Integrand size = 19, antiderivative size = 206 \[ \int \frac {1}{(a+b x)^{11/2} (c+d x)^{3/2}} \, dx=-\frac {2}{9 (b c-a d) (a+b x)^{9/2} \sqrt {c+d x}}+\frac {20 d}{63 (b c-a d)^2 (a+b x)^{7/2} \sqrt {c+d x}}-\frac {32 d^2}{63 (b c-a d)^3 (a+b x)^{5/2} \sqrt {c+d x}}+\frac {64 d^3}{63 (b c-a d)^4 (a+b x)^{3/2} \sqrt {c+d x}}-\frac {256 d^4}{63 (b c-a d)^5 \sqrt {a+b x} \sqrt {c+d x}}-\frac {512 d^5 \sqrt {a+b x}}{63 (b c-a d)^6 \sqrt {c+d x}} \]
-2/9/(-a*d+b*c)/(b*x+a)^(9/2)/(d*x+c)^(1/2)+20/63*d/(-a*d+b*c)^2/(b*x+a)^( 7/2)/(d*x+c)^(1/2)-32/63*d^2/(-a*d+b*c)^3/(b*x+a)^(5/2)/(d*x+c)^(1/2)+64/6 3*d^3/(-a*d+b*c)^4/(b*x+a)^(3/2)/(d*x+c)^(1/2)-256/63*d^4/(-a*d+b*c)^5/(b* x+a)^(1/2)/(d*x+c)^(1/2)-512/63*d^5*(b*x+a)^(1/2)/(-a*d+b*c)^6/(d*x+c)^(1/ 2)
Time = 0.35 (sec) , antiderivative size = 229, normalized size of antiderivative = 1.11 \[ \int \frac {1}{(a+b x)^{11/2} (c+d x)^{3/2}} \, dx=-\frac {2 \left (63 a^5 d^5+315 a^4 b d^4 (c+2 d x)+210 a^3 b^2 d^3 \left (-c^2+4 c d x+8 d^2 x^2\right )+126 a^2 b^3 d^2 \left (c^3-2 c^2 d x+8 c d^2 x^2+16 d^3 x^3\right )+9 a b^4 d \left (-5 c^4+8 c^3 d x-16 c^2 d^2 x^2+64 c d^3 x^3+128 d^4 x^4\right )+b^5 \left (7 c^5-10 c^4 d x+16 c^3 d^2 x^2-32 c^2 d^3 x^3+128 c d^4 x^4+256 d^5 x^5\right )\right )}{63 (b c-a d)^6 (a+b x)^{9/2} \sqrt {c+d x}} \]
(-2*(63*a^5*d^5 + 315*a^4*b*d^4*(c + 2*d*x) + 210*a^3*b^2*d^3*(-c^2 + 4*c* d*x + 8*d^2*x^2) + 126*a^2*b^3*d^2*(c^3 - 2*c^2*d*x + 8*c*d^2*x^2 + 16*d^3 *x^3) + 9*a*b^4*d*(-5*c^4 + 8*c^3*d*x - 16*c^2*d^2*x^2 + 64*c*d^3*x^3 + 12 8*d^4*x^4) + b^5*(7*c^5 - 10*c^4*d*x + 16*c^3*d^2*x^2 - 32*c^2*d^3*x^3 + 1 28*c*d^4*x^4 + 256*d^5*x^5)))/(63*(b*c - a*d)^6*(a + b*x)^(9/2)*Sqrt[c + d *x])
Time = 0.25 (sec) , antiderivative size = 254, normalized size of antiderivative = 1.23, number of steps used = 6, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.316, Rules used = {55, 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.
