3.667 \(\int x^3 \sqrt [3]{a+b x^2} \, dx\)

Optimal. Leaf size=38 \[ \frac{3 \left (a+b x^2\right )^{7/3}}{14 b^2}-\frac{3 a \left (a+b x^2\right )^{4/3}}{8 b^2} \]

[Out]

(-3*a*(a + b*x^2)^(4/3))/(8*b^2) + (3*(a + b*x^2)^(7/3))/(14*b^2)

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Rubi [A]  time = 0.067535, 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 \[ \frac{3 \left (a+b x^2\right )^{7/3}}{14 b^2}-\frac{3 a \left (a+b x^2\right )^{4/3}}{8 b^2} \]

Antiderivative was successfully verified.

[In]  Int[x^3*(a + b*x^2)^(1/3),x]

[Out]

(-3*a*(a + b*x^2)^(4/3))/(8*b^2) + (3*(a + b*x^2)^(7/3))/(14*b^2)

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Rubi in Sympy [A]  time = 7.81525, size = 34, normalized size = 0.89 \[ - \frac{3 a \left (a + b x^{2}\right )^{\frac{4}{3}}}{8 b^{2}} + \frac{3 \left (a + b x^{2}\right )^{\frac{7}{3}}}{14 b^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-3*a*(a + b*x**2)**(4/3)/(8*b**2) + 3*(a + b*x**2)**(7/3)/(14*b**2)

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Mathematica [A]  time = 0.0191875, size = 38, normalized size = 1. \[ \frac{3 \sqrt [3]{a+b x^2} \left (-3 a^2+a b x^2+4 b^2 x^4\right )}{56 b^2} \]

Antiderivative was successfully verified.

[In]  Integrate[x^3*(a + b*x^2)^(1/3),x]

[Out]

(3*(a + b*x^2)^(1/3)*(-3*a^2 + a*b*x^2 + 4*b^2*x^4))/(56*b^2)

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Maple [A]  time = 0.007, size = 25, normalized size = 0.7 \[ -{\frac{-12\,b{x}^{2}+9\,a}{56\,{b}^{2}} \left ( b{x}^{2}+a \right ) ^{{\frac{4}{3}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x^3*(b*x^2+a)^(1/3),x)

[Out]

-3/56*(b*x^2+a)^(4/3)*(-4*b*x^2+3*a)/b^2

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Maxima [A]  time = 1.39595, size = 41, normalized size = 1.08 \[ \frac{3 \,{\left (b x^{2} + a\right )}^{\frac{7}{3}}}{14 \, b^{2}} - \frac{3 \,{\left (b x^{2} + a\right )}^{\frac{4}{3}} a}{8 \, b^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x^2 + a)^(1/3)*x^3,x, algorithm="maxima")

[Out]

3/14*(b*x^2 + a)^(7/3)/b^2 - 3/8*(b*x^2 + a)^(4/3)*a/b^2

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Fricas [A]  time = 0.203093, size = 46, normalized size = 1.21 \[ \frac{3 \,{\left (4 \, b^{2} x^{4} + a b x^{2} - 3 \, a^{2}\right )}{\left (b x^{2} + a\right )}^{\frac{1}{3}}}{56 \, b^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x^2 + a)^(1/3)*x^3,x, algorithm="fricas")

[Out]

3/56*(4*b^2*x^4 + a*b*x^2 - 3*a^2)*(b*x^2 + a)^(1/3)/b^2

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Sympy [A]  time = 4.05133, size = 223, normalized size = 5.87 \[ - \frac{9 a^{\frac{13}{3}} \sqrt [3]{1 + \frac{b x^{2}}{a}}}{56 a^{2} b^{2} + 56 a b^{3} x^{2}} + \frac{9 a^{\frac{13}{3}}}{56 a^{2} b^{2} + 56 a b^{3} x^{2}} - \frac{6 a^{\frac{10}{3}} b x^{2} \sqrt [3]{1 + \frac{b x^{2}}{a}}}{56 a^{2} b^{2} + 56 a b^{3} x^{2}} + \frac{9 a^{\frac{10}{3}} b x^{2}}{56 a^{2} b^{2} + 56 a b^{3} x^{2}} + \frac{15 a^{\frac{7}{3}} b^{2} x^{4} \sqrt [3]{1 + \frac{b x^{2}}{a}}}{56 a^{2} b^{2} + 56 a b^{3} x^{2}} + \frac{12 a^{\frac{4}{3}} b^{3} x^{6} \sqrt [3]{1 + \frac{b x^{2}}{a}}}{56 a^{2} b^{2} + 56 a b^{3} x^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-9*a**(13/3)*(1 + b*x**2/a)**(1/3)/(56*a**2*b**2 + 56*a*b**3*x**2) + 9*a**(13/3)
/(56*a**2*b**2 + 56*a*b**3*x**2) - 6*a**(10/3)*b*x**2*(1 + b*x**2/a)**(1/3)/(56*
a**2*b**2 + 56*a*b**3*x**2) + 9*a**(10/3)*b*x**2/(56*a**2*b**2 + 56*a*b**3*x**2)
+ 15*a**(7/3)*b**2*x**4*(1 + b*x**2/a)**(1/3)/(56*a**2*b**2 + 56*a*b**3*x**2) +
12*a**(4/3)*b**3*x**6*(1 + b*x**2/a)**(1/3)/(56*a**2*b**2 + 56*a*b**3*x**2)

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GIAC/XCAS [A]  time = 0.218817, size = 39, normalized size = 1.03 \[ \frac{3 \,{\left (4 \,{\left (b x^{2} + a\right )}^{\frac{7}{3}} - 7 \,{\left (b x^{2} + a\right )}^{\frac{4}{3}} a\right )}}{56 \, b^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x^2 + a)^(1/3)*x^3,x, algorithm="giac")

[Out]

3/56*(4*(b*x^2 + a)^(7/3) - 7*(b*x^2 + a)^(4/3)*a)/b^2