3.1796 \(\int \frac{(A+B x) \left (a^2+2 a b x+b^2 x^2\right )^2}{\sqrt{d+e x}} \, dx\)

Optimal. Leaf size=216 \[ -\frac{2 b^3 (d+e x)^{9/2} (-4 a B e-A b e+5 b B d)}{9 e^6}+\frac{4 b^2 (d+e x)^{7/2} (b d-a e) (-3 a B e-2 A b e+5 b B d)}{7 e^6}-\frac{4 b (d+e x)^{5/2} (b d-a e)^2 (-2 a B e-3 A b e+5 b B d)}{5 e^6}+\frac{2 (d+e x)^{3/2} (b d-a e)^3 (-a B e-4 A b e+5 b B d)}{3 e^6}-\frac{2 \sqrt{d+e x} (b d-a e)^4 (B d-A e)}{e^6}+\frac{2 b^4 B (d+e x)^{11/2}}{11 e^6} \]

[Out]

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

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Rubi [A]  time = 0.258971, antiderivative size = 216, 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 \[ -\frac{2 b^3 (d+e x)^{9/2} (-4 a B e-A b e+5 b B d)}{9 e^6}+\frac{4 b^2 (d+e x)^{7/2} (b d-a e) (-3 a B e-2 A b e+5 b B d)}{7 e^6}-\frac{4 b (d+e x)^{5/2} (b d-a e)^2 (-2 a B e-3 A b e+5 b B d)}{5 e^6}+\frac{2 (d+e x)^{3/2} (b d-a e)^3 (-a B e-4 A b e+5 b B d)}{3 e^6}-\frac{2 \sqrt{d+e x} (b d-a e)^4 (B d-A e)}{e^6}+\frac{2 b^4 B (d+e x)^{11/2}}{11 e^6} \]

Antiderivative was successfully verified.

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

[Out]

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

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Rubi in Sympy [A]  time = 102.053, size = 219, normalized size = 1.01 \[ \frac{2 B b^{4} \left (d + e x\right )^{\frac{11}{2}}}{11 e^{6}} + \frac{2 b^{3} \left (d + e x\right )^{\frac{9}{2}} \left (A b e + 4 B a e - 5 B b d\right )}{9 e^{6}} + \frac{4 b^{2} \left (d + e x\right )^{\frac{7}{2}} \left (a e - b d\right ) \left (2 A b e + 3 B a e - 5 B b d\right )}{7 e^{6}} + \frac{4 b \left (d + e x\right )^{\frac{5}{2}} \left (a e - b d\right )^{2} \left (3 A b e + 2 B a e - 5 B b d\right )}{5 e^{6}} + \frac{2 \left (d + e x\right )^{\frac{3}{2}} \left (a e - b d\right )^{3} \left (4 A b e + B a e - 5 B b d\right )}{3 e^{6}} + \frac{2 \sqrt{d + e x} \left (A e - B d\right ) \left (a e - b d\right )^{4}}{e^{6}} \]

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

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Mathematica [A]  time = 0.568125, size = 338, normalized size = 1.56 \[ \frac{2 \sqrt{d+e x} \left (1155 a^4 e^4 (3 A e-2 B d+B e x)+924 a^3 b e^3 \left (5 A e (e x-2 d)+B \left (8 d^2-4 d e x+3 e^2 x^2\right )\right )-198 a^2 b^2 e^2 \left (3 B \left (16 d^3-8 d^2 e x+6 d e^2 x^2-5 e^3 x^3\right )-7 A e \left (8 d^2-4 d e x+3 e^2 x^2\right )\right )+44 a b^3 e \left (9 A e \left (-16 d^3+8 d^2 e x-6 d e^2 x^2+5 e^3 x^3\right )+B \left (128 d^4-64 d^3 e x+48 d^2 e^2 x^2-40 d e^3 x^3+35 e^4 x^4\right )\right )+b^4 \left (11 A e \left (128 d^4-64 d^3 e x+48 d^2 e^2 x^2-40 d e^3 x^3+35 e^4 x^4\right )-5 B \left (256 d^5-128 d^4 e x+96 d^3 e^2 x^2-80 d^2 e^3 x^3+70 d e^4 x^4-63 e^5 x^5\right )\right )\right )}{3465 e^6} \]

Antiderivative was successfully verified.

