3.423 \(\int \frac{x^5}{\left (a+b x^3\right )^{3/2}} \, dx\)

Optimal. Leaf size=38 \[ \frac{2 a}{3 b^2 \sqrt{a+b x^3}}+\frac{2 \sqrt{a+b x^3}}{3 b^2} \]

[Out]

(2*a)/(3*b^2*Sqrt[a + b*x^3]) + (2*Sqrt[a + b*x^3])/(3*b^2)

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Rubi [A]  time = 0.0608243, antiderivative size = 38, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.133 \[ \frac{2 a}{3 b^2 \sqrt{a+b x^3}}+\frac{2 \sqrt{a+b x^3}}{3 b^2} \]

Antiderivative was successfully verified.

[In]  Int[x^5/(a + b*x^3)^(3/2),x]

[Out]

(2*a)/(3*b^2*Sqrt[a + b*x^3]) + (2*Sqrt[a + b*x^3])/(3*b^2)

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Rubi in Sympy [A]  time = 7.3163, size = 34, normalized size = 0.89 \[ \frac{2 a}{3 b^{2} \sqrt{a + b x^{3}}} + \frac{2 \sqrt{a + b x^{3}}}{3 b^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [A]  time = 0.0213304, size = 27, normalized size = 0.71 \[ \frac{2 \left (2 a+b x^3\right )}{3 b^2 \sqrt{a+b x^3}} \]

Antiderivative was successfully verified.

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

[Out]

(2*(2*a + b*x^3))/(3*b^2*Sqrt[a + b*x^3])

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Maple [A]  time = 0.008, size = 24, normalized size = 0.6 \[{\frac{2\,b{x}^{3}+4\,a}{3\,{b}^{2}}{\frac{1}{\sqrt{b{x}^{3}+a}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x^5/(b*x^3+a)^(3/2),x)

[Out]

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

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Maxima [A]  time = 1.42177, size = 41, normalized size = 1.08 \[ \frac{2 \, \sqrt{b x^{3} + a}}{3 \, b^{2}} + \frac{2 \, a}{3 \, \sqrt{b x^{3} + a} b^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

2/3*sqrt(b*x^3 + a)/b^2 + 2/3*a/(sqrt(b*x^3 + a)*b^2)

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Fricas [A]  time = 0.228181, size = 31, normalized size = 0.82 \[ \frac{2 \,{\left (b x^{3} + 2 \, a\right )}}{3 \, \sqrt{b x^{3} + a} b^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

2/3*(b*x^3 + 2*a)/(sqrt(b*x^3 + a)*b^2)

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Sympy [A]  time = 3.00017, size = 46, normalized size = 1.21 \[ \begin{cases} \frac{4 a}{3 b^{2} \sqrt{a + b x^{3}}} + \frac{2 x^{3}}{3 b \sqrt{a + b x^{3}}} & \text{for}\: b \neq 0 \\\frac{x^{6}}{6 a^{\frac{3}{2}}} & \text{otherwise} \end{cases} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Piecewise((4*a/(3*b**2*sqrt(a + b*x**3)) + 2*x**3/(3*b*sqrt(a + b*x**3)), Ne(b,
0)), (x**6/(6*a**(3/2)), True))

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GIAC/XCAS [A]  time = 0.245199, size = 35, normalized size = 0.92 \[ \frac{2 \,{\left (\sqrt{b x^{3} + a} + \frac{a}{\sqrt{b x^{3} + a}}\right )}}{3 \, b^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

2/3*(sqrt(b*x^3 + a) + a/sqrt(b*x^3 + a))/b^2