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

Optimal. Leaf size=349 \[ -\frac {3 b^3 e^3 (c+d x)^3 \sqrt {(c+d x)^2+1} \left (a+b \sinh ^{-1}(c+d x)\right )}{32 d}+\frac {45 b^3 e^3 (c+d x) \sqrt {(c+d x)^2+1} \left (a+b \sinh ^{-1}(c+d x)\right )}{64 d}+\frac {3 b^2 e^3 (c+d x)^4 \left (a+b \sinh ^{-1}(c+d x)\right )^2}{16 d}-\frac {45 b^2 e^3 \left (a+b \sinh ^{-1}(c+d x)\right )^2}{128 d}-\frac {9 b^2 e^3 (c+d x)^2 \left (a+b \sinh ^{-1}(c+d x)\right )^2}{16 d}-\frac {b e^3 (c+d x)^3 \sqrt {(c+d x)^2+1} \left (a+b \sinh ^{-1}(c+d x)\right )^3}{4 d}+\frac {3 b e^3 (c+d x) \sqrt {(c+d x)^2+1} \left (a+b \sinh ^{-1}(c+d x)\right )^3}{8 d}+\frac {e^3 (c+d x)^4 \left (a+b \sinh ^{-1}(c+d x)\right )^4}{4 d}-\frac {3 e^3 \left (a+b \sinh ^{-1}(c+d x)\right )^4}{32 d}+\frac {3 b^4 e^3 (c+d x)^4}{128 d}-\frac {45 b^4 e^3 (c+d x)^2}{128 d} \]

[Out]

-45/128*b^4*e^3*(d*x+c)^2/d+3/128*b^4*e^3*(d*x+c)^4/d-45/128*b^2*e^3*(a+b*arcsinh(d*x+c))^2/d-9/16*b^2*e^3*(d*
x+c)^2*(a+b*arcsinh(d*x+c))^2/d+3/16*b^2*e^3*(d*x+c)^4*(a+b*arcsinh(d*x+c))^2/d-3/32*e^3*(a+b*arcsinh(d*x+c))^
4/d+1/4*e^3*(d*x+c)^4*(a+b*arcsinh(d*x+c))^4/d+45/64*b^3*e^3*(d*x+c)*(a+b*arcsinh(d*x+c))*(1+(d*x+c)^2)^(1/2)/
d-3/32*b^3*e^3*(d*x+c)^3*(a+b*arcsinh(d*x+c))*(1+(d*x+c)^2)^(1/2)/d+3/8*b*e^3*(d*x+c)*(a+b*arcsinh(d*x+c))^3*(
1+(d*x+c)^2)^(1/2)/d-1/4*b*e^3*(d*x+c)^3*(a+b*arcsinh(d*x+c))^3*(1+(d*x+c)^2)^(1/2)/d

________________________________________________________________________________________

Rubi [A]  time = 0.67, antiderivative size = 349, normalized size of antiderivative = 1.00, number of steps used = 16, number of rules used = 6, integrand size = 23, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.261, Rules used = {5865, 12, 5661, 5758, 5675, 30} \[ -\frac {3 b^3 e^3 (c+d x)^3 \sqrt {(c+d x)^2+1} \left (a+b \sinh ^{-1}(c+d x)\right )}{32 d}+\frac {45 b^3 e^3 (c+d x) \sqrt {(c+d x)^2+1} \left (a+b \sinh ^{-1}(c+d x)\right )}{64 d}+\frac {3 b^2 e^3 (c+d x)^4 \left (a+b \sinh ^{-1}(c+d x)\right )^2}{16 d}-\frac {45 b^2 e^3 \left (a+b \sinh ^{-1}(c+d x)\right )^2}{128 d}-\frac {9 b^2 e^3 (c+d x)^2 \left (a+b \sinh ^{-1}(c+d x)\right )^2}{16 d}-\frac {b e^3 (c+d x)^3 \sqrt {(c+d x)^2+1} \left (a+b \sinh ^{-1}(c+d x)\right )^3}{4 d}+\frac {3 b e^3 (c+d x) \sqrt {(c+d x)^2+1} \left (a+b \sinh ^{-1}(c+d x)\right )^3}{8 d}+\frac {e^3 (c+d x)^4 \left (a+b \sinh ^{-1}(c+d x)\right )^4}{4 d}-\frac {3 e^3 \left (a+b \sinh ^{-1}(c+d x)\right )^4}{32 d}+\frac {3 b^4 e^3 (c+d x)^4}{128 d}-\frac {45 b^4 e^3 (c+d x)^2}{128 d} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(-45*b^4*e^3*(c + d*x)^2)/(128*d) + (3*b^4*e^3*(c + d*x)^4)/(128*d) + (45*b^3*e^3*(c + d*x)*Sqrt[1 + (c + d*x)
^2]*(a + b*ArcSinh[c + d*x]))/(64*d) - (3*b^3*e^3*(c + d*x)^3*Sqrt[1 + (c + d*x)^2]*(a + b*ArcSinh[c + d*x]))/
(32*d) - (45*b^2*e^3*(a + b*ArcSinh[c + d*x])^2)/(128*d) - (9*b^2*e^3*(c + d*x)^2*(a + b*ArcSinh[c + d*x])^2)/
(16*d) + (3*b^2*e^3*(c + d*x)^4*(a + b*ArcSinh[c + d*x])^2)/(16*d) + (3*b*e^3*(c + d*x)*Sqrt[1 + (c + d*x)^2]*
(a + b*ArcSinh[c + d*x])^3)/(8*d) - (b*e^3*(c + d*x)^3*Sqrt[1 + (c + d*x)^2]*(a + b*ArcSinh[c + d*x])^3)/(4*d)
- (3*e^3*(a + b*ArcSinh[c + d*x])^4)/(32*d) + (e^3*(c + d*x)^4*(a + b*ArcSinh[c + d*x])^4)/(4*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 30

