3.1802 \(\int (A+B x) (d+e x)^{3/2} \left (a^2+2 a b x+b^2 x^2\right )^3 \, dx\)

Optimal. Leaf size=308 \[ -\frac{2 b^5 (d+e x)^{17/2} (-6 a B e-A b e+7 b B d)}{17 e^8}+\frac{2 b^4 (d+e x)^{15/2} (b d-a e) (-5 a B e-2 A b e+7 b B d)}{5 e^8}-\frac{10 b^3 (d+e x)^{13/2} (b d-a e)^2 (-4 a B e-3 A b e+7 b B d)}{13 e^8}+\frac{10 b^2 (d+e x)^{11/2} (b d-a e)^3 (-3 a B e-4 A b e+7 b B d)}{11 e^8}-\frac{2 b (d+e x)^{9/2} (b d-a e)^4 (-2 a B e-5 A b e+7 b B d)}{3 e^8}+\frac{2 (d+e x)^{7/2} (b d-a e)^5 (-a B e-6 A b e+7 b B d)}{7 e^8}-\frac{2 (d+e x)^{5/2} (b d-a e)^6 (B d-A e)}{5 e^8}+\frac{2 b^6 B (d+e x)^{19/2}}{19 e^8} \]

[Out]

(-2*(b*d - a*e)^6*(B*d - A*e)*(d + e*x)^(5/2))/(5*e^8) + (2*(b*d - a*e)^5*(7*b*B
*d - 6*A*b*e - a*B*e)*(d + e*x)^(7/2))/(7*e^8) - (2*b*(b*d - a*e)^4*(7*b*B*d - 5
*A*b*e - 2*a*B*e)*(d + e*x)^(9/2))/(3*e^8) + (10*b^2*(b*d - a*e)^3*(7*b*B*d - 4*
A*b*e - 3*a*B*e)*(d + e*x)^(11/2))/(11*e^8) - (10*b^3*(b*d - a*e)^2*(7*b*B*d - 3
*A*b*e - 4*a*B*e)*(d + e*x)^(13/2))/(13*e^8) + (2*b^4*(b*d - a*e)*(7*b*B*d - 2*A
*b*e - 5*a*B*e)*(d + e*x)^(15/2))/(5*e^8) - (2*b^5*(7*b*B*d - A*b*e - 6*a*B*e)*(
d + e*x)^(17/2))/(17*e^8) + (2*b^6*B*(d + e*x)^(19/2))/(19*e^8)

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Rubi [A]  time = 0.478858, antiderivative size = 308, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 33, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.061 \[ -\frac{2 b^5 (d+e x)^{17/2} (-6 a B e-A b e+7 b B d)}{17 e^8}+\frac{2 b^4 (d+e x)^{15/2} (b d-a e) (-5 a B e-2 A b e+7 b B d)}{5 e^8}-\frac{10 b^3 (d+e x)^{13/2} (b d-a e)^2 (-4 a B e-3 A b e+7 b B d)}{13 e^8}+\frac{10 b^2 (d+e x)^{11/2} (b d-a e)^3 (-3 a B e-4 A b e+7 b B d)}{11 e^8}-\frac{2 b (d+e x)^{9/2} (b d-a e)^4 (-2 a B e-5 A b e+7 b B d)}{3 e^8}+\frac{2 (d+e x)^{7/2} (b d-a e)^5 (-a B e-6 A b e+7 b B d)}{7 e^8}-\frac{2 (d+e x)^{5/2} (b d-a e)^6 (B d-A e)}{5 e^8}+\frac{2 b^6 B (d+e x)^{19/2}}{19 e^8} \]

Antiderivative was successfully verified.

