3.1804 \(\int \frac{(A+B x) \left (a^2+2 a b x+b^2 x^2\right )^3}{\sqrt{d+e x}} \, dx\)

Optimal. Leaf size=306 \[ -\frac{2 b^5 (d+e x)^{13/2} (-6 a B e-A b e+7 b B d)}{13 e^8}+\frac{6 b^4 (d+e x)^{11/2} (b d-a e) (-5 a B e-2 A b e+7 b B d)}{11 e^8}-\frac{10 b^3 (d+e x)^{9/2} (b d-a e)^2 (-4 a B e-3 A b e+7 b B d)}{9 e^8}+\frac{10 b^2 (d+e x)^{7/2} (b d-a e)^3 (-3 a B e-4 A b e+7 b B d)}{7 e^8}-\frac{6 b (d+e x)^{5/2} (b d-a e)^4 (-2 a B e-5 A b e+7 b B d)}{5 e^8}+\frac{2 (d+e x)^{3/2} (b d-a e)^5 (-a B e-6 A b e+7 b B d)}{3 e^8}-\frac{2 \sqrt{d+e x} (b d-a e)^6 (B d-A e)}{e^8}+\frac{2 b^6 B (d+e x)^{15/2}}{15 e^8} \]

[Out]

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

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Rubi [A]  time = 0.467921, antiderivative size = 306, 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 \[ -\frac{2 b^5 (d+e x)^{13/2} (-6 a B e-A b e+7 b B d)}{13 e^8}+\frac{6 b^4 (d+e x)^{11/2} (b d-a e) (-5 a B e-2 A b e+7 b B d)}{11 e^8}-\frac{10 b^3 (d+e x)^{9/2} (b d-a e)^2 (-4 a B e-3 A b e+7 b B d)}{9 e^8}+\frac{10 b^2 (d+e x)^{7/2} (b d-a e)^3 (-3 a B e-4 A b e+7 b B d)}{7 e^8}-\frac{6 b (d+e x)^{5/2} (b d-a e)^4 (-2 a B e-5 A b e+7 b B d)}{5 e^8}+\frac{2 (d+e x)^{3/2} (b d-a e)^5 (-a B e-6 A b e+7 b B d)}{3 e^8}-\frac{2 \sqrt{d+e x} (b d-a e)^6 (B d-A e)}{e^8}+\frac{2 b^6 B (d+e x)^{15/2}}{15 e^8} \]

Antiderivative was successfully verified.

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

[Out]

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

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Rubi in Sympy [A]  time = 161.324, size = 314, normalized size = 1.03 \[ \frac{2 B b^{6} \left (d + e x\right )^{\frac{15}{2}}}{15 e^{8}} + \frac{2 b^{5} \left (d + e x\right )^{\frac{13}{2}} \left (A b e + 6 B a e - 7 B b d\right )}{13 e^{8}} + \frac{6 b^{4} \left (d + e x\right )^{\frac{11}{2}} \left (a e - b d\right ) \left (2 A b e + 5 B a e - 7 B b d\right )}{11 e^{8}} + \frac{10 b^{3} \left (d + e x\right )^{\frac{9}{2}} \left (a e - b d\right )^{2} \left (3 A b e + 4 B a e - 7 B b d\right )}{9 e^{8}} + \frac{10 b^{2} \left (d + e x\right )^{\frac{7}{2}} \left (a e - b d\right )^{3} \left (4 A b e + 3 B a e - 7 B b d\right )}{7 e^{8}} + \frac{6 b \left (d + e x\right )^{\frac{5}{2}} \left (a e - b d\right )^{4} \left (5 A b e + 2 B a e - 7 B b d\right )}{5 e^{8}} + \frac{2 \left (d + e x\right )^{\frac{3}{2}} \left (a e - b d\right )^{5} \left (6 A b e + B a e - 7 B b d\right )}{3 e^{8}} + \frac{2 \sqrt{d + e x} \left (A e - B d\right ) \left (a e - b d\right )^{6}}{e^{8}} \]

