3.2.39 \(\int (c e+d e x)^2 (a+b \sinh ^{-1}(c+d x))^3 \, dx\) [139]

Optimal. Leaf size=227 \[ -\frac {4}{3} a b^2 e^2 x+\frac {14 b^3 e^2 \sqrt {1+(c+d x)^2}}{9 d}-\frac {2 b^3 e^2 \left (1+(c+d x)^2\right )^{3/2}}{27 d}-\frac {4 b^3 e^2 (c+d x) \sinh ^{-1}(c+d x)}{3 d}+\frac {2 b^2 e^2 (c+d x)^3 \left (a+b \sinh ^{-1}(c+d x)\right )}{9 d}+\frac {2 b e^2 \sqrt {1+(c+d x)^2} \left (a+b \sinh ^{-1}(c+d x)\right )^2}{3 d}-\frac {b e^2 (c+d x)^2 \sqrt {1+(c+d x)^2} \left (a+b \sinh ^{-1}(c+d x)\right )^2}{3 d}+\frac {e^2 (c+d x)^3 \left (a+b \sinh ^{-1}(c+d x)\right )^3}{3 d} \]

[Out]

-4/3*a*b^2*e^2*x-2/27*b^3*e^2*(1+(d*x+c)^2)^(3/2)/d-4/3*b^3*e^2*(d*x+c)*arcsinh(d*x+c)/d+2/9*b^2*e^2*(d*x+c)^3
*(a+b*arcsinh(d*x+c))/d+1/3*e^2*(d*x+c)^3*(a+b*arcsinh(d*x+c))^3/d+14/9*b^3*e^2*(1+(d*x+c)^2)^(1/2)/d+2/3*b*e^
2*(a+b*arcsinh(d*x+c))^2*(1+(d*x+c)^2)^(1/2)/d-1/3*b*e^2*(d*x+c)^2*(a+b*arcsinh(d*x+c))^2*(1+(d*x+c)^2)^(1/2)/
d

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Rubi [A]
time = 0.21, antiderivative size = 227, normalized size of antiderivative = 1.00, number of steps used = 12, 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 {2 b^2 e^2 (c+d x)^3 \left (a+b \sinh ^{-1}(c+d x)\right )}{9 d}-\frac {4}{3} a b^2 e^2 x+\frac {2 b e^2 \sqrt {(c+d x)^2+1} \left (a+b \sinh ^{-1}(c+d x)\right )^2}{3 d}-\frac {b e^2 (c+d x)^2 \sqrt {(c+d x)^2+1} \left (a+b \sinh ^{-1}(c+d x)\right )^2}{3 d}+\frac {e^2 (c+d x)^3 \left (a+b \sinh ^{-1}(c+d x)\right )^3}{3 d}-\frac {2 b^3 e^2 \left ((c+d x)^2+1\right )^{3/2}}{27 d}+\frac {14 b^3 e^2 \sqrt {(c+d x)^2+1}}{9 d}-\frac {4 b^3 e^2 (c+d x) \sinh ^{-1}(c+d x)}{3 d} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(c*e + d*e*x)^2*(a + b*ArcSinh[c + d*x])^3,x]

[Out]

(-4*a*b^2*e^2*x)/3 + (14*b^3*e^2*Sqrt[1 + (c + d*x)^2])/(9*d) - (2*b^3*e^2*(1 + (c + d*x)^2)^(3/2))/(27*d) - (
4*b^3*e^2*(c + d*x)*ArcSinh[c + d*x])/(3*d) + (2*b^2*e^2*(c + d*x)^3*(a + b*ArcSinh[c + d*x]))/(9*d) + (2*b*e^
2*Sqrt[1 + (c + d*x)^2]*(a + b*ArcSinh[c + d*x])^2)/(3*d) - (b*e^2*(c + d*x)^2*Sqrt[1 + (c + d*x)^2]*(a + b*Ar
cSinh[c + d*x])^2)/(3*d) + (e^2*(c + d*x)^3*(a + b*ArcSinh[c + d*x])^3)/(3*d)

