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

Optimal. Leaf size=559 \[ -\frac{2 b d^2 e x^2 \sqrt{c^2 x^2+1} \left (a+b \sinh ^{-1}(c x)\right )}{3 c}+\frac{4 b d^2 e \sqrt{c^2 x^2+1} \left (a+b \sinh ^{-1}(c x)\right )}{3 c^3}-\frac{2 b d^3 \sqrt{c^2 x^2+1} \left (a+b \sinh ^{-1}(c x)\right )}{c}-\frac{6 b d e^2 x^4 \sqrt{c^2 x^2+1} \left (a+b \sinh ^{-1}(c x)\right )}{25 c}+\frac{8 b d e^2 x^2 \sqrt{c^2 x^2+1} \left (a+b \sinh ^{-1}(c x)\right )}{25 c^3}-\frac{16 b d e^2 \sqrt{c^2 x^2+1} \left (a+b \sinh ^{-1}(c x)\right )}{25 c^5}-\frac{2 b e^3 x^6 \sqrt{c^2 x^2+1} \left (a+b \sinh ^{-1}(c x)\right )}{49 c}+\frac{12 b e^3 x^4 \sqrt{c^2 x^2+1} \left (a+b \sinh ^{-1}(c x)\right )}{245 c^3}-\frac{16 b e^3 x^2 \sqrt{c^2 x^2+1} \left (a+b \sinh ^{-1}(c x)\right )}{245 c^5}+\frac{32 b e^3 \sqrt{c^2 x^2+1} \left (a+b \sinh ^{-1}(c x)\right )}{245 c^7}+d^2 e x^3 \left (a+b \sinh ^{-1}(c x)\right )^2+d^3 x \left (a+b \sinh ^{-1}(c x)\right )^2+\frac{3}{5} d e^2 x^5 \left (a+b \sinh ^{-1}(c x)\right )^2+\frac{1}{7} e^3 x^7 \left (a+b \sinh ^{-1}(c x)\right )^2-\frac{4 b^2 d^2 e x}{3 c^2}-\frac{8 b^2 d e^2 x^3}{75 c^2}+\frac{16 b^2 d e^2 x}{25 c^4}-\frac{12 b^2 e^3 x^5}{1225 c^2}+\frac{16 b^2 e^3 x^3}{735 c^4}-\frac{32 b^2 e^3 x}{245 c^6}+\frac{2}{9} b^2 d^2 e x^3+2 b^2 d^3 x+\frac{6}{125} b^2 d e^2 x^5+\frac{2}{343} b^2 e^3 x^7 \]

[Out]

2*b^2*d^3*x - (4*b^2*d^2*e*x)/(3*c^2) + (16*b^2*d*e^2*x)/(25*c^4) - (32*b^2*e^3*x)/(245*c^6) + (2*b^2*d^2*e*x^
3)/9 - (8*b^2*d*e^2*x^3)/(75*c^2) + (16*b^2*e^3*x^3)/(735*c^4) + (6*b^2*d*e^2*x^5)/125 - (12*b^2*e^3*x^5)/(122
5*c^2) + (2*b^2*e^3*x^7)/343 - (2*b*d^3*Sqrt[1 + c^2*x^2]*(a + b*ArcSinh[c*x]))/c + (4*b*d^2*e*Sqrt[1 + c^2*x^
2]*(a + b*ArcSinh[c*x]))/(3*c^3) - (16*b*d*e^2*Sqrt[1 + c^2*x^2]*(a + b*ArcSinh[c*x]))/(25*c^5) + (32*b*e^3*Sq
rt[1 + c^2*x^2]*(a + b*ArcSinh[c*x]))/(245*c^7) - (2*b*d^2*e*x^2*Sqrt[1 + c^2*x^2]*(a + b*ArcSinh[c*x]))/(3*c)
+ (8*b*d*e^2*x^2*Sqrt[1 + c^2*x^2]*(a + b*ArcSinh[c*x]))/(25*c^3) - (16*b*e^3*x^2*Sqrt[1 + c^2*x^2]*(a + b*Ar
cSinh[c*x]))/(245*c^5) - (6*b*d*e^2*x^4*Sqrt[1 + c^2*x^2]*(a + b*ArcSinh[c*x]))/(25*c) + (12*b*e^3*x^4*Sqrt[1
+ c^2*x^2]*(a + b*ArcSinh[c*x]))/(245*c^3) - (2*b*e^3*x^6*Sqrt[1 + c^2*x^2]*(a + b*ArcSinh[c*x]))/(49*c) + d^3
*x*(a + b*ArcSinh[c*x])^2 + d^2*e*x^3*(a + b*ArcSinh[c*x])^2 + (3*d*e^2*x^5*(a + b*ArcSinh[c*x])^2)/5 + (e^3*x
^7*(a + b*ArcSinh[c*x])^2)/7

