Integrand size = 19, antiderivative size = 101 \[ \int \frac {(c+d x)^{5/2}}{(a+b x)^{13/2}} \, dx=-\frac {2 (c+d x)^{7/2}}{11 (b c-a d) (a+b x)^{11/2}}+\frac {8 d (c+d x)^{7/2}}{99 (b c-a d)^2 (a+b x)^{9/2}}-\frac {16 d^2 (c+d x)^{7/2}}{693 (b c-a d)^3 (a+b x)^{7/2}} \]
-2/11*(d*x+c)^(7/2)/(-a*d+b*c)/(b*x+a)^(11/2)+8/99*d*(d*x+c)^(7/2)/(-a*d+b *c)^2/(b*x+a)^(9/2)-16/693*d^2*(d*x+c)^(7/2)/(-a*d+b*c)^3/(b*x+a)^(7/2)
Time = 0.22 (sec) , antiderivative size = 77, normalized size of antiderivative = 0.76 \[ \int \frac {(c+d x)^{5/2}}{(a+b x)^{13/2}} \, dx=-\frac {2 (c+d x)^{7/2} \left (99 a^2 d^2+22 a b d (-7 c+2 d x)+b^2 \left (63 c^2-28 c d x+8 d^2 x^2\right )\right )}{693 (b c-a d)^3 (a+b x)^{11/2}} \]
(-2*(c + d*x)^(7/2)*(99*a^2*d^2 + 22*a*b*d*(-7*c + 2*d*x) + b^2*(63*c^2 - 28*c*d*x + 8*d^2*x^2)))/(693*(b*c - a*d)^3*(a + b*x)^(11/2))
Time = 0.18 (sec) , antiderivative size = 114, normalized size of antiderivative = 1.13, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.158, Rules used = {55, 55, 48}
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.
| \(\displaystyle \int \frac {(c+d x)^{5/2}}{(a+b x)^{13/2}} \, dx\) |
| \(\Big \downarrow \) 55 |
| \(\displaystyle -\frac {4 d \int \frac {(c+d x)^{5/2}}{(a+b x)^{11/2}}dx}{11 (b c-a d)}-\frac {2 (c+d x)^{7/2}}{11 (a+b x)^{11/2} (b c-a d)}\) |
| \(\Big \downarrow \) 55 |
| \(\displaystyle -\frac {4 d \left (-\frac {2 d \int \frac {(c+d x)^{5/2}}{(a+b x)^{9/2}}dx}{9 (b c-a d)}-\frac {2 (c+d x)^{7/2}}{9 (a+b x)^{9/2} (b c-a d)}\right )}{11 (b c-a d)}-\frac {2 (c+d x)^{7/2}}{11 (a+b x)^{11/2} (b c-a d)}\) |
| \(\Big \downarrow \) 48 |
| \(\displaystyle -\frac {2 (c+d x)^{7/2}}{11 (a+b x)^{11/2} (b c-a d)}-\frac {4 d \left (\frac {4 d (c+d x)^{7/2}}{63 (a+b x)^{7/2} (b c-a d)^2}-\frac {2 (c+d x)^{7/2}}{9 (a+b x)^{9/2} (b c-a d)}\right )}{11 (b c-a d)}\) |
(-2*(c + d*x)^(7/2))/(11*(b*c - a*d)*(a + b*x)^(11/2)) - (4*d*((-2*(c + d* x)^(7/2))/(9*(b*c - a*d)*(a + b*x)^(9/2)) + (4*d*(c + d*x)^(7/2))/(63*(b*c - a*d)^2*(a + b*x)^(7/2))))/(11*(b*c - a*d))
3.15.90.3.1 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 + 1)/((b*c - a*d)*(m + 1))), x] /; FreeQ[{ a, b, c, d, m, n}, x] && EqQ[m + n + 2, 0] && NeQ[m, -1]
