^2*x^2 + 2*c*d*x + c^2 + 1)*(c^2 + 1)*x^2/d^4 + 1050*(c^2 + 1)*c^3*arcsinh(2*(d^2*x + c*d)
/sqrt(-4*c^2*d^2 + 4*(c^2 + 1)*d^2))/d^6 + 945*sqrt(d^2*x^2 + 2*c*d*x + c^2 + 1)*c^4/d^6 + 161*sqrt(d^2*x^2 +
2*c*d*x + c^2 + 1)*(c^2 + 1)*c*x/d^5 - 225*(c^2 + 1)^2*c*arcsinh(2*(d^2*x + c*d)/sqrt(-4*c^2*d^2 + 4*(c^2 + 1)
*d^2))/d^6 - 735*sqrt(d^2*x^2 + 2*c*d*x + c^2 + 1)*(c^2 + 1)*c^2/d^6 + 64*sqrt(d^2*x^2 + 2*c*d*x + c^2 + 1)*(c
^2 + 1)^2/d^6)*d)*a^2*b*d^4*e^4 + a^3*c^4*x*e^4 + 3*((d*x + c)*arcsinh(d*x + c) - sqrt((d*x + c)^2 + 1))*a^2*b
*c^4*e^4/d + 1/5*(b^3*d^4*x^5*e^4 + 5*b^3*c*d^3*x^4*e^4 + 10*b^3*c^2*d^2*x^3*e^4 + 10*b^3*c^3*d*x^2*e^4 + 5*b^
3*c^4*x*e^4)*log(d*x + c + sqrt(d^2*x^2 + 2*c*d*x + c^2 + 1))^3 + integrate(3/5*((5*a*b^2*d^7 - b^3*d^7)*x^7*e
^4 + 7*(5*a*b^2*c*d^6 - b^3*c*d^6)*x^6*e^4 + (5*(21*c^2*d^5 + d^5)*a*b^2 - (21*c^2*d^5 + d^5)*b^3)*x^5*e^4 + 5
*(5*(7*c^3*d^4 + c*d^4)*a*b^2 - (7*c^3*d^4 + c*d^4)*b^3)*x^4*e^4 + 5*(c^7 + c^5)*a*b^2*e^4 + 5*(5*(7*c^4*d^3 +
2*c^2*d^3)*a*b^2 - (7*c^4*d^3 + 2*c^2*d^3)*b^3)*x^3*e^4 + 5*((21*c^5*d^2 + 10*c^3*d^2)*a*b^2 - 2*(2*c^5*d^2 +
c^3*d^2)*b^3)*x^2*e^4 + 5*((7*c^6*d + 5*c^4*d)*a*b^2 - (c^6*d + c^4*d)*b^3)*x*e^4 + ((5*a*b^2*d^6 - b^3*d^6)*
x^6*e^4 + 6*(5*a*b^2*c*d^5 - b^3*c*d^5)*x^5*e^4 - 5*(3*b^3*c^2*d^4 - (15*c^2*d^4 + d^4)*a*b^2)*x^4*e^4 + 5*(c^
6 + c^4)*a*b^2*e^4 - 20*(b^3*c^3*d^3 - (5*c^3*d^3 + c*d^3)*a*b^2)*x^3*e^4 - 15*(b^3*c^4*d^2 - (5*c^4*d^2 + 2*c
^2*d^2)*a*b^2)*x^2*e^4 - 5*(b^3*c^5*d - 2*(3*c^5*d + 2*c^3*d)*a*b^2)*x*e^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/(d^3*x^3 + 3*c*d^2*x^2 + c^3 + (3*c^2*d + d)*x + (d^2*x^2
+ 2*c*d*x + c^2 + 1)^(3/2) + c), x)

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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 4411 vs. \(2 (281) = 562\).