________________________________________________________________________________________

Rubi [A]  time = 0.965765, antiderivative size = 559, normalized size of antiderivative = 1., number of steps used = 26, number of rules used = 7, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.35, Rules used = {5706, 5653, 5717, 8, 5661, 5758, 30} \[ -\frac{2 b d^2 e x^2 \sqrt{c^2 x^2+1} \left (a+b \sinh ^{-1}(c x)\right )}{3 c}+\frac{4 b d^2 e \sqrt{c^2 x^2+1} \left (a+b \sinh ^{-1}(c x)\right )}{3 c^3}-\frac{2 b d^3 \sqrt{c^2 x^2+1} \left (a+b \sinh ^{-1}(c x)\right )}{c}-\frac{6 b d e^2 x^4 \sqrt{c^2 x^2+1} \left (a+b \sinh ^{-1}(c x)\right )}{25 c}+\frac{8 b d e^2 x^2 \sqrt{c^2 x^2+1} \left (a+b \sinh ^{-1}(c x)\right )}{25 c^3}-\frac{16 b d e^2 \sqrt{c^2 x^2+1} \left (a+b \sinh ^{-1}(c x)\right )}{25 c^5}-\frac{2 b e^3 x^6 \sqrt{c^2 x^2+1} \left (a+b \sinh ^{-1}(c x)\right )}{49 c}+\frac{12 b e^3 x^4 \sqrt{c^2 x^2+1} \left (a+b \sinh ^{-1}(c x)\right )}{245 c^3}-\frac{16 b e^3 x^2 \sqrt{c^2 x^2+1} \left (a+b \sinh ^{-1}(c x)\right )}{245 c^5}+\frac{32 b e^3 \sqrt{c^2 x^2+1} \left (a+b \sinh ^{-1}(c x)\right )}{245 c^7}+d^2 e x^3 \left (a+b \sinh ^{-1}(c x)\right )^2+d^3 x \left (a+b \sinh ^{-1}(c x)\right )^2+\frac{3}{5} d e^2 x^5 \left (a+b \sinh ^{-1}(c x)\right )^2+\frac{1}{7} e^3 x^7 \left (a+b \sinh ^{-1}(c x)\right )^2-\frac{4 b^2 d^2 e x}{3 c^2}-\frac{8 b^2 d e^2 x^3}{75 c^2}+\frac{16 b^2 d e^2 x}{25 c^4}-\frac{12 b^2 e^3 x^5}{1225 c^2}+\frac{16 b^2 e^3 x^3}{735 c^4}-\frac{32 b^2 e^3 x}{245 c^6}+\frac{2}{9} b^2 d^2 e x^3+2 b^2 d^3 x+\frac{6}{125} b^2 d e^2 x^5+\frac{2}{343} b^2 e^3 x^7 \]

Antiderivative was successfully verified.

[In]

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

[Out]

2*b^2*d^3*x - (4*b^2*d^2*e*x)/(3*c^2) + (16*b^2*d*e^2*x)/(25*c^4) - (32*b^2*e^3*x)/(245*c^6) + (2*b^2*d^2*e*x^
3)/9 - (8*b^2*d*e^2*x^3)/(75*c^2) + (16*b^2*e^3*x^3)/(735*c^4) + (6*b^2*d*e^2*x^5)/125 - (12*b^2*e^3*x^5)/(122
5*c^2) + (2*b^2*e^3*x^7)/343 - (2*b*d^3*Sqrt[1 + c^2*x^2]*(a + b*ArcSinh[c*x]))/c + (4*b*d^2*e*Sqrt[1 + c^2*x^
2]*(a + b*ArcSinh[c*x]))/(3*c^3) - (16*b*d*e^2*Sqrt[1 + c^2*x^2]*(a + b*ArcSinh[c*x]))/(25*c^5) + (32*b*e^3*Sq
rt[1 + c^2*x^2]*(a + b*ArcSinh[c*x]))/(245*c^7) - (2*b*d^2*e*x^2*Sqrt[1 + c^2*x^2]*(a + b*ArcSinh[c*x]))/(3*c)
+ (8*b*d*e^2*x^2*Sqrt[1 + c^2*x^2]*(a + b*ArcSinh[c*x]))/(25*c^3) - (16*b*e^3*x^2*Sqrt[1 + c^2*x^2]*(a + b*Ar
cSinh[c*x]))/(245*c^5) - (6*b*d*e^2*x^4*Sqrt[1 + c^2*x^2]*(a + b*ArcSinh[c*x]))/(25*c) + (12*b*e^3*x^4*Sqrt[1
+ c^2*x^2]*(a + b*ArcSinh[c*x]))/(245*c^3) - (2*b*e^3*x^6*Sqrt[1 + c^2*x^2]*(a + b*ArcSinh[c*x]))/(49*c) + d^3
*x*(a + b*ArcSinh[c*x])^2 + d^2*e*x^3*(a + b*ArcSinh[c*x])^2 + (3*d*e^2*x^5*(a + b*ArcSinh[c*x])^2)/5 + (e^3*x
^7*(a + b*ArcSinh[c*x])^2)/7

Rule 5706

Int[((a_.) + ArcSinh[(c_.)*(x_)]*(b_.))^(n_.)*((d_) + (e_.)*(x_)^2)^(p_.), x_Symbol] :> Int[ExpandIntegrand[(a
+ b*ArcSinh[c*x])^n, (d + e*x^2)^p, x], x] /; FreeQ[{a, b, c, d, e, n}, x] && NeQ[e, c^2*d] && IntegerQ[p] &&
(p > 0 || IGtQ[n, 0])

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 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 8

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

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 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 30

