Integrand size = 28, antiderivative size = 421 \[ \int \frac {(b+2 c x) \left (a+b x+c x^2\right )^3}{(d+e x)^{5/2}} \, dx=\frac {2 (2 c d-b e) \left (c d^2-b d e+a e^2\right )^3}{3 e^8 (d+e x)^{3/2}}-\frac {2 \left (c d^2-b d e+a e^2\right )^2 \left (14 c^2 d^2+3 b^2 e^2-2 c e (7 b d-a e)\right )}{e^8 \sqrt {d+e x}}-\frac {6 (2 c d-b e) \left (c d^2-b d e+a e^2\right ) \left (7 c^2 d^2+b^2 e^2-c e (7 b d-3 a e)\right ) \sqrt {d+e x}}{e^8}+\frac {2 \left (70 c^4 d^4+b^4 e^4-4 b^2 c e^3 (5 b d-3 a e)-20 c^3 d^2 e (7 b d-3 a e)+6 c^2 e^2 \left (15 b^2 d^2-10 a b d e+a^2 e^2\right )\right ) (d+e x)^{3/2}}{3 e^8}-\frac {2 c (2 c d-b e) \left (7 c^2 d^2+b^2 e^2-c e (7 b d-3 a e)\right ) (d+e x)^{5/2}}{e^8}+\frac {6 c^2 \left (14 c^2 d^2+3 b^2 e^2-2 c e (7 b d-a e)\right ) (d+e x)^{7/2}}{7 e^8}-\frac {14 c^3 (2 c d-b e) (d+e x)^{9/2}}{9 e^8}+\frac {4 c^4 (d+e x)^{11/2}}{11 e^8} \] Output:
2/3*(-b*e+2*c*d)*(a*e^2-b*d*e+c*d^2)^3/e^8/(e*x+d)^(3/2)-2*(a*e^2-b*d*e+c* d^2)^2*(14*c^2*d^2+3*b^2*e^2-2*c*e*(-a*e+7*b*d))/e^8/(e*x+d)^(1/2)-6*(-b*e +2*c*d)*(a*e^2-b*d*e+c*d^2)*(7*c^2*d^2+b^2*e^2-c*e*(-3*a*e+7*b*d))*(e*x+d) ^(1/2)/e^8+2/3*(70*c^4*d^4+b^4*e^4-4*b^2*c*e^3*(-3*a*e+5*b*d)-20*c^3*d^2*e *(-3*a*e+7*b*d)+6*c^2*e^2*(a^2*e^2-10*a*b*d*e+15*b^2*d^2))*(e*x+d)^(3/2)/e ^8-2*c*(-b*e+2*c*d)*(7*c^2*d^2+b^2*e^2-c*e*(-3*a*e+7*b*d))*(e*x+d)^(5/2)/e ^8+6/7*c^2*(14*c^2*d^2+3*b^2*e^2-2*c*e*(-a*e+7*b*d))*(e*x+d)^(7/2)/e^8-14/ 9*c^3*(-b*e+2*c*d)*(e*x+d)^(9/2)/e^8+4/11*c^4*(e*x+d)^(11/2)/e^8
Time = 0.68 (sec) , antiderivative size = 598, normalized size of antiderivative = 1.42 \[ \int \frac {(b+2 c x) \left (a+b x+c x^2\right )^3}{(d+e x)^{5/2}} \, dx=\frac {-28 c^4 \left (2048 d^7+3072 d^6 e x+768 d^5 e^2 x^2-128 d^4 e^3 x^3+48 d^3 e^4 x^4-24 d^2 e^5 x^5+14 d e^6 x^6-9 e^7 x^7\right )-462 b e^4 \left (a^3 e^3+3 a^2 b e^2 (2 d+3 e x)-3 a b^2 e \left (8 d^2+12 d e x+3 e^2 x^2\right )+b^3 \left (16 d^3+24 d^2 e x+6 d e^2 x^2-e^3 x^3\right )\right )+462 c e^3 \left (-2 a^3 e^3 (2 d+3 e x)+9 a^2 b e^2 \left (8 d^2+12 d e x+3 e^2 x^2\right )+12 a b^2 e \left (-16 d^3-24 d^2 e x-6 d e^2 x^2+e^3 x^3\right )+b^3 \left (128 d^4+192 d^3 e x+48 d^2 e^2 x^2-8 d e^3 x^3+3 e^4 x^4\right )\right )-198 c^2 e^2 \left (14 a^2 e^2 \left (16 d^3+24 d^2 e x+6 d e^2 x^2-e^3 x^3\right )-7 a b e \left (128 d^4+192 d^3 e x+48 d^2 e^2 x^2-8 d e^3 x^3+3 e^4 x^4\right )+3 b^2 \left (256 d^5+384 d^4 e x+96 d^3 e^2 x^2-16 d^2 e^3 x^3+6 d e^4 x^4-3 e^5 x^5\right )\right )+22 c^3 e \left (-18 a e \left (256 d^5+384 d^4 e x+96 d^3 e^2 x^2-16 d^2 e^3 x^3+6 d e^4 x^4-3 e^5 x^5\right )+7 b \left (1024 d^6+1536 d^5 e x+384 d^4 e^2 x^2-64 d^3 e^3 x^3+24 d^2 e^4 x^4-12 d e^5 x^5+7 e^6 x^6\right )\right )}{693 e^8 (d+e x)^{3/2}} \] Input:
Integrate[((b + 2*c*x)*(a + b*x + c*x^2)^3)/(d + e*x)^(5/2),x]
Output:
(-28*c^4*(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) - 462*b*e^4*(a^3 *e^3 + 3*a^2*b*e^2*(2*d + 3*e*x) - 3*a*b^2*e*(8*d^2 + 12*d*e*x + 3*e^2*x^2 ) + b^3*(16*d^3 + 24*d^2*e*x + 6*d*e^2*x^2 - e^3*x^3)) + 462*c*e^3*(-2*a^3 *e^3*(2*d + 3*e*x) + 9*a^2*b*e^2*(8*d^2 + 12*d*e*x + 3*e^2*x^2) + 12*a*b^2 *e*(-16*d^3 - 24*d^2*e*x - 6*d*e^2*x^2 + e^3*x^3) + b^3*(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)) - 198*c^2*e^2*(14*a^2*e^ 2*(16*d^3 + 24*d^2*e*x + 6*d*e^2*x^2 - e^3*x^3) - 7*a*b*e*(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) + 3*b^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)) + 22 *c^3*e*(-18*a*e*(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*b*(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)))/(693*e^8* (d + e*x)^(3/2))
Time = 1.01 (sec) , antiderivative size = 421, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.071, Rules used = {1195, 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.
