Integrand size = 21, antiderivative size = 144 \[ \int \frac {(c+d x)^3}{a+b (c+d x)^3} \, dx=\frac {x}{b}+\frac {\sqrt [3]{a} \arctan \left (\frac {\sqrt [3]{a}-2 \sqrt [3]{b} (c+d x)}{\sqrt {3} \sqrt [3]{a}}\right )}{\sqrt {3} b^{4/3} d}-\frac {\sqrt [3]{a} \log \left (\sqrt [3]{a}+\sqrt [3]{b} (c+d x)\right )}{3 b^{4/3} d}+\frac {\sqrt [3]{a} \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} (c+d x)+b^{2/3} (c+d x)^2\right )}{6 b^{4/3} d} \] Output:
x/b+1/3*a^(1/3)*arctan(1/3*(a^(1/3)-2*b^(1/3)*(d*x+c))*3^(1/2)/a^(1/3))*3^ (1/2)/b^(4/3)/d-1/3*a^(1/3)*ln(a^(1/3)+b^(1/3)*(d*x+c))/b^(4/3)/d+1/6*a^(1 /3)*ln(a^(2/3)-a^(1/3)*b^(1/3)*(d*x+c)+b^(2/3)*(d*x+c)^2)/b^(4/3)/d
Time = 0.02 (sec) , antiderivative size = 142, normalized size of antiderivative = 0.99 \[ \int \frac {(c+d x)^3}{a+b (c+d x)^3} \, dx=\frac {6 \sqrt [3]{b} c+6 \sqrt [3]{b} d x-2 \sqrt {3} \sqrt [3]{a} \arctan \left (\frac {-\sqrt [3]{a}+2 \sqrt [3]{b} (c+d x)}{\sqrt {3} \sqrt [3]{a}}\right )-2 \sqrt [3]{a} \log \left (\sqrt [3]{a}+\sqrt [3]{b} (c+d x)\right )+\sqrt [3]{a} \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} (c+d x)+b^{2/3} (c+d x)^2\right )}{6 b^{4/3} d} \] Input:
Integrate[(c + d*x)^3/(a + b*(c + d*x)^3),x]
Output:
(6*b^(1/3)*c + 6*b^(1/3)*d*x - 2*Sqrt[3]*a^(1/3)*ArcTan[(-a^(1/3) + 2*b^(1 /3)*(c + d*x))/(Sqrt[3]*a^(1/3))] - 2*a^(1/3)*Log[a^(1/3) + b^(1/3)*(c + d *x)] + a^(1/3)*Log[a^(2/3) - a^(1/3)*b^(1/3)*(c + d*x) + b^(2/3)*(c + d*x) ^2])/(6*b^(4/3)*d)
Time = 0.51 (sec) , antiderivative size = 147, normalized size of antiderivative = 1.02, number of steps used = 11, number of rules used = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.476, Rules used = {895, 843, 750, 16, 1142, 25, 27, 1082, 217, 1103}
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.
