3.1795 \(\int (A+B x) \sqrt{d+e x} \left (a^2+2 a b x+b^2 x^2\right )^2 \, dx\)

Optimal. Leaf size=218 \[ -\frac{2 b^3 (d+e x)^{11/2} (-4 a B e-A b e+5 b B d)}{11 e^6}+\frac{4 b^2 (d+e x)^{9/2} (b d-a e) (-3 a B e-2 A b e+5 b B d)}{9 e^6}-\frac{4 b (d+e x)^{7/2} (b d-a e)^2 (-2 a B e-3 A b e+5 b B d)}{7 e^6}+\frac{2 (d+e x)^{5/2} (b d-a e)^3 (-a B e-4 A b e+5 b B d)}{5 e^6}-\frac{2 (d+e x)^{3/2} (b d-a e)^4 (B d-A e)}{3 e^6}+\frac{2 b^4 B (d+e x)^{13/2}}{13 e^6} \]

[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)

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Rubi [A]  time = 0.261069, antiderivative size = 218, 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 \[ -\frac{2 b^3 (d+e x)^{11/2} (-4 a B e-A b e+5 b B d)}{11 e^6}+\frac{4 b^2 (d+e x)^{9/2} (b d-a e) (-3 a B e-2 A b e+5 b B d)}{9 e^6}-\frac{4 b (d+e x)^{7/2} (b d-a e)^2 (-2 a B e-3 A b e+5 b B d)}{7 e^6}+\frac{2 (d+e x)^{5/2} (b d-a e)^3 (-a B e-4 A b e+5 b B d)}{5 e^6}-\frac{2 (d+e x)^{3/2} (b d-a e)^4 (B d-A e)}{3 e^6}+\frac{2 b^4 B (d+e x)^{13/2}}{13 e^6} \]

Antiderivative was successfully verified.

[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)

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Rubi in Sympy [A]  time = 103.865, size = 221, normalized size = 1.01 \[ \frac{2 B b^{4} \left (d + e x\right )^{\frac{13}{2}}}{13 e^{6}} + \frac{2 b^{3} \left (d + e x\right )^{\frac{11}{2}} \left (A b e + 4 B a e - 5 B b d\right )}{11 e^{6}} + \frac{4 b^{2} \left (d + e x\right )^{\frac{9}{2}} \left (a e - b d\right ) \left (2 A b e + 3 B a e - 5 B b d\right )}{9 e^{6}} + \frac{4 b \left (d + e x\right )^{\frac{7}{2}} \left (a e - b d\right )^{2} \left (3 A b e + 2 B a e - 5 B b d\right )}{7 e^{6}} + \frac{2 \left (d + e x\right )^{\frac{5}{2}} \left (a e - b d\right )^{3} \left (4 A b e + B a e - 5 B b d\right )}{5 e^{6}} + \frac{2 \left (d + e x\right )^{\frac{3}{2}} \left (A e - B d\right ) \left (a e - b d\right )^{4}}{3 e^{6}} \]

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)**2*(e*x+d)**(1/2),x)

[Out]

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

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Mathematica [A]  time = 0.619677, size = 339, normalized size = 1.56 \[ \frac{2 (d+e x)^{3/2} \left (3003 a^4 e^4 (5 A e-2 B d+3 B e x)+1716 a^3 b e^3 \left (7 A e (3 e x-2 d)+B \left (8 d^2-12 d e x+15 e^2 x^2\right )\right )-858 a^2 b^2 e^2 \left (B \left (16 d^3-24 d^2 e x+30 d e^2 x^2-35 e^3 x^3\right )-3 A e \left (8 d^2-12 d e x+15 e^2 x^2\right )\right )+52 a b^3 e \left (11 A e \left (-16 d^3+24 d^2 e x-30 d e^2 x^2+35 e^3 x^3\right )+B \left (128 d^4-192 d^3 e x+240 d^2 e^2 x^2-280 d e^3 x^3+315 e^4 x^4\right )\right )+b^4 \left (13 A e \left (128 d^4-192 d^3 e x+240 d^2 e^2 x^2-280 d e^3 x^3+315 e^4 x^4\right )-5 B \left (256 d^5-384 d^4 e x+480 d^3 e^2 x^2-560 d^2 e^3 x^3+630 d e^4 x^4-693 e^5 x^5\right )\right )\right )}{45045 e^6} \]

Antiderivative was successfully verified.

