\(\int (A+B x) \sqrt {d+e x} (a^2+2 a b x+b^2 x^2)^2 \, dx\) [1795]

Optimal result

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]

-2/3*(-a*e+b*d)^4*(-A*e+B*d)*(e*x+d)^(3/2)/e^6+2/5*(-a*e+b*d)^3*(-4*A*b*e-B*a*e+5*B*b*d)*(e*x+d)^(5/2)/e^6-4/7
*b*(-a*e+b*d)^2*(-3*A*b*e-2*B*a*e+5*B*b*d)*(e*x+d)^(7/2)/e^6+4/9*b^2*(-a*e+b*d)*(-2*A*b*e-3*B*a*e+5*B*b*d)*(e*
x+d)^(9/2)/e^6-2/11*b^3*(-A*b*e-4*B*a*e+5*B*b*d)*(e*x+d)^(11/2)/e^6+2/13*b^4*B*(e*x+d)^(13/2)/e^6

Rubi [A] (verified)

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]

Int[(A + B*x)*Sqrt[d + e*x]*(a^2 + 2*a*b*x + b^2*x^2)^2,x]

[Out]

(-2*(b*d - a*e)^4*(B*d - A*e)*(d + e*x)^(3/2))/(3*e^6) + (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) - (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) + (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) - (2*b^3*(5*b*B*d - A*b*e - 4*a*B*e)*(d + e*x)^(11/
2))/(11*e^6) + (2*b^4*B*(d + e*x)^(13/2))/(13*e^6)

Rule 27

Int[(u_.)*((a_) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> Int[u*Cancel[(b/2 + c*x)^(2*p)/c^p], x] /; Fr
eeQ[{a, b, c}, x] && EqQ[b^2 - 4*a*c, 0] && IntegerQ[p]

Rule 78

Int[((a_.) + (b_.)*(x_))*((c_) + (d_.)*(x_))^(n_.)*((e_.) + (f_.)*(x_))^(p_.), x_Symbol] :> Int[ExpandIntegran
d[(a + b*x)*(c + d*x)^n*(e + f*x)^p, x], x] /; FreeQ[{a, b, c, d, e, f, n}, x] && NeQ[b*c - a*d, 0] && ((ILtQ[
n, 0] && ILtQ[p, 0]) || EqQ[p, 1] || (IGtQ[p, 0] && ( !IntegerQ[n] || LeQ[9*p + 5*(n + 2), 0] || GeQ[n + p + 1
, 0] || (GeQ[n + p + 2, 0] && RationalQ[a, b, c, d, e, f]))))

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*}

Mathematica [A] (verified)

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]

Integrate[(A + B*x)*Sqrt[d + e*x]*(a^2 + 2*a*b*x + b^2*x^2)^2,x]

[Out]

(2*(d + e*x)^(3/2)*(3003*a^4*e^4*(-2*B*d + 5*A*e + 3*B*e*x) + 1716*a^3*b*e^3*(7*A*e*(-2*d + 3*e*x) + B*(8*d^2
- 12*d*e*x + 15*e^2*x^2)) - 858*a^2*b^2*e^2*(-3*A*e*(8*d^2 - 12*d*e*x + 15*e^2*x^2) + B*(16*d^3 - 24*d^2*e*x +
30*d*e^2*x^2 - 35*e^3*x^3)) + 52*a*b^3*e*(11*A*e*(-16*d^3 + 24*d^2*e*x - 30*d*e^2*x^2 + 35*e^3*x^3) + B*(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)) + b^4*(13*A*e*(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) - 5*B*(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))))/(45045*e^6)

Maple [A] (verified)

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]

int((B*x+A)*(b^2*x^2+2*a*b*x+a^2)^2*(e*x+d)^(1/2),x,method=_RETURNVERBOSE)

[Out]

2/3*(((3/13*B*x^5+3/11*A*x^4)*b^4+4/3*(9/11*B*x+A)*x^3*a*b^3+18/7*x^2*(7/9*B*x+A)*a^2*b^2+12/5*x*(5/7*B*x+A)*a
^3*b+a^4*(3/5*B*x+A))*e^5-8/5*(5/33*(45/52*B*x+A)*x^3*b^4+5/7*x^2*(28/33*B*x+A)*a*b^3+9/7*x*(5/6*B*x+A)*a^2*b^
2+a^3*(6/7*B*x+A)*b+1/4*B*a^4)*d*e^4+48/35*b*(5/33*x^2*(35/39*B*x+A)*b^3+2/3*(10/11*B*x+A)*x*a*b^2+a^2*(B*x+A)
*b+2/3*B*a^3)*d^2*e^3-64/105*b^2*(3/11*x*(25/26*B*x+A)*b^2+a*(12/11*B*x+A)*b+3/2*B*a^2)*d^3*e^2+128/1155*b^3*d
^4*((15/13*B*x+A)*b+4*B*a)*e-256/3003*b^4*B*d^5)*(e*x+d)^(3/2)/e^6

