Integrand size = 22, antiderivative size = 282 \[ \int \frac {\left (a+b x+c x^2\right )^3}{(d+e x)^{5/2}} \, dx=-\frac {2 \left (c d^2-b d e+a e^2\right )^3}{3 e^7 (d+e x)^{3/2}}+\frac {6 (2 c d-b e) \left (c d^2-b d e+a e^2\right )^2}{e^7 \sqrt {d+e x}}+\frac {6 \left (c d^2-b d e+a e^2\right ) \left (5 c^2 d^2+b^2 e^2-c e (5 b d-a e)\right ) \sqrt {d+e x}}{e^7}-\frac {2 (2 c d-b e) \left (10 c^2 d^2+b^2 e^2-2 c e (5 b d-3 a e)\right ) (d+e x)^{3/2}}{3 e^7}+\frac {6 c \left (5 c^2 d^2+b^2 e^2-c e (5 b d-a e)\right ) (d+e x)^{5/2}}{5 e^7}-\frac {6 c^2 (2 c d-b e) (d+e x)^{7/2}}{7 e^7}+\frac {2 c^3 (d+e x)^{9/2}}{9 e^7} \] Output:
-2/3*(a*e^2-b*d*e+c*d^2)^3/e^7/(e*x+d)^(3/2)+6*(-b*e+2*c*d)*(a*e^2-b*d*e+c *d^2)^2/e^7/(e*x+d)^(1/2)+6*(a*e^2-b*d*e+c*d^2)*(5*c^2*d^2+b^2*e^2-c*e*(-a *e+5*b*d))*(e*x+d)^(1/2)/e^7-2/3*(-b*e+2*c*d)*(10*c^2*d^2+b^2*e^2-2*c*e*(- 3*a*e+5*b*d))*(e*x+d)^(3/2)/e^7+6/5*c*(5*c^2*d^2+b^2*e^2-c*e*(-a*e+5*b*d)) *(e*x+d)^(5/2)/e^7-6/7*c^2*(-b*e+2*c*d)*(e*x+d)^(7/2)/e^7+2/9*c^3*(e*x+d)^ (9/2)/e^7
Time = 0.42 (sec) , antiderivative size = 395, normalized size of antiderivative = 1.40 \[ \int \frac {\left (a+b x+c x^2\right )^3}{(d+e x)^{5/2}} \, dx=\frac {2 \left (5 c^3 \left (1024 d^6+1536 d^5 e x+384 d^4 e^2 x^2-64 d^3 e^3 x^3+24 d^2 e^4 x^4-12 d e^5 x^5+7 e^6 x^6\right )-105 e^3 \left (a^3 e^3+3 a^2 b e^2 (2 d+3 e x)-3 a b^2 e \left (8 d^2+12 d e x+3 e^2 x^2\right )+b^3 \left (16 d^3+24 d^2 e x+6 d e^2 x^2-e^3 x^3\right )\right )+63 c e^2 \left (5 a^2 e^2 \left (8 d^2+12 d e x+3 e^2 x^2\right )+10 a b e \left (-16 d^3-24 d^2 e x-6 d e^2 x^2+e^3 x^3\right )+b^2 \left (128 d^4+192 d^3 e x+48 d^2 e^2 x^2-8 d e^3 x^3+3 e^4 x^4\right )\right )-9 c^2 e \left (-7 a e \left (128 d^4+192 d^3 e x+48 d^2 e^2 x^2-8 d e^3 x^3+3 e^4 x^4\right )+5 b \left (256 d^5+384 d^4 e x+96 d^3 e^2 x^2-16 d^2 e^3 x^3+6 d e^4 x^4-3 e^5 x^5\right )\right )\right )}{315 e^7 (d+e x)^{3/2}} \] Input:
Integrate[(a + b*x + c*x^2)^3/(d + e*x)^(5/2),x]
Output:
(2*(5*c^3*(1024*d^6 + 1536*d^5*e*x + 384*d^4*e^2*x^2 - 64*d^3*e^3*x^3 + 24 *d^2*e^4*x^4 - 12*d*e^5*x^5 + 7*e^6*x^6) - 105*e^3*(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 + 2 4*d^2*e*x + 6*d*e^2*x^2 - e^3*x^3)) + 63*c*e^2*(5*a^2*e^2*(8*d^2 + 12*d*e* x + 3*e^2*x^2) + 10*a*b*e*(-16*d^3 - 24*d^2*e*x - 6*d*e^2*x^2 + e^3*x^3) + b^2*(128*d^4 + 192*d^3*e*x + 48*d^2*e^2*x^2 - 8*d*e^3*x^3 + 3*e^4*x^4)) - 9*c^2*e*(-7*a*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) + 5*b*(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))))/(315*e^7*(d + e*x)^(3/2))
Time = 0.41 (sec) , antiderivative size = 282, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.091, Rules used = {1140, 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.
