Integrand size = 17, antiderivative size = 146 \[ \int \frac {\sqrt {c+d x}}{(a+b x)^4} \, dx=-\frac {\sqrt {c+d x}}{3 b (a+b x)^3}-\frac {d \sqrt {c+d x}}{12 b (b c-a d) (a+b x)^2}+\frac {d^2 \sqrt {c+d x}}{8 b (b c-a d)^2 (a+b x)}-\frac {d^3 \text {arctanh}\left (\frac {\sqrt {b} \sqrt {c+d x}}{\sqrt {b c-a d}}\right )}{8 b^{3/2} (b c-a d)^{5/2}} \]
-1/8*d^3*arctanh(b^(1/2)*(d*x+c)^(1/2)/(-a*d+b*c)^(1/2))/b^(3/2)/(-a*d+b*c )^(5/2)-1/3*(d*x+c)^(1/2)/b/(b*x+a)^3-1/12*d*(d*x+c)^(1/2)/b/(-a*d+b*c)/(b *x+a)^2+1/8*d^2*(d*x+c)^(1/2)/b/(-a*d+b*c)^2/(b*x+a)
Time = 0.57 (sec) , antiderivative size = 131, normalized size of antiderivative = 0.90 \[ \int \frac {\sqrt {c+d x}}{(a+b x)^4} \, dx=\frac {\sqrt {c+d x} \left (-3 a^2 d^2+2 a b d (7 c+4 d x)+b^2 \left (-8 c^2-2 c d x+3 d^2 x^2\right )\right )}{24 b (b c-a d)^2 (a+b x)^3}+\frac {d^3 \arctan \left (\frac {\sqrt {b} \sqrt {c+d x}}{\sqrt {-b c+a d}}\right )}{8 b^{3/2} (-b c+a d)^{5/2}} \]
(Sqrt[c + d*x]*(-3*a^2*d^2 + 2*a*b*d*(7*c + 4*d*x) + b^2*(-8*c^2 - 2*c*d*x + 3*d^2*x^2)))/(24*b*(b*c - a*d)^2*(a + b*x)^3) + (d^3*ArcTan[(Sqrt[b]*Sq rt[c + d*x])/Sqrt[-(b*c) + a*d]])/(8*b^(3/2)*(-(b*c) + a*d)^(5/2))
Time = 0.23 (sec) , antiderivative size = 154, normalized size of antiderivative = 1.05, number of steps used = 6, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.294, Rules used = {51, 52, 52, 73, 221}
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.
| \(\displaystyle \int \frac {\sqrt {c+d x}}{(a+b x)^4} \, dx\) |
| \(\Big \downarrow \) 51 |
| \(\displaystyle \frac {d \int \frac {1}{(a+b x)^3 \sqrt {c+d x}}dx}{6 b}-\frac {\sqrt {c+d x}}{3 b (a+b x)^3}\) |
| \(\Big \downarrow \) 52 |
| \(\displaystyle \frac {d \left (-\frac {3 d \int \frac {1}{(a+b x)^2 \sqrt {c+d x}}dx}{4 (b c-a d)}-\frac {\sqrt {c+d x}}{2 (a+b x)^2 (b c-a d)}\right )}{6 b}-\frac {\sqrt {c+d x}}{3 b (a+b x)^3}\) |
| \(\Big \downarrow \) 52 |
| \(\displaystyle \frac {d \left (-\frac {3 d \left (-\frac {d \int \frac {1}{(a+b x) \sqrt {c+d x}}dx}{2 (b c-a d)}-\frac {\sqrt {c+d x}}{(a+b x) (b c-a d)}\right )}{4 (b c-a d)}-\frac {\sqrt {c+d x}}{2 (a+b x)^2 (b c-a d)}\right )}{6 b}-\frac {\sqrt {c+d x}}{3 b (a+b x)^3}\) |
| \(\Big \downarrow \) 73 |
| \(\displaystyle \frac {d \left (-\frac {3 d \left (-\frac {\int \frac {1}{a+\frac {b (c+d x)}{d}-\frac {b c}{d}}d\sqrt {c+d x}}{b c-a d}-\frac {\sqrt {c+d x}}{(a+b x) (b c-a d)}\right )}{4 (b c-a d)}-\frac {\sqrt {c+d x}}{2 (a+b x)^2 (b c-a d)}\right )}{6 b}-\frac {\sqrt {c+d x}}{3 b (a+b x)^3}\) |
| \(\Big \downarrow \) 221 |
| \(\displaystyle \frac {d \left (-\frac {3 d \left (\frac {d \text {arctanh}\left (\frac {\sqrt {b} \sqrt {c+d x}}{\sqrt {b c-a d}}\right )}{\sqrt {b} (b c-a d)^{3/2}}-\frac {\sqrt {c+d x}}{(a+b x) (b c-a d)}\right )}{4 (b c-a d)}-\frac {\sqrt {c+d x}}{2 (a+b x)^2 (b c-a d)}\right )}{6 b}-\frac {\sqrt {c+d x}}{3 b (a+b x)^3}\) |