[In]  Int[(A + B*x)*(d + e*x)^(3/2)*(a^2 + 2*a*b*x + b^2*x^2)^3,x]

[Out]

(-2*(b*d - a*e)^6*(B*d - A*e)*(d + e*x)^(5/2))/(5*e^8) + (2*(b*d - a*e)^5*(7*b*B
*d - 6*A*b*e - a*B*e)*(d + e*x)^(7/2))/(7*e^8) - (2*b*(b*d - a*e)^4*(7*b*B*d - 5
*A*b*e - 2*a*B*e)*(d + e*x)^(9/2))/(3*e^8) + (10*b^2*(b*d - a*e)^3*(7*b*B*d - 4*
A*b*e - 3*a*B*e)*(d + e*x)^(11/2))/(11*e^8) - (10*b^3*(b*d - a*e)^2*(7*b*B*d - 3
*A*b*e - 4*a*B*e)*(d + e*x)^(13/2))/(13*e^8) + (2*b^4*(b*d - a*e)*(7*b*B*d - 2*A
*b*e - 5*a*B*e)*(d + e*x)^(15/2))/(5*e^8) - (2*b^5*(7*b*B*d - A*b*e - 6*a*B*e)*(
d + e*x)^(17/2))/(17*e^8) + (2*b^6*B*(d + e*x)^(19/2))/(19*e^8)

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Rubi in Sympy [A]  time = 173.116, size = 316, normalized size = 1.03 \[ \frac{2 B b^{6} \left (d + e x\right )^{\frac{19}{2}}}{19 e^{8}} + \frac{2 b^{5} \left (d + e x\right )^{\frac{17}{2}} \left (A b e + 6 B a e - 7 B b d\right )}{17 e^{8}} + \frac{2 b^{4} \left (d + e x\right )^{\frac{15}{2}} \left (a e - b d\right ) \left (2 A b e + 5 B a e - 7 B b d\right )}{5 e^{8}} + \frac{10 b^{3} \left (d + e x\right )^{\frac{13}{2}} \left (a e - b d\right )^{2} \left (3 A b e + 4 B a e - 7 B b d\right )}{13 e^{8}} + \frac{10 b^{2} \left (d + e x\right )^{\frac{11}{2}} \left (a e - b d\right )^{3} \left (4 A b e + 3 B a e - 7 B b d\right )}{11 e^{8}} + \frac{2 b \left (d + e x\right )^{\frac{9}{2}} \left (a e - b d\right )^{4} \left (5 A b e + 2 B a e - 7 B b d\right )}{3 e^{8}} + \frac{2 \left (d + e x\right )^{\frac{7}{2}} \left (a e - b d\right )^{5} \left (6 A b e + B a e - 7 B b d\right )}{7 e^{8}} + \frac{2 \left (d + e x\right )^{\frac{5}{2}} \left (A e - B d\right ) \left (a e - b d\right )^{6}}{5 e^{8}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((B*x+A)*(e*x+d)**(3/2)*(b**2*x**2+2*a*b*x+a**2)**3,x)

[Out]

2*B*b**6*(d + e*x)**(19/2)/(19*e**8) + 2*b**5*(d + e*x)**(17/2)*(A*b*e + 6*B*a*e
- 7*B*b*d)/(17*e**8) + 2*b**4*(d + e*x)**(15/2)*(a*e - b*d)*(2*A*b*e + 5*B*a*e
- 7*B*b*d)/(5*e**8) + 10*b**3*(d + e*x)**(13/2)*(a*e - b*d)**2*(3*A*b*e + 4*B*a*
e - 7*B*b*d)/(13*e**8) + 10*b**2*(d + e*x)**(11/2)*(a*e - b*d)**3*(4*A*b*e + 3*B
*a*e - 7*B*b*d)/(11*e**8) + 2*b*(d + e*x)**(9/2)*(a*e - b*d)**4*(5*A*b*e + 2*B*a
*e - 7*B*b*d)/(3*e**8) + 2*(d + e*x)**(7/2)*(a*e - b*d)**5*(6*A*b*e + B*a*e - 7*
B*b*d)/(7*e**8) + 2*(d + e*x)**(5/2)*(A*e - B*d)*(a*e - b*d)**6/(5*e**8)