| \(\displaystyle \int \frac {(b+2 c x) \left (a+b x+c x^2\right )^3}{(d+e x)^{5/2}} \, dx\) |
| \(\Big \downarrow \) 1195 |
| \(\displaystyle \int \left (\frac {\sqrt {d+e x} \left (6 c^2 e^2 \left (a^2 e^2-10 a b d e+15 b^2 d^2\right )-4 b^2 c e^3 (5 b d-3 a e)-20 c^3 d^2 e (7 b d-3 a e)+b^4 e^4+70 c^4 d^4\right )}{e^7}+\frac {3 c^2 (d+e x)^{5/2} \left (-2 c e (7 b d-a e)+3 b^2 e^2+14 c^2 d^2\right )}{e^7}+\frac {5 c (d+e x)^{3/2} (2 c d-b e) \left (c e (7 b d-3 a e)-b^2 e^2-7 c^2 d^2\right )}{e^7}+\frac {3 (2 c d-b e) \left (a e^2-b d e+c d^2\right ) \left (-3 a c e^2-b^2 e^2+7 b c d e-7 c^2 d^2\right )}{e^7 \sqrt {d+e x}}+\frac {\left (a e^2-b d e+c d^2\right )^2 \left (-2 c e (7 b d-a e)+3 b^2 e^2+14 c^2 d^2\right )}{e^7 (d+e x)^{3/2}}+\frac {(b e-2 c d) \left (a e^2-b d e+c d^2\right )^3}{e^7 (d+e x)^{5/2}}-\frac {7 c^3 (d+e x)^{7/2} (2 c d-b e)}{e^7}+\frac {2 c^4 (d+e x)^{9/2}}{e^7}\right )dx\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {2 (d+e x)^{3/2} \left (6 c^2 e^2 \left (a^2 e^2-10 a b d e+15 b^2 d^2\right )-4 b^2 c e^3 (5 b d-3 a e)-20 c^3 d^2 e (7 b d-3 a e)+b^4 e^4+70 c^4 d^4\right )}{3 e^8}+\frac {6 c^2 (d+e x)^{7/2} \left (-2 c e (7 b d-a e)+3 b^2 e^2+14 c^2 d^2\right )}{7 e^8}-\frac {2 c (d+e x)^{5/2} (2 c d-b e) \left (-c e (7 b d-3 a e)+b^2 e^2+7 c^2 d^2\right )}{e^8}-\frac {6 \sqrt {d+e x} (2 c d-b e) \left (a e^2-b d e+c d^2\right ) \left (-c e (7 b d-3 a e)+b^2 e^2+7 c^2 d^2\right )}{e^8}-\frac {2 \left (a e^2-b d e+c d^2\right )^2 \left (-2 c e (7 b d-a e)+3 b^2 e^2+14 c^2 d^2\right )}{e^8 \sqrt {d+e x}}+\frac {2 (2 c d-b e) \left (a e^2-b d e+c d^2\right )^3}{3 e^8 (d+e x)^{3/2}}-\frac {14 c^3 (d+e x)^{9/2} (2 c d-b e)}{9 e^8}+\frac {4 c^4 (d+e x)^{11/2}}{11 e^8}\) |
Input:
Int[((b + 2*c*x)*(a + b*x + c*x^2)^3)/(d + e*x)^(5/2),x]
Output:
(2*(2*c*d - b*e)*(c*d^2 - b*d*e + a*e^2)^3)/(3*e^8*(d + e*x)^(3/2)) - (2*( c*d^2 - b*d*e + a*e^2)^2*(14*c^2*d^2 + 3*b^2*e^2 - 2*c*e*(7*b*d - a*e)))/( e^8*Sqrt[d + e*x]) - (6*(2*c*d - b*e)*(c*d^2 - b*d*e + a*e^2)*(7*c^2*d^2 + b^2*e^2 - c*e*(7*b*d - 3*a*e))*Sqrt[d + e*x])/e^8 + (2*(70*c^4*d^4 + b^4* e^4 - 4*b^2*c*e^3*(5*b*d - 3*a*e) - 20*c^3*d^2*e*(7*b*d - 3*a*e) + 6*c^2*e ^2*(15*b^2*d^2 - 10*a*b*d*e + a^2*e^2))*(d + e*x)^(3/2))/(3*e^8) - (2*c*(2 *c*d - b*e)*(7*c^2*d^2 + b^2*e^2 - c*e*(7*b*d - 3*a*e))*(d + e*x)^(5/2))/e ^8 + (6*c^2*(14*c^2*d^2 + 3*b^2*e^2 - 2*c*e*(7*b*d - a*e))*(d + e*x)^(7/2) )/(7*e^8) - (14*c^3*(2*c*d - b*e)*(d + e*x)^(9/2))/(9*e^8) + (4*c^4*(d + e *x)^(11/2))/(11*e^8)
Int[((d_.) + (e_.)*(x_))^(m_.)*((f_.) + (g_.)*(x_))^(n_.)*((a_.) + (b_.)*(x _) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> Int[ExpandIntegrand[(d + e*x)^m*(f + g*x)^n*(a + b*x + c*x^2)^p, x], x] /; FreeQ[{a, b, c, d, e, f, g, m, n}, x ] && IGtQ[p, 0]
Time = 1.49 (sec) , antiderivative size = 517, normalized size of antiderivative = 1.23
| method | result | size |
| pseudoelliptic | \(-\frac {2 \left (\left (-\frac {6 c^{4} x^{7}}{11}+\left (-\frac {7}{3} b \,x^{6}-\frac {18}{7} a \,x^{5}\right ) c^{3}+\left (-\frac {27}{7} b^{2} x^{5}-6 a^{2} x^{3}-9 a b \,x^{4}\right ) c^{2}+\left (-3 b^{3} x^{4}-12 a \,b^{2} x^{3}-27 a^{2} b \,x^{2}+6 a^{3} x \right ) c -b^{4} x^{3}-9 a \,b^{3} x^{2}+9 x \,a^{2} b^{2}+a^{3} b \right ) e^{7}+4 d \left (\frac {7 c^{4} x^{6}}{33}+\left (b \,x^{5}+\frac {9}{7} a \,x^{4}\right ) c^{3}+\left (\frac {27}{14} b^{2} x^{4}+6 a b \,x^{3}+9 a^{2} x^{2}\right ) c^{2}+\left (2 b^{3} x^{3}+18 a \,b^{2} x^{2}-27 a^{2} b x +a^{3}\right ) c +\frac {3 b^{2} \left (b^{2} x^{2}-6 a b x +a^{2}\right )}{2}\right ) e^{6}-72 d^{2} \left (\frac {2 c^{4} x^{5}}{99}+\left (\frac {1}{9} b \,x^{4}+\frac {4}{21} a \,x^{3}\right ) c^{3}+\left (\frac {2}{7} b^{2} x^{3}+2 a b \,x^{2}-2 a^{2} x \right ) c^{2}+b \left (\frac {2}{3} b^{2} x^{2}-4 a b x +a^{2}\right ) c +\frac {b^{3} \left (-b x +a \right )}{3}\right ) e^{5}+96 d^{3} \left (\frac {c^{4} x^{4}}{33}+\left (\frac {2}{9} b \,x^{3}+\frac {6}{7} a \,x^{2}\right ) c^{3}+\left (\frac {9}{7} b^{2} x^{2}-6 a b x +a^{2}\right ) c^{2}+\left (-2 b^{3} x +2 a \,b^{2}\right ) c +\frac {b^{4}}{6}\right ) e^{4}-384 \left (\frac {2 c^{3} x^{3}}{99}+\left (\frac {1}{3} b \,x^{2}-\frac {6}{7} a x \right ) c^{2}+b \left (-\frac {9 b x}{7}+a \right ) c +\frac {b^{3}}{3}\right ) d^{4} c \,e^{3}+\frac {1536 d^{5} \left (\frac {7 c^{2} x^{2}}{33}+\left (-\frac {7 b x}{3}+a \right ) c +\frac {3 b^{2}}{2}\right ) c^{2} e^{2}}{7}-\frac {1024 d^{6} \left (-\frac {6 c x}{11}+b \right ) c^{3} e}{3}+\frac {4096 d^{7} c^{4}}{33}\right )}{3 \left (e x +d \right )^{\frac {3}{2}} e^{8}}\) | \(517\) |
