3.728 \(\int \frac{x^3}{\left (a+b x^2\right )^{4/3}} \, dx\)

Optimal. Leaf size=38 \[ \frac{3 a}{2 b^2 \sqrt [3]{a+b x^2}}+\frac{3 \left (a+b x^2\right )^{2/3}}{4 b^2} \]

[Out]

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

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Rubi [A]  time = 0.070699, 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 \[ \frac{3 a}{2 b^2 \sqrt [3]{a+b x^2}}+\frac{3 \left (a+b x^2\right )^{2/3}}{4 b^2} \]

Antiderivative was successfully verified.

[In]  Int[x^3/(a + b*x^2)^(4/3),x]

[Out]

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

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Rubi in Sympy [A]  time = 7.83016, size = 34, normalized size = 0.89 \[ \frac{3 a}{2 b^{2} \sqrt [3]{a + b x^{2}}} + \frac{3 \left (a + b x^{2}\right )^{\frac{2}{3}}}{4 b^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [A]  time = 0.0238519, size = 27, normalized size = 0.71 \[ \frac{3 \left (3 a+b x^2\right )}{4 b^2 \sqrt [3]{a+b x^2}} \]

Antiderivative was successfully verified.

[In]  Integrate[x^3/(a + b*x^2)^(4/3),x]

[Out]

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

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Maple [A]  time = 0.007, size = 24, normalized size = 0.6 \[{\frac{3\,b{x}^{2}+9\,a}{4\,{b}^{2}}{\frac{1}{\sqrt [3]{b{x}^{2}+a}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x^3/(b*x^2+a)^(4/3),x)

[Out]

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

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Maxima [A]  time = 1.47639, size = 41, normalized size = 1.08 \[ \frac{3 \,{\left (b x^{2} + a\right )}^{\frac{2}{3}}}{4 \, b^{2}} + \frac{3 \, a}{2 \,{\left (b x^{2} + a\right )}^{\frac{1}{3}} b^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x^3/(b*x^2 + a)^(4/3),x, algorithm="maxima")

[Out]

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

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Fricas [A]  time = 0.214525, size = 31, normalized size = 0.82 \[ \frac{3 \,{\left (b x^{2} + 3 \, a\right )}}{4 \,{\left (b x^{2} + a\right )}^{\frac{1}{3}} b^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x^3/(b*x^2 + a)^(4/3),x, algorithm="fricas")

[Out]

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

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Sympy [A]  time = 2.46287, size = 46, normalized size = 1.21 \[ \begin{cases} \frac{9 a}{4 b^{2} \sqrt [3]{a + b x^{2}}} + \frac{3 x^{2}}{4 b \sqrt [3]{a + b x^{2}}} & \text{for}\: b \neq 0 \\\frac{x^{4}}{4 a^{\frac{4}{3}}} & \text{otherwise} \end{cases} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Piecewise((9*a/(4*b**2*(a + b*x**2)**(1/3)) + 3*x**2/(4*b*(a + b*x**2)**(1/3)),
Ne(b, 0)), (x**4/(4*a**(4/3)), True))

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GIAC/XCAS [A]  time = 0.213392, size = 36, normalized size = 0.95 \[ \frac{3 \,{\left ({\left (b x^{2} + a\right )}^{\frac{2}{3}} + \frac{2 \, a}{{\left (b x^{2} + a\right )}^{\frac{1}{3}}}\right )}}{4 \, b^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x^3/(b*x^2 + a)^(4/3),x, algorithm="giac")

[Out]

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