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Mathematica [B]  time = 1.45405, size = 629, normalized size = 2.04 \[ \frac{2 (d+e x)^{5/2} \left (138567 a^6 e^6 (7 A e-2 B d+5 B e x)+92378 a^5 b e^5 \left (9 A e (5 e x-2 d)+B \left (8 d^2-20 d e x+35 e^2 x^2\right )\right )-20995 a^4 b^2 e^4 \left (3 B \left (16 d^3-40 d^2 e x+70 d e^2 x^2-105 e^3 x^3\right )-11 A e \left (8 d^2-20 d e x+35 e^2 x^2\right )\right )+6460 a^3 b^3 e^3 \left (13 A e \left (-16 d^3+40 d^2 e x-70 d e^2 x^2+105 e^3 x^3\right )+B \left (128 d^4-320 d^3 e x+560 d^2 e^2 x^2-840 d e^3 x^3+1155 e^4 x^4\right )\right )-1615 a^2 b^4 e^2 \left (B \left (256 d^5-640 d^4 e x+1120 d^3 e^2 x^2-1680 d^2 e^3 x^3+2310 d e^4 x^4-3003 e^5 x^5\right )-3 A e \left (128 d^4-320 d^3 e x+560 d^2 e^2 x^2-840 d e^3 x^3+1155 e^4 x^4\right )\right )+38 a b^5 e \left (17 A e \left (-256 d^5+640 d^4 e x-1120 d^3 e^2 x^2+1680 d^2 e^3 x^3-2310 d e^4 x^4+3003 e^5 x^5\right )+3 B \left (1024 d^6-2560 d^5 e x+4480 d^4 e^2 x^2-6720 d^3 e^3 x^3+9240 d^2 e^4 x^4-12012 d e^5 x^5+15015 e^6 x^6\right )\right )+b^6 \left (19 A e \left (1024 d^6-2560 d^5 e x+4480 d^4 e^2 x^2-6720 d^3 e^3 x^3+9240 d^2 e^4 x^4-12012 d e^5 x^5+15015 e^6 x^6\right )-7 B \left (2048 d^7-5120 d^6 e x+8960 d^5 e^2 x^2-13440 d^4 e^3 x^3+18480 d^3 e^4 x^4-24024 d^2 e^5 x^5+30030 d e^6 x^6-36465 e^7 x^7\right )\right )\right )}{4849845 e^8} \]

Antiderivative was successfully verified.

[In]  Integrate[(A + B*x)*(d + e*x)^(3/2)*(a^2 + 2*a*b*x + b^2*x^2)^3,x]

[Out]

(2*(d + e*x)^(5/2)*(138567*a^6*e^6*(-2*B*d + 7*A*e + 5*B*e*x) + 92378*a^5*b*e^5*
(9*A*e*(-2*d + 5*e*x) + B*(8*d^2 - 20*d*e*x + 35*e^2*x^2)) - 20995*a^4*b^2*e^4*(
-11*A*e*(8*d^2 - 20*d*e*x + 35*e^2*x^2) + 3*B*(16*d^3 - 40*d^2*e*x + 70*d*e^2*x^
2 - 105*e^3*x^3)) + 6460*a^3*b^3*e^3*(13*A*e*(-16*d^3 + 40*d^2*e*x - 70*d*e^2*x^
2 + 105*e^3*x^3) + B*(128*d^4 - 320*d^3*e*x + 560*d^2*e^2*x^2 - 840*d*e^3*x^3 +
1155*e^4*x^4)) - 1615*a^2*b^4*e^2*(-3*A*e*(128*d^4 - 320*d^3*e*x + 560*d^2*e^2*x
^2 - 840*d*e^3*x^3 + 1155*e^4*x^4) + B*(256*d^5 - 640*d^4*e*x + 1120*d^3*e^2*x^2
- 1680*d^2*e^3*x^3 + 2310*d*e^4*x^4 - 3003*e^5*x^5)) + 38*a*b^5*e*(17*A*e*(-256
*d^5 + 640*d^4*e*x - 1120*d^3*e^2*x^2 + 1680*d^2*e^3*x^3 - 2310*d*e^4*x^4 + 3003
*e^5*x^5) + 3*B*(1024*d^6 - 2560*d^5*e*x + 4480*d^4*e^2*x^2 - 6720*d^3*e^3*x^3 +
9240*d^2*e^4*x^4 - 12012*d*e^5*x^5 + 15015*e^6*x^6)) + b^6*(19*A*e*(1024*d^6 -
2560*d^5*e*x + 4480*d^4*e^2*x^2 - 6720*d^3*e^3*x^3 + 9240*d^2*e^4*x^4 - 12012*d*
e^5*x^5 + 15015*e^6*x^6) - 7*B*(2048*d^7 - 5120*d^6*e*x + 8960*d^5*e^2*x^2 - 134
40*d^4*e^3*x^3 + 18480*d^3*e^4*x^4 - 24024*d^2*e^5*x^5 + 30030*d*e^6*x^6 - 36465
*e^7*x^7))))/(4849845*e^8)

