Integrand size = 24, antiderivative size = 346 \[ \int \frac {(A+B x) \left (a+c x^2\right )^3}{(d+e x)^{11/2}} \, dx=\frac {2 (B d-A e) \left (c d^2+a e^2\right )^3}{9 e^8 (d+e x)^{9/2}}-\frac {2 \left (c d^2+a e^2\right )^2 \left (7 B c d^2-6 A c d e+a B e^2\right )}{7 e^8 (d+e x)^{7/2}}+\frac {6 c \left (c d^2+a e^2\right ) \left (7 B c d^3-5 A c d^2 e+3 a B d e^2-a A e^3\right )}{5 e^8 (d+e x)^{5/2}}+\frac {2 c \left (4 A c d e \left (5 c d^2+3 a e^2\right )-B \left (35 c^2 d^4+30 a c d^2 e^2+3 a^2 e^4\right )\right )}{3 e^8 (d+e x)^{3/2}}+\frac {2 c^2 \left (35 B c d^3-15 A c d^2 e+15 a B d e^2-3 a A e^3\right )}{e^8 \sqrt {d+e x}}+\frac {6 c^2 \left (7 B c d^2-2 A c d e+a B e^2\right ) \sqrt {d+e x}}{e^8}-\frac {2 c^3 (7 B d-A e) (d+e x)^{3/2}}{3 e^8}+\frac {2 B c^3 (d+e x)^{5/2}}{5 e^8} \] Output:
2/9*(-A*e+B*d)*(a*e^2+c*d^2)^3/e^8/(e*x+d)^(9/2)-2/7*(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)^(7/2)+6/5*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)^(5/2)+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^8/(e*x+d)^(3/2)+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^8/(e*x+d)^(1/2)+6 *c^2*(-2*A*c*d*e+B*a*e^2+7*B*c*d^2)*(e*x+d)^(1/2)/e^8-2/3*c^3*(-A*e+7*B*d) *(e*x+d)^(3/2)/e^8+2/5*B*c^3*(e*x+d)^(5/2)/e^8
Time = 0.63 (sec) , antiderivative size = 375, normalized size of antiderivative = 1.08 \[ \int \frac {(A+B x) \left (a+c x^2\right )^3}{(d+e x)^{11/2}} \, dx=\frac {-2 A e \left (35 a^3 e^6+3 a^2 c e^4 \left (8 d^2+36 d e x+63 e^2 x^2\right )+3 a c^2 e^2 \left (128 d^4+576 d^3 e x+1008 d^2 e^2 x^2+840 d e^3 x^3+315 e^4 x^4\right )+5 c^3 \left (1024 d^6+4608 d^5 e x+8064 d^4 e^2 x^2+6720 d^3 e^3 x^3+2520 d^2 e^4 x^4+252 d e^5 x^5-21 e^6 x^6\right )\right )+2 B \left (-5 a^3 e^6 (2 d+9 e x)-3 a^2 c e^4 \left (16 d^3+72 d^2 e x+126 d e^2 x^2+105 e^3 x^3\right )+15 a c^2 e^2 \left (256 d^5+1152 d^4 e x+2016 d^3 e^2 x^2+1680 d^2 e^3 x^3+630 d e^4 x^4+63 e^5 x^5\right )+7 c^3 \left (2048 d^7+9216 d^6 e x+16128 d^5 e^2 x^2+13440 d^4 e^3 x^3+5040 d^3 e^4 x^4+504 d^2 e^5 x^5-42 d e^6 x^6+9 e^7 x^7\right )\right )}{315 e^8 (d+e x)^{9/2}} \] Input:
Integrate[((A + B*x)*(a + c*x^2)^3)/(d + e*x)^(11/2),x]
Output:
(-2*A*e*(35*a^3*e^6 + 3*a^2*c*e^4*(8*d^2 + 36*d*e*x + 63*e^2*x^2) + 3*a*c^ 2*e^2*(128*d^4 + 576*d^3*e*x + 1008*d^2*e^2*x^2 + 840*d*e^3*x^3 + 315*e^4* x^4) + 5*c^3*(1024*d^6 + 4608*d^5*e*x + 8064*d^4*e^2*x^2 + 6720*d^3*e^3*x^ 3 + 2520*d^2*e^4*x^4 + 252*d*e^5*x^5 - 21*e^6*x^6)) + 2*B*(-5*a^3*e^6*(2*d + 9*e*x) - 3*a^2*c*e^4*(16*d^3 + 72*d^2*e*x + 126*d*e^2*x^2 + 105*e^3*x^3 ) + 15*a*c^2*e^2*(256*d^5 + 1152*d^4*e*x + 2016*d^3*e^2*x^2 + 1680*d^2*e^3 *x^3 + 630*d*e^4*x^4 + 63*e^5*x^5) + 7*c^3*(2048*d^7 + 9216*d^6*e*x + 1612 8*d^5*e^2*x^2 + 13440*d^4*e^3*x^3 + 5040*d^3*e^4*x^4 + 504*d^2*e^5*x^5 - 4 2*d*e^6*x^6 + 9*e^7*x^7)))/(315*e^8*(d + e*x)^(9/2))
Time = 0.47 (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.
