Integrand size = 26, antiderivative size = 265 \[ \int \frac {(A+B x) \left (b x+c x^2\right )^2}{\sqrt {d+e x}} \, dx=-\frac {2 d^2 (B d-A e) (c d-b e)^2 \sqrt {d+e x}}{e^6}+\frac {2 d (c d-b e) (B d (5 c d-3 b e)-2 A e (2 c d-b e)) (d+e x)^{3/2}}{3 e^6}+\frac {2 \left (A e \left (6 c^2 d^2-6 b c d e+b^2 e^2\right )-B d \left (10 c^2 d^2-12 b c d e+3 b^2 e^2\right )\right ) (d+e x)^{5/2}}{5 e^6}-\frac {2 \left (2 A c e (2 c d-b e)-B \left (10 c^2 d^2-8 b c d e+b^2 e^2\right )\right ) (d+e x)^{7/2}}{7 e^6}-\frac {2 c (5 B c d-2 b B e-A c e) (d+e x)^{9/2}}{9 e^6}+\frac {2 B c^2 (d+e x)^{11/2}}{11 e^6} \] Output:
-2*d^2*(-A*e+B*d)*(-b*e+c*d)^2*(e*x+d)^(1/2)/e^6+2/3*d*(-b*e+c*d)*(B*d*(-3 *b*e+5*c*d)-2*A*e*(-b*e+2*c*d))*(e*x+d)^(3/2)/e^6+2/5*(A*e*(b^2*e^2-6*b*c* d*e+6*c^2*d^2)-B*d*(3*b^2*e^2-12*b*c*d*e+10*c^2*d^2))*(e*x+d)^(5/2)/e^6-2/ 7*(2*A*c*e*(-b*e+2*c*d)-B*(b^2*e^2-8*b*c*d*e+10*c^2*d^2))*(e*x+d)^(7/2)/e^ 6-2/9*c*(-A*c*e-2*B*b*e+5*B*c*d)*(e*x+d)^(9/2)/e^6+2/11*B*c^2*(e*x+d)^(11/ 2)/e^6
Time = 0.29 (sec) , antiderivative size = 273, normalized size of antiderivative = 1.03 \[ \int \frac {(A+B x) \left (b x+c x^2\right )^2}{\sqrt {d+e x}} \, dx=\frac {2 \sqrt {d+e x} \left (11 A e \left (21 b^2 e^2 \left (8 d^2-4 d e x+3 e^2 x^2\right )+18 b c e \left (-16 d^3+8 d^2 e x-6 d e^2 x^2+5 e^3 x^3\right )+c^2 \left (128 d^4-64 d^3 e x+48 d^2 e^2 x^2-40 d e^3 x^3+35 e^4 x^4\right )\right )+B \left (99 b^2 e^2 \left (-16 d^3+8 d^2 e x-6 d e^2 x^2+5 e^3 x^3\right )+22 b c e \left (128 d^4-64 d^3 e x+48 d^2 e^2 x^2-40 d e^3 x^3+35 e^4 x^4\right )-5 c^2 \left (256 d^5-128 d^4 e x+96 d^3 e^2 x^2-80 d^2 e^3 x^3+70 d e^4 x^4-63 e^5 x^5\right )\right )\right )}{3465 e^6} \] Input:
Integrate[((A + B*x)*(b*x + c*x^2)^2)/Sqrt[d + e*x],x]
Output:
(2*Sqrt[d + e*x]*(11*A*e*(21*b^2*e^2*(8*d^2 - 4*d*e*x + 3*e^2*x^2) + 18*b* c*e*(-16*d^3 + 8*d^2*e*x - 6*d*e^2*x^2 + 5*e^3*x^3) + c^2*(128*d^4 - 64*d^ 3*e*x + 48*d^2*e^2*x^2 - 40*d*e^3*x^3 + 35*e^4*x^4)) + B*(99*b^2*e^2*(-16* d^3 + 8*d^2*e*x - 6*d*e^2*x^2 + 5*e^3*x^3) + 22*b*c*e*(128*d^4 - 64*d^3*e* x + 48*d^2*e^2*x^2 - 40*d*e^3*x^3 + 35*e^4*x^4) - 5*c^2*(256*d^5 - 128*d^4 *e*x + 96*d^3*e^2*x^2 - 80*d^2*e^3*x^3 + 70*d*e^4*x^4 - 63*e^5*x^5))))/(34 65*e^6)
Time = 0.70 (sec) , antiderivative size = 265, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.077, 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.