| \(\displaystyle \int \frac {(c+d x)^3}{a+b (c+d x)^3} \, dx\) |
| \(\Big \downarrow \) 895 |
| \(\displaystyle \frac {\int \frac {(c+d x)^3}{b (c+d x)^3+a}d(c+d x)}{d}\) |
| \(\Big \downarrow \) 843 |
| \(\displaystyle \frac {\frac {c+d x}{b}-\frac {a \int \frac {1}{b (c+d x)^3+a}d(c+d x)}{b}}{d}\) |
| \(\Big \downarrow \) 750 |
| \(\displaystyle \frac {\frac {c+d x}{b}-\frac {a \left (\frac {\int \frac {2 \sqrt [3]{a}-\sqrt [3]{b} (c+d x)}{b^{2/3} (c+d x)^2-\sqrt [3]{a} \sqrt [3]{b} (c+d x)+a^{2/3}}d(c+d x)}{3 a^{2/3}}+\frac {\int \frac {1}{\sqrt [3]{b} (c+d x)+\sqrt [3]{a}}d(c+d x)}{3 a^{2/3}}\right )}{b}}{d}\) |
| \(\Big \downarrow \) 16 |
| \(\displaystyle \frac {\frac {c+d x}{b}-\frac {a \left (\frac {\int \frac {2 \sqrt [3]{a}-\sqrt [3]{b} (c+d x)}{b^{2/3} (c+d x)^2-\sqrt [3]{a} \sqrt [3]{b} (c+d x)+a^{2/3}}d(c+d x)}{3 a^{2/3}}+\frac {\log \left (\sqrt [3]{a}+\sqrt [3]{b} (c+d x)\right )}{3 a^{2/3} \sqrt [3]{b}}\right )}{b}}{d}\) |
| \(\Big \downarrow \) 1142 |
| \(\displaystyle \frac {\frac {c+d x}{b}-\frac {a \left (\frac {\frac {3}{2} \sqrt [3]{a} \int \frac {1}{b^{2/3} (c+d x)^2-\sqrt [3]{a} \sqrt [3]{b} (c+d x)+a^{2/3}}d(c+d x)-\frac {\int -\frac {\sqrt [3]{b} \left (\sqrt [3]{a}-2 \sqrt [3]{b} (c+d x)\right )}{b^{2/3} (c+d x)^2-\sqrt [3]{a} \sqrt [3]{b} (c+d x)+a^{2/3}}d(c+d x)}{2 \sqrt [3]{b}}}{3 a^{2/3}}+\frac {\log \left (\sqrt [3]{a}+\sqrt [3]{b} (c+d x)\right )}{3 a^{2/3} \sqrt [3]{b}}\right )}{b}}{d}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {\frac {c+d x}{b}-\frac {a \left (\frac {\frac {3}{2} \sqrt [3]{a} \int \frac {1}{b^{2/3} (c+d x)^2-\sqrt [3]{a} \sqrt [3]{b} (c+d x)+a^{2/3}}d(c+d x)+\frac {\int \frac {\sqrt [3]{b} \left (\sqrt [3]{a}-2 \sqrt [3]{b} (c+d x)\right )}{b^{2/3} (c+d x)^2-\sqrt [3]{a} \sqrt [3]{b} (c+d x)+a^{2/3}}d(c+d x)}{2 \sqrt [3]{b}}}{3 a^{2/3}}+\frac {\log \left (\sqrt [3]{a}+\sqrt [3]{b} (c+d x)\right )}{3 a^{2/3} \sqrt [3]{b}}\right )}{b}}{d}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\frac {c+d x}{b}-\frac {a \left (\frac {\frac {3}{2} \sqrt [3]{a} \int \frac {1}{b^{2/3} (c+d x)^2-\sqrt [3]{a} \sqrt [3]{b} (c+d x)+a^{2/3}}d(c+d x)+\frac {1}{2} \int \frac {\sqrt [3]{a}-2 \sqrt [3]{b} (c+d x)}{b^{2/3} (c+d x)^2-\sqrt [3]{a} \sqrt [3]{b} (c+d x)+a^{2/3}}d(c+d x)}{3 a^{2/3}}+\frac {\log \left (\sqrt [3]{a}+\sqrt [3]{b} (c+d x)\right )}{3 a^{2/3} \sqrt [3]{b}}\right )}{b}}{d}\) |
| \(\Big \downarrow \) 1082 |
