3.2110 \(\int \left (a+b \sqrt{x}\right ) x^3 \, dx\)

Optimal. Leaf size=19 \[ \frac{a x^4}{4}+\frac{2}{9} b x^{9/2} \]

[Out]

(a*x^4)/4 + (2*b*x^(9/2))/9

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Rubi [A]  time = 0.0159518, 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 \[ \frac{a x^4}{4}+\frac{2}{9} b x^{9/2} \]

Antiderivative was successfully verified.

[In]  Int[(a + b*Sqrt[x])*x^3,x]

[Out]

(a*x^4)/4 + (2*b*x^(9/2))/9

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Rubi in Sympy [A]  time = 2.81126, size = 15, normalized size = 0.79 \[ \frac{a x^{4}}{4} + \frac{2 b x^{\frac{9}{2}}}{9} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

a*x**4/4 + 2*b*x**(9/2)/9

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Mathematica [A]  time = 0.00556642, size = 19, normalized size = 1. \[ \frac{a x^4}{4}+\frac{2}{9} b x^{9/2} \]

Antiderivative was successfully verified.

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

[Out]

(a*x^4)/4 + (2*b*x^(9/2))/9

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Maple [A]  time = 0.002, size = 14, normalized size = 0.7 \[{\frac{a{x}^{4}}{4}}+{\frac{2\,b}{9}{x}^{{\frac{9}{2}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x^3*(a+b*x^(1/2)),x)

[Out]

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

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Maxima [A]  time = 1.44506, size = 178, normalized size = 9.37 \[ \frac{2 \,{\left (b \sqrt{x} + a\right )}^{9}}{9 \, b^{8}} - \frac{7 \,{\left (b \sqrt{x} + a\right )}^{8} a}{4 \, b^{8}} + \frac{6 \,{\left (b \sqrt{x} + a\right )}^{7} a^{2}}{b^{8}} - \frac{35 \,{\left (b \sqrt{x} + a\right )}^{6} a^{3}}{3 \, b^{8}} + \frac{14 \,{\left (b \sqrt{x} + a\right )}^{5} a^{4}}{b^{8}} - \frac{21 \,{\left (b \sqrt{x} + a\right )}^{4} a^{5}}{2 \, b^{8}} + \frac{14 \,{\left (b \sqrt{x} + a\right )}^{3} a^{6}}{3 \, b^{8}} - \frac{{\left (b \sqrt{x} + a\right )}^{2} a^{7}}{b^{8}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Fricas [A]  time = 0.23296, size = 18, normalized size = 0.95 \[ \frac{2}{9} \, b x^{\frac{9}{2}} + \frac{1}{4} \, a x^{4} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

2/9*b*x^(9/2) + 1/4*a*x^4

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Sympy [A]  time = 1.68469, size = 15, normalized size = 0.79 \[ \frac{a x^{4}}{4} + \frac{2 b x^{\frac{9}{2}}}{9} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

a*x**4/4 + 2*b*x**(9/2)/9

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GIAC/XCAS [A]  time = 0.215945, size = 18, normalized size = 0.95 \[ \frac{2}{9} \, b x^{\frac{9}{2}} + \frac{1}{4} \, a x^{4} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

2/9*b*x^(9/2) + 1/4*a*x^4