Int[(x_)^(m_.), x_Symbol] :> Simp[x^(m + 1)/(m + 1), x] /; FreeQ[m, x] && NeQ[m, -1]

Rule 5661

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 5675

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

Rule 5758

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

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 (c e+d e x)^3 \left (a+b \sinh ^{-1}(c+d x)\right )^4 \, dx &=\frac {\operatorname {Subst}\left (\int e^3 x^3 \left (a+b \sinh ^{-1}(x)\right )^4 \, dx,x,c+d x\right )}{d}\\ &=\frac {e^3 \operatorname {Subst}\left (\int x^3 \left (a+b \sinh ^{-1}(x)\right )^4 \, dx,x,c+d x\right )}{d}\\ &=\frac {e^3 (c+d x)^4 \left (a+b \sinh ^{-1}(c+d x)\right )^4}{4 d}-\frac {\left (b e^3\right ) \operatorname {Subst}\left (\int \frac {x^4 \left (a+b \sinh ^{-1}(x)\right )^3}{\sqrt {1+x^2}} \, dx,x,c+d x\right )}{d}\\ &=-\frac {b e^3 (c+d x)^3 \sqrt {1+(c+d x)^2} \left (a+b \sinh ^{-1}(c+d x)\right )^3}{4 d}+\frac {e^3 (c+d x)^4 \left (a+b \sinh ^{-1}(c+d x)\right )^4}{4 d}+\frac {\left (3 b e^3\right ) \operatorname {Subst}\left (\int \frac {x^2 \left (a+b \sinh ^{-1}(x)\right )^3}{\sqrt {1+x^2}} \, dx,x,c+d x\right )}{4 d}+\frac {\left (3 b^2 e^3\right ) \operatorname {Subst}\left (\int x^3 \left (a+b \sinh ^{-1}(x)\right )^2 \, dx,x,c+d x\right )}{4 d}\\ &=\frac {3 b^2 e^3 (c+d x)^4 \left (a+b \sinh ^{-1}(c+d x)\right )^2}{16 d}+\frac {3 b e^3 (c+d x) \sqrt {1+(c+d x)^2} \left (a+b \sinh ^{-1}(c+d x)\right )^3}{8 d}-\frac {b e^3 (c+d x)^3 \sqrt {1+(c+d x)^2} \left (a+b \sinh ^{-1}(c+d x)\right )^3}{4 d}+\frac {e^3 (c+d x)^4 \left (a+b \sinh ^{-1}(c+d x)\right )^4}{4 d}-\frac {\left (3 b e^3\right ) \operatorname {Subst}\left (\int \frac {\left (a+b \sinh ^{-1}(x)\right )^3}{\sqrt {1+x^2}} \, dx,x,c+d x\right )}{8 d}-\frac {\left (9 b^2 e^3\right ) \operatorname {Subst}\left (\int x \left (a+b \sinh ^{-1}(x)\right )^2 \, dx,x,c+d x\right )}{8 d}-\frac {\left (3 b^3 e^3\right ) \operatorname {Subst}\left (\int \frac {x^4 \left (a+b \sinh ^{-1}(x)\right )}{\sqrt {1+x^2}} \, dx,x,c+d x\right )}{8 d}\\ &=-\frac {3 b^3 e^3 (c+d x)^3 \sqrt {1+(c+d x)^2} \left (a+b \sinh ^{-1}(c+d x)\right )}{32 d}-\frac {9 b^2 e^3 (c+d x)^2 \left (a+b \sinh ^{-1}(c+d x)\right )^2}{16 d}+\frac {3 b^2 e^3 (c+d x)^4 \left (a+b \sinh ^{-1}(c+d x)\right )^2}{16 d}+\frac {3 b e^3 (c+d x) \sqrt {1+(c+d x)^2} \left (a+b \sinh ^{-1}(c+d x)\right )^3}{8 d}-\frac {b e^3 (c+d x)^3 \sqrt {1+(c+d x)^2} \left (a+b \sinh ^{-1}(c+d x)\right )^3}{4 d}-\frac {3 e^3 \left (a+b \sinh ^{-1}(c+d x)\right )^4}{32 d}+\frac {e^3 (c+d x)^4 \left (a+b \sinh ^{-1}(c+d x)\right )^4}{4 d}+\frac {\left (9 b^3 e^3\right ) \operatorname {Subst}\left (\int \frac {x^2 \left (a+b \sinh ^{-1}(x)\right )}{\sqrt {1+x^2}} \, dx,x,c+d x\right )}{32 d}+\frac {\left (9 b^3 e^3\right ) \operatorname {Subst}\left (\int \frac {x^2 \left (a+b \sinh ^{-1}(x)\right )}{\sqrt {1+x^2}} \, dx,x,c+d x\right )}{8 d}+\frac {\left (3 b^4 e^3\right ) \operatorname {Subst}\left (\int x^3 \, dx,x,c+d x\right )}{32 d}\\ &=\frac {3 b^4 e^3 (c+d x)^4}{128 d}+\frac {45 b^3 e^3 (c+d x) \sqrt {1+(c+d x)^2} \left (a+b \sinh ^{-1}(c+d x)\right )}{64 d}-\frac {3 b^3 e^3 (c+d x)^3 \sqrt {1+(c+d x)^2} \left (a+b \sinh ^{-1}(c+d x)\right )}{32 d}-\frac {9 b^2 e^3 (c+d x)^2 \left (a+b \sinh ^{-1}(c+d x)\right )^2}{16 d}+\frac {3 b^2 e^3 (c+d x)^4 \left (a+b \sinh ^{-1}(c+d x)\right )^2}{16 d}+\frac {3 b e^3 (c+d x) \sqrt {1+(c+d x)^2} \left (a+b \sinh ^{-1}(c+d x)\right )^3}{8 d}-\frac {b e^3 (c+d x)^3 \sqrt {1+(c+d x)^2} \left (a+b \sinh ^{-1}(c+d x)\right )^3}{4 d}-\frac {3 e^3 \left (a+b \sinh ^{-1}(c+d x)\right )^4}{32 d}+\frac {e^3 (c+d x)^4 \left (a+b \sinh ^{-1}(c+d x)\right )^4}{4 d}-\frac {\left (9 b^3 e^3\right ) \operatorname {Subst}\left (\int \frac {a+b \sinh ^{-1}(x)}{\sqrt {1+x^2}} \, dx,x,c+d x\right )}{64 d}-\frac {\left (9 b^3 e^3\right ) \operatorname {Subst}\left (\int \frac {a+b \sinh ^{-1}(x)}{\sqrt {1+x^2}} \, dx,x,c+d x\right )}{16 d}-\frac {\left (9 b^4 e^3\right ) \operatorname {Subst}(\int x \, dx,x,c+d x)}{64 d}-\frac {\left (9 b^4 e^3\right ) \operatorname {Subst}(\int x \, dx,x,c+d x)}{16 d}\\ &=-\frac {45 b^4 e^3 (c+d x)^2}{128 d}+\frac {3 b^4 e^3 (c+d x)^4}{128 d}+\frac {45 b^3 e^3 (c+d x) \sqrt {1+(c+d x)^2} \left (a+b \sinh ^{-1}(c+d x)\right )}{64 d}-\frac {3 b^3 e^3 (c+d x)^3 \sqrt {1+(c+d x)^2} \left (a+b \sinh ^{-1}(c+d x)\right )}{32 d}-\frac {45 b^2 e^3 \left (a+b \sinh ^{-1}(c+d x)\right )^2}{128 d}-\frac {9 b^2 e^3 (c+d x)^2 \left (a+b \sinh ^{-1}(c+d x)\right )^2}{16 d}+\frac {3 b^2 e^3 (c+d x)^4 \left (a+b \sinh ^{-1}(c+d x)\right )^2}{16 d}+\frac {3 b e^3 (c+d x) \sqrt {1+(c+d x)^2} \left (a+b \sinh ^{-1}(c+d x)\right )^3}{8 d}-\frac {b e^3 (c+d x)^3 \sqrt {1+(c+d x)^2} \left (a+b \sinh ^{-1}(c+d x)\right )^3}{4 d}-\frac {3 e^3 \left (a+b \sinh ^{-1}(c+d x)\right )^4}{32 d}+\frac {e^3 (c+d x)^4 \left (a+b \sinh ^{-1}(c+d x)\right )^4}{4 d}\\ \end {align*}

