Integrand size = 24, antiderivative size = 342 \[ \int \frac {(A+B x) \left (a+c x^2\right )^3}{(d+e x)^{9/2}} \, dx=\frac {2 (B d-A e) \left (c d^2+a e^2\right )^3}{7 e^8 (d+e x)^{7/2}}-\frac {2 \left (c d^2+a e^2\right )^2 \left (7 B c d^2-6 A c d e+a B e^2\right )}{5 e^8 (d+e x)^{5/2}}+\frac {2 c \left (c d^2+a e^2\right ) \left (7 B c d^3-5 A c d^2 e+3 a B d e^2-a A e^3\right )}{e^8 (d+e x)^{3/2}}+\frac {2 c \left (4 A c d e \left (5 c d^2+3 a e^2\right )-B \left (35 c^2 d^4+30 a c d^2 e^2+3 a^2 e^4\right )\right )}{e^8 \sqrt {d+e x}}-\frac {2 c^2 \left (35 B c d^3-15 A c d^2 e+15 a B d e^2-3 a A e^3\right ) \sqrt {d+e x}}{e^8}+\frac {2 c^2 \left (7 B c d^2-2 A c d e+a B e^2\right ) (d+e x)^{3/2}}{e^8}-\frac {2 c^3 (7 B d-A e) (d+e x)^{5/2}}{5 e^8}+\frac {2 B c^3 (d+e x)^{7/2}}{7 e^8} \] Output:
2/7*(-A*e+B*d)*(a*e^2+c*d^2)^3/e^8/(e*x+d)^(7/2)-2/5*(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)^(5/2)+2*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^8/(e*x+d)^(3/2)+2*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^8/(e*x+d)^(1/2)-2*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)^(1/2)/e^8+2*c^2 *(-2*A*c*d*e+B*a*e^2+7*B*c*d^2)*(e*x+d)^(3/2)/e^8-2/5*c^3*(-A*e+7*B*d)*(e* x+d)^(5/2)/e^8+2/7*B*c^3*(e*x+d)^(7/2)/e^8
Time = 0.56 (sec) , antiderivative size = 372, normalized size of antiderivative = 1.09 \[ \int \frac {(A+B x) \left (a+c x^2\right )^3}{(d+e x)^{9/2}} \, dx=\frac {2 A e \left (-5 a^3 e^6-a^2 c e^4 \left (8 d^2+28 d e x+35 e^2 x^2\right )+3 a c^2 e^2 \left (128 d^4+448 d^3 e x+560 d^2 e^2 x^2+280 d e^3 x^3+35 e^4 x^4\right )+c^3 \left (1024 d^6+3584 d^5 e x+4480 d^4 e^2 x^2+2240 d^3 e^3 x^3+280 d^2 e^4 x^4-28 d e^5 x^5+7 e^6 x^6\right )\right )-2 B \left (a^3 e^6 (2 d+7 e x)+3 a^2 c e^4 \left (16 d^3+56 d^2 e x+70 d e^2 x^2+35 e^3 x^3\right )+5 a c^2 e^2 \left (256 d^5+896 d^4 e x+1120 d^3 e^2 x^2+560 d^2 e^3 x^3+70 d e^4 x^4-7 e^5 x^5\right )+c^3 \left (2048 d^7+7168 d^6 e x+8960 d^5 e^2 x^2+4480 d^4 e^3 x^3+560 d^3 e^4 x^4-56 d^2 e^5 x^5+14 d e^6 x^6-5 e^7 x^7\right )\right )}{35 e^8 (d+e x)^{7/2}} \] Input:
Integrate[((A + B*x)*(a + c*x^2)^3)/(d + e*x)^(9/2),x]
Output:
(2*A*e*(-5*a^3*e^6 - a^2*c*e^4*(8*d^2 + 28*d*e*x + 35*e^2*x^2) + 3*a*c^2*e ^2*(128*d^4 + 448*d^3*e*x + 560*d^2*e^2*x^2 + 280*d*e^3*x^3 + 35*e^4*x^4) + c^3*(1024*d^6 + 3584*d^5*e*x + 4480*d^4*e^2*x^2 + 2240*d^3*e^3*x^3 + 280 *d^2*e^4*x^4 - 28*d*e^5*x^5 + 7*e^6*x^6)) - 2*B*(a^3*e^6*(2*d + 7*e*x) + 3 *a^2*c*e^4*(16*d^3 + 56*d^2*e*x + 70*d*e^2*x^2 + 35*e^3*x^3) + 5*a*c^2*e^2 *(256*d^5 + 896*d^4*e*x + 1120*d^3*e^2*x^2 + 560*d^2*e^3*x^3 + 70*d*e^4*x^ 4 - 7*e^5*x^5) + c^3*(2048*d^7 + 7168*d^6*e*x + 8960*d^5*e^2*x^2 + 4480*d^ 4*e^3*x^3 + 560*d^3*e^4*x^4 - 56*d^2*e^5*x^5 + 14*d*e^6*x^6 - 5*e^7*x^7))) /(35*e^8*(d + e*x)^(7/2))
Time = 0.47 (sec) , antiderivative size = 342, 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.
