Integrand size = 28, antiderivative size = 246 \[ \int \frac {(b+2 c x) \left (a+b x+c x^2\right )^2}{(d+e x)^{5/2}} \, dx=\frac {2 (2 c d-b e) \left (c d^2-b d e+a e^2\right )^2}{3 e^6 (d+e x)^{3/2}}-\frac {4 \left (c d^2-b d e+a e^2\right ) \left (5 c^2 d^2+b^2 e^2-c e (5 b d-a e)\right )}{e^6 \sqrt {d+e x}}-\frac {2 (2 c d-b e) \left (10 c^2 d^2+b^2 e^2-2 c e (5 b d-3 a e)\right ) \sqrt {d+e x}}{e^6}+\frac {8 c \left (5 c^2 d^2+b^2 e^2-c e (5 b d-a e)\right ) (d+e x)^{3/2}}{3 e^6}-\frac {2 c^2 (2 c d-b e) (d+e x)^{5/2}}{e^6}+\frac {4 c^3 (d+e x)^{7/2}}{7 e^6} \] Output:
2/3*(-b*e+2*c*d)*(a*e^2-b*d*e+c*d^2)^2/e^6/(e*x+d)^(3/2)-4*(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^6/(e*x+d)^(1/2)-2*(-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)^(1/2)/e^6+8/3*c*(5*c^2* d^2+b^2*e^2-c*e*(-a*e+5*b*d))*(e*x+d)^(3/2)/e^6-2*c^2*(-b*e+2*c*d)*(e*x+d) ^(5/2)/e^6+4/7*c^3*(e*x+d)^(7/2)/e^6
Time = 0.30 (sec) , antiderivative size = 289, normalized size of antiderivative = 1.17 \[ \int \frac {(b+2 c x) \left (a+b x+c x^2\right )^2}{(d+e x)^{5/2}} \, dx=-\frac {2 \left (2 c^3 \left (256 d^5+384 d^4 e x+96 d^3 e^2 x^2-16 d^2 e^3 x^3+6 d e^4 x^4-3 e^5 x^5\right )+7 b e^3 \left (a^2 e^2+2 a b e (2 d+3 e x)-b^2 \left (8 d^2+12 d e x+3 e^2 x^2\right )\right )+14 c e^2 \left (a^2 e^2 (2 d+3 e x)-3 a b e \left (8 d^2+12 d e x+3 e^2 x^2\right )+2 b^2 \left (16 d^3+24 d^2 e x+6 d e^2 x^2-e^3 x^3\right )\right )-7 c^2 e \left (4 a e \left (-16 d^3-24 d^2 e x-6 d e^2 x^2+e^3 x^3\right )+b \left (128 d^4+192 d^3 e x+48 d^2 e^2 x^2-8 d e^3 x^3+3 e^4 x^4\right )\right )\right )}{21 e^6 (d+e x)^{3/2}} \] Input:
Integrate[((b + 2*c*x)*(a + b*x + c*x^2)^2)/(d + e*x)^(5/2),x]
Output:
(-2*(2*c^3*(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*e^3*(a^2*e^2 + 2*a*b*e*(2*d + 3*e*x) - b^2*(8*d ^2 + 12*d*e*x + 3*e^2*x^2)) + 14*c*e^2*(a^2*e^2*(2*d + 3*e*x) - 3*a*b*e*(8 *d^2 + 12*d*e*x + 3*e^2*x^2) + 2*b^2*(16*d^3 + 24*d^2*e*x + 6*d*e^2*x^2 - e^3*x^3)) - 7*c^2*e*(4*a*e*(-16*d^3 - 24*d^2*e*x - 6*d*e^2*x^2 + e^3*x^3) + b*(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))))/ (21*e^6*(d + e*x)^(3/2))
Time = 0.68 (sec) , antiderivative size = 246, 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.