| \(\displaystyle \int \frac {(A+B x) \left (b x+c x^2\right )^2}{\sqrt {d+e x}} \, dx\) |
| \(\Big \downarrow \) 1195 |
| \(\displaystyle \int \left (\frac {(d+e x)^{5/2} \left (B \left (b^2 e^2-8 b c d e+10 c^2 d^2\right )-2 A c e (2 c d-b e)\right )}{e^5}+\frac {(d+e x)^{3/2} \left (A e \left (b^2 e^2-6 b c d e+6 c^2 d^2\right )-B d \left (3 b^2 e^2-12 b c d e+10 c^2 d^2\right )\right )}{e^5}-\frac {d^2 (B d-A e) (c d-b e)^2}{e^5 \sqrt {d+e x}}+\frac {c (d+e x)^{7/2} (A c e+2 b B e-5 B c d)}{e^5}+\frac {d \sqrt {d+e x} (c d-b e) (B d (5 c d-3 b e)-2 A e (2 c d-b e))}{e^5}+\frac {B c^2 (d+e x)^{9/2}}{e^5}\right )dx\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle -\frac {2 (d+e x)^{7/2} \left (2 A c e (2 c d-b e)-B \left (b^2 e^2-8 b c d e+10 c^2 d^2\right )\right )}{7 e^6}+\frac {2 (d+e x)^{5/2} \left (A e \left (b^2 e^2-6 b c d e+6 c^2 d^2\right )-B d \left (3 b^2 e^2-12 b c d e+10 c^2 d^2\right )\right )}{5 e^6}-\frac {2 d^2 \sqrt {d+e x} (B d-A e) (c d-b e)^2}{e^6}-\frac {2 c (d+e x)^{9/2} (-A c e-2 b B e+5 B c d)}{9 e^6}+\frac {2 d (d+e x)^{3/2} (c d-b e) (B d (5 c d-3 b e)-2 A e (2 c d-b e))}{3 e^6}+\frac {2 B c^2 (d+e x)^{11/2}}{11 e^6}\) |
Input:
Int[((A + B*x)*(b*x + c*x^2)^2)/Sqrt[d + e*x],x]
Output:
(-2*d^2*(B*d - A*e)*(c*d - b*e)^2*Sqrt[d + e*x])/e^6 + (2*d*(c*d - b*e)*(B *d*(5*c*d - 3*b*e) - 2*A*e*(2*c*d - b*e))*(d + e*x)^(3/2))/(3*e^6) + (2*(A *e*(6*c^2*d^2 - 6*b*c*d*e + b^2*e^2) - B*d*(10*c^2*d^2 - 12*b*c*d*e + 3*b^ 2*e^2))*(d + e*x)^(5/2))/(5*e^6) - (2*(2*A*c*e*(2*c*d - b*e) - B*(10*c^2*d ^2 - 8*b*c*d*e + b^2*e^2))*(d + e*x)^(7/2))/(7*e^6) - (2*c*(5*B*c*d - 2*b* B*e - A*c*e)*(d + e*x)^(9/2))/(9*e^6) + (2*B*c^2*(d + e*x)^(11/2))/(11*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 = 2.32 (sec) , antiderivative size = 256, normalized size of antiderivative = 0.97
| method | result | size |
| pseudoelliptic | \(\frac {16 \sqrt {e x +d}\, \left (\left (\left (\frac {15}{88} e^{5} x^{5}-\frac {25}{132} d \,e^{4} x^{4}-\frac {20}{77} d^{3} e^{2} x^{2}+\frac {50}{231} d^{2} e^{3} x^{3}-\frac {160}{231} d^{5}+\frac {80}{231} d^{4} e x \right ) B +\frac {16 e \left (\frac {35}{128} e^{4} x^{4}-\frac {5}{16} d \,e^{3} x^{3}+\frac {3}{8} d^{2} e^{2} x^{2}-\frac {1}{2} d^{3} e x +d^{4}\right ) A}{21}\right ) c^{2}-\frac {12 e \left (\left (-\frac {35}{144} e^{4} x^{4}+\frac {5}{18} d \,e^{3} x^{3}-\frac {1}{3} d^{2} e^{2} x^{2}+\frac {4}{9} d^{3} e x -\frac {8}{9} d^{4}\right ) B +A e \left (-\frac {5}{16} e^{3} x^{3}+\frac {3}{8} d \,e^{2} x^{2}-\frac {1}{2} d^{2} e x +d^{3}\right )\right ) b c}{7}+e^{2} \left (\left (\frac {15}{56} e^{3} x^{3}-\frac {6}{7} d^{3}-\frac {9}{28} d \,e^{2} x^{2}+\frac {3}{7} d^{2} e x \right ) B +A e \left (\frac {3}{8} e^{2} x^{2}+d^{2}-\frac {1}{2} d e x \right )\right ) b^{2}\right )}{15 e^{6}}\) | \(256\) |
