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

Optimal. Leaf size=36 \[ \frac{2}{3} a^2 x^{3/2}+\frac{4}{11} a c x^{11/2}+\frac{2}{19} c^2 x^{19/2} \]

[Out]

(2*a^2*x^(3/2))/3 + (4*a*c*x^(11/2))/11 + (2*c^2*x^(19/2))/19

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Rubi [A]  time = 0.027979, antiderivative size = 36, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 1, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.067 \[ \frac{2}{3} a^2 x^{3/2}+\frac{4}{11} a c x^{11/2}+\frac{2}{19} c^2 x^{19/2} \]

Antiderivative was successfully verified.

[In]  Int[Sqrt[x]*(a + c*x^4)^2,x]

[Out]

(2*a^2*x^(3/2))/3 + (4*a*c*x^(11/2))/11 + (2*c^2*x^(19/2))/19

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Rubi in Sympy [A]  time = 4.48704, size = 34, normalized size = 0.94 \[ \frac{2 a^{2} x^{\frac{3}{2}}}{3} + \frac{4 a c x^{\frac{11}{2}}}{11} + \frac{2 c^{2} x^{\frac{19}{2}}}{19} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

2*a**2*x**(3/2)/3 + 4*a*c*x**(11/2)/11 + 2*c**2*x**(19/2)/19

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Mathematica [A]  time = 0.0111882, size = 30, normalized size = 0.83 \[ \frac{2}{627} x^{3/2} \left (209 a^2+114 a c x^4+33 c^2 x^8\right ) \]

Antiderivative was successfully verified.

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

[Out]

(2*x^(3/2)*(209*a^2 + 114*a*c*x^4 + 33*c^2*x^8))/627

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Maple [A]  time = 0.008, size = 27, normalized size = 0.8 \[{\frac{66\,{c}^{2}{x}^{8}+228\,ac{x}^{4}+418\,{a}^{2}}{627}{x}^{{\frac{3}{2}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((c*x^4+a)^2*x^(1/2),x)

[Out]

2/627*x^(3/2)*(33*c^2*x^8+114*a*c*x^4+209*a^2)

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Maxima [A]  time = 1.43969, size = 32, normalized size = 0.89 \[ \frac{2}{19} \, c^{2} x^{\frac{19}{2}} + \frac{4}{11} \, a c x^{\frac{11}{2}} + \frac{2}{3} \, a^{2} x^{\frac{3}{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

2/19*c^2*x^(19/2) + 4/11*a*c*x^(11/2) + 2/3*a^2*x^(3/2)

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Fricas [A]  time = 0.229349, size = 36, normalized size = 1. \[ \frac{2}{627} \,{\left (33 \, c^{2} x^{9} + 114 \, a c x^{5} + 209 \, a^{2} x\right )} \sqrt{x} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

2/627*(33*c^2*x^9 + 114*a*c*x^5 + 209*a^2*x)*sqrt(x)

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Sympy [A]  time = 9.64151, size = 34, normalized size = 0.94 \[ \frac{2 a^{2} x^{\frac{3}{2}}}{3} + \frac{4 a c x^{\frac{11}{2}}}{11} + \frac{2 c^{2} x^{\frac{19}{2}}}{19} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

2*a**2*x**(3/2)/3 + 4*a*c*x**(11/2)/11 + 2*c**2*x**(19/2)/19

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GIAC/XCAS [A]  time = 0.214722, size = 32, normalized size = 0.89 \[ \frac{2}{19} \, c^{2} x^{\frac{19}{2}} + \frac{4}{11} \, a c x^{\frac{11}{2}} + \frac{2}{3} \, a^{2} x^{\frac{3}{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

2/19*c^2*x^(19/2) + 4/11*a*c*x^(11/2) + 2/3*a^2*x^(3/2)