Integrand size = 33, antiderivative size = 218 \[ \int (A+B x) \sqrt {d+e x} \left (a^2+2 a b x+b^2 x^2\right )^2 \, dx=-\frac {2 (b d-a e)^4 (B d-A e) (d+e x)^{3/2}}{3 e^6}+\frac {2 (b d-a e)^3 (5 b B d-4 A b e-a B e) (d+e x)^{5/2}}{5 e^6}-\frac {4 b (b d-a e)^2 (5 b B d-3 A b e-2 a B e) (d+e x)^{7/2}}{7 e^6}+\frac {4 b^2 (b d-a e) (5 b B d-2 A b e-3 a B e) (d+e x)^{9/2}}{9 e^6}-\frac {2 b^3 (5 b B d-A b e-4 a B e) (d+e x)^{11/2}}{11 e^6}+\frac {2 b^4 B (d+e x)^{13/2}}{13 e^6} \]
[Out]
Time = 0.06 (sec) , antiderivative size = 218, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.061, Rules used = {27, 78} \[ \int (A+B x) \sqrt {d+e x} \left (a^2+2 a b x+b^2 x^2\right )^2 \, dx=-\frac {2 b^3 (d+e x)^{11/2} (-4 a B e-A b e+5 b B d)}{11 e^6}+\frac {4 b^2 (d+e x)^{9/2} (b d-a e) (-3 a B e-2 A b e+5 b B d)}{9 e^6}-\frac {4 b (d+e x)^{7/2} (b d-a e)^2 (-2 a B e-3 A b e+5 b B d)}{7 e^6}+\frac {2 (d+e x)^{5/2} (b d-a e)^3 (-a B e-4 A b e+5 b B d)}{5 e^6}-\frac {2 (d+e x)^{3/2} (b d-a e)^4 (B d-A e)}{3 e^6}+\frac {2 b^4 B (d+e x)^{13/2}}{13 e^6} \]
[In]
[Out]
Rule 27
Rule 78
Rubi steps \begin{align*} \text {integral}& = \int (a+b x)^4 (A+B x) \sqrt {d+e x} \, dx \\ & = \int \left (\frac {(-b d+a e)^4 (-B d+A e) \sqrt {d+e x}}{e^5}+\frac {(-b d+a e)^3 (-5 b B d+4 A b e+a B e) (d+e x)^{3/2}}{e^5}+\frac {2 b (b d-a e)^2 (-5 b B d+3 A b e+2 a B e) (d+e x)^{5/2}}{e^5}-\frac {2 b^2 (b d-a e) (-5 b B d+2 A b e+3 a B e) (d+e x)^{7/2}}{e^5}+\frac {b^3 (-5 b B d+A b e+4 a B e) (d+e x)^{9/2}}{e^5}+\frac {b^4 B (d+e x)^{11/2}}{e^5}\right ) \, dx \\ & = -\frac {2 (b d-a e)^4 (B d-A e) (d+e x)^{3/2}}{3 e^6}+\frac {2 (b d-a e)^3 (5 b B d-4 A b e-a B e) (d+e x)^{5/2}}{5 e^6}-\frac {4 b (b d-a e)^2 (5 b B d-3 A b e-2 a B e) (d+e x)^{7/2}}{7 e^6}+\frac {4 b^2 (b d-a e) (5 b B d-2 A b e-3 a B e) (d+e x)^{9/2}}{9 e^6}-\frac {2 b^3 (5 b B d-A b e-4 a B e) (d+e x)^{11/2}}{11 e^6}+\frac {2 b^4 B (d+e x)^{13/2}}{13 e^6} \\ \end{align*}
Time = 0.25 (sec) , antiderivative size = 339, normalized size of antiderivative = 1.56 \[ \int (A+B x) \sqrt {d+e x} \left (a^2+2 a b x+b^2 x^2\right )^2 \, dx=\frac {2 (d+e x)^{3/2} \left (3003 a^4 e^4 (-2 B d+5 A e+3 B e x)+1716 a^3 b e^3 \left (7 A e (-2 d+3 e x)+B \left (8 d^2-12 d e x+15 e^2 x^2\right )\right )-858 a^2 b^2 e^2 \left (-3 A e \left (8 d^2-12 d e x+15 e^2 x^2\right )+B \left (16 d^3-24 d^2 e x+30 d e^2 x^2-35 e^3 x^3\right )\right )+52 a b^3 e \left (11 A e \left (-16 d^3+24 d^2 e x-30 d e^2 x^2+35 e^3 x^3\right )+B \left (128 d^4-192 d^3 e x+240 d^2 e^2 x^2-280 d e^3 x^3+315 e^4 x^4\right )\right )+b^4 \left (13 A e \left (128 d^4-192 d^3 e x+240 d^2 e^2 x^2-280 d e^3 x^3+315 e^4 x^4\right )-5 B \left (256 d^5-384 d^4 e x+480 d^3 e^2 x^2-560 d^2 e^3 x^3+630 d e^4 x^4-693 e^5 x^5\right )\right )\right )}{45045 e^6} \]