| \(\displaystyle \int \frac {1}{(a+b x)^{11/2} (c+d x)^{3/2}} \, dx\) |
| \(\Big \downarrow \) 55 |
| \(\displaystyle -\frac {10 d \int \frac {1}{(a+b x)^{9/2} (c+d x)^{3/2}}dx}{9 (b c-a d)}-\frac {2}{9 (a+b x)^{9/2} \sqrt {c+d x} (b c-a d)}\) |
| \(\Big \downarrow \) 55 |
| \(\displaystyle -\frac {10 d \left (-\frac {8 d \int \frac {1}{(a+b x)^{7/2} (c+d x)^{3/2}}dx}{7 (b c-a d)}-\frac {2}{7 (a+b x)^{7/2} \sqrt {c+d x} (b c-a d)}\right )}{9 (b c-a d)}-\frac {2}{9 (a+b x)^{9/2} \sqrt {c+d x} (b c-a d)}\) |
| \(\Big \downarrow \) 55 |
| \(\displaystyle -\frac {10 d \left (-\frac {8 d \left (-\frac {6 d \int \frac {1}{(a+b x)^{5/2} (c+d x)^{3/2}}dx}{5 (b c-a d)}-\frac {2}{5 (a+b x)^{5/2} \sqrt {c+d x} (b c-a d)}\right )}{7 (b c-a d)}-\frac {2}{7 (a+b x)^{7/2} \sqrt {c+d x} (b c-a d)}\right )}{9 (b c-a d)}-\frac {2}{9 (a+b x)^{9/2} \sqrt {c+d x} (b c-a d)}\) |
| \(\Big \downarrow \) 55 |
| \(\displaystyle -\frac {10 d \left (-\frac {8 d \left (-\frac {6 d \left (-\frac {4 d \int \frac {1}{(a+b x)^{3/2} (c+d x)^{3/2}}dx}{3 (b c-a d)}-\frac {2}{3 (a+b x)^{3/2} \sqrt {c+d x} (b c-a d)}\right )}{5 (b c-a d)}-\frac {2}{5 (a+b x)^{5/2} \sqrt {c+d x} (b c-a d)}\right )}{7 (b c-a d)}-\frac {2}{7 (a+b x)^{7/2} \sqrt {c+d x} (b c-a d)}\right )}{9 (b c-a d)}-\frac {2}{9 (a+b x)^{9/2} \sqrt {c+d x} (b c-a d)}\) |
| \(\Big \downarrow \) 55 |
| \(\displaystyle -\frac {10 d \left (-\frac {8 d \left (-\frac {6 d \left (-\frac {4 d \left (-\frac {2 d \int \frac {1}{\sqrt {a+b x} (c+d x)^{3/2}}dx}{b c-a d}-\frac {2}{\sqrt {a+b x} \sqrt {c+d x} (b c-a d)}\right )}{3 (b c-a d)}-\frac {2}{3 (a+b x)^{3/2} \sqrt {c+d x} (b c-a d)}\right )}{5 (b c-a d)}-\frac {2}{5 (a+b x)^{5/2} \sqrt {c+d x} (b c-a d)}\right )}{7 (b c-a d)}-\frac {2}{7 (a+b x)^{7/2} \sqrt {c+d x} (b c-a d)}\right )}{9 (b c-a d)}-\frac {2}{9 (a+b x)^{9/2} \sqrt {c+d x} (b c-a d)}\) |
| \(\Big \downarrow \) 48 |
| \(\displaystyle -\frac {10 d \left (-\frac {8 d \left (-\frac {6 d \left (-\frac {4 d \left (-\frac {4 d \sqrt {a+b x}}{\sqrt {c+d x} (b c-a d)^2}-\frac {2}{\sqrt {a+b x} \sqrt {c+d x} (b c-a d)}\right )}{3 (b c-a d)}-\frac {2}{3 (a+b x)^{3/2} \sqrt {c+d x} (b c-a d)}\right )}{5 (b c-a d)}-\frac {2}{5 (a+b x)^{5/2} \sqrt {c+d x} (b c-a d)}\right )}{7 (b c-a d)}-\frac {2}{7 (a+b x)^{7/2} \sqrt {c+d x} (b c-a d)}\right )}{9 (b c-a d)}-\frac {2}{9 (a+b x)^{9/2} \sqrt {c+d x} (b c-a d)}\) |