Fricas [B] (verification not implemented)

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]

integrate((B*x+A)*(b^2*x^2+2*a*b*x+a^2)^2*(e*x+d)^(1/2),x, algorithm="fricas")

[Out]

2/45045*(3465*B*b^4*e^6*x^6 - 1280*B*b^4*d^6 + 15015*A*a^4*d*e^5 + 1664*(4*B*a*b^3 + A*b^4)*d^5*e - 4576*(3*B*
a^2*b^2 + 2*A*a*b^3)*d^4*e^2 + 6864*(2*B*a^3*b + 3*A*a^2*b^2)*d^3*e^3 - 6006*(B*a^4 + 4*A*a^3*b)*d^2*e^4 + 315
*(B*b^4*d*e^5 + 13*(4*B*a*b^3 + A*b^4)*e^6)*x^5 - 35*(10*B*b^4*d^2*e^4 - 13*(4*B*a*b^3 + A*b^4)*d*e^5 - 286*(3
*B*a^2*b^2 + 2*A*a*b^3)*e^6)*x^4 + 10*(40*B*b^4*d^3*e^3 - 52*(4*B*a*b^3 + A*b^4)*d^2*e^4 + 143*(3*B*a^2*b^2 +
2*A*a*b^3)*d*e^5 + 1287*(2*B*a^3*b + 3*A*a^2*b^2)*e^6)*x^3 - 3*(160*B*b^4*d^4*e^2 - 208*(4*B*a*b^3 + A*b^4)*d^
3*e^3 + 572*(3*B*a^2*b^2 + 2*A*a*b^3)*d^2*e^4 - 858*(2*B*a^3*b + 3*A*a^2*b^2)*d*e^5 - 3003*(B*a^4 + 4*A*a^3*b)
*e^6)*x^2 + (640*B*b^4*d^5*e + 15015*A*a^4*e^6 - 832*(4*B*a*b^3 + A*b^4)*d^4*e^2 + 2288*(3*B*a^2*b^2 + 2*A*a*b
^3)*d^3*e^3 - 3432*(2*B*a^3*b + 3*A*a^2*b^2)*d^2*e^4 + 3003*(B*a^4 + 4*A*a^3*b)*d*e^5)*x)*sqrt(e*x + d)/e^6

Sympy [B] (verification not implemented)

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]

integrate((B*x+A)*(b**2*x**2+2*a*b*x+a**2)**2*(e*x+d)**(1/2),x)

[Out]

Piecewise((2*(B*b**4*(d + e*x)**(13/2)/(13*e**5) + (d + e*x)**(11/2)*(A*b**4*e + 4*B*a*b**3*e - 5*B*b**4*d)/(1
1*e**5) + (d + e*x)**(9/2)*(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)/(9*e**5) + (d + e*x)**(7/2)*(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)/(7*e**5) + (d + e*x)**(5/2)*(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)/(5*e**5) + (d + e*x)**(3/2)*(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)/(3*e**5))/e, Ne(e, 0)), (sqrt(d)*(A*a**4*x + B*b**
4*x**6/6 + x**5*(A*b**4 + 4*B*a*b**3)/5 + x**4*(4*A*a*b**3 + 6*B*a**2*b**2)/4 + x**3*(6*A*a**2*b**2 + 4*B*a**3
*b)/3 + x**2*(4*A*a**3*b + B*a**4)/2), True))

Maxima [B] (verification not implemented)

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]

integrate((B*x+A)*(b^2*x^2+2*a*b*x+a^2)^2*(e*x+d)^(1/2),x, algorithm="maxima")

[Out]