| \(\displaystyle \int \frac {\left (a+c x^2\right )^3 (A+B x)}{(d+e x)^{9/2}} \, dx\) |
| \(\Big \downarrow \) 652 |
| \(\displaystyle \int \left (-\frac {c \left (-3 a^2 B e^4+12 a A c d e^3-30 a B c d^2 e^2+20 A c^2 d^3 e-35 B c^2 d^4\right )}{e^7 (d+e x)^{3/2}}-\frac {3 c^2 \sqrt {d+e x} \left (-a B e^2+2 A c d e-7 B c d^2\right )}{e^7}+\frac {c^2 \left (3 a A e^3-15 a B d e^2+15 A c d^2 e-35 B c d^3\right )}{e^7 \sqrt {d+e x}}+\frac {\left (a e^2+c d^2\right )^2 \left (a B e^2-6 A c d e+7 B c d^2\right )}{e^7 (d+e x)^{7/2}}+\frac {\left (a e^2+c d^2\right )^3 (A e-B d)}{e^7 (d+e x)^{9/2}}+\frac {3 c \left (a e^2+c d^2\right ) \left (a A e^3-3 a B d e^2+5 A c d^2 e-7 B c d^3\right )}{e^7 (d+e x)^{5/2}}+\frac {c^3 (d+e x)^{3/2} (A e-7 B d)}{e^7}+\frac {B c^3 (d+e x)^{5/2}}{e^7}\right )dx\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {2 c \left (4 A c d e \left (3 a e^2+5 c d^2\right )-B \left (3 a^2 e^4+30 a c d^2 e^2+35 c^2 d^4\right )\right )}{e^8 \sqrt {d+e x}}+\frac {2 c^2 (d+e x)^{3/2} \left (a B e^2-2 A c d e+7 B c d^2\right )}{e^8}-\frac {2 c^2 \sqrt {d+e x} \left (-3 a A e^3+15 a B d e^2-15 A c d^2 e+35 B c d^3\right )}{e^8}-\frac {2 \left (a e^2+c d^2\right )^2 \left (a B e^2-6 A c d e+7 B c d^2\right )}{5 e^8 (d+e x)^{5/2}}+\frac {2 \left (a e^2+c d^2\right )^3 (B d-A e)}{7 e^8 (d+e x)^{7/2}}+\frac {2 c \left (a e^2+c d^2\right ) \left (-a A e^3+3 a B d e^2-5 A c d^2 e+7 B c d^3\right )}{e^8 (d+e x)^{3/2}}-\frac {2 c^3 (d+e x)^{5/2} (7 B d-A e)}{5 e^8}+\frac {2 B c^3 (d+e x)^{7/2}}{7 e^8}\) |
Input:
Int[((A + B*x)*(a + c*x^2)^3)/(d + e*x)^(9/2),x]
Output:
(2*(B*d - A*e)*(c*d^2 + a*e^2)^3)/(7*e^8*(d + e*x)^(7/2)) - (2*(c*d^2 + a* e^2)^2*(7*B*c*d^2 - 6*A*c*d*e + a*B*e^2))/(5*e^8*(d + e*x)^(5/2)) + (2*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))/(e^8*(d + e*x)^(3/2)) + (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)))/(e^8*Sqrt[d + e*x]) - (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)*Sqrt[d + e*x])/e^8 + (2*c^2*(7*B*c*d^2 - 2*A*c*d*e + a*B*e^2)*(d + e*x)^(3/2))/e^8 - (2*c^3*(7*B*d - A*e)*(d + e *x)^(5/2))/(5*e^8) + (2*B*c^3*(d + e*x)^(7/2))/(7*e^8)
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 = 1.11 (sec) , antiderivative size = 301, normalized size of antiderivative = 0.88
| method | result | size |
| pseudoelliptic | \(\frac {\left (\left (10 B \,x^{7}+14 A \,x^{6}\right ) c^{3}+210 x^{4} a \left (\frac {B x}{3}+A \right ) c^{2}-70 a^{2} x^{2} \left (3 B x +A \right ) c -10 a^{3} \left (\frac {7 B x}{5}+A \right )\right ) e^{7}-56 d \left (x^{5} \left (\frac {B x}{2}+A \right ) c^{3}-30 \left (-\frac {5 B x}{12}+A \right ) x^{3} a \,c^{2}+a^{2} x \left (\frac {15 B x}{2}+A \right ) c +\frac {B \,a^{3}}{14}\right ) e^{6}-16 c \,d^{2} \left (\left (-7 B \,x^{5}-35 A \,x^{4}\right ) c^{2}-210 x^{2} \left (-\frac {5 B x}{3}+A \right ) a c +a^{2} \left (21 B x +A \right )\right ) e^{5}+2688 c \,d^{3} \left (\frac {5 \left (-\frac {B x}{4}+A \right ) x^{3} c^{2}}{3}+a x \left (-\frac {25 B x}{6}+A \right ) c -\frac {a^{2} B}{28}\right ) e^{4}+768 c^{2} d^{4} \left (\frac {35 x^{2} \left (-B x +A \right ) c}{3}+a \left (-\frac {35 B x}{3}+A \right )\right ) e^{3}+7168 c^{2} d^{5} \left (x \left (-\frac {5 B x}{2}+A \right ) c -\frac {5 B a}{14}\right ) e^{2}+2048 c^{3} d^{6} \left (-7 B x +A \right ) e -4096 B \,c^{3} d^{7}}{35 \left (e x +d \right )^{\frac {7}{2}} e^{8}}\) | \(301\) |
