Integrand size = 32, antiderivative size = 322 \[ \int \frac {(a+b x)^2 \left (A+B x+C x^2+D x^3\right )}{(c+d x)^{3/2}} \, dx=-\frac {2 (b c-a d)^2 \left (c^2 C d-B c d^2+A d^3-c^3 D\right )}{d^6 \sqrt {c+d x}}+\frac {2 (b c-a d) \left (a d \left (2 c C d-B d^2-3 c^2 D\right )-b \left (4 c^2 C d-3 B c d^2+2 A d^3-5 c^3 D\right )\right ) \sqrt {c+d x}}{d^6}+\frac {2 \left (a^2 d^2 (C d-3 c D)-2 a b d \left (3 c C d-B d^2-6 c^2 D\right )+b^2 \left (6 c^2 C d-3 B c d^2+A d^3-10 c^3 D\right )\right ) (c+d x)^{3/2}}{3 d^6}+\frac {2 \left (a^2 d^2 D+2 a b d (C d-4 c D)-b^2 \left (4 c C d-B d^2-10 c^2 D\right )\right ) (c+d x)^{5/2}}{5 d^6}+\frac {2 b (b C d-5 b c D+2 a d D) (c+d x)^{7/2}}{7 d^6}+\frac {2 b^2 D (c+d x)^{9/2}}{9 d^6} \]
2/3*(a^2*d^2*(C*d-3*D*c)-2*a*b*d*(-B*d^2+3*C*c*d-6*D*c^2)+b^2*(A*d^3-3*B*c *d^2+6*C*c^2*d-10*D*c^3))*(d*x+c)^(3/2)/d^6+2/5*(a^2*d^2*D+2*a*b*d*(C*d-4* D*c)-b^2*(-B*d^2+4*C*c*d-10*D*c^2))*(d*x+c)^(5/2)/d^6+2/7*b*(C*b*d+2*D*a*d -5*D*b*c)*(d*x+c)^(7/2)/d^6+2/9*b^2*D*(d*x+c)^(9/2)/d^6-2*(-a*d+b*c)^2*(A* d^3-B*c*d^2+C*c^2*d-D*c^3)/d^6/(d*x+c)^(1/2)+2*(-a*d+b*c)*(a*d*(-B*d^2+2*C *c*d-3*D*c^2)-b*(2*A*d^3-3*B*c*d^2+4*C*c^2*d-5*D*c^3))*(d*x+c)^(1/2)/d^6
Time = 0.32 (sec) , antiderivative size = 326, normalized size of antiderivative = 1.01 \[ \int \frac {(a+b x)^2 \left (A+B x+C x^2+D x^3\right )}{(c+d x)^{3/2}} \, dx=\frac {2 \left (21 a^2 d^2 \left (48 c^3 D-8 c^2 d (5 C-3 D x)+2 c d^2 (15 B-x (10 C+3 D x))+d^3 \left (-15 A+x \left (15 B+5 C x+3 D x^2\right )\right )\right )+6 a b d \left (-384 c^4 D+48 c^3 d (7 C-4 D x)-8 c^2 d^2 (35 B-3 x (7 C+2 D x))+2 c d^3 (105 A-x (70 B+3 x (7 C+4 D x)))+d^4 x (105 A+x (35 B+3 x (7 C+5 D x)))\right )+b^2 \left (1280 c^5 D-128 c^4 d (9 C-5 D x)+16 c^3 d^2 (63 B-2 x (18 C+5 D x))+8 c^2 d^3 (-105 A+x (63 B+2 x (9 C+5 D x)))+d^5 x^2 (105 A+x (63 B+5 x (9 C+7 D x)))-2 c d^4 x (210 A+x (63 B+x (36 C+25 D x)))\right )\right )}{315 d^6 \sqrt {c+d x}} \]
(2*(21*a^2*d^2*(48*c^3*D - 8*c^2*d*(5*C - 3*D*x) + 2*c*d^2*(15*B - x*(10*C + 3*D*x)) + d^3*(-15*A + x*(15*B + 5*C*x + 3*D*x^2))) + 6*a*b*d*(-384*c^4 *D + 48*c^3*d*(7*C - 4*D*x) - 8*c^2*d^2*(35*B - 3*x*(7*C + 2*D*x)) + 2*c*d ^3*(105*A - x*(70*B + 3*x*(7*C + 4*D*x))) + d^4*x*(105*A + x*(35*B + 3*x*( 7*C + 5*D*x)))) + b^2*(1280*c^5*D - 128*c^4*d*(9*C - 5*D*x) + 16*c^3*d^2*( 63*B - 2*x*(18*C + 5*D*x)) + 8*c^2*d^3*(-105*A + x*(63*B + 2*x*(9*C + 5*D* x))) + d^5*x^2*(105*A + x*(63*B + 5*x*(9*C + 7*D*x))) - 2*c*d^4*x*(210*A + x*(63*B + x*(36*C + 25*D*x))))))/(315*d^6*Sqrt[c + d*x])
Time = 0.56 (sec) , antiderivative size = 322, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.062, Rules used = {2123, 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.