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Maple [B]  time = 0.016, size = 913, normalized size = 3. \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((B*x+A)*(e*x+d)^(3/2)*(b^2*x^2+2*a*b*x+a^2)^3,x)

[Out]

2/4849845*(e*x+d)^(5/2)*(255255*B*b^6*e^7*x^7+285285*A*b^6*e^7*x^6+1711710*B*a*b
^5*e^7*x^6-210210*B*b^6*d*e^6*x^6+1939938*A*a*b^5*e^7*x^5-228228*A*b^6*d*e^6*x^5
+4849845*B*a^2*b^4*e^7*x^5-1369368*B*a*b^5*d*e^6*x^5+168168*B*b^6*d^2*e^5*x^5+55
95975*A*a^2*b^4*e^7*x^4-1492260*A*a*b^5*d*e^6*x^4+175560*A*b^6*d^2*e^5*x^4+74613
00*B*a^3*b^3*e^7*x^4-3730650*B*a^2*b^4*d*e^6*x^4+1053360*B*a*b^5*d^2*e^5*x^4-129
360*B*b^6*d^3*e^4*x^4+8817900*A*a^3*b^3*e^7*x^3-4069800*A*a^2*b^4*d*e^6*x^3+1085
280*A*a*b^5*d^2*e^5*x^3-127680*A*b^6*d^3*e^4*x^3+6613425*B*a^4*b^2*e^7*x^3-54264
00*B*a^3*b^3*d*e^6*x^3+2713200*B*a^2*b^4*d^2*e^5*x^3-766080*B*a*b^5*d^3*e^4*x^3+
94080*B*b^6*d^4*e^3*x^3+8083075*A*a^4*b^2*e^7*x^2-5878600*A*a^3*b^3*d*e^6*x^2+27
13200*A*a^2*b^4*d^2*e^5*x^2-723520*A*a*b^5*d^3*e^4*x^2+85120*A*b^6*d^4*e^3*x^2+3
233230*B*a^5*b*e^7*x^2-4408950*B*a^4*b^2*d*e^6*x^2+3617600*B*a^3*b^3*d^2*e^5*x^2
-1808800*B*a^2*b^4*d^3*e^4*x^2+510720*B*a*b^5*d^4*e^3*x^2-62720*B*b^6*d^5*e^2*x^
2+4157010*A*a^5*b*e^7*x-4618900*A*a^4*b^2*d*e^6*x+3359200*A*a^3*b^3*d^2*e^5*x-15
50400*A*a^2*b^4*d^3*e^4*x+413440*A*a*b^5*d^4*e^3*x-48640*A*b^6*d^5*e^2*x+692835*
B*a^6*e^7*x-1847560*B*a^5*b*d*e^6*x+2519400*B*a^4*b^2*d^2*e^5*x-2067200*B*a^3*b^
3*d^3*e^4*x+1033600*B*a^2*b^4*d^4*e^3*x-291840*B*a*b^5*d^5*e^2*x+35840*B*b^6*d^6
*e*x+969969*A*a^6*e^7-1662804*A*a^5*b*d*e^6+1847560*A*a^4*b^2*d^2*e^5-1343680*A*
a^3*b^3*d^3*e^4+620160*A*a^2*b^4*d^4*e^3-165376*A*a*b^5*d^5*e^2+19456*A*b^6*d^6*
e-277134*B*a^6*d*e^6+739024*B*a^5*b*d^2*e^5-1007760*B*a^4*b^2*d^3*e^4+826880*B*a
^3*b^3*d^4*e^3-413440*B*a^2*b^4*d^5*e^2+116736*B*a*b^5*d^6*e-14336*B*b^6*d^7)/e^
8

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Maxima [A]  time = 0.72808, size = 1035, normalized size = 3.36 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b^2*x^2 + 2*a*b*x + a^2)^3*(B*x + A)*(e*x + d)^(3/2),x, algorithm="maxima")

[Out]