| derivativedivides | \(\frac {\frac {2 B \,c^{3} \left (e x +d \right )^{\frac {7}{2}}}{7}+\frac {2 A \,c^{3} e \left (e x +d \right )^{\frac {5}{2}}}{5}-\frac {14 B \,c^{3} d \left (e x +d \right )^{\frac {5}{2}}}{5}-4 A \,c^{3} d e \left (e x +d \right )^{\frac {3}{2}}+2 B a \,c^{2} e^{2} \left (e x +d \right )^{\frac {3}{2}}+14 B \,c^{3} d^{2} \left (e x +d \right )^{\frac {3}{2}}+6 A a \,c^{2} e^{3} \sqrt {e x +d}+30 A \,c^{3} d^{2} e \sqrt {e x +d}-30 B a \,c^{2} d \,e^{2} \sqrt {e x +d}-70 B \,c^{3} d^{3} \sqrt {e x +d}-\frac {2 \left (-6 A \,a^{2} c d \,e^{5}-12 A a \,c^{2} d^{3} e^{3}-6 A \,c^{3} d^{5} e +B \,a^{3} e^{6}+9 B \,a^{2} c \,d^{2} e^{4}+15 B a \,c^{2} d^{4} e^{2}+7 B \,c^{3} d^{6}\right )}{5 \left (e x +d \right )^{\frac {5}{2}}}+\frac {2 c \left (12 A a c d \,e^{3}+20 A \,c^{2} d^{3} e -3 B \,e^{4} a^{2}-30 B a c \,d^{2} e^{2}-35 B \,c^{2} d^{4}\right )}{\sqrt {e x +d}}-\frac {2 \left (A \,a^{3} e^{7}+3 A \,a^{2} c \,d^{2} e^{5}+3 A a \,c^{2} d^{4} e^{3}+A \,c^{3} d^{6} e -B \,a^{3} d \,e^{6}-3 B \,a^{2} c \,d^{3} e^{4}-3 B a \,c^{2} d^{5} e^{2}-B \,c^{3} d^{7}\right )}{7 \left (e x +d \right )^{\frac {7}{2}}}-\frac {2 c \left (A \,a^{2} e^{5}+6 A a c \,d^{2} e^{3}+5 A \,c^{2} d^{4} e -3 B \,a^{2} d \,e^{4}-10 B a c \,d^{3} e^{2}-7 B \,c^{2} d^{5}\right )}{\left (e x +d \right )^{\frac {3}{2}}}}{e^{8}}\) | \(476\) |
| default | \(\frac {\frac {2 B \,c^{3} \left (e x +d \right )^{\frac {7}{2}}}{7}+\frac {2 A \,c^{3} e \left (e x +d \right )^{\frac {5}{2}}}{5}-\frac {14 B \,c^{3} d \left (e x +d \right )^{\frac {5}{2}}}{5}-4 A \,c^{3} d e \left (e x +d \right )^{\frac {3}{2}}+2 B a \,c^{2} e^{2} \left (e x +d \right )^{\frac {3}{2}}+14 B \,c^{3} d^{2} \left (e x +d \right )^{\frac {3}{2}}+6 A a \,c^{2} e^{3} \sqrt {e x +d}+30 A \,c^{3} d^{2} e \sqrt {e x +d}-30 B a \,c^{2} d \,e^{2} \sqrt {e x +d}-70 B \,c^{3} d^{3} \sqrt {e x +d}-\frac {2 \left (-6 A \,a^{2} c d \,e^{5}-12 A a \,c^{2} d^{3} e^{3}-6 A \,c^{3} d^{5} e +B \,a^{3} e^{6}+9 B \,a^{2} c \,d^{2} e^{4}+15 B a \,c^{2} d^{4} e^{2}+7 B \,c^{3} d^{6}\right )}{5 \left (e x +d \right )^{\frac {5}{2}}}+\frac {2 c \left (12 A a c d \,e^{3}+20 A \,c^{2} d^{3} e -3 B \,e^{4} a^{2}-30 B a c \,d^{2} e^{2}-35 B \,c^{2} d^{4}\right )}{\sqrt {e x +d}}-\frac {2 \left (A \,a^{3} e^{7}+3 A \,a^{2} c \,d^{2} e^{5}+3 A a \,c^{2} d^{4} e^{3}+A \,c^{3} d^{6} e -B \,a^{3} d \,e^{6}-3 B \,a^{2} c \,d^{3} e^{4}-3 B a \,c^{2} d^{5} e^{2}-B \,c^{3} d^{7}\right )}{7 \left (e x +d \right )^{\frac {7}{2}}}-\frac {2 c \left (A \,a^{2} e^{5}+6 A a c \,d^{2} e^{3}+5 A \,c^{2} d^{4} e -3 B \,a^{2} d \,e^{4}-10 B a c \,d^{3} e^{2}-7 B \,c^{2} d^{5}\right )}{\left (e x +d \right )^{\frac {3}{2}}}}{e^{8}}\) | \(476\) |
| risch | \(\frac {2 c^{2} \left (5 B c \,x^{3} e^{3}+7 A \,x^{2} c \,e^{3}-34 B \,x^{2} c d \,e^{2}-56 A x c d \,e^{2}+35 B x a \,e^{3}+162 B x c \,d^{2} e +105 A a \,e^{3}+462 A c \,d^{2} e -490 B a d \,e^{2}-1024 B c \,d^{3}\right ) \sqrt {e x +d}}{35 e^{8}}-\frac {2 \left (-420 A \,x^{3} a \,c^{2} d \,e^{6}-700 A \,x^{3} c^{3} d^{3} e^{4}+105 B \,x^{3} a^{2} c \,e^{7}+1050 B \,x^{3} a \,c^{2} d^{2} e^{5}+1225 B \,x^{3} c^{3} d^{4} e^{3}+35 A \,x^{2} a^{2} c \,e^{7}-1050 A \,x^{2} a \,c^{2} d^{2} e^{5}-1925 A \,x^{2} c^{3} d^{4} e^{3}+210 B \,x^{2} a^{2} c d \,e^{6}+2800 B \,x^{2} a \,c^{2} d^{3} e^{4}+3430 B \,x^{2} c^{3} d^{5} e^{2}+28 A x \,a^{2} c d \,e^{6}-924 A x a \,c^{2} d^{3} e^{4}-1792 A x \,c^{3} d^{5} e^{2}+7 B x \,a^{3} e^{7}+168 B x \,a^{2} c \,d^{2} e^{5}+2555 B x a \,c^{2} d^{4} e^{3}+3234 B x \,c^{3} d^{6} e +5 A \,a^{3} e^{7}+8 A \,a^{2} c \,d^{2} e^{5}-279 A a \,c^{2} d^{4} e^{3}-562 A \,c^{3} d^{6} e +2 B \,a^{3} d \,e^{6}+48 B \,a^{2} c \,d^{3} e^{4}+790 B a \,c^{2} d^{5} e^{2}+1024 B \,c^{3} d^{7}\right )}{35 e^{8} \sqrt {e x +d}\, \left (e^{3} x^{3}+3 d \,e^{2} x^{2}+3 d^{2} e x +d^{3}\right )}\) | \(487\) |