| \(\displaystyle \int \frac {(a+b x)^2 \left (A+B x+C x^2+D x^3\right )}{(c+d x)^{3/2}} \, dx\) |
| \(\Big \downarrow \) 2123 |
| \(\displaystyle \int \left (\frac {\sqrt {c+d x} \left (a^2 d^2 (C d-3 c D)-2 a b d \left (-B d^2-6 c^2 D+3 c C d\right )+b^2 \left (A d^3-3 B c d^2-10 c^3 D+6 c^2 C d\right )\right )}{d^5}+\frac {(c+d x)^{3/2} \left (a^2 d^2 D+2 a b d (C d-4 c D)-\left (b^2 \left (-B d^2-10 c^2 D+4 c C d\right )\right )\right )}{d^5}+\frac {(b c-a d) \left (a d \left (-B d^2-3 c^2 D+2 c C d\right )-b \left (2 A d^3-3 B c d^2-5 c^3 D+4 c^2 C d\right )\right )}{d^5 \sqrt {c+d x}}+\frac {(a d-b c)^2 \left (A d^3-B c d^2+c^3 (-D)+c^2 C d\right )}{d^5 (c+d x)^{3/2}}+\frac {b (c+d x)^{5/2} (2 a d D-5 b c D+b C d)}{d^5}+\frac {b^2 D (c+d x)^{7/2}}{d^5}\right )dx\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {2 (c+d x)^{3/2} \left (a^2 d^2 (C d-3 c D)-2 a b d \left (-B d^2-6 c^2 D+3 c C d\right )+b^2 \left (A d^3-3 B c d^2-10 c^3 D+6 c^2 C d\right )\right )}{3 d^6}+\frac {2 (c+d x)^{5/2} \left (a^2 d^2 D+2 a b d (C d-4 c D)-\left (b^2 \left (-B d^2-10 c^2 D+4 c C d\right )\right )\right )}{5 d^6}+\frac {2 \sqrt {c+d x} (b c-a d) \left (a d \left (-B d^2-3 c^2 D+2 c C d\right )-b \left (2 A d^3-3 B c d^2-5 c^3 D+4 c^2 C d\right )\right )}{d^6}-\frac {2 (b c-a d)^2 \left (A d^3-B c d^2+c^3 (-D)+c^2 C d\right )}{d^6 \sqrt {c+d x}}+\frac {2 b (c+d x)^{7/2} (2 a d D-5 b c D+b C d)}{7 d^6}+\frac {2 b^2 D (c+d x)^{9/2}}{9 d^6}\) |
(-2*(b*c - a*d)^2*(c^2*C*d - B*c*d^2 + A*d^3 - c^3*D))/(d^6*Sqrt[c + d*x]) + (2*(b*c - a*d)*(a*d*(2*c*C*d - B*d^2 - 3*c^2*D) - b*(4*c^2*C*d - 3*B*c* d^2 + 2*A*d^3 - 5*c^3*D))*Sqrt[c + d*x])/d^6 + (2*(a^2*d^2*(C*d - 3*c*D) - 2*a*b*d*(3*c*C*d - B*d^2 - 6*c^2*D) + b^2*(6*c^2*C*d - 3*B*c*d^2 + A*d^3 - 10*c^3*D))*(c + d*x)^(3/2))/(3*d^6) + (2*(a^2*d^2*D + 2*a*b*d*(C*d - 4*c *D) - b^2*(4*c*C*d - B*d^2 - 10*c^2*D))*(c + d*x)^(5/2))/(5*d^6) + (2*b*(b *C*d - 5*b*c*D + 2*a*d*D)*(c + d*x)^(7/2))/(7*d^6) + (2*b^2*D*(c + d*x)^(9 /2))/(9*d^6)
3.1.11.3.1 Defintions of rubi rules used
Int[(Px_)*((a_.) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_.), x_Symbol] :> Int[ExpandIntegrand[Px*(a + b*x)^m*(c + d*x)^n, x], x] /; FreeQ[{a, b, c , d, m, n}, x] && PolyQ[Px, x] && (IntegersQ[m, n] || IGtQ[m, -2])
Time = 1.74 (sec) , antiderivative size = 289, normalized size of antiderivative = 0.90
| method | result | size |
| pseudoelliptic | \(\frac {\left (\left (70 D x^{5}+90 C \,x^{4}+126 x^{3} B +210 A \,x^{2}\right ) b^{2}+1260 a \left (A +\frac {1}{7} D x^{3}+\frac {1}{5} C \,x^{2}+\frac {1}{3} B x \right ) x b -630 a^{2} \left (-\frac {1}{5} D x^{3}-\frac {1}{3} C \,x^{2}-B x +A \right )\right ) d^{5}+2520 \left (-\frac {\left (\frac {5}{42} D x^{3}+\frac {6}{35} C \,x^{2}+\frac {3}{10} B x +A \right ) x \,b^{2}}{3}+a \left (-\frac {4}{35} D