2/4849845*(255255*(e*x + d)^(19/2)*B*b^6 - 285285*(7*B*b^6*d - (6*B*a*b^5 + A*b^
6)*e)*(e*x + d)^(17/2) + 969969*(7*B*b^6*d^2 - 2*(6*B*a*b^5 + A*b^6)*d*e + (5*B*
a^2*b^4 + 2*A*a*b^5)*e^2)*(e*x + d)^(15/2) - 1865325*(7*B*b^6*d^3 - 3*(6*B*a*b^5
+ A*b^6)*d^2*e + 3*(5*B*a^2*b^4 + 2*A*a*b^5)*d*e^2 - (4*B*a^3*b^3 + 3*A*a^2*b^4
)*e^3)*(e*x + d)^(13/2) + 2204475*(7*B*b^6*d^4 - 4*(6*B*a*b^5 + A*b^6)*d^3*e + 6
*(5*B*a^2*b^4 + 2*A*a*b^5)*d^2*e^2 - 4*(4*B*a^3*b^3 + 3*A*a^2*b^4)*d*e^3 + (3*B*
a^4*b^2 + 4*A*a^3*b^3)*e^4)*(e*x + d)^(11/2) - 1616615*(7*B*b^6*d^5 - 5*(6*B*a*b
^5 + A*b^6)*d^4*e + 10*(5*B*a^2*b^4 + 2*A*a*b^5)*d^3*e^2 - 10*(4*B*a^3*b^3 + 3*A
*a^2*b^4)*d^2*e^3 + 5*(3*B*a^4*b^2 + 4*A*a^3*b^3)*d*e^4 - (2*B*a^5*b + 5*A*a^4*b
^2)*e^5)*(e*x + d)^(9/2) + 692835*(7*B*b^6*d^6 - 6*(6*B*a*b^5 + A*b^6)*d^5*e + 1
5*(5*B*a^2*b^4 + 2*A*a*b^5)*d^4*e^2 - 20*(4*B*a^3*b^3 + 3*A*a^2*b^4)*d^3*e^3 + 1
5*(3*B*a^4*b^2 + 4*A*a^3*b^3)*d^2*e^4 - 6*(2*B*a^5*b + 5*A*a^4*b^2)*d*e^5 + (B*a
^6 + 6*A*a^5*b)*e^6)*(e*x + d)^(7/2) - 969969*(B*b^6*d^7 - A*a^6*e^7 - (6*B*a*b^
5 + A*b^6)*d^6*e + 3*(5*B*a^2*b^4 + 2*A*a*b^5)*d^5*e^2 - 5*(4*B*a^3*b^3 + 3*A*a^
2*b^4)*d^4*e^3 + 5*(3*B*a^4*b^2 + 4*A*a^3*b^3)*d^3*e^4 - 3*(2*B*a^5*b + 5*A*a^4*
b^2)*d^2*e^5 + (B*a^6 + 6*A*a^5*b)*d*e^6)*(e*x + d)^(5/2))/e^8

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Fricas [A]  time = 0.294129, size = 1509, normalized size = 4.9 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b^2*x^2 + 2*a*b*x + a^2)^3*(B*x + A)*(e*x + d)^(3/2),x, algorithm="fricas")

[Out]