[In]
[Out]
Time = 0.32 (sec) , antiderivative size = 282, normalized size of antiderivative = 1.29
method | result | size |
pseudoelliptic | \(\frac {2 \left (\left (\left (\frac {3}{13} B \,x^{5}+\frac {3}{11} A \,x^{4}\right ) b^{4}+\frac {4 \left (\frac {9 B x}{11}+A \right ) x^{3} a \,b^{3}}{3}+\frac {18 x^{2} \left (\frac {7 B x}{9}+A \right ) a^{2} b^{2}}{7}+\frac {12 x \left (\frac {5 B x}{7}+A \right ) a^{3} b}{5}+a^{4} \left (\frac {3 B x}{5}+A \right )\right ) e^{5}-\frac {8 \left (\frac {5 \left (\frac {45 B x}{52}+A \right ) x^{3} b^{4}}{33}+\frac {5 x^{2} \left (\frac {28 B x}{33}+A \right ) a \,b^{3}}{7}+\frac {9 x \left (\frac {5 B x}{6}+A \right ) a^{2} b^{2}}{7}+a^{3} \left (\frac {6 B x}{7}+A \right ) b +\frac {B \,a^{4}}{4}\right ) d \,e^{4}}{5}+\frac {48 b \left (\frac {5 x^{2} \left (\frac {35 B x}{39}+A \right ) b^{3}}{33}+\frac {2 \left (\frac {10 B x}{11}+A \right ) x a \,b^{2}}{3}+a^{2} \left (B x +A \right ) b +\frac {2 B \,a^{3}}{3}\right ) d^{2} e^{3}}{35}-\frac {64 b^{2} \left (\frac {3 x \left (\frac {25 B x}{26}+A \right ) b^{2}}{11}+a \left (\frac {12 B x}{11}+A \right ) b +\frac {3 B \,a^{2}}{2}\right ) d^{3} e^{2}}{105}+\frac {128 b^{3} d^{4} \left (\left (\frac {15 B x}{13}+A \right ) b +4 B a \right ) e}{1155}-\frac {256 b^{4} B \,d^{5}}{3003}\right ) \left (e x +d \right )^{\frac {3}{2}}}{3 e^{6}}\) | \(282\) |
derivativedivides | \(\frac {\frac {2 b^{4} B \left (e x +d \right )^{\frac {13}{2}}}{13}+\frac {2 \left (\left (A e -B d \right ) b^{4}+2 B \left (2 a b e -2 b^{2} d \right ) b^{2}\right ) \left (e x +d \right )^{\frac {11}{2}}}{11}+\frac {2 \left (2 \left (A e -B d \right ) \left (2 a b e -2 b^{2} d \right ) b^{2}+B \left (2 \left (e^{2} a^{2}-2 a b d e +b^{2} d^{2}\right ) b^{2}+\left (2 a b e -2 b^{2} d \right )^{2}\right )\right ) \left (e x +d \right )^{\frac {9}{2}}}{9}+\frac {2 \left (\left (A e -B d \right ) \left (2 \left (e^{2} a^{2}-2 a b d e +b^{2} d^{2}\right ) b^{2}+\left (2 a b e -2 b^{2} d \right )^{2}\right )+2 B \left (e^{2} a^{2}-2 a b d e +b^{2} d^{2}\right ) \left (2 a b e -2 b^{2} d \right )\right ) \left (e x +d \right )^{\frac {7}{2}}}{7}+\frac {2 \left (2 \left (A e -B d \right ) \left (e^{2} a^{2}-2 a b d e +b^{2} d^{2}\right ) \left (2 a b e -2 b^{2} d \right )+B \left (e^{2} a^{2}-2 a b d e +b^{2} d^{2}\right )^{2}\right ) \left (e x +d \right )^{\frac {5}{2}}}{5}+\frac {2 \left (A e -B d \right ) \left (e^{2} a^{2}-2 a b d e +b^{2} d^{2}\right )^{2} \left (e x +d \right )^{\frac {3}{2}}}{3}}{e^{6}}\) | \(352\) |