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

Rubi steps

\begin{align*} \int \left (d+e x^2\right )^3 \left (a+b \sinh ^{-1}(c x)\right )^2 \, dx &=\int \left (d^3 \left (a+b \sinh ^{-1}(c x)\right )^2+3 d^2 e x^2 \left (a+b \sinh ^{-1}(c x)\right )^2+3 d e^2 x^4 \left (a+b \sinh ^{-1}(c x)\right )^2+e^3 x^6 \left (a+b \sinh ^{-1}(c x)\right )^2\right ) \, dx\\ &=d^3 \int \left (a+b \sinh ^{-1}(c x)\right )^2 \, dx+\left (3 d^2 e\right ) \int x^2 \left (a+b \sinh ^{-1}(c x)\right )^2 \, dx+\left (3 d e^2\right ) \int x^4 \left (a+b \sinh ^{-1}(c x)\right )^2 \, dx+e^3 \int x^6 \left (a+b \sinh ^{-1}(c x)\right )^2 \, dx\\ &=d^3 x \left (a+b \sinh ^{-1}(c x)\right )^2+d^2 e x^3 \left (a+b \sinh ^{-1}(c x)\right )^2+\frac{3}{5} d e^2 x^5 \left (a+b \sinh ^{-1}(c x)\right )^2+\frac{1}{7} e^3 x^7 \left (a+b \sinh ^{-1}(c x)\right )^2-\left (2 b c d^3\right ) \int \frac{x \left (a+b \sinh ^{-1}(c x)\right )}{\sqrt{1+c^2 x^2}} \, dx-\left (2 b c d^2 e\right ) \int \frac{x^3 \left (a+b \sinh ^{-1}(c x)\right )}{\sqrt{1+c^2 x^2}} \, dx-\frac{1}{5} \left (6 b c d e^2\right ) \int \frac{x^5 \left (a+b \sinh ^{-1}(c x)\right )}{\sqrt{1+c^2 x^2}} \, dx-\frac{1}{7} \left (2 b c e^3\right ) \int \frac{x^7 \left (a+b \sinh ^{-1}(c x)\right )}{\sqrt{1+c^2 x^2}} \, dx\\ &=-\frac{2 b d^3 \sqrt{1+c^2 x^2} \left (a+b \sinh ^{-1}(c x)\right )}{c}-\frac{2 b d^2 e x^2 \sqrt{1+c^2 x^2} \left (a+b \sinh ^{-1}(c x)\right )}{3 c}-\frac{6 b d e^2 x^4 \sqrt{1+c^2 x^2} \left (a+b \sinh ^{-1}(c x)\right )}{25 c}-\frac{2 b e^3 x^6 \sqrt{1+c^2 x^2} \left (a+b \sinh ^{-1}(c x)\right )}{49 c}+d^3 x \left (a+b \sinh ^{-1}(c x)\right )^2+d^2 e x^3 \left (a+b \sinh ^{-1}(c x)\right )^2+\frac{3}{5} d e^2 x^5 \left (a+b \sinh ^{-1}(c x)\right )^2+\frac{1}{7} e^3 x^7 \left (a+b \sinh ^{-1}(c x)\right )^2+\left (2 b^2 d^3\right ) \int 1 \, dx+\frac{1}{3} \left (2 b^2 d^2 e\right ) \int x^2 \, dx+\frac{\left (4 b d^2 e\right ) \int \frac{x \left (a+b \sinh ^{-1}(c x)\right )}{\sqrt{1+c^2 x^2}} \, dx}{3 c}+\frac{1}{25} \left (6 b^2 d e^2\right ) \int x^4 \, dx+\frac{\left (24 b d e^2\right ) \int \frac{x^3 \left (a+b \sinh ^{-1}(c x)\right )}{\sqrt{1+c^2 x^2}} \, dx}{25 c}+\frac{1}{49} \left (2 b^2 e^3\right ) \int x^6 \, dx+\frac{\left (12 b e^3\right ) \int \frac{x^5 \left (a+b \sinh ^{-1}(c x)\right )}{\sqrt{1+c^2 x^2}} \, dx}{49 c}\\ &=2 b^2 d^3 x+\frac{2}{9} b^2 d^2 e x^3+\frac{6}{125} b^2 d e^2 x^5+\frac{2}{343} b^2 e^3 x^7-\frac{2 b d^3 \sqrt{1+c^2 x^2} \left (a+b \sinh ^{-1}(c x)\right )}{c}+\frac{4 b d^2 e \sqrt{1+c^2 x^2} \left (a+b \sinh ^{-1}(c x)\right )}{3 c^3}-\frac{2 b d^2 e x^2 \sqrt{1+c^2 x^2} \left (a+b \sinh ^{-1}(c x)\right )}{3 c}+\frac{8 b d e^2 x^2 \sqrt{1+c^2 x^2} \left (a+b \sinh ^{-1}(c x)\right )}{25 c^3}-\frac{6 b d e^2 x^4 \sqrt{1+c^2 x^2} \left (a+b \sinh ^{-1}(c x)\right )}{25 c}+\frac{12 b e^3 x^4 \sqrt{1+c^2 x^2} \left (a+b \sinh ^{-1}(c x)\right )}{245 c^3}-\frac{2 b e^3 x^6 \sqrt{1+c^2 x^2} \left (a+b \sinh ^{-1}(c x)\right )}{49 c}+d^3 x \left (a+b \sinh ^{-1}(c x)\right )^2+d^2 e x^3 \left (a+b \sinh ^{-1}(c x)\right )^2+\frac{3}{5} d e^2 x^5 \left (a+b \sinh ^{-1}(c x)\right )^2+\frac{1}{7} e^3 x^7 \left (a+b \sinh ^{-1}(c x)\right )^2-\frac{\left (4 b^2 d^2 e\right ) \int 1 \, dx}{3 c^2}-\frac{\left (16 b d e^2\right ) \int \frac{x \left (a+b \sinh ^{-1}(c x)\right )}{\sqrt{1+c^2 x^2}} \, dx}{25 c^3}-\frac{\left (8 b^2 d e^2\right ) \int x^2 \, dx}{25 c^2}-\frac{\left (48 b e^3\right ) \int \frac{x^3 \left (a+b \sinh ^{-1}(c x)\right )}{\sqrt{1+c^2 x^2}} \, dx}{245 c^3}-\frac{\left (12 b^2 e^3\right ) \int x^4 \, dx}{245 c^2}\\ &=2 b^2 d^3 x-\frac{4 b^2 d^2 e x}{3 c^2}+\frac{2}{9} b^2 d^2 e x^3-\frac{8 b^2 d e^2 x^3}{75 c^2}+\frac{6}{125} b^2 d e^2 x^5-\frac{12 b^2 e^3 x^5}{1225 c^2}+\frac{2}{343} b^2 e^3 x^7-\frac{2 b d^3 \sqrt{1+c^2 x^2} \left (a+b \sinh ^{-1}(c x)\right )}{c}+\frac{4 b d^2 e \sqrt{1+c^2 x^2} \left (a+b \sinh ^{-1}(c x)\right )}{3 c^3}-\frac{16 b d e^2 \sqrt{1+c^2 x^2} \left (a+b \sinh ^{-1}(c x)\right )}{25 c^5}-\frac{2 b d^2 e x^2 \sqrt{1+c^2 x^2} \left (a+b \sinh ^{-1}(c x)\right )}{3 c}+\frac{8 b d e^2 x^2 \sqrt{1+c^2 x^2} \left (a+b \sinh ^{-1}(c x)\right )}{25 c^3}-\frac{16 b e^3 x^2 \sqrt{1+c^2 x^2} \left (a+b \sinh ^{-1}(c x)\right )}{245 c^5}-\frac{6 b d e^2 x^4 \sqrt{1+c^2 x^2} \left (a+b \sinh ^{-1}(c x)\right )}{25 c}+\frac{12 b e^3 x^4 \sqrt{1+c^2 x^2} \left (a+b \sinh ^{-1}(c x)\right )}{245 c^3}-\frac{2 b e^3 x^6 \sqrt{1+c^2 x^2} \left (a+b \sinh ^{-1}(c x)\right )}{49 c}+d^3 x \left (a+b \sinh ^{-1}(c x)\right )^2+d^2 e x^3 \left (a+b \sinh ^{-1}(c x)\right )^2+\frac{3}{5} d e^2 x^5 \left (a+b \sinh ^{-1}(c x)\right )^2+\frac{1}{7} e^3 x^7 \left (a+b \sinh ^{-1}(c x)\right )^2+\frac{\left (16 b^2 d e^2\right ) \int 1 \, dx}{25 c^4}+\frac{\left (32 b e^3\right ) \int \frac{x \left (a+b \sinh ^{-1}(c x)\right )}{\sqrt{1+c^2 x^2}} \, dx}{245 c^5}+\frac{\left (16 b^2 e^3\right ) \int x^2 \, dx}{245 c^4}\\ &=2 b^2 d^3 x-\frac{4 b^2 d^2 e x}{3 c^2}+\frac{16 b^2 d e^2 x}{25 c^4}+\frac{2}{9} b^2 d^2 e x^3-\frac{8 b^2 d e^2 x^3}{75 c^2}+\frac{16 b^2 e^3 x^3}{735 c^4}+\frac{6}{125} b^2 d e^2 x^5-\frac{12 b^2 e^3 x^5}{1225 c^2}+\frac{2}{343} b^2 e^3 x^7-\frac{2 b d^3 \sqrt{1+c^2 x^2} \left (a+b \sinh ^{-1}(c x)\right )}{c}+\frac{4 b d^2 e \sqrt{1+c^2 x^2} \left (a+b \sinh ^{-1}(c x)\right )}{3 c^3}-\frac{16 b d e^2 \sqrt{1+c^2 x^2} \left (a+b \sinh ^{-1}(c x)\right )}{25 c^5}+\frac{32 b e^3 \sqrt{1+c^2 x^2} \left (a+b \sinh ^{-1}(c x)\right )}{245 c^7}-\frac{2 b d^2 e x^2 \sqrt{1+c^2 x^2} \left (a+b \sinh ^{-1}(c x)\right )}{3 c}+\frac{8 b d e^2 x^2 \sqrt{1+c^2 x^2} \left (a+b \sinh ^{-1}(c x)\right )}{25 c^3}-\frac{16 b e^3 x^2 \sqrt{1+c^2 x^2} \left (a+b \sinh ^{-1}(c x)\right )}{245 c^5}-\frac{6 b d e^2 x^4 \sqrt{1+c^2 x^2} \left (a+b \sinh ^{-1}(c x)\right )}{25 c}+\frac{12 b e^3 x^4 \sqrt{1+c^2 x^2} \left (a+b \sinh ^{-1}(c x)\right )}{245 c^3}-\frac{2 b e^3 x^6 \sqrt{1+c^2 x^2} \left (a+b \sinh ^{-1}(c x)\right )}{49 c}+d^3 x \left (a+b \sinh ^{-1}(c x)\right )^2+d^2 e x^3 \left (a+b \sinh ^{-1}(c x)\right )^2+\frac{3}{5} d e^2 x^5 \left (a+b \sinh ^{-1}(c x)\right )^2+\frac{1}{7} e^3 x^7 \left (a+b \sinh ^{-1}(c x)\right )^2-\frac{\left (32 b^2 e^3\right ) \int 1 \, dx}{245 c^6}\\ &=2 b^2 d^3 x-\frac{4 b^2 d^2 e x}{3 c^2}+\frac{16 b^2 d e^2 x}{25 c^4}-\frac{32 b^2 e^3 x}{245 c^6}+\frac{2}{9} b^2 d^2 e x^3-\frac{8 b^2 d e^2 x^3}{75 c^2}+\frac{16 b^2 e^3 x^3}{735 c^4}+\frac{6}{125} b^2 d e^2 x^5-\frac{12 b^2 e^3 x^5}{1225 c^2}+\frac{2}{343} b^2 e^3 x^7-\frac{2 b d^3 \sqrt{1+c^2 x^2} \left (a+b \sinh ^{-1}(c x)\right )}{c}+\frac{4 b d^2 e \sqrt{1+c^2 x^2} \left (a+b \sinh ^{-1}(c x)\right )}{3 c^3}-\frac{16 b d e^2 \sqrt{1+c^2 x^2} \left (a+b \sinh ^{-1}(c x)\right )}{25 c^5}+\frac{32 b e^3 \sqrt{1+c^2 x^2} \left (a+b \sinh ^{-1}(c x)\right )}{245 c^7}-\frac{2 b d^2 e x^2 \sqrt{1+c^2 x^2} \left (a+b \sinh ^{-1}(c x)\right )}{3 c}+\frac{8 b d e^2 x^2 \sqrt{1+c^2 x^2} \left (a+b \sinh ^{-1}(c x)\right )}{25 c^3}-\frac{16 b e^3 x^2 \sqrt{1+c^2 x^2} \left (a+b \sinh ^{-1}(c x)\right )}{245 c^5}-\frac{6 b d e^2 x^4 \sqrt{1+c^2 x^2} \left (a+b \sinh ^{-1}(c x)\right )}{25 c}+\frac{12 b e^3 x^4 \sqrt{1+c^2 x^2} \left (a+b \sinh ^{-1}(c x)\right )}{245 c^3}-\frac{2 b e^3 x^6 \sqrt{1+c^2 x^2} \left (a+b \sinh ^{-1}(c x)\right )}{49 c}+d^3 x \left (a+b \sinh ^{-1}(c x)\right )^2+d^2 e x^3 \left (a+b \sinh ^{-1}(c x)\right )^2+\frac{3}{5} d e^2 x^5 \left (a+b \sinh ^{-1}(c x)\right )^2+\frac{1}{7} e^3 x^7 \left (a+b \sinh ^{-1}(c x)\right )^2\\ \end{align*}