x^{3}-\frac {1}{5} C \,x^{2}-\frac {2}{3} B x +A \right ) b +\frac {a^{2} \left (-\frac {1}{5} D x^{2}-\frac {2}{3} C x +B \right )}{2}\right ) c \,d^{4}-1680 c^{2} \left (\left (-\frac {2}{21} D x^{3}-\frac {6}{35} C \,x^{2}-\frac {3}{5} B x +A \right ) b^{2}+2 a \left (-\frac {6}{35} D x^{2}-\frac {3}{5} C x +B \right ) b +a^{2} \left (-\frac {3 D x}{5}+C \right )\right ) d^{3}+2016 c^{3} \left (\left (-\frac {10}{63} D x^{2}-\frac {4}{7} C x +B \right ) b^{2}+2 a \left (-\frac {4 D x}{7}+C \right ) b +D a^{2}\right ) d^{2}-2304 \left (\left (-\frac {5 D x}{9}+C \right ) b +2 D a \right ) b \,c^{4} d +2560 D b^{2} c^{5}}{315 \sqrt {d x +c}\, d^{6}}\) | \(289\) |
| gosper | \(-\frac {2 \left (-35 D b^{2} x^{5} d^{5}-45 C \,b^{2} d^{5} x^{4}-90 D a b \,d^{5} x^{4}+50 D b^{2} c \,d^{4} x^{4}-63 B \,b^{2} d^{5} x^{3}-126 C a b \,d^{5} x^{3}+72 C \,b^{2} c \,d^{4} x^{3}-63 D a^{2} d^{5} x^{3}+144 D a b c \,d^{4} x^{3}-80 D b^{2} c^{2} d^{3} x^{3}-105 A \,b^{2} d^{5} x^{2}-210 B a b \,d^{5} x^{2}+126 B \,b^{2} c \,d^{4} x^{2}-105 C \,a^{2} d^{5} x^{2}+252 C a b c \,d^{4} x^{2}-144 C \,b^{2} c^{2} d^{3} x^{2}+126 D a^{2} c \,d^{4} x^{2}-288 D a b \,c^{2} d^{3} x^{2}+160 D b^{2} c^{3} d^{2} x^{2}-630 A a b \,d^{5} x +420 A \,b^{2} c \,d^{4} x -315 B \,a^{2} d^{5} x +840 B a b c \,d^{4} x -504 B \,b^{2} c^{2} d^{3} x +420 C \,a^{2} c \,d^{4} x -1008 C a b \,c^{2} d^{3} x +576 C \,b^{2} c^{3} d^{2} x -504 D a^{2} c^{2} d^{3} x +1152 D a b \,c^{3} d^{2} x -640 D b^{2} c^{4} d x +315 a^{2} A \,d^{5}-1260 A a b c \,d^{4}+840 A \,b^{2} c^{2} d^{3}-630 B \,a^{2} c \,d^{4}+1680 B a b \,c^{2} d^{3}-1008 B \,b^{2} c^{3} d^{2}+840 C \,a^{2} c^{2} d^{3}-2016 C a b \,c^{3} d^{2}+1152 C \,b^{2} c^{4} d -1008 D a^{2} c^{3} d^{2}+2304 D a b \,c^{4} d -1280 D b^{2} c^{5}\right )}{315 \sqrt {d x +c}\, d^{6}}\) | \(505\) |
| trager | \(-\frac {2 \left (-35 D b^{2} x^{5} d^{5}-45 C \,b^{2} d^{5} x^{4}-90 D a b \,d^{5} x^{4}+50 D b^{2} c \,d^{4} x^{4}-63 B \,b^{2} d^{5} x^{3}-126 C a b \,d^{5} x^{3}+72 C \,b^{2} c \,d^{4} x^{3}-63 D a^{2} d^{5} x^{3}+144 D a b c \,d^{4} x^{3}-80 D b^{2} c^{2} d^{3} x^{3}-105 A \,b^{2} d^{5} x^{2}-210 B a b \,d^{5} x^{2}+126 B \,b^{2} c \,d^{4} x^{2}-105 C \,a^{2} d^{5} x^{2}+252 C a b c \,d^{4} x^{2}-144 C \,b^{2} c^{2} d^{3} x^{2}+126 D a^{2} c \,d^{4} x^{2}-288 D a b \,c^{2} d^{3} x^{2}+160 D b^{2} c^{3} d^{2} x^{2}-630 A a b \,d^{5} x +420 A \,b^{2} c \,d^{4} x -315 B \,a^{2} d^{5} x +840 B a b c \,d^{4} x -504 B \,b^{2} c^{2} d^{3} x +420 C \,a^{2} c \,d^{4} x -1008 C a b \,c^{2} d^{3} x +576 C \,b^{2} c^{3} d^{2} x -504 D a^{2} c^{2} d^{3} x +1152 D a b \,c^{3} d^{2} x -640 D b^{2} c^{4} d x +315 a^{2} A \,d^{5}-1260 A a b c \,d^{4}+840 A \,b^{2} c^{2} d^{3}-630 B \,a^{2} c \,d^{4}+1680 B a b \,c^{2} d^{3}-1008 B \,b^{2} c^{3} d^{2}+840 C \,a^{2} c^{2} d^{3}-2016 C a b \,c^{3} d^{2}+1152 C \,b^{2} c^{4} d -1008 D a^{2} c^{3} d^{2}+2304 D a b \,c^{4} d -1280 D b^{2} c^{5}\right )}{315 \sqrt {d x +c}\, d^{6}}\) | \(505\) |