| \(\displaystyle \int \frac {(b+2 c x) \left (a+b x+c x^2\right )^2}{(d+e x)^{5/2}} \, dx\) |
| \(\Big \downarrow \) 1195 |
| \(\displaystyle \int \left (\frac {4 c \sqrt {d+e x} \left (-c e (5 b d-a e)+b^2 e^2+5 c^2 d^2\right )}{e^5}+\frac {(2 c d-b e) \left (2 c e (5 b d-3 a e)-b^2 e^2-10 c^2 d^2\right )}{e^5 \sqrt {d+e x}}+\frac {2 \left (a e^2-b d e+c d^2\right ) \left (a c e^2+b^2 e^2-5 b c d e+5 c^2 d^2\right )}{e^5 (d+e x)^{3/2}}+\frac {(b e-2 c d) \left (a e^2-b d e+c d^2\right )^2}{e^5 (d+e x)^{5/2}}-\frac {5 c^2 (d+e x)^{3/2} (2 c d-b e)}{e^5}+\frac {2 c^3 (d+e x)^{5/2}}{e^5}\right )dx\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {8 c (d+e x)^{3/2} \left (-c e (5 b d-a e)+b^2 e^2+5 c^2 d^2\right )}{3 e^6}-\frac {2 \sqrt {d+e x} (2 c d-b e) \left (-2 c e (5 b d-3 a e)+b^2 e^2+10 c^2 d^2\right )}{e^6}-\frac {4 \left (a e^2-b d e+c d^2\right ) \left (-c e (5 b d-a e)+b^2 e^2+5 c^2 d^2\right )}{e^6 \sqrt {d+e x}}+\frac {2 (2 c d-b e) \left (a e^2-b d e+c d^2\right )^2}{3 e^6 (d+e x)^{3/2}}-\frac {2 c^2 (d+e x)^{5/2} (2 c d-b e)}{e^6}+\frac {4 c^3 (d+e x)^{7/2}}{7 e^6}\) |
Input:
Int[((b + 2*c*x)*(a + b*x + c*x^2)^2)/(d + e*x)^(5/2),x]
Output:
(2*(2*c*d - b*e)*(c*d^2 - b*d*e + a*e^2)^2)/(3*e^6*(d + e*x)^(3/2)) - (4*( 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)))/(e^6*Sqr t[d + e*x]) - (2*(2*c*d - b*e)*(10*c^2*d^2 + b^2*e^2 - 2*c*e*(5*b*d - 3*a* e))*Sqrt[d + e*x])/e^6 + (8*c*(5*c^2*d^2 + b^2*e^2 - c*e*(5*b*d - a*e))*(d + e*x)^(3/2))/(3*e^6) - (2*c^2*(2*c*d - b*e)*(d + e*x)^(5/2))/e^6 + (4*c^ 3*(d + e*x)^(7/2))/(7*e^6)
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 = 1.51 (sec) , antiderivative size = 247, normalized size of antiderivative = 1.00
| method | result | size |
| pseudoelliptic | \(\frac {\left (12 c^{3} x^{5}+\left (42 b \,x^{4}+56 a \,x^{3}\right ) c^{2}+\left (56 b^{2} x^{3}+252 a b \,x^{2}-84 a^{2} x \right ) c +42 b^{3} x^{2}-84 x a \,b^{2}-14 a^{2} b \right ) e^{5}-56 d \left (\frac {3 c^{3} x^{4}}{7}+\left (2 b \,x^{3}+6 a \,x^{2}\right ) c^{2}+\left (6 b^{2} x^{2}-18 a b x +a^{2}\right ) c +b^{2} \left (-3 b x +a \right )\right ) e^{4}+672 d^{2} \left (\frac {2 c^{3} x^{3}}{21}+\left (b \,x^{2}-2 a x \right ) c^{2}+b \left (-2 b x +a \right ) c +\frac {b^{3}}{6}\right ) e^{3}-896 d^{3} \left (\frac {3 c^{2} x^{2}}{7}+\left (-3 b x +a \right ) c +b^{2}\right ) c \,e^{2}+1792 d^{4} \left (-\frac {6 c x}{7}+b \right ) c^{2} e -1024 d^{5} c^{3}}{21 \left (e x +d \right )^{\frac {3}{2}} e^{6}}\) | \(247\) |