[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)

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Maple [B]  time = 0.015, size = 469, normalized size = 2.2 \[{\frac{6930\,{b}^{4}B{x}^{5}{e}^{5}+8190\,A{b}^{4}{e}^{5}{x}^{4}+32760\,Ba{b}^{3}{e}^{5}{x}^{4}-6300\,B{b}^{4}d{e}^{4}{x}^{4}+40040\,Aa{b}^{3}{e}^{5}{x}^{3}-7280\,A{b}^{4}d{e}^{4}{x}^{3}+60060\,B{a}^{2}{b}^{2}{e}^{5}{x}^{3}-29120\,Ba{b}^{3}d{e}^{4}{x}^{3}+5600\,B{b}^{4}{d}^{2}{e}^{3}{x}^{3}+77220\,A{a}^{2}{b}^{2}{e}^{5}{x}^{2}-34320\,Aa{b}^{3}d{e}^{4}{x}^{2}+6240\,A{b}^{4}{d}^{2}{e}^{3}{x}^{2}+51480\,B{a}^{3}b{e}^{5}{x}^{2}-51480\,B{a}^{2}{b}^{2}d{e}^{4}{x}^{2}+24960\,Ba{b}^{3}{d}^{2}{e}^{3}{x}^{2}-4800\,B{b}^{4}{d}^{3}{e}^{2}{x}^{2}+72072\,A{a}^{3}b{e}^{5}x-61776\,A{a}^{2}{b}^{2}d{e}^{4}x+27456\,Aa{b}^{3}{d}^{2}{e}^{3}x-4992\,A{b}^{4}{d}^{3}{e}^{2}x+18018\,B{a}^{4}{e}^{5}x-41184\,B{a}^{3}bd{e}^{4}x+41184\,B{a}^{2}{b}^{2}{d}^{2}{e}^{3}x-19968\,Ba{b}^{3}{d}^{3}{e}^{2}x+3840\,B{b}^{4}{d}^{4}ex+30030\,A{a}^{4}{e}^{5}-48048\,Ad{a}^{3}b{e}^{4}+41184\,A{a}^{2}{b}^{2}{d}^{2}{e}^{3}-18304\,Aa{b}^{3}{d}^{3}{e}^{2}+3328\,A{d}^{4}{b}^{4}e-12012\,B{a}^{4}d{e}^{4}+27456\,B{d}^{2}{a}^{3}b{e}^{3}-27456\,B{d}^{3}{a}^{2}{b}^{2}{e}^{2}+13312\,B{d}^{4}a{b}^{3}e-2560\,{b}^{4}B{d}^{5}}{45045\,{e}^{6}} \left ( ex+d \right ) ^{{\frac{3}{2}}}} \]

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

[Out]

2/45045*(e*x+d)^(3/2)*(3465*B*b^4*e^5*x^5+4095*A*b^4*e^5*x^4+16380*B*a*b^3*e^5*x
^4-3150*B*b^4*d*e^4*x^4+20020*A*a*b^3*e^5*x^3-3640*A*b^4*d*e^4*x^3+30030*B*a^2*b
^2*e^5*x^3-14560*B*a*b^3*d*e^4*x^3+2800*B*b^4*d^2*e^3*x^3+38610*A*a^2*b^2*e^5*x^
2-17160*A*a*b^3*d*e^4*x^2+3120*A*b^4*d^2*e^3*x^2+25740*B*a^3*b*e^5*x^2-25740*B*a
^2*b^2*d*e^4*x^2+12480*B*a*b^3*d^2*e^3*x^2-2400*B*b^4*d^3*e^2*x^2+36036*A*a^3*b*
e^5*x-30888*A*a^2*b^2*d*e^4*x+13728*A*a*b^3*d^2*e^3*x-2496*A*b^4*d^3*e^2*x+9009*
B*a^4*e^5*x-20592*B*a^3*b*d*e^4*x+20592*B*a^2*b^2*d^2*e^3*x-9984*B*a*b^3*d^3*e^2
*x+1920*B*b^4*d^4*e*x+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^3-1
3728*B*a^2*b^2*d^3*e^2+6656*B*a*b^3*d^4*e-1280*B*b^4*d^5)/e^6