| \(\displaystyle \int \frac {\left (a+b x+c x^2\right )^3}{(d+e x)^{5/2}} \, dx\) |
| \(\Big \downarrow \) 1140 |
| \(\displaystyle \int \left (\frac {3 c (d+e x)^{3/2} \left (-c e (5 b d-a e)+b^2 e^2+5 c^2 d^2\right )}{e^6}+\frac {\sqrt {d+e x} (2 c d-b e) \left (2 c e (5 b d-3 a e)-b^2 e^2-10 c^2 d^2\right )}{e^6}+\frac {3 \left (a e^2-b d e+c d^2\right ) \left (a c e^2+b^2 e^2-5 b c d e+5 c^2 d^2\right )}{e^6 \sqrt {d+e x}}+\frac {3 (b e-2 c d) \left (a e^2-b d e+c d^2\right )^2}{e^6 (d+e x)^{3/2}}+\frac {\left (a e^2-b d e+c d^2\right )^3}{e^6 (d+e x)^{5/2}}-\frac {3 c^2 (d+e x)^{5/2} (2 c d-b e)}{e^6}+\frac {c^3 (d+e x)^{7/2}}{e^6}\right )dx\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {6 c (d+e x)^{5/2} \left (-c e (5 b d-a e)+b^2 e^2+5 c^2 d^2\right )}{5 e^7}-\frac {2 (d+e x)^{3/2} (2 c d-b e) \left (-2 c e (5 b d-3 a e)+b^2 e^2+10 c^2 d^2\right )}{3 e^7}+\frac {6 \sqrt {d+e x} \left (a e^2-b d e+c d^2\right ) \left (-c e (5 b d-a e)+b^2 e^2+5 c^2 d^2\right )}{e^7}+\frac {6 (2 c d-b e) \left (a e^2-b d e+c d^2\right )^2}{e^7 \sqrt {d+e x}}-\frac {2 \left (a e^2-b d e+c d^2\right )^3}{3 e^7 (d+e x)^{3/2}}-\frac {6 c^2 (d+e x)^{7/2} (2 c d-b e)}{7 e^7}+\frac {2 c^3 (d+e x)^{9/2}}{9 e^7}\) |
Input:
Int[(a + b*x + c*x^2)^3/(d + e*x)^(5/2),x]
Output:
(-2*(c*d^2 - b*d*e + a*e^2)^3)/(3*e^7*(d + e*x)^(3/2)) + (6*(2*c*d - b*e)* (c*d^2 - b*d*e + a*e^2)^2)/(e^7*Sqrt[d + e*x]) + (6*(c*d^2 - b*d*e + a*e^2 )*(5*c^2*d^2 + b^2*e^2 - c*e*(5*b*d - a*e))*Sqrt[d + e*x])/e^7 - (2*(2*c*d - b*e)*(10*c^2*d^2 + b^2*e^2 - 2*c*e*(5*b*d - 3*a*e))*(d + e*x)^(3/2))/(3 *e^7) + (6*c*(5*c^2*d^2 + b^2*e^2 - c*e*(5*b*d - a*e))*(d + e*x)^(5/2))/(5 *e^7) - (6*c^2*(2*c*d - b*e)*(d + e*x)^(7/2))/(7*e^7) + (2*c^3*(d + e*x)^( 9/2))/(9*e^7)
Int[((d_.) + (e_.)*(x_))^(m_.)*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_.), x _Symbol] :> Int[ExpandIntegrand[(d + e*x)^m*(a + b*x + c*x^2)^p, x], x] /; FreeQ[{a, b, c, d, e, m}, x] && IGtQ[p, 0]
Time = 1.02 (sec) , antiderivative size = 329, normalized size of antiderivative = 1.17
| method | result | size |
| pseudoelliptic | \(-\frac {2 \left (\left (-\frac {c^{3} x^{6}}{3}-\frac {9 x^{4} \left (\frac {5 b x}{7}+a \right ) c^{2}}{5}+\left (-6 a b \,x^{3}-\frac {9}{5} b^{2} x^{4}-9 a^{2} x^{2}\right ) c -b^{3} x^{3}-9 a \,b^{2} x^{2}+9 a^{2} b x +a^{3}\right ) e^{6}+6 d \left (\frac {2 c^{3} x^{5}}{21}+\left (\frac {3}{7} b \,x^{4}+\frac {4}{5} a \,x^{3}\right ) c^{2}+\left (\frac {4}{5} b^{2} x^{3}+6 a b \,x^{2}-6 a^{2} x \right ) c +b \left (b^{2} x^{2}-6 a b x +a^{2}\right )\right ) e^{5}-24 d^{2} \left (\frac {c^{3} x^{4}}{21}+\left (\frac {2}{7} b \,x^{3}+\frac {6}{5} a \,x^{2}\right ) c^{2}+\left (\frac {6}{5} b^{2} x^{2}-6 a b x +a^{2}\right ) c +b^{2} \left (-b x +a \right )\right ) e^{4}+96 \left (\frac {2 c^{3} x^{3}}{63}+\left (\frac {3}{7} b \,x^{2}-\frac {6}{5} a x \right ) c^{2}+b \left (-\frac {6 b x}{5}+a \right ) c +\frac {b^{3}}{6}\right ) d^{3} e^{3}-\frac {384 \left (\frac {5 c^{2} x^{2}}{21}+\left (-\frac {15 b x}{7}+a \right ) c +b^{2}\right ) d^{4} c \,e^{2}}{5}+\frac {768 d^{5} \left (-\frac {2 c x}{3}+b \right ) c^{2} e}{7}-\frac {1024 d^{6} c^{3}}{21}\right )}{3 \left (e x +d \right )^{\frac {3}{2}} e^{7}}\) | \(329\) |
| risch | \(\frac {2 \left (35 c^{3} x^{4} e^{4}+135 b \,c^{2} e^{4} x^{3}-130 c^{3} d \,e^{3} x^{3}+189 a \,c^{2} e^{4} x^{2}+189 b^{2} c \,e^{4} x^{2}-540 b \,c^{2} d \,e^{3} x^{2}+345 c^{3} d^{2} e^{2} x^{2}+630 a b c \,e^{4} x -882 a \,c^{2} d \,e^{3} x +105 b^{3} e^{4} x -882 b^{2} c d \,e^{3} x +1665 b \,c^{2} d^{2} e^{2} x -880 c^{3} d^{3} e x +945 e^{4} a^{2} c +945 a \,b^{2} e^{4}-5040 a b c d \,e^{3}+4599 d^{2} e^{2} a \,c^{2}-840 b^{3} d \,e^{3}+4599 b^{2} c \,d^{2} e^{2}-7110 b \,c^{2} d^{3} e +3335 d^{4} c^{3}\right ) \sqrt {e x +d}}{315 e^{7}}-\frac {2 \left (9 x b \,e^{2}-18 c d x e +a \,e^{2}+8 b d e -17 c \,d^{2}\right ) \left (a^{2} e^{4}-2 d \,e^{3} a b +2 a c \,d^{2} e^{2}+d^{2} e^{2} b^{2}-2 b c \,d^{3} e +c^{2} d^{4}\right )}{3 e^{7} \left (e x +d \right )^{\frac {3}{2}}}\) | \(335\) |