-1/3*Sqrt[c + d*x]/(b*(a + b*x)^3) + (d*(-1/2*Sqrt[c + d*x]/((b*c - a*d)*( a + b*x)^2) - (3*d*(-(Sqrt[c + d*x]/((b*c - a*d)*(a + b*x))) + (d*ArcTanh[ (Sqrt[b]*Sqrt[c + d*x])/Sqrt[b*c - a*d]])/(Sqrt[b]*(b*c - a*d)^(3/2))))/(4 *(b*c - a*d))))/(6*b)
3.14.84.3.1 Defintions of rubi rules used
Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> Simp[ (a + b*x)^(m + 1)*((c + d*x)^n/(b*(m + 1))), x] - Simp[d*(n/(b*(m + 1))) Int[(a + b*x)^(m + 1)*(c + d*x)^(n - 1), x], x] /; FreeQ[{a, b, c, d, n}, x ] && ILtQ[m, -1] && FractionQ[n] && GtQ[n, 0]
Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> Simp[ (a + b*x)^(m + 1)*((c + d*x)^(n + 1)/((b*c - a*d)*(m + 1))), x] - Simp[d*(( m + n + 2)/((b*c - a*d)*(m + 1))) Int[(a + b*x)^(m + 1)*(c + d*x)^n, x], x] /; FreeQ[{a, b, c, d, n}, x] && ILtQ[m, -1] && FractionQ[n] && LtQ[n, 0]
Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> With[ {p = Denominator[m]}, Simp[p/b Subst[Int[x^(p*(m + 1) - 1)*(c - a*(d/b) + d*(x^p/b))^n, x], x, (a + b*x)^(1/p)], x]] /; FreeQ[{a, b, c, d}, x] && Lt Q[-1, m, 0] && LeQ[-1, n, 0] && LeQ[Denominator[n], Denominator[m]] && IntL inearQ[a, b, c, d, m, n, x]
Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(Rt[-a/b, 2]/a)*ArcTanh[x /Rt[-a/b, 2]], x] /; FreeQ[{a, b}, x] && NegQ[a/b]
Time = 0.53 (sec) , antiderivative size = 118, normalized size of antiderivative = 0.81
| method | result | size |
| pseudoelliptic | \(\frac {d^{3} \left (b x +a \right )^{3} \arctan \left (\frac {b \sqrt {d x +c}}{\sqrt {\left (a d -b c \right ) b}}\right )-\left (\frac {\left (d x -2 c \right ) b}{3}+a d \right ) \sqrt {d x +c}\, \left (\left (-3 d x -4 c \right ) b +a d \right ) \sqrt {\left (a d -b c \right ) b}}{8 \sqrt {\left (a d -b c \right ) b}\, \left (b x +a \right )^{3} \left (a d -b c \right )^{2} b}\) | \(118\) |
| derivativedivides | \(2 d^{3} \left (\frac {\frac {b \left (d x +c \right )^{\frac {5}{2}}}{16 a^{2} d^{2}-32 a b c d +16 b^{2} c^{2}}+\frac {\left (d x +c \right )^{\frac {3}{2}}}{6 a d -6 b c}-\frac {\sqrt {d x +c}}{16 b}}{\left (\left (d x +c \right ) b +a d -b c \right )^{3}}+\frac {\arctan \left (\frac {b \sqrt {d x +c}}{\sqrt {\left (a d -b c \right ) b}}\right )}{16 b \left (a^{2} d^{2}-2 a b c d +b^{2} c^{2}\right ) \sqrt {\left (a d -b c \right ) b}}\right )\) | \(152\) |
| default | \(2 d^{3} \left (\frac {\frac {b \left (d x +c \right )^{\frac {5}{2}}}{16 a^{2} d^{2}-32 a b c d +16 b^{2} c^{2}}+\frac {\left (d x +c \right )^{\frac {3}{2}}}{6 a d -6 b c}-\frac {\sqrt {d x +c}}{16 b}}{\left (\left (d x +c \right ) b +a d -b c \right )^{3}}+\frac {\arctan \left (\frac {b \sqrt {d x +c}}{\sqrt {\left (a d -b c \right ) b}}\right )}{16 b \left (a^{2} d^{2}-2 a b c d +b^{2} c^{2}\right ) \sqrt {\left (a d -b c \right ) b}}\right )\) | \(152\) |
1/8*(d^3*(b*x+a)^3*arctan(b*(d*x+c)^(1/2)/((a*d-b*c)*b)^(1/2))-(1/3*(d*x-2 *c)*b+a*d)*(d*x+c)^(1/2)*((-3*d*x-4*c)*b+a*d)*((a*d-b*c)*b)^(1/2))/((a*d-b *c)*b)^(1/2)/(b*x+a)^3/(a*d-b*c)^2/b
Leaf count of result is larger than twice the leaf count of optimal. 386 vs. \(2 (122) = 244\).