default | \(\frac {\frac {2 b^{4} B \left (e x +d \right )^{\frac {13}{2}}}{13}+\frac {2 \left (\left (A e -B d \right ) b^{4}+2 B \left (2 a b e -2 b^{2} d \right ) b^{2}\right ) \left (e x +d \right )^{\frac {11}{2}}}{11}+\frac {2 \left (2 \left (A e -B d \right ) \left (2 a b e -2 b^{2} d \right ) b^{2}+B \left (2 \left (e^{2} a^{2}-2 a b d e +b^{2} d^{2}\right ) b^{2}+\left (2 a b e -2 b^{2} d \right )^{2}\right )\right ) \left (e x +d \right )^{\frac {9}{2}}}{9}+\frac {2 \left (\left (A e -B d \right ) \left (2 \left (e^{2} a^{2}-2 a b d e +b^{2} d^{2}\right ) b^{2}+\left (2 a b e -2 b^{2} d \right )^{2}\right )+2 B \left (e^{2} a^{2}-2 a b d e +b^{2} d^{2}\right ) \left (2 a b e -2 b^{2} d \right )\right ) \left (e x +d \right )^{\frac {7}{2}}}{7}+\frac {2 \left (2 \left (A e -B d \right ) \left (e^{2} a^{2}-2 a b d e +b^{2} d^{2}\right ) \left (2 a b e -2 b^{2} d \right )+B \left (e^{2} a^{2}-2 a b d e +b^{2} d^{2}\right )^{2}\right ) \left (e x +d \right )^{\frac {5}{2}}}{5}+\frac {2 \left (A e -B d \right ) \left (e^{2} a^{2}-2 a b d e +b^{2} d^{2}\right )^{2} \left (e x +d \right )^{\frac {3}{2}}}{3}}{e^{6}}\) | \(352\) |
gosper | \(\frac {2 \left (e x +d \right )^{\frac {3}{2}} \left (3465 B \,x^{5} b^{4} e^{5}+4095 A \,b^{4} e^{5} x^{4}+16380 B \,x^{4} a \,b^{3} e^{5}-3150 B \,x^{4} b^{4} d \,e^{4}+20020 A \,x^{3} a \,b^{3} e^{5}-3640 A \,x^{3} b^{4} d \,e^{4}+30030 B \,x^{3} a^{2} b^{2} e^{5}-14560 B \,x^{3} a \,b^{3} d \,e^{4}+2800 B \,x^{3} b^{4} d^{2} e^{3}+38610 A \,x^{2} a^{2} b^{2} e^{5}-17160 A \,x^{2} a \,b^{3} d \,e^{4}+3120 A \,x^{2} b^{4} d^{2} e^{3}+25740 B \,x^{2} a^{3} b \,e^{5}-25740 B \,x^{2} a^{2} b^{2} d \,e^{4}+12480 B \,x^{2} a \,b^{3} d^{2} e^{3}-2400 B \,x^{2} b^{4} d^{3} e^{2}+36036 A x \,a^{3} b \,e^{5}-30888 A x \,a^{2} b^{2} d \,e^{4}+13728 A x a \,b^{3} d^{2} e^{3}-2496 A x \,b^{4} d^{3} e^{2}+9009 B x \,a^{4} e^{5}-20592 B x \,a^{3} b d \,e^{4}+20592 B x \,a^{2} b^{2} d^{2} e^{3}-9984 B x a \,b^{3} d^{3} e^{2}+1920 B x \,b^{4} d^{4} e +15015 A \,a^{4} e^{5}-24024 A \,a^{3} b d \,e^{4}+20592 A \,a^{2} b^{2} d^{2} e^{3}-9152 A a \,b^{3} d^{3} e^{2}+1664 A \,b^{4} d^{4} e -6006 B \,a^{4} d \,e^{4}+13728 B \,a^{3} b \,d^{2} e^{3}-13728 B \,a^{2} b^{2} d^{3} e^{2}+6656 B a \,b^{3} d^{4} e -1280 b^{4} B \,d^{5}\right )}{45045 e^{6}}\) | \(469\) |
