Integrand size = 22, antiderivative size = 280 \[ \int \frac {\left (a+b x+c x^2\right )^3}{(d+e x)^{3/2}} \, dx=-\frac {2 \left (c d^2-b d e+a e^2\right )^3}{e^7 \sqrt {d+e x}}-\frac {6 (2 c d-b e) \left (c d^2-b d e+a e^2\right )^2 \sqrt {d+e x}}{e^7}+\frac {2 \left (c d^2-b d e+a e^2\right ) \left (5 c^2 d^2+b^2 e^2-c e (5 b d-a e)\right ) (d+e x)^{3/2}}{e^7}-\frac {2 (2 c d-b e) \left (10 c^2 d^2+b^2 e^2-2 c e (5 b d-3 a e)\right ) (d+e x)^{5/2}}{5 e^7}+\frac {6 c \left (5 c^2 d^2+b^2 e^2-c e (5 b d-a e)\right ) (d+e x)^{7/2}}{7 e^7}-\frac {2 c^2 (2 c d-b e) (d+e x)^{9/2}}{3 e^7}+\frac {2 c^3 (d+e x)^{11/2}}{11 e^7} \] Output:
-2*(a*e^2-b*d*e+c*d^2)^3/e^7/(e*x+d)^(1/2)-6*(-b*e+2*c*d)*(a*e^2-b*d*e+c*d ^2)^2*(e*x+d)^(1/2)/e^7+2*(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)^(3/2)/e^7-2/5*(-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)^(5/2)/e^7+6/7*c*(5*c^2*d^2+b^2*e^2-c*e*(-a*e+5*b*d))*( e*x+d)^(7/2)/e^7-2/3*c^2*(-b*e+2*c*d)*(e*x+d)^(9/2)/e^7+2/11*c^3*(e*x+d)^( 11/2)/e^7
Time = 0.36 (sec) , antiderivative size = 394, normalized size of antiderivative = 1.41 \[ \int \frac {\left (a+b x+c x^2\right )^3}{(d+e x)^{3/2}} \, dx=\frac {-10 c^3 \left (1024 d^6+512 d^5 e x-128 d^4 e^2 x^2+64 d^3 e^3 x^3-40 d^2 e^4 x^4+28 d e^5 x^5-21 e^6 x^6\right )+462 e^3 \left (-5 a^3 e^3+15 a^2 b e^2 (2 d+e x)+5 a b^2 e \left (-8 d^2-4 d e x+e^2 x^2\right )+b^3 \left (16 d^3+8 d^2 e x-2 d e^2 x^2+e^3 x^3\right )\right )-66 c e^2 \left (35 a^2 e^2 \left (8 d^2+4 d e x-e^2 x^2\right )-42 a b e \left (16 d^3+8 d^2 e x-2 d e^2 x^2+e^3 x^3\right )+3 b^2 \left (128 d^4+64 d^3 e x-16 d^2 e^2 x^2+8 d e^3 x^3-5 e^4 x^4\right )\right )+22 c^2 e \left (9 a e \left (-128 d^4-64 d^3 e x+16 d^2 e^2 x^2-8 d e^3 x^3+5 e^4 x^4\right )+5 b \left (256 d^5+128 d^4 e x-32 d^3 e^2 x^2+16 d^2 e^3 x^3-10 d e^4 x^4+7 e^5 x^5\right )\right )}{1155 e^7 \sqrt {d+e x}} \] Input:
Integrate[(a + b*x + c*x^2)^3/(d + e*x)^(3/2),x]
Output:
(-10*c^3*(1024*d^6 + 512*d^5*e*x - 128*d^4*e^2*x^2 + 64*d^3*e^3*x^3 - 40*d ^2*e^4*x^4 + 28*d*e^5*x^5 - 21*e^6*x^6) + 462*e^3*(-5*a^3*e^3 + 15*a^2*b*e ^2*(2*d + e*x) + 5*a*b^2*e*(-8*d^2 - 4*d*e*x + e^2*x^2) + b^3*(16*d^3 + 8* d^2*e*x - 2*d*e^2*x^2 + e^3*x^3)) - 66*c*e^2*(35*a^2*e^2*(8*d^2 + 4*d*e*x - e^2*x^2) - 42*a*b*e*(16*d^3 + 8*d^2*e*x - 2*d*e^2*x^2 + e^3*x^3) + 3*b^2 *(128*d^4 + 64*d^3*e*x - 16*d^2*e^2*x^2 + 8*d*e^3*x^3 - 5*e^4*x^4)) + 22*c ^2*e*(9*a*e*(-128*d^4 - 64*d^3*e*x + 16*d^2*e^2*x^2 - 8*d*e^3*x^3 + 5*e^4* x^4) + 5*b*(256*d^5 + 128*d^4*e*x - 32*d^3*e^2*x^2 + 16*d^2*e^3*x^3 - 10*d *e^4*x^4 + 7*e^5*x^5)))/(1155*e^7*Sqrt[d + e*x])
Time = 0.42 (sec) , antiderivative size = 280, 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.
