3.1803 \(\int (A+B x) \sqrt{d+e x} \left (a^2+2 a b x+b^2 x^2\right )^3 \, dx\)

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

[Out]

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

_______________________________________________________________________________________

Rubi [A]  time = 0.483215, 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 \[ -\frac{2 b^5 (d+e x)^{15/2} (-6 a B e-A b e+7 b B d)}{15 e^8}+\frac{6 b^4 (d+e x)^{13/2} (b d-a e) (-5 a B e-2 A b e+7 b B d)}{13 e^8}-\frac{10 b^3 (d+e x)^{11/2} (b d-a e)^2 (-4 a B e-3 A b e+7 b B d)}{11 e^8}+\frac{10 b^2 (d+e x)^{9/2} (b d-a e)^3 (-3 a B e-4 A b e+7 b B d)}{9 e^8}-\frac{6 b (d+e x)^{7/2} (b d-a e)^4 (-2 a B e-5 A b e+7 b B d)}{7 e^8}+\frac{2 (d+e x)^{5/2} (b d-a e)^5 (-a B e-6 A b e+7 b B d)}{5 e^8}-\frac{2 (d+e x)^{3/2} (b d-a e)^6 (B d-A e)}{3 e^8}+\frac{2 b^6 B (d+e x)^{17/2}}{17 e^8} \]

Antiderivative was successfully verified.

[In]  Int[(A + B*x)*Sqrt[d + e*x]*(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)^(3/2))/(3*e^8) + (2*(b*d - a*e)^5*(7*b*B
*d - 6*A*b*e - a*B*e)*(d + e*x)^(5/2))/(5*e^8) - (6*b*(b*d - a*e)^4*(7*b*B*d - 5
*A*b*e - 2*a*B*e)*(d + e*x)^(7/2))/(7*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)^(9/2))/(9*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)^(11/2))/(11*e^8) + (6*b^4*(b*d - a*e)*(7*b*B*d - 2*A*b
*e - 5*a*B*e)*(d + e*x)^(13/2))/(13*e^8) - (2*b^5*(7*b*B*d - A*b*e - 6*a*B*e)*(d
+ e*x)^(15/2))/(15*e^8) + (2*b^6*B*(d + e*x)^(17/2))/(17*e^8)

_______________________________________________________________________________________

Rubi in Sympy [A]  time = 167.542, size = 316, normalized size = 1.03 \[ \frac{2 B b^{6} \left (d + e x\right )^{\frac{17}{2}}}{17 e^{8}} + \frac{2 b^{5} \left (d + e x\right )^{\frac{15}{2}} \left (A b e + 6 B a e - 7 B b d\right )}{15 e^{8}} + \frac{6 b^{4} \left (d + e x\right )^{\frac{13}{2}} \left (a e - b d\right ) \left (2 A b e + 5 B a e - 7 B b d\right )}{13 e^{8}} + \frac{10 b^{3} \left (d + e x\right )^{\frac{11}{2}} \left (a e - b d\right )^{2} \left (3 A b e + 4 B a e - 7 B b d\right )}{11 e^{8}} + \frac{10 b^{2} \left (d + e x\right )^{\frac{9}{2}} \left (a e - b d\right )^{3} \left (4 A b e + 3 B a e - 7 B b d\right )}{9 e^{8}} + \frac{6 b \left (d + e x\right )^{\frac{7}{2}} \left (a e - b d\right )^{4} \left (5 A b e + 2 B a e - 7 B b d\right )}{7 e^{8}} + \frac{2 \left (d + e x\right )^{\frac{5}{2}} \left (a e - b d\right )^{5} \left (6 A b e + B a e - 7 B b d\right )}{5 e^{8}} + \frac{2 \left (d + e x\right )^{\frac{3}{2}} \left (A e - B d\right ) \left (a e - b d\right )^{6}}{3 e^{8}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((B*x+A)*(b**2*x**2+2*a*b*x+a**2)**3*(e*x+d)**(1/2),x)

[Out]

