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

Optimal. Leaf size=34 \[ 2 a^2 \sqrt{x}+\frac{4}{9} a c x^{9/2}+\frac{2}{17} c^2 x^{17/2} \]

[Out]

2*a^2*Sqrt[x] + (4*a*c*x^(9/2))/9 + (2*c^2*x^(17/2))/17

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Rubi [A]  time = 0.0262335, antiderivative size = 34, 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 \[ 2 a^2 \sqrt{x}+\frac{4}{9} a c x^{9/2}+\frac{2}{17} c^2 x^{17/2} \]

Antiderivative was successfully verified.

[In]  Int[(a + c*x^4)^2/Sqrt[x],x]

[Out]

2*a^2*Sqrt[x] + (4*a*c*x^(9/2))/9 + (2*c^2*x^(17/2))/17

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Rubi in Sympy [A]  time = 4.572, size = 32, normalized size = 0.94 \[ 2 a^{2} \sqrt{x} + \frac{4 a c x^{\frac{9}{2}}}{9} + \frac{2 c^{2} x^{\frac{17}{2}}}{17} \]

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*sqrt(x) + 4*a*c*x**(9/2)/9 + 2*c**2*x**(17/2)/17

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Mathematica [A]  time = 0.0115658, size = 30, normalized size = 0.88 \[ \frac{2}{153} \sqrt{x} \left (153 a^2+34 a c x^4+9 c^2 x^8\right ) \]

Antiderivative was successfully verified.

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

[Out]

(2*Sqrt[x]*(153*a^2 + 34*a*c*x^4 + 9*c^2*x^8))/153

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Maple [A]  time = 0.006, size = 27, normalized size = 0.8 \[{\frac{18\,{c}^{2}{x}^{8}+68\,ac{x}^{4}+306\,{a}^{2}}{153}\sqrt{x}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((c*x^4+a)^2/x^(1/2),x)

[Out]

2/153*x^(1/2)*(9*c^2*x^8+34*a*c*x^4+153*a^2)

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Maxima [A]  time = 1.43623, size = 32, normalized size = 0.94 \[ \frac{2}{17} \, c^{2} x^{\frac{17}{2}} + \frac{4}{9} \, a c x^{\frac{9}{2}} + 2 \, a^{2} \sqrt{x} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

2/17*c^2*x^(17/2) + 4/9*a*c*x^(9/2) + 2*a^2*sqrt(x)

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Fricas [A]  time = 0.22316, size = 35, normalized size = 1.03 \[ \frac{2}{153} \,{\left (9 \, c^{2} x^{8} + 34 \, a c x^{4} + 153 \, a^{2}\right )} \sqrt{x} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

2/153*(9*c^2*x^8 + 34*a*c*x^4 + 153*a^2)*sqrt(x)

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Sympy [A]  time = 15.4393, size = 32, normalized size = 0.94 \[ 2 a^{2} \sqrt{x} + \frac{4 a c x^{\frac{9}{2}}}{9} + \frac{2 c^{2} x^{\frac{17}{2}}}{17} \]

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*sqrt(x) + 4*a*c*x**(9/2)/9 + 2*c**2*x**(17/2)/17

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GIAC/XCAS [A]  time = 0.2112, size = 32, normalized size = 0.94 \[ \frac{2}{17} \, c^{2} x^{\frac{17}{2}} + \frac{4}{9} \, a c x^{\frac{9}{2}} + 2 \, a^{2} \sqrt{x} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

2/17*c^2*x^(17/2) + 4/9*a*c*x^(9/2) + 2*a^2*sqrt(x)