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

Optimal. Leaf size=111 \[ \frac{2}{3} a^4 A x^{3/2}+\frac{2}{5} a^3 x^{5/2} (a B+4 A b)+\frac{4}{7} a^2 b x^{7/2} (2 a B+3 A b)+\frac{2}{11} b^3 x^{11/2} (4 a B+A b)+\frac{4}{9} a b^2 x^{9/2} (3 a B+2 A b)+\frac{2}{13} b^4 B x^{13/2} \]

[Out]

(2*a^4*A*x^(3/2))/3 + (2*a^3*(4*A*b + a*B)*x^(5/2))/5 + (4*a^2*b*(3*A*b + 2*a*B)
*x^(7/2))/7 + (4*a*b^2*(2*A*b + 3*a*B)*x^(9/2))/9 + (2*b^3*(A*b + 4*a*B)*x^(11/2
))/11 + (2*b^4*B*x^(13/2))/13

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Rubi [A]  time = 0.133102, antiderivative size = 111, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 29, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.069 \[ \frac{2}{3} a^4 A x^{3/2}+\frac{2}{5} a^3 x^{5/2} (a B+4 A b)+\frac{4}{7} a^2 b x^{7/2} (2 a B+3 A b)+\frac{2}{11} b^3 x^{11/2} (4 a B+A b)+\frac{4}{9} a b^2 x^{9/2} (3 a B+2 A b)+\frac{2}{13} b^4 B x^{13/2} \]

Antiderivative was successfully verified.

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

[Out]

(2*a^4*A*x^(3/2))/3 + (2*a^3*(4*A*b + a*B)*x^(5/2))/5 + (4*a^2*b*(3*A*b + 2*a*B)
*x^(7/2))/7 + (4*a*b^2*(2*A*b + 3*a*B)*x^(9/2))/9 + (2*b^3*(A*b + 4*a*B)*x^(11/2
))/11 + (2*b^4*B*x^(13/2))/13

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Rubi in Sympy [A]  time = 31.9127, size = 114, normalized size = 1.03 \[ \frac{2 A a^{4} x^{\frac{3}{2}}}{3} + \frac{2 B b^{4} x^{\frac{13}{2}}}{13} + \frac{2 a^{3} x^{\frac{5}{2}} \left (4 A b + B a\right )}{5} + \frac{4 a^{2} b x^{\frac{7}{2}} \left (3 A b + 2 B a\right )}{7} + \frac{4 a b^{2} x^{\frac{9}{2}} \left (2 A b + 3 B a\right )}{9} + \frac{2 b^{3} x^{\frac{11}{2}} \left (A b + 4 B a\right )}{11} \]

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*x**(1/2),x)

[Out]

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

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Mathematica [A]  time = 0.0468458, size = 111, normalized size = 1. \[ \frac{2}{3} a^4 A x^{3/2}+\frac{2}{5} a^3 x^{5/2} (a B+4 A b)+\frac{4}{7} a^2 b x^{7/2} (2 a B+3 A b)+\frac{2}{11} b^3 x^{11/2} (4 a B+A b)+\frac{4}{9} a b^2 x^{9/2} (3 a B+2 A b)+\frac{2}{13} b^4 B x^{13/2} \]

Antiderivative was successfully verified.

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

[Out]

(2*a^4*A*x^(3/2))/3 + (2*a^3*(4*A*b + a*B)*x^(5/2))/5 + (4*a^2*b*(3*A*b + 2*a*B)
*x^(7/2))/7 + (4*a*b^2*(2*A*b + 3*a*B)*x^(9/2))/9 + (2*b^3*(A*b + 4*a*B)*x^(11/2
))/11 + (2*b^4*B*x^(13/2))/13

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Maple [A]  time = 0.011, size = 100, normalized size = 0.9 \[{\frac{6930\,{b}^{4}B{x}^{5}+8190\,A{b}^{4}{x}^{4}+32760\,B{x}^{4}a{b}^{3}+40040\,aA{b}^{3}{x}^{3}+60060\,B{x}^{3}{a}^{2}{b}^{2}+77220\,{a}^{2}A{b}^{2}{x}^{2}+51480\,B{x}^{2}{a}^{3}b+72072\,{a}^{3}Abx+18018\,{a}^{4}Bx+30030\,A{a}^{4}}{45045}{x}^{{\frac{3}{2}}}} \]

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*x^(1/2),x)

[Out]

2/45045*x^(3/2)*(3465*B*b^4*x^5+4095*A*b^4*x^4+16380*B*a*b^3*x^4+20020*A*a*b^3*x
^3+30030*B*a^2*b^2*x^3+38610*A*a^2*b^2*x^2+25740*B*a^3*b*x^2+36036*A*a^3*b*x+900
9*B*a^4*x+15015*A*a^4)