Mathematica [A]  time = 0.595735, size = 443, normalized size = 0.79 \[ \frac{11025 a^2 c^7 x \left (35 d^2 e x^2+35 d^3+21 d e^2 x^4+5 e^3 x^6\right )-210 a b \sqrt{c^2 x^2+1} \left (c^6 \left (1225 d^2 e x^2+3675 d^3+441 d e^2 x^4+75 e^3 x^6\right )-2 c^4 e \left (1225 d^2+294 d e x^2+45 e^2 x^4\right )+24 c^2 e^2 \left (49 d+5 e x^2\right )-240 e^3\right )-210 b \sinh ^{-1}(c x) \left (b \sqrt{c^2 x^2+1} \left (c^6 \left (1225 d^2 e x^2+3675 d^3+441 d e^2 x^4+75 e^3 x^6\right )-2 c^4 e \left (1225 d^2+294 d e x^2+45 e^2 x^4\right )+24 c^2 e^2 \left (49 d+5 e x^2\right )-240 e^3\right )-105 a c^7 x \left (35 d^2 e x^2+35 d^3+21 d e^2 x^4+5 e^3 x^6\right )\right )+2 b^2 c x \left (c^6 \left (42875 d^2 e x^2+385875 d^3+9261 d e^2 x^4+1125 e^3 x^6\right )-210 c^4 e \left (1225 d^2+98 d e x^2+9 e^2 x^4\right )+840 c^2 e^2 \left (147 d+5 e x^2\right )-25200 e^3\right )+11025 b^2 c^7 x \sinh ^{-1}(c x)^2 \left (35 d^2 e x^2+35 d^3+21 d e^2 x^4+5 e^3 x^6\right )}{385875 c^7} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(11025*a^2*c^7*x*(35*d^3 + 35*d^2*e*x^2 + 21*d*e^2*x^4 + 5*e^3*x^6) - 210*a*b*Sqrt[1 + c^2*x^2]*(-240*e^3 + 24
*c^2*e^2*(49*d + 5*e*x^2) - 2*c^4*e*(1225*d^2 + 294*d*e*x^2 + 45*e^2*x^4) + c^6*(3675*d^3 + 1225*d^2*e*x^2 + 4
41*d*e^2*x^4 + 75*e^3*x^6)) + 2*b^2*c*x*(-25200*e^3 + 840*c^2*e^2*(147*d + 5*e*x^2) - 210*c^4*e*(1225*d^2 + 98
*d*e*x^2 + 9*e^2*x^4) + c^6*(385875*d^3 + 42875*d^2*e*x^2 + 9261*d*e^2*x^4 + 1125*e^3*x^6)) - 210*b*(-105*a*c^
7*x*(35*d^3 + 35*d^2*e*x^2 + 21*d*e^2*x^4 + 5*e^3*x^6) + b*Sqrt[1 + c^2*x^2]*(-240*e^3 + 24*c^2*e^2*(49*d + 5*
e*x^2) - 2*c^4*e*(1225*d^2 + 294*d*e*x^2 + 45*e^2*x^4) + c^6*(3675*d^3 + 1225*d^2*e*x^2 + 441*d*e^2*x^4 + 75*e
^3*x^6)))*ArcSinh[c*x] + 11025*b^2*c^7*x*(35*d^3 + 35*d^2*e*x^2 + 21*d*e^2*x^4 + 5*e^3*x^6)*ArcSinh[c*x]^2)/(3
85875*c^7)