| risch | \(\frac {2 \left (6 c^{3} x^{3} e^{3}+21 b \,c^{2} e^{3} x^{2}-24 c^{3} d \,e^{2} x^{2}+28 a \,c^{2} e^{3} x +28 x \,b^{2} c \,e^{3}-98 b \,c^{2} d \,e^{2} x +74 d^{2} e \,c^{3} x +126 a b c \,e^{3}-224 d \,e^{2} a \,c^{2}+21 b^{3} e^{3}-224 d \,e^{2} b^{2} c +511 d^{2} e b \,c^{2}-316 d^{3} c^{3}\right ) \sqrt {e x +d}}{21 e^{6}}-\frac {2 \left (6 a c \,e^{3} x +6 e^{3} x \,b^{2}-30 d \,e^{2} c b x +30 d^{2} e \,c^{2} x +a \,e^{3} b +4 a d \,e^{2} c +5 d \,e^{2} b^{2}-27 d^{2} e b c +28 c^{2} d^{3}\right ) \left (a \,e^{2}-b d e +c \,d^{2}\right )}{3 e^{6} \left (e x +d \right )^{\frac {3}{2}}}\) | \(249\) |
| gosper | \(-\frac {2 \left (-6 e^{5} x^{5} c^{3}-21 e^{5} x^{4} c^{2} b +12 x^{4} d \,e^{4} c^{3}-28 e^{5} a \,c^{2} x^{3}-28 e^{5} x^{3} b^{2} c +56 x^{3} d \,e^{4} c^{2} b -32 d^{2} e^{3} c^{3} x^{3}-126 e^{5} a b c \,x^{2}+168 d \,e^{4} a \,c^{2} x^{2}-21 x^{2} e^{5} b^{3}+168 x^{2} d \,e^{4} b^{2} c -336 d^{2} e^{3} b \,c^{2} x^{2}+192 d^{3} e^{2} c^{3} x^{2}+42 e^{5} a^{2} c x +42 e^{5} x a \,b^{2}-504 d \,e^{4} a b c x +672 d^{2} e^{3} a \,c^{2} x -84 x d \,e^{4} b^{3}+672 d^{2} e^{3} b^{2} c x -1344 d^{3} e^{2} c^{2} b x +768 c^{3} d^{4} e x +7 a^{2} b \,e^{5}+28 d \,e^{4} a^{2} c +28 a \,b^{2} d \,e^{4}-336 a b c \,d^{2} e^{3}+448 d^{3} e^{2} a \,c^{2}-56 b^{3} d^{2} e^{3}+448 b^{2} c \,d^{3} e^{2}-896 b \,c^{2} d^{4} e +512 d^{5} c^{3}\right )}{21 \left (e x +d \right )^{\frac {3}{2}} e^{6}}\) | \(359\) |
| trager | \(-\frac {2 \left (-6 e^{5} x^{5} c^{3}-21 e^{5} x^{4} c^{2} b +12 x^{4} d \,e^{4} c^{3}-28 e^{5} a \,c^{2} x^{3}-28 e^{5} x^{3} b^{2} c +56 x^{3} d \,e^{4} c^{2} b -32 d^{2} e^{3} c^{3} x^{3}-126 e^{5} a b c \,x^{2}+168 d \,e^{4} a \,c^{2} x^{2}-21 x^{2} e^{5} b^{3}+168 x^{2} d \,e^{4} b^{2} c -336 d^{2} e^{3} b \,c^{2} x^{2}+192 d^{3} e^{2} c^{3} x^{2}+42 e^{5} a^{2} c x +42 e^{5} x a \,b^{2}-504 d \,e^{4} a b c x +672 d^{2} e^{3} a \,c^{2} x -84 x d \,e^{4} b^{3}+672 d^{2} e^{3} b^{2} c x -1344 d^{3} e^{2} c^{2} b x +768 c^{3} d^{4} e x +7 a^{2} b \,e^{5}+28 d \,e^{4} a^{2} c +28 a \,b^{2} d \,e^{4}-336 a b c \,d^{2} e^{3}+448 d^{3} e^{2} a \,c^{2}-56 b^{3} d^{2} e^{3}+448 b^{2} c \,d^{3} e^{2}-896 b \,c^{2} d^{4} e +512 d^{5} c^{3}\right )}{21 \left (e x +d \right )^{\frac {3}{2}} e^{6}}\) | \(359\) |
| orering | \(-\frac {2 \left (-6 e^{5} x^{5} c^{3}-21 e^{5} x^{4} c^{2} b +12 x^{4} d \,e^{4} c^{3}-28 e^{5} a \,c^{2} x^{3}-28 e^{5} x^{3} b^{2} c +56 x^{3} d \,e^{4} c^{2} b -32 d^{2} e^{3} c^{3} x^{3}-126 e^{5} a b c \,x^{2}+168 d \,e^{4} a \,c^{2} x^{2}-21 x^{2} e^{5} b^{3}+168 x^{2} d \,e^{4} b^{2} c -336 d^{2} e^{3} b \,c^{2} x^{2}+192 d^{3} e^{2} c^{3} x^{2}+42 e^{5} a^{2} c x +42 e^{5} x a \,b^{2}-504 d \,e^{4} a b c x +672 d^{2} e^{3} a \,c^{2} x -84 x d \,e^{4} b^{3}+672 d^{2} e^{3} b^{2} c x -1344 d^{3} e^{2} c^{2} b x +768 c^{3} d^{4} e x +7 a^{2} b \,e^{5}+28 d \,e^{4} a^{2} c +28 a \,b^{2} d \,e^{4}-336 a b c \,d^{2} e^{3}+448 d^{3} e^{2} a \,c^{2}-56 b^{3} d^{2} e^{3}+448 b^{2} c \,d^{3} e^{2}-896 b \,c^{2} d^{4} e +512 d^{5} c^{3}\right )}{21 \left (e x +d \right )^{\frac {3}{2}} e^{6}}\) | \(359\) |
