3.730 \(\int x^{5/2} (A+B x) \left (a^2+2 a b x+b^2 x^2\right ) \, dx\)

Optimal. Leaf size=63 \[ \frac{2}{7} a^2 A x^{7/2}+\frac{2}{11} b x^{11/2} (2 a B+A b)+\frac{2}{9} a x^{9/2} (a B+2 A b)+\frac{2}{13} b^2 B x^{13/2} \]

[Out]

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

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Rubi [A]  time = 0.0807423, antiderivative size = 63, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 27, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.074 \[ \frac{2}{7} a^2 A x^{7/2}+\frac{2}{11} b x^{11/2} (2 a B+A b)+\frac{2}{9} a x^{9/2} (a B+2 A b)+\frac{2}{13} b^2 B x^{13/2} \]

Antiderivative was successfully verified.

[In]  Int[x^(5/2)*(A + B*x)*(a^2 + 2*a*b*x + b^2*x^2),x]

[Out]

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

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Rubi in Sympy [A]  time = 19.4908, size = 63, normalized size = 1. \[ \frac{2 A a^{2} x^{\frac{7}{2}}}{7} + \frac{2 B b^{2} x^{\frac{13}{2}}}{13} + \frac{2 a x^{\frac{9}{2}} \left (2 A b + B a\right )}{9} + \frac{2 b x^{\frac{11}{2}} \left (A b + 2 B a\right )}{11} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(x**(5/2)*(B*x+A)*(b**2*x**2+2*a*b*x+a**2),x)

[Out]

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

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Mathematica [A]  time = 0.0292247, size = 52, normalized size = 0.83 \[ \frac{2 x^{7/2} \left (143 a^2 (9 A+7 B x)+182 a b x (11 A+9 B x)+63 b^2 x^2 (13 A+11 B x)\right )}{9009} \]

Antiderivative was successfully verified.

[In]  Integrate[x^(5/2)*(A + B*x)*(a^2 + 2*a*b*x + b^2*x^2),x]

[Out]

(2*x^(7/2)*(143*a^2*(9*A + 7*B*x) + 182*a*b*x*(11*A + 9*B*x) + 63*b^2*x^2*(13*A
+ 11*B*x)))/9009

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Maple [A]  time = 0.01, size = 52, normalized size = 0.8 \[{\frac{1386\,B{b}^{2}{x}^{3}+1638\,A{b}^{2}{x}^{2}+3276\,B{x}^{2}ab+4004\,aAbx+2002\,{a}^{2}Bx+2574\,A{a}^{2}}{9009}{x}^{{\frac{7}{2}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x^(5/2)*(B*x+A)*(b^2*x^2+2*a*b*x+a^2),x)

[Out]

2/9009*x^(7/2)*(693*B*b^2*x^3+819*A*b^2*x^2+1638*B*a*b*x^2+2002*A*a*b*x+1001*B*a
^2*x+1287*A*a^2)

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Maxima [A]  time = 0.695769, size = 69, normalized size = 1.1 \[ \frac{2}{13} \, B b^{2} x^{\frac{13}{2}} + \frac{2}{7} \, A a^{2} x^{\frac{7}{2}} + \frac{2}{11} \,{\left (2 \, B a b + A b^{2}\right )} x^{\frac{11}{2}} + \frac{2}{9} \,{\left (B a^{2} + 2 \, A a b\right )} x^{\frac{9}{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b^2*x^2 + 2*a*b*x + a^2)*(B*x + A)*x^(5/2),x, algorithm="maxima")

[Out]

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

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Fricas [A]  time = 0.276981, size = 76, normalized size = 1.21 \[ \frac{2}{9009} \,{\left (693 \, B b^{2} x^{6} + 1287 \, A a^{2} x^{3} + 819 \,{\left (2 \, B a b + A b^{2}\right )} x^{5} + 1001 \,{\left (B a^{2} + 2 \, A a b\right )} x^{4}\right )} \sqrt{x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b^2*x^2 + 2*a*b*x + a^2)*(B*x + A)*x^(5/2),x, algorithm="fricas")

[Out]

2/9009*(693*B*b^2*x^6 + 1287*A*a^2*x^3 + 819*(2*B*a*b + A*b^2)*x^5 + 1001*(B*a^2
+ 2*A*a*b)*x^4)*sqrt(x)

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Sympy [A]  time = 9.79915, size = 80, normalized size = 1.27 \[ \frac{2 A a^{2} x^{\frac{7}{2}}}{7} + \frac{4 A a b x^{\frac{9}{2}}}{9} + \frac{2 A b^{2} x^{\frac{11}{2}}}{11} + \frac{2 B a^{2} x^{\frac{9}{2}}}{9} + \frac{4 B a b x^{\frac{11}{2}}}{11} + \frac{2 B b^{2} x^{\frac{13}{2}}}{13} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x**(5/2)*(B*x+A)*(b**2*x**2+2*a*b*x+a**2),x)

[Out]

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

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GIAC/XCAS [A]  time = 0.269227, size = 72, normalized size = 1.14 \[ \frac{2}{13} \, B b^{2} x^{\frac{13}{2}} + \frac{4}{11} \, B a b x^{\frac{11}{2}} + \frac{2}{11} \, A b^{2} x^{\frac{11}{2}} + \frac{2}{9} \, B a^{2} x^{\frac{9}{2}} + \frac{4}{9} \, A a b x^{\frac{9}{2}} + \frac{2}{7} \, A a^{2} x^{\frac{7}{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b^2*x^2 + 2*a*b*x + a^2)*(B*x + A)*x^(5/2),x, algorithm="giac")

[Out]

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