Integrand size = 19, antiderivative size = 171 \[ \int \frac {\sqrt {c+d x}}{(a+b x)^{13/2}} \, dx=-\frac {2 (c+d x)^{3/2}}{11 (b c-a d) (a+b x)^{11/2}}+\frac {16 d (c+d x)^{3/2}}{99 (b c-a d)^2 (a+b x)^{9/2}}-\frac {32 d^2 (c+d x)^{3/2}}{231 (b c-a d)^3 (a+b x)^{7/2}}+\frac {128 d^3 (c+d x)^{3/2}}{1155 (b c-a d)^4 (a+b x)^{5/2}}-\frac {256 d^4 (c+d x)^{3/2}}{3465 (b c-a d)^5 (a+b x)^{3/2}} \]
-2/11*(d*x+c)^(3/2)/(-a*d+b*c)/(b*x+a)^(11/2)+16/99*d*(d*x+c)^(3/2)/(-a*d+ b*c)^2/(b*x+a)^(9/2)-32/231*d^2*(d*x+c)^(3/2)/(-a*d+b*c)^3/(b*x+a)^(7/2)+1 28/1155*d^3*(d*x+c)^(3/2)/(-a*d+b*c)^4/(b*x+a)^(5/2)-256/3465*d^4*(d*x+c)^ (3/2)/(-a*d+b*c)^5/(b*x+a)^(3/2)
Time = 0.33 (sec) , antiderivative size = 170, normalized size of antiderivative = 0.99 \[ \int \frac {\sqrt {c+d x}}{(a+b x)^{13/2}} \, dx=-\frac {2 (c+d x)^{3/2} \left (1155 a^4 d^4+924 a^3 b d^3 (-3 c+2 d x)+198 a^2 b^2 d^2 \left (15 c^2-12 c d x+8 d^2 x^2\right )+44 a b^3 d \left (-35 c^3+30 c^2 d x-24 c d^2 x^2+16 d^3 x^3\right )+b^4 \left (315 c^4-280 c^3 d x+240 c^2 d^2 x^2-192 c d^3 x^3+128 d^4 x^4\right )\right )}{3465 (b c-a d)^5 (a+b x)^{11/2}} \]
(-2*(c + d*x)^(3/2)*(1155*a^4*d^4 + 924*a^3*b*d^3*(-3*c + 2*d*x) + 198*a^2 *b^2*d^2*(15*c^2 - 12*c*d*x + 8*d^2*x^2) + 44*a*b^3*d*(-35*c^3 + 30*c^2*d* x - 24*c*d^2*x^2 + 16*d^3*x^3) + b^4*(315*c^4 - 280*c^3*d*x + 240*c^2*d^2* x^2 - 192*c*d^3*x^3 + 128*d^4*x^4)))/(3465*(b*c - a*d)^5*(a + b*x)^(11/2))
Time = 0.23 (sec) , antiderivative size = 210, normalized size of antiderivative = 1.23, number of steps used = 5, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.263, Rules used = {55, 55, 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.
