Integrand size = 16, antiderivative size = 176 \[ \int (d+e x)^3 (a+b \text {arcsinh}(c x)) \, dx=-\frac {7 b d (d+e x)^2 \sqrt {1+c^2 x^2}}{48 c}-\frac {b (d+e x)^3 \sqrt {1+c^2 x^2}}{16 c}-\frac {b \left (4 d \left (19 c^2 d^2-16 e^2\right )+e \left (26 c^2 d^2-9 e^2\right ) x\right ) \sqrt {1+c^2 x^2}}{96 c^3}-\frac {b \left (8 c^4 d^4-24 c^2 d^2 e^2+3 e^4\right ) \text {arcsinh}(c x)}{32 c^4 e}+\frac {(d+e x)^4 (a+b \text {arcsinh}(c x))}{4 e} \]
-1/32*b*(8*c^4*d^4-24*c^2*d^2*e^2+3*e^4)*arcsinh(c*x)/c^4/e+1/4*(e*x+d)^4* (a+b*arcsinh(c*x))/e-7/48*b*d*(e*x+d)^2*(c^2*x^2+1)^(1/2)/c-1/16*b*(e*x+d) ^3*(c^2*x^2+1)^(1/2)/c-1/96*b*(4*d*(19*c^2*d^2-16*e^2)+e*(26*c^2*d^2-9*e^2 )*x)*(c^2*x^2+1)^(1/2)/c^3
Time = 0.11 (sec) , antiderivative size = 166, normalized size of antiderivative = 0.94 \[ \int (d+e x)^3 (a+b \text {arcsinh}(c x)) \, dx=\frac {24 a c^4 x \left (4 d^3+6 d^2 e x+4 d e^2 x^2+e^3 x^3\right )-b c \sqrt {1+c^2 x^2} \left (-e^2 (64 d+9 e x)+c^2 \left (96 d^3+72 d^2 e x+32 d e^2 x^2+6 e^3 x^3\right )\right )+3 b \left (24 c^2 d^2 e-3 e^3+8 c^4 x \left (4 d^3+6 d^2 e x+4 d e^2 x^2+e^3 x^3\right )\right ) \text {arcsinh}(c x)}{96 c^4} \]
(24*a*c^4*x*(4*d^3 + 6*d^2*e*x + 4*d*e^2*x^2 + e^3*x^3) - b*c*Sqrt[1 + c^2 *x^2]*(-(e^2*(64*d + 9*e*x)) + c^2*(96*d^3 + 72*d^2*e*x + 32*d*e^2*x^2 + 6 *e^3*x^3)) + 3*b*(24*c^2*d^2*e - 3*e^3 + 8*c^4*x*(4*d^3 + 6*d^2*e*x + 4*d* e^2*x^2 + e^3*x^3))*ArcSinh[c*x])/(96*c^4)
Time = 0.39 (sec) , antiderivative size = 202, normalized size of antiderivative = 1.15, number of steps used = 6, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.375, Rules used = {6243, 497, 687, 27, 676, 222}
Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.