2/4849845*(255255*B*b^6*e^9*x^9 - 14336*B*b^6*d^9 + 969969*A*a^6*d^2*e^7 + 19456
*(6*B*a*b^5 + A*b^6)*d^8*e - 82688*(5*B*a^2*b^4 + 2*A*a*b^5)*d^7*e^2 + 206720*(4
*B*a^3*b^3 + 3*A*a^2*b^4)*d^6*e^3 - 335920*(3*B*a^4*b^2 + 4*A*a^3*b^3)*d^5*e^4 +
369512*(2*B*a^5*b + 5*A*a^4*b^2)*d^4*e^5 - 277134*(B*a^6 + 6*A*a^5*b)*d^3*e^6 +
15015*(20*B*b^6*d*e^8 + 19*(6*B*a*b^5 + A*b^6)*e^9)*x^8 + 3003*(B*b^6*d^2*e^7 +
114*(6*B*a*b^5 + A*b^6)*d*e^8 + 323*(5*B*a^2*b^4 + 2*A*a*b^5)*e^9)*x^7 - 231*(1
4*B*b^6*d^3*e^6 - 19*(6*B*a*b^5 + A*b^6)*d^2*e^7 - 5168*(5*B*a^2*b^4 + 2*A*a*b^5
)*d*e^8 - 8075*(4*B*a^3*b^3 + 3*A*a^2*b^4)*e^9)*x^6 + 21*(168*B*b^6*d^4*e^5 - 22
8*(6*B*a*b^5 + A*b^6)*d^3*e^6 + 969*(5*B*a^2*b^4 + 2*A*a*b^5)*d^2*e^7 + 113050*(
4*B*a^3*b^3 + 3*A*a^2*b^4)*d*e^8 + 104975*(3*B*a^4*b^2 + 4*A*a^3*b^3)*e^9)*x^5 -
35*(112*B*b^6*d^5*e^4 - 152*(6*B*a*b^5 + A*b^6)*d^4*e^5 + 646*(5*B*a^2*b^4 + 2*
A*a*b^5)*d^3*e^6 - 1615*(4*B*a^3*b^3 + 3*A*a^2*b^4)*d^2*e^7 - 83980*(3*B*a^4*b^2
+ 4*A*a^3*b^3)*d*e^8 - 46189*(2*B*a^5*b + 5*A*a^4*b^2)*e^9)*x^4 + 5*(896*B*b^6*
d^6*e^3 - 1216*(6*B*a*b^5 + A*b^6)*d^5*e^4 + 5168*(5*B*a^2*b^4 + 2*A*a*b^5)*d^4*
e^5 - 12920*(4*B*a^3*b^3 + 3*A*a^2*b^4)*d^3*e^6 + 20995*(3*B*a^4*b^2 + 4*A*a^3*b
^3)*d^2*e^7 + 461890*(2*B*a^5*b + 5*A*a^4*b^2)*d*e^8 + 138567*(B*a^6 + 6*A*a^5*b
)*e^9)*x^3 - 3*(1792*B*b^6*d^7*e^2 - 323323*A*a^6*e^9 - 2432*(6*B*a*b^5 + A*b^6)
*d^6*e^3 + 10336*(5*B*a^2*b^4 + 2*A*a*b^5)*d^5*e^4 - 25840*(4*B*a^3*b^3 + 3*A*a^
2*b^4)*d^4*e^5 + 41990*(3*B*a^4*b^2 + 4*A*a^3*b^3)*d^3*e^6 - 46189*(2*B*a^5*b +
5*A*a^4*b^2)*d^2*e^7 - 369512*(B*a^6 + 6*A*a^5*b)*d*e^8)*x^2 + (7168*B*b^6*d^8*e
+ 1939938*A*a^6*d*e^8 - 9728*(6*B*a*b^5 + A*b^6)*d^7*e^2 + 41344*(5*B*a^2*b^4 +
2*A*a*b^5)*d^6*e^3 - 103360*(4*B*a^3*b^3 + 3*A*a^2*b^4)*d^5*e^4 + 167960*(3*B*a
^4*b^2 + 4*A*a^3*b^3)*d^4*e^5 - 184756*(2*B*a^5*b + 5*A*a^4*b^2)*d^3*e^6 + 13856
7*(B*a^6 + 6*A*a^5*b)*d^2*e^7)*x)*sqrt(e*x + d)/e^8

_______________________________________________________________________________________

Sympy [A]  time = 23.254, size = 2252, normalized size = 7.31 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((B*x+A)*(e*x+d)**(3/2)*(b**2*x**2+2*a*b*x+a**2)**3,x)

[Out]