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Maple [B]  time = 0.115, size = 1166, normalized size = 2.1 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/c*(a^2/c^6*(1/7*e^3*c^7*x^7+3/5*c^7*d*e^2*x^5+c^7*d^2*e*x^3+x*c^7*d^3)+b^2/c^6*(d^3*c^6*(arcsinh(c*x)^2*c*x-
2*arcsinh(c*x)*(c^2*x^2+1)^(1/2)+2*c*x)+1/9*c^4*d^2*e*(9*arcsinh(c*x)^2*c^3*x^3-6*arcsinh(c*x)*c^2*x^2*(c^2*x^
2+1)^(1/2)+27*arcsinh(c*x)^2*c*x+2*c^3*x^3-42*arcsinh(c*x)*(c^2*x^2+1)^(1/2)+42*c*x)-3*c^4*d^2*e*(arcsinh(c*x)
^2*c*x-2*arcsinh(c*x)*(c^2*x^2+1)^(1/2)+2*c*x)+1/1125*d*e^2*c^2*(675*arcsinh(c*x)^2*c^5*x^5-270*arcsinh(c*x)*(
c^2*x^2+1)^(1/2)*c^4*x^4+2250*arcsinh(c*x)^2*c^3*x^3+54*c^5*x^5-1140*arcsinh(c*x)*c^2*x^2*(c^2*x^2+1)^(1/2)+33
75*arcsinh(c*x)^2*c*x+380*c^3*x^3-4470*arcsinh(c*x)*(c^2*x^2+1)^(1/2)+4470*c*x)-2/9*d*e^2*c^2*(9*arcsinh(c*x)^
2*c^3*x^3-6*arcsinh(c*x)*c^2*x^2*(c^2*x^2+1)^(1/2)+27*arcsinh(c*x)^2*c*x+2*c^3*x^3-42*arcsinh(c*x)*(c^2*x^2+1)
^(1/2)+42*c*x)+3*d*e^2*c^2*(arcsinh(c*x)^2*c*x-2*arcsinh(c*x)*(c^2*x^2+1)^(1/2)+2*c*x)+1/385875*e^3*(55125*arc
sinh(c*x)^2*c^7*x^7-15750*arcsinh(c*x)*(c^2*x^2+1)^(1/2)*c^6*x^6+231525*arcsinh(c*x)^2*c^5*x^5+2250*c^7*x^7-73
710*arcsinh(c*x)*(c^2*x^2+1)^(1/2)*c^4*x^4+385875*arcsinh(c*x)^2*c^3*x^3+14742*c^5*x^5-158970*arcsinh(c*x)*c^2
*x^2*(c^2*x^2+1)^(1/2)+385875*arcsinh(c*x)^2*c*x+52990*c^3*x^3-453810*arcsinh(c*x)*(c^2*x^2+1)^(1/2)+453810*c*
x)-1/1125*e^3*(675*arcsinh(c*x)^2*c^5*x^5-270*arcsinh(c*x)*(c^2*x^2+1)^(1/2)*c^4*x^4+2250*arcsinh(c*x)^2*c^3*x
^3+54*c^5*x^5-1140*arcsinh(c*x)*c^2*x^2*(c^2*x^2+1)^(1/2)+3375*arcsinh(c*x)^2*c*x+380*c^3*x^3-4470*arcsinh(c*x
)*(c^2*x^2+1)^(1/2)+4470*c*x)+1/9*e^3*(9*arcsinh(c*x)^2*c^3*x^3-6*arcsinh(c*x)*c^2*x^2*(c^2*x^2+1)^(1/2)+27*ar
csinh(c*x)^2*c*x+2*c^3*x^3-42*arcsinh(c*x)*(c^2*x^2+1)^(1/2)+42*c*x)-e^3*(arcsinh(c*x)^2*c*x-2*arcsinh(c*x)*(c
^2*x^2+1)^(1/2)+2*c*x))+2*a*b/c^6*(1/7*arcsinh(c*x)*e^3*c^7*x^7+3/5*arcsinh(c*x)*c^7*d*e^2*x^5+arcsinh(c*x)*c^
7*d^2*e*x^3+arcsinh(c*x)*c^7*x*d^3-1/7*e^3*(1/7*c^6*x^6*(c^2*x^2+1)^(1/2)-6/35*c^4*x^4*(c^2*x^2+1)^(1/2)+8/35*
c^2*x^2*(c^2*x^2+1)^(1/2)-16/35*(c^2*x^2+1)^(1/2))-3/5*c^2*d*e^2*(1/5*c^4*x^4*(c^2*x^2+1)^(1/2)-4/15*c^2*x^2*(
c^2*x^2+1)^(1/2)+8/15*(c^2*x^2+1)^(1/2))-c^4*d^2*e*(1/3*c^2*x^2*(c^2*x^2+1)^(1/2)-2/3*(c^2*x^2+1)^(1/2))-d^3*c
^6*(c^2*x^2+1)^(1/2)))