\(\displaystyle \int \frac {\sqrt {c+d x}}{(a+b x)^{13/2}} \, dx\) |
\(\Big \downarrow \) 55 |
\(\displaystyle -\frac {8 d \int \frac {\sqrt {c+d x}}{(a+b x)^{11/2}}dx}{11 (b c-a d)}-\frac {2 (c+d x)^{3/2}}{11 (a+b x)^{11/2} (b c-a d)}\) |
\(\Big \downarrow \) 55 |
\(\displaystyle -\frac {8 d \left (-\frac {2 d \int \frac {\sqrt {c+d x}}{(a+b x)^{9/2}}dx}{3 (b c-a d)}-\frac {2 (c+d x)^{3/2}}{9 (a+b x)^{9/2} (b c-a d)}\right )}{11 (b c-a d)}-\frac {2 (c+d x)^{3/2}}{11 (a+b x)^{11/2} (b c-a d)}\) |
\(\Big \downarrow \) 55 |
\(\displaystyle -\frac {8 d \left (-\frac {2 d \left (-\frac {4 d \int \frac {\sqrt {c+d x}}{(a+b x)^{7/2}}dx}{7 (b c-a d)}-\frac {2 (c+d x)^{3/2}}{7 (a+b x)^{7/2} (b c-a d)}\right )}{3 (b c-a d)}-\frac {2 (c+d x)^{3/2}}{9 (a+b x)^{9/2} (b c-a d)}\right )}{11 (b c-a d)}-\frac {2 (c+d x)^{3/2}}{11 (a+b x)^{11/2} (b c-a d)}\) |
\(\Big \downarrow \) 55 |
\(\displaystyle -\frac {8 d \left (-\frac {2 d \left (-\frac {4 d \left (-\frac {2 d \int \frac {\sqrt {c+d x}}{(a+b x)^{5/2}}dx}{5 (b c-a d)}-\frac {2 (c+d x)^{3/2}}{5 (a+b x)^{5/2} (b c-a d)}\right )}{7 (b c-a d)}-\frac {2 (c+d x)^{3/2}}{7 (a+b x)^{7/2} (b c-a d)}\right )}{3 (b c-a d)}-\frac {2 (c+d x)^{3/2}}{9 (a+b x)^{9/2} (b c-a d)}\right )}{11 (b c-a d)}-\frac {2 (c+d x)^{3/2}}{11 (a+b x)^{11/2} (b c-a d)}\) |
\(\Big \downarrow \) 48 |
\(\displaystyle -\frac {2 (c+d x)^{3/2}}{11 (a+b x)^{11/2} (b c-a d)}-\frac {8 d \left (-\frac {2 (c+d x)^{3/2}}{9 (a+b x)^{9/2} (b c-a d)}-\frac {2 d \left (-\frac {2 (c+d x)^{3/2}}{7 (a+b x)^{7/2} (b c-a d)}-\frac {4 d \left (\frac {4 d (c+d x)^{3/2}}{15 (a+b x)^{3/2} (b c-a d)^2}-\frac {2 (c+d x)^{3/2}}{5 (a+b x)^{5/2} (b c-a d)}\right )}{7 (b c-a d)}\right )}{3 (b c-a d)}\right )}{11 (b c-a d)}\) |
(-2*(c + d*x)^(3/2))/(11*(b*c - a*d)*(a + b*x)^(11/2)) - (8*d*((-2*(c + d* x)^(3/2))/(9*(b*c - a*d)*(a + b*x)^(9/2)) - (2*d*((-2*(c + d*x)^(3/2))/(7* (b*c - a*d)*(a + b*x)^(7/2)) - (4*d*((-2*(c + d*x)^(3/2))/(5*(b*c - a*d)*( a + b*x)^(5/2)) + (4*d*(c + d*x)^(3/2))/(15*(b*c - a*d)^2*(a + b*x)^(3/2)) ))/(7*(b*c - a*d))))/(3*(b*c - a*d))))/(11*(b*c - a*d))