| derivativedivides | \(\frac {\frac {2 B \,c^{2} \left (e x +d \right )^{\frac {11}{2}}}{11}+\frac {2 \left (\left (A e -3 B d \right ) c^{2}+2 B c \left (b e -c d \right )\right ) \left (e x +d \right )^{\frac {9}{2}}}{9}+\frac {2 \left (\left (-2 \left (A e -B d \right ) d +B \,d^{2}\right ) c^{2}+2 \left (A e -3 B d \right ) c \left (b e -c d \right )+B \left (b e -c d \right )^{2}\right ) \left (e x +d \right )^{\frac {7}{2}}}{7}+\frac {2 \left (\left (A e -B d \right ) d^{2} c^{2}+2 \left (-2 \left (A e -B d \right ) d +B \,d^{2}\right ) c \left (b e -c d \right )+\left (A e -3 B d \right ) \left (b e -c d \right )^{2}\right ) \left (e x +d \right )^{\frac {5}{2}}}{5}+\frac {2 \left (2 \left (A e -B d \right ) d^{2} c \left (b e -c d \right )+\left (-2 \left (A e -B d \right ) d +B \,d^{2}\right ) \left (b e -c d \right )^{2}\right ) \left (e x +d \right )^{\frac {3}{2}}}{3}+2 \left (A e -B d \right ) d^{2} \left (b e -c d \right )^{2} \sqrt {e x +d}}{e^{6}}\) | \(277\) |
| default | \(\frac {\frac {2 B \,c^{2} \left (e x +d \right )^{\frac {11}{2}}}{11}+\frac {2 \left (\left (A e -3 B d \right ) c^{2}+2 B c \left (b e -c d \right )\right ) \left (e x +d \right )^{\frac {9}{2}}}{9}+\frac {2 \left (\left (-2 \left (A e -B d \right ) d +B \,d^{2}\right ) c^{2}+2 \left (A e -3 B d \right ) c \left (b e -c d \right )+B \left (b e -c d \right )^{2}\right ) \left (e x +d \right )^{\frac {7}{2}}}{7}+\frac {2 \left (\left (A e -B d \right ) d^{2} c^{2}+2 \left (-2 \left (A e -B d \right ) d +B \,d^{2}\right ) c \left (b e -c d \right )+\left (A e -3 B d \right ) \left (b e -c d \right )^{2}\right ) \left (e x +d \right )^{\frac {5}{2}}}{5}+\frac {2 \left (2 \left (A e -B d \right ) d^{2} c \left (b e -c d \right )+\left (-2 \left (A e -B d \right ) d +B \,d^{2}\right ) \left (b e -c d \right )^{2}\right ) \left (e x +d \right )^{\frac {3}{2}}}{3}+2 \left (A e -B d \right ) d^{2} \left (b e -c d \right )^{2} \sqrt {e x +d}}{e^{6}}\) | \(277\) |
| gosper | \(\frac {2 \left (315 B \,x^{5} c^{2} e^{5}+385 A \,x^{4} c^{2} e^{5}+770 B \,x^{4} b c \,e^{5}-350 B \,x^{4} c^{2} d \,e^{4}+990 A \,x^{3} b c \,e^{5}-440 A \,x^{3} c^{2} d \,e^{4}+495 B \,x^{3} b^{2} e^{5}-880 B \,x^{3} b c d \,e^{4}+400 B \,x^{3} c^{2} d^{2} e^{3}+693 A \,x^{2} b^{2} e^{5}-1188 A \,x^{2} b c d \,e^{4}+528 A \,x^{2} c^{2} d^{2} e^{3}-594 B \,x^{2} b^{2} d \,e^{4}+1056 B \,x^{2} b c \,d^{2} e^{3}-480 B \,x^{2} c^{2} d^{3} e^{2}-924 A x \,b^{2} d \,e^{4}+1584 A x b c \,d^{2} e^{3}-704 A x \,c^{2} d^{3} e^{2}+792 B x \,b^{2} d^{2} e^{3}-1408 B x b c \,d^{3} e^{2}+640 B x \,c^{2} d^{4} e +1848 A \,b^{2} d^{2} e^{3}-3168 A b c \,d^{3} e^{2}+1408 A \,c^{2} d^{4} e -1584 B \,b^{2} d^{3} e^{2}+2816 B b c \,d^{4} e -1280 B \,c^{2} d^{5}\right ) \sqrt {e x +d}}{3465 e^{6}}\) | \(341\) |
