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

Optimal. Leaf size=39 \[ \frac{2}{11} x^{11/2} (A c+b B)+\frac{2}{9} A b x^{9/2}+\frac{2}{13} B c x^{13/2} \]

[Out]

(2*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 = 0.0549209, antiderivative size = 39, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.1 \[ \frac{2}{11} x^{11/2} (A c+b B)+\frac{2}{9} A b x^{9/2}+\frac{2}{13} B c x^{13/2} \]

Antiderivative was successfully verified.

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

[Out]

(2*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 = 6.37807, size = 41, normalized size = 1.05 \[ \frac{2 A b x^{\frac{9}{2}}}{9} + \frac{2 B c x^{\frac{13}{2}}}{13} + x^{\frac{11}{2}} \left (\frac{2 A c}{11} + \frac{2 B b}{11}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [A]  time = 0.0205951, size = 31, normalized size = 0.79 \[ \frac{2 x^{9/2} \left (117 x (A c+b B)+143 A b+99 B c x^2\right )}{1287} \]

Antiderivative was successfully verified.

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

[Out]

(2*x^(9/2)*(143*A*b + 117*(b*B + A*c)*x + 99*B*c*x^2))/1287

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Maple [A]  time = 0.005, size = 28, normalized size = 0.7 \[{\frac{198\,Bc{x}^{2}+234\,Acx+234\,xBb+286\,Ab}{1287}{x}^{{\frac{9}{2}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

2/1287*x^(9/2)*(99*B*c*x^2+117*A*c*x+117*B*b*x+143*A*b)

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Maxima [A]  time = 0.674343, size = 36, normalized size = 0.92 \[ \frac{2}{13} \, B c x^{\frac{13}{2}} + \frac{2}{9} \, A b x^{\frac{9}{2}} + \frac{2}{11} \,{\left (B b + A c\right )} x^{\frac{11}{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Fricas [A]  time = 0.295681, size = 43, normalized size = 1.1 \[ \frac{2}{1287} \,{\left (99 \, B c x^{6} + 143 \, A b x^{4} + 117 \,{\left (B b + A c\right )} x^{5}\right )} \sqrt{x} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

2/1287*(99*B*c*x^6 + 143*A*b*x^4 + 117*(B*b + A*c)*x^5)*sqrt(x)

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Sympy [A]  time = 9.31866, size = 46, normalized size = 1.18 \[ \frac{2 A b x^{\frac{9}{2}}}{9} + \frac{2 A c x^{\frac{11}{2}}}{11} + \frac{2 B b x^{\frac{11}{2}}}{11} + \frac{2 B c x^{\frac{13}{2}}}{13} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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GIAC/XCAS [A]  time = 0.267895, size = 39, normalized size = 1. \[ \frac{2}{13} \, B c x^{\frac{13}{2}} + \frac{2}{11} \, B b x^{\frac{11}{2}} + \frac{2}{11} \, A c x^{\frac{11}{2}} + \frac{2}{9} \, A b x^{\frac{9}{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((c*x^2 + b*x)*(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*A*b*x^(9/2)