| \(\displaystyle \frac {\frac {c+d x}{b}-\frac {a \left (\frac {\frac {1}{2} \int \frac {\sqrt [3]{a}-2 \sqrt [3]{b} (c+d x)}{b^{2/3} (c+d x)^2-\sqrt [3]{a} \sqrt [3]{b} (c+d x)+a^{2/3}}d(c+d x)+\frac {3 \int \frac {1}{-\left (1-\frac {2 \sqrt [3]{b} (c+d x)}{\sqrt [3]{a}}\right )^2-3}d\left (1-\frac {2 \sqrt [3]{b} (c+d x)}{\sqrt [3]{a}}\right )}{\sqrt [3]{b}}}{3 a^{2/3}}+\frac {\log \left (\sqrt [3]{a}+\sqrt [3]{b} (c+d x)\right )}{3 a^{2/3} \sqrt [3]{b}}\right )}{b}}{d}\) |
| \(\Big \downarrow \) 217 |
| \(\displaystyle \frac {\frac {c+d x}{b}-\frac {a \left (\frac {\frac {1}{2} \int \frac {\sqrt [3]{a}-2 \sqrt [3]{b} (c+d x)}{b^{2/3} (c+d x)^2-\sqrt [3]{a} \sqrt [3]{b} (c+d x)+a^{2/3}}d(c+d x)-\frac {\sqrt {3} \arctan \left (\frac {1-\frac {2 \sqrt [3]{b} (c+d x)}{\sqrt [3]{a}}}{\sqrt {3}}\right )}{\sqrt [3]{b}}}{3 a^{2/3}}+\frac {\log \left (\sqrt [3]{a}+\sqrt [3]{b} (c+d x)\right )}{3 a^{2/3} \sqrt [3]{b}}\right )}{b}}{d}\) |
| \(\Big \downarrow \) 1103 |
| \(\displaystyle \frac {\frac {c+d x}{b}-\frac {a \left (\frac {-\frac {\log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} (c+d x)+b^{2/3} (c+d x)^2\right )}{2 \sqrt [3]{b}}-\frac {\sqrt {3} \arctan \left (\frac {1-\frac {2 \sqrt [3]{b} (c+d x)}{\sqrt [3]{a}}}{\sqrt {3}}\right )}{\sqrt [3]{b}}}{3 a^{2/3}}+\frac {\log \left (\sqrt [3]{a}+\sqrt [3]{b} (c+d x)\right )}{3 a^{2/3} \sqrt [3]{b}}\right )}{b}}{d}\) |
Input:
Int[(c + d*x)^3/(a + b*(c + d*x)^3),x]
Output:
((c + d*x)/b - (a*(Log[a^(1/3) + b^(1/3)*(c + d*x)]/(3*a^(2/3)*b^(1/3)) + (-((Sqrt[3]*ArcTan[(1 - (2*b^(1/3)*(c + d*x))/a^(1/3))/Sqrt[3]])/b^(1/3)) - Log[a^(2/3) - a^(1/3)*b^(1/3)*(c + d*x) + b^(2/3)*(c + d*x)^2]/(2*b^(1/3 )))/(3*a^(2/3))))/b)/d
Int[(c_.)/((a_.) + (b_.)*(x_)), x_Symbol] :> Simp[c*(Log[RemoveContent[a + b*x, x]]/b), x] /; FreeQ[{a, b, c}, x]
Int[(a_)*(Fx_), x_Symbol] :> Simp[a Int[Fx, x], x] /; FreeQ[a, x] && !Ma tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(-(Rt[-a, 2]*Rt[-b, 2])^( -1))*ArcTan[Rt[-b, 2]*(x/Rt[-a, 2])], x] /; FreeQ[{a, b}, x] && PosQ[a/b] & & (LtQ[a, 0] || LtQ[b, 0])
Int[((a_) + (b_.)*(x_)^3)^(-1), x_Symbol] :> Simp[1/(3*Rt[a, 3]^2) Int[1/ (Rt[a, 3] + Rt[b, 3]*x), x], x] + Simp[1/(3*Rt[a, 3]^2) Int[(2*Rt[a, 3] - Rt[b, 3]*x)/(Rt[a, 3]^2 - Rt[a, 3]*Rt[b, 3]*x + Rt[b, 3]^2*x^2), x], x] /; FreeQ[{a, b}, x]