| trager | \(\frac {2 \left (315 B \,x^{5} c^{2} e^{5}+385 A \,x^{4} c^{2} e^{5}+770 B \,x^{4} b c \,e^{5}-350 B \,x^{4} c^{2} d \,e^{4}+990 A \,x^{3} b c \,e^{5}-440 A \,x^{3} c^{2} d \,e^{4}+495 B \,x^{3} b^{2} e^{5}-880 B \,x^{3} b c d \,e^{4}+400 B \,x^{3} c^{2} d^{2} e^{3}+693 A \,x^{2} b^{2} e^{5}-1188 A \,x^{2} b c d \,e^{4}+528 A \,x^{2} c^{2} d^{2} e^{3}-594 B \,x^{2} b^{2} d \,e^{4}+1056 B \,x^{2} b c \,d^{2} e^{3}-480 B \,x^{2} c^{2} d^{3} e^{2}-924 A x \,b^{2} d \,e^{4}+1584 A x b c \,d^{2} e^{3}-704 A x \,c^{2} d^{3} e^{2}+792 B x \,b^{2} d^{2} e^{3}-1408 B x b c \,d^{3} e^{2}+640 B x \,c^{2} d^{4} e +1848 A \,b^{2} d^{2} e^{3}-3168 A b c \,d^{3} e^{2}+1408 A \,c^{2} d^{4} e -1584 B \,b^{2} d^{3} e^{2}+2816 B b c \,d^{4} e -1280 B \,c^{2} d^{5}\right ) \sqrt {e x +d}}{3465 e^{6}}\) | \(341\) |
| risch | \(\frac {2 \left (315 B \,x^{5} c^{2} e^{5}+385 A \,x^{4} c^{2} e^{5}+770 B \,x^{4} b c \,e^{5}-350 B \,x^{4} c^{2} d \,e^{4}+990 A \,x^{3} b c \,e^{5}-440 A \,x^{3} c^{2} d \,e^{4}+495 B \,x^{3} b^{2} e^{5}-880 B \,x^{3} b c d \,e^{4}+400 B \,x^{3} c^{2} d^{2} e^{3}+693 A \,x^{2} b^{2} e^{5}-1188 A \,x^{2} b c d \,e^{4}+528 A \,x^{2} c^{2} d^{2} e^{3}-594 B \,x^{2} b^{2} d \,e^{4}+1056 B \,x^{2} b c \,d^{2} e^{3}-480 B \,x^{2} c^{2} d^{3} e^{2}-924 A x \,b^{2} d \,e^{4}+1584 A x b c \,d^{2} e^{3}-704 A x \,c^{2} d^{3} e^{2}+792 B x \,b^{2} d^{2} e^{3}-1408 B x b c \,d^{3} e^{2}+640 B x \,c^{2} d^{4} e +1848 A \,b^{2} d^{2} e^{3}-3168 A b c \,d^{3} e^{2}+1408 A \,c^{2} d^{4} e -1584 B \,b^{2} d^{3} e^{2}+2816 B b c \,d^{4} e -1280 B \,c^{2} d^{5}\right ) \sqrt {e x +d}}{3465 e^{6}}\) | \(341\) |
| orering | \(\frac {2 \left (315 B \,x^{5} c^{2} e^{5}+385 A \,x^{4} c^{2} e^{5}+770 B \,x^{4} b c \,e^{5}-350 B \,x^{4} c^{2} d \,e^{4}+990 A \,x^{3} b c \,e^{5}-440 A \,x^{3} c^{2} d \,e^{4}+495 B \,x^{3} b^{2} e^{5}-880 B \,x^{3} b c d \,e^{4}+400 B \,x^{3} c^{2} d^{2} e^{3}+693 A \,x^{2} b^{2} e^{5}-1188 A \,x^{2} b c d \,e^{4}+528 A \,x^{2} c^{2} d^{2} e^{3}-594 B \,x^{2} b^{2} d \,e^{4}+1056 B \,x^{2} b c \,d^{2} e^{3}-480 B \,x^{2} c^{2} d^{3} e^{2}-924 A x \,b^{2} d \,e^{4}+1584 A x b c \,d^{2} e^{3}-704 A x \,c^{2} d^{3} e^{2}+792 B x \,b^{2} d^{2} e^{3}-1408 B x b c \,d^{3} e^{2}+640 B x \,c^{2} d^{4} e +1848 A \,b^{2} d^{2} e^{3}-3168 A b c \,d^{3} e^{2}+1408 A \,c^{2} d^{4} e -1584 B \,b^{2} d^{3} e^{2}+2816 B b c \,d^{4} e -1280 B \,c^{2} d^{5}\right ) \sqrt {e x +d}\, \left (c \,x^{2}+b x \right )^{2}}{3465 e^{6} \left (c x +b \right )^{2} x^{2}}\) | \(362\) |
Input:
int((B*x+A)*(c*x^2+b*x)^2/(e*x+d)^(1/2),x,method=_RETURNVERBOSE)
Output:
16/15*(e*x+d)^(1/2)*(((15/88*e^5*x^5-25/132*d*e^4*x^4-20/77*d^3*e^2*x^2+50 /231*d^2*e^3*x^3-160/231*d^5+80/231*d^4*e*x)*B+16/21*e*(35/128*e^4*x^4-5/1 6*d*e^3*x^3+3/8*d^2*e^2*x^2-1/2*d^3*e*x+d^4)*A)*c^2-12/7*e*((-35/144*e^4*x ^4+5/18*d*e^3*x^3-1/3*d^2*e^2*x^2+4/9*d^3*e*x-8/9*d^4)*B+A*e*(-5/16*e^3*x^ 3+3/8*d*e^2*x^2-1/2*d^2*e*x+d^3))*b*c+e^2*((15/56*e^3*x^3-6/7*d^3-9/28*d*e ^2*x^2+3/7*d^2*e*x)*B+A*e*(3/8*e^2*x^2+d^2-1/2*d*e*x))*b^2)/e^6
