Integrand size = 24, antiderivative size = 346 \[ \int \frac {(A+B x) \left (a+c x^2\right )^3}{(d+e x)^{5/2}} \, dx=\frac {2 (B d-A e) \left (c d^2+a e^2\right )^3}{3 e^8 (d+e x)^{3/2}}-\frac {2 \left (c d^2+a e^2\right )^2 \left (7 B c d^2-6 A c d e+a B e^2\right )}{e^8 \sqrt {d+e x}}-\frac {6 c \left (c d^2+a e^2\right ) \left (7 B c d^3-5 A c d^2 e+3 a B d e^2-a A e^3\right ) \sqrt {d+e x}}{e^8}-\frac {2 c \left (4 A c d e \left (5 c d^2+3 a e^2\right )-B \left (35 c^2 d^4+30 a c d^2 e^2+3 a^2 e^4\right )\right ) (d+e x)^{3/2}}{3 e^8}-\frac {2 c^2 \left (35 B c d^3-15 A c d^2 e+15 a B d e^2-3 a A e^3\right ) (d+e x)^{5/2}}{5 e^8}+\frac {6 c^2 \left (7 B c d^2-2 A c d e+a B e^2\right ) (d+e x)^{7/2}}{7 e^8}-\frac {2 c^3 (7 B d-A e) (d+e x)^{9/2}}{9 e^8}+\frac {2 B c^3 (d+e x)^{11/2}}{11 e^8} \]
2/3*(-A*e+B*d)*(a*e^2+c*d^2)^3/e^8/(e*x+d)^(3/2)-2/3*c*(4*A*c*d*e*(3*a*e^2 +5*c*d^2)-B*(3*a^2*e^4+30*a*c*d^2*e^2+35*c^2*d^4))*(e*x+d)^(3/2)/e^8-2/5*c ^2*(-3*A*a*e^3-15*A*c*d^2*e+15*B*a*d*e^2+35*B*c*d^3)*(e*x+d)^(5/2)/e^8+6/7 *c^2*(-2*A*c*d*e+B*a*e^2+7*B*c*d^2)*(e*x+d)^(7/2)/e^8-2/9*c^3*(-A*e+7*B*d) *(e*x+d)^(9/2)/e^8+2/11*B*c^3*(e*x+d)^(11/2)/e^8-2*(a*e^2+c*d^2)^2*(-6*A*c *d*e+B*a*e^2+7*B*c*d^2)/e^8/(e*x+d)^(1/2)-6*c*(a*e^2+c*d^2)*(-A*a*e^3-5*A* c*d^2*e+3*B*a*d*e^2+7*B*c*d^3)*(e*x+d)^(1/2)/e^8
Time = 0.26 (sec) , antiderivative size = 375, normalized size of antiderivative = 1.08 \[ \int \frac {(A+B x) \left (a+c x^2\right )^3}{(d+e x)^{5/2}} \, dx=\frac {22 A e \left (-105 a^3 e^6+315 a^2 c e^4 \left (8 d^2+12 d e x+3 e^2 x^2\right )+63 a c^2 e^2 \left (128 d^4+192 d^3 e x+48 d^2 e^2 x^2-8 d e^3 x^3+3 e^4 x^4\right )+5 c^3 \left (1024 d^6+1536 d^5 e x+384 d^4 e^2 x^2-64 d^3 e^3 x^3+24 d^2 e^4 x^4-12 d e^5 x^5+7 e^6 x^6\right )\right )-10 B \left (231 a^3 e^6 (2 d+3 e x)+693 a^2 c e^4 \left (16 d^3+24 d^2 e x+6 d e^2 x^2-e^3 x^3\right )+99 a c^2 e^2 \left (256 d^5+384 d^4 e x+96 d^3 e^2 x^2-16 d^2 e^3 x^3+6 d e^4 x^4-3 e^5 x^5\right )+7 c^3 \left (2048 d^7+3072 d^6 e x+768 d^5 e^2 x^2-128 d^4 e^3 x^3+48 d^3 e^4 x^4-24 d^2 e^5 x^5+14 d e^6 x^6-9 e^7 x^7\right )\right )}{3465 e^8 (d+e x)^{3/2}} \]
(22*A*e*(-105*a^3*e^6 + 315*a^2*c*e^4*(8*d^2 + 12*d*e*x + 3*e^2*x^2) + 63* a*c^2*e^2*(128*d^4 + 192*d^3*e*x + 48*d^2*e^2*x^2 - 8*d*e^3*x^3 + 3*e^4*x^ 4) + 5*c^3*(1024*d^6 + 1536*d^5*e*x + 384*d^4*e^2*x^2 - 64*d^3*e^3*x^3 + 2 4*d^2*e^4*x^4 - 12*d*e^5*x^5 + 7*e^6*x^6)) - 10*B*(231*a^3*e^6*(2*d + 3*e* x) + 693*a^2*c*e^4*(16*d^3 + 24*d^2*e*x + 6*d*e^2*x^2 - e^3*x^3) + 99*a*c^ 2*e^2*(256*d^5 + 384*d^4*e*x + 96*d^3*e^2*x^2 - 16*d^2*e^3*x^3 + 6*d*e^4*x ^4 - 3*e^5*x^5) + 7*c^3*(2048*d^7 + 3072*d^6*e*x + 768*d^5*e^2*x^2 - 128*d ^4*e^3*x^3 + 48*d^3*e^4*x^4 - 24*d^2*e^5*x^5 + 14*d*e^6*x^6 - 9*e^7*x^7))) /(3465*e^8*(d + e*x)^(3/2))
Time = 0.46 (sec) , antiderivative size = 346, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.083, Rules used = {652, 2009}
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.
