3.148 \(\int (c e+d e x)^2 (a+b \sinh ^{-1}(c+d x))^4 \, dx\)

Optimal. Leaf size=281 \[ \frac {160 b^3 e^2 \sqrt {(c+d x)^2+1} \left (a+b \sinh ^{-1}(c+d x)\right )}{27 d}-\frac {8 b^3 e^2 (c+d x)^2 \sqrt {(c+d x)^2+1} \left (a+b \sinh ^{-1}(c+d x)\right )}{27 d}+\frac {4 b^2 e^2 (c+d x)^3 \left (a+b \sinh ^{-1}(c+d x)\right )^2}{9 d}-\frac {8 b^2 e^2 (c+d x) \left (a+b \sinh ^{-1}(c+d x)\right )^2}{3 d}+\frac {8 b e^2 \sqrt {(c+d x)^2+1} \left (a+b \sinh ^{-1}(c+d x)\right )^3}{9 d}-\frac {4 b e^2 (c+d x)^2 \sqrt {(c+d x)^2+1} \left (a+b \sinh ^{-1}(c+d x)\right )^3}{9 d}+\frac {e^2 (c+d x)^3 \left (a+b \sinh ^{-1}(c+d x)\right )^4}{3 d}+\frac {8 b^4 e^2 (c+d x)^3}{81 d}-\frac {160}{27} b^4 e^2 x \]

[Out]

-160/27*b^4*e^2*x+8/81*b^4*e^2*(d*x+c)^3/d-8/3*b^2*e^2*(d*x+c)*(a+b*arcsinh(d*x+c))^2/d+4/9*b^2*e^2*(d*x+c)^3*
(a+b*arcsinh(d*x+c))^2/d+1/3*e^2*(d*x+c)^3*(a+b*arcsinh(d*x+c))^4/d+160/27*b^3*e^2*(a+b*arcsinh(d*x+c))*(1+(d*
x+c)^2)^(1/2)/d-8/27*b^3*e^2*(d*x+c)^2*(a+b*arcsinh(d*x+c))*(1+(d*x+c)^2)^(1/2)/d+8/9*b*e^2*(a+b*arcsinh(d*x+c
))^3*(1+(d*x+c)^2)^(1/2)/d-4/9*b*e^2*(d*x+c)^2*(a+b*arcsinh(d*x+c))^3*(1+(d*x+c)^2)^(1/2)/d

________________________________________________________________________________________

Rubi [A]  time = 0.48, antiderivative size = 281, normalized size of antiderivative = 1.00, number of steps used = 13, number of rules used = 8, integrand size = 23, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.348, Rules used = {5865, 12, 5661, 5758, 5717, 5653, 8, 30} \[ \frac {160 b^3 e^2 \sqrt {(c+d x)^2+1} \left (a+b \sinh ^{-1}(c+d x)\right )}{27 d}-\frac {8 b^3 e^2 (c+d x)^2 \sqrt {(c+d x)^2+1} \left (a+b \sinh ^{-1}(c+d x)\right )}{27 d}+\frac {4 b^2 e^2 (c+d x)^3 \left (a+b \sinh ^{-1}(c+d x)\right )^2}{9 d}-\frac {8 b^2 e^2 (c+d x) \left (a+b \sinh ^{-1}(c+d x)\right )^2}{3 d}+\frac {8 b e^2 \sqrt {(c+d x)^2+1} \left (a+b \sinh ^{-1}(c+d x)\right )^3}{9 d}-\frac {4 b e^2 (c+d x)^2 \sqrt {(c+d x)^2+1} \left (a+b \sinh ^{-1}(c+d x)\right )^3}{9 d}+\frac {e^2 (c+d x)^3 \left (a+b \sinh ^{-1}(c+d x)\right )^4}{3 d}+\frac {8 b^4 e^2 (c+d x)^3}{81 d}-\frac {160}{27} b^4 e^2 x \]

Antiderivative was successfully verified.

[In]

Int[(c*e + d*e*x)^2*(a + b*ArcSinh[c + d*x])^4,x]

[Out]

(-160*b^4*e^2*x)/27 + (8*b^4*e^2*(c + d*x)^3)/(81*d) + (160*b^3*e^2*Sqrt[1 + (c + d*x)^2]*(a + b*ArcSinh[c + d
*x]))/(27*d) - (8*b^3*e^2*(c + d*x)^2*Sqrt[1 + (c + d*x)^2]*(a + b*ArcSinh[c + d*x]))/(27*d) - (8*b^2*e^2*(c +
d*x)*(a + b*ArcSinh[c + d*x])^2)/(3*d) + (4*b^2*e^2*(c + d*x)^3*(a + b*ArcSinh[c + d*x])^2)/(9*d) + (8*b*e^2*
Sqrt[1 + (c + d*x)^2]*(a + b*ArcSinh[c + d*x])^3)/(9*d) - (4*b*e^2*(c + d*x)^2*Sqrt[1 + (c + d*x)^2]*(a + b*Ar
cSinh[c + d*x])^3)/(9*d) + (e^2*(c + d*x)^3*(a + b*ArcSinh[c + d*x])^4)/(3*d)