| \(\displaystyle \int \frac {\left (a+c x^2\right )^3 (A+B x)}{(d+e x)^{11/2}} \, dx\) |
| \(\Big \downarrow \) 652 |
| \(\displaystyle \int \left (-\frac {c \left (-3 a^2 B e^4+12 a A c d e^3-30 a B c d^2 e^2+20 A c^2 d^3 e-35 B c^2 d^4\right )}{e^7 (d+e x)^{5/2}}-\frac {3 c^2 \left (-a B e^2+2 A c d e-7 B c d^2\right )}{e^7 \sqrt {d+e x}}+\frac {c^2 \left (3 a A e^3-15 a B d e^2+15 A c d^2 e-35 B c d^3\right )}{e^7 (d+e x)^{3/2}}+\frac {\left (a e^2+c d^2\right )^2 \left (a B e^2-6 A c d e+7 B c d^2\right )}{e^7 (d+e x)^{9/2}}+\frac {\left (a e^2+c d^2\right )^3 (A e-B d)}{e^7 (d+e x)^{11/2}}+\frac {3 c \left (a e^2+c d^2\right ) \left (a A e^3-3 a B d e^2+5 A c d^2 e-7 B c d^3\right )}{e^7 (d+e x)^{7/2}}+\frac {c^3 \sqrt {d+e x} (A e-7 B d)}{e^7}+\frac {B c^3 (d+e x)^{3/2}}{e^7}\right )dx\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {2 c \left (4 A c d e \left (3 a e^2+5 c d^2\right )-B \left (3 a^2 e^4+30 a c d^2 e^2+35 c^2 d^4\right )\right )}{3 e^8 (d+e x)^{3/2}}+\frac {6 c^2 \sqrt {d+e x} \left (a B e^2-2 A c d e+7 B c d^2\right )}{e^8}+\frac {2 c^2 \left (-3 a A e^3+15 a B d e^2-15 A c d^2 e+35 B c d^3\right )}{e^8 \sqrt {d+e x}}-\frac {2 \left (a e^2+c d^2\right )^2 \left (a B e^2-6 A c d e+7 B c d^2\right )}{7 e^8 (d+e x)^{7/2}}+\frac {2 \left (a e^2+c d^2\right )^3 (B d-A e)}{9 e^8 (d+e x)^{9/2}}+\frac {6 c \left (a e^2+c d^2\right ) \left (-a A e^3+3 a B d e^2-5 A c d^2 e+7 B c d^3\right )}{5 e^8 (d+e x)^{5/2}}-\frac {2 c^3 (d+e x)^{3/2} (7 B d-A e)}{3 e^8}+\frac {2 B c^3 (d+e x)^{5/2}}{5 e^8}\) |
Input:
Int[((A + B*x)*(a + c*x^2)^3)/(d + e*x)^(11/2),x]
Output:
(2*(B*d - A*e)*(c*d^2 + a*e^2)^3)/(9*e^8*(d + e*x)^(9/2)) - (2*(c*d^2 + a* e^2)^2*(7*B*c*d^2 - 6*A*c*d*e + a*B*e^2))/(7*e^8*(d + e*x)^(7/2)) + (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))/(5*e^8*( d + e*x)^(5/2)) + (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)))/(3*e^8*(d + e*x)^(3/2)) + (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))/(e^8*Sqrt[d + e*x]) + (6*c^2*(7 *B*c*d^2 - 2*A*c*d*e + a*B*e^2)*Sqrt[d + e*x])/e^8 - (2*c^3*(7*B*d - A*e)* (d + e*x)^(3/2))/(3*e^8) + (2*B*c^3*(d + e*x)^(5/2))/(5*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 = 1.10 (sec) , antiderivative size = 305, normalized size of antiderivative = 0.88
| method | result | size |
| pseudoelliptic | \(\frac {\left (\left (126 B \,x^{7}+210 A \,x^{6}\right ) c^{3}-1890 a \,x^{4} \left (-B x +A \right ) c^{2}-378 \left (\frac {5 B x}{3}+A \right ) x^{2} a^{2} c -70 a^{3} \left (\frac {9 B x}{7}+A \right )\right ) e^{7}-216 d \left (\frac {35 x^{5} \left (\frac {7 B x}{30}+A \right ) c^{3}}{3}+\frac {70 \left (-\frac {15 B x}{4}+A \right ) x^{3} a \,c^{2}}{3}+a^{2} x \left (\frac {7 B x}{2}+A \right ) c +\frac {5 B \,a^{3}}{54}\right ) e^{6}-48 c \,d^{2} \left (\left (-147 B \,x^{5}+525 A \,x^{4}\right ) c^{2}+126 \left (-\frac {25 B x}{3}+A \right ) x^{2} a c +a^{2} \left (9 B x +A \right )\right ) e^{5}-3456 c \,d^{3} \left (\frac {175 \left (-\frac {21 B x}{20}+A \right ) x^{3} c^{2}}{9}+a x \left (-\frac {35 B x}{2}+A \right ) c +\frac {a^{2} B}{36}\right ) e^{4}-768 c^{2} \left (\left (-245 B \,x^{3}+105 A \,x^{2}\right ) c +a \left (-45 B x +A \right )\right ) d^{4} e^{3}-46080 c^{2} d^{5} \left (x \left (-\frac {49 B x}{10}+A \right ) c -\frac {B a}{6}\right ) e^{2}-10240 c^{3} \left (-\frac {63 B x}{5}+A \right ) d^{6} e +28672 B \,c^{3} d^{7}}{315 \left (e x +d \right )^{\frac {9}{2}} e^{8}}\) | \(305\) |