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*(S implify[m + n + 2]/((b*c - a*d)*(m + 1))) Int[(a + b*x)^Simplify[m + 1]*( c + d*x)^n, x], x] /; FreeQ[{a, b, c, d, m, n}, x] && ILtQ[Simplify[m + n + 2], 0] && NeQ[m, -1] && !(LtQ[m, -1] && LtQ[n, -1] && (EqQ[a, 0] || (NeQ[ c, 0] && LtQ[m - n, 0] && IntegerQ[n]))) && (SumSimplerQ[m, 1] || !SumSimp lerQ[n, 1])
Time = 0.26 (sec) , antiderivative size = 105, normalized size of antiderivative = 1.04
| method | result | size |
| gosper | \(\frac {2 \left (d x +c \right )^{\frac {7}{2}} \left (8 d^{2} x^{2} b^{2}+44 x a b \,d^{2}-28 x \,b^{2} c d +99 a^{2} d^{2}-154 a b c d +63 b^{2} c^{2}\right )}{693 \left (b x +a \right )^{\frac {11}{2}} \left (a^{3} d^{3}-3 a^{2} b c \,d^{2}+3 a \,b^{2} c^{2} d -b^{3} c^{3}\right )}\) | \(105\) |
| default | \(-\frac {\left (d x +c \right )^{\frac {5}{2}}}{3 b \left (b x +a \right )^{\frac {11}{2}}}+\frac {5 \left (a d -b c \right ) \left (-\frac {\left (d x +c \right )^{\frac {3}{2}}}{4 b \left (b x +a \right )^{\frac {11}{2}}}+\frac {3 \left (a d -b c \right ) \left (-\frac {\sqrt {d x +c}}{5 b \left (b x +a \right )^{\frac {11}{2}}}+\frac {\left (a d -b c \right ) \left (-\frac {2 \sqrt {d x +c}}{11 \left (-a d +b c \right ) \left (b x +a \right )^{\frac {11}{2}}}-\frac {10 d \left (-\frac {2 \sqrt {d x +c}}{9 \left (-a d +b c \right ) \left (b x +a \right )^{\frac {9}{2}}}-\frac {8 d \left (-\frac {2 \sqrt {d x +c}}{7 \left (-a d +b c \right ) \left (b x +a \right )^{\frac {7}{2}}}-\frac {6 d \left (-\frac {2 \sqrt {d x +c}}{5 \left (-a d +b c \right ) \left (b x +a \right )^{\frac {5}{2}}}-\frac {4 d \left (-\frac {2 \sqrt {d x +c}}{3 \left (-a d +b c \right ) \left (b x +a \right )^{\frac {3}{2}}}+\frac {4 d \sqrt {d x +c}}{3 \left (-a d +b c \right )^{2} \sqrt {b x +a}}\right )}{5 \left (-a d +b c \right )}\right )}{7 \left (-a d +b c \right )}\right )}{9 \left (-a d +b c \right )}\right )}{11 \left (-a d +b c \right )}\right )}{10 b}\right )}{8 b}\right )}{6 b}\) | \(314\) |
2/693*(d*x+c)^(7/2)*(8*b^2*d^2*x^2+44*a*b*d^2*x-28*b^2*c*d*x+99*a^2*d^2-15 4*a*b*c*d+63*b^2*c^2)/(b*x+a)^(11/2)/(a^3*d^3-3*a^2*b*c*d^2+3*a*b^2*c^2*d- b^3*c^3)
Leaf count of result is larger than twice the leaf count of optimal. 513 vs. \(2 (83) = 166\).