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

_______________________________________________________________________________________

Mathematica [B]  time = 1.25982, size = 628, normalized size = 2.04 \[ \frac{2 (d+e x)^{3/2} \left (51051 a^6 e^6 (5 A e-2 B d+3 B e x)+43758 a^5 b e^5 \left (7 A e (3 e x-2 d)+B \left (8 d^2-12 d e x+15 e^2 x^2\right )\right )-36465 a^4 b^2 e^4 \left (B \left (16 d^3-24 d^2 e x+30 d e^2 x^2-35 e^3 x^3\right )-3 A e \left (8 d^2-12 d e x+15 e^2 x^2\right )\right )+4420 a^3 b^3 e^3 \left (11 A e \left (-16 d^3+24 d^2 e x-30 d e^2 x^2+35 e^3 x^3\right )+B \left (128 d^4-192 d^3 e x+240 d^2 e^2 x^2-280 d e^3 x^3+315 e^4 x^4\right )\right )-255 a^2 b^4 e^2 \left (5 B \left (256 d^5-384 d^4 e x+480 d^3 e^2 x^2-560 d^2 e^3 x^3+630 d e^4 x^4-693 e^5 x^5\right )-13 A e \left (128 d^4-192 d^3 e x+240 d^2 e^2 x^2-280 d e^3 x^3+315 e^4 x^4\right )\right )+102 a b^5 e \left (5 A e \left (-256 d^5+384 d^4 e x-480 d^3 e^2 x^2+560 d^2 e^3 x^3-630 d e^4 x^4+693 e^5 x^5\right )+B \left (1024 d^6-1536 d^5 e x+1920 d^4 e^2 x^2-2240 d^3 e^3 x^3+2520 d^2 e^4 x^4-2772 d e^5 x^5+3003 e^6 x^6\right )\right )+b^6 \left (17 A e \left (1024 d^6-1536 d^5 e x+1920 d^4 e^2 x^2-2240 d^3 e^3 x^3+2520 d^2 e^4 x^4-2772 d e^5 x^5+3003 e^6 x^6\right )-7 B \left (2048 d^7-3072 d^6 e x+3840 d^5 e^2 x^2-4480 d^4 e^3 x^3+5040 d^3 e^4 x^4-5544 d^2 e^5 x^5+6006 d e^6 x^6-6435 e^7 x^7\right )\right )\right )}{765765 e^8} \]

Antiderivative was successfully verified.

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

[Out]

(2*(d + e*x)^(3/2)*(51051*a^6*e^6*(-2*B*d + 5*A*e + 3*B*e*x) + 43758*a^5*b*e^5*(
7*A*e*(-2*d + 3*e*x) + B*(8*d^2 - 12*d*e*x + 15*e^2*x^2)) - 36465*a^4*b^2*e^4*(-
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)) + 4420*a^3*b^3*e^3*(11*A*e*(-16*d^3 + 24*d^2*e*x - 30*d*e^2*x^2 + 3
5*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)) - 255*a^2*b^4*e^2*(-13*A*e*(128*d^4 - 192*d^3*e*x + 240*d^2*e^2*x^2 - 28
0*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)) + 102*a*b^5*e*(5*A*e*(-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) + B*
(1024*d^6 - 1536*d^5*e*x + 1920*d^4*e^2*x^2 - 2240*d^3*e^3*x^3 + 2520*d^2*e^4*x^
4 - 2772*d*e^5*x^5 + 3003*e^6*x^6)) + b^6*(17*A*e*(1024*d^6 - 1536*d^5*e*x + 192
0*d^4*e^2*x^2 - 2240*d^3*e^3*x^3 + 2520*d^2*e^4*x^4 - 2772*d*e^5*x^5 + 3003*e^6*
x^6) - 7*B*(2048*d^7 - 3072*d^6*e*x + 3840*d^5*e^2*x^2 - 4480*d^4*e^3*x^3 + 5040
*d^3*e^4*x^4 - 5544*d^2*e^5*x^5 + 6006*d*e^6*x^6 - 6435*e^7*x^7))))/(765765*e^8)