Rule 12

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

Rule 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)^2 \left (a+b \sinh ^{-1}(c+d x)\right )^3 \, dx &=\frac {\text {Subst}\left (\int e^2 x^2 \left (a+b \sinh ^{-1}(x)\right )^3 \, dx,x,c+d x\right )}{d}\\ &=\frac {e^2 \text {Subst}\left (\int x^2 \left (a+b \sinh ^{-1}(x)\right )^3 \, dx,x,c+d x\right )}{d}\\ &=\frac {e^2 (c+d x)^3 \left (a+b \sinh ^{-1}(c+d x)\right )^3}{3 d}-\frac {\left (b e^2\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 )}{d}\\ &=-\frac {b e^2 (c+d x)^2 \sqrt {1+(c+d x)^2} \left (a+b \sinh ^{-1}(c+d x)\right )^2}{3 d}+\frac {e^2 (c+d x)^3 \left (a+b \sinh ^{-1}(c+d x)\right )^3}{3 d}+\frac {\left (2 b e^2\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 )}{3 d}+\frac {\left (2 b^2 e^2\right ) \text {Subst}\left (\int x^2 \left (a+b \sinh ^{-1}(x)\right ) \, dx,x,c+d x\right )}{3 d}\\ &=\frac {2 b^2 e^2 (c+d x)^3 \left (a+b \sinh ^{-1}(c+d x)\right )}{9 d}+\frac {2 b e^2 \sqrt {1+(c+d x)^2} \left (a+b \sinh ^{-1}(c+d x)\right )^2}{3 d}-\frac {b e^2 (c+d x)^2 \sqrt {1+(c+d x)^2} \left (a+b \sinh ^{-1}(c+d x)\right )^2}{3 d}+\frac {e^2 (c+d x)^3 \left (a+b \sinh ^{-1}(c+d x)\right )^3}{3 d}-\frac {\left (4 b^2 e^2\right ) \text {Subst}\left (\int \left (a+b \sinh ^{-1}(x)\right ) \, dx,x,c+d x\right )}{3 d}-\frac {\left (2 b^3 e^2\right ) \text {Subst}\left (\int \frac {x^3}{\sqrt {1+x^2}} \, dx,x,c+d x\right )}{9 d}\\ &=-\frac {4}{3} a b^2 e^2 x+\frac {2 b^2 e^2 (c+d x)^3 \left (a+b \sinh ^{-1}(c+d x)\right )}{9 d}+\frac {2 b e^2 \sqrt {1+(c+d x)^2} \left (a+b \sinh ^{-1}(c+d x)\right )^2}{3 d}-\frac {b e^2 (c+d x)^2 \sqrt {1+(c+d x)^2} \left (a+b \sinh ^{-1}(c+d x)\right )^2}{3 d}+\frac {e^2 (c+d x)^3 \left (a+b \sinh ^{-1}(c+d x)\right )^3}{3 d}-\frac {\left (b^3 e^2\right ) \text {Subst}\left (\int \frac {x}{\sqrt {1+x}} \, dx,x,(c+d x)^2\right )}{9 d}-\frac {\left (4 b^3 e^2\right ) \text {Subst}\left (\int \sinh ^{-1}(x) \, dx,x,c+d x\right )}{3 d}\\ &=-\frac {4}{3} a b^2 e^2 x-\frac {4 b^3 e^2 (c+d x) \sinh ^{-1}(c+d x)}{3 d}+\frac {2 b^2 e^2 (c+d x)^3 \left (a+b \sinh ^{-1}(c+d x)\right )}{9 d}+\frac {2 b e^2 \sqrt {1+(c+d x)^2} \left (a+b \sinh ^{-1}(c+d x)\right )^2}{3 d}-\frac {b e^2 (c+d x)^2 \sqrt {1+(c+d x)^2} \left (a+b \sinh ^{-1}(c+d x)\right )^2}{3 d}+\frac {e^2 (c+d x)^3 \left (a+b \sinh ^{-1}(c+d x)\right )^3}{3 d}-\frac {\left (b^3 e^2\right ) \text {Subst}\left (\int \left (-\frac {1}{\sqrt {1+x}}+\sqrt {1+x}\right ) \, dx,x,(c+d x)^2\right )}{9 d}+\frac {\left (4 b^3 e^2\right ) \text {Subst}\left (\int \frac {x}{\sqrt {1+x^2}} \, dx,x,c+d x\right )}{3 d}\\ &=-\frac {4}{3} a b^2 e^2 x+\frac {14 b^3 e^2 \sqrt {1+(c+d x)^2}}{9 d}-\frac {2 b^3 e^2 \left (1+(c+d x)^2\right )^{3/2}}{27 d}-\frac {4 b^3 e^2 (c+d x) \sinh ^{-1}(c+d x)}{3 d}+\frac {2 b^2 e^2 (c+d x)^3 \left (a+b \sinh ^{-1}(c+d x)\right )}{9 d}+\frac {2 b e^2 \sqrt {1+(c+d x)^2} \left (a+b \sinh ^{-1}(c+d x)\right )^2}{3 d}-\frac {b e^2 (c+d x)^2 \sqrt {1+(c+d x)^2} \left (a+b \sinh ^{-1}(c+d x)\right )^2}{3 d}+\frac {e^2 (c+d x)^3 \left (a+b \sinh ^{-1}(c+d x)\right )^3}{3 d}\\ \end {align*}