| \(\displaystyle \int \frac {\left (a+b x+c x^2\right )^3}{(d+e x)^{3/2}} \, dx\) |
| \(\Big \downarrow \) 1140 |
| \(\displaystyle \int \left (\frac {3 c (d+e x)^{5/2} \left (-c e (5 b d-a e)+b^2 e^2+5 c^2 d^2\right )}{e^6}+\frac {(d+e x)^{3/2} (2 c d-b e) \left (2 c e (5 b d-3 a e)-b^2 e^2-10 c^2 d^2\right )}{e^6}+\frac {3 \sqrt {d+e x} \left (a e^2-b d e+c d^2\right ) \left (a c e^2+b^2 e^2-5 b c d e+5 c^2 d^2\right )}{e^6}+\frac {3 (b e-2 c d) \left (a e^2-b d e+c d^2\right )^2}{e^6 \sqrt {d+e x}}+\frac {\left (a e^2-b d e+c d^2\right )^3}{e^6 (d+e x)^{3/2}}-\frac {3 c^2 (d+e x)^{7/2} (2 c d-b e)}{e^6}+\frac {c^3 (d+e x)^{9/2}}{e^6}\right )dx\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {6 c (d+e x)^{7/2} \left (-c e (5 b d-a e)+b^2 e^2+5 c^2 d^2\right )}{7 e^7}-\frac {2 (d+e x)^{5/2} (2 c d-b e) \left (-2 c e (5 b d-3 a e)+b^2 e^2+10 c^2 d^2\right )}{5 e^7}+\frac {2 (d+e x)^{3/2} \left (a e^2-b d e+c d^2\right ) \left (-c e (5 b d-a e)+b^2 e^2+5 c^2 d^2\right )}{e^7}-\frac {6 \sqrt {d+e x} (2 c d-b e) \left (a e^2-b d e+c d^2\right )^2}{e^7}-\frac {2 \left (a e^2-b d e+c d^2\right )^3}{e^7 \sqrt {d+e x}}-\frac {2 c^2 (d+e x)^{9/2} (2 c d-b e)}{3 e^7}+\frac {2 c^3 (d+e x)^{11/2}}{11 e^7}\) |
Input:
Int[(a + b*x + c*x^2)^3/(d + e*x)^(3/2),x]
Output:
(-2*(c*d^2 - b*d*e + a*e^2)^3)/(e^7*Sqrt[d + e*x]) - (6*(2*c*d - b*e)*(c*d ^2 - b*d*e + a*e^2)^2*Sqrt[d + e*x])/e^7 + (2*(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))*(d + e*x)^(3/2))/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)^(5/2))/(5*e^7 ) + (6*c*(5*c^2*d^2 + b^2*e^2 - c*e*(5*b*d - a*e))*(d + e*x)^(7/2))/(7*e^7 ) - (2*c^2*(2*c*d - b*e)*(d + e*x)^(9/2))/(3*e^7) + (2*c^3*(d + e*x)^(11/2 ))/(11*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 = 1.04 (sec) , antiderivative size = 324, normalized size of antiderivative = 1.16
| method | result | size |
| pseudoelliptic | \(-\frac {2 \left (\left (-\frac {c^{3} x^{6}}{11}-\frac {3 x^{4} \left (\frac {7 b x}{9}+a \right ) c^{2}}{7}+\left (-\frac {6}{5} a b \,x^{3}-\frac {3}{7} b^{2} x^{4}-a^{2} x^{2}\right ) c -a \,b^{2} x^{2}-3 a^{2} b x +a^{3}-\frac {b^{3} x^{3}}{5}\right ) e^{6}-6 d \left (-\frac {2 c^{3} x^{5}}{99}+\left (-\frac {5}{63} b \,x^{4}-\frac {4}{35} a \,x^{3}\right ) c^{2}+\left (-\frac {4}{35} b^{2} x^{3}-\frac {2}{5} a b \,x^{2}-\frac {2}{3} a^{2} x \right ) c +b \left (-\frac {1}{15} b^{2} x^{2}-\frac {2}{3} a b x +a^{2}\right )\right ) e^{5}+8 d^{2} \left (-\frac {5 c^{3} x^{4}}{231}-\frac {6 x^{2} \left (\frac {5 b x}{9}+a \right ) c^{2}}{35}+\left (-\frac {6}{35} b^{2} x^{2}-\frac {6}{5} a b x +a^{2}\right ) c +b^{2} \left (-\frac {b x}{5}+a \right )\right ) e^{4}-\frac {96 d^{3} \left (-\frac {10 c^{3} x^{3}}{693}-\frac {2 \left (\frac {5 b x}{18}+a \right ) x \,c^{2}}{7}+b \left (-\frac {2 b x}{7}+a \right ) c +\frac {b^{3}}{6}\right ) e^{3}}{5}+\frac {384 d^{4} \left (-\frac {5 c^{2} x^{2}}{99}+\left (-\frac {5 b x}{9}+a \right ) c +b^{2}\right ) c \,e^{2}}{35}-\frac {256 d^{5} \left (-\frac {2 c x}{11}+b \right ) c^{2} e}{21}+\frac {1024 d^{6} c^{3}}{231}\right )}{\sqrt {e x +d}\, e^{7}}\) | \(324\) |
| risch | \(\frac {2 \left (105 e^{5} x^{5} c^{3}+385 e^{5} x^{4} c^{2} b -245 x^{4} d \,e^{4} c^{3}+495 e^{5} a \,c^{2} x^{3}+495 e^{5} x^{3} b^{2} c -935 x^{3} d \,e^{4} c^{2} b +445 d^{2} e^{3} c^{3} x^{3}+1386 e^{5} a b c \,x^{2}-1287 d \,e^{4} a \,c^{2} x^{2}+231 x^{2} e^{5} b^{3}-1287 x^{2} d \,e^{4} b^{2} c +1815 d^{2} e^{3} b \,c^{2} x^{2}-765 d^{3} e^{2} c^{3} x^{2}+1155 e^{5} a^{2} c x +1155 e^{5} x a \,b^{2}-4158 d \,e^{4} a b c x +2871 d^{2} e^{3} a \,c^{2} x -693 x d \,e^{4} b^{3}+2871 d^{2} e^{3} b^{2} c x -3575 d^{3} e^{2} c^{2} b x +1405 c^{3} d^{4} e x +3465 a^{2} b \,e^{5}-5775 d \,e^{4} a^{2} c -5775 a \,b^{2} d \,e^{4}+15246 a b c \,d^{2} e^{3}-9207 d^{3} e^{2} a \,c^{2}+2541 b^{3} d^{2} e^{3}-9207 b^{2} c \,d^{3} e^{2}+10615 b \,c^{2} d^{4} e -3965 d^{5} c^{3}\right ) \sqrt {e x +d}}{1155 e^{7}}-\frac {2 \left (e^{6} a^{3}-3 a^{2} b d \,e^{5}+3 d^{2} e^{4} a^{2} c +3 a \,b^{2} d^{2} e^{4}-6 a b c \,d^{3} e^{3}+3 d^{4} e^{2} a \,c^{2}-b^{3} d^{3} e^{3}+3 b^{2} c \,d^{4} e^{2}-3 b \,c^{2} d^{5} e +d^{6} c^{3}\right )}{e^{7} \sqrt {e x +d}}\) | \(477\) |