-2/(9*(b*c - a*d)*(a + b*x)^(9/2)*Sqrt[c + d*x]) - (10*d*(-2/(7*(b*c - a*d )*(a + b*x)^(7/2)*Sqrt[c + d*x]) - (8*d*(-2/(5*(b*c - a*d)*(a + b*x)^(5/2) *Sqrt[c + d*x]) - (6*d*(-2/(3*(b*c - a*d)*(a + b*x)^(3/2)*Sqrt[c + d*x]) - (4*d*(-2/((b*c - a*d)*Sqrt[a + b*x]*Sqrt[c + d*x]) - (4*d*Sqrt[a + b*x])/ ((b*c - a*d)^2*Sqrt[c + d*x])))/(3*(b*c - a*d))))/(5*(b*c - a*d))))/(7*(b* c - a*d))))/(9*(b*c - a*d))
3.16.11.3.1 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 = 0.53 (sec) , antiderivative size = 225, normalized size of antiderivative = 1.09
| method | result | size |
| default | \(-\frac {2}{9 \left (-a d +b c \right ) \left (b x +a \right )^{\frac {9}{2}} \sqrt {d x +c}}-\frac {10 d \left (-\frac {2}{7 \left (-a d +b c \right ) \left (b x +a \right )^{\frac {7}{2}} \sqrt {d x +c}}-\frac {8 d \left (-\frac {2}{5 \left (-a d +b c \right ) \left (b x +a \right )^{\frac {5}{2}} \sqrt {d x +c}}-\frac {6 d \left (-\frac {2}{3 \left (-a d +b c \right ) \left (b x +a \right )^{\frac {3}{2}} \sqrt {d x +c}}-\frac {4 d \left (-\frac {2}{\left (-a d +b c \right ) \sqrt {b x +a}\, \sqrt {d x +c}}+\frac {4 d \sqrt {b x +a}}{\left (-a d +b c \right ) \sqrt {d x +c}\, \left (a d -b c \right )}\right )}{3 \left (-a d +b c \right )}\right )}{5 \left (-a d +b c \right )}\right )}{7 \left (-a d +b c \right )}\right )}{9 \left (-a d +b c \right )}\) | \(225\) |
| gosper | \(-\frac {2 \left (256 x^{5} b^{5} d^{5}+1152 x^{4} a \,b^{4} d^{5}+128 x^{4} b^{5} c \,d^{4}+2016 x^{3} a^{2} b^{3} d^{5}+576 x^{3} a \,b^{4} c \,d^{4}-32 x^{3} b^{5} c^{2} d^{3}+1680 x^{2} a^{3} b^{2} d^{5}+1008 x^{2} a^{2} b^{3} c \,d^{4}-144 x^{2} a \,b^{4} c^{2} d^{3}+16 x^{2} b^{5} c^{3} d^{2}+630 x \,a^{4} b \,d^{5}+840 x \,a^{3} b^{2} c \,d^{4}-252 x \,a^{2} b^{3} c^{2} d^{3}+72 x a \,b^{4} c^{3} d^{2}-10 x \,b^{5} c^{4} d +63 a^{5} d^{5}+315 a^{4} b c \,d^{4}-210 a^{3} b^{2} c^{2} d^{3}+126 a^{2} b^{3} c^{3} d^{2}-45 a \,b^{4} c^{4} d +7 b^{5} c^{5}\right )}{63 \left (b x +a \right )^{\frac {9}{2}} \sqrt {d x +c}\, \left (a^{6} d^{6}-6 a^{5} b c \,d^{5}+15 a^{4} b^{2} c^{2} d^{4}-20 a^{3} b^{3} c^{3} d^{3}+15 a^{2} b^{4} c^{4} d^{2}-6 a \,b^{5} c^{5} d +b^{6} c^{6}\right )}\) | \(356\) |
-2/9/(-a*d+b*c)/(b*x+a)^(9/2)/(d*x+c)^(1/2)-10/9*d/(-a*d+b*c)*(-2/7/(-a*d+ b*c)/(b*x+a)^(7/2)/(d*x+c)^(1/2)-8/7*d/(-a*d+b*c)*(-2/5/(-a*d+b*c)/(b*x+a) ^(5/2)/(d*x+c)^(1/2)-6/5*d/(-a*d+b*c)*(-2/3/(-a*d+b*c)/(b*x+a)^(3/2)/(d*x+ c)^(1/2)-4/3*d/(-a*d+b*c)*(-2/(-a*d+b*c)/(b*x+a)^(1/2)/(d*x+c)^(1/2)+4*d/( -a*d+b*c)*(b*x+a)^(1/2)/(d*x+c)^(1/2)/(a*d-b*c)))))
Leaf count of result is larger than twice the leaf count of optimal. 955 vs. \(2 (170) = 340\).