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

[Out]

(2*Sqrt[d + e*x]*(1155*a^4*e^4*(-2*B*d + 3*A*e + B*e*x) + 924*a^3*b*e^3*(5*A*e*(
-2*d + e*x) + B*(8*d^2 - 4*d*e*x + 3*e^2*x^2)) - 198*a^2*b^2*e^2*(-7*A*e*(8*d^2
- 4*d*e*x + 3*e^2*x^2) + 3*B*(16*d^3 - 8*d^2*e*x + 6*d*e^2*x^2 - 5*e^3*x^3)) + 4
4*a*b^3*e*(9*A*e*(-16*d^3 + 8*d^2*e*x - 6*d*e^2*x^2 + 5*e^3*x^3) + B*(128*d^4 -
64*d^3*e*x + 48*d^2*e^2*x^2 - 40*d*e^3*x^3 + 35*e^4*x^4)) + b^4*(11*A*e*(128*d^4
- 64*d^3*e*x + 48*d^2*e^2*x^2 - 40*d*e^3*x^3 + 35*e^4*x^4) - 5*B*(256*d^5 - 128
*d^4*e*x + 96*d^3*e^2*x^2 - 80*d^2*e^3*x^3 + 70*d*e^4*x^4 - 63*e^5*x^5))))/(3465
*e^6)

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Maple [B]  time = 0.015, size = 469, normalized size = 2.2 \[{\frac{630\,{b}^{4}B{x}^{5}{e}^{5}+770\,A{b}^{4}{e}^{5}{x}^{4}+3080\,Ba{b}^{3}{e}^{5}{x}^{4}-700\,B{b}^{4}d{e}^{4}{x}^{4}+3960\,Aa{b}^{3}{e}^{5}{x}^{3}-880\,A{b}^{4}d{e}^{4}{x}^{3}+5940\,B{a}^{2}{b}^{2}{e}^{5}{x}^{3}-3520\,Ba{b}^{3}d{e}^{4}{x}^{3}+800\,B{b}^{4}{d}^{2}{e}^{3}{x}^{3}+8316\,A{a}^{2}{b}^{2}{e}^{5}{x}^{2}-4752\,Aa{b}^{3}d{e}^{4}{x}^{2}+1056\,A{b}^{4}{d}^{2}{e}^{3}{x}^{2}+5544\,B{a}^{3}b{e}^{5}{x}^{2}-7128\,B{a}^{2}{b}^{2}d{e}^{4}{x}^{2}+4224\,Ba{b}^{3}{d}^{2}{e}^{3}{x}^{2}-960\,B{b}^{4}{d}^{3}{e}^{2}{x}^{2}+9240\,A{a}^{3}b{e}^{5}x-11088\,A{a}^{2}{b}^{2}d{e}^{4}x+6336\,Aa{b}^{3}{d}^{2}{e}^{3}x-1408\,A{b}^{4}{d}^{3}{e}^{2}x+2310\,B{a}^{4}{e}^{5}x-7392\,B{a}^{3}bd{e}^{4}x+9504\,B{a}^{2}{b}^{2}{d}^{2}{e}^{3}x-5632\,Ba{b}^{3}{d}^{3}{e}^{2}x+1280\,B{b}^{4}{d}^{4}ex+6930\,A{a}^{4}{e}^{5}-18480\,Ad{a}^{3}b{e}^{4}+22176\,A{a}^{2}{b}^{2}{d}^{2}{e}^{3}-12672\,Aa{b}^{3}{d}^{3}{e}^{2}+2816\,A{d}^{4}{b}^{4}e-4620\,B{a}^{4}d{e}^{4}+14784\,B{d}^{2}{a}^{3}b{e}^{3}-19008\,B{d}^{3}{a}^{2}{b}^{2}{e}^{2}+11264\,B{d}^{4}a{b}^{3}e-2560\,{b}^{4}B{d}^{5}}{3465\,{e}^{6}}\sqrt{ex+d}} \]