Time = 0.24 (sec) , antiderivative size = 785, normalized size of antiderivative = 5.38 \[ \int \frac {\sqrt {c+d x}}{(a+b x)^4} \, dx=\left [\frac {3 \, {\left (b^{3} d^{3} x^{3} + 3 \, a b^{2} d^{3} x^{2} + 3 \, a^{2} b d^{3} x + a^{3} d^{3}\right )} \sqrt {b^{2} c - a b d} \log \left (\frac {b d x + 2 \, b c - a d - 2 \, \sqrt {b^{2} c - a b d} \sqrt {d x + c}}{b x + a}\right ) - 2 \, {\left (8 \, b^{4} c^{3} - 22 \, a b^{3} c^{2} d + 17 \, a^{2} b^{2} c d^{2} - 3 \, a^{3} b d^{3} - 3 \, {\left (b^{4} c d^{2} - a b^{3} d^{3}\right )} x^{2} + 2 \, {\left (b^{4} c^{2} d - 5 \, a b^{3} c d^{2} + 4 \, a^{2} b^{2} d^{3}\right )} x\right )} \sqrt {d x + c}}{48 \, {\left (a^{3} b^{5} c^{3} - 3 \, a^{4} b^{4} c^{2} d + 3 \, a^{5} b^{3} c d^{2} - a^{6} b^{2} d^{3} + {\left (b^{8} c^{3} - 3 \, a b^{7} c^{2} d + 3 \, a^{2} b^{6} c d^{2} - a^{3} b^{5} d^{3}\right )} x^{3} + 3 \, {\left (a b^{7} c^{3} - 3 \, a^{2} b^{6} c^{2} d + 3 \, a^{3} b^{5} c d^{2} - a^{4} b^{4} d^{3}\right )} x^{2} + 3 \, {\left (a^{2} b^{6} c^{3} - 3 \, a^{3} b^{5} c^{2} d + 3 \, a^{4} b^{4} c d^{2} - a^{5} b^{3} d^{3}\right )} x\right )}}, \frac {3 \, {\left (b^{3} d^{3} x^{3} + 3 \, a b^{2} d^{3} x^{2} + 3 \, a^{2} b d^{3} x + a^{3} d^{3}\right )} \sqrt {-b^{2} c + a b d} \arctan \left (\frac {\sqrt {-b^{2} c + a b d} \sqrt {d x + c}}{b d x + b c}\right ) - {\left (8 \, b^{4} c^{3} - 22 \, a b^{3} c^{2} d + 17 \, a^{2} b^{2} c d^{2} - 3 \, a^{3} b d^{3} - 3 \, {\left (b^{4} c d^{2} - a b^{3} d^{3}\right )} x^{2} + 2 \, {\left (b^{4} c^{2} d - 5 \, a b^{3} c d^{2} + 4 \, a^{2} b^{2} d^{3}\right )} x\right )} \sqrt {d x + c}}{24 \, {\left (a^{3} b^{5} c^{3} - 3 \, a^{4} b^{4} c^{2} d + 3 \, a^{5} b^{3} c d^{2} - a^{6} b^{2} d^{3} + {\left (b^{8} c^{3} - 3 \, a b^{7} c^{2} d + 3 \, a^{2} b^{6} c d^{2} - a^{3} b^{5} d^{3}\right )} x^{3} + 3 \, {\left (a b^{7} c^{3} - 3 \, a^{2} b^{6} c^{2} d + 3 \, a^{3} b^{5} c d^{2} - a^{4} b^{4} d^{3}\right )} x^{2} + 3 \, {\left (a^{2} b^{6} c^{3} - 3 \, a^{3} b^{5} c^{2} d + 3 \, a^{4} b^{4} c d^{2} - a^{5} b^{3} d^{3}\right )} x\right )}}\right ] \]