| gosper | \(-\frac {2 \left (-35 c^{3} e^{6} x^{6}-135 x^{5} b \,c^{2} e^{6}+60 c^{3} d \,e^{5} x^{5}-189 x^{4} a \,c^{2} e^{6}-189 x^{4} b^{2} c \,e^{6}+270 x^{4} b \,c^{2} d \,e^{5}-120 c^{3} d^{2} e^{4} x^{4}-630 x^{3} a b c \,e^{6}+504 x^{3} a \,c^{2} d \,e^{5}-105 x^{3} b^{3} e^{6}+504 x^{3} b^{2} c d \,e^{5}-720 x^{3} b \,c^{2} d^{2} e^{4}+320 c^{3} d^{3} e^{3} x^{3}-945 x^{2} a^{2} c \,e^{6}-945 x^{2} a \,b^{2} e^{6}+3780 x^{2} a b c d \,e^{5}-3024 x^{2} a \,c^{2} d^{2} e^{4}+630 x^{2} b^{3} d \,e^{5}-3024 x^{2} b^{2} c \,d^{2} e^{4}+4320 x^{2} b \,c^{2} d^{3} e^{3}-1920 c^{3} d^{4} e^{2} x^{2}+945 x \,a^{2} b \,e^{6}-3780 x \,a^{2} c d \,e^{5}-3780 x a \,b^{2} d \,e^{5}+15120 x a b c \,d^{2} e^{4}-12096 x a \,c^{2} d^{3} e^{3}+2520 x \,b^{3} d^{2} e^{4}-12096 x \,b^{2} c \,d^{3} e^{3}+17280 x b \,c^{2} d^{4} e^{2}-7680 c^{3} d^{5} e x +105 e^{6} a^{3}+630 a^{2} b d \,e^{5}-2520 d^{2} e^{4} a^{2} c -2520 a \,b^{2} d^{2} e^{4}+10080 a b c \,d^{3} e^{3}-8064 d^{4} e^{2} a \,c^{2}+1680 b^{3} d^{3} e^{3}-8064 b^{2} c \,d^{4} e^{2}+11520 b \,c^{2} d^{5} e -5120 d^{6} c^{3}\right )}{315 \left (e x +d \right )^{\frac {3}{2}} e^{7}}\) | \(495\) |
| trager | \(-\frac {2 \left (-35 c^{3} e^{6} x^{6}-135 x^{5} b \,c^{2} e^{6}+60 c^{3} d \,e^{5} x^{5}-189 x^{4} a \,c^{2} e^{6}-189 x^{4} b^{2} c \,e^{6}+270 x^{4} b \,c^{2} d \,e^{5}-120 c^{3} d^{2} e^{4} x^{4}-630 x^{3} a b c \,e^{6}+504 x^{3} a \,c^{2} d \,e^{5}-105 x^{3} b^{3} e^{6}+504 x^{3} b^{2} c d \,e^{5}-720 x^{3} b \,c^{2} d^{2} e^{4}+320 c^{3} d^{3} e^{3} x^{3}-945 x^{2} a^{2} c \,e^{6}-945 x^{2} a \,b^{2} e^{6}+3780 x^{2} a b c d \,e^{5}-3024 x^{2} a \,c^{2} d^{2} e^{4}+630 x^{2} b^{3} d \,e^{5}-3024 x^{2} b^{2} c \,d^{2} e^{4}+4320 x^{2} b \,c^{2} d^{3} e^{3}-1920 c^{3} d^{4} e^{2} x^{2}+945 x \,a^{2} b \,e^{6}-3780 x \,a^{2} c d \,e^{5}-3780 x a \,b^{2} d \,e^{5}+15120 x a b c \,d^{2} e^{4}-12096 x a \,c^{2} d^{3} e^{3}+2520 x \,b^{3} d^{2} e^{4}-12096 x \,b^{2} c \,d^{3} e^{3}+17280 x b \,c^{2} d^{4} e^{2}-7680 c^{3} d^{5} e x +105 e^{6} a^{3}+630 a^{2} b d \,e^{5}-2520 d^{2} e^{4} a^{2} c -2520 a \,b^{2} d^{2} e^{4}+10080 a b c \,d^{3} e^{3}-8064 d^{4} e^{2} a \,c^{2}+1680 b^{3} d^{3} e^{3}-8064 b^{2} c \,d^{4} e^{2}+11520 b \,c^{2} d^{5} e -5120 d^{6} c^{3}\right )}{315 \left (e x +d \right )^{\frac {3}{2}} e^{7}}\) | \(495\) |
| orering | \(-\frac {2 \left (-35 c^{3} e^{6} x^{6}-135 x^{5} b \,c^{2} e^{6}+60 c^{3} d \,e^{5} x^{5}-189 x^{4} a \,c^{2} e^{6}-189 x^{4} b^{2} c \,e^{6}+270 x^{4} b \,c^{2} d \,e^{5}-120 c^{3} d^{2} e^{4} x^{4}-630 x^{3} a b c \,e^{6}+504 x^{3} a \,c^{2} d \,e^{5}-105 x^{3} b^{3} e^{6}+504 x^{3} b^{2} c d \,e^{5}-720 x^{3} b \,c^{2} d^{2} e^{4}+320 c^{3} d^{3} e^{3} x^{3}-945 x^{2} a^{2} c \,e^{6}-945 x^{2} a \,b^{2} e^{6}+3780 x^{2} a b c d \,e^{5}-3024 x^{2} a \,c^{2} d^{2} e^{4}+630 x^{2} b^{3} d \,e^{5}-3024 x^{2} b^{2} c \,d^{2} e^{4}+4320 x^{2} b \,c^{2} d^{3} e^{3}-1920 c^{3} d^{4} e^{2} x^{2}+945 x \,a^{2} b \,e^{6}-3780 x \,a^{2} c d \,e^{5}-3780 x a \,b^{2} d \,e^{5}+15120 x a b c \,d^{2} e^{4}-12096 x a \,c^{2} d^{3} e^{3}+2520 x \,b^{3} d^{2} e^{4}-12096 x \,b^{2} c \,d^{3} e^{3}+17280 x b \,c^{2} d^{4} e^{2}-7680 c^{3} d^{5} e x +105 e^{6} a^{3}+630 a^{2} b d \,e^{5}-2520 d^{2} e^{4} a^{2} c -2520 a \,b^{2} d^{2} e^{4}+10080 a b c \,d^{3} e^{3}-8064 d^{4} e^{2} a \,c^{2}+1680 b^{3} d^{3} e^{3}-8064 b^{2} c \,d^{4} e^{2}+11520 b \,c^{2} d^{5} e -5120 d^{6} c^{3}\right )}{315 \left (e x +d \right )^{\frac {3}{2}} e^{7}}\) | \(495\) |