| gosper | \(-\frac {2 \left (-5 B \,x^{7} c^{3} e^{7}-7 A \,x^{6} c^{3} e^{7}+14 B \,x^{6} c^{3} d \,e^{6}+28 A \,x^{5} c^{3} d \,e^{6}-35 B \,x^{5} a \,c^{2} e^{7}-56 B \,x^{5} c^{3} d^{2} e^{5}-105 A \,x^{4} a \,c^{2} e^{7}-280 A \,x^{4} c^{3} d^{2} e^{5}+350 B \,x^{4} a \,c^{2} d \,e^{6}+560 B \,x^{4} c^{3} d^{3} e^{4}-840 A \,x^{3} a \,c^{2} d \,e^{6}-2240 A \,x^{3} c^{3} d^{3} e^{4}+105 B \,x^{3} a^{2} c \,e^{7}+2800 B \,x^{3} a \,c^{2} d^{2} e^{5}+4480 B \,x^{3} c^{3} d^{4} e^{3}+35 A \,x^{2} a^{2} c \,e^{7}-1680 A \,x^{2} a \,c^{2} d^{2} e^{5}-4480 A \,x^{2} c^{3} d^{4} e^{3}+210 B \,x^{2} a^{2} c d \,e^{6}+5600 B \,x^{2} a \,c^{2} d^{3} e^{4}+8960 B \,x^{2} c^{3} d^{5} e^{2}+28 A x \,a^{2} c d \,e^{6}-1344 A x a \,c^{2} d^{3} e^{4}-3584 A x \,c^{3} d^{5} e^{2}+7 B x \,a^{3} e^{7}+168 B x \,a^{2} c \,d^{2} e^{5}+4480 B x a \,c^{2} d^{4} e^{3}+7168 B x \,c^{3} d^{6} e +5 A \,a^{3} e^{7}+8 A \,a^{2} c \,d^{2} e^{5}-384 A a \,c^{2} d^{4} e^{3}-1024 A \,c^{3} d^{6} e +2 B \,a^{3} d \,e^{6}+48 B \,a^{2} c \,d^{3} e^{4}+1280 B a \,c^{2} d^{5} e^{2}+2048 B \,c^{3} d^{7}\right )}{35 \left (e x +d \right )^{\frac {7}{2}} e^{8}}\) | \(489\) |
| trager | \(-\frac {2 \left (-5 B \,x^{7} c^{3} e^{7}-7 A \,x^{6} c^{3} e^{7}+14 B \,x^{6} c^{3} d \,e^{6}+28 A \,x^{5} c^{3} d \,e^{6}-35 B \,x^{5} a \,c^{2} e^{7}-56 B \,x^{5} c^{3} d^{2} e^{5}-105 A \,x^{4} a \,c^{2} e^{7}-280 A \,x^{4} c^{3} d^{2} e^{5}+350 B \,x^{4} a \,c^{2} d \,e^{6}+560 B \,x^{4} c^{3} d^{3} e^{4}-840 A \,x^{3} a \,c^{2} d \,e^{6}-2240 A \,x^{3} c^{3} d^{3} e^{4}+105 B \,x^{3} a^{2} c \,e^{7}+2800 B \,x^{3} a \,c^{2} d^{2} e^{5}+4480 B \,x^{3} c^{3} d^{4} e^{3}+35 A \,x^{2} a^{2} c \,e^{7}-1680 A \,x^{2} a \,c^{2} d^{2} e^{5}-4480 A \,x^{2} c^{3} d^{4} e^{3}+210 B \,x^{2} a^{2} c d \,e^{6}+5600 B \,x^{2} a \,c^{2} d^{3} e^{4}+8960 B \,x^{2} c^{3} d^{5} e^{2}+28 A x \,a^{2} c d \,e^{6}-1344 A x a \,c^{2} d^{3} e^{4}-3584 A x \,c^{3} d^{5} e^{2}+7 B x \,a^{3} e^{7}+168 B x \,a^{2} c \,d^{2} e^{5}+4480 B x a \,c^{2} d^{4} e^{3}+7168 B x \,c^{3} d^{6} e +5 A \,a^{3} e^{7}+8 A \,a^{2} c \,d^{2} e^{5}-384 A a \,c^{2} d^{4} e^{3}-1024 A \,c^{3} d^{6} e +2 B \,a^{3} d \,e^{6}+48 B \,a^{2} c \,d^{3} e^{4}+1280 B a \,c^{2} d^{5} e^{2}+2048 B \,c^{3} d^{7}\right )}{35 \left (e x +d \right )^{\frac {7}{2}} e^{8}}\) | \(489\) |
| orering | \(-\frac {2 \left (-5 B \,x^{7} c^{3} e^{7}-7 A \,x^{6} c^{3} e^{7}+14 B \,x^{6} c^{3} d \,e^{6}+28 A \,x^{5} c^{3} d \,e^{6}-35 B \,x^{5} a \,c^{2} e^{7}-56 B \,x^{5} c^{3} d^{2} e^{5}-105 A \,x^{4} a \,c^{2} e^{7}-280 A \,x^{4} c^{3} d^{2} e^{5}+350 B \,x^{4} a \,c^{2} d \,e^{6}+560 B \,x^{4} c^{3} d^{3} e^{4}-840 A \,x^{3} a \,c^{2} d \,e^{6}-2240 A \,x^{3} c^{3} d^{3} e^{4}+105 B \,x^{3} a^{2} c \,e^{7}+2800 B \,x^{3} a \,c^{2} d^{2} e^{5}+4480 B \,x^{3} c^{3} d^{4} e^{3}+35 A \,x^{2} a^{2} c \,e^{7}-1680 A \,x^{2} a \,c^{2} d^{2} e^{5}-4480 A \,x^{2} c^{3} d^{4} e^{3}+210 B \,x^{2} a^{2} c d \,e^{6}+5600 B \,x^{2} a \,c^{2} d^{3} e^{4}+8960 B \,x^{2} c^{3} d^{5} e^{2}+28 A x \,a^{2} c d \,e^{6}-1344 A x a \,c^{2} d^{3} e^{4}-3584 A x \,c^{3} d^{5} e^{2}+7 B x \,a^{3} e^{7}+168 B x \,a^{2} c \,d^{2} e^{5}+4480 B x a \,c^{2} d^{4} e^{3}+7168 B x \,c^{3} d^{6} e +5 A \,a^{3} e^{7}+8 A \,a^{2} c \,d^{2} e^{5}-384 A a \,c^{2} d^{4} e^{3}-1024 A \,c^{3} d^{6} e +2 B \,a^{3} d \,e^{6}+48 B \,a^{2} c \,d^{3} e^{4}+1280 B a \,c^{2} d^{5} e^{2}+2048 B \,c^{3} d^{7}\right )}{35 \left (e x +d \right )^{\frac {7}{2}} e^{8}}\) | \(489\) |
Input:
int((B*x+A)*(c*x^2+a)^3/(e*x+d)^(9/2),x,method=_RETURNVERBOSE)
Output:
1/35*(((10*B*x^7+14*A*x^6)*c^3+210*x^4*a*(1/3*B*x+A)*c^2-70*a^2*x^2*(3*B*x +A)*c-10*a^3*(7/5*B*x+A))*e^7-56*d*(x^5*(1/2*B*x+A)*c^3-30*(-5/12*B*x+A)*x ^3*a*c^2+a^2*x*(15/2*B*x+A)*c+1/14*B*a^3)*e^6-16*c*d^2*((-7*B*x^5-35*A*x^4 )*c^2-210*x^2*(-5/3*B*x+A)*a*c+a^2*(21*B*x+A))*e^5+2688*c*d^3*(5/3*(-1/4*B *x+A)*x^3*c^2+a*x*(-25/6*B*x+A)*c-1/28*a^2*B)*e^4+768*c^2*d^4*(35/3*x^2*(- B*x+A)*c+a*(-35/3*B*x+A))*e^3+7168*c^2*d^5*(x*(-5/2*B*x+A)*c-5/14*B*a)*e^2 +2048*c^3*d^6*(-7*B*x+A)*e-4096*B*c^3*d^7)/(e*x+d)^(7/2)/e^8
