3.98 \(\int x^2 \sinh (a+b \sqrt [3]{c+d x}) \, dx\)

Optimal. Leaf size=537 \[ -\frac{6 c^2 \sqrt [3]{c+d x} \sinh \left (a+b \sqrt [3]{c+d x}\right )}{b^2 d^3}+\frac{6 c^2 \cosh \left (a+b \sqrt [3]{c+d x}\right )}{b^3 d^3}-\frac{24 (c+d x)^{7/3} \sinh \left (a+b \sqrt [3]{c+d x}\right )}{b^2 d^3}-\frac{1008 (c+d x)^{5/3} \sinh \left (a+b \sqrt [3]{c+d x}\right )}{b^4 d^3}+\frac{30 c (c+d x)^{4/3} \sinh \left (a+b \sqrt [3]{c+d x}\right )}{b^2 d^3}-\frac{20160 (c+d x) \sinh \left (a+b \sqrt [3]{c+d x}\right )}{b^6 d^3}+\frac{360 c (c+d x)^{2/3} \sinh \left (a+b \sqrt [3]{c+d x}\right )}{b^4 d^3}-\frac{120960 \sqrt [3]{c+d x} \sinh \left (a+b \sqrt [3]{c+d x}\right )}{b^8 d^3}+\frac{720 c \sinh \left (a+b \sqrt [3]{c+d x}\right )}{b^6 d^3}+\frac{168 (c+d x)^2 \cosh \left (a+b \sqrt [3]{c+d x}\right )}{b^3 d^3}+\frac{5040 (c+d x)^{4/3} \cosh \left (a+b \sqrt [3]{c+d x}\right )}{b^5 d^3}-\frac{120 c (c+d x) \cosh \left (a+b \sqrt [3]{c+d x}\right )}{b^3 d^3}+\frac{60480 (c+d x)^{2/3} \cosh \left (a+b \sqrt [3]{c+d x}\right )}{b^7 d^3}-\frac{720 c \sqrt [3]{c+d x} \cosh \left (a+b \sqrt [3]{c+d x}\right )}{b^5 d^3}+\frac{120960 \cosh \left (a+b \sqrt [3]{c+d x}\right )}{b^9 d^3}+\frac{3 c^2 (c+d x)^{2/3} \cosh \left (a+b \sqrt [3]{c+d x}\right )}{b d^3}+\frac{3 (c+d x)^{8/3} \cosh \left (a+b \sqrt [3]{c+d x}\right )}{b d^3}-\frac{6 c (c+d x)^{5/3} \cosh \left (a+b \sqrt [3]{c+d x}\right )}{b d^3} \]

[Out]

(120960*Cosh[a + b*(c + d*x)^(1/3)])/(b^9*d^3) + (6*c^2*Cosh[a + b*(c + d*x)^(1/3)])/(b^3*d^3) - (720*c*(c + d
*x)^(1/3)*Cosh[a + b*(c + d*x)^(1/3)])/(b^5*d^3) + (60480*(c + d*x)^(2/3)*Cosh[a + b*(c + d*x)^(1/3)])/(b^7*d^
3) + (3*c^2*(c + d*x)^(2/3)*Cosh[a + b*(c + d*x)^(1/3)])/(b*d^3) - (120*c*(c + d*x)*Cosh[a + b*(c + d*x)^(1/3)
])/(b^3*d^3) + (5040*(c + d*x)^(4/3)*Cosh[a + b*(c + d*x)^(1/3)])/(b^5*d^3) - (6*c*(c + d*x)^(5/3)*Cosh[a + b*
(c + d*x)^(1/3)])/(b*d^3) + (168*(c + d*x)^2*Cosh[a + b*(c + d*x)^(1/3)])/(b^3*d^3) + (3*(c + d*x)^(8/3)*Cosh[
a + b*(c + d*x)^(1/3)])/(b*d^3) + (720*c*Sinh[a + b*(c + d*x)^(1/3)])/(b^6*d^3) - (120960*(c + d*x)^(1/3)*Sinh
[a + b*(c + d*x)^(1/3)])/(b^8*d^3) - (6*c^2*(c + d*x)^(1/3)*Sinh[a + b*(c + d*x)^(1/3)])/(b^2*d^3) + (360*c*(c
+ d*x)^(2/3)*Sinh[a + b*(c + d*x)^(1/3)])/(b^4*d^3) - (20160*(c + d*x)*Sinh[a + b*(c + d*x)^(1/3)])/(b^6*d^3)
+ (30*c*(c + d*x)^(4/3)*Sinh[a + b*(c + d*x)^(1/3)])/(b^2*d^3) - (1008*(c + d*x)^(5/3)*Sinh[a + b*(c + d*x)^(
1/3)])/(b^4*d^3) - (24*(c + d*x)^(7/3)*Sinh[a + b*(c + d*x)^(1/3)])/(b^2*d^3)

________________________________________________________________________________________

Rubi [A]  time = 0.696997, antiderivative size = 537, normalized size of antiderivative = 1., number of steps used = 23, number of rules used = 6, integrand size = 18, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.333, Rules used = {5364, 1593, 5286, 3296, 2638, 2637} \[ -\frac{6 c^2 \sqrt [3]{c+d x} \sinh \left (a+b \sqrt [3]{c+d x}\right )}{b^2 d^3}+\frac{6 c^2 \cosh \left (a+b \sqrt [3]{c+d x}\right )}{b^3 d^3}-\frac{24 (c+d x)^{7/3} \sinh \left (a+b \sqrt [3]{c+d x}\right )}{b^2 d^3}-\frac{1008 (c+d x)^{5/3} \sinh \left (a+b \sqrt [3]{c+d x}\right )}{b^4 d^3}+\frac{30 c (c+d x)^{4/3} \sinh \left (a+b \sqrt [3]{c+d x}\right )}{b^2 d^3}-\frac{20160 (c+d x) \sinh \left (a+b \sqrt [3]{c+d x}\right )}{b^6 d^3}+\frac{360 c (c+d x)^{2/3} \sinh \left (a+b \sqrt [3]{c+d x}\right )}{b^4 d^3}-\frac{120960 \sqrt [3]{c+d x} \sinh \left (a+b \sqrt [3]{c+d x}\right )}{b^8 d^3}+\frac{720 c \sinh \left (a+b \sqrt [3]{c+d x}\right )}{b^6 d^3}+\frac{168 (c+d x)^2 \cosh \left (a+b \sqrt [3]{c+d x}\right )}{b^3 d^3}+\frac{5040 (c+d x)^{4/3} \cosh \left (a+b \sqrt [3]{c+d x}\right )}{b^5 d^3}-\frac{120 c (c+d x) \cosh \left (a+b \sqrt [3]{c+d x}\right )}{b^3 d^3}+\frac{60480 (c+d x)^{2/3} \cosh \left (a+b \sqrt [3]{c+d x}\right )}{b^7 d^3}-\frac{720 c \sqrt [3]{c+d x} \cosh \left (a+b \sqrt [3]{c+d x}\right )}{b^5 d^3}+\frac{120960 \cosh \left (a+b \sqrt [3]{c+d x}\right )}{b^9 d^3}+\frac{3 c^2 (c+d x)^{2/3} \cosh \left (a+b \sqrt [3]{c+d x}\right )}{b d^3}+\frac{3 (c+d x)^{8/3} \cosh \left (a+b \sqrt [3]{c+d x}\right )}{b d^3}-\frac{6 c (c+d x)^{5/3} \cosh \left (a+b \sqrt [3]{c+d x}\right )}{b d^3} \]

Antiderivative was successfully verified.