________________________________________________________________________________________

Mathematica [A]  time = 0.64, size = 475, normalized size = 1.36 \[ \frac {e^3 \left (-9 b^2 \left (8 a^2+5 b^2\right ) (c+d x)^2+2 a b \sqrt {(c+d x)^2+1} (c+d x) \left (-2 \left (8 a^2+3 b^2\right ) (c+d x)^2+24 a^2+45 b^2\right )+3 b^2 \sinh ^{-1}(c+d x)^2 \left (64 a^2 (c+d x)^4-24 a^2-32 a b \sqrt {(c+d x)^2+1} (c+d x)^3+48 a b \sqrt {(c+d x)^2+1} (c+d x)+8 b^2 (c+d x)^4-24 b^2 (c+d x)^2-15 b^2\right )-6 a b \left (8 a^2+15 b^2\right ) \sinh ^{-1}(c+d x)+\left (32 a^4+24 a^2 b^2+3 b^4\right ) (c+d x)^4+2 b (c+d x) \sinh ^{-1}(c+d x) \left (64 a^3 (c+d x)^3+72 a^2 b \sqrt {(c+d x)^2+1}-48 a^2 b (c+d x)^2 \sqrt {(c+d x)^2+1}+24 a b^2 (c+d x)^3-72 a b^2 (c+d x)-6 b^3 (c+d x)^2 \sqrt {(c+d x)^2+1}+45 b^3 \sqrt {(c+d x)^2+1}\right )+16 b^3 \sinh ^{-1}(c+d x)^3 \left (8 a (c+d x)^4-3 a-2 b \sqrt {(c+d x)^2+1} (c+d x)^3+3 b \sqrt {(c+d x)^2+1} (c+d x)\right )+4 b^4 \left (8 (c+d x)^4-3\right ) \sinh ^{-1}(c+d x)^4\right )}{128 d} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(e^3*(-9*b^2*(8*a^2 + 5*b^2)*(c + d*x)^2 + (32*a^4 + 24*a^2*b^2 + 3*b^4)*(c + d*x)^4 + 2*a*b*(c + d*x)*Sqrt[1
+ (c + d*x)^2]*(24*a^2 + 45*b^2 - 2*(8*a^2 + 3*b^2)*(c + d*x)^2) - 6*a*b*(8*a^2 + 15*b^2)*ArcSinh[c + d*x] + 2
*b*(c + d*x)*(-72*a*b^2*(c + d*x) + 64*a^3*(c + d*x)^3 + 24*a*b^2*(c + d*x)^3 + 72*a^2*b*Sqrt[1 + (c + d*x)^2]
+ 45*b^3*Sqrt[1 + (c + d*x)^2] - 48*a^2*b*(c + d*x)^2*Sqrt[1 + (c + d*x)^2] - 6*b^3*(c + d*x)^2*Sqrt[1 + (c +
d*x)^2])*ArcSinh[c + d*x] + 3*b^2*(-24*a^2 - 15*b^2 - 24*b^2*(c + d*x)^2 + 64*a^2*(c + d*x)^4 + 8*b^2*(c + d*
x)^4 + 48*a*b*(c + d*x)*Sqrt[1 + (c + d*x)^2] - 32*a*b*(c + d*x)^3*Sqrt[1 + (c + d*x)^2])*ArcSinh[c + d*x]^2 +
16*b^3*(-3*a + 8*a*(c + d*x)^4 + 3*b*(c + d*x)*Sqrt[1 + (c + d*x)^2] - 2*b*(c + d*x)^3*Sqrt[1 + (c + d*x)^2])
*ArcSinh[c + d*x]^3 + 4*b^4*(-3 + 8*(c + d*x)^4)*ArcSinh[c + d*x]^4))/(128*d)

________________________________________________________________________________________

fricas [B]  time = 0.59, size = 1241, normalized size = 3.56 \[ \text {result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/128*((32*a^4 + 24*a^2*b^2 + 3*b^4)*d^4*e^3*x^4 + 4*(32*a^4 + 24*a^2*b^2 + 3*b^4)*c*d^3*e^3*x^3 - 3*(24*a^2*b
^2 + 15*b^4 - 2*(32*a^4 + 24*a^2*b^2 + 3*b^4)*c^2)*d^2*e^3*x^2 + 2*(2*(32*a^4 + 24*a^2*b^2 + 3*b^4)*c^3 - 9*(8
*a^2*b^2 + 5*b^4)*c)*d*e^3*x + 4*(8*b^4*d^4*e^3*x^4 + 32*b^4*c*d^3*e^3*x^3 + 48*b^4*c^2*d^2*e^3*x^2 + 32*b^4*c
^3*d*e^3*x + (8*b^4*c^4 - 3*b^4)*e^3)*log(d*x + c + sqrt(d^2*x^2 + 2*c*d*x + c^2 + 1))^4 + 16*(8*a*b^3*d^4*e^3
*x^4 + 32*a*b^3*c*d^3*e^3*x^3 + 48*a*b^3*c^2*d^2*e^3*x^2 + 32*a*b^3*c^3*d*e^3*x + (8*a*b^3*c^4 - 3*a*b^3)*e^3
- (2*b^4*d^3*e^3*x^3 + 6*b^4*c*d^2*e^3*x^2 + 3*(2*b^4*c^2 - b^4)*d*e^3*x + (2*b^4*c^3 - 3*b^4*c)*e^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))^3 + 3*(8*(8*a^2*b^2 + b^4)*d^4*e^3
*x^4 + 32*(8*a^2*b^2 + b^4)*c*d^3*e^3*x^3 - 24*(b^4 - 2*(8*a^2*b^2 + b^4)*c^2)*d^2*e^3*x^2 - 16*(3*b^4*c - 2*(
8*a^2*b^2 + b^4)*c^3)*d*e^3*x - (24*b^4*c^2 - 8*(8*a^2*b^2 + b^4)*c^4 + 24*a^2*b^2 + 15*b^4)*e^3 - 16*(2*a*b^3
*d^3*e^3*x^3 + 6*a*b^3*c*d^2*e^3*x^2 + 3*(2*a*b^3*c^2 - a*b^3)*d*e^3*x + (2*a*b^3*c^3 - 3*a*b^3*c)*e^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))^2 + 2*(8*(8*a^3*b + 3*a*b^3)*d^4
*e^3*x^4 + 32*(8*a^3*b + 3*a*b^3)*c*d^3*e^3*x^3 - 24*(3*a*b^3 - 2*(8*a^3*b + 3*a*b^3)*c^2)*d^2*e^3*x^2 - 16*(9
*a*b^3*c - 2*(8*a^3*b + 3*a*b^3)*c^3)*d*e^3*x - (72*a*b^3*c^2 - 8*(8*a^3*b + 3*a*b^3)*c^4 + 24*a^3*b + 45*a*b^
3)*e^3 - 3*(2*(8*a^2*b^2 + b^4)*d^3*e^3*x^3 + 6*(8*a^2*b^2 + b^4)*c*d^2*e^3*x^2 - 3*(8*a^2*b^2 + 5*b^4 - 2*(8*
a^2*b^2 + b^4)*c^2)*d*e^3*x + (2*(8*a^2*b^2 + b^4)*c^3 - 3*(8*a^2*b^2 + 5*b^4)*c)*e^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)) - 2*(2*(8*a^3*b + 3*a*b^3)*d^3*e^3*x^3 + 6*(8*a^3
*b + 3*a*b^3)*c*d^2*e^3*x^2 - 3*(8*a^3*b + 15*a*b^3 - 2*(8*a^3*b + 3*a*b^3)*c^2)*d*e^3*x + (2*(8*a^3*b + 3*a*b
^3)*c^3 - 3*(8*a^3*b + 15*a*b^3)*c)*e^3)*sqrt(d^2*x^2 + 2*c*d*x + c^2 + 1))/d