| \(\displaystyle \int (d+e x)^3 (a+b \text {arcsinh}(c x)) \, dx\) |
| \(\Big \downarrow \) 6243 |
| \(\displaystyle \frac {(d+e x)^4 (a+b \text {arcsinh}(c x))}{4 e}-\frac {b c \int \frac {(d+e x)^4}{\sqrt {c^2 x^2+1}}dx}{4 e}\) |
| \(\Big \downarrow \) 497 |
| \(\displaystyle \frac {(d+e x)^4 (a+b \text {arcsinh}(c x))}{4 e}-\frac {b c \left (\frac {\int \frac {(d+e x)^2 \left (4 d^2 c^2+7 d e x c^2-3 e^2\right )}{\sqrt {c^2 x^2+1}}dx}{4 c^2}+\frac {e \sqrt {c^2 x^2+1} (d+e x)^3}{4 c^2}\right )}{4 e}\) |
| \(\Big \downarrow \) 687 |
| \(\displaystyle \frac {(d+e x)^4 (a+b \text {arcsinh}(c x))}{4 e}-\frac {b c \left (\frac {\frac {\int \frac {c^2 (d+e x) \left (d \left (12 c^2 d^2-23 e^2\right )+e \left (26 c^2 d^2-9 e^2\right ) x\right )}{\sqrt {c^2 x^2+1}}dx}{3 c^2}+\frac {7}{3} d e \sqrt {c^2 x^2+1} (d+e x)^2}{4 c^2}+\frac {e \sqrt {c^2 x^2+1} (d+e x)^3}{4 c^2}\right )}{4 e}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {(d+e x)^4 (a+b \text {arcsinh}(c x))}{4 e}-\frac {b c \left (\frac {\frac {1}{3} \int \frac {(d+e x) \left (d \left (12 c^2 d^2-23 e^2\right )+e \left (26 c^2 d^2-9 e^2\right ) x\right )}{\sqrt {c^2 x^2+1}}dx+\frac {7}{3} d e \sqrt {c^2 x^2+1} (d+e x)^2}{4 c^2}+\frac {e \sqrt {c^2 x^2+1} (d+e x)^3}{4 c^2}\right )}{4 e}\) |
| \(\Big \downarrow \) 676 |
| \(\displaystyle \frac {(d+e x)^4 (a+b \text {arcsinh}(c x))}{4 e}-\frac {b c \left (\frac {\frac {1}{3} \left (\frac {3 \left (8 c^4 d^4-24 c^2 d^2 e^2+3 e^4\right ) \int \frac {1}{\sqrt {c^2 x^2+1}}dx}{2 c^2}+\frac {1}{2} e^2 x \sqrt {c^2 x^2+1} \left (26 d^2-\frac {9 e^2}{c^2}\right )+2 d e \sqrt {c^2 x^2+1} \left (19 d^2-\frac {16 e^2}{c^2}\right )\right )+\frac {7}{3} d e \sqrt {c^2 x^2+1} (d+e x)^2}{4 c^2}+\frac {e \sqrt {c^2 x^2+1} (d+e x)^3}{4 c^2}\right )}{4 e}\) |
| \(\Big \downarrow \) 222 |
| \(\displaystyle \frac {(d+e x)^4 (a+b \text {arcsinh}(c x))}{4 e}-\frac {b c \left (\frac {\frac {1}{3} \left (\frac {3 \text {arcsinh}(c x) \left (8 c^4 d^4-24 c^2 d^2 e^2+3 e^4\right )}{2 c^3}+\frac {1}{2} e^2 x \sqrt {c^2 x^2+1} \left (26 d^2-\frac {9 e^2}{c^2}\right )+2 d e \sqrt {c^2 x^2+1} \left (19 d^2-\frac {16 e^2}{c^2}\right )\right )+\frac {7}{3} d e \sqrt {c^2 x^2+1} (d+e x)^2}{4 c^2}+\frac {e \sqrt {c^2 x^2+1} (d+e x)^3}{4 c^2}\right )}{4 e}\) |
((d + e*x)^4*(a + b*ArcSinh[c*x]))/(4*e) - (b*c*((e*(d + e*x)^3*Sqrt[1 + c ^2*x^2])/(4*c^2) + ((7*d*e*(d + e*x)^2*Sqrt[1 + c^2*x^2])/3 + (2*d*e*(19*d ^2 - (16*e^2)/c^2)*Sqrt[1 + c^2*x^2] + (e^2*(26*d^2 - (9*e^2)/c^2)*x*Sqrt[ 1 + c^2*x^2])/2 + (3*(8*c^4*d^4 - 24*c^2*d^2*e^2 + 3*e^4)*ArcSinh[c*x])/(2 *c^3))/3)/(4*c^2)))/(4*e)