[In]

Int[x^2*Sinh[a + b*(c + d*x)^(1/3)],x]

[Out]

(120960*Cosh[a + b*(c + d*x)^(1/3)])/(b^9*d^3) + (6*c^2*Cosh[a + b*(c + d*x)^(1/3)])/(b^3*d^3) - (720*c*(c + d
*x)^(1/3)*Cosh[a + b*(c + d*x)^(1/3)])/(b^5*d^3) + (60480*(c + d*x)^(2/3)*Cosh[a + b*(c + d*x)^(1/3)])/(b^7*d^
3) + (3*c^2*(c + d*x)^(2/3)*Cosh[a + b*(c + d*x)^(1/3)])/(b*d^3) - (120*c*(c + d*x)*Cosh[a + b*(c + d*x)^(1/3)
])/(b^3*d^3) + (5040*(c + d*x)^(4/3)*Cosh[a + b*(c + d*x)^(1/3)])/(b^5*d^3) - (6*c*(c + d*x)^(5/3)*Cosh[a + b*
(c + d*x)^(1/3)])/(b*d^3) + (168*(c + d*x)^2*Cosh[a + b*(c + d*x)^(1/3)])/(b^3*d^3) + (3*(c + d*x)^(8/3)*Cosh[
a + b*(c + d*x)^(1/3)])/(b*d^3) + (720*c*Sinh[a + b*(c + d*x)^(1/3)])/(b^6*d^3) - (120960*(c + d*x)^(1/3)*Sinh
[a + b*(c + d*x)^(1/3)])/(b^8*d^3) - (6*c^2*(c + d*x)^(1/3)*Sinh[a + b*(c + d*x)^(1/3)])/(b^2*d^3) + (360*c*(c
+ d*x)^(2/3)*Sinh[a + b*(c + d*x)^(1/3)])/(b^4*d^3) - (20160*(c + d*x)*Sinh[a + b*(c + d*x)^(1/3)])/(b^6*d^3)
+ (30*c*(c + d*x)^(4/3)*Sinh[a + b*(c + d*x)^(1/3)])/(b^2*d^3) - (1008*(c + d*x)^(5/3)*Sinh[a + b*(c + d*x)^(
1/3)])/(b^4*d^3) - (24*(c + d*x)^(7/3)*Sinh[a + b*(c + d*x)^(1/3)])/(b^2*d^3)

Rule 5364

Int[(x_)^(m_.)*((a_.) + (b_.)*Sinh[(c_.) + (d_.)*(u_)^(n_)])^(p_.), x_Symbol] :> Dist[1/Coefficient[u, x, 1]^(
m + 1), Subst[Int[(x - Coefficient[u, x, 0])^m*(a + b*Sinh[c + d*x^n])^p, x], x, u], x] /; FreeQ[{a, b, c, d,
n, p}, x] && LinearQ[u, x] && NeQ[u, x] && IntegerQ[m]

Rule 1593

Int[(u_.)*((a_.)*(x_)^(p_.) + (b_.)*(x_)^(q_.))^(n_.), x_Symbol] :> Int[u*x^(n*p)*(a + b*x^(q - p))^n, x] /; F
reeQ[{a, b, p, q}, x] && IntegerQ[n] && PosQ[q - p]

Rule 5286

Int[((e_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_.)*Sinh[(c_.) + (d_.)*(x_)], x_Symbol] :> Int[ExpandIntegra
nd[Sinh[c + d*x], (e*x)^m*(a + b*x^n)^p, x], x] /; FreeQ[{a, b, c, d, e, m, n}, x] && IGtQ[p, 0]

Rule 3296

Int[((c_.) + (d_.)*(x_))^(m_.)*sin[(e_.) + (f_.)*(x_)], x_Symbol] :> -Simp[((c + d*x)^m*Cos[e + f*x])/f, x] +
Dist[(d*m)/f, Int[(c + d*x)^(m - 1)*Cos[e + f*x], x], x] /; FreeQ[{c, d, e, f}, x] && GtQ[m, 0]

Rule 2638

Int[sin[(c_.) + (d_.)*(x_)], x_Symbol] :> -Simp[Cos[c + d*x]/d, x] /; FreeQ[{c, d}, x]

Rule 2637

Int[sin[Pi/2 + (c_.) + (d_.)*(x_)], x_Symbol] :> Simp[Sin[c + d*x]/d, x] /; FreeQ[{c, d}, x]