Time = 0.08 (sec) , antiderivative size = 290, normalized size of antiderivative = 1.09 \[ \int \frac {(A+B x) \left (b x+c x^2\right )^2}{\sqrt {d+e x}} \, dx=\frac {2 \, {\left (315 \, B c^{2} e^{5} x^{5} - 1280 \, B c^{2} d^{5} + 1848 \, A b^{2} d^{2} e^{3} + 1408 \, {\left (2 \, B b c + A c^{2}\right )} d^{4} e - 1584 \, {\left (B b^{2} + 2 \, A b c\right )} d^{3} e^{2} - 35 \, {\left (10 \, B c^{2} d e^{4} - 11 \, {\left (2 \, B b c + A c^{2}\right )} e^{5}\right )} x^{4} + 5 \, {\left (80 \, B c^{2} d^{2} e^{3} - 88 \, {\left (2 \, B b c + A c^{2}\right )} d e^{4} + 99 \, {\left (B b^{2} + 2 \, A b c\right )} e^{5}\right )} x^{3} - 3 \, {\left (160 \, B c^{2} d^{3} e^{2} - 231 \, A b^{2} e^{5} - 176 \, {\left (2 \, B b c + A c^{2}\right )} d^{2} e^{3} + 198 \, {\left (B b^{2} + 2 \, A b c\right )} d e^{4}\right )} x^{2} + 4 \, {\left (160 \, B c^{2} d^{4} e - 231 \, A b^{2} d e^{4} - 176 \, {\left (2 \, B b c + A c^{2}\right )} d^{3} e^{2} + 198 \, {\left (B b^{2} + 2 \, A b c\right )} d^{2} e^{3}\right )} x\right )} \sqrt {e x + d}}{3465 \, e^{6}} \] Input:
integrate((B*x+A)*(c*x^2+b*x)^2/(e*x+d)^(1/2),x, algorithm="fricas")
Output:
2/3465*(315*B*c^2*e^5*x^5 - 1280*B*c^2*d^5 + 1848*A*b^2*d^2*e^3 + 1408*(2* B*b*c + A*c^2)*d^4*e - 1584*(B*b^2 + 2*A*b*c)*d^3*e^2 - 35*(10*B*c^2*d*e^4 - 11*(2*B*b*c + A*c^2)*e^5)*x^4 + 5*(80*B*c^2*d^2*e^3 - 88*(2*B*b*c + A*c ^2)*d*e^4 + 99*(B*b^2 + 2*A*b*c)*e^5)*x^3 - 3*(160*B*c^2*d^3*e^2 - 231*A*b ^2*e^5 - 176*(2*B*b*c + A*c^2)*d^2*e^3 + 198*(B*b^2 + 2*A*b*c)*d*e^4)*x^2 + 4*(160*B*c^2*d^4*e - 231*A*b^2*d*e^4 - 176*(2*B*b*c + A*c^2)*d^3*e^2 + 1 98*(B*b^2 + 2*A*b*c)*d^2*e^3)*x)*sqrt(e*x + d)/e^6
Time = 1.52 (sec) , antiderivative size = 435, normalized size of antiderivative = 1.64 \[ \int \frac {(A+B x) \left (b x+c x^2\right )^2}{\sqrt {d+e x}} \, dx=\begin {cases} \frac {2 \left (\frac {B c^{2} \left (d + e x\right )^{\frac {11}{2}}}{11 e^{5}} + \frac {\left (d + e x\right )^{\frac {9}{2}} \left (A c^{2} e + 2 B b c e - 5 B c^{2} d\right )}{9 e^{5}} + \frac {\left (d + e x\right )^{\frac {7}{2}} \cdot \left (2 A b c e^{2} - 4 A c^{2} d e + B b^{2} e^{2} - 8 B b c d e + 10 B c^{2} d^{2}\right )}{7 e^{5}} + \frac {\left (d + e x\right )^{\frac {5}{2}} \left (A b^{2} e^{3} - 6 A b c d e^{2} + 6 A c^{2} d^{2} e - 3 B b^{2} d e^{2} + 12 B b c d^{2} e - 10 B c^{2} d^{3}\right )}{5 e^{5}} + \frac {\left (d + e x\right )^{\frac {3}{2}} \left (- 2 A b^{2} d e^{3} + 6 A b c d^{2} e^{2} - 4 A c^{2} d^{3} e + 3 B b^{2} d^{2} e^{2} - 8 B b c d^{3} e + 5 B c^{2} d^{4}\right )}{3 e^{5}} + \frac {\sqrt {d + e x} \left (A b^{2} d^{2} e^{3} - 2 A b c d^{3} e^{2} + A c^{2} d^{4} e - B b^{2} d^{3} e^{2} + 2 B b c d^{4} e - B c^{2} d^{5}\right )}{e^{5}}\right )}{e} & \text {for}\: e \neq 0 \\\frac {\frac {A b^{2} x^{3}}{3} + \frac {B c^{2} x^{6}}{6} + \frac {x^{5} \left (A c^{2} + 2 B b c\right )}{5} + \frac {x^{4} \cdot \left (2 A b c + B b^{2}\right )}{4}}{\sqrt {d}} & \text {otherwise} \end {cases} \] Input:
integrate((B*x+A)*(c*x**2+b*x)**2/(e*x+d)**(1/2),x)
Output:
Piecewise((2*(B*c**2*(d + e*x)**(11/2)/(11*e**5) + (d + e*x)**(9/2)*(A*c** 2*e + 2*B*b*c*e - 5*B*c**2*d)/(9*e**5) + (d + e*x)**(7/2)*(2*A*b*c*e**2 - 4*A*c**2*d*e + B*b**2*e**2 - 8*B*b*c*d*e + 10*B*c**2*d**2)/(7*e**5) + (d + e*x)**(5/2)*(A*b**2*e**3 - 6*A*b*c*d*e**2 + 6*A*c**2*d**2*e - 3*B*b**2*d* e**2 + 12*B*b*c*d**2*e - 10*B*c**2*d**3)/(5*e**5) + (d + e*x)**(3/2)*(-2*A *b**2*d*e**3 + 6*A*b*c*d**2*e**2 - 4*A*c**2*d**3*e + 3*B*b**2*d**2*e**2 - 8*B*b*c*d**3*e + 5*B*c**2*d**4)/(3*e**5) + sqrt(d + e*x)*(A*b**2*d**2*e**3 - 2*A*b*c*d**3*e**2 + A*c**2*d**4*e - B*b**2*d**3*e**2 + 2*B*b*c*d**4*e - B*c**2*d**5)/e**5)/e, Ne(e, 0)), ((A*b**2*x**3/3 + B*c**2*x**6/6 + x**5*( A*c**2 + 2*B*b*c)/5 + x**4*(2*A*b*c + B*b**2)/4)/sqrt(d), True))
Time = 0.03 (sec) , antiderivative size = 291, normalized size of antiderivative = 1.10 \[ \int \frac {(A+B x) \left (b x+c x^2\right )^2}{\sqrt {d+e x}} \, dx=\frac {2 \, {\left (315 \, {\left (e x + d\right )}^{\frac {11}{2}} B c^{2} - 385 \, {\left (5 \, B c^{2} d - {\left (2 \, B b c + A c^{2}\right )} e\right )} {\left (e x + d\right )}^{\frac {9}{2}} + 495 \, {\left (10 \, B c^{2} d^{2} - 4 \, {\left (2 \, B b c + A c^{2}\right )} d e + {\left (B b^{2} + 2 \, A b c\right )} e^{2}\right )} {\left (e x + d\right )}^{\frac {7}{2}} - 693 \, {\left (10 \, B c^{2} d^{3} - A b^{2} e^{3} - 6 \, {\left (2 \, B b c + A c^{2}\right )} d^{2} e + 3 \, {\left (B b^{2} + 2 \, A b c\right )} d e^{2}\right )} {\left (e x + d\right )}^{\frac {5}{2}} + 1155 \, {\left (5 \, B c^{2} d^{4} - 2 \, A b^{2} d e^{3} - 4 \, {\left (2 \, B b c + A c^{2}\right )} d^{3} e + 3 \, {\left (B b^{2} + 2 \, A b c\right )} d^{2} e^{2}\right )} {\left (e x + d\right )}^{\frac {3}{2}} - 3465 \, {\left (B c^{2} d^{5} - A b^{2} d^{2} e^{3} - {\left (2 \, B b c + A c^{2}\right )} d^{4} e + {\left (B b^{2} + 2 \, A b c\right )} d^{3} e^{2}\right )} \sqrt {e x + d}\right )}}{3465 \, e^{6}} \] Input:
integrate((B*x+A)*(c*x^2+b*x)^2/(e*x+d)^(1/2),x, algorithm="maxima")
Output:
2/3465*(315*(e*x + d)^(11/2)*B*c^2 - 385*(5*B*c^2*d - (2*B*b*c + A*c^2)*e) *(e*x + d)^(9/2) + 495*(10*B*c^2*d^2 - 4*(2*B*b*c + A*c^2)*d*e + (B*b^2 + 2*A*b*c)*e^2)*(e*x + d)^(7/2) - 693*(10*B*c^2*d^3 - A*b^2*e^3 - 6*(2*B*b*c + A*c^2)*d^2*e + 3*(B*b^2 + 2*A*b*c)*d*e^2)*(e*x + d)^(5/2) + 1155*(5*B*c ^2*d^4 - 2*A*b^2*d*e^3 - 4*(2*B*b*c + A*c^2)*d^3*e + 3*(B*b^2 + 2*A*b*c)*d ^2*e^2)*(e*x + d)^(3/2) - 3465*(B*c^2*d^5 - A*b^2*d^2*e^3 - (2*B*b*c + A*c ^2)*d^4*e + (B*b^2 + 2*A*b*c)*d^3*e^2)*sqrt(e*x + d))/e^6