| derivativedivides | \(\frac {\frac {2 c^{3} \left (e x +d \right )^{\frac {9}{2}}}{9}+\frac {6 b \,c^{2} e \left (e x +d \right )^{\frac {7}{2}}}{7}-\frac {12 c^{3} d \left (e x +d \right )^{\frac {7}{2}}}{7}+\frac {6 a \,c^{2} e^{2} \left (e x +d \right )^{\frac {5}{2}}}{5}+\frac {6 b^{2} c \,e^{2} \left (e x +d \right )^{\frac {5}{2}}}{5}-6 b \,c^{2} d e \left (e x +d \right )^{\frac {5}{2}}+6 c^{3} d^{2} \left (e x +d \right )^{\frac {5}{2}}+4 a b c \,e^{3} \left (e x +d \right )^{\frac {3}{2}}-8 a \,c^{2} d \,e^{2} \left (e x +d \right )^{\frac {3}{2}}+\frac {2 b^{3} e^{3} \left (e x +d \right )^{\frac {3}{2}}}{3}-8 b^{2} c d \,e^{2} \left (e x +d \right )^{\frac {3}{2}}+20 b \,c^{2} d^{2} e \left (e x +d \right )^{\frac {3}{2}}-\frac {40 c^{3} d^{3} \left (e x +d \right )^{\frac {3}{2}}}{3}+6 a^{2} c \,e^{4} \sqrt {e x +d}+6 a \,b^{2} e^{4} \sqrt {e x +d}-36 a b c d \,e^{3} \sqrt {e x +d}+36 a \,c^{2} d^{2} e^{2} \sqrt {e x +d}-6 b^{3} d \,e^{3} \sqrt {e x +d}+36 b^{2} c \,d^{2} e^{2} \sqrt {e x +d}-60 b \,c^{2} d^{3} e \sqrt {e x +d}+30 c^{3} d^{4} \sqrt {e x +d}-\frac {2 \left (3 a^{2} b \,e^{5}-6 d \,e^{4} a^{2} c -6 a \,b^{2} d \,e^{4}+18 a b c \,d^{2} e^{3}-12 d^{3} e^{2} a \,c^{2}+3 b^{3} d^{2} e^{3}-12 b^{2} c \,d^{3} e^{2}+15 b \,c^{2} d^{4} e -6 d^{5} c^{3}\right )}{\sqrt {e x +d}}-\frac {2 \left (e^{6} a^{3}-3 a^{2} b d \,e^{5}+3 d^{2} e^{4} a^{2} c +3 a \,b^{2} d^{2} e^{4}-6 a b c \,d^{3} e^{3}+3 d^{4} e^{2} a \,c^{2}-b^{3} d^{3} e^{3}+3 b^{2} c \,d^{4} e^{2}-3 b \,c^{2} d^{5} e +d^{6} c^{3}\right )}{3 \left (e x +d \right )^{\frac {3}{2}}}}{e^{7}}\) | \(555\) |
| default | \(\frac {\frac {2 c^{3} \left (e x +d \right )^{\frac {9}{2}}}{9}+\frac {6 b \,c^{2} e \left (e x +d \right )^{\frac {7}{2}}}{7}-\frac {12 c^{3} d \left (e x +d \right )^{\frac {7}{2}}}{7}+\frac {6 a \,c^{2} e^{2} \left (e x +d \right )^{\frac {5}{2}}}{5}+\frac {6 b^{2} c \,e^{2} \left (e x +d \right )^{\frac {5}{2}}}{5}-6 b \,c^{2} d e \left (e x +d \right )^{\frac {5}{2}}+6 c^{3} d^{2} \left (e x +d \right )^{\frac {5}{2}}+4 a b c \,e^{3} \left (e x +d \right )^{\frac {3}{2}}-8 a \,c^{2} d \,e^{2} \left (e x +d \right )^{\frac {3}{2}}+\frac {2 b^{3} e^{3} \left (e x +d \right )^{\frac {3}{2}}}{3}-8 b^{2} c d \,e^{2} \left (e x +d \right )^{\frac {3}{2}}+20 b \,c^{2} d^{2} e \left (e x +d \right )^{\frac {3}{2}}-\frac {40 c^{3} d^{3} \left (e x +d \right )^{\frac {3}{2}}}{3}+6 a^{2} c \,e^{4} \sqrt {e x +d}+6 a \,b^{2} e^{4} \sqrt {e x +d}-36 a b c d \,e^{3} \sqrt {e x +d}+36 a \,c^{2} d^{2} e^{2} \sqrt {e x +d}-6 b^{3} d \,e^{3} \sqrt {e x +d}+36 b^{2} c \,d^{2} e^{2} \sqrt {e x +d}-60 b \,c^{2} d^{3} e \sqrt {e x +d}+30 c^{3} d^{4} \sqrt {e x +d}-\frac {2 \left (3 a^{2} b \,e^{5}-6 d \,e^{4} a^{2} c -6 a \,b^{2} d \,e^{4}+18 a b c \,d^{2} e^{3}-12 d^{3} e^{2} a \,c^{2}+3 b^{3} d^{2} e^{3}-12 b^{2} c \,d^{3} e^{2}+15 b \,c^{2} d^{4} e -6 d^{5} c^{3}\right )}{\sqrt {e x +d}}-\frac {2 \left (e^{6} a^{3}-3 a^{2} b d \,e^{5}+3 d^{2} e^{4} a^{2} c +3 a \,b^{2} d^{2} e^{4}-6 a b c \,d^{3} e^{3}+3 d^{4} e^{2} a \,c^{2}-b^{3} d^{3} e^{3}+3 b^{2} c \,d^{4} e^{2}-3 b \,c^{2} d^{5} e +d^{6} c^{3}\right )}{3 \left (e x +d \right )^{\frac {3}{2}}}}{e^{7}}\) | \(555\) |
Input:
int((c*x^2+b*x+a)^3/(e*x+d)^(5/2),x,method=_RETURNVERBOSE)
Output:
-2/3*((-1/3*c^3*x^6-9/5*x^4*(5/7*b*x+a)*c^2+(-6*a*b*x^3-9/5*b^2*x^4-9*a^2* x^2)*c-b^3*x^3-9*a*b^2*x^2+9*a^2*b*x+a^3)*e^6+6*d*(2/21*c^3*x^5+(3/7*b*x^4 +4/5*a*x^3)*c^2+(4/5*b^2*x^3+6*a*b*x^2-6*a^2*x)*c+b*(b^2*x^2-6*a*b*x+a^2)) *e^5-24*d^2*(1/21*c^3*x^4+(2/7*b*x^3+6/5*a*x^2)*c^2+(6/5*b^2*x^2-6*a*b*x+a ^2)*c+b^2*(-b*x+a))*e^4+96*(2/63*c^3*x^3+(3/7*b*x^2-6/5*a*x)*c^2+b*(-6/5*b *x+a)*c+1/6*b^3)*d^3*e^3-384/5*(5/21*c^2*x^2+(-15/7*b*x+a)*c+b^2)*d^4*c*e^ 2+768/7*d^5*(-2/3*c*x+b)*c^2*e-1024/21*d^6*c^3)/(e*x+d)^(3/2)/e^7
Time = 0.10 (sec) , antiderivative size = 429, normalized size of antiderivative = 1.52 \[ \int \frac {\left (a+b x+c x^2\right )^3}{(d+e x)^{5/2}} \, dx=\frac {2 \, {\left (35 \, c^{3} e^{6} x^{6} + 5120 \, c^{3} d^{6} - 11520 \, b c^{2} d^{5} e - 630 \, a^{2} b d e^{5} - 105 \, a^{3} e^{6} + 8064 \, {\left (b^{2} c + a c^{2}\right )} d^{4} e^{2} - 1680 \, {\left (b^{3} + 6 \, a b c\right )} d^{3} e^{3} + 2520 \, {\left (a b^{2} + a^{2} c\right )} d^{2} e^{4} - 15 \, {\left (4 \, c^{3} d e^{5} - 9 \, b c^{2} e^{6}\right )} x^{5} + 3 \, {\left (40 \, c^{3} d^{2} e^{4} - 90 \, b c^{2} d e^{5} + 63 \, {\left (b^{2} c + a c^{2}\right )} e^{6}\right )} x^{4} - {\left (320 \, c^{3} d^{3} e^{3} - 720 \, b c^{2} d^{2} e^{4} + 504 \, {\left (b^{2} c + a c^{2}\right )} d e^{5} - 105 \, {\left (b^{3} + 6 \, a b c\right )} e^{6}\right )} x^{3} + 3 \, {\left (640 \, c^{3} d^{4} e^{2} - 1440 \, b c^{2} d^{3} e^{3} + 1008 \, {\left (b^{2} c + a c^{2}\right )} d^{2} e^{4} - 210 \, {\left (b^{3} + 6 \, a b c\right )} d e^{5} + 315 \, {\left (a b^{2} + a^{2} c\right )} e^{6}\right )} x^{2} + 3 \, {\left (2560 \, c^{3} d^{5} e - 5760 \, b c^{2} d^{4} e^{2} - 315 \, a^{2} b e^{6} + 4032 \, {\left (b^{2} c + a c^{2}\right )} d^{3} e^{3} - 840 \, {\left (b^{3} + 6 \, a b c\right )} d^{2} e^{4} + 1260 \, {\left (a b^{2} + a^{2} c\right )} d e^{5}\right )} x\right )} \sqrt {e x + d}}{315 \, {\left (e^{9} x^{2} + 2 \, d e^{8} x + d^{2} e^{7}\right )}} \] Input:
integrate((c*x^2+b*x+a)^3/(e*x+d)^(5/2),x, algorithm="fricas")
Output:
2/315*(35*c^3*e^6*x^6 + 5120*c^3*d^6 - 11520*b*c^2*d^5*e - 630*a^2*b*d*e^5 - 105*a^3*e^6 + 8064*(b^2*c + a*c^2)*d^4*e^2 - 1680*(b^3 + 6*a*b*c)*d^3*e ^3 + 2520*(a*b^2 + a^2*c)*d^2*e^4 - 15*(4*c^3*d*e^5 - 9*b*c^2*e^6)*x^5 + 3 *(40*c^3*d^2*e^4 - 90*b*c^2*d*e^5 + 63*(b^2*c + a*c^2)*e^6)*x^4 - (320*c^3 *d^3*e^3 - 720*b*c^2*d^2*e^4 + 504*(b^2*c + a*c^2)*d*e^5 - 105*(b^3 + 6*a* b*c)*e^6)*x^3 + 3*(640*c^3*d^4*e^2 - 1440*b*c^2*d^3*e^3 + 1008*(b^2*c + a* c^2)*d^2*e^4 - 210*(b^3 + 6*a*b*c)*d*e^5 + 315*(a*b^2 + a^2*c)*e^6)*x^2 + 3*(2560*c^3*d^5*e - 5760*b*c^2*d^4*e^2 - 315*a^2*b*e^6 + 4032*(b^2*c + a*c ^2)*d^3*e^3 - 840*(b^3 + 6*a*b*c)*d^2*e^4 + 1260*(a*b^2 + a^2*c)*d*e^5)*x) *sqrt(e*x + d)/(e^9*x^2 + 2*d*e^8*x + d^2*e^7)
Time = 24.99 (sec) , antiderivative size = 449, normalized size of antiderivative = 1.59 \[ \int \frac {\left (a+b x+c x^2\right )^3}{(d+e x)^{5/2}} \, dx=\begin {cases} \frac {2 \left (\frac {c^{3} \left (d + e x\right )^{\frac {9}{2}}}{9 e^{6}} + \frac {\left (d + e x\right )^{\frac {7}{2}} \cdot \left (3 b c^{2} e - 6 c^{3} d\right )}{7 e^{6}} + \frac {\left (d + e x\right )^{\frac {5}{2}} \cdot \left (3 a c^{2} e^{2} + 3 b^{2} c e^{2} - 15 b c^{2} d e + 15 c^{3} d^{2}\right )}{5 e^{6}} + \frac {\left (d + e x\right )^{\frac {3}{2}} \cdot \left (6 a b c e^{3} - 12 a c^{2} d e^{2} + b^{3} e^{3} - 12 b^{2} c d e^{2} + 30 b c^{2} d^{2} e - 20 c^{3} d^{3}\right )}{3 e^{6}} + \frac {\sqrt {d + e x} \left (3 a^{2} c e^{4} + 3 a b^{2} e^{4} - 18 a b c d e^{3} + 18 a c^{2} d^{2} e^{2} - 3 b^{3} d e^{3} + 18 b^{2} c d^{2} e^{2} - 30 b c^{2} d^{3} e + 15 c^{3} d^{4}\right )}{e^{6}} - \frac {3 \left (b e - 2 c d\right ) \left (a e^{2} - b d e + c d^{2}\right )^{2}}{e^{6} \sqrt {d + e x}} - \frac {\left (a e^{2} - b d e + c d^{2}\right )^{3}}{3 e^{6} \left (d + e x\right )^{\frac {3}{2}}}\right )}{e} & \text {for}\: e \neq 0 \\\frac {a^{3} x + \frac {3 a^{2} b x^{2}}{2} + \frac {b c^{2} x^{6}}{2} + \frac {c^{3} x^{7}}{7} + \frac {x^{5} \cdot \left (3 a c^{2} + 3 b^{2} c\right )}{5} + \frac {x^{4} \cdot \left (6 a b c + b^{3}\right )}{4} + \frac {x^{3} \cdot \left (3 a^{2} c + 3 a b^{2}\right )}{3}}{d^{\frac {5}{2}}} & \text {otherwise} \end {cases} \] Input:
integrate((c*x**2+b*x+a)**3/(e*x+d)**(5/2),x)
Output:
Piecewise((2*(c**3*(d + e*x)**(9/2)/(9*e**6) + (d + e*x)**(7/2)*(3*b*c**2* e - 6*c**3*d)/(7*e**6) + (d + e*x)**(5/2)*(3*a*c**2*e**2 + 3*b**2*c*e**2 - 15*b*c**2*d*e + 15*c**3*d**2)/(5*e**6) + (d + e*x)**(3/2)*(6*a*b*c*e**3 - 12*a*c**2*d*e**2 + b**3*e**3 - 12*b**2*c*d*e**2 + 30*b*c**2*d**2*e - 20*c **3*d**3)/(3*e**6) + sqrt(d + e*x)*(3*a**2*c*e**4 + 3*a*b**2*e**4 - 18*a*b *c*d*e**3 + 18*a*c**2*d**2*e**2 - 3*b**3*d*e**3 + 18*b**2*c*d**2*e**2 - 30 *b*c**2*d**3*e + 15*c**3*d**4)/e**6 - 3*(b*e - 2*c*d)*(a*e**2 - b*d*e + c* d**2)**2/(e**6*sqrt(d + e*x)) - (a*e**2 - b*d*e + c*d**2)**3/(3*e**6*(d + e*x)**(3/2)))/e, Ne(e, 0)), ((a**3*x + 3*a**2*b*x**2/2 + b*c**2*x**6/2 + c **3*x**7/7 + x**5*(3*a*c**2 + 3*b**2*c)/5 + x**4*(6*a*b*c + b**3)/4 + x**3 *(3*a**2*c + 3*a*b**2)/3)/d**(5/2), True))
Time = 0.05 (sec) , antiderivative size = 413, normalized size of antiderivative = 1.46 \[ \int \frac {\left (a+b x+c x^2\right )^3}{(d+e x)^{5/2}} \, dx=\frac {2 \, {\left (\frac {35 \, {\left (e x + d\right )}^{\frac {9}{2}} c^{3} - 135 \, {\left (2 \, c^{3} d - b c^{2} e\right )} {\left (e x + d\right )}^{\frac {7}{2}} + 189 \, {\left (5 \, c^{3} d^{2} - 5 \, b c^{2} d e + {\left (b^{2} c + a c^{2}\right )} e^{2}\right )} {\left (e x + d\right )}^{\frac {5}{2}} - 105 \, {\left (20 \, c^{3} d^{3} - 30 \, b c^{2} d^{2} e + 12 \, {\left (b^{2} c + a c^{2}\right )} d e^{2} - {\left (b^{3} + 6 \, a b c\right )} e^{3}\right )} {\left (e x + d\right )}^{\frac {3}{2}} + 945 \, {\left (5 \, c^{3} d^{4} - 10 \, b c^{2} d^{3} e + 6 \, {\left (b^{2} c + a c^{2}\right )} d^{2} e^{2} - {\left (b^{3} + 6 \, a b c\right )} d e^{3} + {\left (a b^{2} + a^{2} c\right )} e^{4}\right )} \sqrt {e x + d}}{e^{6}} - \frac {105 \, {\left (c^{3} d^{6} - 3 \, b c^{2} d^{5} e - 3 \, a^{2} b d e^{5} + a^{3} e^{6} + 3 \, {\left (b^{2} c + a c^{2}\right )} d^{4} e^{2} - {\left (b^{3} + 6 \, a b c\right )} d^{3} e^{3} + 3 \, {\left (a b^{2} + a^{2} c\right )} d^{2} e^{4} - 9 \, {\left (2 \, c^{3} d^{5} - 5 \, b c^{2} d^{4} e - a^{2} b e^{5} + 4 \, {\left (b^{2} c + a c^{2}\right )} d^{3} e^{2} - {\left (b^{3} + 6 \, a b c\right )} d^{2} e^{3} + 2 \, {\left (a b^{2} + a^{2} c\right )} d e^{4}\right )} {\left (e x + d\right )}\right )}}{{\left (e x + d\right )}^{\frac {3}{2}} e^{6}}\right )}}{315 \, e} \] Input:
integrate((c*x^2+b*x+a)^3/(e*x+d)^(5/2),x, algorithm="maxima")
Output:
2/315*((35*(e*x + d)^(9/2)*c^3 - 135*(2*c^3*d - b*c^2*e)*(e*x + d)^(7/2) + 189*(5*c^3*d^2 - 5*b*c^2*d*e + (b^2*c + a*c^2)*e^2)*(e*x + d)^(5/2) - 105 *(20*c^3*d^3 - 30*b*c^2*d^2*e + 12*(b^2*c + a*c^2)*d*e^2 - (b^3 + 6*a*b*c) *e^3)*(e*x + d)^(3/2) + 945*(5*c^3*d^4 - 10*b*c^2*d^3*e + 6*(b^2*c + a*c^2 )*d^2*e^2 - (b^3 + 6*a*b*c)*d*e^3 + (a*b^2 + a^2*c)*e^4)*sqrt(e*x + d))/e^ 6 - 105*(c^3*d^6 - 3*b*c^2*d^5*e - 3*a^2*b*d*e^5 + a^3*e^6 + 3*(b^2*c + a* c^2)*d^4*e^2 - (b^3 + 6*a*b*c)*d^3*e^3 + 3*(a*b^2 + a^2*c)*d^2*e^4 - 9*(2* c^3*d^5 - 5*b*c^2*d^4*e - a^2*b*e^5 + 4*(b^2*c + a*c^2)*d^3*e^2 - (b^3 + 6 *a*b*c)*d^2*e^3 + 2*(a*b^2 + a^2*c)*d*e^4)*(e*x + d))/((e*x + d)^(3/2)*e^6 ))/e
Leaf count of result is larger than twice the leaf count of optimal. 617 vs. \(2 (258) = 516\).