3.15.70.3.1 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 = 248, normalized size of antiderivative = 1.45
method | result | size |
default | \(-\frac {\sqrt {d x +c}}{5 b \left (b x +a \right )^{\frac {11}{2}}}+\frac {\left (a d -b c \right ) \left (-\frac {2 \sqrt {d x +c}}{11 \left (-a d +b c \right ) \left (b x +a \right )^{\frac {11}{2}}}-\frac {10 d \left (-\frac {2 \sqrt {d x +c}}{9 \left (-a d +b c \right ) \left (b x +a \right )^{\frac {9}{2}}}-\frac {8 d \left (-\frac {2 \sqrt {d x +c}}{7 \left (-a d +b c \right ) \left (b x +a \right )^{\frac {7}{2}}}-\frac {6 d \left (-\frac {2 \sqrt {d x +c}}{5 \left (-a d +b c \right ) \left (b x +a \right )^{\frac {5}{2}}}-\frac {4 d \left (-\frac {2 \sqrt {d x +c}}{3 \left (-a d +b c \right ) \left (b x +a \right )^{\frac {3}{2}}}+\frac {4 d \sqrt {d x +c}}{3 \left (-a d +b c \right )^{2} \sqrt {b x +a}}\right )}{5 \left (-a d +b c \right )}\right )}{7 \left (-a d +b c \right )}\right )}{9 \left (-a d +b c \right )}\right )}{11 \left (-a d +b c \right )}\right )}{10 b}\) | \(248\) |
gosper | \(\frac {2 \left (d x +c \right )^{\frac {3}{2}} \left (128 d^{4} x^{4} b^{4}+704 a \,b^{3} d^{4} x^{3}-192 b^{4} c \,d^{3} x^{3}+1584 a^{2} b^{2} d^{4} x^{2}-1056 a \,b^{3} c \,d^{3} x^{2}+240 b^{4} c^{2} d^{2} x^{2}+1848 a^{3} b \,d^{4} x -2376 a^{2} b^{2} c \,d^{3} x +1320 a \,b^{3} c^{2} d^{2} x -280 b^{4} c^{3} d x +1155 a^{4} d^{4}-2772 a^{3} b c \,d^{3}+2970 a^{2} b^{2} c^{2} d^{2}-1540 a \,b^{3} c^{3} d +315 b^{4} c^{4}\right )}{3465 \left (b x +a \right )^{\frac {11}{2}} \left (a^{5} d^{5}-5 a^{4} b c \,d^{4}+10 a^{3} b^{2} c^{2} d^{3}-10 a^{2} b^{3} c^{3} d^{2}+5 a \,b^{4} c^{4} d -b^{5} c^{5}\right )}\) | \(256\) |
-1/5/b*(d*x+c)^(1/2)/(b*x+a)^(11/2)+1/10*(a*d-b*c)/b*(-2/11/(-a*d+b*c)/(b* x+a)^(11/2)*(d*x+c)^(1/2)-10/11*d/(-a*d+b*c)*(-2/9*(d*x+c)^(1/2)/(-a*d+b*c )/(b*x+a)^(9/2)-8/9*d/(-a*d+b*c)*(-2/7*(d*x+c)^(1/2)/(-a*d+b*c)/(b*x+a)^(7 /2)-6/7*d/(-a*d+b*c)*(-2/5*(d*x+c)^(1/2)/(-a*d+b*c)/(b*x+a)^(5/2)-4/5*d/(- a*d+b*c)*(-2/3*(d*x+c)^(1/2)/(-a*d+b*c)/(b*x+a)^(3/2)+4/3*d*(d*x+c)^(1/2)/ (-a*d+b*c)^2/(b*x+a)^(1/2))))))
Leaf count of result is larger than twice the leaf count of optimal. 781 vs. \(2 (141) = 282\).