| derivativedivides | \(\frac {\frac {2 B \,c^{3} \left (e x +d \right )^{\frac {5}{2}}}{5}+\frac {2 A \,c^{3} e \left (e x +d \right )^{\frac {3}{2}}}{3}-\frac {14 B \,c^{3} d \left (e x +d \right )^{\frac {3}{2}}}{3}-12 A \,c^{3} d e \sqrt {e x +d}+6 B a \,c^{2} e^{2} \sqrt {e x +d}+42 B \,c^{3} d^{2} \sqrt {e x +d}-\frac {2 \left (A \,a^{3} e^{7}+3 A \,a^{2} c \,d^{2} e^{5}+3 A a \,c^{2} d^{4} e^{3}+A \,c^{3} d^{6} e -B \,a^{3} d \,e^{6}-3 B \,a^{2} c \,d^{3} e^{4}-3 B a \,c^{2} d^{5} e^{2}-B \,c^{3} d^{7}\right )}{9 \left (e x +d \right )^{\frac {9}{2}}}-\frac {6 c \left (A \,a^{2} e^{5}+6 A a c \,d^{2} e^{3}+5 A \,c^{2} d^{4} e -3 B \,a^{2} d \,e^{4}-10 B a c \,d^{3} e^{2}-7 B \,c^{2} d^{5}\right )}{5 \left (e x +d \right )^{\frac {5}{2}}}-\frac {2 c^{2} \left (3 A a \,e^{3}+15 A c \,d^{2} e -15 B a d \,e^{2}-35 B c \,d^{3}\right )}{\sqrt {e x +d}}-\frac {2 \left (-6 A \,a^{2} c d \,e^{5}-12 A a \,c^{2} d^{3} e^{3}-6 A \,c^{3} d^{5} e +B \,a^{3} e^{6}+9 B \,a^{2} c \,d^{2} e^{4}+15 B a \,c^{2} d^{4} e^{2}+7 B \,c^{3} d^{6}\right )}{7 \left (e x +d \right )^{\frac {7}{2}}}+\frac {2 c \left (12 A a c d \,e^{3}+20 A \,c^{2} d^{3} e -3 B \,e^{4} a^{2}-30 B a c \,d^{2} e^{2}-35 B \,c^{2} d^{4}\right )}{3 \left (e x +d \right )^{\frac {3}{2}}}}{e^{8}}\) | \(453\) |
| default | \(\frac {\frac {2 B \,c^{3} \left (e x +d \right )^{\frac {5}{2}}}{5}+\frac {2 A \,c^{3} e \left (e x +d \right )^{\frac {3}{2}}}{3}-\frac {14 B \,c^{3} d \left (e x +d \right )^{\frac {3}{2}}}{3}-12 A \,c^{3} d e \sqrt {e x +d}+6 B a \,c^{2} e^{2} \sqrt {e x +d}+42 B \,c^{3} d^{2} \sqrt {e x +d}-\frac {2 \left (A \,a^{3} e^{7}+3 A \,a^{2} c \,d^{2} e^{5}+3 A a \,c^{2} d^{4} e^{3}+A \,c^{3} d^{6} e -B \,a^{3} d \,e^{6}-3 B \,a^{2} c \,d^{3} e^{4}-3 B a \,c^{2} d^{5} e^{2}-B \,c^{3} d^{7}\right )}{9 \left (e x +d \right )^{\frac {9}{2}}}-\frac {6 c \left (A \,a^{2} e^{5}+6 A a c \,d^{2} e^{3}+5 A \,c^{2} d^{4} e -3 B \,a^{2} d \,e^{4}-10 B a c \,d^{3} e^{2}-7 B \,c^{2} d^{5}\right )}{5 \left (e x +d \right )^{\frac {5}{2}}}-\frac {2 c^{2} \left (3 A a \,e^{3}+15 A c \,d^{2} e -15 B a d \,e^{2}-35 B c \,d^{3}\right )}{\sqrt {e x +d}}-\frac {2 \left (-6 A \,a^{2} c d \,e^{5}-12 A a \,c^{2} d^{3} e^{3}-6 A \,c^{3} d^{5} e +B \,a^{3} e^{6}+9 B \,a^{2} c \,d^{2} e^{4}+15 B a \,c^{2} d^{4} e^{2}+7 B \,c^{3} d^{6}\right )}{7 \left (e x +d \right )^{\frac {7}{2}}}+\frac {2 c \left (12 A a c d \,e^{3}+20 A \,c^{2} d^{3} e -3 B \,e^{4} a^{2}-30 B a c \,d^{2} e^{2}-35 B \,c^{2} d^{4}\right )}{3 \left (e x +d \right )^{\frac {3}{2}}}}{e^{8}}\) | \(453\) |
| gosper | \(-\frac {2 \left (-63 B \,x^{7} c^{3} e^{7}-105 A \,x^{6} c^{3} e^{7}+294 B \,x^{6} c^{3} d \,e^{6}+1260 A \,x^{5} c^{3} d \,e^{6}-945 B \,x^{5} a \,c^{2} e^{7}-3528 B \,x^{5} c^{3} d^{2} e^{5}+945 A \,x^{4} a \,c^{2} e^{7}+12600 A \,x^{4} c^{3} d^{2} e^{5}-9450 B \,x^{4} a \,c^{2} d \,e^{6}-35280 B \,x^{4} c^{3} d^{3} e^{4}+2520 A \,x^{3} a \,c^{2} d \,e^{6}+33600 A \,x^{3} c^{3} d^{3} e^{4}+315 B \,x^{3} a^{2} c \,e^{7}-25200 B \,x^{3} a \,c^{2} d^{2} e^{5}-94080 B \,x^{3} c^{3} d^{4} e^{3}+189 A \,x^{2} a^{2} c \,e^{7}+3024 A \,x^{2} a \,c^{2} d^{2} e^{5}+40320 A \,x^{2} c^{3} d^{4} e^{3}+378 B \,x^{2} a^{2} c d \,e^{6}-30240 B \,x^{2} a \,c^{2} d^{3} e^{4}-112896 B \,x^{2} c^{3} d^{5} e^{2}+108 A x \,a^{2} c d \,e^{6}+1728 A x a \,c^{2} d^{3} e^{4}+23040 A x \,c^{3} d^{5} e^{2}+45 B x \,a^{3} e^{7}+216 B x \,a^{2} c \,d^{2} e^{5}-17280 B x a \,c^{2} d^{4} e^{3}-64512 B x \,c^{3} d^{6} e +35 A \,a^{3} e^{7}+24 A \,a^{2} c \,d^{2} e^{5}+384 A a \,c^{2} d^{4} e^{3}+5120 A \,c^{3} d^{6} e +10 B \,a^{3} d \,e^{6}+48 B \,a^{2} c \,d^{3} e^{4}-3840 B a \,c^{2} d^{5} e^{2}-14336 B \,c^{3} d^{7}\right )}{315 \left (e x +d \right )^{\frac {9}{2}} e^{8}}\) | \(489\) |
| trager | \(-\frac {2 \left (-63 B \,x^{7} c^{3} e^{7}-105 A \,x^{6} c^{3} e^{7}+294 B \,x^{6} c^{3} d \,e^{6}+1260 A \,x^{5} c^{3} d \,e^{6}-945 B \,x^{5} a \,c^{2} e^{7}-3528 B \,x^{5} c^{3} d^{2} e^{5}+945 A \,x^{4} a \,c^{2} e^{7}+12600 A \,x^{4} c^{3} d^{2} e^{5}-9450 B \,x^{4} a \,c^{2} d \,e^{6}-35280 B \,x^{4} c^{3} d^{3} e^{4}+2520 A \,x^{3} a \,c^{2} d \,e^{6}+33600 A \,x^{3} c^{3} d^{3} e^{4}+315 B \,x^{3} a^{2} c \,e^{7}-25200 B \,x^{3} a \,c^{2} d^{2} e^{5}-94080 B \,x^{3} c^{3} d^{4} e^{3}+189 A \,x^{2} a^{2} c \,e^{7}+3024 A \,x^{2} a \,c^{2} d^{2} e^{5}+40320 A \,x^{2} c^{3} d^{4} e^{3}+378 B \,x^{2} a^{2} c d \,e^{6}-30240 B \,x^{2} a \,c^{2} d^{3} e^{4}-112896 B \,x^{2} c^{3} d^{5} e^{2}+108 A x \,a^{2} c d \,e^{6}+1728 A x a \,c^{2} d^{3} e^{4}+23040 A x \,c^{3} d^{5} e^{2}+45 B x \,a^{3} e^{7}+216 B x \,a^{2} c \,d^{2} e^{5}-17280 B x a \,c^{2} d^{4} e^{3}-64512 B x \,c^{3} d^{6} e +35 A \,a^{3} e^{7}+24 A \,a^{2} c \,d^{2} e^{5}+384 A a \,c^{2} d^{4} e^{3}+5120 A \,c^{3} d^{6} e +10 B \,a^{3} d \,e^{6}+48 B \,a^{2} c \,d^{3} e^{4}-3840 B a \,c^{2} d^{5} e^{2}-14336 B \,c^{3} d^{7}\right )}{315 \left (e x +d \right )^{\frac {9}{2}} e^{8}}\) | \(489\) |