| derivativedivides | \(\frac {\frac {4 c^{3} \left (e x +d \right )^{\frac {7}{2}}}{7}+2 b \,c^{2} e \left (e x +d \right )^{\frac {5}{2}}-4 c^{3} d \left (e x +d \right )^{\frac {5}{2}}+\frac {8 a \,c^{2} e^{2} \left (e x +d \right )^{\frac {3}{2}}}{3}+\frac {8 b^{2} c \,e^{2} \left (e x +d \right )^{\frac {3}{2}}}{3}-\frac {40 b \,c^{2} d e \left (e x +d \right )^{\frac {3}{2}}}{3}+\frac {40 c^{3} d^{2} \left (e x +d \right )^{\frac {3}{2}}}{3}+12 a b c \,e^{3} \sqrt {e x +d}-24 a \,c^{2} d \,e^{2} \sqrt {e x +d}+2 b^{3} e^{3} \sqrt {e x +d}-24 b^{2} c d \,e^{2} \sqrt {e x +d}+60 b \,c^{2} d^{2} e \sqrt {e x +d}-40 c^{3} d^{3} \sqrt {e x +d}-\frac {2 \left (2 e^{4} a^{2} c +2 a \,b^{2} e^{4}-12 a b c d \,e^{3}+12 d^{2} e^{2} a \,c^{2}-2 b^{3} d \,e^{3}+12 b^{2} c \,d^{2} e^{2}-20 b \,c^{2} d^{3} e +10 d^{4} c^{3}\right )}{\sqrt {e x +d}}-\frac {2 \left (a^{2} b \,e^{5}-2 d \,e^{4} a^{2} c -2 a \,b^{2} d \,e^{4}+6 a b c \,d^{2} e^{3}-4 d^{3} e^{2} a \,c^{2}+b^{3} d^{2} e^{3}-4 b^{2} c \,d^{3} e^{2}+5 b \,c^{2} d^{4} e -2 d^{5} c^{3}\right )}{3 \left (e x +d \right )^{\frac {3}{2}}}}{e^{6}}\) | \(391\) |
| default | \(\frac {\frac {4 c^{3} \left (e x +d \right )^{\frac {7}{2}}}{7}+2 b \,c^{2} e \left (e x +d \right )^{\frac {5}{2}}-4 c^{3} d \left (e x +d \right )^{\frac {5}{2}}+\frac {8 a \,c^{2} e^{2} \left (e x +d \right )^{\frac {3}{2}}}{3}+\frac {8 b^{2} c \,e^{2} \left (e x +d \right )^{\frac {3}{2}}}{3}-\frac {40 b \,c^{2} d e \left (e x +d \right )^{\frac {3}{2}}}{3}+\frac {40 c^{3} d^{2} \left (e x +d \right )^{\frac {3}{2}}}{3}+12 a b c \,e^{3} \sqrt {e x +d}-24 a \,c^{2} d \,e^{2} \sqrt {e x +d}+2 b^{3} e^{3} \sqrt {e x +d}-24 b^{2} c d \,e^{2} \sqrt {e x +d}+60 b \,c^{2} d^{2} e \sqrt {e x +d}-40 c^{3} d^{3} \sqrt {e x +d}-\frac {2 \left (2 e^{4} a^{2} c +2 a \,b^{2} e^{4}-12 a b c d \,e^{3}+12 d^{2} e^{2} a \,c^{2}-2 b^{3} d \,e^{3}+12 b^{2} c \,d^{2} e^{2}-20 b \,c^{2} d^{3} e +10 d^{4} c^{3}\right )}{\sqrt {e x +d}}-\frac {2 \left (a^{2} b \,e^{5}-2 d \,e^{4} a^{2} c -2 a \,b^{2} d \,e^{4}+6 a b c \,d^{2} e^{3}-4 d^{3} e^{2} a \,c^{2}+b^{3} d^{2} e^{3}-4 b^{2} c \,d^{3} e^{2}+5 b \,c^{2} d^{4} e -2 d^{5} c^{3}\right )}{3 \left (e x +d \right )^{\frac {3}{2}}}}{e^{6}}\) | \(391\) |
Input:
int((2*c*x+b)*(c*x^2+b*x+a)^2/(e*x+d)^(5/2),x,method=_RETURNVERBOSE)
Output:
1/21*((12*c^3*x^5+(42*b*x^4+56*a*x^3)*c^2+(56*b^2*x^3+252*a*b*x^2-84*a^2*x )*c+42*b^3*x^2-84*x*a*b^2-14*a^2*b)*e^5-56*d*(3/7*c^3*x^4+(2*b*x^3+6*a*x^2 )*c^2+(6*b^2*x^2-18*a*b*x+a^2)*c+b^2*(-3*b*x+a))*e^4+672*d^2*(2/21*c^3*x^3 +(b*x^2-2*a*x)*c^2+b*(-2*b*x+a)*c+1/6*b^3)*e^3-896*d^3*(3/7*c^2*x^2+(-3*b* x+a)*c+b^2)*c*e^2+1792*d^4*(-6/7*c*x+b)*c^2*e-1024*d^5*c^3)/(e*x+d)^(3/2)/ e^6