Time = 4.73 (sec) , antiderivative size = 781, normalized size of antiderivative = 4.57 \[ \int \frac {\sqrt {c+d x}}{(a+b x)^{13/2}} \, dx=-\frac {2 \, {\left (128 \, b^{4} d^{5} x^{5} + 315 \, b^{4} c^{5} - 1540 \, a b^{3} c^{4} d + 2970 \, a^{2} b^{2} c^{3} d^{2} - 2772 \, a^{3} b c^{2} d^{3} + 1155 \, a^{4} c d^{4} - 64 \, {\left (b^{4} c d^{4} - 11 \, a b^{3} d^{5}\right )} x^{4} + 16 \, {\left (3 \, b^{4} c^{2} d^{3} - 22 \, a b^{3} c d^{4} + 99 \, a^{2} b^{2} d^{5}\right )} x^{3} - 8 \, {\left (5 \, b^{4} c^{3} d^{2} - 33 \, a b^{3} c^{2} d^{3} + 99 \, a^{2} b^{2} c d^{4} - 231 \, a^{3} b d^{5}\right )} x^{2} + {\left (35 \, b^{4} c^{4} d - 220 \, a b^{3} c^{3} d^{2} + 594 \, a^{2} b^{2} c^{2} d^{3} - 924 \, a^{3} b c d^{4} + 1155 \, a^{4} d^{5}\right )} x\right )} \sqrt {b x + a} \sqrt {d x + c}}{3465 \, {\left (a^{6} b^{5} c^{5} - 5 \, a^{7} b^{4} c^{4} d + 10 \, a^{8} b^{3} c^{3} d^{2} - 10 \, a^{9} b^{2} c^{2} d^{3} + 5 \, a^{10} b c d^{4} - a^{11} d^{5} + {\left (b^{11} c^{5} - 5 \, a b^{10} c^{4} d + 10 \, a^{2} b^{9} c^{3} d^{2} - 10 \, a^{3} b^{8} c^{2} d^{3} + 5 \, a^{4} b^{7} c d^{4} - a^{5} b^{6} d^{5}\right )} x^{6} + 6 \, {\left (a b^{10} c^{5} - 5 \, a^{2} b^{9} c^{4} d + 10 \, a^{3} b^{8} c^{3} d^{2} - 10 \, a^{4} b^{7} c^{2} d^{3} + 5 \, a^{5} b^{6} c d^{4} - a^{6} b^{5} d^{5}\right )} x^{5} + 15 \, {\left (a^{2} b^{9} c^{5} - 5 \, a^{3} b^{8} c^{4} d + 10 \, a^{4} b^{7} c^{3} d^{2} - 10 \, a^{5} b^{6} c^{2} d^{3} + 5 \, a^{6} b^{5} c d^{4} - a^{7} b^{4} d^{5}\right )} x^{4} + 20 \, {\left (a^{3} b^{8} c^{5} - 5 \, a^{4} b^{7} c^{4} d + 10 \, a^{5} b^{6} c^{3} d^{2} - 10 \, a^{6} b^{5} c^{2} d^{3} + 5 \, a^{7} b^{4} c d^{4} - a^{8} b^{3} d^{5}\right )} x^{3} + 15 \, {\left (a^{4} b^{7} c^{5} - 5 \, a^{5} b^{6} c^{4} d + 10 \, a^{6} b^{5} c^{3} d^{2} - 10 \, a^{7} b^{4} c^{2} d^{3} + 5 \, a^{8} b^{3} c d^{4} - a^{9} b^{2} d^{5}\right )} x^{2} + 6 \, {\left (a^{5} b^{6} c^{5} - 5 \, a^{6} b^{5} c^{4} d + 10 \, a^{7} b^{4} c^{3} d^{2} - 10 \, a^{8} b^{3} c^{2} d^{3} + 5 \, a^{9} b^{2} c d^{4} - a^{10} b d^{5}\right )} x\right )}} \]
-2/3465*(128*b^4*d^5*x^5 + 315*b^4*c^5 - 1540*a*b^3*c^4*d + 2970*a^2*b^2*c ^3*d^2 - 2772*a^3*b*c^2*d^3 + 1155*a^4*c*d^4 - 64*(b^4*c*d^4 - 11*a*b^3*d^ 5)*x^4 + 16*(3*b^4*c^2*d^3 - 22*a*b^3*c*d^4 + 99*a^2*b^2*d^5)*x^3 - 8*(5*b ^4*c^3*d^2 - 33*a*b^3*c^2*d^3 + 99*a^2*b^2*c*d^4 - 231*a^3*b*d^5)*x^2 + (3 5*b^4*c^4*d - 220*a*b^3*c^3*d^2 + 594*a^2*b^2*c^2*d^3 - 924*a^3*b*c*d^4 + 1155*a^4*d^5)*x)*sqrt(b*x + a)*sqrt(d*x + c)/(a^6*b^5*c^5 - 5*a^7*b^4*c^4* d + 10*a^8*b^3*c^3*d^2 - 10*a^9*b^2*c^2*d^3 + 5*a^10*b*c*d^4 - a^11*d^5 + (b^11*c^5 - 5*a*b^10*c^4*d + 10*a^2*b^9*c^3*d^2 - 10*a^3*b^8*c^2*d^3 + 5*a ^4*b^7*c*d^4 - a^5*b^6*d^5)*x^6 + 6*(a*b^10*c^5 - 5*a^2*b^9*c^4*d + 10*a^3 *b^8*c^3*d^2 - 10*a^4*b^7*c^2*d^3 + 5*a^5*b^6*c*d^4 - a^6*b^5*d^5)*x^5 + 1 5*(a^2*b^9*c^5 - 5*a^3*b^8*c^4*d + 10*a^4*b^7*c^3*d^2 - 10*a^5*b^6*c^2*d^3 + 5*a^6*b^5*c*d^4 - a^7*b^4*d^5)*x^4 + 20*(a^3*b^8*c^5 - 5*a^4*b^7*c^4*d + 10*a^5*b^6*c^3*d^2 - 10*a^6*b^5*c^2*d^3 + 5*a^7*b^4*c*d^4 - a^8*b^3*d^5) *x^3 + 15*(a^4*b^7*c^5 - 5*a^5*b^6*c^4*d + 10*a^6*b^5*c^3*d^2 - 10*a^7*b^4 *c^2*d^3 + 5*a^8*b^3*c*d^4 - a^9*b^2*d^5)*x^2 + 6*(a^5*b^6*c^5 - 5*a^6*b^5 *c^4*d + 10*a^7*b^4*c^3*d^2 - 10*a^8*b^3*c^2*d^3 + 5*a^9*b^2*c*d^4 - a^10* b*d^5)*x)
\[ \int \frac {\sqrt {c+d x}}{(a+b x)^{13/2}} \, dx=\int \frac {\sqrt {c + d x}}{\left (a + b x\right )^{\frac {13}{2}}}\, dx \]
Exception generated. \[ \int \frac {\sqrt {c+d x}}{(a+b x)^{13/2}} \, dx=\text {Exception raised: ValueError} \]
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. 1345 vs. \(2 (141) = 282\).
Time = 0.46 (sec) , antiderivative size = 1345, normalized size of antiderivative = 7.87 \[ \int \frac {\sqrt {c+d x}}{(a+b x)^{13/2}} \, dx=\text {Too large to display} \]
-512/3465*(sqrt(b*d)*b^16*c^6*d^5 - 6*sqrt(b*d)*a*b^15*c^5*d^6 + 15*sqrt(b *d)*a^2*b^14*c^4*d^7 - 20*sqrt(b*d)*a^3*b^13*c^3*d^8 + 15*sqrt(b*d)*a^4*b^ 12*c^2*d^9 - 6*sqrt(b*d)*a^5*b^11*c*d^10 + sqrt(b*d)*a^6*b^10*d^11 - 11*sq rt(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^5*d^5 + 55*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^4*d^6 - 110*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^3*d^7 + 110*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^2*d^8 - 55*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*d^9 + 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^5*b^9*d^10 + 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^4*d^5 - 220*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^3*d^6 + 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^2*b^10*c^2*d^7 - 220*sqrt(b*d)*(sqrt(b*d)* sqrt(b*x + a) - sqrt(b^2*c + (b*x + a)*b*d - a*b*d))^4*a^3*b^9*c*d^8 + 55* sqrt(b*d)*(sqrt(b*d)*sqrt(b*x + a) - sqrt(b^2*c + (b*x + a)*b*d - a*b*d))^ 4*a^4*b^8*d^9 - 165*sqrt(b*d)*(sqrt(b*d)*sqrt(b*x + a) - sqrt(b^2*c + (b*x + a)*b*d - a*b*d))^6*b^10*c^3*d^5 + 495*sqrt(b*d)*(sqrt(b*d)*sqrt(b*x + a ) - sqrt(b^2*c + (b*x + a)*b*d - a*b*d))^6*a*b^9*c^2*d^6 - 495*sqrt(b*d...