| orering | \(-\frac {2 \left (-63 B \,x^{7} c^{3} e^{7}-105 A \,x^{6} c^{3} e^{7}+294 B \,x^{6} c^{3} d \,e^{6}+1260 A \,x^{5} c^{3} d \,e^{6}-945 B \,x^{5} a \,c^{2} e^{7}-3528 B \,x^{5} c^{3} d^{2} e^{5}+945 A \,x^{4} a \,c^{2} e^{7}+12600 A \,x^{4} c^{3} d^{2} e^{5}-9450 B \,x^{4} a \,c^{2} d \,e^{6}-35280 B \,x^{4} c^{3} d^{3} e^{4}+2520 A \,x^{3} a \,c^{2} d \,e^{6}+33600 A \,x^{3} c^{3} d^{3} e^{4}+315 B \,x^{3} a^{2} c \,e^{7}-25200 B \,x^{3} a \,c^{2} d^{2} e^{5}-94080 B \,x^{3} c^{3} d^{4} e^{3}+189 A \,x^{2} a^{2} c \,e^{7}+3024 A \,x^{2} a \,c^{2} d^{2} e^{5}+40320 A \,x^{2} c^{3} d^{4} e^{3}+378 B \,x^{2} a^{2} c d \,e^{6}-30240 B \,x^{2} a \,c^{2} d^{3} e^{4}-112896 B \,x^{2} c^{3} d^{5} e^{2}+108 A x \,a^{2} c d \,e^{6}+1728 A x a \,c^{2} d^{3} e^{4}+23040 A x \,c^{3} d^{5} e^{2}+45 B x \,a^{3} e^{7}+216 B x \,a^{2} c \,d^{2} e^{5}-17280 B x a \,c^{2} d^{4} e^{3}-64512 B x \,c^{3} d^{6} e +35 A \,a^{3} e^{7}+24 A \,a^{2} c \,d^{2} e^{5}+384 A a \,c^{2} d^{4} e^{3}+5120 A \,c^{3} d^{6} e +10 B \,a^{3} d \,e^{6}+48 B \,a^{2} c \,d^{3} e^{4}-3840 B a \,c^{2} d^{5} e^{2}-14336 B \,c^{3} d^{7}\right )}{315 \left (e x +d \right )^{\frac {9}{2}} e^{8}}\) | \(489\) |
| risch | \(-\frac {2 c^{2} \left (-3 e^{2} B c \,x^{2}-5 A c \,e^{2} x +29 B c d e x +85 A c d e -45 B a \,e^{2}-283 B c \,d^{2}\right ) \sqrt {e x +d}}{15 e^{8}}-\frac {2 \left (945 A \,x^{4} a \,c^{2} e^{7}+4725 A \,x^{4} c^{3} d^{2} e^{5}-4725 B \,x^{4} a \,c^{2} d \,e^{6}-11025 B \,x^{4} c^{3} d^{3} e^{4}+2520 A \,x^{3} a \,c^{2} d \,e^{6}+16800 A \,x^{3} c^{3} d^{3} e^{4}+315 B \,x^{3} a^{2} c \,e^{7}-15750 B \,x^{3} a \,c^{2} d^{2} e^{5}-40425 B \,x^{3} c^{3} d^{4} e^{3}+189 A \,x^{2} a^{2} c \,e^{7}+3024 A \,x^{2} a \,c^{2} d^{2} e^{5}+22995 A \,x^{2} c^{3} d^{4} e^{3}+378 B \,x^{2} a^{2} c d \,e^{6}-20790 B \,x^{2} a \,c^{2} d^{3} e^{4}-56448 B \,x^{2} c^{3} d^{5} e^{2}+108 A x \,a^{2} c d \,e^{6}+1728 A x a \,c^{2} d^{3} e^{4}+14220 A x \,c^{3} d^{5} e^{2}+45 B x \,a^{3} e^{7}+216 B x \,a^{2} c \,d^{2} e^{5}-12555 B x a \,c^{2} d^{4} e^{3}-35406 B x \,c^{3} d^{6} e +35 A \,a^{3} e^{7}+24 A \,a^{2} c \,d^{2} e^{5}+384 A a \,c^{2} d^{4} e^{3}+3335 A \,c^{3} d^{6} e +10 B \,a^{3} d \,e^{6}+48 B \,a^{2} c \,d^{3} e^{4}-2895 B a \,c^{2} d^{5} e^{2}-8393 B \,c^{3} d^{7}\right )}{315 e^{8} \sqrt {e x +d}\, \left (e^{4} x^{4}+4 d \,e^{3} x^{3}+6 d^{2} e^{2} x^{2}+4 d^{3} e x +d^{4}\right )}\) | \(513\) |
Input:
int((B*x+A)*(c*x^2+a)^3/(e*x+d)^(11/2),x,method=_RETURNVERBOSE)
Output:
1/315*(((126*B*x^7+210*A*x^6)*c^3-1890*a*x^4*(-B*x+A)*c^2-378*(5/3*B*x+A)* x^2*a^2*c-70*a^3*(9/7*B*x+A))*e^7-216*d*(35/3*x^5*(7/30*B*x+A)*c^3+70/3*(- 15/4*B*x+A)*x^3*a*c^2+a^2*x*(7/2*B*x+A)*c+5/54*B*a^3)*e^6-48*c*d^2*((-147* B*x^5+525*A*x^4)*c^2+126*(-25/3*B*x+A)*x^2*a*c+a^2*(9*B*x+A))*e^5-3456*c*d ^3*(175/9*(-21/20*B*x+A)*x^3*c^2+a*x*(-35/2*B*x+A)*c+1/36*a^2*B)*e^4-768*c ^2*((-245*B*x^3+105*A*x^2)*c+a*(-45*B*x+A))*d^4*e^3-46080*c^2*d^5*(x*(-49/ 10*B*x+A)*c-1/6*B*a)*e^2-10240*c^3*(-63/5*B*x+A)*d^6*e+28672*B*c^3*d^7)/(e *x+d)^(9/2)/e^8
