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

Optimal. Leaf size=36 \[ \frac{2}{7} a^2 x^{7/2}+\frac{4}{9} a b x^{9/2}+\frac{2}{11} b^2 x^{11/2} \]

[Out]

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

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Rubi [A]  time = 0.0219416, antiderivative size = 36, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 1, integrand size = 13, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.077 \[ \frac{2}{7} a^2 x^{7/2}+\frac{4}{9} a b x^{9/2}+\frac{2}{11} b^2 x^{11/2} \]

Antiderivative was successfully verified.

[In]  Int[x^(5/2)*(a + b*x)^2,x]

[Out]

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

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Rubi in Sympy [A]  time = 3.90524, size = 34, normalized size = 0.94 \[ \frac{2 a^{2} x^{\frac{7}{2}}}{7} + \frac{4 a b x^{\frac{9}{2}}}{9} + \frac{2 b^{2} x^{\frac{11}{2}}}{11} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [A]  time = 0.00914127, size = 28, normalized size = 0.78 \[ \frac{2}{693} x^{7/2} \left (99 a^2+154 a b x+63 b^2 x^2\right ) \]

Antiderivative was successfully verified.

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

[Out]

(2*x^(7/2)*(99*a^2 + 154*a*b*x + 63*b^2*x^2))/693

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Maple [A]  time = 0.007, size = 25, normalized size = 0.7 \[{\frac{126\,{b}^{2}{x}^{2}+308\,abx+198\,{a}^{2}}{693}{x}^{{\frac{7}{2}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x^(5/2)*(b*x+a)^2,x)

[Out]

2/693*x^(7/2)*(63*b^2*x^2+154*a*b*x+99*a^2)

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Maxima [A]  time = 1.34051, size = 32, normalized size = 0.89 \[ \frac{2}{11} \, b^{2} x^{\frac{11}{2}} + \frac{4}{9} \, a b x^{\frac{9}{2}} + \frac{2}{7} \, a^{2} x^{\frac{7}{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Fricas [A]  time = 0.207593, size = 39, normalized size = 1.08 \[ \frac{2}{693} \,{\left (63 \, b^{2} x^{5} + 154 \, a b x^{4} + 99 \, a^{2} x^{3}\right )} \sqrt{x} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

2/693*(63*b^2*x^5 + 154*a*b*x^4 + 99*a^2*x^3)*sqrt(x)

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Sympy [A]  time = 5.09984, size = 34, normalized size = 0.94 \[ \frac{2 a^{2} x^{\frac{7}{2}}}{7} + \frac{4 a b x^{\frac{9}{2}}}{9} + \frac{2 b^{2} x^{\frac{11}{2}}}{11} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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GIAC/XCAS [A]  time = 0.203648, size = 32, normalized size = 0.89 \[ \frac{2}{11} \, b^{2} x^{\frac{11}{2}} + \frac{4}{9} \, a b x^{\frac{9}{2}} + \frac{2}{7} \, a^{2} x^{\frac{7}{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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