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]

2/3465*(315*B*b^4*e^5*x^5+385*A*b^4*e^5*x^4+1540*B*a*b^3*e^5*x^4-350*B*b^4*d*e^4
*x^4+1980*A*a*b^3*e^5*x^3-440*A*b^4*d*e^4*x^3+2970*B*a^2*b^2*e^5*x^3-1760*B*a*b^
3*d*e^4*x^3+400*B*b^4*d^2*e^3*x^3+4158*A*a^2*b^2*e^5*x^2-2376*A*a*b^3*d*e^4*x^2+
528*A*b^4*d^2*e^3*x^2+2772*B*a^3*b*e^5*x^2-3564*B*a^2*b^2*d*e^4*x^2+2112*B*a*b^3
*d^2*e^3*x^2-480*B*b^4*d^3*e^2*x^2+4620*A*a^3*b*e^5*x-5544*A*a^2*b^2*d*e^4*x+316
8*A*a*b^3*d^2*e^3*x-704*A*b^4*d^3*e^2*x+1155*B*a^4*e^5*x-3696*B*a^3*b*d*e^4*x+47
52*B*a^2*b^2*d^2*e^3*x-2816*B*a*b^3*d^3*e^2*x+640*B*b^4*d^4*e*x+3465*A*a^4*e^5-9
240*A*a^3*b*d*e^4+11088*A*a^2*b^2*d^2*e^3-6336*A*a*b^3*d^3*e^2+1408*A*b^4*d^4*e-
2310*B*a^4*d*e^4+7392*B*a^3*b*d^2*e^3-9504*B*a^2*b^2*d^3*e^2+5632*B*a*b^3*d^4*e-
1280*B*b^4*d^5)*(e*x+d)^(1/2)/e^6

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Maxima [A]  time = 0.732245, size = 552, normalized size = 2.56 \[ \frac{2 \,{\left (315 \,{\left (e x + d\right )}^{\frac{11}{2}} B b^{4} - 385 \,{\left (5 \, B b^{4} d -{\left (4 \, B a b^{3} + A b^{4}\right )} e\right )}{\left (e x + d\right )}^{\frac{9}{2}} + 990 \,{\left (5 \, B b^{4} d^{2} - 2 \,{\left (4 \, B a b^{3} + A b^{4}\right )} d e +{\left (3 \, B a^{2} b^{2} + 2 \, A a b^{3}\right )} e^{2}\right )}{\left (e x + d\right )}^{\frac{7}{2}} - 1386 \,{\left (5 \, B b^{4} d^{3} - 3 \,{\left (4 \, B a b^{3} + A b^{4}\right )} d^{2} e + 3 \,{\left (3 \, B a^{2} b^{2} + 2 \, A a b^{3}\right )} d e^{2} -{\left (2 \, B a^{3} b + 3 \, A a^{2} b^{2}\right )} e^{3}\right )}{\left (e x + d\right )}^{\frac{5}{2}} + 1155 \,{\left (5 \, B b^{4} d^{4} - 4 \,{\left (4 \, B a b^{3} + A b^{4}\right )} d^{3} e + 6 \,{\left (3 \, B a^{2} b^{2} + 2 \, A a b^{3}\right )} d^{2} e^{2} - 4 \,{\left (2 \, B a^{3} b + 3 \, A a^{2} b^{2}\right )} d e^{3} +{\left (B a^{4} + 4 \, A a^{3} b\right )} e^{4}\right )}{\left (e x + d\right )}^{\frac{3}{2}} - 3465 \,{\left (B b^{4} d^{5} - A a^{4} e^{5} -{\left (4 \, B a b^{3} + A b^{4}\right )} d^{4} e + 2 \,{\left (3 \, B a^{2} b^{2} + 2 \, A a b^{3}\right )} d^{3} e^{2} - 2 \,{\left (2 \, B a^{3} b + 3 \, A a^{2} b^{2}\right )} d^{2} e^{3} +{\left (B a^{4} + 4 \, A a^{3} b\right )} d e^{4}\right )} \sqrt{e x + d}\right )}}{3465 \, e^{6}} \]