3.1.4.3.1 Defintions of rubi rules used
Int[(a_)*(Fx_), x_Symbol] :> Simp[a Int[Fx, x], x] /; FreeQ[a, x] && !Ma tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
Int[1/Sqrt[(a_) + (b_.)*(x_)^2], x_Symbol] :> Simp[ArcSinh[Rt[b, 2]*(x/Sqrt [a])]/Rt[b, 2], x] /; FreeQ[{a, b}, x] && GtQ[a, 0] && PosQ[b]
Int[((c_) + (d_.)*(x_))^(n_)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> Simp[ d*(c + d*x)^(n - 1)*((a + b*x^2)^(p + 1)/(b*(n + 2*p + 1))), x] + Simp[1/(b *(n + 2*p + 1)) Int[(c + d*x)^(n - 2)*(a + b*x^2)^p*Simp[b*c^2*(n + 2*p + 1) - a*d^2*(n - 1) + 2*b*c*d*(n + p)*x, x], x], x] /; FreeQ[{a, b, c, d, n , p}, x] && If[RationalQ[n], GtQ[n, 1], SumSimplerQ[n, -2]] && NeQ[n + 2*p + 1, 0] && IntQuadraticQ[a, 0, b, c, d, n, p, x]
Int[((d_.) + (e_.)*(x_))*((f_.) + (g_.)*(x_))*((a_) + (c_.)*(x_)^2)^(p_), x _Symbol] :> Simp[(e*f + d*g)*((a + c*x^2)^(p + 1)/(2*c*(p + 1))), x] + (Sim p[e*g*x*((a + c*x^2)^(p + 1)/(c*(2*p + 3))), x] - Simp[(a*e*g - c*d*f*(2*p + 3))/(c*(2*p + 3)) Int[(a + c*x^2)^p, x], x]) /; FreeQ[{a, c, d, e, f, g , p}, x] && !LeQ[p, -1]
Int[((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))*((a_) + (c_.)*(x_)^2)^(p _.), x_Symbol] :> Simp[g*(d + e*x)^m*((a + c*x^2)^(p + 1)/(c*(m + 2*p + 2)) ), x] + Simp[1/(c*(m + 2*p + 2)) Int[(d + e*x)^(m - 1)*(a + c*x^2)^p*Simp [c*d*f*(m + 2*p + 2) - a*e*g*m + c*(e*f*(m + 2*p + 2) + d*g*m)*x, x], x], x ] /; FreeQ[{a, c, d, e, f, g, p}, x] && GtQ[m, 0] && NeQ[m + 2*p + 2, 0] && (IntegerQ[m] || IntegerQ[p] || IntegersQ[2*m, 2*p]) && !(IGtQ[m, 0] && Eq Q[f, 0])
Int[((a_.) + ArcSinh[(c_.)*(x_)]*(b_.))^(n_.)*((d_.) + (e_.)*(x_))^(m_.), x _Symbol] :> Simp[(d + e*x)^(m + 1)*((a + b*ArcSinh[c*x])^n/(e*(m + 1))), x] - Simp[b*c*(n/(e*(m + 1))) Int[(d + e*x)^(m + 1)*((a + b*ArcSinh[c*x])^( n - 1)/Sqrt[1 + c^2*x^2]), x], x] /; FreeQ[{a, b, c, d, e, m}, x] && IGtQ[n , 0] && NeQ[m, -1]
Time = 0.31 (sec) , antiderivative size = 242, normalized size of antiderivative = 1.38
| method | result | size |
| parts | \(\frac {a \left (e x +d \right )^{4}}{4 e}+\frac {b \left (\frac {c \,e^{3} \operatorname {arcsinh}\left (c x \right ) x^{4}}{4}+c \,e^{2} \operatorname {arcsinh}\left (c x \right ) x^{3} d +\frac {3 c \,\operatorname {arcsinh}\left (c x \right ) d^{2} e \,x^{2}}{2}+\operatorname {arcsinh}\left (c x \right ) c x \,d^{3}+\frac {c \,\operatorname {arcsinh}\left (c x \right ) d^{4}}{4 e}-\frac {c^{4} d^{4} \operatorname {arcsinh}\left (c x \right )+e^{4} \left (\frac {c^{3} x^{3} \sqrt {c^{2} x^{2}+1}}{4}-\frac {3 c x \sqrt {c^{2} x^{2}+1}}{8}+\frac {3 \,\operatorname {arcsinh}\left (c x \right )}{8}\right )+4 d^{3} c^{3} e \sqrt {c^{2} x^{2}+1}+6 d^{2} c^{2} e^{2} \left (\frac {c x \sqrt {c^{2} x^{2}+1}}{2}-\frac {\operatorname {arcsinh}\left (c x \right )}{2}\right )+4 d c \,e^{3} \left (\frac {c^{2} x^{2} \sqrt {c^{2} x^{2}+1}}{3}-\frac {2 \sqrt {c^{2} x^{2}+1}}{3}\right )}{4 c^{3} e}\right )}{c}\) | \(242\) |
| derivativedivides | \(\frac {\frac {a \left (e c x +c d \right )^{4}}{4 c^{3} e}+\frac {b \left (\frac {\operatorname {arcsinh}\left (c x \right ) c^{4} d^{4}}{4 e}+\operatorname {arcsinh}\left (c x \right ) c^{4} d^{3} x +\frac {3 e \,\operatorname {arcsinh}\left (c x \right ) c^{4} d^{2} x^{2}}{2}+e^{2} \operatorname {arcsinh}\left (c x \right ) c^{4} d \,x^{3}+\frac {e^{3} \operatorname {arcsinh}\left (c x \right ) c^{4} x^{4}}{4}-\frac {c^{4} d^{4} \operatorname {arcsinh}\left (c x \right )+e^{4} \left (\frac {c^{3} x^{3} \sqrt {c^{2} x^{2}+1}}{4}-\frac {3 c x \sqrt {c^{2} x^{2}+1}}{8}+\frac {3 \,\operatorname {arcsinh}\left (c x \right )}{8}\right )+4 d^{3} c^{3} e \sqrt {c^{2} x^{2}+1}+6 d^{2} c^{2} e^{2} \left (\frac {c x \sqrt {c^{2} x^{2}+1}}{2}-\frac {\operatorname {arcsinh}\left (c x \right )}{2}\right )+4 d c \,e^{3} \left (\frac {c^{2} x^{2} \sqrt {c^{2} x^{2}+1}}{3}-\frac {2 \sqrt {c^{2} x^{2}+1}}{3}\right )}{4 e}\right )}{c^{3}}}{c}\) | \(259\) |
| default | \(\frac {\frac {a \left (e c x +c d \right )^{4}}{4 c^{3} e}+\frac {b \left (\frac {\operatorname {arcsinh}\left (c x \right ) c^{4} d^{4}}{4 e}+\operatorname {arcsinh}\left (c x \right ) c^{4} d^{3} x +\frac {3 e \,\operatorname {arcsinh}\left (c x \right ) c^{4} d^{2} x^{2}}{2}+e^{2} \operatorname {arcsinh}\left (c x \right ) c^{4} d \,x^{3}+\frac {e^{3} \operatorname {arcsinh}\left (c x \right ) c^{4} x^{4}}{4}-\frac {c^{4} d^{4} \operatorname {arcsinh}\left (c x \right )+e^{4} \left (\frac {c^{3} x^{3} \sqrt {c^{2} x^{2}+1}}{4}-\frac {3 c x \sqrt {c^{2} x^{2}+1}}{8}+\frac {3 \,\operatorname {arcsinh}\left (c x \right )}{8}\right )+4 d^{3} c^{3} e \sqrt {c^{2} x^{2}+1}+6 d^{2} c^{2} e^{2} \left (\frac {c x \sqrt {c^{2} x^{2}+1}}{2}-\frac {\operatorname {arcsinh}\left (c x \right )}{2}\right )+4 d c \,e^{3} \left (\frac {c^{2} x^{2} \sqrt {c^{2} x^{2}+1}}{3}-\frac {2 \sqrt {c^{2} x^{2}+1}}{3}\right )}{4 e}\right )}{c^{3}}}{c}\) | \(259\) |