| gosper | \(-\frac {2 \left (-105 c^{3} e^{6} x^{6}-385 x^{5} b \,c^{2} e^{6}+140 c^{3} d \,e^{5} x^{5}-495 x^{4} a \,c^{2} e^{6}-495 x^{4} b^{2} c \,e^{6}+550 x^{4} b \,c^{2} d \,e^{5}-200 c^{3} d^{2} e^{4} x^{4}-1386 x^{3} a b c \,e^{6}+792 x^{3} a \,c^{2} d \,e^{5}-231 x^{3} b^{3} e^{6}+792 x^{3} b^{2} c d \,e^{5}-880 x^{3} b \,c^{2} d^{2} e^{4}+320 c^{3} d^{3} e^{3} x^{3}-1155 x^{2} a^{2} c \,e^{6}-1155 x^{2} a \,b^{2} e^{6}+2772 x^{2} a b c d \,e^{5}-1584 x^{2} a \,c^{2} d^{2} e^{4}+462 x^{2} b^{3} d \,e^{5}-1584 x^{2} b^{2} c \,d^{2} e^{4}+1760 x^{2} b \,c^{2} d^{3} e^{3}-640 c^{3} d^{4} e^{2} x^{2}-3465 x \,a^{2} b \,e^{6}+4620 x \,a^{2} c d \,e^{5}+4620 x a \,b^{2} d \,e^{5}-11088 x a b c \,d^{2} e^{4}+6336 x a \,c^{2} d^{3} e^{3}-1848 x \,b^{3} d^{2} e^{4}+6336 x \,b^{2} c \,d^{3} e^{3}-7040 x b \,c^{2} d^{4} e^{2}+2560 c^{3} d^{5} e x +1155 e^{6} a^{3}-6930 a^{2} b d \,e^{5}+9240 d^{2} e^{4} a^{2} c +9240 a \,b^{2} d^{2} e^{4}-22176 a b c \,d^{3} e^{3}+12672 d^{4} e^{2} a \,c^{2}-3696 b^{3} d^{3} e^{3}+12672 b^{2} c \,d^{4} e^{2}-14080 b \,c^{2} d^{5} e +5120 d^{6} c^{3}\right )}{1155 \sqrt {e x +d}\, e^{7}}\) | \(495\) |
| trager | \(-\frac {2 \left (-105 c^{3} e^{6} x^{6}-385 x^{5} b \,c^{2} e^{6}+140 c^{3} d \,e^{5} x^{5}-495 x^{4} a \,c^{2} e^{6}-495 x^{4} b^{2} c \,e^{6}+550 x^{4} b \,c^{2} d \,e^{5}-200 c^{3} d^{2} e^{4} x^{4}-1386 x^{3} a b c \,e^{6}+792 x^{3} a \,c^{2} d \,e^{5}-231 x^{3} b^{3} e^{6}+792 x^{3} b^{2} c d \,e^{5}-880 x^{3} b \,c^{2} d^{2} e^{4}+320 c^{3} d^{3} e^{3} x^{3}-1155 x^{2} a^{2} c \,e^{6}-1155 x^{2} a \,b^{2} e^{6}+2772 x^{2} a b c d \,e^{5}-1584 x^{2} a \,c^{2} d^{2} e^{4}+462 x^{2} b^{3} d \,e^{5}-1584 x^{2} b^{2} c \,d^{2} e^{4}+1760 x^{2} b \,c^{2} d^{3} e^{3}-640 c^{3} d^{4} e^{2} x^{2}-3465 x \,a^{2} b \,e^{6}+4620 x \,a^{2} c d \,e^{5}+4620 x a \,b^{2} d \,e^{5}-11088 x a b c \,d^{2} e^{4}+6336 x a \,c^{2} d^{3} e^{3}-1848 x \,b^{3} d^{2} e^{4}+6336 x \,b^{2} c \,d^{3} e^{3}-7040 x b \,c^{2} d^{4} e^{2}+2560 c^{3} d^{5} e x +1155 e^{6} a^{3}-6930 a^{2} b d \,e^{5}+9240 d^{2} e^{4} a^{2} c +9240 a \,b^{2} d^{2} e^{4}-22176 a b c \,d^{3} e^{3}+12672 d^{4} e^{2} a \,c^{2}-3696 b^{3} d^{3} e^{3}+12672 b^{2} c \,d^{4} e^{2}-14080 b \,c^{2} d^{5} e +5120 d^{6} c^{3}\right )}{1155 \sqrt {e x +d}\, e^{7}}\) | \(495\) |
| orering | \(-\frac {2 \left (-105 c^{3} e^{6} x^{6}-385 x^{5} b \,c^{2} e^{6}+140 c^{3} d \,e^{5} x^{5}-495 x^{4} a \,c^{2} e^{6}-495 x^{4} b^{2} c \,e^{6}+550 x^{4} b \,c^{2} d \,e^{5}-200 c^{3} d^{2} e^{4} x^{4}-1386 x^{3} a b c \,e^{6}+792 x^{3} a \,c^{2} d \,e^{5}-231 x^{3} b^{3} e^{6}+792 x^{3} b^{2} c d \,e^{5}-880 x^{3} b \,c^{2} d^{2} e^{4}+320 c^{3} d^{3} e^{3} x^{3}-1155 x^{2} a^{2} c \,e^{6}-1155 x^{2} a \,b^{2} e^{6}+2772 x^{2} a b c d \,e^{5}-1584 x^{2} a \,c^{2} d^{2} e^{4}+462 x^{2} b^{3} d \,e^{5}-1584 x^{2} b^{2} c \,d^{2} e^{4}+1760 x^{2} b \,c^{2} d^{3} e^{3}-640 c^{3} d^{4} e^{2} x^{2}-3465 x \,a^{2} b \,e^{6}+4620 x \,a^{2} c d \,e^{5}+4620 x a \,b^{2} d \,e^{5}-11088 x a b c \,d^{2} e^{4}+6336 x a \,c^{2} d^{3} e^{3}-1848 x \,b^{3} d^{2} e^{4}+6336 x \,b^{2} c \,d^{3} e^{3}-7040 x b \,c^{2} d^{4} e^{2}+2560 c^{3} d^{5} e x +1155 e^{6} a^{3}-6930 a^{2} b d \,e^{5}+9240 d^{2} e^{4} a^{2} c +9240 a \,b^{2} d^{2} e^{4}-22176 a b c \,d^{3} e^{3}+12672 d^{4} e^{2} a \,c^{2}-3696 b^{3} d^{3} e^{3}+12672 b^{2} c \,d^{4} e^{2}-14080 b \,c^{2} d^{5} e +5120 d^{6} c^{3}\right )}{1155 \sqrt {e x +d}\, e^{7}}\) | \(495\) |