[1/48*(3*(b^3*d^3*x^3 + 3*a*b^2*d^3*x^2 + 3*a^2*b*d^3*x + a^3*d^3)*sqrt(b^ 2*c - a*b*d)*log((b*d*x + 2*b*c - a*d - 2*sqrt(b^2*c - a*b*d)*sqrt(d*x + c ))/(b*x + a)) - 2*(8*b^4*c^3 - 22*a*b^3*c^2*d + 17*a^2*b^2*c*d^2 - 3*a^3*b *d^3 - 3*(b^4*c*d^2 - a*b^3*d^3)*x^2 + 2*(b^4*c^2*d - 5*a*b^3*c*d^2 + 4*a^ 2*b^2*d^3)*x)*sqrt(d*x + c))/(a^3*b^5*c^3 - 3*a^4*b^4*c^2*d + 3*a^5*b^3*c* d^2 - a^6*b^2*d^3 + (b^8*c^3 - 3*a*b^7*c^2*d + 3*a^2*b^6*c*d^2 - a^3*b^5*d ^3)*x^3 + 3*(a*b^7*c^3 - 3*a^2*b^6*c^2*d + 3*a^3*b^5*c*d^2 - a^4*b^4*d^3)* x^2 + 3*(a^2*b^6*c^3 - 3*a^3*b^5*c^2*d + 3*a^4*b^4*c*d^2 - a^5*b^3*d^3)*x) , 1/24*(3*(b^3*d^3*x^3 + 3*a*b^2*d^3*x^2 + 3*a^2*b*d^3*x + a^3*d^3)*sqrt(- b^2*c + a*b*d)*arctan(sqrt(-b^2*c + a*b*d)*sqrt(d*x + c)/(b*d*x + b*c)) - (8*b^4*c^3 - 22*a*b^3*c^2*d + 17*a^2*b^2*c*d^2 - 3*a^3*b*d^3 - 3*(b^4*c*d^ 2 - a*b^3*d^3)*x^2 + 2*(b^4*c^2*d - 5*a*b^3*c*d^2 + 4*a^2*b^2*d^3)*x)*sqrt (d*x + c))/(a^3*b^5*c^3 - 3*a^4*b^4*c^2*d + 3*a^5*b^3*c*d^2 - a^6*b^2*d^3 + (b^8*c^3 - 3*a*b^7*c^2*d + 3*a^2*b^6*c*d^2 - a^3*b^5*d^3)*x^3 + 3*(a*b^7 *c^3 - 3*a^2*b^6*c^2*d + 3*a^3*b^5*c*d^2 - a^4*b^4*d^3)*x^2 + 3*(a^2*b^6*c ^3 - 3*a^3*b^5*c^2*d + 3*a^4*b^4*c*d^2 - a^5*b^3*d^3)*x)]
Timed out. \[ \int \frac {\sqrt {c+d x}}{(a+b x)^4} \, dx=\text {Timed out} \]
Exception generated. \[ \int \frac {\sqrt {c+d x}}{(a+b x)^4} \, dx=\text {Exception raised: ValueError} \]
Exception raised: ValueError >> Computation failed since Maxima requested additional constraints; using the 'assume' command before evaluation *may* help (example of legal syntax is 'assume(a*d-b*c>0)', see `assume?` for m ore detail
Time = 0.30 (sec) , antiderivative size = 207, normalized size of antiderivative = 1.42 \[ \int \frac {\sqrt {c+d x}}{(a+b x)^4} \, dx=\frac {d^{3} \arctan \left (\frac {\sqrt {d x + c} b}{\sqrt {-b^{2} c + a b d}}\right )}{8 \, {\left (b^{3} c^{2} - 2 \, a b^{2} c d + a^{2} b d^{2}\right )} \sqrt {-b^{2} c + a b d}} + \frac {3 \, {\left (d x + c\right )}^{\frac {5}{2}} b^{2} d^{3} - 8 \, {\left (d x + c\right )}^{\frac {3}{2}} b^{2} c d^{3} - 3 \, \sqrt {d x + c} b^{2} c^{2} d^{3} + 8 \, {\left (d x + c\right )}^{\frac {3}{2}} a b d^{4} + 6 \, \sqrt {d x + c} a b c d^{4} - 3 \, \sqrt {d x + c} a^{2} d^{5}}{24 \, {\left (b^{3} c^{2} - 2 \, a b^{2} c d + a^{2} b d^{2}\right )} {\left ({\left (d x + c\right )} b - b c + a d\right )}^{3}} \]