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

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Mathematica [B]  time = 1.32576, size = 628, normalized size = 2.05 \[ \frac{2 \sqrt{d+e x} \left (15015 a^6 e^6 (3 A e-2 B d+B e x)+18018 a^5 b e^5 \left (5 A e (e x-2 d)+B \left (8 d^2-4 d e x+3 e^2 x^2\right )\right )-6435 a^4 b^2 e^4 \left (3 B \left (16 d^3-8 d^2 e x+6 d e^2 x^2-5 e^3 x^3\right )-7 A e \left (8 d^2-4 d e x+3 e^2 x^2\right )\right )+2860 a^3 b^3 e^3 \left (9 A e \left (-16 d^3+8 d^2 e x-6 d e^2 x^2+5 e^3 x^3\right )+B \left (128 d^4-64 d^3 e x+48 d^2 e^2 x^2-40 d e^3 x^3+35 e^4 x^4\right )\right )-195 a^2 b^4 e^2 \left (5 B \left (256 d^5-128 d^4 e x+96 d^3 e^2 x^2-80 d^2 e^3 x^3+70 d e^4 x^4-63 e^5 x^5\right )-11 A e \left (128 d^4-64 d^3 e x+48 d^2 e^2 x^2-40 d e^3 x^3+35 e^4 x^4\right )\right )+30 a b^5 e \left (13 A e \left (-256 d^5+128 d^4 e x-96 d^3 e^2 x^2+80 d^2 e^3 x^3-70 d e^4 x^4+63 e^5 x^5\right )+3 B \left (1024 d^6-512 d^5 e x+384 d^4 e^2 x^2-320 d^3 e^3 x^3+280 d^2 e^4 x^4-252 d e^5 x^5+231 e^6 x^6\right )\right )+b^6 \left (15 A e \left (1024 d^6-512 d^5 e x+384 d^4 e^2 x^2-320 d^3 e^3 x^3+280 d^2 e^4 x^4-252 d e^5 x^5+231 e^6 x^6\right )-7 B \left (2048 d^7-1024 d^6 e x+768 d^5 e^2 x^2-640 d^4 e^3 x^3+560 d^3 e^4 x^4-504 d^2 e^5 x^5+462 d e^6 x^6-429 e^7 x^7\right )\right )\right )}{45045 e^8} \]

Antiderivative was successfully verified.

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

[Out]

(2*Sqrt[d + e*x]*(15015*a^6*e^6*(-2*B*d + 3*A*e + B*e*x) + 18018*a^5*b*e^5*(5*A*
e*(-2*d + e*x) + B*(8*d^2 - 4*d*e*x + 3*e^2*x^2)) - 6435*a^4*b^2*e^4*(-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))
+ 2860*a^3*b^3*e^3*(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)) - 195*a^2*b^
4*e^2*(-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)) + 30*a*b^5*e*(13*A*e*(-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) + 3*B*(1024*d^6 - 512*d^5*e*x + 384*
d^4*e^2*x^2 - 320*d^3*e^3*x^3 + 280*d^2*e^4*x^4 - 252*d*e^5*x^5 + 231*e^6*x^6))
+ b^6*(15*A*e*(1024*d^6 - 512*d^5*e*x + 384*d^4*e^2*x^2 - 320*d^3*e^3*x^3 + 280*
d^2*e^4*x^4 - 252*d*e^5*x^5 + 231*e^6*x^6) - 7*B*(2048*d^7 - 1024*d^6*e*x + 768*
d^5*e^2*x^2 - 640*d^4*e^3*x^3 + 560*d^3*e^4*x^4 - 504*d^2*e^5*x^5 + 462*d*e^6*x^
6 - 429*e^7*x^7))))/(45045*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)*(b^2*x^2+2*a*b*x+a^2)^3/(e*x+d)^(1/2),x)

[Out]

2/45045*(3003*B*b^6*e^7*x^7+3465*A*b^6*e^7*x^6+20790*B*a*b^5*e^7*x^6-3234*B*b^6*
d*e^6*x^6+24570*A*a*b^5*e^7*x^5-3780*A*b^6*d*e^6*x^5+61425*B*a^2*b^4*e^7*x^5-226
80*B*a*b^5*d*e^6*x^5+3528*B*b^6*d^2*e^5*x^5+75075*A*a^2*b^4*e^7*x^4-27300*A*a*b^
5*d*e^6*x^4+4200*A*b^6*d^2*e^5*x^4+100100*B*a^3*b^3*e^7*x^4-68250*B*a^2*b^4*d*e^
6*x^4+25200*B*a*b^5*d^2*e^5*x^4-3920*B*b^6*d^3*e^4*x^4+128700*A*a^3*b^3*e^7*x^3-
85800*A*a^2*b^4*d*e^6*x^3+31200*A*a*b^5*d^2*e^5*x^3-4800*A*b^6*d^3*e^4*x^3+96525
*B*a^4*b^2*e^7*x^3-114400*B*a^3*b^3*d*e^6*x^3+78000*B*a^2*b^4*d^2*e^5*x^3-28800*
B*a*b^5*d^3*e^4*x^3+4480*B*b^6*d^4*e^3*x^3+135135*A*a^4*b^2*e^7*x^2-154440*A*a^3
*b^3*d*e^6*x^2+102960*A*a^2*b^4*d^2*e^5*x^2-37440*A*a*b^5*d^3*e^4*x^2+5760*A*b^6
*d^4*e^3*x^2+54054*B*a^5*b*e^7*x^2-115830*B*a^4*b^2*d*e^6*x^2+137280*B*a^3*b^3*d
^2*e^5*x^2-93600*B*a^2*b^4*d^3*e^4*x^2+34560*B*a*b^5*d^4*e^3*x^2-5376*B*b^6*d^5*
e^2*x^2+90090*A*a^5*b*e^7*x-180180*A*a^4*b^2*d*e^6*x+205920*A*a^3*b^3*d^2*e^5*x-
137280*A*a^2*b^4*d^3*e^4*x+49920*A*a*b^5*d^4*e^3*x-7680*A*b^6*d^5*e^2*x+15015*B*
a^6*e^7*x-72072*B*a^5*b*d*e^6*x+154440*B*a^4*b^2*d^2*e^5*x-183040*B*a^3*b^3*d^3*
e^4*x+124800*B*a^2*b^4*d^4*e^3*x-46080*B*a*b^5*d^5*e^2*x+7168*B*b^6*d^6*e*x+4504
5*A*a^6*e^7-180180*A*a^5*b*d*e^6+360360*A*a^4*b^2*d^2*e^5-411840*A*a^3*b^3*d^3*e
^4+274560*A*a^2*b^4*d^4*e^3-99840*A*a*b^5*d^5*e^2+15360*A*b^6*d^6*e-30030*B*a^6*
d*e^6+144144*B*a^5*b*d^2*e^5-308880*B*a^4*b^2*d^3*e^4+366080*B*a^3*b^3*d^4*e^3-2
49600*B*a^2*b^4*d^5*e^2+92160*B*a*b^5*d^6*e-14336*B*b^6*d^7)*(e*x+d)^(1/2)/e^8