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Maxima [A]  time = 1.25873, size = 923, normalized size = 1.65 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((e*x^2+d)^3*(a+b*arcsinh(c*x))^2,x, algorithm="maxima")

[Out]

1/7*b^2*e^3*x^7*arcsinh(c*x)^2 + 1/7*a^2*e^3*x^7 + 3/5*b^2*d*e^2*x^5*arcsinh(c*x)^2 + 3/5*a^2*d*e^2*x^5 + b^2*
d^2*e*x^3*arcsinh(c*x)^2 + a^2*d^2*e*x^3 + b^2*d^3*x*arcsinh(c*x)^2 + 2/3*(3*x^3*arcsinh(c*x) - c*(sqrt(c^2*x^
2 + 1)*x^2/c^2 - 2*sqrt(c^2*x^2 + 1)/c^4))*a*b*d^2*e - 2/9*(3*c*(sqrt(c^2*x^2 + 1)*x^2/c^2 - 2*sqrt(c^2*x^2 +
1)/c^4)*arcsinh(c*x) - (c^2*x^3 - 6*x)/c^2)*b^2*d^2*e + 2/25*(15*x^5*arcsinh(c*x) - (3*sqrt(c^2*x^2 + 1)*x^4/c
^2 - 4*sqrt(c^2*x^2 + 1)*x^2/c^4 + 8*sqrt(c^2*x^2 + 1)/c^6)*c)*a*b*d*e^2 - 2/375*(15*(3*sqrt(c^2*x^2 + 1)*x^4/
c^2 - 4*sqrt(c^2*x^2 + 1)*x^2/c^4 + 8*sqrt(c^2*x^2 + 1)/c^6)*c*arcsinh(c*x) - (9*c^4*x^5 - 20*c^2*x^3 + 120*x)
/c^4)*b^2*d*e^2 + 2/245*(35*x^7*arcsinh(c*x) - (5*sqrt(c^2*x^2 + 1)*x^6/c^2 - 6*sqrt(c^2*x^2 + 1)*x^4/c^4 + 8*
sqrt(c^2*x^2 + 1)*x^2/c^6 - 16*sqrt(c^2*x^2 + 1)/c^8)*c)*a*b*e^3 - 2/25725*(105*(5*sqrt(c^2*x^2 + 1)*x^6/c^2 -
6*sqrt(c^2*x^2 + 1)*x^4/c^4 + 8*sqrt(c^2*x^2 + 1)*x^2/c^6 - 16*sqrt(c^2*x^2 + 1)/c^8)*c*arcsinh(c*x) - (75*c^
6*x^7 - 126*c^4*x^5 + 280*c^2*x^3 - 1680*x)/c^6)*b^2*e^3 + 2*b^2*d^3*(x - sqrt(c^2*x^2 + 1)*arcsinh(c*x)/c) +
a^2*d^3*x + 2*(c*x*arcsinh(c*x) - sqrt(c^2*x^2 + 1))*a*b*d^3/c