| derivativedivides | \(\frac {\frac {2 D b^{2} \left (d x +c \right )^{\frac {9}{2}}}{9}-\frac {2 \left (a^{2} A \,d^{5}-2 A a b c \,d^{4}+A \,b^{2} c^{2} d^{3}-B \,a^{2} c \,d^{4}+2 B a b \,c^{2} d^{3}-B \,b^{2} c^{3} d^{2}+C \,a^{2} c^{2} d^{3}-2 C a b \,c^{3} d^{2}+C \,b^{2} c^{4} d -D a^{2} c^{3} d^{2}+2 D a b \,c^{4} d -D b^{2} c^{5}\right )}{\sqrt {d x +c}}-16 D a b \,c^{3} d \sqrt {d x +c}-\frac {16 D a b c d \left (d x +c \right )^{\frac {5}{2}}}{5}-4 C a b c \,d^{2} \left (d x +c \right )^{\frac {3}{2}}+8 D a b \,c^{2} d \left (d x +c \right )^{\frac {3}{2}}-8 B a b c \,d^{3} \sqrt {d x +c}+12 C a b \,c^{2} d^{2} \sqrt {d x +c}+10 D b^{2} c^{4} \sqrt {d x +c}+\frac {2 A \,b^{2} d^{3} \left (d x +c \right )^{\frac {3}{2}}}{3}+\frac {2 C \,a^{2} d^{3} \left (d x +c \right )^{\frac {3}{2}}}{3}-\frac {10 D b^{2} c \left (d x +c \right )^{\frac {7}{2}}}{7}+\frac {2 B \,b^{2} d^{2} \left (d x +c \right )^{\frac {5}{2}}}{5}+\frac {2 D a^{2} d^{2} \left (d x +c \right )^{\frac {5}{2}}}{5}+4 D b^{2} c^{2} \left (d x +c \right )^{\frac {5}{2}}+\frac {2 C \,b^{2} d \left (d x +c \right )^{\frac {7}{2}}}{7}+2 B \,a^{2} d^{4} \sqrt {d x +c}-\frac {20 D b^{2} c^{3} \left (d x +c \right )^{\frac {3}{2}}}{3}+\frac {4 B a b \,d^{3} \left (d x +c \right )^{\frac {3}{2}}}{3}-2 B \,b^{2} c \,d^{2} \left (d x +c \right )^{\frac {3}{2}}+4 C \,b^{2} c^{2} d \left (d x +c \right )^{\frac {3}{2}}-2 D a^{2} c \,d^{2} \left (d x +c \right )^{\frac {3}{2}}+6 B \,b^{2} c^{2} d^{2} \sqrt {d x +c}-4 C \,a^{2} c \,d^{3} \sqrt {d x +c}-8 C \,b^{2} c^{3} d \sqrt {d x +c}+6 D a^{2} c^{2} d^{2} \sqrt {d x +c}-4 A \,b^{2} c \,d^{3} \sqrt {d x +c}+4 A a b \,d^{4} \sqrt {d x +c}+\frac {4 D a b d \left (d x +c \right )^{\frac {7}{2}}}{7}+\frac {4 C a b \,d^{2} \left (d x +c \right )^{\frac {5}{2}}}{5}-\frac {8 C \,b^{2} c d \left (d x +c \right )^{\frac {5}{2}}}{5}}{d^{6}}\) | \(616\) |
| default | \(\frac {\frac {2 D b^{2} \left (d x +c \right )^{\frac {9}{2}}}{9}-\frac {2 \left (a^{2} A \,d^{5}-2 A a b c \,d^{4}+A \,b^{2} c^{2} d^{3}-B \,a^{2} c \,d^{4}+2 B a b \,c^{2} d^{3}-B \,b^{2} c^{3} d^{2}+C \,a^{2} c^{2} d^{3}-2 C a b \,c^{3} d^{2}+C \,b^{2} c^{4} d -D a^{2} c^{3} d^{2}+2 D a b \,c^{4} d -D b^{2} c^{5}\right )}{\sqrt {d x +c}}-16 D a b \,c^{3} d \sqrt {d x +c}-\frac {16 D a b c d \left (d x +c \right )^{\frac {5}{2}}}{5}-4 C a b c \,d^{2} \left (d x +c \right )^{\frac {3}{2}}+8 D a b \,c^{2} d \left (d x +c \right )^{\frac {3}{2}}-8 B a b c \,d^{3} \sqrt {d x +c}+12 C a b \,c^{2} d^{2} \sqrt {d x +c}+10 D b^{2} c^{4} \sqrt {d x +c}+\frac {2 A \,b^{2} d^{3} \left (d x +c \right )^{\frac {3}{2}}}{3}+\frac {2 C \,a^{2} d^{3} \left (d x +c \right )^{\frac {3}{2}}}{3}-\frac {10 D b^{2} c \left (d x +c \right )^{\frac {7}{2}}}{7}+\frac {2 B \,b^{2} d^{2} \left (d x +c \right )^{\frac {5}{2}}}{5}+\frac {2 D a^{2} d^{2} \left (d x +c \right )^{\frac {5}{2}}}{5}+4 D b^{2} c^{2} \left (d x +c \right )^{\frac {5}{2}}+\frac {2 C \,b^{2} d \left (d x +c \right )^{\frac {7}{2}}}{7}+2 B \,a^{2} d^{4} \sqrt {d x +c}-\frac {20 D b^{2} c^{3} \left (d x +c \right )^{\frac {3}{2}}}{3}+\frac {4 B a b \,d^{3} \left (d x +c \right )^{\frac {3}{2}}}{3}-2 B \,b^{2} c \,d^{2} \left (d x +c \right )^{\frac {3}{2}}+4 C \,b^{2} c^{2} d \left (d x +c \right )^{\frac {3}{2}}-2 D a^{2} c \,d^{2} \left (d x +c \right )^{\frac {3}{2}}+6 B \,b^{2} c^{2} d^{2} \sqrt {d x +c}-4 C \,a^{2} c \,d^{3} \sqrt {d x +c}-8 C \,b^{2} c^{3} d \sqrt {d x +c}+6 D a^{2} c^{2} d^{2} \sqrt {d x +c}-4 A \,b^{2} c \,d^{3} \sqrt {d x +c}+4 A a b \,d^{4} \sqrt {d x +c}+\frac {4 D a b d \left (d x +c \right )^{\frac {7}{2}}}{7}+\frac {4 C a b \,d^{2} \left (d x +c \right )^{\frac {5}{2}}}{5}-\frac {8 C \,b^{2} c d \left (d x +c \right )^{\frac {5}{2}}}{5}}{d^{6}}\) | \(616\) |
1/315*(((70*D*x^5+90*C*x^4+126*B*x^3+210*A*x^2)*b^2+1260*a*(A+1/7*D*x^3+1/ 5*C*x^2+1/3*B*x)*x*b-630*a^2*(-1/5*D*x^3-1/3*C*x^2-B*x+A))*d^5+2520*(-1/3* (5/42*D*x^3+6/35*C*x^2+3/10*B*x+A)*x*b^2+a*(-4/35*D*x^3-1/5*C*x^2-2/3*B*x+ A)*b+1/2*a^2*(-1/5*D*x^2-2/3*C*x+B))*c*d^4-1680*c^2*((-2/21*D*x^3-6/35*C*x ^2-3/5*B*x+A)*b^2+2*a*(-6/35*D*x^2-3/5*C*x+B)*b+a^2*(-3/5*D*x+C))*d^3+2016 *c^3*((-10/63*D*x^2-4/7*C*x+B)*b^2+2*a*(-4/7*D*x+C)*b+D*a^2)*d^2-2304*((-5 /9*D*x+C)*b+2*D*a)*b*c^4*d+2560*D*b^2*c^5)/(d*x+c)^(1/2)/d^6
Time = 0.31 (sec) , antiderivative size = 419, normalized size of antiderivative = 1.30 \[ \int \frac {(a+b x)^2 \left (A+B x+C x^2+D x^3\right )}{(c+d x)^{3/2}} \, dx=\frac {2 \, {\left (35 \, D b^{2} d^{5} x^{5} + 1280 \, D b^{2} c^{5} - 315 \, A a^{2} d^{5} - 840 \, {\left (C a^{2} + 2 \, B a b + A b^{2}\right )} c^{2} d^{3} + 630 \, {\left (B a^{2} + 2 \, A a b\right )} c d^{4} - 5 \, {\left (10 \, D b^{2} c d^{4} - 9 \, {\left (2 \, D a b + C b^{2}\right )} d^{5}\right )} x^{4} + {\left (80 \, D b^{2} c^{2} d^{3} + 63 \, {\left (D a^{2} + 2 \, C a b + B b^{2}\right )} d^{5} - 72 \, {\left (2 \, D a b c + C b^{2} c\right )} d^{4}\right )} x^{3} + 1008 \, {\left (D a^{2} c^{3} + {\left (2 \, C a b + B b^{2}\right )} c^{3}\right )} d^{2} - {\left (160 \, D b^{2} c^{3} d^{2} - 105 \, {\left (C a^{2} + 2 \, B a b + A b^{2}\right )} d^{5} + 126 \, {\left (D a^{2} c + {\left (2 \, C a b + B b^{2}\right )} c\right )} d^{4} - 144 \, {\left (2 \, D a b c^{2} + C b^{2} c^{2}\right )} d^{3}\right )} x^{2} - 1152 \, {\left (2 \, D a b c^{4} + C b^{2} c^{4}\right )} d + {\left (640 \, D b^{2} c^{4} d - 420 \, {\left (C a^{2} + 2 \, B a b + A b^{2}\right )} c d^{4} + 315 \, {\left (B a^{2} + 2 \, A a b\right )} d^{5} + 504 \, {\left (D a^{2} c^{2} + {\left (2 \, C a b + B b^{2}\right )} c^{2}\right )} d^{3} - 576 \, {\left (2 \, D a b c^{3} + C b^{2} c^{3}\right )} d^{2}\right )} x\right )} \sqrt {d x + c}}{315 \, {\left (d^{7} x + c d^{6}\right )}} \]
