Integrand size = 23, antiderivative size = 64 \[ \int \frac {\sqrt {c+d x^3}}{4 c+d x^3} \, dx=\frac {x \sqrt {c+d x^3} \operatorname {AppellF1}\left (\frac {1}{3},1,-\frac {1}{2},\frac {4}{3},-\frac {d x^3}{4 c},-\frac {d x^3}{c}\right )}{4 c \sqrt {1+\frac {d x^3}{c}}} \] Output:
1/4*x*(d*x^3+c)^(1/2)*AppellF1(1/3,-1/2,1,4/3,-d*x^3/c,-1/4*d*x^3/c)/c/(1+ d*x^3/c)^(1/2)
Leaf count is larger than twice the leaf count of optimal. \(165\) vs. \(2(64)=128\).
Time = 10.18 (sec) , antiderivative size = 165, normalized size of antiderivative = 2.58 \[ \int \frac {\sqrt {c+d x^3}}{4 c+d x^3} \, dx=\frac {16 c x \sqrt {c+d x^3} \operatorname {AppellF1}\left (\frac {1}{3},-\frac {1}{2},1,\frac {4}{3},-\frac {d x^3}{c},-\frac {d x^3}{4 c}\right )}{\left (4 c+d x^3\right ) \left (16 c \operatorname {AppellF1}\left (\frac {1}{3},-\frac {1}{2},1,\frac {4}{3},-\frac {d x^3}{c},-\frac {d x^3}{4 c}\right )-3 d x^3 \left (\operatorname {AppellF1}\left (\frac {4}{3},-\frac {1}{2},2,\frac {7}{3},-\frac {d x^3}{c},-\frac {d x^3}{4 c}\right )-2 \operatorname {AppellF1}\left (\frac {4}{3},\frac {1}{2},1,\frac {7}{3},-\frac {d x^3}{c},-\frac {d x^3}{4 c}\right )\right )\right )} \] Input:
Integrate[Sqrt[c + d*x^3]/(4*c + d*x^3),x]
Output:
(16*c*x*Sqrt[c + d*x^3]*AppellF1[1/3, -1/2, 1, 4/3, -((d*x^3)/c), -1/4*(d* x^3)/c])/((4*c + d*x^3)*(16*c*AppellF1[1/3, -1/2, 1, 4/3, -((d*x^3)/c), -1 /4*(d*x^3)/c] - 3*d*x^3*(AppellF1[4/3, -1/2, 2, 7/3, -((d*x^3)/c), -1/4*(d *x^3)/c] - 2*AppellF1[4/3, 1/2, 1, 7/3, -((d*x^3)/c), -1/4*(d*x^3)/c])))
Time = 0.31 (sec) , antiderivative size = 64, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.087, Rules used = {937, 936}
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.
| \(\displaystyle \int \frac {\sqrt {c+d x^3}}{4 c+d x^3} \, dx\) |
| \(\Big \downarrow \) 937 |
| \(\displaystyle \frac {\sqrt {c+d x^3} \int \frac {\sqrt {\frac {d x^3}{c}+1}}{d x^3+4 c}dx}{\sqrt {\frac {d x^3}{c}+1}}\) |
| \(\Big \downarrow \) 936 |
| \(\displaystyle \frac {x \sqrt {c+d x^3} \operatorname {AppellF1}\left (\frac {1}{3},1,-\frac {1}{2},\frac {4}{3},-\frac {d x^3}{4 c},-\frac {d x^3}{c}\right )}{4 c \sqrt {\frac {d x^3}{c}+1}}\) |
Input:
Int[Sqrt[c + d*x^3]/(4*c + d*x^3),x]
Output:
(x*Sqrt[c + d*x^3]*AppellF1[1/3, 1, -1/2, 4/3, -1/4*(d*x^3)/c, -((d*x^3)/c )])/(4*c*Sqrt[1 + (d*x^3)/c])
Int[((a_) + (b_.)*(x_)^(n_))^(p_)*((c_) + (d_.)*(x_)^(n_))^(q_), x_Symbol] :> Simp[a^p*c^q*x*AppellF1[1/n, -p, -q, 1 + 1/n, (-b)*(x^n/a), (-d)*(x^n/c) ], x] /; FreeQ[{a, b, c, d, n, p, q}, x] && NeQ[b*c - a*d, 0] && NeQ[n, -1] && (IntegerQ[p] || GtQ[a, 0]) && (IntegerQ[q] || GtQ[c, 0])
Int[((a_) + (b_.)*(x_)^(n_))^(p_)*((c_) + (d_.)*(x_)^(n_))^(q_), x_Symbol] :> Simp[a^IntPart[p]*((a + b*x^n)^FracPart[p]/(1 + b*(x^n/a))^FracPart[p]) Int[(1 + b*(x^n/a))^p*(c + d*x^n)^q, x], x] /; FreeQ[{a, b, c, d, n, p, q }, x] && NeQ[b*c - a*d, 0] && NeQ[n, -1] && !(IntegerQ[p] || GtQ[a, 0])
Result contains higher order function than in optimal. Order 9 vs. order 6.