Rule 8

Int[a_, x_Symbol] :> Simp[a*x, x] /; FreeQ[a, x]

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 5653

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 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 5717

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*d^IntPart[p]*(d + e*x^2)^FracPart[p])/(2*c*(p +
1)*(1 + c^2*x^2)^FracPart[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 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)^2 \left (a+b \sinh ^{-1}(c+d x)\right )^4 \, dx &=\frac {\operatorname {Subst}\left (\int e^2 x^2 \left (a+b \sinh ^{-1}(x)\right )^4 \, dx,x,c+d x\right )}{d}\\ &=\frac {e^2 \operatorname {Subst}\left (\int x^2 \left (a+b \sinh ^{-1}(x)\right )^4 \, dx,x,c+d x\right )}{d}\\ &=\frac {e^2 (c+d x)^3 \left (a+b \sinh ^{-1}(c+d x)\right )^4}{3 d}-\frac {\left (4 b e^2\right ) \operatorname {Subst}\left (\int \frac {x^3 \left (a+b \sinh ^{-1}(x)\right )^3}{\sqrt {1+x^2}} \, dx,x,c+d x\right )}{3 d}\\ &=-\frac {4 b e^2 (c+d x)^2 \sqrt {1+(c+d x)^2} \left (a+b \sinh ^{-1}(c+d x)\right )^3}{9 d}+\frac {e^2 (c+d x)^3 \left (a+b \sinh ^{-1}(c+d x)\right )^4}{3 d}+\frac {\left (8 b e^2\right ) \operatorname {Subst}\left (\int \frac {x \left (a+b \sinh ^{-1}(x)\right )^3}{\sqrt {1+x^2}} \, dx,x,c+d x\right )}{9 d}+\frac {\left (4 b^2 e^2\right ) \operatorname {Subst}\left (\int x^2 \left (a+b \sinh ^{-1}(x)\right )^2 \, dx,x,c+d x\right )}{3 d}\\ &=\frac {4 b^2 e^2 (c+d x)^3 \left (a+b \sinh ^{-1}(c+d x)\right )^2}{9 d}+\frac {8 b e^2 \sqrt {1+(c+d x)^2} \left (a+b \sinh ^{-1}(c+d x)\right )^3}{9 d}-\frac {4 b e^2 (c+d x)^2 \sqrt {1+(c+d x)^2} \left (a+b \sinh ^{-1}(c+d x)\right )^3}{9 d}+\frac {e^2 (c+d x)^3 \left (a+b \sinh ^{-1}(c+d x)\right )^4}{3 d}-\frac {\left (8 b^2 e^2\right ) \operatorname {Subst}\left (\int \left (a+b \sinh ^{-1}(x)\right )^2 \, dx,x,c+d x\right )}{3 d}-\frac {\left (8 b^3 e^2\right ) \operatorname {Subst}\left (\int \frac {x^3 \left (a+b \sinh ^{-1}(x)\right )}{\sqrt {1+x^2}} \, dx,x,c+d x\right )}{9 d}\\ &=-\frac {8 b^3 e^2 (c+d x)^2 \sqrt {1+(c+d x)^2} \left (a+b \sinh ^{-1}(c+d x)\right )}{27 d}-\frac {8 b^2 e^2 (c+d x) \left (a+b \sinh ^{-1}(c+d x)\right )^2}{3 d}+\frac {4 b^2 e^2 (c+d x)^3 \left (a+b \sinh ^{-1}(c+d x)\right )^2}{9 d}+\frac {8 b e^2 \sqrt {1+(c+d x)^2} \left (a+b \sinh ^{-1}(c+d x)\right )^3}{9 d}-\frac {4 b e^2 (c+d x)^2 \sqrt {1+(c+d x)^2} \left (a+b \sinh ^{-1}(c+d x)\right )^3}{9 d}+\frac {e^2 (c+d x)^3 \left (a+b \sinh ^{-1}(c+d x)\right )^4}{3 d}+\frac {\left (16 b^3 e^2\right ) \operatorname {Subst}\left (\int \frac {x \left (a+b \sinh ^{-1}(x)\right )}{\sqrt {1+x^2}} \, dx,x,c+d x\right )}{27 d}+\frac {\left (16 b^3 e^2\right ) \operatorname {Subst}\left (\int \frac {x \left (a+b \sinh ^{-1}(x)\right )}{\sqrt {1+x^2}} \, dx,x,c+d x\right )}{3 d}+\frac {\left (8 b^4 e^2\right ) \operatorname {Subst}\left (\int x^2 \, dx,x,c+d x\right )}{27 d}\\ &=\frac {8 b^4 e^2 (c+d x)^3}{81 d}+\frac {160 b^3 e^2 \sqrt {1+(c+d x)^2} \left (a+b \sinh ^{-1}(c+d x)\right )}{27 d}-\frac {8 b^3 e^2 (c+d x)^2 \sqrt {1+(c+d x)^2} \left (a+b \sinh ^{-1}(c+d x)\right )}{27 d}-\frac {8 b^2 e^2 (c+d x) \left (a+b \sinh ^{-1}(c+d x)\right )^2}{3 d}+\frac {4 b^2 e^2 (c+d x)^3 \left (a+b \sinh ^{-1}(c+d x)\right )^2}{9 d}+\frac {8 b e^2 \sqrt {1+(c+d x)^2} \left (a+b \sinh ^{-1}(c+d x)\right )^3}{9 d}-\frac {4 b e^2 (c+d x)^2 \sqrt {1+(c+d x)^2} \left (a+b \sinh ^{-1}(c+d x)\right )^3}{9 d}+\frac {e^2 (c+d x)^3 \left (a+b \sinh ^{-1}(c+d x)\right )^4}{3 d}-\frac {\left (16 b^4 e^2\right ) \operatorname {Subst}(\int 1 \, dx,x,c+d x)}{27 d}-\frac {\left (16 b^4 e^2\right ) \operatorname {Subst}(\int 1 \, dx,x,c+d x)}{3 d}\\ &=-\frac {160}{27} b^4 e^2 x+\frac {8 b^4 e^2 (c+d x)^3}{81 d}+\frac {160 b^3 e^2 \sqrt {1+(c+d x)^2} \left (a+b \sinh ^{-1}(c+d x)\right )}{27 d}-\frac {8 b^3 e^2 (c+d x)^2 \sqrt {1+(c+d x)^2} \left (a+b \sinh ^{-1}(c+d x)\right )}{27 d}-\frac {8 b^2 e^2 (c+d x) \left (a+b \sinh ^{-1}(c+d x)\right )^2}{3 d}+\frac {4 b^2 e^2 (c+d x)^3 \left (a+b \sinh ^{-1}(c+d x)\right )^2}{9 d}+\frac {8 b e^2 \sqrt {1+(c+d x)^2} \left (a+b \sinh ^{-1}(c+d x)\right )^3}{9 d}-\frac {4 b e^2 (c+d x)^2 \sqrt {1+(c+d x)^2} \left (a+b \sinh ^{-1}(c+d x)\right )^3}{9 d}+\frac {e^2 (c+d x)^3 \left (a+b \sinh ^{-1}(c+d x)\right )^4}{3 d}\\ \end {align*}