A*a**6*d*Piecewise((sqrt(d)*x, Eq(e, 0)), (2*(d + e*x)**(3/2)/(3*e), True)) + 2*
A*a**6*(-d*(d + e*x)**(3/2)/3 + (d + e*x)**(5/2)/5)/e + 12*A*a**5*b*d*(-d*(d + e
*x)**(3/2)/3 + (d + e*x)**(5/2)/5)/e**2 + 12*A*a**5*b*(d**2*(d + e*x)**(3/2)/3 -
2*d*(d + e*x)**(5/2)/5 + (d + e*x)**(7/2)/7)/e**2 + 30*A*a**4*b**2*d*(d**2*(d +
e*x)**(3/2)/3 - 2*d*(d + e*x)**(5/2)/5 + (d + e*x)**(7/2)/7)/e**3 + 30*A*a**4*b
**2*(-d**3*(d + e*x)**(3/2)/3 + 3*d**2*(d + e*x)**(5/2)/5 - 3*d*(d + e*x)**(7/2)
/7 + (d + e*x)**(9/2)/9)/e**3 + 40*A*a**3*b**3*d*(-d**3*(d + e*x)**(3/2)/3 + 3*d
**2*(d + e*x)**(5/2)/5 - 3*d*(d + e*x)**(7/2)/7 + (d + e*x)**(9/2)/9)/e**4 + 40*
A*a**3*b**3*(d**4*(d + e*x)**(3/2)/3 - 4*d**3*(d + e*x)**(5/2)/5 + 6*d**2*(d + e
*x)**(7/2)/7 - 4*d*(d + e*x)**(9/2)/9 + (d + e*x)**(11/2)/11)/e**4 + 30*A*a**2*b
**4*d*(d**4*(d + e*x)**(3/2)/3 - 4*d**3*(d + e*x)**(5/2)/5 + 6*d**2*(d + e*x)**(
7/2)/7 - 4*d*(d + e*x)**(9/2)/9 + (d + e*x)**(11/2)/11)/e**5 + 30*A*a**2*b**4*(-
d**5*(d + e*x)**(3/2)/3 + d**4*(d + e*x)**(5/2) - 10*d**3*(d + e*x)**(7/2)/7 + 1
0*d**2*(d + e*x)**(9/2)/9 - 5*d*(d + e*x)**(11/2)/11 + (d + e*x)**(13/2)/13)/e**
5 + 12*A*a*b**5*d*(-d**5*(d + e*x)**(3/2)/3 + d**4*(d + e*x)**(5/2) - 10*d**3*(d
+ e*x)**(7/2)/7 + 10*d**2*(d + e*x)**(9/2)/9 - 5*d*(d + e*x)**(11/2)/11 + (d +
e*x)**(13/2)/13)/e**6 + 12*A*a*b**5*(d**6*(d + e*x)**(3/2)/3 - 6*d**5*(d + e*x)*
*(5/2)/5 + 15*d**4*(d + e*x)**(7/2)/7 - 20*d**3*(d + e*x)**(9/2)/9 + 15*d**2*(d
+ e*x)**(11/2)/11 - 6*d*(d + e*x)**(13/2)/13 + (d + e*x)**(15/2)/15)/e**6 + 2*A*
b**6*d*(d**6*(d + e*x)**(3/2)/3 - 6*d**5*(d + e*x)**(5/2)/5 + 15*d**4*(d + e*x)*
*(7/2)/7 - 20*d**3*(d + e*x)**(9/2)/9 + 15*d**2*(d + e*x)**(11/2)/11 - 6*d*(d +
e*x)**(13/2)/13 + (d + e*x)**(15/2)/15)/e**7 + 2*A*b**6*(-d**7*(d + e*x)**(3/2)/
3 + 7*d**6*(d + e*x)**(5/2)/5 - 3*d**5*(d + e*x)**(7/2) + 35*d**4*(d + e*x)**(9/
2)/9 - 35*d**3*(d + e*x)**(11/2)/11 + 21*d**2*(d + e*x)**(13/2)/13 - 7*d*(d + e*
x)**(15/2)/15 + (d + e*x)**(17/2)/17)/e**7 + 2*B*a**6*d*(-d*(d + e*x)**(3/2)/3 +
(d + e*x)**(5/2)/5)/e**2 + 2*B*a**6*(d**2*(d + e*x)**(3/2)/3 - 2*d*(d + e*x)**(
5/2)/5 + (d + e*x)**(7/2)/7)/e**2 + 12*B*a**5*b*d*(d**2*(d + e*x)**(3/2)/3 - 2*d
*(d + e*x)**(5/2)/5 + (d + e*x)**(7/2)/7)/e**3 + 12*B*a**5*b*(-d**3*(d + e*x)**(
3/2)/3 + 3*d**2*(d + e*x)**(5/2)/5 - 3*d*(d + e*x)**(7/2)/7 + (d + e*x)**(9/2)/9