Time = 1.02 (sec) , antiderivative size = 696, normalized size of antiderivative = 10.88
| method | result | size |
| default | \(-\frac {2 i \sqrt {3}\, \left (-c \,d^{2}\right )^{\frac {1}{3}} \sqrt {\frac {i \left (x +\frac {\left (-c \,d^{2}\right )^{\frac {1}{3}}}{2 d}-\frac {i \sqrt {3}\, \left (-c \,d^{2}\right )^{\frac {1}{3}}}{2 d}\right ) \sqrt {3}\, d}{\left (-c \,d^{2}\right )^{\frac {1}{3}}}}\, \sqrt {\frac {x -\frac {\left (-c \,d^{2}\right )^{\frac {1}{3}}}{d}}{-\frac {3 \left (-c \,d^{2}\right )^{\frac {1}{3}}}{2 d}+\frac {i \sqrt {3}\, \left (-c \,d^{2}\right )^{\frac {1}{3}}}{2 d}}}\, \sqrt {-\frac {i \left (x +\frac {\left (-c \,d^{2}\right )^{\frac {1}{3}}}{2 d}+\frac {i \sqrt {3}\, \left (-c \,d^{2}\right )^{\frac {1}{3}}}{2 d}\right ) \sqrt {3}\, d}{\left (-c \,d^{2}\right )^{\frac {1}{3}}}}\, \operatorname {EllipticF}\left (\frac {\sqrt {3}\, \sqrt {\frac {i \left (x +\frac {\left (-c \,d^{2}\right )^{\frac {1}{3}}}{2 d}-\frac {i \sqrt {3}\, \left (-c \,d^{2}\right )^{\frac {1}{3}}}{2 d}\right ) \sqrt {3}\, d}{\left (-c \,d^{2}\right )^{\frac {1}{3}}}}}{3}, \sqrt {\frac {i \sqrt {3}\, \left (-c \,d^{2}\right )^{\frac {1}{3}}}{d \left (-\frac {3 \left (-c \,d^{2}\right )^{\frac {1}{3}}}{2 d}+\frac {i \sqrt {3}\, \left (-c \,d^{2}\right )^{\frac {1}{3}}}{2 d}\right )}}\right )}{3 d \sqrt {d \,x^{3}+c}}+\frac {i \sqrt {2}\, \left (\munderset {\underline {\hspace {1.25 ex}}\alpha =\operatorname {RootOf}\left (d \,\textit {\_Z}^{3}+4 c \right )}{\sum }\frac {\left (-c \,d^{2}\right )^{\frac {1}{3}} \sqrt {2}\, \sqrt {\frac {i d \left (2 x +\frac {-i \sqrt {3}\, \left (-c \,d^{2}\right )^{\frac {1}{3}}+\left (-c \,d^{2}\right )^{\frac {1}{3}}}{d}\right )}{\left (-c \,d^{2}\right )^{\frac {1}{3}}}}\, \sqrt {\frac {d \left (x -\frac {\left (-c \,d^{2}\right )^{\frac {1}{3}}}{d}\right )}{-3 \left (-c \,d^{2}\right )^{\frac {1}{3}}+i \sqrt {3}\, \left (-c \,d^{2}\right )^{\frac {1}{3}}}}\, \sqrt {-\frac {i d \left (2 x +\frac {i \sqrt {3}\, \left (-c \,d^{2}\right )^{\frac {1}{3}}+\left (-c \,d^{2}\right )^{\frac {1}{3}}}{d}\right )}{2 \left (-c \,d^{2}\right )^{\frac {1}{3}}}}\, \left (i \left (-c \,d^{2}\right )^{\frac {1}{3}} \underline {\hspace {1.25 ex}}\alpha \sqrt {3}\, d -i \sqrt {3}\, \left (-c \,d^{2}\right )^{\frac {2}{3}}+2 \underline {\hspace {1.25 ex}}\alpha ^{2} d^{2}-\left (-c \,d^{2}\right )^{\frac {1}{3}} \underline {\hspace {1.25 ex}}\alpha d -\left (-c \,d^{2}\right )^{\frac {2}{3}}\right ) \operatorname {EllipticPi}\left (\frac {\sqrt {3}\, \sqrt {\frac {i \left (x +\frac {\left (-c \,d^{2}\right )^{\frac {1}{3}}}{2 d}-\frac {i \sqrt {3}\, \left (-c \,d^{2}\right )^{\frac {1}{3}}}{2 d}\right ) \sqrt {3}\, d}{\left (-c \,d^{2}\right )^{\frac {1}{3}}}}}{3}, \frac {2 i \left (-c \,d^{2}\right )^{\frac {1}{3}} \underline {\hspace {1.25 ex}}\alpha ^{2} \sqrt {3}\, d -i \left (-c \,d^{2}\right )^{\frac {2}{3}} \underline {\hspace {1.25 ex}}\alpha \sqrt {3}+i \sqrt {3}\, c d -3 \left (-c \,d^{2}\right )^{\frac {2}{3}} \underline {\hspace {1.25 ex}}\alpha -3 c d}{6 d c}, \sqrt {\frac {i \sqrt {3}\, \left (-c \,d^{2}\right )^{\frac {1}{3}}}{d \left (-\frac {3 \left (-c \,d^{2}\right )^{\frac {1}{3}}}{2 d}+\frac {i \sqrt {3}\, \left (-c \,d^{2}\right )^{\frac {1}{3}}}{2 d}\right )}}\right )}{2 \underline {\hspace {1.25 ex}}\alpha ^{2} \sqrt {d \,x^{3}+c}}\right )}{3 d^{3}}\) | \(696\) |
| elliptic | \(-\frac {2 i \sqrt {3}\, \left (-c \,d^{2}\right )^{\frac {1}{3}} \sqrt {\frac {i \left (x +\frac {\left (-c \,d^{2}\right )^{\frac {1}{3}}}{2 d}-\frac {i \sqrt {3}\, \left (-c \,d^{2}\right )^{\frac {1}{3}}}{2 d}\right ) \sqrt {3}\, d}{\left (-c \,d^{2}\right )^{\frac {1}{3}}}}\, \sqrt {\frac {x -\frac {\left (-c \,d^{2}\right )^{\frac {1}{3}}}{d}}{-\frac {3 \left (-c \,d^{2}\right )^{\frac {1}{3}}}{2 d}+\frac {i \sqrt {3}\, \left (-c \,d^{2}\right )^{\frac {1}{3}}}{2 d}}}\, \sqrt {-\frac {i \left (x +\frac {\left (-c \,d^{2}\right )^{\frac {1}{3}}}{2 d}+\frac {i \sqrt {3}\, \left (-c \,d^{2}\right )^{\frac {1}{3}}}{2 d}\right ) \sqrt {3}\, d}{\left (-c \,d^{2}\right )^{\frac {1}{3}}}}\, \operatorname {EllipticF}\left (\frac {\sqrt {3}\, \sqrt {\frac {i \left (x +\frac {\left (-c \,d^{2}\right )^{\frac {1}{3}}}{2 d}-\frac {i \sqrt {3}\, \left (-c \,d^{2}\right )^{\frac {1}{3}}}{2 d}\right ) \sqrt {3}\, d}{\left (-c \,d^{2}\right )^{\frac {1}{3}}}}}{3}, \sqrt {\frac {i \sqrt {3}\, \left (-c \,d^{2}\right )^{\frac {1}{3}}}{d \left (-\frac {3 \left (-c \,d^{2}\right )^{\frac {1}{3}}}{2 d}+\frac {i \sqrt {3}\, \left (-c \,d^{2}\right )^{\frac {1}{3}}}{2 d}\right )}}\right )}{3 d \sqrt {d \,x^{3}+c}}+\frac {i \sqrt {2}\, \left (\munderset {\underline {\hspace {1.25 ex}}\alpha =\operatorname {RootOf}\left (d \,\textit {\_Z}^{3}+4 c \right )}{\sum }\frac {\left (-c \,d^{2}\right )^{\frac {1}{3}} \sqrt {2}\, \sqrt {\frac {i d \left (2 x +\frac {-i \sqrt {3}\, \left (-c \,d^{2}\right )^{\frac {1}{3}}+\left (-c \,d^{2}\right )^{\frac {1}{3}}}{d}\right )}{\left (-c \,d^{2}\right )^{\frac {1}{3}}}}\, \sqrt {\frac {d \left (x -\frac {\left (-c \,d^{2}\right )^{\frac {1}{3}}}{d}\right )}{-3 \left (-c \,d^{2}\right )^{\frac {1}{3}}+i \sqrt {3}\, \left (-c \,d^{2}\right )^{\frac {1}{3}}}}\, \sqrt {-\frac {i d \left (2 x +\frac {i \sqrt {3}\, \left (-c \,d^{2}\right )^{\frac {1}{3}}+\left (-c \,d^{2}\right )^{\frac {1}{3}}}{d}\right )}{2 \left (-c \,d^{2}\right )^{\frac {1}{3}}}}\, \left (i \left (-c \,d^{2}\right )^{\frac {1}{3}} \underline {\hspace {1.25 ex}}\alpha \sqrt {3}\, d -i \sqrt {3}\, \left (-c \,d^{2}\right )^{\frac {2}{3}}+2 \underline {\hspace {1.25 ex}}\alpha ^{2} d^{2}-\left (-c \,d^{2}\right )^{\frac {1}{3}} \underline {\hspace {1.25 ex}}\alpha d -\left (-c \,d^{2}\right )^{\frac {2}{3}}\right ) \operatorname {EllipticPi}\left (\frac {\sqrt {3}\, \sqrt {\frac {i \left (x +\frac {\left (-c \,d^{2}\right )^{\frac {1}{3}}}{2 d}-\frac {i \sqrt {3}\, \left (-c \,d^{2}\right )^{\frac {1}{3}}}{2 d}\right ) \sqrt {3}\, d}{\left (-c \,d^{2}\right )^{\frac {1}{3}}}}}{3}, \frac {2 i \left (-c \,d^{2}\right )^{\frac {1}{3}} \underline {\hspace {1.25 ex}}\alpha ^{2} \sqrt {3}\, d -i \left (-c \,d^{2}\right )^{\frac {2}{3}} \underline {\hspace {1.25 ex}}\alpha \sqrt {3}+i \sqrt {3}\, c d -3 \left (-c \,d^{2}\right )^{\frac {2}{3}} \underline {\hspace {1.25 ex}}\alpha -3 c d}{6 d c}, \sqrt {\frac {i \sqrt {3}\, \left (-c \,d^{2}\right )^{\frac {1}{3}}}{d \left (-\frac {3 \left (-c \,d^{2}\right )^{\frac {1}{3}}}{2 d}+\frac {i \sqrt {3}\, \left (-c \,d^{2}\right )^{\frac {1}{3}}}{2 d}\right )}}\right )}{2 \underline {\hspace {1.25 ex}}\alpha ^{2} \sqrt {d \,x^{3}+c}}\right )}{3 d^{3}}\) | \(696\) |
