3.2.48 \(\int (c e+d e x)^2 (a+b \sinh ^{-1}(c+d x))^4 \, dx\) [148]

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

[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

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Rubi [A]
time = 0.33, 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 = {5859, 12, 5776, 5812, 5798, 5772, 8, 30} \begin {gather*} \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 \end {gather*}

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 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)^2 \left (a+b \sinh ^{-1}(c+d x)\right )^4 \, dx &=\frac {\text {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 \text {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 ) \text {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 ) \text {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 ) \text {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 ) \text {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 ) \text {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 ) \text {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 ) \text {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 ) \text {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 ) \text {Subst}(\int 1 \, dx,x,c+d x)}{27 d}-\frac {\left (16 b^4 e^2\right ) \text {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*}

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Mathematica [A]
time = 0.28, size = 412, normalized size = 1.47 \begin {gather*} \frac {e^2 \left (-24 b^2 \left (9 a^2+20 b^2\right ) (c+d x)+\left (27 a^4+36 a^2 b^2+8 b^4\right ) (c+d x)^3+12 a b \sqrt {1+(c+d x)^2} \left (6 a^2+40 b^2-\left (3 a^2+2 b^2\right ) (c+d x)^2\right )+12 b \left (-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}\right ) \sinh ^{-1}(c+d x)+18 b^2 \left (-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}\right ) \sinh ^{-1}(c+d x)^2-36 b^3 \left (-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}\right ) \sinh ^{-1}(c+d x)^3+27 b^4 (c+d x)^3 \sinh ^{-1}(c+d x)^4\right )}{81 d} \end {gather*}

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)

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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(1123\) vs. \(2(255)=510\).
time = 4.78, size = 1124, normalized size = 4.00

method	result	size
default	\(\text {Expression too large to display}\)	\(1124\)

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,method=_RETURNVERBOSE)

[Out]

1/3*e^2*(d*x+c)^3*a^4/d+1/81*e^2*b^4*(27*arcsinh(d*x+c)^4*x^3*d^3+81*arcsinh(d*x+c)^4*x^2*c*d^2+81*arcsinh(d*x
+c)^4*x*c^2*d-36*arcsinh(d*x+c)^3*(d^2*x^2+2*c*d*x+c^2+1)^(1/2)*x^2*d^2+36*arcsinh(d*x+c)^2*x^3*d^3+27*arcsinh
(d*x+c)^4*c^3-72*arcsinh(d*x+c)^3*(d^2*x^2+2*c*d*x+c^2+1)^(1/2)*x*c*d+108*arcsinh(d*x+c)^2*x^2*c*d^2-36*arcsin
h(d*x+c)^3*(d^2*x^2+2*c*d*x+c^2+1)^(1/2)*c^2+108*arcsinh(d*x+c)^2*x*c^2*d-24*arcsinh(d*x+c)*(d^2*x^2+2*c*d*x+c
^2+1)^(1/2)*x^2*d^2+8*d^3*x^3+36*arcsinh(d*x+c)^2*c^3-48*arcsinh(d*x+c)*(d^2*x^2+2*c*d*x+c^2+1)^(1/2)*x*c*d+24
*x^2*c*d^2+72*(d^2*x^2+2*c*d*x+c^2+1)^(1/2)*arcsinh(d*x+c)^3-216*arcsinh(d*x+c)^2*x*d-24*arcsinh(d*x+c)*(d^2*x
^2+2*c*d*x+c^2+1)^(1/2)*c^2+24*x*c^2*d-216*arcsinh(d*x+c)^2*c+8*c^3+480*arcsinh(d*x+c)*(d^2*x^2+2*c*d*x+c^2+1)
^(1/2)-480*d*x-480*c)/d+4/27*e^2*a*b^3*(9*arcsinh(d*x+c)^3*x^3*d^3+27*arcsinh(d*x+c)^3*x^2*c*d^2+27*arcsinh(d*
x+c)^3*x*c^2*d-9*arcsinh(d*x+c)^2*(d^2*x^2+2*c*d*x+c^2+1)^(1/2)*x^2*d^2+6*arcsinh(d*x+c)*x^3*d^3+9*arcsinh(d*x
+c)^3*c^3-18*arcsinh(d*x+c)^2*(d^2*x^2+2*c*d*x+c^2+1)^(1/2)*x*c*d+18*arcsinh(d*x+c)*x^2*c*d^2-9*arcsinh(d*x+c)
^2*(d^2*x^2+2*c*d*x+c^2+1)^(1/2)*c^2+18*arcsinh(d*x+c)*x*c^2*d-2*(d^2*x^2+2*c*d*x+c^2+1)^(1/2)*x^2*d^2+6*arcsi
nh(d*x+c)*c^3-4*(d^2*x^2+2*c*d*x+c^2+1)^(1/2)*x*c*d+18*(d^2*x^2+2*c*d*x+c^2+1)^(1/2)*arcsinh(d*x+c)^2-36*arcsi
nh(d*x+c)*x*d-2*(d^2*x^2+2*c*d*x+c^2+1)^(1/2)*c^2-36*arcsinh(d*x+c)*c+40*(d^2*x^2+2*c*d*x+c^2+1)^(1/2))/d+2/9*
e^2*a^2*b^2*(9*arcsinh(d*x+c)^2*x^3*d^3+27*arcsinh(d*x+c)^2*x^2*c*d^2+27*arcsinh(d*x+c)^2*x*c^2*d-6*arcsinh(d*
x+c)*(d^2*x^2+2*c*d*x+c^2+1)^(1/2)*x^2*d^2+2*d^3*x^3+9*arcsinh(d*x+c)^2*c^3-12*arcsinh(d*x+c)*(d^2*x^2+2*c*d*x
+c^2+1)^(1/2)*x*c*d+6*x^2*c*d^2-6*arcsinh(d*x+c)*(d^2*x^2+2*c*d*x+c^2+1)^(1/2)*c^2+6*x*c^2*d+2*c^3+12*arcsinh(
d*x+c)*(d^2*x^2+2*c*d*x+c^2+1)^(1/2)-12*d*x-12*c)/d+4*e^2*a^3*b/d*(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))