)/e**3 + 30*B*a**4*b**2*d*(-d**3*(d + e*x)**(3/2)/3 + 3*d**2*(d + e*x)**(5/2)/5
- 3*d*(d + e*x)**(7/2)/7 + (d + e*x)**(9/2)/9)/e**4 + 30*B*a**4*b**2*(d**4*(d +
e*x)**(3/2)/3 - 4*d**3*(d + e*x)**(5/2)/5 + 6*d**2*(d + e*x)**(7/2)/7 - 4*d*(d +
e*x)**(9/2)/9 + (d + e*x)**(11/2)/11)/e**4 + 40*B*a**3*b**3*d*(d**4*(d + e*x)**
(3/2)/3 - 4*d**3*(d + e*x)**(5/2)/5 + 6*d**2*(d + e*x)**(7/2)/7 - 4*d*(d + e*x)*
*(9/2)/9 + (d + e*x)**(11/2)/11)/e**5 + 40*B*a**3*b**3*(-d**5*(d + e*x)**(3/2)/3
+ d**4*(d + e*x)**(5/2) - 10*d**3*(d + e*x)**(7/2)/7 + 10*d**2*(d + e*x)**(9/2)
/9 - 5*d*(d + e*x)**(11/2)/11 + (d + e*x)**(13/2)/13)/e**5 + 30*B*a**2*b**4*d*(-
d**5*(d + e*x)**(3/2)/3 + d**4*(d + e*x)**(5/2) - 10*d**3*(d + e*x)**(7/2)/7 + 1
0*d**2*(d + e*x)**(9/2)/9 - 5*d*(d + e*x)**(11/2)/11 + (d + e*x)**(13/2)/13)/e**
6 + 30*B*a**2*b**4*(d**6*(d + e*x)**(3/2)/3 - 6*d**5*(d + e*x)**(5/2)/5 + 15*d**
4*(d + e*x)**(7/2)/7 - 20*d**3*(d + e*x)**(9/2)/9 + 15*d**2*(d + e*x)**(11/2)/11
- 6*d*(d + e*x)**(13/2)/13 + (d + e*x)**(15/2)/15)/e**6 + 12*B*a*b**5*d*(d**6*(
d + e*x)**(3/2)/3 - 6*d**5*(d + e*x)**(5/2)/5 + 15*d**4*(d + e*x)**(7/2)/7 - 20*
d**3*(d + e*x)**(9/2)/9 + 15*d**2*(d + e*x)**(11/2)/11 - 6*d*(d + e*x)**(13/2)/1
3 + (d + e*x)**(15/2)/15)/e**7 + 12*B*a*b**5*(-d**7*(d + e*x)**(3/2)/3 + 7*d**6*
(d + e*x)**(5/2)/5 - 3*d**5*(d + e*x)**(7/2) + 35*d**4*(d + e*x)**(9/2)/9 - 35*d
**3*(d + e*x)**(11/2)/11 + 21*d**2*(d + e*x)**(13/2)/13 - 7*d*(d + e*x)**(15/2)/
15 + (d + e*x)**(17/2)/17)/e**7 + 2*B*b**6*d*(-d**7*(d + e*x)**(3/2)/3 + 7*d**6*
(d + e*x)**(5/2)/5 - 3*d**5*(d + e*x)**(7/2) + 35*d**4*(d + e*x)**(9/2)/9 - 35*d
**3*(d + e*x)**(11/2)/11 + 21*d**2*(d + e*x)**(13/2)/13 - 7*d*(d + e*x)**(15/2)/
15 + (d + e*x)**(17/2)/17)/e**8 + 2*B*b**6*(d**8*(d + e*x)**(3/2)/3 - 8*d**7*(d
+ e*x)**(5/2)/5 + 4*d**6*(d + e*x)**(7/2) - 56*d**5*(d + e*x)**(9/2)/9 + 70*d**4
*(d + e*x)**(11/2)/11 - 56*d**3*(d + e*x)**(13/2)/13 + 28*d**2*(d + e*x)**(15/2)
/15 - 8*d*(d + e*x)**(17/2)/17 + (d + e*x)**(19/2)/19)/e**8

_______________________________________________________________________________________

GIAC/XCAS [A]  time = 0.337061, size = 1, normalized size = 0. \[ \mathit{Done} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b^2*x^2 + 2*a*b*x + a^2)^3*(B*x + A)*(e*x + d)^(3/2),x, algorithm="giac")

[Out]

Done