Int[((c_.)*(x_))^(m_)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[c^(n - 1)*(c*x)^(m - n + 1)*((a + b*x^n)^(p + 1)/(b*(m + n*p + 1))), x] - Simp[ a*c^n*((m - n + 1)/(b*(m + n*p + 1))) Int[(c*x)^(m - n)*(a + b*x^n)^p, x] , x] /; FreeQ[{a, b, c, p}, x] && IGtQ[n, 0] && GtQ[m, n - 1] && NeQ[m + n* p + 1, 0] && IntBinomialQ[a, b, c, n, m, p, x]
Int[(u_)^(m_.)*((a_) + (b_.)*(v_)^(n_))^(p_.), x_Symbol] :> Simp[u^m/(Coeff icient[v, x, 1]*v^m) Subst[Int[x^m*(a + b*x^n)^p, x], x, v], x] /; FreeQ[ {a, b, m, n, p}, x] && LinearPairQ[u, v, x]
Int[((a_) + (b_.)*(x_) + (c_.)*(x_)^2)^(-1), x_Symbol] :> With[{q = 1 - 4*S implify[a*(c/b^2)]}, Simp[-2/b Subst[Int[1/(q - x^2), x], x, 1 + 2*c*(x/b )], x] /; RationalQ[q] && (EqQ[q^2, 1] || !RationalQ[b^2 - 4*a*c])] /; Fre eQ[{a, b, c}, x]
Int[((d_) + (e_.)*(x_))/((a_.) + (b_.)*(x_) + (c_.)*(x_)^2), x_Symbol] :> S imp[d*(Log[RemoveContent[a + b*x + c*x^2, x]]/b), x] /; FreeQ[{a, b, c, d, e}, x] && EqQ[2*c*d - b*e, 0]
Int[((d_.) + (e_.)*(x_))/((a_) + (b_.)*(x_) + (c_.)*(x_)^2), x_Symbol] :> S imp[(2*c*d - b*e)/(2*c) Int[1/(a + b*x + c*x^2), x], x] + Simp[e/(2*c) Int[(b + 2*c*x)/(a + b*x + c*x^2), x], x] /; FreeQ[{a, b, c, d, e}, x]
Result contains higher order function than in optimal. Order 9 vs. order 3.
Time = 0.55 (sec) , antiderivative size = 78, normalized size of antiderivative = 0.54
| method | result | size |
| default | \(\frac {x}{b}-\frac {\left (\munderset {\textit {\_R} =\operatorname {RootOf}\left (b \,d^{3} \textit {\_Z}^{3}+3 b c \,d^{2} \textit {\_Z}^{2}+3 b \,c^{2} d \textit {\_Z} +c^{3} b +a \right )}{\sum }\frac {\ln \left (x -\textit {\_R} \right )}{d^{2} \textit {\_R}^{2}+2 c d \textit {\_R} +c^{2}}\right ) a}{3 b^{2} d}\) | \(78\) |
| risch | \(\frac {x}{b}-\frac {\left (\munderset {\textit {\_R} =\operatorname {RootOf}\left (b \,d^{3} \textit {\_Z}^{3}+3 b c \,d^{2} \textit {\_Z}^{2}+3 b \,c^{2} d \textit {\_Z} +c^{3} b +a \right )}{\sum }\frac {\ln \left (x -\textit {\_R} \right )}{d^{2} \textit {\_R}^{2}+2 c d \textit {\_R} +c^{2}}\right ) a}{3 b^{2} d}\) | \(78\) |
Input:
int((d*x+c)^3/(a+b*(d*x+c)^3),x,method=_RETURNVERBOSE)
Output:
x/b-1/3/b^2/d*sum(1/(_R^2*d^2+2*_R*c*d+c^2)*ln(x-_R),_R=RootOf(_Z^3*b*d^3+ 3*_Z^2*b*c*d^2+3*_Z*b*c^2*d+b*c^3+a))*a
Time = 0.08 (sec) , antiderivative size = 135, normalized size of antiderivative = 0.94 \[ \int \frac {(c+d x)^3}{a+b (c+d x)^3} \, dx=\frac {6 \, d x + 2 \, \sqrt {3} \left (-\frac {a}{b}\right )^{\frac {1}{3}} \arctan \left (\frac {2 \, \sqrt {3} {\left (b d x + b c\right )} \left (-\frac {a}{b}\right )^{\frac {2}{3}} - \sqrt {3} a}{3 \, a}\right ) - \left (-\frac {a}{b}\right )^{\frac {1}{3}} \log \left (d^{2} x^{2} + 2 \, c d x + c^{2} + {\left (d x + c\right )} \left (-\frac {a}{b}\right )^{\frac {1}{3}} + \left (-\frac {a}{b}\right )^{\frac {2}{3}}\right ) + 2 \, \left (-\frac {a}{b}\right )^{\frac {1}{3}} \log \left (d x + c - \left (-\frac {a}{b}\right )^{\frac {1}{3}}\right )}{6 \, b d} \] Input:
integrate((d*x+c)^3/(a+b*(d*x+c)^3),x, algorithm="fricas")
Output:
1/6*(6*d*x + 2*sqrt(3)*(-a/b)^(1/3)*arctan(1/3*(2*sqrt(3)*(b*d*x + b*c)*(- a/b)^(2/3) - sqrt(3)*a)/a) - (-a/b)^(1/3)*log(d^2*x^2 + 2*c*d*x + c^2 + (d *x + c)*(-a/b)^(1/3) + (-a/b)^(2/3)) + 2*(-a/b)^(1/3)*log(d*x + c - (-a/b) ^(1/3)))/(b*d)
Time = 0.19 (sec) , antiderivative size = 27, normalized size of antiderivative = 0.19 \[ \int \frac {(c+d x)^3}{a+b (c+d x)^3} \, dx=\frac {\operatorname {RootSum} {\left (27 t^{3} b^{4} + a, \left ( t \mapsto t \log {\left (x + \frac {- 3 t b + c}{d} \right )} \right )\right )}}{d} + \frac {x}{b} \] Input:
integrate((d*x+c)**3/(a+b*(d*x+c)**3),x)
Output:
RootSum(27*_t**3*b**4 + a, Lambda(_t, _t*log(x + (-3*_t*b + c)/d)))/d + x/ b
\[ \int \frac {(c+d x)^3}{a+b (c+d x)^3} \, dx=\int { \frac {{\left (d x + c\right )}^{3}}{{\left (d x + c\right )}^{3} b + a} \,d x } \] Input:
integrate((d*x+c)^3/(a+b*(d*x+c)^3),x, algorithm="maxima")
Output:
-a*integrate(1/(b*d^3*x^3 + 3*b*c*d^2*x^2 + 3*b*c^2*d*x + b*c^3 + a), x)/b + x/b
Time = 0.13 (sec) , antiderivative size = 180, normalized size of antiderivative = 1.25 \[ \int \frac {(c+d x)^3}{a+b (c+d x)^3} \, dx=\frac {x}{b} + \frac {2 \, \sqrt {3} \left (-\frac {a d^{6}}{b}\right )^{\frac {1}{3}} \arctan \left (-\frac {b d x + b c - \left (-a b^{2}\right )^{\frac {1}{3}}}{\sqrt {3} b d x + \sqrt {3} b c + \sqrt {3} \left (-a b^{2}\right )^{\frac {1}{3}}}\right ) - \left (-\frac {a d^{6}}{b}\right )^{\frac {1}{3}} \log \left (4 \, {\left (\sqrt {3} b d x + \sqrt {3} b c + \sqrt {3} \left (-a b^{2}\right )^{\frac {1}{3}}\right )}^{2} + 4 \, {\left (b d x + b c - \left (-a b^{2}\right )^{\frac {1}{3}}\right )}^{2}\right ) + 2 \, \left (-\frac {a d^{6}}{b}\right )^{\frac {1}{3}} \log \left ({\left | -b d x - b c + \left (-a b^{2}\right )^{\frac {1}{3}} \right |}\right )}{6 \, b d^{3}} \] Input:
integrate((d*x+c)^3/(a+b*(d*x+c)^3),x, algorithm="giac")
Output:
x/b + 1/6*(2*sqrt(3)*(-a*d^6/b)^(1/3)*arctan(-(b*d*x + b*c - (-a*b^2)^(1/3 ))/(sqrt(3)*b*d*x + sqrt(3)*b*c + sqrt(3)*(-a*b^2)^(1/3))) - (-a*d^6/b)^(1 /3)*log(4*(sqrt(3)*b*d*x + sqrt(3)*b*c + sqrt(3)*(-a*b^2)^(1/3))^2 + 4*(b* d*x + b*c - (-a*b^2)^(1/3))^2) + 2*(-a*d^6/b)^(1/3)*log(abs(-b*d*x - b*c + (-a*b^2)^(1/3))))/(b*d^3)