_______________________________________________________________________________________

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)*(b^2*x^2+2*a*b*x+a^2)^3*(e*x+d)^(1/2),x)

[Out]

2/765765*(e*x+d)^(3/2)*(45045*B*b^6*e^7*x^7+51051*A*b^6*e^7*x^6+306306*B*a*b^5*e
^7*x^6-42042*B*b^6*d*e^6*x^6+353430*A*a*b^5*e^7*x^5-47124*A*b^6*d*e^6*x^5+883575
*B*a^2*b^4*e^7*x^5-282744*B*a*b^5*d*e^6*x^5+38808*B*b^6*d^2*e^5*x^5+1044225*A*a^
2*b^4*e^7*x^4-321300*A*a*b^5*d*e^6*x^4+42840*A*b^6*d^2*e^5*x^4+1392300*B*a^3*b^3
*e^7*x^4-803250*B*a^2*b^4*d*e^6*x^4+257040*B*a*b^5*d^2*e^5*x^4-35280*B*b^6*d^3*e
^4*x^4+1701700*A*a^3*b^3*e^7*x^3-928200*A*a^2*b^4*d*e^6*x^3+285600*A*a*b^5*d^2*e
^5*x^3-38080*A*b^6*d^3*e^4*x^3+1276275*B*a^4*b^2*e^7*x^3-1237600*B*a^3*b^3*d*e^6
*x^3+714000*B*a^2*b^4*d^2*e^5*x^3-228480*B*a*b^5*d^3*e^4*x^3+31360*B*b^6*d^4*e^3
*x^3+1640925*A*a^4*b^2*e^7*x^2-1458600*A*a^3*b^3*d*e^6*x^2+795600*A*a^2*b^4*d^2*
e^5*x^2-244800*A*a*b^5*d^3*e^4*x^2+32640*A*b^6*d^4*e^3*x^2+656370*B*a^5*b*e^7*x^
2-1093950*B*a^4*b^2*d*e^6*x^2+1060800*B*a^3*b^3*d^2*e^5*x^2-612000*B*a^2*b^4*d^3
*e^4*x^2+195840*B*a*b^5*d^4*e^3*x^2-26880*B*b^6*d^5*e^2*x^2+918918*A*a^5*b*e^7*x
-1312740*A*a^4*b^2*d*e^6*x+1166880*A*a^3*b^3*d^2*e^5*x-636480*A*a^2*b^4*d^3*e^4*
x+195840*A*a*b^5*d^4*e^3*x-26112*A*b^6*d^5*e^2*x+153153*B*a^6*e^7*x-525096*B*a^5
*b*d*e^6*x+875160*B*a^4*b^2*d^2*e^5*x-848640*B*a^3*b^3*d^3*e^4*x+489600*B*a^2*b^
4*d^4*e^3*x-156672*B*a*b^5*d^5*e^2*x+21504*B*b^6*d^6*e*x+255255*A*a^6*e^7-612612
*A*a^5*b*d*e^6+875160*A*a^4*b^2*d^2*e^5-777920*A*a^3*b^3*d^3*e^4+424320*A*a^2*b^
4*d^4*e^3-130560*A*a*b^5*d^5*e^2+17408*A*b^6*d^6*e-102102*B*a^6*d*e^6+350064*B*a
^5*b*d^2*e^5-583440*B*a^4*b^2*d^3*e^4+565760*B*a^3*b^3*d^4*e^3-326400*B*a^2*b^4*
d^5*e^2+104448*B*a*b^5*d^6*e-14336*B*b^6*d^7)/e^8

_______________________________________________________________________________________

Maxima [A]  time = 0.722685, 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)*sqrt(e*x + d),x, algorithm="maxima")

[Out]