| derivativedivides | \(\frac {\frac {12 a b c \,e^{3} \left (e x +d \right )^{\frac {5}{2}}}{5}-\frac {24 a \,c^{2} d \,e^{2} \left (e x +d \right )^{\frac {5}{2}}}{5}-24 b^{2} c \,d^{3} e^{2} \sqrt {e x +d}-\frac {24 b^{2} c d \,e^{2} \left (e x +d \right )^{\frac {5}{2}}}{5}-24 a \,c^{2} d^{3} e^{2} \sqrt {e x +d}+12 b \,c^{2} d^{2} e \left (e x +d \right )^{\frac {5}{2}}+\frac {6 b^{2} c \,e^{2} \left (e x +d \right )^{\frac {7}{2}}}{7}+30 b \,c^{2} d^{4} e \sqrt {e x +d}-20 b \,c^{2} d^{3} e \left (e x +d \right )^{\frac {3}{2}}+12 a \,c^{2} d^{2} e^{2} \left (e x +d \right )^{\frac {3}{2}}+12 b^{2} c \,d^{2} e^{2} \left (e x +d \right )^{\frac {3}{2}}-\frac {30 b \,c^{2} d e \left (e x +d \right )^{\frac {7}{2}}}{7}-12 a^{2} c d \,e^{4} \sqrt {e x +d}-12 a \,b^{2} d \,e^{4} \sqrt {e x +d}+2 a^{2} c \,e^{4} \left (e x +d \right )^{\frac {3}{2}}+36 a b c \,d^{2} e^{3} \sqrt {e x +d}-12 a b c d \,e^{3} \left (e x +d \right )^{\frac {3}{2}}+2 a \,b^{2} e^{4} \left (e x +d \right )^{\frac {3}{2}}+\frac {2 b \,c^{2} e \left (e x +d \right )^{\frac {9}{2}}}{3}-2 b^{3} d \,e^{3} \left (e x +d \right )^{\frac {3}{2}}+\frac {6 a \,c^{2} e^{2} \left (e x +d \right )^{\frac {7}{2}}}{7}+6 a^{2} b \,e^{5} \sqrt {e x +d}+6 b^{3} d^{2} e^{3} \sqrt {e x +d}+\frac {2 c^{3} \left (e x +d \right )^{\frac {11}{2}}}{11}-8 c^{3} d^{3} \left (e x +d \right )^{\frac {5}{2}}+10 c^{3} d^{4} \left (e x +d \right )^{\frac {3}{2}}-\frac {2 \left (e^{6} a^{3}-3 a^{2} b d \,e^{5}+3 d^{2} e^{4} a^{2} c +3 a \,b^{2} d^{2} e^{4}-6 a b c \,d^{3} e^{3}+3 d^{4} e^{2} a \,c^{2}-b^{3} d^{3} e^{3}+3 b^{2} c \,d^{4} e^{2}-3 b \,c^{2} d^{5} e +d^{6} c^{3}\right )}{\sqrt {e x +d}}+\frac {2 b^{3} e^{3} \left (e x +d \right )^{\frac {5}{2}}}{5}-\frac {4 c^{3} d \left (e x +d \right )^{\frac {9}{2}}}{3}-12 c^{3} d^{5} \sqrt {e x +d}+\frac {30 c^{3} d^{2} \left (e x +d \right )^{\frac {7}{2}}}{7}}{e^{7}}\) | \(606\) |
| default | \(\frac {\frac {12 a b c \,e^{3} \left (e x +d \right )^{\frac {5}{2}}}{5}-\frac {24 a \,c^{2} d \,e^{2} \left (e x +d \right )^{\frac {5}{2}}}{5}-24 b^{2} c \,d^{3} e^{2} \sqrt {e x +d}-\frac {24 b^{2} c d \,e^{2} \left (e x +d \right )^{\frac {5}{2}}}{5}-24 a \,c^{2} d^{3} e^{2} \sqrt {e x +d}+12 b \,c^{2} d^{2} e \left (e x +d \right )^{\frac {5}{2}}+\frac {6 b^{2} c \,e^{2} \left (e x +d \right )^{\frac {7}{2}}}{7}+30 b \,c^{2} d^{4} e \sqrt {e x +d}-20 b \,c^{2} d^{3} e \left (e x +d \right )^{\frac {3}{2}}+12 a \,c^{2} d^{2} e^{2} \left (e x +d \right )^{\frac {3}{2}}+12 b^{2} c \,d^{2} e^{2} \left (e x +d \right )^{\frac {3}{2}}-\frac {30 b \,c^{2} d e \left (e x +d \right )^{\frac {7}{2}}}{7}-12 a^{2} c d \,e^{4} \sqrt {e x +d}-12 a \,b^{2} d \,e^{4} \sqrt {e x +d}+2 a^{2} c \,e^{4} \left (e x +d \right )^{\frac {3}{2}}+36 a b c \,d^{2} e^{3} \sqrt {e x +d}-12 a b c d \,e^{3} \left (e x +d \right )^{\frac {3}{2}}+2 a \,b^{2} e^{4} \left (e x +d \right )^{\frac {3}{2}}+\frac {2 b \,c^{2} e \left (e x +d \right )^{\frac {9}{2}}}{3}-2 b^{3} d \,e^{3} \left (e x +d \right )^{\frac {3}{2}}+\frac {6 a \,c^{2} e^{2} \left (e x +d \right )^{\frac {7}{2}}}{7}+6 a^{2} b \,e^{5} \sqrt {e x +d}+6 b^{3} d^{2} e^{3} \sqrt {e x +d}+\frac {2 c^{3} \left (e x +d \right )^{\frac {11}{2}}}{11}-8 c^{3} d^{3} \left (e x +d \right )^{\frac {5}{2}}+10 c^{3} d^{4} \left (e x +d \right )^{\frac {3}{2}}-\frac {2 \left (e^{6} a^{3}-3 a^{2} b d \,e^{5}+3 d^{2} e^{4} a^{2} c +3 a \,b^{2} d^{2} e^{4}-6 a b c \,d^{3} e^{3}+3 d^{4} e^{2} a \,c^{2}-b^{3} d^{3} e^{3}+3 b^{2} c \,d^{4} e^{2}-3 b \,c^{2} d^{5} e +d^{6} c^{3}\right )}{\sqrt {e x +d}}+\frac {2 b^{3} e^{3} \left (e x +d \right )^{\frac {5}{2}}}{5}-\frac {4 c^{3} d \left (e x +d \right )^{\frac {9}{2}}}{3}-12 c^{3} d^{5} \sqrt {e x +d}+\frac {30 c^{3} d^{2} \left (e x +d \right )^{\frac {7}{2}}}{7}}{e^{7}}\) | \(606\) |