time = 0.58, size = 4411, normalized size = 13.53 \begin {gather*} \text {Too large to display} \end {gather*}

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]

1/5625*(15*(3*(25*a^3 + 6*a*b^2)*d^5*x^5 + 15*(25*a^3 + 6*a*b^2)*c*d^4*x^4 - 10*(4*a*b^2 - 3*(25*a^3 + 6*a*b^2
)*c^2)*d^3*x^3 - 30*(4*a*b^2*c - (25*a^3 + 6*a*b^2)*c^3)*d^2*x^2 - 15*(8*a*b^2*c^2 - (25*a^3 + 6*a*b^2)*c^4 -
16*a*b^2)*d*x)*cosh(1)^4 + 60*(3*(25*a^3 + 6*a*b^2)*d^5*x^5 + 15*(25*a^3 + 6*a*b^2)*c*d^4*x^4 - 10*(4*a*b^2 -
3*(25*a^3 + 6*a*b^2)*c^2)*d^3*x^3 - 30*(4*a*b^2*c - (25*a^3 + 6*a*b^2)*c^3)*d^2*x^2 - 15*(8*a*b^2*c^2 - (25*a^
3 + 6*a*b^2)*c^4 - 16*a*b^2)*d*x)*cosh(1)^3*sinh(1) + 90*(3*(25*a^3 + 6*a*b^2)*d^5*x^5 + 15*(25*a^3 + 6*a*b^2)
*c*d^4*x^4 - 10*(4*a*b^2 - 3*(25*a^3 + 6*a*b^2)*c^2)*d^3*x^3 - 30*(4*a*b^2*c - (25*a^3 + 6*a*b^2)*c^3)*d^2*x^2
- 15*(8*a*b^2*c^2 - (25*a^3 + 6*a*b^2)*c^4 - 16*a*b^2)*d*x)*cosh(1)^2*sinh(1)^2 + 60*(3*(25*a^3 + 6*a*b^2)*d^
5*x^5 + 15*(25*a^3 + 6*a*b^2)*c*d^4*x^4 - 10*(4*a*b^2 - 3*(25*a^3 + 6*a*b^2)*c^2)*d^3*x^3 - 30*(4*a*b^2*c - (2
5*a^3 + 6*a*b^2)*c^3)*d^2*x^2 - 15*(8*a*b^2*c^2 - (25*a^3 + 6*a*b^2)*c^4 - 16*a*b^2)*d*x)*cosh(1)*sinh(1)^3 +
15*(3*(25*a^3 + 6*a*b^2)*d^5*x^5 + 15*(25*a^3 + 6*a*b^2)*c*d^4*x^4 - 10*(4*a*b^2 - 3*(25*a^3 + 6*a*b^2)*c^2)*d
^3*x^3 - 30*(4*a*b^2*c - (25*a^3 + 6*a*b^2)*c^3)*d^2*x^2 - 15*(8*a*b^2*c^2 - (25*a^3 + 6*a*b^2)*c^4 - 16*a*b^2
)*d*x)*sinh(1)^4 + 1125*((b^3*d^5*x^5 + 5*b^3*c*d^4*x^4 + 10*b^3*c^2*d^3*x^3 + 10*b^3*c^3*d^2*x^2 + 5*b^3*c^4*
d*x + b^3*c^5)*cosh(1)^4 + 4*(b^3*d^5*x^5 + 5*b^3*c*d^4*x^4 + 10*b^3*c^2*d^3*x^3 + 10*b^3*c^3*d^2*x^2 + 5*b^3*
c^4*d*x + b^3*c^5)*cosh(1)^3*sinh(1) + 6*(b^3*d^5*x^5 + 5*b^3*c*d^4*x^4 + 10*b^3*c^2*d^3*x^3 + 10*b^3*c^3*d^2*
x^2 + 5*b^3*c^4*d*x + b^3*c^5)*cosh(1)^2*sinh(1)^2 + 4*(b^3*d^5*x^5 + 5*b^3*c*d^4*x^4 + 10*b^3*c^2*d^3*x^3 + 1