Rubi steps

\begin{align*} \int x^2 \sinh \left (a+b \sqrt [3]{c+d x}\right ) \, dx &=\frac{\operatorname{Subst}\left (\int (-c+x)^2 \sinh \left (a+b \sqrt [3]{x}\right ) \, dx,x,c+d x\right )}{d^3}\\ &=\frac{3 \operatorname{Subst}\left (\int \left (-c x+x^4\right )^2 \sinh (a+b x) \, dx,x,\sqrt [3]{c+d x}\right )}{d^3}\\ &=\frac{3 \operatorname{Subst}\left (\int x^2 \left (-c+x^3\right )^2 \sinh (a+b x) \, dx,x,\sqrt [3]{c+d x}\right )}{d^3}\\ &=\frac{3 \operatorname{Subst}\left (\int \left (c^2 x^2 \sinh (a+b x)-2 c x^5 \sinh (a+b x)+x^8 \sinh (a+b x)\right ) \, dx,x,\sqrt [3]{c+d x}\right )}{d^3}\\ &=\frac{3 \operatorname{Subst}\left (\int x^8 \sinh (a+b x) \, dx,x,\sqrt [3]{c+d x}\right )}{d^3}-\frac{(6 c) \operatorname{Subst}\left (\int x^5 \sinh (a+b x) \, dx,x,\sqrt [3]{c+d x}\right )}{d^3}+\frac{\left (3 c^2\right ) \operatorname{Subst}\left (\int x^2 \sinh (a+b x) \, dx,x,\sqrt [3]{c+d x}\right )}{d^3}\\ &=\frac{3 c^2 (c+d x)^{2/3} \cosh \left (a+b \sqrt [3]{c+d x}\right )}{b d^3}-\frac{6 c (c+d x)^{5/3} \cosh \left (a+b \sqrt [3]{c+d x}\right )}{b d^3}+\frac{3 (c+d x)^{8/3} \cosh \left (a+b \sqrt [3]{c+d x}\right )}{b d^3}-\frac{24 \operatorname{Subst}\left (\int x^7 \cosh (a+b x) \, dx,x,\sqrt [3]{c+d x}\right )}{b d^3}+\frac{(30 c) \operatorname{Subst}\left (\int x^4 \cosh (a+b x) \, dx,x,\sqrt [3]{c+d x}\right )}{b d^3}-\frac{\left (6 c^2\right ) \operatorname{Subst}\left (\int x \cosh (a+b x) \, dx,x,\sqrt [3]{c+d x}\right )}{b d^3}\\ &=\frac{3 c^2 (c+d x)^{2/3} \cosh \left (a+b \sqrt [3]{c+d x}\right )}{b d^3}-\frac{6 c (c+d x)^{5/3} \cosh \left (a+b \sqrt [3]{c+d x}\right )}{b d^3}+\frac{3 (c+d x)^{8/3} \cosh \left (a+b \sqrt [3]{c+d x}\right )}{b d^3}-\frac{6 c^2 \sqrt [3]{c+d x} \sinh \left (a+b \sqrt [3]{c+d x}\right )}{b^2 d^3}+\frac{30 c (c+d x)^{4/3} \sinh \left (a+b \sqrt [3]{c+d x}\right )}{b^2 d^3}-\frac{24 (c+d x)^{7/3} \sinh \left (a+b \sqrt [3]{c+d x}\right )}{b^2 d^3}+\frac{168 \operatorname{Subst}\left (\int x^6 \sinh (a+b x) \, dx,x,\sqrt [3]{c+d x}\right )}{b^2 d^3}-\frac{(120 c) \operatorname{Subst}\left (\int x^3 \sinh (a+b x) \, dx,x,\sqrt [3]{c+d x}\right )}{b^2 d^3}+\frac{\left (6 c^2\right ) \operatorname{Subst}\left (\int \sinh (a+b x) \, dx,x,\sqrt [3]{c+d x}\right )}{b^2 d^3}\\ &=\frac{6 c^2 \cosh \left (a+b \sqrt [3]{c+d x}\right )}{b^3 d^3}+\frac{3 c^2 (c+d x)^{2/3} \cosh \left (a+b \sqrt [3]{c+d x}\right )}{b d^3}-\frac{120 c (c+d x) \cosh \left (a+b \sqrt [3]{c+d x}\right )}{b^3 d^3}-\frac{6 c (c+d x)^{5/3} \cosh \left (a+b \sqrt [3]{c+d x}\right )}{b d^3}+\frac{168 (c+d x)^2 \cosh \left (a+b \sqrt [3]{c+d x}\right )}{b^3 d^3}+\frac{3 (c+d x)^{8/3} \cosh \left (a+b \sqrt [3]{c+d x}\right )}{b d^3}-\frac{6 c^2 \sqrt [3]{c+d x} \sinh \left (a+b \sqrt [3]{c+d x}\right )}{b^2 d^3}+\frac{30 c (c+d x)^{4/3} \sinh \left (a+b \sqrt [3]{c+d x}\right )}{b^2 d^3}-\frac{24 (c+d x)^{7/3} \sinh \left (a+b \sqrt [3]{c+d x}\right )}{b^2 d^3}-\frac{1008 \operatorname{Subst}\left (\int x^5 \cosh (a+b x) \, dx,x,\sqrt [3]{c+d x}\right )}{b^3 d^3}+\frac{(360 c) \operatorname{Subst}\left (\int x^2 \cosh (a+b x) \, dx,x,\sqrt [3]{c+d x}\right )}{b^3 d^3}\\ &=\frac{6 c^2 \cosh \left (a+b \sqrt [3]{c+d x}\right )}{b^3 d^3}+\frac{3 c^2 (c+d x)^{2/3} \cosh \left (a+b \sqrt [3]{c+d x}\right )}{b d^3}-\frac{120 c (c+d x) \cosh \left (a+b \sqrt [3]{c+d x}\right )}{b^3 d^3}-\frac{6 c (c+d x)^{5/3} \cosh \left (a+b \sqrt [3]{c+d x}\right )}{b d^3}+\frac{168 (c+d x)^2 \cosh \left (a+b \sqrt [3]{c+d x}\right )}{b^3 d^3}+\frac{3 (c+d x)^{8/3} \cosh \left (a+b \sqrt [3]{c+d x}\right )}{b d^3}-\frac{6 c^2 \sqrt [3]{c+d x} \sinh \left (a+b \sqrt [3]{c+d x}\right )}{b^2 d^3}+\frac{360 c (c+d x)^{2/3} \sinh \left (a+b \sqrt [3]{c+d x}\right )}{b^4 d^3}+\frac{30 c (c+d x)^{4/3} \sinh \left (a+b \sqrt [3]{c+d x}\right )}{b^2 d^3}-\frac{1008 (c+d x)^{5/3} \sinh \left (a+b \sqrt [3]{c+d x}\right )}{b^4 d^3}-\frac{24 (c+d x)^{7/3} \sinh \left (a+b \sqrt [3]{c+d x}\right )}{b^2 d^3}+\frac{5040 \operatorname{Subst}\left (\int x^4 \sinh (a+b x) \, dx,x,\sqrt [3]{c+d x}\right )}{b^4 d^3}-\frac{(720 c) \operatorname{Subst}\left (\int x \sinh (a+b x) \, dx,x,\sqrt [3]{c+d x}\right )}{b^4 d^3}\\ &=\frac{6 c^2 \cosh \left (a+b \sqrt [3]{c+d x}\right )}{b^3 d^3}-\frac{720 c \sqrt [3]{c+d x} \cosh \left (a+b \sqrt [3]{c+d x}\right )}{b^5 d^3}+\frac{3 c^2 (c+d x)^{2/3} \cosh \left (a+b \sqrt [3]{c+d x}\right )}{b d^3}-\frac{120 c (c+d x) \cosh \left (a+b \sqrt [3]{c+d x}\right )}{b^3 d^3}+\frac{5040 (c+d x)^{4/3} \cosh \left (a+b \sqrt [3]{c+d x}\right )}{b^5 d^3}-\frac{6 c (c+d x)^{5/3} \cosh \left (a+b \sqrt [3]{c+d x}\right )}{b d^3}+\frac{168 (c+d x)^2 \cosh \left (a+b \sqrt [3]{c+d x}\right )}{b^3 d^3}+\frac{3 (c+d x)^{8/3} \cosh \left (a+b \sqrt [3]{c+d x}\right )}{b d^3}-\frac{6 c^2 \sqrt [3]{c+d x} \sinh \left (a+b \sqrt [3]{c+d x}\right )}{b^2 d^3}+\frac{360 c (c+d x)^{2/3} \sinh \left (a+b \sqrt [3]{c+d x}\right )}{b^4 d^3}+\frac{30 c (c+d x)^{4/3} \sinh \left (a+b \sqrt [3]{c+d x}\right )}{b^2 d^3}-\frac{1008 (c+d x)^{5/3} \sinh \left (a+b \sqrt [3]{c+d x}\right )}{b^4 d^3}-\frac{24 (c+d x)^{7/3} \sinh \left (a+b \sqrt [3]{c+d x}\right )}{b^2 d^3}-\frac{20160 \operatorname{Subst}\left (\int x^3 \cosh (a+b x) \, dx,x,\sqrt [3]{c+d x}\right )}{b^5 d^3}+\frac{(720 c) \operatorname{Subst}\left (\int \cosh (a+b x) \, dx,x,\sqrt [3]{c+d x}\right )}{b^5 d^3}\\ &=\frac{6 c^2 \cosh \left (a+b \sqrt [3]{c+d x}\right )}{b^3 d^3}-\frac{720 c \sqrt [3]{c+d x} \cosh \left (a+b \sqrt [3]{c+d x}\right )}{b^5 d^3}+\frac{3 c^2 (c+d x)^{2/3} \cosh \left (a+b \sqrt [3]{c+d x}\right )}{b d^3}-\frac{120 c (c+d x) \cosh \left (a+b \sqrt [3]{c+d x}\right )}{b^3 d^3}+\frac{5040 (c+d x)^{4/3} \cosh \left (a+b \sqrt [3]{c+d x}\right )}{b^5 d^3}-\frac{6 c (c+d x)^{5/3} \cosh \left (a+b \sqrt [3]{c+d x}\right )}{b d^3}+\frac{168 (c+d x)^2 \cosh \left (a+b \sqrt [3]{c+d x}\right )}{b^3 d^3}+\frac{3 (c+d x)^{8/3} \cosh \left (a+b \sqrt [3]{c+d x}\right )}{b d^3}+\frac{720 c \sinh \left (a+b \sqrt [3]{c+d x}\right )}{b^6 d^3}-\frac{6 c^2 \sqrt [3]{c+d x} \sinh \left (a+b \sqrt [3]{c+d x}\right )}{b^2 d^3}+\frac{360 c (c+d x)^{2/3} \sinh \left (a+b \sqrt [3]{c+d x}\right )}{b^4 d^3}-\frac{20160 (c+d x) \sinh \left (a+b \sqrt [3]{c+d x}\right )}{b^6 d^3}+\frac{30 c (c+d x)^{4/3} \sinh \left (a+b \sqrt [3]{c+d x}\right )}{b^2 d^3}-\frac{1008 (c+d x)^{5/3} \sinh \left (a+b \sqrt [3]{c+d x}\right )}{b^4 d^3}-\frac{24 (c+d x)^{7/3} \sinh \left (a+b \sqrt [3]{c+d x}\right )}{b^2 d^3}+\frac{60480 \operatorname{Subst}\left (\int x^2 \sinh (a+b x) \, dx,x,\sqrt [3]{c+d x}\right )}{b^6 d^3}\\ &=\frac{6 c^2 \cosh \left (a+b \sqrt [3]{c+d x}\right )}{b^3 d^3}-\frac{720 c \sqrt [3]{c+d x} \cosh \left (a+b \sqrt [3]{c+d x}\right )}{b^5 d^3}+\frac{60480 (c+d x)^{2/3} \cosh \left (a+b \sqrt [3]{c+d x}\right )}{b^7 d^3}+\frac{3 c^2 (c+d x)^{2/3} \cosh \left (a+b \sqrt [3]{c+d x}\right )}{b d^3}-\frac{120 c (c+d x) \cosh \left (a+b \sqrt [3]{c+d x}\right )}{b^3 d^3}+\frac{5040 (c+d x)^{4/3} \cosh \left (a+b \sqrt [3]{c+d x}\right )}{b^5 d^3}-\frac{6 c (c+d x)^{5/3} \cosh \left (a+b \sqrt [3]{c+d x}\right )}{b d^3}+\frac{168 (c+d x)^2 \cosh \left (a+b \sqrt [3]{c+d x}\right )}{b^3 d^3}+\frac{3 (c+d x)^{8/3} \cosh \left (a+b \sqrt [3]{c+d x}\right )}{b d^3}+\frac{720 c \sinh \left (a+b \sqrt [3]{c+d x}\right )}{b^6 d^3}-\frac{6 c^2 \sqrt [3]{c+d x} \sinh \left (a+b \sqrt [3]{c+d x}\right )}{b^2 d^3}+\frac{360 c (c+d x)^{2/3} \sinh \left (a+b \sqrt [3]{c+d x}\right )}{b^4 d^3}-\frac{20160 (c+d x) \sinh \left (a+b \sqrt [3]{c+d x}\right )}{b^6 d^3}+\frac{30 c (c+d x)^{4/3} \sinh \left (a+b \sqrt [3]{c+d x}\right )}{b^2 d^3}-\frac{1008 (c+d x)^{5/3} \sinh \left (a+b \sqrt [3]{c+d x}\right )}{b^4 d^3}-\frac{24 (c+d x)^{7/3} \sinh \left (a+b \sqrt [3]{c+d x}\right )}{b^2 d^3}-\frac{120960 \operatorname{Subst}\left (\int x \cosh (a+b x) \, dx,x,\sqrt [3]{c+d x}\right )}{b^7 d^3}\\ &=\frac{6 c^2 \cosh \left (a+b \sqrt [3]{c+d x}\right )}{b^3 d^3}-\frac{720 c \sqrt [3]{c+d x} \cosh \left (a+b \sqrt [3]{c+d x}\right )}{b^5 d^3}+\frac{60480 (c+d x)^{2/3} \cosh \left (a+b \sqrt [3]{c+d x}\right )}{b^7 d^3}+\frac{3 c^2 (c+d x)^{2/3} \cosh \left (a+b \sqrt [3]{c+d x}\right )}{b d^3}-\frac{120 c (c+d x) \cosh \left (a+b \sqrt [3]{c+d x}\right )}{b^3 d^3}+\frac{5040 (c+d x)^{4/3} \cosh \left (a+b \sqrt [3]{c+d x}\right )}{b^5 d^3}-\frac{6 c (c+d x)^{5/3} \cosh \left (a+b \sqrt [3]{c+d x}\right )}{b d^3}+\frac{168 (c+d x)^2 \cosh \left (a+b \sqrt [3]{c+d x}\right )}{b^3 d^3}+\frac{3 (c+d x)^{8/3} \cosh \left (a+b \sqrt [3]{c+d x}\right )}{b d^3}+\frac{720 c \sinh \left (a+b \sqrt [3]{c+d x}\right )}{b^6 d^3}-\frac{120960 \sqrt [3]{c+d x} \sinh \left (a+b \sqrt [3]{c+d x}\right )}{b^8 d^3}-\frac{6 c^2 \sqrt [3]{c+d x} \sinh \left (a+b \sqrt [3]{c+d x}\right )}{b^2 d^3}+\frac{360 c (c+d x)^{2/3} \sinh \left (a+b \sqrt [3]{c+d x}\right )}{b^4 d^3}-\frac{20160 (c+d x) \sinh \left (a+b \sqrt [3]{c+d x}\right )}{b^6 d^3}+\frac{30 c (c+d x)^{4/3} \sinh \left (a+b \sqrt [3]{c+d x}\right )}{b^2 d^3}-\frac{1008 (c+d x)^{5/3} \sinh \left (a+b \sqrt [3]{c+d x}\right )}{b^4 d^3}-\frac{24 (c+d x)^{7/3} \sinh \left (a+b \sqrt [3]{c+d x}\right )}{b^2 d^3}+\frac{120960 \operatorname{Subst}\left (\int \sinh (a+b x) \, dx,x,\sqrt [3]{c+d x}\right )}{b^8 d^3}\\ &=\frac{120960 \cosh \left (a+b \sqrt [3]{c+d x}\right )}{b^9 d^3}+\frac{6 c^2 \cosh \left (a+b \sqrt [3]{c+d x}\right )}{b^3 d^3}-\frac{720 c \sqrt [3]{c+d x} \cosh \left (a+b \sqrt [3]{c+d x}\right )}{b^5 d^3}+\frac{60480 (c+d x)^{2/3} \cosh \left (a+b \sqrt [3]{c+d x}\right )}{b^7 d^3}+\frac{3 c^2 (c+d x)^{2/3} \cosh \left (a+b \sqrt [3]{c+d x}\right )}{b d^3}-\frac{120 c (c+d x) \cosh \left (a+b \sqrt [3]{c+d x}\right )}{b^3 d^3}+\frac{5040 (c+d x)^{4/3} \cosh \left (a+b \sqrt [3]{c+d x}\right )}{b^5 d^3}-\frac{6 c (c+d x)^{5/3} \cosh \left (a+b \sqrt [3]{c+d x}\right )}{b d^3}+\frac{168 (c+d x)^2 \cosh \left (a+b \sqrt [3]{c+d x}\right )}{b^3 d^3}+\frac{3 (c+d x)^{8/3} \cosh \left (a+b \sqrt [3]{c+d x}\right )}{b d^3}+\frac{720 c \sinh \left (a+b \sqrt [3]{c+d x}\right )}{b^6 d^3}-\frac{120960 \sqrt [3]{c+d x} \sinh \left (a+b \sqrt [3]{c+d x}\right )}{b^8 d^3}-\frac{6 c^2 \sqrt [3]{c+d x} \sinh \left (a+b \sqrt [3]{c+d x}\right )}{b^2 d^3}+\frac{360 c (c+d x)^{2/3} \sinh \left (a+b \sqrt [3]{c+d x}\right )}{b^4 d^3}-\frac{20160 (c+d x) \sinh \left (a+b \sqrt [3]{c+d x}\right )}{b^6 d^3}+\frac{30 c (c+d x)^{4/3} \sinh \left (a+b \sqrt [3]{c+d x}\right )}{b^2 d^3}-\frac{1008 (c+d x)^{5/3} \sinh \left (a+b \sqrt [3]{c+d x}\right )}{b^4 d^3}-\frac{24 (c+d x)^{7/3} \sinh \left (a+b \sqrt [3]{c+d x}\right )}{b^2 d^3}\\ \end{align*}

