\(\int \frac {\sqrt [3]{a+b x^3}}{x^{11} (c+d x^3)} \, dx\) [686]

Optimal result

Integrand size = 24, antiderivative size = 318 \[ \int \frac {\sqrt [3]{a+b x^3}}{x^{11} \left (c+d x^3\right )} \, dx=-\frac {\sqrt [3]{a+b x^3}}{10 c x^{10}}-\frac {(b c-10 a d) \sqrt [3]{a+b x^3}}{70 a c^2 x^7}+\frac {\left (3 b^2 c^2+5 a b c d-35 a^2 d^2\right ) \sqrt [3]{a+b x^3}}{140 a^2 c^3 x^4}-\frac {\left (9 b^3 c^3+15 a b^2 c^2 d+35 a^2 b c d^2-140 a^3 d^3\right ) \sqrt [3]{a+b x^3}}{140 a^3 c^4 x}+\frac {d^3 \sqrt [3]{b c-a d} \arctan \left (\frac {1+\frac {2 \sqrt [3]{b c-a d} x}{\sqrt [3]{c} \sqrt [3]{a+b x^3}}}{\sqrt {3}}\right )}{\sqrt {3} c^{13/3}}-\frac {d^3 \sqrt [3]{b c-a d} \log \left (c+d x^3\right )}{6 c^{13/3}}+\frac {d^3 \sqrt [3]{b c-a d} \log \left (\frac {\sqrt [3]{b c-a d} x}{\sqrt [3]{c}}-\sqrt [3]{a+b x^3}\right )}{2 c^{13/3}} \] Output:

-1/10*(b*x^3+a)^(1/3)/c/x^10-1/70*(-10*a*d+b*c)*(b*x^3+a)^(1/3)/a/c^2/x^7+ 
1/140*(-35*a^2*d^2+5*a*b*c*d+3*b^2*c^2)*(b*x^3+a)^(1/3)/a^2/c^3/x^4-1/140* 
(-140*a^3*d^3+35*a^2*b*c*d^2+15*a*b^2*c^2*d+9*b^3*c^3)*(b*x^3+a)^(1/3)/a^3 
/c^4/x+1/3*d^3*(-a*d+b*c)^(1/3)*arctan(1/3*(1+2*(-a*d+b*c)^(1/3)*x/c^(1/3) 
/(b*x^3+a)^(1/3))*3^(1/2))*3^(1/2)/c^(13/3)-1/6*d^3*(-a*d+b*c)^(1/3)*ln(d* 
x^3+c)/c^(13/3)+1/2*d^3*(-a*d+b*c)^(1/3)*ln((-a*d+b*c)^(1/3)*x/c^(1/3)-(b* 
x^3+a)^(1/3))/c^(13/3)
 

Mathematica [C] (warning: unable to verify)

Result contains complex when optimal does not.

Time = 2.75 (sec) , antiderivative size = 419, normalized size of antiderivative = 1.32 \[ \int \frac {\sqrt [3]{a+b x^3}}{x^{11} \left (c+d x^3\right )} \, dx=\frac {\frac {3 \sqrt [3]{c} \sqrt [3]{a+b x^3} \left (-9 b^3 c^3 x^9+3 a b^2 c^2 x^6 \left (c-5 d x^3\right )+a^2 b c x^3 \left (-2 c^2+5 c d x^3-35 d^2 x^6\right )+a^3 \left (-14 c^3+20 c^2 d x^3-35 c d^2 x^6+140 d^3 x^9\right )\right )}{a^3 x^{10}}-70 \sqrt {-6-6 i \sqrt {3}} d^3 \sqrt [3]{b c-a d} \arctan \left (\frac {3 \sqrt [3]{b c-a d} x}{\sqrt {3} \sqrt [3]{b c-a d} x-\left (3 i+\sqrt {3}\right ) \sqrt [3]{c} \sqrt [3]{a+b x^3}}\right )+70 i \left (i+\sqrt {3}\right ) d^3 \sqrt [3]{b c-a d} \log \left (2 \sqrt [3]{b c-a d} x+\left (1+i \sqrt {3}\right ) \sqrt [3]{c} \sqrt [3]{a+b x^3}\right )+35 \left (1-i \sqrt {3}\right ) d^3 \sqrt [3]{b c-a d} \log \left (2 (b c-a d)^{2/3} x^2+\left (-1-i \sqrt {3}\right ) \sqrt [3]{c} \sqrt [3]{b c-a d} x \sqrt [3]{a+b x^3}+i \left (i+\sqrt {3}\right ) c^{2/3} \left (a+b x^3\right )^{2/3}\right )}{420 c^{13/3}} \] Input:

Integrate[(a + b*x^3)^(1/3)/(x^11*(c + d*x^3)),x]
 

Output:

((3*c^(1/3)*(a + b*x^3)^(1/3)*(-9*b^3*c^3*x^9 + 3*a*b^2*c^2*x^6*(c - 5*d*x 
^3) + a^2*b*c*x^3*(-2*c^2 + 5*c*d*x^3 - 35*d^2*x^6) + a^3*(-14*c^3 + 20*c^ 
2*d*x^3 - 35*c*d^2*x^6 + 140*d^3*x^9)))/(a^3*x^10) - 70*Sqrt[-6 - (6*I)*Sq 
rt[3]]*d^3*(b*c - a*d)^(1/3)*ArcTan[(3*(b*c - a*d)^(1/3)*x)/(Sqrt[3]*(b*c 
- a*d)^(1/3)*x - (3*I + Sqrt[3])*c^(1/3)*(a + b*x^3)^(1/3))] + (70*I)*(I + 
 Sqrt[3])*d^3*(b*c - a*d)^(1/3)*Log[2*(b*c - a*d)^(1/3)*x + (1 + I*Sqrt[3] 
)*c^(1/3)*(a + b*x^3)^(1/3)] + 35*(1 - I*Sqrt[3])*d^3*(b*c - a*d)^(1/3)*Lo 
g[2*(b*c - a*d)^(2/3)*x^2 + (-1 - I*Sqrt[3])*c^(1/3)*(b*c - a*d)^(1/3)*x*( 
a + b*x^3)^(1/3) + I*(I + Sqrt[3])*c^(2/3)*(a + b*x^3)^(2/3)])/(420*c^(13/ 
3))
 

Rubi [A] (verified)

Time = 1.01 (sec) , antiderivative size = 352, normalized size of antiderivative = 1.11, number of steps used = 7, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.292, Rules used = {975, 1053, 27, 1053, 1053, 27, 992}

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.

\(\displaystyle \int \frac {\sqrt [3]{a+b x^3}}{x^{11} \left (c+d x^3\right )} \, dx\)

\(\Big \downarrow \) 975

\(\displaystyle \frac {\int \frac {-9 b d x^3+b c-10 a d}{x^8 \left (b x^3+a\right )^{2/3} \left (d x^3+c\right )}dx}{10 c}-\frac {\sqrt [3]{a+b x^3}}{10 c x^{10}}\)

\(\Big \downarrow \) 1053

\(\displaystyle \frac {-\frac {\int \frac {2 \left (3 b d (b c-10 a d) x^3+3 b^2 c^2-35 a^2 d^2+5 a b c d\right )}{x^5 \left (b x^3+a\right )^{2/3} \left (d x^3+c\right )}dx}{7 a c}-\frac {\sqrt [3]{a+b x^3} (b c-10 a d)}{7 a c x^7}}{10 c}-\frac {\sqrt [3]{a+b x^3}}{10 c x^{10}}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {-\frac {2 \int \frac {3 b d (b c-10 a d) x^3+3 b^2 c^2-35 a^2 d^2+5 a b c d}{x^5 \left (b x^3+a\right )^{2/3} \left (d x^3+c\right )}dx}{7 a c}-\frac {\sqrt [3]{a+b x^3} (b c-10 a d)}{7 a c x^7}}{10 c}-\frac {\sqrt [3]{a+b x^3}}{10 c x^{10}}\)