________________________________________________________________________________________

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

maple [A]  time = 0.05, size = 573, normalized size = 1.64 \[ \frac {\frac {\left (d x +c \right )^{4} e^{3} a^{4}}{4}+e^{3} b^{4} \left (\frac {\left (d x +c \right )^{4} \arcsinh \left (d x +c \right )^{4}}{4}-\frac {\left (d x +c \right )^{3} \arcsinh \left (d x +c \right )^{3} \sqrt {1+\left (d x +c \right )^{2}}}{4}+\frac {3 \arcsinh \left (d x +c \right )^{3} \left (d x +c \right ) \sqrt {1+\left (d x +c \right )^{2}}}{8}-\frac {3 \arcsinh \left (d x +c \right )^{4}}{32}+\frac {3 \left (d x +c \right )^{4} \arcsinh \left (d x +c \right )^{2}}{16}-\frac {3 \left (d x +c \right )^{3} \arcsinh \left (d x +c \right ) \sqrt {1+\left (d x +c \right )^{2}}}{32}+\frac {45 \arcsinh \left (d x +c \right ) \sqrt {1+\left (d x +c \right )^{2}}\, \left (d x +c \right )}{64}+\frac {27 \arcsinh \left (d x +c \right )^{2}}{128}+\frac {3 \left (d x +c \right )^{4}}{128}-\frac {45 \left (d x +c \right )^{2}}{128}-\frac {45}{128}-\frac {9 \arcsinh \left (d x +c \right )^{2} \left (1+\left (d x +c \right )^{2}\right )}{16}\right )+4 e^{3} a \,b^{3} \left (\frac {\left (d x +c \right )^{4} \arcsinh \left (d x +c \right )^{3}}{4}-\frac {3 \left (d x +c \right )^{3} \arcsinh \left (d x +c \right )^{2} \sqrt {1+\left (d x +c \right )^{2}}}{16}+\frac {9 \arcsinh \left (d x +c \right )^{2} \sqrt {1+\left (d x +c \right )^{2}}\, \left (d x +c \right )}{32}-\frac {3 \arcsinh \left (d x +c \right )^{3}}{32}+\frac {3 \left (d x +c \right )^{4} \arcsinh \left (d x +c \right )}{32}-\frac {3 \left (d x +c \right )^{3} \sqrt {1+\left (d x +c \right )^{2}}}{128}+\frac {45 \left (d x +c \right ) \sqrt {1+\left (d x +c \right )^{2}}}{256}+\frac {27 \arcsinh \left (d x +c \right )}{256}-\frac {9 \arcsinh \left (d x +c \right ) \left (1+\left (d x +c \right )^{2}\right )}{32}\right )+6 e^{3} a^{2} b^{2} \left (\frac {\left (d x +c \right )^{4} \arcsinh \left (d x +c \right )^{2}}{4}-\frac {\left (d x +c \right )^{3} \arcsinh \left (d x +c \right ) \sqrt {1+\left (d x +c \right )^{2}}}{8}+\frac {3 \arcsinh \left (d x +c \right ) \sqrt {1+\left (d x +c \right )^{2}}\, \left (d x +c \right )}{16}-\frac {3 \arcsinh \left (d x +c \right )^{2}}{32}+\frac {\left (d x +c \right )^{4}}{32}-\frac {3 \left (d x +c \right )^{2}}{32}-\frac {3}{32}\right )+4 e^{3} a^{3} b \left (\frac {\left (d x +c \right )^{4} \arcsinh \left (d x +c \right )}{4}-\frac {\left (d x +c \right )^{3} \sqrt {1+\left (d x +c \right )^{2}}}{16}+\frac {3 \left (d x +c \right ) \sqrt {1+\left (d x +c \right )^{2}}}{32}-\frac {3 \arcsinh \left (d x +c \right )}{32}\right )}{d} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/d*(1/4*(d*x+c)^4*e^3*a^4+e^3*b^4*(1/4*(d*x+c)^4*arcsinh(d*x+c)^4-1/4*(d*x+c)^3*arcsinh(d*x+c)^3*(1+(d*x+c)^2
)^(1/2)+3/8*arcsinh(d*x+c)^3*(d*x+c)*(1+(d*x+c)^2)^(1/2)-3/32*arcsinh(d*x+c)^4+3/16*(d*x+c)^4*arcsinh(d*x+c)^2
-3/32*(d*x+c)^3*arcsinh(d*x+c)*(1+(d*x+c)^2)^(1/2)+45/64*arcsinh(d*x+c)*(1+(d*x+c)^2)^(1/2)*(d*x+c)+27/128*arc
sinh(d*x+c)^2+3/128*(d*x+c)^4-45/128*(d*x+c)^2-45/128-9/16*arcsinh(d*x+c)^2*(1+(d*x+c)^2))+4*e^3*a*b^3*(1/4*(d
*x+c)^4*arcsinh(d*x+c)^3-3/16*(d*x+c)^3*arcsinh(d*x+c)^2*(1+(d*x+c)^2)^(1/2)+9/32*arcsinh(d*x+c)^2*(1+(d*x+c)^
2)^(1/2)*(d*x+c)-3/32*arcsinh(d*x+c)^3+3/32*(d*x+c)^4*arcsinh(d*x+c)-3/128*(d*x+c)^3*(1+(d*x+c)^2)^(1/2)+45/25
6*(d*x+c)*(1+(d*x+c)^2)^(1/2)+27/256*arcsinh(d*x+c)-9/32*arcsinh(d*x+c)*(1+(d*x+c)^2))+6*e^3*a^2*b^2*(1/4*(d*x
+c)^4*arcsinh(d*x+c)^2-1/8*(d*x+c)^3*arcsinh(d*x+c)*(1+(d*x+c)^2)^(1/2)+3/16*arcsinh(d*x+c)*(1+(d*x+c)^2)^(1/2
)*(d*x+c)-3/32*arcsinh(d*x+c)^2+1/32*(d*x+c)^4-3/32*(d*x+c)^2-3/32)+4*e^3*a^3*b*(1/4*(d*x+c)^4*arcsinh(d*x+c)-
1/16*(d*x+c)^3*(1+(d*x+c)^2)^(1/2)+3/32*(d*x+c)*(1+(d*x+c)^2)^(1/2)-3/32*arcsinh(d*x+c)))