Mathematica [A]  time = 2.8738, size = 378, normalized size = 0.7 \[ \frac{3 \left ((\sinh (a)+\cosh (a)) \left (2 b^6 \left (9 c^2+36 c d x+28 d^2 x^2\right )+b^8 d^2 x^2 (c+d x)^{2/3}-2 b^7 d x \sqrt [3]{c+d x} (3 c+4 d x)-24 b^5 (c+d x)^{2/3} (9 c+14 d x)+240 b^4 \sqrt [3]{c+d x} (6 c+7 d x)-240 b^3 (27 c+28 d x)+20160 b^2 (c+d x)^{2/3}-40320 b \sqrt [3]{c+d x}+40320\right ) \left (\sinh \left (b \sqrt [3]{c+d x}\right )+\cosh \left (b \sqrt [3]{c+d x}\right )\right )+\left (2 b^6 \left (9 c^2+36 c d x+28 d^2 x^2\right )+b^8 d^2 x^2 (c+d x)^{2/3}+2 b^7 d x \sqrt [3]{c+d x} (3 c+4 d x)+24 b^5 (c+d x)^{2/3} (9 c+14 d x)+240 b^4 \sqrt [3]{c+d x} (6 c+7 d x)+240 b^3 (27 c+28 d x)+20160 b^2 (c+d x)^{2/3}+40320 b \sqrt [3]{c+d x}+40320\right ) \left (\cosh \left (a+b \sqrt [3]{c+d x}\right )-\sinh \left (a+b \sqrt [3]{c+d x}\right )\right )\right )}{2 b^9 d^3} \]

Antiderivative was successfully verified.

[In]

Integrate[x^2*Sinh[a + b*(c + d*x)^(1/3)],x]

[Out]

(3*((40320 - 40320*b*(c + d*x)^(1/3) + 20160*b^2*(c + d*x)^(2/3) + b^8*d^2*x^2*(c + d*x)^(2/3) - 2*b^7*d*x*(c
+ d*x)^(1/3)*(3*c + 4*d*x) + 240*b^4*(c + d*x)^(1/3)*(6*c + 7*d*x) - 24*b^5*(c + d*x)^(2/3)*(9*c + 14*d*x) - 2
40*b^3*(27*c + 28*d*x) + 2*b^6*(9*c^2 + 36*c*d*x + 28*d^2*x^2))*(Cosh[a] + Sinh[a])*(Cosh[b*(c + d*x)^(1/3)] +
Sinh[b*(c + d*x)^(1/3)]) + (40320 + 40320*b*(c + d*x)^(1/3) + 20160*b^2*(c + d*x)^(2/3) + b^8*d^2*x^2*(c + d*
x)^(2/3) + 2*b^7*d*x*(c + d*x)^(1/3)*(3*c + 4*d*x) + 240*b^4*(c + d*x)^(1/3)*(6*c + 7*d*x) + 24*b^5*(c + d*x)^
(2/3)*(9*c + 14*d*x) + 240*b^3*(27*c + 28*d*x) + 2*b^6*(9*c^2 + 36*c*d*x + 28*d^2*x^2))*(Cosh[a + b*(c + d*x)^
(1/3)] - Sinh[a + b*(c + d*x)^(1/3)])))/(2*b^9*d^3)

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Maple [B]  time = 0.01, size = 1815, normalized size = 3.4 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^2*sinh(a+b*(d*x+c)^(1/3)),x)

[Out]