Time = 0.28 (sec) , antiderivative size = 158, normalized size of antiderivative = 1.10 \[ \int \frac {(c+d x)^3}{a+b (c+d x)^3} \, dx=\frac {x}{b}+\frac {{\left (-a\right )}^{1/3}\,\ln \left ({\left (-a\right )}^{4/3}+a\,b^{1/3}\,c+a\,b^{1/3}\,d\,x\right )}{3\,b^{4/3}\,d}-\frac {{\left (-a\right )}^{1/3}\,\ln \left (3\,a\,b\,c\,d^5-3\,{\left (-a\right )}^{4/3}\,b^{2/3}\,d^5\,\left (\frac {1}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}\right )+3\,a\,b\,d^6\,x\right )\,\left (\frac {1}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}\right )}{3\,b^{4/3}\,d}+\frac {{\left (-a\right )}^{1/3}\,\ln \left (9\,{\left (-a\right )}^{4/3}\,b^{2/3}\,d^5\,\left (-\frac {1}{6}+\frac {\sqrt {3}\,1{}\mathrm {i}}{6}\right )+3\,a\,b\,c\,d^5+3\,a\,b\,d^6\,x\right )\,\left (-\frac {1}{6}+\frac {\sqrt {3}\,1{}\mathrm {i}}{6}\right )}{b^{4/3}\,d} \] Input:
int((c + d*x)^3/(a + b*(c + d*x)^3),x)
Output:
x/b + ((-a)^(1/3)*log((-a)^(4/3) + a*b^(1/3)*c + a*b^(1/3)*d*x))/(3*b^(4/3 )*d) - ((-a)^(1/3)*log(3*a*b*c*d^5 - 3*(-a)^(4/3)*b^(2/3)*d^5*((3^(1/2)*1i )/2 + 1/2) + 3*a*b*d^6*x)*((3^(1/2)*1i)/2 + 1/2))/(3*b^(4/3)*d) + ((-a)^(1 /3)*log(9*(-a)^(4/3)*b^(2/3)*d^5*((3^(1/2)*1i)/6 - 1/6) + 3*a*b*c*d^5 + 3* a*b*d^6*x)*((3^(1/2)*1i)/6 - 1/6))/(b^(4/3)*d)
Time = 0.24 (sec) , antiderivative size = 123, normalized size of antiderivative = 0.85 \[ \int \frac {(c+d x)^3}{a+b (c+d x)^3} \, dx=\frac {2 a^{\frac {1}{3}} \sqrt {3}\, \mathit {atan} \left (\frac {a^{\frac {1}{3}}-2 b^{\frac {1}{3}} c -2 b^{\frac {1}{3}} d x}{a^{\frac {1}{3}} \sqrt {3}}\right )+a^{\frac {1}{3}} \mathrm {log}\left (a^{\frac {2}{3}}-b^{\frac {1}{3}} a^{\frac {1}{3}} c -b^{\frac {1}{3}} a^{\frac {1}{3}} d x +b^{\frac {2}{3}} c^{2}+2 b^{\frac {2}{3}} c d x +b^{\frac {2}{3}} d^{2} x^{2}\right )-2 a^{\frac {1}{3}} \mathrm {log}\left (a^{\frac {1}{3}}+b^{\frac {1}{3}} c +b^{\frac {1}{3}} d x \right )+6 b^{\frac {1}{3}} d x}{6 b^{\frac {4}{3}} d} \] Input:
int((d*x+c)^3/(a+b*(d*x+c)^3),x)
Output:
(2*a**(1/3)*sqrt(3)*atan((a**(1/3) - 2*b**(1/3)*c - 2*b**(1/3)*d*x)/(a**(1 /3)*sqrt(3))) + a**(1/3)*log(a**(2/3) - b**(1/3)*a**(1/3)*c - b**(1/3)*a** (1/3)*d*x + b**(2/3)*c**2 + 2*b**(2/3)*c*d*x + b**(2/3)*d**2*x**2) - 2*a** (1/3)*log(a**(1/3) + b**(1/3)*c + b**(1/3)*d*x) + 6*b**(1/3)*d*x)/(6*b**(1 /3)*b*d)