Time = 3.41 (sec) , antiderivative size = 955, normalized size of antiderivative = 4.64 \[ \int \frac {1}{(a+b x)^{11/2} (c+d x)^{3/2}} \, dx=-\frac {2 \, {\left (256 \, b^{5} d^{5} x^{5} + 7 \, b^{5} c^{5} - 45 \, a b^{4} c^{4} d + 126 \, a^{2} b^{3} c^{3} d^{2} - 210 \, a^{3} b^{2} c^{2} d^{3} + 315 \, a^{4} b c d^{4} + 63 \, a^{5} d^{5} + 128 \, {\left (b^{5} c d^{4} + 9 \, a b^{4} d^{5}\right )} x^{4} - 32 \, {\left (b^{5} c^{2} d^{3} - 18 \, a b^{4} c d^{4} - 63 \, a^{2} b^{3} d^{5}\right )} x^{3} + 16 \, {\left (b^{5} c^{3} d^{2} - 9 \, a b^{4} c^{2} d^{3} + 63 \, a^{2} b^{3} c d^{4} + 105 \, a^{3} b^{2} d^{5}\right )} x^{2} - 2 \, {\left (5 \, b^{5} c^{4} d - 36 \, a b^{4} c^{3} d^{2} + 126 \, a^{2} b^{3} c^{2} d^{3} - 420 \, a^{3} b^{2} c d^{4} - 315 \, a^{4} b d^{5}\right )} x\right )} \sqrt {b x + a} \sqrt {d x + c}}{63 \, {\left (a^{5} b^{6} c^{7} - 6 \, a^{6} b^{5} c^{6} d + 15 \, a^{7} b^{4} c^{5} d^{2} - 20 \, a^{8} b^{3} c^{4} d^{3} + 15 \, a^{9} b^{2} c^{3} d^{4} - 6 \, a^{10} b c^{2} d^{5} + a^{11} c d^{6} + {\left (b^{11} c^{6} d - 6 \, a b^{10} c^{5} d^{2} + 15 \, a^{2} b^{9} c^{4} d^{3} - 20 \, a^{3} b^{8} c^{3} d^{4} + 15 \, a^{4} b^{7} c^{2} d^{5} - 6 \, a^{5} b^{6} c d^{6} + a^{6} b^{5} d^{7}\right )} x^{6} + {\left (b^{11} c^{7} - a b^{10} c^{6} d - 15 \, a^{2} b^{9} c^{5} d^{2} + 55 \, a^{3} b^{8} c^{4} d^{3} - 85 \, a^{4} b^{7} c^{3} d^{4} + 69 \, a^{5} b^{6} c^{2} d^{5} - 29 \, a^{6} b^{5} c d^{6} + 5 \, a^{7} b^{4} d^{7}\right )} x^{5} + 5 \, {\left (a b^{10} c^{7} - 4 \, a^{2} b^{9} c^{6} d + 3 \, a^{3} b^{8} c^{5} d^{2} + 10 \, a^{4} b^{7} c^{4} d^{3} - 25 \, a^{5} b^{6} c^{3} d^{4} + 24 \, a^{6} b^{5} c^{2} d^{5} - 11 \, a^{7} b^{4} c d^{6} + 2 \, a^{8} b^{3} d^{7}\right )} x^{4} + 10 \, {\left (a^{2} b^{9} c^{7} - 5 \, a^{3} b^{8} c^{6} d + 9 \, a^{4} b^{7} c^{5} d^{2} - 5 \, a^{5} b^{6} c^{4} d^{3} - 5 \, a^{6} b^{5} c^{3} d^{4} + 9 \, a^{7} b^{4} c^{2} d^{5} - 