Time = 0.09 (sec) , antiderivative size = 496, normalized size of antiderivative = 1.45 \[ \int \frac {(A+B x) \left (a+c x^2\right )^3}{(d+e x)^{9/2}} \, dx=\frac {2 \, {\left (5 \, B c^{3} e^{7} x^{7} - 2048 \, B c^{3} d^{7} + 1024 \, A c^{3} d^{6} e - 1280 \, B a c^{2} d^{5} e^{2} + 384 \, A a c^{2} d^{4} e^{3} - 48 \, B a^{2} c d^{3} e^{4} - 8 \, A a^{2} c d^{2} e^{5} - 2 \, B a^{3} d e^{6} - 5 \, A a^{3} e^{7} - 7 \, {\left (2 \, B c^{3} d e^{6} - A c^{3} e^{7}\right )} x^{6} + 7 \, {\left (8 \, B c^{3} d^{2} e^{5} - 4 \, A c^{3} d e^{6} + 5 \, B a c^{2} e^{7}\right )} x^{5} - 35 \, {\left (16 \, B c^{3} d^{3} e^{4} - 8 \, A c^{3} d^{2} e^{5} + 10 \, B a c^{2} d e^{6} - 3 \, A a c^{2} e^{7}\right )} x^{4} - 35 \, {\left (128 \, B c^{3} d^{4} e^{3} - 64 \, A c^{3} d^{3} e^{4} + 80 \, B a c^{2} d^{2} e^{5} - 24 \, A a c^{2} d e^{6} + 3 \, B a^{2} c e^{7}\right )} x^{3} - 35 \, {\left (256 \, B c^{3} d^{5} e^{2} - 128 \, A c^{3} d^{4} e^{3} + 160 \, B a c^{2} d^{3} e^{4} - 48 \, A a c^{2} d^{2} e^{5} + 6 \, B a^{2} c d e^{6} + A a^{2} c e^{7}\right )} x^{2} - 7 \, {\left (1024 \, B c^{3} d^{6} e - 512 \, A c^{3} d^{5} e^{2} + 640 \, B a c^{2} d^{4} e^{3} - 192 \, A a c^{2} d^{3} e^{4} + 24 \, B a^{2} c d^{2} e^{5} + 4 \, A a^{2} c d e^{6} + B a^{3} e^{7}\right )} x\right )} \sqrt {e x + d}}{35 \, {\left (e^{12} x^{4} + 4 \, d e^{11} x^{3} + 6 \, d^{2} e^{10} x^{2} + 4 \, d^{3} e^{9} x + d^{4} e^{8}\right )}} \] Input:
integrate((B*x+A)*(c*x^2+a)^3/(e*x+d)^(9/2),x, algorithm="fricas")
Output:
2/35*(5*B*c^3*e^7*x^7 - 2048*B*c^3*d^7 + 1024*A*c^3*d^6*e - 1280*B*a*c^2*d ^5*e^2 + 384*A*a*c^2*d^4*e^3 - 48*B*a^2*c*d^3*e^4 - 8*A*a^2*c*d^2*e^5 - 2* B*a^3*d*e^6 - 5*A*a^3*e^7 - 7*(2*B*c^3*d*e^6 - A*c^3*e^7)*x^6 + 7*(8*B*c^3 *d^2*e^5 - 4*A*c^3*d*e^6 + 5*B*a*c^2*e^7)*x^5 - 35*(16*B*c^3*d^3*e^4 - 8*A *c^3*d^2*e^5 + 10*B*a*c^2*d*e^6 - 3*A*a*c^2*e^7)*x^4 - 35*(128*B*c^3*d^4*e ^3 - 64*A*c^3*d^3*e^4 + 80*B*a*c^2*d^2*e^5 - 24*A*a*c^2*d*e^6 + 3*B*a^2*c* e^7)*x^3 - 35*(256*B*c^3*d^5*e^2 - 128*A*c^3*d^4*e^3 + 160*B*a*c^2*d^3*e^4 - 48*A*a*c^2*d^2*e^5 + 6*B*a^2*c*d*e^6 + A*a^2*c*e^7)*x^2 - 7*(1024*B*c^3 *d^6*e - 512*A*c^3*d^5*e^2 + 640*B*a*c^2*d^4*e^3 - 192*A*a*c^2*d^3*e^4 + 2 4*B*a^2*c*d^2*e^5 + 4*A*a^2*c*d*e^6 + B*a^3*e^7)*x)*sqrt(e*x + d)/(e^12*x^ 4 + 4*d*e^11*x^3 + 6*d^2*e^10*x^2 + 4*d^3*e^9*x + d^4*e^8)
Leaf count of result is larger than twice the leaf count of optimal. 3218 vs. \(2 (359) = 718\).
Time = 0.99 (sec) , antiderivative size = 3218, normalized size of antiderivative = 9.41 \[ \int \frac {(A+B x) \left (a+c x^2\right )^3}{(d+e x)^{9/2}} \, dx=\text {Too large to display} \] Input:
integrate((B*x+A)*(c*x**2+a)**3/(e*x+d)**(9/2),x)
Output:
Piecewise((-10*A*a**3*e**7/(35*d**3*e**8*sqrt(d + e*x) + 105*d**2*e**9*x*s qrt(d + e*x) + 105*d*e**10*x**2*sqrt(d + e*x) + 35*e**11*x**3*sqrt(d + e*x )) - 16*A*a**2*c*d**2*e**5/(35*d**3*e**8*sqrt(d + e*x) + 105*d**2*e**9*x*s qrt(d + e*x) + 105*d*e**10*x**2*sqrt(d + e*x) + 35*e**11*x**3*sqrt(d + e*x )) - 56*A*a**2*c*d*e**6*x/(35*d**3*e**8*sqrt(d + e*x) + 105*d**2*e**9*x*sq rt(d + e*x) + 105*d*e**10*x**2*sqrt(d + e*x) + 35*e**11*x**3*sqrt(d + e*x) ) - 70*A*a**2*c*e**7*x**2/(35*d**3*e**8*sqrt(d + e*x) + 105*d**2*e**9*x*sq rt(d + e*x) + 105*d*e**10*x**2*sqrt(d + e*x) + 35*e**11*x**3*sqrt(d + e*x) ) + 768*A*a*c**2*d**4*e**3/(35*d**3*e**8*sqrt(d + e*x) + 105*d**2*e**9*x*s qrt(d + e*x) + 105*d*e**10*x**2*sqrt(d + e*x) + 35*e**11*x**3*sqrt(d + e*x )) + 2688*A*a*c**2*d**3*e**4*x/(35*d**3*e**8*sqrt(d + e*x) + 105*d**2*e**9 *x*sqrt(d + e*x) + 105*d*e**10*x**2*sqrt(d + e*x) + 35*e**11*x**3*sqrt(d + e*x)) + 3360*A*a*c**2*d**2*e**5*x**2/(35*d**3*e**8*sqrt(d + e*x) + 105*d* *2*e**9*x*sqrt(d + e*x) + 105*d*e**10*x**2*sqrt(d + e*x) + 35*e**11*x**3*s qrt(d + e*x)) + 1680*A*a*c**2*d*e**6*x**3/(35*d**3*e**8*sqrt(d + e*x) + 10 5*d**2*e**9*x*sqrt(d + e*x) + 105*d*e**10*x**2*sqrt(d + e*x) + 35*e**11*x* *3*sqrt(d + e*x)) + 210*A*a*c**2*e**7*x**4/(35*d**3*e**8*sqrt(d + e*x) + 1 05*d**2*e**9*x*sqrt(d + e*x) + 105*d*e**10*x**2*sqrt(d + e*x) + 35*e**11*x **3*sqrt(d + e*x)) + 2048*A*c**3*d**6*e/(35*d**3*e**8*sqrt(d + e*x) + 105* d**2*e**9*x*sqrt(d + e*x) + 105*d*e**10*x**2*sqrt(d + e*x) + 35*e**11*x...