Input:
int((c*x^2+b*x+a)^3/(e*x+d)^(3/2),x,method=_RETURNVERBOSE)
Output:
-2*((-1/11*c^3*x^6-3/7*x^4*(7/9*b*x+a)*c^2+(-6/5*a*b*x^3-3/7*b^2*x^4-a^2*x ^2)*c-a*b^2*x^2-3*a^2*b*x+a^3-1/5*b^3*x^3)*e^6-6*d*(-2/99*c^3*x^5+(-5/63*b *x^4-4/35*a*x^3)*c^2+(-4/35*b^2*x^3-2/5*a*b*x^2-2/3*a^2*x)*c+b*(-1/15*b^2* x^2-2/3*a*b*x+a^2))*e^5+8*d^2*(-5/231*c^3*x^4-6/35*x^2*(5/9*b*x+a)*c^2+(-6 /35*b^2*x^2-6/5*a*b*x+a^2)*c+b^2*(-1/5*b*x+a))*e^4-96/5*d^3*(-10/693*c^3*x ^3-2/7*(5/18*b*x+a)*x*c^2+b*(-2/7*b*x+a)*c+1/6*b^3)*e^3+384/35*d^4*(-5/99* c^2*x^2+(-5/9*b*x+a)*c+b^2)*c*e^2-256/21*d^5*(-2/11*c*x+b)*c^2*e+1024/231* d^6*c^3)/(e*x+d)^(1/2)/e^7
Time = 0.08 (sec) , antiderivative size = 417, normalized size of antiderivative = 1.49 \[ \int \frac {\left (a+b x+c x^2\right )^3}{(d+e x)^{3/2}} \, dx=\frac {2 \, {\left (105 \, c^{3} e^{6} x^{6} - 5120 \, c^{3} d^{6} + 14080 \, b c^{2} d^{5} e + 6930 \, a^{2} b d e^{5} - 1155 \, a^{3} e^{6} - 12672 \, {\left (b^{2} c + a c^{2}\right )} d^{4} e^{2} + 3696 \, {\left (b^{3} + 6 \, a b c\right )} d^{3} e^{3} - 9240 \, {\left (a b^{2} + a^{2} c\right )} d^{2} e^{4} - 35 \, {\left (4 \, c^{3} d e^{5} - 11 \, b c^{2} e^{6}\right )} x^{5} + 5 \, {\left (40 \, c^{3} d^{2} e^{4} - 110 \, b c^{2} d e^{5} + 99 \, {\left (b^{2} c + a c^{2}\right )} e^{6}\right )} x^{4} - {\left (320 \, c^{3} d^{3} e^{3} - 880 \, b c^{2} d^{2} e^{4} + 792 \, {\left (b^{2} c + a c^{2}\right )} d e^{5} - 231 \, {\left (b^{3} + 6 \, a b c\right )} e^{6}\right )} x^{3} + {\left (640 \, c^{3} d^{4} e^{2} - 1760 \, b c^{2} d^{3} e^{3} + 1584 \, {\left (b^{2} c + a c^{2}\right )} d^{2} e^{4} - 462 \, {\left (b^{3} + 6 \, a b c\right )} d e^{5} + 1155 \, {\left (a b^{2} + a^{2} c\right )} e^{6}\right )} x^{2} - {\left (2560 \, c^{3} d^{5} e - 7040 \, b c^{2} d^{4} e^{2} - 3465 \, a^{2} b e^{6} + 6336 \, {\left (b^{2} c + a c^{2}\right )} d^{3} e^{3} - 1848 \, {\left (b^{3} + 6 \, a b c\right )} d^{2} e^{4} + 4620 \, {\left (a b^{2} + a^{2} c\right )} d e^{5}\right )} x\right )} \sqrt {e x + d}}{1155 \, {\left (e^{8} x + d e^{7}\right )}} \] Input:
integrate((c*x^2+b*x+a)^3/(e*x+d)^(3/2),x, algorithm="fricas")
Output:
2/1155*(105*c^3*e^6*x^6 - 5120*c^3*d^6 + 14080*b*c^2*d^5*e + 6930*a^2*b*d* e^5 - 1155*a^3*e^6 - 12672*(b^2*c + a*c^2)*d^4*e^2 + 3696*(b^3 + 6*a*b*c)* d^3*e^3 - 9240*(a*b^2 + a^2*c)*d^2*e^4 - 35*(4*c^3*d*e^5 - 11*b*c^2*e^6)*x ^5 + 5*(40*c^3*d^2*e^4 - 110*b*c^2*d*e^5 + 99*(b^2*c + a*c^2)*e^6)*x^4 - ( 320*c^3*d^3*e^3 - 880*b*c^2*d^2*e^4 + 792*(b^2*c + a*c^2)*d*e^5 - 231*(b^3 + 6*a*b*c)*e^6)*x^3 + (640*c^3*d^4*e^2 - 1760*b*c^2*d^3*e^3 + 1584*(b^2*c + a*c^2)*d^2*e^4 - 462*(b^3 + 6*a*b*c)*d*e^5 + 1155*(a*b^2 + a^2*c)*e^6)* x^2 - (2560*c^3*d^5*e - 7040*b*c^2*d^4*e^2 - 3465*a^2*b*e^6 + 6336*(b^2*c + a*c^2)*d^3*e^3 - 1848*(b^3 + 6*a*b*c)*d^2*e^4 + 4620*(a*b^2 + a^2*c)*d*e ^5)*x)*sqrt(e*x + d)/(e^8*x + d*e^7)
Time = 25.51 (sec) , antiderivative size = 530, normalized size of antiderivative = 1.89 \[ \int \frac {\left (a+b x+c x^2\right )^3}{(d+e x)^{3/2}} \, dx=\begin {cases} \frac {2 \left (\frac {c^{3} \left (d + e x\right )^{\frac {11}{2}}}{11 e^{6}} + \frac {\left (d + e x\right )^{\frac {9}{2}} \cdot \left (3 b c^{2} e - 6 c^{3} d\right )}{9 e^{6}} + \frac {\left (d + e x\right )^{\frac {7}{2}} \cdot \left (3 a c^{2} e^{2} + 3 b^{2} c e^{2} - 15 b c^{2} d e + 15 c^{3} d^{2}\right )}{7 e^{6}} + \frac {\left (d + e x\right )^{\frac {5}{2}} \cdot \left (6 a b c e^{3} - 12 a c^{2} d e^{2} + b^{3} e^{3} - 12 b^{2} c d e^{2} + 30 b c^{2} d^{2} e - 20 c^{3} d^{3}\right )}{5 e^{6}} + \frac {\left (d + e x\right )^{\frac {3}{2}} \cdot \left (3 