3.1043 \(\int \sqrt{x} \left (a+b x^2+c x^4\right ) \, dx\)

Optimal. Leaf size=31 \[ \frac{2}{3} a x^{3/2}+\frac{2}{7} b x^{7/2}+\frac{2}{11} c x^{11/2} \]

[Out]

(2*a*x^(3/2))/3 + (2*b*x^(7/2))/7 + (2*c*x^(11/2))/11

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Rubi [A]  time = 0.0160379, antiderivative size = 31, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 1, integrand size = 18, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.056 \[ \frac{2}{3} a x^{3/2}+\frac{2}{7} b x^{7/2}+\frac{2}{11} c x^{11/2} \]

Antiderivative was successfully verified.

[In]  Int[Sqrt[x]*(a + b*x^2 + c*x^4),x]

[Out]

(2*a*x^(3/2))/3 + (2*b*x^(7/2))/7 + (2*c*x^(11/2))/11

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Rubi in Sympy [A]  time = 4.40565, size = 29, normalized size = 0.94 \[ \frac{2 a x^{\frac{3}{2}}}{3} + \frac{2 b x^{\frac{7}{2}}}{7} + \frac{2 c x^{\frac{11}{2}}}{11} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(x**(1/2)*(c*x**4+b*x**2+a),x)

[Out]

2*a*x**(3/2)/3 + 2*b*x**(7/2)/7 + 2*c*x**(11/2)/11

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Mathematica [A]  time = 0.00910448, size = 25, normalized size = 0.81 \[ \frac{2}{231} x^{3/2} \left (77 a+33 b x^2+21 c x^4\right ) \]

Antiderivative was successfully verified.

[In]  Integrate[Sqrt[x]*(a + b*x^2 + c*x^4),x]

[Out]

(2*x^(3/2)*(77*a + 33*b*x^2 + 21*c*x^4))/231

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Maple [A]  time = 0.004, size = 22, normalized size = 0.7 \[{\frac{42\,c{x}^{4}+66\,b{x}^{2}+154\,a}{231}{x}^{{\frac{3}{2}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x^(1/2)*(c*x^4+b*x^2+a),x)

[Out]

2/231*x^(3/2)*(21*c*x^4+33*b*x^2+77*a)

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Maxima [A]  time = 0.772612, size = 26, normalized size = 0.84 \[ \frac{2}{11} \, c x^{\frac{11}{2}} + \frac{2}{7} \, b x^{\frac{7}{2}} + \frac{2}{3} \, a x^{\frac{3}{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((c*x^4 + b*x^2 + a)*sqrt(x),x, algorithm="maxima")

[Out]

2/11*c*x^(11/2) + 2/7*b*x^(7/2) + 2/3*a*x^(3/2)

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Fricas [A]  time = 0.269389, size = 30, normalized size = 0.97 \[ \frac{2}{231} \,{\left (21 \, c x^{5} + 33 \, b x^{3} + 77 \, a x\right )} \sqrt{x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((c*x^4 + b*x^2 + a)*sqrt(x),x, algorithm="fricas")

[Out]

2/231*(21*c*x^5 + 33*b*x^3 + 77*a*x)*sqrt(x)

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Sympy [A]  time = 2.50494, size = 29, normalized size = 0.94 \[ \frac{2 a x^{\frac{3}{2}}}{3} + \frac{2 b x^{\frac{7}{2}}}{7} + \frac{2 c x^{\frac{11}{2}}}{11} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x**(1/2)*(c*x**4+b*x**2+a),x)

[Out]

2*a*x**(3/2)/3 + 2*b*x**(7/2)/7 + 2*c*x**(11/2)/11

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GIAC/XCAS [A]  time = 0.261633, size = 26, normalized size = 0.84 \[ \frac{2}{11} \, c x^{\frac{11}{2}} + \frac{2}{7} \, b x^{\frac{7}{2}} + \frac{2}{3} \, a x^{\frac{3}{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((c*x^4 + b*x^2 + a)*sqrt(x),x, algorithm="giac")

[Out]

2/11*c*x^(11/2) + 2/7*b*x^(7/2) + 2/3*a*x^(3/2)