trager | \(\frac {2 \left (3465 b^{4} B \,e^{6} x^{6}+4095 A \,b^{4} e^{6} x^{5}+16380 B a \,b^{3} e^{6} x^{5}+315 b^{4} B d \,e^{5} x^{5}+20020 A a \,b^{3} e^{6} x^{4}+455 A \,b^{4} d \,e^{5} x^{4}+30030 B \,a^{2} b^{2} e^{6} x^{4}+1820 B a \,b^{3} d \,e^{5} x^{4}-350 b^{4} B \,d^{2} e^{4} x^{4}+38610 A \,a^{2} b^{2} e^{6} x^{3}+2860 A a \,b^{3} d \,e^{5} x^{3}-520 A \,b^{4} d^{2} e^{4} x^{3}+25740 B \,a^{3} b \,e^{6} x^{3}+4290 B \,a^{2} b^{2} d \,e^{5} x^{3}-2080 B a \,b^{3} d^{2} e^{4} x^{3}+400 B \,b^{4} d^{3} e^{3} x^{3}+36036 A \,a^{3} b \,e^{6} x^{2}+7722 A \,a^{2} b^{2} d \,e^{5} x^{2}-3432 A a \,b^{3} d^{2} e^{4} x^{2}+624 A \,b^{4} d^{3} e^{3} x^{2}+9009 B \,e^{6} a^{4} x^{2}+5148 B \,a^{3} b d \,e^{5} x^{2}-5148 B \,a^{2} b^{2} d^{2} e^{4} x^{2}+2496 B a \,b^{3} d^{3} e^{3} x^{2}-480 b^{4} B \,d^{4} e^{2} x^{2}+15015 A \,a^{4} e^{6} x +12012 A \,a^{3} b d \,e^{5} x -10296 A \,a^{2} b^{2} d^{2} e^{4} x +4576 A a \,b^{3} d^{3} e^{3} x -832 A \,b^{4} d^{4} e^{2} x +3003 B \,a^{4} d \,e^{5} x -6864 B \,a^{3} b \,d^{2} e^{4} x +6864 B \,a^{2} b^{2} d^{3} e^{3} x -3328 B a \,b^{3} d^{4} e^{2} x +640 b^{4} B \,d^{5} e x +15015 A \,a^{4} d \,e^{5}-24024 A \,a^{3} b \,d^{2} e^{4}+20592 A \,a^{2} b^{2} d^{3} e^{3}-9152 A a \,b^{3} d^{4} e^{2}+1664 A \,b^{4} d^{5} e -6006 B \,a^{4} d^{2} e^{4}+13728 B \,a^{3} b \,d^{3} e^{3}-13728 B \,a^{2} b^{2} d^{4} e^{2}+6656 B a \,b^{3} d^{5} e -1280 b^{4} B \,d^{6}\right ) \sqrt {e x +d}}{45045 e^{6}}\) | \(625\) |
risch | \(\frac {2 \left (3465 b^{4} B \,e^{6} x^{6}+4095 A \,b^{4} e^{6} x^{5}+16380 B a \,b^{3} e^{6} x^{5}+315 b^{4} B d \,e^{5} x^{5}+20020 A a \,b^{3} e^{6} x^{4}+455 A \,b^{4} d \,e^{5} x^{4}+30030 B \,a^{2} b^{2} e^{6} x^{4}+1820 B a \,b^{3} d \,e^{5} x^{4}-350 b^{4} B \,d^{2} e^{4} x^{4}+38610 A \,a^{2} b^{2} e^{6} x^{3}+2860 A a \,b^{3} d \,e^{5} x^{3}-520 A \,b^{4} d^{2} e^{4} x^{3}+25740 B \,a^{3} b \,e^{6} x^{3}+4290 B \,a^{2} b^{2} d \,e^{5} x^{3}-2080 B a \,b^{3} d^{2} e^{4} x^{3}+400 B \,b^{4} d^{3} e^{3} x^{3}+36036 A \,a^{3} b \,e^{6} x^{2}+7722 A \,a^{2} b^{2} d \,e^{5} x^{2}-3432 A a \,b^{3} d^{2} e^{4} x^{2}+624 A \,b^{4} d^{3} e^{3} x^{2}+9009 B \,e^{6} a^{4} x^{2}+5148 B \,a^{3} b d \,e^{5} x^{2}-5148 B \,a^{2} b^{2} d^{2} e^{4} x^{2}+2496 B a \,b^{3} d^{3} e^{3} x^{2}-480 b^{4} B \,d^{4} e^{2} x^{2}+15015 A \,a^{4} e^{6} x +12012 A \,a^{3} b d \,e^{5} x -10296 A \,a^{2} b^{2} d^{2} e^{4} x +4576 A a \,b^{3} d^{3} e^{3} x -832 A \,b^{4} d^{4} e^{2} x +3003 B \,a^{4} d \,e^{5} x -6864 B \,a^{3} b \,d^{2} e^{4} x +6864 B \,a^{2} b^{2} d^{3} e^{3} x -3328 B a \,b^{3} d^{4} e^{2} x +640 b^{4} B \,d^{5} e x +15015 A \,a^{4} d \,e^{5}-24024 A \,a^{3} b \,d^{2} e^{4}+20592 A \,a^{2} b^{2} d^{3} e^{3}-9152 A a \,b^{3} d^{4} e^{2}+1664 A \,b^{4} d^{5} e -6006 B \,a^{4} d^{2} e^{4}+13728 B \,a^{3} b \,d^{3} e^{3}-13728 B \,a^{2} b^{2} d^{4} e^{2}+6656 B a \,b^{3} d^{5} e -1280 b^{4} B \,d^{6}\right ) \sqrt {e x +d}}{45045 e^{6}}\) | \(625\) |
[In]
[Out]
Leaf count of result is larger than twice the leaf count of optimal. 527 vs. \(2 (194) = 388\).
Time = 0.30 (sec) , antiderivative size = 527, normalized size of antiderivative = 2.42 \[ \int (A+B x) \sqrt {d+e x} \left (a^2+2 a b x+b^2 x^2\right )^2 \, dx=\frac {2 \, {\left (3465 \, B b^{4} e^{6} x^{6} - 1280 \, B b^{4} d^{6} + 15015 \, A a^{4} d e^{5} + 1664 \, {\left (4 \, B a b^{3} + A b^{4}\right )} d^{5} e - 4576 \, {\left (3 \, B a^{2} b^{2} + 2 \, A a b^{3}\right )} d^{4} e^{2} + 6864 \, {\left (2 \, B a^{3} b + 3 \, A a^{2} b^{2}\right )} d^{3} e^{3} - 6006 \, {\left (B a^{4} + 4 \, A a^{3} b\right )} d^{2} e^{4} + 315 \, {\left (B b^{4} d e^{5} + 13 \, {\left (4 \, B a b^{3} + A b^{4}\right )} e^{6}\right )} x^{5} - 35 \, {\left (10 \, B b^{4} d^{2} e^{4} - 13 \, {\left (4 \, B a b^{3} + A b^{4}\right )} d e^{5} - 286 \, {\left (3 \, B a^{2} b^{2} + 2 \, A a b^{3}\right )} e^{6}\right )} x^{4} + 10 \, {\left (40 \, B b^{4} d^{3} e^{3} - 52 \, {\left (4 \, B a b^{3} + A b^{4}\right )} d^{2} e^{4} + 143 \, {\left (3 \, B a^{2} b^{2} + 2 \, A a b^{3}\right )} d e^{5} + 1287 \, {\left (2 \, B a^{3} b + 3 \, A a^{2} b^{2}\right )} e^{6}\right )} x^{3} - 3 \, {\left (160 \, B b^{4} d^{4} e^{2} - 208 \, {\left (4 \, B a b^{3} + A b^{4}\right )} d^{3} e^{3} + 572 \, {\left (3 \, B a^{2} b^{2} + 2 \, A a b^{3}\right )} d^{2} e^{4} - 858 \, {\left (2 \, B a^{3} b + 3 \, A a^{2} b^{2}\right )} d e^{5} - 3003 \, {\left (B a^{4} + 4 \, A a^{3} b\right )} e^{6}\right )} x^{2} + {\left (640 \, B b^{4} d^{5} e + 15015 \, A a^{4} e^{6} - 832 \, {\left (4 \, B a b^{3} + A b^{4}\right )} d^{4} e^{2} + 2288 \, {\left (3 \, B a^{2} b^{2} + 2 \, A a b^{3}\right )} d^{3} e^{3} - 3432 \, {\left (2 \, B a^{3} b + 3 \, A a^{2} b^{2}\right )} d^{2} e^{4} + 3003 \, {\left (B a^{4} + 4 \, A a^{3} b\right )} d e^{5}\right )} x\right )} \sqrt {e x + d}}{45045 \, e^{6}} \]
[In]
[Out]
Leaf count of result is larger than twice the leaf count of optimal. 624 vs. \(2 (221) = 442\).