| risch | \(\frac {2 \left (126 c^{4} x^{5} e^{5}+539 b \,c^{3} e^{5} x^{4}-448 c^{4} d \,e^{4} x^{4}+594 a \,c^{3} e^{5} x^{3}+891 b^{2} c^{2} e^{5} x^{3}-2002 b \,c^{3} d \,e^{4} x^{3}+1106 c^{4} d^{2} e^{3} x^{3}+2079 a b \,c^{2} e^{5} x^{2}-2376 a \,c^{3} d \,e^{4} x^{2}+693 b^{3} c \,e^{5} x^{2}-3564 b^{2} c^{2} d \,e^{4} x^{2}+5313 b \,c^{3} d^{2} e^{3} x^{2}-2436 c^{4} d^{3} e^{2} x^{2}+1386 a^{2} c^{2} e^{5} x +2772 a \,b^{2} c \,e^{5} x -9702 a b \,c^{2} d \,e^{4} x +7326 a \,c^{3} d^{2} e^{3} x +231 b^{4} e^{5} x -3234 b^{3} c d \,e^{4} x +10989 b^{2} c^{2} d^{2} e^{3} x -13552 b \,c^{3} d^{3} e^{2} x +5558 c^{4} d^{4} e x +6237 a^{2} b c \,e^{5}-11088 a^{2} c^{2} d \,e^{4}+2079 a \,b^{3} e^{5}-22176 a \,b^{2} c d \,e^{4}+50589 a b \,c^{2} d^{2} e^{3}-31284 a \,c^{3} d^{3} e^{2}-1848 b^{4} d \,e^{4}+16863 b^{3} c \,d^{2} e^{3}-46926 b^{2} c^{2} d^{3} e^{2}+51359 b \,c^{3} d^{4} e -19432 c^{4} d^{5}\right ) \sqrt {e x +d}}{693 e^{8}}-\frac {2 \left (6 a c \,e^{3} x +9 e^{3} x \,b^{2}-42 d \,e^{2} c b x +42 d^{2} e \,c^{2} x +a \,e^{3} b +4 a d \,e^{2} c +8 d \,e^{2} b^{2}-39 d^{2} e b c +40 c^{2} d^{3}\right ) \left (a^{2} e^{4}-2 d \,e^{3} a b +2 a c \,d^{2} e^{2}+d^{2} e^{2} b^{2}-2 b c \,d^{3} e +c^{2} d^{4}\right )}{3 e^{8} \left (e x +d \right )^{\frac {3}{2}}}\) | \(550\) |
| gosper | \(-\frac {2 \left (-126 x^{7} c^{4} e^{7}-539 x^{6} b \,c^{3} e^{7}+196 x^{6} c^{4} d \,e^{6}-594 x^{5} a \,c^{3} e^{7}-891 x^{5} b^{2} c^{2} e^{7}+924 x^{5} b \,c^{3} d \,e^{6}-336 x^{5} c^{4} d^{2} e^{5}-2079 x^{4} a b \,c^{2} e^{7}+1188 x^{4} a \,c^{3} d \,e^{6}-693 x^{4} b^{3} c \,e^{7}+1782 x^{4} b^{2} c^{2} d \,e^{6}-1848 x^{4} b \,c^{3} d^{2} e^{5}+672 x^{4} c^{4} d^{3} e^{4}-1386 x^{3} a^{2} c^{2} e^{7}-2772 x^{3} a \,b^{2} c \,e^{7}+5544 x^{3} a b \,c^{2} d \,e^{6}-3168 x^{3} a \,c^{3} d^{2} e^{5}-231 x^{3} b^{4} e^{7}+1848 x^{3} b^{3} c d \,e^{6}-4752 x^{3} b^{2} c^{2} d^{2} e^{5}+4928 x^{3} b \,c^{3} d^{3} e^{4}-1792 x^{3} c^{4} d^{4} e^{3}-6237 x^{2} a^{2} b c \,e^{7}+8316 x^{2} a^{2} c^{2} d \,e^{6}-2079 x^{2} a \,b^{3} e^{7}+16632 x^{2} a \,b^{2} c d \,e^{6}-33264 x^{2} a b \,c^{2} d^{2} e^{5}+19008 x^{2} a \,c^{3} d^{3} e^{4}+1386 x^{2} b^{4} d \,e^{6}-11088 x^{2} b^{3} c \,d^{2} e^{5}+28512 x^{2} b^{2} c^{2} d^{3} e^{4}-29568 x^{2} b \,c^{3} d^{4} e^{3}+10752 x^{2} c^{4} d^{5} e^{2}+1386 x \,a^{3} c \,e^{7}+2079 x \,a^{2} b^{2} e^{7}-24948 x \,a^{2} b c d \,e^{6}+33264 x \,a^{2} c^{2} d^{2} e^{5}-8316 x a \,b^{3} d \,e^{6}+66528 x a \,b^{2} c \,d^{2} e^{5}-133056 x a b \,c^{2} d^{3} e^{4}+76032 x a \,c^{3} d^{4} e^{3}+5544 x \,b^{4} d^{2} e^{5}-44352 x \,b^{3} c \,d^{3} e^{4}+114048 x \,b^{2} c^{2} d^{4} e^{3}-118272 x b \,c^{3} d^{5} e^{2}+43008 x \,c^{4} d^{6} e +231 a^{3} b \,e^{7}+924 d \,e^{6} c \,a^{3}+1386 a^{2} b^{2} d \,e^{6}-16632 a^{2} b c \,d^{2} e^{5}+22176 d^{3} e^{4} a^{2} c^{2}-5544 a \,b^{3} d^{2} e^{5}+44352 a \,b^{2} c \,d^{3} e^{4}-88704 a b \,c^{2} d^{4} e^{3}+50688 d^{5} e^{2} a \,c^{3}+3696 b^{4} d^{3} e^{4}-29568 b^{3} c \,d^{4} e^{3}+76032 b^{2} c^{2} d^{5} e^{2}-78848 b \,c^{3} d^{6} e +28672 d^{7} c^{4}\right )}{693 \left (e x +d \right )^{\frac {3}{2}} e^{8}}\) | \(795\) |
| trager | \(-\frac {2 \left (-126 x^{7} c^{4} e^{7}-539 x^{6} b \,c^{3} e^{7}+196 x^{6} c^{4} d \,e^{6}-594 x^{5} a \,c^{3} e^{7}-891 x^{5} b^{2} c^{2} e^{7}+924 x^{5} b \,c^{3} d \,e^{6}-336 x^{5} c^{4} d^{2} e^{5}-2079 x^{4} a b \,c^{2} e^{7}+1188 x^{4} a \,c^{3} d \,e^{6}-693 x^{4} b^{3} c \,e^{7}+1782 x^{4} b^{2} c^{2} d \,e^{6}-1848 x^{4} b \,c^{3} d^{2} e^{5}+672 x^{4} c^{4} d^{3} e^{4}-1386 x^{3} a^{2} c^{2} e^{7}-2772 x^{3} a \,b^{2} c \,e^{7}+5544 x^{3} a b \,c^{2} d \,e^{6}-3168 x^{3} a \,c^{3} d^{2} e^{5}-231 x^{3} b^{4} e^{7}+1848 x^{3} b^{3} c d \,e^{6}-4752 x^{3} b^{2} c^{2} d^{2} e^{5}+4928 x^{3} b \,c^{3} d^{3} e^{4}-1792 x^{3} c^{4} d^{4} e^{3}-6237 x^{2} a^{2} b c \,e^{7}+8316 x^{2} a^{2} c^{2} d \,e^{6}-2079 x^{2} a \,b^{3} e^{7}+16632 x^{2} a \,b^{2} c d \,e^{6}-33264 x^{2} a b \,c^{2} d^{2} e^{5}+19008 x^{2} a \,c^{3} d^{3} e^{4}+1386 x^{2} b^{4} d \,e^{6}-11088 x^{2} b^{3} c \,d^{2} e^{5}+28512 x^{2} b^{2} c^{2} d^{3} e^{4}-29568 x^{2} b \,c^{3} d^{4} e^{3}+10752 x^{2} c^{4} d^{5} e^{2}+1386 x \,a^{3} c \,e^{7}+2079 x \,a^{2} b^{2} e^{7}-24948 x \,a^{2} b c d \,e^{6}+33264 x \,a^{2} c^{2} d^{2} e^{5}-8316 x a \,b^{3} d \,e^{6}+66528 x a \,b^{2} c \,d^{2} e^{5}-133056 x a b \,c^{2} d^{3} e^{4}+76032 x a \,c^{3} d^{4} e^{3}+5544 x \,b^{4} d^{2} e^{5}-44352 x \,b^{3} c \,d^{3} e^{4}+114048 x \,b^{2} c^{2} d^{4} e^{3}-118272 x b \,c^{3} d^{5} e^{2}+43008 x \,c^{4} d^{6} e +231 a^{3} b \,e^{7}+924 d \,e^{6} c \,a^{3}+1386 a^{2} b^{2} d \,e^{6}-16632 a^{2} b c \,d^{2} e^{5}+22176 d^{3} e^{4} a^{2} c^{2}-5544 a \,b^{3} d^{2} e^{5}+44352 a \,b^{2} c \,d^{3} e^{4}-88704 a b \,c^{2} d^{4} e^{3}+50688 d^{5} e^{2} a \,c^{3}+3696 b^{4} d^{3} e^{4}-29568 b^{3} c \,d^{4} e^{3}+76032 b^{2} c^{2} d^{5} e^{2}-78848 b \,c^{3} d^{6} e +28672 d^{7} c^{4}\right )}{693 \left (e x +d \right )^{\frac {3}{2}} e^{8}}\) | \(795\) |