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Maxima [A]  time = 0.740947, size = 552, normalized size = 2.53 \[ \frac{2 \,{\left (3465 \,{\left (e x + d\right )}^{\frac{13}{2}} B b^{4} - 4095 \,{\left (5 \, B b^{4} d -{\left (4 \, B a b^{3} + A b^{4}\right )} e\right )}{\left (e x + d\right )}^{\frac{11}{2}} + 10010 \,{\left (5 \, B b^{4} d^{2} - 2 \,{\left (4 \, B a b^{3} + A b^{4}\right )} d e +{\left (3 \, B a^{2} b^{2} + 2 \, A a b^{3}\right )} e^{2}\right )}{\left (e x + d\right )}^{\frac{9}{2}} - 12870 \,{\left (5 \, B b^{4} d^{3} - 3 \,{\left (4 \, B a b^{3} + A b^{4}\right )} d^{2} e + 3 \,{\left (3 \, B a^{2} b^{2} + 2 \, A a b^{3}\right )} d e^{2} -{\left (2 \, B a^{3} b + 3 \, A a^{2} b^{2}\right )} e^{3}\right )}{\left (e x + d\right )}^{\frac{7}{2}} + 9009 \,{\left (5 \, B b^{4} d^{4} - 4 \,{\left (4 \, B a b^{3} + A b^{4}\right )} d^{3} e + 6 \,{\left (3 \, B a^{2} b^{2} + 2 \, A a b^{3}\right )} d^{2} e^{2} - 4 \,{\left (2 \, B a^{3} b + 3 \, A a^{2} b^{2}\right )} d e^{3} +{\left (B a^{4} + 4 \, A a^{3} b\right )} e^{4}\right )}{\left (e x + d\right )}^{\frac{5}{2}} - 15015 \,{\left (B b^{4} d^{5} - A a^{4} e^{5} -{\left (4 \, B a b^{3} + A b^{4}\right )} d^{4} e + 2 \,{\left (3 \, B a^{2} b^{2} + 2 \, A a b^{3}\right )} d^{3} e^{2} - 2 \,{\left (2 \, B a^{3} b + 3 \, A a^{2} b^{2}\right )} d^{2} e^{3} +{\left (B a^{4} + 4 \, A a^{3} b\right )} d e^{4}\right )}{\left (e x + d\right )}^{\frac{3}{2}}\right )}}{45045 \, e^{6}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