Time = 1.61 (sec) , antiderivative size = 624, normalized size of antiderivative = 2.86 \[ \int (A+B x) \sqrt {d+e x} \left (a^2+2 a b x+b^2 x^2\right )^2 \, dx=\begin {cases} \frac {2 \left (\frac {B b^{4} \left (d + e x\right )^{\frac {13}{2}}}{13 e^{5}} + \frac {\left (d + e x\right )^{\frac {11}{2}} \left (A b^{4} e + 4 B a b^{3} e - 5 B b^{4} d\right )}{11 e^{5}} + \frac {\left (d + e x\right )^{\frac {9}{2}} \cdot \left (4 A a b^{3} e^{2} - 4 A b^{4} d e + 6 B a^{2} b^{2} e^{2} - 16 B a b^{3} d e + 10 B b^{4} d^{2}\right )}{9 e^{5}} + \frac {\left (d + e x\right )^{\frac {7}{2}} \cdot \left (6 A a^{2} b^{2} e^{3} - 12 A a b^{3} d e^{2} + 6 A b^{4} d^{2} e + 4 B a^{3} b e^{3} - 18 B a^{2} b^{2} d e^{2} + 24 B a b^{3} d^{2} e - 10 B b^{4} d^{3}\right )}{7 e^{5}} + \frac {\left (d + e x\right )^{\frac {5}{2}} \cdot \left (4 A a^{3} b e^{4} - 12 A a^{2} b^{2} d e^{3} + 12 A a b^{3} d^{2} e^{2} - 4 A b^{4} d^{3} e + B a^{4} e^{4} - 8 B a^{3} b d e^{3} + 18 B a^{2} b^{2} d^{2} e^{2} - 16 B a b^{3} d^{3} e + 5 B b^{4} d^{4}\right )}{5 e^{5}} + \frac {\left (d + e x\right )^{\frac {3}{2}} \left (A a^{4} e^{5} - 4 A a^{3} b d e^{4} + 6 A a^{2} b^{2} d^{2} e^{3} - 4 A a b^{3} d^{3} e^{2} + A b^{4} d^{4} e - B a^{4} d e^{4} + 4 B a^{3} b d^{2} e^{3} - 6 B a^{2} b^{2} d^{3} e^{2} + 4 B a b^{3} d^{4} e - B b^{4} d^{5}\right )}{3 e^{5}}\right )}{e} & \text {for}\: e \neq 0 \\\sqrt {d} \left (A a^{4} x + \frac {B b^{4} x^{6}}{6} + \frac {x^{5} \left (A b^{4} + 4 B a b^{3}\right )}{5} + \frac {x^{4} \cdot \left (4 A a b^{3} + 6 B a^{2} b^{2}\right )}{4} + \frac {x^{3} \cdot \left (6 A a^{2} b^{2} + 4 B a^{3} b\right )}{3} + \frac {x^{2} \cdot \left (4 A a^{3} b + B a^{4}\right )}{2}\right ) & \text {otherwise} \end {cases} \]
[In]
[Out]
Leaf count of result is larger than twice the leaf count of optimal. 409 vs. \(2 (194) = 388\).