Time = 5.48 (sec) , antiderivative size = 513, normalized size of antiderivative = 5.08 \[ \int \frac {(c+d x)^{5/2}}{(a+b x)^{13/2}} \, dx=-\frac {2 \, {\left (8 \, b^{2} d^{5} x^{5} + 63 \, b^{2} c^{5} - 154 \, a b c^{4} d + 99 \, a^{2} c^{3} d^{2} - 4 \, {\left (b^{2} c d^{4} - 11 \, a b d^{5}\right )} x^{4} + {\left (3 \, b^{2} c^{2} d^{3} - 22 \, a b c d^{4} + 99 \, a^{2} d^{5}\right )} x^{3} + {\left (113 \, b^{2} c^{3} d^{2} - 330 \, a b c^{2} d^{3} + 297 \, a^{2} c d^{4}\right )} x^{2} + {\left (161 \, b^{2} c^{4} d - 418 \, a b c^{3} d^{2} + 297 \, a^{2} c^{2} d^{3}\right )} x\right )} \sqrt {b x + a} \sqrt {d x + c}}{693 \, {\left (a^{6} b^{3} c^{3} - 3 \, a^{7} b^{2} c^{2} d + 3 \, a^{8} b c d^{2} - a^{9} d^{3} + {\left (b^{9} c^{3} - 3 \, a b^{8} c^{2} d + 3 \, a^{2} b^{7} c d^{2} - a^{3} b^{6} d^{3}\right )} x^{6} + 6 \, {\left (a b^{8} c^{3} - 3 \, a^{2} b^{7} c^{2} d + 3 \, a^{3} b^{6} c d^{2} - a^{4} b^{5} d^{3}\right )} x^{5} + 15 \, {\left (a^{2} b^{7} c^{3} - 3 \, a^{3} b^{6} c^{2} d + 3 \, a^{4} b^{5} c d^{2} - a^{5} b^{4} d^{3}\right )} x^{4} + 20 \, {\left (a^{3} b^{6} c^{3} - 3 \, a^{4} b^{5} c^{2} d + 3 \, a^{5} b^{4} c d^{2} - a^{6} b^{3} d^{3}\right )} x^{3} + 15 \, {\left (a^{4} b^{5} c^{3} - 3 \, a^{5} b^{4} c^{2} d + 3 \, a^{6} b^{3} c d^{2} - a^{7} b^{2} d^{3}\right )} x^{2} + 6 \, {\left (a^{5} b^{4} c^{3} - 3 \, a^{6} b^{3} c^{2} d + 3 \, a^{7} b^{2} c d^{2} - a^{8} b d^{3}\right )} x\right )}} \]
-2/693*(8*b^2*d^5*x^5 + 63*b^2*c^5 - 154*a*b*c^4*d + 99*a^2*c^3*d^2 - 4*(b ^2*c*d^4 - 11*a*b*d^5)*x^4 + (3*b^2*c^2*d^3 - 22*a*b*c*d^4 + 99*a^2*d^5)*x ^3 + (113*b^2*c^3*d^2 - 330*a*b*c^2*d^3 + 297*a^2*c*d^4)*x^2 + (161*b^2*c^ 4*d - 418*a*b*c^3*d^2 + 297*a^2*c^2*d^3)*x)*sqrt(b*x + a)*sqrt(d*x + c)/(a ^6*b^3*c^3 - 3*a^7*b^2*c^2*d + 3*a^8*b*c*d^2 - a^9*d^3 + (b^9*c^3 - 3*a*b^ 8*c^2*d + 3*a^2*b^7*c*d^2 - a^3*b^6*d^3)*x^6 + 6*(a*b^8*c^3 - 3*a^2*b^7*c^ 2*d + 3*a^3*b^6*c*d^2 - a^4*b^5*d^3)*x^5 + 15*(a^2*b^7*c^3 - 3*a^3*b^6*c^2 *d + 3*a^4*b^5*c*d^2 - a^5*b^4*d^3)*x^4 + 20*(a^3*b^6*c^3 - 3*a^4*b^5*c^2* d + 3*a^5*b^4*c*d^2 - a^6*b^3*d^3)*x^3 + 15*(a^4*b^5*c^3 - 3*a^5*b^4*c^2*d + 3*a^6*b^3*c*d^2 - a^7*b^2*d^3)*x^2 + 6*(a^5*b^4*c^3 - 3*a^6*b^3*c^2*d + 3*a^7*b^2*c*d^2 - a^8*b*d^3)*x)
\[ \int \frac {(c+d x)^{5/2}}{(a+b x)^{13/2}} \, dx=\int \frac {\left (c + d x\right )^{\frac {5}{2}}}{\left (a + b x\right )^{\frac {13}{2}}}\, dx \]
Exception generated. \[ \int \frac {(c+d x)^{5/2}}{(a+b x)^{13/2}} \, dx=\text {Exception raised: ValueError} \]
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
Leaf count of result is larger than twice the leaf count of optimal. 2316 vs. \(2 (83) = 166\).