3/d^3/b^3*(1/b^6*((a+b*(d*x+c)^(1/3))^8*cosh(a+b*(d*x+c)^(1/3))-8*(a+b*(d*x+c)^(1/3))^7*sinh(a+b*(d*x+c)^(1/3)
)+56*(a+b*(d*x+c)^(1/3))^6*cosh(a+b*(d*x+c)^(1/3))-336*(a+b*(d*x+c)^(1/3))^5*sinh(a+b*(d*x+c)^(1/3))+1680*(a+b
*(d*x+c)^(1/3))^4*cosh(a+b*(d*x+c)^(1/3))-6720*(a+b*(d*x+c)^(1/3))^3*sinh(a+b*(d*x+c)^(1/3))+20160*(a+b*(d*x+c
)^(1/3))^2*cosh(a+b*(d*x+c)^(1/3))-40320*(a+b*(d*x+c)^(1/3))*sinh(a+b*(d*x+c)^(1/3))+40320*cosh(a+b*(d*x+c)^(1
/3)))+1/b^6*a^8*cosh(a+b*(d*x+c)^(1/3))+c^2*((a+b*(d*x+c)^(1/3))^2*cosh(a+b*(d*x+c)^(1/3))-2*(a+b*(d*x+c)^(1/3
))*sinh(a+b*(d*x+c)^(1/3))+2*cosh(a+b*(d*x+c)^(1/3)))+c^2*a^2*cosh(a+b*(d*x+c)^(1/3))+2/b^3*a^5*c*cosh(a+b*(d*
x+c)^(1/3))-56/b^6*a^3*((a+b*(d*x+c)^(1/3))^5*cosh(a+b*(d*x+c)^(1/3))-5*(a+b*(d*x+c)^(1/3))^4*sinh(a+b*(d*x+c)
^(1/3))+20*(a+b*(d*x+c)^(1/3))^3*cosh(a+b*(d*x+c)^(1/3))-60*(a+b*(d*x+c)^(1/3))^2*sinh(a+b*(d*x+c)^(1/3))+120*
(a+b*(d*x+c)^(1/3))*cosh(a+b*(d*x+c)^(1/3))-120*sinh(a+b*(d*x+c)^(1/3)))-2/b^3*c*((a+b*(d*x+c)^(1/3))^5*cosh(a
+b*(d*x+c)^(1/3))-5*(a+b*(d*x+c)^(1/3))^4*sinh(a+b*(d*x+c)^(1/3))+20*(a+b*(d*x+c)^(1/3))^3*cosh(a+b*(d*x+c)^(1
/3))-60*(a+b*(d*x+c)^(1/3))^2*sinh(a+b*(d*x+c)^(1/3))+120*(a+b*(d*x+c)^(1/3))*cosh(a+b*(d*x+c)^(1/3))-120*sinh
(a+b*(d*x+c)^(1/3)))-8/b^6*a^7*((a+b*(d*x+c)^(1/3))*cosh(a+b*(d*x+c)^(1/3))-sinh(a+b*(d*x+c)^(1/3)))-2*c^2*a*(
(a+b*(d*x+c)^(1/3))*cosh(a+b*(d*x+c)^(1/3))-sinh(a+b*(d*x+c)^(1/3)))+70/b^6*a^4*((a+b*(d*x+c)^(1/3))^4*cosh(a+
b*(d*x+c)^(1/3))-4*(a+b*(d*x+c)^(1/3))^3*sinh(a+b*(d*x+c)^(1/3))+12*(a+b*(d*x+c)^(1/3))^2*cosh(a+b*(d*x+c)^(1/
3))-24*(a+b*(d*x+c)^(1/3))*sinh(a+b*(d*x+c)^(1/3))+24*cosh(a+b*(d*x+c)^(1/3)))-56/b^6*a^5*((a+b*(d*x+c)^(1/3))
^3*cosh(a+b*(d*x+c)^(1/3))-3*(a+b*(d*x+c)^(1/3))^2*sinh(a+b*(d*x+c)^(1/3))+6*(a+b*(d*x+c)^(1/3))*cosh(a+b*(d*x
+c)^(1/3))-6*sinh(a+b*(d*x+c)^(1/3)))-8/b^6*a*((a+b*(d*x+c)^(1/3))^7*cosh(a+b*(d*x+c)^(1/3))-7*(a+b*(d*x+c)^(1
/3))^6*sinh(a+b*(d*x+c)^(1/3))+42*(a+b*(d*x+c)^(1/3))^5*cosh(a+b*(d*x+c)^(1/3))-210*(a+b*(d*x+c)^(1/3))^4*sinh
(a+b*(d*x+c)^(1/3))+840*(a+b*(d*x+c)^(1/3))^3*cosh(a+b*(d*x+c)^(1/3))-2520*(a+b*(d*x+c)^(1/3))^2*sinh(a+b*(d*x
+c)^(1/3))+5040*(a+b*(d*x+c)^(1/3))*cosh(a+b*(d*x+c)^(1/3))-5040*sinh(a+b*(d*x+c)^(1/3)))+28/b^6*a^2*((a+b*(d*
x+c)^(1/3))^6*cosh(a+b*(d*x+c)^(1/3))-6*(a+b*(d*x+c)^(1/3))^5*sinh(a+b*(d*x+c)^(1/3))+30*(a+b*(d*x+c)^(1/3))^4
*cosh(a+b*(d*x+c)^(1/3))-120*(a+b*(d*x+c)^(1/3))^3*sinh(a+b*(d*x+c)^(1/3))+360*(a+b*(d*x+c)^(1/3))^2*cosh(a+b*
(d*x+c)^(1/3))-720*(a+b*(d*x+c)^(1/3))*sinh(a+b*(d*x+c)^(1/3))+720*cosh(a+b*(d*x+c)^(1/3)))+28/b^6*a^6*((a+b*(
d*x+c)^(1/3))^2*cosh(a+b*(d*x+c)^(1/3))-2*(a+b*(d*x+c)^(1/3))*sinh(a+b*(d*x+c)^(1/3))+2*cosh(a+b*(d*x+c)^(1/3)
))+20/b^3*a^3*c*((a+b*(d*x+c)^(1/3))^2*cosh(a+b*(d*x+c)^(1/3))-2*(a+b*(d*x+c)^(1/3))*sinh(a+b*(d*x+c)^(1/3))+2
*cosh(a+b*(d*x+c)^(1/3)))-10/b^3*a^4*c*((a+b*(d*x+c)^(1/3))*cosh(a+b*(d*x+c)^(1/3))-sinh(a+b*(d*x+c)^(1/3)))+1
0/b^3*c*a*((a+b*(d*x+c)^(1/3))^4*cosh(a+b*(d*x+c)^(1/3))-4*(a+b*(d*x+c)^(1/3))^3*sinh(a+b*(d*x+c)^(1/3))+12*(a
+b*(d*x+c)^(1/3))^2*cosh(a+b*(d*x+c)^(1/3))-24*(a+b*(d*x+c)^(1/3))*sinh(a+b*(d*x+c)^(1/3))+24*cosh(a+b*(d*x+c)
^(1/3)))-20/b^3*c*a^2*((a+b*(d*x+c)^(1/3))^3*cosh(a+b*(d*x+c)^(1/3))-3*(a+b*(d*x+c)^(1/3))^2*sinh(a+b*(d*x+c)^
(1/3))+6*(a+b*(d*x+c)^(1/3))*cosh(a+b*(d*x+c)^(1/3))-6*sinh(a+b*(d*x+c)^(1/3))))

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Maxima [A]  time = 1.18272, size = 867, normalized size = 1.61 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^2*sinh(a+b*(d*x+c)^(1/3)),x, algorithm="maxima")

[Out]