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

[Out]

1/3*a^4*d^2*x^3*e^2 + a^4*c*d*x^2*e^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*x*e^2 + 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*x^3*e^2 + 3*b^4*c*d*x^2*e^2 + 3*b^4*c^2*x*e^2)*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 - b^4*d^5)*x^5*e^2 + 3*(c^5 + c^3)*a*b^3*e^
2 + 5*(3*a*b^3*c*d^4 - b^4*c*d^4)*x^4*e^2 + (3*(10*c^2*d^3 + d^3)*a*b^3 - (10*c^2*d^3 + d^3)*b^4)*x^3*e^2 + 3*
((10*c^3*d^2 + 3*c*d^2)*a*b^3 - (3*c^3*d^2 + c*d^2)*b^4)*x^2*e^2 + 3*((5*c^4*d + 3*c^2*d)*a*b^3 - (c^4*d + c^2
*d)*b^4)*x*e^2 + (3*(c^4 + c^2)*a*b^3*e^2 + (3*a*b^3*d^4 - b^4*d^4)*x^4*e^2 + 4*(3*a*b^3*c*d^3 - b^4*c*d^3)*x^
3*e^2 - 3*(2*b^4*c^2*d^2 - (6*c^2*d^2 + d^2)*a*b^3)*x^2*e^2 - 3*(b^4*c^3*d - 2*(2*c^3*d + c*d)*a*b^3)*x*e^2)*s
qrt(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*x^5*e^2
+ 5*a^2*b^2*c*d^4*x^4*e^2 + (10*c^2*d^3 + d^3)*a^2*b^2*x^3*e^2 + (10*c^3*d^2 + 3*c*d^2)*a^2*b^2*x^2*e^2 + (5*c
^4*d + 3*c^2*d)*a^2*b^2*x*e^2 + (c^5 + c^3)*a^2*b^2*e^2 + (a^2*b^2*d^4*x^4*e^2 + 4*a^2*b^2*c*d^3*x^3*e^2 + (6*
c^2*d^2 + d^2)*a^2*b^2*x^2*e^2 + 2*(2*c^3*d + c*d)*a^2*b^2*x*e^2 + (c^4 + c^2)*a^2*b^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))^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. 2103 vs. \(2 (246) = 492\).
time = 0.46, size = 2103, normalized size = 7.48 \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)^2*(a+b*arcsinh(d*x+c))^4,x, algorithm="fricas")

