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

Optimal. Leaf size=39 \[ \frac{2}{9} x^{9/2} (a B+A b)+\frac{2}{7} a A x^{7/2}+\frac{2}{11} b B x^{11/2} \]

[Out]

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

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Rubi [A]  time = 0.0438114, antiderivative size = 39, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 1, integrand size = 16, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.062 \[ \frac{2}{9} x^{9/2} (a B+A b)+\frac{2}{7} a A x^{7/2}+\frac{2}{11} b B x^{11/2} \]

Antiderivative was successfully verified.

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

[Out]

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

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Rubi in Sympy [A]  time = 5.21104, size = 41, normalized size = 1.05 \[ \frac{2 A a x^{\frac{7}{2}}}{7} + \frac{2 B b x^{\frac{11}{2}}}{11} + x^{\frac{9}{2}} \left (\frac{2 A b}{9} + \frac{2 B a}{9}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [A]  time = 0.0159246, size = 31, normalized size = 0.79 \[ \frac{2}{693} x^{7/2} \left (77 x (a B+A b)+99 a A+63 b B x^2\right ) \]

Antiderivative was successfully verified.

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

[Out]

(2*x^(7/2)*(99*a*A + 77*(A*b + a*B)*x + 63*b*B*x^2))/693

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Maple [A]  time = 0.006, size = 28, normalized size = 0.7 \[{\frac{126\,bB{x}^{2}+154\,Abx+154\,Bax+198\,Aa}{693}{x}^{{\frac{7}{2}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x^(5/2)*(b*x+a)*(B*x+A),x)

[Out]

2/693*x^(7/2)*(63*B*b*x^2+77*A*b*x+77*B*a*x+99*A*a)

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Maxima [A]  time = 1.33149, size = 36, normalized size = 0.92 \[ \frac{2}{11} \, B b x^{\frac{11}{2}} + \frac{2}{7} \, A a x^{\frac{7}{2}} + \frac{2}{9} \,{\left (B a + A b\right )} x^{\frac{9}{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Fricas [A]  time = 0.210153, size = 43, normalized size = 1.1 \[ \frac{2}{693} \,{\left (63 \, B b x^{5} + 99 \, A a x^{3} + 77 \,{\left (B a + A b\right )} x^{4}\right )} \sqrt{x} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

2/693*(63*B*b*x^5 + 99*A*a*x^3 + 77*(B*a + A*b)*x^4)*sqrt(x)

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Sympy [A]  time = 7.43079, size = 46, normalized size = 1.18 \[ \frac{2 A a x^{\frac{7}{2}}}{7} + \frac{2 A b x^{\frac{9}{2}}}{9} + \frac{2 B a x^{\frac{9}{2}}}{9} + \frac{2 B b x^{\frac{11}{2}}}{11} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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GIAC/XCAS [A]  time = 0.261211, size = 39, normalized size = 1. \[ \frac{2}{11} \, B b x^{\frac{11}{2}} + \frac{2}{9} \, B a x^{\frac{9}{2}} + \frac{2}{9} \, A b x^{\frac{9}{2}} + \frac{2}{7} \, A a x^{\frac{7}{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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