Time = 0.09 (sec) , antiderivative size = 509, normalized size of antiderivative = 1.47 \[ \int \frac {(A+B x) \left (a+c x^2\right )^3}{(d+e x)^{11/2}} \, dx=\frac {2 \, {\left (63 \, B c^{3} e^{7} x^{7} + 14336 \, B c^{3} d^{7} - 5120 \, A c^{3} d^{6} e + 3840 \, B a c^{2} d^{5} e^{2} - 384 \, A a c^{2} d^{4} e^{3} - 48 \, B a^{2} c d^{3} e^{4} - 24 \, A a^{2} c d^{2} e^{5} - 10 \, B a^{3} d e^{6} - 35 \, A a^{3} e^{7} - 21 \, {\left (14 \, B c^{3} d e^{6} - 5 \, A c^{3} e^{7}\right )} x^{6} + 63 \, {\left (56 \, B c^{3} d^{2} e^{5} - 20 \, A c^{3} d e^{6} + 15 \, B a c^{2} e^{7}\right )} x^{5} + 315 \, {\left (112 \, B c^{3} d^{3} e^{4} - 40 \, A c^{3} d^{2} e^{5} + 30 \, B a c^{2} d e^{6} - 3 \, A a c^{2} e^{7}\right )} x^{4} + 105 \, {\left (896 \, B c^{3} d^{4} e^{3} - 320 \, A c^{3} d^{3} e^{4} + 240 \, B a c^{2} d^{2} e^{5} - 24 \, A a c^{2} d e^{6} - 3 \, B a^{2} c e^{7}\right )} x^{3} + 63 \, {\left (1792 \, B c^{3} d^{5} e^{2} - 640 \, A c^{3} d^{4} e^{3} + 480 \, B a c^{2} d^{3} e^{4} - 48 \, A a c^{2} d^{2} e^{5} - 6 \, B a^{2} c d e^{6} - 3 \, A a^{2} c e^{7}\right )} x^{2} + 9 \, {\left (7168 \, B c^{3} d^{6} e - 2560 \, A c^{3} d^{5} e^{2} + 1920 \, B a c^{2} d^{4} e^{3} - 192 \, A a c^{2} d^{3} e^{4} - 24 \, B a^{2} c d^{2} e^{5} - 12 \, A a^{2} c d e^{6} - 5 \, B a^{3} e^{7}\right )} x\right )} \sqrt {e x + d}}{315 \, {\left (e^{13} x^{5} + 5 \, d e^{12} x^{4} + 10 \, d^{2} e^{11} x^{3} + 10 \, d^{3} e^{10} x^{2} + 5 \, d^{4} e^{9} x + d^{5} e^{8}\right )}} \] Input:
integrate((B*x+A)*(c*x^2+a)^3/(e*x+d)^(11/2),x, algorithm="fricas")
Output:
2/315*(63*B*c^3*e^7*x^7 + 14336*B*c^3*d^7 - 5120*A*c^3*d^6*e + 3840*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 - 24*A*a^2*c*d^2*e^5 - 10*B*a^3*d*e^6 - 35*A*a^3*e^7 - 21*(14*B*c^3*d*e^6 - 5*A*c^3*e^7)*x^6 + 63*(56*B*c^3*d^2*e^5 - 20*A*c^3*d*e^6 + 15*B*a*c^2*e^7)*x^5 + 315*(112*B*c ^3*d^3*e^4 - 40*A*c^3*d^2*e^5 + 30*B*a*c^2*d*e^6 - 3*A*a*c^2*e^7)*x^4 + 10 5*(896*B*c^3*d^4*e^3 - 320*A*c^3*d^3*e^4 + 240*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 + 63*(1792*B*c^3*d^5*e^2 - 640*A*c^3*d^4*e^3 + 480*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 - 3*A*a^2*c*e ^7)*x^2 + 9*(7168*B*c^3*d^6*e - 2560*A*c^3*d^5*e^2 + 1920*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 - 12*A*a^2*c*d*e^6 - 5*B*a^3*e^ 7)*x)*sqrt(e*x + d)/(e^13*x^5 + 5*d*e^12*x^4 + 10*d^2*e^11*x^3 + 10*d^3*e^ 10*x^2 + 5*d^4*e^9*x + d^5*e^8)
Leaf count of result is larger than twice the leaf count of optimal. 3952 vs. \(2 (362) = 724\).
Time = 1.27 (sec) , antiderivative size = 3952, normalized size of antiderivative = 11.42 \[ \int \frac {(A+B x) \left (a+c x^2\right )^3}{(d+e x)^{11/2}} \, dx=\text {Too large to display} \] Input:
integrate((B*x+A)*(c*x**2+a)**3/(e*x+d)**(11/2),x)
Output:
Piecewise((-70*A*a**3*e**7/(315*d**4*e**8*sqrt(d + e*x) + 1260*d**3*e**9*x *sqrt(d + e*x) + 1890*d**2*e**10*x**2*sqrt(d + e*x) + 1260*d*e**11*x**3*sq rt(d + e*x) + 315*e**12*x**4*sqrt(d + e*x)) - 48*A*a**2*c*d**2*e**5/(315*d **4*e**8*sqrt(d + e*x) + 1260*d**3*e**9*x*sqrt(d + e*x) + 1890*d**2*e**10* x**2*sqrt(d + e*x) + 1260*d*e**11*x**3*sqrt(d + e*x) + 315*e**12*x**4*sqrt (d + e*x)) - 216*A*a**2*c*d*e**6*x/(315*d**4*e**8*sqrt(d + e*x) + 1260*d** 3*e**9*x*sqrt(d + e*x) + 1890*d**2*e**10*x**2*sqrt(d + e*x) + 1260*d*e**11 *x**3*sqrt(d + e*x) + 315*e**12*x**4*sqrt(d + e*x)) - 378*A*a**2*c*e**7*x* *2/(315*d**4*e**8*sqrt(d + e*x) + 1260*d**3*e**9*x*sqrt(d + e*x) + 1890*d* *2*e**10*x**2*sqrt(d + e*x) + 1260*d*e**11*x**3*sqrt(d + e*x) + 315*e**12* x**4*sqrt(d + e*x)) - 768*A*a*c**2*d**4*e**3/(315*d**4*e**8*sqrt(d + e*x) + 1260*d**3*e**9*x*sqrt(d + e*x) + 1890*d**2*e**10*x**2*sqrt(d + e*x) + 12 60*d*e**11*x**3*sqrt(d + e*x) + 315*e**12*x**4*sqrt(d + e*x)) - 3456*A*a*c **2*d**3*e**4*x/(315*d**4*e**8*sqrt(d + e*x) + 1260*d**3*e**9*x*sqrt(d + e *x) + 1890*d**2*e**10*x**2*sqrt(d + e*x) + 1260*d*e**11*x**3*sqrt(d + e*x) + 315*e**12*x**4*sqrt(d + e*x)) - 6048*A*a*c**2*d**2*e**5*x**2/(315*d**4* e**8*sqrt(d + e*x) + 1260*d**3*e**9*x*sqrt(d + e*x) + 1890*d**2*e**10*x**2 *sqrt(d + e*x) + 1260*d*e**11*x**3*sqrt(d + e*x) + 315*e**12*x**4*sqrt(d + e*x)) - 5040*A*a*c**2*d*e**6*x**3/(315*d**4*e**8*sqrt(d + e*x) + 1260*d** 3*e**9*x*sqrt(d + e*x) + 1890*d**2*e**10*x**2*sqrt(d + e*x) + 1260*d*e*...