Time = 0.28 (sec) , antiderivative size = 358, normalized size of antiderivative = 1.35 \[ \int \frac {(A+B x) \left (b x+c x^2\right )^2}{\sqrt {d+e x}} \, dx=\frac {2 \, {\left (\frac {231 \, {\left (3 \, {\left (e x + d\right )}^{\frac {5}{2}} - 10 \, {\left (e x + d\right )}^{\frac {3}{2}} d + 15 \, \sqrt {e x + d} d^{2}\right )} A b^{2}}{e^{2}} + \frac {99 \, {\left (5 \, {\left (e x + d\right )}^{\frac {7}{2}} - 21 \, {\left (e x + d\right )}^{\frac {5}{2}} d + 35 \, {\left (e x + d\right )}^{\frac {3}{2}} d^{2} - 35 \, \sqrt {e x + d} d^{3}\right )} B b^{2}}{e^{3}} + \frac {198 \, {\left (5 \, {\left (e x + d\right )}^{\frac {7}{2}} - 21 \, {\left (e x + d\right )}^{\frac {5}{2}} d + 35 \, {\left (e x + d\right )}^{\frac {3}{2}} d^{2} - 35 \, \sqrt {e x + d} d^{3}\right )} A b c}{e^{3}} + \frac {22 \, {\left (35 \, {\left (e x + d\right )}^{\frac {9}{2}} - 180 \, {\left (e x + d\right )}^{\frac {7}{2}} d + 378 \, {\left (e x + d\right )}^{\frac {5}{2}} d^{2} - 420 \, {\left (e x + d\right )}^{\frac {3}{2}} d^{3} + 315 \, \sqrt {e x + d} d^{4}\right )} B b c}{e^{4}} + \frac {11 \, {\left (35 \, {\left (e x + d\right )}^{\frac {9}{2}} - 180 \, {\left (e x + d\right )}^{\frac {7}{2}} d + 378 \, {\left (e x + d\right )}^{\frac {5}{2}} d^{2} - 420 \, {\left (e x + d\right )}^{\frac {3}{2}} d^{3} + 315 \, \sqrt {e x + d} d^{4}\right )} A c^{2}}{e^{4}} + \frac {5 \, {\left (63 \, {\left (e x + d\right )}^{\frac {11}{2}} - 385 \, {\left (e x + d\right )}^{\frac {9}{2}} d + 990 \, {\left (e x + d\right )}^{\frac {7}{2}} d^{2} - 1386 \, {\left (e x + d\right )}^{\frac {5}{2}} d^{3} + 1155 \, {\left (e x + d\right )}^{\frac {3}{2}} d^{4} - 693 \, \sqrt {e x + d} d^{5}\right )} B c^{2}}{e^{5}}\right )}}{3465 \, e} \] Input:
integrate((B*x+A)*(c*x^2+b*x)^2/(e*x+d)^(1/2),x, algorithm="giac")
Output:
2/3465*(231*(3*(e*x + d)^(5/2) - 10*(e*x + d)^(3/2)*d + 15*sqrt(e*x + d)*d ^2)*A*b^2/e^2 + 99*(5*(e*x + d)^(7/2) - 21*(e*x + d)^(5/2)*d + 35*(e*x + d )^(3/2)*d^2 - 35*sqrt(e*x + d)*d^3)*B*b^2/e^3 + 198*(5*(e*x + d)^(7/2) - 2 1*(e*x + d)^(5/2)*d + 35*(e*x + d)^(3/2)*d^2 - 35*sqrt(e*x + d)*d^3)*A*b*c /e^3 + 22*(35*(e*x + d)^(9/2) - 180*(e*x + d)^(7/2)*d + 378*(e*x + d)^(5/2 )*d^2 - 420*(e*x + d)^(3/2)*d^3 + 315*sqrt(e*x + d)*d^4)*B*b*c/e^4 + 11*(3 5*(e*x + d)^(9/2) - 180*(e*x + d)^(7/2)*d + 378*(e*x + d)^(5/2)*d^2 - 420* (e*x + d)^(3/2)*d^3 + 315*sqrt(e*x + d)*d^4)*A*c^2/e^4 + 5*(63*(e*x + d)^( 11/2) - 385*(e*x + d)^(9/2)*d + 990*(e*x + d)^(7/2)*d^2 - 1386*(e*x + d)^( 5/2)*d^3 + 1155*(e*x + d)^(3/2)*d^4 - 693*sqrt(e*x + d)*d^5)*B*c^2/e^5)/e