1/8*d^3*arctan(sqrt(d*x + c)*b/sqrt(-b^2*c + a*b*d))/((b^3*c^2 - 2*a*b^2*c *d + a^2*b*d^2)*sqrt(-b^2*c + a*b*d)) + 1/24*(3*(d*x + c)^(5/2)*b^2*d^3 - 8*(d*x + c)^(3/2)*b^2*c*d^3 - 3*sqrt(d*x + c)*b^2*c^2*d^3 + 8*(d*x + c)^(3 /2)*a*b*d^4 + 6*sqrt(d*x + c)*a*b*c*d^4 - 3*sqrt(d*x + c)*a^2*d^5)/((b^3*c ^2 - 2*a*b^2*c*d + a^2*b*d^2)*((d*x + c)*b - b*c + a*d)^3)
Time = 0.59 (sec) , antiderivative size = 207, normalized size of antiderivative = 1.42 \[ \int \frac {\sqrt {c+d x}}{(a+b x)^4} \, dx=\frac {\frac {d^3\,{\left (c+d\,x\right )}^{3/2}}{3\,\left (a\,d-b\,c\right )}-\frac {d^3\,\sqrt {c+d\,x}}{8\,b}+\frac {b\,d^3\,{\left (c+d\,x\right )}^{5/2}}{8\,{\left (a\,d-b\,c\right )}^2}}{\left (c+d\,x\right )\,\left (3\,a^2\,b\,d^2-6\,a\,b^2\,c\,d+3\,b^3\,c^2\right )+b^3\,{\left (c+d\,x\right )}^3-\left (3\,b^3\,c-3\,a\,b^2\,d\right )\,{\left (c+d\,x\right )}^2+a^3\,d^3-b^3\,c^3+3\,a\,b^2\,c^2\,d-3\,a^2\,b\,c\,d^2}+\frac {d^3\,\mathrm {atan}\left (\frac {\sqrt {b}\,\sqrt {c+d\,x}}{\sqrt {a\,d-b\,c}}\right )}{8\,b^{3/2}\,{\left (a\,d-b\,c\right )}^{5/2}} \]
((d^3*(c + d*x)^(3/2))/(3*(a*d - b*c)) - (d^3*(c + d*x)^(1/2))/(8*b) + (b* d^3*(c + d*x)^(5/2))/(8*(a*d - b*c)^2))/((c + d*x)*(3*b^3*c^2 + 3*a^2*b*d^ 2 - 6*a*b^2*c*d) + b^3*(c + d*x)^3 - (3*b^3*c - 3*a*b^2*d)*(c + d*x)^2 + a ^3*d^3 - b^3*c^3 + 3*a*b^2*c^2*d - 3*a^2*b*c*d^2) + (d^3*atan((b^(1/2)*(c + d*x)^(1/2))/(a*d - b*c)^(1/2)))/(8*b^(3/2)*(a*d - b*c)^(5/2))
Time = 0.00 (sec) , antiderivative size = 536, normalized size of antiderivative = 3.67 \[ \int \frac {\sqrt {c+d x}}{(a+b x)^4} \, dx=\frac {3 \sqrt {b}\, \sqrt {a d -b c}\, \mathit {atan} \left (\frac {\sqrt {d x +c}\, b}{\sqrt {b}\, \sqrt {a d -b c}}\right ) a^{3} d^{3}+9 \sqrt {b}\, \sqrt {a d -b c}\, \mathit {atan} \left (\frac {\sqrt {d x +c}\, b}{\sqrt {b}\, \sqrt {a d -b c}}\right ) a^{2} b \,d^{3} x +9 \sqrt {b}\, \sqrt {a d -b c}\, \mathit {atan} \left (\frac {\sqrt {d x +c}\, b}{\sqrt {b}\, \sqrt {a d -b c}}\right ) a \,b^{2} d^{3} x^{2}+3 \sqrt {b}\, \sqrt {a d -b c}\, \mathit {atan} \left (\frac {\sqrt {d x +c}\, b}{\sqrt {b}\, \sqrt {a d -b c}}\right ) b^{3} d^{3} x^{3}-3 \sqrt {d x +c}\, a^{3} b \,d^{3}+17 \sqrt {d x +c}\, a^{2} b^{2} c \,d^{2}+8 \sqrt {d x +c}\, a^{2} b^{2} d^{3} x -22 \sqrt {d x +c}\, a \,b^{3} c^{2} d -10 \sqrt {d x +c}\, a \,b^{3} c \,d^{2} x +3 \sqrt {d x +c}\, a \,b^{3} d^{3} x^{2}+8 \sqrt {d x +c}\, b^{4} c^{3}+2 \sqrt {d x +c}\, b^{4} c^{2} d x -3 \sqrt {d x +c}\, b^{4} c \,d^{2} x^{2}}{24 b^{2} \left (a^{3} b^{3} d^{3} x^{3}-3 a^{2} b^{4} c \,d^{2} x^{3}+3 a \,b^{5} c^{2} d \,x^{3}-b^{6} c^{3} x^{3}+3 a^{4} b^{2} d^{3} x^{2}-9 a^{3} b^{3} c \,d^{2} x^{2}+9 a^{2} b^{4} c^{2} d \,x^{2}-3 a \,b^{5} c^{3} x^{2}+3 a^{5} b \,d^{3} x -9 a^{4} b^{2} c \,d^{2} x +9 a^{3} b^{3} c^{2} d x -3 a^{2} b^{4} c^{3} x +a^{6} d^{3}-3 a^{5} b c \,d^{2}+3 a^{4} b^{2} c^{2} d -a^{3} b^{3} c^{3}\right )} \]
(3*sqrt(b)*sqrt(a*d - b*c)*atan((sqrt(c + d*x)*b)/(sqrt(b)*sqrt(a*d - b*c) ))*a**3*d**3 + 9*sqrt(b)*sqrt(a*d - b*c)*atan((sqrt(c + d*x)*b)/(sqrt(b)*s qrt(a*d - b*c)))*a**2*b*d**3*x + 9*sqrt(b)*sqrt(a*d - b*c)*atan((sqrt(c + d*x)*b)/(sqrt(b)*sqrt(a*d - b*c)))*a*b**2*d**3*x**2 + 3*sqrt(b)*sqrt(a*d - b*c)*atan((sqrt(c + d*x)*b)/(sqrt(b)*sqrt(a*d - b*c)))*b**3*d**3*x**3 - 3 *sqrt(c + d*x)*a**3*b*d**3 + 17*sqrt(c + d*x)*a**2*b**2*c*d**2 + 8*sqrt(c + d*x)*a**2*b**2*d**3*x - 22*sqrt(c + d*x)*a*b**3*c**2*d - 10*sqrt(c + d*x )*a*b**3*c*d**2*x + 3*sqrt(c + d*x)*a*b**3*d**3*x**2 + 8*sqrt(c + d*x)*b** 4*c**3 + 2*sqrt(c + d*x)*b**4*c**2*d*x - 3*sqrt(c + d*x)*b**4*c*d**2*x**2) /(24*b**2*(a**6*d**3 - 3*a**5*b*c*d**2 + 3*a**5*b*d**3*x + 3*a**4*b**2*c** 2*d - 9*a**4*b**2*c*d**2*x + 3*a**4*b**2*d**3*x**2 - a**3*b**3*c**3 + 9*a* *3*b**3*c**2*d*x - 9*a**3*b**3*c*d**2*x**2 + a**3*b**3*d**3*x**3 - 3*a**2* b**4*c**3*x + 9*a**2*b**4*c**2*d*x**2 - 3*a**2*b**4*c*d**2*x**3 - 3*a*b**5 *c**3*x**2 + 3*a*b**5*c**2*d*x**3 - b**6*c**3*x**3))