a^{2} c e^{4} + 3 a b^{2} e^{4} - 18 a b c d e^{3} + 18 a c^{2} d^{2} e^{2} - 3 b^{3} d e^{3} + 18 b^{2} c d^{2} e^{2} - 30 b c^{2} d^{3} e + 15 c^{3} d^{4}\right )}{3 e^{6}} + \frac {\sqrt {d + e x} \left (3 a^{2} b e^{5} - 6 a^{2} c d e^{4} - 6 a b^{2} d e^{4} + 18 a b c d^{2} e^{3} - 12 a c^{2} d^{3} e^{2} + 3 b^{3} d^{2} e^{3} - 12 b^{2} c d^{3} e^{2} + 15 b c^{2} d^{4} e - 6 c^{3} d^{5}\right )}{e^{6}} - \frac {\left (a e^{2} - b d e + c d^{2}\right )^{3}}{e^{6} \sqrt {d + e x}}\right )}{e} & \text {for}\: e \neq 0 \\\frac {a^{3} x + \frac {3 a^{2} b x^{2}}{2} + \frac {b c^{2} x^{6}}{2} + \frac {c^{3} x^{7}}{7} + \frac {x^{5} \cdot \left (3 a c^{2} + 3 b^{2} c\right )}{5} + \frac {x^{4} \cdot \left (6 a b c + b^{3}\right )}{4} + \frac {x^{3} \cdot \left (3 a^{2} c + 3 a b^{2}\right )}{3}}{d^{\frac {3}{2}}} & \text {otherwise} \end {cases} \] Input:
integrate((c*x**2+b*x+a)**3/(e*x+d)**(3/2),x)
Output:
Piecewise((2*(c**3*(d + e*x)**(11/2)/(11*e**6) + (d + e*x)**(9/2)*(3*b*c** 2*e - 6*c**3*d)/(9*e**6) + (d + e*x)**(7/2)*(3*a*c**2*e**2 + 3*b**2*c*e**2 - 15*b*c**2*d*e + 15*c**3*d**2)/(7*e**6) + (d + e*x)**(5/2)*(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)/(5*e**6) + (d + e*x)**(3/2)*(3*a**2*c*e**4 + 3*a*b**2*e**4 - 1 8*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)/(3*e**6) + sqrt(d + e*x)*(3*a**2*b*e** 5 - 6*a**2*c*d*e**4 - 6*a*b**2*d*e**4 + 18*a*b*c*d**2*e**3 - 12*a*c**2*d** 3*e**2 + 3*b**3*d**2*e**3 - 12*b**2*c*d**3*e**2 + 15*b*c**2*d**4*e - 6*c** 3*d**5)/e**6 - (a*e**2 - b*d*e + c*d**2)**3/(e**6*sqrt(d + e*x)))/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**(3/2), True))
Time = 0.04 (sec) , antiderivative size = 415, normalized size of antiderivative = 1.48 \[ \int \frac {\left (a+b x+c x^2\right )^3}{(d+e x)^{3/2}} \, dx=\frac {2 \, {\left (\frac {105 \, {\left (e x + d\right )}^{\frac {11}{2}} c^{3} - 385 \, {\left (2 \, c^{3} d - b c^{2} e\right )} {\left (e x + d\right )}^{\frac {9}{2}} + 495 \, {\left (5 \, c^{3} d^{2} - 5 \, b c^{2} d e + {\left (b^{2} c + a c^{2}\right )} e^{2}\right )} {\left (e x + d\right )}^{\frac {7}{2}} - 231 \, {\left (20 \, c^{3} d^{3} - 30 \, b c^{2} d^{2} e + 12 \, {\left (b^{2} c + a c^{2}\right )} d e^{2} - {\left (b^{3} + 6 \, a b c\right )} e^{3}\right )} {\left (e x + d\right )}^{\frac {5}{2}} + 1155 \, {\left (5 \, c^{3} d^{4} - 10 \, b c^{2} d^{3} e + 6 \, {\left (b^{2} c + a c^{2}\right )} d^{2} e^{2} - {\left (b^{3} + 6 \, a b c\right )} d e^{3} + {\left (a b^{2} + a^{2} c\right )} e^{4}\right )} {\left (e x + d\right )}^{\frac {3}{2}} - 3465 \, {\left (2 \, c^{3} d^{5} - 5 \, b c^{2} d^{4} e - a^{2} b e^{5} + 4 \, {\left (b^{2} c + a c^{2}\right )} d^{3} e^{2} - {\left (b^{3} + 6 \, a b c\right )} d^{2} e^{3} + 2 \, {\left (a b^{2} + a^{2} c\right )} d e^{4}\right )} \sqrt {e x + d}}{e^{6}} - \frac {1155 \, {\left (c^{3} d^{6} - 3 \, b c^{2} d^{5} e - 3 \, a^{2} b d e^{5} + a^{3} e^{6} + 3 \, {\left (b^{2} c + a c^{2}\right )} d^{4} e^{2} - {\left (b^{3} + 6 \, a b c\right )} d^{3} e^{3} + 3 \, {\left (a b^{2} + a^{2} c\right )} d^{2} e^{4}\right )}}{\sqrt {e x + d} e^{6}}\right )}}{1155 \, e} \] Input:
integrate((c*x^2+b*x+a)^3/(e*x+d)^(3/2),x, algorithm="maxima")
Output:
2/1155*((105*(e*x + d)^(11/2)*c^3 - 385*(2*c^3*d - b*c^2*e)*(e*x + d)^(9/2 ) + 495*(5*c^3*d^2 - 5*b*c^2*d*e + (b^2*c + a*c^2)*e^2)*(e*x + d)^(7/2) - 231*(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)^(5/2) + 1155*(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)*(e*x + d)^(3/ 2) - 3465*(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)*sqrt(e*x + d))/e^6 - 1155*(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)/(sqrt( e*x + d)*e^6))/e
Leaf count of result is larger than twice the leaf count of optimal. 639 vs. \(2 (258) = 516\).