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Fricas [A]  time = 2.58636, size = 1338, normalized size = 2.39 \begin{align*} \frac{1125 \,{\left (49 \, a^{2} + 2 \, b^{2}\right )} c^{7} e^{3} x^{7} + 189 \,{\left (49 \,{\left (25 \, a^{2} + 2 \, b^{2}\right )} c^{7} d e^{2} - 20 \, b^{2} c^{5} e^{3}\right )} x^{5} + 35 \,{\left (1225 \,{\left (9 \, a^{2} + 2 \, b^{2}\right )} c^{7} d^{2} e - 1176 \, b^{2} c^{5} d e^{2} + 240 \, b^{2} c^{3} e^{3}\right )} x^{3} + 11025 \,{\left (5 \, b^{2} c^{7} e^{3} x^{7} + 21 \, b^{2} c^{7} d e^{2} x^{5} + 35 \, b^{2} c^{7} d^{2} e x^{3} + 35 \, b^{2} c^{7} d^{3} x\right )} \log \left (c x + \sqrt{c^{2} x^{2} + 1}\right )^{2} + 105 \,{\left (3675 \,{\left (a^{2} + 2 \, b^{2}\right )} c^{7} d^{3} - 4900 \, b^{2} c^{5} d^{2} e + 2352 \, b^{2} c^{3} d e^{2} - 480 \, b^{2} c e^{3}\right )} x + 210 \,{\left (525 \, a b c^{7} e^{3} x^{7} + 2205 \, a b c^{7} d e^{2} x^{5} + 3675 \, a b c^{7} d^{2} e x^{3} + 3675 \, a b c^{7} d^{3} x -{\left (75 \, b^{2} c^{6} e^{3} x^{6} + 3675 \, b^{2} c^{6} d^{3} - 2450 \, b^{2} c^{4} d^{2} e + 1176 \, b^{2} c^{2} d e^{2} - 240 \, b^{2} e^{3} + 9 \,{\left (49 \, b^{2} c^{6} d e^{2} - 10 \, b^{2} c^{4} e^{3}\right )} x^{4} +{\left (1225 \, b^{2} c^{6} d^{2} e - 588 \, b^{2} c^{4} d e^{2} + 120 \, b^{2} c^{2} e^{3}\right )} x^{2}\right )} \sqrt{c^{2} x^{2} + 1}\right )} \log \left (c x + \sqrt{c^{2} x^{2} + 1}\right ) - 210 \,{\left (75 \, a b c^{6} e^{3} x^{6} + 3675 \, a b c^{6} d^{3} - 2450 \, a b c^{4} d^{2} e + 1176 \, a b c^{2} d e^{2} - 240 \, a b e^{3} + 9 \,{\left (49 \, a b c^{6} d e^{2} - 10 \, a b c^{4} e^{3}\right )} x^{4} +{\left (1225 \, a b c^{6} d^{2} e - 588 \, a b c^{4} d e^{2} + 120 \, a b c^{2} e^{3}\right )} x^{2}\right )} \sqrt{c^{2} x^{2} + 1}}{385875 \, c^{7}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((e*x^2+d)^3*(a+b*arcsinh(c*x))^2,x, algorithm="fricas")

[Out]

1/385875*(1125*(49*a^2 + 2*b^2)*c^7*e^3*x^7 + 189*(49*(25*a^2 + 2*b^2)*c^7*d*e^2 - 20*b^2*c^5*e^3)*x^5 + 35*(1
225*(9*a^2 + 2*b^2)*c^7*d^2*e - 1176*b^2*c^5*d*e^2 + 240*b^2*c^3*e^3)*x^3 + 11025*(5*b^2*c^7*e^3*x^7 + 21*b^2*
c^7*d*e^2*x^5 + 35*b^2*c^7*d^2*e*x^3 + 35*b^2*c^7*d^3*x)*log(c*x + sqrt(c^2*x^2 + 1))^2 + 105*(3675*(a^2 + 2*b
^2)*c^7*d^3 - 4900*b^2*c^5*d^2*e + 2352*b^2*c^3*d*e^2 - 480*b^2*c*e^3)*x + 210*(525*a*b*c^7*e^3*x^7 + 2205*a*b
*c^7*d*e^2*x^5 + 3675*a*b*c^7*d^2*e*x^3 + 3675*a*b*c^7*d^3*x - (75*b^2*c^6*e^3*x^6 + 3675*b^2*c^6*d^3 - 2450*b
^2*c^4*d^2*e + 1176*b^2*c^2*d*e^2 - 240*b^2*e^3 + 9*(49*b^2*c^6*d*e^2 - 10*b^2*c^4*e^3)*x^4 + (1225*b^2*c^6*d^
2*e - 588*b^2*c^4*d*e^2 + 120*b^2*c^2*e^3)*x^2)*sqrt(c^2*x^2 + 1))*log(c*x + sqrt(c^2*x^2 + 1)) - 210*(75*a*b*
c^6*e^3*x^6 + 3675*a*b*c^6*d^3 - 2450*a*b*c^4*d^2*e + 1176*a*b*c^2*d*e^2 - 240*a*b*e^3 + 9*(49*a*b*c^6*d*e^2 -
10*a*b*c^4*e^3)*x^4 + (1225*a*b*c^6*d^2*e - 588*a*b*c^4*d*e^2 + 120*a*b*c^2*e^3)*x^2)*sqrt(c^2*x^2 + 1))/c^7