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Mathematica [A]
time = 0.20, size = 258, normalized size = 1.14 \begin {gather*} \frac {e^2 \left (-12 a b^2 (c+d x)+a \left (3 a^2+2 b^2\right ) (c+d x)^3+\frac {1}{3} b \sqrt {1+(c+d x)^2} \left (18 a^2+40 b^2-\left (9 a^2+2 b^2\right ) (c+d x)^2\right )-b \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)-3 b^2 \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)^2+3 b^3 (c+d x)^3 \sinh ^{-1}(c+d x)^3\right )}{9 d} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(c*e + d*e*x)^2*(a + b*ArcSinh[c + d*x])^3,x]

[Out]

(e^2*(-12*a*b^2*(c + d*x) + a*(3*a^2 + 2*b^2)*(c + d*x)^3 + (b*Sqrt[1 + (c + d*x)^2]*(18*a^2 + 40*b^2 - (9*a^2
+ 2*b^2)*(c + d*x)^2))/3 - b*(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] - 3*b^2*(-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])*ArcSinh[c + d*x]^2 + 3*b^3*(c + d*x)^3*ArcSinh[c + d*x]^3)
)/(9*d)

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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(679\) vs. \(2(203)=406\).
time = 3.35, size = 680, normalized size = 3.00

method	result	size
default	\(\frac {e^{2} \left (d x +c \right )^{3} a^{3}}{3 d}+\frac {b^{3} e^{2} \left (9 \arcsinh \left (d x +c \right )^{3} x^{3} d^{3}+27 \arcsinh \left (d x +c \right )^{3} x^{2} c \,d^{2}+27 \arcsinh \left (d x +c \right )^{3} x \,c^{2} d -9 \arcsinh \left (d x +c \right )^{2} \sqrt {d^{2} x^{2}+2 c d x +c^{2}+1}\, x^{2} d^{2}+6 \arcsinh \left (d x +c \right ) x^{3} d^{3}+9 \arcsinh \left (d x +c \right )^{3} c^{3}-18 \arcsinh \left (d x +c \right )^{2} \sqrt {d^{2} x^{2}+2 c d x +c^{2}+1}\, x c d +18 \arcsinh \left (d x +c \right ) x^{2} c \,d^{2}-9 \arcsinh \left (d x +c \right )^{2} \sqrt {d^{2} x^{2}+2 c d x +c^{2}+1}\, c^{2}+18 \arcsinh \left (d x +c \right ) x \,c^{2} d -2 \sqrt {d^{2} x^{2}+2 c d x +c^{2}+1}\, x^{2} d^{2}+6 \arcsinh \left (d x +c \right ) c^{3}-4 \sqrt {d^{2} x^{2}+2 c d x +c^{2}+1}\, x c d +18 \sqrt {d^{2} x^{2}+2 c d x +c^{2}+1}\, \arcsinh \left (d x +c \right )^{2}-36 \arcsinh \left (d x +c \right ) x d -2 \sqrt {d^{2} x^{2}+2 c d x +c^{2}+1}\, c^{2}-36 \arcsinh \left (d x +c \right ) c +40 \sqrt {d^{2} x^{2}+2 c d x +c^{2}+1}\right )}{27 d}+\frac {a \,b^{2} e^{2} \left (9 \arcsinh \left (d x +c \right )^{2} x^{3} d^{3}+27 \arcsinh \left (d x +c \right )^{2} x^{2} c \,d^{2}+27 \arcsinh \left (d x +c \right )^{2} x \,c^{2} d -6 \arcsinh \left (d x +c \right ) \sqrt {d^{2} x^{2}+2 c d x +c^{2}+1}\, x^{2} d^{2}+2 d^{3} x^{3}+9 \arcsinh \left (d x +c \right )^{2} c^{3}-12 \arcsinh \left (d x +c \right ) \sqrt {d^{2} x^{2}+2 c d x +c^{2}+1}\, x c d +6 x^{2} c \,d^{2}-6 \arcsinh \left (d x +c \right ) \sqrt {d^{2} x^{2}+2 c d x +c^{2}+1}\, c^{2}+6 x \,c^{2} d +2 c^{3}+12 \arcsinh \left (d x +c \right ) \sqrt {d^{2} x^{2}+2 c d x +c^{2}+1}-12 d x -12 c \right )}{9 d}+\frac {3 a^{2} b \,e^{2} \left (\frac {\left (d x +c \right )^{3} \arcsinh \left (d x +c \right )}{3}-\frac {\left (d x +c \right )^{2} \sqrt {1+\left (d x +c \right )^{2}}}{9}+\frac {2 \sqrt {1+\left (d x +c \right )^{2}}}{9}\right )}{d}\)	\(680\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((d*e*x+c*e)^2*(a+b*arcsinh(d*x+c))^3,x,method=_RETURNVERBOSE)