________________________________________________________________________________________

Mathematica [A]  time = 0.45, size = 412, normalized size = 1.47 \[ \frac {e^2 \left (-24 b^2 \left (9 a^2+20 b^2\right ) (c+d x)+12 a b \sqrt {(c+d x)^2+1} \left (-\left (3 a^2+2 b^2\right ) (c+d x)^2+6 a^2+40 b^2\right )+18 b^2 \sinh ^{-1}(c+d x)^2 \left (9 a^2 (c+d x)^3-6 a b \sqrt {(c+d x)^2+1} (c+d x)^2+12 a b \sqrt {(c+d x)^2+1}+2 b^2 (c+d x)^3-12 b^2 (c+d x)\right )+\left (27 a^4+36 a^2 b^2+8 b^4\right ) (c+d x)^3+12 b \sinh ^{-1}(c+d x) \left (9 a^3 (c+d x)^3+18 a^2 b \sqrt {(c+d x)^2+1}-9 a^2 b (c+d x)^2 \sqrt {(c+d x)^2+1}+6 a b^2 (c+d x)^3-36 a b^2 (c+d x)-2 b^3 (c+d x)^2 \sqrt {(c+d x)^2+1}+40 b^3 \sqrt {(c+d x)^2+1}\right )-36 b^3 \sinh ^{-1}(c+d x)^3 \left (-3 a (c+d x)^3+b \sqrt {(c+d x)^2+1} (c+d x)^2-2 b \sqrt {(c+d x)^2+1}\right )+27 b^4 (c+d x)^3 \sinh ^{-1}(c+d x)^4\right )}{81 d} \]

Antiderivative was successfully verified.

[In]

Integrate[(c*e + d*e*x)^2*(a + b*ArcSinh[c + d*x])^4,x]

[Out]