2/315*(35*D*b^2*d^5*x^5 + 1280*D*b^2*c^5 - 315*A*a^2*d^5 - 840*(C*a^2 + 2* B*a*b + A*b^2)*c^2*d^3 + 630*(B*a^2 + 2*A*a*b)*c*d^4 - 5*(10*D*b^2*c*d^4 - 9*(2*D*a*b + C*b^2)*d^5)*x^4 + (80*D*b^2*c^2*d^3 + 63*(D*a^2 + 2*C*a*b + B*b^2)*d^5 - 72*(2*D*a*b*c + C*b^2*c)*d^4)*x^3 + 1008*(D*a^2*c^3 + (2*C*a* b + B*b^2)*c^3)*d^2 - (160*D*b^2*c^3*d^2 - 105*(C*a^2 + 2*B*a*b + A*b^2)*d ^5 + 126*(D*a^2*c + (2*C*a*b + B*b^2)*c)*d^4 - 144*(2*D*a*b*c^2 + C*b^2*c^ 2)*d^3)*x^2 - 1152*(2*D*a*b*c^4 + C*b^2*c^4)*d + (640*D*b^2*c^4*d - 420*(C *a^2 + 2*B*a*b + A*b^2)*c*d^4 + 315*(B*a^2 + 2*A*a*b)*d^5 + 504*(D*a^2*c^2 + (2*C*a*b + B*b^2)*c^2)*d^3 - 576*(2*D*a*b*c^3 + C*b^2*c^3)*d^2)*x)*sqrt (d*x + c)/(d^7*x + c*d^6)
Time = 27.76 (sec) , antiderivative size = 534, normalized size of antiderivative = 1.66 \[ \int \frac {(a+b x)^2 \left (A+B x+C x^2+D x^3\right )}{(c+d x)^{3/2}} \, dx=\begin {cases} \frac {2 \left (\frac {D b^{2} \left (c + d x\right )^{\frac {9}{2}}}{9 d^{5}} + \frac {\left (c + d x\right )^{\frac {7}{2}} \left (C b^{2} d + 2 D a b d - 5 D b^{2} c\right )}{7 d^{5}} + \frac {\left (c + d x\right )^{\frac {5}{2}} \left (B b^{2} d^{2} + 2 C a b d^{2} - 4 C b^{2} c d + D a^{2} d^{2} - 8 D a b c d + 10 D b^{2} c^{2}\right )}{5 d^{5}} + \frac {\left (c + d x\right )^{\frac {3}{2}} \left (A b^{2} d^{3} + 2 B a b d^{3} - 3 B b^{2} c d^{2} + C a^{2} d^{3} - 6 C a b c d^{2} + 6 C b^{2} c^{2} d - 3 D a^{2} c d^{2} + 12 D a b c^{2} d - 10 D b^{2} c^{3}\right )}{3 d^{5}} + \frac {\sqrt {c + d x} \left (2 A a b d^{4} - 2 A b^{2} c d^{3} + B a^{2} d^{4} - 4 B a b c d^{3} + 3 B b^{2} c^{2} d^{2} - 2 C a^{2} c d^{3} + 6 C a b c^{2} d^{2} - 4 C b^{2} c^{3} d + 3 D a^{2} c^{2} d^{2} - 8 D a b c^{3} d + 5 D b^{2} c^{4}\right )}{d^{5}} + \frac {\left (a d - b c\right )^{2} \left (- A d^{3} + B c d^{2} - C c^{2} d + D c^{3}\right )}{d^{5} \sqrt {c + d x}}\right )}{d} & \text {for}\: d \neq 0 \\\frac {A a^{2} x + \frac {D b^{2} x^{6}}{6} + \frac {x^{5} \left (C b^{2} + 2 D a b\right )}{5} + \frac {x^{4} \left (B b^{2} + 2 C a b + D a^{2}\right )}{4} + \frac {x^{3} \left (A b^{2} + 2 B a b + C a^{2}\right )}{3} + \frac {x^{2} \cdot \left (2 A a b + B a^{2}\right )}{2}}{c^{\frac {3}{2}}} & \text {otherwise} \end {cases} \]