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Maxima [A]  time = 0.727744, size = 1035, normalized size = 3.38 \[ \text{result too large to display} \]

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]

2/45045*(3003*(e*x + d)^(15/2)*B*b^6 - 3465*(7*B*b^6*d - (6*B*a*b^5 + A*b^6)*e)*
(e*x + d)^(13/2) + 12285*(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)^(11/2) - 25025*(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)^(9/2) + 32175*(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)^(7/2) - 27027*(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)^(5/2) + 15015*(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)^(3/2) - 45045*(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)*sqrt(e*x + d))/e^8

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Fricas [A]  time = 0.290328, size = 1038, normalized size = 3.39 \[ \text{result too large to display} \]

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]

2/45045*(3003*B*b^6*e^7*x^7 - 14336*B*b^6*d^7 + 45045*A*a^6*e^7 + 15360*(6*B*a*b
^5 + A*b^6)*d^6*e - 49920*(5*B*a^2*b^4 + 2*A*a*b^5)*d^5*e^2 + 91520*(4*B*a^3*b^3
+ 3*A*a^2*b^4)*d^4*e^3 - 102960*(3*B*a^4*b^2 + 4*A*a^3*b^3)*d^3*e^4 + 72072*(2*
B*a^5*b + 5*A*a^4*b^2)*d^2*e^5 - 30030*(B*a^6 + 6*A*a^5*b)*d*e^6 - 231*(14*B*b^6
*d*e^6 - 15*(6*B*a*b^5 + A*b^6)*e^7)*x^6 + 63*(56*B*b^6*d^2*e^5 - 60*(6*B*a*b^5
+ A*b^6)*d*e^6 + 195*(5*B*a^2*b^4 + 2*A*a*b^5)*e^7)*x^5 - 35*(112*B*b^6*d^3*e^4
- 120*(6*B*a*b^5 + A*b^6)*d^2*e^5 + 390*(5*B*a^2*b^4 + 2*A*a*b^5)*d*e^6 - 715*(4
*B*a^3*b^3 + 3*A*a^2*b^4)*e^7)*x^4 + 5*(896*B*b^6*d^4*e^3 - 960*(6*B*a*b^5 + A*b
^6)*d^3*e^4 + 3120*(5*B*a^2*b^4 + 2*A*a*b^5)*d^2*e^5 - 5720*(4*B*a^3*b^3 + 3*A*a
^2*b^4)*d*e^6 + 6435*(3*B*a^4*b^2 + 4*A*a^3*b^3)*e^7)*x^3 - 3*(1792*B*b^6*d^5*e^
2 - 1920*(6*B*a*b^5 + A*b^6)*d^4*e^3 + 6240*(5*B*a^2*b^4 + 2*A*a*b^5)*d^3*e^4 -
11440*(4*B*a^3*b^3 + 3*A*a^2*b^4)*d^2*e^5 + 12870*(3*B*a^4*b^2 + 4*A*a^3*b^3)*d*
e^6 - 9009*(2*B*a^5*b + 5*A*a^4*b^2)*e^7)*x^2 + (7168*B*b^6*d^6*e - 7680*(6*B*a*
b^5 + A*b^6)*d^5*e^2 + 24960*(5*B*a^2*b^4 + 2*A*a*b^5)*d^4*e^3 - 45760*(4*B*a^3*
b^3 + 3*A*a^2*b^4)*d^3*e^4 + 51480*(3*B*a^4*b^2 + 4*A*a^3*b^3)*d^2*e^5 - 36036*(
2*B*a^5*b + 5*A*a^4*b^2)*d*e^6 + 15015*(B*a^6 + 6*A*a^5*b)*e^7)*x)*sqrt(e*x + d)
/e^8

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Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \[ \text{Timed out} \]

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]

Timed out

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GIAC/XCAS [A]  time = 0.29713, 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