Time = 1.39 (sec) , antiderivative size = 397, normalized size of antiderivative = 2.32 \[ \int \frac {\sqrt {c+d x}}{(a+b x)^{13/2}} \, dx=\frac {\sqrt {c+d\,x}\,\left (\frac {2310\,a^4\,c\,d^4-5544\,a^3\,b\,c^2\,d^3+5940\,a^2\,b^2\,c^3\,d^2-3080\,a\,b^3\,c^4\,d+630\,b^4\,c^5}{3465\,b^5\,{\left (a\,d-b\,c\right )}^5}+\frac {x\,\left (2310\,a^4\,d^5-1848\,a^3\,b\,c\,d^4+1188\,a^2\,b^2\,c^2\,d^3-440\,a\,b^3\,c^3\,d^2+70\,b^4\,c^4\,d\right )}{3465\,b^5\,{\left (a\,d-b\,c\right )}^5}+\frac {256\,d^5\,x^5}{3465\,b\,{\left (a\,d-b\,c\right )}^5}+\frac {16\,d^2\,x^2\,\left (231\,a^3\,d^3-99\,a^2\,b\,c\,d^2+33\,a\,b^2\,c^2\,d-5\,b^3\,c^3\right )}{3465\,b^4\,{\left (a\,d-b\,c\right )}^5}+\frac {128\,d^4\,x^4\,\left (11\,a\,d-b\,c\right )}{3465\,b^2\,{\left (a\,d-b\,c\right )}^5}+\frac {32\,d^3\,x^3\,\left (99\,a^2\,d^2-22\,a\,b\,c\,d+3\,b^2\,c^2\right )}{3465\,b^3\,{\left (a\,d-b\,c\right )}^5}\right )}{x^5\,\sqrt {a+b\,x}+\frac {a^5\,\sqrt {a+b\,x}}{b^5}+\frac {10\,a^2\,x^3\,\sqrt {a+b\,x}}{b^2}+\frac {10\,a^3\,x^2\,\sqrt {a+b\,x}}{b^3}+\frac {5\,a\,x^4\,\sqrt {a+b\,x}}{b}+\frac {5\,a^4\,x\,\sqrt {a+b\,x}}{b^4}} \]
((c + d*x)^(1/2)*((630*b^4*c^5 + 2310*a^4*c*d^4 - 5544*a^3*b*c^2*d^3 + 594 0*a^2*b^2*c^3*d^2 - 3080*a*b^3*c^4*d)/(3465*b^5*(a*d - b*c)^5) + (x*(2310* a^4*d^5 + 70*b^4*c^4*d - 440*a*b^3*c^3*d^2 + 1188*a^2*b^2*c^2*d^3 - 1848*a ^3*b*c*d^4))/(3465*b^5*(a*d - b*c)^5) + (256*d^5*x^5)/(3465*b*(a*d - b*c)^ 5) + (16*d^2*x^2*(231*a^3*d^3 - 5*b^3*c^3 + 33*a*b^2*c^2*d - 99*a^2*b*c*d^ 2))/(3465*b^4*(a*d - b*c)^5) + (128*d^4*x^4*(11*a*d - b*c))/(3465*b^2*(a*d - b*c)^5) + (32*d^3*x^3*(99*a^2*d^2 + 3*b^2*c^2 - 22*a*b*c*d))/(3465*b^3* (a*d - b*c)^5)))/(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)