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]

2/3465*(315*(e*x + d)^(11/2)*B*b^4 - 385*(5*B*b^4*d - (4*B*a*b^3 + A*b^4)*e)*(e*
x + d)^(9/2) + 990*(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)^(7/2) - 1386*(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)^(5
/2) + 1155*(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)^(3/2) - 3465*(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)*sqrt(e*x + d))/e^6

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Fricas [A]  time = 0.292938, size = 551, normalized size = 2.55 \[ \frac{2 \,{\left (315 \, B b^{4} e^{5} x^{5} - 1280 \, B b^{4} d^{5} + 3465 \, A a^{4} e^{5} + 1408 \,{\left (4 \, B a b^{3} + A b^{4}\right )} d^{4} e - 3168 \,{\left (3 \, B a^{2} b^{2} + 2 \, A a b^{3}\right )} d^{3} e^{2} + 3696 \,{\left (2 \, B a^{3} b + 3 \, A a^{2} b^{2}\right )} d^{2} e^{3} - 2310 \,{\left (B a^{4} + 4 \, A a^{3} b\right )} d e^{4} - 35 \,{\left (10 \, B b^{4} d e^{4} - 11 \,{\left (4 \, B a b^{3} + A b^{4}\right )} e^{5}\right )} x^{4} + 10 \,{\left (40 \, B b^{4} d^{2} e^{3} - 44 \,{\left (4 \, B a b^{3} + A b^{4}\right )} d e^{4} + 99 \,{\left (3 \, B a^{2} b^{2} + 2 \, A a b^{3}\right )} e^{5}\right )} x^{3} - 6 \,{\left (80 \, B b^{4} d^{3} e^{2} - 88 \,{\left (4 \, B a b^{3} + A b^{4}\right )} d^{2} e^{3} + 198 \,{\left (3 \, B a^{2} b^{2} + 2 \, A a b^{3}\right )} d e^{4} - 231 \,{\left (2 \, B a^{3} b + 3 \, A a^{2} b^{2}\right )} e^{5}\right )} x^{2} +{\left (640 \, B b^{4} d^{4} e - 704 \,{\left (4 \, B a b^{3} + A b^{4}\right )} d^{3} e^{2} + 1584 \,{\left (3 \, B a^{2} b^{2} + 2 \, A a b^{3}\right )} d^{2} e^{3} - 1848 \,{\left (2 \, B a^{3} b + 3 \, A a^{2} b^{2}\right )} d e^{4} + 1155 \,{\left (B a^{4} + 4 \, A a^{3} b\right )} e^{5}\right )} x\right )} \sqrt{e x + d}}{3465 \, e^{6}} \]

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]

2/3465*(315*B*b^4*e^5*x^5 - 1280*B*b^4*d^5 + 3465*A*a^4*e^5 + 1408*(4*B*a*b^3 +
A*b^4)*d^4*e - 3168*(3*B*a^2*b^2 + 2*A*a*b^3)*d^3*e^2 + 3696*(2*B*a^3*b + 3*A*a^
2*b^2)*d^2*e^3 - 2310*(B*a^4 + 4*A*a^3*b)*d*e^4 - 35*(10*B*b^4*d*e^4 - 11*(4*B*a
*b^3 + A*b^4)*e^5)*x^4 + 10*(40*B*b^4*d^2*e^3 - 44*(4*B*a*b^3 + A*b^4)*d*e^4 + 9
9*(3*B*a^2*b^2 + 2*A*a*b^3)*e^5)*x^3 - 6*(80*B*b^4*d^3*e^2 - 88*(4*B*a*b^3 + A*b
^4)*d^2*e^3 + 198*(3*B*a^2*b^2 + 2*A*a*b^3)*d*e^4 - 231*(2*B*a^3*b + 3*A*a^2*b^2
)*e^5)*x^2 + (640*B*b^4*d^4*e - 704*(4*B*a*b^3 + A*b^4)*d^3*e^2 + 1584*(3*B*a^2*
b^2 + 2*A*a*b^3)*d^2*e^3 - 1848*(2*B*a^3*b + 3*A*a^2*b^2)*d*e^4 + 1155*(B*a^4 +
4*A*a^3*b)*e^5)*x)*sqrt(e*x + d)/e^6