\(\Big \downarrow \) 1053

\(\displaystyle \frac {-\frac {2 \left (-\frac {\int \frac {9 b^3 c^3+15 a b^2 d c^2+35 a^2 b d^2 c-140 a^3 d^3+3 b d \left (3 b^2 c^2+5 a b d c-35 a^2 d^2\right ) x^3}{x^2 \left (b x^3+a\right )^{2/3} \left (d x^3+c\right )}dx}{4 a c}-\frac {\sqrt [3]{a+b x^3} \left (\frac {3 b^2 c}{a}-\frac {35 a d^2}{c}+5 b d\right )}{4 x^4}\right )}{7 a c}-\frac {\sqrt [3]{a+b x^3} (b c-10 a d)}{7 a c x^7}}{10 c}-\frac {\sqrt [3]{a+b x^3}}{10 c x^{10}}\)

\(\Big \downarrow \) 1053

\(\displaystyle \frac {-\frac {2 \left (-\frac {-\frac {\int \frac {140 a^3 d^3 (b c-a d) x}{\left (b x^3+a\right )^{2/3} \left (d x^3+c\right )}dx}{a c}-\frac {\sqrt [3]{a+b x^3} \left (-140 a^3 d^3+35 a^2 b c d^2+15 a b^2 c^2 d+9 b^3 c^3\right )}{a c x}}{4 a c}-\frac {\sqrt [3]{a+b x^3} \left (\frac {3 b^2 c}{a}-\frac {35 a d^2}{c}+5 b d\right )}{4 x^4}\right )}{7 a c}-\frac {\sqrt [3]{a+b x^3} (b c-10 a d)}{7 a c x^7}}{10 c}-\frac {\sqrt [3]{a+b x^3}}{10 c x^{10}}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {-\frac {2 \left (-\frac {-\frac {140 a^2 d^3 (b c-a d) \int \frac {x}{\left (b x^3+a\right )^{2/3} \left (d x^3+c\right )}dx}{c}-\frac {\sqrt [3]{a+b x^3} \left (-140 a^3 d^3+35 a^2 b c d^2+15 a b^2 c^2 d+9 b^3 c^3\right )}{a c x}}{4 a c}-\frac {\sqrt [3]{a+b x^3} \left (\frac {3 b^2 c}{a}-\frac {35 a d^2}{c}+5 b d\right )}{4 x^4}\right )}{7 a c}-\frac {\sqrt [3]{a+b x^3} (b c-10 a d)}{7 a c x^7}}{10 c}-\frac {\sqrt [3]{a+b x^3}}{10 c x^{10}}\)

\(\Big \downarrow \) 992

\(\displaystyle \frac {-\frac {2 \left (-\frac {-\frac {140 a^2 d^3 (b c-a d) \left (-\frac {\arctan \left (\frac {\frac {2 x \sqrt [3]{b c-a d}}{\sqrt [3]{c} \sqrt [3]{a+b x^3}}+1}{\sqrt {3}}\right )}{\sqrt {3} \sqrt [3]{c} (b c-a d)^{2/3}}+\frac {\log \left (c+d x^3\right )}{6 \sqrt [3]{c} (b c-a d)^{2/3}}-\frac {\log \left (\frac {x \sqrt [3]{b c-a d}}{\sqrt [3]{c}}-\sqrt [3]{a+b x^3}\right )}{2 \sqrt [3]{c} (b c-a d)^{2/3}}\right )}{c}-\frac {\sqrt [3]{a+b x^3} \left (-140 a^3 d^3+35 a^2 b c d^2+15 a b^2 c^2 d+9 b^3 c^3\right )}{a c x}}{4 a c}-\frac {\sqrt [3]{a+b x^3} \left (\frac {3 b^2 c}{a}-\frac {35 a d^2}{c}+5 b d\right )}{4 x^4}\right )}{7 a c}-\frac {\sqrt [3]{a+b x^3} (b c-10 a d)}{7 a c x^7}}{10 c}-\frac {\sqrt [3]{a+b x^3}}{10 c x^{10}}\)

Input:

Int[(a + b*x^3)^(1/3)/(x^11*(c + d*x^3)),x]
 

Output:

-1/10*(a + b*x^3)^(1/3)/(c*x^10) + (-1/7*((b*c - 10*a*d)*(a + b*x^3)^(1/3) 
)/(a*c*x^7) - (2*(-1/4*(((3*b^2*c)/a + 5*b*d - (35*a*d^2)/c)*(a + b*x^3)^( 
1/3))/x^4 - (-(((9*b^3*c^3 + 15*a*b^2*c^2*d + 35*a^2*b*c*d^2 - 140*a^3*d^3 
)*(a + b*x^3)^(1/3))/(a*c*x)) - (140*a^2*d^3*(b*c - a*d)*(-(ArcTan[(1 + (2 
*(b*c - a*d)^(1/3)*x)/(c^(1/3)*(a + b*x^3)^(1/3)))/Sqrt[3]]/(Sqrt[3]*c^(1/ 
3)*(b*c - a*d)^(2/3))) + Log[c + d*x^3]/(6*c^(1/3)*(b*c - a*d)^(2/3)) - Lo 
g[((b*c - a*d)^(1/3)*x)/c^(1/3) - (a + b*x^3)^(1/3)]/(2*c^(1/3)*(b*c - a*d 
)^(2/3))))/c)/(4*a*c)))/(7*a*c))/(10*c)
 

Defintions of rubi rules used

rule 27

Int[(a_)*(Fx_), x_Symbol] :> Simp[a   Int[Fx, x], x] /; FreeQ[a, x] &&  !Ma 
tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
 

rule 975

Int[((e_.)*(x_))^(m_)*((a_) + (b_.)*(x_)^(n_))^(p_)*((c_) + (d_.)*(x_)^(n_) 
)^(q_), x_Symbol] :> Simp[(e*x)^(m + 1)*(a + b*x^n)^(p + 1)*((c + d*x^n)^q/ 
(a*e*(m + 1))), x] - Simp[1/(a*e^n*(m + 1))   Int[(e*x)^(m + n)*(a + b*x^n) 
^p*(c + d*x^n)^(q - 1)*Simp[c*b*(m + 1) + n*(b*c*(p + 1) + a*d*q) + d*(b*(m 
 + 1) + b*n*(p + q + 1))*x^n, x], x], x] /; FreeQ[{a, b, c, d, e, p}, x] && 
 NeQ[b*c - a*d, 0] && IGtQ[n, 0] && LtQ[0, q, 1] && LtQ[m, -1] && IntBinomi 
alQ[a, b, c, d, e, m, n, p, q, x]
 

rule 992

Int[(x_)/(((a_) + (b_.)*(x_)^3)^(2/3)*((c_) + (d_.)*(x_)^3)), x_Symbol] :> 
With[{q = Rt[(b*c - a*d)/c, 3]}, Simp[-ArcTan[(1 + (2*q*x)/(a + b*x^3)^(1/3 
))/Sqrt[3]]/(Sqrt[3]*c*q^2), x] + (-Simp[Log[q*x - (a + b*x^3)^(1/3)]/(2*c* 
q^2), x] + Simp[Log[c + d*x^3]/(6*c*q^2), x])] /; FreeQ[{a, b, c, d}, x] && 
 NeQ[b*c - a*d, 0]
 

rule 1053

Int[((g_.)*(x_))^(m_)*((a_) + (b_.)*(x_)^(n_))^(p_.)*((c_) + (d_.)*(x_)^(n_ 
))^(q_.)*((e_) + (f_.)*(x_)^(n_)), x_Symbol] :> Simp[e*(g*x)^(m + 1)*(a + b 
*x^n)^(p + 1)*((c + d*x^n)^(q + 1)/(a*c*g*(m + 1))), x] + Simp[1/(a*c*g^n*( 
m + 1))   Int[(g*x)^(m + n)*(a + b*x^n)^p*(c + d*x^n)^q*Simp[a*f*c*(m + 1) 
- e*(b*c + a*d)*(m + n + 1) - e*n*(b*c*p + a*d*q) - b*e*d*(m + n*(p + q + 2 
) + 1)*x^n, x], x], x] /; FreeQ[{a, b, c, d, e, f, g, p, q}, x] && IGtQ[n, 
0] && LtQ[m, -1]
 

Maple [A] (verified)