Time = 0.21 (sec) , antiderivative size = 409, normalized size of antiderivative = 1.88 \[ \int (A+B x) \sqrt {d+e x} \left (a^2+2 a b x+b^2 x^2\right )^2 \, dx=\frac {2 \, {\left (3465 \, {\left (e x + d\right )}^{\frac {13}{2}} B b^{4} - 4095 \, {\left (5 \, B b^{4} d - {\left (4 \, B a b^{3} + A b^{4}\right )} e\right )} {\left (e x + d\right )}^{\frac {11}{2}} + 10010 \, {\left (5 \, B b^{4} d^{2} - 2 \, {\left (4 \, B a b^{3} + A b^{4}\right )} d e + {\left (3 \, B a^{2} b^{2} + 2 \, A a b^{3}\right )} e^{2}\right )} {\left (e x + d\right )}^{\frac {9}{2}} - 12870 \, {\left (5 \, B b^{4} d^{3} - 3 \, {\left (4 \, B a b^{3} + A b^{4}\right )} d^{2} e + 3 \, {\left (3 \, B a^{2} b^{2} + 2 \, A a b^{3}\right )} d e^{2} - {\left (2 \, B a^{3} b + 3 \, A a^{2} b^{2}\right )} e^{3}\right )} {\left (e x + d\right )}^{\frac {7}{2}} + 9009 \, {\left (5 \, B b^{4} d^{4} - 4 \, {\left (4 \, B a b^{3} + A b^{4}\right )} d^{3} e + 6 \, {\left (3 \, B a^{2} b^{2} + 2 \, A a b^{3}\right )} d^{2} e^{2} - 4 \, {\left (2 \, B a^{3} b + 3 \, A a^{2} b^{2}\right )} d e^{3} + {\left (B a^{4} + 4 \, A a^{3} b\right )} e^{4}\right )} {\left (e x + d\right )}^{\frac {5}{2}} - 15015 \, {\left (B b^{4} d^{5} - A a^{4} e^{5} - {\left (4 \, B a b^{3} + A b^{4}\right )} d^{4} e + 2 \, {\left (3 \, B a^{2} b^{2} + 2 \, A a b^{3}\right )} d^{3} e^{2} - 2 \, {\left (2 \, B a^{3} b + 3 \, A a^{2} b^{2}\right )} d^{2} e^{3} + {\left (B a^{4} + 4 \, A a^{3} b\right )} d e^{4}\right )} {\left (e x + d\right )}^{\frac {3}{2}}\right )}}{45045 \, e^{6}} \]
[In]
[Out]
Leaf count of result is larger than twice the leaf count of optimal. 1083 vs. \(2 (194) = 388\).
Time = 0.28 (sec) , antiderivative size = 1083, normalized size of antiderivative = 4.97 \[ \int (A+B x) \sqrt {d+e x} \left (a^2+2 a b x+b^2 x^2\right )^2 \, dx=\text {Too large to display} \]
[In]
[Out]
Time = 10.84 (sec) , antiderivative size = 197, normalized size of antiderivative = 0.90 \[ \int (A+B x) \sqrt {d+e x} \left (a^2+2 a b x+b^2 x^2\right )^2 \, dx=\frac {{\left (d+e\,x\right )}^{11/2}\,\left (2\,A\,b^4\,e-10\,B\,b^4\,d+8\,B\,a\,b^3\,e\right )}{11\,e^6}+\frac {2\,{\left (a\,e-b\,d\right )}^3\,{\left (d+e\,x\right )}^{5/2}\,\left (4\,A\,b\,e+B\,a\,e-5\,B\,b\,d\right )}{5\,e^6}+\frac {2\,B\,b^4\,{\left (d+e\,x\right )}^{13/2}}{13\,e^6}+\frac {2\,\left (A\,e-B\,d\right )\,{\left (a\,e-b\,d\right )}^4\,{\left (d+e\,x\right )}^{3/2}}{3\,e^6}+\frac {4\,b\,{\left (a\,e-b\,d\right )}^2\,{\left (d+e\,x\right )}^{7/2}\,\left (3\,A\,b\,e+2\,B\,a\,e-5\,B\,b\,d\right )}{7\,e^6}+\frac {4\,b^2\,\left (a\,e-b\,d\right )\,{\left (d+e\,x\right )}^{9/2}\,\left (2\,A\,b\,e+3\,B\,a\,e-5\,B\,b\,d\right )}{9\,e^6} \]
[In]
[Out]