Time = 0.04 (sec) , antiderivative size = 461, normalized size of antiderivative = 1.33 \[ \int \frac {(A+B x) \left (a+c x^2\right )^3}{(d+e x)^{11/2}} \, dx=\frac {2 \, {\left (\frac {21 \, {\left (3 \, {\left (e x + d\right )}^{\frac {5}{2}} B c^{3} - 5 \, {\left (7 \, B c^{3} d - A c^{3} e\right )} {\left (e x + d\right )}^{\frac {3}{2}} + 45 \, {\left (7 \, B c^{3} d^{2} - 2 \, A c^{3} d e + B a c^{2} e^{2}\right )} \sqrt {e x + d}\right )}}{e^{7}} + \frac {35 \, B c^{3} d^{7} - 35 \, A c^{3} d^{6} e + 105 \, B a c^{2} d^{5} e^{2} - 105 \, A a c^{2} d^{4} e^{3} + 105 \, B a^{2} c d^{3} e^{4} - 105 \, A a^{2} c d^{2} e^{5} + 35 \, B a^{3} d e^{6} - 35 \, A a^{3} e^{7} + 315 \, {\left (35 \, B c^{3} d^{3} - 15 \, A c^{3} d^{2} e + 15 \, B a c^{2} d e^{2} - 3 \, A a c^{2} e^{3}\right )} {\left (e x + d\right )}^{4} - 105 \, {\left (35 \, B c^{3} d^{4} - 20 \, A c^{3} d^{3} e + 30 \, B a c^{2} d^{2} e^{2} - 12 \, A a c^{2} d e^{3} + 3 \, B a^{2} c e^{4}\right )} {\left (e x + d\right )}^{3} + 189 \, {\left (7 \, B c^{3} d^{5} - 5 \, A c^{3} d^{4} e + 10 \, B a c^{2} d^{3} e^{2} - 6 \, A a c^{2} d^{2} e^{3} + 3 \, B a^{2} c d e^{4} - A a^{2} c e^{5}\right )} {\left (e x + d\right )}^{2} - 45 \, {\left (7 \, B c^{3} d^{6} - 6 \, A c^{3} d^{5} e + 15 \, B a c^{2} d^{4} e^{2} - 12 \, A a c^{2} d^{3} e^{3} + 9 \, B a^{2} c d^{2} e^{4} - 6 \, A a^{2} c d e^{5} + B a^{3} e^{6}\right )} {\left (e x + d\right )}}{{\left (e x + d\right )}^{\frac {9}{2}} e^{7}}\right )}}{315 \, e} \] Input:
integrate((B*x+A)*(c*x^2+a)^3/(e*x+d)^(11/2),x, algorithm="maxima")
Output:
2/315*(21*(3*(e*x + d)^(5/2)*B*c^3 - 5*(7*B*c^3*d - A*c^3*e)*(e*x + d)^(3/ 2) + 45*(7*B*c^3*d^2 - 2*A*c^3*d*e + B*a*c^2*e^2)*sqrt(e*x + d))/e^7 + (35 *B*c^3*d^7 - 35*A*c^3*d^6*e + 105*B*a*c^2*d^5*e^2 - 105*A*a*c^2*d^4*e^3 + 105*B*a^2*c*d^3*e^4 - 105*A*a^2*c*d^2*e^5 + 35*B*a^3*d*e^6 - 35*A*a^3*e^7 + 315*(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)^4 - 105*(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 + 189*(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 - 45*(7*B*c^3*d^6 - 6*A*c^3*d^5*e + 15*B*a*c^2*d^4*e^2 - 1 2*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)^(9/2)*e^7))/e
Time = 0.15 (sec) , antiderivative size = 589, normalized size of antiderivative = 1.70 \[ \int \frac {(A+B x) \left (a+c x^2\right )^3}{(d+e x)^{11/2}} \, dx=\frac {2 \, {\left (11025 \, {\left (e x + d\right )}^{4} B c^{3} d^{3} - 3675 \, {\left (e x + d\right )}^{3} B c^{3} d^{4} + 1323 \, {\left (e x + d\right )}^{2} B c^{3} d^{5} - 315 \, {\left (e x + d\right )} B c^{3} d^{6} + 35 \, B c^{3} d^{7} - 4725 \, {\left (e x + d\right )}^{4} A c^{3} d^{2} e + 2100 \, {\left (e x + d\right )}^{3} A c^{3} d^{3} e - 945 \, {\left (e x + d\right )}^{2} A c^{3} d^{4} e + 270 \, {\left (e x + d\right )} A c^{3} d^{5} e - 35 \, A c^{3} d^{6} e + 4725 \, {\left (e x + d\right )}^{4} B a c^{2} d e^{2} - 3150 \, {\left (e x + d\right )}^{3} B a c^{2} d^{2} e^{2} + 1890 \, {\left (e x + d\right )}^{2} B a c^{2} d^{3} e^{2} - 675 \, {\left (e x + d\right )} B a c^{2} d^{4} e^{2} + 105 \, B a c^{2} d^{5} e^{2} - 945 \, {\left (e x + d\right )}^{4} A a c^{2} e^{3} + 1260 \, {\left (e x + d\right )}^{3} A a c^{2} d e^{3} - 1134 \, {\left (e x + d\right )}^{2} A a c^{2} d^{2} e^{3} + 540 \, {\left (e x + d\right )} A a c^{2} d^{3} e^{3} - 105 \, A a c^{2} d^{4} e^{3} - 315 \, {\left (e x + d\right )}^{3} B a^{2} c e^{4} + 567 \, {\left (e x + d\right )}^{2} B