[Out]

1/3*e^2*(d*x+c)^3*a^3/d+1/27*b^3*e^2*(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*arcsinh
(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*arcsinh
(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+1/9*a*
b^2*e^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+3*a^2*b*e^2/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))^3,x, algorithm="maxima")

[Out]

1/3*a^3*d^2*x^3*e^2 + a^3*c*d*x^2*e^2 + 3/2*(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)
/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*d*e^2 + 1/6*(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*d^2*e^2 + a^3*c^2*x*e^2 + 3*((d*x + c)*arcsinh(d*x + c) - sqrt((
d*x + c)^2 + 1))*a^2*b*c^2*e^2/d + 1/3*(b^3*d^2*x^3*e^2 + 3*b^3*c*d*x^2*e^2 + 3*b^3*c^2*x*e^2)*log(d*x + c + s
qrt(d^2*x^2 + 2*c*d*x + c^2 + 1))^3 + integrate(((3*a*b^2*d^5 - b^3*d^5)*x^5*e^2 + 5*(3*a*b^2*c*d^4 - b^3*c*d^
4)*x^4*e^2 + 3*(c^5 + c^3)*a*b^2*e^2 + (3*(10*c^2*d^3 + d^3)*a*b^2 - (10*c^2*d^3 + d^3)*b^3)*x^3*e^2 + 3*((10*
c^3*d^2 + 3*c*d^2)*a*b^2 - (3*c^3*d^2 + c*d^2)*b^3)*x^2*e^2 + 3*((5*c^4*d + 3*c^2*d)*a*b^2 - (c^4*d + c^2*d)*b
^3)*x*e^2 + ((3*a*b^2*d^4 - b^3*d^4)*x^4*e^2 + 3*(c^4 + c^2)*a*b^2*e^2 + 4*(3*a*b^2*c*d^3 - b^3*c*d^3)*x^3*e^2
- 3*(2*b^3*c^2*d^2 - (6*c^2*d^2 + d^2)*a*b^2)*x^2*e^2 - 3*(b^3*c^3*d - 2*(2*c^3*d + c*d)*a*b^2)*x*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. 1395 vs. \(2 (195) = 390\).
time = 0.40, size = 1395, normalized size = 6.15 \begin {gather*} \frac {9 \, {\left ({\left (b^{3} d^{3} x^{3} + 3 \, b^{3} c d^{2} x^{2} + 3 \, b^{3} c^{2} d x + b^{3} c^{3}\right )} \cosh \left (1\right )^{2} + 2 \, {\left (b^{3} d^{3} x^{3} + 3 \, b^{3} c d^{2} x^{2} + 3 \, b^{3} c^{2} d x + b^{3} c^{3}\right )} \cosh \left (1\right ) \sinh \left (1\right ) + {\left (b^{3} d^{3} x^{3} + 3 \, b^{3} c d^{2} x^{2} + 3 \, b^{3} c^{2} d x + b^{3} c^{3}\right )} \sinh \left (1\right )^{2}\right )} \log \left (d x + c + \sqrt {d^{2} x^{2} + 2 \, c d x + c^{2} + 1}\right )^{3} + 3 \, {\left ({\left (3 \, a^{3} + 2 \, a b^{2}\right )} d^{3} x^{3} + 3 \, {\left (3 \, a^{3} + 2 \, a b^{2}\right )} c d^{2} x^{2} - 3 \, {\left (4 \, a b^{2} - {\left (3 \, a^{3} + 2 \, a b^{2}\right )} c^{2}\right )} d x\right )} \cosh \left (1\right )^{2} + 9 \, {\left (3 \, {\left (a b^{2} d^{3} x^{3} + 3 \, a b^{2} c d^{2} x^{2} + 3 \, a b^{2} c^{2} d x + a b^{2} c^{3}\right )} \cosh \left (1\right )^{2} + 6 \, {\left (a b^{2} d^{3} x^{3} + 3 \, a b^{2} c d^{2} x^{2} + 3 \, a b^{2} c^{2} d x + a b^{2} c^{3}\right )} \cosh \left (1\right ) \sinh \left (1\right ) + 3 \, {\left (a b^{2} d^{3} x^{3} + 3 \, a b^{2} c d^{2} x^{2} + 3 \, a b^{2} c^{2} d x + a b^{2} c^{3}\right )} \sinh \left (1\right )^{2} - \sqrt {d^{2} x^{2} + 2 \, c d x + c^{2} + 1} {\left ({\left (b^{3} d^{2} x^{2} + 2 \, b^{3} c d x + b^{3} c^{2} - 2 \, b^{3}\right )} \cosh \left (1\right )^{2} + 2 \, {\left (b^{3} d^{2} x^{2} + 2 \, b^{3} c d x + b^{3} c^{2} - 2 \, b^{3}\right )} \cosh \left (1\right ) \sinh \left (1\right ) + {\left (b^{3} d^{2} x^{2} + 2 \, b^{3} c d x + b^{3} c^{2} - 2 \, b^{3}\right )} \sinh \left (1\right )^{2}\right )}\right )} \log \left (d x + c + \sqrt {d^{2} x^{2} + 2 \, c d x + c^{2} + 1}\right )^{2} + 6 \, {\left ({\left (3 \, a^{3} + 2 \, a b^{2}\right )} d^{3} x^{3} + 3 \, {\left (3 \, a^{3} + 2 \, a b^{2}\right )} c d^{2} x^{2} - 3 \, {\left (4 \, a b^{2} - {\left (3 \, a^{3} + 2 \, a b^{2}\right )} c^{2}\right )} d x\right )} \cosh \left (1\right ) \sinh \left (1\right ) + 3 \, {\left ({\left (3 \, a^{3} + 2 \, a b^{2}\right )} d^{3} x^{3} + 3 \, {\left (3 \, a^{3} + 2 \, a b^{2}\right )} c d^{2} x^{2} - 3 \, {\left (4 \, a b^{2} - {\left (3 \, a^{3} + 2 \, a b^{2}\right )} c^{2}\right )} d x\right )} \sinh \left (1\right )^{2} + 3 \, {\left ({\left ({\left (9 \, a^{2} b + 2 \, b^{3}\right )} d^{3} x^{3} + 3 \, {\left (9 \, a^{2} b + 2 \, b^{3}\right )} c d^{2} x^{2} - 12 \, b^{3} c + {\left (9 \, a^{2} b + 2 \, b^{3}\right )} c^{3} - 3 \, {\left (4 \, b^{3} - {\left (9 \, a^{2} b + 2 \, b^{3}\right )} c^{2}\right )} d x\right )} \cosh \left (1\right )^{2} + 2 \, {\left ({\left (9 \, a^{2} b + 2 \, b^{3}\right )} d^{3} x^{3} + 3 \, {\left (9 \, a^{2} b + 2 \, b^{3}\right )} c d^{2} x^{2} - 12 \, b^{3} c + {\left (9 \, a^{2} b + 2 \, b^{3}\right )} c^{3} - 3 \, {\left (4 \, b^{3} - {\left (9 \, a^{2} b + 2 \, b^{3}\right )} c^{2}\right )} d x\right )} \cosh \left (1\right ) \sinh \left (1\right ) + {\left ({\left (9 \, a^{2} b + 2 \, b^{3}\right )} d^{3} x^{3} + 3 \, {\left (9 \, a^{2} b + 2 \, b^{3}\right )} c d^{2} x^{2} - 12 \, b^{3} c + {\left (9 \, a^{2} b + 2 \, b^{3}\right )} c^{3} - 3 \, {\left (4 \, b^{3} - {\left (9 \, a^{2} b + 2 \, b^{3}\right )} c^{2}\right )} d x\right )} \sinh \left (1\right )^{2} - 6 \, \sqrt {d^{2} x^{2} + 2 \, c d x + c^{2} + 1} {\left ({\left (a b^{2} d^{2} x^{2} + 2 \, a b^{2} c d x + a b^{2} c^{2} - 2 \, a b^{2}\right )} \cosh \left (1\right )^{2} + 2 \, {\left (a b^{2} d^{2} x^{2} + 2 \, a b^{2} c d x + a b^{2} c^{2} - 2 \, a b^{2}\right )} \cosh \left (1\right ) \sinh \left (1\right ) + {\left (a b^{2} d^{2} x^{2} + 2 \, a b^{2} c d x + a b^{2} c^{2} - 2 \, a b^{2}\right )} \sinh \left (1\right )^{2}\right )}\right )} \log \left (d x + c + \sqrt {d^{2} x^{2} + 2 \, c d x + c^{2} + 1}\right ) - \sqrt {d^{2} x^{2} + 2 \, c d x + c^{2} + 1} {\left ({\left ({\left (9 \, a^{2} b + 2 \, b^{3}\right )} d^{2} x^{2} + 2 \, {\left (9 \, a^{2} b + 2 \, b^{3}\right )} c d x - 18 \, a^{2} b - 40 \, b^{3} + {\left (9 \, a^{2} b + 2 \, b^{3}\right )} c^{2}\right )} \cosh \left (1\right )^{2} + 2 \, {\left ({\left (9 \, a^{2} b + 2 \, b^{3}\right )} d^{2} x^{2} + 2 \, {\left (9 \, a^{2} b + 2 \, b^{3}\right )} c d x - 18 \, a^{2} b - 40 \, b^{3} + {\left (9 \, a^{2} b + 2 \, b^{3}\right )} c^{2}\right )} \cosh \left (1\right ) \sinh \left (1\right ) + {\left ({\left (9 \, a^{2} b + 2 \, b^{3}\right )} d^{2} x^{2} + 2 \, {\left (9 \, a^{2} b + 2 \, b^{3}\right )} c d x - 18 \, a^{2} b - 40 \, b^{3} + {\left (9 \, a^{2} b + 2 \, b^{3}\right )} c^{2}\right )} \sinh \left (1\right )^{2}\right )}}{27 \, d} \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))^3,x, algorithm="fricas")