Time = 0.39 (sec) , antiderivative size = 617, normalized size of antiderivative = 2.19 \[ \int \frac {\left (a+b x+c x^2\right )^3}{(d+e x)^{5/2}} \, dx=\frac {2 \, {\left (18 \, {\left (e x + d\right )} c^{3} d^{5} - c^{3} d^{6} - 45 \, {\left (e x + d\right )} b c^{2} d^{4} e + 3 \, b c^{2} d^{5} e + 36 \, {\left (e x + d\right )} b^{2} c d^{3} e^{2} + 36 \, {\left (e x + d\right )} a c^{2} d^{3} e^{2} - 3 \, b^{2} c d^{4} e^{2} - 3 \, a c^{2} d^{4} e^{2} - 9 \, {\left (e x + d\right )} b^{3} d^{2} e^{3} - 54 \, {\left (e x + d\right )} a b c d^{2} e^{3} + b^{3} d^{3} e^{3} + 6 \, a b c d^{3} e^{3} + 18 \, {\left (e x + d\right )} a b^{2} d e^{4} + 18 \, {\left (e x + d\right )} a^{2} c d e^{4} - 3 \, a b^{2} d^{2} e^{4} - 3 \, a^{2} c d^{2} e^{4} - 9 \, {\left (e x + d\right )} a^{2} b e^{5} + 3 \, a^{2} b d e^{5} - a^{3} e^{6}\right )}}{3 \, {\left (e x + d\right )}^{\frac {3}{2}} e^{7}} + \frac {2 \, {\left (35 \, {\left (e x + d\right )}^{\frac {9}{2}} c^{3} e^{56} - 270 \, {\left (e x + d\right )}^{\frac {7}{2}} c^{3} d e^{56} + 945 \, {\left (e x + d\right )}^{\frac {5}{2}} c^{3} d^{2} e^{56} - 2100 \, {\left (e x + d\right )}^{\frac {3}{2}} c^{3} d^{3} e^{56} + 4725 \, \sqrt {e x + d} c^{3} d^{4} e^{56} + 135 \, {\left (e x + d\right )}^{\frac {7}{2}} b c^{2} e^{57} - 945 \, {\left (e x + d\right )}^{\frac {5}{2}} b c^{2} d e^{57} + 3150 \, {\left (e x + d\right )}^{\frac {3}{2}} b c^{2} d^{2} e^{57} - 9450 \, \sqrt {e x + d} b c^{2} d^{3} e^{57} + 189 \, {\left (e x + d\right )}^{\frac {5}{2}} b^{2} c e^{58} + 189 \, {\left (e x + d\right )}^{\frac {5}{2}} a c^{2} e^{58} - 1260 \, {\left (e x + d\right )}^{\frac {3}{2}} b^{2} c d e^{58} - 1260 \, {\left (e x + d\right )}^{\frac {3}{2}} a c^{2} d e^{58} + 5670 \, \sqrt {e x + d} b^{2} c d^{2} e^{58} + 5670 \, \sqrt {e x + d} a c^{2} d^{2} e^{58} + 105 \, {\left (e x + d\right )}^{\frac {3}{2}} b^{3} e^{59} + 630 \, {\left (e x + d\right )}^{\frac {3}{2}} a b c e^{59} - 945 \, \sqrt {e x + d} b^{3} d e^{59} - 5670 \, \sqrt {e x + d} a b c d e^{59} + 945 \, \sqrt {e x + d} a b^{2} e^{60} + 945 \, \sqrt {e x + d} a^{2} c e^{60}\right )}}{315 \, e^{63}} \] Input:
integrate((c*x^2+b*x+a)^3/(e*x+d)^(5/2),x, algorithm="giac")
Output:
2/3*(18*(e*x + d)*c^3*d^5 - c^3*d^6 - 45*(e*x + d)*b*c^2*d^4*e + 3*b*c^2*d ^5*e + 36*(e*x + d)*b^2*c*d^3*e^2 + 36*(e*x + d)*a*c^2*d^3*e^2 - 3*b^2*c*d ^4*e^2 - 3*a*c^2*d^4*e^2 - 9*(e*x + d)*b^3*d^2*e^3 - 54*(e*x + d)*a*b*c*d^ 2*e^3 + b^3*d^3*e^3 + 6*a*b*c*d^3*e^3 + 18*(e*x + d)*a*b^2*d*e^4 + 18*(e*x + d)*a^2*c*d*e^4 - 3*a*b^2*d^2*e^4 - 3*a^2*c*d^2*e^4 - 9*(e*x + d)*a^2*b* e^5 + 3*a^2*b*d*e^5 - a^3*e^6)/((e*x + d)^(3/2)*e^7) + 2/315*(35*(e*x + d) ^(9/2)*c^3*e^56 - 270*(e*x + d)^(7/2)*c^3*d*e^56 + 945*(e*x + d)^(5/2)*c^3 *d^2*e^56 - 2100*(e*x + d)^(3/2)*c^3*d^3*e^56 + 4725*sqrt(e*x + d)*c^3*d^4 *e^56 + 135*(e*x + d)^(7/2)*b*c^2*e^57 - 945*(e*x + d)^(5/2)*b*c^2*d*e^57 + 3150*(e*x + d)^(3/2)*b*c^2*d^2*e^57 - 9450*sqrt(e*x + d)*b*c^2*d^3*e^57 + 189*(e*x + d)^(5/2)*b^2*c*e^58 + 189*(e*x + d)^(5/2)*a*c^2*e^58 - 1260*( e*x + d)^(3/2)*b^2*c*d*e^58 - 1260*(e*x + d)^(3/2)*a*c^2*d*e^58 + 5670*sqr t(e*x + d)*b^2*c*d^2*e^58 + 5670*sqrt(e*x + d)*a*c^2*d^2*e^58 + 105*(e*x + d)^(3/2)*b^3*e^59 + 630*(e*x + d)^(3/2)*a*b*c*e^59 - 945*sqrt(e*x + d)*b^ 3*d*e^59 - 5670*sqrt(e*x + d)*a*b*c*d*e^59 + 945*sqrt(e*x + d)*a*b^2*e^60 + 945*sqrt(e*x + d)*a^2*c*e^60)/e^63
Time = 0.05 (sec) , antiderivative size = 448, normalized size of antiderivative = 1.59 \[ \int \frac {\left (a+b x+c x^2\right )^3}{(d+e x)^{5/2}} \, dx=\frac {\sqrt {d+e\,x}\,\left (6\,a^2\,c\,e^4+6\,a\,b^2\,e^4-36\,a\,b\,c\,d\,e^3+36\,a\,c^2\,d^2\,e^2-6\,b^3\,d\,e^3+36\,b^2\,c\,d^2\,e^2-60\,b\,c^2\,d^3\,e+30\,c^3\,d^4\right )}{e^7}+\frac {2\,c^3\,{\left (d+e\,x\right )}^{9/2}}{9\,e^7}-\frac {\left (12\,c^3\,d-6\,b\,c^2\,e\right )\,{\left (d+e\,x\right )}^{7/2}}{7\,e^7}-\frac {\frac {2\,a^3\,e^6}{3}-\left (d+e\,x\right )\,\left (-6\,a^2\,b\,e^5+12\,a^2\,c\,d\,e^4+12\,a\,b^2\,d\,e^4-36\,a\,b\,c\,d^2\,e^3+24\,a\,c^2\,d^3\,e^2-6\,b^3\,d^2\,e^3+24\,b^2\,c\,d^3\,e^2-30\,b\,c^2\,d^4\,e+12\,c^3\,d^5\right )+\frac {2\,c^3\,d^6}{3}-\frac {2\,b^3\,d^3\,e^3}{3}+2\,a\,b^2\,d^2\,e^4+2\,a\,c^2\,d^4\,e^2+2\,a^2\,c\,d^2\,e^4+2\,b^2\,c\,d^4\,e^2-2\,a^2\,b\,d\,e^5-2\,b\,c^2\,d^5\,e-4\,a\,b\,c\,d^3\,e^3}{e^7\,{\left (d+e\,x\right )}^{3/2}}+\frac {{\left (d+e\,x\right )}^{5/2}\,\left (6\,b^2\,c\,e^2-30\,b\,c^2\,d\,e+30\,c^3\,d^2+6\,a\,c^2\,e^2\right )}{5\,e^7}+\frac {2\,\left (b\,e-2\,c\,d\right )\,{\left (d+e\,x\right )}^{3/2}\,\left (b^2\,e^2-10\,b\,c\,d\,e+10\,c^2\,d^2+6\,a\,c\,e^2\right )}{3\,e^7} \] Input:
int((a + b*x + c*x^2)^3/(d + e*x)^(5/2),x)
Output:
((d + e*x)^(1/2)*(30*c^3*d^4 + 6*a*b^2*e^4 + 6*a^2*c*e^4 - 6*b^3*d*e^3 + 3 6*a*c^2*d^2*e^2 + 36*b^2*c*d^2*e^2 - 60*b*c^2*d^3*e - 36*a*b*c*d*e^3))/e^7 + (2*c^3*(d + e*x)^(9/2))/(9*e^7) - ((12*c^3*d - 6*b*c^2*e)*(d + e*x)^(7/ 2))/(7*e^7) - ((2*a^3*e^6)/3 - (d + e*x)*(12*c^3*d^5 - 6*a^2*b*e^5 - 6*b^3 *d^2*e^3 + 24*a*c^2*d^3*e^2 + 24*b^2*c*d^3*e^2 + 12*a*b^2*d*e^4 + 12*a^2*c *d*e^4 - 30*b*c^2*d^4*e - 36*a*b*c*d^2*e^3) + (2*c^3*d^6)/3 - (2*b^3*d^3*e ^3)/3 + 2*a*b^2*d^2*e^4 + 2*a*c^2*d^4*e^2 + 2*a^2*c*d^2*e^4 + 2*b^2*c*d^4* e^2 - 2*a^2*b*d*e^5 - 2*b*c^2*d^5*e - 4*a*b*c*d^3*e^3)/(e^7*(d + e*x)^(3/2 )) + ((d + e*x)^(5/2)*(30*c^3*d^2 + 6*a*c^2*e^2 + 6*b^2*c*e^2 - 30*b*c^2*d *e))/(5*e^7) + (2*(b*e - 2*c*d)*(d + e*x)^(3/2)*(b^2*e^2 + 10*c^2*d^2 + 6* a*c*e^2 - 10*b*c*d*e))/(3*e^7)
Time = 0.19 (sec) , antiderivative size = 502, normalized size of antiderivative = 1.78 \[ \int \frac {\left (a+b x+c x^2\right )^3}{(d+e x)^{5/2}} \, dx=\frac {24 a \,b^{2} d \,e^{5} x +4 a b c \,e^{6} x^{3}-\frac {16}{5} a \,c^{2} d \,e^{5} x^{3}+\frac {384}{5} b^{2} c \,d^{3} e^{3} x +\frac {96}{5} b^{2} c \,d^{2} e^{4} x^{2}-\frac {16}{5} b^{2} c d \,e^{5} x^{3}-\frac {768}{7} b \,c^{2} d^{4} e^{2} x -\frac {192}{7} b \,c^{2} d^{3} e^{3} x^{2}+\frac {32}{7} b \,c^{2} d^{2} e^{4} x^{3}-\frac {12}{7} b \,c^{2} d \,e^{5} x^{4}+24 a^{2} c d \,e^{5} x +\frac {384}{5} a \,c^{2} d^{3} e^{3} x +\frac {96}{5} a \,c^{2} d^{2} e^{4} x^{2}+\frac {2}{3} b^{3} e^{6} x^{3}+\frac {2}{9} c^{3} e^{6} x^{6}+\frac {1024}{21} c^{3} d^{5} e x +\frac {256}{21} c^{3} d^{4} e^{2} x^{2}-\frac {128}{63} c^{3} d^{3} e^{3} x^{3}-96 a b c \,d^{2} e^{4} x -24 a b c d \,e^{5} x^{2}-4 a^{2} b d \,e^{5}+16 a^{2} c \,d^{2} e^{4}+16 a \,b^{2} d^{2} e^{4}+\frac {256}{5} a \,c^{2} d^{4} e^{2}-64 a b c \,d^{3} e^{3}-\frac {32}{3} b^{3} d^{3} e^{3}-6 a^{2} b \,e^{6} x +6 a^{2} c \,e^{6} x^{2}+6 a \,b^{2} e^{6} x^{2}+\frac {6}{5} a \,c^{2} e^{6} x^{4}-16 b^{3} d^{2} e^{4} x -4 b^{3} d \,e^{5} x^{2}+\frac {6}{5} b^{2} c \,e^{6} x^{4}+\frac {6}{7} b \,c^{2} e^{6} x^{5}+\frac {16}{21} c^{3} d^{2} e^{4} x^{4}-\frac {8}{21} c^{3} d \,e^{5} x^{5}-\frac {2}{3} a^{3} e^{6}+\frac {2048}{63} c^{3} d^{6}+\frac {256}{5} b^{2} c \,d^{4} e^{2}-\frac {512}{7} b \,c^{2} d^{5} e}{\sqrt {e x +d}\, e^{7} \left (e x +d \right )} \] Input:
int((c*x^2+b*x+a)^3/(e*x+d)^(5/2),x)
Output:
(2*( - 105*a**3*e**6 - 630*a**2*b*d*e**5 - 945*a**2*b*e**6*x + 2520*a**2*c *d**2*e**4 + 3780*a**2*c*d*e**5*x + 945*a**2*c*e**6*x**2 + 2520*a*b**2*d** 2*e**4 + 3780*a*b**2*d*e**5*x + 945*a*b**2*e**6*x**2 - 10080*a*b*c*d**3*e* *3 - 15120*a*b*c*d**2*e**4*x - 3780*a*b*c*d*e**5*x**2 + 630*a*b*c*e**6*x** 3 + 8064*a*c**2*d**4*e**2 + 12096*a*c**2*d**3*e**3*x + 3024*a*c**2*d**2*e* *4*x**2 - 504*a*c**2*d*e**5*x**3 + 189*a*c**2*e**6*x**4 - 1680*b**3*d**3*e **3 - 2520*b**3*d**2*e**4*x - 630*b**3*d*e**5*x**2 + 105*b**3*e**6*x**3 + 8064*b**2*c*d**4*e**2 + 12096*b**2*c*d**3*e**3*x + 3024*b**2*c*d**2*e**4*x **2 - 504*b**2*c*d*e**5*x**3 + 189*b**2*c*e**6*x**4 - 11520*b*c**2*d**5*e - 17280*b*c**2*d**4*e**2*x - 4320*b*c**2*d**3*e**3*x**2 + 720*b*c**2*d**2* e**4*x**3 - 270*b*c**2*d*e**5*x**4 + 135*b*c**2*e**6*x**5 + 5120*c**3*d**6 + 7680*c**3*d**5*e*x + 1920*c**3*d**4*e**2*x**2 - 320*c**3*d**3*e**3*x**3 + 120*c**3*d**2*e**4*x**4 - 60*c**3*d*e**5*x**5 + 35*c**3*e**6*x**6))/(31 5*sqrt(d + e*x)*e**7*(d + e*x))