Time = 0.08 (sec) , antiderivative size = 327, normalized size of antiderivative = 1.33 \[ \int \frac {(b+2 c x) \left (a+b x+c x^2\right )^2}{(d+e x)^{5/2}} \, dx=\frac {2 \, {\left (6 \, c^{3} e^{5} x^{5} - 512 \, c^{3} d^{5} + 896 \, b c^{2} d^{4} e - 7 \, a^{2} b e^{5} - 448 \, {\left (b^{2} c + a c^{2}\right )} d^{3} e^{2} + 56 \, {\left (b^{3} + 6 \, a b c\right )} d^{2} e^{3} - 28 \, {\left (a b^{2} + a^{2} c\right )} d e^{4} - 3 \, {\left (4 \, c^{3} d e^{4} - 7 \, b c^{2} e^{5}\right )} x^{4} + 4 \, {\left (8 \, c^{3} d^{2} e^{3} - 14 \, b c^{2} d e^{4} + 7 \, {\left (b^{2} c + a c^{2}\right )} e^{5}\right )} x^{3} - 3 \, {\left (64 \, c^{3} d^{3} e^{2} - 112 \, b c^{2} d^{2} e^{3} + 56 \, {\left (b^{2} c + a c^{2}\right )} d e^{4} - 7 \, {\left (b^{3} + 6 \, a b c\right )} e^{5}\right )} x^{2} - 6 \, {\left (128 \, c^{3} d^{4} e - 224 \, b c^{2} d^{3} e^{2} + 112 \, {\left (b^{2} c + a c^{2}\right )} d^{2} e^{3} - 14 \, {\left (b^{3} + 6 \, a b c\right )} d e^{4} + 7 \, {\left (a b^{2} + a^{2} c\right )} e^{5}\right )} x\right )} \sqrt {e x + d}}{21 \, {\left (e^{8} x^{2} + 2 \, d e^{7} x + d^{2} e^{6}\right )}} \] Input:
integrate((2*c*x+b)*(c*x^2+b*x+a)^2/(e*x+d)^(5/2),x, algorithm="fricas")
Output:
2/21*(6*c^3*e^5*x^5 - 512*c^3*d^5 + 896*b*c^2*d^4*e - 7*a^2*b*e^5 - 448*(b ^2*c + a*c^2)*d^3*e^2 + 56*(b^3 + 6*a*b*c)*d^2*e^3 - 28*(a*b^2 + a^2*c)*d* e^4 - 3*(4*c^3*d*e^4 - 7*b*c^2*e^5)*x^4 + 4*(8*c^3*d^2*e^3 - 14*b*c^2*d*e^ 4 + 7*(b^2*c + a*c^2)*e^5)*x^3 - 3*(64*c^3*d^3*e^2 - 112*b*c^2*d^2*e^3 + 5 6*(b^2*c + a*c^2)*d*e^4 - 7*(b^3 + 6*a*b*c)*e^5)*x^2 - 6*(128*c^3*d^4*e - 224*b*c^2*d^3*e^2 + 112*(b^2*c + a*c^2)*d^2*e^3 - 14*(b^3 + 6*a*b*c)*d*e^4 + 7*(a*b^2 + a^2*c)*e^5)*x)*sqrt(e*x + d)/(e^8*x^2 + 2*d*e^7*x + d^2*e^6)
Time = 15.26 (sec) , antiderivative size = 298, normalized size of antiderivative = 1.21 \[ \int \frac {(b+2 c x) \left (a+b x+c x^2\right )^2}{(d+e x)^{5/2}} \, dx=\begin {cases} \frac {2 \cdot \left (\frac {2 c^{3} \left (d + e x\right )^{\frac {7}{2}}}{7 e^{5}} + \frac {\left (d + e x\right )^{\frac {5}{2}} \cdot \left (5 b c^{2} e - 10 c^{3} d\right )}{5 e^{5}} + \frac {\left (d + e x\right )^{\frac {3}{2}} \cdot \left (4 a c^{2} e^{2} + 4 b^{2} c e^{2} - 20 b c^{2} d e + 20 c^{3} d^{2}\right )}{3 e^{5}} + \frac {\sqrt {d + e x} \left (6 a b c e^{3} - 12 a c^{2} d e^{2} + b^{3} e^{3} - 12 b^{2} c d e^{2} + 30 b c^{2} d^{2} e - 20 c^{3} d^{3}\right )}{e^{5}} - \frac {2 \left (a e^{2} - b d e + c d^{2}\right ) \left (a c e^{2} + b^{2} e^{2} - 5 b c d e + 5 c^{2} d^{2}\right )}{e^{5} \sqrt {d + e x}} - \frac {\left (b e - 2 c d\right ) \left (a e^{2} - b d e + c d^{2}\right )^{2}}{3 e^{5} \left (d + e x\right )^{\frac {3}{2}}}\right )}{e} & \text {for}\: e \neq 0 \\\frac {\left (a + b x + c x^{2}\right )^{3}}{3 d^{\frac {5}{2}}} & \text {otherwise} \end {cases} \] Input:
integrate((2*c*x+b)*(c*x**2+b*x+a)**2/(e*x+d)**(5/2),x)
Output:
Piecewise((2*(2*c**3*(d + e*x)**(7/2)/(7*e**5) + (d + e*x)**(5/2)*(5*b*c** 2*e - 10*c**3*d)/(5*e**5) + (d + e*x)**(3/2)*(4*a*c**2*e**2 + 4*b**2*c*e** 2 - 20*b*c**2*d*e + 20*c**3*d**2)/(3*e**5) + sqrt(d + e*x)*(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)/e**5 - 2*(a*e**2 - b*d*e + c*d**2)*(a*c*e**2 + b**2*e**2 - 5*b*c *d*e + 5*c**2*d**2)/(e**5*sqrt(d + e*x)) - (b*e - 2*c*d)*(a*e**2 - b*d*e + c*d**2)**2/(3*e**5*(d + e*x)**(3/2)))/e, Ne(e, 0)), ((a + b*x + c*x**2)** 3/(3*d**(5/2)), True))
Time = 0.03 (sec) , antiderivative size = 314, normalized size of antiderivative = 1.28 \[ \int \frac {(b+2 c x) \left (a+b x+c x^2\right )^2}{(d+e x)^{5/2}} \, dx=\frac {2 \, {\left (\frac {6 \, {\left (e x + d\right )}^{\frac {7}{2}} c^{3} - 21 \, {\left (2 \, c^{3} d - b c^{2} e\right )} {\left (e x + d\right )}^{\frac {5}{2}} + 28 \, {\left (5 \, c^{3} d^{2} - 5 \, b c^{2} d e + {\left (b^{2} c + a c^{2}\right )} e^{2}\right )} {\left (e x + d\right )}^{\frac {3}{2}} - 21 \, {\left (20 \, c^{3} d^{3} - 30 \, b c^{2} d^{2} e + 12 \, {\left (b^{2} c + a c^{2}\right )} d e^{2} - {\left (b^{3} + 6 \, a b c\right )} e^{3}\right )} \sqrt {e x + d}}{e^{5}} + \frac {7 \, {\left (2 \, c^{3} d^{5} - 5 \, b c^{2} d^{4} e - a^{2} b e^{5} + 4 \, {\left (b^{2} c + a c^{2}\right )} d^{3} e^{2} - {\left (b^{3} + 6 \, a b c\right )} d^{2} e^{3} + 2 \, {\left (a b^{2} + a^{2} c\right )} d e^{4} - 6 \, {\left (5 \, c^{3} d^{4} - 10 \, b c^{2} d^{3} e + 6 \, {\left (b^{2} c + a c^{2}\right )} d^{2} e^{2} - {\left (b^{3} + 6 \, a b c\right )} d e^{3} + {\left (a b^{2} + a^{2} c\right )} e^{4}\right )} {\left (e x + d\right )}\right )}}{{\left (e x + d\right )}^{\frac {3}{2}} e^{5}}\right )}}{21 \, e} \] Input:
integrate((2*c*x+b)*(c*x^2+b*x+a)^2/(e*x+d)^(5/2),x, algorithm="maxima")
Output:
2/21*((6*(e*x + d)^(7/2)*c^3 - 21*(2*c^3*d - b*c^2*e)*(e*x + d)^(5/2) + 28 *(5*c^3*d^2 - 5*b*c^2*d*e + (b^2*c + a*c^2)*e^2)*(e*x + d)^(3/2) - 21*(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) *sqrt(e*x + d))/e^5 + 7*(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 - 6*( 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))/((e*x + d)^(3/2)*e^5))/e
Time = 0.79 (sec) , antiderivative size = 444, normalized size of antiderivative = 1.80 \[ \int \frac {(b+2 c x) \left (a+b x+c x^2\right )^2}{(d+e x)^{5/2}} \, dx=-\frac {2 \, {\left (30 \, {\left (e x + d\right )} c^{3} d^{4} - 2 \, c^{3} d^{5} - 60 \, {\left (e x + d\right )} b c^{2} d^{3} e + 5 \, b c^{2} d^{4} e + 36 \, {\left (e x + d\right )} b^{2} c d^{2} e^{2} + 36 \, {\left (e x + d\right )} a c^{2} d^{2} e^{2} - 4 \, b^{2} c d^{3} e^{2} - 4 \, a c^{2} d^{3} e^{2} - 6 \, {\left (e x + d\right )} b^{3} d e^{3} - 36 \, {\left (e x + d\right )} a b c d e^{3} + b^{3} d^{2} e^{3} + 6 \, a b c d^{2} e^{3} + 6 \, {\left (e x + d\right )} a b^{2} e^{4} + 6 \, {\left (e x + d\right )} a^{2} c e^{4} - 2 \, a b^{2} d e^{4} - 2 \, a^{2} c d e^{4} + a^{2} b e^{5}\right )}}{3 \, {\left (e x + d\right )}^{\frac {3}{2}} e^{6}} + \frac {2 \, {\left (6 \, {\left (e x + d\right )}^{\frac {7}{2}} c^{3} e^{36} - 42 \, {\left (e x + d\right )}^{\frac {5}{2}} c^{3} d e^{36} + 140 \, {\left (e x + d\right )}^{\frac {3}{2}} c^{3} d^{2} e^{36} - 420 \, \sqrt {e x + d} c^{3} d^{3} e^{36} + 21 \, {\left (e x + d\right )}^{\frac {5}{2}} b c^{2} e^{37} - 140 \, {\left (e x + d\right )}^{\frac {3}{2}} b c^{2} d e^{37} + 630 \, \sqrt {e x + d} b c^{2} d^{2} e^{37} + 28 \, {\left (e x + d\right )}^{\frac {3}{2}} b^{2} c e^{38} + 28 \, {\left (e x + d\right )}^{\frac {3}{2}} a c^{2} e^{38} - 252 \, \sqrt {e x + d} b^{2} c d e^{38} - 252 \, \sqrt {e x + d} a c^{2} d e^{38} + 21 \, \sqrt {e x + d} b^{3} e^{39} + 126 \, \sqrt {e x + d} a b c e^{39}\right )}}{21 \, e^{42}} \] Input:
integrate((2*c*x+b)*(c*x^2+b*x+a)^2/(e*x+d)^(5/2),x, algorithm="giac")
Output:
-2/3*(30*(e*x + d)*c^3*d^4 - 2*c^3*d^5 - 60*(e*x + d)*b*c^2*d^3*e + 5*b*c^ 2*d^4*e + 36*(e*x + d)*b^2*c*d^2*e^2 + 36*(e*x + d)*a*c^2*d^2*e^2 - 4*b^2* c*d^3*e^2 - 4*a*c^2*d^3*e^2 - 6*(e*x + d)*b^3*d*e^3 - 36*(e*x + d)*a*b*c*d *e^3 + b^3*d^2*e^3 + 6*a*b*c*d^2*e^3 + 6*(e*x + d)*a*b^2*e^4 + 6*(e*x + d) *a^2*c*e^4 - 2*a*b^2*d*e^4 - 2*a^2*c*d*e^4 + a^2*b*e^5)/((e*x + d)^(3/2)*e ^6) + 2/21*(6*(e*x + d)^(7/2)*c^3*e^36 - 42*(e*x + d)^(5/2)*c^3*d*e^36 + 1 40*(e*x + d)^(3/2)*c^3*d^2*e^36 - 420*sqrt(e*x + d)*c^3*d^3*e^36 + 21*(e*x + d)^(5/2)*b*c^2*e^37 - 140*(e*x + d)^(3/2)*b*c^2*d*e^37 + 630*sqrt(e*x + d)*b*c^2*d^2*e^37 + 28*(e*x + d)^(3/2)*b^2*c*e^38 + 28*(e*x + d)^(3/2)*a* c^2*e^38 - 252*sqrt(e*x + d)*b^2*c*d*e^38 - 252*sqrt(e*x + d)*a*c^2*d*e^38 + 21*sqrt(e*x + d)*b^3*e^39 + 126*sqrt(e*x + d)*a*b*c*e^39)/e^42
Time = 11.39 (sec) , antiderivative size = 329, normalized size of antiderivative = 1.34 \[ \int \frac {(b+2 c x) \left (a+b x+c x^2\right )^2}{(d+e x)^{5/2}} \, dx=\frac {4\,c^3\,{\left (d+e\,x\right )}^{7/2}}{7\,e^6}-\frac {\left (20\,c^3\,d-10\,b\,c^2\,e\right )\,{\left (d+e\,x\right )}^{5/2}}{5\,e^6}+\frac {{\left (d+e\,x\right )}^{3/2}\,\left (8\,b^2\,c\,e^2-40\,b\,c^2\,d\,e+40\,c^3\,d^2+8\,a\,c^2\,e^2\right )}{3\,e^6}+\frac {\frac {4\,c^3\,d^5}{3}-\left (d+e\,x\right )\,\left (4\,a^2\,c\,e^4+4\,a\,b^2\,e^4-24\,a\,b\,c\,d\,e^3+24\,a\,c^2\,d^2\,e^2-4\,b^3\,d\,e^3+24\,b^2\,c\,d^2\,e^2-40\,b\,c^2\,d^3\,e+20\,c^3\,d^4\right )-\frac {2\,a^2\,b\,e^5}{3}-\frac {2\,b^3\,d^2\,e^3}{3}+\frac {8\,a\,c^2\,d^3\,e^2}{3}+\frac {8\,b^2\,c\,d^3\,e^2}{3}+\frac {4\,a\,b^2\,d\,e^4}{3}+\frac {4\,a^2\,c\,d\,e^4}{3}-\frac {10\,b\,c^2\,d^4\,e}{3}-4\,a\,b\,c\,d^2\,e^3}{e^6\,{\left (d+e\,x\right )}^{3/2}}+\frac {2\,\left (b\,e-2\,c\,d\right )\,\sqrt {d+e\,x}\,\left (b^2\,e^2-10\,b\,c\,d\,e+10\,c^2\,d^2+6\,a\,c\,e^2\right )}{e^6} \] Input:
int(((b + 2*c*x)*(a + b*x + c*x^2)^2)/(d + e*x)^(5/2),x)
Output:
(4*c^3*(d + e*x)^(7/2))/(7*e^6) - ((20*c^3*d - 10*b*c^2*e)*(d + e*x)^(5/2) )/(5*e^6) + ((d + e*x)^(3/2)*(40*c^3*d^2 + 8*a*c^2*e^2 + 8*b^2*c*e^2 - 40* b*c^2*d*e))/(3*e^6) + ((4*c^3*d^5)/3 - (d + e*x)*(20*c^3*d^4 + 4*a*b^2*e^4 + 4*a^2*c*e^4 - 4*b^3*d*e^3 + 24*a*c^2*d^2*e^2 + 24*b^2*c*d^2*e^2 - 40*b* c^2*d^3*e - 24*a*b*c*d*e^3) - (2*a^2*b*e^5)/3 - (2*b^3*d^2*e^3)/3 + (8*a*c ^2*d^3*e^2)/3 + (8*b^2*c*d^3*e^2)/3 + (4*a*b^2*d*e^4)/3 + (4*a^2*c*d*e^4)/ 3 - (10*b*c^2*d^4*e)/3 - 4*a*b*c*d^2*e^3)/(e^6*(d + e*x)^(3/2)) + (2*(b*e - 2*c*d)*(d + e*x)^(1/2)*(b^2*e^2 + 10*c^2*d^2 + 6*a*c*e^2 - 10*b*c*d*e))/ e^6
Time = 0.21 (sec) , antiderivative size = 366, normalized size of antiderivative = 1.49 \[ \int \frac {(b+2 c x) \left (a+b x+c x^2\right )^2}{(d+e x)^{5/2}} \, dx=\frac {-\frac {1024}{21} c^{3} d^{5}+32 a b c \,d^{2} e^{3}+2 b^{3} e^{5} x^{2}+\frac {4}{7} c^{3} e^{5} x^{5}+12 a b c \,e^{5} x^{2}-64 a \,c^{2} d^{2} e^{3} x -16 a \,c^{2} d \,e^{4} x^{2}-64 b^{2} c \,d^{2} e^{3} x -16 b^{2} c d \,e^{4} x^{2}+128 b \,c^{2} d^{3} e^{2} x +32 b \,c^{2} d^{2} e^{3} x^{2}-\frac {16}{3} b \,c^{2} d \,e^{4} x^{3}-\frac {8}{3} a^{2} c d \,e^{4}-\frac {8}{3} a \,b^{2} d \,e^{4}-\frac {128}{3} a \,c^{2} d^{3} e^{2}-\frac {128}{3} b^{2} c \,d^{3} e^{2}+\frac {256}{3} b \,c^{2} d^{4} e -\frac {2}{3} a^{2} b \,e^{5}+\frac {16}{3} b^{3} d^{2} e^{3}+48 a b c d \,e^{4} x -4 a^{2} c \,e^{5} x -4 a \,b^{2} e^{5} x +\frac {8}{3} a \,c^{2} e^{5} x^{3}+8 b^{3} d \,e^{4} x +\frac {8}{3} b^{2} c \,e^{5} x^{3}+2 b \,c^{2} e^{5} x^{4}-\frac {512}{7} c^{3} d^{4} e x -\frac {128}{7} c^{3} d^{3} e^{2} x^{2}+\frac {64}{21} c^{3} d^{2} e^{3} x^{3}-\frac {8}{7} c^{3} d \,e^{4} x^{4}}{\sqrt {e x +d}\, e^{6} \left (e x +d \right )} \] Input:
int((2*c*x+b)*(c*x^2+b*x+a)^2/(e*x+d)^(5/2),x)
Output:
(2*( - 7*a**2*b*e**5 - 28*a**2*c*d*e**4 - 42*a**2*c*e**5*x - 28*a*b**2*d*e **4 - 42*a*b**2*e**5*x + 336*a*b*c*d**2*e**3 + 504*a*b*c*d*e**4*x + 126*a* b*c*e**5*x**2 - 448*a*c**2*d**3*e**2 - 672*a*c**2*d**2*e**3*x - 168*a*c**2 *d*e**4*x**2 + 28*a*c**2*e**5*x**3 + 56*b**3*d**2*e**3 + 84*b**3*d*e**4*x + 21*b**3*e**5*x**2 - 448*b**2*c*d**3*e**2 - 672*b**2*c*d**2*e**3*x - 168* b**2*c*d*e**4*x**2 + 28*b**2*c*e**5*x**3 + 896*b*c**2*d**4*e + 1344*b*c**2 *d**3*e**2*x + 336*b*c**2*d**2*e**3*x**2 - 56*b*c**2*d*e**4*x**3 + 21*b*c* *2*e**5*x**4 - 512*c**3*d**5 - 768*c**3*d**4*e*x - 192*c**3*d**3*e**2*x**2 + 32*c**3*d**2*e**3*x**3 - 12*c**3*d*e**4*x**4 + 6*c**3*e**5*x**5))/(21*s qrt(d + e*x)*e**6*(d + e*x))