2/765765*(45045*(e*x + d)^(17/2)*B*b^6 - 51051*(7*B*b^6*d - (6*B*a*b^5 + A*b^6)*
e)*(e*x + d)^(15/2) + 176715*(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)^(13/2) - 348075*(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)^(11/2) + 425425*(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)^(9/2) - 328185*(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)^(7/2) + 153153*(7*B*b^6*d^6 - 6*(6*B*a*b^5 + A*b^6)*d^5*e + 15*(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 + 15*(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)^(5/2) - 255255*(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)^(3/2))/e^8

_______________________________________________________________________________________

Fricas [A]  time = 0.28996, size = 1272, normalized size = 4.13 \[ \text{result too large to display} \]

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)*sqrt(e*x + d),x, algorithm="fricas")

[Out]

2/765765*(45045*B*b^6*e^8*x^8 - 14336*B*b^6*d^8 + 255255*A*a^6*d*e^7 + 17408*(6*
B*a*b^5 + A*b^6)*d^7*e - 65280*(5*B*a^2*b^4 + 2*A*a*b^5)*d^6*e^2 + 141440*(4*B*a
^3*b^3 + 3*A*a^2*b^4)*d^5*e^3 - 194480*(3*B*a^4*b^2 + 4*A*a^3*b^3)*d^4*e^4 + 175
032*(2*B*a^5*b + 5*A*a^4*b^2)*d^3*e^5 - 102102*(B*a^6 + 6*A*a^5*b)*d^2*e^6 + 300
3*(B*b^6*d*e^7 + 17*(6*B*a*b^5 + A*b^6)*e^8)*x^7 - 231*(14*B*b^6*d^2*e^6 - 17*(6
*B*a*b^5 + A*b^6)*d*e^7 - 765*(5*B*a^2*b^4 + 2*A*a*b^5)*e^8)*x^6 + 63*(56*B*b^6*
d^3*e^5 - 68*(6*B*a*b^5 + A*b^6)*d^2*e^6 + 255*(5*B*a^2*b^4 + 2*A*a*b^5)*d*e^7 +
5525*(4*B*a^3*b^3 + 3*A*a^2*b^4)*e^8)*x^5 - 35*(112*B*b^6*d^4*e^4 - 136*(6*B*a*
b^5 + A*b^6)*d^3*e^5 + 510*(5*B*a^2*b^4 + 2*A*a*b^5)*d^2*e^6 - 1105*(4*B*a^3*b^3
+ 3*A*a^2*b^4)*d*e^7 - 12155*(3*B*a^4*b^2 + 4*A*a^3*b^3)*e^8)*x^4 + 5*(896*B*b^
6*d^5*e^3 - 1088*(6*B*a*b^5 + A*b^6)*d^4*e^4 + 4080*(5*B*a^2*b^4 + 2*A*a*b^5)*d^
3*e^5 - 8840*(4*B*a^3*b^3 + 3*A*a^2*b^4)*d^2*e^6 + 12155*(3*B*a^4*b^2 + 4*A*a^3*
b^3)*d*e^7 + 65637*(2*B*a^5*b + 5*A*a^4*b^2)*e^8)*x^3 - 3*(1792*B*b^6*d^6*e^2 -
2176*(6*B*a*b^5 + A*b^6)*d^5*e^3 + 8160*(5*B*a^2*b^4 + 2*A*a*b^5)*d^4*e^4 - 1768
0*(4*B*a^3*b^3 + 3*A*a^2*b^4)*d^3*e^5 + 24310*(3*B*a^4*b^2 + 4*A*a^3*b^3)*d^2*e^
6 - 21879*(2*B*a^5*b + 5*A*a^4*b^2)*d*e^7 - 51051*(B*a^6 + 6*A*a^5*b)*e^8)*x^2 +
(7168*B*b^6*d^7*e + 255255*A*a^6*e^8 - 8704*(6*B*a*b^5 + A*b^6)*d^6*e^2 + 32640
*(5*B*a^2*b^4 + 2*A*a*b^5)*d^5*e^3 - 70720*(4*B*a^3*b^3 + 3*A*a^2*b^4)*d^4*e^4 +
97240*(3*B*a^4*b^2 + 4*A*a^3*b^3)*d^3*e^5 - 87516*(2*B*a^5*b + 5*A*a^4*b^2)*d^2
*e^6 + 51051*(B*a^6 + 6*A*a^5*b)*d*e^7)*x)*sqrt(e*x + d)/e^8