1/6*(2*d^3*x^3*sinh((d*x + c)^(1/3)*b + a) + (c^3*e^((d*x + c)^(1/3)*b + a)/b - c^3*e^(-(d*x + c)^(1/3)*b - a)
/b - 3*((d*x + c)*b^3*e^a - 3*(d*x + c)^(2/3)*b^2*e^a + 6*(d*x + c)^(1/3)*b*e^a - 6*e^a)*c^2*e^((d*x + c)^(1/3
)*b)/b^4 + 3*((d*x + c)*b^3 + 3*(d*x + c)^(2/3)*b^2 + 6*(d*x + c)^(1/3)*b + 6)*c^2*e^(-(d*x + c)^(1/3)*b - a)/
b^4 + 3*((d*x + c)^2*b^6*e^a - 6*(d*x + c)^(5/3)*b^5*e^a + 30*(d*x + c)^(4/3)*b^4*e^a - 120*(d*x + c)*b^3*e^a
+ 360*(d*x + c)^(2/3)*b^2*e^a - 720*(d*x + c)^(1/3)*b*e^a + 720*e^a)*c*e^((d*x + c)^(1/3)*b)/b^7 - 3*((d*x + c
)^2*b^6 + 6*(d*x + c)^(5/3)*b^5 + 30*(d*x + c)^(4/3)*b^4 + 120*(d*x + c)*b^3 + 360*(d*x + c)^(2/3)*b^2 + 720*(
d*x + c)^(1/3)*b + 720)*c*e^(-(d*x + c)^(1/3)*b - a)/b^7 - ((d*x + c)^3*b^9*e^a - 9*(d*x + c)^(8/3)*b^8*e^a +
72*(d*x + c)^(7/3)*b^7*e^a - 504*(d*x + c)^2*b^6*e^a + 3024*(d*x + c)^(5/3)*b^5*e^a - 15120*(d*x + c)^(4/3)*b^
4*e^a + 60480*(d*x + c)*b^3*e^a - 181440*(d*x + c)^(2/3)*b^2*e^a + 362880*(d*x + c)^(1/3)*b*e^a - 362880*e^a)*
e^((d*x + c)^(1/3)*b)/b^10 + ((d*x + c)^3*b^9 + 9*(d*x + c)^(8/3)*b^8 + 72*(d*x + c)^(7/3)*b^7 + 504*(d*x + c)
^2*b^6 + 3024*(d*x + c)^(5/3)*b^5 + 15120*(d*x + c)^(4/3)*b^4 + 60480*(d*x + c)*b^3 + 181440*(d*x + c)^(2/3)*b
^2 + 362880*(d*x + c)^(1/3)*b + 362880)*e^(-(d*x + c)^(1/3)*b - a)/b^10)*b)/d^3

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Fricas [A]  time = 2.0585, size = 467, normalized size = 0.87 \begin{align*} \frac{3 \,{\left ({\left (56 \, b^{6} d^{2} x^{2} + 72 \, b^{6} c d x + 18 \, b^{6} c^{2} +{\left (b^{8} d^{2} x^{2} + 20160 \, b^{2}\right )}{\left (d x + c\right )}^{\frac{2}{3}} + 240 \,{\left (7 \, b^{4} d x + 6 \, b^{4} c\right )}{\left (d x + c\right )}^{\frac{1}{3}} + 40320\right )} \cosh \left ({\left (d x + c\right )}^{\frac{1}{3}} b + a\right ) - 2 \,{\left (3360 \, b^{3} d x + 3240 \, b^{3} c + 12 \,{\left (14 \, b^{5} d x + 9 \, b^{5} c\right )}{\left (d x + c\right )}^{\frac{2}{3}} +{\left (4 \, b^{7} d^{2} x^{2} + 3 \, b^{7} c d x + 20160 \, b\right )}{\left (d x + c\right )}^{\frac{1}{3}}\right )} \sinh \left ({\left (d x + c\right )}^{\frac{1}{3}} b + a\right )\right )}}{b^{9} d^{3}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^2*sinh(a+b*(d*x+c)^(1/3)),x, algorithm="fricas")

[Out]

3*((56*b^6*d^2*x^2 + 72*b^6*c*d*x + 18*b^6*c^2 + (b^8*d^2*x^2 + 20160*b^2)*(d*x + c)^(2/3) + 240*(7*b^4*d*x +
6*b^4*c)*(d*x + c)^(1/3) + 40320)*cosh((d*x + c)^(1/3)*b + a) - 2*(3360*b^3*d*x + 3240*b^3*c + 12*(14*b^5*d*x
+ 9*b^5*c)*(d*x + c)^(2/3) + (4*b^7*d^2*x^2 + 3*b^7*c*d*x + 20160*b)*(d*x + c)^(1/3))*sinh((d*x + c)^(1/3)*b +
a))/(b^9*d^3)

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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int x^{2} \sinh{\left (a + b \sqrt [3]{c + d x} \right )}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x**2*sinh(a+b*(d*x+c)**(1/3)),x)

[Out]

Integral(x**2*sinh(a + b*(c + d*x)**(1/3)), x)

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Giac [B]  time = 3.70743, size = 2919, normalized size = 5.44 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^2*sinh(a+b*(d*x+c)^(1/3)),x, algorithm="giac")

[Out]