0*b^3*c^3*d^2*x^2 + 5*b^3*c^4*d*x + b^3*c^5)*cosh(1)*sinh(1)^3 + (b^3*d^5*x^5 + 5*b^3*c*d^4*x^4 + 10*b^3*c^2*d
^3*x^3 + 10*b^3*c^3*d^2*x^2 + 5*b^3*c^4*d*x + b^3*c^5)*sinh(1)^4)*log(d*x + c + sqrt(d^2*x^2 + 2*c*d*x + c^2 +
1))^3 + 225*(15*(a*b^2*d^5*x^5 + 5*a*b^2*c*d^4*x^4 + 10*a*b^2*c^2*d^3*x^3 + 10*a*b^2*c^3*d^2*x^2 + 5*a*b^2*c^
4*d*x + a*b^2*c^5)*cosh(1)^4 + 60*(a*b^2*d^5*x^5 + 5*a*b^2*c*d^4*x^4 + 10*a*b^2*c^2*d^3*x^3 + 10*a*b^2*c^3*d^2
*x^2 + 5*a*b^2*c^4*d*x + a*b^2*c^5)*cosh(1)^3*sinh(1) + 90*(a*b^2*d^5*x^5 + 5*a*b^2*c*d^4*x^4 + 10*a*b^2*c^2*d
^3*x^3 + 10*a*b^2*c^3*d^2*x^2 + 5*a*b^2*c^4*d*x + a*b^2*c^5)*cosh(1)^2*sinh(1)^2 + 60*(a*b^2*d^5*x^5 + 5*a*b^2
*c*d^4*x^4 + 10*a*b^2*c^2*d^3*x^3 + 10*a*b^2*c^3*d^2*x^2 + 5*a*b^2*c^4*d*x + a*b^2*c^5)*cosh(1)*sinh(1)^3 + 15
*(a*b^2*d^5*x^5 + 5*a*b^2*c*d^4*x^4 + 10*a*b^2*c^2*d^3*x^3 + 10*a*b^2*c^3*d^2*x^2 + 5*a*b^2*c^4*d*x + a*b^2*c^
5)*sinh(1)^4 - ((3*b^3*d^4*x^4 + 12*b^3*c*d^3*x^3 + 3*b^3*c^4 - 4*b^3*c^2 + 2*(9*b^3*c^2 - 2*b^3)*d^2*x^2 + 8*
b^3 + 4*(3*b^3*c^3 - 2*b^3*c)*d*x)*cosh(1)^4 + 4*(3*b^3*d^4*x^4 + 12*b^3*c*d^3*x^3 + 3*b^3*c^4 - 4*b^3*c^2 + 2
*(9*b^3*c^2 - 2*b^3)*d^2*x^2 + 8*b^3 + 4*(3*b^3*c^3 - 2*b^3*c)*d*x)*cosh(1)^3*sinh(1) + 6*(3*b^3*d^4*x^4 + 12*
b^3*c*d^3*x^3 + 3*b^3*c^4 - 4*b^3*c^2 + 2*(9*b^3*c^2 - 2*b^3)*d^2*x^2 + 8*b^3 + 4*(3*b^3*c^3 - 2*b^3*c)*d*x)*c
osh(1)^2*sinh(1)^2 + 4*(3*b^3*d^4*x^4 + 12*b^3*c*d^3*x^3 + 3*b^3*c^4 - 4*b^3*c^2 + 2*(9*b^3*c^2 - 2*b^3)*d^2*x
^2 + 8*b^3 + 4*(3*b^3*c^3 - 2*b^3*c)*d*x)*cosh(1)*sinh(1)^3 + (3*b^3*d^4*x^4 + 12*b^3*c*d^3*x^3 + 3*b^3*c^4 -
4*b^3*c^2 + 2*(9*b^3*c^2 - 2*b^3)*d^2*x^2 + 8*b^3 + 4*(3*b^3*c^3 - 2*b^3*c)*d*x)*sinh(1)^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 + 15*((9*(25*a^2*b + 2*b^3)*d^5*x^5 + 45*(