Time = 1.98 (sec) , antiderivative size = 306, normalized size of antiderivative = 0.96

method	result	size
pseudoelliptic	\(-\frac {c \left (\left (\frac {9}{14} b^{2} x^{6}-\frac {6}{7} a b \,x^{3}+a^{2}\right ) \left (b \,x^{3}+a \right ) c^{3}-\frac {10 a d \left (b \,x^{3}+a \right ) \left (-\frac {3 b \,x^{3}}{4}+a \right ) x^{3} c^{2}}{7}+\frac {5 \left (b \,x^{3}+a \right ) a^{2} c \,d^{2} x^{6}}{2}-10 a^{3} d^{3} x^{9}\right ) \left (\frac {a d -b c}{c}\right )^{\frac {2}{3}} \left (b \,x^{3}+a \right )^{\frac {1}{3}}-\frac {5 a^{3} d^{3} x^{10} \left (a d -b c \right ) \left (2 \sqrt {3}\, \arctan \left (\frac {\sqrt {3}\, \left (\left (\frac {a d -b c}{c}\right )^{\frac {1}{3}} x -2 \left (b \,x^{3}+a \right )^{\frac {1}{3}}\right )}{3 \left (\frac {a d -b c}{c}\right )^{\frac {1}{3}} x}\right )+\ln \left (\frac {\left (\frac {a d -b c}{c}\right )^{\frac {2}{3}} x^{2}-\left (\frac {a d -b c}{c}\right )^{\frac {1}{3}} \left (b \,x^{3}+a \right )^{\frac {1}{3}} x +\left (b \,x^{3}+a \right )^{\frac {2}{3}}}{x^{2}}\right )-2 \ln \left (\frac {\left (\frac {a d -b c}{c}\right )^{\frac {1}{3}} x +\left (b \,x^{3}+a \right )^{\frac {1}{3}}}{x}\right )\right )}{3}}{10 \left (\frac {a d -b c}{c}\right )^{\frac {2}{3}} x^{10} c^{5} a^{3}}\)	\(306\)

Input:

int((b*x^3+a)^(1/3)/x^11/(d*x^3+c),x,method=_RETURNVERBOSE)
 

Output:

-1/10*(c*((9/14*b^2*x^6-6/7*a*b*x^3+a^2)*(b*x^3+a)*c^3-10/7*a*d*(b*x^3+a)* 
(-3/4*b*x^3+a)*x^3*c^2+5/2*(b*x^3+a)*a^2*c*d^2*x^6-10*a^3*d^3*x^9)*((a*d-b 
*c)/c)^(2/3)*(b*x^3+a)^(1/3)-5/3*a^3*d^3*x^10*(a*d-b*c)*(2*3^(1/2)*arctan( 
1/3*3^(1/2)*(((a*d-b*c)/c)^(1/3)*x-2*(b*x^3+a)^(1/3))/((a*d-b*c)/c)^(1/3)/ 
x)+ln((((a*d-b*c)/c)^(2/3)*x^2-((a*d-b*c)/c)^(1/3)*(b*x^3+a)^(1/3)*x+(b*x^ 
3+a)^(2/3))/x^2)-2*ln((((a*d-b*c)/c)^(1/3)*x+(b*x^3+a)^(1/3))/x)))/((a*d-b 
*c)/c)^(2/3)/x^10/c^5/a^3
 

Fricas [F(-1)]

Timed out. \[ \int \frac {\sqrt [3]{a+b x^3}}{x^{11} \left (c+d x^3\right )} \, dx=\text {Timed out} \] Input:

integrate((b*x^3+a)^(1/3)/x^11/(d*x^3+c),x, algorithm="fricas")
 

Output:

Timed out
 

Sympy [F]

\[ \int \frac {\sqrt [3]{a+b x^3}}{x^{11} \left (c+d x^3\right )} \, dx=\int \frac {\sqrt [3]{a + b x^{3}}}{x^{11} \left (c + d x^{3}\right )}\, dx \] Input:

integrate((b*x**3+a)**(1/3)/x**11/(d*x**3+c),x)
 

Output:

Integral((a + b*x**3)**(1/3)/(x**11*(c + d*x**3)), x)
 

Maxima [F]

\[ \int \frac {\sqrt [3]{a+b x^3}}{x^{11} \left (c+d x^3\right )} \, dx=\int { \frac {{\left (b x^{3} + a\right )}^{\frac {1}{3}}}{{\left (d x^{3} + c\right )} x^{11}} \,d x } \] Input:

integrate((b*x^3+a)^(1/3)/x^11/(d*x^3+c),x, algorithm="maxima")
 