3/2*((((d*x + c)^(1/3)*b + a)^2*b^6*c^2 - 2*((d*x + c)^(1/3)*b + a)*a*b^6*c^2 + a^2*b^6*c^2 - 2*((d*x + c)^(1/
3)*b + a)^5*b^3*c + 10*((d*x + c)^(1/3)*b + a)^4*a*b^3*c - 20*((d*x + c)^(1/3)*b + a)^3*a^2*b^3*c + 20*((d*x +
c)^(1/3)*b + a)^2*a^3*b^3*c - 10*((d*x + c)^(1/3)*b + a)*a^4*b^3*c + 2*a^5*b^3*c - 2*((d*x + c)^(1/3)*b + a)*
b^6*c^2 + 2*a*b^6*c^2 + ((d*x + c)^(1/3)*b + a)^8 - 8*((d*x + c)^(1/3)*b + a)^7*a + 28*((d*x + c)^(1/3)*b + a)
^6*a^2 - 56*((d*x + c)^(1/3)*b + a)^5*a^3 + 70*((d*x + c)^(1/3)*b + a)^4*a^4 - 56*((d*x + c)^(1/3)*b + a)^3*a^
5 + 28*((d*x + c)^(1/3)*b + a)^2*a^6 - 8*((d*x + c)^(1/3)*b + a)*a^7 + a^8 + 10*((d*x + c)^(1/3)*b + a)^4*b^3*
c - 40*((d*x + c)^(1/3)*b + a)^3*a*b^3*c + 60*((d*x + c)^(1/3)*b + a)^2*a^2*b^3*c - 40*((d*x + c)^(1/3)*b + a)
*a^3*b^3*c + 10*a^4*b^3*c + 2*b^6*c^2 - 8*((d*x + c)^(1/3)*b + a)^7 + 56*((d*x + c)^(1/3)*b + a)^6*a - 168*((d
*x + c)^(1/3)*b + a)^5*a^2 + 280*((d*x + c)^(1/3)*b + a)^4*a^3 - 280*((d*x + c)^(1/3)*b + a)^3*a^4 + 168*((d*x
+ c)^(1/3)*b + a)^2*a^5 - 56*((d*x + c)^(1/3)*b + a)*a^6 + 8*a^7 - 40*((d*x + c)^(1/3)*b + a)^3*b^3*c + 120*(
(d*x + c)^(1/3)*b + a)^2*a*b^3*c - 120*((d*x + c)^(1/3)*b + a)*a^2*b^3*c + 40*a^3*b^3*c + 56*((d*x + c)^(1/3)*
b + a)^6 - 336*((d*x + c)^(1/3)*b + a)^5*a + 840*((d*x + c)^(1/3)*b + a)^4*a^2 - 1120*((d*x + c)^(1/3)*b + a)^
3*a^3 + 840*((d*x + c)^(1/3)*b + a)^2*a^4 - 336*((d*x + c)^(1/3)*b + a)*a^5 + 56*a^6 + 120*((d*x + c)^(1/3)*b
+ a)^2*b^3*c - 240*((d*x + c)^(1/3)*b + a)*a*b^3*c + 120*a^2*b^3*c - 336*((d*x + c)^(1/3)*b + a)^5 + 1680*((d*
x + c)^(1/3)*b + a)^4*a - 3360*((d*x + c)^(1/3)*b + a)^3*a^2 + 3360*((d*x + c)^(1/3)*b + a)^2*a^3 - 1680*((d*x
+ c)^(1/3)*b + a)*a^4 + 336*a^5 - 240*((d*x + c)^(1/3)*b + a)*b^3*c + 240*a*b^3*c + 1680*((d*x + c)^(1/3)*b +
a)^4 - 6720*((d*x + c)^(1/3)*b + a)^3*a + 10080*((d*x + c)^(1/3)*b + a)^2*a^2 - 6720*((d*x + c)^(1/3)*b + a)*
a^3 + 1680*a^4 + 240*b^3*c - 6720*((d*x + c)^(1/3)*b + a)^3 + 20160*((d*x + c)^(1/3)*b + a)^2*a - 20160*((d*x
+ c)^(1/3)*b + a)*a^2 + 6720*a^3 + 20160*((d*x + c)^(1/3)*b + a)^2 - 40320*((d*x + c)^(1/3)*b + a)*a + 20160*a
^2 - 40320*(d*x + c)^(1/3)*b + 40320)*e^((d*x + c)^(1/3)*b + a)/(b^8*d^2) + (((d*x + c)^(1/3)*b + a)^2*b^6*c^2
- 2*((d*x + c)^(1/3)*b + a)*a*b^6*c^2 + a^2*b^6*c^2 - 2*((d*x + c)^(1/3)*b + a)^5*b^3*c + 10*((d*x + c)^(1/3)
*b + a)^4*a*b^3*c - 20*((d*x + c)^(1/3)*b + a)^3*a^2*b^3*c + 20*((d*x + c)^(1/3)*b + a)^2*a^3*b^3*c - 10*((d*x
+ c)^(1/3)*b + a)*a^4*b^3*c + 2*a^5*b^3*c + 2*((d*x + c)^(1/3)*b + a)*b^6*c^2 - 2*a*b^6*c^2 + ((d*x + c)^(1/3
)*b + a)^8 - 8*((d*x + c)^(1/3)*b + a)^7*a + 28*((d*x + c)^(1/3)*b + a)^6*a^2 - 56*((d*x + c)^(1/3)*b + a)^5*a
^3 + 70*((d*x + c)^(1/3)*b + a)^4*a^4 - 56*((d*x + c)^(1/3)*b + a)^3*a^5 + 28*((d*x + c)^(1/3)*b + a)^2*a^6 -
8*((d*x + c)^(1/3)*b + a)*a^7 + a^8 - 10*((d*x + c)^(1/3)*b + a)^4*b^3*c + 40*((d*x + c)^(1/3)*b + a)^3*a*b^3*
c - 60*((d*x + c)^(1/3)*b + a)^2*a^2*b^3*c + 40*((d*x + c)^(1/3)*b + a)*a^3*b^3*c - 10*a^4*b^3*c + 2*b^6*c^2 +
8*((d*x + c)^(1/3)*b + a)^7 - 56*((d*x + c)^(1/3)*b + a)^6*a + 168*((d*x + c)^(1/3)*b + a)^5*a^2 - 280*((d*x
+ c)^(1/3)*b + a)^4*a^3 + 280*((d*x + c)^(1/3)*b + a)^3*a^4 - 168*((d*x + c)^(1/3)*b + a)^2*a^5 + 56*((d*x + c
)^(1/3)*b + a)*a^6 - 8*a^7 - 40*((d*x + c)^(1/3)*b + a)^3*b^3*c + 120*((d*x + c)^(1/3)*b + a)^2*a*b^3*c - 120*
((d*x + c)^(1/3)*b + a)*a^2*b^3*c + 40*a^3*b^3*c + 56*((d*x + c)^(1/3)*b + a)^6 - 336*((d*x + c)^(1/3)*b + a)^
5*a + 840*((d*x + c)^(1/3)*b + a)^4*a^2 - 1120*((d*x + c)^(1/3)*b + a)^3*a^3 + 840*((d*x + c)^(1/3)*b + a)^2*a
^4 - 336*((d*x + c)^(1/3)*b + a)*a^5 + 56*a^6 - 120*((d*x + c)^(1/3)*b + a)^2*b^3*c + 240*((d*x + c)^(1/3)*b +
a)*a*b^3*c - 120*a^2*b^3*c + 336*((d*x + c)^(1/3)*b + a)^5 - 1680*((d*x + c)^(1/3)*b + a)^4*a + 3360*((d*x +
c)^(1/3)*b + a)^3*a^2 - 3360*((d*x + c)^(1/3)*b + a)^2*a^3 + 1680*((d*x + c)^(1/3)*b + a)*a^4 - 336*a^5 - 240*
((d*x + c)^(1/3)*b + a)*b^3*c + 240*a*b^3*c + 1680*((d*x + c)^(1/3)*b + a)^4 - 6720*((d*x + c)^(1/3)*b + a)^3*
a + 10080*((d*x + c)^(1/3)*b + a)^2*a^2 - 6720*((d*x + c)^(1/3)*b + a)*a^3 + 1680*a^4 - 240*b^3*c + 6720*((d*x
+ c)^(1/3)*b + a)^3 - 20160*((d*x + c)^(1/3)*b + a)^2*a + 20160*((d*x + c)^(1/3)*b + a)*a^2 - 6720*a^3 + 2016
0*((d*x + c)^(1/3)*b + a)^2 - 40320*((d*x + c)^(1/3)*b + a)*a + 20160*a^2 + 40320*(d*x + c)^(1/3)*b + 40320)*e
^(-(d*x + c)^(1/3)*b - a)/(b^8*d^2))/(b*d)