| \(\displaystyle \int \frac {\left (a+c x^2\right )^3 (A+B x)}{(d+e x)^{5/2}} \, dx\) |
| \(\Big \downarrow \) 652 |
| \(\displaystyle \int \left (-\frac {c \sqrt {d+e x} \left (-3 a^2 B e^4+12 a A c d e^3-30 a B c d^2 e^2+20 A c^2 d^3 e-35 B c^2 d^4\right )}{e^7}-\frac {3 c^2 (d+e x)^{5/2} \left (-a B e^2+2 A c d e-7 B c d^2\right )}{e^7}+\frac {c^2 (d+e x)^{3/2} \left (3 a A e^3-15 a B d e^2+15 A c d^2 e-35 B c d^3\right )}{e^7}+\frac {\left (a e^2+c d^2\right )^2 \left (a B e^2-6 A c d e+7 B c d^2\right )}{e^7 (d+e x)^{3/2}}+\frac {\left (a e^2+c d^2\right )^3 (A e-B d)}{e^7 (d+e x)^{5/2}}+\frac {3 c \left (a e^2+c d^2\right ) \left (a A e^3-3 a B d e^2+5 A c d^2 e-7 B c d^3\right )}{e^7 \sqrt {d+e x}}+\frac {c^3 (d+e x)^{7/2} (A e-7 B d)}{e^7}+\frac {B c^3 (d+e x)^{9/2}}{e^7}\right )dx\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle -\frac {2 c (d+e x)^{3/2} \left (4 A c d e \left (3 a e^2+5 c d^2\right )-B \left (3 a^2 e^4+30 a c d^2 e^2+35 c^2 d^4\right )\right )}{3 e^8}+\frac {6 c^2 (d+e x)^{7/2} \left (a B e^2-2 A c d e+7 B c d^2\right )}{7 e^8}-\frac {2 c^2 (d+e x)^{5/2} \left (-3 a A e^3+15 a B d e^2-15 A c d^2 e+35 B c d^3\right )}{5 e^8}-\frac {2 \left (a e^2+c d^2\right )^2 \left (a B e^2-6 A c d e+7 B c d^2\right )}{e^8 \sqrt {d+e x}}+\frac {2 \left (a e^2+c d^2\right )^3 (B d-A e)}{3 e^8 (d+e x)^{3/2}}-\frac {6 c \sqrt {d+e x} \left (a e^2+c d^2\right ) \left (-a A e^3+3 a B d e^2-5 A c d^2 e+7 B c d^3\right )}{e^8}-\frac {2 c^3 (d+e x)^{9/2} (7 B d-A e)}{9 e^8}+\frac {2 B c^3 (d+e x)^{11/2}}{11 e^8}\) |
(2*(B*d - A*e)*(c*d^2 + a*e^2)^3)/(3*e^8*(d + e*x)^(3/2)) - (2*(c*d^2 + a* e^2)^2*(7*B*c*d^2 - 6*A*c*d*e + a*B*e^2))/(e^8*Sqrt[d + e*x]) - (6*c*(c*d^ 2 + a*e^2)*(7*B*c*d^3 - 5*A*c*d^2*e + 3*a*B*d*e^2 - a*A*e^3)*Sqrt[d + e*x] )/e^8 - (2*c*(4*A*c*d*e*(5*c*d^2 + 3*a*e^2) - B*(35*c^2*d^4 + 30*a*c*d^2*e ^2 + 3*a^2*e^4))*(d + e*x)^(3/2))/(3*e^8) - (2*c^2*(35*B*c*d^3 - 15*A*c*d^ 2*e + 15*a*B*d*e^2 - 3*a*A*e^3)*(d + e*x)^(5/2))/(5*e^8) + (6*c^2*(7*B*c*d ^2 - 2*A*c*d*e + a*B*e^2)*(d + e*x)^(7/2))/(7*e^8) - (2*c^3*(7*B*d - A*e)* (d + e*x)^(9/2))/(9*e^8) + (2*B*c^3*(d + e*x)^(11/2))/(11*e^8)
3.15.43.3.1 Defintions of rubi rules used
Int[((d_.) + (e_.)*(x_))^(m_.)*((f_.) + (g_.)*(x_))^(n_.)*((a_) + (c_.)*(x_ )^2)^(p_.), x_Symbol] :> Int[ExpandIntegrand[(d + e*x)^m*(f + g*x)^n*(a + c *x^2)^p, x], x] /; FreeQ[{a, c, d, e, f, g, m, n}, x] && IGtQ[p, 0]
Time = 0.45 (sec) , antiderivative size = 295, normalized size of antiderivative = 0.85
| method | result | size |
| pseudoelliptic | \(-\frac {2 \left (\left (-\frac {x^{6} \left (\frac {9 B x}{11}+A \right ) c^{3}}{3}-\frac {9 \left (\frac {5 B x}{7}+A \right ) x^{4} a \,c^{2}}{5}-9 \left (\frac {B x}{3}+A \right ) x^{2} a^{2} c +a^{3} \left (3 B x +A \right )\right ) e^{7}-36 \left (-\frac {\left (\frac {49 B x}{66}+A \right ) x^{5} c^{3}}{63}-\frac {2 \left (\frac {15 B x}{28}+A \right ) x^{3} a \,c^{2}}{15}+a^{2} x \left (-\frac {B x}{2}+A \right ) c -\frac {B \,a^{3}}{18}\right ) d \,e^{6}-24 c \,d^{2} \left (\frac {x^{4} \left (\frac {7 B x}{11}+A \right ) c^{2}}{21}+\frac {6 \left (\frac {5 B x}{21}+A \right ) x^{2} a c}{5}+a^{2} \left (-3 B x +A \right )\right ) e^{5}-\frac {576 c \,d^{3} \left (-\frac {5 \left (\frac {21 B x}{44}+A \right ) x^{3} c^{2}}{189}+a x \left (-\frac {5 B x}{14}+A \right ) c -\frac {5 B \,a^{2}}{12}\right ) e^{4}}{5}-\frac {384 c^{2} \left (\frac {5 \left (\frac {7 B x}{33}+A \right ) x^{2} c}{21}+a \left (-\frac {15 B x}{7}+A \right )\right ) d^{4} e^{3}}{5}-\frac {512 c^{2} d^{5} \left (x \left (-\frac {7 B x}{22}+A \right ) c -\frac {3 B a}{2}\right ) e^{2}}{7}-\frac {1024 c^{3} d^{6} \left (-\frac {21 B x}{11}+A \right ) e}{21}+\frac {2048 B \,c^{3} d^{7}}{33}\right )}{3 \left (e x +d \right )^{\frac {3}{2}} e^{8}}\) | \(295\) |