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Sympy [A]  time = 18.8257, size = 989, normalized size = 1.77 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Piecewise((a**2*d**3*x + a**2*d**2*e*x**3 + 3*a**2*d*e**2*x**5/5 + a**2*e**3*x**7/7 + 2*a*b*d**3*x*asinh(c*x)
+ 2*a*b*d**2*e*x**3*asinh(c*x) + 6*a*b*d*e**2*x**5*asinh(c*x)/5 + 2*a*b*e**3*x**7*asinh(c*x)/7 - 2*a*b*d**3*sq
rt(c**2*x**2 + 1)/c - 2*a*b*d**2*e*x**2*sqrt(c**2*x**2 + 1)/(3*c) - 6*a*b*d*e**2*x**4*sqrt(c**2*x**2 + 1)/(25*
c) - 2*a*b*e**3*x**6*sqrt(c**2*x**2 + 1)/(49*c) + 4*a*b*d**2*e*sqrt(c**2*x**2 + 1)/(3*c**3) + 8*a*b*d*e**2*x**
2*sqrt(c**2*x**2 + 1)/(25*c**3) + 12*a*b*e**3*x**4*sqrt(c**2*x**2 + 1)/(245*c**3) - 16*a*b*d*e**2*sqrt(c**2*x*
*2 + 1)/(25*c**5) - 16*a*b*e**3*x**2*sqrt(c**2*x**2 + 1)/(245*c**5) + 32*a*b*e**3*sqrt(c**2*x**2 + 1)/(245*c**
7) + b**2*d**3*x*asinh(c*x)**2 + 2*b**2*d**3*x + b**2*d**2*e*x**3*asinh(c*x)**2 + 2*b**2*d**2*e*x**3/9 + 3*b**
2*d*e**2*x**5*asinh(c*x)**2/5 + 6*b**2*d*e**2*x**5/125 + b**2*e**3*x**7*asinh(c*x)**2/7 + 2*b**2*e**3*x**7/343
- 2*b**2*d**3*sqrt(c**2*x**2 + 1)*asinh(c*x)/c - 2*b**2*d**2*e*x**2*sqrt(c**2*x**2 + 1)*asinh(c*x)/(3*c) - 6*
b**2*d*e**2*x**4*sqrt(c**2*x**2 + 1)*asinh(c*x)/(25*c) - 2*b**2*e**3*x**6*sqrt(c**2*x**2 + 1)*asinh(c*x)/(49*c
) - 4*b**2*d**2*e*x/(3*c**2) - 8*b**2*d*e**2*x**3/(75*c**2) - 12*b**2*e**3*x**5/(1225*c**2) + 4*b**2*d**2*e*sq
rt(c**2*x**2 + 1)*asinh(c*x)/(3*c**3) + 8*b**2*d*e**2*x**2*sqrt(c**2*x**2 + 1)*asinh(c*x)/(25*c**3) + 12*b**2*
e**3*x**4*sqrt(c**2*x**2 + 1)*asinh(c*x)/(245*c**3) + 16*b**2*d*e**2*x/(25*c**4) + 16*b**2*e**3*x**3/(735*c**4
) - 16*b**2*d*e**2*sqrt(c**2*x**2 + 1)*asinh(c*x)/(25*c**5) - 16*b**2*e**3*x**2*sqrt(c**2*x**2 + 1)*asinh(c*x)
/(245*c**5) - 32*b**2*e**3*x/(245*c**6) + 32*b**2*e**3*sqrt(c**2*x**2 + 1)*asinh(c*x)/(245*c**7), Ne(c, 0)), (
a**2*(d**3*x + d**2*e*x**3 + 3*d*e**2*x**5/5 + e**3*x**7/7), True))

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Giac [A]  time = 3.02436, size = 986, normalized size = 1.76 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((e*x^2+d)^3*(a+b*arcsinh(c*x))^2,x, algorithm="giac")

[Out]

2*(x*log(c*x + sqrt(c^2*x^2 + 1)) - sqrt(c^2*x^2 + 1)/c)*a*b*d^3 + (x*log(c*x + sqrt(c^2*x^2 + 1))^2 + 2*c*(x/
c - sqrt(c^2*x^2 + 1)*log(c*x + sqrt(c^2*x^2 + 1))/c^2))*b^2*d^3 + a^2*d^3*x + 1/25725*(3675*a^2*x^7 + 210*(35
*x^7*log(c*x + sqrt(c^2*x^2 + 1)) - (5*(c^2*x^2 + 1)^(7/2) - 21*(c^2*x^2 + 1)^(5/2) + 35*(c^2*x^2 + 1)^(3/2) -
35*sqrt(c^2*x^2 + 1))/c^7)*a*b + (3675*x^7*log(c*x + sqrt(c^2*x^2 + 1))^2 + 2*c*((75*c^6*x^7 - 126*c^4*x^5 +
280*c^2*x^3 - 1680*x)/c^7 - 105*(5*(c^2*x^2 + 1)^(7/2) - 21*(c^2*x^2 + 1)^(5/2) + 35*(c^2*x^2 + 1)^(3/2) - 35*
sqrt(c^2*x^2 + 1))*log(c*x + sqrt(c^2*x^2 + 1))/c^8))*b^2)*e^3 + 1/375*(225*a^2*d*x^5 + 30*(15*x^5*log(c*x + s
qrt(c^2*x^2 + 1)) - (3*(c^2*x^2 + 1)^(5/2) - 10*(c^2*x^2 + 1)^(3/2) + 15*sqrt(c^2*x^2 + 1))/c^5)*a*b*d + (225*
x^5*log(c*x + sqrt(c^2*x^2 + 1))^2 + 2*c*((9*c^4*x^5 - 20*c^2*x^3 + 120*x)/c^5 - 15*(3*(c^2*x^2 + 1)^(5/2) - 1
0*(c^2*x^2 + 1)^(3/2) + 15*sqrt(c^2*x^2 + 1))*log(c*x + sqrt(c^2*x^2 + 1))/c^6))*b^2*d)*e^2 + 1/9*(9*a^2*d^2*x
^3 + 6*(3*x^3*log(c*x + sqrt(c^2*x^2 + 1)) - ((c^2*x^2 + 1)^(3/2) - 3*sqrt(c^2*x^2 + 1))/c^3)*a*b*d^2 + (9*x^3
*log(c*x + sqrt(c^2*x^2 + 1))^2 + 2*c*((c^2*x^3 - 6*x)/c^3 - 3*((c^2*x^2 + 1)^(3/2) - 3*sqrt(c^2*x^2 + 1))*log
(c*x + sqrt(c^2*x^2 + 1))/c^4))*b^2*d^2)*e