________________________________________________________________________________________

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

[Out]

1/4*a^4*d^3*e^3*x^4 + a^4*c*d^2*e^3*x^3 + 3/2*a^4*c^2*d*e^3*x^2 + 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^3*b*c^2*d*e^3 + 2/3*(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*arcsi
nh(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^3*b*c*d^2*e^3 + 1/24*(24*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^3
*b*d^3*e^3 + a^4*c^3*e^3*x + 4*((d*x + c)*arcsinh(d*x + c) - sqrt((d*x + c)^2 + 1))*a^3*b*c^3*e^3/d + 1/4*(b^4
*d^3*e^3*x^4 + 4*b^4*c*d^2*e^3*x^3 + 6*b^4*c^2*d*e^3*x^2 + 4*b^4*c^3*e^3*x)*log(d*x + c + sqrt(d^2*x^2 + 2*c*d
*x + c^2 + 1))^4 + integrate((((4*a*b^3*d^6*e^3 - b^4*d^6*e^3)*x^6 + 6*(4*a*b^3*c*d^5*e^3 - b^4*c*d^5*e^3)*x^5
+ 4*(c^6*e^3 + c^4*e^3)*a*b^3 + (4*(15*c^2*d^4*e^3 + d^4*e^3)*a*b^3 - (15*c^2*d^4*e^3 + d^4*e^3)*b^4)*x^4 + 4
*(4*(5*c^3*d^3*e^3 + c*d^3*e^3)*a*b^3 - (5*c^3*d^3*e^3 + c*d^3*e^3)*b^4)*x^3 + 2*(6*(5*c^4*d^2*e^3 + 2*c^2*d^2
*e^3)*a*b^3 - (7*c^4*d^2*e^3 + 3*c^2*d^2*e^3)*b^4)*x^2 + 4*(2*(3*c^5*d*e^3 + 2*c^3*d*e^3)*a*b^3 - (c^5*d*e^3 +
c^3*d*e^3)*b^4)*x + ((4*a*b^3*d^5*e^3 - b^4*d^5*e^3)*x^5 + 4*(c^5*e^3 + c^3*e^3)*a*b^3 + 5*(4*a*b^3*c*d^4*e^3
- b^4*c*d^4*e^3)*x^4 - 2*(5*b^4*c^2*d^3*e^3 - 2*(10*c^2*d^3*e^3 + d^3*e^3)*a*b^3)*x^3 - 2*(5*b^4*c^3*d^2*e^3
- 2*(10*c^3*d^2*e^3 + 3*c*d^2*e^3)*a*b^3)*x^2 - 4*(b^4*c^4*d*e^3 - (5*c^4*d*e^3 + 3*c^2*d*e^3)*a*b^3)*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 + 6*(a^2*b^2*d^6*e^3*x^6 + 6*
a^2*b^2*c*d^5*e^3*x^5 + (15*c^2*d^4*e^3 + d^4*e^3)*a^2*b^2*x^4 + 4*(5*c^3*d^3*e^3 + c*d^3*e^3)*a^2*b^2*x^3 + 3
*(5*c^4*d^2*e^3 + 2*c^2*d^2*e^3)*a^2*b^2*x^2 + 2*(3*c^5*d*e^3 + 2*c^3*d*e^3)*a^2*b^2*x + (c^6*e^3 + c^4*e^3)*a
^2*b^2 + (a^2*b^2*d^5*e^3*x^5 + 5*a^2*b^2*c*d^4*e^3*x^4 + (10*c^2*d^3*e^3 + d^3*e^3)*a^2*b^2*x^3 + (10*c^3*d^2
*e^3 + 3*c*d^2*e^3)*a^2*b^2*x^2 + (5*c^4*d*e^3 + 3*c^2*d*e^3)*a^2*b^2*x + (c^5*e^3 + c^3*e^3)*a^2*b^2)*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)