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Sympy [A]  time = 92.1398, size = 1311, normalized size = 6.07 \[ \text{result too large to display} \]

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

[Out]

Piecewise((-(2*A*a**4*d/sqrt(d + e*x) + 2*A*a**4*(-d/sqrt(d + e*x) - sqrt(d + e*
x)) + 8*A*a**3*b*d*(-d/sqrt(d + e*x) - sqrt(d + e*x))/e + 8*A*a**3*b*(d**2/sqrt(
d + e*x) + 2*d*sqrt(d + e*x) - (d + e*x)**(3/2)/3)/e + 12*A*a**2*b**2*d*(d**2/sq
rt(d + e*x) + 2*d*sqrt(d + e*x) - (d + e*x)**(3/2)/3)/e**2 + 12*A*a**2*b**2*(-d*
*3/sqrt(d + e*x) - 3*d**2*sqrt(d + e*x) + d*(d + e*x)**(3/2) - (d + e*x)**(5/2)/
5)/e**2 + 8*A*a*b**3*d*(-d**3/sqrt(d + e*x) - 3*d**2*sqrt(d + e*x) + d*(d + e*x)
**(3/2) - (d + e*x)**(5/2)/5)/e**3 + 8*A*a*b**3*(d**4/sqrt(d + e*x) + 4*d**3*sqr
t(d + e*x) - 2*d**2*(d + e*x)**(3/2) + 4*d*(d + e*x)**(5/2)/5 - (d + e*x)**(7/2)
/7)/e**3 + 2*A*b**4*d*(d**4/sqrt(d + e*x) + 4*d**3*sqrt(d + e*x) - 2*d**2*(d + e
*x)**(3/2) + 4*d*(d + e*x)**(5/2)/5 - (d + e*x)**(7/2)/7)/e**4 + 2*A*b**4*(-d**5
/sqrt(d + e*x) - 5*d**4*sqrt(d + e*x) + 10*d**3*(d + e*x)**(3/2)/3 - 2*d**2*(d +
e*x)**(5/2) + 5*d*(d + e*x)**(7/2)/7 - (d + e*x)**(9/2)/9)/e**4 + 2*B*a**4*d*(-
d/sqrt(d + e*x) - sqrt(d + e*x))/e + 2*B*a**4*(d**2/sqrt(d + e*x) + 2*d*sqrt(d +
e*x) - (d + e*x)**(3/2)/3)/e + 8*B*a**3*b*d*(d**2/sqrt(d + e*x) + 2*d*sqrt(d +
e*x) - (d + e*x)**(3/2)/3)/e**2 + 8*B*a**3*b*(-d**3/sqrt(d + e*x) - 3*d**2*sqrt(
d + e*x) + d*(d + e*x)**(3/2) - (d + e*x)**(5/2)/5)/e**2 + 12*B*a**2*b**2*d*(-d*
*3/sqrt(d + e*x) - 3*d**2*sqrt(d + e*x) + d*(d + e*x)**(3/2) - (d + e*x)**(5/2)/
5)/e**3 + 12*B*a**2*b**2*(d**4/sqrt(d + e*x) + 4*d**3*sqrt(d + e*x) - 2*d**2*(d
+ e*x)**(3/2) + 4*d*(d + e*x)**(5/2)/5 - (d + e*x)**(7/2)/7)/e**3 + 8*B*a*b**3*d
*(d**4/sqrt(d + e*x) + 4*d**3*sqrt(d + e*x) - 2*d**2*(d + e*x)**(3/2) + 4*d*(d +
e*x)**(5/2)/5 - (d + e*x)**(7/2)/7)/e**4 + 8*B*a*b**3*(-d**5/sqrt(d + e*x) - 5*
d**4*sqrt(d + e*x) + 10*d**3*(d + e*x)**(3/2)/3 - 2*d**2*(d + e*x)**(5/2) + 5*d*
(d + e*x)**(7/2)/7 - (d + e*x)**(9/2)/9)/e**4 + 2*B*b**4*d*(-d**5/sqrt(d + e*x)
- 5*d**4*sqrt(d + e*x) + 10*d**3*(d + e*x)**(3/2)/3 - 2*d**2*(d + e*x)**(5/2) +
5*d*(d + e*x)**(7/2)/7 - (d + e*x)**(9/2)/9)/e**5 + 2*B*b**4*(d**6/sqrt(d + e*x)
+ 6*d**5*sqrt(d + e*x) - 5*d**4*(d + e*x)**(3/2) + 4*d**3*(d + e*x)**(5/2) - 15
*d**2*(d + e*x)**(7/2)/7 + 2*d*(d + e*x)**(9/2)/3 - (d + e*x)**(11/2)/11)/e**5)/
e, Ne(e, 0)), ((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)/sqrt(d), True))

