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

Optimal. Leaf size=21 \[ \frac{2}{3} a x^{3/2}+\frac{2}{11} c x^{11/2} \]

[Out]

(2*a*x^(3/2))/3 + (2*c*x^(11/2))/11

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Rubi [A]  time = 0.0162625, antiderivative size = 21, 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 \[ \frac{2}{3} a x^{3/2}+\frac{2}{11} c x^{11/2} \]

Antiderivative was successfully verified.

[In]  Int[Sqrt[x]*(a + c*x^4),x]

[Out]

(2*a*x^(3/2))/3 + (2*c*x^(11/2))/11

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Rubi in Sympy [A]  time = 2.65604, size = 19, normalized size = 0.9 \[ \frac{2 a x^{\frac{3}{2}}}{3} + \frac{2 c x^{\frac{11}{2}}}{11} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

2*a*x**(3/2)/3 + 2*c*x**(11/2)/11

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Mathematica [A]  time = 0.0071129, size = 21, normalized size = 1. \[ \frac{2}{3} a x^{3/2}+\frac{2}{11} c x^{11/2} \]

Antiderivative was successfully verified.

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

[Out]

(2*a*x^(3/2))/3 + (2*c*x^(11/2))/11

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Maple [A]  time = 0.005, size = 16, normalized size = 0.8 \[{\frac{6\,c{x}^{4}+22\,a}{33}{x}^{{\frac{3}{2}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((c*x^4+a)*x^(1/2),x)

[Out]

2/33*x^(3/2)*(3*c*x^4+11*a)

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Maxima [A]  time = 1.50853, size = 18, normalized size = 0.86 \[ \frac{2}{11} \, c x^{\frac{11}{2}} + \frac{2}{3} \, a x^{\frac{3}{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

2/11*c*x^(11/2) + 2/3*a*x^(3/2)

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Fricas [A]  time = 0.224774, size = 22, normalized size = 1.05 \[ \frac{2}{33} \,{\left (3 \, c x^{5} + 11 \, a x\right )} \sqrt{x} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

2/33*(3*c*x^5 + 11*a*x)*sqrt(x)

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Sympy [A]  time = 2.30966, size = 19, normalized size = 0.9 \[ \frac{2 a x^{\frac{3}{2}}}{3} + \frac{2 c x^{\frac{11}{2}}}{11} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

2*a*x**(3/2)/3 + 2*c*x**(11/2)/11

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GIAC/XCAS [A]  time = 0.213894, size = 18, normalized size = 0.86 \[ \frac{2}{11} \, c x^{\frac{11}{2}} + \frac{2}{3} \, a x^{\frac{3}{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

2/11*c*x^(11/2) + 2/3*a*x^(3/2)