Input:
int((d*x^3+c)^(1/2)/(d*x^3+4*c),x,method=_RETURNVERBOSE)
Output:
-2/3*I*3^(1/2)/d*(-c*d^2)^(1/3)*(I*(x+1/2/d*(-c*d^2)^(1/3)-1/2*I*3^(1/2)/d *(-c*d^2)^(1/3))*3^(1/2)*d/(-c*d^2)^(1/3))^(1/2)*((x-1/d*(-c*d^2)^(1/3))/( -3/2/d*(-c*d^2)^(1/3)+1/2*I*3^(1/2)/d*(-c*d^2)^(1/3)))^(1/2)*(-I*(x+1/2/d* (-c*d^2)^(1/3)+1/2*I*3^(1/2)/d*(-c*d^2)^(1/3))*3^(1/2)*d/(-c*d^2)^(1/3))^( 1/2)/(d*x^3+c)^(1/2)*EllipticF(1/3*3^(1/2)*(I*(x+1/2/d*(-c*d^2)^(1/3)-1/2* I*3^(1/2)/d*(-c*d^2)^(1/3))*3^(1/2)*d/(-c*d^2)^(1/3))^(1/2),(I*3^(1/2)/d*( -c*d^2)^(1/3)/(-3/2/d*(-c*d^2)^(1/3)+1/2*I*3^(1/2)/d*(-c*d^2)^(1/3)))^(1/2 ))+1/3*I/d^3*2^(1/2)*sum(1/_alpha^2*(-c*d^2)^(1/3)*(1/2*I*d*(2*x+1/d*(-I*3 ^(1/2)*(-c*d^2)^(1/3)+(-c*d^2)^(1/3)))/(-c*d^2)^(1/3))^(1/2)*(d*(x-1/d*(-c *d^2)^(1/3))/(-3*(-c*d^2)^(1/3)+I*3^(1/2)*(-c*d^2)^(1/3)))^(1/2)*(-1/2*I*d *(2*x+1/d*(I*3^(1/2)*(-c*d^2)^(1/3)+(-c*d^2)^(1/3)))/(-c*d^2)^(1/3))^(1/2) /(d*x^3+c)^(1/2)*(I*(-c*d^2)^(1/3)*_alpha*3^(1/2)*d-I*3^(1/2)*(-c*d^2)^(2/ 3)+2*_alpha^2*d^2-(-c*d^2)^(1/3)*_alpha*d-(-c*d^2)^(2/3))*EllipticPi(1/3*3 ^(1/2)*(I*(x+1/2/d*(-c*d^2)^(1/3)-1/2*I*3^(1/2)/d*(-c*d^2)^(1/3))*3^(1/2)* d/(-c*d^2)^(1/3))^(1/2),1/6/d*(2*I*(-c*d^2)^(1/3)*_alpha^2*3^(1/2)*d-I*(-c *d^2)^(2/3)*_alpha*3^(1/2)+I*3^(1/2)*c*d-3*(-c*d^2)^(2/3)*_alpha-3*c*d)/c, (I*3^(1/2)/d*(-c*d^2)^(1/3)/(-3/2/d*(-c*d^2)^(1/3)+1/2*I*3^(1/2)/d*(-c*d^2 )^(1/3)))^(1/2)),_alpha=RootOf(_Z^3*d+4*c))
Leaf count of result is larger than twice the leaf count of optimal. 2240 vs. \(2 (50) = 100\).
Time = 0.57 (sec) , antiderivative size = 2240, normalized size of antiderivative = 35.00 \[ \int \frac {\sqrt {c+d x^3}}{4 c+d x^3} \, dx=\text {Too large to display} \] Input:
integrate((d*x^3+c)^(1/2)/(d*x^3+4*c),x, algorithm="fricas")
Output:
1/24*((1/108)^(1/6)*(sqrt(-3)*d + d)*(-1/(c*d^2))^(1/6)*log((d^3*x^9 - 66* c*d^2*x^6 - 72*c^2*d*x^3 - 32*c^3 + 12*(1/4)^(2/3)*(c*d^4*x^8 - 7*c^2*d^3* x^5 - 8*c^3*d^2*x^2 + sqrt(-3)*(c*d^4*x^8 - 7*c^2*d^3*x^5 - 8*c^3*d^2*x^2) )*(-1/(c*d^2))^(2/3) + 24*(1/4)^(1/3)*(c*d^3*x^7 - c^2*d^2*x^4 - 2*c^3*d*x - sqrt(-3)*(c*d^3*x^7 - c^2*d^2*x^4 - 2*c^3*d*x))*(-1/(c*d^2))^(1/3) + 6* sqrt(d*x^3 + c)*(18*(1/108)^(5/6)*(c*d^4*x^7 - 16*c^2*d^3*x^4 - 8*c^3*d^2* x - sqrt(-3)*(c*d^4*x^7 - 16*c^2*d^3*x^4 - 8*c^3*d^2*x))*(-1/(c*d^2))^(5/6 ) - sqrt(1/3)*(5*c*d^3*x^6 - 20*c^2*d^2*x^3 - 16*c^3*d)*sqrt(-1/(c*d^2)) + 9*(1/108)^(1/6)*(sqrt(-3)*c*d^2*x^5 + c*d^2*x^5)*(-1/(c*d^2))^(1/6)))/(d^ 3*x^9 + 12*c*d^2*x^6 + 48*c^2*d*x^3 + 64*c^3)) - (1/108)^(1/6)*(sqrt(-3)*d + d)*(-1/(c*d^2))^(1/6)*log((d^3*x^9 - 66*c*d^2*x^6 - 72*c^2*d*x^3 - 32*c ^3 + 12*(1/4)^(2/3)*(c*d^4*x^8 - 7*c^2*d^3*x^5 - 8*c^3*d^2*x^2 + sqrt(-3)* (c*d^4*x^8 - 7*c^2*d^3*x^5 - 8*c^3*d^2*x^2))*(-1/(c*d^2))^(2/3) + 24*(1/4) ^(1/3)*(c*d^3*x^7 - c^2*d^2*x^4 - 2*c^3*d*x - sqrt(-3)*(c*d^3*x^7 - c^2*d^ 2*x^4 - 2*c^3*d*x))*(-1/(c*d^2))^(1/3) - 6*sqrt(d*x^3 + c)*(18*(1/108)^(5/ 6)*(c*d^4*x^7 - 16*c^2*d^3*x^4 - 8*c^3*d^2*x - sqrt(-3)*(c*d^4*x^7 - 16*c^ 2*d^3*x^4 - 8*c^3*d^2*x))*(-1/(c*d^2))^(5/6) - sqrt(1/3)*(5*c*d^3*x^6 - 20 *c^2*d^2*x^3 - 16*c^3*d)*sqrt(-1/(c*d^2)) + 9*(1/108)^(1/6)*(sqrt(-3)*c*d^ 2*x^5 + c*d^2*x^5)*(-1/(c*d^2))^(1/6)))/(d^3*x^9 + 12*c*d^2*x^6 + 48*c^2*d *x^3 + 64*c^3)) - (1/108)^(1/6)*(sqrt(-3)*d - d)*(-1/(c*d^2))^(1/6)*log...
\[ \int \frac {\sqrt {c+d x^3}}{4 c+d x^3} \, dx=\int \frac {\sqrt {c + d x^{3}}}{4 c + d x^{3}}\, dx \] Input:
integrate((d*x**3+c)**(1/2)/(d*x**3+4*c),x)
Output:
Integral(sqrt(c + d*x**3)/(4*c + d*x**3), x)
\[ \int \frac {\sqrt {c+d x^3}}{4 c+d x^3} \, dx=\int { \frac {\sqrt {d x^{3} + c}}{d x^{3} + 4 \, c} \,d x } \] Input:
integrate((d*x^3+c)^(1/2)/(d*x^3+4*c),x, algorithm="maxima")
Output:
integrate(sqrt(d*x^3 + c)/(d*x^3 + 4*c), x)
\[ \int \frac {\sqrt {c+d x^3}}{4 c+d x^3} \, dx=\int { \frac {\sqrt {d x^{3} + c}}{d x^{3} + 4 \, c} \,d x } \] Input:
integrate((d*x^3+c)^(1/2)/(d*x^3+4*c),x, algorithm="giac")
Output:
integrate(sqrt(d*x^3 + c)/(d*x^3 + 4*c), x)
Timed out. \[ \int \frac {\sqrt {c+d x^3}}{4 c+d x^3} \, dx=\int \frac {\sqrt {d\,x^3+c}}{d\,x^3+4\,c} \,d x \] Input:
int((c + d*x^3)^(1/2)/(4*c + d*x^3),x)
Output:
int((c + d*x^3)^(1/2)/(4*c + d*x^3), x)
\[ \int \frac {\sqrt {c+d x^3}}{4 c+d x^3} \, dx=\int \frac {\sqrt {d \,x^{3}+c}}{d \,x^{3}+4 c}d x \] Input:
int((d*x^3+c)^(1/2)/(d*x^3+4*c),x)
Output:
int(sqrt(c + d*x**3)/(4*c + d*x**3),x)