1/4*a*(e*x+d)^4/e+b/c*(1/4*c*e^3*arcsinh(c*x)*x^4+c*e^2*arcsinh(c*x)*x^3*d +3/2*c*arcsinh(c*x)*d^2*e*x^2+arcsinh(c*x)*c*x*d^3+1/4*c/e*arcsinh(c*x)*d^ 4-1/4/c^3/e*(c^4*d^4*arcsinh(c*x)+e^4*(1/4*c^3*x^3*(c^2*x^2+1)^(1/2)-3/8*c *x*(c^2*x^2+1)^(1/2)+3/8*arcsinh(c*x))+4*d^3*c^3*e*(c^2*x^2+1)^(1/2)+6*d^2 *c^2*e^2*(1/2*c*x*(c^2*x^2+1)^(1/2)-1/2*arcsinh(c*x))+4*d*c*e^3*(1/3*c^2*x ^2*(c^2*x^2+1)^(1/2)-2/3*(c^2*x^2+1)^(1/2))))
Time = 0.28 (sec) , antiderivative size = 214, normalized size of antiderivative = 1.22 \[ \int (d+e x)^3 (a+b \text {arcsinh}(c x)) \, dx=\frac {24 \, a c^{4} e^{3} x^{4} + 96 \, a c^{4} d e^{2} x^{3} + 144 \, a c^{4} d^{2} e x^{2} + 96 \, a c^{4} d^{3} x + 3 \, {\left (8 \, b c^{4} e^{3} x^{4} + 32 \, b c^{4} d e^{2} x^{3} + 48 \, b c^{4} d^{2} e x^{2} + 32 \, b c^{4} d^{3} x + 24 \, b c^{2} d^{2} e - 3 \, b e^{3}\right )} \log \left (c x + \sqrt {c^{2} x^{2} + 1}\right ) - {\left (6 \, b c^{3} e^{3} x^{3} + 32 \, b c^{3} d e^{2} x^{2} + 96 \, b c^{3} d^{3} - 64 \, b c d e^{2} + 9 \, {\left (8 \, b c^{3} d^{2} e - b c e^{3}\right )} x\right )} \sqrt {c^{2} x^{2} + 1}}{96 \, c^{4}} \]
1/96*(24*a*c^4*e^3*x^4 + 96*a*c^4*d*e^2*x^3 + 144*a*c^4*d^2*e*x^2 + 96*a*c ^4*d^3*x + 3*(8*b*c^4*e^3*x^4 + 32*b*c^4*d*e^2*x^3 + 48*b*c^4*d^2*e*x^2 + 32*b*c^4*d^3*x + 24*b*c^2*d^2*e - 3*b*e^3)*log(c*x + sqrt(c^2*x^2 + 1)) - (6*b*c^3*e^3*x^3 + 32*b*c^3*d*e^2*x^2 + 96*b*c^3*d^3 - 64*b*c*d*e^2 + 9*(8 *b*c^3*d^2*e - b*c*e^3)*x)*sqrt(c^2*x^2 + 1))/c^4
Time = 0.30 (sec) , antiderivative size = 316, normalized size of antiderivative = 1.80 \[ \int (d+e x)^3 (a+b \text {arcsinh}(c x)) \, dx=\begin {cases} a d^{3} x + \frac {3 a d^{2} e x^{2}}{2} + a d e^{2} x^{3} + \frac {a e^{3} x^{4}}{4} + b d^{3} x \operatorname {asinh}{\left (c x \right )} + \frac {3 b d^{2} e x^{2} \operatorname {asinh}{\left (c x \right )}}{2} + b d e^{2} x^{3} \operatorname {asinh}{\left (c x \right )} + \frac {b e^{3} x^{4} \operatorname {asinh}{\left (c x \right )}}{4} - \frac {b d^{3} \sqrt {c^{2} x^{2} + 1}}{c} - \frac {3 b d^{2} e x \sqrt {c^{2} x^{2} + 1}}{4 c} - \frac {b d e^{2} x^{2} \sqrt {c^{2} x^{2} + 1}}{3 c} - \frac {b e^{3} x^{3} \sqrt {c^{2} x^{2} + 1}}{16 c} + \frac {3 b d^{2} e \operatorname {asinh}{\left (c x \right )}}{4 c^{2}} + \frac {2 b d e^{2} \sqrt {c^{2} x^{2} + 1}}{3 c^{3}} + \frac {3 b e^{3} x \sqrt {c^{2} x^{2} + 1}}{32 c^{3}} - \frac {3 b e^{3} \operatorname {asinh}{\left (c x \right )}}{32 c^{4}} & \text {for}\: c \neq 0 \\a \left (d^{3} x + \frac {3 d^{2} e x^{2}}{2} + d e^{2} x^{3} + \frac {e^{3} x^{4}}{4}\right ) & \text {otherwise} \end {cases} \]