| risch | \(\frac {2 c \left (315 B \,x^{5} c^{2} e^{5}+385 A \,x^{4} c^{2} e^{5}-1120 B \,x^{4} c^{2} d \,e^{4}-1430 A \,x^{3} c^{2} d \,e^{4}+1485 B \,x^{3} a c \,e^{5}+2765 B \,x^{3} c^{2} d^{2} e^{3}+2079 A \,x^{2} a c \,e^{5}+3795 A \,x^{2} c^{2} d^{2} e^{3}-5940 B \,x^{2} a c d \,e^{4}-6090 B \,x^{2} c^{2} d^{3} e^{2}-9702 A x a c d \,e^{4}-9680 A x \,c^{2} d^{3} e^{2}+3465 B x \,a^{2} e^{5}+18315 B x a c \,d^{2} e^{3}+13895 B x \,c^{2} d^{4} e +10395 A \,a^{2} e^{5}+50589 A a c \,d^{2} e^{3}+36685 A \,c^{2} d^{4} e -27720 B \,a^{2} d \,e^{4}-78210 B a c \,d^{3} e^{2}-48580 B \,c^{2} d^{5}\right ) \sqrt {e x +d}}{3465 e^{8}}-\frac {2 \left (-18 A c d \,e^{2} x +3 B x a \,e^{3}+21 B c \,d^{2} e x +A a \,e^{3}-17 A c \,d^{2} e +2 B a d \,e^{2}+20 B c \,d^{3}\right ) \left (a^{2} e^{4}+2 a c \,d^{2} e^{2}+c^{2} d^{4}\right )}{3 e^{8} \left (e x +d \right )^{\frac {3}{2}}}\) | \(354\) |
| gosper | \(-\frac {2 \left (-315 B \,x^{7} c^{3} e^{7}-385 A \,x^{6} c^{3} e^{7}+490 B \,x^{6} c^{3} d \,e^{6}+660 A \,x^{5} c^{3} d \,e^{6}-1485 B \,x^{5} a \,c^{2} e^{7}-840 B \,x^{5} c^{3} d^{2} e^{5}-2079 A \,x^{4} a \,c^{2} e^{7}-1320 A \,x^{4} c^{3} d^{2} e^{5}+2970 B \,x^{4} a \,c^{2} d \,e^{6}+1680 B \,x^{4} c^{3} d^{3} e^{4}+5544 A \,x^{3} a \,c^{2} d \,e^{6}+3520 A \,x^{3} c^{3} d^{3} e^{4}-3465 B \,x^{3} a^{2} c \,e^{7}-7920 B \,x^{3} a \,c^{2} d^{2} e^{5}-4480 B \,x^{3} c^{3} d^{4} e^{3}-10395 A \,x^{2} a^{2} c \,e^{7}-33264 A \,x^{2} a \,c^{2} d^{2} e^{5}-21120 A \,x^{2} c^{3} d^{4} e^{3}+20790 B \,x^{2} a^{2} c d \,e^{6}+47520 B \,x^{2} a \,c^{2} d^{3} e^{4}+26880 B \,x^{2} c^{3} d^{5} e^{2}-41580 A x \,a^{2} c d \,e^{6}-133056 A x a \,c^{2} d^{3} e^{4}-84480 A x \,c^{3} d^{5} e^{2}+3465 B x \,a^{3} e^{7}+83160 B x \,a^{2} c \,d^{2} e^{5}+190080 B x a \,c^{2} d^{4} e^{3}+107520 B x \,c^{3} d^{6} e +1155 A \,a^{3} e^{7}-27720 A \,a^{2} c \,d^{2} e^{5}-88704 A a \,c^{2} d^{4} e^{3}-56320 A \,c^{3} d^{6} e +2310 B \,a^{3} d \,e^{6}+55440 B \,a^{2} c \,d^{3} e^{4}+126720 B a \,c^{2} d^{5} e^{2}+71680 B \,c^{3} d^{7}\right )}{3465 \left (e x +d \right )^{\frac {3}{2}} e^{8}}\) | \(489\) |
| trager | \(-\frac {2 \left (-315 B \,x^{7} c^{3} e^{7}-385 A \,x^{6} c^{3} e^{7}+490 B \,x^{6} c^{3} d \,e^{6}+660 A \,x^{5} c^{3} d \,e^{6}-1485 B \,x^{5} a \,c^{2} e^{7}-840 B \,x^{5} c^{3} d^{2} e^{5}-2079 A \,x^{4} a \,c^{2} e^{7}-1320 A \,x^{4} c^{3} d^{2} e^{5}+2970 B \,x^{4} a \,c^{2} d \,e^{6}+1680 B \,x^{4} c^{3} d^{3} e^{4}+5544 A \,x^{3} a \,c^{2} d \,e^{6}+3520 A \,x^{3} c^{3} d^{3} e^{4}-3465 B \,x^{3} a^{2} c \,e^{7}-7920 B \,x^{3} a \,c^{2} d^{2} e^{5}-4480 B \,x^{3} c^{3} d^{4} e^{3}-10395 A \,x^{2} a^{2} c \,e^{7}-33264 A \,x^{2} a \,c^{2} d^{2} e^{5}-21120 A \,x^{2} c^{3} d^{4} e^{3}+20790 B \,x^{2} a^{2} c d \,e^{6}+47520 B \,x^{2} a \,c^{2} d^{3} e^{4}+26880 B \,x^{2} c^{3} d^{5} e^{2}-41580 A x \,a^{2} c d \,e^{6}-133056 A x a \,c^{2} d^{3} e^{4}-84480 A x \,c^{3} d^{5} e^{2}+3465 B x \,a^{3} e^{7}+83160 B x \,a^{2} c \,d^{2} e^{5}+190080 B x a \,c^{2} d^{4} e^{3}+107520 B x \,c^{3} d^{6} e +1155 A \,a^{3} e^{7}-27720 A \,a^{2} c \,d^{2} e^{5}-88704 A a \,c^{2} d^{4} e^{3}-56320 A \,c^{3} d^{6} e +2310 B \,a^{3} d \,e^{6}+55440 B \,a^{2} c \,d^{3} e^{4}+126720 B a \,c^{2} d^{5} e^{2}+71680 B \,c^{3} d^{7}\right )}{3465 \left (e x +d \right )^{\frac {3}{2}} e^{8}}\) | \(489\) |