Piecewise((2*(D*b**2*(c + d*x)**(9/2)/(9*d**5) + (c + d*x)**(7/2)*(C*b**2* d + 2*D*a*b*d - 5*D*b**2*c)/(7*d**5) + (c + d*x)**(5/2)*(B*b**2*d**2 + 2*C *a*b*d**2 - 4*C*b**2*c*d + D*a**2*d**2 - 8*D*a*b*c*d + 10*D*b**2*c**2)/(5* d**5) + (c + d*x)**(3/2)*(A*b**2*d**3 + 2*B*a*b*d**3 - 3*B*b**2*c*d**2 + C *a**2*d**3 - 6*C*a*b*c*d**2 + 6*C*b**2*c**2*d - 3*D*a**2*c*d**2 + 12*D*a*b *c**2*d - 10*D*b**2*c**3)/(3*d**5) + sqrt(c + d*x)*(2*A*a*b*d**4 - 2*A*b** 2*c*d**3 + B*a**2*d**4 - 4*B*a*b*c*d**3 + 3*B*b**2*c**2*d**2 - 2*C*a**2*c* d**3 + 6*C*a*b*c**2*d**2 - 4*C*b**2*c**3*d + 3*D*a**2*c**2*d**2 - 8*D*a*b* c**3*d + 5*D*b**2*c**4)/d**5 + (a*d - b*c)**2*(-A*d**3 + B*c*d**2 - C*c**2 *d + D*c**3)/(d**5*sqrt(c + d*x)))/d, Ne(d, 0)), ((A*a**2*x + D*b**2*x**6/ 6 + x**5*(C*b**2 + 2*D*a*b)/5 + x**4*(B*b**2 + 2*C*a*b + D*a**2)/4 + x**3* (A*b**2 + 2*B*a*b + C*a**2)/3 + x**2*(2*A*a*b + B*a**2)/2)/c**(3/2), True) )
Time = 0.20 (sec) , antiderivative size = 395, normalized size of antiderivative = 1.23 \[ \int \frac {(a+b x)^2 \left (A+B x+C x^2+D x^3\right )}{(c+d x)^{3/2}} \, dx=\frac {2 \, {\left (\frac {35 \, {\left (d x + c\right )}^{\frac {9}{2}} D b^{2} - 45 \, {\left (5 \, D b^{2} c - {\left (2 \, D a b + C b^{2}\right )} d\right )} {\left (d x + c\right )}^{\frac {7}{2}} + 63 \, {\left (10 \, D b^{2} c^{2} - 4 \, {\left (2 \, D a b + C b^{2}\right )} c d + {\left (D a^{2} + 2 \, C a b + B b^{2}\right )} d^{2}\right )} {\left (d x + c\right )}^{\frac {5}{2}} - 105 \, {\left (10 \, D b^{2} c^{3} - 6 \, {\left (2 \, D a b + C b^{2}\right )} c^{2} d + 3 \, {\left (D a^{2} + 2 \, C a b + B b^{2}\right )} c d^{2} - {\left (C a^{2} + 2 \, B a b + A b^{2}\right )} d^{3}\right )} {\left (d x + c\right )}^{\frac {3}{2}} + 315 \, {\left (5 \, D b^{2} c^{4} - 4 \, {\left (2 \, D a b + C b^{2}\right )} c^{3} d + 3 \, {\left (D a^{2} + 2 \, C a b + B b^{2}\right )} c^{2} d^{2} - 2 \, {\left (C a^{2} + 2 \, B a b + A b^{2}\right )} c d^{3} + {\left (B a^{2} + 2 \, A a b\right )} d^{4}\right )} \sqrt {d x + c}}{d^{5}} + \frac {315 \, {\left (D b^{2} c^{5} - A a^{2} d^{5} - {\left (2 \, D a b + C b^{2}\right )} c^{4} d + {\left (D a^{2} + 2 \, C a b + B b^{2}\right )} c^{3} d^{2} - {\left (C a^{2} + 2 \, B a b + A b^{2}\right )} c^{2} d^{3} + {\left (B a^{2} + 2 \, A a b\right )} c d^{4}\right )}}{\sqrt {d x + c} d^{5}}\right )}}{315 \, d} \]
2/315*((35*(d*x + c)^(9/2)*D*b^2 - 45*(5*D*b^2*c - (2*D*a*b + C*b^2)*d)*(d *x + c)^(7/2) + 63*(10*D*b^2*c^2 - 4*(2*D*a*b + C*b^2)*c*d + (D*a^2 + 2*C* a*b + B*b^2)*d^2)*(d*x + c)^(5/2) - 105*(10*D*b^2*c^3 - 6*(2*D*a*b + C*b^2 )*c^2*d + 3*(D*a^2 + 2*C*a*b + B*b^2)*c*d^2 - (C*a^2 + 2*B*a*b + A*b^2)*d^ 3)*(d*x + c)^(3/2) + 315*(5*D*b^2*c^4 - 4*(2*D*a*b + C*b^2)*c^3*d + 3*(D*a ^2 + 2*C*a*b + B*b^2)*c^2*d^2 - 2*(C*a^2 + 2*B*a*b + A*b^2)*c*d^3 + (B*a^2 + 2*A*a*b)*d^4)*sqrt(d*x + c))/d^5 + 315*(D*b^2*c^5 - A*a^2*d^5 - (2*D*a* b + C*b^2)*c^4*d + (D*a^2 + 2*C*a*b + B*b^2)*c^3*d^2 - (C*a^2 + 2*B*a*b + A*b^2)*c^2*d^3 + (B*a^2 + 2*A*a*b)*c*d^4)/(sqrt(d*x + c)*d^5))/d
Leaf count of result is larger than twice the leaf count of optimal. 651 vs. \(2 (302) = 604\).