5 \, a^{8} b^{3} c d^{6} + a^{9} b^{2} d^{7}\right )} x^{3} + 5 \, {\left (2 \, a^{3} b^{8} c^{7} - 11 \, a^{4} b^{7} c^{6} d + 24 \, a^{5} b^{6} c^{5} d^{2} - 25 \, a^{6} b^{5} c^{4} d^{3} + 10 \, a^{7} b^{4} c^{3} d^{4} + 3 \, a^{8} b^{3} c^{2} d^{5} - 4 \, a^{9} b^{2} c d^{6} + a^{10} b d^{7}\right )} x^{2} + {\left (5 \, a^{4} b^{7} c^{7} - 29 \, a^{5} b^{6} c^{6} d + 69 \, a^{6} b^{5} c^{5} d^{2} - 85 \, a^{7} b^{4} c^{4} d^{3} + 55 \, a^{8} b^{3} c^{3} d^{4} - 15 \, a^{9} b^{2} c^{2} d^{5} - a^{10} b c d^{6} + a^{11} d^{7}\right )} x\right )}} \]
-2/63*(256*b^5*d^5*x^5 + 7*b^5*c^5 - 45*a*b^4*c^4*d + 126*a^2*b^3*c^3*d^2 - 210*a^3*b^2*c^2*d^3 + 315*a^4*b*c*d^4 + 63*a^5*d^5 + 128*(b^5*c*d^4 + 9* a*b^4*d^5)*x^4 - 32*(b^5*c^2*d^3 - 18*a*b^4*c*d^4 - 63*a^2*b^3*d^5)*x^3 + 16*(b^5*c^3*d^2 - 9*a*b^4*c^2*d^3 + 63*a^2*b^3*c*d^4 + 105*a^3*b^2*d^5)*x^ 2 - 2*(5*b^5*c^4*d - 36*a*b^4*c^3*d^2 + 126*a^2*b^3*c^2*d^3 - 420*a^3*b^2* c*d^4 - 315*a^4*b*d^5)*x)*sqrt(b*x + a)*sqrt(d*x + c)/(a^5*b^6*c^7 - 6*a^6 *b^5*c^6*d + 15*a^7*b^4*c^5*d^2 - 20*a^8*b^3*c^4*d^3 + 15*a^9*b^2*c^3*d^4 - 6*a^10*b*c^2*d^5 + a^11*c*d^6 + (b^11*c^6*d - 6*a*b^10*c^5*d^2 + 15*a^2* b^9*c^4*d^3 - 20*a^3*b^8*c^3*d^4 + 15*a^4*b^7*c^2*d^5 - 6*a^5*b^6*c*d^6 + a^6*b^5*d^7)*x^6 + (b^11*c^7 - a*b^10*c^6*d - 15*a^2*b^9*c^5*d^2 + 55*a^3* b^8*c^4*d^3 - 85*a^4*b^7*c^3*d^4 + 69*a^5*b^6*c^2*d^5 - 29*a^6*b^5*c*d^6 + 5*a^7*b^4*d^7)*x^5 + 5*(a*b^10*c^7 - 4*a^2*b^9*c^6*d + 3*a^3*b^8*c^5*d^2 + 10*a^4*b^7*c^4*d^3 - 25*a^5*b^6*c^3*d^4 + 24*a^6*b^5*c^2*d^5 - 11*a^7*b^ 4*c*d^6 + 2*a^8*b^3*d^7)*x^4 + 10*(a^2*b^9*c^7 - 5*a^3*b^8*c^6*d + 9*a^4*b ^7*c^5*d^2 - 5*a^5*b^6*c^4*d^3 - 5*a^6*b^5*c^3*d^4 + 9*a^7*b^4*c^2*d^5 - 5 *a^8*b^3*c*d^6 + a^9*b^2*d^7)*x^3 + 5*(2*a^3*b^8*c^7 - 11*a^4*b^7*c^6*d + 24*a^5*b^6*c^5*d^2 - 25*a^6*b^5*c^4*d^3 + 10*a^7*b^4*c^3*d^4 + 3*a^8*b^3*c ^2*d^5 - 4*a^9*b^2*c*d^6 + a^10*b*d^7)*x^2 + (5*a^4*b^7*c^7 - 29*a^5*b^6*c ^6*d + 69*a^6*b^5*c^5*d^2 - 85*a^7*b^4*c^4*d^3 + 55*a^8*b^3*c^3*d^4 - 15*a ^9*b^2*c^2*d^5 - a^10*b*c*d^6 + a^11*d^7)*x)