| derivativedivides | \(\frac {\frac {2 B \,c^{3} \left (e x +d \right )^{\frac {11}{2}}}{11}+\frac {2 A \,c^{3} e \left (e x +d \right )^{\frac {9}{2}}}{9}-\frac {14 B \,c^{3} d \left (e x +d \right )^{\frac {9}{2}}}{9}-\frac {12 A \,c^{3} d e \left (e x +d \right )^{\frac {7}{2}}}{7}+\frac {6 B a \,c^{2} e^{2} \left (e x +d \right )^{\frac {7}{2}}}{7}+6 B \,c^{3} d^{2} \left (e x +d \right )^{\frac {7}{2}}+\frac {6 A a \,c^{2} e^{3} \left (e x +d \right )^{\frac {5}{2}}}{5}+6 A \,c^{3} d^{2} e \left (e x +d \right )^{\frac {5}{2}}-6 B a \,c^{2} d \,e^{2} \left (e x +d \right )^{\frac {5}{2}}-14 B \,c^{3} d^{3} \left (e x +d \right )^{\frac {5}{2}}-8 A a \,c^{2} d \,e^{3} \left (e x +d \right )^{\frac {3}{2}}-\frac {40 A \,c^{3} d^{3} e \left (e x +d \right )^{\frac {3}{2}}}{3}+2 B \,a^{2} c \,e^{4} \left (e x +d \right )^{\frac {3}{2}}+20 B a \,c^{2} d^{2} e^{2} \left (e x +d \right )^{\frac {3}{2}}+\frac {70 B \,c^{3} d^{4} \left (e x +d \right )^{\frac {3}{2}}}{3}+6 A \,a^{2} c \,e^{5} \sqrt {e x +d}+36 A a \,c^{2} d^{2} e^{3} \sqrt {e x +d}+30 A \,c^{3} d^{4} e \sqrt {e x +d}-18 B \,a^{2} c d \,e^{4} \sqrt {e x +d}-60 B a \,c^{2} d^{3} e^{2} \sqrt {e x +d}-42 B \,c^{3} d^{5} \sqrt {e x +d}-\frac {2 \left (A \,a^{3} e^{7}+3 A \,a^{2} c \,d^{2} e^{5}+3 A a \,c^{2} d^{4} e^{3}+A \,c^{3} d^{6} e -B \,a^{3} d \,e^{6}-3 B \,a^{2} c \,d^{3} e^{4}-3 B a \,c^{2} d^{5} e^{2}-B \,c^{3} d^{7}\right )}{3 \left (e x +d \right )^{\frac {3}{2}}}-\frac {2 \left (-6 A \,a^{2} c d \,e^{5}-12 A a \,c^{2} d^{3} e^{3}-6 A \,c^{3} d^{5} e +B \,a^{3} e^{6}+9 B \,a^{2} c \,d^{2} e^{4}+15 B a \,c^{2} d^{4} e^{2}+7 B \,c^{3} d^{6}\right )}{\sqrt {e x +d}}}{e^{8}}\) | \(544\) |
| default | \(\frac {\frac {2 B \,c^{3} \left (e x +d \right )^{\frac {11}{2}}}{11}+\frac {2 A \,c^{3} e \left (e x +d \right )^{\frac {9}{2}}}{9}-\frac {14 B \,c^{3} d \left (e x +d \right )^{\frac {9}{2}}}{9}-\frac {12 A \,c^{3} d e \left (e x +d \right )^{\frac {7}{2}}}{7}+\frac {6 B a \,c^{2} e^{2} \left (e x +d \right )^{\frac {7}{2}}}{7}+6 B \,c^{3} d^{2} \left (e x +d \right )^{\frac {7}{2}}+\frac {6 A a \,c^{2} e^{3} \left (e x +d \right )^{\frac {5}{2}}}{5}+6 A \,c^{3} d^{2} e \left (e x +d \right )^{\frac {5}{2}}-6 B a \,c^{2} d \,e^{2} \left (e x +d \right )^{\frac {5}{2}}-14 B \,c^{3} d^{3} \left (e x +d \right )^{\frac {5}{2}}-8 A a \,c^{2} d \,e^{3} \left (e x +d \right )^{\frac {3}{2}}-\frac {40 A \,c^{3} d^{3} e \left (e x +d \right )^{\frac {3}{2}}}{3}+2 B \,a^{2} c \,e^{4} \left (e x +d \right )^{\frac {3}{2}}+20 B a \,c^{2} d^{2} e^{2} \left (e x +d \right )^{\frac {3}{2}}+\frac {70 B \,c^{3} d^{4} \left (e x +d \right )^{\frac {3}{2}}}{3}+6 A \,a^{2} c \,e^{5} \sqrt {e x +d}+36 A a \,c^{2} d^{2} e^{3} \sqrt {e x +d}+30 A \,c^{3} d^{4} e \sqrt {e x +d}-18 B \,a^{2} c d \,e^{4} \sqrt {e x +d}-60 B a \,c^{2} d^{3} e^{2} \sqrt {e x +d}-42 B \,c^{3} d^{5} \sqrt {e x +d}-\frac {2 \left (A \,a^{3} e^{7}+3 A \,a^{2} c \,d^{2} e^{5}+3 A a \,c^{2} d^{4} e^{3}+A \,c^{3} d^{6} e -B \,a^{3} d \,e^{6}-3 B \,a^{2} c \,d^{3} e^{4}-3 B a \,c^{2} d^{5} e^{2}-B \,c^{3} d^{7}\right )}{3 \left (e x +d \right )^{\frac {3}{2}}}-\frac {2 \left (-6 A \,a^{2} c d \,e^{5}-12 A a \,c^{2} d^{3} e^{3}-6 A \,c^{3} d^{5} e +B \,a^{3} e^{6}+9 B \,a^{2} c \,d^{2} e^{4}+15 B a \,c^{2} d^{4} e^{2}+7 B \,c^{3} d^{6}\right )}{\sqrt {e x +d}}}{e^{8}}\) | \(544\) |