a^{2} c d e^{4} - 405 \, {\left (e x + d\right )} B a^{2} c d^{2} e^{4} + 105 \, B a^{2} c d^{3} e^{4} - 189 \, {\left (e x + d\right )}^{2} A a^{2} c e^{5} + 270 \, {\left (e x + d\right )} A a^{2} c d e^{5} - 105 \, A a^{2} c d^{2} e^{5} - 45 \, {\left (e x + d\right )} B a^{3} e^{6} + 35 \, B a^{3} d e^{6} - 35 \, A a^{3} e^{7}\right )}}{315 \, {\left (e x + d\right )}^{\frac {9}{2}} e^{8}} + \frac {2 \, {\left (3 \, {\left (e x + d\right )}^{\frac {5}{2}} B c^{3} e^{32} - 35 \, {\left (e x + d\right )}^{\frac {3}{2}} B c^{3} d e^{32} + 315 \, \sqrt {e x + d} B c^{3} d^{2} e^{32} + 5 \, {\left (e x + d\right )}^{\frac {3}{2}} A c^{3} e^{33} - 90 \, \sqrt {e x + d} A c^{3} d e^{33} + 45 \, \sqrt {e x + d} B a c^{2} e^{34}\right )}}{15 \, e^{40}} \] Input:
integrate((B*x+A)*(c*x^2+a)^3/(e*x+d)^(11/2),x, algorithm="giac")
Output:
2/315*(11025*(e*x + d)^4*B*c^3*d^3 - 3675*(e*x + d)^3*B*c^3*d^4 + 1323*(e* x + d)^2*B*c^3*d^5 - 315*(e*x + d)*B*c^3*d^6 + 35*B*c^3*d^7 - 4725*(e*x + d)^4*A*c^3*d^2*e + 2100*(e*x + d)^3*A*c^3*d^3*e - 945*(e*x + d)^2*A*c^3*d^ 4*e + 270*(e*x + d)*A*c^3*d^5*e - 35*A*c^3*d^6*e + 4725*(e*x + d)^4*B*a*c^ 2*d*e^2 - 3150*(e*x + d)^3*B*a*c^2*d^2*e^2 + 1890*(e*x + d)^2*B*a*c^2*d^3* e^2 - 675*(e*x + d)*B*a*c^2*d^4*e^2 + 105*B*a*c^2*d^5*e^2 - 945*(e*x + d)^ 4*A*a*c^2*e^3 + 1260*(e*x + d)^3*A*a*c^2*d*e^3 - 1134*(e*x + d)^2*A*a*c^2* d^2*e^3 + 540*(e*x + d)*A*a*c^2*d^3*e^3 - 105*A*a*c^2*d^4*e^3 - 315*(e*x + d)^3*B*a^2*c*e^4 + 567*(e*x + d)^2*B*a^2*c*d*e^4 - 405*(e*x + d)*B*a^2*c* d^2*e^4 + 105*B*a^2*c*d^3*e^4 - 189*(e*x + d)^2*A*a^2*c*e^5 + 270*(e*x + d )*A*a^2*c*d*e^5 - 105*A*a^2*c*d^2*e^5 - 45*(e*x + d)*B*a^3*e^6 + 35*B*a^3* d*e^6 - 35*A*a^3*e^7)/((e*x + d)^(9/2)*e^8) + 2/15*(3*(e*x + d)^(5/2)*B*c^ 3*e^32 - 35*(e*x + d)^(3/2)*B*c^3*d*e^32 + 315*sqrt(e*x + d)*B*c^3*d^2*e^3 2 + 5*(e*x + d)^(3/2)*A*c^3*e^33 - 90*sqrt(e*x + d)*A*c^3*d*e^33 + 45*sqrt (e*x + d)*B*a*c^2*e^34)/e^40
Time = 6.65 (sec) , antiderivative size = 454, normalized size of antiderivative = 1.31 \[ \int \frac {(A+B x) \left (a+c x^2\right )^3}{(d+e x)^{11/2}} \, dx=\frac {{\left (d+e\,x\right )}^4\,\left (70\,B\,c^3\,d^3-30\,A\,c^3\,d^2\,e+30\,B\,a\,c^2\,d\,e^2-6\,A\,a\,c^2\,e^3\right )-\left (d+e\,x\right )\,\left (\frac {2\,B\,a^3\,e^6}{7}+\frac {18\,B\,a^2\,c\,d^2\,e^4}{7}-\frac {12\,A\,a^2\,c\,d\,e^5}{7}+\frac {30\,B\,a\,c^2\,d^4\,e^2}{7}-\frac {24\,A\,a\,c^2\,d^3\,e^3}{7}+2\,B\,c^3\,d^6-\frac {12\,A\,c^3\,d^5\,e}{7}\right )-{\left (d+e\,x\right )}^3\,\left (2\,B\,a^2\,c\,e^4+20\,B\,a\,c^2\,d^2\,e^2-8\,A\,a\,c^2\,d\,e^3+\frac {70\,B\,c^3\,d^4}{3}-\frac {40\,A\,c^3\,d^3\,e}{3}\right )+{\left (d+e\,x\right )}^2\,\left (\frac {18\,B\,a^2\,c\,d\,e^4}{5}-\frac {6\,A\,a^2\,c\,e^5}{5}+12\,B\,a\,c^2\,d^3\,e^2-\frac {36\,A\,a\,c^2\,d^2\,e^3}{5}+\frac {42\,B\,c^3\,d^5}{5}-6\,A\,c^3\,d^4\,e\right )-\frac {2\,A\,a^3\,e^7}{9}+\frac {2\,B\,c^3\,d^7}{9}+\frac {2\,B\,a^3\,d\,e^6}{9}-\frac {2\,A\,c^3\,d^6\,e}{9}-\frac {2\,A\,a\,c^2\,d^4\,e^3}{3}-\frac {2\,A\,a^2\,c\,d^2\,e^5}{3}+\frac {2\,B\,a\,c^2\,d^5\,e^2}{3}+\frac {2\,B\,a^2\,c\,d^3\,e^4}{3}}{e^8\,{\left (d+e\,x\right )}^{9/2}}+\frac {\sqrt {d+e\,x}\,\left (42\,B\,c^3\,d^2-12\,A\,c^3\,d\,e+6\,B\,a\,c^2\,e^2\right )}{e^8}+\frac {2\,B\,c^3\,{\left (d+e\,x\right )}^{5/2}}{5\,e^8}+\frac {2\,c^3\,\left (A\,e-7\,B\,d\right )\,{\left (d+e\,x\right )}^{3/2}}{3\,e^8} \] Input:
int(((a + c*x^2)^3*(A + B*x))/(d + e*x)^(11/2),x)
Output:
((d + e*x)^4*(70*B*c^3*d^3 - 6*A*a*c^2*e^3 - 30*A*c^3*d^2*e + 30*B*a*c^2*d *e^2) - (d + e*x)*((2*B*a^3*e^6)/7 + 2*B*c^3*d^6 - (12*A*c^3*d^5*e)/7 - (2 4*A*a*c^2*d^3*e^3)/7 + (30*B*a*c^2*d^4*e^2)/7 + (18*B*a^2*c*d^2*e^4)/7 - ( 12*A*a^2*c*d*e^5)/7) - (d + e*x)^3*((70*B*c^3*d^4)/3 + 2*B*a^2*c*e^4 - (40 *A*c^3*d^3*e)/3 + 20*B*a*c^2*d^2*e^2 - 8*A*a*c^2*d*e^3) + (d + e*x)^2*((42 *B*c^3*d^5)/5 - (6*A*a^2*c*e^5)/5 - 6*A*c^3*d^4*e - (36*A*a*c^2*d^2*e^3)/5 + 12*B*a*c^2*d^3*e^2 + (18*B*a^2*c*d*e^4)/5) - (2*A*a^3*e^7)/9 + (2*B*c^3 *d^7)/9 + (2*B*a^3*d*e^6)/9 - (2*A*c^3*d^6*e)/9 - (2*A*a*c^2*d^4*e^3)/3 - (2*A*a^2*c*d^2*e^5)/3 + (2*B*a*c^2*d^5*e^2)/3 + (2*B*a^2*c*d^3*e^4)/3)/(e^ 8*(d + e*x)^(9/2)) + ((d + e*x)^(1/2)*(42*B*c^3*d^2 - 12*A*c^3*d*e + 6*B*a *c^2*e^2))/e^8 + (2*B*c^3*(d + e*x)^(5/2))/(5*e^8) + (2*c^3*(A*e - 7*B*d)* (d + e*x)^(3/2))/(3*e^8)
Time = 0.23 (sec) , antiderivative size = 530, normalized size of antiderivative = 1.53 \[ \int \frac {(A+B x) \left (a+c x^2\right )^3}{(d+e x)^{11/2}} \, dx=\frac {\frac {4096}{45} b \,c^{3} d^{7}-\frac {24}{35} a^{3} c d \,e^{6} x -2 a^{2} b c \,e^{7} x^{3}-\frac {4}{63} a^{3} b d \,e^{6}-\frac {16}{105} a^{3} c \,d^{2} e^{5}-\frac {256}{105} a^{2} c^{2} d^{4} e^{3}-\frac {2048}{63} a \,c^{3} d^{6} e -\frac {2}{9} a^{4} e^{7}-\frac {48}{35} a^{2} b c \,d^{2} e^{5} x -\frac {12}{5} a^{2} b c d \,e^{6} x^{2}+\frac {768}{7} a b \,c^{2} d^{4} e^{3} x +192 a b \,c^{2} d^{3} e^{4} x^{2}+160 a b \,c^{2} d^{2} e^{5} x^{3}+60 a b \,c^{2} d \,e^{6} x^{4}-\frac {2}{7} a^{3} b \,e^{7} x -\frac {6}{5} a^{3} c \,e^{7} x^{2}-6 a^{2} c^{2} e^{7} x^{4}+\frac {2}{3} a \,c^{3} e^{7} x^{6}+\frac {2}{5} b \,c^{3} e^{7} x^{7}-\frac {384}{35} a^{2} c^{2} d^{3} e^{4} x -\frac {96}{5} a^{2} c^{2} d^{2} e^{5} x^{2}-16 a^{2} c^{2} d \,e^{6} x^{3}+6 a b \,c^{2} e^{7} x^{5}-\frac {1024}{7} a \,c^{3} d^{5} e^{2} x -256 a \,c^{3} d^{4} e^{3} x^{2}-\frac {640}{3} a \,c^{3} d^{3} e^{4} x^{3}-80 a \,c^{3} d^{2} e^{5} x^{4}-8 a \,c^{3} d \,e^{6} x^{5}+\frac {2048}{5} b \,c^{3} d^{6} e x +\frac {3584}{5} b \,c^{3} d^{5} e^{2} x^{2}+\frac {1792}{3} b \,c^{3} d^{4} e^{3} x^{3}+224 b \,c^{3} d^{3} e^{4} x^{4}+\frac {112}{5} b \,c^{3} d^{2} e^{5} x^{5}-\frac {28}{15} b \,c^{3} d \,e^{6} x^{6}-\frac {32}{105} a^{2} b c \,d^{3} e^{4}+\frac {512}{21} a b \,c^{2} d^{5} e^{2}}{\sqrt {e x +d}\, e^{8} \left (e^{4} x^{4}+4 d \,e^{3} x^{3}+6 d^{2} e^{2} x^{2}+4 d^{3} e x +d^{4}\right )} \] Input:
int((B*x+A)*(c*x^2+a)^3/(e*x+d)^(11/2),x)
Output:
(2*( - 35*a**4*e**7 - 10*a**3*b*d*e**6 - 45*a**3*b*e**7*x - 24*a**3*c*d**2 *e**5 - 108*a**3*c*d*e**6*x - 189*a**3*c*e**7*x**2 - 48*a**2*b*c*d**3*e**4 - 216*a**2*b*c*d**2*e**5*x - 378*a**2*b*c*d*e**6*x**2 - 315*a**2*b*c*e**7 *x**3 - 384*a**2*c**2*d**4*e**3 - 1728*a**2*c**2*d**3*e**4*x - 3024*a**2*c **2*d**2*e**5*x**2 - 2520*a**2*c**2*d*e**6*x**3 - 945*a**2*c**2*e**7*x**4 + 3840*a*b*c**2*d**5*e**2 + 17280*a*b*c**2*d**4*e**3*x + 30240*a*b*c**2*d* *3*e**4*x**2 + 25200*a*b*c**2*d**2*e**5*x**3 + 9450*a*b*c**2*d*e**6*x**4 + 945*a*b*c**2*e**7*x**5 - 5120*a*c**3*d**6*e - 23040*a*c**3*d**5*e**2*x - 40320*a*c**3*d**4*e**3*x**2 - 33600*a*c**3*d**3*e**4*x**3 - 12600*a*c**3*d **2*e**5*x**4 - 1260*a*c**3*d*e**6*x**5 + 105*a*c**3*e**7*x**6 + 14336*b*c **3*d**7 + 64512*b*c**3*d**6*e*x + 112896*b*c**3*d**5*e**2*x**2 + 94080*b* c**3*d**4*e**3*x**3 + 35280*b*c**3*d**3*e**4*x**4 + 3528*b*c**3*d**2*e**5* x**5 - 294*b*c**3*d*e**6*x**6 + 63*b*c**3*e**7*x**7))/(315*sqrt(d + e*x)*e **8*(d**4 + 4*d**3*e*x + 6*d**2*e**2*x**2 + 4*d*e**3*x**3 + e**4*x**4))