| orering | \(-\frac {2 \left (-126 x^{7} c^{4} e^{7}-539 x^{6} b \,c^{3} e^{7}+196 x^{6} c^{4} d \,e^{6}-594 x^{5} a \,c^{3} e^{7}-891 x^{5} b^{2} c^{2} e^{7}+924 x^{5} b \,c^{3} d \,e^{6}-336 x^{5} c^{4} d^{2} e^{5}-2079 x^{4} a b \,c^{2} e^{7}+1188 x^{4} a \,c^{3} d \,e^{6}-693 x^{4} b^{3} c \,e^{7}+1782 x^{4} b^{2} c^{2} d \,e^{6}-1848 x^{4} b \,c^{3} d^{2} e^{5}+672 x^{4} c^{4} d^{3} e^{4}-1386 x^{3} a^{2} c^{2} e^{7}-2772 x^{3} a \,b^{2} c \,e^{7}+5544 x^{3} a b \,c^{2} d \,e^{6}-3168 x^{3} a \,c^{3} d^{2} e^{5}-231 x^{3} b^{4} e^{7}+1848 x^{3} b^{3} c d \,e^{6}-4752 x^{3} b^{2} c^{2} d^{2} e^{5}+4928 x^{3} b \,c^{3} d^{3} e^{4}-1792 x^{3} c^{4} d^{4} e^{3}-6237 x^{2} a^{2} b c \,e^{7}+8316 x^{2} a^{2} c^{2} d \,e^{6}-2079 x^{2} a \,b^{3} e^{7}+16632 x^{2} a \,b^{2} c d \,e^{6}-33264 x^{2} a b \,c^{2} d^{2} e^{5}+19008 x^{2} a \,c^{3} d^{3} e^{4}+1386 x^{2} b^{4} d \,e^{6}-11088 x^{2} b^{3} c \,d^{2} e^{5}+28512 x^{2} b^{2} c^{2} d^{3} e^{4}-29568 x^{2} b \,c^{3} d^{4} e^{3}+10752 x^{2} c^{4} d^{5} e^{2}+1386 x \,a^{3} c \,e^{7}+2079 x \,a^{2} b^{2} e^{7}-24948 x \,a^{2} b c d \,e^{6}+33264 x \,a^{2} c^{2} d^{2} e^{5}-8316 x a \,b^{3} d \,e^{6}+66528 x a \,b^{2} c \,d^{2} e^{5}-133056 x a b \,c^{2} d^{3} e^{4}+76032 x a \,c^{3} d^{4} e^{3}+5544 x \,b^{4} d^{2} e^{5}-44352 x \,b^{3} c \,d^{3} e^{4}+114048 x \,b^{2} c^{2} d^{4} e^{3}-118272 x b \,c^{3} d^{5} e^{2}+43008 x \,c^{4} d^{6} e +231 a^{3} b \,e^{7}+924 d \,e^{6} c \,a^{3}+1386 a^{2} b^{2} d \,e^{6}-16632 a^{2} b c \,d^{2} e^{5}+22176 d^{3} e^{4} a^{2} c^{2}-5544 a \,b^{3} d^{2} e^{5}+44352 a \,b^{2} c \,d^{3} e^{4}-88704 a b \,c^{2} d^{4} e^{3}+50688 d^{5} e^{2} a \,c^{3}+3696 b^{4} d^{3} e^{4}-29568 b^{3} c \,d^{4} e^{3}+76032 b^{2} c^{2} d^{5} e^{2}-78848 b \,c^{3} d^{6} e +28672 d^{7} c^{4}\right )}{693 \left (e x +d \right )^{\frac {3}{2}} e^{8}}\) | \(795\) |
| derivativedivides | \(\frac {-12 a \,c^{3} d \,e^{2} \left (e x +d \right )^{\frac {5}{2}}+60 b^{2} c^{2} d^{2} e^{2} \left (e x +d \right )^{\frac {3}{2}}-\frac {280 b \,c^{3} d^{3} e \left (e x +d \right )^{\frac {3}{2}}}{3}+\frac {18 b^{2} c^{2} e^{2} \left (e x +d \right )^{\frac {7}{2}}}{7}+6 a \,b^{3} e^{5} \sqrt {e x +d}-6 b^{4} d \,e^{4} \sqrt {e x +d}+4 a^{2} c^{2} e^{4} \left (e x +d \right )^{\frac {3}{2}}+18 a^{2} b c \,e^{5} \sqrt {e x +d}-36 a^{2} c^{2} d \,e^{4} \sqrt {e x +d}+8 a \,b^{2} c \,e^{4} \left (e x +d \right )^{\frac {3}{2}}-\frac {40 b^{3} c d \,e^{3} \left (e x +d \right )^{\frac {3}{2}}}{3}+40 a \,c^{3} d^{2} e^{2} \left (e x +d \right )^{\frac {3}{2}}+210 b \,c^{3} d^{4} e \sqrt {e x +d}-120 a \,c^{3} d^{3} e^{2} \sqrt {e x +d}+60 b^{3} c \,d^{2} e^{3} \sqrt {e x +d}-18 b^{2} c^{2} d \,e^{2} \left (e x +d \right )^{\frac {5}{2}}+42 b \,c^{3} d^{2} e \left (e x +d \right )^{\frac {5}{2}}+6 a b \,c^{2} e^{3} \left (e x +d \right )^{\frac {5}{2}}+\frac {14 b \,c^{3} e \left (e x +d \right )^{\frac {9}{2}}}{9}+\frac {12 a \,c^{3} e^{2} \left (e x +d \right )^{\frac {7}{2}}}{7}-12 b \,c^{3} d e \left (e x +d \right )^{\frac {7}{2}}-180 b^{2} c^{2} d^{3} e^{2} \sqrt {e x +d}+2 b^{3} c \,e^{3} \left (e x +d \right )^{\frac {5}{2}}-\frac {2 \left (2 e^{6} c \,a^{3}+3 a^{2} b^{2} e^{6}-18 a^{2} b c d \,e^{5}+18 d^{2} e^{4} a^{2} c^{2}-6 a \,b^{3} d \,e^{5}+36 a \,b^{2} c \,d^{2} e^{4}-60 a b \,c^{2} d^{3} e^{3}+30 d^{4} e^{2} a \,c^{3}+3 b^{4} d^{2} e^{4}-20 b^{3} c \,d^{3} e^{3}+45 b^{2} c^{2} d^{4} e^{2}-42 b \,c^{3} d^{5} e +14 d^{6} c^{4}\right )}{\sqrt {e x +d}}-\frac {28 c^{4} d \left (e x +d \right )^{\frac {9}{2}}}{9}+12 c^{4} d^{2} \left (e x +d \right )^{\frac {7}{2}}-40 a b \,c^{2} d \,e^{3} \left (e x +d \right )^{\frac {3}{2}}-72 a \,b^{2} c d \,e^{4} \sqrt {e x +d}+180 a b \,c^{2} d^{2} e^{3} \sqrt {e x +d}-\frac {2 \left (a^{3} b \,e^{7}-2 d \,e^{6} c \,a^{3}-3 a^{2} b^{2} d \,e^{6}+9 a^{2} b c \,d^{2} e^{5}-6 d^{3} e^{4} a^{2} c^{2}+3 a \,b^{3} d^{2} e^{5}-12 a \,b^{2} c \,d^{3} e^{4}+15 a b \,c^{2} d^{4} e^{3}-6 d^{5} e^{2} a \,c^{3}-b^{4} d^{3} e^{4}+5 b^{3} c \,d^{4} e^{3}-9 b^{2} c^{2} d^{5} e^{2}+7 b \,c^{3} d^{6} e -2 d^{7} c^{4}\right )}{3 \left (e x +d \right )^{\frac {3}{2}}}+\frac {4 c^{4} \left (e x +d \right )^{\frac {11}{2}}}{11}+\frac {140 c^{4} d^{4} \left (e x +d \right )^{\frac {3}{2}}}{3}-28 c^{4} d^{3} \left (e x +d \right )^{\frac {5}{2}}-84 c^{4} d^{5} \sqrt {e x +d}+\frac {2 b^{4} e^{4} \left (e x +d \right )^{\frac {3}{2}}}{3}}{e^{8}}\) | \(894\) |