-2/3*((-1/3*x^6*(9/11*B*x+A)*c^3-9/5*(5/7*B*x+A)*x^4*a*c^2-9*(1/3*B*x+A)*x ^2*a^2*c+a^3*(3*B*x+A))*e^7-36*(-1/63*(49/66*B*x+A)*x^5*c^3-2/15*(15/28*B* x+A)*x^3*a*c^2+a^2*x*(-1/2*B*x+A)*c-1/18*B*a^3)*d*e^6-24*c*d^2*(1/21*x^4*( 7/11*B*x+A)*c^2+6/5*(5/21*B*x+A)*x^2*a*c+a^2*(-3*B*x+A))*e^5-576/5*c*d^3*( -5/189*(21/44*B*x+A)*x^3*c^2+a*x*(-5/14*B*x+A)*c-5/12*B*a^2)*e^4-384/5*c^2 *(5/21*(7/33*B*x+A)*x^2*c+a*(-15/7*B*x+A))*d^4*e^3-512/7*c^2*d^5*(x*(-7/22 *B*x+A)*c-3/2*B*a)*e^2-1024/21*c^3*d^6*(-21/11*B*x+A)*e+2048/33*B*c^3*d^7) /(e*x+d)^(3/2)/e^8
Time = 0.39 (sec) , antiderivative size = 475, normalized size of antiderivative = 1.37 \[ \int \frac {(A+B x) \left (a+c x^2\right )^3}{(d+e x)^{5/2}} \, dx=\frac {2 \, {\left (315 \, B c^{3} e^{7} x^{7} - 71680 \, B c^{3} d^{7} + 56320 \, A c^{3} d^{6} e - 126720 \, B a c^{2} d^{5} e^{2} + 88704 \, A a c^{2} d^{4} e^{3} - 55440 \, B a^{2} c d^{3} e^{4} + 27720 \, A a^{2} c d^{2} e^{5} - 2310 \, B a^{3} d e^{6} - 1155 \, A a^{3} e^{7} - 35 \, {\left (14 \, B c^{3} d e^{6} - 11 \, A c^{3} e^{7}\right )} x^{6} + 15 \, {\left (56 \, B c^{3} d^{2} e^{5} - 44 \, A c^{3} d e^{6} + 99 \, B a c^{2} e^{7}\right )} x^{5} - 3 \, {\left (560 \, B c^{3} d^{3} e^{4} - 440 \, A c^{3} d^{2} e^{5} + 990 \, B a c^{2} d e^{6} - 693 \, A a c^{2} e^{7}\right )} x^{4} + {\left (4480 \, B c^{3} d^{4} e^{3} - 3520 \, A c^{3} d^{3} e^{4} + 7920 \, B a c^{2} d^{2} e^{5} - 5544 \, A a c^{2} d e^{6} + 3465 \, B a^{2} c e^{7}\right )} x^{3} - 3 \, {\left (8960 \, B c^{3} d^{5} e^{2} - 7040 \, A c^{3} d^{4} e^{3} + 15840 \, B a c^{2} d^{3} e^{4} - 11088 \, A a c^{2} d^{2} e^{5} + 6930 \, B a^{2} c d e^{6} - 3465 \, A a^{2} c e^{7}\right )} x^{2} - 3 \, {\left (35840 \, B c^{3} d^{6} e - 28160 \, A c^{3} d^{5} e^{2} + 63360 \, B a c^{2} d^{4} e^{3} - 44352 \, A a c^{2} d^{3} e^{4} + 27720 \, B a^{2} c d^{2} e^{5} - 13860 \, A a^{2} c d e^{6} + 1155 \, B a^{3} e^{7}\right )} x\right )} \sqrt {e x + d}}{3465 \, {\left (e^{10} x^{2} + 2 \, d e^{9} x + d^{2} e^{8}\right )}} \]
2/3465*(315*B*c^3*e^7*x^7 - 71680*B*c^3*d^7 + 56320*A*c^3*d^6*e - 126720*B *a*c^2*d^5*e^2 + 88704*A*a*c^2*d^4*e^3 - 55440*B*a^2*c*d^3*e^4 + 27720*A*a ^2*c*d^2*e^5 - 2310*B*a^3*d*e^6 - 1155*A*a^3*e^7 - 35*(14*B*c^3*d*e^6 - 11 *A*c^3*e^7)*x^6 + 15*(56*B*c^3*d^2*e^5 - 44*A*c^3*d*e^6 + 99*B*a*c^2*e^7)* x^5 - 3*(560*B*c^3*d^3*e^4 - 440*A*c^3*d^2*e^5 + 990*B*a*c^2*d*e^6 - 693*A *a*c^2*e^7)*x^4 + (4480*B*c^3*d^4*e^3 - 3520*A*c^3*d^3*e^4 + 7920*B*a*c^2* d^2*e^5 - 5544*A*a*c^2*d*e^6 + 3465*B*a^2*c*e^7)*x^3 - 3*(8960*B*c^3*d^5*e ^2 - 7040*A*c^3*d^4*e^3 + 15840*B*a*c^2*d^3*e^4 - 11088*A*a*c^2*d^2*e^5 + 6930*B*a^2*c*d*e^6 - 3465*A*a^2*c*e^7)*x^2 - 3*(35840*B*c^3*d^6*e - 28160* A*c^3*d^5*e^2 + 63360*B*a*c^2*d^4*e^3 - 44352*A*a*c^2*d^3*e^4 + 27720*B*a^ 2*c*d^2*e^5 - 13860*A*a^2*c*d*e^6 + 1155*B*a^3*e^7)*x)*sqrt(e*x + d)/(e^10 *x^2 + 2*d*e^9*x + d^2*e^8)