Piecewise((a*d**3*x + 3*a*d**2*e*x**2/2 + a*d*e**2*x**3 + a*e**3*x**4/4 + b*d**3*x*asinh(c*x) + 3*b*d**2*e*x**2*asinh(c*x)/2 + b*d*e**2*x**3*asinh(c *x) + b*e**3*x**4*asinh(c*x)/4 - b*d**3*sqrt(c**2*x**2 + 1)/c - 3*b*d**2*e *x*sqrt(c**2*x**2 + 1)/(4*c) - b*d*e**2*x**2*sqrt(c**2*x**2 + 1)/(3*c) - b *e**3*x**3*sqrt(c**2*x**2 + 1)/(16*c) + 3*b*d**2*e*asinh(c*x)/(4*c**2) + 2 *b*d*e**2*sqrt(c**2*x**2 + 1)/(3*c**3) + 3*b*e**3*x*sqrt(c**2*x**2 + 1)/(3 2*c**3) - 3*b*e**3*asinh(c*x)/(32*c**4), Ne(c, 0)), (a*(d**3*x + 3*d**2*e* x**2/2 + d*e**2*x**3 + e**3*x**4/4), True))
Time = 0.18 (sec) , antiderivative size = 230, normalized size of antiderivative = 1.31 \[ \int (d+e x)^3 (a+b \text {arcsinh}(c x)) \, dx=\frac {1}{4} \, a e^{3} x^{4} + a d e^{2} x^{3} + \frac {3}{2} \, a d^{2} e x^{2} + \frac {3}{4} \, {\left (2 \, x^{2} \operatorname {arsinh}\left (c x\right ) - c {\left (\frac {\sqrt {c^{2} x^{2} + 1} x}{c^{2}} - \frac {\operatorname {arsinh}\left (c x\right )}{c^{3}}\right )}\right )} b d^{2} e + \frac {1}{3} \, {\left (3 \, x^{3} \operatorname {arsinh}\left (c x\right ) - c {\left (\frac {\sqrt {c^{2} x^{2} + 1} x^{2}}{c^{2}} - \frac {2 \, \sqrt {c^{2} x^{2} + 1}}{c^{4}}\right )}\right )} b d e^{2} + \frac {1}{32} \, {\left (8 \, x^{4} \operatorname {arsinh}\left (c x\right ) - {\left (\frac {2 \, \sqrt {c^{2} x^{2} + 1} x^{3}}{c^{2}} - \frac {3 \, \sqrt {c^{2} x^{2} + 1} x}{c^{4}} + \frac {3 \, \operatorname {arsinh}\left (c x\right )}{c^{5}}\right )} c\right )} b e^{3} + a d^{3} x + \frac {{\left (c x \operatorname {arsinh}\left (c x\right ) - \sqrt {c^{2} x^{2} + 1}\right )} b d^{3}}{c} \]
1/4*a*e^3*x^4 + a*d*e^2*x^3 + 3/2*a*d^2*e*x^2 + 3/4*(2*x^2*arcsinh(c*x) - c*(sqrt(c^2*x^2 + 1)*x/c^2 - arcsinh(c*x)/c^3))*b*d^2*e + 1/3*(3*x^3*arcsi nh(c*x) - c*(sqrt(c^2*x^2 + 1)*x^2/c^2 - 2*sqrt(c^2*x^2 + 1)/c^4))*b*d*e^2 + 1/32*(8*x^4*arcsinh(c*x) - (2*sqrt(c^2*x^2 + 1)*x^3/c^2 - 3*sqrt(c^2*x^ 2 + 1)*x/c^4 + 3*arcsinh(c*x)/c^5)*c)*b*e^3 + a*d^3*x + (c*x*arcsinh(c*x) - sqrt(c^2*x^2 + 1))*b*d^3/c
Exception generated. \[ \int (d+e x)^3 (a+b \text {arcsinh}(c x)) \, dx=\text {Exception raised: RuntimeError} \]
Exception raised: RuntimeError >> an error occurred running a Giac command :INPUT:sage2OUTPUT:sym2poly/r2sym(const gen & e,const index_m & i,const ve cteur & l) Error: Bad Argument Value
Timed out. \[ \int (d+e x)^3 (a+b \text {arcsinh}(c x)) \, dx=\int \left (a+b\,\mathrm {asinh}\left (c\,x\right )\right )\,{\left (d+e\,x\right )}^3 \,d x \]