3.2111 \(\int \left (a+b \sqrt{x}\right ) x^2 \, dx\)

Optimal. Leaf size=19 \[ \frac{a x^3}{3}+\frac{2}{7} b x^{7/2} \]

[Out]

(a*x^3)/3 + (2*b*x^(7/2))/7

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Rubi [A]  time = 0.0158827, antiderivative size = 19, 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 \[ \frac{a x^3}{3}+\frac{2}{7} b x^{7/2} \]

Antiderivative was successfully verified.

[In]  Int[(a + b*Sqrt[x])*x^2,x]

[Out]

(a*x^3)/3 + (2*b*x^(7/2))/7

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Rubi in Sympy [A]  time = 2.78548, size = 15, normalized size = 0.79 \[ \frac{a x^{3}}{3} + \frac{2 b x^{\frac{7}{2}}}{7} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

a*x**3/3 + 2*b*x**(7/2)/7

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Mathematica [A]  time = 0.00542435, size = 19, normalized size = 1. \[ \frac{a x^3}{3}+\frac{2}{7} b x^{7/2} \]

Antiderivative was successfully verified.

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

[Out]

(a*x^3)/3 + (2*b*x^(7/2))/7

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Maple [A]  time = 0.002, size = 14, normalized size = 0.7 \[{\frac{a{x}^{3}}{3}}+{\frac{2\,b}{7}{x}^{{\frac{7}{2}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x^2*(a+b*x^(1/2)),x)

[Out]

1/3*a*x^3+2/7*b*x^(7/2)

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Maxima [A]  time = 1.44746, size = 132, normalized size = 6.95 \[ \frac{2 \,{\left (b \sqrt{x} + a\right )}^{7}}{7 \, b^{6}} - \frac{5 \,{\left (b \sqrt{x} + a\right )}^{6} a}{3 \, b^{6}} + \frac{4 \,{\left (b \sqrt{x} + a\right )}^{5} a^{2}}{b^{6}} - \frac{5 \,{\left (b \sqrt{x} + a\right )}^{4} a^{3}}{b^{6}} + \frac{10 \,{\left (b \sqrt{x} + a\right )}^{3} a^{4}}{3 \, b^{6}} - \frac{{\left (b \sqrt{x} + a\right )}^{2} a^{5}}{b^{6}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

2/7*(b*sqrt(x) + a)^7/b^6 - 5/3*(b*sqrt(x) + a)^6*a/b^6 + 4*(b*sqrt(x) + a)^5*a^
2/b^6 - 5*(b*sqrt(x) + a)^4*a^3/b^6 + 10/3*(b*sqrt(x) + a)^3*a^4/b^6 - (b*sqrt(x
) + a)^2*a^5/b^6

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Fricas [A]  time = 0.236269, size = 18, normalized size = 0.95 \[ \frac{2}{7} \, b x^{\frac{7}{2}} + \frac{1}{3} \, a x^{3} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

2/7*b*x^(7/2) + 1/3*a*x^3

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Sympy [A]  time = 1.46637, size = 15, normalized size = 0.79 \[ \frac{a x^{3}}{3} + \frac{2 b x^{\frac{7}{2}}}{7} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

a*x**3/3 + 2*b*x**(7/2)/7

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GIAC/XCAS [A]  time = 0.216112, size = 18, normalized size = 0.95 \[ \frac{2}{7} \, b x^{\frac{7}{2}} + \frac{1}{3} \, a x^{3} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

2/7*b*x^(7/2) + 1/3*a*x^3