(e^2*(-24*b^2*(9*a^2 + 20*b^2)*(c + d*x) + (27*a^4 + 36*a^2*b^2 + 8*b^4)*(c + d*x)^3 + 12*a*b*Sqrt[1 + (c + d*
x)^2]*(6*a^2 + 40*b^2 - (3*a^2 + 2*b^2)*(c + d*x)^2) + 12*b*(-36*a*b^2*(c + d*x) + 9*a^3*(c + d*x)^3 + 6*a*b^2
*(c + d*x)^3 + 18*a^2*b*Sqrt[1 + (c + d*x)^2] + 40*b^3*Sqrt[1 + (c + d*x)^2] - 9*a^2*b*(c + d*x)^2*Sqrt[1 + (c
+ d*x)^2] - 2*b^3*(c + d*x)^2*Sqrt[1 + (c + d*x)^2])*ArcSinh[c + d*x] + 18*b^2*(-12*b^2*(c + d*x) + 9*a^2*(c
+ d*x)^3 + 2*b^2*(c + d*x)^3 + 12*a*b*Sqrt[1 + (c + d*x)^2] - 6*a*b*(c + d*x)^2*Sqrt[1 + (c + d*x)^2])*ArcSinh
[c + d*x]^2 - 36*b^3*(-3*a*(c + d*x)^3 - 2*b*Sqrt[1 + (c + d*x)^2] + b*(c + d*x)^2*Sqrt[1 + (c + d*x)^2])*ArcS
inh[c + d*x]^3 + 27*b^4*(c + d*x)^3*ArcSinh[c + d*x]^4))/(81*d)

________________________________________________________________________________________

fricas [B]  time = 0.75, size = 900, normalized size = 3.20 \[ \frac {{\left (27 \, a^{4} + 36 \, a^{2} b^{2} + 8 \, b^{4}\right )} d^{3} e^{2} x^{3} + 3 \, {\left (27 \, a^{4} + 36 \, a^{2} b^{2} + 8 \, b^{4}\right )} c d^{2} e^{2} x^{2} - 3 \, {\left (72 \, a^{2} b^{2} + 160 \, b^{4} - {\left (27 \, a^{4} + 36 \, a^{2} b^{2} + 8 \, b^{4}\right )} c^{2}\right )} d e^{2} x + 27 \, {\left (b^{4} d^{3} e^{2} x^{3} + 3 \, b^{4} c d^{2} e^{2} x^{2} + 3 \, b^{4} c^{2} d e^{2} x + b^{4} c^{3} e^{2}\right )} \log \left (d x + c + \sqrt {d^{2} x^{2} + 2 \, c d x + c^{2} + 1}\right )^{4} + 36 \, {\left (3 \, a b^{3} d^{3} e^{2} x^{3} + 9 \, a b^{3} c d^{2} e^{2} x^{2} + 9 \, a b^{3} c^{2} d e^{2} x + 3 \, a b^{3} c^{3} e^{2} - {\left (b^{4} d^{2} e^{2} x^{2} + 2 \, b^{4} c d e^{2} x + {\left (b^{4} c^{2} - 2 \, b^{4}\right )} e^{2}\right )} \sqrt {d^{2} x^{2} + 2 \, c d x + c^{2} + 1}\right )} \log \left (d x + c + \sqrt {d^{2} x^{2} + 2 \, c d x + c^{2} + 1}\right )^{3} + 18 \, {\left ({\left (9 \, a^{2} b^{2} + 2 \, b^{4}\right )} d^{3} e^{2} x^{3} + 3 \, {\left (9 \, a^{2} b^{2} + 2 \, b^{4}\right )} c d^{2} e^{2} x^{2} - 3 \, {\left (4 \, b^{4} - {\left (9 \, a^{2} b^{2} + 2 \, b^{4}\right )} c^{2}\right )} d e^{2} x - {\left (12 \, b^{4} c - {\left (9 \, a^{2} b^{2} + 2 \, b^{4}\right )} c^{3}\right )} e^{2} - 6 \, {\left (a b^{3} d^{2} e^{2} x^{2} + 2 \, a b^{3} c d e^{2} x + {\left (a b^{3} c^{2} - 2 \, a b^{3}\right )} e^{2}\right )} \sqrt {d^{2} x^{2} + 2 \, c d x + c^{2} + 1}\right )} \log \left (d x + c + \sqrt {d^{2} x^{2} + 2 \, c d x + c^{2} + 1}\right )^{2} + 12 \, {\left (3 \, {\left (3 \, a^{3} b + 2 \, a b^{3}\right )} d^{3} e^{2} x^{3} + 9 \, {\left (3 \, a^{3} b + 2 \, a b^{3}\right )} c d^{2} e^{2} x^{2} - 9 \, {\left (4 \, a b^{3} - {\left (3 \, a^{3} b + 2 \, a b^{3}\right )} c^{2}\right )} d e^{2} x - 3 \, {\left (12 \, a b^{3} c - {\left (3 \, a^{3} b + 2 \, a b^{3}\right )} c^{3}\right )} e^{2} - {\left ({\left (9 \, a^{2} b^{2} + 2 \, b^{4}\right )} d^{2} e^{2} x^{2} + 2 \, {\left (9 \, a^{2} b^{2} + 2 \, b^{4}\right )} c d e^{2} x - {\left (18 \, a^{2} b^{2} + 40 \, b^{4} - {\left (9 \, a^{2} b^{2} + 2 \, b^{4}\right )} c^{2}\right )} e^{2}\right )} \sqrt {d^{2} x^{2} + 2 \, c d x + c^{2} + 1}\right )} \log \left (d x + c + \sqrt {d^{2} x^{2} + 2 \, c d x + c^{2} + 1}\right ) - 12 \, {\left ({\left (3 \, a^{3} b + 2 \, a b^{3}\right )} d^{2} e^{2} x^{2} + 2 \, {\left (3 \, a^{3} b + 2 \, a b^{3}\right )} c d e^{2} x - {\left (6 \, a^{3} b + 40 \, a b^{3} - {\left (3 \, a^{3} b + 2 \, a b^{3}\right )} c^{2}\right )} e^{2}\right )} \sqrt {d^{2} x^{2} + 2 \, c d x + c^{2} + 1}}{81 \, d} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((d*e*x+c*e)^2*(a+b*arcsinh(d*x+c))^4,x, algorithm="fricas")