Time = 0.81 (sec) , antiderivative size = 2316, normalized size of antiderivative = 22.93 \[ \int \frac {(c+d x)^{5/2}}{(a+b x)^{13/2}} \, dx=\text {Too large to display} \]
-32/693*(sqrt(b*d)*b^16*c^8*d^5*abs(b) - 8*sqrt(b*d)*a*b^15*c^7*d^6*abs(b) + 28*sqrt(b*d)*a^2*b^14*c^6*d^7*abs(b) - 56*sqrt(b*d)*a^3*b^13*c^5*d^8*ab s(b) + 70*sqrt(b*d)*a^4*b^12*c^4*d^9*abs(b) - 56*sqrt(b*d)*a^5*b^11*c^3*d^ 10*abs(b) + 28*sqrt(b*d)*a^6*b^10*c^2*d^11*abs(b) - 8*sqrt(b*d)*a^7*b^9*c* d^12*abs(b) + sqrt(b*d)*a^8*b^8*d^13*abs(b) - 11*sqrt(b*d)*(sqrt(b*d)*sqrt (b*x + a) - sqrt(b^2*c + (b*x + a)*b*d - a*b*d))^2*b^14*c^7*d^5*abs(b) + 7 7*sqrt(b*d)*(sqrt(b*d)*sqrt(b*x + a) - sqrt(b^2*c + (b*x + a)*b*d - a*b*d) )^2*a*b^13*c^6*d^6*abs(b) - 231*sqrt(b*d)*(sqrt(b*d)*sqrt(b*x + a) - sqrt( b^2*c + (b*x + a)*b*d - a*b*d))^2*a^2*b^12*c^5*d^7*abs(b) + 385*sqrt(b*d)* (sqrt(b*d)*sqrt(b*x + a) - sqrt(b^2*c + (b*x + a)*b*d - a*b*d))^2*a^3*b^11 *c^4*d^8*abs(b) - 385*sqrt(b*d)*(sqrt(b*d)*sqrt(b*x + a) - sqrt(b^2*c + (b *x + a)*b*d - a*b*d))^2*a^4*b^10*c^3*d^9*abs(b) + 231*sqrt(b*d)*(sqrt(b*d) *sqrt(b*x + a) - sqrt(b^2*c + (b*x + a)*b*d - a*b*d))^2*a^5*b^9*c^2*d^10*a bs(b) - 77*sqrt(b*d)*(sqrt(b*d)*sqrt(b*x + a) - sqrt(b^2*c + (b*x + a)*b*d - a*b*d))^2*a^6*b^8*c*d^11*abs(b) + 11*sqrt(b*d)*(sqrt(b*d)*sqrt(b*x + a) - sqrt(b^2*c + (b*x + a)*b*d - a*b*d))^2*a^7*b^7*d^12*abs(b) + 55*sqrt(b* d)*(sqrt(b*d)*sqrt(b*x + a) - sqrt(b^2*c + (b*x + a)*b*d - a*b*d))^4*b^12* c^6*d^5*abs(b) - 330*sqrt(b*d)*(sqrt(b*d)*sqrt(b*x + a) - sqrt(b^2*c + (b* x + a)*b*d - a*b*d))^4*a*b^11*c^5*d^6*abs(b) + 825*sqrt(b*d)*(sqrt(b*d)*sq rt(b*x + a) - sqrt(b^2*c + (b*x + a)*b*d - a*b*d))^4*a^2*b^10*c^4*d^7*a...
Time = 1.30 (sec) , antiderivative size = 333, normalized size of antiderivative = 3.30 \[ \int \frac {(c+d x)^{5/2}}{(a+b x)^{13/2}} \, dx=\frac {\sqrt {c+d\,x}\,\left (\frac {198\,a^2\,c^3\,d^2-308\,a\,b\,c^4\,d+126\,b^2\,c^5}{693\,b^5\,{\left (a\,d-b\,c\right )}^3}+\frac {x^3\,\left (198\,a^2\,d^5-44\,a\,b\,c\,d^4+6\,b^2\,c^2\,d^3\right )}{693\,b^5\,{\left (a\,d-b\,c\right )}^3}+\frac {16\,d^5\,x^5}{693\,b^3\,{\left (a\,d-b\,c\right )}^3}+\frac {8\,d^4\,x^4\,\left (11\,a\,d-b\,c\right )}{693\,b^4\,{\left (a\,d-b\,c\right )}^3}+\frac {2\,c\,d^2\,x^2\,\left (297\,a^2\,d^2-330\,a\,b\,c\,d+113\,b^2\,c^2\right )}{693\,b^5\,{\left (a\,d-b\,c\right )}^3}+\frac {2\,c^2\,d\,x\,\left (297\,a^2\,d^2-418\,a\,b\,c\,d+161\,b^2\,c^2\right )}{693\,b^5\,{\left (a\,d-b\,c\right )}^3}\right )}{x^5\,\sqrt {a+b\,x}+\frac {a^5\,\sqrt {a+b\,x}}{b^5}+\frac {10\,a^2\,x^3\,\sqrt {a+b\,x}}{b^2}+\frac {10\,a^3\,x^2\,\sqrt {a+b\,x}}{b^3}+\frac {5\,a\,x^4\,\sqrt {a+b\,x}}{b}+\frac {5\,a^4\,x\,\sqrt {a+b\,x}}{b^4}} \]
