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

Optimal. Leaf size=55 \[ \frac{2}{9} x^{9/2} (a B+A b)+\frac{2}{7} a A x^{7/2}+\frac{2}{11} x^{11/2} (A c+b B)+\frac{2}{13} B c x^{13/2} \]

[Out]

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

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Rubi [A]  time = 0.0658864, antiderivative size = 55, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 1, integrand size = 21, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.048 \[ \frac{2}{9} x^{9/2} (a B+A b)+\frac{2}{7} a A x^{7/2}+\frac{2}{11} x^{11/2} (A c+b B)+\frac{2}{13} B c x^{13/2} \]

Antiderivative was successfully verified.

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

[Out]

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

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Rubi in Sympy [A]  time = 8.34451, size = 60, normalized size = 1.09 \[ \frac{2 A a x^{\frac{7}{2}}}{7} + \frac{2 B c x^{\frac{13}{2}}}{13} + x^{\frac{11}{2}} \left (\frac{2 A c}{11} + \frac{2 B b}{11}\right ) + x^{\frac{9}{2}} \left (\frac{2 A b}{9} + \frac{2 B a}{9}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [A]  time = 0.033109, size = 43, normalized size = 0.78 \[ \frac{2 x^{7/2} \left (1001 x (a B+A b)+1287 a A+819 x^2 (A c+b B)+693 B c x^3\right )}{9009} \]

Antiderivative was successfully verified.

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

[Out]

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

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Maple [A]  time = 0.004, size = 42, normalized size = 0.8 \[{\frac{1386\,Bc{x}^{3}+1638\,Ac{x}^{2}+1638\,Bb{x}^{2}+2002\,Abx+2002\,aBx+2574\,aA}{9009}{x}^{{\frac{7}{2}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Maxima [A]  time = 0.713571, size = 53, normalized size = 0.96 \[ \frac{2}{13} \, B c x^{\frac{13}{2}} + \frac{2}{11} \,{\left (B b + A c\right )} x^{\frac{11}{2}} + \frac{2}{7} \, A a x^{\frac{7}{2}} + \frac{2}{9} \,{\left (B a + A b\right )} x^{\frac{9}{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

2/13*B*c*x^(13/2) + 2/11*(B*b + A*c)*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 = 0.276998, size = 59, normalized size = 1.07 \[ \frac{2}{9009} \,{\left (693 \, B c x^{6} + 819 \,{\left (B b + A c\right )} x^{5} + 1287 \, A a x^{3} + 1001 \,{\left (B a + A b\right )} x^{4}\right )} \sqrt{x} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Sympy [A]  time = 9.45183, size = 70, normalized size = 1.27 \[ \frac{2 A a x^{\frac{7}{2}}}{7} + \frac{2 A b x^{\frac{9}{2}}}{9} + \frac{2 A c x^{\frac{11}{2}}}{11} + \frac{2 B a x^{\frac{9}{2}}}{9} + \frac{2 B b x^{\frac{11}{2}}}{11} + \frac{2 B c x^{\frac{13}{2}}}{13} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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GIAC/XCAS [A]  time = 0.268541, size = 58, normalized size = 1.05 \[ \frac{2}{13} \, B c x^{\frac{13}{2}} + \frac{2}{11} \, B b x^{\frac{11}{2}} + \frac{2}{11} \, A c x^{\frac{11}{2}} + \frac{2}{9} \, B a x^{\frac{9}{2}} + \frac{2}{9} \, A b x^{\frac{9}{2}} + \frac{2}{7} \, A a x^{\frac{7}{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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