3.224 \(\int \frac{\sqrt{a x^4}}{\sqrt{1+x^3}} \, dx\)

Optimal. Leaf size=25 \[ \frac{2 \sqrt{x^3+1} \sqrt{a x^4}}{3 x^2} \]

[Out]

(2*Sqrt[a*x^4]*Sqrt[1 + x^3])/(3*x^2)

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Rubi [A]  time = 0.00999083, antiderivative size = 25, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.105 \[ \frac{2 \sqrt{x^3+1} \sqrt{a x^4}}{3 x^2} \]

Antiderivative was successfully verified.

[In]  Int[Sqrt[a*x^4]/Sqrt[1 + x^3],x]

[Out]

(2*Sqrt[a*x^4]*Sqrt[1 + x^3])/(3*x^2)

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Rubi in Sympy [A]  time = 6.52583, size = 22, normalized size = 0.88 \[ \frac{2 \sqrt{a x^{4}} \sqrt{x^{3} + 1}}{3 x^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((a*x**4)**(1/2)/(x**3+1)**(1/2),x)

[Out]

2*sqrt(a*x**4)*sqrt(x**3 + 1)/(3*x**2)

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Mathematica [A]  time = 0.0087445, size = 25, normalized size = 1. \[ \frac{2 \sqrt{x^3+1} \sqrt{a x^4}}{3 x^2} \]

Antiderivative was successfully verified.

[In]  Integrate[Sqrt[a*x^4]/Sqrt[1 + x^3],x]

[Out]

(2*Sqrt[a*x^4]*Sqrt[1 + x^3])/(3*x^2)

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Maple [A]  time = 0.007, size = 31, normalized size = 1.2 \[{\frac{ \left ( 2+2\,x \right ) \left ({x}^{2}-x+1 \right ) }{3\,{x}^{2}}\sqrt{a{x}^{4}}{\frac{1}{\sqrt{{x}^{3}+1}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((a*x^4)^(1/2)/(x^3+1)^(1/2),x)

[Out]

2/3*(1+x)*(x^2-x+1)/x^2*(a*x^4)^(1/2)/(x^3+1)^(1/2)

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Maxima [A]  time = 0.794808, size = 38, normalized size = 1.52 \[ \frac{2 \,{\left (\sqrt{a} x^{3} + \sqrt{a}\right )}}{3 \, \sqrt{x^{2} - x + 1} \sqrt{x + 1}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(sqrt(a*x^4)/sqrt(x^3 + 1),x, algorithm="maxima")

[Out]

2/3*(sqrt(a)*x^3 + sqrt(a))/(sqrt(x^2 - x + 1)*sqrt(x + 1))

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Fricas [A]  time = 0.274466, size = 26, normalized size = 1.04 \[ \frac{2 \, \sqrt{a x^{4}} \sqrt{x^{3} + 1}}{3 \, x^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(sqrt(a*x^4)/sqrt(x^3 + 1),x, algorithm="fricas")

[Out]

2/3*sqrt(a*x^4)*sqrt(x^3 + 1)/x^2

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Sympy [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{\sqrt{a x^{4}}}{\sqrt{\left (x + 1\right ) \left (x^{2} - x + 1\right )}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((a*x**4)**(1/2)/(x**3+1)**(1/2),x)

[Out]

Integral(sqrt(a*x**4)/sqrt((x + 1)*(x**2 - x + 1)), x)

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GIAC/XCAS [A]  time = 0.26073, size = 16, normalized size = 0.64 \[ \frac{2}{3} \, \sqrt{x^{3} + 1} \sqrt{a} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(sqrt(a*x^4)/sqrt(x^3 + 1),x, algorithm="giac")

[Out]

2/3*sqrt(x^3 + 1)*sqrt(a)