[Out]

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

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Sympy [B] Leaf count of result is larger than twice the leaf count of optimal. 1173 vs. \(2 (211) = 422\).
time = 0.56, size = 1173, normalized size = 5.17 \begin {gather*} \begin {cases} a^{3} c^{2} e^{2} x + a^{3} c d e^{2} x^{2} + \frac {a^{3} d^{2} e^{2} x^{3}}{3} + \frac {a^{2} b c^{3} e^{2} \operatorname {asinh}{\left (c + d x \right )}}{d} + 3 a^{2} b c^{2} e^{2} x \operatorname {asinh}{\left (c + d x \right )} - \frac {a^{2} b c^{2} e^{2} \sqrt {c^{2} + 2 c d x + d^{2} x^{2} + 1}}{3 d} + 3 a^{2} b c d e^{2} x^{2} \operatorname {asinh}{\left (c + d x \right )} - \frac {2 a^{2} b c e^{2} x \sqrt {c^{2} + 2 c d x + d^{2} x^{2} + 1}}{3} + a^{2} b d^{2} e^{2} x^{3} \operatorname {asinh}{\left (c + d x \right )} - \frac {a^{2} b d e^{2} x^{2} \sqrt {c^{2} + 2 c d x + d^{2} x^{2} + 1}}{3} + \frac {2 a^{2} b e^{2} \sqrt {c^{2} + 2 c d x + d^{2} x^{2} + 1}}{3 d} + \frac {a b^{2} c^{3} e^{2} \operatorname {asinh}^{2}{\left (c + d x \right )}}{d} + 3 a b^{2} c^{2} e^{2} x \operatorname {asinh}^{2}{\left (c + d x \right )} + \frac {2 a b^{2} c^{2} e^{2} x}{3} - \frac {2 a b^{2} c^{2} e^{2} \sqrt {c^{2} + 2 c d x + d^{2} x^{2} + 1} \operatorname {asinh}{\left (c + d x \right )}}{3 d} + 3 a b^{2} c d e^{2} x^{2} \operatorname {asinh}^{2}{\left (c + d x \right )} + \frac {2 a b^{2} c d e^{2} x^{2}}{3} - \frac {4 a b^{2} c e^{2} x \sqrt {c^{2} + 2 c d x + d^{2} x^{2} + 1} \operatorname {asinh}{\left (c + d x \right )}}{3} + a b^{2} d^{2} e^{2} x^{3} \operatorname {asinh}^{2}{\left (c + d x \right )} + \frac {2 a b^{2} d^{2} e^{2} x^{3}}{9} - \frac {2 a b^{2} d e^{2} x^{2} \sqrt {c^{2} + 2 c d x + d^{2} x^{2} + 1} \operatorname {asinh}{\left (c + d x \right )}}{3} - \frac {4 a b^{2} e^{2} x}{3} + \frac {4 a b^{2} e^{2} \sqrt {c^{2} + 2 c d x + d^{2} x^{2} + 1} \operatorname {asinh}{\left (c + d x \right )}}{3 d} + \frac {b^{3} c^{3} e^{2} \operatorname {asinh}^{3}{\left (c + d x \right )}}{3 d} + \frac {2 b^{3} c^{3} e^{2} \operatorname {asinh}{\left (c + d x \right )}}{9 d} + b^{3} c^{2} e^{2} x \operatorname {asinh}^{3}{\left (c + d x \right )} + \frac {2 b^{3} c^{2} e^{2} x \operatorname {asinh}{\left (c + d x \right )}}{3} - \frac {b^{3} c^{2} e^{2} \sqrt {c^{2} + 2 c d x + d^{2} x^{2} + 1} \operatorname {asinh}^{2}{\left (c + d x \right )}}{3 d} - \frac {2 b^{3} c^{2} e^{2} \sqrt {c^{2} + 2 c d x + d^{2} x^{2} + 1}}{27 d} + b^{3} c d e^{2} x^{2} \operatorname {asinh}^{3}{\left (c + d x \right )} + \frac {2 b^{3} c d e^{2} x^{2} \operatorname {asinh}{\left (c + d x \right )}}{3} - \frac {2 b^{3} c e^{2} x \sqrt {c^{2} + 2 c d x + d^{2} x^{2} + 1} \operatorname {asinh}^{2}{\left (c + d x \right )}}{3} - \frac {4 b^{3} c e^{2} x \sqrt {c^{2} + 2 c d x + d^{2} x^{2} + 1}}{27} - \frac {4 b^{3} c e^{2} \operatorname {asinh}{\left (c + d x \right )}}{3 d} + \frac {b^{3} d^{2} e^{2} x^{3} \operatorname {asinh}^{3}{\left (c + d x \right )}}{3} + \frac {2 b^{3} d^{2} e^{2} x^{3} \operatorname {asinh}{\left (c + d x \right )}}{9} - \frac {b^{3} d e^{2} x^{2} \sqrt {c^{2} + 2 c d x + d^{2} x^{2} + 1} \operatorname {asinh}^{2}{\left (c + d x \right )}}{3} - \frac {2 b^{3} d e^{2} x^{2} \sqrt {c^{2} + 2 c d x + d^{2} x^{2} + 1}}{27} - \frac {4 b^{3} e^{2} x \operatorname {asinh}{\left (c + d x \right )}}{3} + \frac {2 b^{3} e^{2} \sqrt {c^{2} + 2 c d x + d^{2} x^{2} + 1} \operatorname {asinh}^{2}{\left (c + d x \right )}}{3 d} + \frac {40 b^{3} e^{2} \sqrt {c^{2} + 2 c d x + d^{2} x^{2} + 1}}{27 d} & \text {for}\: d \neq 0 \\c^{2} e^{2} x \left (a + b \operatorname {asinh}{\left (c \right )}\right )^{3} & \text {otherwise} \end {cases} \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))**3,x)

[Out]