2/45045*(3465*(e*x + d)^(13/2)*B*b^4 - 4095*(5*B*b^4*d - (4*B*a*b^3 + A*b^4)*e)*(e*x + d)^(11/2) + 10010*(5*B*
b^4*d^2 - 2*(4*B*a*b^3 + A*b^4)*d*e + (3*B*a^2*b^2 + 2*A*a*b^3)*e^2)*(e*x + d)^(9/2) - 12870*(5*B*b^4*d^3 - 3*
(4*B*a*b^3 + A*b^4)*d^2*e + 3*(3*B*a^2*b^2 + 2*A*a*b^3)*d*e^2 - (2*B*a^3*b + 3*A*a^2*b^2)*e^3)*(e*x + d)^(7/2)
+ 9009*(5*B*b^4*d^4 - 4*(4*B*a*b^3 + A*b^4)*d^3*e + 6*(3*B*a^2*b^2 + 2*A*a*b^3)*d^2*e^2 - 4*(2*B*a^3*b + 3*A*
a^2*b^2)*d*e^3 + (B*a^4 + 4*A*a^3*b)*e^4)*(e*x + d)^(5/2) - 15015*(B*b^4*d^5 - A*a^4*e^5 - (4*B*a*b^3 + A*b^4)
*d^4*e + 2*(3*B*a^2*b^2 + 2*A*a*b^3)*d^3*e^2 - 2*(2*B*a^3*b + 3*A*a^2*b^2)*d^2*e^3 + (B*a^4 + 4*A*a^3*b)*d*e^4
)*(e*x + d)^(3/2))/e^6

Giac [B] (verification not implemented)

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]

integrate((B*x+A)*(b^2*x^2+2*a*b*x+a^2)^2*(e*x+d)^(1/2),x, algorithm="giac")

[Out]

2/45045*(45045*sqrt(e*x + d)*A*a^4*d + 15015*((e*x + d)^(3/2) - 3*sqrt(e*x + d)*d)*A*a^4 + 15015*((e*x + d)^(3
/2) - 3*sqrt(e*x + d)*d)*B*a^4*d/e + 60060*((e*x + d)^(3/2) - 3*sqrt(e*x + d)*d)*A*a^3*b*d/e + 12012*(3*(e*x +
d)^(5/2) - 10*(e*x + d)^(3/2)*d + 15*sqrt(e*x + d)*d^2)*B*a^3*b*d/e^2 + 18018*(3*(e*x + d)^(5/2) - 10*(e*x +
d)^(3/2)*d + 15*sqrt(e*x + d)*d^2)*A*a^2*b^2*d/e^2 + 3003*(3*(e*x + d)^(5/2) - 10*(e*x + d)^(3/2)*d + 15*sqrt(
e*x + d)*d^2)*B*a^4/e + 12012*(3*(e*x + d)^(5/2) - 10*(e*x + d)^(3/2)*d + 15*sqrt(e*x + d)*d^2)*A*a^3*b/e + 77
22*(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*a^2*b^2*d/e^3
+ 5148*(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)*A*a*b^3*d/e^
3 + 5148*(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*a^3*b/e^
2 + 7722*(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)*A*a^2*b^2/
e^2 + 572*(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 + 31
5*sqrt(e*x + d)*d^4)*B*a*b^3*d/e^4 + 143*(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)*A*b^4*d/e^4 + 858*(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*a^2*b^2/e^3 + 572*(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
)*A*a*b^3/e^3 + 65*(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*b^4*d/e^5 + 260*(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*a*b^3/e^4 + 65*(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)*A*b^4/e^4 + 15*(231*(e*x + d)^(13/2) - 1638*(e
*x + d)^(11/2)*d + 5005*(e*x + d)^(9/2)*d^2 - 8580*(e*x + d)^(7/2)*d^3 + 9009*(e*x + d)^(5/2)*d^4 - 6006*(e*x
+ d)^(3/2)*d^5 + 3003*sqrt(e*x + d)*d^6)*B*b^4/e^5)/e

Mupad [B] (verification not implemented)

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]

int((A + B*x)*(d + e*x)^(1/2)*(a^2 + b^2*x^2 + 2*a*b*x)^2,x)

[Out]

((d + e*x)^(11/2)*(2*A*b^4*e - 10*B*b^4*d + 8*B*a*b^3*e))/(11*e^6) + (2*(a*e - b*d)^3*(d + e*x)^(5/2)*(4*A*b*e
+ B*a*e - 5*B*b*d))/(5*e^6) + (2*B*b^4*(d + e*x)^(13/2))/(13*e^6) + (2*(A*e - B*d)*(a*e - b*d)^4*(d + e*x)^(3
/2))/(3*e^6) + (4*b*(a*e - b*d)^2*(d + e*x)^(7/2)*(3*A*b*e + 2*B*a*e - 5*B*b*d))/(7*e^6) + (4*b^2*(a*e - b*d)*
(d + e*x)^(9/2)*(2*A*b*e + 3*B*a*e - 5*B*b*d))/(9*e^6)