_______________________________________________________________________________________

Sympy [A]  time = 16.975, size = 969, normalized size = 3.15 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((B*x+A)*(b**2*x**2+2*a*b*x+a**2)**3*(e*x+d)**(1/2),x)

[Out]

2*(B*b**6*(d + e*x)**(17/2)/(17*e**7) + (d + e*x)**(15/2)*(A*b**6*e + 6*B*a*b**5
*e - 7*B*b**6*d)/(15*e**7) + (d + e*x)**(13/2)*(6*A*a*b**5*e**2 - 6*A*b**6*d*e +
15*B*a**2*b**4*e**2 - 36*B*a*b**5*d*e + 21*B*b**6*d**2)/(13*e**7) + (d + e*x)**
(11/2)*(15*A*a**2*b**4*e**3 - 30*A*a*b**5*d*e**2 + 15*A*b**6*d**2*e + 20*B*a**3*
b**3*e**3 - 75*B*a**2*b**4*d*e**2 + 90*B*a*b**5*d**2*e - 35*B*b**6*d**3)/(11*e**
7) + (d + e*x)**(9/2)*(20*A*a**3*b**3*e**4 - 60*A*a**2*b**4*d*e**3 + 60*A*a*b**5
*d**2*e**2 - 20*A*b**6*d**3*e + 15*B*a**4*b**2*e**4 - 80*B*a**3*b**3*d*e**3 + 15
0*B*a**2*b**4*d**2*e**2 - 120*B*a*b**5*d**3*e + 35*B*b**6*d**4)/(9*e**7) + (d +
e*x)**(7/2)*(15*A*a**4*b**2*e**5 - 60*A*a**3*b**3*d*e**4 + 90*A*a**2*b**4*d**2*e
**3 - 60*A*a*b**5*d**3*e**2 + 15*A*b**6*d**4*e + 6*B*a**5*b*e**5 - 45*B*a**4*b**
2*d*e**4 + 120*B*a**3*b**3*d**2*e**3 - 150*B*a**2*b**4*d**3*e**2 + 90*B*a*b**5*d
**4*e - 21*B*b**6*d**5)/(7*e**7) + (d + e*x)**(5/2)*(6*A*a**5*b*e**6 - 30*A*a**4
*b**2*d*e**5 + 60*A*a**3*b**3*d**2*e**4 - 60*A*a**2*b**4*d**3*e**3 + 30*A*a*b**5
*d**4*e**2 - 6*A*b**6*d**5*e + B*a**6*e**6 - 12*B*a**5*b*d*e**5 + 45*B*a**4*b**2
*d**2*e**4 - 80*B*a**3*b**3*d**3*e**3 + 75*B*a**2*b**4*d**4*e**2 - 36*B*a*b**5*d
**5*e + 7*B*b**6*d**6)/(5*e**7) + (d + e*x)**(3/2)*(A*a**6*e**7 - 6*A*a**5*b*d*e
**6 + 15*A*a**4*b**2*d**2*e**5 - 20*A*a**3*b**3*d**3*e**4 + 15*A*a**2*b**4*d**4*
e**3 - 6*A*a*b**5*d**5*e**2 + A*b**6*d**6*e - B*a**6*d*e**6 + 6*B*a**5*b*d**2*e*
*5 - 15*B*a**4*b**2*d**3*e**4 + 20*B*a**3*b**3*d**4*e**3 - 15*B*a**2*b**4*d**5*e
**2 + 6*B*a*b**5*d**6*e - B*b**6*d**7)/(3*e**7))/e

_______________________________________________________________________________________

GIAC/XCAS [A]  time = 0.298343, 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)*sqrt(e*x + d),x, algorithm="giac")

[Out]

Done