3.286 \(\int \frac{x^3}{\sqrt{a x^2+b x^5}} \, dx\)

Optimal. Leaf size=25 \[ \frac{2 \sqrt{a x^2+b x^5}}{3 b x} \]

[Out]

(2*Sqrt[a*x^2 + b*x^5])/(3*b*x)

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Rubi [A]  time = 0.0149602, antiderivative size = 25, normalized size of antiderivative = 1., number of steps used = 1, number of rules used = 1, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.053 \[ \frac{2 \sqrt{a x^2+b x^5}}{3 b x} \]

Antiderivative was successfully verified.

[In]  Int[x^3/Sqrt[a*x^2 + b*x^5],x]

[Out]

(2*Sqrt[a*x^2 + b*x^5])/(3*b*x)

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Rubi in Sympy [A]  time = 7.05235, size = 19, normalized size = 0.76 \[ \frac{2 \sqrt{a x^{2} + b x^{5}}}{3 b x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(x**3/(b*x**5+a*x**2)**(1/2),x)

[Out]

2*sqrt(a*x**2 + b*x**5)/(3*b*x)

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Mathematica [A]  time = 0.0148376, size = 25, normalized size = 1. \[ \frac{2 \sqrt{x^2 \left (a+b x^3\right )}}{3 b x} \]

Antiderivative was successfully verified.

[In]  Integrate[x^3/Sqrt[a*x^2 + b*x^5],x]

[Out]

(2*Sqrt[x^2*(a + b*x^3)])/(3*b*x)

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Maple [A]  time = 0.008, size = 27, normalized size = 1.1 \[{\frac{2\,x \left ( b{x}^{3}+a \right ) }{3\,b}{\frac{1}{\sqrt{b{x}^{5}+a{x}^{2}}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x^3/(b*x^5+a*x^2)^(1/2),x)

[Out]

2/3*x*(b*x^3+a)/b/(b*x^5+a*x^2)^(1/2)

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Maxima [A]  time = 1.39405, size = 19, normalized size = 0.76 \[ \frac{2 \, \sqrt{b x^{3} + a}}{3 \, b} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x^3/sqrt(b*x^5 + a*x^2),x, algorithm="maxima")

[Out]

2/3*sqrt(b*x^3 + a)/b

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Fricas [A]  time = 0.215296, size = 28, normalized size = 1.12 \[ \frac{2 \, \sqrt{b x^{5} + a x^{2}}}{3 \, b x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x^3/sqrt(b*x^5 + a*x^2),x, algorithm="fricas")

[Out]

2/3*sqrt(b*x^5 + a*x^2)/(b*x)

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Sympy [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{x^{3}}{\sqrt{x^{2} \left (a + b x^{3}\right )}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x**3/(b*x**5+a*x**2)**(1/2),x)

[Out]

Integral(x**3/sqrt(x**2*(a + b*x**3)), x)

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GIAC/XCAS [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{x^{3}}{\sqrt{b x^{5} + a x^{2}}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x^3/sqrt(b*x^5 + a*x^2),x, algorithm="giac")

[Out]

integrate(x^3/sqrt(b*x^5 + a*x^2), x)