\[ \int \frac {1}{(a+b x)^{11/2} (c+d x)^{3/2}} \, dx=\int \frac {1}{\left (a + b x\right )^{\frac {11}{2}} \left (c + d x\right )^{\frac {3}{2}}}\, dx \]
Exception generated. \[ \int \frac {1}{(a+b x)^{11/2} (c+d x)^{3/2}} \, dx=\text {Exception raised: ValueError} \]
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. 2438 vs. \(2 (170) = 340\).
Time = 1.13 (sec) , antiderivative size = 2438, normalized size of antiderivative = 11.83 \[ \int \frac {1}{(a+b x)^{11/2} (c+d x)^{3/2}} \, dx=\text {Too large to display} \]
-2*sqrt(b*x + a)*b^2*d^5/((b^6*c^6*abs(b) - 6*a*b^5*c^5*d*abs(b) + 15*a^2* b^4*c^4*d^2*abs(b) - 20*a^3*b^3*c^3*d^3*abs(b) + 15*a^4*b^2*c^2*d^4*abs(b) - 6*a^5*b*c*d^5*abs(b) + a^6*d^6*abs(b))*sqrt(b^2*c + (b*x + a)*b*d - a*b *d)) - 4/63*(193*sqrt(b*d)*b^18*c^8*d^4 - 1544*sqrt(b*d)*a*b^17*c^7*d^5 + 5404*sqrt(b*d)*a^2*b^16*c^6*d^6 - 10808*sqrt(b*d)*a^3*b^15*c^5*d^7 + 13510 *sqrt(b*d)*a^4*b^14*c^4*d^8 - 10808*sqrt(b*d)*a^5*b^13*c^3*d^9 + 5404*sqrt (b*d)*a^6*b^12*c^2*d^10 - 1544*sqrt(b*d)*a^7*b^11*c*d^11 + 193*sqrt(b*d)*a ^8*b^10*d^12 - 1674*sqrt(b*d)*(sqrt(b*d)*sqrt(b*x + a) - sqrt(b^2*c + (b*x + a)*b*d - a*b*d))^2*b^16*c^7*d^4 + 11718*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^15*c^6*d^5 - 35154*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^14*c^5*d^6 + 58590*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^13*c^4*d^7 - 58590*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^12*c^3*d^8 + 35154 *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^11*c^2*d^9 - 11718*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^10*c*d^10 + 1674*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^9*d^11 + 6318 *sqrt(b*d)*(sqrt(b*d)*sqrt(b*x + a) - sqrt(b^2*c + (b*x + a)*b*d - a*b*d)) ^4*b^14*c^6*d^4 - 37908*sqrt(b*d)*(sqrt(b*d)*sqrt(b*x + a) - sqrt(b^2*c...