Time = 0.04 (sec) , antiderivative size = 460, normalized size of antiderivative = 1.35 \[ \int \frac {(A+B x) \left (a+c x^2\right )^3}{(d+e x)^{9/2}} \, dx=\frac {2 \, {\left (\frac {5 \, {\left (e x + d\right )}^{\frac {7}{2}} B c^{3} - 7 \, {\left (7 \, B c^{3} d - A c^{3} e\right )} {\left (e x + d\right )}^{\frac {5}{2}} + 35 \, {\left (7 \, B c^{3} d^{2} - 2 \, A c^{3} d e + B a c^{2} e^{2}\right )} {\left (e x + d\right )}^{\frac {3}{2}} - 35 \, {\left (35 \, B c^{3} d^{3} - 15 \, A c^{3} d^{2} e + 15 \, B a c^{2} d e^{2} - 3 \, A a c^{2} e^{3}\right )} \sqrt {e x + d}}{e^{7}} + \frac {5 \, B c^{3} d^{7} - 5 \, A c^{3} d^{6} e + 15 \, B a c^{2} d^{5} e^{2} - 15 \, A a c^{2} d^{4} e^{3} + 15 \, B a^{2} c d^{3} e^{4} - 15 \, A a^{2} c d^{2} e^{5} + 5 \, B a^{3} d e^{6} - 5 \, A a^{3} e^{7} - 35 \, {\left (35 \, B c^{3} d^{4} - 20 \, A c^{3} d^{3} e + 30 \, B a c^{2} d^{2} e^{2} - 12 \, A a c^{2} d e^{3} + 3 \, B a^{2} c e^{4}\right )} {\left (e x + d\right )}^{3} + 35 \, {\left (7 \, B c^{3} d^{5} - 5 \, A c^{3} d^{4} e + 10 \, B a c^{2} d^{3} e^{2} - 6 \, A a c^{2} d^{2} e^{3} + 3 \, B a^{2} c d e^{4} - A a^{2} c e^{5}\right )} {\left (e x + d\right )}^{2} - 7 \, {\left (7 \, B c^{3} d^{6} - 6 \, A c^{3} d^{5} e + 15 \, B a c^{2} d^{4} e^{2} - 12 \, A a c^{2} d^{3} e^{3} + 9 \, B a^{2} c d^{2} e^{4} - 6 \, A a^{2} c d e^{5} + B a^{3} e^{6}\right )} {\left (e x + d\right )}}{{\left (e x + d\right )}^{\frac {7}{2}} e^{7}}\right )}}{35 \, e} \] Input:
integrate((B*x+A)*(c*x^2+a)^3/(e*x+d)^(9/2),x, algorithm="maxima")
Output:
2/35*((5*(e*x + d)^(7/2)*B*c^3 - 7*(7*B*c^3*d - A*c^3*e)*(e*x + d)^(5/2) + 35*(7*B*c^3*d^2 - 2*A*c^3*d*e + B*a*c^2*e^2)*(e*x + d)^(3/2) - 35*(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)*sqrt(e*x + d)) /e^7 + (5*B*c^3*d^7 - 5*A*c^3*d^6*e + 15*B*a*c^2*d^5*e^2 - 15*A*a*c^2*d^4* e^3 + 15*B*a^2*c*d^3*e^4 - 15*A*a^2*c*d^2*e^5 + 5*B*a^3*d*e^6 - 5*A*a^3*e^ 7 - 35*(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)*(e*x + d)^3 + 35*(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)*(e*x + d)^2 - 7*(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)^(7/2)*e^7))/e
Time = 0.16 (sec) , antiderivative size = 594, normalized size of antiderivative = 1.74 \[ \int \frac {(A+B x) \left (a+c x^2\right )^3}{(d+e x)^{9/2}} \, dx=-\frac {2 \, {\left (1225 \, {\left (e x + d\right )}^{3} B c^{3} d^{4} - 245 \, {\left (e x + d\right )}^{2} B c^{3} d^{5} + 49 \, {\left (e x + d\right )} B c^{3} d^{6} - 5 \, B c^{3} d^{7} - 700 \, {\left (e x + d\right )}^{3} A c^{3} d^{3} e + 175 \, {\left (e x + d\right )}^{2} A c^{3} d^{4} e - 42 \, {\left (e x + d\right )} A c^{3} d^{5} e + 5 \, A c^{3} d^{6} e + 1050 \, {\left (e x + d\right )}^{3} B a c^{2} d^{2} e^{2} - 350 \, {\left (e x + d\right )}^{2} B a c^{2} d^{3} e^{2} + 105 \, {\left (e x + d\right )} B a c^{2} d^{4} e^{2} - 15 \, B a c^{2} d^{5} e^{2} - 420 \, {\left (e x + d\right )}^{3} A a c^{2} d e^{3} + 210 \, {\left (e x + d\right )}^{2} A a c^{2} d^{2} e^{3} - 84 \, {\left (e x + d\right )} A a c^{2} d^{3} e^{3} + 15 \, A a c^{2} d^{4} e^{3} + 105 \, {\left (e x + d\right )}^{3} B a^{2} c e^{4} - 105 \, {\left (e x + d\right )}^{2} B a^{2} c d e^{4} + 63 \, {\left (e x + d\right )} B a^{2} c d^{2} e^{4} - 15 \, B a^{2} c d^{3} e^{4} + 35 \, {\left (e x + d\right )}^{2} A a^{2} c e^{5} - 42 \, {\left (e x + d\right )} A