[Out]

1/81*(27*((b^4*d^3*x^3 + 3*b^4*c*d^2*x^2 + 3*b^4*c^2*d*x + b^4*c^3)*cosh(1)^2 + 2*(b^4*d^3*x^3 + 3*b^4*c*d^2*x
^2 + 3*b^4*c^2*d*x + b^4*c^3)*cosh(1)*sinh(1) + (b^4*d^3*x^3 + 3*b^4*c*d^2*x^2 + 3*b^4*c^2*d*x + b^4*c^3)*sinh
(1)^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*x^3 + 3*a*b^3*c*d^2*x^2 + 3*a*b^3
*c^2*d*x + a*b^3*c^3)*cosh(1)^2 + 6*(a*b^3*d^3*x^3 + 3*a*b^3*c*d^2*x^2 + 3*a*b^3*c^2*d*x + a*b^3*c^3)*cosh(1)*
sinh(1) + 3*(a*b^3*d^3*x^3 + 3*a*b^3*c*d^2*x^2 + 3*a*b^3*c^2*d*x + a*b^3*c^3)*sinh(1)^2 - sqrt(d^2*x^2 + 2*c*d
*x + c^2 + 1)*((b^4*d^2*x^2 + 2*b^4*c*d*x + b^4*c^2 - 2*b^4)*cosh(1)^2 + 2*(b^4*d^2*x^2 + 2*b^4*c*d*x + b^4*c^
2 - 2*b^4)*cosh(1)*sinh(1) + (b^4*d^2*x^2 + 2*b^4*c*d*x + b^4*c^2 - 2*b^4)*sinh(1)^2))*log(d*x + c + sqrt(d^2*
x^2 + 2*c*d*x + c^2 + 1))^3 + ((27*a^4 + 36*a^2*b^2 + 8*b^4)*d^3*x^3 + 3*(27*a^4 + 36*a^2*b^2 + 8*b^4)*c*d^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*x)*cosh(1)^2 + 18*(((9*a^2*b^2 + 2*b^4)*d^
3*x^3 + 3*(9*a^2*b^2 + 2*b^4)*c*d^2*x^2 - 12*b^4*c + (9*a^2*b^2 + 2*b^4)*c^3 - 3*(4*b^4 - (9*a^2*b^2 + 2*b^4)*
c^2)*d*x)*cosh(1)^2 + 2*((9*a^2*b^2 + 2*b^4)*d^3*x^3 + 3*(9*a^2*b^2 + 2*b^4)*c*d^2*x^2 - 12*b^4*c + (9*a^2*b^2
+ 2*b^4)*c^3 - 3*(4*b^4 - (9*a^2*b^2 + 2*b^4)*c^2)*d*x)*cosh(1)*sinh(1) + ((9*a^2*b^2 + 2*b^4)*d^3*x^3 + 3*(9
*a^2*b^2 + 2*b^4)*c*d^2*x^2 - 12*b^4*c + (9*a^2*b^2 + 2*b^4)*c^3 - 3*(4*b^4 - (9*a^2*b^2 + 2*b^4)*c^2)*d*x)*si
nh(1)^2 - 6*sqrt(d^2*x^2 + 2*c*d*x + c^2 + 1)*((a*b^3*d^2*x^2 + 2*a*b^3*c*d*x + a*b^3*c^2 - 2*a*b^3)*cosh(1)^2
+ 2*(a*b^3*d^2*x^2 + 2*a*b^3*c*d*x + a*b^3*c^2 - 2*a*b^3)*cosh(1)*sinh(1) + (a*b^3*d^2*x^2 + 2*a*b^3*c*d*x +
a*b^3*c^2 - 2*a*b^3)*sinh(1)^2))*log(d*x + c + sqrt(d^2*x^2 + 2*c*d*x + c^2 + 1))^2 + 2*((27*a^4 + 36*a^2*b^2
+ 8*b^4)*d^3*x^3 + 3*(27*a^4 + 36*a^2*b^2 + 8*b^4)*c*d^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*x)*cosh(1)*sinh(1) + ((27*a^4 + 36*a^2*b^2 + 8*b^4)*d^3*x^3 + 3*(27*a^4 + 36*a^2*b^2 + 8*b^4)*
c*d^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*x)*sinh(1)^2 + 12*(3*((3*a^3*b + 2*