________________________________________________________________________________________

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

sympy [A]  time = 20.60, size = 2876, normalized size = 8.24 \[ \text {result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Piecewise((a**4*c**3*e**3*x + 3*a**4*c**2*d*e**3*x**2/2 + a**4*c*d**2*e**3*x**3 + a**4*d**3*e**3*x**4/4 + a**3
*b*c**4*e**3*asinh(c + d*x)/d + 4*a**3*b*c**3*e**3*x*asinh(c + d*x) - a**3*b*c**3*e**3*sqrt(c**2 + 2*c*d*x + d
**2*x**2 + 1)/(4*d) + 6*a**3*b*c**2*d*e**3*x**2*asinh(c + d*x) - 3*a**3*b*c**2*e**3*x*sqrt(c**2 + 2*c*d*x + d*
*2*x**2 + 1)/4 + 4*a**3*b*c*d**2*e**3*x**3*asinh(c + d*x) - 3*a**3*b*c*d*e**3*x**2*sqrt(c**2 + 2*c*d*x + d**2*
x**2 + 1)/4 + 3*a**3*b*c*e**3*sqrt(c**2 + 2*c*d*x + d**2*x**2 + 1)/(8*d) + a**3*b*d**3*e**3*x**4*asinh(c + d*x
) - a**3*b*d**2*e**3*x**3*sqrt(c**2 + 2*c*d*x + d**2*x**2 + 1)/4 + 3*a**3*b*e**3*x*sqrt(c**2 + 2*c*d*x + d**2*
x**2 + 1)/8 - 3*a**3*b*e**3*asinh(c + d*x)/(8*d) + 3*a**2*b**2*c**4*e**3*asinh(c + d*x)**2/(2*d) + 6*a**2*b**2
*c**3*e**3*x*asinh(c + d*x)**2 + 3*a**2*b**2*c**3*e**3*x/4 - 3*a**2*b**2*c**3*e**3*sqrt(c**2 + 2*c*d*x + d**2*
x**2 + 1)*asinh(c + d*x)/(4*d) + 9*a**2*b**2*c**2*d*e**3*x**2*asinh(c + d*x)**2 + 9*a**2*b**2*c**2*d*e**3*x**2
/8 - 9*a**2*b**2*c**2*e**3*x*sqrt(c**2 + 2*c*d*x + d**2*x**2 + 1)*asinh(c + d*x)/4 + 6*a**2*b**2*c*d**2*e**3*x
**3*asinh(c + d*x)**2 + 3*a**2*b**2*c*d**2*e**3*x**3/4 - 9*a**2*b**2*c*d*e**3*x**2*sqrt(c**2 + 2*c*d*x + d**2*
x**2 + 1)*asinh(c + d*x)/4 - 9*a**2*b**2*c*e**3*x/8 + 9*a**2*b**2*c*e**3*sqrt(c**2 + 2*c*d*x + d**2*x**2 + 1)*
asinh(c + d*x)/(8*d) + 3*a**2*b**2*d**3*e**3*x**4*asinh(c + d*x)**2/2 + 3*a**2*b**2*d**3*e**3*x**4/16 - 3*a**2
*b**2*d**2*e**3*x**3*sqrt(c**2 + 2*c*d*x + d**2*x**2 + 1)*asinh(c + d*x)/4 - 9*a**2*b**2*d*e**3*x**2/16 + 9*a*
*2*b**2*e**3*x*sqrt(c**2 + 2*c*d*x + d**2*x**2 + 1)*asinh(c + d*x)/8 - 9*a**2*b**2*e**3*asinh(c + d*x)**2/(16*
d) + a*b**3*c**4*e**3*asinh(c + d*x)**3/d + 3*a*b**3*c**4*e**3*asinh(c + d*x)/(8*d) + 4*a*b**3*c**3*e**3*x*asi
nh(c + d*x)**3 + 3*a*b**3*c**3*e**3*x*asinh(c + d*x)/2 - 3*a*b**3*c**3*e**3*sqrt(c**2 + 2*c*d*x + d**2*x**2 +
1)*asinh(c + d*x)**2/(4*d) - 3*a*b**3*c**3*e**3*sqrt(c**2 + 2*c*d*x + d**2*x**2 + 1)/(32*d) + 6*a*b**3*c**2*d*
e**3*x**2*asinh(c + d*x)**3 + 9*a*b**3*c**2*d*e**3*x**2*asinh(c + d*x)/4 - 9*a*b**3*c**2*e**3*x*sqrt(c**2 + 2*
c*d*x + d**2*x**2 + 1)*asinh(c + d*x)**2/4 - 9*a*b**3*c**2*e**3*x*sqrt(c**2 + 2*c*d*x + d**2*x**2 + 1)/32 - 9*
a*b**3*c**2*e**3*asinh(c + d*x)/(8*d) + 4*a*b**3*c*d**2*e**3*x**3*asinh(c + d*x)**3 + 3*a*b**3*c*d**2*e**3*x**