Time = 16.82 (sec) , antiderivative size = 507, normalized size of antiderivative = 1.47 \[ \int \frac {(A+B x) \left (a+c x^2\right )^3}{(d+e x)^{5/2}} \, dx=\begin {cases} \frac {2 \left (\frac {B c^{3} \left (d + e x\right )^{\frac {11}{2}}}{11 e^{7}} + \frac {\left (d + e x\right )^{\frac {9}{2}} \left (A c^{3} e - 7 B c^{3} d\right )}{9 e^{7}} + \frac {\left (d + e x\right )^{\frac {7}{2}} \left (- 6 A c^{3} d e + 3 B a c^{2} e^{2} + 21 B c^{3} d^{2}\right )}{7 e^{7}} + \frac {\left (d + e x\right )^{\frac {5}{2}} \cdot \left (3 A a c^{2} e^{3} + 15 A c^{3} d^{2} e - 15 B a c^{2} d e^{2} - 35 B c^{3} d^{3}\right )}{5 e^{7}} + \frac {\left (d + e x\right )^{\frac {3}{2}} \left (- 12 A a c^{2} d e^{3} - 20 A c^{3} d^{3} e + 3 B a^{2} c e^{4} + 30 B a c^{2} d^{2} e^{2} + 35 B c^{3} d^{4}\right )}{3 e^{7}} + \frac {\sqrt {d + e x} \left (3 A a^{2} c e^{5} + 18 A a c^{2} d^{2} e^{3} + 15 A c^{3} d^{4} e - 9 B a^{2} c d e^{4} - 30 B a c^{2} d^{3} e^{2} - 21 B c^{3} d^{5}\right )}{e^{7}} - \frac {\left (a e^{2} + c d^{2}\right )^{2} \left (- 6 A c d e + B a e^{2} + 7 B c d^{2}\right )}{e^{7} \sqrt {d + e x}} + \frac {\left (- A e + B d\right ) \left (a e^{2} + c d^{2}\right )^{3}}{3 e^{7} \left (d + e x\right )^{\frac {3}{2}}}\right )}{e} & \text {for}\: e \neq 0 \\\frac {A a^{3} x + A a^{2} c x^{3} + \frac {3 A a c^{2} x^{5}}{5} + \frac {A c^{3} x^{7}}{7} + \frac {B a^{3} x^{2}}{2} + \frac {3 B a^{2} c x^{4}}{4} + \frac {B a c^{2} x^{6}}{2} + \frac {B c^{3} x^{8}}{8}}{d^{\frac {5}{2}}} & \text {otherwise} \end {cases} \]
Piecewise((2*(B*c**3*(d + e*x)**(11/2)/(11*e**7) + (d + e*x)**(9/2)*(A*c** 3*e - 7*B*c**3*d)/(9*e**7) + (d + e*x)**(7/2)*(-6*A*c**3*d*e + 3*B*a*c**2* e**2 + 21*B*c**3*d**2)/(7*e**7) + (d + e*x)**(5/2)*(3*A*a*c**2*e**3 + 15*A *c**3*d**2*e - 15*B*a*c**2*d*e**2 - 35*B*c**3*d**3)/(5*e**7) + (d + e*x)** (3/2)*(-12*A*a*c**2*d*e**3 - 20*A*c**3*d**3*e + 3*B*a**2*c*e**4 + 30*B*a*c **2*d**2*e**2 + 35*B*c**3*d**4)/(3*e**7) + sqrt(d + e*x)*(3*A*a**2*c*e**5 + 18*A*a*c**2*d**2*e**3 + 15*A*c**3*d**4*e - 9*B*a**2*c*d*e**4 - 30*B*a*c* *2*d**3*e**2 - 21*B*c**3*d**5)/e**7 - (a*e**2 + c*d**2)**2*(-6*A*c*d*e + B *a*e**2 + 7*B*c*d**2)/(e**7*sqrt(d + e*x)) + (-A*e + B*d)*(a*e**2 + c*d**2 )**3/(3*e**7*(d + e*x)**(3/2)))/e, Ne(e, 0)), ((A*a**3*x + A*a**2*c*x**3 + 3*A*a*c**2*x**5/5 + A*c**3*x**7/7 + B*a**3*x**2/2 + 3*B*a**2*c*x**4/4 + B *a*c**2*x**6/2 + B*c**3*x**8/8)/d**(5/2), True))
Time = 0.21 (sec) , antiderivative size = 459, normalized size of antiderivative = 1.33 \[ \int \frac {(A+B x) \left (a+c x^2\right )^3}{(d+e x)^{5/2}} \, dx=\frac {2 \, {\left (\frac {315 \, {\left (e x + d\right )}^{\frac {11}{2}} B c^{3} - 385 \, {\left (7 \, B c^{3} d - A c^{3} e\right )} {\left (e x + d\right )}^{\frac {9}{2}} + 1485 \, {\left (7 \, B c^{3} d^{2} - 2 \, A c^{3} d e + B a c^{2} e^{2}\right )} {\left (e x + d\right )}^{\frac {7}{2}} - 693 \, {\left (35 \, B c^{3} d^{3} - 15 \, A c^{3} d^{2} e + 15 \, B a c^{2} d e^{2} - 3 \, A a c^{2} e^{3}\right )} {\left (e x + d\right )}^{\frac {5}{2}} + 1155 \, {\left (35 \, B c^{3} d^{4} - 20 \, A c^{3} d^{3} e + 30 \, B a c^{2} d^{2} e^{2} - 12 \, A a c^{2} d e^{3} + 3 \, B a^{2} c e^{4}\right )} {\left (e x + d\right )}^{\frac {3}{2}} - 10395 \, {\left (7 \, B c^{3} d^{5} - 5 \, A c^{3} d^{4} e + 10 \, B a c^{2} d^{3} e^{2} - 6 \, A a c^{2} d^{2} e^{3} + 3 \, B a^{2} c d e^{4} - A a^{2} c e^{5}\right )} \sqrt {e x + d}}{e^{7}} + \frac {1155 \, {\left (B c^{3} d^{7} - A c^{3} d^{6} e + 3 \, B a c^{2} d^{5} e^{2} - 3 \, A a c^{2} d^{4} e^{3} + 3 \, B a^{2} c d^{3} e^{4} - 3 \, A a^{2} c d^{2} e^{5} + B a^{3} d e^{6} - A a^{3} e^{7} - 3 \, {\left (7 \, B c^{3} d^{6} - 6 \, A c^{3} d^{5} e + 15 \, B a c^{2} d^{4} e^{2} - 12 \, A a c^{2} d^{3} e^{3} + 9 \, B a^{2} c d^{2} e^{4} - 6 \, A a^{2} c d e^{5} + B a^{3} e^{6}\right )} {\left (e x + d\right )}\right )}}{{\left (e x + d\right )}^{\frac {3}{2}} e^{7}}\right )}}{3465 \, e} \]
2/3465*((315*(e*x + d)^(11/2)*B*c^3 - 385*(7*B*c^3*d - A*c^3*e)*(e*x + d)^ (9/2) + 1485*(7*B*c^3*d^2 - 2*A*c^3*d*e + B*a*c^2*e^2)*(e*x + d)^(7/2) - 6 93*(35*B*c^3*d^3 - 15*A*c^3*d^2*e + 15*B*a*c^2*d*e^2 - 3*A*a*c^2*e^3)*(e*x + d)^(5/2) + 1155*(35*B*c^3*d^4 - 20*A*c^3*d^3*e + 30*B*a*c^2*d^2*e^2 - 1 2*A*a*c^2*d*e^3 + 3*B*a^2*c*e^4)*(e*x + d)^(3/2) - 10395*(7*B*c^3*d^5 - 5* A*c^3*d^4*e + 10*B*a*c^2*d^3*e^2 - 6*A*a*c^2*d^2*e^3 + 3*B*a^2*c*d*e^4 - A *a^2*c*e^5)*sqrt(e*x + d))/e^7 + 1155*(B*c^3*d^7 - A*c^3*d^6*e + 3*B*a*c^2 *d^5*e^2 - 3*A*a*c^2*d^4*e^3 + 3*B*a^2*c*d^3*e^4 - 3*A*a^2*c*d^2*e^5 + B*a ^3*d*e^6 - A*a^3*e^7 - 3*(7*B*c^3*d^6 - 6*A*c^3*d^5*e + 15*B*a*c^2*d^4*e^2 - 12*A*a*c^2*d^3*e^3 + 9*B*a^2*c*d^2*e^4 - 6*A*a^2*c*d*e^5 + B*a^3*e^6)*( e*x + d))/((e*x + d)^(3/2)*e^7))/e
Time = 0.29 (sec) , antiderivative size = 602, normalized size of antiderivative = 1.74 \[ \int \frac {(A+B x) \left (a+c x^2\right )^3}{(d+e x)^{5/2}} \, dx=-\frac {2 \, {\left (21 \, {\left (e x + d\right )} B c^{3} d^{6} - B c^{3} d^{7} - 18 \, {\left (e x + d\right )} A c^{3} d^{5} e + A c^{3} d^{6} e + 45 \, {\left (e x + d\right )} B a c^{2} d^{4} e^{2} - 3 \, B a c^{2} d^{5} e^{2} - 36 \, {\left (e x + d\right )} A a c^{2} d^{3} e^{3} + 3 \, A a c^{2} d^{4} e^{3} + 27 \, {\left (e x + d\right )} B a^{2} c d^{2} e^{4} - 3 \, B a^{2} c d^{3} e^{4} - 18 \, {\left (e x + d\right )} A a^{2} c d e^{5} + 3 \, A a^{2} c d^{2} e^{5} + 3 \, {\left (e x + d\right )} B a^{3} e^{6} - B a^{3} d e^{6} + A a^{3} e^{7}\right )}}{3 \, {\left (e x + d\right )}^{\frac {3}{2}} e^{8}} + \frac {2 \, {\left (315 \, {\left (e x + d\right )}^{\frac {11}{2}} B c^{3} e^{80} - 2695 \, {\left (e x + d\right )}^{\frac {9}{2}} B c^{3} d e^{80} + 10395 \, {\left (e x + d\right )}^{\frac {7}{2}} B c^{3} d^{2} e^{80} - 24255 \, {\left (e x + d\right )}^{\frac {5}{2}} B c^{3} d^{3} e^{80} + 40425 \, {\left (e x + d\right )}^{\frac {3}{2}} B c^{3} d^{4} e^{80} - 72765 \, \sqrt {e x + d} B c^{3} d^{5} e^{80} + 385 \, {\left (e x + d\right )}^{\frac {9}{2}} A c^{3} e^{81} - 2970 \, {\left (e x + d\right )}^{\frac {7}{2}} A c^{3} d e^{81} + 10395 \, {\left (e x + d\right )}^{\frac {5}{2}} A c^{3} d^{2} e^{81} - 23100 \, {\left (e x + d\right )}^{\frac {3}{2}} A c^{3} d^{3} e^{81} + 51975 \, \sqrt {e x + d} A c^{3} d^{4} e^{81} + 1485 \, {\left (e x + d\right )}^{\frac {7}{2}} B a c^{2} e^{82} - 10395 \, {\left (e x + d\right )}^{\frac {5}{2}} B a c^{2} d e^{82} + 34650 \, {\left (e x + d\right )}^{\frac {3}{2}} B a c^{2} d^{2} e^{82} - 103950 \, \sqrt {e x + d} B a c^{2} d^{3} e^{82} + 2079 \, {\left (e x + d\right )}^{\frac {5}{2}} A a c^{2} e^{83} - 13860 \, {\left (e x + d\right )}^{\frac {3}{2}} A a c^{2} d e^{83} + 62370 \, \sqrt {e x + d} A a c^{2} d^{2} e^{83} + 3465 \, {\left (e x + d\right )}^{\frac {3}{2}} B a^{2} c e^{84} - 31185 \, \sqrt {e x + d} B a^{2} c d e^{84} + 10395 \, \sqrt {e x + d} A a^{2} c e^{85}\right )}}{3465 \, e^{88}} \]