a^{2} c d e^{5} + 15 \, A a^{2} c d^{2} e^{5} + 7 \, {\left (e x + d\right )} B a^{3} e^{6} - 5 \, B a^{3} d e^{6} + 5 \, A a^{3} e^{7}\right )}}{35 \, {\left (e x + d\right )}^{\frac {7}{2}} e^{8}} + \frac {2 \, {\left (5 \, {\left (e x + d\right )}^{\frac {7}{2}} B c^{3} e^{48} - 49 \, {\left (e x + d\right )}^{\frac {5}{2}} B c^{3} d e^{48} + 245 \, {\left (e x + d\right )}^{\frac {3}{2}} B c^{3} d^{2} e^{48} - 1225 \, \sqrt {e x + d} B c^{3} d^{3} e^{48} + 7 \, {\left (e x + d\right )}^{\frac {5}{2}} A c^{3} e^{49} - 70 \, {\left (e x + d\right )}^{\frac {3}{2}} A c^{3} d e^{49} + 525 \, \sqrt {e x + d} A c^{3} d^{2} e^{49} + 35 \, {\left (e x + d\right )}^{\frac {3}{2}} B a c^{2} e^{50} - 525 \, \sqrt {e x + d} B a c^{2} d e^{50} + 105 \, \sqrt {e x + d} A a c^{2} e^{51}\right )}}{35 \, e^{56}} \] Input:
integrate((B*x+A)*(c*x^2+a)^3/(e*x+d)^(9/2),x, algorithm="giac")
Output:
-2/35*(1225*(e*x + d)^3*B*c^3*d^4 - 245*(e*x + d)^2*B*c^3*d^5 + 49*(e*x + d)*B*c^3*d^6 - 5*B*c^3*d^7 - 700*(e*x + d)^3*A*c^3*d^3*e + 175*(e*x + d)^2 *A*c^3*d^4*e - 42*(e*x + d)*A*c^3*d^5*e + 5*A*c^3*d^6*e + 1050*(e*x + d)^3 *B*a*c^2*d^2*e^2 - 350*(e*x + d)^2*B*a*c^2*d^3*e^2 + 105*(e*x + d)*B*a*c^2 *d^4*e^2 - 15*B*a*c^2*d^5*e^2 - 420*(e*x + d)^3*A*a*c^2*d*e^3 + 210*(e*x + d)^2*A*a*c^2*d^2*e^3 - 84*(e*x + d)*A*a*c^2*d^3*e^3 + 15*A*a*c^2*d^4*e^3 + 105*(e*x + d)^3*B*a^2*c*e^4 - 105*(e*x + d)^2*B*a^2*c*d*e^4 + 63*(e*x + d)*B*a^2*c*d^2*e^4 - 15*B*a^2*c*d^3*e^4 + 35*(e*x + d)^2*A*a^2*c*e^5 - 42* (e*x + d)*A*a^2*c*d*e^5 + 15*A*a^2*c*d^2*e^5 + 7*(e*x + d)*B*a^3*e^6 - 5*B *a^3*d*e^6 + 5*A*a^3*e^7)/((e*x + d)^(7/2)*e^8) + 2/35*(5*(e*x + d)^(7/2)* B*c^3*e^48 - 49*(e*x + d)^(5/2)*B*c^3*d*e^48 + 245*(e*x + d)^(3/2)*B*c^3*d ^2*e^48 - 1225*sqrt(e*x + d)*B*c^3*d^3*e^48 + 7*(e*x + d)^(5/2)*A*c^3*e^49 - 70*(e*x + d)^(3/2)*A*c^3*d*e^49 + 525*sqrt(e*x + d)*A*c^3*d^2*e^49 + 35 *(e*x + d)^(3/2)*B*a*c^2*e^50 - 525*sqrt(e*x + d)*B*a*c^2*d*e^50 + 105*sqr t(e*x + d)*A*a*c^2*e^51)/e^56
Time = 6.53 (sec) , antiderivative size = 452, normalized size of antiderivative = 1.32 \[ \int \frac {(A+B x) \left (a+c x^2\right )^3}{(d+e x)^{9/2}} \, dx=\frac {{\left (d+e\,x\right )}^{3/2}\,\left (42\,B\,c^3\,d^2-12\,A\,c^3\,d\,e+6\,B\,a\,c^2\,e^2\right )}{3\,e^8}-\frac {\left (d+e\,x\right )\,\left (\frac {2\,B\,a^3\,e^6}{5}+\frac {18\,B\,a^2\,c\,d^2\,e^4}{5}-\frac {12\,A\,a^2\,c\,d\,e^5}{5}+6\,B\,a\,c^2\,d^4\,e^2-\frac {24\,A\,a\,c^2\,d^3\,e^3}{5}+\frac {14\,B\,c^3\,d^6}{5}-\frac {12\,A\,c^3\,d^5\,e}{5}\right )+{\left (d+e\,x\right )}^3\,\left (6\,B\,a^2\,c\,e^4+60\,B\,a\,c^2\,d^2\,e^2-24\,A\,a\,c^2\,d\,e^3+70\,B\,c^3\,d^4-40\,A\,c^3\,d^3\,e\right )-{\left (d+e\,x\right )}^2\,\left (6\,B\,a^2\,c\,d\,e^4-2\,A\,a^2\,c\,e^5+20\,B\,a\,c^2\,d^3\,e^2-12\,A\,a\,c^2\,d^2\,e^3+14\,B\,c^3\,d^5-10\,A\,c^3\,d^4\,e\right )+\frac {2\,A\,a^3\,e^7}{7}-\frac {2\,B\,c^3\,d^7}{7}-\frac {2\,B\,a^3\,d\,e^6}{7}+\frac {2\,A\,c^3\,d^6\,e}{7}+\frac {6\,A\,a\,c^2\,d^4\,e^3}{7}+\frac {6\,A\,a^2\,c\,d^2\,e^5}{7}-\frac {6\,B\,a\,c^2\,d^5\,e^2}{7}-\frac {6\,B\,a^2\,c\,d^3\,e^4}{7}}{e^8\,{\left (d+e\,x\right )}^{7/2}}+\frac {2\,B\,c^3\,{\left (d+e\,x\right )}^{7/2}}{7\,e^8}+\frac {2\,c^2\,\sqrt {d+e\,x}\,\left (-35\,B\,c\,d^3+15\,A\,c\,d^2\,e-15\,B\,a\,d\,e^2+3\,A\,a\,e^3\right )}{e^8}+\frac {2\,c^3\,\left (A\,e-7\,B\,d\right )\,{\left (d+e\,x\right )}^{5/2}}{5\,e^8} \] Input:
int(((a + c*x^2)^3*(A + B*x))/(d + e*x)^(9/2),x)
Output:
((d + e*x)^(3/2)*(42*B*c^3*d^2 - 12*A*c^3*d*e + 6*B*a*c^2*e^2))/(3*e^8) - ((d + e*x)*((2*B*a^3*e^6)/5 + (14*B*c^3*d^6)/5 - (12*A*c^3*d^5*e)/5 - (24* A*a*c^2*d^3*e^3)/5 + 6*B*a*c^2*d^4*e^2 + (18*B*a^2*c*d^2*e^4)/5 - (12*A*a^ 2*c*d*e^5)/5) + (d + e*x)^3*(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) - (d + e*x)^2*(14*B*c^3*d^5 - 2* A*a^2*c*e^5 - 10*A*c^3*d^4*e - 12*A*a*c^2*d^2*e^3 + 20*B*a*c^2*d^3*e^2 + 6 *B*a^2*c*d*e^4) + (2*A*a^3*e^7)/7 - (2*B*c^3*d^7)/7 - (2*B*a^3*d*e^6)/7 + (2*A*c^3*d^6*e)/7 + (6*A*a*c^2*d^4*e^3)/7 + (6*A*a^2*c*d^2*e^5)/7 - (6*B*a *c^2*d^5*e^2)/7 - (6*B*a^2*c*d^3*e^4)/7)/(e^8*(d + e*x)^(7/2)) + (2*B*c^3* (d + e*x)^(7/2))/(7*e^8) + (2*c^2*(d + e*x)^(1/2)*(3*A*a*e^3 - 35*B*c*d^3 - 15*B*a*d*e^2 + 15*A*c*d^2*e))/e^8 + (2*c^3*(A*e - 7*B*d)*(d + e*x)^(5/2) )/(5*e^8)
Time = 0.27 (sec) , antiderivative size = 519, normalized size of antiderivative = 1.52 \[ \int \frac {(A+B x) \left (a+c x^2\right )^3}{(d+e x)^{9/2}} \, dx=\frac {-\frac {4096}{35} b \,c^{3} d^{7}-\frac {8}{5} a^{3} c d \,e^{6} x -6 a^{2} b c \,e^{7} x^{3}-\frac {4}{35} a^{3} b d \,e^{6}-\frac {16}{35} a^{3} c \,d^{2} e^{5}+\frac {768}{35} a^{2} c^{2} d^{4} e^{3}+\frac {2048}{35} a \,c^{3} d^{6} e -\frac {2}{7} a^{4} e^{7}-\frac {48}{5} a^{2} b c \,d^{2} e^{5} x -12 a^{2} b c d \,e^{6} x^{2}-256 a b \,c^{2} d^{4} e^{3} x -320 a b \,c^{2} d^{3} e^{4} x^{2}-160 a b \,c^{2} d^{2} e^{5} x^{3}-20 a b \,c^{2} d \,e^{6} x^{4}-\frac {2}{5} a^{3} b \,e^{7} x -2 a^{3} c \,e^{7} x^{2}+6 a^{2} c^{2} e^{7} x^{4}+\frac {2}{5} a \,c^{3} e^{7} x^{6}+\frac {2}{7} b \,c^{3} e^{7} x^{7}+\frac {384}{5} a^{2} c^{2} d^{3} e^{4} x +96 a^{2} c^{2} d^{2} e^{5} x^{2}+48 a^{2} c^{2} d \,e^{6} x^{3}+2 a b \,c^{2} e^{7} x^{5}+\frac {1024}{5} a \,c^{3} d^{5} e^{2} x +256 a \,c^{3} d^{4} e^{3} x^{2}+128 a \,c^{3} d^{3} e^{4} x^{3}+16 a \,c^{3} d^{2} e^{5} x^{4}-\frac {8}{5} a \,c^{3} d \,e^{6} x^{5}-\frac {2048}{5} b \,c^{3} d^{6} e x -512 b \,c^{3} d^{5} e^{2} x^{2}-256 b \,c^{3} d^{4} e^{3} x^{3}-32 b \,c^{3} d^{3} e^{4} x^{4}+\frac {16}{5} b \,c^{3} d^{2} e^{5} x^{5}-\frac {4}{5} b \,c^{3} d \,e^{6} x^{6}-\frac {96}{35} a^{2} b c \,d^{3} e^{4}-\frac {512}{7} a b \,c^{2} d^{5} e^{2}}{\sqrt {e x +d}\, e^{8} \left (e^{3} x^{3}+3 d \,e^{2} x^{2}+3 d^{2} e x +d^{3}\right )} \] Input:
int((B*x+A)*(c*x^2+a)^3/(e*x+d)^(9/2),x)
Output:
(2*( - 5*a**4*e**7 - 2*a**3*b*d*e**6 - 7*a**3*b*e**7*x - 8*a**3*c*d**2*e** 5 - 28*a**3*c*d*e**6*x - 35*a**3*c*e**7*x**2 - 48*a**2*b*c*d**3*e**4 - 168 *a**2*b*c*d**2*e**5*x - 210*a**2*b*c*d*e**6*x**2 - 105*a**2*b*c*e**7*x**3 + 384*a**2*c**2*d**4*e**3 + 1344*a**2*c**2*d**3*e**4*x + 1680*a**2*c**2*d* *2*e**5*x**2 + 840*a**2*c**2*d*e**6*x**3 + 105*a**2*c**2*e**7*x**4 - 1280* a*b*c**2*d**5*e**2 - 4480*a*b*c**2*d**4*e**3*x - 5600*a*b*c**2*d**3*e**4*x **2 - 2800*a*b*c**2*d**2*e**5*x**3 - 350*a*b*c**2*d*e**6*x**4 + 35*a*b*c** 2*e**7*x**5 + 1024*a*c**3*d**6*e + 3584*a*c**3*d**5*e**2*x + 4480*a*c**3*d **4*e**3*x**2 + 2240*a*c**3*d**3*e**4*x**3 + 280*a*c**3*d**2*e**5*x**4 - 2 8*a*c**3*d*e**6*x**5 + 7*a*c**3*e**7*x**6 - 2048*b*c**3*d**7 - 7168*b*c**3 *d**6*e*x - 8960*b*c**3*d**5*e**2*x**2 - 4480*b*c**3*d**4*e**3*x**3 - 560* b*c**3*d**3*e**4*x**4 + 56*b*c**3*d**2*e**5*x**5 - 14*b*c**3*d*e**6*x**6 + 5*b*c**3*e**7*x**7))/(35*sqrt(d + e*x)*e**8*(d**3 + 3*d**2*e*x + 3*d*e**2 *x**2 + e**3*x**3))