25*a^2*b + 2*b^3)*c*d^4*x^4 - 10*(4*b^3 - 9*(25*a^2*b + 2*b^3)*c^2)*d^3*x^3 - 40*b^3*c^3 + 9*(25*a^2*b + 2*b^3
)*c^5 - 30*(4*b^3*c - 3*(25*a^2*b + 2*b^3)*c^3)*d^2*x^2 + 240*b^3*c - 15*(8*b^3*c^2 - 3*(25*a^2*b + 2*b^3)*c^4
- 16*b^3)*d*x)*cosh(1)^4 + 4*(9*(25*a^2*b + 2*b^3)*d^5*x^5 + 45*(25*a^2*b + 2*b^3)*c*d^4*x^4 - 10*(4*b^3 - 9*
(25*a^2*b + 2*b^3)*c^2)*d^3*x^3 - 40*b^3*c^3 + 9*(25*a^2*b + 2*b^3)*c^5 - 30*(4*b^3*c - 3*(25*a^2*b + 2*b^3)*c
^3)*d^2*x^2 + 240*b^3*c - 15*(8*b^3*c^2 - 3*(25*a^2*b + 2*b^3)*c^4 - 16*b^3)*d*x)*cosh(1)^3*sinh(1) + 6*(9*(25
*a^2*b + 2*b^3)*d^5*x^5 + 45*(25*a^2*b + 2*b^3)*c*d^4*x^4 - 10*(4*b^3 - 9*(25*a^2*b + 2*b^3)*c^2)*d^3*x^3 - 40
*b^3*c^3 + 9*(25*a^2*b + 2*b^3)*c^5 - 30*(4*b^3*c - 3*(25*a^2*b + 2*b^3)*c^3)*d^2*x^2 + 240*b^3*c - 15*(8*b^3*
c^2 - 3*(25*a^2*b + 2*b^3)*c^4 - 16*b^3)*d*x)*cosh(1)^2*sinh(1)^2 + 4*(9*(25*a^2*b + 2*b^3)*d^5*x^5 + 45*(25*a
^2*b + 2*b^3)*c*d^4*x^4 - 10*(4*b^3 - 9*(25*a^2*b + 2*b^3)*c^2)*d^3*x^3 - 40*b^3*c^3 + 9*(25*a^2*b + 2*b^3)*c^
5 - 30*(4*b^3*c - 3*(25*a^2*b + 2*b^3)*c^3)*d^2*x^2 + 240*b^3*c - 15*(8*b^3*c^2 - 3*(25*a^2*b + 2*b^3)*c^4 - 1
6*b^3)*d*x)*cosh(1)*sinh(1)^3 + (9*(25*a^2*b + 2*b^3)*d^5*x^5 + 45*(25*a^2*b + 2*b^3)*c*d^4*x^4 - 10*(4*b^3 -
9*(25*a^2*b + 2*b^3)*c^2)*d^3*x^3 - 40*b^3*c^3 + 9*(25*a^2*b + 2*b^3)*c^5 - 30*(4*b^3*c - 3*(25*a^2*b + 2*b^3)
*c^3)*d^2*x^2 + 240*b^3*c - 15*(8*b^3*c^2 - 3*(25*a^2*b + 2*b^3)*c^4 - 16*b^3)*d*x)*sinh(1)^4 - 30*((3*a*b^2*d
^4*x^4 + 12*a*b^2*c*d^3*x^3 + 3*a*b^2*c^4 - 4*a*b^2*c^2 + 2*(9*a*b^2*c^2 - 2*a*b^2)*d^2*x^2 + 8*a*b^2 + 4*(3*a
*b^2*c^3 - 2*a*b^2*c)*d*x)*cosh(1)^4 + 4*(3*a*b...

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Sympy [B] Leaf count of result is larger than twice the leaf count of optimal. 2518 vs. \(2 (306) = 612\).