Time = 10.72 (sec) , antiderivative size = 254, normalized size of antiderivative = 0.96 \[ \int \frac {(A+B x) \left (b x+c x^2\right )^2}{\sqrt {d+e x}} \, dx=\frac {{\left (d+e\,x\right )}^{9/2}\,\left (2\,A\,c^2\,e-10\,B\,c^2\,d+4\,B\,b\,c\,e\right )}{9\,e^6}+\frac {{\left (d+e\,x\right )}^{5/2}\,\left (-6\,B\,b^2\,d\,e^2+2\,A\,b^2\,e^3+24\,B\,b\,c\,d^2\,e-12\,A\,b\,c\,d\,e^2-20\,B\,c^2\,d^3+12\,A\,c^2\,d^2\,e\right )}{5\,e^6}+\frac {{\left (d+e\,x\right )}^{7/2}\,\left (2\,B\,b^2\,e^2-16\,B\,b\,c\,d\,e+4\,A\,b\,c\,e^2+20\,B\,c^2\,d^2-8\,A\,c^2\,d\,e\right )}{7\,e^6}+\frac {2\,B\,c^2\,{\left (d+e\,x\right )}^{11/2}}{11\,e^6}-\frac {2\,d\,\left (b\,e-c\,d\right )\,{\left (d+e\,x\right )}^{3/2}\,\left (2\,A\,b\,e^2+5\,B\,c\,d^2-4\,A\,c\,d\,e-3\,B\,b\,d\,e\right )}{3\,e^6}+\frac {2\,d^2\,\left (A\,e-B\,d\right )\,{\left (b\,e-c\,d\right )}^2\,\sqrt {d+e\,x}}{e^6} \] Input:
int(((b*x + c*x^2)^2*(A + B*x))/(d + e*x)^(1/2),x)
Output:
((d + e*x)^(9/2)*(2*A*c^2*e - 10*B*c^2*d + 4*B*b*c*e))/(9*e^6) + ((d + e*x )^(5/2)*(2*A*b^2*e^3 - 20*B*c^2*d^3 + 12*A*c^2*d^2*e - 6*B*b^2*d*e^2 - 12* A*b*c*d*e^2 + 24*B*b*c*d^2*e))/(5*e^6) + ((d + e*x)^(7/2)*(2*B*b^2*e^2 + 2 0*B*c^2*d^2 + 4*A*b*c*e^2 - 8*A*c^2*d*e - 16*B*b*c*d*e))/(7*e^6) + (2*B*c^ 2*(d + e*x)^(11/2))/(11*e^6) - (2*d*(b*e - c*d)*(d + e*x)^(3/2)*(2*A*b*e^2 + 5*B*c*d^2 - 4*A*c*d*e - 3*B*b*d*e))/(3*e^6) + (2*d^2*(A*e - B*d)*(b*e - c*d)^2*(d + e*x)^(1/2))/e^6
Time = 0.24 (sec) , antiderivative size = 340, normalized size of antiderivative = 1.28 \[ \int \frac {(A+B x) \left (b x+c x^2\right )^2}{\sqrt {d+e x}} \, dx=\frac {2 \sqrt {e x +d}\, \left (315 b \,c^{2} e^{5} x^{5}+385 a \,c^{2} e^{5} x^{4}+770 b^{2} c \,e^{5} x^{4}-350 b \,c^{2} d \,e^{4} x^{4}+990 a b c \,e^{5} x^{3}-440 a \,c^{2} d \,e^{4} x^{3}+495 b^{3} e^{5} x^{3}-880 b^{2} c d \,e^{4} x^{3}+400 b \,c^{2} d^{2} e^{3} x^{3}+693 a \,b^{2} e^{5} x^{2}-1188 a b c d \,e^{4} x^{2}+528 a \,c^{2} d^{2} e^{3} x^{2}-594 b^{3} d \,e^{4} x^{2}+1056 b^{2} c \,d^{2} e^{3} x^{2}-480 b \,c^{2} d^{3} e^{2} x^{2}-924 a \,b^{2} d \,e^{4} x +1584 a b c \,d^{2} e^{3} x -704 a \,c^{2} d^{3} e^{2} x +792 b^{3} d^{2} e^{3} x -1408 b^{2} c \,d^{3} e^{2} x +640 b \,c^{2} d^{4} e x +1848 a \,b^{2} d^{2} e^{3}-3168 a b c \,d^{3} e^{2}+1408 a \,c^{2} d^{4} e -1584 b^{3} d^{3} e^{2}+2816 b^{2} c \,d^{4} e -1280 b \,c^{2} d^{5}\right )}{3465 e^{6}} \] Input:
int((B*x+A)*(c*x^2+b*x)^2/(e*x+d)^(1/2),x)
Output:
(2*sqrt(d + e*x)*(1848*a*b**2*d**2*e**3 - 924*a*b**2*d*e**4*x + 693*a*b**2 *e**5*x**2 - 3168*a*b*c*d**3*e**2 + 1584*a*b*c*d**2*e**3*x - 1188*a*b*c*d* e**4*x**2 + 990*a*b*c*e**5*x**3 + 1408*a*c**2*d**4*e - 704*a*c**2*d**3*e** 2*x + 528*a*c**2*d**2*e**3*x**2 - 440*a*c**2*d*e**4*x**3 + 385*a*c**2*e**5 *x**4 - 1584*b**3*d**3*e**2 + 792*b**3*d**2*e**3*x - 594*b**3*d*e**4*x**2 + 495*b**3*e**5*x**3 + 2816*b**2*c*d**4*e - 1408*b**2*c*d**3*e**2*x + 1056 *b**2*c*d**2*e**3*x**2 - 880*b**2*c*d*e**4*x**3 + 770*b**2*c*e**5*x**4 - 1 280*b*c**2*d**5 + 640*b*c**2*d**4*e*x - 480*b*c**2*d**3*e**2*x**2 + 400*b* c**2*d**2*e**3*x**3 - 350*b*c**2*d*e**4*x**4 + 315*b*c**2*e**5*x**5))/(346 5*e**6)