-2/3*(21*(e*x + d)*B*c^3*d^6 - B*c^3*d^7 - 18*(e*x + d)*A*c^3*d^5*e + A*c^ 3*d^6*e + 45*(e*x + d)*B*a*c^2*d^4*e^2 - 3*B*a*c^2*d^5*e^2 - 36*(e*x + d)* A*a*c^2*d^3*e^3 + 3*A*a*c^2*d^4*e^3 + 27*(e*x + d)*B*a^2*c*d^2*e^4 - 3*B*a ^2*c*d^3*e^4 - 18*(e*x + d)*A*a^2*c*d*e^5 + 3*A*a^2*c*d^2*e^5 + 3*(e*x + d )*B*a^3*e^6 - B*a^3*d*e^6 + A*a^3*e^7)/((e*x + d)^(3/2)*e^8) + 2/3465*(315 *(e*x + d)^(11/2)*B*c^3*e^80 - 2695*(e*x + d)^(9/2)*B*c^3*d*e^80 + 10395*( e*x + d)^(7/2)*B*c^3*d^2*e^80 - 24255*(e*x + d)^(5/2)*B*c^3*d^3*e^80 + 404 25*(e*x + d)^(3/2)*B*c^3*d^4*e^80 - 72765*sqrt(e*x + d)*B*c^3*d^5*e^80 + 3 85*(e*x + d)^(9/2)*A*c^3*e^81 - 2970*(e*x + d)^(7/2)*A*c^3*d*e^81 + 10395* (e*x + d)^(5/2)*A*c^3*d^2*e^81 - 23100*(e*x + d)^(3/2)*A*c^3*d^3*e^81 + 51 975*sqrt(e*x + d)*A*c^3*d^4*e^81 + 1485*(e*x + d)^(7/2)*B*a*c^2*e^82 - 103 95*(e*x + d)^(5/2)*B*a*c^2*d*e^82 + 34650*(e*x + d)^(3/2)*B*a*c^2*d^2*e^82 - 103950*sqrt(e*x + d)*B*a*c^2*d^3*e^82 + 2079*(e*x + d)^(5/2)*A*a*c^2*e^ 83 - 13860*(e*x + d)^(3/2)*A*a*c^2*d*e^83 + 62370*sqrt(e*x + d)*A*a*c^2*d^ 2*e^83 + 3465*(e*x + d)^(3/2)*B*a^2*c*e^84 - 31185*sqrt(e*x + d)*B*a^2*c*d *e^84 + 10395*sqrt(e*x + d)*A*a^2*c*e^85)/e^88
Time = 10.86 (sec) , antiderivative size = 434, normalized size of antiderivative = 1.25 \[ \int \frac {(A+B x) \left (a+c x^2\right )^3}{(d+e x)^{5/2}} \, dx=\frac {{\left (d+e\,x\right )}^{3/2}\,\left (6\,B\,a^2\,c\,e^4+60\,B\,a\,c^2\,d^2\,e^2-24\,A\,a\,c^2\,d\,e^3+70\,B\,c^3\,d^4-40\,A\,c^3\,d^3\,e\right )}{3\,e^8}+\frac {{\left (d+e\,x\right )}^{7/2}\,\left (42\,B\,c^3\,d^2-12\,A\,c^3\,d\,e+6\,B\,a\,c^2\,e^2\right )}{7\,e^8}-\frac {\left (d+e\,x\right )\,\left (2\,B\,a^3\,e^6+18\,B\,a^2\,c\,d^2\,e^4-12\,A\,a^2\,c\,d\,e^5+30\,B\,a\,c^2\,d^4\,e^2-24\,A\,a\,c^2\,d^3\,e^3+14\,B\,c^3\,d^6-12\,A\,c^3\,d^5\,e\right )+\frac {2\,A\,a^3\,e^7}{3}-\frac {2\,B\,c^3\,d^7}{3}-\frac {2\,B\,a^3\,d\,e^6}{3}+\frac {2\,A\,c^3\,d^6\,e}{3}+2\,A\,a\,c^2\,d^4\,e^3+2\,A\,a^2\,c\,d^2\,e^5-2\,B\,a\,c^2\,d^5\,e^2-2\,B\,a^2\,c\,d^3\,e^4}{e^8\,{\left (d+e\,x\right )}^{3/2}}+\frac {2\,B\,c^3\,{\left (d+e\,x\right )}^{11/2}}{11\,e^8}+\frac {2\,c^2\,{\left (d+e\,x\right )}^{5/2}\,\left (-35\,B\,c\,d^3+15\,A\,c\,d^2\,e-15\,B\,a\,d\,e^2+3\,A\,a\,e^3\right )}{5\,e^8}+\frac {2\,c^3\,\left (A\,e-7\,B\,d\right )\,{\left (d+e\,x\right )}^{9/2}}{9\,e^8}+\frac {6\,c\,\left (c\,d^2+a\,e^2\right )\,\sqrt {d+e\,x}\,\left (-7\,B\,c\,d^3+5\,A\,c\,d^2\,e-3\,B\,a\,d\,e^2+A\,a\,e^3\right )}{e^8} \]
((d + e*x)^(3/2)*(70*B*c^3*d^4 + 6*B*a^2*c*e^4 - 40*A*c^3*d^3*e + 60*B*a*c ^2*d^2*e^2 - 24*A*a*c^2*d*e^3))/(3*e^8) + ((d + e*x)^(7/2)*(42*B*c^3*d^2 - 12*A*c^3*d*e + 6*B*a*c^2*e^2))/(7*e^8) - ((d + e*x)*(2*B*a^3*e^6 + 14*B*c ^3*d^6 - 12*A*c^3*d^5*e - 24*A*a*c^2*d^3*e^3 + 30*B*a*c^2*d^4*e^2 + 18*B*a ^2*c*d^2*e^4 - 12*A*a^2*c*d*e^5) + (2*A*a^3*e^7)/3 - (2*B*c^3*d^7)/3 - (2* B*a^3*d*e^6)/3 + (2*A*c^3*d^6*e)/3 + 2*A*a*c^2*d^4*e^3 + 2*A*a^2*c*d^2*e^5 - 2*B*a*c^2*d^5*e^2 - 2*B*a^2*c*d^3*e^4)/(e^8*(d + e*x)^(3/2)) + (2*B*c^3 *(d + e*x)^(11/2))/(11*e^8) + (2*c^2*(d + e*x)^(5/2)*(3*A*a*e^3 - 35*B*c*d ^3 - 15*B*a*d*e^2 + 15*A*c*d^2*e))/(5*e^8) + (2*c^3*(A*e - 7*B*d)*(d + e*x )^(9/2))/(9*e^8) + (6*c*(a*e^2 + c*d^2)*(d + e*x)^(1/2)*(A*a*e^3 - 7*B*c*d ^3 - 3*B*a*d*e^2 + 5*A*c*d^2*e))/e^8