((c + d*x)^(1/2)*((126*b^2*c^5 + 198*a^2*c^3*d^2 - 308*a*b*c^4*d)/(693*b^5 *(a*d - b*c)^3) + (x^3*(198*a^2*d^5 + 6*b^2*c^2*d^3 - 44*a*b*c*d^4))/(693* b^5*(a*d - b*c)^3) + (16*d^5*x^5)/(693*b^3*(a*d - b*c)^3) + (8*d^4*x^4*(11 *a*d - b*c))/(693*b^4*(a*d - b*c)^3) + (2*c*d^2*x^2*(297*a^2*d^2 + 113*b^2 *c^2 - 330*a*b*c*d))/(693*b^5*(a*d - b*c)^3) + (2*c^2*d*x*(297*a^2*d^2 + 1 61*b^2*c^2 - 418*a*b*c*d))/(693*b^5*(a*d - b*c)^3)))/(x^5*(a + b*x)^(1/2) + (a^5*(a + b*x)^(1/2))/b^5 + (10*a^2*x^3*(a + b*x)^(1/2))/b^2 + (10*a^3*x ^2*(a + b*x)^(1/2))/b^3 + (5*a*x^4*(a + b*x)^(1/2))/b + (5*a^4*x*(a + b*x) ^(1/2))/b^4)
Time = 0.05 (sec) , antiderivative size = 738, normalized size of antiderivative = 7.31 \[ \int \frac {(c+d x)^{5/2}}{(a+b x)^{13/2}} \, dx=\frac {-\frac {16 \sqrt {d}\, \sqrt {b}\, \sqrt {b x +a}\, a^{5} d^{5}}{693}-\frac {80 \sqrt {d}\, \sqrt {b}\, \sqrt {b x +a}\, a^{4} b \,d^{5} x}{693}-\frac {160 \sqrt {d}\, \sqrt {b}\, \sqrt {b x +a}\, a^{3} b^{2} d^{5} x^{2}}{693}-\frac {160 \sqrt {d}\, \sqrt {b}\, \sqrt {b x +a}\, a^{2} b^{3} d^{5} x^{3}}{693}-\frac {80 \sqrt {d}\, \sqrt {b}\, \sqrt {b x +a}\, a \,b^{4} d^{5} x^{4}}{693}-\frac {16 \sqrt {d}\, \sqrt {b}\, \sqrt {b x +a}\, b^{5} d^{5} x^{5}}{693}+\frac {2 \sqrt {d x +c}\, a^{2} b^{4} c^{3} d^{2}}{7}+\frac {6 \sqrt {d x +c}\, a^{2} b^{4} c^{2} d^{3} x}{7}+\frac {6 \sqrt {d x +c}\, a^{2} b^{4} c \,d^{4} x^{2}}{7}+\frac {2 \sqrt {d x +c}\, a^{2} b^{4} d^{5} x^{3}}{7}-\frac {4 \sqrt {d x +c}\, a \,b^{5} c^{4} d}{9}-\frac {76 \sqrt {d x +c}\, a \,b^{5} c^{3} d^{2} x}{63}-\frac {20 \sqrt {d x +c}\, a \,b^{5} c^{2} d^{3} x^{2}}{21}-\frac {4 \sqrt {d x +c}\, a \,b^{5} c \,d^{4} x^{3}}{63}+\frac {8 \sqrt {d x +c}\, a \,b^{5} d^{5} x^{4}}{63}+\frac {2 \sqrt {d x +c}\, b^{6} c^{5}}{11}+\frac {46 \sqrt {d x +c}\, b^{6} c^{4} d x}{99}+\frac {226 \sqrt {d x +c}\, b^{6} c^{3} d^{2} x^{2}}{693}+\frac {2 \sqrt {d x +c}\, b^{6} c^{2} d^{3} x^{3}}{231}-\frac {8 \sqrt {d x +c}\, b^{6} c \,d^{4} x^{4}}{693}+\frac {16 \sqrt {d x +c}\, b^{6} d^{5} x^{5}}{693}}{\sqrt {b x +a}\, b^{4} \left (a^{3} b^{5} d^{3} x^{5}-3 a^{2} b^{6} c \,d^{2} x^{5}+3 a \,b^{7} c^{2} d \,x^{5}-b^{8} c^{3} x^{5}+5 a^{4} b^{4} d^{3} x^{4}-15 a^{3} b^{5} c \,d^{2} x^{4}+15 a^{2} b^{6} c^{2} d \,x^{4}-5 a \,b^{7} c^{3} x^{4}+10 a^{5} b^{3} d^{3} x^{3}-30 a^{4} b^{4} c \,d^{2} x^{3}+30 a^{3} b^{5} c^{2} d \,x^{3}-10 a^{2} b^{6} c^{3} x^{3}+10 a^{6} b^{2} d^{3} x^{2}-30 a^{5} b^{3} c \,d^{2} x^{2}+30 a^{4} b^{4} c^{2} d \,x^{2}-10 a^{3} b^{5} c^{3} x^{2}+5 a^{7} b \,d^{3} x -15 a^{6} b^{2} c \,d^{2} x +15 a^{5} b^{3} c^{2} d x -5 a^{4} b^{4} c^{3} x +a^{8} d^{3}-3 a^{7} b c \,d^{2}+3 a^{6} b^{2} c^{2} d -a^{5} b^{3} c^{3}\right )} \]
int((sqrt(c + d*x)*(c**2 + 2*c*d*x + d**2*x**2))/(sqrt(a + b*x)*(a**6 + 6* a**5*b*x + 15*a**4*b**2*x**2 + 20*a**3*b**3*x**3 + 15*a**2*b**4*x**4 + 6*a *b**5*x**5 + b**6*x**6)),x)
(2*( - 8*sqrt(d)*sqrt(b)*sqrt(a + b*x)*a**5*d**5 - 40*sqrt(d)*sqrt(b)*sqrt (a + b*x)*a**4*b*d**5*x - 80*sqrt(d)*sqrt(b)*sqrt(a + b*x)*a**3*b**2*d**5* x**2 - 80*sqrt(d)*sqrt(b)*sqrt(a + b*x)*a**2*b**3*d**5*x**3 - 40*sqrt(d)*s qrt(b)*sqrt(a + b*x)*a*b**4*d**5*x**4 - 8*sqrt(d)*sqrt(b)*sqrt(a + b*x)*b* *5*d**5*x**5 + 99*sqrt(c + d*x)*a**2*b**4*c**3*d**2 + 297*sqrt(c + d*x)*a* *2*b**4*c**2*d**3*x + 297*sqrt(c + d*x)*a**2*b**4*c*d**4*x**2 + 99*sqrt(c + d*x)*a**2*b**4*d**5*x**3 - 154*sqrt(c + d*x)*a*b**5*c**4*d - 418*sqrt(c + d*x)*a*b**5*c**3*d**2*x - 330*sqrt(c + d*x)*a*b**5*c**2*d**3*x**2 - 22*s qrt(c + d*x)*a*b**5*c*d**4*x**3 + 44*sqrt(c + d*x)*a*b**5*d**5*x**4 + 63*s qrt(c + d*x)*b**6*c**5 + 161*sqrt(c + d*x)*b**6*c**4*d*x + 113*sqrt(c + d* x)*b**6*c**3*d**2*x**2 + 3*sqrt(c + d*x)*b**6*c**2*d**3*x**3 - 4*sqrt(c + d*x)*b**6*c*d**4*x**4 + 8*sqrt(c + d*x)*b**6*d**5*x**5))/(693*sqrt(a + b*x )*b**4*(a**8*d**3 - 3*a**7*b*c*d**2 + 5*a**7*b*d**3*x + 3*a**6*b**2*c**2*d - 15*a**6*b**2*c*d**2*x + 10*a**6*b**2*d**3*x**2 - a**5*b**3*c**3 + 15*a* *5*b**3*c**2*d*x - 30*a**5*b**3*c*d**2*x**2 + 10*a**5*b**3*d**3*x**3 - 5*a **4*b**4*c**3*x + 30*a**4*b**4*c**2*d*x**2 - 30*a**4*b**4*c*d**2*x**3 + 5* a**4*b**4*d**3*x**4 - 10*a**3*b**5*c**3*x**2 + 30*a**3*b**5*c**2*d*x**3 - 15*a**3*b**5*c*d**2*x**4 + a**3*b**5*d**3*x**5 - 10*a**2*b**6*c**3*x**3 + 15*a**2*b**6*c**2*d*x**4 - 3*a**2*b**6*c*d**2*x**5 - 5*a*b**7*c**3*x**4 + 3*a*b**7*c**2*d*x**5 - b**8*c**3*x**5))