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Maxima [A]  time = 0.680768, size = 134, normalized size = 1.21 \[ \frac{2}{13} \, B b^{4} x^{\frac{13}{2}} + \frac{2}{3} \, A a^{4} x^{\frac{3}{2}} + \frac{2}{11} \,{\left (4 \, B a b^{3} + A b^{4}\right )} x^{\frac{11}{2}} + \frac{4}{9} \,{\left (3 \, B a^{2} b^{2} + 2 \, A a b^{3}\right )} x^{\frac{9}{2}} + \frac{4}{7} \,{\left (2 \, B a^{3} b + 3 \, A a^{2} b^{2}\right )} x^{\frac{7}{2}} + \frac{2}{5} \,{\left (B a^{4} + 4 \, A a^{3} b\right )} x^{\frac{5}{2}} \]

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(x),x, algorithm="maxima")

[Out]

2/13*B*b^4*x^(13/2) + 2/3*A*a^4*x^(3/2) + 2/11*(4*B*a*b^3 + A*b^4)*x^(11/2) + 4/
9*(3*B*a^2*b^2 + 2*A*a*b^3)*x^(9/2) + 4/7*(2*B*a^3*b + 3*A*a^2*b^2)*x^(7/2) + 2/
5*(B*a^4 + 4*A*a^3*b)*x^(5/2)

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Fricas [A]  time = 0.283573, size = 138, normalized size = 1.24 \[ \frac{2}{45045} \,{\left (3465 \, B b^{4} x^{6} + 15015 \, A a^{4} x + 4095 \,{\left (4 \, B a b^{3} + A b^{4}\right )} x^{5} + 10010 \,{\left (3 \, B a^{2} b^{2} + 2 \, A a b^{3}\right )} x^{4} + 12870 \,{\left (2 \, B a^{3} b + 3 \, A a^{2} b^{2}\right )} x^{3} + 9009 \,{\left (B a^{4} + 4 \, A a^{3} b\right )} x^{2}\right )} \sqrt{x} \]

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(x),x, algorithm="fricas")

[Out]

2/45045*(3465*B*b^4*x^6 + 15015*A*a^4*x + 4095*(4*B*a*b^3 + A*b^4)*x^5 + 10010*(
3*B*a^2*b^2 + 2*A*a*b^3)*x^4 + 12870*(2*B*a^3*b + 3*A*a^2*b^2)*x^3 + 9009*(B*a^4
+ 4*A*a^3*b)*x^2)*sqrt(x)

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Sympy [A]  time = 4.53789, size = 124, normalized size = 1.12 \[ \frac{2 A a^{4} x^{\frac{3}{2}}}{3} + \frac{2 B b^{4} x^{\frac{13}{2}}}{13} + \frac{2 x^{\frac{11}{2}} \left (A b^{4} + 4 B a b^{3}\right )}{11} + \frac{2 x^{\frac{9}{2}} \left (4 A a b^{3} + 6 B a^{2} b^{2}\right )}{9} + \frac{2 x^{\frac{7}{2}} \left (6 A a^{2} b^{2} + 4 B a^{3} b\right )}{7} + \frac{2 x^{\frac{5}{2}} \left (4 A a^{3} b + B a^{4}\right )}{5} \]

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*x**(1/2),x)

[Out]

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

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GIAC/XCAS [A]  time = 0.269164, size = 136, normalized size = 1.23 \[ \frac{2}{13} \, B b^{4} x^{\frac{13}{2}} + \frac{8}{11} \, B a b^{3} x^{\frac{11}{2}} + \frac{2}{11} \, A b^{4} x^{\frac{11}{2}} + \frac{4}{3} \, B a^{2} b^{2} x^{\frac{9}{2}} + \frac{8}{9} \, A a b^{3} x^{\frac{9}{2}} + \frac{8}{7} \, B a^{3} b x^{\frac{7}{2}} + \frac{12}{7} \, A a^{2} b^{2} x^{\frac{7}{2}} + \frac{2}{5} \, B a^{4} x^{\frac{5}{2}} + \frac{8}{5} \, A a^{3} b x^{\frac{5}{2}} + \frac{2}{3} \, A a^{4} x^{\frac{3}{2}} \]

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(x),x, algorithm="giac")

[Out]

2/13*B*b^4*x^(13/2) + 8/11*B*a*b^3*x^(11/2) + 2/11*A*b^4*x^(11/2) + 4/3*B*a^2*b^
2*x^(9/2) + 8/9*A*a*b^3*x^(9/2) + 8/7*B*a^3*b*x^(7/2) + 12/7*A*a^2*b^2*x^(7/2) +
2/5*B*a^4*x^(5/2) + 8/5*A*a^3*b*x^(5/2) + 2/3*A*a^4*x^(3/2)