[Out]

1/81*((27*a^4 + 36*a^2*b^2 + 8*b^4)*d^3*e^2*x^3 + 3*(27*a^4 + 36*a^2*b^2 + 8*b^4)*c*d^2*e^2*x^2 - 3*(72*a^2*b^
2 + 160*b^4 - (27*a^4 + 36*a^2*b^2 + 8*b^4)*c^2)*d*e^2*x + 27*(b^4*d^3*e^2*x^3 + 3*b^4*c*d^2*e^2*x^2 + 3*b^4*c
^2*d*e^2*x + b^4*c^3*e^2)*log(d*x + c + sqrt(d^2*x^2 + 2*c*d*x + c^2 + 1))^4 + 36*(3*a*b^3*d^3*e^2*x^3 + 9*a*b
^3*c*d^2*e^2*x^2 + 9*a*b^3*c^2*d*e^2*x + 3*a*b^3*c^3*e^2 - (b^4*d^2*e^2*x^2 + 2*b^4*c*d*e^2*x + (b^4*c^2 - 2*b
^4)*e^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))^3 + 18*((9*a^2*b^
2 + 2*b^4)*d^3*e^2*x^3 + 3*(9*a^2*b^2 + 2*b^4)*c*d^2*e^2*x^2 - 3*(4*b^4 - (9*a^2*b^2 + 2*b^4)*c^2)*d*e^2*x - (
12*b^4*c - (9*a^2*b^2 + 2*b^4)*c^3)*e^2 - 6*(a*b^3*d^2*e^2*x^2 + 2*a*b^3*c*d*e^2*x + (a*b^3*c^2 - 2*a*b^3)*e^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 + 12*(3*(3*a^3*b + 2*a
*b^3)*d^3*e^2*x^3 + 9*(3*a^3*b + 2*a*b^3)*c*d^2*e^2*x^2 - 9*(4*a*b^3 - (3*a^3*b + 2*a*b^3)*c^2)*d*e^2*x - 3*(1
2*a*b^3*c - (3*a^3*b + 2*a*b^3)*c^3)*e^2 - ((9*a^2*b^2 + 2*b^4)*d^2*e^2*x^2 + 2*(9*a^2*b^2 + 2*b^4)*c*d*e^2*x
- (18*a^2*b^2 + 40*b^4 - (9*a^2*b^2 + 2*b^4)*c^2)*e^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)) - 12*((3*a^3*b + 2*a*b^3)*d^2*e^2*x^2 + 2*(3*a^3*b + 2*a*b^3)*c*d*e^2*x - (6*a^3*
b + 40*a*b^3 - (3*a^3*b + 2*a*b^3)*c^2)*e^2)*sqrt(d^2*x^2 + 2*c*d*x + c^2 + 1))/d

________________________________________________________________________________________

giac [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int {\left (d e x + c e\right )}^{2} {\left (b \operatorname {arsinh}\left (d x + c\right ) + a\right )}^{4}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((d*e*x+c*e)^2*(a+b*arcsinh(d*x+c))^4,x, algorithm="giac")

[Out]

integrate((d*e*x + c*e)^2*(b*arcsinh(d*x + c) + a)^4, x)