Time = 0.37 (sec) , antiderivative size = 639, normalized size of antiderivative = 2.28 \[ \int \frac {\left (a+b x+c x^2\right )^3}{(d+e x)^{3/2}} \, dx =\text {Too large to display} \] Input:
integrate((c*x^2+b*x+a)^3/(e*x+d)^(3/2),x, algorithm="giac")
Output:
-2*(c^3*d^6 - 3*b*c^2*d^5*e + 3*b^2*c*d^4*e^2 + 3*a*c^2*d^4*e^2 - b^3*d^3* e^3 - 6*a*b*c*d^3*e^3 + 3*a*b^2*d^2*e^4 + 3*a^2*c*d^2*e^4 - 3*a^2*b*d*e^5 + a^3*e^6)/(sqrt(e*x + d)*e^7) + 2/1155*(105*(e*x + d)^(11/2)*c^3*e^70 - 7 70*(e*x + d)^(9/2)*c^3*d*e^70 + 2475*(e*x + d)^(7/2)*c^3*d^2*e^70 - 4620*( e*x + d)^(5/2)*c^3*d^3*e^70 + 5775*(e*x + d)^(3/2)*c^3*d^4*e^70 - 6930*sqr t(e*x + d)*c^3*d^5*e^70 + 385*(e*x + d)^(9/2)*b*c^2*e^71 - 2475*(e*x + d)^ (7/2)*b*c^2*d*e^71 + 6930*(e*x + d)^(5/2)*b*c^2*d^2*e^71 - 11550*(e*x + d) ^(3/2)*b*c^2*d^3*e^71 + 17325*sqrt(e*x + d)*b*c^2*d^4*e^71 + 495*(e*x + d) ^(7/2)*b^2*c*e^72 + 495*(e*x + d)^(7/2)*a*c^2*e^72 - 2772*(e*x + d)^(5/2)* b^2*c*d*e^72 - 2772*(e*x + d)^(5/2)*a*c^2*d*e^72 + 6930*(e*x + d)^(3/2)*b^ 2*c*d^2*e^72 + 6930*(e*x + d)^(3/2)*a*c^2*d^2*e^72 - 13860*sqrt(e*x + d)*b ^2*c*d^3*e^72 - 13860*sqrt(e*x + d)*a*c^2*d^3*e^72 + 231*(e*x + d)^(5/2)*b ^3*e^73 + 1386*(e*x + d)^(5/2)*a*b*c*e^73 - 1155*(e*x + d)^(3/2)*b^3*d*e^7 3 - 6930*(e*x + d)^(3/2)*a*b*c*d*e^73 + 3465*sqrt(e*x + d)*b^3*d^2*e^73 + 20790*sqrt(e*x + d)*a*b*c*d^2*e^73 + 1155*(e*x + d)^(3/2)*a*b^2*e^74 + 115 5*(e*x + d)^(3/2)*a^2*c*e^74 - 6930*sqrt(e*x + d)*a*b^2*d*e^74 - 6930*sqrt (e*x + d)*a^2*c*d*e^74 + 3465*sqrt(e*x + d)*a^2*b*e^75)/e^77
Time = 5.61 (sec) , antiderivative size = 386, normalized size of antiderivative = 1.38 \[ \int \frac {\left (a+b x+c x^2\right )^3}{(d+e x)^{3/2}} \, dx=\frac {{\left (d+e\,x\right )}^{3/2}\,\left (6\,a^2\,c\,e^4+6\,a\,b^2\,e^4-36\,a\,b\,c\,d\,e^3+36\,a\,c^2\,d^2\,e^2-6\,b^3\,d\,e^3+36\,b^2\,c\,d^2\,e^2-60\,b\,c^2\,d^3\,e+30\,c^3\,d^4\right )}{3\,e^7}+\frac {2\,c^3\,{\left (d+e\,x\right )}^{11/2}}{11\,e^7}-\frac {\left (12\,c^3\,d-6\,b\,c^2\,e\right )\,{\left (d+e\,x\right )}^{9/2}}{9\,e^7}+\frac {{\left (d+e\,x\right )}^{7/2}\,\left (6\,b^2\,c\,e^2-30\,b\,c^2\,d\,e+30\,c^3\,d^2+6\,a\,c^2\,e^2\right )}{7\,e^7}-\frac {2\,a^3\,e^6-6\,a^2\,b\,d\,e^5+6\,a^2\,c\,d^2\,e^4+6\,a\,b^2\,d^2\,e^4-12\,a\,b\,c\,d^3\,e^3+6\,a\,c^2\,d^4\,e^2-2\,b^3\,d^3\,e^3+6\,b^2\,c\,d^4\,e^2-6\,b\,c^2\,d^5\,e+2\,c^3\,d^6}{e^7\,\sqrt {d+e\,x}}+\frac {2\,\left (b\,e-2\,c\,d\right )\,{\left (d+e\,x\right )}^{5/2}\,\left (b^2\,e^2-10\,b\,c\,d\,e+10\,c^2\,d^2+6\,a\,c\,e^2\right )}{5\,e^7}+\frac {6\,\left (b\,e-2\,c\,d\right )\,\sqrt {d+e\,x}\,{\left (c\,d^2-b\,d\,e+a\,e^2\right )}^2}{e^7} \] Input:
int((a + b*x + c*x^2)^3/(d + e*x)^(3/2),x)
Output:
((d + e*x)^(3/2)*(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))/(3* e^7) + (2*c^3*(d + e*x)^(11/2))/(11*e^7) - ((12*c^3*d - 6*b*c^2*e)*(d + e* x)^(9/2))/(9*e^7) + ((d + e*x)^(7/2)*(30*c^3*d^2 + 6*a*c^2*e^2 + 6*b^2*c*e ^2 - 30*b*c^2*d*e))/(7*e^7) - (2*a^3*e^6 + 2*c^3*d^6 - 2*b^3*d^3*e^3 + 6*a *b^2*d^2*e^4 + 6*a*c^2*d^4*e^2 + 6*a^2*c*d^2*e^4 + 6*b^2*c*d^4*e^2 - 6*a^2 *b*d*e^5 - 6*b*c^2*d^5*e - 12*a*b*c*d^3*e^3)/(e^7*(d + e*x)^(1/2)) + (2*(b *e - 2*c*d)*(d + e*x)^(5/2)*(b^2*e^2 + 10*c^2*d^2 + 6*a*c*e^2 - 10*b*c*d*e ))/(5*e^7) + (6*(b*e - 2*c*d)*(d + e*x)^(1/2)*(a*e^2 + c*d^2 - b*d*e)^2)/e ^7
Time = 0.20 (sec) , antiderivative size = 495, normalized size of antiderivative = 1.77 \[ \int \frac {\left (a+b x+c x^2\right )^3}{(d+e x)^{3/2}} \, dx=\frac {-8 a \,b^{2} d \,e^{5} x +\frac {12}{5} a b c \,e^{6} x^{3}-\frac {48}{35} a \,c^{2} d \,e^{5} x^{3}-\frac {384}{35} b^{2} c \,d^{3} e^{3} x +\frac {96}{35} b^{2} c \,d^{2} e^{4} x^{2}-\frac {48}{35} b^{2} c d \,e^{5} x^{3}+\frac {256}{21} b \,c^{2} d^{4} e^{2} x -\frac {64}{21} b \,c^{2} d^{3} e^{3} x^{2}+\frac {32}{21} b \,c^{2} d^{2} e^{4} x^{3}-\frac {20}{21} b \,c^{2} d \,e^{5} x^{4}-8 a^{2} c d \,e^{5} x -\frac {384}{35} a \,c^{2} d^{3} e^{3} x +\frac {96}{35} a \,c^{2} d^{2} e^{4} x^{2}+\frac {2}{5} b^{3} e^{6} x^{3}+\frac {2}{11} c^{3} e^{6} x^{6}-\frac {1024}{231} c^{3} d^{5} e x +\frac {256}{231} c^{3} d^{4} e^{2} x^{2}-\frac {128}{231} c^{3} d^{3} e^{3} x^{3}+\frac {96}{5} a b c \,d^{2} e^{4} x -\frac {24}{5} a b c d \,e^{5} x^{2}+12 a^{2} b d \,e^{5}-16 a^{2} c \,d^{2} e^{4}-16 a \,b^{2} d^{2} e^{4}-\frac {768}{35} a \,c^{2} d^{4} e^{2}+\frac {192}{5} a b c \,d^{3} e^{3}+\frac {32}{5} b^{3} d^{3} e^{3}+6 a^{2} b \,e^{6} x +2 a^{2} c \,e^{6} x^{2}+2 a \,b^{2} e^{6} x^{2}+\frac {6}{7} a \,c^{2} e^{6} x^{4}+\frac {16}{5} b^{3} d^{2} e^{4} x -\frac {4}{5} b^{3} d \,e^{5} x^{2}+\frac {6}{7} b^{2} c \,e^{6} x^{4}+\frac {2}{3} b \,c^{2} e^{6} x^{5}+\frac {80}{231} c^{3} d^{2} e^{4} x^{4}-\frac {8}{33} c^{3} d \,e^{5} x^{5}-2 a^{3} e^{6}-\frac {2048}{231} c^{3} d^{6}-\frac {768}{35} b^{2} c \,d^{4} e^{2}+\frac {512}{21} b \,c^{2} d^{5} e}{\sqrt {e x +d}\, e^{7}} \] Input:
int((c*x^2+b*x+a)^3/(e*x+d)^(3/2),x)
Output:
(2*( - 1155*a**3*e**6 + 6930*a**2*b*d*e**5 + 3465*a**2*b*e**6*x - 9240*a** 2*c*d**2*e**4 - 4620*a**2*c*d*e**5*x + 1155*a**2*c*e**6*x**2 - 9240*a*b**2 *d**2*e**4 - 4620*a*b**2*d*e**5*x + 1155*a*b**2*e**6*x**2 + 22176*a*b*c*d* *3*e**3 + 11088*a*b*c*d**2*e**4*x - 2772*a*b*c*d*e**5*x**2 + 1386*a*b*c*e* *6*x**3 - 12672*a*c**2*d**4*e**2 - 6336*a*c**2*d**3*e**3*x + 1584*a*c**2*d **2*e**4*x**2 - 792*a*c**2*d*e**5*x**3 + 495*a*c**2*e**6*x**4 + 3696*b**3* d**3*e**3 + 1848*b**3*d**2*e**4*x - 462*b**3*d*e**5*x**2 + 231*b**3*e**6*x **3 - 12672*b**2*c*d**4*e**2 - 6336*b**2*c*d**3*e**3*x + 1584*b**2*c*d**2* e**4*x**2 - 792*b**2*c*d*e**5*x**3 + 495*b**2*c*e**6*x**4 + 14080*b*c**2*d **5*e + 7040*b*c**2*d**4*e**2*x - 1760*b*c**2*d**3*e**3*x**2 + 880*b*c**2* d**2*e**4*x**3 - 550*b*c**2*d*e**5*x**4 + 385*b*c**2*e**6*x**5 - 5120*c**3 *d**6 - 2560*c**3*d**5*e*x + 640*c**3*d**4*e**2*x**2 - 320*c**3*d**3*e**3* x**3 + 200*c**3*d**2*e**4*x**4 - 140*c**3*d*e**5*x**5 + 105*c**3*e**6*x**6 ))/(1155*sqrt(d + e*x)*e**7)