time = 1.33, size = 2518, normalized size = 7.72 \begin {gather*} \text {Too large to display} \end {gather*}

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]

Piecewise((a**3*c**4*e**4*x + 2*a**3*c**3*d*e**4*x**2 + 2*a**3*c**2*d**2*e**4*x**3 + a**3*c*d**3*e**4*x**4 + a
**3*d**4*e**4*x**5/5 + 3*a**2*b*c**5*e**4*asinh(c + d*x)/(5*d) + 3*a**2*b*c**4*e**4*x*asinh(c + d*x) - 3*a**2*
b*c**4*e**4*sqrt(c**2 + 2*c*d*x + d**2*x**2 + 1)/(25*d) + 6*a**2*b*c**3*d*e**4*x**2*asinh(c + d*x) - 12*a**2*b
*c**3*e**4*x*sqrt(c**2 + 2*c*d*x + d**2*x**2 + 1)/25 + 6*a**2*b*c**2*d**2*e**4*x**3*asinh(c + d*x) - 18*a**2*b
*c**2*d*e**4*x**2*sqrt(c**2 + 2*c*d*x + d**2*x**2 + 1)/25 + 4*a**2*b*c**2*e**4*sqrt(c**2 + 2*c*d*x + d**2*x**2
+ 1)/(25*d) + 3*a**2*b*c*d**3*e**4*x**4*asinh(c + d*x) - 12*a**2*b*c*d**2*e**4*x**3*sqrt(c**2 + 2*c*d*x + d**
2*x**2 + 1)/25 + 8*a**2*b*c*e**4*x*sqrt(c**2 + 2*c*d*x + d**2*x**2 + 1)/25 + 3*a**2*b*d**4*e**4*x**5*asinh(c +
d*x)/5 - 3*a**2*b*d**3*e**4*x**4*sqrt(c**2 + 2*c*d*x + d**2*x**2 + 1)/25 + 4*a**2*b*d*e**4*x**2*sqrt(c**2 + 2
*c*d*x + d**2*x**2 + 1)/25 - 8*a**2*b*e**4*sqrt(c**2 + 2*c*d*x + d**2*x**2 + 1)/(25*d) + 3*a*b**2*c**5*e**4*as
inh(c + d*x)**2/(5*d) + 3*a*b**2*c**4*e**4*x*asinh(c + d*x)**2 + 6*a*b**2*c**4*e**4*x/25 - 6*a*b**2*c**4*e**4*
sqrt(c**2 + 2*c*d*x + d**2*x**2 + 1)*asinh(c + d*x)/(25*d) + 6*a*b**2*c**3*d*e**4*x**2*asinh(c + d*x)**2 + 12*
a*b**2*c**3*d*e**4*x**2/25 - 24*a*b**2*c**3*e**4*x*sqrt(c**2 + 2*c*d*x + d**2*x**2 + 1)*asinh(c + d*x)/25 + 6*
a*b**2*c**2*d**2*e**4*x**3*asinh(c + d*x)**2 + 12*a*b**2*c**2*d**2*e**4*x**3/25 - 36*a*b**2*c**2*d*e**4*x**2*s
qrt(c**2 + 2*c*d*x + d**2*x**2 + 1)*asinh(c + d*x)/25 - 8*a*b**2*c**2*e**4*x/25 + 8*a*b**2*c**2*e**4*sqrt(c**2
+ 2*c*d*x + d**2*x**2 + 1)*asinh(c + d*x)/(25*d) + 3*a*b**2*c*d**3*e**4*x**4*asinh(c + d*x)**2 + 6*a*b**2*c*d
**3*e**4*x**4/25 - 24*a*b**2*c*d**2*e**4*x**3*sqrt(c**2 + 2*c*d*x + d**2*x**2 + 1)*asinh(c + d*x)/25 - 8*a*b**
2*c*d*e**4*x**2/25 + 16*a*b**2*c*e**4*x*sqrt(c**2 + 2*c*d*x + d**2*x**2 + 1)*asinh(c + d*x)/25 + 3*a*b**2*d**4
*e**4*x**5*asinh(c + d*x)**2/5 + 6*a*b**2*d**4*e**4*x**5/125 - 6*a*b**2*d**3*e**4*x**4*sqrt(c**2 + 2*c*d*x + d
**2*x**2 + 1)*asinh(c + d*x)/25 - 8*a*b**2*d**2*e**4*x**3/75 + 8*a*b**2*d*e**4*x**2*sqrt(c**2 + 2*c*d*x + d**2
*x**2 + 1)*asinh(c + d*x)/25 + 16*a*b**2*e**4*x/25 - 16*a*b**2*e**4*sqrt(c**2 + 2*c*d*x + d**2*x**2 + 1)*asinh