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Fricas [A]  time = 0.289206, size = 711, normalized size = 3.26 \[ \frac{2 \,{\left (3465 \, B b^{4} e^{6} x^{6} - 1280 \, B b^{4} d^{6} + 15015 \, A a^{4} d e^{5} + 1664 \,{\left (4 \, B a b^{3} + A b^{4}\right )} d^{5} e - 4576 \,{\left (3 \, B a^{2} b^{2} + 2 \, A a b^{3}\right )} d^{4} e^{2} + 6864 \,{\left (2 \, B a^{3} b + 3 \, A a^{2} b^{2}\right )} d^{3} e^{3} - 6006 \,{\left (B a^{4} + 4 \, A a^{3} b\right )} d^{2} e^{4} + 315 \,{\left (B b^{4} d e^{5} + 13 \,{\left (4 \, B a b^{3} + A b^{4}\right )} e^{6}\right )} x^{5} - 35 \,{\left (10 \, B b^{4} d^{2} e^{4} - 13 \,{\left (4 \, B a b^{3} + A b^{4}\right )} d e^{5} - 286 \,{\left (3 \, B a^{2} b^{2} + 2 \, A a b^{3}\right )} e^{6}\right )} x^{4} + 10 \,{\left (40 \, B b^{4} d^{3} e^{3} - 52 \,{\left (4 \, B a b^{3} + A b^{4}\right )} d^{2} e^{4} + 143 \,{\left (3 \, B a^{2} b^{2} + 2 \, A a b^{3}\right )} d e^{5} + 1287 \,{\left (2 \, B a^{3} b + 3 \, A a^{2} b^{2}\right )} e^{6}\right )} x^{3} - 3 \,{\left (160 \, B b^{4} d^{4} e^{2} - 208 \,{\left (4 \, B a b^{3} + A b^{4}\right )} d^{3} e^{3} + 572 \,{\left (3 \, B a^{2} b^{2} + 2 \, A a b^{3}\right )} d^{2} e^{4} - 858 \,{\left (2 \, B a^{3} b + 3 \, A a^{2} b^{2}\right )} d e^{5} - 3003 \,{\left (B a^{4} + 4 \, A a^{3} b\right )} e^{6}\right )} x^{2} +{\left (640 \, B b^{4} d^{5} e + 15015 \, A a^{4} e^{6} - 832 \,{\left (4 \, B a b^{3} + A b^{4}\right )} d^{4} e^{2} + 2288 \,{\left (3 \, B a^{2} b^{2} + 2 \, A a b^{3}\right )} d^{3} e^{3} - 3432 \,{\left (2 \, B a^{3} b + 3 \, A a^{2} b^{2}\right )} d^{2} e^{4} + 3003 \,{\left (B a^{4} + 4 \, A a^{3} b\right )} d e^{5}\right )} x\right )} \sqrt{e x + d}}{45045 \, e^{6}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b^2*x^2 + 2*a*b*x + a^2)^2*(B*x + A)*sqrt(e*x + d),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 + 57
2*(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

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Sympy [A]  time = 10.6131, size = 517, normalized size = 2.37 \[ \frac{2 \left (\frac{B b^{4} \left (d + e x\right )^{\frac{13}{2}}}{13 e^{5}} + \frac{\left (d + e x\right )^{\frac{11}{2}} \left (A b^{4} e + 4 B a b^{3} e - 5 B b^{4} d\right )}{11 e^{5}} + \frac{\left (d + e x\right )^{\frac{9}{2}} \left (4 A a b^{3} e^{2} - 4 A b^{4} d e + 6 B a^{2} b^{2} e^{2} - 16 B a b^{3} d e + 10 B b^{4} d^{2}\right )}{9 e^{5}} + \frac{\left (d + e x\right )^{\frac{7}{2}} \left (6 A a^{2} b^{2} e^{3} - 12 A a b^{3} d e^{2} + 6 A b^{4} d^{2} e + 4 B a^{3} b e^{3} - 18 B a^{2} b^{2} d e^{2} + 24 B a b^{3} d^{2} e - 10 B b^{4} d^{3}\right )}{7 e^{5}} + \frac{\left (d + e x\right )^{\frac{5}{2}} \left (4 A a^{3} b e^{4} - 12 A a^{2} b^{2} d e^{3} + 12 A a b^{3} d^{2} e^{2} - 4 A b^{4} d^{3} e + B a^{4} e^{4} - 8 B a^{3} b d e^{3} + 18 B a^{2} b^{2} d^{2} e^{2} - 16 B a b^{3} d^{3} e + 5 B b^{4} d^{4}\right )}{5 e^{5}} + \frac{\left (d + e x\right )^{\frac{3}{2}} \left (A a^{4} e^{5} - 4 A a^{3} b d e^{4} + 6 A a^{2} b^{2} d^{2} e^{3} - 4 A a b^{3} d^{3} e^{2} + A b^{4} d^{4} e - B a^{4} d e^{4} + 4 B a^{3} b d^{2} e^{3} - 6 B a^{2} b^{2} d^{3} e^{2} + 4 B a b^{3} d^{4} e - B b^{4} d^{5}\right )}{3 e^{5}}\right )}{e} \]

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)**2*(e*x+d)**(1/2),x)

[Out]

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)/(11*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

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GIAC/XCAS [A]  time = 0.296675, 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)^2*(B*x + A)*sqrt(e*x + d),x, algorithm="giac")

[Out]

Done