a*b^3)*d^3*x^3 + 3*(3*a^3*b + 2*a*b^3)*c*d^2*x^2 - 12*a*b^3*c + (3*a^3*b + 2*a*b^3)*c^3 - 3*(4*a*b^3 - (3*a^3*
b + 2*a*b^3)*c^2)*d*x)*cosh(1)^2 + 6*((3*a^3*b + 2*a*b^3)*d^3*x^3 + 3*(3*a^3*b + 2*a*b^3)*c*d^2*x^2 - 12*a*b^3
*c + (3*a^3*b + 2*a*b^3)*c^3 - 3*(4*a*b^3 - (3*a^3*b + 2*a*b^3)*c^2)*d*x)*cosh(1)*sinh(1) + 3*((3*a^3*b + 2*a*
b^3)*d^3*x^3 + 3*(3*a^3*b + 2*a*b^3)*c*d^2*x^2 - 12*a*b^3*c + (3*a^3*b + 2*a*b^3)*c^3 - 3*(4*a*b^3 - (3*a^3*b
+ 2*a*b^3)*c^2)*d*x)*sinh(1)^2 - sqrt(d^2*x^2 + 2*c*d*x + c^2 + 1)*(((9*a^2*b^2 + 2*b^4)*d^2*x^2 - 18*a^2*b^2
- 40*b^4 + 2*(9*a^2*b^2 + 2*b^4)*c*d*x + (9*a^2*b^2 + 2*b^4)*c^2)*cosh(1)^2 + 2*((9*a^2*b^2 + 2*b^4)*d^2*x^2 -
18*a^2*b^2 - 40*b^4 + 2*(9*a^2*b^2 + 2*b^4)*c*d*x + (9*a^2*b^2 + 2*b^4)*c^2)*cosh(1)*sinh(1) + ((9*a^2*b^2 +
2*b^4)*d^2*x^2 - 18*a^2*b^2 - 40*b^4 + 2*(9*a^2*b^2 + 2*b^4)*c*d*x + (9*a^2*b^2 + 2*b^4)*c^2)*sinh(1)^2))*log(
d*x + c + sqrt(d^2*x^2 + 2*c*d*x + c^2 + 1)) - 12*sqrt(d^2*x^2 + 2*c*d*x + c^2 + 1)*(((3*a^3*b + 2*a*b^3)*d^2*
x^2 - 6*a^3*b - 40*a*b^3 + 2*(3*a^3*b + 2*a*b^3)*c*d*x + (3*a^3*b + 2*a*b^3)*c^2)*cosh(1)^2 + 2*((3*a^3*b + 2*
a*b^3)*d^2*x^2 - 6*a^3*b - 40*a*b^3 + 2*(3*a^3*b + 2*a*b^3)*c*d*x + (3*a^3*b + 2*a*b^3)*c^2)*cosh(1)*sinh(1) +
((3*a^3*b + 2*a*b^3)*d^2*x^2 - 6*a^3*b - 40*a*b^3 + 2*(3*a^3*b + 2*a*b^3)*c*d*x + (3*a^3*b + 2*a*b^3)*c^2)*si
nh(1)^2))/d

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Sympy [B] Leaf count of result is larger than twice the leaf count of optimal. 1889 vs. \(2 (264) = 528\).
time = 0.92, size = 1889, normalized size = 6.72 \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)**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))

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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)^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)

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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 )}^2\,{\left (a+b\,\mathrm {asinh}\left (c+d\,x\right )\right )}^4 \,d x \end {gather*}

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)

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