________________________________________________________________________________________

maple [A]  time = 0.05, size = 473, normalized size = 1.68 \[ \frac {\frac {\left (d x +c \right )^{3} e^{2} a^{4}}{3}+e^{2} b^{4} \left (\frac {\left (d x +c \right )^{3} \arcsinh \left (d x +c \right )^{4}}{3}+\frac {8 \arcsinh \left (d x +c \right )^{3} \sqrt {1+\left (d x +c \right )^{2}}}{9}-\frac {4 \left (d x +c \right )^{2} \arcsinh \left (d x +c \right )^{3} \sqrt {1+\left (d x +c \right )^{2}}}{9}-\frac {8 \left (d x +c \right ) \arcsinh \left (d x +c \right )^{2}}{3}+\frac {160 \arcsinh \left (d x +c \right ) \sqrt {1+\left (d x +c \right )^{2}}}{27}-\frac {160 d x}{27}-\frac {160 c}{27}+\frac {4 \left (d x +c \right )^{3} \arcsinh \left (d x +c \right )^{2}}{9}-\frac {8 \arcsinh \left (d x +c \right ) \sqrt {1+\left (d x +c \right )^{2}}\, \left (d x +c \right )^{2}}{27}+\frac {8 \left (d x +c \right )^{3}}{81}\right )+4 e^{2} a \,b^{3} \left (\frac {\left (d x +c \right )^{3} \arcsinh \left (d x +c \right )^{3}}{3}+\frac {2 \arcsinh \left (d x +c \right )^{2} \sqrt {1+\left (d x +c \right )^{2}}}{3}-\frac {\left (d x +c \right )^{2} \arcsinh \left (d x +c \right )^{2} \sqrt {1+\left (d x +c \right )^{2}}}{3}-\frac {4 \left (d x +c \right ) \arcsinh \left (d x +c \right )}{3}+\frac {40 \sqrt {1+\left (d x +c \right )^{2}}}{27}+\frac {2 \left (d x +c \right )^{3} \arcsinh \left (d x +c \right )}{9}-\frac {2 \left (d x +c \right )^{2} \sqrt {1+\left (d x +c \right )^{2}}}{27}\right )+6 e^{2} a^{2} b^{2} \left (\frac {\left (d x +c \right )^{3} \arcsinh \left (d x +c \right )^{2}}{3}+\frac {4 \arcsinh \left (d x +c \right ) \sqrt {1+\left (d x +c \right )^{2}}}{9}-\frac {2 \arcsinh \left (d x +c \right ) \sqrt {1+\left (d x +c \right )^{2}}\, \left (d x +c \right )^{2}}{9}-\frac {4 d x}{9}-\frac {4 c}{9}+\frac {2 \left (d x +c \right )^{3}}{27}\right )+4 e^{2} a^{3} b \left (\frac {\left (d x +c \right )^{3} \arcsinh \left (d x +c \right )}{3}-\frac {\left (d x +c \right )^{2} \sqrt {1+\left (d x +c \right )^{2}}}{9}+\frac {2 \sqrt {1+\left (d x +c \right )^{2}}}{9}\right )}{d} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((d*e*x+c*e)^2*(a+b*arcsinh(d*x+c))^4,x)

[Out]

1/d*(1/3*(d*x+c)^3*e^2*a^4+e^2*b^4*(1/3*(d*x+c)^3*arcsinh(d*x+c)^4+8/9*arcsinh(d*x+c)^3*(1+(d*x+c)^2)^(1/2)-4/
9*(d*x+c)^2*arcsinh(d*x+c)^3*(1+(d*x+c)^2)^(1/2)-8/3*(d*x+c)*arcsinh(d*x+c)^2+160/27*arcsinh(d*x+c)*(1+(d*x+c)
^2)^(1/2)-160/27*d*x-160/27*c+4/9*(d*x+c)^3*arcsinh(d*x+c)^2-8/27*arcsinh(d*x+c)*(1+(d*x+c)^2)^(1/2)*(d*x+c)^2
+8/81*(d*x+c)^3)+4*e^2*a*b^3*(1/3*(d*x+c)^3*arcsinh(d*x+c)^3+2/3*arcsinh(d*x+c)^2*(1+(d*x+c)^2)^(1/2)-1/3*(d*x
+c)^2*arcsinh(d*x+c)^2*(1+(d*x+c)^2)^(1/2)-4/3*(d*x+c)*arcsinh(d*x+c)+40/27*(1+(d*x+c)^2)^(1/2)+2/9*(d*x+c)^3*
arcsinh(d*x+c)-2/27*(d*x+c)^2*(1+(d*x+c)^2)^(1/2))+6*e^2*a^2*b^2*(1/3*(d*x+c)^3*arcsinh(d*x+c)^2+4/9*arcsinh(d
*x+c)*(1+(d*x+c)^2)^(1/2)-2/9*arcsinh(d*x+c)*(1+(d*x+c)^2)^(1/2)*(d*x+c)^2-4/9*d*x-4/9*c+2/27*(d*x+c)^3)+4*e^2
*a^3*b*(1/3*(d*x+c)^3*arcsinh(d*x+c)-1/9*(d*x+c)^2*(1+(d*x+c)^2)^(1/2)+2/9*(1+(d*x+c)^2)^(1/2)))

________________________________________________________________________________________

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)^2*(a+b*arcsinh(d*x+c))^4,x, algorithm="maxima")

[Out]

