3.443 \(\int x^{5/2} (a+b x)^3 \, dx\)

Optimal. Leaf size=51 \[ \frac{2}{7} a^3 x^{7/2}+\frac{2}{3} a^2 b x^{9/2}+\frac{6}{11} a b^2 x^{11/2}+\frac{2}{13} b^3 x^{13/2} \]

[Out]

(2*a^3*x^(7/2))/7 + (2*a^2*b*x^(9/2))/3 + (6*a*b^2*x^(11/2))/11 + (2*b^3*x^(13/2
))/13

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Rubi [A]  time = 0.0310032, antiderivative size = 51, 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 \[ \frac{2}{7} a^3 x^{7/2}+\frac{2}{3} a^2 b x^{9/2}+\frac{6}{11} a b^2 x^{11/2}+\frac{2}{13} b^3 x^{13/2} \]

Antiderivative was successfully verified.

[In]  Int[x^(5/2)*(a + b*x)^3,x]

[Out]

(2*a^3*x^(7/2))/7 + (2*a^2*b*x^(9/2))/3 + (6*a*b^2*x^(11/2))/11 + (2*b^3*x^(13/2
))/13

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Rubi in Sympy [A]  time = 5.16145, size = 49, normalized size = 0.96 \[ \frac{2 a^{3} x^{\frac{7}{2}}}{7} + \frac{2 a^{2} b x^{\frac{9}{2}}}{3} + \frac{6 a b^{2} x^{\frac{11}{2}}}{11} + \frac{2 b^{3} x^{\frac{13}{2}}}{13} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(x**(5/2)*(b*x+a)**3,x)

[Out]

2*a**3*x**(7/2)/7 + 2*a**2*b*x**(9/2)/3 + 6*a*b**2*x**(11/2)/11 + 2*b**3*x**(13/
2)/13

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Mathematica [A]  time = 0.0113069, size = 39, normalized size = 0.76 \[ \frac{2 x^{7/2} \left (429 a^3+1001 a^2 b x+819 a b^2 x^2+231 b^3 x^3\right )}{3003} \]

Antiderivative was successfully verified.

[In]  Integrate[x^(5/2)*(a + b*x)^3,x]

[Out]

(2*x^(7/2)*(429*a^3 + 1001*a^2*b*x + 819*a*b^2*x^2 + 231*b^3*x^3))/3003

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Maple [A]  time = 0.006, size = 36, normalized size = 0.7 \[{\frac{462\,{b}^{3}{x}^{3}+1638\,a{b}^{2}{x}^{2}+2002\,{a}^{2}bx+858\,{a}^{3}}{3003}{x}^{{\frac{7}{2}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x^(5/2)*(b*x+a)^3,x)

[Out]

2/3003*x^(7/2)*(231*b^3*x^3+819*a*b^2*x^2+1001*a^2*b*x+429*a^3)

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Maxima [A]  time = 1.33094, size = 47, normalized size = 0.92 \[ \frac{2}{13} \, b^{3} x^{\frac{13}{2}} + \frac{6}{11} \, a b^{2} x^{\frac{11}{2}} + \frac{2}{3} \, a^{2} b x^{\frac{9}{2}} + \frac{2}{7} \, a^{3} x^{\frac{7}{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x + a)^3*x^(5/2),x, algorithm="maxima")

[Out]

2/13*b^3*x^(13/2) + 6/11*a*b^2*x^(11/2) + 2/3*a^2*b*x^(9/2) + 2/7*a^3*x^(7/2)

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Fricas [A]  time = 0.204472, size = 54, normalized size = 1.06 \[ \frac{2}{3003} \,{\left (231 \, b^{3} x^{6} + 819 \, a b^{2} x^{5} + 1001 \, a^{2} b x^{4} + 429 \, a^{3} x^{3}\right )} \sqrt{x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x + a)^3*x^(5/2),x, algorithm="fricas")

[Out]

2/3003*(231*b^3*x^6 + 819*a*b^2*x^5 + 1001*a^2*b*x^4 + 429*a^3*x^3)*sqrt(x)

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Sympy [A]  time = 6.71063, size = 49, normalized size = 0.96 \[ \frac{2 a^{3} x^{\frac{7}{2}}}{7} + \frac{2 a^{2} b x^{\frac{9}{2}}}{3} + \frac{6 a b^{2} x^{\frac{11}{2}}}{11} + \frac{2 b^{3} x^{\frac{13}{2}}}{13} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x**(5/2)*(b*x+a)**3,x)

[Out]

2*a**3*x**(7/2)/7 + 2*a**2*b*x**(9/2)/3 + 6*a*b**2*x**(11/2)/11 + 2*b**3*x**(13/
2)/13

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GIAC/XCAS [A]  time = 0.206502, size = 47, normalized size = 0.92 \[ \frac{2}{13} \, b^{3} x^{\frac{13}{2}} + \frac{6}{11} \, a b^{2} x^{\frac{11}{2}} + \frac{2}{3} \, a^{2} b x^{\frac{9}{2}} + \frac{2}{7} \, a^{3} x^{\frac{7}{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x + a)^3*x^(5/2),x, algorithm="giac")

[Out]

2/13*b^3*x^(13/2) + 6/11*a*b^2*x^(11/2) + 2/3*a^2*b*x^(9/2) + 2/7*a^3*x^(7/2)