| default | \(\frac {-12 a \,c^{3} d \,e^{2} \left (e x +d \right )^{\frac {5}{2}}+60 b^{2} c^{2} d^{2} e^{2} \left (e x +d \right )^{\frac {3}{2}}-\frac {280 b \,c^{3} d^{3} e \left (e x +d \right )^{\frac {3}{2}}}{3}+\frac {18 b^{2} c^{2} e^{2} \left (e x +d \right )^{\frac {7}{2}}}{7}+6 a \,b^{3} e^{5} \sqrt {e x +d}-6 b^{4} d \,e^{4} \sqrt {e x +d}+4 a^{2} c^{2} e^{4} \left (e x +d \right )^{\frac {3}{2}}+18 a^{2} b c \,e^{5} \sqrt {e x +d}-36 a^{2} c^{2} d \,e^{4} \sqrt {e x +d}+8 a \,b^{2} c \,e^{4} \left (e x +d \right )^{\frac {3}{2}}-\frac {40 b^{3} c d \,e^{3} \left (e x +d \right )^{\frac {3}{2}}}{3}+40 a \,c^{3} d^{2} e^{2} \left (e x +d \right )^{\frac {3}{2}}+210 b \,c^{3} d^{4} e \sqrt {e x +d}-120 a \,c^{3} d^{3} e^{2} \sqrt {e x +d}+60 b^{3} c \,d^{2} e^{3} \sqrt {e x +d}-18 b^{2} c^{2} d \,e^{2} \left (e x +d \right )^{\frac {5}{2}}+42 b \,c^{3} d^{2} e \left (e x +d \right )^{\frac {5}{2}}+6 a b \,c^{2} e^{3} \left (e x +d \right )^{\frac {5}{2}}+\frac {14 b \,c^{3} e \left (e x +d \right )^{\frac {9}{2}}}{9}+\frac {12 a \,c^{3} e^{2} \left (e x +d \right )^{\frac {7}{2}}}{7}-12 b \,c^{3} d e \left (e x +d \right )^{\frac {7}{2}}-180 b^{2} c^{2} d^{3} e^{2} \sqrt {e x +d}+2 b^{3} c \,e^{3} \left (e x +d \right )^{\frac {5}{2}}-\frac {2 \left (2 e^{6} c \,a^{3}+3 a^{2} b^{2} e^{6}-18 a^{2} b c d \,e^{5}+18 d^{2} e^{4} a^{2} c^{2}-6 a \,b^{3} d \,e^{5}+36 a \,b^{2} c \,d^{2} e^{4}-60 a b \,c^{2} d^{3} e^{3}+30 d^{4} e^{2} a \,c^{3}+3 b^{4} d^{2} e^{4}-20 b^{3} c \,d^{3} e^{3}+45 b^{2} c^{2} d^{4} e^{2}-42 b \,c^{3} d^{5} e +14 d^{6} c^{4}\right )}{\sqrt {e x +d}}-\frac {28 c^{4} d \left (e x +d \right )^{\frac {9}{2}}}{9}+12 c^{4} d^{2} \left (e x +d \right )^{\frac {7}{2}}-40 a b \,c^{2} d \,e^{3} \left (e x +d \right )^{\frac {3}{2}}-72 a \,b^{2} c d \,e^{4} \sqrt {e x +d}+180 a b \,c^{2} d^{2} e^{3} \sqrt {e x +d}-\frac {2 \left (a^{3} b \,e^{7}-2 d \,e^{6} c \,a^{3}-3 a^{2} b^{2} d \,e^{6}+9 a^{2} b c \,d^{2} e^{5}-6 d^{3} e^{4} a^{2} c^{2}+3 a \,b^{3} d^{2} e^{5}-12 a \,b^{2} c \,d^{3} e^{4}+15 a b \,c^{2} d^{4} e^{3}-6 d^{5} e^{2} a \,c^{3}-b^{4} d^{3} e^{4}+5 b^{3} c \,d^{4} e^{3}-9 b^{2} c^{2} d^{5} e^{2}+7 b \,c^{3} d^{6} e -2 d^{7} c^{4}\right )}{3 \left (e x +d \right )^{\frac {3}{2}}}+\frac {4 c^{4} \left (e x +d \right )^{\frac {11}{2}}}{11}+\frac {140 c^{4} d^{4} \left (e x +d \right )^{\frac {3}{2}}}{3}-28 c^{4} d^{3} \left (e x +d \right )^{\frac {5}{2}}-84 c^{4} d^{5} \sqrt {e x +d}+\frac {2 b^{4} e^{4} \left (e x +d \right )^{\frac {3}{2}}}{3}}{e^{8}}\) | \(894\) |
Input:
int((2*c*x+b)*(c*x^2+b*x+a)^3/(e*x+d)^(5/2),x,method=_RETURNVERBOSE)
Output:
-2/3*((-6/11*c^4*x^7+(-7/3*b*x^6-18/7*a*x^5)*c^3+(-27/7*b^2*x^5-6*a^2*x^3- 9*a*b*x^4)*c^2+(-3*b^3*x^4-12*a*b^2*x^3-27*a^2*b*x^2+6*a^3*x)*c-b^4*x^3-9* a*b^3*x^2+9*x*a^2*b^2+a^3*b)*e^7+4*d*(7/33*c^4*x^6+(b*x^5+9/7*a*x^4)*c^3+( 27/14*b^2*x^4+6*a*b*x^3+9*a^2*x^2)*c^2+(2*b^3*x^3+18*a*b^2*x^2-27*a^2*b*x+ a^3)*c+3/2*b^2*(b^2*x^2-6*a*b*x+a^2))*e^6-72*d^2*(2/99*c^4*x^5+(1/9*b*x^4+ 4/21*a*x^3)*c^3+(2/7*b^2*x^3+2*a*b*x^2-2*a^2*x)*c^2+b*(2/3*b^2*x^2-4*a*b*x +a^2)*c+1/3*b^3*(-b*x+a))*e^5+96*d^3*(1/33*c^4*x^4+(2/9*b*x^3+6/7*a*x^2)*c ^3+(9/7*b^2*x^2-6*a*b*x+a^2)*c^2+(-2*b^3*x+2*a*b^2)*c+1/6*b^4)*e^4-384*(2/ 99*c^3*x^3+(1/3*b*x^2-6/7*a*x)*c^2+b*(-9/7*b*x+a)*c+1/3*b^3)*d^4*c*e^3+153 6/7*d^5*(7/33*c^2*x^2+(-7/3*b*x+a)*c+3/2*b^2)*c^2*e^2-1024/3*d^6*(-6/11*c* x+b)*c^3*e+4096/33*d^7*c^4)/(e*x+d)^(3/2)/e^8