_______________________________________________________________________________________

GIAC/XCAS [A]  time = 0.283203, size = 760, normalized size = 3.52 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

2/3465*(1155*((x*e + d)^(3/2) - 3*sqrt(x*e + d)*d)*B*a^4*e^(-1) + 4620*((x*e + d
)^(3/2) - 3*sqrt(x*e + d)*d)*A*a^3*b*e^(-1) + 924*(3*(x*e + d)^(5/2)*e^8 - 10*(x
*e + d)^(3/2)*d*e^8 + 15*sqrt(x*e + d)*d^2*e^8)*B*a^3*b*e^(-10) + 1386*(3*(x*e +
d)^(5/2)*e^8 - 10*(x*e + d)^(3/2)*d*e^8 + 15*sqrt(x*e + d)*d^2*e^8)*A*a^2*b^2*e
^(-10) + 594*(5*(x*e + d)^(7/2)*e^18 - 21*(x*e + d)^(5/2)*d*e^18 + 35*(x*e + d)^
(3/2)*d^2*e^18 - 35*sqrt(x*e + d)*d^3*e^18)*B*a^2*b^2*e^(-21) + 396*(5*(x*e + d)
^(7/2)*e^18 - 21*(x*e + d)^(5/2)*d*e^18 + 35*(x*e + d)^(3/2)*d^2*e^18 - 35*sqrt(
x*e + d)*d^3*e^18)*A*a*b^3*e^(-21) + 44*(35*(x*e + d)^(9/2)*e^32 - 180*(x*e + d)
^(7/2)*d*e^32 + 378*(x*e + d)^(5/2)*d^2*e^32 - 420*(x*e + d)^(3/2)*d^3*e^32 + 31
5*sqrt(x*e + d)*d^4*e^32)*B*a*b^3*e^(-36) + 11*(35*(x*e + d)^(9/2)*e^32 - 180*(x
*e + d)^(7/2)*d*e^32 + 378*(x*e + d)^(5/2)*d^2*e^32 - 420*(x*e + d)^(3/2)*d^3*e^
32 + 315*sqrt(x*e + d)*d^4*e^32)*A*b^4*e^(-36) + 5*(63*(x*e + d)^(11/2)*e^50 - 3
85*(x*e + d)^(9/2)*d*e^50 + 990*(x*e + d)^(7/2)*d^2*e^50 - 1386*(x*e + d)^(5/2)*
d^3*e^50 + 1155*(x*e + d)^(3/2)*d^4*e^50 - 693*sqrt(x*e + d)*d^5*e^50)*B*b^4*e^(
-55) + 3465*sqrt(x*e + d)*A*a^4)*e^(-1)