(c + d*x)/(25*d) + b**3*c**5*e**4*asinh(c + d*x)**3/(5*d) + 6*b**3*c**5*e**4*asinh(c + d*x)/(125*d) + b**3*c**
4*e**4*x*asinh(c + d*x)**3 + 6*b**3*c**4*e**4*x*asinh(c + d*x)/25 - 3*b**3*c**4*e**4*sqrt(c**2 + 2*c*d*x + d**
2*x**2 + 1)*asinh(c + d*x)**2/(25*d) - 6*b**3*c**4*e**4*sqrt(c**2 + 2*c*d*x + d**2*x**2 + 1)/(625*d) + 2*b**3*
c**3*d*e**4*x**2*asinh(c + d*x)**3 + 12*b**3*c**3*d*e**4*x**2*asinh(c + d*x)/25 - 12*b**3*c**3*e**4*x*sqrt(c**
2 + 2*c*d*x + d**2*x**2 + 1)*asinh(c + d*x)**2/25 - 24*b**3*c**3*e**4*x*sqrt(c**2 + 2*c*d*x + d**2*x**2 + 1)/6
25 - 8*b**3*c**3*e**4*asinh(c + d*x)/(75*d) + 2*b**3*c**2*d**2*e**4*x**3*asinh(c + d*x)**3 + 12*b**3*c**2*d**2
*e**4*x**3*asinh(c + d*x)/25 - 18*b**3*c**2*d*e**4*x**2*sqrt(c**2 + 2*c*d*x + d**2*x**2 + 1)*asinh(c + d*x)**2
/25 - 36*b**3*c**2*d*e**4*x**2*sqrt(c**2 + 2*c*d*x + d**2*x**2 + 1)/625 - 8*b**3*c**2*e**4*x*asinh(c + d*x)/25
+ 4*b**3*c**2*e**4*sqrt(c**2 + 2*c*d*x + d**2*x**2 + 1)*asinh(c + d*x)**2/(25*d) + 272*b**3*c**2*e**4*sqrt(c*
*2 + 2*c*d*x + d**2*x**2 + 1)/(5625*d) + b**3*c*d**3*e**4*x**4*asinh(c + d*x)**3 + 6*b**3*c*d**3*e**4*x**4*asi
nh(c + d*x)/25 - 12*b**3*c*d**2*e**4*x**3*sqrt(c**2 + 2*c*d*x + d**2*x**2 + 1)*asinh(c + d*x)**2/25 - 24*b**3*
c*d**2*e**4*x**3*sqrt(c**2 + 2*c*d*x + d**2*x**2 + 1)/625 - 8*b**3*c*d*e**4*x**2*asinh(c + d*x)/25 + 8*b**3*c*
e**4*x*sqrt(c**2 + 2*c*d*x + d**2*x**2 + 1)*asinh(c + d*x)**2/25 + 544*b**3*c*e**4*x*sqrt(c**2 + 2*c*d*x + d**
2*x**2 + 1)/5625 + 16*b**3*c*e**4*asinh(c + d*x)/(25*d) + b**3*d**4*e**4*x**5*asinh(c + d*x)**3/5 + 6*b**3*d**
4*e**4*x**5*asinh(c + d*x)/125 - 3*b**3*d**3*e**4*x**4*sqrt(c**2 + 2*c*d*x + d**2*x**2 + 1)*asinh(c + d*x)**2/
25 - 6*b**3*d**3*e**4*x**4*sqrt(c**2 + 2*c*d*x + d**2*x**2 + 1)/625 - 8*b**3*d**2*e**4*x**3*asinh(c + d*x)/75
+ 4*b**3*d*e**4*x**2*sqrt(c**2 + 2*c*d*x + d**2*x**2 + 1)*asinh(c + d*x)**2/25 + 272*b**3*d*e**4*x**2*sqrt(c**
2 + 2*c*d*x + d**2*x**2 + 1)/5625 + 16*b**3*e**4*x*asinh(c + d*x)/25 - 8*b**3*e**4*sqrt(c**2 + 2*c*d*x + d**2*
x**2 + 1)*asinh(c + d*x)**2/(25*d) - 4144*b**3*e**4*sqrt(c**2 + 2*c*d*x + d**2*x**2 + 1)/(5625*d), Ne(d, 0)),
(c**4*e**4*x*(a + b*asinh(c))**3, True))

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Giac [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}

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

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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