Time = 0.30 (sec) , antiderivative size = 651, normalized size of antiderivative = 2.02 \[ \int \frac {(a+b x)^2 \left (A+B x+C x^2+D x^3\right )}{(c+d x)^{3/2}} \, dx=\frac {2 \, {\left (D b^{2} c^{5} - 2 \, D a b c^{4} d - C b^{2} c^{4} d + D a^{2} c^{3} d^{2} + 2 \, C a b c^{3} d^{2} + B b^{2} c^{3} d^{2} - C a^{2} c^{2} d^{3} - 2 \, B a b c^{2} d^{3} - A b^{2} c^{2} d^{3} + B a^{2} c d^{4} + 2 \, A a b c d^{4} - A a^{2} d^{5}\right )}}{\sqrt {d x + c} d^{6}} + \frac {2 \, {\left (35 \, {\left (d x + c\right )}^{\frac {9}{2}} D b^{2} d^{48} - 225 \, {\left (d x + c\right )}^{\frac {7}{2}} D b^{2} c d^{48} + 630 \, {\left (d x + c\right )}^{\frac {5}{2}} D b^{2} c^{2} d^{48} - 1050 \, {\left (d x + c\right )}^{\frac {3}{2}} D b^{2} c^{3} d^{48} + 1575 \, \sqrt {d x + c} D b^{2} c^{4} d^{48} + 90 \, {\left (d x + c\right )}^{\frac {7}{2}} D a b d^{49} + 45 \, {\left (d x + c\right )}^{\frac {7}{2}} C b^{2} d^{49} - 504 \, {\left (d x + c\right )}^{\frac {5}{2}} D a b c d^{49} - 252 \, {\left (d x + c\right )}^{\frac {5}{2}} C b^{2} c d^{49} + 1260 \, {\left (d x + c\right )}^{\frac {3}{2}} D a b c^{2} d^{49} + 630 \, {\left (d x + c\right )}^{\frac {3}{2}} C b^{2} c^{2} d^{49} - 2520 \, \sqrt {d x + c} D a b c^{3} d^{49} - 1260 \, \sqrt {d x + c} C b^{2} c^{3} d^{49} + 63 \, {\left (d x + c\right )}^{\frac {5}{2}} D a^{2} d^{50} + 126 \, {\left (d x + c\right )}^{\frac {5}{2}} C a b d^{50} + 63 \, {\left (d x + c\right )}^{\frac {5}{2}} B b^{2} d^{50} - 315 \, {\left (d x + c\right )}^{\frac {3}{2}} D a^{2} c d^{50} - 630 \, {\left (d x + c\right )}^{\frac {3}{2}} C a b c d^{50} - 315 \, {\left (d x + c\right )}^{\frac {3}{2}} B b^{2} c d^{50} + 945 \, \sqrt {d x + c} D a^{2} c^{2} d^{50} + 1890 \, \sqrt {d x + c} C a b c^{2} d^{50} + 945 \, \sqrt {d x + c} B b^{2} c^{2} d^{50} + 105 \, {\left (d x + c\right )}^{\frac {3}{2}} C a^{2} d^{51} + 210 \, {\left (d x + c\right )}^{\frac {3}{2}} B a b d^{51} + 105 \, {\left (d x + c\right )}^{\frac {3}{2}} A b^{2} d^{51} - 630 \, \sqrt {d x + c} C a^{2} c d^{51} - 1260 \, \sqrt {d x + c} B a b c d^{51} - 630 \, \sqrt {d x + c} A b^{2} c d^{51} + 315 \, \sqrt {d x + c} B a^{2} d^{52} + 630 \, \sqrt {d x + c} A a b d^{52}\right )}}{315 \, d^{54}} \]
2*(D*b^2*c^5 - 2*D*a*b*c^4*d - C*b^2*c^4*d + D*a^2*c^3*d^2 + 2*C*a*b*c^3*d ^2 + B*b^2*c^3*d^2 - C*a^2*c^2*d^3 - 2*B*a*b*c^2*d^3 - A*b^2*c^2*d^3 + B*a ^2*c*d^4 + 2*A*a*b*c*d^4 - A*a^2*d^5)/(sqrt(d*x + c)*d^6) + 2/315*(35*(d*x + c)^(9/2)*D*b^2*d^48 - 225*(d*x + c)^(7/2)*D*b^2*c*d^48 + 630*(d*x + c)^ (5/2)*D*b^2*c^2*d^48 - 1050*(d*x + c)^(3/2)*D*b^2*c^3*d^48 + 1575*sqrt(d*x + c)*D*b^2*c^4*d^48 + 90*(d*x + c)^(7/2)*D*a*b*d^49 + 45*(d*x + c)^(7/2)* C*b^2*d^49 - 504*(d*x + c)^(5/2)*D*a*b*c*d^49 - 252*(d*x + c)^(5/2)*C*b^2* c*d^49 + 1260*(d*x + c)^(3/2)*D*a*b*c^2*d^49 + 630*(d*x + c)^(3/2)*C*b^2*c ^2*d^49 - 2520*sqrt(d*x + c)*D*a*b*c^3*d^49 - 1260*sqrt(d*x + c)*C*b^2*c^3 *d^49 + 63*(d*x + c)^(5/2)*D*a^2*d^50 + 126*(d*x + c)^(5/2)*C*a*b*d^50 + 6 3*(d*x + c)^(5/2)*B*b^2*d^50 - 315*(d*x + c)^(3/2)*D*a^2*c*d^50 - 630*(d*x + c)^(3/2)*C*a*b*c*d^50 - 315*(d*x + c)^(3/2)*B*b^2*c*d^50 + 945*sqrt(d*x + c)*D*a^2*c^2*d^50 + 1890*sqrt(d*x + c)*C*a*b*c^2*d^50 + 945*sqrt(d*x + c)*B*b^2*c^2*d^50 + 105*(d*x + c)^(3/2)*C*a^2*d^51 + 210*(d*x + c)^(3/2)*B *a*b*d^51 + 105*(d*x + c)^(3/2)*A*b^2*d^51 - 630*sqrt(d*x + c)*C*a^2*c*d^5 1 - 1260*sqrt(d*x + c)*B*a*b*c*d^51 - 630*sqrt(d*x + c)*A*b^2*c*d^51 + 315 *sqrt(d*x + c)*B*a^2*d^52 + 630*sqrt(d*x + c)*A*a*b*d^52)/d^54
Timed out. \[ \int \frac {(a+b x)^2 \left (A+B x+C x^2+D x^3\right )}{(c+d x)^{3/2}} \, dx=\int \frac {{\left (a+b\,x\right )}^2\,\left (A+B\,x+C\,x^2+x^3\,D\right )}{{\left (c+d\,x\right )}^{3/2}} \,d x \]