1/3*a^4*d^2*e^2*x^3 + a^4*c*d*e^2*x^2 + 2*(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)/s
qrt(-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*d*e^2 + 2/9*(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^3*b*d^2*e^2 + a^4*c^2*e^2*x + 4*((d*x + c)*arcsinh(d*x + c) - sqrt((d*
x + c)^2 + 1))*a^3*b*c^2*e^2/d + 1/3*(b^4*d^2*e^2*x^3 + 3*b^4*c*d*e^2*x^2 + 3*b^4*c^2*e^2*x)*log(d*x + c + sqr
t(d^2*x^2 + 2*c*d*x + c^2 + 1))^4 + integrate(2/3*(2*((3*a*b^3*d^5*e^2 - b^4*d^5*e^2)*x^5 + 3*(c^5*e^2 + c^3*e
^2)*a*b^3 + 5*(3*a*b^3*c*d^4*e^2 - b^4*c*d^4*e^2)*x^4 + (3*(10*c^2*d^3*e^2 + d^3*e^2)*a*b^3 - (10*c^2*d^3*e^2
+ d^3*e^2)*b^4)*x^3 + 3*((10*c^3*d^2*e^2 + 3*c*d^2*e^2)*a*b^3 - (3*c^3*d^2*e^2 + c*d^2*e^2)*b^4)*x^2 + 3*((5*c
^4*d*e^2 + 3*c^2*d*e^2)*a*b^3 - (c^4*d*e^2 + c^2*d*e^2)*b^4)*x + (3*(c^4*e^2 + c^2*e^2)*a*b^3 + (3*a*b^3*d^4*e
^2 - b^4*d^4*e^2)*x^4 + 4*(3*a*b^3*c*d^3*e^2 - b^4*c*d^3*e^2)*x^3 - 3*(2*b^4*c^2*d^2*e^2 - (6*c^2*d^2*e^2 + d^
2*e^2)*a*b^3)*x^2 - 3*(b^4*c^3*d*e^2 - 2*(2*c^3*d*e^2 + c*d*e^2)*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 + 9*(a^2*b^2*d^5*e^2*x^5 + 5*a^2*b^2*c*d^4*e^2*x^4 + (10*c^
2*d^3*e^2 + d^3*e^2)*a^2*b^2*x^3 + (10*c^3*d^2*e^2 + 3*c*d^2*e^2)*a^2*b^2*x^2 + (5*c^4*d*e^2 + 3*c^2*d*e^2)*a^
2*b^2*x + (c^5*e^2 + c^3*e^2)*a^2*b^2 + (a^2*b^2*d^4*e^2*x^4 + 4*a^2*b^2*c*d^3*e^2*x^3 + (6*c^2*d^2*e^2 + d^2*
e^2)*a^2*b^2*x^2 + 2*(2*c^3*d*e^2 + c*d*e^2)*a^2*b^2*x + (c^4*e^2 + c^2*e^2)*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 \[ \int {\left (c\,e+d\,e\,x\right )}^2\,{\left (a+b\,\mathrm {asinh}\left (c+d\,x\right )\right )}^4 \,d x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((c*e + d*e*x)^2*(a + b*asinh(c + d*x))^4,x)

[Out]

int((c*e + d*e*x)^2*(a + b*asinh(c + d*x))^4, x)

________________________________________________________________________________________

sympy [A]  time = 9.65, size = 1889, normalized size = 6.72 \[ \text {result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((d*e*x+c*e)**2*(a+b*asinh(d*x+c))**4,x)

[Out]