Time = 0.09 (sec) , antiderivative size = 669, normalized size of antiderivative = 1.59 \[ \int \frac {(b+2 c x) \left (a+b x+c x^2\right )^3}{(d+e x)^{5/2}} \, dx=\frac {2 \, {\left (126 \, c^{4} e^{7} x^{7} - 28672 \, c^{4} d^{7} + 78848 \, b c^{3} d^{6} e - 231 \, a^{3} b e^{7} - 25344 \, {\left (3 \, b^{2} c^{2} + 2 \, a c^{3}\right )} d^{5} e^{2} + 29568 \, {\left (b^{3} c + 3 \, a b c^{2}\right )} d^{4} e^{3} - 3696 \, {\left (b^{4} + 12 \, a b^{2} c + 6 \, a^{2} c^{2}\right )} d^{3} e^{4} + 5544 \, {\left (a b^{3} + 3 \, a^{2} b c\right )} d^{2} e^{5} - 462 \, {\left (3 \, a^{2} b^{2} + 2 \, a^{3} c\right )} d e^{6} - 49 \, {\left (4 \, c^{4} d e^{6} - 11 \, b c^{3} e^{7}\right )} x^{6} + 3 \, {\left (112 \, c^{4} d^{2} e^{5} - 308 \, b c^{3} d e^{6} + 99 \, {\left (3 \, b^{2} c^{2} + 2 \, a c^{3}\right )} e^{7}\right )} x^{5} - 3 \, {\left (224 \, c^{4} d^{3} e^{4} - 616 \, b c^{3} d^{2} e^{5} + 198 \, {\left (3 \, b^{2} c^{2} + 2 \, a c^{3}\right )} d e^{6} - 231 \, {\left (b^{3} c + 3 \, a b c^{2}\right )} e^{7}\right )} x^{4} + {\left (1792 \, c^{4} d^{4} e^{3} - 4928 \, b c^{3} d^{3} e^{4} + 1584 \, {\left (3 \, b^{2} c^{2} + 2 \, a c^{3}\right )} d^{2} e^{5} - 1848 \, {\left (b^{3} c + 3 \, a b c^{2}\right )} d e^{6} + 231 \, {\left (b^{4} + 12 \, a b^{2} c + 6 \, a^{2} c^{2}\right )} e^{7}\right )} x^{3} - 3 \, {\left (3584 \, c^{4} d^{5} e^{2} - 9856 \, b c^{3} d^{4} e^{3} + 3168 \, {\left (3 \, b^{2} c^{2} + 2 \, a c^{3}\right )} d^{3} e^{4} - 3696 \, {\left (b^{3} c + 3 \, a b c^{2}\right )} d^{2} e^{5} + 462 \, {\left (b^{4} + 12 \, a b^{2} c + 6 \, a^{2} c^{2}\right )} d e^{6} - 693 \, {\left (a b^{3} + 3 \, a^{2} b c\right )} e^{7}\right )} x^{2} - 3 \, {\left (14336 \, c^{4} d^{6} e - 39424 \, b c^{3} d^{5} e^{2} + 12672 \, {\left (3 \, b^{2} c^{2} + 2 \, a c^{3}\right )} d^{4} e^{3} - 14784 \, {\left (b^{3} c + 3 \, a b c^{2}\right )} d^{3} e^{4} + 1848 \, {\left (b^{4} + 12 \, a b^{2} c + 6 \, a^{2} c^{2}\right )} d^{2} e^{5} - 2772 \, {\left (a b^{3} + 3 \, a^{2} b c\right )} d e^{6} + 231 \, {\left (3 \, a^{2} b^{2} + 2 \, a^{3} c\right )} e^{7}\right )} x\right )} \sqrt {e x + d}}{693 \, {\left (e^{10} x^{2} + 2 \, d e^{9} x + d^{2} e^{8}\right )}} \] Input:
integrate((2*c*x+b)*(c*x^2+b*x+a)^3/(e*x+d)^(5/2),x, algorithm="fricas")
Output:
2/693*(126*c^4*e^7*x^7 - 28672*c^4*d^7 + 78848*b*c^3*d^6*e - 231*a^3*b*e^7 - 25344*(3*b^2*c^2 + 2*a*c^3)*d^5*e^2 + 29568*(b^3*c + 3*a*b*c^2)*d^4*e^3 - 3696*(b^4 + 12*a*b^2*c + 6*a^2*c^2)*d^3*e^4 + 5544*(a*b^3 + 3*a^2*b*c)* d^2*e^5 - 462*(3*a^2*b^2 + 2*a^3*c)*d*e^6 - 49*(4*c^4*d*e^6 - 11*b*c^3*e^7 )*x^6 + 3*(112*c^4*d^2*e^5 - 308*b*c^3*d*e^6 + 99*(3*b^2*c^2 + 2*a*c^3)*e^ 7)*x^5 - 3*(224*c^4*d^3*e^4 - 616*b*c^3*d^2*e^5 + 198*(3*b^2*c^2 + 2*a*c^3 )*d*e^6 - 231*(b^3*c + 3*a*b*c^2)*e^7)*x^4 + (1792*c^4*d^4*e^3 - 4928*b*c^ 3*d^3*e^4 + 1584*(3*b^2*c^2 + 2*a*c^3)*d^2*e^5 - 1848*(b^3*c + 3*a*b*c^2)* d*e^6 + 231*(b^4 + 12*a*b^2*c + 6*a^2*c^2)*e^7)*x^3 - 3*(3584*c^4*d^5*e^2 - 9856*b*c^3*d^4*e^3 + 3168*(3*b^2*c^2 + 2*a*c^3)*d^3*e^4 - 3696*(b^3*c + 3*a*b*c^2)*d^2*e^5 + 462*(b^4 + 12*a*b^2*c + 6*a^2*c^2)*d*e^6 - 693*(a*b^3 + 3*a^2*b*c)*e^7)*x^2 - 3*(14336*c^4*d^6*e - 39424*b*c^3*d^5*e^2 + 12672* (3*b^2*c^2 + 2*a*c^3)*d^4*e^3 - 14784*(b^3*c + 3*a*b*c^2)*d^3*e^4 + 1848*( b^4 + 12*a*b^2*c + 6*a^2*c^2)*d^2*e^5 - 2772*(a*b^3 + 3*a^2*b*c)*d*e^6 + 2 31*(3*a^2*b^2 + 2*a^3*c)*e^7)*x)*sqrt(e*x + d)/(e^10*x^2 + 2*d*e^9*x + d^2 *e^8)
Time = 66.12 (sec) , antiderivative size = 585, normalized size of antiderivative = 1.39 \[ \int \frac {(b+2 c x) \left (a+b x+c x^2\right )^3}{(d+e x)^{5/2}} \, dx=\begin {cases} \frac {2 \cdot \left (\frac {2 c^{4} \left (d + e x\right )^{\frac {11}{2}}}{11 e^{7}} + \frac {\left (d + e x\right )^{\frac {9}{2}} \cdot \left (7 b c^{3} e - 14 c^{4} d\right )}{9 e^{7}} + \frac {\left (d + e x\right )^{\frac {7}{2}} \cdot \left (6 a c^{3} e^{2} + 9 b^{2} c^{2} e^{2} - 42 b c^{3} d e + 42 c^{4} d^{2}\right )}{7 e^{7}} + \frac {\left (d + e x\right )^{\frac {5}{2}} \cdot \left (15 a b c^{2} e^{3} - 30 a c^{3} d e^{2} + 5 b^{3} c e^{3} - 45 b^{2} c^{2} d e^{2} + 105 b c^{3} d^{2} e - 70 c^{4} d^{3}\right )}{5 e^{7}} + \frac {\left (d + e x\right )^{\frac {3}{2}} \cdot \left (6 a^{2} c^{2} e^{4} + 12 a b^{2} c e^{4} - 60 a b c^{2} d e^{3} + 60 a c^{3} d^{2} e^{2} + b^{4} e^{4} - 20 b^{3} c d e^{3} + 90 b^{2} c^{2} d^{2} e^{2} - 140 b c^{3} d^{3} e + 70 c^{4} d^{4}\right )}{3 e^{7}} + \frac {\sqrt {d + e x} \left (9 a^{2} b c e^{5} - 18 a^{2} c^{2} d e^{4} + 3 a b^{3} e^{5} - 36 a b^{2} c d e^{4} + 90 a b c^{2} d^{2} e^{3} - 60 a c^{3} d^{3} e^{2} - 3 b^{4} d e^{4} + 30 b^{3} c d^{2} e^{3} - 90 b^{2} c^{2} d^{3} e^{2} + 105 b c^{3} d^{4} e - 42 c^{4} d^{5}\right )}{e^{7}} - \frac {\left (a e^{2} - b d e + c d^{2}\right )^{2} \cdot \left (2 a c e^{2} + 3 b^{2} e^{2} - 14 b c d e + 14 c^{2} d^{2}\right )}{e^{7} \sqrt {d + e x}} - \frac {\left (b e - 2 c d\right ) \left (a e^{2} - b d e + c d^{2}\right )^{3}}{3 e^{7} \left (d + e x\right )^{\frac {3}{2}}}\right )}{e} & \text {for}\: e \neq 0 \\\frac {\left (a + b x + c x^{2}\right )^{4}}{4 d^{\frac {5}{2}}} & \text {otherwise} \end {cases} \] Input:
integrate((2*c*x+b)*(c*x**2+b*x+a)**3/(e*x+d)**(5/2),x)
Output:
Piecewise((2*(2*c**4*(d + e*x)**(11/2)/(11*e**7) + (d + e*x)**(9/2)*(7*b*c **3*e - 14*c**4*d)/(9*e**7) + (d + e*x)**(7/2)*(6*a*c**3*e**2 + 9*b**2*c** 2*e**2 - 42*b*c**3*d*e + 42*c**4*d**2)/(7*e**7) + (d + e*x)**(5/2)*(15*a*b *c**2*e**3 - 30*a*c**3*d*e**2 + 5*b**3*c*e**3 - 45*b**2*c**2*d*e**2 + 105* b*c**3*d**2*e - 70*c**4*d**3)/(5*e**7) + (d + e*x)**(3/2)*(6*a**2*c**2*e** 4 + 12*a*b**2*c*e**4 - 60*a*b*c**2*d*e**3 + 60*a*c**3*d**2*e**2 + b**4*e** 4 - 20*b**3*c*d*e**3 + 90*b**2*c**2*d**2*e**2 - 140*b*c**3*d**3*e + 70*c** 4*d**4)/(3*e**7) + sqrt(d + e*x)*(9*a**2*b*c*e**5 - 18*a**2*c**2*d*e**4 + 3*a*b**3*e**5 - 36*a*b**2*c*d*e**4 + 90*a*b*c**2*d**2*e**3 - 60*a*c**3*d** 3*e**2 - 3*b**4*d*e**4 + 30*b**3*c*d**2*e**3 - 90*b**2*c**2*d**3*e**2 + 10 5*b*c**3*d**4*e - 42*c**4*d**5)/e**7 - (a*e**2 - b*d*e + c*d**2)**2*(2*a*c *e**2 + 3*b**2*e**2 - 14*b*c*d*e + 14*c**2*d**2)/(e**7*sqrt(d + e*x)) - (b *e - 2*c*d)*(a*e**2 - b*d*e + c*d**2)**3/(3*e**7*(d + e*x)**(3/2)))/e, Ne( e, 0)), ((a + b*x + c*x**2)**4/(4*d**(5/2)), True))
Time = 0.04 (sec) , antiderivative size = 651, normalized size of antiderivative = 1.55 \[ \int \frac {(b+2 c x) \left (a+b x+c x^2\right )^3}{(d+e x)^{5/2}} \, dx =\text {Too large to display} \] Input:
integrate((2*c*x+b)*(c*x^2+b*x+a)^3/(e*x+d)^(5/2),x, algorithm="maxima")
Output:
2/693*((126*(e*x + d)^(11/2)*c^4 - 539*(2*c^4*d - b*c^3*e)*(e*x + d)^(9/2) + 297*(14*c^4*d^2 - 14*b*c^3*d*e + (3*b^2*c^2 + 2*a*c^3)*e^2)*(e*x + d)^( 7/2) - 693*(14*c^4*d^3 - 21*b*c^3*d^2*e + 3*(3*b^2*c^2 + 2*a*c^3)*d*e^2 - (b^3*c + 3*a*b*c^2)*e^3)*(e*x + d)^(5/2) + 231*(70*c^4*d^4 - 140*b*c^3*d^3 *e + 30*(3*b^2*c^2 + 2*a*c^3)*d^2*e^2 - 20*(b^3*c + 3*a*b*c^2)*d*e^3 + (b^ 4 + 12*a*b^2*c + 6*a^2*c^2)*e^4)*(e*x + d)^(3/2) - 2079*(14*c^4*d^5 - 35*b *c^3*d^4*e + 10*(3*b^2*c^2 + 2*a*c^3)*d^3*e^2 - 10*(b^3*c + 3*a*b*c^2)*d^2 *e^3 + (b^4 + 12*a*b^2*c + 6*a^2*c^2)*d*e^4 - (a*b^3 + 3*a^2*b*c)*e^5)*sqr t(e*x + d))/e^7 + 231*(2*c^4*d^7 - 7*b*c^3*d^6*e - a^3*b*e^7 + 3*(3*b^2*c^ 2 + 2*a*c^3)*d^5*e^2 - 5*(b^3*c + 3*a*b*c^2)*d^4*e^3 + (b^4 + 12*a*b^2*c + 6*a^2*c^2)*d^3*e^4 - 3*(a*b^3 + 3*a^2*b*c)*d^2*e^5 + (3*a^2*b^2 + 2*a^3*c )*d*e^6 - 3*(14*c^4*d^6 - 42*b*c^3*d^5*e + 15*(3*b^2*c^2 + 2*a*c^3)*d^4*e^ 2 - 20*(b^3*c + 3*a*b*c^2)*d^3*e^3 + 3*(b^4 + 12*a*b^2*c + 6*a^2*c^2)*d^2* e^4 - 6*(a*b^3 + 3*a^2*b*c)*d*e^5 + (3*a^2*b^2 + 2*a^3*c)*e^6)*(e*x + d))/ ((e*x + d)^(3/2)*e^7))/e
Leaf count of result is larger than twice the leaf count of optimal. 981 vs. \(2 (395) = 790\).
Time = 0.22 (sec) , antiderivative size = 981, normalized size of antiderivative = 2.33 \[ \int \frac {(b+2 c x) \left (a+b x+c x^2\right )^3}{(d+e x)^{5/2}} \, dx=\text {Too large to display} \] Input:
integrate((2*c*x+b)*(c*x^2+b*x+a)^3/(e*x+d)^(5/2),x, algorithm="giac")
Output:
-2/3*(42*(e*x + d)*c^4*d^6 - 2*c^4*d^7 - 126*(e*x + d)*b*c^3*d^5*e + 7*b*c ^3*d^6*e + 135*(e*x + d)*b^2*c^2*d^4*e^2 + 90*(e*x + d)*a*c^3*d^4*e^2 - 9* b^2*c^2*d^5*e^2 - 6*a*c^3*d^5*e^2 - 60*(e*x + d)*b^3*c*d^3*e^3 - 180*(e*x + d)*a*b*c^2*d^3*e^3 + 5*b^3*c*d^4*e^3 + 15*a*b*c^2*d^4*e^3 + 9*(e*x + d)* b^4*d^2*e^4 + 108*(e*x + d)*a*b^2*c*d^2*e^4 + 54*(e*x + d)*a^2*c^2*d^2*e^4 - b^4*d^3*e^4 - 12*a*b^2*c*d^3*e^4 - 6*a^2*c^2*d^3*e^4 - 18*(e*x + d)*a*b ^3*d*e^5 - 54*(e*x + d)*a^2*b*c*d*e^5 + 3*a*b^3*d^2*e^5 + 9*a^2*b*c*d^2*e^ 5 + 9*(e*x + d)*a^2*b^2*e^6 + 6*(e*x + d)*a^3*c*e^6 - 3*a^2*b^2*d*e^6 - 2* a^3*c*d*e^6 + a^3*b*e^7)/((e*x + d)^(3/2)*e^8) + 2/693*(126*(e*x + d)^(11/ 2)*c^4*e^80 - 1078*(e*x + d)^(9/2)*c^4*d*e^80 + 4158*(e*x + d)^(7/2)*c^4*d ^2*e^80 - 9702*(e*x + d)^(5/2)*c^4*d^3*e^80 + 16170*(e*x + d)^(3/2)*c^4*d^ 4*e^80 - 29106*sqrt(e*x + d)*c^4*d^5*e^80 + 539*(e*x + d)^(9/2)*b*c^3*e^81 - 4158*(e*x + d)^(7/2)*b*c^3*d*e^81 + 14553*(e*x + d)^(5/2)*b*c^3*d^2*e^8 1 - 32340*(e*x + d)^(3/2)*b*c^3*d^3*e^81 + 72765*sqrt(e*x + d)*b*c^3*d^4*e ^81 + 891*(e*x + d)^(7/2)*b^2*c^2*e^82 + 594*(e*x + d)^(7/2)*a*c^3*e^82 - 6237*(e*x + d)^(5/2)*b^2*c^2*d*e^82 - 4158*(e*x + d)^(5/2)*a*c^3*d*e^82 + 20790*(e*x + d)^(3/2)*b^2*c^2*d^2*e^82 + 13860*(e*x + d)^(3/2)*a*c^3*d^2*e ^82 - 62370*sqrt(e*x + d)*b^2*c^2*d^3*e^82 - 41580*sqrt(e*x + d)*a*c^3*d^3 *e^82 + 693*(e*x + d)^(5/2)*b^3*c*e^83 + 2079*(e*x + d)^(5/2)*a*b*c^2*e^83 - 4620*(e*x + d)^(3/2)*b^3*c*d*e^83 - 13860*(e*x + d)^(3/2)*a*b*c^2*d*...