3*asinh(c + d*x)/2 - 9*a*b**3*c*d*e**3*x**2*sqrt(c**2 + 2*c*d*x + d**2*x**2 + 1)*asinh(c + d*x)**2/4 - 9*a*b**
3*c*d*e**3*x**2*sqrt(c**2 + 2*c*d*x + d**2*x**2 + 1)/32 - 9*a*b**3*c*e**3*x*asinh(c + d*x)/4 + 9*a*b**3*c*e**3
*sqrt(c**2 + 2*c*d*x + d**2*x**2 + 1)*asinh(c + d*x)**2/(8*d) + 45*a*b**3*c*e**3*sqrt(c**2 + 2*c*d*x + d**2*x*
*2 + 1)/(64*d) + a*b**3*d**3*e**3*x**4*asinh(c + d*x)**3 + 3*a*b**3*d**3*e**3*x**4*asinh(c + d*x)/8 - 3*a*b**3
*d**2*e**3*x**3*sqrt(c**2 + 2*c*d*x + d**2*x**2 + 1)*asinh(c + d*x)**2/4 - 3*a*b**3*d**2*e**3*x**3*sqrt(c**2 +
2*c*d*x + d**2*x**2 + 1)/32 - 9*a*b**3*d*e**3*x**2*asinh(c + d*x)/8 + 9*a*b**3*e**3*x*sqrt(c**2 + 2*c*d*x + d
**2*x**2 + 1)*asinh(c + d*x)**2/8 + 45*a*b**3*e**3*x*sqrt(c**2 + 2*c*d*x + d**2*x**2 + 1)/64 - 3*a*b**3*e**3*a
sinh(c + d*x)**3/(8*d) - 45*a*b**3*e**3*asinh(c + d*x)/(64*d) + b**4*c**4*e**3*asinh(c + d*x)**4/(4*d) + 3*b**
4*c**4*e**3*asinh(c + d*x)**2/(16*d) + b**4*c**3*e**3*x*asinh(c + d*x)**4 + 3*b**4*c**3*e**3*x*asinh(c + d*x)*
*2/4 + 3*b**4*c**3*e**3*x/32 - b**4*c**3*e**3*sqrt(c**2 + 2*c*d*x + d**2*x**2 + 1)*asinh(c + d*x)**3/(4*d) - 3
*b**4*c**3*e**3*sqrt(c**2 + 2*c*d*x + d**2*x**2 + 1)*asinh(c + d*x)/(32*d) + 3*b**4*c**2*d*e**3*x**2*asinh(c +
d*x)**4/2 + 9*b**4*c**2*d*e**3*x**2*asinh(c + d*x)**2/8 + 9*b**4*c**2*d*e**3*x**2/64 - 3*b**4*c**2*e**3*x*sqr
t(c**2 + 2*c*d*x + d**2*x**2 + 1)*asinh(c + d*x)**3/4 - 9*b**4*c**2*e**3*x*sqrt(c**2 + 2*c*d*x + d**2*x**2 + 1
)*asinh(c + d*x)/32 - 9*b**4*c**2*e**3*asinh(c + d*x)**2/(16*d) + b**4*c*d**2*e**3*x**3*asinh(c + d*x)**4 + 3*
b**4*c*d**2*e**3*x**3*asinh(c + d*x)**2/4 + 3*b**4*c*d**2*e**3*x**3/32 - 3*b**4*c*d*e**3*x**2*sqrt(c**2 + 2*c*
d*x + d**2*x**2 + 1)*asinh(c + d*x)**3/4 - 9*b**4*c*d*e**3*x**2*sqrt(c**2 + 2*c*d*x + d**2*x**2 + 1)*asinh(c +
d*x)/32 - 9*b**4*c*e**3*x*asinh(c + d*x)**2/8 - 45*b**4*c*e**3*x/64 + 3*b**4*c*e**3*sqrt(c**2 + 2*c*d*x + d**
2*x**2 + 1)*asinh(c + d*x)**3/(8*d) + 45*b**4*c*e**3*sqrt(c**2 + 2*c*d*x + d**2*x**2 + 1)*asinh(c + d*x)/(64*d
) + b**4*d**3*e**3*x**4*asinh(c + d*x)**4/4 + 3*b**4*d**3*e**3*x**4*asinh(c + d*x)**2/16 + 3*b**4*d**3*e**3*x*
*4/128 - b**4*d**2*e**3*x**3*sqrt(c**2 + 2*c*d*x + d**2*x**2 + 1)*asinh(c + d*x)**3/4 - 3*b**4*d**2*e**3*x**3*
sqrt(c**2 + 2*c*d*x + d**2*x**2 + 1)*asinh(c + d*x)/32 - 9*b**4*d*e**3*x**2*asinh(c + d*x)**2/16 - 45*b**4*d*e
**3*x**2/128 + 3*b**4*e**3*x*sqrt(c**2 + 2*c*d*x + d**2*x**2 + 1)*asinh(c + d*x)**3/8 + 45*b**4*e**3*x*sqrt(c*
*2 + 2*c*d*x + d**2*x**2 + 1)*asinh(c + d*x)/64 - 3*b**4*e**3*asinh(c + d*x)**4/(32*d) - 45*b**4*e**3*asinh(c
+ d*x)**2/(128*d), Ne(d, 0)), (c**3*e**3*x*(a + b*asinh(c))**4, True))

________________________________________________________________________________________