Output:

integrate((b*x^3 + a)^(1/3)/((d*x^3 + c)*x^11), x)
 

Giac [F]

\[ \int \frac {\sqrt [3]{a+b x^3}}{x^{11} \left (c+d x^3\right )} \, dx=\int { \frac {{\left (b x^{3} + a\right )}^{\frac {1}{3}}}{{\left (d x^{3} + c\right )} x^{11}} \,d x } \] Input:

integrate((b*x^3+a)^(1/3)/x^11/(d*x^3+c),x, algorithm="giac")
 

Output:

integrate((b*x^3 + a)^(1/3)/((d*x^3 + c)*x^11), x)
 

Mupad [F(-1)]

Timed out. \[ \int \frac {\sqrt [3]{a+b x^3}}{x^{11} \left (c+d x^3\right )} \, dx=\int \frac {{\left (b\,x^3+a\right )}^{1/3}}{x^{11}\,\left (d\,x^3+c\right )} \,d x \] Input:

int((a + b*x^3)^(1/3)/(x^11*(c + d*x^3)),x)
 

Output:

int((a + b*x^3)^(1/3)/(x^11*(c + d*x^3)), x)
 

Reduce [F]

\[ \int \frac {\sqrt [3]{a+b x^3}}{x^{11} \left (c+d x^3\right )} \, dx=\frac {-14 \left (b \,x^{3}+a \right )^{\frac {1}{3}} a^{3} c^{2}+20 \left (b \,x^{3}+a \right )^{\frac {1}{3}} a^{3} c d \,x^{3}-35 \left (b \,x^{3}+a \right )^{\frac {1}{3}} a^{3} d^{2} x^{6}-2 \left (b \,x^{3}+a \right )^{\frac {1}{3}} a^{2} b \,c^{2} x^{3}+5 \left (b \,x^{3}+a \right )^{\frac {1}{3}} a^{2} b c d \,x^{6}+105 \left (b \,x^{3}+a \right )^{\frac {1}{3}} a^{2} b \,d^{2} x^{9}+3 \left (b \,x^{3}+a \right )^{\frac {1}{3}} a \,b^{2} c^{2} x^{6}-15 \left (b \,x^{3}+a \right )^{\frac {1}{3}} a \,b^{2} c d \,x^{9}-9 \left (b \,x^{3}+a \right )^{\frac {1}{3}} b^{3} c^{2} x^{9}-140 \left (\int \frac {\left (b \,x^{3}+a \right )^{\frac {1}{3}}}{b d \,x^{8}+a d \,x^{5}+b c \,x^{5}+a c \,x^{2}}d x \right ) a^{4} d^{3} x^{10}+140 \left (\int \frac {\left (b \,x^{3}+a \right )^{\frac {1}{3}}}{b d \,x^{8}+a d \,x^{5}+b c \,x^{5}+a c \,x^{2}}d x \right ) a^{3} b c \,d^{2} x^{10}}{140 a^{3} c^{3} x^{10}} \] Input:

int((b*x^3+a)^(1/3)/x^11/(d*x^3+c),x)
 

Output:

( - 14*(a + b*x**3)**(1/3)*a**3*c**2 + 20*(a + b*x**3)**(1/3)*a**3*c*d*x** 
3 - 35*(a + b*x**3)**(1/3)*a**3*d**2*x**6 - 2*(a + b*x**3)**(1/3)*a**2*b*c 
**2*x**3 + 5*(a + b*x**3)**(1/3)*a**2*b*c*d*x**6 + 105*(a + b*x**3)**(1/3) 
*a**2*b*d**2*x**9 + 3*(a + b*x**3)**(1/3)*a*b**2*c**2*x**6 - 15*(a + b*x** 
3)**(1/3)*a*b**2*c*d*x**9 - 9*(a + b*x**3)**(1/3)*b**3*c**2*x**9 - 140*int 
((a + b*x**3)**(1/3)/(a*c*x**2 + a*d*x**5 + b*c*x**5 + b*d*x**8),x)*a**4*d 
**3*x**10 + 140*int((a + b*x**3)**(1/3)/(a*c*x**2 + a*d*x**5 + b*c*x**5 + 
b*d*x**8),x)*a**3*b*c*d**2*x**10)/(140*a**3*c**3*x**10)