Time = 1.83 (sec) , antiderivative size = 454, normalized size of antiderivative = 2.20 \[ \int \frac {1}{(a+b x)^{11/2} (c+d x)^{3/2}} \, dx=-\frac {\sqrt {c+d\,x}\,\left (\frac {126\,a^5\,d^5+630\,a^4\,b\,c\,d^4-420\,a^3\,b^2\,c^2\,d^3+252\,a^2\,b^3\,c^3\,d^2-90\,a\,b^4\,c^4\,d+14\,b^5\,c^5}{63\,b^4\,d\,{\left (a\,d-b\,c\right )}^6}+\frac {512\,b\,d^4\,x^5}{63\,{\left (a\,d-b\,c\right )}^6}+\frac {256\,d^3\,x^4\,\left (9\,a\,d+b\,c\right )}{63\,{\left (a\,d-b\,c\right )}^6}+\frac {x\,\left (1260\,a^4\,b\,d^5+1680\,a^3\,b^2\,c\,d^4-504\,a^2\,b^3\,c^2\,d^3+144\,a\,b^4\,c^3\,d^2-20\,b^5\,c^4\,d\right )}{63\,b^4\,d\,{\left (a\,d-b\,c\right )}^6}+\frac {64\,d^2\,x^3\,\left (63\,a^2\,d^2+18\,a\,b\,c\,d-b^2\,c^2\right )}{63\,b\,{\left (a\,d-b\,c\right )}^6}+\frac {32\,d\,x^2\,\left (105\,a^3\,d^3+63\,a^2\,b\,c\,d^2-9\,a\,b^2\,c^2\,d+b^3\,c^3\right )}{63\,b^2\,{\left (a\,d-b\,c\right )}^6}\right )}{x^5\,\sqrt {a+b\,x}+\frac {a^4\,c\,\sqrt {a+b\,x}}{b^4\,d}+\frac {x^4\,\left (4\,a\,d+b\,c\right )\,\sqrt {a+b\,x}}{b\,d}+\frac {2\,a\,x^3\,\left (3\,a\,d+2\,b\,c\right )\,\sqrt {a+b\,x}}{b^2\,d}+\frac {a^3\,x\,\left (a\,d+4\,b\,c\right )\,\sqrt {a+b\,x}}{b^4\,d}+\frac {2\,a^2\,x^2\,\left (2\,a\,d+3\,b\,c\right )\,\sqrt {a+b\,x}}{b^3\,d}} \]
-((c + d*x)^(1/2)*((126*a^5*d^5 + 14*b^5*c^5 + 252*a^2*b^3*c^3*d^2 - 420*a ^3*b^2*c^2*d^3 - 90*a*b^4*c^4*d + 630*a^4*b*c*d^4)/(63*b^4*d*(a*d - b*c)^6 ) + (512*b*d^4*x^5)/(63*(a*d - b*c)^6) + (256*d^3*x^4*(9*a*d + b*c))/(63*( a*d - b*c)^6) + (x*(1260*a^4*b*d^5 - 20*b^5*c^4*d + 144*a*b^4*c^3*d^2 + 16 80*a^3*b^2*c*d^4 - 504*a^2*b^3*c^2*d^3))/(63*b^4*d*(a*d - b*c)^6) + (64*d^ 2*x^3*(63*a^2*d^2 - b^2*c^2 + 18*a*b*c*d))/(63*b*(a*d - b*c)^6) + (32*d*x^ 2*(105*a^3*d^3 + b^3*c^3 - 9*a*b^2*c^2*d + 63*a^2*b*c*d^2))/(63*b^2*(a*d - b*c)^6)))/(x^5*(a + b*x)^(1/2) + (a^4*c*(a + b*x)^(1/2))/(b^4*d) + (x^4*( 4*a*d + b*c)*(a + b*x)^(1/2))/(b*d) + (2*a*x^3*(3*a*d + 2*b*c)*(a + b*x)^( 1/2))/(b^2*d) + (a^3*x*(a*d + 4*b*c)*(a + b*x)^(1/2))/(b^4*d) + (2*a^2*x^2 *(2*a*d + 3*b*c)*(a + b*x)^(1/2))/(b^3*d))