Time = 11.69 (sec) , antiderivative size = 677, normalized size of antiderivative = 1.61 \[ \int \frac {(b+2 c x) \left (a+b x+c x^2\right )^3}{(d+e x)^{5/2}} \, dx=\frac {{\left (d+e\,x\right )}^{7/2}\,\left (18\,b^2\,c^2\,e^2-84\,b\,c^3\,d\,e+84\,c^4\,d^2+12\,a\,c^3\,e^2\right )}{7\,e^8}+\frac {4\,c^4\,{\left (d+e\,x\right )}^{11/2}}{11\,e^8}-\frac {\left (28\,c^4\,d-14\,b\,c^3\,e\right )\,{\left (d+e\,x\right )}^{9/2}}{9\,e^8}+\frac {\frac {4\,c^4\,d^7}{3}-\left (d+e\,x\right )\,\left (4\,a^3\,c\,e^6+6\,a^2\,b^2\,e^6-36\,a^2\,b\,c\,d\,e^5+36\,a^2\,c^2\,d^2\,e^4-12\,a\,b^3\,d\,e^5+72\,a\,b^2\,c\,d^2\,e^4-120\,a\,b\,c^2\,d^3\,e^3+60\,a\,c^3\,d^4\,e^2+6\,b^4\,d^2\,e^4-40\,b^3\,c\,d^3\,e^3+90\,b^2\,c^2\,d^4\,e^2-84\,b\,c^3\,d^5\,e+28\,c^4\,d^6\right )-\frac {2\,a^3\,b\,e^7}{3}+\frac {2\,b^4\,d^3\,e^4}{3}-2\,a\,b^3\,d^2\,e^5+2\,a^2\,b^2\,d\,e^6+4\,a\,c^3\,d^5\,e^2-\frac {10\,b^3\,c\,d^4\,e^3}{3}+4\,a^2\,c^2\,d^3\,e^4+6\,b^2\,c^2\,d^5\,e^2+\frac {4\,a^3\,c\,d\,e^6}{3}-\frac {14\,b\,c^3\,d^6\,e}{3}-10\,a\,b\,c^2\,d^4\,e^3+8\,a\,b^2\,c\,d^3\,e^4-6\,a^2\,b\,c\,d^2\,e^5}{e^8\,{\left (d+e\,x\right )}^{3/2}}+\frac {{\left (d+e\,x\right )}^{3/2}\,\left (12\,a^2\,c^2\,e^4+24\,a\,b^2\,c\,e^4-120\,a\,b\,c^2\,d\,e^3+120\,a\,c^3\,d^2\,e^2+2\,b^4\,e^4-40\,b^3\,c\,d\,e^3+180\,b^2\,c^2\,d^2\,e^2-280\,b\,c^3\,d^3\,e+140\,c^4\,d^4\right )}{3\,e^8}+\frac {6\,\left (b\,e-2\,c\,d\right )\,\sqrt {d+e\,x}\,\left (3\,a^2\,c\,e^4+a\,b^2\,e^4-10\,a\,b\,c\,d\,e^3+10\,a\,c^2\,d^2\,e^2-b^3\,d\,e^3+8\,b^2\,c\,d^2\,e^2-14\,b\,c^2\,d^3\,e+7\,c^3\,d^4\right )}{e^8}+\frac {2\,c\,\left (b\,e-2\,c\,d\right )\,{\left (d+e\,x\right )}^{5/2}\,\left (b^2\,e^2-7\,b\,c\,d\,e+7\,c^2\,d^2+3\,a\,c\,e^2\right )}{e^8} \] Input:
int(((b + 2*c*x)*(a + b*x + c*x^2)^3)/(d + e*x)^(5/2),x)
Output:
((d + e*x)^(7/2)*(84*c^4*d^2 + 12*a*c^3*e^2 + 18*b^2*c^2*e^2 - 84*b*c^3*d* e))/(7*e^8) + (4*c^4*(d + e*x)^(11/2))/(11*e^8) - ((28*c^4*d - 14*b*c^3*e) *(d + e*x)^(9/2))/(9*e^8) + ((4*c^4*d^7)/3 - (d + e*x)*(28*c^4*d^6 + 4*a^3 *c*e^6 + 6*a^2*b^2*e^6 + 6*b^4*d^2*e^4 + 60*a*c^3*d^4*e^2 - 40*b^3*c*d^3*e ^3 + 36*a^2*c^2*d^2*e^4 + 90*b^2*c^2*d^4*e^2 - 12*a*b^3*d*e^5 - 84*b*c^3*d ^5*e - 36*a^2*b*c*d*e^5 - 120*a*b*c^2*d^3*e^3 + 72*a*b^2*c*d^2*e^4) - (2*a ^3*b*e^7)/3 + (2*b^4*d^3*e^4)/3 - 2*a*b^3*d^2*e^5 + 2*a^2*b^2*d*e^6 + 4*a* c^3*d^5*e^2 - (10*b^3*c*d^4*e^3)/3 + 4*a^2*c^2*d^3*e^4 + 6*b^2*c^2*d^5*e^2 + (4*a^3*c*d*e^6)/3 - (14*b*c^3*d^6*e)/3 - 10*a*b*c^2*d^4*e^3 + 8*a*b^2*c *d^3*e^4 - 6*a^2*b*c*d^2*e^5)/(e^8*(d + e*x)^(3/2)) + ((d + e*x)^(3/2)*(2* b^4*e^4 + 140*c^4*d^4 + 12*a^2*c^2*e^4 + 120*a*c^3*d^2*e^2 + 180*b^2*c^2*d ^2*e^2 + 24*a*b^2*c*e^4 - 280*b*c^3*d^3*e - 40*b^3*c*d*e^3 - 120*a*b*c^2*d *e^3))/(3*e^8) + (6*(b*e - 2*c*d)*(d + e*x)^(1/2)*(7*c^3*d^4 + a*b^2*e^4 + 3*a^2*c*e^4 - b^3*d*e^3 + 10*a*c^2*d^2*e^2 + 8*b^2*c*d^2*e^2 - 14*b*c^2*d ^3*e - 10*a*b*c*d*e^3))/e^8 + (2*c*(b*e - 2*c*d)*(d + e*x)^(5/2)*(b^2*e^2 + 7*c^2*d^2 + 3*a*c*e^2 - 7*b*c*d*e))/e^8
Time = 0.24 (sec) , antiderivative size = 802, normalized size of antiderivative = 1.90 \[ \int \frac {(b+2 c x) \left (a+b x+c x^2\right )^3}{(d+e x)^{5/2}} \, dx =\text {Too large to display} \] Input:
int((2*c*x+b)*(c*x^2+b*x+a)^3/(e*x+d)^(5/2),x)
Output:
(2*( - 231*a**3*b*e**7 - 924*a**3*c*d*e**6 - 1386*a**3*c*e**7*x - 1386*a** 2*b**2*d*e**6 - 2079*a**2*b**2*e**7*x + 16632*a**2*b*c*d**2*e**5 + 24948*a **2*b*c*d*e**6*x + 6237*a**2*b*c*e**7*x**2 - 22176*a**2*c**2*d**3*e**4 - 3 3264*a**2*c**2*d**2*e**5*x - 8316*a**2*c**2*d*e**6*x**2 + 1386*a**2*c**2*e **7*x**3 + 5544*a*b**3*d**2*e**5 + 8316*a*b**3*d*e**6*x + 2079*a*b**3*e**7 *x**2 - 44352*a*b**2*c*d**3*e**4 - 66528*a*b**2*c*d**2*e**5*x - 16632*a*b* *2*c*d*e**6*x**2 + 2772*a*b**2*c*e**7*x**3 + 88704*a*b*c**2*d**4*e**3 + 13 3056*a*b*c**2*d**3*e**4*x + 33264*a*b*c**2*d**2*e**5*x**2 - 5544*a*b*c**2* d*e**6*x**3 + 2079*a*b*c**2*e**7*x**4 - 50688*a*c**3*d**5*e**2 - 76032*a*c **3*d**4*e**3*x - 19008*a*c**3*d**3*e**4*x**2 + 3168*a*c**3*d**2*e**5*x**3 - 1188*a*c**3*d*e**6*x**4 + 594*a*c**3*e**7*x**5 - 3696*b**4*d**3*e**4 - 5544*b**4*d**2*e**5*x - 1386*b**4*d*e**6*x**2 + 231*b**4*e**7*x**3 + 29568 *b**3*c*d**4*e**3 + 44352*b**3*c*d**3*e**4*x + 11088*b**3*c*d**2*e**5*x**2 - 1848*b**3*c*d*e**6*x**3 + 693*b**3*c*e**7*x**4 - 76032*b**2*c**2*d**5*e **2 - 114048*b**2*c**2*d**4*e**3*x - 28512*b**2*c**2*d**3*e**4*x**2 + 4752 *b**2*c**2*d**2*e**5*x**3 - 1782*b**2*c**2*d*e**6*x**4 + 891*b**2*c**2*e** 7*x**5 + 78848*b*c**3*d**6*e + 118272*b*c**3*d**5*e**2*x + 29568*b*c**3*d* *4*e**3*x**2 - 4928*b*c**3*d**3*e**4*x**3 + 1848*b*c**3*d**2*e**5*x**4 - 9 24*b*c**3*d*e**6*x**5 + 539*b*c**3*e**7*x**6 - 28672*c**4*d**7 - 43008*c** 4*d**6*e*x - 10752*c**4*d**5*e**2*x**2 + 1792*c**4*d**4*e**3*x**3 - 672...