Piecewise((a**4*c**2*e**2*x + a**4*c*d*e**2*x**2 + a**4*d**2*e**2*x**3/3 + 4*a**3*b*c**3*e**2*asinh(c + d*x)/(
3*d) + 4*a**3*b*c**2*e**2*x*asinh(c + d*x) - 4*a**3*b*c**2*e**2*sqrt(c**2 + 2*c*d*x + d**2*x**2 + 1)/(9*d) + 4
*a**3*b*c*d*e**2*x**2*asinh(c + d*x) - 8*a**3*b*c*e**2*x*sqrt(c**2 + 2*c*d*x + d**2*x**2 + 1)/9 + 4*a**3*b*d**
2*e**2*x**3*asinh(c + d*x)/3 - 4*a**3*b*d*e**2*x**2*sqrt(c**2 + 2*c*d*x + d**2*x**2 + 1)/9 + 8*a**3*b*e**2*sqr
t(c**2 + 2*c*d*x + d**2*x**2 + 1)/(9*d) + 2*a**2*b**2*c**3*e**2*asinh(c + d*x)**2/d + 6*a**2*b**2*c**2*e**2*x*
asinh(c + d*x)**2 + 4*a**2*b**2*c**2*e**2*x/3 - 4*a**2*b**2*c**2*e**2*sqrt(c**2 + 2*c*d*x + d**2*x**2 + 1)*asi
nh(c + d*x)/(3*d) + 6*a**2*b**2*c*d*e**2*x**2*asinh(c + d*x)**2 + 4*a**2*b**2*c*d*e**2*x**2/3 - 8*a**2*b**2*c*
e**2*x*sqrt(c**2 + 2*c*d*x + d**2*x**2 + 1)*asinh(c + d*x)/3 + 2*a**2*b**2*d**2*e**2*x**3*asinh(c + d*x)**2 +
4*a**2*b**2*d**2*e**2*x**3/9 - 4*a**2*b**2*d*e**2*x**2*sqrt(c**2 + 2*c*d*x + d**2*x**2 + 1)*asinh(c + d*x)/3 -
8*a**2*b**2*e**2*x/3 + 8*a**2*b**2*e**2*sqrt(c**2 + 2*c*d*x + d**2*x**2 + 1)*asinh(c + d*x)/(3*d) + 4*a*b**3*
c**3*e**2*asinh(c + d*x)**3/(3*d) + 8*a*b**3*c**3*e**2*asinh(c + d*x)/(9*d) + 4*a*b**3*c**2*e**2*x*asinh(c + d
*x)**3 + 8*a*b**3*c**2*e**2*x*asinh(c + d*x)/3 - 4*a*b**3*c**2*e**2*sqrt(c**2 + 2*c*d*x + d**2*x**2 + 1)*asinh
(c + d*x)**2/(3*d) - 8*a*b**3*c**2*e**2*sqrt(c**2 + 2*c*d*x + d**2*x**2 + 1)/(27*d) + 4*a*b**3*c*d*e**2*x**2*a
sinh(c + d*x)**3 + 8*a*b**3*c*d*e**2*x**2*asinh(c + d*x)/3 - 8*a*b**3*c*e**2*x*sqrt(c**2 + 2*c*d*x + d**2*x**2
+ 1)*asinh(c + d*x)**2/3 - 16*a*b**3*c*e**2*x*sqrt(c**2 + 2*c*d*x + d**2*x**2 + 1)/27 - 16*a*b**3*c*e**2*asin
h(c + d*x)/(3*d) + 4*a*b**3*d**2*e**2*x**3*asinh(c + d*x)**3/3 + 8*a*b**3*d**2*e**2*x**3*asinh(c + d*x)/9 - 4*
a*b**3*d*e**2*x**2*sqrt(c**2 + 2*c*d*x + d**2*x**2 + 1)*asinh(c + d*x)**2/3 - 8*a*b**3*d*e**2*x**2*sqrt(c**2 +
2*c*d*x + d**2*x**2 + 1)/27 - 16*a*b**3*e**2*x*asinh(c + d*x)/3 + 8*a*b**3*e**2*sqrt(c**2 + 2*c*d*x + d**2*x*
*2 + 1)*asinh(c + d*x)**2/(3*d) + 160*a*b**3*e**2*sqrt(c**2 + 2*c*d*x + d**2*x**2 + 1)/(27*d) + b**4*c**3*e**2
*asinh(c + d*x)**4/(3*d) + 4*b**4*c**3*e**2*asinh(c + d*x)**2/(9*d) + b**4*c**2*e**2*x*asinh(c + d*x)**4 + 4*b
**4*c**2*e**2*x*asinh(c + d*x)**2/3 + 8*b**4*c**2*e**2*x/27 - 4*b**4*c**2*e**2*sqrt(c**2 + 2*c*d*x + d**2*x**2
+ 1)*asinh(c + d*x)**3/(9*d) - 8*b**4*c**2*e**2*sqrt(c**2 + 2*c*d*x + d**2*x**2 + 1)*asinh(c + d*x)/(27*d) +
b**4*c*d*e**2*x**2*asinh(c + d*x)**4 + 4*b**4*c*d*e**2*x**2*asinh(c + d*x)**2/3 + 8*b**4*c*d*e**2*x**2/27 - 8*
b**4*c*e**2*x*sqrt(c**2 + 2*c*d*x + d**2*x**2 + 1)*asinh(c + d*x)**3/9 - 16*b**4*c*e**2*x*sqrt(c**2 + 2*c*d*x
+ d**2*x**2 + 1)*asinh(c + d*x)/27 - 8*b**4*c*e**2*asinh(c + d*x)**2/(3*d) + b**4*d**2*e**2*x**3*asinh(c + d*x
)**4/3 + 4*b**4*d**2*e**2*x**3*asinh(c + d*x)**2/9 + 8*b**4*d**2*e**2*x**3/81 - 4*b**4*d*e**2*x**2*sqrt(c**2 +
2*c*d*x + d**2*x**2 + 1)*asinh(c + d*x)**3/9 - 8*b**4*d*e**2*x**2*sqrt(c**2 + 2*c*d*x + d**2*x**2 + 1)*asinh(
c + d*x)/27 - 8*b**4*e**2*x*asinh(c + d*x)**2/3 - 160*b**4*e**2*x/27 + 8*b**4*e**2*sqrt(c**2 + 2*c*d*x + d**2*
x**2 + 1)*asinh(c + d*x)**3/(9*d) + 160*b**4*e**2*sqrt(c**2 + 2*c*d*x + d**2*x**2 + 1)*asinh(c + d*x)/(27*d),
Ne(d, 0)), (c**2*e**2*x*(a + b*asinh(c))**4, True))

________________________________________________________________________________________