3.2119 \(\int \left (a+b \sqrt{x}\right )^2 x^3 \, dx\)

Optimal. Leaf size=32 \[ \frac{a^2 x^4}{4}+\frac{4}{9} a b x^{9/2}+\frac{b^2 x^5}{5} \]

[Out]

(a^2*x^4)/4 + (4*a*b*x^(9/2))/9 + (b^2*x^5)/5

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Rubi [A]  time = 0.0672598, antiderivative size = 32, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.133 \[ \frac{a^2 x^4}{4}+\frac{4}{9} a b x^{9/2}+\frac{b^2 x^5}{5} \]

Antiderivative was successfully verified.

[In]  Int[(a + b*Sqrt[x])^2*x^3,x]

[Out]

(a^2*x^4)/4 + (4*a*b*x^(9/2))/9 + (b^2*x^5)/5

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Rubi in Sympy [A]  time = 8.67306, size = 27, normalized size = 0.84 \[ \frac{a^{2} x^{4}}{4} + \frac{4 a b x^{\frac{9}{2}}}{9} + \frac{b^{2} x^{5}}{5} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(x**3*(a+b*x**(1/2))**2,x)

[Out]

a**2*x**4/4 + 4*a*b*x**(9/2)/9 + b**2*x**5/5

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Mathematica [A]  time = 0.00815925, size = 32, normalized size = 1. \[ \frac{a^2 x^4}{4}+\frac{4}{9} a b x^{9/2}+\frac{b^2 x^5}{5} \]

Antiderivative was successfully verified.

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

[Out]

(a^2*x^4)/4 + (4*a*b*x^(9/2))/9 + (b^2*x^5)/5

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Maple [A]  time = 0.004, size = 25, normalized size = 0.8 \[{\frac{{x}^{4}{a}^{2}}{4}}+{\frac{4\,ab}{9}{x}^{{\frac{9}{2}}}}+{\frac{{b}^{2}{x}^{5}}{5}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x^3*(a+b*x^(1/2))^2,x)

[Out]

1/4*x^4*a^2+4/9*a*b*x^(9/2)+1/5*b^2*x^5

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Maxima [A]  time = 1.44538, size = 178, normalized size = 5.56 \[ \frac{{\left (b \sqrt{x} + a\right )}^{10}}{5 \, b^{8}} - \frac{14 \,{\left (b \sqrt{x} + a\right )}^{9} a}{9 \, b^{8}} + \frac{21 \,{\left (b \sqrt{x} + a\right )}^{8} a^{2}}{4 \, b^{8}} - \frac{10 \,{\left (b \sqrt{x} + a\right )}^{7} a^{3}}{b^{8}} + \frac{35 \,{\left (b \sqrt{x} + a\right )}^{6} a^{4}}{3 \, b^{8}} - \frac{42 \,{\left (b \sqrt{x} + a\right )}^{5} a^{5}}{5 \, b^{8}} + \frac{7 \,{\left (b \sqrt{x} + a\right )}^{4} a^{6}}{2 \, b^{8}} - \frac{2 \,{\left (b \sqrt{x} + a\right )}^{3} a^{7}}{3 \, b^{8}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/5*(b*sqrt(x) + a)^10/b^8 - 14/9*(b*sqrt(x) + a)^9*a/b^8 + 21/4*(b*sqrt(x) + a)
^8*a^2/b^8 - 10*(b*sqrt(x) + a)^7*a^3/b^8 + 35/3*(b*sqrt(x) + a)^6*a^4/b^8 - 42/
5*(b*sqrt(x) + a)^5*a^5/b^8 + 7/2*(b*sqrt(x) + a)^4*a^6/b^8 - 2/3*(b*sqrt(x) + a
)^3*a^7/b^8

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Fricas [A]  time = 0.230737, size = 32, normalized size = 1. \[ \frac{1}{5} \, b^{2} x^{5} + \frac{4}{9} \, a b x^{\frac{9}{2}} + \frac{1}{4} \, a^{2} x^{4} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/5*b^2*x^5 + 4/9*a*b*x^(9/2) + 1/4*a^2*x^4

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Sympy [A]  time = 2.17268, size = 27, normalized size = 0.84 \[ \frac{a^{2} x^{4}}{4} + \frac{4 a b x^{\frac{9}{2}}}{9} + \frac{b^{2} x^{5}}{5} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x**3*(a+b*x**(1/2))**2,x)

[Out]

a**2*x**4/4 + 4*a*b*x**(9/2)/9 + b**2*x**5/5

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GIAC/XCAS [A]  time = 0.216202, size = 32, normalized size = 1. \[ \frac{1}{5} \, b^{2} x^{5} + \frac{4}{9} \, a b x^{\frac{9}{2}} + \frac{1}{4} \, a^{2} x^{4} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/5*b^2*x^5 + 4/9*a*b*x^(9/2) + 1/4*a^2*x^4