Piecewise((a**3*c**2*e**2*x + a**3*c*d*e**2*x**2 + a**3*d**2*e**2*x**3/3 + a**2*b*c**3*e**2*asinh(c + d*x)/d +
3*a**2*b*c**2*e**2*x*asinh(c + d*x) - a**2*b*c**2*e**2*sqrt(c**2 + 2*c*d*x + d**2*x**2 + 1)/(3*d) + 3*a**2*b*
c*d*e**2*x**2*asinh(c + d*x) - 2*a**2*b*c*e**2*x*sqrt(c**2 + 2*c*d*x + d**2*x**2 + 1)/3 + a**2*b*d**2*e**2*x**
3*asinh(c + d*x) - a**2*b*d*e**2*x**2*sqrt(c**2 + 2*c*d*x + d**2*x**2 + 1)/3 + 2*a**2*b*e**2*sqrt(c**2 + 2*c*d
*x + d**2*x**2 + 1)/(3*d) + a*b**2*c**3*e**2*asinh(c + d*x)**2/d + 3*a*b**2*c**2*e**2*x*asinh(c + d*x)**2 + 2*
a*b**2*c**2*e**2*x/3 - 2*a*b**2*c**2*e**2*sqrt(c**2 + 2*c*d*x + d**2*x**2 + 1)*asinh(c + d*x)/(3*d) + 3*a*b**2
*c*d*e**2*x**2*asinh(c + d*x)**2 + 2*a*b**2*c*d*e**2*x**2/3 - 4*a*b**2*c*e**2*x*sqrt(c**2 + 2*c*d*x + d**2*x**
2 + 1)*asinh(c + d*x)/3 + a*b**2*d**2*e**2*x**3*asinh(c + d*x)**2 + 2*a*b**2*d**2*e**2*x**3/9 - 2*a*b**2*d*e**
2*x**2*sqrt(c**2 + 2*c*d*x + d**2*x**2 + 1)*asinh(c + d*x)/3 - 4*a*b**2*e**2*x/3 + 4*a*b**2*e**2*sqrt(c**2 + 2
*c*d*x + d**2*x**2 + 1)*asinh(c + d*x)/(3*d) + b**3*c**3*e**2*asinh(c + d*x)**3/(3*d) + 2*b**3*c**3*e**2*asinh
(c + d*x)/(9*d) + b**3*c**2*e**2*x*asinh(c + d*x)**3 + 2*b**3*c**2*e**2*x*asinh(c + d*x)/3 - b**3*c**2*e**2*sq
rt(c**2 + 2*c*d*x + d**2*x**2 + 1)*asinh(c + d*x)**2/(3*d) - 2*b**3*c**2*e**2*sqrt(c**2 + 2*c*d*x + d**2*x**2
+ 1)/(27*d) + b**3*c*d*e**2*x**2*asinh(c + d*x)**3 + 2*b**3*c*d*e**2*x**2*asinh(c + d*x)/3 - 2*b**3*c*e**2*x*s
qrt(c**2 + 2*c*d*x + d**2*x**2 + 1)*asinh(c + d*x)**2/3 - 4*b**3*c*e**2*x*sqrt(c**2 + 2*c*d*x + d**2*x**2 + 1)
/27 - 4*b**3*c*e**2*asinh(c + d*x)/(3*d) + b**3*d**2*e**2*x**3*asinh(c + d*x)**3/3 + 2*b**3*d**2*e**2*x**3*asi
nh(c + d*x)/9 - 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 - 2*b**3*d*e**2*x**2
*sqrt(c**2 + 2*c*d*x + d**2*x**2 + 1)/27 - 4*b**3*e**2*x*asinh(c + d*x)/3 + 2*b**3*e**2*sqrt(c**2 + 2*c*d*x +
d**2*x**2 + 1)*asinh(c + d*x)**2/(3*d) + 40*b**3*e**2*sqrt(c**2 + 2*c*d*x + d**2*x**2 + 1)/(27*d), Ne(d, 0)),
(c**2*e**2*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)^2*(a+b*arcsinh(d*x+c))^3,x, algorithm="giac")

[Out]

integrate((d*e*x + c*e)^2*(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 )}^2\,{\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)^2*(a + b*asinh(c + d*x))^3,x)

[Out]

int((c*e + d*e*x)^2*(a + b*asinh(c + d*x))^3, x)

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